Nonlinear Dynamics
after Inflation

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9/30/19, 11:36 AM

Mustafa\ A. Amin

synopsis(results more general than end of inflation)
theoretical/numerical results

obs. implications

1. instability in oscillating fields

1. expansion history

2. formation of solitons (oscillons)

2. gravitational waves

3. eq. of state (with & without solitons)

3. structure formation*+some novel connections toaxions, early dark energy & condensed matter systems

what we “know” about inflation
Planck Collaboration: Cosmological parameters
0.25

Planck TT+lowP
Planck TT+lowP+BKP+lensing+ext

tensor-to-scalarr0.002 ratio (r)

p=4

0

p<2

0.20

5
N=

V( )/

p

60
N=

4

2

0.15

p=2

Co
nv
Co ex
nc
ave

0.10

p=1
0.05

for example:
Starobinsky Inflation (1979/80)
Silverstein & Westhpal (2008)
Kallosh & Linde (2013)

0.00

0.95

0.96

0.97

ns

0.98

primordial tilt

0.99

1.00

(ns )

Left: Constraints on the tensor-to-scalar ratio r0.002 in the ⇤CDM model, using Planck TT+lowP and Planck+lensing+BAO+JLA+H
but also see talk by Cumrun
Vafablue,nextrespectively) assuming negligible running and the inflationary consistency rela0 (red andduring inflation
V( )end of inflation

end of inflationp<2flattened potential
Planck Collaboration: Cosmological parameters
0.25

60

N=
50

2

0.15
Co
nv
Co ex
nc
ave

r0.002

V( )/| |

2n

N=

0.20

Planck TT+lowP
Planck TT+lowP+BKP+lensing+ext

0.10

0.05

0.00

0.95

0.96

0.97

0.98

0.99

1.00

ns

Fig. 21. Left: Constraints on the tensor-to-scalar ratio r0.002 in the ⇤CDM model, using Planck TT+lowP and Planck
TT+lowP+lensing+BAO+JLA+H0 (red and blue, respectively) assuming negligible running and the inflationary consistency relation. The result is model-dependent; for example, the grey contours show how the results change if there were additional relativistic
degrees of freedom with Ne↵ = 0.39 (disfavoured, but not excluded, by Planck). Dotted lines show loci of approximately constant e-folding number N, assuming simple V / ( /mPl ) p single-field inflation. Solid lines show the approximate ns –r relation for
quadratic and linear potentials to first order in slow roll; red lines show the approximate allowed range assuming 50 < N < 60 anda power-law potential for the duration of inflation. The solid black line (corresponding to a linear potential) separates concave andconvex potentials. Right: Equivalent constraints in the ⇤CDM model when adding B-mode polarization results corresponding to thedefault configuration of the BICEP2/Keck Array+Planck (BKP) likelihood. These exclude the quadratic potential at a higher levelof significance compared to the Planck-alone constraints.
limited by cosmic variance of the dominant scalar anisotropies,
and it is also model dependent. In polarization, in addition to Bmodes, the EE and T E spectra also contain a signal from tensor
modes coming from reionization and last scattering. However,in this release the addition of Planck polarization constraints at` 30 do not significantly change the results from temperatureand low-` polarization (see Table 5).
Figure 21 shows the 2015 Planck constraint in the ns –r plane,
adding r as a one-parameter extension to base ⇤CDM. Note thatfor base ⇤CDM (r = 0), the value of ns is| |⇠M

ns = 0.9655 ± 0.0062,(38)

We highlight this number here since ns , a key parameter for inflationary cosmology, shows one of the largest shifts of any parameter in base ⇤CDM between the Planck 2013 and Planck2015 analyses (about 0.7 ). As explained in Sect. 3.1, part ofthis shift was caused by the ` ⇡ 1800 systematic in the nominalmission 217 ⇥ 217 spectrum used in PCP13.
The red contours in Fig. 21 show the constraints from Planck
TT+lowP. These are similar to the constraints shown in Fig. 23
of PCP13, but with ns shifted to slightly higher values. The addition of BAO or the Planck lensing data to Planck TT+lowP
lowers the value of ⌦c h2 , which at fixed ✓⇤ increases the smallscale CMB power. To maintain the fit to the Planck temperature power spectrum for models with r = 0, these parameter
shifts are compensated by a change in amplitude As and the tiltns (by about 0.4 ). The increase in ns to match the observed
power on small scales leads to a decrease in the scalar poweron large scales, allowing room for a slightly larger contribution• shape of the potential (self couplings)• couplings to other fields

Planck TT+lowP.

34

,

Aµ

from tensor modes. The constraints shown by the blue contours
in Fig. 21, which add Planck lensing, BAO, and other astrophysical data, are therefore tighter in the ns direction and shifted toslightly higher values, but marginally weaker in the r-direction.
The 95 % limits on r0.002 are
r0.002 < 0.10,
r0.002 < 0.11,

Planck TT+lowP,
Planck TT+lowP+lensing+ext,

(39a)
(39b)

consistent with the results reported in PCP13. Note that we assume the second-order slow-roll consistency relation for the tensor spectral index. The result in Eqs. (39a) and (39b) are mildly
scale dependent, with equivalent limits on r0.05 being weaker by
about 5 %.
PCP13 noted a mismatch between the best-fit base ⇤CDM
model and the temperature power spectrum at multipoles ` <⇠ 40,partly driven by the dip in the multipole range 20 <⇠`<⇠ 30. If
this mismatch is simply a statistical fluctuation of the ⇤CDM
model (and there is no compelling evidence to think otherwise),
the strong Planck limit (compared to forecasts) is the result ofchance low levels of scalar mode confusion. On the other hand if
the dip represents a failure of the ⇤CDM model, the 95 % limitsof Eqs. (39a) and (39b) may be underestimates. These issues are
considered at greater length in Planck Collaboration XX (2015)
and will not be discussed further in this paper.
As mentioned above, the Planck temperature constraints onr are model-dependent and extensions to ⇤CDM can give significantly di↵erent results. For example, extra relativistic degrees of freedom increase the small-scale damping of the CMB
anisotropies at a fixed angular scale, which can be compensatedend of inflation in “simple” models
flattened potentialp<22
VV(( ) )//| |2ndistance from minimum
where potential flattens

| |⇠
⇠M
M

• shape of the potential (self couplings)• couplings to other fields,

Aµ

See P. Adshead’s talk about very strong coupling to gauge fields

instability of oscillating fields
expansion
self-interactions
gravitational int.

⇤ =V ( )(t, x)/M0

MA (2010)
1006.3075

instabilities in an expanding universe
growth-rate of fluctuations<[µk ]=
expansion rateH(t, x)/M

generalmpl⇠
M

V ( ) / | |2n1

dominant contributor
to energy density

| |=
⇠M
M

result 1: instability in oscillating fields

oscillating, cosmologically relevant, (almost)
homogeneous scalar fields are often unstable tospatial perturbations

Khlopov, Malomed, Zeldovich (1985)* there are timescales associated with the instability, typically the longest is Hubble time scalewhat drives the instabilities ?

•self-interactions-•

2

fields can cluster/become inhomogeneous(can be) much faster than Hubble (due to self-resonance)

gravity-V( )/fields cluster, Hubble time scales*for this talk, I will ignore interactions with other fields, which can also be important.
Review :MA, Kaiser, Karouby, Herzberg [1410.3808]

gravity only: Gilmore, Flauger & Easther (2012)2

Early Works : Kofman, Linde, Starobinsky (1994/97)
Shatnov, Traschen & Brandenburger (1990/95)
Khlebnikov & Tkachev (1996), Bellido & Linde (1997)

instability — formation of oscillons(non-topological “solitons” in real scalar fields)
expansion
self-interactions
gravitational int.

⇤ =V ( )(t, x)/M0

MA (2010)
1006.3075

insensitive to initial conditions

simulation of “quasi-thermal” example in Farhi, Guth, Iqbal, Graham 2008

oscillon formation at the end of inflation
expansion
self-interactions
gravitational int.

1
0.25Hend

MA, Easther, Finkel, Flauger & Hertzberg (2011)1106.3335

solitons ?
(1) oscillatory (2) spatially localized (3) very long lived

For example:
Segur & Kruskal (1987)
MA & Shirokoff (2010)
Hertzberg (2011)
MA (2013)
Mukaida et. al (2016)
Salmi & Hindmarsh (2014)
Sakstein & Trodden (2018)

i
c
os

!
n
llo

Bogolubsky & Makhankov (1976), Gleiser (1994), Copeland, Gleiser and Mueller et al. (1995) …

lifetimes (without gravity)
3

Tmax ⇠ 10 m

1

Tmax > 10 m

V ( ) = m2 M 2 1

2

2

cos

m M
V( )=
tanh2
2

Tmax ⇠ 106 m 1

7



1

V ( ) = m2 M 2

"r

✓

M

M
2

1+

M2

◆

#

1

Zhang, MA, Copeland, Saffin, Lozanov (in progress)

family of scalar field solitons

gravityoscillaton

boson staroscillon

p
m
co

lex

self-interaction

Q

balls

oscillon formation at the end of inflation
expansion
self-interactions
gravitational int.

1. oscillons dominate
the energy densityof the field
2. eq. of state w = 0

long term dynamics ?
expansion
self-interactions
gravitational int.

assuming coupling to other fields is sufficiently weak

self-interactions+ gravity*(Schrodinger-Poisson)
expansion
self-interactions
gravitational int.
relativistic?

MA & Mocz (2019)1902.07261
qualitative comparison
with relativistic system
Lozanov & MA (2019) 1902.06736

self interaction more important than gravity initiallyself-interactions+ gravity*(Schrodinger-Poisson)
expansion
self-interactions
gravitational int.
relativistic?

MA & Mocz (2019)1902.07261
qualitative comparison
with relativistic system
Lozanov & MA (2019) 1902.06736

self interaction more important than gravity initiallyself-interactions+ gravity*(Schrodinger-Poisson)
expansion
self-interactions
gravitational int.
relativistic?

MA & Mocz (2019)1902.07261
qualitative comparison
with relativistic system
Lozanov & MA (2019) 1902.06736

self interaction more important than gravity initiallyself-interactions+ gravity*(Schrodinger-Poisson)
expansion
self-interactions
gravitational int.
relativistic?

MA & Mocz (2019)1902.07261
qualitative comparison
with relativistic system
Lozanov & MA (2019) 1902.06736

self interaction more important than gravity initiallygravitational clustering of solitons7

ING

cted to beer the soliate times,
dominatedg our newarger thanectation is
ons shouldnding unional forces
er than 2reh the force

ion of solio quantitaclustering.
unction of/ r 2 consistent with nonlinear

clustering of “point” masses[co-moving separation]*theoretical arguments for r-2 in Saslaw 1980

FIG. 5. The two point correlation functionof soliton
loca- the velocity distribution*we don’t
fully understand/ <[ ]a (t, x) =|✓1

✓2 | ' ⇡|✓1✓2 | ' 0“binary”

mergerbouncephasestrong
dependent
interactions
interactionsi⌫a t+✓a(x)earesult 2: oscillon formation (solitons)

reson

ant growt

h

V( )
↵<2

1.oscillons dominate the energy density2⇠ M ⌧ mpl

2.they cluster gravitationally
3. can undergo complex scatteringso far, quadratic minima with wings …
flattened potential

2
VV(( ) )//| |2nquadratic minimum| |⇠M• shape of the potential (self couplings)• couplings to other fields

p<2,non-quadratic, power-law minima ?
Planck Collaboration: Cosmological parameters
0.25

50
N=

flattened potential

60
N=

0.20

Planck TT+lowP
Planck TT+lowP+BKP+lensing+ext

2

0.15

r0.002

Co
nv
Co ex
nc
ave

0.10

0.05

V( )/| |

2n

0.00

0.95

0.96

0.97

0.98

0.99

1.00

ns

Fig. 21. Left: Constraints on the tensor-to-scalar ratio r0.002 in the ⇤CDM model, using Planck TT+lowP and Planck
TT+lowP+lensing+BAO+JLA+H0 (red and blue, respectively) assuming negligible running and the inflationary consistency relation. The result is model-dependent; for example, the grey contours show how the results change if there were additional relativistic
degrees of freedom with Ne↵ = 0.39 (disfavoured, but not excluded, by Planck). Dotted lines show loci of approximately constant e-folding number N, assuming simple V / ( /mPl ) p single-field inflation. Solid lines show the approximate ns –r relation for
quadratic and linear potentials to first order in slow roll; red lines show the approximate allowed range assuming 50 < N < 60 anda power-law potential for the duration of inflation. The solid black line (corresponding to a linear potential) separates concave andconvex potentials. Right: Equivalent constraints in the ⇤CDM model when adding B-mode polarization results corresponding to thedefault configuration of the BICEP2/Keck Array+Planck (BKP) likelihood. These exclude the quadratic potential at a higher levelof significance compared to the Planck-alone constraints.

power law at the minimum(fractional powers also)
limited by cosmic variance of the dominant scalar anisotropies,
and it is also model dependent. In polarization, in addition to Bmodes, the EE and T E spectra also contain a signal from tensor
modes coming from reionization and last scattering. However,in this release the addition of Planck polarization constraints at` 30 do not significantly change the results from temperatureand low-` polarization (see Table 5).
Figure 21 shows the 2015 Planck constraint in the ns –r plane,
adding r as a one-parameter extension to base ⇤CDM. Note thatfor base ⇤CDM (r = 0), the value of ns is
ns = 0.9655 ± 0.0062,(38)

We highlight this number here since ns , a key parameter for inflationary cosmology, shows one of the largest shifts of any parameter in base ⇤CDM between the Planck 2013 and Planck2015 analyses (about 0.7 ). As explained in Sect. 3.1, part ofthis shift was caused by the ` ⇡ 1800 systematic in the nominalmission 217 ⇥ 217 spectrum used in PCP13.
The red contours in Fig. 21 show the constraints from Planck
TT+lowP. These are similar to the constraints shown in Fig. 23
of PCP13, but with ns shifted to slightly higher values. The addition of BAO or the Planck lensing data to Planck TT+lowP
lowers the value of ⌦c h2 , which at fixed ✓⇤ increases the smallscale CMB power. To maintain the fit to the Planck temperature power spectrum for models with r = 0, these parameter
shifts are compensated by a change in amplitude As and the tiltns (by about 0.4 ). The increase in ns to match the observed
power on small scales leads to a decrease in the scalar poweron large scales, allowing room for a slightly larger contribution

| |⇠M

34

• shape of the potential (self couplings)• couplings to other fields

Planck TT+lowP.

,from tensor modes. The constraints shown by the blue contours
in Fig. 21, which add Planck lensing, BAO, and other astrophysical data, are therefore tighter in the ns direction and shifted toslightly higher values, but marginally weaker in the r-direction.
The 95 % limits on r0.002 are
r0.002 < 0.10,
r0.002 < 0.11,

Planck TT+lowP,
Planck TT+lowP+lensing+ext,

(39a)
(39b)

consistent with the results reported in PCP13. Note that we assume the second-order slow-roll consistency relation for the tensor spectral index. The result in Eqs. (39a) and (39b) are mildly
scale dependent, with equivalent limits on r0.05 being weaker by
about 5 %.
PCP13 noted a mismatch between the best-fit base ⇤CDM
model and the temperature power spectrum at multipoles ` <⇠ 40,partly driven by the dip in the multipole range 20 <⇠`<⇠ 30. If
this mismatch is simply a statistical fluctuation of the ⇤CDM
model (and there is no compelling evidence to think otherwise),
the strong Planck limit (compared to forecasts) is the result ofchance low levels of scalar mode confusion. On the other hand if
the dip represents a failure of the ⇤CDM model, the 95 % limitsof Eqs. (39a) and (39b) may be underestimates. These issues are
considered at greater length in Planck Collaboration XX (2015)
and will not be discussed further in this paper.
As mentioned above, the Planck temperature constraints onr are model-dependent and extensions to ⇤CDM can give significantly di↵erent results. For example, extra relativistic degrees of freedom increase the small-scale damping of the CMB
anisotropies at a fixed angular scale, which can be compensateddynamics in different power law minima
Homogeneous oscillations
n 1
w=
n+1

s

on
t
i
l
o
os

n
t
u
ts, b

Turner (1983)

n

al
u
t
n

inflaton potential

ve
e
ld

ly

e
m
g
fra

eq. of statepressurew⌘! 1/3
densityfie
V ( ) / | |2n

| |⇠M

n&1

radiation
domination

n=1
fiel
d

frag
me

eq. of statents

Lozanov & MA (2016/17)
1608.01213, 1710.06851

into

sol
iton
s

w!0
(for

M<

<m

pl )

matterdomination

result 3: “equation-of-state”

the spatially averaged equation-of-state of fields- (n = 1) quadratic minima
w=0- (n > 1) non-quadratic minima w = 1/3 (after sufficient time)
why? µk /H /1

power law at the minimum

V ( ) / | |2n| |⇠MLozanov & MA (2016/17)

1608.01213, 1710.06851

implications

eq. of state & CMB observables

tensor-to-scalar ratio(r)power law at the minimum
V( )/

0

V ( ) / | |2n

M = 5mpl| |⇠M

scale fo flattening

primordial tilt

MA & Lozanov 2017 [1608.01213, 1710.06851]
also see: Kamionkowski & Munoz (2014)(ns )n 6= 1* non-quadratic minimumimplications

the remnant condensate continues to oscillate withslowly decaying amplitude, continuing its particle
production. After sufficiently long times, we findthat the kinetic and gradient energies are approximately equal and much greater than the potential
energy with the field again virialized. This yields
power
at theminimum
an equation of state parameterw ⇡law
1/3.
Note
that
the n = 2 case would yield w = 1/3 for the homogeneous and inhomogeneous field. A summary of 0V( )/the asymptotic equation of state is shown in Fig. 4.

reduction in uncertainty

tensor-to-scalar ratio(r)

Figure 4. A summary for the asymptotic equation ofstate without coupling to additional fields. The numerical results from lattice simulations are shown as
green circles for M ⇡ 2.45 mPl , and orange squares forM ⇡ 7.75 ⇥ 10 3 mPl . The dotted blue line is the expectation from a homogeneous, oscillating condensate.

V ( ) / | |2ne-folds to Radiation Domination — Our linear
the transients decay, the inflaton is completely
analysis of the instabilities allows us to estimate
fragmented with almost no energy remaining in thethe number of e-folds after inflation required to
R aradhomogeneous condensate. The field configuration
reach radiation domination, Nrad ⌘ aend d ln a,
M = 5mpl| |⇠M
now evolves freely in a turbulent manner (asby calculating the time of backreaction of the flucdiscussed for n = 2 in [29]). Numerically, we findtuations. First, note that for n = 2, Nrad ⌧ 1that the kinetic and gradient energies are approxiscale fo
flattening
since in this case w ! 1/3 withandwithout
mately equal to each other and much greater thanfragmentation. For all other n & 1, the universe
the potential energy, implying
w ! 1/3 tilt
(cf. Fig.
primordial(ns ) becomes radiation dominated within3), and that the field is virialized in the sense that
8
2
2
2
˙
h /2is,t = h(r ) /2a is,t + nhV is,t holds. We<1
M . 10 2 mPl ,✓◆Nrad ⇠ n + 1
can then get an estimate of the deviation of w
 10M
:
ln
M & 10 2 mPl .
from 1/3: w 1/3 ! (2/3)(n 2)⇥ the fraction
3
 mPl
(5)
of energy density in the potential energy. For
n 6= 1 width of theHere, / ⇠ 10 2 is the fractional
inefficient
M & 2.51710.06851]⇥ 10 2 mPl and
Lozanov initial& MA resonance(2017) [1608.01213,* non-quadratic
minimumfields areinitiallyassumedsome
to besmall
light excitations
and massless. first k 6= 0 narrow resonanceband (cf. Fig.
2).
n *all= 1,other
we observeimplications

primordial black holeformation from solitons?

Lozanov & MA
1902.06736. few ⇥ 10 3

Not easy to form PHBs
from individual solitons from self resonanceHowever accidental over-densities in solitons
more likely to form PBHs (Cotner et. al
2018/19)⇥ 104

Also see talk by Juan Garcia Bellido on Monday on PBHs from inflation

⇥ 104

⇥ 104

implications

quantitative?
gravitational clustering of solitons7

ING

cted to beer the soliate times,
dominatedg our newarger thanectation is
ons shouldnding unional forces
er than 2reh the force

ion of solio quantitaclustering.
unction of/ r 2 consistent with nonlinear

clustering of “point” masses[co-moving separation]

FIG. 5. The two point correlation function of soliton loca-

implications: gravitational waves

��-��

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��-�

��

���

���

����

����

��-�
��-�

BBN + CMB

��-�aLIGO/Virgo

-�

��

��-��
��-��
��-��
��-��

��-��

��-��

��-�

��-�

�

���

�
��
p

f⇠

also potential constraints from Neff from CMB S4

HH0

limits adapted from Lasky et. al (2015)

caveat* early universe g-waves amplitude depend on assumptions of expansion history
Earlier work on g-waves from end of inflation: Khlebnikov & Tkachev (1996), Easther, Giblin, Lim (2006/07), Dufaux et. al (2007)

another example: Higgs - modulus system
MA, J. Fan, K. Lozanov & M. Reece (2018) [1802.00444]
time !

electroweak symmetry breaking and early universe cosmology. It also motivates further studies on the potentialof gravitational wave probes for new physics beyond the
SM.

2

a3/2

modulus :V ( , h)00

II A Simple Model A simplified potential capturing the most salient features of a Higgs field, h, coupled
to a modulus, , is h✓◆2
1 2 2 M2v†m+(
+ (h† h)2 . (1)0) h h2
f2

Higgs : ha3/2

M4
! 1 (first, rapid
FIG. S4. Snapshots of the values of the Modulusrow) and fragmentation
Higgs (second row) fields on a two-dimensional
FIG. S4. Snapshots of the values ofthe2Modulus (first row) and Higgs (second row) fields
on a two-dimensional22fm
the simulation box at four di↵erent
times.
Aroundthetimeofbackreaction,
23m 1 (second column), the Hig
p times. Around the time of backreaction, tt ⇡the simulation box at four di↵erent⇡ 23m 1 (second column), the Hig
x/m 1

x/m 1

x/m 1

x/m 1

0
The fine
globalminimumthedomains (‘bubbles’) with = ±p2| |f /q. They disappear within t ⇠ 10m 11, due to collisions, as well as osci
1 potential lies at = 0, where domains
tuning, of⌧(‘bubbles’) with = ± 2| |f /q. They disappear within t ⇠ 10m , due to collisions, as well as osci
remnant condensate. The used parameters are b = 1, q = 1022 , M = 10 12m , f = mpl .
f simply the Standard Model Higgs remnant
condensate. The used parameters are b = 1, q = 10 , M = 10 12 mpl
the potential becomes
pl , f = mpl .
FIG.
2.
The
ratio
of
the
spatially
averagedenergydensitypotential. The constant v 2 = M 2 0 /( f ). Placingthethe Higgs andmodulus fields
as a function
time obnon-trivial
eq. of state: 1/4 . w . 1/3
stochastic ingravitationalwaves
S2 Gravitational
Gravitational Waves
and
Lattice
Simulations
minimum at = 0 is a pure convention; in particular, S2
Waves
and Lattice
Simulations
tained from ourlatticesimulations.
This dynamics of energy
carries no charges and can be shifted by10a-8 constant. We
0.3and Higgs fields is representransfer between the modulus
2take the mass scale M to be the natural value of the
1. Equationsof Motion i.e. whentative of the case where the modulus
fragments,
1. Equationsof Motion42 2
b ⌘ M /2 f m ! 1. For 0.2
the above plot we have chosen
Higgs mass and f to be the natural scaleof the modulus
10-10
2 gravitational
2
2
12
We
calculatethewaves
generated
by
the
field
dynamics
the /mgravitationalby 10
the nonlinear
nonlinear
field
dynamics using
using
field . That is, we suppose that quantum corrections to Web calculate= 1, M= 10 waves
andgenerated
M/f =
. The
interaction
2
term is not included in the above
energydensities.
the Higgs mass would be of order M 2 andthat generic
r
2 TT
2 TT
T0.1
T
TT
-12
r
ḧ
+
3H
ḣ
h
T
T
T
T
T T = 22 ⇧ij
10
ij
ij
ij
2
ḧij + 3H ḣij
hij = m2 ⇧TijT
avalues ⇠ f produce Higgs masses of this order.
2
a
mpl
pl
The e↵ective Higgs boson mass
0.0
TT
FRW

10-14

m2h; e↵ ( ) = M 2
is positive at0f

0 and negative atwhere hhij
T T is the spatial, transverse, traceless part of the metric perturbations (gµ⌫ = gµ⌫
FRW + hµ⌫ ), andwhere+ hµ⌫ ), an
ij is the spatial, transverse, traceless part of the metric perturbations (gµ⌫ = gµ⌫transverse-traceless
part
of
the
energymomentumtensorofthefieldswhich
sourcesthegravitationaltransverse-traceless part of the energy momentum tensor of the fields which sources the gravitational w
w
lead
to
non-adiabatic,
out-of-equilibrium
productionofa“passive
calculation”
where
the
(small)backreactionofthemetricperturbations
onthefieldsis
ignoreof the metricon the fields(2) a “passive calculation” where the (small) backreaction
050
100 perturbations
150
200
250is ignore

100

1000
104 particles.
105
106 By considering tachyonic resonancethe Higgs⌧ 0, transitioning[15], and for f ⇠ in ⇠ mpl , thesuch particle
2. efficiency
Characteristicof
Scales2. Characteristic
2
2 Scalesproduction is controlled by q ⌘ M /m . In particular,

1

10

end of inflation & dark matter abundance?
Can the abundance of dark matter depend on the non-thermal
conditions at the end of inflation ?
*assuming only gravitational strength couplings between inflaton to dark matter

Yes, If:

inflaton! radiation(s) / sn*

! dark mattern>2

radiation is SM particles
Garcia & MA (2018)arXiv:1806.01865

There is a lot to learn here
— theoretical progress and upcoming observations.
universe gets!
populatedwith particles

hot thermal soupwith nuclei

CMB

Cosmology@

Reheating !
after inflationinflation
Inflation?

few minutes
106 eV
400,000 years
10 1 eV

14 billion yearsinflation

10

16

GeV

• inflation ends
• populate the universe(reheating — Standard Model) ?
• matter-antimatter asymmetry ?
• dark matter ?
• EW symmetry breaking• QCD phase transition

TeV
MeV
eV

generality & novel connections

universe gets!
populatedwith particles

Inflation

Cosmology@

Reheating !
after inflation

๏ Axionic dark matter [Jens Neimeyer’s talk on Monday]๏ Hubble tension ? [seeTristan Smith’s talk on Monday]๏ Condensed matter solitons๏ Stochastic particle production (and connection toAnderson localization)

dark matter:axion-like fields
(a)

DM

(b)

CDM

NATURE PHYSICS DOI: 10.1038/NPHYS2996

10

Core

10

5 kpc

10

105
103

10

200 kpc

ρ (r)/〈ρ 〉101
50 kpcSchive et. al (2014)

for example:
Peccei & Quinn (1977)
Hogan & Reece (1988)
Kolb & Tkachev (1994)
Hu, Barkana & Gruzinov (2000)
Marsh & Silk (2014)
Niemeyer & Engels (2016)
Hui et. al (2016)
Arvanitaki et. al (2009/19)
Mocz et. al (2019)

Figure 2 | A slice of the density field of the DM simulation on various
scales at z = 0.1. This scaled sequence (each of thickness 60 pc) shows
how quantum interference patterns can be clearly seen everywhere fromthe large-scale filaments, tangential fringes near the virial boundaries, tothe granular structure inside the haloes. Distinct solitonic cores with radii⇠0.3–1.6 kpc are found within collapsed haloes (which have virial masses
Mvir ⇠ 109 ˘1011 M ). The density shown here spans over nine orders ofmagnitude, from 10 1 to 108 (normalized to the cosmic mean density). The
colour map scales logarithmically, with cyan corresponding to density .10.

10

10

10

10

V( )

giving rise to a co-moving Jeans length, J / (1 + z)1/4 mB 1/2 , during
the matter-dominated epoch17 . The insensitivity of J to redshift, z,
generates a sharp cutoff mass below which structures are suppressed.
Cosmological simulations in this context turn out to be much
more challenging than standard N-body simulations, as the highest
frequency oscillations, !, given approximately by the matter wave
dispersion relation, ! / mB 1 2 , where is the wavelength, occur
on the smallest scales, requiring very fine temporal resolution evenfor moderate spatial resolution (Supplementary Fig. 1). In this work,
we optimize an adaptive-mesh-refinement (AMR) scheme, withgraphic processing unit acceleration, improving performance by

FigureDashe
norma
distinc
A detamean(NFW(blackwell th⇢300 owith thimplications

structure formation with lightstronginteractions
scalarMocz, +MA, et. al (2019)(a) box fields(b) projection(c) slice7dm

subhalos“usual” cold
dark matter

gas

stars

caustics

warm
dark matterinterference10

1×10 M“fuzzy”
dark matter9

5×10 M 2×10 7M

9

5×10 M
1.5 Mpc

0.5 Mpc

“Hubble Tension” resolution— some novel implications
10 6

ant growt

narrow resonancereson

gravitational potential

z = 350z = 300z = 250

h
10 7
0.33

y
r
a
in

0.34

0.35

0.36

0.37

0.38

k [h/Mpc]

1.4 × 10-12
1.2 × 10-12

V( )1. × 10-12
8. × 10-13
6. × 10-13

il m
e
r
p

gravitational waves

4. × 10-13
2. × 10-13
2n6=2

0

1. × 10-16

2. × 10-16

5. × 10-16

1. × 10-15*no-solitons
Also see: Karwal & Kamionkowski (2016), Poulin et. al (2018), Agrawal et. al (2019).

Smith, Poulin & MA (2019)
MA, Lozanov & Smith (in progress)

A Novel Connections

universe gets!
populatedwith particles

Cosmology@

Reheating !
after inflationinflation

cold-atom
Bose-Einstein Condensates

Inflation?

few minutes
106 eV

14 billion yearsrelated solitons in BECs

Nguyen, Luo & Hulet (2017)

nonlinear Klein Gordon — nonlinear Schrodinger eq.

@t2

@t2 '

c2 r2 + @ V ( ) = 0

c2s r2 ' + @' V(') = 0

relative phase between different condensates

i@t

=



non-relativistic

1 2
r + U 0 (| |2 )2m

(Stochastic) Particle Production in Cosmology
appropriate for sufficiently complex models of inflation•

Wires to Cosmology(w/ Baumann 1512.02637)•

Multifield Stochastic Particle Production(w/ Garcia, Wen & Xie 1706.02319)•

Stochastic Particle Production in deSitter Space(w/ Garcia, Carlsten & Green 1902.06736)•

Curvature Perturbations from Stochastic Particle Production(in progress)

And related works by Brandenberger & B, Basset

temporal
complexity

d2 k
2
2
+
k
+
m
( )
2
d

spatialcomplexity

d2
2
+
k
dx2

k =0

•

time

•

position along wire•

number of particles•

resistance•complicated temporal behavior•

impurities in wires

V (x)

=0

curvature perturbations
from particle production

IN S
K
S
R
E
O R
W OG
PR

Garcia, MA, Green, Baumann &Chia (in progress)
2⇣ (k) =

observable range
1

2(0)
(k) +
⇣

2
⇣ (k)

kf ⌧ k

2
⇣

2
⇣ (k)

2
⇣ +

observable?
2
⇣

2(0)
(k)
⇣

kf
1
2 2
2
⇣ + ⇣ ⇣

2
⇣ (k)

22
⇣,0
⇣

k

⌧kk ⌧
⌧kkf
kk00 ⌧f

Features ?
2(0)
(k)
⇣

kk00

kk

kkff

PBH ?

2
⇣

Figure 16: Relative position of the observable co-moving momentum band relative to the domain spannedk ⌧ k0
1
2
2
by the stochastically sourced power spectrum, in the case⇣ '⇣,0 . Here k? denotes the (Planck) pivot
scale,
k0 = |⌧0 | 1 and kf = |⌧f | 1 . The tilt of the vacuum contribution 2⇣,0 has been omitted for clarity.
2
2
⇣ +

⇣ (k)

2(0) physics is time translation invariant. In particular, the probability distribution for
the underlying(k)⇣2
the stochastic
masses is independent of time and thus a large change to the power spectrum on small⇣
scales occurs only by chance. In this sense, indirect observations of the power spectrum may stillk
k0
yield dramatic surprises, even if inflation itself made no e↵ort to hide these e↵ects.

SpectralDistortions ?

Also see: Flauger, Mirbabayi, Senatore, Silverstein (2016)Nonlinear Dynamics
of Cosmological Fields (and novel connections)
3 theoretical/numerical results

3 obs. implications

1. instability in oscillating fields

1. gravitational waves

2. formation of solitons

2. structure formation

3. eq. of state

3. expansion historythanks