Please%20provide%20a%20simplified%20and%20condensed%20explanation.%0A%0A%23%23%23%0A%20MNRAS%20535,%201449%E2%80%931474%20(2024)%20https://doi.org/10.1093/mnras/stae2236%20Advance%20Access%20publication%202024%20November%2013%0AThe%20impact%20of%20the%20cosmological%20constant%20on%20past%20and%20future%20star%20formation%0ADaniele%20Sorini%20,1,2,3%E2%80%B9%20John%20A.%20Peacock%202%20and%20Lucas%20Lombriser3%0A1Institute%20for%20Computational%20Cosmology,%20Department%20of%20Physics,%20Durham%20University,%20South%20Road,%20Durham%20DH1%203LE,%20UK%0A2Institute%20for%20Astronomy,%20University%20of%20Edinburgh,%20Royal%20Observatory,%20Edinburgh%20EH9%203HJ,%20UK%0A3De%20%CC%81partement%20de%20Physique%20The%20%CC%81orique,%20Universite%20%CC%81%20de%20Gene%60ve,%2024%20quai%20Ernest%20Ansermet,%20CH-1211%20Gene%60ve%204,%20Switzerland%0AAccepted%202024%20September%2017.%20Received%202024%20September%2013;%20in%20original%20form%202023%20September%2022%0AABSTRACT%0AWe%20present%20an%20extended%20analytical%20model%20for%20cosmic%20star%20formation,%20with%20the%20aim%20of%20investigating%20the%20impact%20of%20cosmological%20parameters%20on%20the%20star%20formation%20history%20within%20the%20%02CDM%20paradigm.%20Constructing%20an%20ensemble%20of%20flat%20%02CDM%20models%20where%20the%20cosmological%20constant%20varies%20between%20%02%20%3D%200%20and%20105%20times%20the%20observed%20value,%20%02obs,%20we%20find%20that%20the%20fraction%20of%20cosmic%20baryons%20that%20are%20converted%20into%20stars%20over%20the%20entire%20history%20of%20the%20universe%20peaks%20at%20%E2%88%BC%2027%20per%20cent%20for%200.01%20%02%20%02/%02obs%20%02%201.%20We%20explain,%20from%20first%20principles,%20that%20the%20decline%20of%20this%20asymptotic%20star%20formation%20efficiency%20for%20lower%20and%20higher%20values%20of%20%02%20is%20driven,%20respectively,%20by%20the%20astrophysics%20of%20star%20formation,%20and%20by%20the%20suppression%20of%20cosmic%20structure%20formation.%20However,%20the%20asymptotic%20efficiency%20declines%20slowly%20as%20%02%20increases,%20falling%20below%205%20per%20cent%20only%20for%20%02%20%3E%20100%20%02obs%20.%20Making%20the%20minimal%20assumption%20that%20the%20probability%20of%20generating%20observers%20is%20proportional%20to%20this%20efficiency,%20and%20following%20Weinberg%20in%20adopting%20a%20flat%20prior%20on%20%02,%20the%20median%20posterior%20value%20of%20%02%20is%20539%20%02obs%20.%20Furthermore,%20the%20probability%20of%20observing%20%02%20%E2%89%A4%20%02obs%20is%20only%200.5%20per%20cent.%20Although%20this%20work%20has%20not%20considered%20recollapsing%20models%20with%20%02%20%3C%200,%20the%20indication%20is%20thus%20that%20%02obs%20appears%20to%20be%20unreasonably%20small%20compared%20to%20the%20predictions%20of%20the%20simplest%20multiverse%20ensemble.%20This%20poses%20a%20challenge%20for%20anthropic%20reasoning%20as%20a%20viable%20explanation%20for%20cosmic%20coincidences%20and%20the%20apparent%20fine-tuning%20of%20the%20Universe:%20either%20the%20approach%20is%20invalid%20or%20more%20parameters%20than%20%02%20alone%20must%20vary%20within%20the%20ensemble.%0AKey%20words:%20methods:%20analytical%20%E2%80%93%20galaxies:%20formation%20%E2%80%93%20galaxies:%20star%20formation%20%E2%80%93%20cosmological%20parameters%20%E2%80%93%20cosmology:%20the-%20ory.%0A1%20INTRODUCTION%0AGalaxies%20are%20the%20most%20obvious%20tracers%20of%20large-scale%20structure%20in%20the%20Universe,%20and%20understanding%20how%20they%20emerge%20and%20evolve%20through-%20out%20cosmic%20history%20has%20been%20a%20longstanding%20goal%20of%20cosmology%20and%20astrophysics.%20The%20%02CDM%20paradigm%20approaches%20this%20via%20the%20gravitational%20collapse%20of%20dark%20matter%20haloes%20from%20an%20initial%20quasi-%20homogeneous%20Universe.%20This%20process%20is%20well%20understood,%20thanks%20to%20early%20analytical%20models%20of%20halo%20assembly%20(Lacey%20%26%20Cole%201993),%20which%20were%20subsequently%20validated%20numerically%20with%20full%20N-body%20simulations%20(Springel%20et%20al.%202005;%20Klypin,%20Trujillo-Gomez%20%26%20Primack%202011;%20Angulo%20et%20al.%202012;%20Fosalba%20et%20al.%202015).%20However,%20many%20of%20the%20detailed%20baryonic%20processes%20that%20govern%20the%20build-up%20of%20galaxies,%20such%20as%20gas%20accretion,%20star%20formation,%20and%20outflows,%20are%20not%20yet%20fully%20understood.%20A%20successful%20theory%20of%20galaxy%20formation%20needs%20to%20account%20for%20these%20processes%20in%20order%20to%20predict%20the%20main%20observed%20properties%20of%20galaxies.%20In%20particular,%20reproducing%20the%20observed%20efficiency%20of%20star%20formation,%20both%20locally%20within%20individual%20galaxies%20and%20globally%20over%20a%20cosmologically%20representative%20volume,%20constitutes%20a%20crucial%20test%20for%20any%20model%20of%20galaxy%20formation%20(see%20the%20review%20by%20Madau%20%26%20Dickin-%20son%202014).%20In%20this%20paper,%20we%20consider%20predictions%20for%20this%20efficiency%0A%E2%8B%86%20E-mail:%<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>%0Aand%20how%20it%20depends%20on%20cosmological%20parameters,%20specifically%20the%20cosmological%20constant.%0AEarly%20fully%20analytical%20models%20of%20star%20formation%20were%20based%20on%20simple%20prescriptions%20for%20the%20cooling%20time%20for%20the%20gas%20within%20galaxies%20and%20the%20typical%20time-scale%20for%20converting%20it%20into%20stars%20(e.g.%20Hernquist%20%26%20Springel%202003),%20while%20following%20the%20growth%20of%20structures%20with%20well-%20established%20analytical%20forms%20for%20the%20halo%20mass%20function%20(Press%20%26%20Schechter%201974;%20Sheth%20%26%20Tormen%201999,%202002).%20However,%20these%20insightful%20models%20neglected%20more%20sophisticated%20mechanisms%20such%20as%20feedback%20processes%20driven%20by%20stars%20or%20active%20galactic%20nuclei%20(AGNs).%20With%20a%20slightly%20different%20approach,%20White%20%26%20Frenk%20(1991)%20implemented%20the%20effect%20of%20subgalactic%20physics%20through%20a%20series%20of%20approximate%20formulae,%20while%20keeping%20the%20treatment%20of%20structure%20formation%20nearly%20analytical.%20This%20seminal%20work%20set%20the%20stage%20for%20a%20sequence%20of%20refinements%20of%20the%20modelling%20(e.g.%20Kauffmann,%20White%20%26%20Guiderdoni%201993;%20Cole%20et%20al.%201994,%202000;%20Guiderdoni%20et%20al.%201998;%20Kauffmann%20et%20al.%201999).%20Further%20extensions%20included%20the%20assembly%20of%20the%20central%20black%20hole,%20which%20enabled%20a%20description%20of%20the%20co-%20evolution%20of%20galaxies%20and%20quasars%20(e.g.%20Kauffmann%20%26%20Haehnelt%202000;%20Somerville%20et%20al.%202008;%20Henriques%20et%20al.%202015;%20Lacey%20et%20al.%202016).%20Other%20semi-analytical%20techniques%20followed%20the%20formation%20of%20dark%20matter%20haloes%20in%20full%20N-body%20simulations,%20coupled%20with%20analytical%20recipes%20for%20the%20baryonic%20physics%20(e.g.%20Croton%20et%20al.%202006;%20Henriques%20et%20al.%202020).%0A%C2%A9%202024%20The%20Author(s).%0APublished%20by%20Oxford%20University%20Press%20on%20behalf%20of%20Royal%20Astronomical%20Society.%20This%20is%20an%20Open%20Access%20article%20distributed%20under%20the%20terms%20of%20the%20Creative%20Commons%20Attribution%20License%20(https://creativecommons.org/licenses/by/4.0/),%20which%20permits%20unrestricted%20reuse,%20distribution,%20and%20reproduction%20in%20any%20medium,%20provided%20the%20original%20work%20is%20properly%20cited.%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1450%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0AFull%20hydrodynamical%20simulations%20incorporate%20several%20physical%20processes%20into%20the%20modelling%20and%20follow%20the%20evolution%20of%20baryons%20and%20dark%20matter%20from%20first%20principles.%20Several%20cosmological%20simulations%20(e.g.%20Schaye%20et%20al.%202010;%20Dubois%20et%20al.%202014;%20Hopkins%20et%20al.%202014;%20Vogelsberger%20et%20al.%202014;%20Lukic%20%CC%81%20et%20al.%202015;%20Schaye%20et%20al.%202015;%20Dave%20%CC%81,%20Thompson%20%26%20Hopkins%202016;%20McCarthy%20et%20al.%202017;%20Pillepich%20et%20al.%202018;%20Dave%20%CC%81%20et%20al.%202019),%20often%20based%20on%20different%20numerical%20approaches%20(Springel,%20Yoshida%20%26%20White%202001a;%20Springel%20et%20al.%202005,%202021;%20Almgren%20et%20al.%202013;%20Bryan%20et%20al.%202014;%20Hopkins%202015),%20generally%20managed%20to%20reproduce%20a%20plethora%20of%20observations%20to%20a%20satisfactory%20level%20of%20accuracy.%20However,%20feedback%20processes%20are%20implemented%20via%20various%20different%20numerical%20%E2%80%98subgrid%E2%80%99%20prescriptions,%20whose%20parameters%20are%20tuned%20to%20reproduce%20certain%20observations.%20It%20is%20therefore%20important%20to%20ask%20if%20the%20predictions%20of%20these%20calculations%20are%20robust,%20or%20whether%20they%20are%20fine-tuned%20to%20our%20Universe%20and%20have%20a%20high%20sensitivity%20to%20model%20parameters.%20While%20a%20substantial%20body%20of%20literature%20has%20focused%20on%20the%20effect%20on%20star%20formation%20history%20of%20varying%20the%20subgrid%20parameters%20(see%20the%20review%20by%20Somerville%20%26%20Dave%20%CC%81%202015),%20the%20impact%20of%20changing%20the%20cosmological%20parameters%20has%20historically%20received%20less%20attention.%20This%20may%20partially%20reflect%20the%20tremendous%20progress%20in%20constraining%20the%20parameters%20of%20the%20%02CDM%20paradigm%20(e.g.%20Planck%20Collaboration%20VI%202020;%20but%20see%20also%20Verde,%20Treu%20%26%20Riess%202019%20for%20a%20discussion%20on%20recent%20tensions).%20However,%20studying%20the%20predictions%20of%20hydrodynamical%20simulations%20regarding%20the%20star%20formation%20history%20in%20counter-factual%20cosmological%20models%20would%20constitute%20an%20interesting%20%E2%80%98stress-test%E2%80%99%20of%20our%20understanding%20of%20galaxy%20formation.%0AThere%20is%20also%20a%20more%20fundamental%20reason%20to%20explore%20this%20question.%20Although%20the%20%02CDM%20model%20is%20highly%20successful%20in%20many%20ways,%20it%20has%20theoretical%20issues%20that%20are%20hard%20to%20ignore%20(see%20e.g.%20the%20review%20by%20Bull%20et%20al.%202016).%20While%20the%20cosmological%20constant%20%02%20is%20required%20to%20explain%20the%20accelerating%20expansion%20of%20the%20universe,%20there%20is%20no%20consensus%20on%20its%20physical%20meaning.%20If%20%02%20is%20a%20manifestation%20of%20the%20energy%20of%20the%20quantum%20vacuum,%20then%20one%20can%20estimate%20its%20value%20by%20integrating%20over%20the%20zero-point%20energy%20of%20all%20possible%20modes.%20Adopting%20a%20cut-off%20energy%20Ev%20yields%20a%20vacuum%20density%20of%20Ev4%20in%20natural%20units;%20but%20the%20observed%20%02%20requires%20a%20cut-off%20at%20rather%20low%20energies,%20Ev%20%E2%88%BC%201%20meV,%20in%20gross%20conflict%20with%20the%20scale%20of%20new%20physics%20at%20perhaps%2010%20TeV%20or%20above%20(Martin%202012).%20This%20common%20calculation%20is%20in%20fact%20deeply%20flawed%20as%20it%20is%20non-relativistic%20and%20yields%20the%20wrong%20vacuum%20equation%20of%20state%20(Koksma%20%26%20Prokopec%202011):%20a%20more%20sophisticated%20calculation%20yields%20a%20vacuum%20density%20of%20order%20M%204%20,%20in%20terms%20of%20the%20particle%20mass.%20However,%20since%20particles%20exist%20with%20M%20near%20to%20the%20TeV%20scale%20(0.17%20TeV%20for%20the%20top%20quark),%20the%20discrepancy%20in%20the%20estimated%20%02%20is%20much%20the%20same%20as%20in%20the%20naive%20approach.%20For%20a%20more%20detailed%20discussion%20of%20this%20%E2%80%98cosmological%20constant%20problem%E2%80%99%20see,%20for%20example,%20Weinberg%20(2000a)%20or%20Abel,%20Bryan%20%26%20Norman%20(2002).%0AMany%20attempts%20have%20been%20made%20to%20move%20beyond%20the%20idea%20of%20%02%20representing%20a%20simple%20vacuum%20density,%20and%20to%20hypothesize%20some%20more%20general%20%E2%80%98dark%20energy%E2%80%99%20contributing%20to%20an%20effective%20%02%20that%20may%20vary%20with%20time.%20Ratra%20%26%20Peebles%20(1988)%20proposed%20that%20a%20dynamical%20scalar%20field%20could%20cause%20the%20accelerating%20expansion%20of%20the%20universe.%20While%20the%20scalar%20field%20proposed%20by%20Ratra%20%26%20Peebles%20(1988)%20was%20not%20motivated%20by%20fundamental%20physics,%20later%20models%20tried%20to%20connect%20it%20to%20extensions%20of%20the%20standard%20model%20of%20particle%20physics.%20In%20practice,%20the%20kinetic%20or%20potential%20terms%20of%20the%20Lagrangian%20of%20the%20scalar%20field%20depend%20on%20some%20fundamental%20mass%20scale%20(e.g.%20Zlatev,%20Wang%20%26%20Steinhardt%201999;%20Armendariz-Picon,%20Mukhanov%20%26%20Steinhardt%202001).%20An%20alternative%20view%20is%20that%20the%20cosmic%20acceleration%20in%20fact%20shows%20the%20need%20for%20some%20modified%20theory%20of%20gravity%20(see%20e.g.%20Joyce,%20Lombriser%20%26%20Schmidt%202016).%20Other%20approaches%20include%20mechanisms%20that%20prevent%20vacuum%20energy%20from%20gravitating%20(Kaloper%20%26%20Padilla%202014).%20However,%20all%20these%0Amodels%20effectively%20move%20the%20problem%20of%20the%20value%20of%20the%20%02%20to%20the%20fine-%20tuning%20of%20some%20other%20parameter%20of%20the%20theory,%20such%20as%20the%20mass%20scale%20associated%20with%20the%20scalar%20field%20(see%20e.g.%20Amendola%20%26%20Tsujikawa%202010).%20There%20is%20also%20a%20more%20radical%20position%20asserting%20that%20the%20debate%20on%20the%20physical%20nature%20of%20the%20cosmological%20constant%20is%20moot,%20and%20that%20%02%20is%20simply%20a%20fundamental%20constant%20emerging%20within%20the%20theory%20of%20General%20Relativity%20(Bianchi%20%26%20Rovelli%202010).%0ARegardless%20of%20the%20physical%20interpretation%20of%20the%20cosmological%20constant,%20the%20oddly%20small%20non-null%20value%20of%20%02%20gives%20rise%20to%20a%20number%20of%20coincidences%20that%20cry%20out%20for%20an%20explanation.%20Perhaps%20the%20most%20well-known%20of%20these%20is%20that%20the%20Universe%20became%20%02-dominated%20at%20z%20%E2%89%88%200.39,%20near%20to%20the%20formation%20time%20of%20the%20Sun.%20We%20therefore%20appear%20to%20live%20near%20the%20unique%20era%20when%20%02%20transitions%20from%20being%20negligible%20to%20dominating%20the%20Universe:%20this%20is%20the%20%E2%80%98why-now%20problem%E2%80%99%20(Velten,%20vom%20Marttens%20%26%20Zimdahl%202014).%20A%20further%20time%20coincidence%20is%20that%20recombination%20occurred%20at%20the%20same%20time%20of%20baryon-radiation%20equality.%20In%20addition,%20Lombriser%20%26%20Smer-Barreto%20(2017)%20pointed%20out%20that%20the%20equality%20between%20%02%20and%20radiation%20occurs%20around%20the%20mid-%20point%20of%20cosmic%20reionization.%20All%20these%20time-scale%20puzzles%20are%20in%20principle%20distinct%20from%20the%20cosmological%20constant%20problem%20described%20earlier%20(but%20see%20also%20the%20discussion%20in%20Lombriser%202023).%0AA%20possible%20explanation%20for%20the%20%E2%80%98why%20now%E2%80%99%20problem%20was%20suggested%20by%20Weinberg%20(1987).%20He%20noted%20that%20if%20the%20cosmological%20constant%20had%20been%20much%20larger%20than%20observed,%20the%20accelerating%20expansion%20of%20the%20Universe%20would%20have%20set%20in%20at%20earlier%20times,%20freezing%20out%20the%20growth%20of%20structure%20before%20galaxy-scale%20haloes%20had%20been%20able%20to%20form.%20Thus%20the%20star%20formation%20in%20galaxies%20that%20is%20necessary%20for%20the%20creation%20of%20observers%20would%20not%20occur%20if%20%02%20was%20substantially%20larger%20than%20the%20observed%20value.%20Such%20an%20argument%20is%20an%20example%20of%20anthropic%20reasoning%20(Carter%201974),%20which%20effectively%20considers%20the%20existence%20of%20observers%20(such%20as%20ourselves)%20as%20a%20%E2%80%98data%20point%E2%80%99%20and%20explores%20the%20implications%20for%20the%20cosmological%20parameters%20conditional%20on%20this%20information.%0ATo%20make%20this%20Bayesian%20argument%20(probability%20of%20%02%20given%20that%20it%20is%20observed),%20we%20need%20there%20to%20be%20some%20physical%20mechanism%20that%20allows%20%02%20to%20vary.%20It%20is%20also%20common%20to%20invoke%20a%20multiverse:%20an%20ensemble%20of%20different%20universes.%20The%20probability%20calculus%20is%20the%20same%20whether%20or%20not%20the%20members%20of%20the%20ensemble%20actually%20exist,%20or%20merely%20have%20the%20potential%20to%20do%20so.%20However,%20the%20idea%20of%20a%20concrete%20multiverse%20underlying%20Weinberg%E2%80%99s%20argument%20was%20given%20stronger%20motivation%20by%20the%20theory%20of%20inflation.%20Here,%20a%20multiverse%20of%20causally%20disconnected%20%E2%80%98bubble%20universes%E2%80%99%20arises%20in%20models%20of%20stochastic%20inflation%20where%20inflation%20proceeds%20eternally%20(Vilenkin%201983;%20Linde%201986;%20Guth%202007;%20Freivogel%202011).%20All%20causally%20disconnected%20bubbles%20evolve%20as%20independent%20universes,%20each%20characterized%20by%20a%20different%20set%20of%20constants,%20including%20%02.%20The%20advantage%20of%20this%20picture%20is%20that%20for%20a%20sufficiently%20large%20ensemble%20there%20are%20guaranteed%20to%20exist%20universes%20suitable%20for%20the%20formation%20of%20structure.%20Thus,%20that%20would%20explain%20the%20existence%20of%20our%20Universe,%20no%20matter%20how%20atypical%20it%20is%20within%20the%20ensemble.%0AAnthropic%20arguments%20often%20encounter%20significant%20resistance,%20with%20many%20physicists%20arguing%20that%20efforts%20should%20be%20focused%20on%20finding%20solutions%20to%20cosmological%20puzzles%20from%20first%20principles%20(e.g.%20Kane,%20Perry%20%26%20Zytkow%202002).%20Of%20course,%20such%20efforts%20should%20always%20be%20pursued.%20However,%20we%20can%20note%20that%20anthropic%20approaches%20pervade%20other%20fields%20of%20astronomy,%20without%20generating%20controversy.%20A%20prime%20example%20is%20the%20concept%20of%20circumstellar%20%E2%80%98habitable%20zone%E2%80%99,%20which%20is%20defined%20based%20on%20the%20conditions%20that%20can%20sustain%20life%20on%20a%20planet%20(see%20e.g.%20Kasting,%20Whitmire%20%26%20Reynolds%201993).%20Of%20course,%20unlike%20with%20exoplanets,%20there%20is%20only%20one%20Universe%20that%20can%20actually%20be%20observed%20(although%20see%20Aguirre%20%26%20Kozaczuk%202013;%20Wainwright%20et%20al.%202014;%20Johnson%20et%20al.%202016).%20As%20such,%20anthropic%20arguments%20cannot%20be%20tested%20in%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0Athe%20Galilean%20sense%20ingrained%20in%20the%20scientific%20method.%20However,%20they%20can%20still%20be%20tested,%20since%20we%20can%20in%20principle%20predict%20the%20probability%20distribution%20of%20observed%20values%20of%20cosmological%20parameters%20over%20the%20ensemble.%20This%20prediction%20can%20be%20compared%20with%20the%20single%20%E2%80%98data%20point%E2%80%99%20of%20our%20Universe,%20and%20a%20sufficiently%20large%20deviation%20from%20the%20average%20serves%20to%20rule%20out%20the%20model%20on%20which%20the%20prediction%20was%20based.%0AAfter%20the%20initial%20formulation%20given%20by%20Weinberg%20(1987),%20investiga-%20tions%20of%20the%20anthropic%20approach%20became%20progressively%20more%20refined.%20Weinberg%20(1989)%20extended%20his%20argument%20by%20noting%20that%20observer%20selection%20does%20not%20require%20%02%20to%20be%20exactly%20zero,%20and%20thus%20a%20small%20non-%20zero%20(in%20principle%20even%20negative)%20value%20of%20%02%20would%20be%20anthropically%20predicted.%20A%20critical%20element%20of%20this%20argument%20is%20the%20idea%20that%20the%20prior%20distribution%20of%20%02%20should%20be%20flat%20because%20%02%20%3D%200%20is%20not%20a%20spatial%20point,%20and%20therefore%20any%20continuous%20prior%20must%20be%20treatable%20as%20a%20constant%20in%20a%20narrow%20range%20near%20zero.%20Efstathiou%20(1995)%20revisited%20the%20argument%20by%20making%20more%20detailed%20estimates%20of%20the%20abundance%20of%20galaxies%20in%20universes%20with%20%02%20%E2%89%A5%200,%20subject%20to%20the%20constraint%20that%20the%20observed%20temperature%20of%20the%20cosmic%20microwave%20background%20(CMB)%20is%20equal%20to%202.73K.%20He%20concluded%20that%20anthropic%20arguments%20may%20explain%20a%20value%20of%20%02%20close%20to%20the%20one%20that%20is%20observed.%20Later%20works%20allowed%20for%20more%20than%20one%20cosmological%20parameter%20to%20vary.%20Garriga,%20Livio%20%26%20Vilenkin%20(1999)%20considered%20simultaneous%20variations%20of%20%02%20and%20the%20density%20contrast%20at%20the%20time%20of%20recombination.%20Peacock%20(2007)%20explored%20anthropic%20arguments%20treating%20both%20the%20CMB%20temperature%20and%20%02%20as%20free%20parameters,%20and%20further%20allowing%20%02%20to%20assume%20negative%20values.%20Bousso%20%26%20Leichenauer%20(2010)%20also%20considered%20the%20variation%20of%20multiple%20cosmological%20parameters,%20as%20well%20as%20negative%20values%20of%20%02.%20In%20their%20work,%20the%20generation%20of%20observers%20is%20tied%20to%20the%20global%20efficiency%20of%20star%20formation%20in%20the%20different%20members%20of%20the%20multiverse%20ensemble.%20The%20star%20formation%20history%20is%20predicted%20with%20a%20semi-analytical%20model,%20under%20the%20assumption%20that%20the%20astrophysics%20of%20star%20formation%20does%20not%20vary%20throughout%20the%20multiverse%20(Bousso%20%26%20Leichenauer%202009).%20Other%20semi-analytical%20work%20by%20Sudoh%20et%20al.%20(2017)%20showed%20that%20incorporating%20the%20astrophysics%20of%20galaxy%20formation%20in%20anthropic%20reasoning%20can%20affect%20the%20range%20of%20the%20anthropically%20favoured%20values%20of%20%02%20by%20almost%20an%20order%20of%20magnitude.%0AMore%20recently,%20progress%20in%20numerical%20power%20has%20enabled%20the%20testing%20of%20anthropic%20reasoning%20with%20full%20hydrodynamical%20simulations,%20exploring%20the%20simulated%20past%20and%20future%20star%20formation%20history%20for%20a%20wide%20range%20of%20%02%20(Barnes%20et%20al.%202018;%20Salcido%20et%20al.%202018;%20Oh%20et%20al.%202022).%20While%20such%20simulations%20include%20several%20astrophysical%20processes%20(albeit%20in%20an%20approximate%20parametrized%20form),%20it%20is%20hard%20to%20probe%20a%20wide%20parameter%20space,%20or%20to%20explore%20the%20far%20future%20of%20the%20universe%20(beyond%20%E2%88%BC%20100%20Gyr%20cosmic%20time),%20without%20a%20massive%20commitment%20of%20computational%20resources.%0AAnalytic%20models%20of%20star%20formation%20therefore%20represent%20an%20attractive%20complementary%20approach%20as%20they%20are%20not%20subject%20to%20the%20same%20limi-%20tations%20as%20hydrodynamical%20simulations.%20While%20inevitably%20simplified%20in%20terms%20of%20astrophysics,%20they%20are%20an%20efficient%20technique%20that%20can%20offer%20a%20more%20intuitive%20picture%20of%20the%20evolution%20of%20star%20formation%20(e.g.%20Rasera%20%26%20Teyssier%202006;%20Dave%20%CC%81,%20Finlator%20%26%20Oppenheimer%202012;%20Sharma%20%26%20Theuns%202019;%20Salcido,%20Bower%20%26%20Theuns%202020;%20Fukugita%20%26%20Kawasaki%202022).%20Recently,%20Sorini%20%26%20Peacock%20(2021)%20generalized%20the%20Hernquist%20%26%20Springel%20(2003)%20model%20of%20cosmic%20star%20formation%20such%20that%20it%20can%20be%20applied%20to%20arbitrarily%20large%20times.%20In%20this%20work,%20we%20further%20adapt%20the%20Sorini%20%26%20Peacock%20(2021)%20formalism%20to%20explore%20the%20impact%20of%20different%20values%20of%20%02%20on%20past%20and%20future%20star%20formation.%20The%20main%20result%20of%20our%20study%20will%20be%20that%20values%20of%20%02%20in%20the%20range%200.01%20%02%20%02/%02obs%20%02%20%02obs%20maximize%20the%20global%20efficiency%20of%20star%20formation.%20However,%20the%20interesting%20question%20is%20how%20effectively%20star%20formation%20is%20truncated%20by%20large%20values%20of%20%02.%20If%20this%20suppression%0Ais%20not%20effective%20until%20%02%20is%20vastly%20beyond%20the%20observed%20value,%20then%20Weinberg%E2%80%99s%20flat%20prior%20will%20mean%20that%20the%20observed%20Universe%20risks%20being%20an%20implausibly%20rare%20outlier.%20The%20main%20aim%20of%20this%20paper%20is%20to%20quantify%20just%20how%20unusual%20our%20Universe%20is%20in%20this%20respect.%0AThis%20manuscript%20is%20organized%20as%20follows.%20In%20Section%202,%20we%20give%20an%20overview%20of%20the%20formalism%20in%20Sorini%20%26%20Peacock%20(2021)%20and%20improve%20it%20by%20introducing%20extra%20features.%20In%20Section%203,%20we%20explain%20how%20we%20adapt%20it%20to%20%02CDM%20models%20with%20arbitrary%20non-negative%20values%20of%20%02%20and%20discuss%20the%20impact%20of%20changing%20the%20cosmological%20constant%20on%20the%20halo%20mass%20function%20and%20on%20the%20efficiency%20of%20star%20formation%20within%20haloes.%20In%20Section%204,%20we%20present%20our%20predictions%20for%20the%20long-term%20efficiency%20of%20star%20formation%20for%20different%20values%20of%20%02.%20In%20Section%205,%20we%20discuss%20the%20implications%20of%20our%20results%20for%20anthropic%20reasoning.%20We%20mainly%20focus%20on%20the%20cosmological%20constant%20problem%20but%20also%20marginally%20consider%20the%20why-now%20problem.%20We%20also%20compare%20our%20results%20with%20previous%20literature%20on%20the%20subject%20and%20discuss%20the%20limitations%20of%20our%20model.%20Finally,%20in%20Section%206%20we%20summarize%20the%20main%20conclusions%20of%20our%20work%20and%20discuss%20the%20future%20developments%20of%20our%20line%20of%20research.%0AThroughout%20this%20work,%20unless%20otherwise%20indicated,%20units%20of%20distance%20are%20understood%20to%20be%20proper%20units.%20Comoving%20units%20are%20designated%20with%20a%20%E2%80%98c%E2%80%99%20prefix%20(e.g.%20ckpc,%20cMpc).%0A2%20FORMALISM%0AIn%20this%20work,%20we%20aim%20to%20exploit%20the%20analytical%20model%20for%20cosmic%20star%20formation%20developed%20by%20Sorini%20%26%20Peacock%20(2021)%20%E2%80%93%20hereafter%20SP21.%20We%20therefore%20summarize%20the%20main%20aspects%20of%20the%20formalism%20in%20Section%202.1.%20In%20Section%202.2,%20we%20will%20show%20how%20we%20extended%20the%20SP21%20model%20to%20obtain%20greater%20accuracy%20in%20the%20predictions%20of%20the%20late-time%20behaviour%20of%20star%20formation.%20As%20we%20will%20explain,%20this%20generalization%20will%20be%20crucial%20in%20answering%20the%20main%20scientific%20questions%20addressed%20in%20this%20work.%0A2.1%20Summary%20of%20the%20SP21%20model%0AThe%20SP21%20model%20predicts%20the%20cosmic%20star%20formation%20rate%20density%20(CSFRD)%20from%20first%20principles,%20generalizing%20the%20seminal%20work%20by%20Hernquist%20%26%20Springel%20(2003)%20%E2%80%93%20hereafter%20HS03.%20The%20basic%20idea%20of%20the%20formalism%20is%20that%20the%20CSFRD%20is%20obtained%20by%20integrating%20the%20star%20formation%20rate%20(SFR)%20in%20all%20haloes%20within%20a%20given%20comoving%20volume,%20weighted%20by%20the%20halo%20multiplicity%20function:%0A%CF%81%20%CC%87%E2%88%97(z)%3D%CF%81%20%CC%840%0A%02%20dF(M,%20z)%0As(M,z)%20dlnM%20dlnM,%20(1)%0AImpact%20of%20%02%20on%20star%20formation%20history%201451%0Awhere%20the%20%CF%81%20%CC%840%20is%20the%20comoving%20mean%20matter%20density%20of%20the%20universe%20and%20dF%20(M%20,%20z)/d%20ln%20M%20is%20the%20halo%20multiplicity%20function,%20with%20F%20(M%20,%20z)%20being%20the%20collapsed%20mass%20fraction%20in%20haloes%20with%20total%20mass%20%3E%20M.%20These%20quantities%20encode%20the%20impact%20of%20background%20cosmology%20and%20of%20the%20growth%20of%20large-scale%20structure%20on%20the%20CSFRD.%20The%20specifically%20astrophysical%20component%20of%20the%20CSFRD%20is%20encapsulated%20in%20the%20term%20s%20(M%20,%20z)%20%3D%20SFR/M%20,%20which%20represents%20the%20average%20SFR%20over%20a%20population%20of%20haloes%20of%20mass%20M%20at%20redshift%20z,%20normalized%20by%20the%20total%20halo%20mass%20M%20.%20We%20will%20refer%20to%20s%20(M%20,%20z)%20as%20%E2%80%98normalized%20SFR%E2%80%99%20(nSFR).%0AIn%20this%20work,%20we%20follow%20SP21%E2%80%99s%20choice%20of%20modelling%20F(M,%20z)%20via%20the%20Sheth%E2%80%93Tormen%20formalism%20(Sheth%20%26%20Tormen%201999,%202002).%20We%20also%20adopt%20the%20same%20definitions%20of%20the%20virial%20radius,%20mass,%20and%20temperature%20as%20in%20SP21.%20The%20virial%20radius%20R%20of%20a%20halo%20of%20virial%20mass%20M%20at%20redshift%20z%20is%20defined%20as%20the%20radius%20of%20the%20sphere%20that%20contains%20a%20matter%20density%20equal%20to%20%05%20%CF%81c%20(z),%20where%20%CF%81c%20(z)%20is%20the%20critical%20density%20of%20the%20universe%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1452%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%20and%20%05%20a%20suitable%20multiplying%20factor:%0A4%0AM%3D%053%CF%80R3%CF%81c(z);%20(2)%20SP21%20adopted%20%05%20%3D%20200.%20One%20can%20then%20define%20a%20characteristic%20virial%0AV%3DR,%20(3)%20where%20G%20is%20the%20universal%20gravitational%20constant.%20We%20additionally%0Adefine%20the%20virial%20temperature%20T%20such%20that%0AV2%20%3D%202kBT%20,%20(4)%0A%CE%BC%0Awhere%20kB%20is%20the%20Boltzmann%20constant%20and%20%CE%BC%20is%20the%20mean%20molecular%20weight.%20We%20assume%20that%20%CE%BC%20%E2%89%88%200.6mp,%20which%20is%20a%20valid%20approximation%20for%20a%20fully%20ionized%20plasma%20of%20primordial%20composition.%20From%20equations%20(2)%E2%80%93(4),%20it%20follows%20that%0Adensity,%20which%20is%20assumed%20to%20follow%20a%20spherically%20symmetric%20power-%20law%20profile:%0A%04%20%05%CE%B7%0AR%20,%20(7)%0Ar%0Awhere%20Mgas%20is%20the%20total%20gas%20mass%20enclosed%20in%20the%20halo%20and%20the%20slope%0A%CE%B7%20is%20a%20free%20parameter%20of%20the%20model.%20Clearly,%20one%20must%20have%20%CE%B7%20%3C%203%20to%20prevent%20the%20halo%20gas%20mass%20from%20diverging.%20We%20also%20assume%20that%20%CE%B7%20%3E%200,%20so%20that%20density%20falls%20with%20radius.%20The%20assumption%20of%20a%20power-%20law%20gas%20density%20profile%20is%20generally%20supported%20by%20full%20cosmological%20simulations%20(e.g.%20Sorini%20et%20al.%202024b),%20but%20more%20complex%20models%20may%20provide%20a%20more%20realistic%20physical%20picture%20(see%20e.g.%20Mathews%20%26%20Prochaska%202017;%20Prochaska%20%26%20Zheng%202019;%20Khrykin%20et%20al.%202024;%20Sorini%20et%20al.%202024a).%0AThe%20cooling%20time%20at%20distance%20r%20from%20the%20centre%20of%20the%20halo%20is%20given%20by%0Avelocity%0A2%20GM%0A%032%201%20%042kT%0532%20M%3D%20B%20.%0A%05%20GH(z)%20%CE%BC%0A(5)%0Atcool%20%3D%0A3kBT%20%CF%81gas(r)%20,%20(8)%202%CE%BCnH(r)2C(T)%0AAlthough%20the%20CSFRD%20in%20equation%20(1)%20is%20expressed%20as%20an%20integral%20over%20the%20halo%20mass,%20it%20may%20be%20at%20times%20more%20convenient%20to%20switch%20the%20integration%20variable%20to%20the%20virial%20temperature.%20In%20particular,%20within%20the%20HS03%20formalism,%20the%20nSFR%20is%20more%20naturally%20expressed%20for%20a%20population%20of%20haloes%20at%20given%20T%20,%20as%20we%20shall%20now%20explain.%0AAt%20any%20given%20z,%20the%20nSFR%20is%20set%20by%20a%20characteristic%20time-scale.%20At%20high%20redshift,%20haloes%20are%20denser%20and%20hence%20gas%20cooling%20is%20more%20efficient.%20The%20bottleneck%20of%20star%20formation%20is%20therefore%20represented%20by%20the%20typical%20time-scale%20that%20governs%20the%20conversion%20of%20cool%20gas%20into%20stars.%20This%20%E2%80%98average%20gas%20consumption%20time-scale%E2%80%99,%20%E2%9F%A8t%E2%88%97%E2%9F%A9,%20is%20assumed%20to%20be%20a%20constant,%20with%20no%20dependence%20on%20the%20virial%20temperature%20of%20the%20halo%20or%20on%20the%20cosmological%20epoch.%20This%20assumption%20was%20originally%20introduced%20by%20HS03%20following%20the%20results%20of%20full%20hydrodynamical%20cosmological%20simulations%20(Springel%20%26%20Hernquist%202003a,%20b),%20but%20really%20asserts%20that%20%E2%9F%A8t%E2%88%97%E2%9F%A9%20is%20a%20time-scale%20set%20by%20the%20local%20microphysics%20of%20molecular%20clouds.%20The%20same%20prescription%20was%20adopted%20by%20SP21,%20who%20determined%20a%20physically%20reasonable%20range%20of%20%E2%9F%A8t%E2%88%97%E2%9F%A9%20by%20matching%20their%20predictions%20of%20the%20Kennicutt%E2%80%93Schmidt%20law%20(Kennicutt%201998)%20with%20measurements%20by%20Genzel%20et%20al.%20(2010).%20The%20value%20of%20%E2%9F%A8t%E2%88%97%E2%9F%A9%20that%20best%20reproduces%20the%20observations%20was%20found%20to%20be%202.39%20Gyr,%20but%20values%20in%20the%20range%20(1.63%20%E2%88%92%203.87)%20Gyr%20are%20consistent%20with%20the%20data%20within%203%CF%83.%20SP21%20showed%20that%20values%20of%20%E2%9F%A8t%E2%88%97%E2%9F%A9%20within%20this%20range%20can%20predict%20the%20CSFRD%20at%20high%20redshift%20within%20a%20factor%20of%20two,%20and%20we%20adopt%20the%20same%20prescription%20for%20the%20average%20high-redshift%20nSFR%20of%20a%20halo%20population%20at%20given%20virial%20temperature:%0Awhere%20nH%20is%20the%20number%20density%20of%20hydrogen%20and%20C(T%20)%20is%20the%20cooling%20function.%20As%20in%20HS03%20and%20SP21,%20we%20assume%20a%20primordial%20cooling%20function%20(Sutherland%20%26%20Dopita%201993),%20effectively%20neglecting%20metal%20cooling.%20However,%20this%20limitation%20is%20expected%20to%20alter%20the%20CSFRD%20at%20low%20redshift%20in%20a%20manner%20that%20does%20not%20significantly%20affect%20conclusions%20regarding%20the%20long-term%20star%20formation%20history%20(see%20the%20discussion%20in%20SP21%20and%20HS03),%20so%20that%20metallicity%20evolution%20is%20unimportant%20for%20the%20scope%20of%20this%20work.%20The%20cooling%20function%20depends%20purely%20on%20temperature,%20so%20there%20is%20some%20advantage%20in%20considering%20the%20evolution%20of%20the%20nSFR%20for%20haloes%20with%20given%20T%20rather%20than%20given%20M.%0AThe%20cooling%20rate%20of%20gas%20within%20haloes%20of%20virial%20temperature%20T%20at%20a%20given%20time%20is%20then%20estimated%20by%20following%20the%20expansion%20of%20a%20cooling%20front%20from%20the%20centre%20of%20the%20halo%20outwards.%20At%20any%20time%20t,%20the%20cooling%20front%20reaches%20the%20cooling%20radius%20rcool(t),%20defined%20by%20the%20criterion%20tcool%20(rcool%20(t%20))%20%3D%20t%20.%20The%20gas%20mass%20Mcool%20within%20rcool%20cools%20down%20and%20remains%20cool%20thereafter.%20The%20cooling%20rate%20is%20then%20determined%20by%20solving%20the%20equation%0AdMcool%202%20drcool%0Adt%20%3D%204%CF%80%CF%81gas(rcool)rcool%20dt%20.%20(9)%0AHere,%20the%20meaning%20of%20the%20time%20t%20needs%20to%20be%20defined%20quite%20carefully.%20In%20principle,%20this%20would%20be%20the%20time%20since%20the%20formation%20of%20the%20halo,%20i.e.%20since%20the%20last%20major%20merger.%20In%20the%20matter-dominated%20era,%20low-mass%20haloes%20have%20a%20life%20span%20comparable%20to%20the%20age%20of%20the%20universe,%20whereas%20high-mass%20haloes%20beyond%20the%20exponential%20cut-off%20survive%20for%20shorter%20times.%20However,%20massive%20haloes%20are%20rare,%20so%20one%20can%20effectively%20make%20the%20reasonable%20assumption%20that%20t%20(and%20tcool)%20are%20comparable%20to%20the%20cosmic%20time.%20However,%20it%20has%20been%20suggested%20that%20tcool%20should%20be%20of%20the%20order%20of%20the%20dynamical%20time%20of%20the%20halo%20tdyn%20%3D%20R/V%20,%20as%20it%20is%20on%20this%20time-scale%20that%20the%20gas%20profile%20reacts%20to%20pressure%20losses,%20and%20hence%20should%20set%20the%20extent%20of%20the%20cooling%20radius%20(Springel%20et%20al.%202001b).%20Because%20this%20prescription%20provides%20good%20agreement%20with%20simulations%20(Yoshidaetal.2002),HS03solvedequation(9)imposingtcool%20%3Dtdyn.%20However,%20SP21%20noted%20that%20this%20assumption%20breaks%20down%20in%20the%20far%20future,%20when%20the%20universe%20becomes%20%02-dominated.%20In%20this%20regime,%20merging%20eventually%20ceases%20and%20haloes%20are%20isolated.%20Therefore,%20haloes%20can%20in%20principle%20cool%20down%20for%20arbitrarily%20large%20times.%20Thus,%20SP21%20adopted%20the%20prescription%0A%06%20%04t%05m%07m1%0Atcool(t)%3Dfdyntdyn%201%E2%88%92E+%20f%20t%20,%20(10)%20dyn%20dyn%0A(1%E2%88%92%CE%B2)xfgas(T,%20z)%20shigh(T,z)%3D%20%E2%9F%A8t%E2%88%97%E2%9F%A9%20,%0A(6)%0Awhere%20x%20is%20the%20fraction%20of%20cold%20gas%20clouds,%20%CE%B2%20is%20the%20mass%20fraction%20of%20massive%20(%3E%208%20M%E2%8A%99%20)%20short-lived%20stars,%20and%20fgas%20(T%20,%20z)%20is%20the%20mass%20fraction%20of%20gas%20within%20the%20haloes%20considered.%20Following%20the%20same%20choice%20as%20in%20HS03%20and%20SP21,%20we%20set%20x%20%3D%200.95.%20This%20value%20is%20again%20motivated%20by%20the%20subgrid%20model%20for%20star%20formation%20by%20Springel%20%26%20Hernquist%20(2003b),%20which%20was%20utilized%20in%20hydrodynamical%20simula-%20tions%20by%20Springel%20%26%20Hernquist%20(2003a).%20As%20in%20SP21,%20we%20set%20%CE%B2%20%3D%200.21.%20This%20is%20determined%20by%20assuming%20a%20Chabrier%20(2003)%20initial%20mass%20function%20(IMF)%20over%20a%20stellar%20mass%20range%20between%200.1%20and%2080%20M%E2%8A%99%20.%0AAt%20low%20redshift,%20haloes%20are%20less%20dense%20and%20gas%20cooling%20thus%20proceeds%20more%20slowly.%20The%20nSFR%20is%20then%20no%20longer%20dictated%20by%20an%20internal%20time-%20scale,%20but%20is%20limited%20by%20the%20supply%20of%20new%20cold%20gas,%20which%20is%20set%20by%20the%20cooling%20time-scale%20tcool.%20This%20quantity%20depends%20on%20the%20local%20gas%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%CF%81gas(r)%3D(3%E2%88%92%CE%B7)%20Mgas%204%CF%80R3%0A%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%20%0Awhere%0A%08%03m%0AE%3D2%05.%20(11)%0A3fdyn%202%0ABy%20construction,%20equation%20(10)%20yields%20tcool%20%E2%89%88%20fdyntdyn%20for%20t%20%E2%89%AA%20tdyn,%20where%20fdyn%20is%20a%20constant%20of%20order%20unity,%20and%20tcool%20%E2%89%88%20t%20for%20t%20%E2%89%AB%20tdyn.%20SP21%20showed%20that%20the%20values%20of%20the%20parameters%20fdyn%20and%20m%20have%20minimal%20impact%20on%20the%20predicted%20CSFRD.%20We%20will%20then%20adopt%20their%20fiducial%20values,%20fdyn%20%3D%201%20and%20m%20%3D%202.%0AIn%20a%20flat%20%02CDM%20universe,%20the%20correspondence%20between%20redshift%20and%20cosmic%20time%20can%20be%20obtained%20analytically%20outside%20the%20radiation-%20dominated%20era:%0AbTFR%20by%20Lelli,%20McGaugh%20%26%20Schombert%20(2016),%20SP21%20conclude%20that%20values%20of%20%CE%B7%20in%20the%20range%201.9%20%E2%88%92%202.4%20are%20compatible%20with%20the%20data%20within%203%CF%83%20.%20The%20same%20values%20are%20simultaneously%20compatible,%20within%20a%20similar%20level%20of%20precision,%20with%20the%20Genzel%20et%20al.%20(2010)%20observations%20of%20the%20Kennicutt%E2%80%93Schmidt%20relationship.%0AThe%20final%20expression%20of%20the%20nSFR%20as%20a%20function%20of%20time%20is%20then%20obtained%20by%20simply%20interpolating%20the%20high-redshift%20(equation%206)%20and%20low-redshift%20solutions%20(equation%2013)%20with%20a%20smooth%20function,%20such%20that%0Aa(t)%3D%0Asinh%202%20%02H0t%20,%20m%20,%20and%20%02%20are%20the%20Hubble%0A(12)%0Aparameter%20on%20the%20predicted%20CSFRD%20is%20minimal.%20Equation%20(16),%20combined%20with%20the%20Sheth%E2%80%93Tormen%20formalism%20for%20the%20halo%20multiplicity%20function,%20allows%20one%20to%20obtain%20an%20analytic%20expression%20for%20the%20CSFRD%20given%20by%20equation%20(1).%0A2.2%20Improving%20the%20accuracy%20of%20the%20SFR%20at%20late%20times%0AAs%20explained%20in%20Section%202.1,%20one%20simplifying%20assumption%20adopted%20in%20SP21%20is%20that%20the%20baryon%20mass%20fraction%20in%20haloes%20is%20roughly%20equal%20to%20the%20gas%20mass%20fraction.%20This%20approximation%20can%20be%20easily%20justified%20at%20high%20redshift,%20shortly%20after%20the%20onset%20of%20star%20formation.%20However,%20this%20assumption%20can%20be%20too%20strong%20at%20later%20times,%20when%20the%20stellar%20mass%20fraction%20of%20haloes%20is%20not%20negligible%20with%20respect%20to%20their%20gas%20mass%20fraction%20(e.g.%20McGaugh%20et%20al.%202010).%20This%20could%20become%20an%20even%20more%20important%20issue%20when%20considering%20the%20future%20of%20the%20universe.%20Since%20the%20major%20focus%20of%20this%20work%20is%20understanding%20the%20impact%20of%20%02%20on%20both%20the%20past%20and%20future%20star%20formation%20history,%20we%20need%20to%20first%20extend%20the%20SP21%20formalism%20by%20keeping%20fgas%20(T%20,%20z)%20and%20fb,%20halo%20(T%20,%20z)%20distinct.%0AOne%20would%20be%20tempted%20to%20compute%20fgas%20from%20equation%20(15),%20replacing%20the%20cosmic%20baryon%20mass%20fraction%20with%20fb%20%E2%88%92%20f%E2%88%97%20(T%20,%20z),%20where%20f%E2%88%97(T%20,%20z)%20is%20the%20stellar%20mass%20fraction%20for%20a%20halo%20with%20virial%20temperature%20T%20at%20redshift%20z.%20The%20stellar%20mass%20fraction%20is%20related%20to%20the%20time%20integral%20of%20the%20nSFR%20provided%20by%20equation%20(16).%20The%20trouble%20is%20that%20the%20nSFR%20depends%20on%20fgas,%20while%20fgas%20in%20turn%20depends%20on%20the%20nSFR.%0ATo%20break%20this%20circularity,%20we%20adopt%20the%20following%20iterative%20method%20to%20calculate%20the%20nSFR:%0A(i)%20we%20assume%20that,%20at%20any%20fixed%20virial%20temperature,%20fgas(T%20,%20z)%20%E2%89%88%20fb,%20halo%20(T%20,%20z)%20at%20a%20sufficiently%20high%20redshift%20zin%20,%20and%20compute%20the%20corresponding%20nSFR%20following%20the%20original%20SP21%20model;%0A(ii)%20we%20obtain%20the%20stellar%20mass%20fraction%20as%20a%20function%20of%20redshift%20by%20integrating%20the%20nSFR%20over%20time,%20or,%20in%20terms%20of%20redshift:%0A1%20%02zin%20M(T,z%E2%80%B2)s(T,z%E2%80%B2)%20%E2%80%B2%20f%E2%88%97(T,z)%3DM(T,z)%20z%20(1+z%E2%80%B2)H(z%E2%80%B2)%20dz;%20(17)%0A(iii)%20we%20compute%20the%20cooling%20radius%20as%20a%20function%20of%20redshift%20from%20the%20criterion%20tcool%20(rcool%20(t%20))%20%3D%20t%20explained%20in%20Section%202.1;%20the%20cooling%20time%20is%20given%20by%20equation%20(8),%20with%20the%20gas%20density%20given%20by%20equation%20(7),%20where%20we%20further%20impose%20that%20Mgas%20(T%20,%20z)%20%3D%20(fb%20%E2%88%92%20f%E2%88%97%20(T%20,%20z))M%20.%20We%20then%20compare%20the%20evolution%20of%20the%20cooling%20radius%20with%20that%20of%20the%20critical%20radius%20described%20in%20the%20previous%20section%20to%20determine%20the%20critical%20temperature%20Tcrit(z)%20(see%20the%20appendix%20in%20SP21%20for%20more%20details);%0A(iv)%20we%20update%20fgas(T%20,%20z)%20via%20equation%20(15),%20where%20we%20now%20replace%20fbwithfb%E2%88%92f%E2%88%97(T,z);%0A(v)%20we%20use%20the%20new%20gas%20mass%20fraction%20to%20re-calculate%20the%20nSFR%20via%20equations%20(6),%20(13),%20and%20(16);%0A(vi)%20we%20repeat%20the%20above%20protocol%20restarting%20from%20point%20(ii).%0A%04%051%06%04%20%05%072%20m333%0As(T%20,%20z)%20%3D%0ASP21%20set%20m%20%3D%202,%20although%20they%20showed%20that%20the%20impact%20of%20this%0A%02%0Awhere%0Aparameters%20of%20matter%20and%20%02%20at%20z%20%3D%200,%20respectively.%20A%20principal%20interest%20of%20this%20paper%20is%20the%20impact%20of%20changing%20the%20value%20of%20%02,%20which%20then%20alters%20the%20expansion%20history%20of%20the%20universe%20and%20changes%20all%20cosmological%20parameters.%20We%20discuss%20below%20how%20to%20handle%20these%20changes%20in%20a%20consistent%20fashion.%20Combining%20equation%20(12)%20with%20equations%20(9)%20and%20(10),%20one%20can%20then%20obtain%20the%20cooling%20rate%20as%20a%20function%20of%20the%20virial%20temperature%20and%20redshift.%20Using%20the%20cooling%20rate%20as%20a%20proxy%20for%20the%20low-redshift%20nSFR,%20SP21%20found%0AH0%20,%0A%20%CC%83%20slow(T,z)%3DS(T)%0Aparameter%0A3%E2%88%92%CE%B7%E2%88%921%201%E2%88%92E+A(z)m%20m%CE%B7%0A%04%053%20H(z)fgas(T,z)%20%CE%B7%0AH0%20%03fb%0A%0E%0A(1+z)3%20,%20(13)%0A3f%202%04H%052%20A(z)m%E2%88%921+(1%E2%88%92E)%20dyn%200%0Am%0Awhere%20A(z)%3D%202%0A3fdyn%0A2%20%05%20H(z)%20%0F%20%08%0F%0A%05%20H(z)arcsinh%202%20%02%20H0%0A%02%0Am(1%20+%20z)3%0A(14)%0Aand%20S%20%CC%83%20(T%20)%20is%20a%20temperature-dependent%20proportionality%20factor%20with%20the%20units%20of%20an%20SFR%20per%20unit%20mass.%0AClearly,%20to%20determine%20the%20nSFR%20both%20at%20high%20and%20low%20redshift,%20it%20is%20crucial%20to%20have%20an%20expression%20for%20fgas(T%20,%20z).%20SP21%20adopt%20the%20approximation%20fgas%20(T%20,%20z)%20%E2%89%88%20fb,%20halo%20(T%20,%20z),%20where%20fb,%20halo%20(T%20,%20z)%20is%20the%20baryon%20mass%20fraction%20in%20haloes%20of%20temperature%20T%20at%20redshift%20z.%20This%20is%20in%20turn%20determined%20by%20studying%20the%20balance%20between%20the%20gravitational%20force%20and%20outward%20pressure%20exerted%20by%20supernova-driven%20winds%20on%20a%20gas%20parcel%20within%20the%20virial%20radius,%20as%20a%20function%20of%20its%20distance%20from%20the%20centre.%20One%20can%20then%20define%20a%20critical%20radius%20within%20which%20baryons%20are%20bound%20to%20the%20halo%20and%20escape%20otherwise.%20The%20question%20is%20whether%20such%20a%20critical%20radius%20falls%20within%20the%20cooling%20radius,%20which%20is%20the%20region%20of%20the%20halo%20that%20can%20produce%20stars%20in%20the%20simplified%20picture%20considered%20by%20SP21%20and%20HS03.%20SP21%20show%20that%20above%20a%20certain%20critical%20virial%20temperature%20Tcrit(z)%20the%20critical%20radius%20falls%20outside%20the%20cooling%20radius,%20hence%20all%20haloes%20retain%20their%20cosmic%20share%20of%20baryons.%20Below%20Tcrit(z),%20this%20approach%20yields%20a%20baryon%20mass%20fraction%20enclosed%20in%20the%20halo%20that%20scales%20as%20a%20power%20of%20its%20virial%20temperature:%0A(15)%0A%E2%8E%A8%20T%20fb,%20halo(T%20,%20z)%20%3D%20%E2%8E%A9%20Tcrit(z)%0Afb%0Ab%20crit%20.%20otherwise%0A%E2%8E%A7%13%0A%14%203%E2%88%92%CE%B7%0A2(%CE%B7%E2%88%921)f%20ifT%3CT(z)%0AThe%20transition%20between%20the%20two-temperature%20regimes%20is%20smoothed%20with%20a%20suitable%20analytic%20function.%20Equation%20(15)%20effectively%20provides%20a%20prediction%20for%20the%20baryonic%20Tully%E2%80%93Fisher%20relationship%20(bTFR),%20an%20empirical%20correlation%20between%20the%20baryonic%20mass%20and%20total%20mass%20of%20haloes.%20Comparing%20the%20predicted%20bTFR%20with%20observations%20of%20the%0Aand%20density%0AImpact%20of%20%02%20on%20star%20formation%20history%201453%0Ashigh(T%20,%20z)slow(T%20,%20z)%20.%20(16)%20(shigh(T,%20z)m%20+slow(T,%20z)m)m1%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1454%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0ATo%20ensure%20that%20the%20iterative%20procedure%20described%20above%20produces%20converging%20results,%20we%20examined%20the%20nSFR,%20stellar%20mass%20fraction,%20and%20critical%20temperature%20computed%20at%20each%20iteration.%20We%20plot%20the%20evolution%20of%20these%20quantities%20in%20Fig.%201.%20The%20nSFR%20and%20stellar%20mass%20fraction%20refer%20to%20a%20halo%20with%20virial%20temperature%20T%20%3D%20105%20K%20in%20our%20%02CDM%20Universe.%20For%20all%20quantities%20plotted,%20convergence%20is%20achieved%20very%20quickly%20%E2%80%93%20within%20five%20iterations.%20As%20expected,%20a%20non-null%20stellar%20mass%20fraction%20reduces%20the%20gas%20available%20for%20star%20formation,%20so%20that%20the%20nSFR%20is%20reduced.%20This%20in%20turns%20makes%20the%20cooling%20radius%20smaller:%20haloes%20that%20were%20at%20the%20critical%20temperature%20in%20the%20SP21%20work%20are%20therefore%20now%20above%20the%20critical%20temperature%20so%20that%20the%20critical%20temperature%20is%20lowered%20with%20respect%20to%20the%20original%20formalism.%0AAs%20we%20can%20see%20in%20the%20middle%20panel%20of%20Fig.%201,%20the%20stellar%20mass%20fraction%20asymptotes%20to%20a%20constant%20in%20the%20far%20future.%20That%20is%20a%20consequence%20of%20the%20decay%20in%20the%20nSFR,%20which%20tends%20to%20zero%20in%20the%20limit%20of%20infinitely%20large%20cosmic%20time.%20We%20will%20discuss%20this%20in%20detail%20in%20Section%204.1.%20The%20stellar%20mass%20fraction%20is%20a%20monotonic%20function%20of%20time,%20because%20the%20nSFR%20is%20always%20positive.%20There%20is%20no%20explicit%20mechanism%20in%20the%20model%20that%20removes%20stars%20that%20have%20gone%20supernova%20or%20became%20stellar%20remnants.%20The%20contribution%20of%20short-lived%20and%20massive%20stars%20to%20the%20nSFR%20is%20removed%20through%20the%20%CE%B2%20factor%20in%20equations%20(6)%20and%20(13),%20but%20all%20other%20stars%20are%20%E2%80%98eternal%E2%80%99%20once%20formed.%20This%20is%20a%20reasonably%20good%20approximation,%20since%20the%20time-scale%20for%20white%20dwarfs%20to%20turn%20into%20black%20dwarfs%20has%20been%20estimated%20to%20be%20%E2%88%BC%20105%20Gyr%20(Dyson%201979;%20Adams%20%26%20Laughlin%201997;%20Caplan%202020);%20we%20will%20discuss%20the%20subsequent%20implications%20for%20anthropic%20reasoning%20in%20Section%205.1.%0AWe%20verified%20that%20convergence%20is%20achieved%20in%20the%20range%20of%20interest%20for%20this%20study,%20that%20is%20104%20K%20%3C%20T%20%3C%201010%20K%20(as%20explained%20in%20SP21,%20the%20contribution%20of%20haloes%20with%20T%20%3E%201010%20K%20to%20the%20CSFRD%20is%20negligible;%20we%20verified%20that%20this%20holds%20also%20for%20the%20cosmological%20models%20consid-%20ered%20in%20this%20work).%20At%20higher%20temperatures,%20convergence%20is%20faster:%20at%20T%20%3D%20107%20K,%20two%20iterations%20are%20sufficient.%20For%20all%20quantities%20shown%20in%20Fig.%201,%20we%20chose%20zin%20%3D%2020%20as%20the%20initial%20redshift,%20but%20we%20verified%20that%20the%20exact%20value%20is%20unimportant%20as%20far%20as%20our%20analysis%20is%20concerned%20(see%20Section%204.1%20for%20a%20more%20detailed%20discussion).%0ATo%20assess%20whether%20the%20stellar%20mass%20fractions%20predicted%20by%20our%0Arevised%20SP21%20method%20are%20physically%20sensible,%20let%20us%20consider%20the%0Amass%20of%20the%20Milky%20Way.%20Recent%20estimates%20point%20to%20a%20stellar%20mass%0Aof%20(6.08%20%C2%B1%201.14)%20%C3%97%201010%20M%E2%8A%99%20(Licquia%20%26%20Newman%202015)%20and%20a%0Avirial%20mass%20of%201.54+0.75%20%C3%97%201012%20M%20(Watkins%20et%20al.%202019).%20There-%20%E2%88%920.44%20%E2%8A%99%0Afore,%20the%20stellar-to-total%20mass%20ratio%20is%20expected%20to%20be%20in%20the%20range%202.2%20%E2%88%92%206.6%20per%20cent.%20Applying%20equation%20(5),%20we%20can%20derive%20the%20virial%20temperatures%20corresponding%20to%20the%20virial%20mass%20found%20by%20Watkins%20et%20al.%20(2019),%20and%20then%20compute%20the%20corresponding%20stellar%20mass%20fraction%20at%20z%20%3D%200%20via%20equation%20(17),%20obtaining%206.7%20per%20cent.%20Thus,%20the%20predictions%20of%20our%20model%20are%20compatible%20with%20the%20upper%20bound%20of%20the%20stellar-to-%20total%20halo%20mass%20range%20provided%20by%20the%20observations.%20Considering%20the%20simplifications%20made%20in%20the%20formalism,%20this%20is%20a%20reassuring%20result.%0AThe%20key%20quantity%20under%20study%20in%20this%20paper%20is%20the%20CSFRD,%20so%20it%20is%20important%20to%20see%20how%20this%20is%20affected%20by%20the%20changes%20in%20normalized%20SFR%20shown%20in%20Fig.%201,%20which%20result%20from%20our%20generalization%20of%20the%20SP21%20model.%20In%20the%20upper-left%20panel%20of%20Fig.%202%20we%20show%20the%20CSFRD%20in%20the%20redshift%20range%200%20%3C%20z%20%3C%2010,%20computed%20both%20with%20the%20original%20and%20revised%20formalism.%20It%20is%20clear%20that%20our%20procedure%20yields%20a%20slightly%20lower%20CSFRD%20at%20low%20redshift,%20with%20little%20change%20in%20the%20slope%20at%20late%20times.%20In%20the%20upper-right%20panel,%20we%20show%20the%20behaviour%20of%20the%20two%20models%20in%20the%20future%20of%20the%20Universe.%20As%20future%20times%20correspond%20to%20negative%20redshifts,%20and%20the%20CSFRD%20approaches%20zero%20as%20time%20tends%20to%20infinity,%20the%20scale%20of%20both%20axes%20switches%20from%20logarithmic%20to%20linear%20when%20moving%20from%20the%20upper-left%20to%20the%20upper-right%20panel.%20We%20quantify%20the%20relative%20difference%20between%20the%20original%20SP21%20model%20and%20our%20updated%20formalism%20in%20the%20lower%20panels%20of%20Fig.%202%20(dashed%20black%20line).%0AThe%20differences%20are%20at%20the%20per%20cent%20level%20for%20z%20%3E%205,%20and%20then%20grow%20progressively,%20reaching%20%E2%88%BC%2010%20per%20cent%20at%20the%20peak%20of%20star%20formation%20and%20%E2%88%BC%2020%20per%20cent%20at%20z%20%3D%200.%20The%20discrepancy%20grows%20slowly%20but%20steadily%20in%20the%20far%20future%20of%20the%20Universe,%20stopping%20short%20of%2030%20per%20cent%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E.%20Such%20differences%20are%20subdominant%20with%20respect%20to%20the%20uncertainties%20of%20the%20parameters%20of%20the%20model.%20The%20original%20SP21%20model%20is%20thus%20still%20suitable%20for%20rapidly%20predicting%20the%20past%20star%20formation%20history%20of%20our%20Universe,%20but%20our%20generalization%20becomes%20indispensable%20for%20the%20future%20star%20formation%20history.%20This%20is%20even%20more%20important%20when%20one%20considers%20larger%20values%20of%20the%20cosmological%20constant:%20we%20verified%20that%20the%20relative%20difference%20with%20respect%20to%20the%20SP21%20model%20grows%20as%20%02%20increases.%0AIn%20the%20upper-left%20panel%20of%20Fig.%202,%20we%20also%20show%20the%20compilation%20of%20observational%20data%20provided%20by%20Madau%20%26%20Dickinson%202014,%20alongside%20their%20empirical%20fit%20to%20the%20data.%20Both%20data%20and%20fitting%20function%20have%20been%20properly%20rescaled%20from%20a%20Salpeter%20(1955)%20to%20a%20Chabrier%20(2003)%20IMF,%20for%20consistency%20with%20the%20SP21%20formalism%20and%20the%20model%20presented%20in%20this%20work%20(the%20data%20points%20were%20not%20rescaled%20in%20this%20way%20in%20the%20SP21%20paper).%20The%20relative%20difference%20between%20the%20SP21%20model%20and%20the%20Madau%20%26%20Dickinson%20(2014)%20fit%20are%20quantified%20in%20the%20lower-%20left%20panel,%20confirming%20that%20the%20SP21%20predictions%20recover%20the%20fitting%20function%20within%20a%20factor%20of%20%E2%88%BC%202%20for%200%20%3C%20z%20%3C%2010.%20Since%20our%20updated%20formalism%20deviates%20from%20SP21%20by%20at%20most%2020%20per%20cent%20in%20the%20same%20redshift%20range,%20we%20can%20conclude%20that%20our%20model%20also%20matches%20the%20Madau%20%26%20Dickinson%20(2014)%20fit%20within%20a%20factor%20of%20%E2%88%BC2.%20This%20level%20of%20agreement%20is%20remarkable,%20considering%20the%20simplicity%20of%20the%20SP21%20model.%20As%20a%20reference,%20sophisticated%20hydrodynamic%20cosmological%20simulations,%20while%20providing%20an%20overall%20better%20agreement%20with%20the%20data,%20exhibit%20discrepancies%20of%20a%20factor%20of%20%E2%88%BC%202%20or%20larger%20around%20cosmic%20noon%20(McCarthy%20et%20al.%202017;%20Dave%20%CC%81%20et%20al.%202019)%20and%20within%20a%20factor%20of%20%E2%88%BC%201.6%20at%20z%20%02%203%20(Salcido%20et%20al.%202018).%20However,%20other%20state-of-the-%20art%20cosmological%20simulations,%20such%20as%20IllustrisTNG%20(Pillepich%20et%20al.%202018),%20achieve%20a%20significantly%20more%20accurate%20match%20with%20observations%20of%20the%20CSFRD%20(Weinberger%20et%20al.%202017).%20Thus,%20whereas%20the%20SP21%20model%20manages%20to%20broadly%20reproduce%20both%20observed%20and%20simulated%20trends%20of%20the%20CSFRD%20(Scharre%20%CC%81,%20Sorini%20%26%20Dave%20%CC%81%202024),%20there%20is%20certainly%20room%20for%20further%20improvement.%20A%20detailed%20discussion%20on%20the%20limitations%20of%20the%20SP21%20model%20and%20the%20resulting%20impact%20on%20the%20predicted%20CSFRD%20can%20be%20found%20in%20Sorini%20%26%20Peacock%20(2021).%0AIt%20is%20also%20worth%20noting%20that%20the%20data%20in%20the%20Madau%20%26%20Dickinson%20(2014)%20compilation%20might%20be%20affected%20by%20biases%20introduced%20by%20the%20parametric%20models%20underlying%20the%20estimation%20of%20the%20CSFRD%20from%20observations%20of%20galaxy%20spectral%20energy%20distributions%20(Carnall%20et%20al.%202019).%20Additionally,%20subsequent%20estimates%20of%20the%20CSFRD%20deviate%20from%20the%20Madau%20%26%20Dickinson%20(2014)%20fit,%20both%20at%20high%20(Gruppioni%20et%20al.%202015;%20Rowan-Robinson%20et%20al.%202016)%20and%20low%20(Gruppioni%20et%20al.%202015)%20redshifts.%20Other%20measurements%20exhibited%20a%20lower%20normalization%20for%20the%20CSFRD%20at%20the%20peak%20of%20star%20formation%20(Liu%20et%20al.%202018),%20or%20a%20plateau%20at%20cosmic%20noon%20(Traina%20et%20al.%202024).%20The%20spread%20of%20some%20of%20the%20data%20points%20from%20these%20more%20recent%20measurements%20is%20generally%20comparable%20to,%20or%20even%20larger%20than,%20the%20discrepancy%20between%20the%20SP21%20model%20and%20the%20Madau%20%26%20Dickinson%20(2014)%20fit.%20Given%20this%20range%20of%20predictions,%20the%20precision%20of%20the%20CSFRD%20given%20by%20the%20SP21%20model%20seems%20satisfactory%20for%20the%20purpose%20of%20the%20present%20investigation.%0AAs%20for%20the%20performance%20of%20the%20model%20in%20predicting%20the%20future%20star%20formation%20history,%20we%20note%20that%20the%20shape%20of%20the%20future%20CSFRD%20is%20essentially%20dictated%20by%20the%20evolution%20of%20the%20cooling%20radius%20and%20of%20the%20critical%20temperature%20(see%20Appendix%20A%20for%20a%20detailed%20derivation%20of%20the%20asymptotic%20scaling%20of%20the%20CSFRD).%20The%20normalization%20of%20the%20CSFRD%20for%20z%20%3C%200%20will%20follow%20the%20overall%20normalization%20of%20the%20CSFRD%20at%20earlier%20times.%20Thus,%20given%20a%20factor%20of%20%E2%88%BC%202%20uncertainty%20in%20the%20past%20star%20formation%20history%20at%20z%20%E2%89%88%200,%20we%20can%20reasonably%20expect%20a%20similar%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0AFigure%201.%20Redshift%20evolution%20of%20the%20nSFR%20(left%20panel)%20and%20stellar%20mass%20fraction%20(middle%20panel)%20for%20a%20halo%20with%20virial%20temperature%20T%20%3D%20105%20K,%20as%20predicted%20at%20every%20iteration%20of%20our%20updated%20version%20of%20Sorini%20%26%20Peacock%20(2021)%20described%20in%20Section%202.2.%20The%20right%20panel%20shows%20the%20prediction%20for%20the%20redshift%20evolution%20of%20the%20critical%20temperature.%20The%20black%20lines%20labelled%20%E2%80%98iteration%200%E2%80%99%20refer%20to%20the%20predictions%20of%20the%20original%20SP21%20model%20with%20f%E2%88%97(T%20,%20z)%20%3D%200;%20thus%20no%20black%20line%20appears%20in%20the%20middle%20panel,%20which%20adopts%20a%20logarithmic%20scale%20on%20the%20y-axis.%20For%20all%20quantities%20plotted%20in%20this%20figure,%20full%20convergence%20is%20achieved%20after%20five%20iterations.%0AFigure2.%20Upperpanels:CSFRD,ascomputedwithourextensionoftheSP21model(dashedblackline)andasgivenbytheoriginalSP21model(dot%E2%80%93dashed%20red%20line).%20In%20the%20left%20panel,%20we%20further%20show%20with%20the%20empirical%20fit%20(dotted%20black%20line)%20to%20a%20compilation%20of%20observational%20data%20provided%20by%20Madau%20%26%20Dickinson%20(2014)%20(grey%20data%20points).%20The%20right%20panel%20shows%20the%20behaviour%20of%20the%20SP21%20model%20and%20our%20extended%20formalism%20in%20the%20future%20of%20the%20Universe.%20As%20future%20times%20correspond%20to%20negative%20redshifts,%20and%20the%20CSFRD%20becomes%20asymptotically%20null%20at%20arbitrarily%20large%20times,%20both%20axes%20switch%20from%20logarithmic%20to%20linear%20scale%20when%20moving%20from%20the%20left%20to%20the%20right%20panel.%20The%20grey%20hatched%20shaded%20area%20in%20the%20right%20panel%20excludes%20the%20non-physical%20region%20corresponding%20to%20%CF%81%20%CC%87%E2%88%97%20%3C%200.%20Lower%20panels:%20Relative%20difference%20between%20the%20two%20models%20showed%20above%20(dashed%20black%20line),%20and%20between%20our%20formalism%20and%20the%20fit%20from%20Madau%20%26%20Dickinson%20(2014)%20(dotted%20black%20line).%20The%20latter%20comparison%20is%20available%20only%20for%20positive%20redshifts%20(i.e.%20past%20cosmic%20times),%20as%20the%20fit%20represents%20a%20purely%20empirical%20fit%20to%20the%20data,%20and%20not%20a%20predictive%20theoretical%20model.%20The%20SP21%20model%20agrees%20well%20with%20the%20improved%20method%20introduced%20in%20this%20work%20at%20high%20redshift,%20but%20overestimates%20the%20CSFRD%20at%20lower%20redshift%20and%20in%20the%20future.%20In%20the%20redshift%20range%200%20%3C%20z%20%3C%2010,%20both%20the%20original%20SP21%20model%20and%20our%20updated%20formalism%20agree%20with%20the%20empirical%20fit%20within%20a%20factor%20of%20%E2%88%BC%202.%0Asystematic%20imprecision%20in%20the%20future%20CSFRD.%20As%20it%20will%20become%20apparent%20in%20Section%204.1,%20the%20future%20behaviour%20of%20the%20CSFRD%20may%20contribute%20significantly%20to%20the%20overall%20production%20of%20stars%20in%20the%20entire%20history%20of%20the%20universe.%20However,%20as%20we%20will%20show,%20the%20differences%20in%20the%20total%20stellar%20mass%20produced%20in%20the%20universe%20for%20different%20values%20of%20the%20cosmological%20constant%20can%20span%20several%20orders%20of%20magnitude,%20thus%0Aour%20main%20conclusions%20will%20be%20largely%20unaffected%20by%20an%20uncertainty%20of%20a%20factor%202%20in%20the%20CSFRD.%0AWe%20also%20highlight%20that%20our%20analysis%20is%20relative,%20i.e.%20based%20on%20the%20ratio%20of%20the%20star%20formation%20histories%20produced%20in%20different%20cosmologies,%20and%20will%20not%20depend%20on%20the%20absolute%20value%20of%20the%20star%20formation%20efficiency%20in%20a%20single%20cosmology.%20Of%20course,%20our%20posterior%20for%20%02%20will%0AImpact%20of%20%02%20on%20star%20formation%20history%201455%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1456%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0Astill%20be%20subject%20to%20accepting%20the%20underlying%20simplified%20astrophysical%20model%20for%20cosmic%20star%20formation.%20We%20discuss%20the%20impact%20of%20these%20limitations%20of%20our%20study,%20and%20prospects%20for%20their%20future%20amelioration,%20in%20Section%205.4.%0A3%20IMPACT%20OF%20CHANGING%20%02%0AHaving%20introduced%20the%20basic%20formalism%20in%20the%20previous%20section,%20we%20will%20now%20explain%20how%20varying%20%02%20would%20affect%20the%20star%20formation%20history.%20Changing%20the%20cosmological%20constant%20obviously%20affects%20the%20evolution%20of%20large-scale%20structure;%20this%20is%20the%20subject%20of%20Section%203.1.%20However,%20a%20different%20value%20of%20%02%20can%20also%20directly%20affect%20the%20cooling%20rate%20within%20haloes.%20The%20astrophysical%20impact%20of%20changing%20%02%20will%20be%20discussed%20in%20Section%203.2.%0A3.1%20Cosmological%20impact%0ATo%20begin%20with,%20we%20need%20to%20understand%20how%20the%20empirical%20z%20%3D%200%20cosmological%20parameters%20such%20as%20H0%20are%20altered%20when%20we%20vary%20%02.%20We%20will%20restrict%20our%20discussion%20to%20arbitrary%20flat%02CDM%20universes%20only%20%E2%80%93%20implicitly%20assuming%20an%20inflationary%20multiverse.%20In%20addition,%20all%20the%20dimensionless%20parameters%20of%20the%20%02CDM%20model%20are%20assumed%20to%20be%20unchanged:%20the%20horizon-scale%20amplitude%20As,%20the%20ratio%20of%20baryonic%20and%20CDM%20densities,%20and%20the%20ratio%20of%20photon%20and%20baryon%20number%20densities.%20The%20consequence%20is%20that%20all%20model%20universes%20under%20study%20are%20indistinguishable%20copies%20of%20our%20own%20Universe%20at%20very%20high%20redshifts%20where%20%02%20is%20dynamically%20unimportant.%20However,%20in%20endowing%20them%20with%20different%20values%20of%20%02,%20the%20histories%20of%20these%20copies%20become%20diverse%20at%20later%20times.%20The%20reason%20for%20this%20restriction%20is%20the%20measure%20problem:%20for%20%02%20we%20can%20appeal%20to%20Weinberg%E2%80%99s%20argument%20for%20a%20flat%20prior,%20but%20if%20other%20parameters%20were%20to%20vary%20then%20we%20have%20little%20idea%20what%20the%20appropriate%20prior%20would%20be.%20Therefore,%20we%20concentrate%20on%20this%20simplest%20ensemble.%0AIt%20is%20convenient%20to%20quantify%20these%20alternative%20models%20using%20the%20standard%20set%20of%20parameters%20for%20the%20%02CDM%20model:%20m,%20%02,%20etc.%20We%20will%20add%20the%20subscript%20%E2%80%98ref%E2%80%99%20when%20we%20indicate%20the%20corresponding%20parameters%20in%20our%20own%20Universe.%20This%20requires%20a%20definition%20of%20the%20%E2%80%98present%E2%80%99,%20which%20we%20take%20to%20be%20the%20point%20at%20which%20the%20CMB%20temperature%20equals%20the%20standard%20value%20(T0%20%3D%202.7255%20K;%20see%20Planck%20Collaboration%20VI%202020).%20We%20normalize%20the%20scale%20factor%20such%20that%20a%20%3D%201%20at%20this%20point,%20so%20that%20the%20radiation%20density%20in%20all%20models%20scales%20%E2%88%9D%20a%E2%88%924,%20with%20the%20identical%20constant%20of%20proportionality%20in%20all%20cases.%20However,%20changing%20the%20value%20of%20%02%20means%20that%20the%20time%20corresponding%20to%20T%20%3D%20T0%20and%20the%20Hubble%20parameter%20at%20that%20point%20take%20values%20that%20are%20different%20from%20those%20in%20the%20observed%20Universe,%20as%20we%20now%20explain.%0AWe%20start%20by%20noting%20that%20the%20Friedmann%20equation%20for%20a%20flat%20universe%20applies%20in%20all%20cases,%20so%20at%20times%20relevant%20for%20star%20formation%20we%20have%0AH2%3DH02(ma%E2%88%923+%02).%20(18)%0AWe%20preserve%20the%20matter%20density%20and%20re-scale%20the%20vacuum%20density%20by%20a%20factor%20%CE%B1%02%20%3D%20%02/%02ref%20,%20where%20%02ref%20is%20the%20cosmological%20constant%20in%20thereferencecosmology,sothat%CF%81m%20%3D%CF%81m,ref%20and%CF%81%02%20%3D%CE%B1%02%CF%81%02,ref%20in%20all%20universes.%20The%20Hubble%20parameter%20in%20the%20generic%20universe%20of%20the%20ensemble%20can%20be%20then%20be%20re-written%20as%0AH2%3DH2%20a%E2%88%923+%CE%B1H2%20.%20(19)%200,ref%20m,ref%20%02%200,ref%20%02,ref%0AWriting%20the%20ratio%20of%20the%20two%20terms%20on%20the%20right%20hand%20side%20of%20equations%0AFigure3.%20EvolutionoftheHubbleparameterH(z)fordifferentre-scalingsof%20the%20cosmological%20constant.%20For%20larger%20values%20of%20%02,%20the%20asymptotic%20value%20of%20the%20Hubble%20constant%20in%20the%20far%20future%20of%20the%20universe%20(negative%20redshift)%20increases.%20In%20an%20EdS%20universe,%20the%20Hubble%20constant%20becomes%20indefinitely%20small%20as%20the%20cosmic%20time%20tends%20to%20infinity.%0AHowever,%20the%20unchanged%20matter%20density%20tells%20us%20that%20H02%20%3D%20H%202%20/%20,%20so%20that%0A(18)%20and%20(19),%20and%20remembering%20that%20flatness%20always%20ensures%201%20%E2%88%92%20m,%20we%20deduce%0Am%20%3D%20m,ref%20.%201%E2%88%92%20m%20%CE%B1%02(1%E2%88%92%20m,ref)%0A%02%20%3D%20(20)%0Aat%20earlier%20redshifts,%20and%20this%20value%20increases%20with%20%02.%20This%20reflects%20the%20fact%20that%20the%20early%20phase%20of%20all%20models%20is%20Einstein%E2%80%93de%20Sitter%20(EdS),%20with%20H%20%E2%88%9D%20a%E2%88%923/2,%20while%20H%20freezes%20out%20at%20the%20value%20of%20the%20EdS%20model%20at%20the%20point%20where%20%02%20comes%20to%20dominate.%0ACalculating%20the%20new%20value%20of%20%CF%838%20and%20its%20scaling%20with%20%CE%B1%02%20is%20more%0Acomplicated:%20%CF%838%20is%20affected%20both%20by%20the%20new%20m,%20which%20changes%0Athe%20power%20spectrum%20shape%20and%20evolution,%20and%20by%20the%20new%20Hubble%0Aconstant,%20which%20changes%20the%20scale%208%20h%E2%88%921%20cMpc.%20These%20effects%20can%0Abe%20allowed%20for%20by%20recalling%20that%20conditions%20at%20very%20high%20z%20are%20the%0Asame%20in%20all%20models,%20so%20that%20%CF%83(8h%E2%88%921cMpc)%20is%20known%20at%20z%20%E2%89%AB%201.%20We%20ref%0Athen%20compute%20the%20scale-dependent%20correction%20numerically,%20using%20the%20code%20CAMB%20(Lewis,%20Challinor%20%26%20Lasenby%202000;%20Lewis%20%26%20Challinor%202011).%20This%20dependence%20of%20%CF%838%20on%20%02%20was%20studied%20by%20Oh%20et%20al.%20(2022),%20who%20showed%20that%20as%20the%20universe%20approaches%20an%20EdS%20solution,%20%CF%838%20exhibits%20an%20asymptotic%20behaviour%20(see%20their%20fig.%201).%20For%20a%20Planck-%202018%20cosmology,%20the%20asymptotic%20value%20of%20%CF%838%20for%20low%20values%20of%20%02%20is%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0Am,%20ref%200,ref%20m%2022%0AH0%20%3DH0,ref%5B%CE%B1%02%20+(1%E2%88%92%CE%B1%02)%20m,%20ref%5D.%20(21)%0AHence,%20we%20readily%20obtain%20the%20new%20Hubble%20parameter%20and%20density%20parameter%20at%20a%20%3D%201,%20for%20a%20given%20scaling%20of%20%02.%20The%20cosmic%20baryon%20mass%20fraction%20is%20taken%20as%20unchanged.%20Similar%20reasoning%20was%20given%20by%20Oh%20et%20al.%20(2022).%0AThis%20argument%20applies%20for%20any%20sign%20of%20%CE%B1%02,%20but%20rescaling%20to%20negative%20values%20complicates%20things.%20The%20expansion%20ceases%20at%20some%20maximum%20value%20of%20a%20and%20the%20universe%20subsequently%20undergoes%20recollapse.%20In%20principle,%20the%20SP21%20code%20should%20be%20able%20to%20handle%20such%20changes,%20although%20in%20practice%20it%20will%20require%20amendment%20in%20order%20to%20cope%20with%20a%20non-monotonic%20a(t)%20relation.%20However,%20in%20any%20case,%20the%20physical%20effect%20on%20the%20CSFRD%20of%20a%20negative%20%02%20is%20qualitatively%20different%20to%20that%20of%20a%20positive%20value.%20In%20the%20latter%20case,%20a%20large%20value%20of%20%02%20is%20expected%20to%20suppress%20star%20formation%20because%20structure%20growth%20ceases%20before%20galaxy-scale%20haloes%20can%20be%20generated.%20The%20main%20aim%20of%20this%20paper%20is%20to%20make%20a%20quantitative%20estimate%20of%20this%20suppression.%20In%20recollapsing%20models,%20however,%20growth%20does%20not%20freeze%20out%20and%20the%20relevant%20question%20is%20how%20much%20star%20formation%20can%20occur%20in%20the%20limited%20time%20before%20the%20big%20crunch.%20We%20aim%20to%20consider%20this%20question%20elsewhere.%0AIn%20Fig.%203,%20we%20show%20the%20resulting%20redshift%20evolution%20of%20the%20Hubble%0Aparameter%20for%20different%20values%20of%20%CE%B1%02%20.%20As%20expected,%20for%20larger%20values%20of%0A%02%20the%20Hubble%20parameter%20reaches%20its%20asymptotic%20value%20H%20%E2%86%92%20H%20%E2%88%9A%200%02%0A%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%20%20%0AFigure4.%20Sheth%E2%80%93Tormenhalomassfunctionatdifferentredshift,represented%20by%20different%20colours,%20and%20for%20different%20values%20of%20%02%20(different%20line%20styles).%20For%20a%20given%20redshift,%20larger%20values%20of%20%02%20result%20in%20a%20mass%20cut%20at%20smaller%20halo%20masses.%20In%20an%20EdS%20universe,%20the%20mass%20cut%20grows%20indefinitely%20in%20the%20future%20(z%20%3C%200).%20This%20is%20not%20the%20case%20for%20%02%20%3E%200,%20where%20the%20mass%20cut%20approaches%20an%20asymptotic%20value%20as%20z%20%E2%86%92%20%E2%88%921%20(i.e.%20t%20%E2%86%92%20%E2%88%9E).%20This%20is%20a%20consequence%20of%20the%20freeze%20out%20of%20structure%20formation%20in%20the%20%02CDM%20model.%0A0.6786%20(Oh%20et%20al.%202022).%20If%20one%20increases%20the%20value%20of%20%02%20with%20respect%20to%20the%20fiducial%20one,%20%CF%838%20reaches%20a%20maximum%20of%20%CF%838%20%E2%88%BC%200.9%20at%20around%20%CE%B1%02%20%E2%88%BC%208.%20For%20larger%20values%20of%20%02,%20it%20keeps%20declining.%20As%20a%20reference,%20for%20%CE%B1%02%20%3D%201000%20one%20obtains%20%CF%838%20%E2%89%88%200.5.%0AWe%20are%20now%20fully%20equipped%20to%20study%20the%20impact%20of%20%02%20on%20the%20CSFRD.%20The%20most%20obvious%20effect%20of%20altering%20%02%20is%20a%20modification%20of%20the%20cosmological%20term%20in%20the%20integrand%20of%20equation%20(1),%20i.e.%20the%20halo%20mass%20function.%20A%20larger%20cosmological%20constant%20causes%20the%20%02-dominated%20accelerating%20phase%20to%20begin%20earlier.%20Consequently,%20the%20freeze-out%20of%20structure%20formation%20would%20also%20occur%20earlier,%20thus%20shifting%20the%20truncation%20of%20the%20halo%20mass%20function%20to%20smaller%20halo%20masses,%20and%20decreasing%20the%20merger%20rate.%20This%20is%20exactly%20what%20we%20observe%20in%20Fig.%204,%20which%20plots%20the%20halo%20multiplicity%20function%20for%20different%20values%20of%20%CE%B1%02%20and%20redshift.%20For%20a%20cosmological%20constant%20100%20times%20larger%20than%20the%20observed%20value%20(dashed%20lines),%20the%20cut-off%20in%20the%20HMF%20drops%20by%20two%20orders%20of%20magnitude%20at%20z%20%3D%200.%20For%20reduced%20values%20of%20%02,%20the%20trend%20is%20opposite.%20At%20any%20given%20redshift,%20models%20with%20%CE%B1%02%20%3C%201%20overproduce%20massive%20haloes%20relative%20to%20the%20reference%20cosmology.%20In%20the%20limit%20of%20an%20EdS%20universe%20(not%20plotted%20in%20the%20figure),%20there%20is%20no%20freeze-out%20and%20the%20halo%20mass%20cut-off%20becomes%20arbitrarily%20large%20at%20sufficiently%20great%20times.%0ATo%20summarize,%20lower%20positive%20values%20of%20%02%20promote%20structure%20forma-%20tion.%20However,%20it%20would%20be%20premature%20to%20assume%20that%20the%20probability%20of%20generating%20observers%20is%20solely%20dependent%20on%20the%20collapsed%20mass%20fraction%20in%20the%20universe,%20and%20thus%20to%20conclude%20that%20the%20EdS%20model%20is%20anthropically%20favoured.%20Rather,%20we%20must%20consider%20the%20effect%20of%20%02%20on%20the%20astrophysical%20processes%20that%20govern%20cosmological%20star%20formation.%20This%20will%20be%20the%20focus%20of%20the%20next%20section.%0A3.2%20Astrophysical%20impact%0AAltering%20%02%20affects%20the%20astrophysics%20of%20star%20formation%20because%20both%20the%20gas%20mass%20fraction%20in%20haloes%20and%20the%20gas%20cooling%20rate%20depend%20on%20the%20virial%20quantities%20of%20haloes,%20which%20in%20turn%20are%20sensitive%20to%20the%20background%20cosmology%20via%20the%20Hubble%20parameter%20(see%20Section%202).%20In%20this%20section,%20we%20will%20illustrate%20these%20points%20in%20detail.%0AFigure5.%20Redshiftevolutionofthecoolingradiusrcool,inunitsofthevirial%20radius,%20at%20fixed%20virial%20temperature.%20Different%20colours%20refer%20to%20different%20values%20of%20%02,%20while%20different%20line%20styles%20represent%20different%20virial%20temperatures.%20In%20an%20EdS%20universe,%20the%20cooling%20radius%20shrinks%20indefinitely%20in%20the%20far%20future.%20For%20%02%20%3E%200,%20it%20reaches%20a%20minimum%20value,%20and%20then%20slowly%20resumes%20increasing.%20This%20is%20a%20consequence%20of%20the%20freeze-out%20of%20structure%20formation:%20see%20Section%203.2.1%20for%20further%20details.%0AIn%20this%20work,%20we%20will%20fix%20the%20astrophysical%20parameters%20to%20the%20%E2%80%98best-%20fitting%E2%80%99%20choice%20in%20SP21,%20i.e.:%20%CE%B7%20%3D%201.9,%20%E2%9F%A8t%E2%88%97%20%E2%9F%A9%20%3D%203.87%20Gyr%20and%20Tmin%20%3D%20104.5%20K.%0A3.2.1%20Cooling%20rate%0AAs%20we%20recalled%20in%20Section%202,%20within%20the%20SP21%20model%20the%20low-redshift%20SFR%20is%20set%20by%20the%20gas%20cooling%20rate.%20This%20is%20in%20turn%20determined%20by%20the%20extent%20of%20the%20cooling%20radius%20rcool.%20We%20will%20now%20analyse%20how%20the%20virial%20temperature%20of%20the%20halo%20affects%20the%20cooling%20radius,%20and%20how%20this%20evolves%20depending%20on%20the%20cosmological%20model.%20A%20particular%20emphasis%20will%20be%20given%20to%20the%20asymptotic%20behaviour%20of%20the%20cooling%20radius%20for%20arbitrarily%20large%20cosmic%20times.%0AWe%20can%20express%20the%20cooling%20radius%20in%20terms%20of%20the%20cooling%20time%20by%20obtaining%20the%20gas%20density%20at%20rcool%20via%20equation%20(7),%20and%20then%20inserting%20it%20into%20equation%20(8):%0A%13rcool%14%CE%B7%20%3D%20(3%E2%88%92%CE%B7)%CE%BCX2C(T)Mgas%20t%20.%20(22)%20R%206%CF%80kBTm2HR3%20cool%0ATo%20obtain%20an%20explicit%20dependence%20of%20rcool%20on%20redshift,%20we%20can%20simply%20replace%20tcool%20with%20the%20prescription%20for%20the%20%E2%80%98effective%20cooling%20time%E2%80%99%20given%20by%20equation%20(10),%20making%20use%20of%20the%20cosmic%20time-redshift%20relationship%20defined%20by%20equation%20(12).%20We%20can%20further%20apply%20the%20definition%20Mgas%20%3D%20fgas(T%20,%20z)M,%20and%20then%20express%20all%20virial%20quantities%20in%20terms%20of%20T%20through%20equations%20(2)%E2%80%93(5).%0A%13rcool%20%14%CE%B7%20R%0A%03%05%20(3%20%E2%88%92%20%CE%B7)X2%CE%BC%20C(T%20)%0AImpact%20of%20%02%20on%20star%20formation%20history%201457%0A%3D%202%206%CF%80kBGm2H%0AT%20t%20(%20z%20)%0Afgas(T,%20z)H(z)%20%05%20m%20%07%20m1%0A%06%0A%C3%97%201%E2%88%92E+%0A%04%0Afdyn%20tdyn%20(z)%0A.%20(23)%0AWe%20can%20therefore%20follow%20the%20redshift%20evolution%20of%20rcool;%20this%20is%20shown%20in%20Fig.%205%20for%20different%20virial%20temperatures%20and%20different%20values%20of%20%02.%0AFor%20a%20fixed%20value%20of%20%02,%20rcool/R%20decreases%20with%20increasing%20virial%20temperature.%20In%20haloes%20above%20the%20critical%20temperature%20Tcrit%20(see%20Section%202),%20this%20behaviour%20can%20be%20readily%20understood%20from%20equation%20(23).%20If%20T%20%3E%20Tcrit,%20then%20the%20gas%20mass%20fraction%20in%20haloes%20reaches%20its%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%20%20%20%0A1458%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0AFigure%206.%20Redshift%20evolution%20of%20the%20critical%20virial%20temperature%20Tcrit%20above%20which%20the%20baryon%20mass%20fraction%20in%20haloes%20saturates%20to%20the%20cosmic%20value%20fb%20%3D%20b/%20m,%20according%20to%20the%20SP21%20model,%20for%20different%20values%20of%20the%20cosmological%20constant%20(see%20colour%20bar).%20The%20scaling%20of%20Tcrit%20with%20redshift%20mirrors%20the%20relationship%20between%20cosmic%20time%20and%20redshift.%20In%20an%20EdS%20universe,%20the%20long-term%20behaviour%20resembles%20a%20power%20law.%20As%20%02%20increases,%20Tcrit(z)%20deviates%20from%20a%20power%20law%20at%20progressively%20earlier%20redshifts%20(see%20Section%203.2.2%20for%20further%20details).%0Amaximum%20value,%20and%20no%20longer%20depends%20on%20the%20virial%20temperature.%20The%20cooling%20radius%20is%20then%20simply%20proportional%20to%20(C(T%20)/T%20)1/%CE%B7.%20For%20T%20%03%20106%20K,%20a%20Sutherland%20%26%20Dopita%20(1993)%20cooling%20function%20for%20an%20H/He%20plasma%20with%20primordial%20abundances%20scales%20approximately%20as%20%E2%88%9D%20T%201/2%20.%20Therefore,%20rcool%20/R%20%E2%88%9D%20T%20%E2%88%921/2%CE%B7%20,%20so%20the%20cooling%20radius%20decreases%20with%20temperature%20in%20the%20regime%20considered.%20For%20T%20%3C%20Tcrit,%20there%20is%20an%20extra%20temperature-dependence%20carried%20by%20the%20fgas(T%20,%20z)%20factor,%20but%20we%20verified%20that%20this%20does%20not%20qualitatively%20impact%20the%20scaling%20of%20rcool%20with%20temperature.%0AWe%20can%20now%20examine%20the%20redshift%20evolution%20of%20rcool/R%20for%20a%20fixed%20virial%20temperature,%20for%20different%20cosmological%20models.%20To%20begin%20with,%20we%20restrict%20the%20discussion%20to%20universes%20with%20%02%20%3E%200.%20At%20high%20redshift,%20the%20effective%20cooling%20time%20is%20approximately%20equal%20to%20the%20dynamical%20time%20(see%20equation%2010).%20Thus,%20the%20factor%20containing%20the%20cosmic%20time%20in%20equation%20(23)%20is%20very%20close%20to%20unity.%20Also,%20the%20critical%20temperature%20drops%20at%20sufficiently%20high%20redshift%20(see%20SP21%20and%20Fig.%206).%20Therefore,%20at%20early%20times,%20we%20have%20that%20fgas%20%E2%89%88%20fb,%20halo%20%E2%89%88%20fb%20,%20so%20that%20the%20evolution%20of%20the%20cooling%20radius%20is%20dictated%20primarily%20by%20the%20Hubble%20parameter%20in%20equation%20(23).%20In%20this%20regime,%20the%20universe%20is%20dominated%20by%20matter,%20thus%20the%20cooling%20radius%20scales%20as%20H%20(z)1/%CE%B7%20%E2%88%9D%20(1%20+%20z)3/2%CE%B7%20.%20Recalling%20that%20we%20set%20%CE%B7%20%3D%201.9,%20we%20would%20then%20expect%20that%20the%20redshift-evolution%20of%20rcool/R%20to%20be%20a%20power%20law%20with%20index%20close%20to%203/4%20at%20high%20redshift,%20where%20all%20models%20are%20matter%20dominated.%20This%20is%20exactly%20what%20we%20observe%20in%20Fig.%205.%0AConversely,%20as%20the%20universe%20becomes%20dominated%20by%20%02,%20the%20effective%20cooling%20time%20given%20by%20equation%20(10)%20approaches%20the%20cosmic%20time,%20so%20that%20the%20cooling%20radius%20scales%20as%20%E2%88%9D%20(fgas%20t)1/%CE%B7.%20To%20determine%20the%20evolution%20of%20the%20cooling%20radius,%20we%20then%20need%20to%20understand%20how%20fgas%20scales%20with%20redshift%20or,%20equivalently,%20cosmic%20time.%20Because%20for%20%02%20%3E%200%20the%20critical%20temperature%20is%20monotonically%20increasing%20with%20cosmic%20time%20after%20the%20point%20of%20%02-domination%20(see%20Fig.%206),%20there%20is%20always%20a%20sufficiently%20late%20cosmic%20time%20after%20which%20the%20virial%20temperature%20of%20a%20halo%20drops%20below%20Tcrit(z).%20Thereafter,%20the%20gas%20mass%20fraction%20within%20haloes%20of%20a%20given%20virial%20temperature%20scales%20with%20T%20/Tcrit%20(z)%20as%20in%20equation%20(15).%0AThe%20redshift%20evolution%20of%20Tcrit%20(z)%20needs%20to%20be%20computed%20numerically,%20but%20we%20can%20find%20an%20analytical%20approximation%20by%20simplifying%20the%20Sutherland%20%26%20Dopita%20(1993)%20cooling%20function%20with%20a%20piece-wise%20power%20law.%20Under%20this%20assumption,%20the%20critical%20temperature%20itself%0Ascales%20as%20a%20power%20of%20cosmic%20time,%20hence%20so%20does%20the%20cooling%20radius%20(see%20Appendix%20A2%20for%20details).%20We%20verified%20that%20the%20index%20of%20the%20power%20law%20is%20positive%20and%20below%20unity,%20explaining%20why%20the%20cooling%20radius%20increases%20after%20the%20onset%20of%20%02-domination,%20while%20its%20time%20derivative%20flattens%20out%20to%20zero.%20This%20is%20indeed%20the%20trend%20that%20we%20observe%20in%20Fig.%205%20at%20z%20%3C%200%20for%20all%20models%20with%20%02%20%3E%200.%20For%20a%20fixed%20redshift%20and%20virial%20temperature%20and%20after%20the%20point%20of%20%02-domination,%20rcool/R%20becomes%20larger%20as%20%02%20increases.%20To%20prevent%20the%20cooling%20radius%20from%20overflowing%20beyond%20the%20virial%20radius,%20we%20cap%20the%20ratio%20rcool/R%20to%20unity%20with%20a%20suitably%20smooth%20function,%20similar%20to%20the%20one%20used%20for%20the%20nSFR%20in%20equation%20(16).%0AThe%20behaviour%20that%20we%20just%20described%20descends%20from%20the%20definition%20of%20the%20boundaries%20of%20haloes:%20for%20a%20given%20virial%20temperature,%20it%20follows%20from%20equations%20(2)%E2%80%93(5)%20that%20both%20the%20virial%20radius%20and%20virial%20mass%20of%20a%20halo%20scale%20as%201/H%20(z).%20Thus,%20in%20a%20universe%20with%20a%20larger%20value%20of%20%02,%20haloes%20of%20a%20given%20temperature%20become%20more%20concentrated%20(see%20Fig.%203).%20Therefore,%20the%20gas%20density%20becomes%20overall%20larger,%20and%20consequently%20cooling%20becomes%20more%20efficient.%20However,%20in%20the%20far%20future%20of%20the%20universe%20structure%20formation%20freezes%20out,%20and%20haloes%20become%20isolated.%20Eventually,%20the%20SP21%20framework%20assumes%20that%20haloes%20will%20expel%20their%20baryon%20content%20through%20feedback%20processes,%20progressively%20becoming%20devoid%20of%20gas%20and%20unable%20to%20form%20stars%20(see%20Fig.%201).%0ATo%20summarize,%20we%20saw%20that%20in%20a%20universe%20with%20%02%20%3E%200%20the%20cooling%20radius%20decreases%20as%20a%20power%20of%20time%20at%20high%20redshift,%20but%20that%20it%20eventually%20switches%20to%20evolve%20in%20the%20opposite%20direction.%20The%20point%20of%20turnaround%20between%20these%20two%20regimes%20occurs%20at%20the%20transition%20to%20%02-domination,%20when%20the%20effective%20cooling%20time%20in%20equation%20(10)%20switches%20from%20being%20approximated%20by%20the%20dynamical%20time%20to%20tracking%20the%20cosmic%20time.%20As%20can%20be%20seen%20in%20Fig.%205,%20increasing%20%02%20moves%20the%20turnaround%20of%20the%20cooling%20radius%20to%20earlier%20redshifts.%20In%20contrast,%20there%20is%20no%20such%20turnaround%20in%20an%20EdS%20universe,%20and%20the%20cooling%20radius%20(in%20units%20of%20the%20virial%20radius)%20shrinks%20indefinitely%20as%20time%20goes%20by.%20Because%20there%20is%20no%20lower%20limit%20to%20the%20Hubble%20parameter,%20the%20size%20of%20haloes%20with%20a%20given%20virial%20temperature%20can%20become%20arbitrarily%20large.%20This%20makes%20the%20diffuseness%20of%20haloes%20and%20consequent%20inefficiency%20of%20gas%20cooling%20more%20dramatic%20than%20in%20any%20universe%20with%20positive%20%02.%0A3.2.2%20Gas%20mass%20fraction%20in%20haloes%0AAs%20explained%20in%20Section%202,%20the%20SP21%20model%20assumes%20that%20the%20baryon%20mass%20fraction%20in%20haloes%20scales%20with%20the%20virial%20temperature%20below%20a%20certain%20critical%20threshold%20Tcrit,%20and%20saturates%20to%20fb%20above%20this%20value.%20In%20our%20improved%20variation%20laid%20out%20in%20Section%202.2,%20the%20gas%20mass%20in%20haloes%20above%20the%20critical%20temperature%20saturates%20to%20fb%20%E2%88%92%20f%E2%88%97(T%20,%20z),%20where%20f%E2%88%97(T%20,%20z)%20is%20the%20stellar%20mass%20fraction%20of%20a%20halo%20of%20virial%20temperature%20T%20at%20redshift%20z.%20In%20either%20case,%20the%20physical%20significance%20of%20the%20critical%20temperature%20is%20the%20same.%0AIn%20haloes%20with%20T%20%3C%20Tcrit,%20the%20gaseous%20component%20is%20gravitationally%20bound%20only%20within%20a%20certain%20critical%20radius%20rcrit.%20Beyond%20this%20scale,%20the%20momentum%20injected%20by%20supernova-driven%20winds%20into%20the%20surrounding%20gas%20is%20sufficient%20to%20overcome%20gravity%20and%20eject%20the%20gas.%20In%20practice%20then,%20Tcrit%20is%20the%20temperature%20at%20which%20rcrit%20coincides%20with%20the%20boundary%20of%20the%20star-forming%20region%20within%20the%20halo.%20At%20high%20redshift,%20this%20is%20the%20entirety%20of%20the%20halo,%20so%20that%20Tcrit%20is%20determined%20by%20the%20condition%20rcrit%20%3C%20R.%20At%20low%20redshift,%20star%20formation%20is%20cooling-driven,%20and%20the%20relevant%20condition%20is%20rcrit%20%3C%20rcool%20.%0ASince%20the%20cosmological%20model%20affects%20both%20the%20virial%20radius%20and%20the%20cooling%20radius%20(see%20Section%203.2.1),%20the%20critical%20temperature%20is%20also%20sensitive%20to%20the%20cosmological%20parameters.%20In%20particular,%20in%20Fig.%206%20we%20show%20the%20redshift%20evolution%20of%20Tcrit%20for%20different%20values%20of%20%CE%B1%02%20.%20The%20late-%20time%20behaviour%20of%20the%20critical%20temperature%20can%20be%20understood%20starting%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0Afrom%20the%20condition%20rcrit%20%3D%20rcool,%20which%20defines%20Tcrit.%20As%20discussed%20in%20the%20previous%20section,%20the%20cooling%20radius%20explicitly%20depends%20on%20the%20cooling%20time;%20in%20the%20far%20future%20of%20the%20universe,%20rcool%20corresponds%20to%20the%20point%20at%20which%20the%20cooling%20time%20is%20approximately%20equal%20to%20the%20cosmic%20time.%0AIn%20a%20universe%20with%20%02%20%3E%200,%20under%20the%20reasonable%20approximation%20of%20the%20cooling%20function%20as%20a%20piece-wise%20power%20law,%20the%20critical%20temperature%20scales%20as%20a%20power%20of%20cosmic%20time%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E%20(see%20Appendix%20A2%20for%20details).%20We%20verified%20that%20the%20index%20of%20this%20power%20law%20is%20positive%20in%20all%20physically%20relevant%20cases.%20Thus,%20for%20%02%20%3E%200,%20the%20critical%20temperature%20increases%20at%20arbitrarily%20large%20times.%20The%20mapping%20between%20the%20scale%20factor%20and%20cosmic%20time%20becomes%20exponential%20for%20t%20%E2%86%92%20%E2%88%9E%20in%20models%20with%20%02%20%3E%200.%20Therefore,%20in%20the%20far%20future,%20the%20critical%20temperature%20scales%20with%20redshift%20as%20a%20power%20of%20%E2%88%92%20ln(1%20+%20z),%20and%20that%20is%20why%20the%20increase%20of%20Tcrit%20with%20redshift%20is%20rather%20slow%20(see%20Fig.%206).%0AIn%20an%20EdS%20universe,%20we%20observe%20a%20qualitatively%20different%20behaviour%20for%20the%20critical%20temperature,%20which%20decreases%20rather%20than%20increasing%20at%20late%20times.%20This%20can%20be%20understood%20by%20considering%20that%20for%20a%20fixed%20virial%20temperature,%20the%20virial%20radius%20scales%20%E2%88%9D%20H%20(z)%E2%88%921%20%3D%20(1%20+%20z)%E2%88%923/2%20,%20hence%20diverging%20for%20z%20%E2%86%92%20%E2%88%921.%20Therefore,%20all%20haloes%20of%20a%20given%20T%20become%20arbitrarily%20large%20in%20the%20far%20future,%20so%20that%20their%20baryonic%20mass%20fraction%20must%20eventually%20match%20the%20cosmic%20value.%20Within%20the%20SP21%20formalism,%20this%20means%20that%20all%20haloes%20must%20asymptotically%20exceed%20the%20critical%20temperature.%20This%20condition%20is%20achieved%20if%20Tcrit%20%E2%86%92%200%20for%20t%20%E2%86%92%20%E2%88%9E,%20in%20accord%20with%20our%20numerical%20results.%0AAt%20the%20high-redshift%20end%20of%20Fig.%206,%20all%20models%20start%20out%20with%20the%20same%0ATcrit(z)%20relation,%20because%20all%20universes%20are%20matter%20dominated%20at%20early%0Atimes.%20As%20we%20saw%20in%20Section%203.2.1,%20rcool%20in%20the%20far%20future%20increases%0Afor%20larger%20values%20of%20%02%20(see%20Fig.%205).%20The%20critical%20temperature%20increases%0Aaccordingly%20for%20%CE%B1%02%20%E2%89%A4%20100.%20For%20%CE%B1%02%20%3E%20100,%20the%20critical%20temperature%0Aappears%20to%20invert%20this%20trend,%20and%20although%20it%20still%20increases%20at%20later%0Atimes,%20its%20value%20is%20overall%20lower%20than%20in%20universes%20with%20a%20smaller%0Acosmological%20constant.%20The%20reason%20is%20that%20in%20universes%20with%20%CE%B1%02%20%E2%89%AB%0A1,%20haloes%20reach%20overall%20higher%20densities:%20for%20a%20fixed%20halo%20mass%20M%0A(or,%20equivalently,%20for%20a%20fixed%20virial%20temperature%20at%20future%20times),%20the%0AvirialradiusscalesasR%20%E2%88%9D%20M1/3H(z)%E2%88%922/3%20%E2%88%9D%20M1/3%CE%B1%E2%88%921/3.Ahighergas%20%02%0Adensity%20makes%20the%20cooling%20process%20more%20efficient%20(see%20equation%208),%20so%20that%20the%20cooling%20radius%20extends%20to%20the%20virial%20radius.%20This%20means%20that%20in%20practice%20the%20condition%20for%20determining%20the%20critical%20temperature,%20rcrit%20%3C%20rcool,%20translates%20to%20rcrit%20%3C%20R.%20It%20easily%20follows%20that%20Tcrit(z)%20%E2%88%9D%20H(z)%E2%88%922,%20regardless%20of%20the%20value%20of%20%CE%B7.%20At%20late%20times,%20H(z)2%20%E2%88%9D%20%CE%B1%02,%20therefore%20the%20critical%20temperature%20diminishes%20as%20%02%20increases%20(see%20Fig.%203).%0AThe%20behaviour%20of%20Tcrit%20shown%20in%20Fig.%206%20directly%20affects%20the%20nSFR%20both%20at%20high%20and%20low%20redshift,%20as%20it%20affects%20the%20evolution%20of%20the%20baryon%20mass%20fraction%20in%20haloes.%20Piecing%20together%20the%20discussion%20in%20Section%203.2.1%20and%20in%20this%20section,%20we%20gain%20two%20important%20physical%20insights%20into%20the%20astrophysical%20impact%20of%20%02%20on%20the%20SFR.%20For%20small%20values%20of%20%02,%20there%20is%20a%20larger%20gas%20mass%20fraction%20available%20in%20haloes%20at%20earlier%20times,%20but%20a%20progressively%20smaller%20fraction%20will%20cool%20down%20at%20later%20times.%20For%20larger%20values%20of%20%02,%20there%20is%20an%20overall%20smaller%20mass%20fraction%20of%20gas%20that%20resides%20within%20haloes%20at%20later%20times,%20but%20a%20larger%20fraction%20will%20cool%20down%20as%20time%20goes%20by.%20These%20features%20will%20play%20a%20prominent%20role%20in%20shaping%20the%20dependence%20of%20the%20cosmic%20star%20formation%20efficiency%20as%20a%20function%20of%20%02,%20as%20we%20will%20discuss%20in%20Sections%204.1%20and%204.2.%0A4%20RESULTS%0A4.1%20Long-term%20efficiency%20of%20star%20formation%0AIn%20this%20section,%20we%20will%20show%20our%20results%20for%20the%20star%20formation%20history%20for%20different%20values%20of%20%02,%20focusing%20especially%20on%20the%20long-term%20behaviour%20of%20star%20formation.%20Determining%20the%20asymptotic%20behaviour%0Aof%20the%20nSFR%20will%20tell%20us%20whether%20there%20is%20a%20well-defined%20final%20cumulative%20efficiency%20of%20cosmic%20star%20formation%20%E2%80%93%20both%20in%20individual%20haloes%20and%20in%20the%20universe%20as%20a%20whole.%0ALet%20us%20then%20start%20with%20the%20evolution%20of%20the%20nSFR.%20In%20the%20original%20SP21%20formalism,%20the%20nSFR%20is%20most%20naturally%20expressed%20for%20a%20fixed%20virial%20temperature,%20as%20that%20is%20the%20independent%20variable%20in%20the%20cooling%20function.%20Nevertheless,%20as%20we%20explained%20earlier%20in%20Section%202,%20the%20Sheth%E2%80%93Tormen%20formalism%20for%20the%20clustering%20of%20haloes%20allows%20us%20to%20convert%20straightforwardly%20between%20the%20fixed-T%20and%20the%20fixed-M%20views,%20making%20use%20of%20the%20definitions%20expressed%20in%20equations%20(2)%E2%80%93%20(5).%20For%20the%20remainder%20of%20this%20paper,%20the%20fixed-M%20view%20will%20be%20more%20convenient.%0AIn%20the%20upper%20panels%20in%20Fig.%207,%20we%20show%20the%20time%20evolution%20of%20the%20nSFR%20for%20populations%20of%20haloes%20at%20given%20virial%20mass.%20From%20left%20to%20right,%20we%20show%20the%20results%20for%20a%20typical%20dwarf%20galaxy%20(1010%20M%E2%8A%99),%20a%20Milky%20Way-like%20galaxy%20(1012%20M%E2%8A%99),%20and%20a%20galaxy%20cluster%20(1014%20M%E2%8A%99),%20respectively.%20The%20main%20independent%20variable%20in%20the%20plot%20is%20cosmic%20time;%20the%20upper%20x-axis%20shows%20the%20redshift%20that%20would%20be%20measured%20by%20an%20observer%20in%20the%20reference%20Universe%20for%20a%20given%20cosmic%20time.%20Therefore,%20it%20applies%20only%20to%20the%20case%20%CE%B1%02%20%3D%201.%0AWe%20note%20that%20for%20all%20halo%20masses%20and%20all%20cosmologies,%20the%20nSFR%20tends%20to%20a%20universal%20value%20at%20sufficiently%20early%20times.%20This%20happens%20because,%20for%20a%20given%20virial%20mass,%20the%20virial%20temperature%20increases%20at%20earlier%20times,%20while%20the%20critical%20temperature%20is%20lower%20(see%20Fig.%206).%20There%20will%20thus%20always%20be%20a%20sufficiently%20early%20time%20when%20the%20baryon%20mass%20fraction%20within%20haloes%20saturates%20to%20fb.%20Equation%20(6)%20then%20tells%20us%20that%20the%20nSFR%20reaches%20a%20fixed%20constant%20value%20in%20this%20regime.%0ASince%20all%20universes%20in%20our%20ensemble%20are%20identical%20and%20matter-%20dominated%20at%20a%20sufficiently%20high%20redshift,%20they%20are%20naturally%20indis-%20tinguishable%20at%20early%20times.%20But%20for%20all%20halo%20masses%20considered,%20the%20nSFR%20drops%20steeply%20in%20the%20future,%20and%20the%20departure%20from%20the%20EdS%20solution%20occurs%20earlier%20for%20universes%20with%20a%20larger%20value%20of%20%02.%20The%20asymptotic%20behaviour%20can%20be%20deduced%20by%20studying%20each%20factor%20in%20the%20r.h.s.%20of%20equation%20(13).%20As%20discussed%20in%20Section%203.2,%20the%20main%20challenge%20is%20understanding%20the%20late-time%20behaviour%20of%20the%20critical%20temperature,%20which%20needs%20to%20be%20determined%20numerically.%20However,%20under%20the%20reasonable%20assumption%20of%20a%20piece-wise%20cooling%20function,%20this%20asymptotic%20behaviour%20can%20be%20approximated%20analytically%20with%20a%20power%20of%20time.%20The%20index%20of%20the%20power%20law%20depends%20on%20the%20virial%20temperature%20of%20the%20halo%20and%20on%20the%20slope%20of%20the%20gas%20density%20profile.%20It%20is%20then%20possible%20to%20prove%20that%20the%20time%20integral%20of%20the%20nSFR%20at%20fixed%20mass%20(i.e.%20the%20stellar%20mass%20fraction)%20is%20convergent%20(although%20the%20argument%20is%20relatively%20detailed,%20and%20is%20presented%20in%20Appendix%20A).%20This%20convergence%20holds%20both%20in%20an%20EdS%20universe%20and%20for%20%02%20%3E%200,%20and%20is%20seen%20in%20Fig.%207,%20where%20the%20cumulative%20stellar%20mass%20fraction%20reaches%20a%20plateau%20at%20late%20times.%20We%20plot%20this%20asymptotic%20stellar%20mass%20fraction,%20f%E2%88%97%20%E2%88%9E%20,%20as%20a%20function%20of%20halo%20mass%20in%20Fig.%208.%0AFor%20all%20values%20of%20%02%20considered,%20we%20notice%20that%20f%E2%88%97%E2%88%9E%20increases%20with%20mass%20for%20very%20low%20masses.%20For%20%CE%B1%02%20%3C%201,%20the%20efficiency%20then%20peaks%20at%20M%20%E2%89%88%201011%20M%E2%8A%99,%20and%20then%20decreases%20for%20larger%20halo%20masses,%20with%20the%20drop%20being%20faster%20in%20an%20EdS%20universe.%20However,%20even%20for%20still%20relatively%20small%20values%20of%20%02,%20such%20as%20%CE%B1%02%20%3D%200.1,%20the%20asymptotic%20efficiency%20decreases%20more%20slowly%20after%20reaching%20its%20maximum,%20and%20for%20M%20%E2%89%88%201018%20M%E2%8A%99%20it%20is%20almost%20one%20order%20of%20magnitude%20larger%20than%20in%20an%20EdS%20universe.%20For%20%02%20at%20the%20observed%20level%20or%20above,%20the%20initial%20decline%20in%20efficiency%20above%20M%20%3D%201011%20M%E2%8A%99%20is%20rather%20slow,%20with%20a%20plateau%20of%20nearly%20constant%20efficiency,%20followed%20by%20an%20eventual%20sharper%20drop.%20In%20the%20case%20of%20%CE%B1%02%20%3D%20104,%20f%E2%88%97%E2%88%9E%20exhibits%20little%20variation%20over%20the%20nine%20decades%20of%20halo%20mass%20shown%20in%20Fig.%208.%20In%20such%20an%20extreme%20cosmology,%20the%20freeze%20out%20of%20structure%20formation%20occurs%20very%20early%20and%20the%20critical%20temperature%20is%20so%20low%20(see%20Fig.%206)%20that%20most%20haloes%20retain%20their%20cosmic%0AImpact%20of%20%02%20on%20star%20formation%20history%201459%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A%201460%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0AFigure%207.%20Upper%20panels:%20Time%20evolution%20of%20the%20SFR,%20normalized%20by%20the%20total%20halo%20mass.%20Each%20panel%20refers%20to%20a%20population%20of%20haloes%20with%20fixed%20virial%20mass%20M,%20as%20annotated%20in%20the%20figure.%20Different%20colours%20refer%20to%20different%20values%20of%20%02%20in%20units%20of%20the%20observed%20value%20%02obs,%20as%20reported%20in%20the%20colour%20bar.%20The%20lower%20x-axis%20reports%20the%20cosmic%20time,%20while%20the%20upper%20x-axis%20the%20corresponding%20redshift%20that%20would%20be%20observed%20in%20our%20Universe%20(%02%20%3D%20%02obs).%20The%20break%20in%20the%20upper%20x-axis%20indicates%20that%20the%20ticks%20are%20omitted%20for%20z%20%3E%20%E2%88%921%20+%2010%E2%88%9210,%20since%20the%20exponential%20growth%20of%20the%20scale%20factor%20in%20the%20future%20makes%20it%20increasingly%20hard%20to%20represent%20the%20redshift%20on%20a%20meaningful%20scale.%20Regardless%20of%20the%20halo%20mass,%20the%20normalized%20SFR%20tends%20to%20zero%20in%20the%20limit%20of%20t%20%E2%86%92%20%E2%88%9E.%20Lower%20panels:%20Fraction%20of%20the%20stellar%20mass%20produced%20up%20to%20a%20certain%20cosmic%20time,%20with%20respect%20to%20the%20total%20halo%20mass,%20for%20haloes%20with%20fixed%20virial%20mass.%20This%20is%20the%20time%20integral%20of%20the%20quantity%20shown%20in%20the%20corresponding%20upper%20panels.%20The%20colour%20coding%20and%20the%20upper%20and%20lower%20x-axes%20are%20the%20same%20as%20in%20the%20upper%20panels.%20In%20all%20haloes%20and%20cosmological%20models%20considered,%20the%20cumulative%20stellar%20mass%20fraction%20reaches%20an%20asymptotic%20value%20for%20t%20%E2%86%92%20%E2%88%9E.%0AD%0Aow%0Anloaded%0Afrom%0Ahtps:/academ%0Aic.oup.com%0A/m%0Anras/article/535/2/149/7896079%0Aby%0Aguest%20on%0A14%0AN%0Aovem%0Aber%202024%0AFigure%208.%20Asymptotic%20value%20(i.e.%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E)%20of%20the%20stellar%20mass%20fraction%20for%20a%20halo%20population%20with%20given%20total%20mass%20M.%20Different%20colours%20refer%20to%20different%20values%20of%20%02,%20as%20indicated%20in%20the%20colour%20bar.%20Even%20for%20large%20values%20of%20%02,%20massive%20haloes%20can%20still%20be%20very%20efficient%20(see%20Section%204.1%20for%20the%20detailed%20explanation).%0Ashare%20of%20baryons;%20thus%20they%20are%20always%20efficient%20at%20forming%20stars,%20regardless%20of%20their%20mass.%0AThe%20absence%20of%20a%20sharp%20peak%20in%20the%20stellar%20mass%20fraction%20and%20subsequent%20steep%20decline%20at%20the%20higher%20mass%20end%20might%20seem%20puzzling,%20at%20least%20for%20the%20case%20%CE%B1%20%3D%201,%20as%20it%20appears%20to%20contradict%20the%20current%20paradigm%20of%20galaxy%20formation%20(see%20e.g.%20Behroozi%20et%20al.%202019).%20However,%20we%20stress%20that%20Fig.%208%20shows%20the%20asymptotic%20stellar%20mass%20fraction%20(i.e.%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E),%20whereas%20the%20consensus%20on%20the%20mass%20dependence%20of%20the%20stellar%20mass%20fraction%20is%20based%20on%20observations%20and%20models%20of%20the%20past%20star%20formation%20history.%20We%20verified%20that%20our%20formalism%20would%20also%20predict%20a%20sharper%20peak%20and%20a%20steeper%20decline%20of%20the%20stellar%20mass%20fraction%20for%20M%20%03%201012%20M%E2%8A%99%20at%20z%20%3D%200,%0Ain%20qualitative%20agreement%20with%20the%20accepted%20framework%20for%20galaxy%20formation.%20Therefore,%20Fig.%208%20tells%20us%20that,%20according%20to%20our%20model,%20high-mass%20haloes%20produce%20a%20larger%20fraction%20of%20their%20asymptotic%20stellar%20mass%20in%20the%20future%20of%20the%20universe%20compared%20to%20low-mass%20haloes.%20Higher%20mass%20haloes%20appear%20to%20be%20especially%20efficient%20at%20forming%20stars%20in%20cosmologies%20with%20larger%20values%20of%20%02,%20where%20haloes%20of%20a%20given%20virial%20mass%20are%20overall%20denser,%20hence%20more%20efficient%20at%20cooling%20(see%20Section%203.2.2).%20While%20this%20argument%20provides%20us%20with%20a%20basic%20understanding%20of%20the%20trend%20observed%20in%20Fig.%205,%20one%20should%20bear%20in%20mind%20that%20gas%20cooling%20is%20not%20the%20only%20relevant%20process%20for%20star%20formation,%20even%20at%20late%20times.%20While%20we%20incorporate%20the%20SP21%20model%20of%20supernova-driven%20winds%20(see%20their%20Appendix%20A)%20in%20our%20calculations,%20we%20do%20not%20include%20an%20explicit%20self-regulation%20mechanism%20due%20to%20AGNs.%20This%20may%20have%20a%20noteworthy%20impact%20on%20the%20future%20SFR.%20We%20will%20discuss%20the%20subsequent%20implications%20for%20our%20conclusions%20in%20Section%205.3.%0AIt%20is%20important%20to%20bear%20in%20mind%20that%20Fig.%208%20ignores%20the%20cosmology%20dependence%20of%20halo%20multiplicities.%20We%20then%20need%20to%20weight%20the%20efficiencies%20shown%20in%20Fig.%208%20by%20the%20HMF.%20If%20we%20then%20further%20integrate%20over%20halo%20mass,%20that%20gives%20us%20the%20stellar%20mass%20density%20(SMD)%20of%20all%20stars%20formed%20up%20to%20time%20t%20from%20the%20time%20corresponding%20to%20the%20onset%20of%20star%20formation%20tin.%20This%20quantity%20is%20by%20definition%20the%20time%20integral%20of%20the%20CSFRD%20or,%20in%20terms%20of%20redshift:%0A%02%20zin%0A%E2%80%B2%0A%CF%81%E2%88%97(z)%3D%0A%CF%81%20%CC%87%E2%88%97(z)%20dz%E2%80%B2%20(24)%20z%0A(1%20+%20z%E2%80%B2)H(z%E2%80%B2)%0Awhere%20%CF%81%20%CC%87%E2%88%97(z)%20is%20given%20by%20equation%20(1).%20To%20study%20the%20asymptotic%20behaviour%20of%20%CF%81%E2%88%97(z),%20we%20should%20thus%20study%20the%20convergence%20of%20the%20integrand,%20which%20depends%20both%20on%20the%20nSFR%20and%20the%20HMF.%20We%20have%20already%20discussed%20the%20convergence%20of%20the%20nSFR%20for%20%02%20%3E%200%20above,%20and%20we%20now%20focus%20on%20the%20behaviour%20of%20the%20HMF%20for%20t%20%E2%86%92%E2%88%9E.%20In%20a%20universe%20with%20%02%20%3E%200,%20the%20asymptotic%20behaviour%20of%20%CF%83%20(M%20,%20z)%20is%20a%20non-%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%0Anull%20constant,%20for%20any%20M.%20Therefore,%20the%20integral%20in%20equation%20(24)%20converges%20as%20long%20as%20the%20stellar%20mass%20fraction%20converges%20%E2%80%93%20and%20we%20have%20already%20seen%20that%20this%20quantity%20does%20indeed%20asymptote%20to%20a%20finite%20constant%20in%20the%20far%20future.%20In%20an%20EdS%20model,%20the%20HMF%20continues%20to%20evolve%20indefinitely,%20but%20in%20fact%20the%20total%20stellar%20density%20still%20converges%20(see%20Appendix%20A1%20for%20the%20detailed%20argument).%0AThe%20asymptotic%20study%20of%20the%20CSFRD%20described%20above%20also%20allows%20us%20to%20define%20physically%20motivated%20analytic%20approximations%20for%20how%20the%20CSFRD%20approaches%20its%20asymptotic%20value%20in%20the%20far%20future%20(see%20Appendix%20A%20for%20details).%20The%20results%20of%20this%20exercise%20are%20shown%20in%20the%20upper%20panel%20of%20Fig.%209,%20plotting%20the%20stellar%20density%20as%20a%20function%20of%20time%20for%20various%20values%20of%20%CE%B1%02.%20In%20an%20EdS%20universe,%20the%20asymptotic%20SMD%20is%20reached%20at%20t%20%E2%88%BC%20100%20Gyr;%20but%20as%20%02%20is%20increased%20from%20zero%20the%20asymptotic%20SMD%20becomes%20larger,%20because%20now%20the%20cooling%20radius%20will%20not%20shrink%20indefinitely%20at%20late%20times,%20hence%20increasing%20the%20star%20formation%20efficiency%20of%20haloes.%20However,%20for%20%CE%B1%02%20%3E%200.1,%20the%20trend%20is%20reversed.%20This%20might%20seem%20somewhat%20at%20odds%20with%20Fig.%208,%20which%20shows%20that%20the%20stellar%20mass%20fraction%20within%20haloes%20keeps%20increasing%20for%20%CE%B1%02%20%3E%200.1.%20However,%20Fig.%208%20focuses%20on%20isolated%20haloes,%20without%20accounting%20for%20their%20multiplicity.%20Instead,%20the%20stellar%20production%20needs%20to%20be%20weighted%20by%20the%20HMF%20when%20computing%20the%20SMD.%20Thus,%20Fig.%209%20tells%20us%20that%20the%20lower%20asymptotic%20SMD%20for%20large%20values%20of%20%02%20is%20a%20consequence%20of%20the%20suppression%20of%20structure%20due%20to%20the%20cut-off%20of%20the%20HMF%20occurring%20at%20lower%20masses.%0AIn%20the%20lower%20panel%20of%20Fig.%209,%20we%20normalize%20the%20SMD%20by%20the%20asymptotic%20value.%20For%20the%20observed%20value%20of%20%02%20(green%20line),%20about%2032%20per%20cent%20of%20the%20stellar%20mass%20that%20will%20ever%20exist%20in%20our%20Universe%20has%20already%20been%20formed.%20In%20the%20original%20SP21%20paper,%20this%20quantity%20was%20quoted%20to%20be%20close%20to%2050%20per%20cent,%20but%20in%20that%20case%20the%20SMD%20was%20computed%20differently.%20Namely,%20SP21%20did%20not%20go%20through%20the%20detailed%20convergence%20study%20undertaken%20here,%20but%20simply%20integrated%20the%20CSFRD%20up%20to%20z%20%3D%20%E2%88%920.9999,%20which%20corresponds%20to%20t%20%E2%88%BC%20200%20Gyr%20in%20our%20Universe;%20Fig.%209%20teaches%20us%20that%20this%20is%20not%20always%20sufficiently%20late%20to%20reach%20the%20true%20asymptotic%20stellar%20mass%20density.%0AFor%20small%20positive%20%02,%20the%20fractional%20SMD%20in%20the%20lower%20panel%20of%20Fig.%209%20departs%20from%20the%20EdS%20solution%20much%20earlier%20than%20for%20the%20SMD%20evolution%20shown%20in%20the%20upper%20panel,%20reflecting%20the%20larger%20asymptotic%20values%20reached%20for%200%20%3C%20%02%20%3C%200.1.%20Thus,%20similar%20values%20of%20the%20SMD%20at%20early%20times%20can%20still%20differ%20considerably%20in%20fractional%20terms.%20Instead,%20for%20large%20values%20of%20%02,%20haloes%20are%20more%20efficient%20at%20producing%20stars,%20therefore%20the%20asymptotic%20SMD%20is%20reached%20earlier.%20Fig.%208%20shows%20that%20for%20%02%20%02%20103,%20star%20formation%20is%20driven%20by%20higher%20mass%20haloes.%20For%20extreme%20cosmologies%20such%20as%20%CE%B1%02%20%3D%20104,%20haloes%20are%20at%20nearly%20full%20star%20formation%20efficiency%20irrespective%20of%20their%20mass,%20but%20the%20cut-off%20of%20the%20HMF%20at%20late%20times%20occurs%20at%20M%20%E2%88%BC%20109%20M%E2%8A%99.%20Recalling%20that%20we%20consider%20only%20haloes%20with%20T%20%3E%20104.5%20K%20as%20eligible%20for%20star%20formation,%20this%20means%20that%20stars%20are%20produced%20only%20by%20haloes%20in%20the%20narrow%20mass%20range%20108%20M%E2%8A%99%20%02%20M%20%02%20109%20M%E2%8A%99.%20Therefore,%20it%20takes%20longer%20to%20reach%20the%20asymptotic%20SMD.%0AIn%20conclusion,%20if%20%02%20is%20higher,%20halo%20formation%20ceases%20at%20early%20times:%20haloes%20are%20unable%20to%20attain%20masses%20as%20large%20as%20in%20our%20Universe%20(see%20Section%203.1),%20and%20are%20less%20efficient%20at%20forming%20stars.%20The%20dominant%20fraction%20of%20stars%20is%20formed%20in%20the%20future%20(i.e.%20at%20negative%20redshift),%20when%20haloes%20are%20isolated%20and%20efficient%20at%20forming%20stars%20due%20to%20their%20larger%20cooling%20rate%20with%20respect%20to%20their%20counterparts%20in%20our%20own%20Universe%20(see%20Section%203.2.1).%20This%20long-term%20future%20of%20star%20formation%20is%20expected%20to%20play%20an%20important%20role%20in%20determining%20the%20asymptotic%20efficiency%20of%20star%20formation%20in%20universes%20with%20a%20large%20cosmological%20constant.%0AImpact%20of%20%02%20on%20star%20formation%20history%201461%204.2%20Cosmic%20star%20formation%20efficiency%0AIn%20the%20previous%20section,%20we%20showed%20that%20the%20comoving%20stellar%20mass%20density%20produced%20in%20a%20universe%20with%20%02%20%E2%89%A5%200%20asymptotes%20to%20a%20constant%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E.%20We%20now%20want%20to%20understand%20how%20this%20asymptotic%20density,%20%CF%81%E2%88%97%E2%88%9E,%20varies%20as%20a%20function%20of%20%02.%20In%20practice,%20we%20consider%20the%20cosmic%20star%20formation%20efficiency,%20defined%20as%20%CE%B5%20%3D%20%CF%81%E2%88%97%20%E2%88%9E%20/%CF%81%20%CC%84b%20,%20where%20%CF%81%20%CC%84b%20is%20the%20mean%20comoving%20baryon%20density%20in%20the%20universe.%0AWe%20plot%20this%20cosmic%20stellar%20efficiency%20in%20Fig.%2010,%20where%20we%20see%20the%20striking%20result%20that%20the%20peak%20of%20cosmic%20star%20formation%20efficiency%20occurs%20within%20one%20order%20of%20magnitude%20of%20%02obs.%20At%20the%20peak%20efficiency,%20%02%20%E2%89%88%200.1%02obs%20,%20the%20model%20predicts%20that%20about%2027%20per%20cent%20of%20the%20baryonic%20mass%20in%20the%20universe%20will%20eventually%20be%20converted%20into%20stars.%20In%20our%20Universe,%20this%20figure%20is%2023%20per%20cent.%20For%20larger%20values%20of%20%02,%20the%20cosmic%20efficiency%20declines,%20becoming%20negligibly%20small%20as%20%CE%B1%02%20approaches%20105%20.%0AIt%20might%20seem%20surprising%20that%20the%20cosmic%20stellar%20efficiency%20does%20not%20fall%20monotonically%20as%20%02%20increases.%20A%20larger%20cosmological%20constant%20would%20raise%20the%20acceleration%20of%20the%20cosmic%20expansion%20at%20earlier%20times,%20hence%20suppressing%20the%20gravitational%20collapse%20of%20galaxy-scale%20haloes;%20higher%20values%20of%20%02%20thus%20correspond%20to%20smaller%20fractions%20of%20matter%20in%20collapsed%20structures%20(e.g.%20Weinberg%201987;%20Martel,%20Shapiro%20%26%20Weinberg%201998).%20However,%20the%20picture%20is%20more%20complicated%20when%20considering%20the%20impact%20of%20%02%20on%20the%20fraction%20of%20baryonic%20rather%20than%20total%20matter%20within%20haloes%20(see%20e.g.%20the%20discussion%20in%20Page%202011).%20Furthermore,%20in%20Fig.%2010%20we%20are%20concerned%20with%20the%20stellar%20component%20only,%20which%20is%20subject%20to%20the%20complex%20interplay%20of%20astrophysical%20processes%20and%20the%20evolution%20of%20the%20large-scale%20structure%20of%20the%20universe.%0AThe%20suppression%20of%20the%20star%20formation%20efficiency%20at%20high%20%02%20cannot%20be%20attributed%20to%20astrophysical%20processes,%20since%20Fig.%208%20shows%20that%20high-mass%20haloes%20remain%20highly%20efficient%20at%20producing%20stars,%20even%20for%20large%20values%20of%20%02.%20Therefore,%20the%20small%20%CE%B5%20found%20for%20large%20%02%20must%20be%20caused%20by%20the%20cut-off%20of%20the%20HMF%20at%20small%20halo%20masses%20(see%20Section%203.1).%20Ultimately,%20this%20is%20a%20consequence%20of%20the%20earlier%20freeze-out%20that%20results%20from%20a%20large%20cosmological%20constant.%20In%20short,%20our%20model%20predicts%20that%20haloes%20are%20generally%20efficient%20at%20forming%20stars%20when%20%02%20is%20high,%20but%20their%20number%20density%20is%20then%20so%20low%20that%20the%20overall%20effect%20is%20a%20reduction%20in%20the%20cosmic%20star%20formation%20efficiency.%0AFor%20small%20values%20of%20%02,%20the%20opposite%20is%20true.%20There%20is%20now%20stronger%0Aclustering%20and%20haloes%20can%20attain%20larger%20masses.%20However,%20for%20a%20given%0Avirial%20mass,%20lower%20values%20of%20%02%20correspond%20to%20a%20larger%20virial%20radius,%0Agiven%20that%20R%20%E2%88%9D%20M%201/3%20H%20%E2%88%922/3%20and%20H%20is%20smaller%20for%20lower%20values%20of%2000%0A%CE%B1%02%20(see%20equation%2021%20and%20discussion%20in%20Section%203.2.2).%20Thus,%20a%20halo%20of%20a%20given%20virial%20mass%20has%20a%20lower%20internal%20density%20in%20a%20universe%20with%20a%20smaller%20cosmological%20constant.%20This%20reflects%20the%20continued%20progress%20of%20mergers,%20which%20dilute%20the%20internal%20density%20of%20haloes%20to%20a%20multiple%20of%20the%20cosmic%20density,%20up%20to%20the%20point%20where%20%02%20dominates.%20The%20reduced%20gas%20density%20results%20in%20a%20smaller%20cooling%20rate,%20making%20star%20formation%20less%20efficient%20at%20late%20times.%20The%20cosmic%20stellar%20efficiency%20at%20low%20values%20of%20%02%20in%20Fig.%2010%20is%20therefore%20suppressed%20for%20astrophysical%20reasons.%0AThese%20astrophysics-driven%20and%20cosmology-driven%20suppression%20ef-%20fects%20on%20star%20formation%20balance%20out%20around%20%CE%B1%02%20%E2%88%BC%200.1.%20Fig.%2010%20suggests%20that%20the%20observed%20value%20of%20%02%20is%20somewhat%20peculiar,%20in%20the%20sense%20that%20it%20is%20close%20to%20the%20optimal%20value%20for%20maximum%20cosmic%20star%20formation%20efficiency.%20However,%20this%20is%20not%20the%20same%20as%20saying%20that%20it%20is%20the%20most%20likely%20value%20of%20%02%20for%20observers%20to%20experience.%20In%20the%20next%20section,%20we%20will%20address%20the%20implications%20of%20Fig.%2010%20for%20such%20anthropic%20issues.%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1462%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0AFigure9.%20Upperpanel:Cumulativestellarmassproducedinaunitcomovingvolumeuptoacertaincosmictime.Differentlinescorrespondtodifferentvalues%20of%20%02,%20as%20indicated%20in%20the%20colour%20bar.%20The%20upper%20x-axis%20shows%20the%20redshift%20that%20corresponds%20to%20the%20cosmic%20time,%20for%20the%20reference%20cosmology.%20Therefore,%20the%20correspondence%20between%20the%20upper%20and%20lower%20x-axes%20applies%20only%20to%20the%20case%20%02%20%3D%20%02obs.%20The%20break%20in%20the%20upper%20x-axis%20indicates%20that%20the%20ticks%20are%20omitted%20for%20z%20%3E%20%E2%88%921%20+%2010%E2%88%9210,%20since%20the%20exponential%20growth%20of%20the%20scale%20factor%20in%20the%20future%20makes%20it%20increasingly%20hard%20to%20represent%20the%20redshift%20on%20a%20meaningful%20scale.%20Lower%20panel:%20Same%20as%20in%20the%20upper%20panel,%20but%20normalized%20to%20the%20total%20stellar%20mass%20density%20produced%20over%20the%20entire%20history%20of%20the%20universe.%20The%20stellar%20mass%20fraction%20produced%20up%20to%20present%20time%20in%20our%20Universe%20is%20%E2%88%BC%2032%20per%20cent,%20hence%20we%20are%20typical%20observers.%20However,%20in%20general%20the%20mid-point%20of%20star%20formation%20history%20does%20not%20coincide%20with%20%02-domination%20in%20other%20universes,%20meaning%20that%20the%20%E2%80%98why-now%E2%80%99%20problem%20is%20specific%20to%20%02%20%3D%20%02obs%20(see%20discussion%20in%20Section%205.1.1).%0A5%20DISCUSSION%0A5.1%20Implications%20for%20anthropic%20arguments%0A5.1.1%20Typicality%20and%20the%20%E2%80%98why%20now%E2%80%99%20problem%0AThe%20first%20aspect%20of%20anthropic%20arguments%20that%20we%20will%20address%20concerns%20the%20so-called%20why-now%20problem,%20i.e.%20the%20fact%20that%20we%20live%20at%20an%20usual%20epoch%20when%20the%20density%20of%20matter%20and%20dark%20energy%20are%20of%20the%20same%20order%20of%20magnitude.%20The%20%E2%80%98weak%E2%80%99%20anthropic%20approach%20to%20this%20puzzle%20asserts%20that%20typical%20observers%20should%20live%20when%20the%20age%0Aof%20the%20universe%20corresponds%20roughly%20to%20the%20lifetime%20of%20Sun-like%20stars,%20assuming%20these%20to%20be%20necessary%20for%20forming%20and%20sustaining%20habitable%20planets%20(Carter%201974).%20We%20can%20test%20this%20principle%20to%20some%20extent,%20by%20asking%20how%20typical%20we%20are%20as%20observers.%20Consider%20the%20fraction%20of%20stellar%20mass%20density%20produced%20up%20to%20present%20time%20in%20our%20own%20Universe%20only%20(green%20line%20in%20Fig.%209):%20this%20predicts%20that%20roughly%2032%20per%20cent%20of%20all%20the%20stars%20that%20will%20ever%20exist%20are%20already%20in%20place,%20so%20it%20appears%20that%20the%20present%20time%20is%20indeed%20typical,%20lying%20neither%20at%20the%20beginning%20nor%20at%20the%20end%20of%20the%20cosmic%20star%20formation%20history.%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0AFigure10.%20Rightpanel:Fractionofbaryonicmass(perunitcomovingvolume)thatisconvertedintostarsovertheentirehistoryoftheuniverse,asafunction%20of%20%02.%20This%20global%20stellar%20efficiency%20peaks%20around%201/10%20of%20the%20observed%20value%20of%20%02%20(thin%20black%20vertical%20line),%20and%20becomes%20negligibly%20small%20for%20large%20values%20of%20%02.%20This%20is%20due%20to%20the%20suppression%20of%20cosmological%20structure%20formation%20caused%20by%20the%20earlier%20and%20larger%20acceleration%20of%20the%20expansion%20of%20the%20universe.%20The%20horizontal%20thin%20black%20line%20represents%20the%20global%20stellar%20efficiency%20in%20an%20EdS%20universe,%20which%20is%20smaller%20than%20in%20our%20Universe.%20The%20decrease%20in%20the%20cosmic%20star%20formation%20efficiency%20at%20small%20positive%20%02%20is%20driven%20by%20astrophysical%20rather%20than%20cosmological%20factors:%20haloes%20of%20a%20given%20virial%20mass%20become%20larger%20as%20%02%20decreases,%20and%20the%20consequent%20reduction%20in%20internal%20gas%20density%20diminishes%20the%20gas%20cooling%20rate,%20which%20in%20turn%20determines%20the%20SFR%20at%20low%20redshift%20(see%20discussion%20in%20Section%204.2).%0AHowever,%20this%20argument%20neglects%20the%20question%20of%20how%20long%20intel-%20ligent%20life%20may%20to%20take%20to%20emerge%20following%20the%20formation%20of%20a%20star.%20This%20is%20an%20issue%20on%20which%20cosmologists%20are%20unlikely%20to%20have%20a%20reliable%20opinion.%20However,%20we%20can%20note%20that%20if%20the%20emergence%20of%20life%20is%20rare,%20then%20indeed%20an%20offset%20of%20the%20order%20of%20the%20typical%20stellar%20lifetime%20might%20be%20expected,%20and%20this%20is%20the%20basis%20of%20the%20standard%20anthropic%20argument%20for%20the%20current%20age%20of%20the%20Universe.%20Alternatively,%20we%20can%20follow%20Peacock%20(2007)%20and%20eliminate%20the%20biological%20element%20by%20noting%20that%20the%20Sun%20formed%20at%20z%20%E2%89%88%200.45.%20From%20our%20results,%20we%20find%20that%20%E2%88%BC%2025%20per%20cent%20of%20the%20asymptotic%20stellar%20density%20was%20in%20place%20at%20that%20point,%20so%20the%20Sun%20is%20a%20slight%20%E2%80%98early%20adopter%E2%80%99%20but%20by%20no%20means%20atypical.%0AThe%20Sun%20thus%20formed%20almost%20exactly%20at%20the%20point%20of%20matter-%02%20equality,%20and%20the%20interesting%20question%20is%20whether%20this%20coincidence%20would%20still%20have%20held%20for%20a%20typical%20star%20if%20%02%20had%20taken%20a%20very%20different%20value.%20It%20is%20apparent%20from%20the%20predictions%20in%20Fig.%209%20that%20the%20observed%20coincidence%20is%20not%20to%20be%20expected%20in%20all%20cases.%20For%20%CE%B1%02%20%3D%20100,%20the%2050%20per%20cent%20point%20of%20stellar%20production%20is%20reached%20at%20t%20%E2%89%88%2020%20Gyr.%20That%20corresponds%20to%201%20+%20z%20%E2%89%88%2010%E2%88%926%20in%20that%20universe,%20at%20which%20point%20the%20density%20from%20%02%20exceeds%20that%20of%20matter%20by%20a%20factor%20of%20approximately%201014.%20We%20also%20note%20that%20the%20predicted%20CSFRD%20is%20not%20so%20very%20different%20from%20the%20observed%20one%20when%20we%20carry%20out%20the%20prediction%20for%20the%20EdS%20universe.%20In%20that%20case,%20there%20is%20still%20a%20well-defined%20typical%20era%20of%20star%20formation,%20which%20is%20clearly%20not%20dictated%20by%20the%20cosmological%20constant.%0AOur%20conclusion%20is%20then%20that%20the%20why-now%20puzzle%20is%20specific%20to%20the%20actual%20value%20of%20%02.%20Weak%20anthropic%20selection%20that%20biases%20observers%20towards%20living%20at%20a%20special%20time%20in%20the%20universe%20would%20not%20have%20produced%20such%20a%20coincidence%20in%20general.%20We%20are%20therefore%20driven%20to%20ask%20if%20there%20is%20some%20more%20general%20selection%20effect%20that%20favours%20the%20observation%20of%20this%20particular%20value%20of%20%02.%0A5.1.2%20Ensembles%20and%20the%20observed%20value%20of%20%02%0AThe%20multiverse%20approach%20to%20%02%20considers%20an%20ensemble%20of%20different%20universes,%20each%20with%20its%20own%20cosmological%20constant,%20and%20argues%20that%20only%20the%20universes%20that%20have%20a%20small%20enough%20value%20of%20%02%20would%20be%0Aable%20to%20host%20observers.%20As%20was%20emphasized%20by%20Efstathiou%20(1995),%20this%20situation%20can%20be%20analysed%20in%20a%20Bayesian%20framework.%20If%20we%20consider%20the%20existence%20of%20observers%20as%20the%20%E2%80%98data%E2%80%99,%20then%20the%20posterior%20on%20%02%20is%0Ap(%02%7CO)%20%E2%88%9D%20p(O%7C%02)p(%02)%20.%20(25)%0AIn%20the%20equation%20above,%20p(O%7C%02)%20is%20the%20likelihood,%20i.e.%20the%20probability%20of%20observers%20to%20emerge%20for%20a%20given%20value%20of%20%02,%20and%20p(%02)%20is%20the%20prior%20on%20%02.%0ABoth%20the%20prior%20on%20%02%20and%20the%20likelihood%20are%20non-trivial%20to%20determine.%20The%20main%20conceptual%20difficulty%20is%20that%20one%20would%20need%20to%20choose%20a%20well-defined%20measure%20for%20weighting%20different%20members%20of%20the%20ensem-%20ble.%20However,%20for%20an%20inflationary%20multiverse,%20the%20bubble%20universes%20are%20formally%20of%20infinite%20volume,%20and%20it%20is%20therefore%20not%20clear%20how%20to%20define%20the%20measure%20(see,%20e.g.,%20Wenmackers%202024).%20However,%20if%20%02%20is%20the%20sole%20parameter%20that%20varies%20across%20the%20ensemble,%20then%20the%20problem%20can%20be%20evaded%20following%20Weinberg%20(1987,%201989,%202000a,%20b),%20who%20pointed%20out%20that%20there%20is%20no%20known%20physical%20mechanism%20that%20preferences%20%02%20%3D%200.%20Therefore,%20the%20prior%20on%20%02%20should%20be%20flat%20at%20least%20in%20a%20neighbourhood%20of%20%02%20%3D%200.%20We%20will%20adopt%20Weinberg%E2%80%99s%20convention%20of%20a%20flat%20prior,%20and%20extend%20it%20to%20the%20full%20range%20of%20values%20of%20%02%20considered%20in%20our%20work,%20i.e.%200%E2%89%A4%CE%B1%02%20%E2%89%A4105.%0ARegarding%20the%20likelihood%20p(O%7C%02),%20we%20will%20adhere%20to%20the%20view%20expressed%20in%20the%20previous%20section%20that%20stars%20are%20a%20precondition%20for%20observers.%20Thus,%20we%20will%20simply%20assume%20that%20the%20likelihood%20is%20proportional%20to%20the%20cosmic%20star%20formation%20efficiency%20%CE%B5.%20We%20stress%20that%20we%20are%20agnostic%20concerning%20the%20time%20at%20which%20observers%20appear,%20unlike%20in%20the%20previous%20section:%20%CE%B5%20is%20an%20integrated%20quantity%20over%20the%20entire%20history%20of%20the%20universe.%0AWe%20can%20now%20write%20the%20posterior%20probability%20density%20in%20the%20interval%20(%02,%20%02%20+%20d%02)%20as:%0Adp(%02%7CO)%20%E2%88%9D%20%CE%B5(%02)%20(26)%20d%02%0AImpact%20of%20%02%20on%20star%20formation%20history%201463%0Aor,%20more%20conveniently,%20as%20the%20distribution%20of%20ln%20%02:%0Adp(%02%7CO)%0Ad%20ln%20%02%20%E2%88%9D%20%CE%B1%02%CE%B5(%CE%B1%02)%20.%0A(27)%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%20%20%0A1464%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0AThis%20probability%20distribution%20is%20shown%20in%20the%20left%20panel%20of%20Fig.%2011%20(see%20also%20Sorini%202022).%20We%20notice%20immediately%20that%20the%20peak%20corresponds%20to%20%CE%B1%02%20%E2%89%88%20800:%20the%20most%20likely%20order%20of%20magnitude%20of%20%02,%20weighted%20by%20observer%20number%20across%20the%20multiverse,%20is%20vastly%20greater%20than%20the%20observed%20value.%20Furthermore,%20%CE%B1%02%20%3D%201%20(indicated%20with%20a%20vertical%20black%20line%20in%20the%20plot)%20appears%20to%20be%20an%20outlier%20of%20the%20distribution.%20To%20quantify%20this%20more%20precisely,%20we%20also%20show%20the%20corresponding%20cumulative%20probability%20distribution%20in%20the%20right%20panel%20of%20Fig.%2011%20%E2%80%93%20emphasizing%20once%20again%20that%20this%20neglects%20the%20entirely%20distinct%20case%20of%20recollapsing%20models.%20We%20then%20find%20that%20the%20probability%20of%20observing%20a%20cosmological%20constant%20equal%20to%20or%20below%20the%20value%20of%20%02%20observed%20in%20our%20Universe%20is%20around%200.5%20per%20cent.%20Thus,%20with%20our%20stated%20assumptions%20of%20(1)%20the%20extension%20to%20the%20SP21%20model%20of%20cosmic%20star%20formation%20presented%20in%20this%20work;%20(2)%20a%20flat%20prior%20on%20%02;%20and%20(3)%20assuming%20a%20direct%20proportionality%20between%20the%20cosmic%20star-forming%20efficiency%20and%20the%20likelihood%20of%20generating%20observers%20at%20a%20given%20%02,%20the%20simplest%20multiverse%20framework%20is%20not%20able%20to%20account%20naturally%20for%20the%20small%20observed%20value%20of%20%02.%0A5.2%20Alternative%20anthropic%20assumptions%0AThe%20above%20failure%20to%20account%20for%20the%20magnitude%20of%20%02%20can%20mean%20one%20of%20three%20things:%20(i)%20there%20is%20no%20multiverse;%20(ii)%20members%20of%20the%20ensemble%20vary%20in%20a%20more%20complicated%20way%20than%20simply%20altering%20%02;%20and%20(iii)%20some%20of%20our%20basic%20assumptions%20are%20inappropriate.%0AUndoubtedly,%20it%20is%20interesting%20to%20consider%20a%20more%20complex%20multi-%20verse%20ensemble,%20where%20more%20fundamental%20parameters%20are%20varied%20at%20the%20same%20time%20(up%20to%2031:%20Tegmark%20et%20al.%202006).%20Indeed,%20other%20works%20have%20studied%20the%20simultaneous%20impact%20of%20varying%20two%20or%20three%20cosmological%20parameters%20on%20the%20star%20formation%20history%20(Cline,%20Frey%20%26%20Holder%202008;%20Bousso%20%26%20Hall%202013).%20For%20instance,%20a%20universe%20with%20both%20a%20larger%20value%20of%20%02%20and%20a%20larger%20amount%20of%20matter%20may%20well%20be%20evolving%20in%20a%20similar%20fashion%20as%20our%20own%20Universe,%20as%20the%20increased%20acceleration%20of%20the%20expansion%20induced%20by%20a%20higher%20cosmological%20constant%20would%20be%20compensated%20by%20a%20stronger%20clustering%20of%20matter.%20More%20radically,%20we%20may%20consider%20altering%20other%20fundamental%20parameters,%20such%20as%20the%20gravitational%20constant.%20Increasing%20G%20would%20obviously%20favour%20the%20collapse%20of%20structures,%20and%20may%20counteract%20the%20effect%20of%20a%20larger%20%02.%20At%20the%20same%20time,%20such%20a%20change%20would%20also%20affect%20the%20structure%20of%20haloes%20and%20subsequently%20the%20cooling%20rate.%20If%20one%20changes%20other%20coupling%20constants,%20such%20as%20the%20fine%20structure%20constant,%20then%20the%20picture%20becomes%20increasingly%20more%20complex.%20That%20would%20affect%20the%20very%20structure%20of%20matter,%20and%20modify%20cooling%20rates%20in%20unusual%20ways.%20A%20comprehensive%20discussion%20on%20how%20star%20formation%20would%20proceed%20in%20such%20exotic%20scenarios%20is%20provided%20by%20Adams%20(2019).%20We%20will%20explore%20these%20generalizations%20and%20other%20matters%20in%20future%20work.%0AThe%20problem%20with%20all%20these%20more%20elaborate%20ensembles%20is%20the%20difficulty%20in%20deciding%20on%20the%20prior%20for%20the%20additional%20parameters.%20We%20have%20adopted%20Weinberg%E2%80%99s%20prior%20on%20%02,%20arguing%20that%20it%20is%20a%20special%20case.%20Nevertheless,%20it%20is%20interesting%20to%20consider%20what%20would%20happen%20if%20Weinberg%E2%80%99s%20argument%20was%20in%20error,%20since%20a%20prior%20that%20gives%20more%20weight%20to%20lower%20values%20of%20%02%20could%20radically%20change%20the%20conclusions.%20A%20prior%20that%20is%20flat%20in%20ln(%CE%B1%02)%20would%20yield%20by%20construction%20a%20posterior%20for%20%02%20that%20is%20identical%20to%20the%20cosmic%20efficiency%20%CE%B5,%20properly%20renormalized.%20However,%20a%20logarithmic%20prior%20cannot%20be%20applied%20if%20we%20allow%20for%20%02%20%3D%200,%20let%20alone%20if%20we%20accept%20that%20%02%20%3C%200%20is%20possible.%20A%20power-law%20prior%20on%20%02%20would%20be%20allowed%20if%20we%20insist%20that%20%02%20%E2%89%A5%200,%20and%20we%20show%20posteriors%20on%20%02%20for%20various%20power-law%20slopes%20in%20Fig.%2012.%20As%20the%20power%20law%20becomes%20steeper,%20the%20probability%20distribution%20becomes%20broader,%20and%20the%20peak%20moves%20towards%20smaller%20values%20of%20%CE%B1%02.%0AOf%20particular%20interest%20is%20the%20case%20of%20a%20prior%20in%20the%20form%20%CE%B1%E2%88%920.75.%20If%20%02%0Aone%20assumes%20that%20the%20cosmological%20constant%20is%20set%20by%20the%20energy%20of%20MNRAS%20535,%201449%E2%80%931474%20(2024)%0Athe%20vacuum%20Ev,%20then%20%02%20%E2%88%9D%20Ev4.%20One%20might%20then%20opt%20for%20a%20flat%20prior%20on%20E%20,%20corresponding%20to%20a%20prior%20on%20%02%20in%20the%20form%20%CE%B1%E2%88%920.75.%20In%20this%20case%0Av%02%0Aour%20model%20would%20yield%20a%20probability%20of%20observing%20%02%20%E2%89%A4%20%02obs%20of%2039%20per%20cent,%20very%20well%20consistent%20with%20observation.%20However,%20the%20prior%20should%20be%20motivated%20by%20a%20priori%20theoretical%20considerations,%20rather%20than%20being%20chosen%20in%20order%20to%20achieve%20by%20construction%20some%20preferred%20conclusion.%20At%20present,%20we%20do%20not%20have%20a%20compelling%20argument%20that%20would%20justify%20a%20uniform%20prior%20in%20Ev%20,%20nor%20for%20the%20associated%20assumption%20that%02%3E0.%0AThere%20have%20been%20several%20attempts%20at%20solving%20the%20cosmological%20constant%20problem%20that%20predict%20a%20non-negative%20value.%20Supersymme-%20try%20generates%20a%20strictly%20positive%20vacuum%20energy%20density%20if%20it%20is%20spontaneously%20broken%20(see%20e.g.%20Martin%202012).%20Kaloper%20%26%20Padilla%20(2014)%20proposed%20a%20reformulation%20of%20General%20Relativity%20whereby%20all%20vacuum%20energy%20is%20sequestered%20from%20the%20matter%20sector.%20This%20prevents%20the%20vacuum%20energy%20from%20sourcing%20spatial%20curvature%20and%20predicts%20a%20positive%20cosmological%20constant,%20which%20is%20expressed%20as%20the%20space-time%20average%20of%20the%20trace%20of%20the%20energy%E2%80%93momentum%20tensor.%20However,%20the%20model%20requires%20the%20universe%20to%20end%20in%20order%20to%20produce%20a%20positive%20%02.%20This%20is%20not%20necessary%20in%20a%20later%20generalization%20of%20the%20sequestering%20model%20by%20Lombriser%20(2019b),%20who%20revisited%20the%20work%20of%20Kaloper%20%26%20Padilla%20(2014)%20by%20explicitly%20modelling%20the%20effect%20of%20collapsed%20structures.%20A%20positive%20%02%20is%20also%20predicted%20by%20models%20attempting%20to%20solve%20the%20cosmological%20constant%20problem%20by%20varying%20the%20Planck%20mass%20(Lombriser%202019a;%20Sobral-Blanco%20%26%20Lombriser%202020,%202021),%20at%20least%20in%20their%20simplest%20incarnation.%20The%20basic%20idea%20is%20to%20vary%20the%20Einstein%E2%80%93Hilbert%20action%20with%20respect%20to%20both%20the%20metric%20and%20the%20Planck%20mass%20over%20a%20manifold.%20The%20resulting%20cosmological%20constant%20can%20be%20expressed%20in%20terms%20of%20the%20average%20energy%E2%80%93momentum%20tensor,%20and%20can%20be%20interpreted%20as%20a%20%E2%80%98backreaction%E2%80%99%20of%20structures%20on%20the%20expanding%20universe.%20Assuming%20the%20typical%20size%20of%20collapsed%20structures,%20the%20model%20predicts%20%02%20%3D%200.704,%20very%20close%20to%20the%20observed%20value.%20Recently,%20Gaztan%20%CC%83%20aga%20(2021)%20proposed%20that%20%02%20represents%20a%20zero-%20action%20boundary%20term%20in%20the%20Einstein%E2%80%93Hilbert%20action.%20It%20can%20be%20derived%20that%20%02%20is%20proportional%20to%20the%20average%20density%20of%20the%20universe%20and%20is%20thus%20a%20non-negative%20quantity.%0AThere%20are,%20however,%20other%20strong%20theoretical%20grounds%20for%20consid-%20ering%20a%20negative%20cosmological%20constant.%20In%20supergravity,%20the%20vacuum%20energy%20can%20be%20negative%20(Martin%202012).%20In%20the%20landscape%20of%20string%20theory,%20negative%20values%20of%20%02%20emerge%20naturally,%20and%20predicting%20a%20small%20positive%20value%20remains%20a%20challenge%20(see%20e.g.%20Lerh%20Feng%20et%20al.%202021,%20Demirtas%20et%20al.%202022).%20These%20scenarios%20would%20then%20not%20be%20compatible%20with%20a%20power-law%20or%20logarithmic%20prior%20on%20%02.%0ALastly,%20our%20conclusions%20would%20certainly%20be%20subject%20to%20change%20if%20we%20were%20to%20alter%20the%20model%20of%20cosmic%20star%20formation.%20We%20next%20discuss%20how%20the%20limitations%20of%20our%20present%20modelling%20could%20affect%20our%20conclusions%20in%20Section%205.3.%20In%20Section%205.4,%20we%20will%20then%20make%20a%20detailed%20comparison%20of%20our%20findings%20with%20other%20related%20works%20in%20the%20literature.%0A5.3%20Limitations%20of%20our%20study%0AThis%20paper%20is%20based%20on%20the%20SP21%20model%20of%20cosmological%20star%20formation.%20This%20aimed%20to%20retain%20the%20physical%20transparency%20of%20the%20original%20Hernquist%20%26%20Springel%20(2003)%20model,%20which%20comes%20at%20the%20price%20of%20some%20simplifications.%20Chief%20among%20these%20is%20the%20fact%20that%20the%20model%20formalism%20does%20not%20explicitly%20include%20AGN%20feedback,%20but%20only%20supernova-driven%20winds.%20Nevertheless,%20the%20parameters%20underlying%20the%20stellar%20feedback%20model%20in%20the%20SP21%20formalism%20are%20tuned%20to%20reproduce%20observations%20of%20the%20baryonic%20Tully%E2%80%93Fisher%20relationship%20(McGaugh%20et%20al.%202000)%20and%20of%20the%20Kennicutt%E2%80%93Schmidt%20relationship%20(Kennicutt%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0AFigure%2011.%20Left%20panel:%20probability%20of%20generating%20observers%20in%20a%20given%20universe,%20at%20any%20point%20in%20time,%20as%20a%20function%20of%20%02,%20assuming%20a%20direct%20proportionality%20to%20the%20stellar%20efficiency%20shown%20in%20Fig.%2010%20(right%20panel)%20and%20imposing%20a%20flat%20prior%20on%20%02.%20The%20peak%20occurs%20around%20%02%20%E2%89%88%20800%02obs.%20The%20observed%20value%20of%20%02%20(thin%20black%20vertical%20line)%20is%20an%20outlier%20in%20the%20distribution.%20Right%20panel:%20cumulative%20probability%20function%20of%20the%20distribution%20shown%20in%20the%20left%20panel.%20The%20probability%20that%20%02%20%E2%89%A4%20%02obs%20is%20%E2%88%BC%200.5%20per%20cent,%20meaning%20that%20this%20value%20is%20not%20anthropically%20favoured.%0AFigure12.%20SameasintheleftpanelofFig.11,butfordifferentpriorson%02.%20The%20red%20line%20corresponds%20to%20the%20flat%20prior,%20and%20is%20therefore%20the%20same%20quantity%20plotted%20in%20the%20left%20panel%20of%20Fig.%2011.%20The%20other%20lines%20refer%20to%20a%20power-law%20prior,%20as%20reported%20in%20the%20legend.%20A%20power-law%20prior%20with%20an%20index%20in%20the%20range%20(%E2%88%921,%20%E2%88%920.9%5D%20would%20strongly%20favour%20the%20observed%20value%20of%20%02,%20but%20of%20course%20such%20a%20prior%20would%20need%20to%20be%20motivated%20by%20a%20theory%20and%20not%20chosen%20a%20posteriori%20in%20order%20to%20validate%20anthropic%20reasoning%20(see%20discussion%20in%20the%20main%20text).%0A1998)%20in%20our%20Universe,%20which%20does%20harbour%20AGN%20feedback.%20Thus,%20the%20effect%20of%20AGN%20feedback%20is%20implicitly%20allowed%20for,%20as%20in%20the%20case%20of%20cosmological%20simulations%20that%20contain%20stellar%20feedback%20only,%20but%20which%20calibrate%20the%20parameters%20of%20their%20subgrid%20prescriptions%20to%20reproduce%20observed%20data.%20This%20approach%20was%20followed,%20for%20example,%20by%20Oh%20et%20al.%20(2022),%20who%20analysed%20the%20impact%20of%20%02%20in%20a%20suite%20of%20hy-%20drodynamical%20simulations%20that%20extended%20up%20to%20t%20%3D%2097%20Gyr,%20including%20only%20stellar%20feedback.%20They%20argued%20that%20any%20energy%20fed%20back%20into%20the%20environment%20by%20black%20holes%20at%20late%20times%20will%20happen%20on%20a%20time-scale%20correlated%20to%20the%20feedback%20from%20star%20formation.%20Then,%20even%20without%20including%20an%20AGN%20feedback%20model,%20their%20feedback%20parameters%20would%20capture%20the%20global%20energy%20input%20that%20would%20be%20expected%20from%20both%20star%20formation%20and%20black%20holes.%20The%20empirical%20correlation%20of%20stellar%20mass%20and%20black%20hole%20mass%20in%20galaxies%20is%20commonly%20taken%20as%20evidence%0Afor%20co-evolution%20of%20these%20two%20populations%20(see%20e.g.%20Kormendy%20%26%20Ho%202013;%20Heckman%20%26%20Best%202014),%20and%20the%20neglect%20of%20explicit%20AGN%20feedback%20here%20is%20justifiable%20as%20long%20as%20co-evolution%20holds%20exactly.%0AHowever,%20several%20hydrodynamical%20simulations%20do%20indicate%20that%20AGN-driven%20jets%20are%20an%20important%20mechanism%20in%20the%20quenching%20of%20star%20formation%20after%20z%20%3D%202%20(see%20e.g.%20Vogelsberger%20et%20al.%202013;%20Weinberger%20et%20al.%202017;%20Pillepich%20et%20al.%202018;%20Sorini%20et%20al.%202022;%20Scharre%20%CC%81%20et%20al.%202024).%20While%20other%20groups%20found%20a%20more%20limited%20impact%20of%20AGN%20feedback%20on%20the%20low-redshift%20CSFRD%20(McCarthy%20et%20al.%202017;%20Salcido%20et%20al.%202018;%20see%20also%20the%20discussion%20in%20Salcido%20et%20al.%202020),%20the%20general%20consensus%20is%20that%20AGN%20feedback%20should%20be%20explicitly%20included%20in%20realistic%20models%20of%20the%20star%20formation%20history.%20Indeed,%20the%20total%20energy%20input%20from%20feedback%20processes%20is%20not%20the%20sole%20important%20factor%20in%20determining%20the%20quenching%20of%20star%20formation.%20Cosmological%20simulations%20show%20that%20the%20way%20such%20energy%20propagates,%20for%20example%20through%20diffuse%20winds%20or%20collimated%20AGN-driven%20jets,%20has%20a%20potentially%20significant%20impact%20on%20the%20star%20formation%20history%20and%20the%20diverse%20gaseous%20phases%20within%20haloes%20(e.g.%20Weinberger%20et%20al.%202017;%20Sorini%20et%20al.%202022;%20Scharre%20%CC%81%20et%20al.%202024;%20Yang%20et%20al.%202024).%20However,%20such%20considerations%20are%20beyond%20the%20scope%20of%20this%20work,%20and%20their%20incorporation%20would%20require%20a%20major%20expansion%20of%20the%20SP21%20model.%0AAs%20SP21%20pointed%20out,%20the%20introduction%20of%20AGN%20feedback%20in%20their%20formalism%20could%20plausibly%20diminish%20the%20low-z%20CSFRD%20and%20hence%20also%20the%20long-term%20efficiency%20of%20star%20formation.%20Because%20most%20of%20the%20stellar%20mass%20in%20universes%20with%20large%20values%20of%20%02%20is%20formed%20in%20the%20far%20future%20(see%20Fig.%209),%20AGN%20feedback%20may%20suppress%20the%20cosmic%20star%20formation%20efficiency%20at%20the%20high-%02%20end%20more%20rapidly.%20This%20would%20in%20turn%20increase%20the%20probability%20of%20observing%20a%20value%20of%20%02%20that%20is%20%E2%89%A4%20%02obs,%20reducing%20the%20discrepancy%20with%20observation.%20It%20remains%20to%20be%20seen%20whether%20the%20explicit%20inclusion%20of%20AGN%20feedback%20would%20significantly%20alter%20the%20conclusions%20in%20our%20work%20regarding%20the%20anthropically%20favoured%20value%20of%20%02,%20or%20whether%20its%20impact%20would%20be%20sub-dominant.%20We%20leave%20this%20important%20analysis,%20which%20requires%20incorporating%20a%20first-principles%20model%20of%20AGN%20feedback%20within%20our%20formalism,%20for%20future%20work.%0AOther%20potentially%20important%20physical%20ingredients%20omitted%20from%20the%20SP21%20model%20are%20chemical%20evolution%20and%20metal%20enrichment%20of%20the%20interstellar%20and%20intergalactic%20media.%20Following%20the%20evolution%20of%20the%20metallicity%20of%20stars%20and%20gas%20within%20and%20outside%20haloes%20would%20alter%20the%20cooling%20function%20over%20time.%20At%20later%20times,%20and%20reasonably%20also%20in%0AImpact%20of%20%02%20on%20star%20formation%20history%201465%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1466%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0Athe%20future%20of%20the%20universe,%20the%20metallicity%20would%20increase.%20For%20virial%20temperatures%20above%20T%20%3D%20104%20K,%20which%20are%20those%20of%20interest%20for%20the%20SP21%20model,%20the%20cooling%20function%20becomes%20larger%20as%20the%20metallicity%20increases%20(see%20Sutherland%20%26%20Dopita%201993).%20Hence,%20in%20the%20future%20of%20the%20universe%20the%20cooling%20rate%20would%20become%20higher,%20increasing%20the%20star%20formation%20efficiency.%0AWe%20also%20acknowledge%20that%20more%20exotic%20effects%20might%20become%20significant%20in%20the%20far%20future%20of%20the%20universe.%20For%20instance,%20gas%20cooling%20via%20inverse%20Compton%20scattering%20of%20CMB%20photons%20might%20become%20an%20important%20additional%20cooling%20mechanism%20in%20the%20far%20future%20(Adams%20%26%20Laughlin%201997;%20Bousso%20%26%20Leichenauer%202010;%20Bousso%20%26%20Hall%202013).%20As%20a%20further%20example,%20Adams%20%26%20Laughlin%20(1997)%20argued%20that%20grav-%20itational%20instabilities%20within%20galaxies%20could%20deviate%20the%20trajectory%20of%20stars%20from%20their%20orbits,%20so%20that%20stars%20either%20collapse%20towards%20the%20centre%20of%20the%20galaxy,%20or%20drift%20away%20from%20it,%20becoming%20isolated%20and%20unbound%20objects.%20In%20the%20former%20scenario,%20there%20can%20be%20a%20secondary%20spark%20of%20star%20formation,%20triggered%20by%20the%20clash%20between%20brown%20dwarfs%20around%20the%20galactic%20centre.%20Such%20exotic%20possible%20channels%20of%20star%20formation%20occur%20in%20the%20extreme%20far%20future,%20t%20%E2%89%88%201015%20Gyr.%20The%20ejection%20of%20stars%20from%20the%20galaxy%20due%20to%20the%20accumulation%20of%20stellar%20encounters%20would%20occur%20on%20a%20smaller,%20but%20still%20colossal,%20time-scale,%20t%20%E2%89%88%201010%20%E2%88%92%201011%20Gyr%20(Adams%20%26%20Laughlin%201997).%20These%20processes%20occur%20well%20after%20the%20point%20where%20our%20calculations%20of%20star%20formation%20have%20converged,%20and%20there%20is%20no%20reason%20to%20believe%20that%20they%20will%20dominate%20over%20previous%20activity%20in%20terms%20of%20total%20stellar%20production.%0AOne%20can%20also%20ask%20whether%20it%20is%20reasonable%20to%20believe%20that%20%E2%80%98con-%20ventional%20star%20formation%E2%80%99%20proceeds%20all%20the%20way%20up%20to%20the%201010%20Gyr%20time-scale%20represented%20in%20Fig.%209.%20Adams%20%26%20Laughlin%20(1997)%20argued%20that,%20based%20on%20the%20life-time%20of%20typical%20red%20dwarfs,%20%E2%80%98conventional%E2%80%99%20star%20formation%20should%20halt%20on%20a%20time-scale%20of%20t%20%E2%88%BC%20104%20Gyr.%20If%20we%20imposed%20such%20a%20cut-off%20in%20the%20star%20formation%20history,%20that%20would%20lead%20to%20a%20lower%20star%20formation%20efficiency%20in%20universes%20with%20lower%20%02.%20On%20the%20contrary,%20it%20would%20not%20affect%20as%20significantly%20universes%20with%20higher%20%02,%20where%20the%20bulk%20of%20star%20formation%20occurs%20at%20earlier%20times%20(see%20Fig.%209).%20Therefore,%20the%20posterior%20on%20%02%20would%20be%20more%20skewed%20towards%20higher%20values,%20hence%20making%20the%20anthropic%20prediction%20even%20less%20consistent%20with%20observation.%20However,%20arguments%20on%20the%20time-%20scales%20of%20different%20channels%20of%20star%20formation%20in%20the%20far%20future%20of%20the%20universe%20are%20themselves%20fairly%20uncertain.%20We%20therefore%20opted%20for%20not%20introducing%20an%20additional%20ill-defined%20parameter%20in%20our%20model,%20and%20simply%20integrated%20the%20star%20formation%20history%20up%20to%20t%20%E2%86%92%20%E2%88%9E.%0AThroughout%20this%20work,%20we%20have%20assumed%20that%20generation%20of%20ob-%20servers%20is%20simply%20proportional%20to%20the%20cosmic%20star%20formation%20efficiency.%20More%20realistically,%20we%20might%20consider%20what%20is%20known%20about%20statistics%20of%20exoplanets%20as%20a%20function%20of%20stellar%20mass.%20This%20variation%20would%20not%20matter%20provided%20the%20IMF%20was%20invariant,%20but%20given%20that%20the%20typical%20density%20and%20temperature%20of%20star-forming%20gas%20is%20likely%20to%20change%20with%20epoch,%20there%20is%20ample%20scope%20for%20the%20IMF%20to%20alter.%20We%20have%20neglected%20this%20complication,%20as%20have%20most%20authors%20in%20this%20area.%0AAnother%20option%20could%20be%20to%20weight%20the%20CSFRD%20by%20the%20metallicity%20of%20the%20stars%20formed.%20Indeed,%20Dayal%20et%20al.%20(2015)%20argued%20that%20the%20stellar%20mass,%20total%20metal%20mass,%20and%20SFR%20all%20conspire%20in%20enhancing%20the%20habitability%20of%20galaxies%20such%20as%20metal-rich%20giant%20elliptical%20galaxies.%20Furthermore,%20accounting%20for%20the%20effect%20of%20radiation%20on%20the%20emergence%20of%20life%20on%20potentially%20habitable%20planets%20(e.g.%20Totani%20et%20al.%202019;%20Gobat%20et%20al.%202021)%20could%20bring%20in%20the%20structure%20of%20the%20galaxy%20as%20a%20relevant%20factor.%20Indeed,%20recent%20works%20defined%20a%20%E2%80%98galactic%20habitable%20zone%E2%80%99%20(Gobat%20%26%20Hong%202016),%20in%20analogy%20to%20the%20circumstellar%20habitable%20zone%20(Huang%201959;%20Kasting%20et%20al.%201993).%20However,%20these%20variations%20in%20the%20generation%20of%20observers%20within%20and%20between%20galaxies%20are%20not%20a%20concern%20for%20the%20main%20focus%20of%20this%20paper,%20which%20is%20on%20the%20global%20production%0Aof%20observers%20summed%20over%20the%20universe%20and%20over%20all%20cosmic%20time.%20Therefore,%20it%20seems%20reasonable%20to%20neglect%20such%20issues%20for%20the%20present.%20In%20conclusion,%20our%20model%20can%20certainly%20be%20improved%20in%20different%20ways,%20and%20we%20hope%20to%20address%20some%20of%20them%20in%20the%20future.%20However,%20despite%20its%20simplicity%20the%20model%20is%20capable%20of%20capturing%20the%20main%20physical%20processes%20that%20are%20relevant%20for%20a%20sound%20description%20of%20the%20star%20formation%20history%20in%20the%20universe,%20both%20in%20the%20past%20and%20the%20future,%0Aand%20in%20different%20cosmologies.%0A5.4%20Comparison%20with%20previous%20work%0A5.4.1%20Posterior%20on%20%02%20and%20anthropic%20considerations%0AIn%20this%20section,%20we%20will%20discuss%20our%20results%20on%20the%20posterior%20distribution%20on%20%02%20and%20the%20implications%20for%20anthropic%20arguments%20in%20the%20context%20of%20previous%20relevant%20work%20in%20the%20literature.%0AOh%20et%20al.%20(2022)%20ran%20a%20suite%20of%20hydrodynamical%20simulations%20based%20on%20the%20ENZO%20code%20(Bryan%20et%20al.%202014),%20for%20an%20EdS%20universe%20and%20for%20%02CDM%20universes%20with%20different%20values%20of%20%02,%20up%20to%20100%20times%20the%20observed%20value.%20The%20simulations%20included%20a%20subgrid%20model%20for%20stellar%20feedback%20(Oh%20et%20al.%202020),%20but%20no%20AGN%20feedback.%20This%20model%20was%20shown%20to%20yield%20physically%20sensible%20results%20even%20when%20run%20up%20to%20a%20cosmic%20time%20of%20100%20Gyr%20(Oh%20et%20al.%202021).%20Using%20the%20same%20assumptions%20adopted%20in%20this%20work%20concerning%20the%20probability%20of%20generating%20observers%20in%20a%20given%20universe,%20Oh%20et%20al.%20(2022)%20found%20that%20the%20median%20of%20the%20posterior%20distribution%20on%20%02%20corresponds%20to%208.0%02obs,%20with%2095%20percentofthedistributionspanningtherange%CE%B1%02%20%3D0.32%E2%88%92105,and%20thus%20consistent%20with%20the%20observed%20%02.%20They%20find%20that%20the%20probability%20that%20%02%20%E2%89%A4%20%02obs%20is%2012.9%20per%20cent.%20Oh%20et%20al.%20(2022)%20also%20consider%20the%20traditional%20assumption%20for%20the%20anthropic%20weighting,%20based%20on%20the%20single-scale%20approximation%20for%20the%20collapsed%20fraction%20of%20matter%20in%20galaxy-scale%20haloes%20(see%20Peacock%202007).%20Under%20this%20assumption,%20the%20median%20of%20the%20posterior%20distribution%20on%20%02%20rises%20to%2012.6%20%02obs%20,%20and%20the%20corresponding%20probability%20of%20an%20observer%20measuring%20%02%20%E2%89%A4%20%02obs%20is%208.2%20per%20cent.%20The%20overall%20conclusions%20from%20the%20detailed%20modelling%20in%20Oh%20et%20al.%20(2022)%20are%20thus%20not%20hugely%20different%20from%20the%20simple%20single-%20scale%20approach.%20These%20conclusions%20are%20qualitatively%20in%20accord%20with%20our%20own,%20to%20the%20extent%20that%20the%20observed%20value%20of%20%02%20is%20a%20low%20outlier%20in%20the%20ensemble.%20However,%20we%20find%20an%20unacceptably%20small%20probability%20of%20%02%20%E2%89%A4%20%02obs%20%E2%80%93%20whereas%20Oh%20et%20al.%20(2022)%20find%20a%20probability%20that,%20while%20small,%20does%20not%20strongly%20rule%20out%20the%20multiverse%20hypothesis.%0AAn%20earlier%20example%20of%20a%20large%20%E2%80%98multiverse%20simulation%E2%80%99%20is%20given%20by%20Barnes%20et%20al.%20(2018).%20They%20ran%20a%20suite%20of%20variants%20of%20the%20EAGLE%20hydrodynamical%20simulation%20(Schaye%20et%20al.%202015),%20for%20values%20of%20%CE%B1%02%20in%20the%20range%200%20%E2%88%92%20300,%20until%20cosmic%20time%20t%20%3D%2020.7%20Gyr.%20These%20simulations%20include%20recipes%20for%20both%20stellar%20feedback%20and%20AGN%20feedback.%20Barnes%20et%20al.%20(2018)%20consider%20three%20possible%20models%20for%20the%20generation%20of%20observers:%0A(i)%20Observers%20are%20generated%20following%20the%20formation%20of%20stars,%20after%20a%20fixed%20delay%20of%205%20Gyr,%20which%20would%20represent%20the%20time-scale%20for%20the%20evolution%20of%20intelligent%20life;%0A(ii)%20The%20rate%20of%20generation%20of%20observers%20at%20a%20given%20time%20around%20a%20certain%20stellar%20population%20is%20proportional%20to%20the%20fraction%20of%20stars%20that%20are%20still%20on%20the%20main%20sequence,%20to%20account%20for%20the%20fact%20that%20life%20appears%20to%20be%20a%20rare%20phenomenon%20in%20the%20universe,%20and%20therefore%20should%20occur%20towards%20the%20end%20of%20the%20main-sequence%20lifetime%20of%20a%20star%20(Carter%201983;%20Barrow%20%26%20Tipler%201986);%0A(iii)%20The%20rate%20of%20generation%20of%20observers%20at%20a%20given%20time%20is%20proportional%20to%20the%20metallicity%20of%20the%20fraction%20of%20stars%20that%20are%20still%20on%20the%20main%20sequence%20for%20the%20stellar%20population%20considered,%20to%20reflect%20the%20argument%20that%20life%20would%20more%20easily%20emerge%20on%20rocky%20planets,%20which%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0Awould%20be%20more%20likely%20to%20form%20around%20stars%20with%20higher%20metallicity%20(Gonzalez%201997;%20Fischer%20%26%20Valenti%202005;%20but%20see%20also%20Buchhave%20%26%20Latham%202015;%20Wang%20%26%20Fischer%202015).%0AThe%20main%20impact%20of%20these%20different%20choices%20is%20to%20change%20when%20the%20observers%20form,%20rather%20than%20the%20total%20eventual%20numbers,%20which%20is%20our%20main%20concern.%20The%20results%20for%20all%20these%20models%20are%20reported%20in%20Table%201,%20in%20correspondence%20of%20the%20cells%20in%20the%20second%20column,%20labelled%20%E2%80%98star%20formation%20+%20delay%E2%80%99,%20%E2%80%98star%20formation%20+%20lifetime%E2%80%99,%20and%20%E2%80%98star%20formation%20+%20metals%E2%80%99,%20respectively.%20For%20each%20of%20the%20aforementioned%20models,%20Barnes%20et%20al.%20(2018)%20further%20considered%20three%20possible%20ways%20of%20constructing%20a%20measure%20in%20the%20multiverse:%0A(i)%20a%20mass-weighted%20measure,%20whereby%20a%20given%20mass%20element%20in%20the%20universe%20can%20inhabit%20a%20region%20with%20any%20value%20of%20%02%20with%20equal%20probability;%0A(ii)%20a%20causal-patch%20measure,%20in%20which%20regions%20of%20the%20universe%20that%20have%20equal%20comoving%20volume%20of%20their%20causal%20patch%20at%20a%20certain%20cosmic%20time%20host%20the%20same%20number%20of%20observers,%20for%20any%20value%20of%20%02%20considered;%0A(iii)%20a%20causal-diamond%20measure,%20analogous%20to%20the%20previous%20mea-%20sure,%20except%20that%20the%20comoving%20volume%20of%20the%20region%20(at%20given%20cosmic%20time%20t)%20that%20is%20enclosed%20by%20a%20photon%20that%20departs%20from%20a%20world%20line%20at%20reheating%20and%20returns%20at%20t%20is%20considered%20instead%20of%20the%20volume%20of%20the%20causal%20patch.%0AIt%20is%20important%20to%20note%20that%20in%20all%20cases%20described%20above%20Barnes%20et%20al.%20(2018)%20consider%20a%20flat%20prior%20on%20%02.%20Thus%20their%20posterior%20on%20%02%20is%20the%20product%20of%20the%20measure%20times%20the%20weighting%20based%20on%20star%20formation.%20The%20mass-weighted%20measure%20is%20essentially%20what%20we%20adopt%20in%20this%20work.%0AThe%20results%20that%20Barnes%20et%20al.%20(2018)%20obtain%20for%20all%20observer%20generation%20models%20and%20all%20multiverse%20measures%20are%20listed%20in%20Table%201.%20The%20mass-weighted%20measures%20yields%20a%20median%20of%20the%20posterior%20distribution%20on%20%02%20of%2045%20or%2059,%20depending%20on%20the%20observer%20generation%20prescription.%20The%2084th%20percentile%20of%20the%20distribution%20can%20reach%20values%20as%20large%20as%20194%20%02obs%20.%20On%20the%20other%20hand,%20the%20causal%20patch%20and%20causal%20diamond%20measures%20move%20the%20median%20%02%20to%20much%20smaller%20values,%20and%20always%20below%20%02obs%20.%20Consequently,%20the%20mass-weighted%20measure%20would%20predict%20a%20significantly%20smaller%20probability%20of%20observing%20%02%20%E2%89%A4%20%02obs%20(1.9%20per%20cent%E2%80%932.5%20per%20cent)%20than%20the%20other%20two%20measures%20(73%20per%20cent%E2%80%9393%20per%20cent).%20Thus,%20the%20choice%20of%20the%20measure%20appears%20to%20be%20crucial.%0AHowever,%20the%20measure%20is%20not%20a%20degree%20of%20freedom%20that%20can%20be%20adjusted%20at%20will%20(see%20the%20discussion%20in%20Barnes%20et%20al.%202018).%20We%20confess%20that%20we%20find%20the%20motivation%20for%20the%20causal%20measures%20obscure:%20in%20our%20own%20Universe%20there%20are%20definitely%20pairs%20of%20galaxies%20that%20are%20causally%20disconnected%20(those%20in%20opposite%20directions%20on%20the%20sky%20at%20high%20redshift),%20and%20yet%20each%20can%20potentially%20host%20observers.%20Barnes%20et%20al.%20(2018)%20mention%20theoretical%20motivations%20for%20choosing%20the%20causal%20patch%20and%20diamond%20measures%20in%20the%20context%20of%20the%20quantum%20xeroxing%20paradox%20(Susskind,%20Thorlacius%20%26%20Uglum%201993;%20Bousso%202006;%20Bousso,%20Freivogel%20%26%20Yang%202006)%20and%20holographic%20probability%20in%20eternal%20inflation%20(Bousso%202006).%20However,%20it%20seems%20to%20us%20that%20the%20results%20are%20biased,%20and%20systematically%20exclude%20potential%20observers%20at%20early%20times%20when%20the%20horizon%20is%20small.%20We%20will%20therefore%20compare%20our%20predictions%20with%20the%20results%20in%20Barnes%20et%20al.%20(2018)%20arising%20from%20their%20mass-weighted%20measure%20only.%0AWe%20can%20see%20that%20our%20results%20predict%20a%20median%20value%20of%20%02%20that%20is%209%20%E2%88%92%2012%20times%20larger%20than%20that%20of%20Barnes%20et%20al.%20(2018),%20and%20our%20probability%20for%20%02%20%E2%89%A4%20%02obs%20is%20about%20%E2%88%BC%204%20%E2%88%92%205%20times%20smaller.%20However,%20we%20integrate%20the%20CSFRD%20all%20the%20way%20to%20t%20%E2%86%92%20%E2%88%9E,%20whereas%20Barnes%20et%20al.%20(2018)%20halt%20at%2020.7%20Gyr%20in%20all%20cases.%20Nevertheless,%20we%20verified%20that%20even%20if%20we%20stopped%20our%20integration%20at%20this%20same%20early%20time,%20the%0Aprobability%20of%20measuring%20%02%20%E2%89%A4%20%02obs%20increases%20only%20slightly%20to%200.7%20per%20cent.%20This%20increase%20does%20not%20qualitatively%20affect%20our%20conclusions.%20Anthropic%20calculations%20have%20also%20been%20pursued%20using%20semi-%20analytical%20methods.%20Sudoh%20et%20al.%20(2017)%20adopted%20the%20semi-analytical%20model%20of%20galaxy%20formation%20by%20Nagashima%20%26%20Yoshii%20(2004),%20coupled%20with%20a%20Monte%20Carlo%20calculation%20for%20the%20merger%20history%20of%20dark%20matter%20haloes.%20Adopting%20a%20flat%20prior%20on%20%02%20and%20using%20stellar%20mass%20density%20as%20a%20proxy%20for%20observer%20weighting,%20they%20obtained%20a%20median%20value%20of%20%02%20in%20the%20range%20%CE%B1%02%20%3D%209.1%20%E2%88%92%2012,%20depending%20on%20the%20exact%20details%20of%20their%20model.%20Overall,%20the%20distribution%20is%20broad,%20and%20their%20probability%20of%20observing%20%02%20%E2%89%A4%20%02obs%20is%20between%206.7%20per%20cent%20and%209.7%20per%20cent.%20Such%20values%20are%20in%20agreement%20with%20the%20results%20by%20Oh%20et%20al.%20(2022),%20but%20are%20in%20tension%20with%20our%20conclusions%20and%20with%20Barnes%20et%20al.%20(2018).%20It%20is%20not%20surprising%20that%20Sudoh%20et%20al.%20(2017)%20obtain%20a%20lower%20median%20%CE%B1%02%20compared%20with%20us.%20Rather%20than%20integrating%20the%20mass%20of%20cool%20gas%20that%20is%20converted%20into%20stars%20over%20the%20entire%20history%20of%20the%20universe,%20they%20estimate%20the%20number%20of%20observers%20by%20calculating%20the%20total%20stellar%20mass%20density%20at%20the%20fixed%20cosmic%20time%20t%20%3D%2015%20Gyr.%20The%20authors%20argue%20that%20this%20is%20a%20reasonable%20approximation,%20but%20it%20is%20not%20an%20assumption%20that%20is%20supported%20by%20our%20model:%20rather,%20we%20predict%20that%20in%20universes%20with%20a%20large%20value%20of%20%02%20the%20majority%20of%20stars%20form%20at%20t%20%E2%89%AB%2015%20Gyr%20(see%20e.g.%0AFig.%209).%0ATo%20summarize,%20a%20common%20feature%20of%20all%20work%20on%20the%20simplest%0Aanthropic%20ensemble%20is%20that%20the%20posterior%20probability%20distribution%20on%20%02%20is%20broad,%20with%20a%20median%20that%20is%20substantially%20larger%20than%20%02obs.%20The%20exact%20value%20is%20heavily%20dependent%20on%20the%20underlying%20star%20formation%20model%20and%20the%20probability%20measure,%20but%20in%20all%20works%20considered%20the%20observed%20value%20of%20the%20cosmological%20constant%20seems%20to%20be%20located%20in%20the%20tail%20of%20the%20posterior%20for%20%02%20(except%20for%20the%20causal%20patch%20and%20diamond%20measures%20in%20Barnes%20et%20al.%202018).%20Nevertheless,%20in%20the%20previous%20literature%20discussed%20here%20the%20probability%20of%20measuring%20%02%20%E2%89%A4%20%02obs%20is%20still%20of%20the%20order%20of%20a%20few%20percent%20or%20even%20%E2%88%BC10%20percent,%20thus%20failing%20to%20reject%20the%20multiverse%20model.%20In%20contrast,%20our%20results%20pose%20a%20challenge%20in%20this%20respect,%20given%20that%20this%20probability%20is%20only%200.5%20per%20cent.%20The%20viability%20of%20Weinberg%E2%80%99s%20simple%20anthropic%20explanation%20for%20%02%20thus%20depends%20on%20which%20of%20these%20calculations%20is%20closest%20to%20a%20correct%20physical%20prediction%20of%20star%20formation%20in%20counter-factual%20universes.%20All%20existing%20studies%20have%20limitations,%20and%20ours%20is%20definitely%20no%20exception.%20However,%20in%20yielding%20a%20result%20that%20is%20so%20far%20from%20the%20observed%20level%20of%20%02,%20it%20raises%20the%20motivation%20for%20continuing%20to%20seek%20a%20more%20realistic%20treatment%20of%20this%20problem.%0A5.4.2%20Dependence%20of%20the%20cosmic%20stellar%20efficiency%20on%20%02%0AThe%20conclusions%20on%20the%20posterior%20distribution%20of%20%02%20reached%20by%20the%20different%20works%20discussed%20in%20the%20previous%20section%20are%20predicated%20on%20the%20dependence%20of%20the%20comic%20star%20formation%20efficiency%20on%20%02.%20Analysing%20this%20key%20quantity%20is%20informative%20regardless%20of%20any%20anthropic%20consideration,%20as%20it%20explores%20the%20predictions%20of%20different%20cosmic%20star%20formation%20models%20outside%20the%20canonical%20values%20of%20the%20underlying%20cosmological%20parameters.%20In%20this%20section,%20we%20will%20discuss%20similarities%20and%20discrepancies%20of%20our%20results%20on%20the%20cosmic%20star%20formation%20efficiency%20(Fig.%2010)%20with%20respect%20to%20the%20results%20from%20direct%20simulations.%0AFig.%2013%20shows%20the%20cosmic%20star%20formation%20efficiency%20predicted%20by%20our%20model,%20i.e.%20the%20same%20as%20in%20Fig.%2010,%20in%20comparison%20with%20the%20efficiencies%20obtained%20with%20the%20Eagle%20simulations%20by%20Barnes%20et%20al.%20(2018)%20and%20the%20Enzo%20simulations%20by%20Oh%20et%20al.%20(2022).%20For%20%02%20%3E%20%02obs,%20there%20is%20general%20qualitative%20agreement,%20with%20all%20works%20considered%20showing%20a%20suppression%20of%20the%20cosmic%20star%20formation%20efficiency%20for%20high%20values%20of%20%02.%20Our%20model%20predicts%20a%20peak%20at%20%02%20%E2%89%88%200.1%20%02obs%20,%20and%20lower%20efficiencies%20as%20we%20approach%20an%20EdS%20universe.%20By%20contrast,%20the%20fitting%20functions%0AImpact%20of%20%02%20on%20star%20formation%20history%201467%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1468%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0ATable%201.%20Median,%2016th%20and%2084th%20percentiles%20of%20the%20posterior%20distribution%20on%20%CE%B1%02%20for%20different%20choices%20of%20the%20prior%20on%20%02%20or%20of%20the%20measure%20of%20observers%20in%20the%20multiverse,%20and%20for%20different%20models.%20The%20table%20also%20shows%20the%20probability%20of%20observers%20measuring%20a%20value%20of%20%02%20equal%20to%20or%20smaller%20than%20%02obs.%20Unless%20otherwise%20explicitly%20stated%20in%20the%20main%20text,%20results%20from%20other%20works%20are%20quoted%20directly%20from%20the%20respective%20manuscripts,%20or%20derived%20from%20fitting%20formulae%20therein.%0AStar%20formation%20model%20This%20work%0AOh%20et%20al.%20(2022)%20Barnes%20et%20al.%20(2018)%0AObservers%20model%20Star%20formation%0AStar%20formation%20Star%20formation%20+%20delay%0AStar%20formation%20+%20lifetime%20Star%20formation%20+%20metals%0APrior/measure%20Flat%20in%20%CE%B1%02%0A%CE%B1%E2%88%920.1%20%02%20%CE%B1%E2%88%920.75%0A%02%20%CE%B1%E2%88%920.9%0A%02%0ASingle-scale%20approximation%20Simulation%0AMass%20weighted%20Causal%20patch%20Causal%20diamond%20Mass%20weighted%20Causal%20patch%20Causal%20diamond%20Mass%20weighted%20Causal%20patch%20Causal%20diamond%0A16th%20percentile%0A73%0A44%200.041%206.9%20%C3%97%2010%E2%88%924%202.3%0A1.4%0A10%200.04%200.13%0A10%200.009%200.01%0A8%200.004%200.01%0AMedian%2084th%20percentile%0A539%202306%20389%201868%202.8%2084%200.18%2012%2012.6%2043%208.0%2034%2059%20194%200.34%200.96%200.65%203.15%2059%20194%200.089%200.849%200.25%200.96%2045%20163%200.07%200.71%200.17%200.87%0Ap(%02%20%E2%89%A4%20%02obs)%200.52%20per%20cent%0A1.0%20per%20cent%2039%20per%20cent%2065%20per%20cent%208.2%20per%20cent%2012.9%20per%20cent%201.9%20per%20cent%2086%20per%20cent%2073%20per%20cent%201.9%20per%20cent%2090%20per%20cent%2086%20per%20cent%202.5%20per%20cent%2093%20per%20cent%2090%20per%20cent%0AFigure%2013.%20Same%20as%20Fig.%2010,%20for%20the%20model%20presented%20in%20this%20work%20(red%20line),%20and%20the%20cosmological%20simulations%20by%20Barnes%20et%20al.%20(2018)%20and%20Oh%20et%20al.%20(2022)%20(blue%20circles%20and%20purple%20squares,%20respectively).%20The%20corresponding%20fitting%20formulae%20to%20the%20numerical%20results,%20proposed%20by%20Oh%20et%20al.%20(2022),%20are%20represented%20with%20blue%20and%20purple%20solid%20lines,%20respectively.%20Integrating%20the%20CSFRD%20given%20by%20our%20model%20up%20to%20the%20same%20finite%20cosmic%20time%20as%20in%20the%20simulations%20(tmax%20%3D%2020.7%20Gyr%20and%20tmax%20%3D%2097%20Gyr%20for%20Barnes%20et%20al.%202018%20and%20Oh%20et%20al.%202022,%20respectively)%20improves%20the%20agreement%20with%20the%20numerical%20results%20(dashed%20and%20solid%20salmon%20lines,%20respectively).%20The%20match%20with%20Barnes%20et%20al.%20(2018)%20is%20further%20improved%20by%20properly%20changing%20the%20slope%20of%20the%20gas%20density%20profiles%20within%20haloes,%20%CE%B7,%20which%20is%20a%20free%20parameter%20in%20our%20model%20(dotted%20salmon%20line;%20see%20discussion%20in%20Section%205.4.2%20for%20details).%0Ato%20the%20numerical%20results%20appear%20to%20indicate%20a%20monotonic%20increase%20in%20the%20asymptotic%20cosmic%20star%20formation%20efficiency%20as%20%02%20%E2%86%92%200.%20While%20this%20trend%20is%20qualitatively%20different%20from%20our%20result,%20no%20simulations%20actually%20cover%20the%20range%200%20%3C%20%02%20%3C%20%02obs.%20It%20would%20be%20interesting%20to%20run%20further%20simulations%20in%20this%20range%20in%20order%20to%20establish%20whether%20the%20simulated%20dependence%20of%20the%20cosmic%20star%20formation%20efficiency%20on%20%02%20is%20genuinely%20monotonic.%0ANevertheless,%20assuming%20that%20the%20fitting%20functions%20proposed%20by%20Oh%20et%20al.%20(2022)%20do%20represent%20the%20actual%20trend%20of%20the%20efficiency%20in%20the%20range%200%20%3C%20%02%20%3C%20%02obs,%20it%20is%20worth%20speculating%20on%20the%20possible%20origins%20of%20the%20different%20result%20that%20we%20obtain.%20We%20should,%20however,%20first%20make%20sure%20that%20we%20make%20a%20fair%20comparison%20across%20all%20models%20considered.%20The%20simulations%20are%20inevitably%20stopped%20at%20some%20finite%20future%20time;%20for%20our%20cosmology%20and%20an%20EdS%20universe,%20this%20maximum%20time%20is%20tmax%20%3D%2097%20Gyr%20and%20tmax%20%3D%2020.7%20Gyr%20in%20Barnes%20et%20al.%20(2018)%20and%20Oh%20et%20al.%20(2022),%0Arespectively.%20To%20account%20for%20the%20star%20formation%20history%20at%20arbitrarily%20large%20times,%20Barnes%20et%20al.%20(2018)%20extrapolate%20the%20time%20evolution%20of%20the%20CSFRD%20beyond%20tmax%20with%20a%20decaying%20exponential,%20and%20conclude%20that%20the%20contribution%20from%20times%20t%20%3E%20tmax%20to%20the%20total%20cosmic%20star%20formation%20efficiency%20is%20negligible.%20Therefore,%20for%20a%20fairer%20comparison%20between%20our%20model%20and%20the%20Oh%20et%20al.%20(2022)%20fitting%20formula%20to%20the%20Barnes%20et%20al.%20(2018)%20simulations%20(blue%20line%20in%20Fig.%2013),%20we%20should%20re-compute%20our%20cosmic%20star%20formation%20efficiencies%20by%20integrating%20the%20CSFRD%20up%20to%20tmax%3D20.7Gyr,andnotinthelimitt%E2%86%92%E2%88%9E.%0AThe%20result%20is%20represented%20by%20the%20dashed%20salmon%20line%20in%20Fig.%2013,%20which%20considerably%20improves%20the%20agreement%20with%20the%20fitting%20formula%20to%20the%20Barnes%20et%20al.%20(2018)%20results.%20Our%20model%20recovers%20the%20cosmic%20efficiency%20in%20the%20Barnes%20et%20al.%20(2018)%20simulations%20within%2050%20per%20cent%20for%20an%20EdS%20universe,%20and%20yields%20an%20excellent%20match%20at%20the%20high-%02%20end.%20The%20largest%20discrepancy%20occurs%20for%20%02%20%E2%89%88%200.3%02obs,%20and%20amounts%20to%20less%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0Athan%20a%20factor%20of%20%E2%88%BC%203.%20This%20deviation%20is%20smaller%20than%20the%20discrepancy%20between%20Barnes%20et%20al.%20(2018)%20and%20Oh%20et%20al.%20(2022).%20However,%20the%20non-%20monotonic%20dependence%20of%20the%20cosmic%20star%20formation%20efficiency%20on%20%02%20for%20%02%20%3C%20%02obs%20does%20not%20disappear%20by%20simply%20imposing%20a%20cut-off%20in%20the%20time%20integral%20of%20the%20CSFRD.%20However,%20the%20peak%20becomes%20much%20less%20prominent,%20and%20the%20discrepancy%20with%20the%20numerical%20results%20is%20reduced.%0AIf%20we%20apply%20the%20same%20reasoning%20for%20a%20fair%20comparison%20between%20our%20model%20and%20the%20Oh%20et%20al.%20(2022)%20simulations,%20we%20reach%20overall%20similar%20conclusions.%20To%20estimate%20the%20integral%20of%20the%20CSFRD%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E,%20Oh%20et%20al.%20(2022)%20extrapolated%20their%20numerical%20results%20beyond%20tmax%20by%20fitting%20their%20CSFRD%20with%20the%20same%20functional%20shape%20used%20by%20Madau%20%26%20Dickinson%20(2014).%20This%20analytical%20formula%20allowed%20us%20to%20subtract%20the%20contribution%20from%20cosmic%20times%20t%20%3E%20tmax%20from%20the%20cosmic%20star%20formation%20efficiencies%20computed%20by%20Oh%20et%20al.%20(2022),%20and%20we%20verified%20that%20the%20change%20would%20be%20in%20any%20case%20negligible.%20We%20can%20then%20simply%20cut%20the%20time%20integral%20of%20the%20CSFRD%20at%20tmax%20%3D%2097%20Gyr.%20The%20result%20is%20given%20by%20the%20salmon%20solid%20line%20in%20Fig.%2013.%20The%20peak%20is%20less%20conspicuous,%20and%20the%20agreement%20with%20the%20fit%20to%20the%20numerical%20results%20(purple%20line)%20improves.%20However,%20the%20decay%20at%20large%20values%20of%20%02%20is%20less%20steep%20compared%20to%20Oh%20et%20al.%20(2022)%20work,%20and%20the%20efficiency%20in%20the%20EdS%20universe%20is%20recovered%20within%20a%20factor%20of%20%E2%88%BC%204.5.%0AWithout%20adding%20any%20new%20physics%20to%20our%20model,%20we%20can%20check%20whether%20a%20change%20in%20the%20underlying%20parameters%20would%20yield%20a%20better%20match%20with%20cosmological%20simulations.%20This%20can%20be%20done%20by%20consider-%20ing%20the%20late-time%20behaviour%20of%20the%20CSFRD.%20As%20mentioned%20earlier,%20Oh%20et%20al.%20(2022)%20showed%20that%20their%20CSFRDs%20can%20be%20well%20described%20with%20the%20same%20fitting%20formula%20adopted%20by%20Madau%20%26%20Dickinson%20(2014).%20We%20verified%20that%20this%20is%20the%20case%20for%20the%20Barnes%20et%20al.%20(2018)%20CSFRDs%20too.%20At%20late%20times,%20the%20Madau%20%26%20Dickinson%20(2014)%20curve%20can%20be%20in%20turn%20approximated%20with%20a%20power-law%20scaling%20of%20the%20CSFRD%20with%20(1%20+%20z).%20Our%20model%20also%20predicts%20a%20power-law%20scaling%20in%20an%20EdS%20universe%20at%20late%20times,%20with%20an%20index%20that%20depends%20on%20the%20slope%20of%20the%20gas%20density%20profile%20within%20haloes,%20%CE%B7%20(see%20Appendix%20A1).%20We%20can%20therefore%20find%20the%20value%20of%20%CE%B7%20that%20yields%20the%20same%20late-time%20power-law%20scaling%20in%20an%20EdS%20universe%20as%20in%20Barnes%20et%20al.%20(2018)%20and%20Oh%20et%20al.%20(2022).%0AIn%20the%20first%20case,%20we%20find%20that%20the%20desired%20value%20is%20%CE%B7%20%3D%202.75.%20We%20then%20run%20our%20model%20for%20such%20value%20of%20%CE%B7,%20while%20leaving%20all%20other%20parameters%20the%20same.%20We%20integrate%20the%20CSFRD%20for%20any%20value%20of%20%02%20up%20to%20tmax%20%3D%2020.7%20Gyr,%20and%20obtain%20the%20cosmic%20star%20formation%20efficiency.%20Moving%20from%20%CE%B7%20%3D%201.9%20to%20%CE%B7%20%3D%202.75%20further%20reduces%20the%20prominence%20of%20the%20peak%20in%20the%20cosmic%20star%20formation%20efficiency.%20However,%20the%20overall%20agreement%20with%20Barnes%20et%20al.%20(2018)%20does%20not%20significantly%20improve.%20Also,%20the%20value%20%CE%B7%20%3D%202.75,%20while%20mathematically%20acceptable,%20would%20fall%20outside%20the%20range%20that%20provides%20a%20consistency%20within%203%CF%83%20with%20observational%20constraints%20from%20the%20Kennicutt%E2%80%93Schmidt%20relationship%20and%20the%20baryonic%20Tully%E2%80%93Fisher%20relationship%20(see%20SP21%20for%20details).%20If%20we%20apply%20the%20same%20logic%20to%20the%20Oh%20et%20al.%20(2022)%20simulations,%20we%20would%20find%20%CE%B7%20%3D%203.21.%20This%20is%20not%20even%20mathematically%20acceptable%20in%20our%20formalism,%20as%20values%20of%20%CE%B7%20%3E%203%20would%20cause%20the%20gas%20mass%20of%20haloes%20to%20diverge%20(see%20equation%207).%20Therefore,%20we%20cannot%20fine%20tune%20%CE%B7%20to%20match%20the%20late-time%20behaviour%20of%20the%20CSFRD%20found%20by%20Oh%20et%20al.%20(2022)%20for%20an%20EdS%20universe.%0ATo%20summarize,%20the%20qualitative%20discrepancy%20between%20the%20cosmic%20star%20formation%20efficiency%20predicted%20by%20our%20model%20and%20by%20analytical%20fits%20to%20cosmological%20simulations%20is%20considerably%20mitigated%20once%20a%20cut-off%20time%20is%20taken%20into%20consideration,%20for%20a%20fair%20comparison%20with%20the%20numerical%20results.%20Where%20possible,%20the%20significance%20of%20the%20peak%20in%20the%20cosmic%20star%20formation%20efficiency%20predicted%20by%20our%20model%20can%20be%20further%20reduced%20by%20regulating%20the%20value%20of%20%CE%B7%20to%20match%20the%20late-time%20behaviour%20of%20the%20simulated%20CSFRD%20in%20an%20EdS%20universe.%20However,%20the%20peak%20does%20not%20disappear%20completely,%20therefore%20some%20change%20in%20the%20physics%20included%20in%20our%20model%20may%20be%20necessary%20in%20order%20to%20obtain%20a%0Acosmic%20star%20formation%20efficiency%20that%20monotonically%20decreases%20with%20%02.%0AThe%20residual%20qualitative%20discrepancies%20between%20our%20model%20and%20numerical%20simulation%20occur%20in%20the%20range%200%20%3C%20%02%20%3C%20%02obs.%20Within%20our%20formalism,%20the%20asymptotic%20stellar%20mass%20produced%20in%20a%20halo%20of%20a%20fixed%20virial%20mass%20M%20%02%201011%20M%E2%8A%99%20does%20not%20vary%20appreciably%20with%20the%20cosmological%20constant%20for%20%02%20%3C%20%02obs.%20However,%20for%20the%20same%20cosmologies,%20the%20stellar%20mass%20produced%20in%20high-mass%20haloes%20(M%20%03%201011%20M%E2%8A%99%20)%20can%20vary%20by%20up%20to%20one%20order%20of%20magnitude%20(see%20Fig.%208).%20Therefore,%20the%20treatment%20of%20star%20formation%20within%20high-mass%20haloes%20is%20probably%20the%20main%20reason%20behind%20the%20discrepancies%20between%20our%20model%20and%20simulations.%20In%20this%20respect,%20an%20obvious%20missing%20piece%20of%20physics%20in%20our%20model%20is%20AGN%20feedback,%20as%20discussed%20in%20Section%205.3.%20An%20explicit%20AGN%20feedback%20mechanism%20could%20reduce%20the%20contribution%20of%20high-mass%20haloes%20to%20the%20star%20formation%20history,%20and%20may%20bring%20the%20cosmic%20efficiency%20at%200%20%3C%20%02%20%3C%20%02obs%20closer%20to%20that%20of%20an%20EdS%20universe.%0AIt%20is%20worth%20noting%20that%20Oh%20et%20al.%20(2022)%20predict%20a%20decreasing%20efficiency%20even%20in%20the%20absence%20of%20AGN%20feedback.%20There%20may%20therefore%20be%20other%20ways%20to%20achieve%20the%20same%20qualitative%20trend%20without%20(or%20in%20addition%20to)%20AGN%20feedback.%20For%20example,%20one%20may%20include%20chemical%20evolution%20models,%20or%20an%20explicit%20shutdown%20of%20star%20formation%20at%20very%20late%20times,%20as%20detailed%20in%20Section%205.3.%20We%20intend%20to%20address%20these%20issues%20in%20future%20work.%20At%20the%20same%20time,%20we%20point%20out%20that%20it%20would%20be%20very%20informative%20to%20run%20cosmological%20simulations%20in%20the%20range%200%20%3C%20%02%20%3C%20%02obs,%20which%20is%20the%20range%20where%20our%20predictions%20differ%20from%20the%20fits%20to%20the%20numerical%20results%20by%20Barnes%20et%20al.%20(2018)%20and%20Oh%20et%20al.%20(2022).%0A6%20CONCLUSIONS%20AND%20PERSPECTIVES%0AWe%20have%20improved%20the%20SP21%20analytical%20model%20for%20cosmic%20star%20formation%20(Sorini%20%26%20Peacock%202021),%20which%20was%20itself%20based%20on%20the%20work%20by%20Hernquist%20%26%20Springel%20(2003).%20We%20have%20then%20used%20this%20model%20to%20predict%20the%20cosmic%20star%20formation%20history%20in%20flat%20%02CDM%20cosmological%20models%20where%20the%20cosmological%20constant%20is%20varied%20between%20%02%20%3D%200%20and%20%02%20%3D%20105%20%02obs.%20This%20ensemble%20of%20universes%20was%20designed%20so%20that%20all%20members%20are%20indistinguishable%20at%20high%20redshift,%20but%20later%20part%20company%20as%20their%20different%20cosmological%20constants%20come%20to%20dominate.%0AThe%20CSFRD%20is%20obtained%20by%20integrating%20the%20SFR%20within%20haloes%20over%20the%20halo%20mass%20function.%20We%20assumed%20that%20the%20same%20astrophysical%20processes%20operate%20in%20each%20member%20of%20the%20ensemble%20as%20in%20our%20own%20universe.%20At%20high%20redshift,%20the%20SFR%20is%20determined%20by%20the%20typical%20time%20taken%20to%20convert%20cool%20gas%20into%20stars,%20whereas%20at%20low%20redshift%20it%20is%20set%20by%20the%20cooling%20time.%20We%20were%20then%20able%20to%20compute%20the%20total%20stellar%20mass%20generated%20by%20a%20given%20time,%20as%20a%20fraction%20of%20the%20total%20baryonic%20mass,%20per%20comoving%20volume.%20We%20showed%20that%20this%20cumulative%20star%20formation%20efficiency%20tends%20to%20a%20well-defined%20limit%20as%20t%20%E2%86%92%20%E2%88%9E,%20and%20we%20determined%20the%20dependence%20of%20this%20asymptotic%20efficiency%20on%20the%20value%20of%20%02.%20We%20then%20investigated%20the%20implications%20of%20our%20results%20for%20anthropic%20attempts%20to%20explain%20the%20observed%20value%20of%20%02.%20Our%20main%20conclusions%20are%20as%20follows:%0A(i)%20The%20SP21%20formalism%20predicts%20a%20converging%20total%20comoving%20stellar%20mass%20density%20as%20the%20cosmic%20time%20tends%20to%20infinity.%20In%20our%20universe,%20about%2032%20percent%20of%20the%20eventual%20stellar%20mass%20has%20been%20formed%20by%20z%20%3D%200%20(Fig.%209).%20Thus,%20we%20are%20typical%20observers%20in%20terms%20of%20the%20cosmological%20epoch%20in%20which%20we%20live,%20which%20is%20marginally%20%02-%20dominated.%20However,%20we%20find%20that%20this%20coincidence%20between%20typical%20eras%20of%20star%20formation%20and%20%02-domination%20does%20not%20hold%20for%20other%20values%20of%20%02;%20hence%20single-universe%20anthropic%20arguments%20do%20not%20account%20for%20the%20%E2%80%98why-now%E2%80%99%20problem.%0AImpact%20of%20%02%20on%20star%20formation%20history%201469%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1470%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0A(ii)%20Our%20model%20predicts%20that%20the%20cosmic%20star%20formation%20efficiency%20peaks%20at%20around%2027%20per%20cent%20for%20%02%20%E2%89%88%200.1%02obs%20(Fig.%2010).%20In%20an%20Einstein%E2%80%93%20de%20Sitter%20universe%20with%20%02%20%3D%200,%20the%20efficiency%20is%20approximately%20a%20factor%20of%203%20lower%20than%20this;%20but%20the%20efficiency%20is%20much%20more%20strongly%20suppressed%20at%20high%20values%20of%20%02.%20The%20decline%20is,%20however,%20relatively%20slow,%20and%20the%20efficiency%20only%20becomes%20negligibly%20small%20as%20%02%20approaches%20105%20%02obs.%0A(iii)%20For%20a%20given%20%02,%20there%20is%20a%20characteristic%20halo%20mass%20above%20which%20the%20stellar%20efficiency%20within%20haloes%20is%20maximized%20(Fig.%208).%20A%20halo%20with%20a%20given%20virial%20mass%20is%20smaller%20in%20universes%20with%20larger%20%02,%20so%20that%20its%20typical%20density%20is%20higher.%20This%20in%20turn%20makes%20gas%20cooling,%20and%20subsequently%20star%20formation,%20more%20efficient.%20Thus,%20the%20suppression%20of%20the%20cosmic%20star%20formation%20efficiency%20at%20high%20%02%20must%20be%20cosmological%20in%20nature,%20and%20not%20astrophysical:%20it%20is%20induced%20by%20the%20cut-off%20in%20the%20halo%20mass%20function%20occurring%20at%20lower%20halo%20masses.%20Conversely,%20the%20reason%20for%20the%20reduced%20cosmic%20star%20formation%20efficiency%20at%20low%20%02%20is%20astrophysical:%20haloes%20are%20more%20diffuse%20as%20the%20cosmological%20model%20approximates%20an%20Einstein%E2%80%93de%20Sitter%20universe,%20so%20that%20the%20cooling%20rate%20is%20reduced.%20Thus,%20according%20to%20our%20model,%20astrophysics%20favours%20star%20formation%20in%20those%20haloes%20that%20exist%20when%20%02%20is%20high,%20whereas%20cosmology%20increases%20the%20abundance%20of%20haloes%20when%20%02%20is%20low.%20The%20observed%20value%20of%20the%20cosmological%20constant%20finds%20itself%20near%20the%20sweet%20spot%20of%20these%20two%20opposite%20trends%20(Fig.%2010).%20However,%20that%20does%20not%20imply%20that%20the%20generation%20of%20observers%20is%20maximized%20around%20the%20same%20value%20of%20%02.%0A(iv)%20Assuming%20a%20flat%20prior%20on%20%02%20and%20that%20the%20observer%20weighting%20of%20a%20universe%20with%20a%20given%20%02%20is%20proportional%20to%20the%20cosmic%20star%20formation%20efficiency%20in%20that%20universe,%20the%20peak%20of%20the%20posterior%20on%20%02%20occurs%20at%20%02%20%E2%89%88%20800%20%02obs%20(Fig.%2011).%20The%20median%20of%20the%20distribution%20is%20%02%20%3D%20539,%20and%20the%2016th%20%E2%88%92%2084th%20percentile%20range%20corresponds%20to%20%02%20%3D%2073%20%E2%88%92%202306%20(Table%201).%20The%20probability%20that%20an%20observer%20measures%20a%20value%20of%20the%20cosmological%20constant%20%02%20%E2%89%A4%20%02obs%20is%20%E2%88%BC%200.5%20per%20cent.%20Thus,%20Weinberg%E2%80%99s%20anthropic%20reasoning%20would%20be%20disfavoured%20by%20the%20SP21%20model,%20under%20the%20given%20assumptions.%0AIt%20is%20striking%20that%20the%20observed%20value%20of%20%02%20is%20within%20one%20order%20of%20magnitude%20of%20the%20optimal%20value%20that%20maximizes%20the%20cosmic%20star%20formation%20efficiency.%20We%20also%20succeed%20in%20providing%20an%20explanation%20for%20why%20this%20is%20the%20case,%20based%20on%20calculations%20from%20first%20principles.%20However,%20we%20also%20found%20that%20this%20asymptotic%20efficiency%20of%20star%20formation%20declines%20rather%20slowly%20as%20%02%20increases,%20falling%20below%201/10%20of%20the%20peak%20value%20only%20for%20%02%20%3E%20500%20%02obs%20(Fig.%2010).%20Hence,%20with%20Weinberg%E2%80%99s%20assumption%20of%20a%20uniform%20prior%20on%20%02,%20it%20is%20inevitable%20that%20typical%20observers%20would%20expect%20to%20experience%20values%20of%20%02%20much%20greater%20than%20the%20one%20we%20observe,%20and%20this%20is%20why%20we%20find%20that%20values%20of%20%02%20as%20small%20as%20observed%20are%20highly%20unlikely.%20In%20principle%20this%20rules%20out%20the%20simple%20anthropic%20ensemble%20considered%20here,%20subject%20to%20the%20assumptions%20of%20our%20model.%20But%20we%20have%20reviewed%20alternative%20calculations%20and%20found%20that%20some%20give%20a%20less%20severe%20disagreement%20%E2%80%93%20although%20there%20is%20unanimity%20that%20the%20observed%20%02%20is%20unusually%20small%20at%20some%20level.%0AFuture%20progress%20on%20this%20question%20requires%20more%20detailed%20and%20realis-%20tic%20modelling:%20we%20will%20endeavour%20to%20generalize%20our%20model%20of%20cosmic%20star%20formation%20by%20including%20features%20such%20as%20chemical%20evolution%20of%20the%20interstellar/intergalactic%20medium%20and%20AGN%20feedback.%20In%20particular,%20the%20absence%20of%20an%20explicit%20AGN%20feedback%20mechanism%20may%20lead%20to%20an%20overproduction%20of%20stars%20in%20massive%20haloes,%20which,%20according%20to%20our%20model,%20are%20especially%20efficient%20in%20universes%20with%20a%20high%20value%20of%20%02.%20Whether%20or%20not%20this%20is%20important%20depends%20on%20the%20applicability%20of%20exact%20co-evolution%20between%20black%20holes%20and%20stellar%20content%20in%20galaxies.%20If%20this%20concept%20holds%20exactly,%20the%20modelled%20effective%20energy%20feedback%20from%20star%20formation%20may%20be%20able%20to%20capture%20the%20effect%20of%0AAGN%20feedback.%20However,%20if%20co-evolution%20does%20not%20hold%20exactly,%20then%20the%20inclusion%20of%20separate%20AGN%20feedback%20in%20our%20formalism%20might%20alter%20the%20resulting%20conclusions%20on%20the%20anthropically%20favoured%20range%20of%20%02.%20We%20plan%20to%20address%20this%20question%20in%20future%20work.%0AOne%20would%20also%20like%20to%20see%20further%20direct%20calculations%20via%20sim-%20ulations.%20However,%20hydrodynamical%20simulations%20are%20expensive%20to%20run,%20especially%20beyond%20z%20%3D%200,%20thus%20it%20is%20important%20to%20invest%20the%20computational%20resources%20to%20study%20models%20that%20promise%20to%20bring%20the%20most%20valuable%20physical%20insights.%20Our%20analytical%20model%20enables%20a%20fast%20exploration%20of%20the%20dependence%20of%20star%20formation%20history%20on%20the%20cosmological%20parameters,%20so%20it%20can%20aid%20in%20designing%20a%20suite%20of%20simulations%20exploring%20cosmological%20models%20that%20differ%20from%20the%20fiducial%20%02CDM%20cosmology%20in%20especially%20interesting%20ways.%0AEventually,%20we%20can%20hope%20for%20a%20convergence%20of%20these%20studies%20and%20a%20consensus%20on%20the%20viability%20of%20Weinberg%E2%80%99s%20anthropic%20explanation%20for%20%02.%20However,%20even%20if%20the%20verdict%20is%20negative,%20this%20would%20be%20far%20from%20a%20rejection%20of%20anthropic%20reasoning%20in%20general.%20This%20manuscript%20focuses%20on%20changing%20only%20one%20fundamental%20parameter,%20i.e.%20the%20cosmological%20constant%20%E2%80%93%20and%20even%20there%20we%20have%20restricted%20ourselves%20to%20the%20case%20of%20%02%20%E2%89%A5%200,%20leaving%20the%20whole%20topic%20of%20recollapsing%20models%20for%20a%20further%20investigation.%20Beyond%20this,%20one%20should%20in%20principle%20vary%20multiple%20parameters%20simultaneously,%20and%20explore%20their%20joint%20impact%20on%20star%20formation%20history%20and%20the%20generation%20of%20observers.%20We%20are%20already%20working%20on%20this%20kind%20of%20study%20(Lombriser%20et%20al.,%20in%20preparation).%0AACKNOWLEDGEMENTS%0AWe%20thank%20the%20anonymous%20reviewers%20for%20helpful%20comments.%20The%20authors%20thank%20Sownak%20Bose%20and%20Boon%20Kiat%20Oh%20for%20useful%20discussions,%20and%20Ed%20Wheeler%20for%20comments%20on%20a%20draft%20of%20this%20manuscript.%20DS%20and%20LL%20acknowledge%20support%20from%20theSwiss%20National%20Science%20Founda-%20tion%20Professorship%20grant%20no.%20202671.%20DS%20and%20JAP%20were%20supported%20by%20the%20European%20Research%20Council,%20under%20grant%20no.%20670193%20and%20by%20the%20Science%20and%20Technology%20Facilities%20Council%20conso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%20P.%20F.,%20Cox%20T.%20J.,%20Robertson%20B.%20E.,%20Hernquist%20L.,%0A2008,%20MNRAS,%20391,%20481%0ASorini%20D.,%202022,preprint%20(arXiv:2204.07509)%0ASorini%20D.,%20Bose%20S.,%20Dave%20%CC%81%20R.,%20Angle%20%CC%81%20s%20Alca%20%CC%81%20zar%20D.,%202024b,%20preprint%0A(arXiv:2409.05815)%0ASorini%20D.,%20Bose%20S.,%20Pakmor%20R.,%20Hernquist%20L.,%20Springel%20V.,%20Hadzhiyska%20B.,%0AHerna%20%CC%81ndez-Aguayo%20C.,%20Kannan%20R.,%202024a,%20preprint%20(arXiv:2409.01758)%0AThomas%20P.%20A.,%20White%20S.%20D.%20M.,%202020,%20MNRAS,%20491,%205795%20Hernquist%20L.,%20Springel%20V.,%202003,%20MNRAS,%20341,%201253%20Hopkins%20P.%20F.,%202015,%20MNRAS,%20450,%2053%0AHopkins%20P.%20F.,%20Keres%CB%87%20D.,%20On%20%CC%83%20orbe%20J.,%20Faucher-Gigue%60%20re%0AMurray%20N.,%20Bullock%20J.%20S.,%202014,%20MNRAS,%20445,%20581%0AHuang%20S.%20S.,%201959,%20PASP,%2071,%20220%0AJohnson%20M.%20C.,%20Wainwright%20C.%20L.,%20Aguirre%20A.,%20Peiris%20H.%20V.,%202016,%20J.%20Cosmol.%0AAstropart.%20Phys.,%202016,%20020%0AJoyce%20A.,%20Lombriser%20L.,%20Schmidt%20F.,%202016,%20Annu.%20Rev.%20Nucl.%20Part.%20Sci.,%2066,%2095%20Kaloper%20N.,%20Padilla%20A.,%202014,%20Phys.%20Rev.%20Lett.,%20112,%20091304%0AKane%20G.%20L.,%20Perry%20M.%20J.,%20Zytkow%20A.%20N.,%202002,%20New%20Astron.,%207,%2045%0AKasting%20J.%20F.,%20Whitmire%20D.%20P.,%20Reynolds%20R.%20T.,%201993,%20Icarus,%20101,%20108%0AC.-A.,%20Quataert%0AE.,%0AM.%20H.,%0AImpact%20of%20%02%20on%20star%20formation%20history%201471%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0ADownloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%0A1472%20D.%20Sorini,%20J.%20A.%20Peacock%20and%20L.%20Lombriser%0ASorini%20D.,%20Dave%20%CC%81%20R.,%20Cui%20W.,%20Appleby%20S.,%202022,%20MNRAS,%20516,%20883%20Sorini%20D.,%20Peacock%20J.%20A.,%202021,%20MNRAS,%20508,%205802%0ASpringel%20V.%20et%20al.,%202005,%20Nature,%20435,%20629%0ASpringel%20V.,%20Hernquist%20L.,%202003a,%20MNRAS,%20339,%20312%0ASpringel%20V.,%20Hernquist%20L.,%202003b,%20MNRAS,%20339,%20289%0ASpringel%20V.,%20Pakmor%20R.,%20Zier%20O.,%20Reinecke%20M.,%202021,%20MNRAS,%20506,%202871%20Springel%20V.,%20White%20S.%20D.%20M.,%20Tormen%20G.,%20Kauffmann%20G.,%202001b,%20MNRAS,%0A328,%20726%0ASpringel%20V.,%20Yoshida%20N.,%20White%20S.%20D.%20M.,%202001a,%20New%20Astron.,%206,%2079%0ASudoh%20T.,%20Totani%20T.,%20Makiya%20R.,%20Nagashima%20M.,%202017,%20MNRAS,%20464,%201563%20Susskind%20L.,%20Thorlacius%20L.,%20Uglum%20J.,%201993,%20Phys.%20Rev.%20D,%2048,%203743%20Sutherland%20R.%20S.,%20Dopita%20M.%20A.,%201993,%20ApJS,%2088,%20253%0ATegmark%20M.,%20Aguirre%20A.,%20Rees%20M.%20J.,%20Wilczek%20F.,%202006,%20Phys.%20Rev.%20D,%2073,%0A023505%0ATotani%20T.,%20Omiya%20H.,%20Sudoh%20T.,%20Kobayashi%20M.%20A.%20R.,%20Nagashima%20M.,%202019,%0AAstrobiology,%2019,%20126%0ATraina%20A.%20et%20al.,%202024,%20A%26A,%20681,%20A118%0AVelten%20H.%20E.%20S.,%20vom%20Marttens%20R.%20F.,%20Zimdahl%20W.,%202014,%20Eur.%20Phys.%20J.%20C,%2074,%0A3160%0AVerde%20L.,%20Treu%20T.,%20Riess%20A.%20G.,%202019,%20Nat.%20Astron.,%203,%20891%0AVilenkin%20A.,%201983,%20Phys.%20Rev.%20D,%2027,%202848%0AVogelsberger%20M.%20et%20al.,%202014,%20MNRAS,%20444,%201518%0AVogelsberger%20M.,%20Genel%20S.,%20Sijacki%20D.,%20Torrey%20P.,%20Springel%20V.,%20Hernquist%20L.,%0A2013,%20MNRAS,%20436,%203031%0AWainwright%20C.%20L.,%20Johnson%20M.%20C.,%20Aguirre%20A.,%20Peiris%20H.%20V.,%202014,%20J.%20Cosmol.%0AAstropart.%20Phys.,%202014,%20024%0AWang%20J.,%20Fischer%20D.%20A.,%202015,%20AJ,%20149,%2014%0AWatkins%20L.%20L.,%20van%20der%20Marel%20R.%20P.,%20Sohn%20S.%20T.,%20Evans%20N.%20W.,%202019,%20ApJ,%20873,%0A118%0AWeinberg%20S.,%201987,%20Phys.%20Rev.%20Lett.,%2059,%202607%0AWeinberg%20S.,%201989,%20Rev.%20Mod.%20Phys.,%2061,%201%0AWeinberg%20S.,%202000a,%20preprint%20(astro-ph/0005265)%0AWeinberg%20S.,%202000b,%20Phys.%20Rev.%20D,%2061,%20103505%0AWeinberger%20R.%20et%20al.,%202017,%20MNRAS,%20465,%203291%0AWenmackers%20S.%20,%202024,%20preprint%20(%20arXiv:2308.12229)%0AWhite%20S.%20D.%20M.,%20Frenk%20C.%20S.,%201991,%20ApJ,%20379,%2052%0AWolfram%20Research,%20Inc.,%202023,%20Mathematica,%20Version%2013.2.1.0,%20https://www.%0Awolfram.com/mathematica%0AYang%20T.,%20Dave%20%CC%81%20R.,%20Cui%20W.,%20Cai%20Y.-C.,%20Peacock%20J.%20A.,%20Sorini%20D.,%202024,%20MNRAS,%0A527,%201612%0AYoshida%20N.,%20Stoehr%20F.,%20Springel%20V.,%20White%20S.%20D.%20M.,%202002,%20MNRAS,%20335,%20762%20Zlatev%20I.,%20Wang%20L.,%20Steinhardt%20P.%20J.,%201999,%20Phys.%20Rev.%20Lett.,%2082,%20896%0AAPPENDIX%20A:%20DERIVATION%20OF%20THE%20ASYMPTOTIC%20BEHAVIOUR%20OF%20STAR%20FORMATION%20HISTORY%0AIn%20Section%204.1,%20we%20showed%20that%20the%20stellar%20mass%20fraction%20in%20haloes%20with%20fixed%20mass%20M%20asymptotes%20to%20a%20constant%20in%20the%20far%20future%20of%20the%20universe.%20This%20means%20that%20the%20corresponding%20nSFR,%20and%20consequently%20the%20CSFRD,%20tend%20to%20zero%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E.%20These%20conclusions%20result%20from%20the%20numerical%20implementation%20of%20the%20iterative%20procedure%20that%20we%20introduced%20to%20extend%20the%20SP21%20analytic%20model%20(see%20Section%202.2).%20In%20this%20Appendix,%20we%20will%20show%20that,%20under%20well-motivated%20approximations,%20it%20is%20possible%20to%20derive%20the%20asymptotic%20behaviour%20of%20the%20nSFR%20and%20CSFRD%20analytically.%20Crucially,%20this%20will%20enable%20us%20to%20prove%20that%20the%20SMD,%20which%20is%20the%20time%20integral%20of%20the%20CSFRD,%20converges%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E.%0ASince%20the%20CSFRD%20depends%20on%20the%20HMF%20and%20the%20nSFR%20at%20fixed%20M%20(see%20equation%201),%20we%20will%20first%20need%20to%20study%20the%20limit%20of%20these%20two%20quantities%20for%20arbitrarily%20large%20cosmic%20times.%20Once%20we%20obtain%20the%20asymptotic%20behaviour%20of%20the%20CSFRD,%20it%20will%20be%20straightforward%20to%20study%20the%20convergence%20of%20the%20SMD.%20We%20will%20need%20to%20distinguish%20between%20the%20two%20qualitatively%20different%20cases%20of%20%02%20%3D%200%20and%20%02%20%3E%200.%0AAll%20computations%20in%20this%20section%20have%20been%20verified%20with%20the%20software%20MATHEMATICA%20(Wolfram%20Research,%20Inc.%202023).%0AA1%20Einstein%E2%80%93de%20Sitter%20universe%0AA1.1%20Normalized%20SFR%0AThe%20evolution%20of%20the%20nSFR%20of%20a%20halo%20with%20fixed%20virial%20temperature%20T%20in%20the%20low-redshift%20regime%20is%20given%20by%20equation%20(13).%20In%20an%20EdS%20universe,%20the%20effective%20cooling%20time%20coincides%20with%20the%20dynamical%20time%20at%20any%20cosmic%20epoch%20(see%20equation%20(10)).%20Therefore,%20equation%20(13)%20can%20be%20simplified%20to%0AMNRAS%20535,%201449%E2%80%931474%20(2024)%0A%04%053%20H(z)fgas(T(M,z),z)%20%CE%B7%0A%20%CC%83%0Aslow(M,z)%E2%89%88S(T(M,z))%20H%20f%20,%20(A1)%0A0b%0Aswitching%20from%20fixed%20virial%20temperature%20to%20fixed%20virial%20mass%20as%20the%20independent%20variable.%20Thus,%20the%20first%20term%20in%20the%20r.h.s%20of%20the%20equa-%20tion%20above%20acquires%20a%20redshift%20dependence,%20which%20can%20be%20determined%20by%20replacing%20the%20virial%20temperature%20in%20SP21%20equation%20(25)%20with%20the%20expression%20given%20by%20equation%20(5).%20Recalling%20the%20definition%20of%20S%20%CC%83%20(T%20)%20from%20SP21,%0AS%20%CC%83(T%20)%20%3D%201%20%CE%B7%0A%05%20(3%20%E2%88%92%20%CE%B7)fdynfbX2%CE%BCH0C(T%20)%202%206%CF%80Gm2kT%0A6%CF%80Gm2HkBT%20,%20f2%20X2%CE%BCC(T)%0A(A2)%0A%03%0A%0E%20%CE%B73%0AHB%20dyn%0Ait%20becomes%20obvious%20that%20this%20task%20is%20complicated%20by%20the%20presence%20of%20the%20cooling%20function%20C(T%20),%20which%20has%20a%20non-trivial%20dependence%20on%20temperature.%20However,%20we%20can%20reasonably%20approximate%20the%20Suther-%20land%20%26%20Dopita%20(1993)%20cooling%20function%20for%20a%20plasma%20of%20primordial%20composition,%20in%20the%20temperature%20range%20of%20our%20interest%20(T%20%3E%20104.5%20K),%20with%20a%20piece-wise%20power%20law:%0A%E2%8E%A7%E2%8E%AA%CE%B8%20%3D%2010.75%20%E2%8E%AA%0A%E2%8E%A8%CE%B8%20%3D%20%E2%88%921.5%0AC(T)%E2%88%9DT%CE%B8,where%E2%8E%AA%CE%B8%3D1.7%20%E2%8E%AA%E2%8E%A9%CE%B8%20%3D%20%E2%88%921%20%CE%B8%20%3D%200.5%0A(A3)%20Within%20this%20approximation,%20and%20considering%20that%20in%20an%20EdS%20universe%0AH(z)%20%E2%88%9D%20(1%20+%20z)3/2,%20it%20follows%20that%202(%CE%B8%20%E2%88%921)(3%E2%88%92%CE%B7)%0AS%20%CC%83%20(%20M%20(%20T%20,%20z%20)%20)%20%E2%88%9D%20(%201%20+%20z%20)%203%20%CE%B7%20.%20(%20A%204%20)%0ALet%20us%20now%20turn%20to%20the%20second%20factor%20in%20the%20r.h.s.%20of%20equation%20(A1).%20As%20discussed%20in%20Section%203.2.2,%20the%20critical%20temperature%20decreases%20with%20cosmic%20time%20in%20an%20EdS%20universe.%20Recalling%20that%20for%20the%20computation%20of%20the%20CSFRD%20we%20only%20consider%20haloes%20above%20a%20certain%20virial%20temper-%20ature%20threshold%20(in%20this%20work,%20104.5%20K),%20it%20is%20apparent%20that%20all%20haloes%20that%20contribute%20to%20the%20CSFRD%20will%20exceed%20the%20critical%20temperature%20at%20a%20sufficiently%20late%20time.%20Thus,%20fgas%20will%20asymptotically%20reach%20a%20non-%20null%20constant,%20so%20that%20the%20only%20remaining%20redshift%20dependence%20in%20the%20r.h.s.%20of%20equation%20(A1)%20is%20H%20(z)3/%CE%B7%20%E2%88%9D%20(1%20+%20z)9/2%CE%B7%20.%20Combining%20it%20with%20equation%20(A4),%20it%20follows%20that%20slow(M,%20z)%20%E2%88%9D%20(1%20+%20z)%CE%B6%20,%20with%0A27+4(%CE%B8%20%E2%88%921)(3%E2%88%92%CE%B7)%0A%CE%B6%3D%206%CE%B7%20.%20(A5)%0AThe%20nSFR%20then%20converges%20as%20z%20%E2%86%92%20%E2%88%921%20(or,%20equivalently,%20t%20%E2%86%92%20%E2%88%9E),%20as%20long%20as%20%CE%B6%20%3E%20%E2%88%921.%20In%20the%20physically%20relevant%20range%203/2%20%3C%20%CE%B7%20%3C%203,%20this%20condition%20is%20satisfied%20for%20all%20values%20of%20%CE%B8%20quoted%20in%20equation%20(A3).%0Aif%20T%20%3C%20104.5%20K%0A4.5%204.7%0Aif10K%3CT%3C10K%20if%20104.7K%3CT%20%3C105K%20.%20if%20105K%3CT%20%3C106K%20ifT%3E106K%0A%20%20%20%20%20Downloaded%20from%20https://academic.oup.com/mnras/article/535/2/1449/7896079%20by%20guest%20on%2014%20November%202024%0A%20%0AA1.2%20Halo%20mass%20function%0AWithin%20the%20Sheth%E2%80%93Tormen%20formalism%20(Sheth%20%26%20Tormen%201999,%202002),%20the%20halo%20multiplicity%20function%20is%0AdF%202Ndln%CF%830(M)%13%20%E2%88%9A%14%E2%88%9235%202%20dlnM%3D%E2%88%92%E2%88%9A%CF%80dlnMy1+2y%20exp%E2%88%92y,%20(A6)%0Awhere%20%CF%830(M)%20is%20the%20linear-theory%20fractional%20variance%20of%20matter%20density%20fluctuations%20averaged%20over%20spheres%20containing%20a%20mass%20M%20at%20z%20%3D%200,%20and%20N%20is%20a%20normalization%20constant%20chosen%20such%20that%20all%20cosmic%20mass%20is%20contained%20in%20haloes.%20We%20also%20defined%0Ay%3D%CE%B4c,%20(A7)%20234%20D(z)%CF%830(M)%0Awhere%20D(z)%20is%20the%20growth%20function,%20which,%20in%20an%20EdS%20universe,%20scales%20as%20D(z)%20%E2%88%9D%20(1%20+%20z)%E2%88%921.%20Therefore,%20the%20scaling%20of%20y%20with%20redshift%20is%20y%20%E2%88%9D%20(1%20+%20z).%0AIt%20follows%20that%20the%20redshift%20dependence%20of%20the%20halo%20multiplicity%20function%20is%20given%20by%0AImpact%20of%20%02%20on%20star%20formation%20history%201473%20fixed%20virial%20temperature:%0Aslow(T(M,z),z)%3D%20M%0A%04%20H%20(%20z%20)%20%05%20%CE%B76%20%E2%88%92%203%0AH0%0AIn%20a%20universe%20with%20%02%20%3E%200%20the%20Hubble%20parameter%20asymptotes%20to%20a%20positive%20constant,%20so%20we%20can%20ignore%20any%20H(z)%20factor%20in%20the%20equation%20above%20for%20our%20convergence%20study.%20From%20equation%20(4),%20it%20also%20follows%20that%20haloes%20with%20a%20fixed%20mass%20will%20have%20an%20approximately%20constant%20virial%20temperature%20in%20the%20far%20future,%20so%20we%20can%20disregard%20any%20redshift%20dependence%20originating%20from%20the%20relationship%20between%20virial%20temperature%20and%20virial%20mass.%20Finally,%20we%20notice%20that%20our%20prescription%20for%20the%20cooling%20time%20(equation%2010)%20yields%20tcool%20%E2%88%BC%20t%20in%20the%20far%20future.%20Therefore,%20we%20can%20immediately%20remove%20the%20last%20factor%20on%20the%20r.h.s.%20of%20equation%20(A9).%20Its%20asymptotic%20behaviour%20in%20the%20limit%20t%20%E2%86%92%20%E2%88%9E%20is%20therefore%0Aslow(T(M,%20z),z)%E2%88%9D%0AWe%20then%20need%20to%20study%20the%20scaling%20of%20fgas%20with%20cosmic%20time%20in%20this%20limit.%0AOur%20numerical%20results%20in%20Fig.%206%20suggest%20that%20the%20critical%20temper-%20ature%20increases%20with%20time%20once%20%02%20dominates.%20Therefore,%20there%20will%20eventually%20be%20a%20time%20when%20the%20virial%20temperature%20falls%20below%20the%20critical%20temperature.%20At%20this%20point,%20the%20scaling%20for%20fgas%20is%20set%20by%20the%20critical%20temperature%20%E2%80%93%20see%20the%20case%20T%20%3C%20Tcrit%20in%20equation%20(15).%20We%20then%20need%20to%20determine%20the%20asymptotic%20time%20dependence%20of%20the%20critical%20temperature.%0AAs%20mentioned%20in%20Section%203.2.1,%20the%20critical%20temperature%20is%20the%20virial%20temperature%20above%20which%20haloes%20retain%20a%20baryon%20mass%20fraction%20equal%20to%20the%20cosmic%20value%20fb.%20At%20low%20redshift,%20this%20temperature%20is%20the%20one%20that%20satisfies%20the%20following%20condition%20(see%20SP21):%0AdF%20%E2%88%9A%20%E2%88%923%20%16%202%17%20dlnM%20%E2%88%9D%20(1+z)%5B1+%202(1+z)%5D%205%20exp%20%E2%88%92(1+z)%20.%0A%04%053%0Af%20(T(M,z),z)%20%CE%B7%203%0A(A8)%20In%20the%20limit%20z%20%E2%86%92%20%E2%88%921,%20the%20scaling%20above%20reduces%20to%20dF%20/d%20ln%20M%20%E2%88%9D%0Agas%20t%CE%B7%E2%88%922%20.%20(A10)%20fb%0A(1%20+%20z).%0AA1.3%20Stellar%20mass%20density%0AThe%20convergence%20of%20the%20CSFRD%20depends%20on%20the%20asymptotic%20behaviour%20of%20the%20integrand%20in%20equation%20(1).%20As%20explained%20in%20Section%203.2.2,%20the%20critical%20temperature%20becomes%20arbitrarily%20low%20at%20large%20times,%20therefore%20the%20relevant%20regime%20in%20our%20piece-wise%20approximation%20of%20the%20cooling%20function%20is%20Tcrit%20%3C%20104.5%20K,%20which%20corresponds%20to%20%CE%B8%20%3D%2010.75.%20If%20we%20further%20set%20%CE%B7%20%3D%201.9,%20we%20obtain%20that%20%CE%B6%20%E2%89%88%206.1.%20We%20saw%20in%20Section%20A1.2%20that%20the%20asymptotic%20redshift%20dependence%20of%20the%20halo%20multiplicity%20function%20in%20equation%20(1)%20is%20%E2%88%9D%20(1%20+%20z).%20Therefore,%20the%20integrand%20scales%20approximately%20as%20(1%20+%20z)%CE%B6%20+1%20%E2%89%88%20(1%20+%20z)7.1%20.%20This%20is%20then%20the%20redshift-%20dependence%20of%20the%20CSFRD%20in%20the%20far%20future.%0AThe%20SMD%20is%20simply%20the%20time%20integral%20of%20the%20CSFRD.%20Changing%20the%0Aintegration%20variable%20from%20cosmic%20time%20to%20redshift,%20the%20SMD%20is%20defined%0Aby%20equation%20(24).%20For%20an%20EdS%20model,%20the%20integrand%20in%20that%20equation%20is%0Aapproxima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