The ultraviolet (UV) upturn or excess is an unexpected rise in flux in the spectral energy distributions (SEDs) of early-type galaxies shortwards of 2500 Å. It appears to be a nearly ubiquitous property of spheroids and bulge-dominated galaxies (see e.g. Yi 2008, 2010 for a recent review), although their generally old, metal-rich, and quiescent stellar populations (e.g. Thomas et al. 2005, 2010) should contain no sources capable of providing significant flux below the 4000 Å break. While many candidates have been proposed, the source population is generally agreed to consist of hot horizontal branch (HB) stars (Greggio & Renzini 1990; Brown et al. 1998a) and it now appears most likely that this population is helium-rich and formed in situ at high redshift (Ali et al. 2018a,b,c). While such stars are now known to exist in globular clusters in our Galaxy (e.g. see Norris 2004; Piotto et al. 2005, 2007) and likely elsewhere (Kaviraj et al. 2007b; Mieske et al. 2008; Peacock et al. 2017), the origin of such high helium abundances is unclear.1 It has been suggested that the effect may depend on the environment: stratification of helium in the centres of clusters might create populations of galaxies with high helium abundances (Peng & Nagai 2009). On the other hand, it is also possible that the helium enrichment depends on the details of early star formation, as in globular clusters. Differences in the originating mechanisms or time-scales may be reflected in different evolutionary histories for galaxies in clusters and the field. Atlee, Assef & Kochanek (2009) find a slow decrease in the strength of the upturn for a sample of bright field elliptical galaxies at 0  z  0.65 compared to the nearly constant colour for very bright ellipticals in the works of Brown et al. (1998b, 2000, 2003) and brightest cluster galaxies (Loubser & Sánchez-Blázquez 2011; Boissier et al. 2018). In our work on much larger samples of cluster early types with luminosities down to L*, we also see no evolution at z  0.55 (Ali et al. 2018a,b,c) but then detect a rapid reddening in the UV colour at z = 0.7; this may be consistent with the last data point in Atlee et al. (2009), despite the large errors and small number statistics. Le Cras et al. (2016) also find evidence for evolution in the UV upturn at z > 0.6, albeit from stacked spectra and using spectroscopic indices sensitive to the HB population, for a sample of very luminous BOSS galaxies (as opposed to the UV photometry used by other studies).


In our work on much larger samples of cluster early types with luminosities down to L*, we also see no evolution at z  0.55

but then detect a rapid reddening in the UV colour at z = 0.7


It may be appropriate to single out at this point the recent and interesting works by Lin & Ishak (2017a,b) in which the authors run a so-called (dis)cordance test based on using a proposed index of inconsistency (IOI) tailored at finding possible inconsistencies/tensions between two or more data sets in a systematic and efficient way. For instance, it is well known that there is a persistent discrepancy between the Planck CMB measurements of H0 and the local measurements based on distance ladder (Riess et al. 2016, 2018b). At the same time, if one compares what is inferred from Planck 2015 best-fitting values, the LSS/RSD measurements generally assign smaller power to the LSS data parametrized in terms of the weighted linear growth rate f(z)σ8(z). This feature is of course nothing but the σ8-tension we have been addressing in this paper. It is therefore natural to run the IOI test for the different kinds of H0 measurements and also to study the consistency between the H0 and the growth data. For example, upon comparing the constraints on H0 from different methods, Lin & Ishak (2017b) observe a decrease of the IOI when the local H0 measurement is removed. From this fact they conclude that the local measurement of H0 is an outlier compared to the others, what would favour a systematics-based explanation. This situation is compatible with the observed improvement in the statistical quality of the fitting analysis by Solà, Gómez-Valent & de Cruz Pérez (2017b,c) when the local H0 measurement is removed from the overall fit of the data using the RVM and the ΛCDM. In this respect, let us mention that a recent model-independent analysis of data on cosmic chronometers and an updated compilation of SNIa seem to favour the lower range of H0 (Gómez-Valent & Amendola 2018), what would be more along the line of the results found here, which favour a theoretical interpretation of the observed σ8 and H0 tensions in terms of vacuum dynamics and in general of DDE (cf. Fig. 10).


For example, upon comparing the constraints on H0 from different methods,

observe a decrease of the IOI when the local H0 measurement is removed.

From this fact they conclude that the local measurement of H0 is an outlier compared to the others, what would favour a systematics-based explanation.


It may be appropriate to single out at this point the recent and interesting works by

in which the authors run a so-called (dis)cordance test based on using a proposed index of inconsistency (IOI) tailored at finding possible inconsistencies/tensions between two or more data sets in a systematic and efficient way.


These results show that the shape of the mass function and the position of the mass peak of massive clusters have little evolution over the course of the galaxy collision of more than 1 Gyr. We note that the evolution of SCs is subjected to several destructive processes (e.g., Gnedin et al. 1999a; Fall & Zhang 2001). For low-mass clusters (

), the destruction is mainly dominated by two-body relaxation processes, in which the mass of a cluster linearly decreases with time until it is destroyed. For more massive clusters, the evolution is primarily influenced by stellar evolution at early times (

 Myr) and by gravitational shocks at later times. These effects are included in our hydrodynamic simulations, but the mass resolution is not high enough to resolve the processes realistically, since the two-body relaxation and stellar evolution depend on individual stars. However, the cluster disruption timescale due to two-body relaxation is proportional to the cluster mass, trlx ∼ 1.7 Gyr × (Mclus/104

)0.62 × (T/104 Gyr−2)−0.5, where Mclus is the cluster mass and T is the tidal strength around the clusters (Kruijssen et al. 2012). Using the minimum cluster mass of

 in our simulation and a typical range of tidal strength in the nuclear region (since most of these clusters are concentrated around galaxy nuclei) of

 (Renaud 2010), we find that the range of the disruption timescale is

 Gyr. For more massive SCs, the timescale is even longer, beyond our run time of 1 Gyr. Therefore, the two-body relaxation may not have a major disruptive effect on these SCs. Furthermore, the mass loss timescale due to gravitational shocks (tsh) depends strongly on the cluster density,

 (Kruijssen et al. 2012). The majority of our clusters have a density range of

, as shown in Figure 9, which suggests

 Gyr, much longer than the Hubble time. So gravitational shocks may not have a significant impact on the clusters in our simulation. In addition, as demonstrated by Renaud & Gieles (2013), SCs formed in galaxy mergers are also affected by the intense tidal field of the galaxies, more so for clusters in the merger remnant than for the ejected ones. We note, however, the clusters in their simulations have masses

, and mass loss decreases as the cluster mass increases. For example, for a cluster to increase its mass by a factor of 2, from

 to

, its survival rate (fraction of initial mass survived) after 1 Gyr increases from 0.6 to 0.7. Extrapolating this trend to our clusters, which are ∼10 times more massive than those in Renaud & Gieles (2013), we argue that destruction from tidal fields likely has negligible effects on the clusters we consider here.


Therefore, the two-body relaxation may not have a major disruptive effect on these SCs. Furthermore, the mass loss timescale due to gravitational shocks (tsh) depends strongly on the cluster density,


The second significant source of energy dissipation is stream–stream collisions near apocenter from GR apsidal precssion. As in Dai, McKinney & Miller (2015) and Bonnerot et al. (2017), we calculate the ingoing collision stream velocities by finding the intersection point of two elliptical orbits, shifted by an instantaneous pericenter shift δϖ from GR apsidal precession. Since an eccentric orbit undergoes apsidal precession at a rate (assuming af ≫ Rg)
(24)$$\begin{eqnarray*}
\left. \frac{\partial \varpi }{\partial t} \right|_{\rm GR} = \frac{3 R_{\rm g}n}{a (1-e^2)},
\end{eqnarray*}$$we have after one orbital period
(25)$$\begin{eqnarray*}
\delta \varpi \simeq \frac{3 \pi R_{\rm g}}{r_{\rm p}} = 11.5^\circ \, \beta \frac{\bar{M}_\bullet ^{2/3} \bar{M}_\star ^{1/3}}{\bar{R}_\star }.
\end{eqnarray*}$$With δϖ, one can calculate the relative stream velocities during the collision, leading to a loss of energy of (Dai et al. 2015; Bonnerot et al. 2017)
(26)$$\begin{eqnarray*}
\delta E_{\rm ss} &=& \frac{e_{\rm f}^2 G M_\bullet }{2a_{\rm f}(1-e_{\rm f}^2)} \sin ^2 \left(\frac{\delta \varpi }{2} \right) \simeq \frac{G M_\bullet }{4 r_{\rm p}} \sin ^2 \left(\frac{\delta \varpi }{2} \right),
\end{eqnarray*}$$hence
(27)$$\begin{eqnarray*}
\mathcal {V}_{\rm ss} = \frac{\delta E_{\rm ss}}{E_{\rm f}} = \frac{a_{\rm f}}{2 r_{\rm p}} \sin ^2 \left(\frac{\delta \varpi }{2} \right).
\end{eqnarray*}$$When δϖ ≪ 1, equation (27) reduces to
(28)$$\begin{eqnarray*}
\mathcal {V}_{\rm ss} \simeq \frac{9 \pi ^2}{16} \frac{R_{\rm g}^2 a_{\rm f}}{r_{\rm p}^3} = 0.125 \, \beta ^3 \frac{\bar{M}_\bullet ^{5/3} \bar{M}_\star ^{1/3}}{\bar{R}_\star ^2} \left(\frac{t}{t_{\rm f0}} \right)^{2/3}.
\end{eqnarray*}$$Different sources of energy dissipation need to be compared with the total loss of energy required to completely circularize the disc. If the debris circularizes with constant orbital angular momentum, the final semimajor axis is given by $a_{\rm circ} = (1 - e_{\rm f}^2)a_{\rm f}\simeq 2 r_{\rm p}$. Hence the orbital energy of a completely circularized debris stream is given by
(29)$$\begin{eqnarray*}
E_{\rm circ} = - \frac{G M_\bullet }{4 r_{\rm p}},
\end{eqnarray*}$$so the fractional binding energy which must be lost to circularize the stream is
(30)$$\begin{eqnarray*}
\mathcal {V}_{\rm circ} = \frac{E_{\rm circ}}{E_{\rm f}} = \frac{R_{\rm t}^2}{4 r_{\rm p}R_\star } \left(\frac{t}{t_{\rm f0}} \right)^{2/3} = \frac{25}{\beta } \frac{\bar{M}_\bullet ^{1/3}}{\bar{M}_\star ^{1/3}} \left(\frac{t}{t_{\rm f0}} \right)^{2/3}.
\end{eqnarray*}$$We see that $\mathcal {V}_{\rm circ} \gg 1$ for typical parameters.


While a non-negligible fraction of stars have disc-like kinematics, the majority of the UIP stars, as well as the VMP and EMP stars, have halo-like kinematics. This does not necessarily mean that they are all accreted, since a fraction of the halo can be made of stars formerly in the disc, but later kinematically heated to halo kinematics by one or several satellite accretions. This has indeed proven to be the dominant in-situ mode of formation of the Galactic halo for stars at higher metallicities, and at few kpc from the Sun (see Di Matteo et al. 2019), as we discuss more extensively in the next section. It is, however, interesting to note that, compared to stars at higher metallicities, such as stars in the Nissen & Schuster sample and stars in the Gaia DR2-APOGEE sample (middle and right columns in Fig. 5), in the Toomre diagram, the UIP and LP samples seem to lack stars with null angular momentum, meaning along the Lz = 0 line. At values of

(

V
R

2

+

V
Z

2

)

≳
200

$ \sqrt{({V_R}^2+{V_Z}^2)} \gtrsim 200 $

 km s−1, stars with [Fe/H]   −4 have rather prograde or retrograde motions, but none seem to lie along the sequence of accreted halo stars discovered by Nissen & Schuster (2010), and later confirmed in Gaia DR1 and DR2 data by Belokurov et al. (2018) (Gaia Sausage), Haywood et al. (2018), Helmi et al. (2018) (Gaia Enceladus). The Gaia Sausage, which is this group of halo stars with very radial orbits, and null VΦ, seems to indeed disappear at [Fe/H]   −2 in this plane. We emphasise that this apparent difference between the kinematics of VMP and EMP stars, on the one side, and stars with [Fe/H] >  −2, on the other side, is simply the consequence of these stars probing different regions and distances from the Galactic centre. Indeed, when one compares the kinematics of these different samples of stars in the quasi-integral-of-motion space Lz − Lperp plane (see Fig. 5, second row), rather than in the Toomre diagram, the kinematic properties of these samples are the same, over the whole [Fe/H] interval. Before moving further, we need, however, to emphasise two points of the comparison with samples at higher metallicity. Firstly, when compared to the Nissen & Schuster (2010) sample, which, we remind the reader, is a kinematically selected sample of thick disc and halo stars, the VMP and EMP stars in the LP First stars sample show an excess of stars at retrograde motions (positive Lz, i.e. Lz >  5) and high values of Lperp (Lperp >  15). None of the Nissen & Schuster (2010) stars occupy this region of the Lz − Lperp diagram, and we suggest this is a consequence of the “local” character of stars in the Nissen & Schuster (2010) study, which are all limited to a few hundred parsecs from the Sun. Indeed, when VMP and EMP stars are compared to stars in the Gaia DR2-APOGEE sample, one can see that stars with Lz and Lperp as extreme as Lz >  5 and Lperp >  15 are found also in the latter. Secondly, because in the comparison shown in this figure we used all stars in the Gaia DR2-APOGEE sample, not restricting ourself to stars from the kinematically defined thick disc and halo, the reader will not be surprised to find that the majority of stars in Gaia DR2-APOGEE sample are stars with cold (i.e. thin) disc-like kinematics, their distribution peaking at Lz ∼ 20 and Lperp ≤ 5. At this stage, what is important to retain is that the region occupied by all these samples, independently of their [Fe/H] ratio, is the same: the relative fraction of stars in one or in another region of the space under analysis can change from one sample to another, but not their overall distribution.


Before moving further, we need, however, to emphasise two points of the comparison with samples at higher metallicity. Firstly, when compared to the

sample, which, we remind the reader, is a kinematically selected sample of thick disc and halo stars, the VMP and EMP stars in the LP First stars sample show an excess of stars at retrograde motions (positive Lz, i.e. Lz >  5) and high values of Lperp (Lperp >  15).


stars occupy this region of the Lz − Lperp diagram,


and we suggest this is a consequence of the “local” character of stars in the

study, which are all limited to a few hundred parsecs from the Sun.


In this Letter we propose to address these apparent difficulties and account for many of the observed differences between the properties of planets in cool and hot stars by postulating that, in addition to the tidal interaction with existing close-in planets, a large fraction of the stellar hosts—both cool and hot—ingest a hot Jupiter early on in their evolution. This proposal is motivated by the expectation that a large fraction (up to 80% according to Trilling et al. 2002) of solar-type stars possess giant planets during their pre-MS phase, and that a large fraction of the giant planets that form in a protoplanetary disk on scales

 migrate close to their host star before the disk is dispersed (e.g., Ida & Lin 2004). Numerical simulations incorporating an N-body code and a 1D α-viscosity disk model (Thommes et al. 2008) demonstrated that this behavior can be expected for disks with

 that are sufficiently massive. The inward planet migration is likely stopped by the strong (

 kG) protostellar magnetic field that truncates the disk at a radius (rin) of a few stellar radii (e.g., Lin et al. 1996). Gravitational interaction with the disk causes a planet reaching rin to penetrate into the magnetospheric cavity and, if it is massive enough, to undergo eccentricity excitation that can rapidly lead to a collision with the star (e.g., Rice et al. 2008). It was, however, inferred that if Mp is sufficiently small (

), the planet would remain stranded at a distance where its orbital period is ∼0.5 of that at rin until well after the gas disk disappears (on a timescale of ∼ 106–107 years). In our proposed scenario, the primordial disk orientations span a broad angular range that is reflected in the orbital orientations of the stranded planets. When the latter are ingested by tidal interaction with the host star (on a timescale

 Gyr), the absorbed angular momentum is sufficient to align a solar-mass star in that general direction, but not an MS star with

 K. This is because cool stars have significantly lower angular momenta at the time of ingestion than hot stars as a result of a more efficient magnetic braking process and of a lower moment of inertia. Given the proximity of the stranded HJs (SHJs) to their host stars (

days), they can be expected to have been ingested by the time their parent planetary systems are observed; however, giant planets farther out can continue to interact with their host stars and potentially affect their measured obliquities. In our simplified formulation, we model the SHJs using as parameters their characteristic mass MSHJ and the fraction p of systems that initially harbored an SHJ. By comparing the predictions of this model with the observational data, we infer

 and

.


Low-frequency observation of radio sky is also essential for the study of the astrophysics at play in various evolutionary stages of different Galactic and extragalactic sources. Radio emission at low frequencies, together with their redshift information, can be used to infer several astrophysical properties associated with the sources. In general, the source distribution is assumed Poissonian (with the possibility of clustering following single power law) (Ali et al. 2008; Jelić et al. 2010; Trott et al. 2016). Source counts are also modelled via a single power-law distribution (Hurley-Walker et al. 2016; Intema et al. 2017; Franzen et al. 2019). However, several recent studies have shown deviation from the single power-law model (Williams et al. 2016; Prandoni et al. 2018; Hale et al. 2019). Thus more detailed studies both in wide field as well as deep fields are required for generating a fiducial model of sources at low frequencies. The differential source counts at these are also useful for constraining the nature of sources. At frequencies in the GHz range and upwards, the source properties are well characterized. However, at lower frequencies, there is a lack of consensus for the same. It has been seen in previous studies that at these frequencies, AGNs dominate in the flux density scales down to few 100 μJy while SFGs and radio quite AGNs become dominant below ∼100 μJy (Simpson et al. 2006; Mignano et al. 2008; Seymour et al. 2008; Smolčić et al. 2008; Padovani et al. 2009, 2011, 2015; Prandoni et al. 2018; Hale et al. 2019). This is inferred from the flattening of the source counts below 1 mJy. However, such studies are very few on account of the limitations in reaching the required SNR. Thus empirical constraints at frequencies ≲1.4 GHz are limited. Therefore the study of the low-frequency radio sky is vital for fiducial modelling of foregrounds for 21-cm cosmology as well as constraining the physics and the astrophysics of the sources.


It has been seen in previous studies that at these frequencies, AGNs dominate in the flux density scales down to few 100 μJy while SFGs and radio quite AGNs become dominant below ∼100 μJy


In general, the source distribution is assumed Poissonian (with the possibility of clustering following single power law)


Applying the P&T profiles defined in Sect. 2 for the three regions of IRAS 16293 our computations imply that 56 mechanisms possess valid reaction thermodynamic function values to proceed with the formation of H2CO. Twenty-six of these can produce H2CO in region I, 32 in region II, and 33 in region III (please see Table 7). These are the amended reactions with detected reagents in every region of the disk. The presence of methane (CH4) and neutral atomic carbon (CI) in this astronomical object, in addition to trans-HONO in region III (detected at ~ 150 AU from the core – Coutens et al. 2019), confirms the hypothetical feasibility of these mechanisms. Moreover, because nitrosyl hydride (HNO) has not been detected in IRAS 16293, R84 cannot be included in any circumstellar envelope region, as it was in DCDMCs. In Fig. 3 we show the abundance of reactants and formaldehyde itself already detected in the three regions of the disk. They have been calculated for the methylene (⋅ 3CH2) (Vasyunin & Herbst 2013), methyl radical (⋅CH3) (Sakai et al. 2012; Woon 2002), and methane (CH4) (Sakai et al. 2012) ground electronic state abundances rather than detected. The reason is that they are planar (or linear) symmetric-top molecules and therefore have a small permanent electric dipole moment, which makes them undetectable through terrestrial radio telescopes by rotational spectroscopy. On the other side of the spectrum, neither high-resolution infrared absorption nor ultraviolet-visible spectroscopy telescopes can identify them in the gas-phase because they lack of a powerful enough source of electromagnetic radiation. The abundance values for the triplet ground-state of methylene (⋅CH2(X3B1)) have therefore been estimated by using Monte Carlo algorithms (Vasyunin & Herbst 2013) which in turn are based on known abundances of derived species. The presence of the methyl radical (⋅CH3) in region I just before the CO depletion zone from icy grains should be negligible because ⋅CH3 is thought tobe mainly produced in this region by CH4 photodissociation (Sakai et al. 2012) and CH3OH (Woon 2002) desorbs from the grains in region II. Consequently, reactions based on ⋅CH3 were considered only for regions II and III, assuming for both an abundance of 6.40 × 10−9 cm−3 (Sakai et al. 2012).


The presence of the methyl radical (⋅CH3) in region I just before the CO depletion zone from icy grains should be negligible because ⋅CH3 is thought tobe mainly produced in this region by CH4 photodissociation


Consequently, reactions based on ⋅CH3 were considered only for regions II and III, assuming for both an abundance of 6.40 × 10−9 cm−3


Fukui et al. (2014, 2015) compared the Planck dust data and the gas emission data such as

 and

 for the solar neighborhood. The

 emission is generally assumed to be optically thin, as written in textbooks. If the dust properties are uniform and DGR is a constant, it is expected that the velocity-integrated intensity of the

 spectrum (

) is proportional to

 for the data points where the

 emission is not detected. Fukui et al. (2014, 2015), however, found that the correlation between them is not so good. By introducing

 into the

 correlation plot, these authors discovered that the poor correlation in the

 plot is mainly due to the data points where the density is high and

 is low. There are two main possibilities to explain this bad correlation between

 and

 one is the presence of optically thick

 gas, and the other is the presence of “

-dark

 gas,” which is

 gas without the

 emission (e.g., Wolfire et al. 2010; Planck Collaboration et al. 2011c; Langer et al. 2014). Fukui et al. (2014, 2015) investigated the

 fractions in the hydrogen gas by referring to the UV measurements (Gillmon et al. 2006), and found that the fractions are typically

 toward the lines of sight whose column densities are up to at least

.1

1
Note that in Figure 16 of Fukui et al. (2015), the

 fractions are somewhat larger than 10% toward two Galactic B-type stars, HD 210121 and HD 102065 (Rachford et al. 2002). These two stars may be contaminated by their own localized gas, and therefore the

 fractions for the local ISM are possibly not reliable toward them. For this reason, Fukui et al. (2017; see Section 4.3) did not use the results obtained in Rachford et al. (2002).
 That is,

 dominates

, and the “

-dark

 gas” would not be a dominant component in the local ISM. Therefore, these authors concluded that there exists a large amount of optically thick

 gas in the local ISM, whose typical optical depth is

, and the amount of the

 gas is underestimated by

 if a correction for the opacity effect is not applied. In Fukui et al. (2014, 2015), these analyses were made for the high Galactic latitude at

 where the gas density is low. An open issue discussed in Fukui et al. (2014, 2015) is if

 obeys a simple linear relationship with the

 or not. A study in the Orion A molecular cloud (Roy et al. 2013) indicates that the dust optical depth is proportional to the 1.28th power of

, rather than a simple linear relation. If correct, this nonlinearity may be ascribed to the dust evolution at high column density. This suggests that it is possible to apply a minor modification of the method of Fukui et al. (2014, 2015) in order to improve the accuracy in

.


Half a century ago, a series of theorems laid the ground for the Kerr hypothesis (Israel 1967; Carter 1971; Robinson 1975); according to these no-hair theorems, the only stationary, axisymmetric, asymptotically flat, regular outside of the horizon solution to four-dimensional GR when the matter fields feature the same isometries as the spacetime is the Kerr BH. Notwithstanding their significance, there are many ways with which to circumvent them and discover different solutions. Still, in four dimensions, hairy BHs have been described in different theories of gravity, such as Einstein–Yang–Mills (Bizon 1990; Künzle & Masood-ul-Alam 1990; Volkov & Galtsov 1990; Breitenlohner et al. 1992; Kleihaus & Kunz 1998, 2001; Kleihaus et al. 2004), scalar-tensor (Bocharova et al. 1970; Bekenstein 1974; Kleihaus et al. 2015; Collodel et al. 2020a), and Gauss–Bonnet theories (Kanti et al. 1996; Kleihaus et al. 2011, 2016; Antoniou et al. 2018; Doneva & Yazadjiev 2018; Silva et al. 2018; Cunha et al. 2019; Collodel et al. 2020b; Herdeiro et al. 2021; Berti et al. 2021). Remarkably, by dropping the assumption that the matter fields must be stationary and axisymmetric, Herdeiro and Radu found solutions in the context of GR where BHs have hair (Herdeiro & Radu 2014b, 2015), by minimally coupling to gravity a complex scalar field that depends on time and on the axial coordinate while its energy-momentum tensor still possesses the respective isometries; see Herdeiro et al. (2015, 2016a, 2016b), Brihaye et al. (2016), and Delgado et al. (2016) for generalizations. These are known as scalarized Kerr black holes (KBHsSH) and they are the object of study of this paper. In their domain of existence, they connect Kerr BHs (that is, with no hair) with pure solitonic solutions, also known as boson stars (BS), which are regular everywhere and feature no horizons. In this sense, one can think of the KBHsSH indeed as a combined system of a BS with a horizon at its center, and therefore it shares traits of both objects.


Half a century ago, a series of theorems laid the ground for the Kerr hypothesis

according to these no-hair theorems, the only stationary, axisymmetric, asymptotically flat, regular outside of the horizon solution to four-dimensional GR when the matter fields feature the same isometries as the spacetime is the Kerr BH.


Remarkably, by dropping the assumption that the matter fields must be stationary and axisymmetric, Herdeiro and Radu found solutions in the context of GR where BHs have hair

by minimally coupling to gravity a complex scalar field that depends on time and on the axial coordinate while its energy-momentum tensor still possesses the respective isometries;


Column 11.—Adopted distance DH and its uncertainty σD, both in Mpc. For objects with cz⊙ > 6000

, the distance is simply estimated as czcmb/H, where czcmb is the recessional velocity measured in the cosmic microwave background reference frame (Lineweaver et al. 1996) and H is the Hubble constant, adopted to be 70

 Mpc−1. For objects with czcmb  6000

, we use the local peculiar velocity model of Masters (2005), which is based in large part on the SFI++ catalog of galaxies (Springob et al. 2007) and results from analysis of the peculiar motions of galaxies, groups, and clusters using a combination of primary distances from the literature and secondary distances from the Tully–Fisher relation. The resulting model includes two attractors with infall onto the Virgo cluster and the Hydra-Centaurus supercluster, as well as a quadrupole and a dipole component. The transition from one distance estimation method to the other is selected to be at cz⊙ = 6000

  because the uncertainties in each method become comparable at that distance. Where available, primary distances in the published literature are adopted; we also use secondary distances, mainly from Tully et al. (2013), for galaxies with czcmb  6000

. When the galaxy is a known member of a group (Springob et al. 2007), the group systemic recessional velocity czcmb is used to determine the distance estimate according to the general prescription just described. Where primary distances are not available, objects in the Virgo region are assigned to likely Virgo substructures (Mei et al. 2007), and then the distances to those subclusters are adopted following Hallenbeck et al. (2012). Errors on the distance are generated by running 1000 Monte Carlo iterations where peculiar velocities are drawn from a normal distribution each time, as described in Section 4.1 of Jones et al. (2018). Such errors are likely underestimates in the vicinity of major attractors such as Virgo. It should be noted that the values quoted here are Hubble distances, not comoving or luminosity distances.


When the galaxy is a known member of a group

the group systemic recessional velocity czcmb is used to determine the distance estimate according to the general prescription just described.


The above formulation in principle applies to any planet with a Dungey cycle-type stellar wind–magnetosphere interaction, and we thus consider here the appropriate parameters for exoplanets orbiting at arbitrary distances, with an emphasis on close-orbiting hot Jupiters. As discussed above, whereas for the RBL the radio powers are computed as functions of incident kinetic or Poynting flux, in our model, the powers are principally functions of the motional electric field of the solar wind, the dynamic pressure of the stellar wind and the Pedersen conductance of the ionosphere, all of which are dependent on further stellar and planetary parameters as described below. We examine results for both a solar-like stellar wind, and that representative of a young Sun-like star with high-mass-loss rate and magnetic field strength relative to the Sun. Considering first the Sun-like stellar wind, the relevant parameters are shown in Fig. 4 versus radial distance d normalized by the solar radius Rs [we truncate the inner radial distance of the plot at 2 $\mathrm{{\rm {\it R}}_s}$, being the canonical location of the heliospheric magnetic field ‘source surface’ (Owens & Forsyth 2013)]. Absolute distances in au are shown on the top axis for information, although we recognize that in reality the conversion from $\mathrm{{\rm {\it R}}_s}$ to au depends on the individual star. Specifically, Fig. 4(a) shows with the solid line the incident velocity of the solar wind on the magnetosphere vm, which is a function both of the stellar wind speed and the planet's orbital speed. For simplicity, we employ Parker's isothermal solution for the stellar wind speed vsw (Parker 1958), which is fully parametrized by the sound speed cs, and which, as shown by Cranmer (2004) has the closed-form solution

(18)

\begin{equation}
{v_{{\rm sw}}}^2= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}-v_{\rm c}^2W_0[-D(d)] & {\rm if } d \le d_{\rm c} ,\\
-v_{\rm c}^2W_{-1}[-D(d)] & {\rm if } d \ge d_{\rm c} ,\\
\end{array}\right.
\end{equation}

where W0 and W−1 are branches of the Lambert W function, dc is the critical distance at which vsw passes through the sound speed cs, given by

(19)

\begin{equation}
d_{\rm c}=\frac{GM_{\rm s}}{2c_{\rm s}^2},
\end{equation}

where Ms = 1.9891 × 1030 kg is the solar mass and D(d) is given by

(20)

\begin{equation}
D(d)=\left(\frac{d}{d_{\rm c}}\right)^{-4}\exp \left[4 \left(1-\frac{d_{\rm c}}{d}\right)-1\right].
\end{equation}

For the Sun-like wind, we employ a sound speed cs = 130 km s−1 (which, for a Sun-like average particle mass of 1.92 × 10−27 kg corresponds to a temperature of ∼1.18 MK, though note for the present Sun calculation we actually make no assumptions in this regard), yielding a velocity at 1 au of ∼480 km s−1, consistent with observations, and ∼50–200 km s−1 in the hot Jupiter region of 3–10 Rs (indicated by the grey region). The dotted line indicates the Keplerian speed of a planet in a circular orbit vorb, and the solid line is the sum in quadrature of the two, giving the resultant incident stellar wind speed vm. Note that the two speeds are comparable in the inner region associated with hot Jupiters, and although this will modify the orientation of the magnetosphere with respect to the radial vector, it will not significantly alter the magnetospheric dynamics. We further show with the loosely dotted and dot–dashed lines the (constant) sound speed cs and the Alfvén speed vA given by equation (5). It is evident that the interaction is everywhere supersonic (modestly so in the hot Jupiter region, with a Mach number of ∼2) but becomes sub-Alfvénic inside of ∼15 $\mathrm{{\rm {\it R}}_S}$, such that Alfvén wings will form along the IMF field lines, as discussed by Saur et al. (2013), effectively shielding the stellar wind motional electric field and is related to KR saturation of the convection potential.


A simpler model with sinusoids only will be used when analyzing time series of activity indices. That is,
5

6

7

where

 is some measurement of an activity index (e.g., emission in the

 line, S-index, etc.) obtained with instrument INS. The parentheses in

 mean that fits with and without long-term trends will also be investigated. The significance assessments here are performed using frequentist estimations of the false alarm probability (FAP) following the recipes by Baluev (2009). Detecting a periodic signal in a time series will consist of  comparing a null model (e.g., no

 term) with a model with one more periodic signal (e.g., one sin/cos term). Since the period is a highly nonlinear parameter, many periods (about 20,000 equally spaced in 1/P or frequency domain) will be tested between 2 and 50,000 days. A plot of the improvement of the logarithm of the maximum likelihood statistic (

) against the period of the test signal produces a plot that is often called a periodogram. Periodograms for unevenly sampled data were developed in the late 1970s (e.g., Lomb 1976; Scargle 1982). In those cases, the merit statistic plotted against the test period was the improvement in

 or the F-ratio statistic. These early methods only accounted for a model with a single sin/cos component. A further refinement was later implemented by also including an offset (as our γ) as a free parameter to acknowledge the fact that the mean of the series was not known a priori. These periodograms were called floating mean periodograms (Ferraz-Mello 1981) or generalized least-squares (GLS, Zechmeister & Kürster 2009) periodograms. RRM15 relies on the use of GLS periodograms. GLS is adequate to identify and quantify the significance of periodicities if the data contain a single sinusoidal signal and uncertainties in the time series are realistic. Otherwise, analytic significance estimates of the favored solutions using GLS can only be considered to be approximate. In this sense, GLS is a very efficient method to guess the most likely signals left in residual data, but it cannot be used to provide a reliable assessment of its significance. This is discussed in the context of the basic model comparison in Anglada-Escudé & Tuomi (2015). Further refinements of the model (e.g., simultaneous fit to a trend, combining different instruments with different offsets, include basic models for unaccounted noise) require the use of more sophisticated tools beyond GLS, like the in-likelihood periodograms presented here. Note that the

 statistic used by GLS and previous periodograms is a particular case of the ln-likelihood statistic with

,

, and only one instrument INS. Finding the maximum likelihood statistic requires adjusting all of the free parameters of the model at each test period. For these reasons, it produces much more robust significance estimates than its predecessors, especially when the description of the data requires models more complex than a single sinusoid. Reviewing the properties of the likelihood function in the context of periodic signal detection is not the topic of this article. We refer the interested reader to Baluev (2009, 2013) for more detailed discussions and formal proofs.


We stress the uncertainty of our present interpretation, given that the density and temperature structures of our target sources are not yet constrained in a sufficiently detailed way on our resolved spatial scales. In addition, our 44 GHz observationsare still subject to missing short-spacing, which nevertheless should not impact regions with high intensity. To provide a quantitative sense of how missing short-spacing biased the observed Stokes I intensity fractionally, we assume that exterior to the disks (or disk-like structures) around IRAS 16293–2422 A and B there is ~2 M⊙ of gas on ≳10′′ scales (Mundy et al. 1990), and assume that the gas to dust mass ratio is 100. The averaged dust mass column density is Σdust ~ 0.1 g cm−2. Assuming the dust opacity at 230 GHz is κ230 GHz = 0.5

$^{0.5}_{-0.4}$−0.4+0.5
 cm2 g−1 (c.f., Draine 2006), and the dust opacity spectral index is β ~1−1.75 (Li et al. 2017), the derived averaged dust optical depth at 44 GHz is

$\tau_{\mbox{\tiny 44 GHz}}=\kappa_{\mbox{\tiny 230 GHz}}(44/230)^{\beta}\mathrm{\Sigma}_{\mbox{\tiny dust}} = $τ 44 GHz=κ 230 GHz(44/230)βΣ dust=
2.8

$^{16}_{-2.2}\,\times$−2.2+16 ×
 10−3. Assuming a nominal 20–30 K dust temperature on 10′′ scales (van Dishoeck et al. 1995; Ceccarelli et al. 2000; Chandler et al. 2005), the brightness temperature of the extended dust emission is ~ (25 ± 5) × (1 − e−τ44 GHz) = 0.069

$^{0.57}_{-0.058}$−0.058+0.57
 K. This is approximately how much missing short-spacing can bias the Stokes I intensity. If we take regions where the observed brightness temperature in the Robust = 0 weighted 44 GHz image is above 1.4 K (i.e., 6σ; the second contour from bottom in the right panel of Fig. 1) for example, missing flux can bias Stokes I intensity by at most ~(0.64 K)/(1.4 K + 0.64 K) ~ 31%. For structures detected at the 6σ significance, clean errors and thermal noise (e.g., 3σ) may dominate the errors in the measured polarization percentages. The brightness temperature of the missing flux can be comparable with the 6σ detection limit of our Robust = 2 weighted 44 GHz image (~0.6 K), and therefore the observed polarization percentages can be biased by over ~50% in the low intensity (3–6σ) area in that image (Table B.1). Our assessment is that for our present 44 GHz images, missing short-spacing biased the observed polarization percentages of the detected, spatially-extended, dust emission component. Since the brightness temperature of the extended component is low (Fig. 1) such that dust is likely to be optically thin (i.e., Tbg = TCMB), missing short-spacing does not dramatically flip the observed polarization position angles by 90° (Sect. 4.1; Figs. 2–4).


In this work we first used the data points of low-redshift Hubble diagrams for Pantheons, quasars, and GRBs to put constraints on the present value of cosmographic parameters in an independent cosmography approach. To do this, we used different combinations of data samples including Pantheon, Pantheon + quasars, Pantheon + GRB, finally Pantheon + quasars + GRB. In the context of a cosmography approach, we obtained the best-fit values of cosmographic parameters as well as their confidence regions up to 3 − σ uncertainties for different combinations of data samples. Our results showed that the best-fit value of the deceleration parameter q0 varies in the range of −0.844 to −0.702 and the best fit of jerk parameter j0 varies in the range of 1.60 to 2.61 for different combinations of data samples. Note that here we used the Hubble diagrams of quasars and GRBs, respectively, derived in Lusso & Risaliti (2016a) and Demianski et al. (2017a). In the calibration procedure to form the Hubble diagrams of both quasars and GRBs, they have used the SNIa data at low redshifts. Their results for quasars and GRBs samples are consistent with that of the SNIa samples at the low-redshift universe. Hence we adopted their calibrations and used their Hubble diagrams for quasars and GRBs. In the case of concordance ΛCDM cosmology, our results are also compatible with recent work in Lusso et al. (2019). They confirmed the presence of a tension between Λ cosmology and the best-fit cosmographic parameters ∼4σ with SnIa+quasars, at ∼2σ with SnIa+GRBs, and at  4σ with the whole SnIa+quasars+GRB data set (Lusso et al. 2019). Furthermore, we studied some relevant DE parameterizations as well as the concordance ΛCDM cosmology using the Hubble diagrams of Pantheons, quasars, and GRB observations in the context of a cosmography approach. The DE parameterizations studied in our analysis are wCDM, CPL, and Pade parameterizations. First, using the different combinations of data samples and in the context of the MCMC algorithm, we calculate the χ2 function of the distance modulus to find the best-fit values and also the 1 − σ uncertainty of cosmological parameters for each DE parameterization. Using the chain of data obtained for cosmological parameters, we found the best-fit values and the 1 − σ confidence region of the cosmographic parameters of DE parameterizations. Comparing the results for DE models with those obtained for the model-independent approach leads us to conclude that the model is in better (worse) agreement with Hubble diagrams of Pantheons, quasars, and GRBs. In the first stage, using the solely Pantheon sample, we found that the wCDM model is the most compatible model with the result of model-independent constraints and on the other hand the concordance ΛCDM model is the worst model. In the second step, by combining the GRB data to the Pantheon sample, we obtained disappointing results for the ΛCDM model. In this case the q0 parameter of the ΛCDM has a 3.8 − σ tension with that of the model-independent cosmography approach. Moreover, the j0 parameter of ΛCDM cosmology has roughly 5 − σ tension with that of the model-independent approach. These results will be more complicated when we see the results of other DE models and parameterizations that we studied in this work. We observed that wCDM, CPL, and Pade parameterizations are in better agreement with the results of the model-independent cosmography approach rather than the concordance model. In the third and fourth steps, using the combinations Pantheon+quasars and Pantheon+GRB+quasars data points, we obtained the same results again, supporting our results in previous steps. So we conclude that the concordance ΛCDM cosmology has much tension with observations of quasars and GRBs at higher redshift. Note that the DE parameterizations studied in this work are in better agreement with Hubble diagrams of high-redshift quasars and GRB observations. Finally, we reconstructed the Hubble parameter using the best-fit value of cosmographic parameters for both model-independent approaches, the ΛCDM model, and DE parameterizations. We observed that for different data sample combinations, the evolution of reconstructed H(z) in a concordance ΛCDM model has a maximum deviation from the confidence region compared to different DE parameterizations. Upon this result, we can conclude that among the different cosmological models studied in this work, the ΛCDM has the minimum compatibility with the predictions of a model-independent approach and thus it is falsified by a cosmography approach. The large value of tensions (between 3σ to 6σ for different data combinations) that we observed between the cosmographic parameters of ΛCDM and those we obtained in the model-independent approach support this claim again that we should explore other alternatives for standard ΛCDM cosmology. We observed that other DE parameterizations in this study cannot be refuted in the context of a cosmography approach. Our results for ΛCDM cosmology are in agreement with the results of recent work (Yang et al. 2019; Khadka & Ratra 2020), which were obtained using a different approach and different data sets. Although in the literature it has been thoroughly affirmed that the ΛCDM describes the evolution of the universe until recent times, our conclusion confirms the result of Benetti & Capozziello (2019) that show large tensions emerge at higher redshifts for ΛCDM. Our analysis can be extended by considering other cosmic observations in the context of a cosmography approach.


They confirmed the presence of a tension between Λ cosmology and the best-fit cosmographic parameters ∼4σ with SnIa+quasars, at ∼2σ with SnIa+GRBs, and at  4σ with the whole SnIa+quasars+GRB data set


Note that here we used the Hubble diagrams of quasars and GRBs, respectively, derived in Lusso & Risaliti (2016a) and


Our results for ΛCDM cosmology are in agreement with the results of recent work

which were obtained using a different approach and different data sets.


that show large tensions emerge at higher redshifts for ΛCDM.

Although in the literature it has been thoroughly affirmed that the ΛCDM describes the evolution of the universe until recent times, our conclusion confirms the result of


To definitively assess whether or not the growth of the vertical kinetic energy observed in our simulations is associated with the disc eccentricity, we performed three additional simulations using the parameters of the fiducial model, except that: (i) In the first simulation, the disc orbits a single star and the cooling time-scale is set to tc = 0.1Ω−1; (ii) In a second run, the disc orbits a single star and the cooling time-scale is set to tc = 0.001Ω−1; and (iii) In the third run, we restarted the fiducial model at time t0 = 955 but with the system slowly transitioning from a central binary system into a single star located at the centre of mass. This is done by treating the gravitational potential as the sum of two terms, one corresponding to the binary plus one corresponding to a single star, with weighting factors that change with time. More precisely, the gravitational potential for this simulation is given by (Mutter et al. 2017)
(23)$$\begin{eqnarray}
\Phi _{\rm trans}=(1-W (t))\Phi _{\rm bin}-\frac{GM_\star }{r} W(t)
\end{eqnarray}$$with
(24)$$\begin{eqnarray}
W(t) &= \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1 & \text{if } \,\, t-t_0 \ge 100T_{\rm bin}\\
\frac{1}{2}\left[1-\cos \left(\frac{\pi (t-t_0)}{100T_{\rm bin}}\right) \right] & \text{if} \,\, t-t_0 \lt 100T_{\rm bin}\end{array}.\right.
\end{eqnarray}$$The aim of simulations (i) and (ii) is to check that our fiducial disc is stable to the VSI whereas run (iii) is useful to examine whether the turbulent flow is maintained under unforced conditions. For these three calculations, the time evolution of the vertical kinetic energy is shown in Fig. 5. Considering simulations (i) and (ii), exponential growth of the vertical kinetic energy due to growth of the VSI only occurs for the case with tc = 0.001 Ω−1. This is consistent with the results of Richard, Nelson & Umurhan (2016), who found that for h = 0.05 the disc remains stable to the VSI provided tc > 0.05 Ω−1. It also demonstrates that our fiducial model, which has tc = Ω−1, is stable to the VSI. For simulation (iii), the saturation level of the vertical kinetic energy remains essentially unchanged after the switch (t = 1055). Contours of the normalized density perturbation one scale height above the mid-plane, and of the meridional velocity in a [R, Z] plane located at azimuth φ = π, are presented in Fig. 6 at t = 1500. They clearly reveal that both the eccentric mode and the turbulence can be maintained under unforced conditions. The continued existence of the eccentric mode at that time is not surprising since m = 1 free eccentric modes can be long lived (Papaloizou 2005b). The persistence of the turbulent flow is a strong indication that the instability originates from the presence of an eccentric mode in the disc.


Another, probably more plausible, explanation for the systematic trend in Figure 5 might be that, in deriving SN explosion energy from the global parameters, it is difficult to take into account the complex environments of SNRs, and this is compounded by the parameter sensitivity of Equation (10) (and to a lesser extent, Equation (8)). As we have already seen in many observed SNRs (e.g., Figure 1) and simulated SNRs (e.g., Figure 3), the real ISM provides a very complex environment for SNRs. In this case, the background medium density parameter

 for describing SNR evolution is not well-defined. Fortunately, as demonstrated in 3D simulations considering inhomogeneous background medium (e.g., Figure 4; see also Cho et al. 2015; Iffrig & Hennebelle 2015; Kim & Ostriker 2015; Li et al. 2015; Martizzi et al. 2015; Walch & Naab 2015; Zhang & Chevalier 2019), the evolutionary tracks of integrated quantities such as total momentum and kinetic energy in the radiative SNRs (i.e., after the shell formation) are not sensitive to the complexity of the background medium. In particular, the total momentum does not evolve in time after reaching a terminal value. The model uncertainty due to the ISM inhomogeneity is less than a factor of two. However, a unitary background medium density

 does not seem be applicable to both

 and

. In our TS models, where the ambient medium has a large-scale nonuniformity, the thermal energy is considerably smaller than that of the uniform medium case with the same mean density (Figure 4(e)). This also happens for some SNRs in TI models (see Section 4.2). On the other hand, the observed thermal energies of the SNRs appear to be generally larger than those of the uniform medium cases with the same mean densities (Figure 4(f)). For the two prototypical SNRs interacting with MCs, W44 and IC 443, for example, Table 5 shows that

/

 = 6–8 when we adopt the mean ambient density

(H i+H2)(=50–70 cm−3). It is, however, clear that this density is much higher than the density of the material filling most of the volume of the ISM where the SN blast wave propagates. For example, the ambient density derived from an analysis of the X-ray surface brightness profile of W44 is ∼3 cm−3 (Harrus et al. 1997). An overestimated ambient density would predict a smaller

 or a larger

 to match the observed

 (see Equation (10)). On the other hand, Equation (10) is based on 1D simulations. In real SNRs in inhomogeneous/nonuniform media, the cooling could be significantly enhanced due to the mixing and diffusion between the engulfed dense material and the hot plasma, in which case the thermal energy of real SNRs might be smaller than the

 predicted from 1D simulations. Therefore, it is not obvious whether Equation (10) with

(H i+H2) would overpredict or underpredict

 for SNRs in complex environments (see also Section 4.2). High-resolution simulations of SNRs in realistic environments, including the complex physics at the hot/cool interface, are needed in oder to understand the environmental dependency of the evolution of thermal energy.


It has been clear since many years now (Matthews et al. 1964) that the majority of powerful radio-AGN is hosted within the most massive galaxies (stellar masses M∗∼1011M⊙\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$M_* \sim 10^{11}\, M_\odot$$\end{document} and above) known in the universe. This result has been confirmed at least up to z∼1\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$z\sim 1$$\end{document} (and in some cases beyond) both by direct estimates (e.g., Heeschen 1970; Ekers and Ekers 1973; Auriemma et al. 1977; Jenkins 1982; Best et al. 2005a; Mauch and Sadler 2007; Smolčić et al. 2009; Magliocchetti et al. 2016; Sabater et al. 2019; Capetti et al. 2022) and also by the extremely tight correlation in the MK-z\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$M_K-z$$\end{document} plane exhibited by the hosts of radio-AGN (e.g., Lilly and Longair 1984; Eales et al. 1997; Jarvis et al. 2001; De Breuck et al. 2002; Willott et al. 2003) that can be explained by assuming that the observed K-band light is dominated by emission from the old stellar population and that radio-AGN have very similar (large) host masses all across half of the age of the universe. More recent results based on rest-frame 1.6 and 4.5 μ\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\upmu$$\end{document}m photometry also report very high masses for the hosts of radio-AGN, at least up to z∼3\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$z\sim 3$$\end{document} if not beyond (Seymour et al. 2007; De Breuck et al. 2010; Gürkan et al. 2014; Drouart et al. 2016).


On a ∼30000 au scale, a bipolar outflow associated with this source was detected in the east-west direction by FCRAO single-dish observations in 12CO

J
=
1
−
0

 molecular line emission (Narayanan et al. 2012) and by James Clerk Maxwell Telescope (JCMT) single-dish observations in 12CO

J
=
3
−
2

 molecular line emission (Hogerheijde et al. 1998). Their results show that the blue and red robes of the outflows are on the eastern and western sides, respectively. On the other hand, inner parts (∼8000 au scale) of the outflow mapped with the Nobeyama Millimeter Array (NMA) in 12CO

J
=
1
−
0

 show the opposite distribution, i.e., stronger blueshifted emission on the western side and stronger redshifted emission on the eastern side (Tamura et al. 1996). Mid-infrared observations toward L1527 IRS with Spitzer Space Telescope shows bright bipolar scattered light nebulae along the outflow axis in the ∼20000 au scale (Tobin et al. 2008). They fitted a protostellar envelope model to near- and mid-infrared scattered light images and spectral energy distribution (SED). As a result, the inclination angle of the envelope around L1527 IRS was estimated to be

i
=
85
°

, where

i
=
90
°

 is the edge-on configuration. In addition, Oya et al. (2015) found that the western side is closer to observers. This inclination angle is consistent with a disk-like structure of dust highly elongated along the north-south direction, which was spatially resolved for the first time in 7-mm continuum emission by the Very Large Array (VLA; Loinard et al. 2002). By expanding the studies by Tobin et al. (2008, 2013), we fitted a model composed of an envelope and a disk to (sub)millimeter continuum emissions and visibilities observed with the Submillimeter Array (SMA) and the Combined Array for Research in Millimeter Astronomy (CARMA), as well as infrared images and SED. Their best-fitting model suggests a highly flared disk structure (

H
∝

R

1.3

, H = 48 au at R = 100 au) with a radius of 125 au. This study has geometrically distinguished the protostellar disk and envelope around L1527 IRS, although they were not kinematically distinguished from one another.


The error budget associated with Δtij was estimated by considering three sources of uncertainties. Firstly, we propagated the statistical error inferred from the MCMC sampling. This was simply done by constructing histograms for Δtij h−1 from the independent samples of SIEg model parameters reported in Figs. 3 and 4. An example of these histograms is illustrated in Fig. 7 for the case of zl = 0.7. Secondly, we considered the impact of hypothetical massive objects lying on the line-of-sight by scaling the theoretical time delay distance with an external convergence term κext such that

D

Δ
t

=

D

Δ
t

theory

/

(
1
−

κ

ext

)

$ D_{\Delta t} = D_{\Delta t}^{\text{ theory}} / (1 - \kappa_{\text{ ext}}) $

 (see, e.g., Keeton 2003). We applied a different scaling for each set of model parameters used to construct the histograms (see Fig. 7). The κext values were randomly drawn from a zero mean normal distribution and characterized by a conservative standard deviation σκ = 0.03 (see, e.g., Wong et al. 2017). When combined, the typical errors are ∼7.7% for ΔtCB, ∼7.8% for ΔtCA, and ∼7.6% for ΔtCD, compared to the median values. Thirdly, we considered the impact of the Source Position Transformation (SPT), a degeneracy existing between different lens density profiles that reproduce equally well the lensing observables, except for the product Δt H0 (Schneider & Sluse 2013, 2014). A short summary is given in Appendix A. We SPT-transformed the SIEg model using the modified deflection law

α

̂

=

α

SIEg

+

β

−

β

̂

(

β

)

$ {\hat{\boldsymbol{\alpha}}} = \boldsymbol{\alpha}_{\text{ SIEg}} + \boldsymbol{\beta} - {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) $

 along with a radial stretching of the source plane defined by

β

̂

(

β

)

=

[
1
+
f
(
|

β

|
)
]

β

$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = [1 + f(|\boldsymbol{\beta}|)]\ \boldsymbol{\beta} $

. In particular, we considered the special case where the deformation function f(|β|) is the lowest-order expansion of more general functions, f(|β|) = f0 + f2|β|2/(2θE), where the constants f0 = f(0) and

f
2

=
θ
E
2

f

′′

(0)

$ f_2 = \theta_{{\rm E}}^2 f^{\prime\prime}(0) $

 quantify the magnitude of the deformation. This choice explores a large variety of degeneracies characterized by an isotropic transformation of the source plane. In the end, this at least defines a lower limit on the impact of the SPT on the time delays. In the particular case of f2 = 0, the SPT reduces to the well-known mass-sheet degeneracy (MSD) characterized by

β

̂

(

β

)

=
λ

β

≡

(
1
+

f
0

)

β

$ {\hat{\boldsymbol{\beta}}}(\boldsymbol{\beta}) = \lambda \boldsymbol{\beta} \equiv (1 + f_0) \boldsymbol{\beta} $

. In our analysis, we only explored degeneracies that cannot be explained by a MSD, hence f0 = 0.


For completeness and ease of our discussions, here we summarize briefly the main features of the NS EOS-metamodel we use in solving the NS inverse-structure problem. More details can be found in our previous publications (Zhang et al. 2018; Xie & Li 2019, 2020; Zhang & Li 2019a, 2019b, 2019c, 2020). We assume the cores of NSs are made of totally charged neutral neutrons, protons, electrons, and muons (the npeμ model) at β-equilibrium. Unlike the widely used composition-degenerate spectral functions and/or piecewise polytropes that directly parameterize the pressure as a function of energy or baryon density, to probe the high-density symmetry energy, we have to keep the isospin dependence of the EOS and retain explicitly the composition information at all densities. For this reason, we metamodel NS EOS by starting at the single-nucleon energy level with explicitly isospin dependence. Specifically, we parameterize the E0(ρ) and Esym(ρ) according to
2

E

0

(
ρ
)
=

E

0

(

ρ

0

)
+

K

0

2

ρ
−

ρ

0

3

ρ

0

2

+

J

0

6

ρ
−

ρ

0

3

ρ

0

3

,

3

E

sym

(
ρ
)

=

E

sym

ρ

0

+
L

ρ
−

ρ

0

3

ρ

0

+

K

sym

2

ρ
−

ρ

0

3

ρ

0

2

+

J

sym

6

ρ
−

ρ

0

3

ρ

0

3

where E0(ρ0) = −15.9 ± 0.4 MeV is the binding energy and K0 ≈ 230 ± 20 MeV (Shlomo et al. 2006; Piekarewicz 2010; Garg & Colò 2018) is the incompressibility at the saturation density ρ0 of SNM, while Esym(ρ0) = 31.7 ± 3.2 MeV is the magnitude and L ≈ 58.7 ± 28.1 MeV is the slope of symmetry energy at ρ0 (Li & Han 2013; Oertel et al. 2017) based on earlier surveys of over 50 analyses of both terrestrial experiments and astrophysical observations, respectively. A very recent survey of 24 new analyses of NS observations since GW179817 indicates that L ≈ 57.7 ± 19 MeV and Ksym ≈ −107 ± 88 MeV at 68% confidence level (Li et al. 2021). In this study, we use L = 58.7 as its most probable value and vary it within ±20 MeV. We keep the E0(ρ0), K0, and Esym(ρ0) at their most probable values given above as they have been relatively well determined and their variations within their remaining uncertain ranges have been shown to have little effects on the masses, radii, and tidal deformabilities of NSs Fattoyev et al. (2013, 2014), Xie & Li (2019, 2020). As a reference, we notice that very recent calculations based on a set of commonly used Hamiltonian operators including two- and three-nucleon forces derived from chiral effective field theory predicted that L = 47 ± 3 MeV, Ksym = −146 ± 43 MeV, Jsym = 90 ± 334 MeV (Somasundaram et al. 2021). While in the same calculations, with a prior J0 = −300 ± 400 MeV, the posterior ranges between J0 =−573 ± 133 MeV and J0 = −172 ± 243 MeV depending on the scaling schemes used. The EOS parameter ranges we used are consistent with these calculations.


Next, we estimated the distance and reddening. As in Brown et al. (2014), we compared the Eri II main sequence to a synthetic Victoria-Regina theoretical isochrone from VandenBerg et al. (2014) and the horizontal branch of Eri II to the horizontal branch of M92, which was observed with the same ACS filters by Brown et al. (2005). The synthetic main-sequence isochrone was constructed assuming an age of 13 Gyr, the metallicity distribution displayed in Figure 3, and a binary fraction of 0.48 (Geha et al. 2013). For M92, we assumed a distance modulus of 14.62 mag (the mean of literature measurements by Del Principe et al. 2005, Sollima et al. 2006, and Paust et al. 2007) and reddening of E(B − V) = 0.023 mag (Schlegel et al. 1998). We included main-sequence stars more than 0.5 mag below the main-sequence turnoff (m814 > 27.67) in order to avoid portions that are age-sensitive, and we used horizontal branch stars that are bluer than m606 − m814 = −0.68 in order to avoid the RR Lyrae instability strip. Fitting the main sequence and the horizontal branch simultaneously, we found m − M = 22.65 mag (d = 339 kpc) and E(B − V) = 0.03 mag, as shown in Figure 4. This distance modulus is consistent, within the uncertainties, with the distance derived from the Eri II RR Lyrae stars by A. Garofalo et al. (2021, in preparation). The statistical uncertainties on the distance fit are very small (0.015 mag in distance modulus, 0.003 mag in reddening), and they are certainly dominated by systematics in the choice of comparison cluster/isochrone and the magnitude and color range of stars to include. By analogy to Brown et al. (2014), we assume overall uncertainties of 0.07 mag in m − M and 0.01 mag in E(B − V). Similar to a few of the ultra-faint dwarfs studied by Brown et al. (2012, 2014), the distance modulus determined with this approach is in modest disagreement with some literature results (Crnojević et al. 2016 measured m − M = 22.8 ± 0.1 mag and Koposov et al. 2015 estimated m − M = 22.9 mag, although Bechtol et al. 2015 found a comparable value of m − M = 22.6 mag) and the derived reddening is larger than the E(B − V) = 0.009 mag value obtained by Schlafly & Finkbeiner (2011) (also see Mutlu-Pakdil et al. 2019). However, for consistency with previous star formation history analyses, we adopt the derived values for the remainder of this study.


The synthetic main-sequence isochrone was constructed assuming an age of 13 Gyr, the metallicity distribution displayed in Figure 3, and a binary fraction of 0.48


A. Garofalo et al. (2021, in preparation)

This distance modulus is consistent, within the uncertainties, with the distance derived from the Eri II RR Lyrae stars by


Similar to a few of the ultra-faint dwarfs studied by Brown et al. (2012, 2014), the distance modulus determined with this approach is in modest disagreement with some literature results


To model the inner coma plasma, we use the simplified model by Eriksson et al. (2017). Neutral gas flows outward from the comet with a velocity of about 0.7 km s−1 (Gulkis et al. 2015), with the density decreasing as r−2(1)
\begin{equation}
n_{{\rm n}} (r) \approx n_{0} \left( \frac{R}{r} \right)^{2},
\end{equation}
where n0 is the neutral gas density at the surface of a comet with radius R, and nn is the density at the cometocentric distance r. Assuming that steady ionization by solar EUV radiation dominates, this gives a peak in plasma density at some cometocentric distance, and a decrease of this density as r−1 at large distances (since the density of neutrals is typically much higher than the plasma density):
(2)
\begin{equation}
n_{{\rm e}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where ne is the electron density at some distance r, ν is an ionization frequency depending on the distance to the sun and u is the outflow velocity of the plasma (Haser 1957; Mandt et al. 2016; Eriksson et al. 2017; Vigren & Eriksson 2017; Vigren 2017). Assuming that the plasma is dominated by water ions from the comet, ion–electron pairs are created with particle energies of less than 0.1 and about 12 eV, respectively (due to conservation of momentum). When the density is high enough close to the nucleus, neutral-electron collisions will cool the electrons to temperatures of 0.1 eV or less. Beyond some distance rce the electrons are not efficiently cooled, and a mixture of cool (0.1 eV) and warm (5–10 eV) electrons are observed (Edberg et al. 2015; Mandt et al. 2016; Eriksson et al. 2017). Collisions between neutrals and ions are important out to larger distances. We investigate the region where electron-neutral collisions can be neglected, and warm electrons are present and magnetized, while the ions are unmagnetized due to collisions and their large gyroradius. This is a region favourable for lower hybrid waves and is where Rosetta spend much of its time during the 2 yr investigation of 67P (Mandt et al. 2016). We use three populations to model the plasma, cold and warm electrons, and H2O + ions
(3)
\begin{equation}
n_{{\rm ew}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{r_{{\rm ce}}}{r}\right) \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(4)
\begin{equation}
n_{{\rm ec}} (r) \approx n_{{\rm e}} (r_{{\rm ce}}) \frac{r^2_{{\rm ce}}}{r^2} = n_0 \frac{\nu R}{u} \frac{Rr_{{\rm ce}}}{r^2} \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(5)
\begin{equation}
n_{{\rm i}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where new, nec and ni are the densities of warm and cold electrons, and ions, respectively (Eriksson et al. 2017). For distances smaller than rce, the electron motion is disrupted by collisions and lower hybrid waves are likely to be heavily damped.


Assuming that steady ionization by solar EUV radiation dominates, this gives a peak in plasma density at some cometocentric distance, and a decrease of this density as r−1 at large distances (since the density of neutrals is typically much higher than the plasma density):
(2)
\begin{equation}
n_{{\rm e}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where ne is the electron density at some distance r, ν is an ionization frequency depending on the distance to the sun and u is the outflow velocity of the plasma


Beyond some distance rce the electrons are not efficiently cooled, and a mixture of cool (0.1 eV) and warm (5–10 eV) electrons are observed


We use three populations to model the plasma, cold and warm electrons, and H2O + ions
(3)
\begin{equation}
n_{{\rm ew}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{r_{{\rm ce}}}{r}\right) \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(4)
\begin{equation}
n_{{\rm ec}} (r) \approx n_{{\rm e}} (r_{{\rm ce}}) \frac{r^2_{{\rm ce}}}{r^2} = n_0 \frac{\nu R}{u} \frac{Rr_{{\rm ce}}}{r^2} \quad \quad r \ge r_{{\rm ce}},
\end{equation}
(5)
\begin{equation}
n_{{\rm i}} (r) \approx n_{0} \frac{\nu R}{u} \frac{R}{r} \left(1-\frac{R}{r} \right),
\end{equation}
where new, nec and ni are the densities of warm and cold electrons, and ions, respectively


Multiwavelength searches for the donor stars have been pursued for more than a decade. Some have observed ULXs in the optical range (e.g. Gutiérrez & López-Corredoira 2006; Ptak et al. 2006; Roberts et al. 2011; Gladstone et al. 2013; Fabrika et al. 2015). As several of the ULXs are located in or near young star clusters (e.g. Fabbiano, Zezas & Murray 2001; Roberts et al. 2002; Poutanen et al. 2013), some donor stars of ULXs might as well be RSGs (Copperwheat et al. 2005, 2007; Patruno & Zampieri 2008), which are bright in the near-infrared (NIR) band. In light of this, we started an observing campaign of ULXs in the NIR (Heida et al. 2014; López et al. 2017), tailored to detect RSG donor stars (−8  H, Ks  −11; Elias, Frogel & Humphreys 1985; Drilling & Landolt 2000). We targeted sources at distances of up to 10 Mpc, within uncertainties. Of the 97 ULXs observed by Heida et al. (2014) and López et al. (2017), we detected a NIR candidate counterpart for 33 ULXs. Of these 33 NIR sources, 19 had the absolute magnitude consistent with an RSG. So far, we performed spectroscopic follow-up observations for 7 out of these 19 candidate RSGs and confirmed the RSG nature for 3 counterparts: ULX RX J004722.4−252051 (in NGC 253; Heida et al. 2015a), ULX J022721+333500 (in NGC 925), and ULX J120922+295559 (in NGC 4136; Heida et al. 2016). Additionally, we discovered that four counterparts, initially classified as RSG based on photometry, are actually nebulae partially powered by the X-ray emission of the ULX: ULX J022727+333443 (in NGC 925), ULX J120922+295551 (in NGC 4136), ULX Ho II X-1 (in Holmberg II; Heida et al. 2016), and [SST2011] J110545.62+000016.2 (in NGC 3521; López et al. 2019). This means that, at the time of writing this manuscript, the photometric and spectroscopic observations provided 3 confirmed RSGs, 12 candidate RSGs, 4 confirmed nebulae, and 14 candidate stellar clusters/active galactic nuclei (AGNs). Recently, Lau et al. (2019) observed 96 ULXs within 10 Mpc in the mid-IR with the Spitzer Space Telescope. They detected and identified 12 ULXs with candidate counterparts whose absolute mid-IR magnitude is consistent with being RSGs.


We first focus on the properties of the PDLA obtained through the VPFit code. Together with the fitted absorption profile, both the column density and the redshift of the PDLA are derived: (8)\begin{equation} \log(N) = 20.9290 \pm 0.0024;~~~ z_{\rm abs} = 3.082097 \pm 0.000053. \label{eq:col_dens_zabs} \end{equation}log(N)=20.9290±0.0024;zabs=3.082097±0.000053.The redshift value obtained here differs by no more than 0.000163 from the value obtained based on metallic lines and is thus entirely consistent with it (see Sect. 3.2 above). We took the refractive index n of air into account that causes a shift in the observed wavelengths: λair = λvac/n. We used the Ciddor equation (Ciddor 1996), taking the atmospheric conditions at the time of the measurement into account (air temperature 12°C, pressure 744.9 hPa, and relative humidity 3%, as reported in the header of the original FITS file). The errors are the formal errors provided by VPFit and are certainly underestimated, as they do not include uncertainties linked to the continuum determination. In their pioneer study, Leibundgut & Robertson (1999) obtained a column density and redshift \begin{eqnarray*} \log(N) = 20.85 \pm 0.03;~~~ z_{\rm abs} = 3.0825. \end{eqnarray*}log(N)=20.85±0.03;zabs=3.0825.Our estimate of the column density is higher than theirs by 2.6σ (referring to their error estimate), and we find a slightly lower redshift (the difference amounts to 7.6σ according to our own error estimate). The latter discrepancy might be due to our lower spectral resolution (R ~ 1840 instead of 3300), which might cause the metallic absorption lines in the blue wing of the Lyα absorption trough to be less easily recognized and be partly included in the Voigt profile fit. This would result in a slightly overestimated column densitiy and in a slight blue shift. However, our VPFit estimate of the redshift, based on the Lyα line alone, agrees within only 0.5σ with the PDLA redshift estimate based on metallic lines.


obtained a column density and redshift \begin{eqnarray*} \log(N) = 20.85 \pm 0.03;~~~ z_{\rm abs} = 3.0825. \end{eqnarray*}log(N)=20.85±0.03;zabs=3.0825.

Our estimate of the column density is higher than theirs by 2.6σ (referring to their error estimate), and we find a slightly lower redshift (the difference amounts to 7.6σ according to our own error estimate).


We quantitatively assess the possibility of bulge formation in our sample of the 12 massive galaxies with reliable size measurements of dust continuum emission. Quiescent galaxies generally have a dense core with high stellar mass surface densities within 1 kpc of galaxy centers of

, while star-forming galaxies mostly do not (van Dokkum et al. 2014; Barro et al. 2015). For our sample, we create stellar mass maps by spatially resolved SED modeling with multiband HST data (Wuyts et al. 2012; Lang et al. 2014) to calculate stellar mass surface densities within 1 kpc from the 870  μm center. None of the members of our sample satisfy the criterion of a dense core at the current moment (Table 2). The spatial distribution of star formation within galaxies allows us to understand when the dense core is formed by subsequent star formation. Exploiting the geometric information of the best-fit exponential models at 870  μm, we derive the SFR surface densities within the central 1 kpc (ΣSFR1 kpc) from the Spitzer/Herschel-based total SFRs over galaxies. For nine galaxies with compact dust emission of

, they are intensely forming stars in the central region with ΣSFR1 kpc = 40 (19–65) M⊙ yr−1 kpc−2 (Table 2). Then, bulge formation timescales to reach

 kpc−2) = 10 are estimated by
3

taking into account mass loss due to stellar winds (w = 0.6 in Chabrier IMF; see also van Dokkum et al. 2014). The estimated bulge formation timescales are

 (8.16–8.79) for the nine galaxies with R1/2,870 μm  1. 5 kpc. They can complete the dense core formation by z = 2 when the current level of star formation is maintained for several hundred megayears. Galaxies forming stars in disks as extended as the rest-optical light would have to keep the current star formation for a longer time (∼2 Gyr). This is not consistent with stellar populations obtained in high-redshift quiescent galaxies, where timescales for star formation are τ  1 Gyr (e.g., Belli et al. 2015; Onodera et al. 2015).


For Region 1, we adopt a thick, uniform disk approximation and assume that we view this disk edge-on. In this case, the density can be obtained by dividing the central/maximum column density toward the axis by the diameter/width of the disk, i.e., the depth. The average of color excess E(H â Ks) is â¼1.5 Â± 0.7 mag for 20 sources detected toward the central part of Region 1 (within 60â³ of the symmetric axis), and  cmâ2 is roughly obtained, using the reddening law, E(H â Ks) = 0.065 Ã AV (Chini & Wargau 1998), and the standard gas-to-extinction ratio,  cmâ2 magâ1 (Dickman 1978). This estimated column density seems to be consistent with that estimated from the CO data, â¼1.3â4.1 Ã 1022 cmâ2. Adopting a width of â¼30 (â¼1.4Â pc) as the depth of Region 1, we can derive a number density of  cmâ3. Since Region 2 is located around the symmetric axis of the cloud, the column density in Region 2 is expected to be close to the central/maximum column density if the local density is uniform. The average of color excess E(H â Ks) is â¼1.1 Â± 0.4 mag for 41 sources toward Region 2, and  cmâ2 is roughly obtained. This column density also seems to be consistent with that estimated from the CO data, â² 2.1 Ã 1022 cmâ2. Adopting the width of â¼80 (â¼3.5Â pc) as the depth of Region 2,  cmâ3 can be obtained in Region 2. The estimated number density in Region 1 is â¼3â4Â times higher than that in Region 2, supporting the idea that Region 1 was strongly compressed by UV and then became higher in density. This result does not seem to match the simulation results for the nonmagnetized clouds that the number density in the tip is expected to be higher by more than one order of magnitude than that of the area away from the tip (e.g., Figures 3, 7, and 10 of Miao etÂ al. 2009). However, the simulation taking into account the magnetic pressure predicts a smaller difference of a factor of 10 (e.g., Figure 8 of Miao etÂ al. 2006), which is more consistent with our observations. Motoyama etÂ al. (2013) also suggests flatter column density profiles in the magnetized cases than in the nonmagnetized ones.


In the interpretation of clumpy star formation in a disk-like galaxy, as usually observed in high-z galaxies (e.g., Elmegreen et al. 2009; Förster Schreiber et al. 2011; Murata et al. 2014; Tadaki et al. 2014), the ellipticity of a source may be an intrinsic property related to the viewing angle of the disk. That is, large ellipticity implies that its viewing angle is close to edge-on and that the stellar disk lies in the elongated direction. If we consider that Lyα is emitted in directions perpendicular to the disk, as predicted by the recent theoretical studies for Lyα line transfer (e.g., Verhamme et al. 2012; Yajima et al. 2012b), the pitch angle of Lyα emission will be at right angles to that of UV continuum. Moreover, in such a case, it is also expected that the size in Lyα emission aHL(NB711) shows a positive correlation with ellipticity measured in rest-frame UV continuum because the Lyα-emitting region in bipolar directions perpendicular to the disk can be viewed from longer distances if the viewing angle of the disk is closer to edge-on, that is, larger ellipticity. However, as presented in the top panel of Figure 17, we do not find such positive correlation between aHL(NB711) and ellipticity for the 54 ACS-detected LAEs. Furthermore, as shown in the bottom panel of Figure 17,14 the observed distribution of the LAEs in the EW0– plane seems not to be quantitatively consistent with the interpretation of clumpy star formation in a disk-like galaxy, where EW0 is expected to decrease significantly toward the edge-on direction (i.e., larger ellipticity) via radiative transfer effects for Lyα resonance photons (e.g., Verhamme et al. 2012; Yajima et al. 2012b); this result is consistent with Shibuya et al. (2014). Therefore, the interpretation of clumpy star formation in a disk-like galaxy seems not to be preferred for our LAE sample. This conclusion can be reinforced with the absence of an ACS source with a large size and round shape; if there are multiple clumpy star-forming regions in a disk-like galaxy, some such galaxies will be viewed face-on, resulting in alarge size and round shape. We emphasize again that this result is not affected by a selection bias against them if they are bright enough (i.e., I814 ≲ 26 mag), as shown in Figures 7 and 18.
14
Note that, considering the strong positive correlation between aHL and  shown in Figure 8, this plot is qualitatively identical to the distribution in the EW0–aHL plane shown in the top panel of Figure 16.

Verhamme et al. 2012
Verhamme et al. 2012


In the interpretation of clumpy star formation in a disk-like galaxy, as usually observed in high-z galaxies (e.g.,

the ellipticity of a source may be an intrinsic property related to the viewing angle of the disk.


On the basis of the absolute PM estimate derived in the previous section, we performed a numerical integration of the orbit of Terzan 5 in the Galactic potential. We used the three-component (bulge, disk, and halo) axisymmetric model from Allen & Santillan (1991), which has been extensively used and discussed in the literature to study orbits and dynamical environmental effects on Galactic stellar systems (e.g., Allen et al. 2006; Montuori et al. 2007; Ortolani et al. 2011; Moreno et al. 2014; Zonoozi et al. 2014), thanks to its relative simplicity and fully analytic nature. We adjusted the various model parameters to make the rotation velocity curve match the value of 243 km s−1 measured at the Solar Galactocentric distance of 8.4 kpc (see above). The cluster PMs were reported in the Cartesian Galactocentric reference frame, resulting in a velocity vector (vx, vy, vz) = (−60.4 ± 1.3, 85.7 ± 11.1, 35.0 ± 6.0) km s−1 at the position (x, y, z) = (−2.51 ± 0.30, 0.39 ± 0.02, 0.17 ± 0.01) kpc. We adopted the convention in which the X axis points opposite to the Sun (i.e., the Sun position is (−8.4,0,0)). The orbit was then time-integrated backwards for 12 Gyr, starting from the given initial (current) conditions and using a second-order leapfrog integrator (e.g., Hockney & Eastwood 1988) with a rather small and constant time step (∼100 kyr, corresponding to

 of the dynamical time at the Sun distance, computed as

, where

 is a characteristic Galactic mass parameter). During the >105 time steps used to describe the entire orbit evolution, the errors on the conservation of both the energy and the Z component of angular momentum were kept under control and never exceeded one part over 105 and 1013, respectively. To take into account the uncertainties on the kinematic data, we generated a set of 1,000 orbits starting from the phase-space initial conditions normally distributed within a 3σ range around the cluster velocity vector components and the current position, with σ being equal to the quoted uncertainties on these parameters. For all of these orbits we repeated the backward time integration. The probability densities of the resulting orbits projected on the equatorial and meridional Galactic planes are shown in Figures 13 and 14, respectively. Darker colors correspond to more probable regions of the space, that is, to Galactic coordinates crossed more frequently by the simulated orbits. As is apparent, the larger distances reached by the system during its evolution are R = 3.5 kpc and

 kpc at a 3σ level of significance, which roughly correspond to a region having the current size of the bulge.


A scenario that could potentially support superdiffusive parallel transport for energetic ions interacting with SMFRs in the large-scale solar wind is when (1) these structures can be viewed as coherent, quasi-helical magnetic field structures with a typical width L

I
 of intermediate size and sharp boundaries (Greco et al. 2009), and (2) energetic particles have gyroradii r

g
 that are small compared to the typical SMFR width L

I
. There is observational support at 1 au for all these assumptions (e.g., Greco et al. 2009; Cartwright & Moldwin 2010; le Roux et al. 2015). Thus, it is reasonable to envisage that the particle guiding centers follow the helical magnetic field lines (negligible cross-field drifts) to propagate through SMFR structures in a coherent fashion for a considerable distance before scattering during exit of the structure through the narrow magnetic boundaries. To be more specific, SMFRs identified at 1 au typically have widths L

I
 ≈ 0.01 − 0.001 au (Cartwright & Moldwin 2010; Khabarova et al. 2015), while data analysis of low-frequency 2D turbulence suggests elongated SMFR structures with an average length L

I∣∣ so that L

I∣∣/L

I
 ≈ 2 − 3 (Weygand et al. 2011). Based on the expression for the path length s

I
 of a SMFR helical field line (Equation (9)), we estimate that the distance a particle guiding center travel through a SMFR structure by following magnetic field lines can be considerable because s

I
 ≈ 0.004 − 0.04 au. Furthermore, the protons accelerated by SMFRs reach kinetic energies of ≲5 MeV (Khabarova & Zank 2017) in the solar wind at 1 au, thus easily fulfilling the requirement r

g
 ≪ L

I
. Consequently, the transport through SMFRs can be characterized as involving an enhanced probability of relatively long transport distances or step sizes along the background/guide field perpendicular to the shock surface (in the z-direction) between scattering events indicating Lévy walks as discussed by Zimbardo & Perri (2020) and Zimbardo et al. (2021).


To be more specific, SMFRs identified at 1 au typically have widths L

I
 ≈ 0.01 − 0.001 au


In recent years, there has been an increasing interest in identifying embedded filaments at an early stage of fragmentation, where star formation activities have not yet started. Kainulainen et al. (2016) studied the Musca molecular cloud using the NIR, 870 μm dust continuum and molecular line data, and suggested that the Musca cloud is a very promising candidate for a filament at an early stage of fragmentation and shows very few signs of ongoing star formation. However, the identification of such filaments is still limited in the literature (e.g., Kainulainen et al. 2016). In the I05463+2652 site, we have selected two elongated filamentary features (“s-fl” and “nw-fl”). The Herschel temperature map reveals these filaments with a temperature of ∼11 K. It has been suggested that the thermally supercritical filaments (i.e.,

M

line

>

M

line
,
crit

) can be unstable to radial collapse and fragmentation (André et al. 2010). However, thermally subcritical filaments (i.e.,

M

line

M

line
,
crit

) may lack prestellar clumps/cores and embedded protostars (André et al. 2010). In the I05463+2652 site, the observed masses per unit length of the filaments are computed to be

∼
60

M

⊙

 pc−1 (for filament “s-fl”) and

∼
200

M

⊙

 pc−1 (for filament “nw-fl”), which are much higher than the critical mass per unit length at T = 10 K (i.e.,

M

line
,
crit

=
16

M

⊙

 pc−1; see Section 3.2). For the purpose of comparison, we also provide the observed masses per unit length of some well known filaments (such as DR21 (

M

line

∼
4500

M

⊙

 pc−1), Serpens South (

M

line

∼
290

M

⊙

 pc−1), Taurus B211/B213 (

M

line

∼
50

M

⊙

 pc−1), and Musca (

M

line

∼
20

M

⊙

 pc−1)) (see Table 1 in André et al. 2016). Based on the analysis of masses per unit length, the elongated filaments (“s-fl” and “nw-fl”) are thermally supercritical (see Section 3.2). Furthermore, these two filaments contain clumps (or fragments), indicating the signature of gravitational fragmentation. Arzoumanian et al. (2013) pointed out that supercritical filaments may undergo gravitational contraction and increase in mass per unit length through the accretion of background material.


It has been suggested that the thermally supercritical filaments (i.e.,

M

line

>

M

line
,
crit

) can be unstable to radial collapse and fragmentation


One of the theoretical models to account for different depletion times is the turbulence model of Krumholz & McKee (2005). According to this model, turbulence will determine the density PDF of the clouds. The most important parameter is the SFE per free-fall time, which is the ratio between the free-fall time and the depletion time. The general equation for free-fall time is
(15)$$\begin{eqnarray*}
t_{\text{ff}}=\sqrt{\frac{3\pi }{32G\rho _{\mathrm{mol, mid}}}},
\end{eqnarray*}$$where ρmol, mid is the volume density of the molecular gas in the middle of the disc. If we assume the galaxy is filled with gas, then the volume density can be calculated as
(16)$$\begin{eqnarray*}
\rho _{\mathrm{mol, mid}}=\frac{\Sigma _{\text{mol}}}{2 H_{\mathrm{mol}}} ,
\end{eqnarray*}$$where Hmol is the scale height of the disc and Σmol is the surface density of the molecular gas measured from the 12CO J = 2–1 cube (see Section 3.2). Assuming the gas disc is in equilibrium and that vertical gravity is dominated by gas self-gravity, then we can calculate the scale height as (Wilson et al. 2019)
(17)$$\begin{eqnarray*}
H_{\mathrm{mol}}= 0.5\frac{\sigma _v^2}{\pi G \Sigma _{\text{mol}}} ,
\end{eqnarray*}$$where σv is the velocity dispersion of the molecular gas. Combining all the equations above, we can write the free-fall time as
(18)$$\begin{eqnarray*}
t_{\text{ff}}=\frac{\sqrt{3}}{4G} \frac{\sigma _v}{\Sigma _{\text{mol}}} .
\end{eqnarray*}$$Therefore, the SFE per free-fall time can be calculated as (Wilson et al. 2019)
(19)$$\begin{eqnarray*}
\epsilon _{\text{ff}}=\frac{t_{\text{ff}}}{t_{\text{dep}}}=\frac{\sqrt{3}}{4G}\frac{\sigma _v \Sigma _{\text{SFR}}}{\Sigma ^2_{\mathrm{mol}}} .
\end{eqnarray*}$$Assuming a constant SFE per free-fall time, the depletion time will decrease as the surface density of the gas increases. To calculate ϵff, we need maps of Σmol, ΣSFR, and σv. We plot ϵff versus Σmol in Fig. 11 (left plot). As we can see, there is a clear decreasing trend for ϵff versus Σmol for Arp 240, while ϵff stays relatively constant for the other U/LIRGs. Furthermore, in the low surface density regions, ϵff can be above 1.0. For normal clouds, ϵff can be as high as several 10 per cent (Lee et al. 2016). Therefore, our method probably overestimates the true efficiency in these galaxies.


Therefore, the SFE per free-fall time can be calculated as

(19)$$\begin{eqnarray*}
\epsilon _{\text{ff}}=\frac{t_{\text{ff}}}{t_{\text{dep}}}=\frac{\sqrt{3}}{4G}\frac{\sigma _v \Sigma _{\text{SFR}}}{\Sigma ^2_{\mathrm{mol}}} .
\end{eqnarray*}$$


Since the HDE model is obtained by choosing the event horizon length-scale, an obvious drawback concerning causality appears in this scenario. Recently, a new DE model, dubbed agegraphic dark energy (ADE) model, has been suggested by Cai (2007) in order to alleviate the above problem. In particular, combining the uncertainty principle in quantum mechanics and the gravitational effects of GR Karolyhazy and his collaborators (Karolyhazy 1966; Karolyhazy & Lukacs 1982, 1986) made an interesting observation concerning the distance measurement for the Minkowski space–time through a light-clock Gedanken experiment (see also Maziashvili 2007a). They found that the distance t in Minkowski space–time cannot be known to a better accuracy than $\delta t=\beta t_{\rm p}^{2/3}t^{1/3}$, where β is a dimensionless constant of the order of ${\cal O}(1)$ (see also Maziashvili 2007a).1 Based on the Karolyhazy relation, Maziashvili (2007a) argued that the energy density of metric fluctuations in the Minkowski space–time is written as (see also Maziashvili 2007b)

(2)

\begin{equation}
\rho _{{\rm d}}\sim \frac{1}{t_{{\rm p}}^{2}t^{2}}\sim \frac{m_{{\rm p}}}{t^{2}},
\end{equation}

where mp and tp are the reduced Plank mass and the Plank time, respectively (see also Ng & Van Dam 1994, 1995; Sasakura 1999; Krauss & Turner 2004; Christiansen, Ng & van Dam 2006; Arzano, Kephart & Ng 2007; Ng 2007). Using equation (2), Cai (2007) proposed another version of holographic DE, the so-called ADE, in which the time-scale t is chosen to be equal with the age of the universe $T= \int ^{t}_0{\rm d}t=\int _{0}^{a}\frac{{\rm d}a}{aH}$, with a the scalefactor of the universe and H the Hubble parameter. Therefore, the ADE energy density is given by (Cai 2007)

(3)

\begin{equation}
\rho _{{\rm d}}=\frac{3n^{2}m_{{\rm p}}^{2}}{T^{2}},
\end{equation}

where n is a free parameter and the coefficient 3 appears for convenience. The present value of the age of universe ($T_{\rm 0}\sim H_{\rm 0}^{-1}$) implies that n is of the order of ${\cal O}(1)$. It has been shown that the condition n > 1 is required in order to have cosmic acceleration (Cai 2007). Although, the ADE scenario does not suffer from the causality problem (Cai 2007), it faces some problems towards describing the matter-dominated epoch (Neupane 2007; Wei & Cai 2008a,b). To overcome this issue, Wei & Cai (2008a) proposed a new agegraphic dark energy (NADE) model, in which the cosmic time t is replaced by the conformal time $\eta = \int ^{t}_0{\frac{{\rm d}t}{a(t)}}=\int ^{a}_0{\frac{{\rm d}a}{a^{2}H}}$ and thus the energy density in this case becomes (Wei & Cai 2008a)

(4)

\begin{equation}
\rho _{{\rm d}}=\frac{3n^{2}m_{{\rm p}}^{2}}{\eta ^{2}}.
\end{equation}

It is interesting to mention that Kim, Lee & Myung (2008b) showed that the NADE model provides the proper matter-dominated and radiation-dominated epochs, in the case of n > 2.68 and n > 2.51, respectively. Also, Wei & Cai (2008b) found that the coincidence problem can be alleviated naturally in this model and using the cosmological data (SNIa, CMB, etc.), they obtained $n=2.716^{+0.111}_{-0.109}$. We would like to point out that the cosmological properties of the ADE and the NADE models can be found in Kim et al. (2008a), Setare & Jamil (2011), Karami et al. (2011), Sheykhi & Setare (2010), Sheykhi (2009), Sheykhi (2010a), Sheykhi (2010b), Sheykhi (2010c), Lee, Kim & Myung (2008), Jawad, Chattopadhyay & Pasqua (2013), Liu et al. (2012), Zhang, Li & Zhang (2013), Farajollahi, Ravanpak & Fadakar (2012), Zhai, Zhang & Liu (2011), Chen et al. (2011), Sun & Yue (2011), Lemets, Yerokhin & Zazunov (2011), Zhang, Zhang & Zhang (2010), Liu, Zhang & Zhang (2010), Karami & Khaledian (2011), Malekjani & Khodam-Mohammadi (2010), Jamil & Saridakis (2010), Karami et al. (2010), Sheykhi (2010a), Sheykhi (2010b), Sheykhi (2009), Wu, Ma & Ling (2008), Zhang, Zhang & Liu (2008), Kim et al. (2008b), Neupane (2009) and Wei & Cai (2008b).


proposed another version of holographic DE, the so-called ADE, in which the time-scale t is chosen to be equal with the age of the universe $T= \int ^{t}_0{\rm d}t=\int _{0}^{a}\frac{{\rm d}a}{aH}$, with a the scalefactor of the universe and H the Hubble parameter.


Therefore, the ADE energy density is given by

(3)

\begin{equation}
\rho _{{\rm d}}=\frac{3n^{2}m_{{\rm p}}^{2}}{T^{2}},
\end{equation}

where n is a free parameter and the coefficient 3 appears for convenience.


Although, the ADE scenario does not suffer from the causality problem

it faces some problems towards describing the matter-dominated epoch


As mentioned in Sections 2, six sources (G035.19–00.74, G049.48–00.36, G133.94+01.06, G031.28+00.06, G043.89–00.78 and G059.78+00.06) were observed by both the ARO 12 m and IRAM 30 m telescopes, which can be used to check beam size effects potentially influencing our abundance ratios. A comparison between the spectra from the ARO 12 m and the IRAM 30 m is shown in Figure 3. Measured line parameters are given in Tables 1 and 2. If the scale of our targets is smaller than the beam size, filling factors are less than unity. So, measured line intensities are diluted. In this case, the intensity of detected lines has to be adjusted for the beam dilution effect

/(

 +

) (ηBD), where θs and θbeam are source size and beam size, respectively (Zhang et al. 2017). Thus, the brightness temperatures (TB) of the sources can be derived from the main beam brightness temperature (Tmb) dilution:
5

There are two Tmb values, measured by the IRAM 30 m and the ARO 12 m with beam sizes of ∼25″ and ∼63″. Therefore, the source size can be estimated using Equation (5). We employed CS, C34S, 13CS, and C33S line data to estimate the size of those six sources, and the results are listed in Table 4. Source sizes turn out to be in the range 24″–126″ and are quite different between estimations from different isotopes. To the contrary, the sulfur isotope abundance ratios do not show significant differences between the ARO 12 m and the IRAM 30 m measurements. Within the error ranges, the average abundance ratios determined from the ARO 12 m (17.6 ± 2.7 and 4.9 ± 0.7 for 32S/34S and 34S/33S, respectively) are consistent with those from the IRAM 30 m observations (18.1 ± 4.1 and 5.5 ± 0.8; the errors given in this subsection represent standard deviations for individual sources). In addition, two sources (Sgr B2, G111.54+00.77 (NGC 7538)) among our sample were also observed by the 7 m antenna of the Crawford Hill Observatory (Frerking et al. 1980). The I(13CS)/I(C34S) line intensity ratios measured by Frerking et al. (1980) yield 0.62 ± 0.04 and 0.43 ± 0.3 with a beam size of ∼22, which is also consistent with our results from the ARO 12 m (0.60 ± 0.03 and 0.38 ± 0.04). Thus, our measurements of sulfur isotope ratios are not seriously affected by beam size effects.


For all galaxies the extended profiles of $\Sigma _\mathrm{H\,\small {I}}$ obtained from observations at the WSRT telescope were found in Yim & van der Hulst (2016) and Noordermeer et al. (2005). The surface density of $\Sigma _\mathrm{H_2}$ is usually calculated from observations of CO molecule lines, but the exact value of the conversion factor XCO remains debatable. If CO intensity profile is available, we follow the procedure described in Yim & van der Hulst (2016) to obtain $\Sigma _\mathrm{H_2}$. For NGC 4258, NGC 4725 (Yim & van der Hulst 2016), and NGC 2985 (Young et al. 1995), the CO(J = 1 → 0) line was used to calculate $\Sigma _\mathrm{H_2}$:

\begin{equation*}
\Sigma _\mathrm{H_2} (\text{M}_{\odot }\,\mathrm{pc}^{-2}) = 3.2\times I_\mathrm{CO}(\mathrm{K\,km\,s^{-1}})\,.
\end{equation*}
However, for three of four remaining galaxies only the total mass of molecular hydrogen $M_\mathrm{H_2}$ was found. Regan et al. (2001) make a conclusion based on the BIMA survey data that $\Sigma _\mathrm{H_2}(R)$ profile decreases exponentially with a scale equal to the scale length of the disc h on the average. Therefore for these remaining galaxies we found central surface densities from the formula of the total mass of the molecular disc $\displaystyle \Sigma _\mathrm{H_2}(0) = \frac{M_\mathrm{H_2}}{2 {\pi} \, h^2}$ and then extended them exponentially with the appropriate h scale length. If more than one photometry is used for an examined galaxy this approach gives several possible profiles of $\Sigma _\mathrm{H_2}(R)$ and several estimates of Q, respectively both for one-fluid and two-component cases. We approximate the $\Sigma _\mathrm{H_2}(R)$ and $\Sigma _\mathrm{H\, \small {I}}(R)$ profiles using linear interpolation between points. The total gas surface density was corrected for the presence of helium and other heavy elements using $\Sigma _\mathrm{g} = 1.36\,(\Sigma _\mathrm{H\,\small {I}} + \Sigma _\mathrm{H_2})$ formula (see for example Leroy et al. 2008).