The evolution of cold neutral gas and the star formation history

TL;DR

This paper investigates how the fraction of cold neutral gas in the universe evolves at high redshift and its relation to star formation history, using 21-cm absorption data to extend previous studies beyond redshift 2.

Contribution

It provides new insights into the evolution of cold neutral gas at z > 2 using archival 21-cm absorption data, revealing a steeper decline than star formation rate density.

Findings

Cold gas fraction decreases steeply at z ~ 3

Mean spin temperature is around 3000 K at high redshift

Temperatures are consistent with high neutral hydrogen column densities

Abstract

There is a well known disparity between the evolution the star formation rate density, {\psi}*, and the abundance of neutral hydrogen (HI), the raw material for star formation. Recently, however, we have shown that {\psi}* may be correlated with the fraction of cool atomic gas, as traced through the 21-cm absorption of HI. This is expected since star formation requires cold (T ~ 10 K) gas and so this could address the issue of why the star formation rate density does not trace the bulk atomic gas. The data are, however, limited to redshifts of z < 2, where both {\psi}* and the cold gas fraction exhibit a similar steep climb from the present day (z = 0), and so it is unknown whether the cold gas fraction follows the same decline as {\psi}* at higher redshift. In order to address this, we have used unpublished archival observations of 21-cm absorption in high redshift damped Lyman-{\alpha} absorption systems to increase the sample at z > 2. The data suggest that the cold gas fraction does exhibit a decrease, although this is significantly steeper than {\psi}* at z ~ 3. This is, however, degenerate with the extents of the absorbing galaxy and the background continuum emission and upon removing these, via canonical evolution models, we find the mean spin temperature of the gas to be <T> ~ 3000 K, compared to the ~2000 K expected from the fit at z < 2. These temperatures are consistent with the observed high neutral hydrogen column densities, which require T < 4000 K in order for the gas not to be highly ionised.

Tables

Table 1. Table 1: The unpublished archival data not completely corrupted by RFI. The name of the background continuum source is followed by its redshift. The redshift and neutral hydrogen column density of the absorber are then given, followed by the observed frequency, GMRT project number, the date of the observation and the total observing time, t obs subscript 𝑡 obs t_{\rm obs} . Δ ​ S Δ 𝑆 \Delta S is the rms noise reached per 10 km s − 1 superscript s 1 {\rm s}^{-1}  channel, S obs subscript 𝑆 obs S_{\rm obs} is the observed flux density and τ 3 ​ σ subscript 𝜏 3 𝜎 \tau_{3\sigma} the derived optical depth limit, where τ 3 ​ σ = − ln ⁡ ( 1 − 3 ​ Δ ​ S / S meas ) subscript 𝜏 3 𝜎 1 3 Δ 𝑆 subscript 𝑆 meas \tau_{3\sigma}=-\ln(1-3\Delta S/S_{\rm meas}) is quoted for these non-detections.

QSO	$z_{\mathrm{QSO}}$	$z_{\mathrm{abs}}$	$N_{\text{H}\text{ i}}$ [${\text{cm}}^{-2}$]	${\nu }_{\mathrm{obs}}$ [MHz]	Proj.	Date	$t_{\mathrm{obs}}$ [h]	$\Delta S$ [mJy]	$S_{\mathrm{obs}}$ [Jy]	${\tau }_{3\sigma }$
SDSS J003843.98+031120.8	3.674	3.582	21.60	310.00	24_025	1/9/2013	6.17	20.1	0.338
…	…	3.263	20.29	333.19	24_025	9/8/2013	4.41	17.6	0.345
SDSS J021435.77+015702.8	3.285	2.488	20.88	407.23	30_068	2/9/2016	1.53	21.8	0.125
SDSS J021437.02+063251.3	2.311	2.107	20.80	457.16	30_068	25/8/2016	4.07	8.87	0.058
2MASS J04071807–4410141	3.020	1.9130	20.60	487.61	30_068	2/9/2016	1.97	20.1	0.134
[HB89] 0528–250	2.813	2.1404	21.00	452.30	30_068	5/9/2016	2.00	8.53	0.479
B2 0931+31	2.895	2.390	—	419.00	30_068	1/10/2016	2.31	114	1.050
PKS 1251–407	4.460	3.752	20.30	298.91	24_055	16/9/2013	3.98	33.4	0.454
SDSS J152219.67+211957.3	3.225	3.103	20.55	346.19	24_055	14/5/2013	6.59	129	2.100
SDSS J164208.62+184859.4	3.333	3.150	20.68	342.27	24_055	12/7/2013	3.52	30.7	0.299
…	…	3.223	20.51	336.35	24_055	13/7/2013	6.25	22.5	0.252
WISE J164558.54+633010.8	2.379	2.1253	20.55	454.49	30_068	25/8/2016	1.17	25.7	0.210

Equations

$$N_{\mathrm{HI}}=1.823\times 10^{18}{T}_{\mathrm{spin}}\int \tau dv,$$

$$\tau \equiv -\ln \left(1-\frac{{\tau }_{\mathrm{obs}}}{f}\right)\approx \frac{{\tau }_{\mathrm{obs}}}{f},\mathrm{for}{\tau }_{\mathrm{obs}}\equiv \frac{\Delta S}{S_{\mathrm{obs}}}\stackrel{<}{{}_{\sim }}0.3,$$

$$N_{\text{Hsci}}\approx 1.823\times 10^{18}\frac{T_{\mathrm{spin}}}{f}\int {\tau }_{\mathrm{obs}}dv.$$

$$f=\left\{\begin{array}[]{l l}\left(\frac{d_{\rm abs}DA_{\rm QSO}}{d_{\rm QSO}DA_{\rm abs}}\right)^{2}&\text{ if }\theta_{\rm abs}<\theta_{\rm QSO}\\ 1&\text{ if }\theta_{\rm abs}\geq\theta_{\rm QSO},\\ \end{array}\right.$$

$$DA=\frac{DC}{z+1},\mathrm{where}$$

$$DC=\frac{c}{H_0}{\int }_0^z\frac{dz}{\sqrt{{\Omega }_{\mathrm{m}}(z+1{)}^3+(1-{\Omega }_{\mathrm{m}}-{\Omega }_{\Lambda })(z+1{)}^2+{\Omega }_{\Lambda }}}$$

$$\frac{1}{T_{\mathrm{spin}}}{\left(\frac{d_{\mathrm{abs}}}{d_{\mathrm{QSO}}}\right)}^2=\frac{1.823\times 10^{18}}{N_{\text{Hsci}}}{\left(\frac{D{A}_{\mathrm{abs}}}{D{A}_{\mathrm{QSO}}}\right)}^2\int {\tau }_{\mathrm{obs}}dv$$

$$=\frac{1.823\times 10^{18}}{N_{\text{Hsci}}}{\left(\frac{D{A}_{\mathrm{abs}}}{D{A}_{\mathrm{QSO}}}\right)}^2\frac{\Delta S}{S_{\mathrm{obs}}}\Delta v,$$

$$\left<T_{\rm spin}\right>=\left\{\begin{array}[]{l l}\frac{N_{\text{H{\sc\,i}}}}{1.823\times 10^{18}}\left(\frac{d_{\rm abs}DA_{\rm QSO}}{d_{\rm QSO}DA_{\rm abs}}\right)^{2}\frac{S_{\rm obs}}{\Delta S\Delta v}&\text{ if }\theta_{\rm abs}<\theta_{\rm QSO}\\ &\\ \frac{N_{\text{H{\sc\,i}}}}{1.823\times 10^{18}}\frac{S_{\rm obs}}{\Delta S\Delta v}&\text{ if }\theta_{\rm abs}\geq\theta_{\rm QSO},\\ \end{array}\right.$$

$$\frac{x^2}{1-x}=\frac{1}{n}{\left(\frac{2\pi m_{\mathrm{e}}kT}{h^2}\right)}^{3/2}{e}^{-\varphi /kT},\text{ where }x\equiv \frac{{\mathcal{N}}_{\mathrm{II}}}{{\mathcal{N}}_{\mathrm{I}}+{\mathcal{N}}_{\mathrm{II}}},$$

Full text

††pagerange: The evolution of cold neutral gas and the star formation history–References††pubyear: 2019

1 Introduction

Galaxies intervening the sight-lines to more distant Quasi-Stellar Objects (QSOs) allow study of the neutral gas in the distant Universe, through the absorption of the background continuum radiation by the Lyman-$\alpha$ ($\lambda=1215.67$ Å) transition of hydrogen. These so-called damped Lyman-$\alpha$ absorbers (DLAs), where the neutral hydrogen column density exceeds $N_{\rm HI}\geq 2\times 10^{20}$ atoms $\hbox{{\rm cm}}^{-2}$, may contain up to 80% of the neutral gas mass density in the Universe (Prochaska et al., 2005). The detection of this transition is restricted to redshifts of $z\stackrel{{\scriptstyle>}}{{{}_{\sim}}}1.7$ (the first four billion years of the Universe’s history) by ground-bases telescopes, where this ultra-violet band transition is redshifted into the atmospheric window at visible wavelengths. Apart from space-based observations (e.g. Rao et al. 2017), hydrogen at lower redshifts can be detected through the spin-flip ($\lambda=21.1$ cm) transition, which occurs in the radio band.

Observations of both H i 21-cm emission (Zwaan et al., 2005b; Lah et al., 2007; Braun, 2012; Delhaize et al., 2013; Rhee et al., 2013; Hoppmann et al., 2015) and Lyman-$\alpha$ absorption (Hopkins & Beacom, 2006; Burgarella et al., 2013; Behroozi et al., 2013; Sobral et al., 2013; Madau & Dickinson, 2014; Zwart et al., 2014) show that there is little evolution in the mass density of neutral hydrogen, which is in stark contrast to the steep evolution in the cosmic star formation rate (Fig. 1).

However, both Lyman-$\alpha$ absorption and 21-cm emission trace all of the neutral gas ($T\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10\,000$ K), whereas only the gas clouds which are cool enough ($T\sim 10$ K) to collapse under their own gravity can initiate star formation. This cool component of the gas is detected through the absorption of the 21-cm radiation from a background radio-loud QSO (quasar). If the Lyman-$\alpha$ and 21-cm absorption trace the same sight-line, comparison of the 21-cm absorption strength with the total neutral hydrogen column density can, in principle, yield the spin temperature of the gas, $T_{\rm spin}$, although this is degenerate with the fraction of the background radio flux intercepted by the absorbing gas.

Although this covering factor is difficult to quantify, by accounting for the angular diameter distances to the absorbing galaxy and background continuum source, we can at least remove any bias introduced by an expanding Universe from the flux coverage. This yields the spin temperature degenerate with the ratio of the projected sizes of the absorber and the continuum source, $T_{\rm spin}(d_{\rm QSO}/d_{\rm abs})^{2}$. Using this method, we (Curran, 2017a, b) have shown that the reciprocal of this, $(1/T_{\rm spin})(d_{\rm abs}/d_{\rm QSO})^{2}$, increases by a similar amount as the star formation rate from $z_{\rm abs}=0$ to the peak of star formation at $z_{\rm abs}\sim 2$. That is, for a non-evolving $d_{\rm abs}/d_{\rm QSO}$ ratio, the spin temperature is anti-correlated with $\psi_{*}$, as would be expected upon the basis that stars can only form out of the coldest gas. Being limited to $z_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2$, however, means that we do not know whether $(1/T_{\rm spin})(d_{\rm abs}/d_{\rm QSO})^{2}$ exhibits a downturn at higher redshift, thus truly tracing the star formation history, rather than a coincidental similar factor of increase over $0\stackrel{{\scriptstyle<}}{{{}_{\sim}}}z_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2$. In order to address this, here we add a dozen previously unpublished high redshift searches for 21-cm absorption in DLAs from the data archive of the Giant Metrewave Radio Telescope (GMRT).

2 Data acquisition and reduction

The GMRT is the longest serving large interferometer capable of observing at the required frequencies ($\stackrel{{\scriptstyle<}}{{{}_{\sim}}}470$ MHz for 21-cm at $z\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$), thus having the most comprehensive archive, in addition to being able to reach the required sensitivities (Curran, 2018). Radio frequency interference (RFI) meant that many of the sources in the archive could not be reduced and some flagging of badly affected data were required on the dozen remaining sight-lines (Table 1). These had all been observed using the full 30 antenna array, with 3C 48, 3C 147 and 3C 298 used for bandpass calibration and a nearby bright, unresolved radio source for phase calibration. Upon downloading the data, these were calibrated and flagged with the miriad interferometry reduction package.

For each of these sources, after calibration, the two polarisations were averaged and a cube produced, from which a spectrum was extracted (Fig. 2).

The addition of these absorbers, takes the total number to 85, cf. the 74 previously (Curran, 2017a).111From 11 additional sight-lines. No value for the column density could be found for the $z_{\rm abs}=2.390$ absorber towards B2 0931+31. All of the new absorbers have redshifts of $z_{\rm abs}\geq 1.913$, where there are currently 36 DLAs with published searches for 21-cm absorption.222These have been compiled from Davis & May (1978); Brown & Spencer (1979); Briggs & Wolfe (1983); Chengalur & Kanekar (2000); Kanekar et al. (2001a, b, 2009b, 2013, 2014); Briggs et al. (2001); Kanekar & Chengalur (2001, 2003); Curran et al. (2005, 2007, 2010); York et al. (2007); Gupta et al. (2009a, b); Ellison et al. (2012); Srianand et al. (2012); Roy et al. (2013); Kanekar (2014).

3 Analysis

3.1 Spin temperature–covering factor degeneracy

The total neutral atomic hydrogen column density, $N_{\rm HI}$ [$\hbox{{\rm cm}}^{-2}$], is related to the velocity integrated optical depth of the H i 21-cm absorption [km ${\rm s}^{-1}$] via (Wolfe & Burbidge, 1975)

[TABLE]

where $T_{\rm spin}$ [K] is the harmonic mean spin temperature – the density weighted average of the spin temperature of the absorbing gas along the sight-line. This is a measure of the population of the lower hyperfine level ($F=1$), where the gas can absorb 21-cm photons (Purcell & Field, 1956), relative to the upper hyperfine level ($F=2$). However, we cannot measure $\int\!\tau\,dv$ directly, since the observed optical depth, which is the ratio of the line depth, $\Delta S$, to the observed background flux, $S_{\rm obs}$, is related to the intrinsic optical depth via

[TABLE]

where the covering factor, $f$, is the fraction of $S_{\rm obs}$ intercepted by the absorber. In the optically thin regime (where $\tau_{\rm obs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}0.3$)333This applies to all of the DLAs detected in 21-cm absorption, where the observed optical depths span $\tau_{\rm obs}=0.05-0.26$., Equ. 1 can be approximated as

[TABLE]

That is, comparison of the 21-cm line strength with the total column density yields the spin temperature degenerate with the covering factor ($T_{\rm spin}/f$).

In order to obtain the physically interesting spin temperature the covering factor must be known. This requires knowledge of the absorbing cross-section, the extent of the background continuum, as well as the alignment between the absorber and continuum source (see Curran 2017a) and so is generally unknown. Estimates of the covering factor assume that this is equal to the ratio of the compact unresolved component’s flux to the total radio flux (e.g. Briggs & Wolfe 1983; Kanekar et al. 2014). However, this gives no information on the depth of the absorption when the extended continuum emission is resolved out nor can it yield information on the absorber and how effectively it covers the emission. This requires high resolution, highly sensitive observations of the absorption across the emission region (e.g. Lane et al. 2000) at low frequencies, thus requiring the Square Kilometre Array (SKA): Phase-1 will be an order of magnitude more sensitive than the GMRT at $z\stackrel{{\scriptstyle>}}{{{}_{\sim}}}3$, with phase-2 increasing the sensitivity by another order of magnitude (Curran, 2018).

Until such high resolution imaging of the absorption becomes available at the required frequencies, we can use a statistical approach: In the small angle approximation, the covering factor can be obtained from

[TABLE]

(Curran, 2012), where the angular diameter distance to a source is

[TABLE]

[TABLE]

is the line-of-sight co-moving distance (e.g. Peacock 1999), in which $c$ is the speed of light and $H_{0}$ the Hubble constant.

For a standard $\Lambda$ cosmology with $H_{0}=71$ km s*-1* Mpc*-1*, $\Omega_{\rm matter}=0.27$ and $\Omega_{\Lambda}=0.73$, this gives a peak in the angular diameter distance at $z\approx 1.6$, which has the consequence that below this redshift both $DA_{\rm DLA}\ll DA_{\rm QSO}$ and $DA_{\rm DLA}\sim DA_{\rm QSO}$ are possible, whereas above $z_{\rm abs}\sim 1.6$, only $DA_{\rm DLA}\sim DA_{\rm QSO}$ is possible. This leads a mix of angular diameter distance ratios ($DA_{\rm abs}/DA_{\rm QSO}$) at low redshift but exclusively high ratios ($DA_{\rm abs}/DA_{\rm QSO}\sim 1$) at high redshift (Curran & Webb, 2006). Thus, there is a clear bias introduced by the geometry of the Universe, which must be accounted for when evaluating the spin temperature. Otherwise this leads to an apparent mix of spin temperatures at $z\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ and exclusively high spin temperatures at $z\stackrel{{\scriptstyle>}}{{{}_{\sim}}}1$ (Kanekar & Chengalur, 2003).

For covering factors of less than unity, which is generally expected to be the case (Curran 2017a, see also Kanekar et al. 2014), this gives the spin temperature degenerate with the ratio of the absorber–emitter size,

[TABLE]

[TABLE]

where $\Delta S$ is the line depth in the case of a detection or $3\sigma_{\rm rms}$, where $\sigma_{\rm rms}$ is the r.m.s. noise, per $\Delta v$ channel for a non-detection.444In the literature the spectral resolutions span a large range of values (Curran, 2017b), and so we re-sample the r.m.s. noise levels to a common channel width, which is then used as full-width half maximum (FWHM) of the putative absorption profile.

Adding the new data to the previous gives the distribution shown in Fig. 3. In the binned data, the limits are included via the Kaplan–Meier estimator, a fundamental tenet in non-parametric survival analysis (Kaplan & Meier, 1958), which gives a maximum-likelihood estimate based upon the parent population (Feigelson & Nelson, 1985).

From the binning, we see that the increase in $(T_{\rm spin})(d_{\rm QSO}/d_{\rm abs})^{2}$ does indeed continue with redshift, although this is somewhat steeper than the decline of the star formation density. However, the decline in $\psi_{*}$ has also steepened since previous estimates (Hopkins & Beacom 2006 and references therein), due to newer estimates of the dust obscuration of the high redshift ultra-violet photometry (Behroozi et al., 2013). This is confirmed by far-infrared photometry, which is much less attenuated by dust (Burgarella et al., 2013). While the radio-band data are impervious to the effects of intervening dust, the UV emission from the QSO will certainly be attenuated and depletion of metallic species does indicate the presence of dust in DLAs (e.g. Ledoux et al. 2002, but see also Wild et al. 2006), although the degree of reddening suggests that this is low (Murphy & Bernet, 2016). Alternatively, in the binning of the data, the high redshift bin contains only a single detection, upon which the censored data are estimated. That is, the high redshift bin may be better considered a lower limit and so further 21-cm detections are required in order to rule out further corrections to the high redshift evolution of $\psi_{*}$.

3.2 Absorber size

Asides from this last high redshift bin, the bins in Fig. 3 exhibit some scatter around the star formation rate density fits. This is not unexpected, given that the mean $d_{\rm abs}/d_{\rm QSO}$ is assumed to be constant with redshift, which may not be the case: For massive ($\geq 10^{11}$ M⊙) galaxies, which are the easiest to resolve at high redshift, there is a well documented evolution in size, where large galaxies dominate the low redshift population (Baker et al., 2000) and dwarf galaxies the high redshift population (Lanfranchi & Friaça, 2003), although massive DLA hosts have been observed out to $z\sim 4$ (Djorgovski, 1998). Due to hierarchical build-up, one may also expect a size evolution in DLAs and, if similar to the massive galaxies (several low redshift DLAs have been identified as spirals), we expect a similar decrease in size with redshift. Furthermore, both disk and spheroid like galaxies exhibit a decrease in size with redshift (Fig. 4) and, while imaging of DLA hosts is difficult due to the bright background QSO, at low redshift, where this is less challenging, some DLAs have been identified to be spirals (see e.g. Fig. 3).

Since it is close to a mean of the other curves, in addition to providing an absolute size, we use the fit of Bouwens et al. (2004) to evolve the absorber sizes with redshift, i.e. $r_{\rm HL}=7.6(z_{\rm abs}+1)^{-1.05}$ kpc. This corresponds to $r_{\rm HL}\approx 8$ kpc at $z\approx 0$ for the optical extent of the galaxy, although the H i disk is generally considerably larger than this (Walter et al., 2008). For instance, DLA column densities ($N_{\rm HI}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10^{20}$ $\hbox{{\rm cm}}^{-2}$) have been mapped to radii of up $r\approx 40$ kpc in nearby galaxies (Bowen et al., 2002; Zwaan et al., 2005a; Curran et al., 2008; Reeves et al., 2015, 2016; Rhodin et al., 2018), including the Milky way (Kalberla & Kerp, 2009), as well as in distant ($z\sim 1$) galaxies (Péroux et al. 2011; Bouché et al. 2013). We therefore use an H i radius of $r_{\rm abs}=5r_{\rm HL}$ and evolve this according to Bouwens et al. (2004).

3.3 Radio source size

Quasars also exhibit a redshift evolution, although the size of the source is also dependent upon the observed frequency:

The curves in Fig. 5 show the evolution of quasar size at 408 and 1400 MHz.555Where we have converted the former (Singal, 1988) from the angular sizes using contemporary cosmological parameters (Equ. 5). Converting the power-law fits of the low (Singal, 1988) and high (Chyży & Ziȩba, 1995) frequency fits (Fig. 5), we obtain $d_{\rm QSO}=97.7\,z_{\rm QSO}^{-0.307}$ at 408 MHz and $d_{\rm QSO}=58.2\,z_{\rm QSO}^{-0.520}$ at 1400 MHz, which we interpolate between in order to determine the QSO size at the observed frequency.

4 Results and discussion

4.1 Canonical values

Applying the derived $d_{\rm abs}/d_{\rm QSO}$ (canonical) ratios to Equ. 4 we obtain the covering factors shown in Fig. 6 and

from Equ. 6 the statistical spin temperature,

[TABLE]

where $\theta_{\rm abs}=d_{\rm abs}/DA_{\rm abs}$ and $\theta_{\rm QSO}=d_{\rm QSO}/DA_{\rm QSO}$. Applying this to the data gives the distribution shown in Fig. 7,

from which we see the binned values of $\left<1/T_{\rm spin}\right>$ tighten around the evolution of $\psi_{*}$. This includes the errant high redshift bin, which now agrees with the star formation density of Behroozi et al. (2013) to within $1\sigma$. Note that there is one point with $T_{\rm spin}=3.1$ K, which is lower than the temperature of the Cosmic Microwave Background (CMB) expected at $z_{\rm abs}=1.191$ ($T_{\rm spin}=6.0$ K, Muller et al. 2013). This is unphysical, although it should be borne in mind that this is a statistical correction, with the binned values occupying the same range as other studies (Lane & Briggs 2001; Kanekar & Chengalur 2003; Srianand et al. 2012; Roy et al. 2013; Kanekar et al. 2014; Curran et al. 2016), where the covering factor is estimated/assumed.

4.2 Measured radio extents

While several studies suggest that radio sources evolve from $\sim 100-10$ kpc over $z=0-5$ (Nilsson et al., 1993; Singal, 1993; Willott et al., 1999; Onah et al., 2018), we now explore the possibility that the source size may be significantly smaller. For example, for radio selected AGN and star-forming galaxies, Bondi et al. (2018) find sizes of $\approx 6-3$ kpc over $z=0-3$ for $\mu$Jy radio selected AGN and star-forming galaxies, although at 3 GHz. Furthermore, 49 of the background sources of the DLAs searched in 21-cm absorption have been imaged at high resolution (Kanekar et al., 2009a, 2013, 2014; Ellison et al., 2012) and found to have sizes of $\approx 0-700$ pc, which are significantly smaller than those obtained from the source size evolution (Fig. 8).

Applying the deconvolved sizes gives the statistical spin temperature distribution shown in Fig. 9,

where the covering factor is unity in all cases ($\theta_{\rm abs}>\theta_{\rm QSO}$), when using the evolving galaxy model (Fig. 4). While this gives a similar result to the canonical model with $\psi_{*}\approx 100/T_{\rm spin}$  M⊙ yr*-1* Mpc*-3* (Fig. 7) up to the peak of $\psi_{*}$, we see a rapid rise in the spin temperature at $z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$ resulting in a much more severe departure from the evolution of the star formation density.

These sizes are, however, obtained from the deconvolution of the radio images in which the emission is clearly resolved in the convolved synthesised images, leading, in four cases, to unphysical sizes of $\approx 0$ pc. Therefore, in Fig. 8 we also show the measured source sizes as obtained directly from the resolved radio images, in conjunction with others from the high resolution Very Large Baseline/Array (VLBI/VLA) observations.666Compiled from Ulvestad et al. (1981); Perley (1982); Schilizzi et al. (1982); Hintzen et al. (1983); Gower & Hutchings (1984); Stocke et al. (1984); Antonucci & Ulvestad (1985); Rogora et al. (1987); Barthel et al. (1988); Briggs et al. (1989); Neff & Hutchings (1990); Stanghellini et al. (1990); Fejes et al. (1992); van Breugel et al. (1992); Lonsdale et al. (1993); Murphy et al. (1993); Price et al. (1993); Gurvits et al. (1994); Lister et al. (1994); Perlman et al. (1994); Campbell et al. (1995); Polatidis et al. (1995); Reid et al. (1995); Bondi et al. (1996); Chu et al. (1996); Fey et al. (1996); Fey & Charlot (1997); Harvanek et al. (1997); Stanghellini et al. (1997); Shen et al. (1997); Browne et al. (1998); Dallacasa et al. (1998); Saikia et al. (1998); Shen et al. (1998); Tingay et al. (1998); Wilkinson et al. (1998); Reid et al. (1999); Barthel et al. (2000); Fey & Charlot (2000); Fomalont et al. (2000); Beasley et al. (2002); Helmboldt et al. (2007); Gupta et al. (2012); Srianand et al. (2012). These give the statistical spin temperature distribution shown in Fig. 10,

which is similar to that of the deconvolved sizes, due to the fact that $f=1$ for the majority (48 out of 55) of the DLAs.

The difference in the sizes between the general radio source ($\sim 100$ kpc) and DLA background ($\sim 10$ kpc) populations (Fig.8) may be due to the high resolution radio images being obtained at high frequencies, where the radio source size will be smaller. This could be particularly acute for the 21-cm absorption, where $\nu_{\rm 21-cm}\ll 1420$ MHz at high redshift. From Fig. 11,

we see that this is often the case. Furthermore, high resolution imaging may resolve out all of the extended flux, again underestimating the source size as seen by the absorbing gas.

4.3 Physical implications

For an evolving absorber size, the canonical model and deconvolved/measured radio extents predict very different spin temperatures at high redshift, which may be evident through the degree of ionisation of the neutral gas. For a given number of baryons, an increase in the ionised gas, $N_{\text{H{\sc\,ii}}}$, will be matched by a decrease in the neutral gas. However, like the general DLA population, there is no evidence of an evolution in the mean column density (Fig. 12),

and so we do not expect any significant ionisation over and above that at lower redshift.777Unlike the atomic gas within the host galaxies of high redshift active sources, which is completely ionised at $\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10^{56}$ ionising photons s*-1* (Curran & Whiting, 2012; Curran et al., 2019).

The Saha equation (Saha, 1921; Fridman, 2008) gives the ionisation fraction, the ratio of the number of ions, ${\cal N}_{\rm II}$, to the total number of ions plus atoms, ${\cal N}_{\rm I}$, at temperature $T$, as

[TABLE]

with $n$ being the number density of particles [m*-3*], $m_{\rm e}$ the mass of the electron, $k$ the Boltzmann constant, $h$ the Planck constant and $\phi=13.6$ eV the ionisation potential.

Assuming thermodynamic equilibrium across the disk and solving the quadratic in Equ. 7, for an exponential gas disk, $n=n_{0}e^{-r/R}$, we obtain the ionisation fraction profiles in Fig. 13.

These are shown for the gas distribution of the Milky Way ($n_{0}=13.4$ $\hbox{{\rm cm}}^{-3}$ and $R_{\rm MW}=3.15$ kpc, Kalberla & Kerp 2009), as well as for an evolved absorber size, where we may expect the scale-length of the atomic gas density to decrease as $R=R_{\rm MW}(z_{\rm abs}+1)^{-1.05}$, giving $R=0.73$ kpc at $z\sim 3$ (Sect. 3.2). In either case, the gas completely is ionised at all radii for $T\stackrel{{\scriptstyle>}}{{{}_{\sim}}}4000$ K, whereas at $T=3250$ K (the canonical values, Sect. 4.1) the gas is mostly neutral out to radii of $\sim 1$ kpc, beyond which the density drops to below $n\sim 1$ $\hbox{{\rm cm}}^{-3}$.

Although similar to the canonical values at $z_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2$, the deconvolved/measured source sizes (Sect. 4.2) in conjunction with the evolved absorber sizes (Sect. 3.2) give much higher statistical spin temperatures at $z_{\rm abs}\sim 3$ ($T_{\rm spin}\approx 60\,000$ K). This implies a very high ionisation fraction ($x=1$), to the point that the strength of the Lyman-$\alpha$ absorption would be below that of Lyman Limit Systems ($N_{\text{H{\sc\,i}}}=10^{17.2}-10^{20.3}$ $\hbox{{\rm cm}}^{-2}$) at the level of the Lyman-$\alpha$ Forest, contrary to what is observed (Fig. 12).

Dispensing with the evolved absorber sizes, we can use the fact that the gas is mostly neutral to estimate the covering factor required to ensure the gas is below this temperature (Equ. 3), and thus the maximum permitted absorber extent (Equ 4). From Fig. 14,

we see that most of the deconvolved source sizes and many of the measured sizes require absorber extents of $d_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ kpc. This is significantly smaller than those predicted from the evolution of large galaxies, $d_{\rm abs}\approx 10$ kpc at $z\sim 3$ (Sect. 3.2). Taking the raw values (Fig. 3), the $z\sim 3$ bin gives $T_{\rm spin}(d_{\rm QSO}/d_{\rm abs})^{2}\approx 10\,000$ K, which compared with the $T\stackrel{{\scriptstyle<}}{{{}_{\sim}}}4000$ K limit imposed by the ionisation fraction, gives $d_{\rm QSO}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2d_{\rm abs}$.

High redshift imaging of the H i 21-cm emission from DLA hosts is required to verify that the H i disk extends as far past the optical emission at $z\stackrel{{\scriptstyle>}}{{{}_{\sim}}}1$ as it does at lower redshift ($d_{\rm abs}\sim 5d_{\rm HL}$), although even the SKA will be limited to $z\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ in the detection of 21-cm spectral line emission (Staveley-Smith & Oosterloo, 2015). If the case, the above absorption cross-sections imply $d_{\rm HL}\approx 2$ kpc for the canonical model and $d_{\rm HL}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}200$ pc for the measured radio source sizes. DLAs are hypothesised to arise in a multitude of sources, including galactic disks, dwarf galaxies, rapidly rotating proto-disks, merging sub-galactic systems and low surface brightness galaxies (Wolfe et al. 1986; Matteucci et al. 1997; Prochaska & Wolfe 1997; Haehnelt et al. 1998; Jimenez et al. 1999), which does not constrain the above absorption cross-sections. Note, however, that impact parameters of $b\stackrel{{\scriptstyle>}}{{{}_{\sim}}}20$ kpc have been found at $z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$, although identification of DLA hosts is very rare at high redshift (see Fumagalli et al. 2015 and references therein).

5 Summary

We have used unpublished archival GMRT data of H i 21-cm absorption searches in high redshift ($z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$) damped Lyman-$\alpha$ absorption systems to determine whether the fraction of cold neutral gas continues to trace the star formation density at these redshifts. This adds a dozen new data points, albeit limits, from which the spin temperature, degenerate with the ratio of the absorber and the continuum source sizes, does increase at high redshift. This is, however, higher than predicted by the star formation density [$T_{\rm spin}(d_{\rm QSO}/d_{\rm abs})^{2}\approx 10\,000$ cf. $\approx 2000$ K]. This could be explained by an evolution in $d_{\rm QSO}/d_{\rm abs}$ and estimating this from the known evolution of galaxy and radio source sizes (the canonical model), gives $\left<T_{\rm spin}\right>\approx 3000$ K, cf. $\left<T_{\rm spin}\right>\approx 2000$ K expected from the star formation density. This may suggest that the UV data, from which $\psi_{*}$ is derived at high redshift, is still over-corrected for dust obscuration (Behroozi et al. 2013, cf. Hopkins & Beacom 2006). However, given that there is only a single 21-cm detection in this bin, which forms the parent population with which to incorporate the limits, further $z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}3$ detections are required to verify this. Until then, the high redshift bin is best treated as a lower limit to $\left<T_{\rm spin}\right>$.

There exists high resolution radio imaging of many of the background QSOs, although applying the galaxy size evolution yields $\left<T_{\rm spin}\right>\approx 60\,000$ K at $z_{\rm abs}\sim 3$, where all of the neutral gas would be ionised, which is inconsistent with the observed flat evolution of the column density. We have used the limiting $T\approx 4000$ K, above which all of the gas is expected to be ionised, to constrain the required absorber extents based upon the high resolution images. These require H i extents of $d_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ kpc, or half-light radii of $r_{\rm HL}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}100$ pc, which would support the hypothesis that high redshift DLAs arise predominately in dwarf galaxies, although large impact parameters ($b\stackrel{{\scriptstyle>}}{{{}_{\sim}}}20$ kpc) have been found at $z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}3$.

The raw values [$T_{\rm spin}(d_{\rm QSO}/d_{\rm abs})^{2}$], which remove the bias between the low ($z_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$) and high redshift samples introduced by the geometry of an expanding Universe, do suggest that the spin temperature, degenerate with the background source–absorber size ratio, traces the star formation density, with the rogue $z_{\rm abs}\sim 3$ bin being “reigned in” when canonical values of $d_{\rm abs}$ and $d_{\rm QSO}$ are applied. This suggests that $\psi_{*}T_{\rm spin}\approx 100$  M⊙ yr*-1* Mpc*-3* K, at least at $z_{\rm abs}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}3$. Again, this may be due to the single detection in the parent population or could suggest over-corrections to the UV data, as well as a high redshift variation in the $d_{\rm QSO}/d_{\rm abs}$ ratio, perhaps due to a differing ratio in the half-light/H i radii. This could be addressed with further H i 21-cm detections in $z_{\rm abs}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}3$ DLAs, which, although difficult, is feasible with current instruments (Curran, 2018).

Acknowledgements

I wish to thank the anonymous referee for their helpful comments. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and NASA’s Astrophysics Data System Bibliographic Service. This research has also made use of NASA’s Astrophysics Data System Bibliographic Service and asurv Rev 1.2 (Lavalley et al., 1992), which implements the methods presented in Isobe et al. (1986).