A statistical measurement of the HI spin temperature in DLAs at cosmological distances
MMNRAS , 1–12 (2020) Preprint 22 February 2021 Compiled using MNRAS L A TEX style file v3.0
A statistical measurement of the H i spin temperature in DLAs atcosmological distances James R. Allison , (cid:63) Sub-Dept. of Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Rd., Oxford, OX1 3RH, UK ARC Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO 3D)
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Evolution of the cosmic star formation rate (SFR) and molecular mass density is expected tobe matched by a similarly strong evolution of the fraction of atomic hydrogen (H i) in the coldneutral medium (CNM). We use results from a recent commissioning survey for intervening 21-cm absorbers with the Australian Square Kilometre Array Pathfinder (ASKAP) to constructa Bayesian statistical model of the N HI -weighted harmonic mean spin temperature ( T s ) atredshifts between z = 0 . and . . We find that T s ≤ K with 95 per cent probability,suggesting that at these redshifts the typical H i gas in galaxies at equivalent DLA columndensities may be colder than the Milky Way interstellar medium ( T s , MW ∼ K). This resultis consistent with an evolving CNM fraction that mirrors the molecular gas towards the peakin SFR at z ∼ . We expect that future surveys for H i 21-cm absorption with the current SKApathfinder telescopes will be able to provide constraints on the CNM fraction that are an orderof magnitude greater than presented here. Key words: methods: statistical - galaxies: evolution - galaxies: ISM - quasars: absorptionlines - radio lines: galaxies
The coldest ( T k < K) interstellar gas has a fundamental rolein forming stars and fuelling galaxy evolution throughout cosmichistory. Understanding why the star formation rate (SFR) densityof the Universe has declined rapidly since peaking at z ≈ (e.g.Hopkins & Beacom 2006; Madau & Dickinson 2014; Driver et al.2018) is intimately tied to determining how the mass fraction ofcold gas in galaxies has evolved.Stars form in the dusty molecular gas of the interstellar medium(ISM) and so it is believed that this phase is most important in driv-ing evolution of the SFR density throughout cosmic history. Recentsurveys for the tracers of the dense molecular gas, principally COemission (e.g. Decarli et al. 2019, 2020; Lenkić et al. 2020; Riecherset al. 2020a,b; Fletcher et al. 2021) and the far-infrared and mm-wavelength dust continuum (e.g. Berta et al. 2013; Scoville et al.2017; Magnelli et al. 2020), provide evidence that the mass densityevolves strongly and roughly mirrors that of the SFR density (albeitwith a slightly delayed peak; Tacconi et al. 2020). Observations ofthe diffuse atomic gas, principally using H i 21-cm emission (e.g.Zwaan et al. 2005a; Braun 2012; Rhee et al. 2018; Jones et al. 2018;Chowdhury et al. 2020) and Lyman- α absorption (e.g. Noterdaemeet al. 2012; Zafar et al. 2013; Crighton et al. 2015; Sanchez-Ramirezet al. 2016; Bird et al. 2017), reveal a mass density that evolves com- (cid:63) E-mail: [email protected] paratively slowly and declines by only a factor ∼ from peak SFRto the present day. Recently, Walter et al. (2020) showed that thisglobal behaviour could be described by a simple phenomenologicalmodel of the gas parametrised by the net infall rate of ionised in-ter/circumgalactic gas, which replenishes the H i reservoir, and theconversion of H i to H , both of which have declined by an ordermagnitude since their peak.Evolution of the physical state of the atomic gas in galaxiesis as yet unknown and could well deviate from the cosmologicalmass density measured from 21-cm emission and Lyman- α ab-sorption surveys. Observations of the Milky Way ISM reveal amulti-phased diffuse neutral medium that spans two orders of mag-nitude in temperature and density (e.g. Heiles & Troland 2003;Murray et al. 2018). Two distinct stable phases, the denser cold neu-tral medium (CNM; T k ∼ K) and more diffuse warm neutralmedium (WNM; T k ∼
10 000
K), co-exist in a pressure equilib-rium that is determined by heating and cooling processes that aredependent on the star formation rate, dust abundance and gas-phasemetallicity (Wolfire et al. 2003). Further dynamical processes, suchas turbulence and supernova shocks, are thought to generate an un-stable third phase at intermediate temperatures (UNM; e.g. Murrayet al. 2018). If the presence of CNM is a pre-requisite for molec-ular cloud and star formation (e.g. Krumholz et al. 2009), then asimilarly strong redshift evolution could be observed in the CNMfraction.The H i 21-cm absorption line detected in the spectra of back- © a r X i v : . [ a s t r o - ph . GA ] F e b J. R. Allison ground radio sources is an effective tracer of the cold phase atomicgas to large redshifts. The equivalent width is inversely sensitiveto the excitation (spin) temperature, thereby providing a means bywhich the relative mass fractions of distinct thermal phases can beinferred. In the few cases where the H i column density can deter-mined independently from either 21-cm emission (e.g. Reeves et al.2016; Borthakur 2016; Gupta et al. 2018) or Lyman- α absorption(see Kanekar et al. 2014 and references therein; hereafter K14), the N HI -weighted harmonic mean of the spin temperature can be mea-sured directly. K14 show that the distribution of spin temperaturesfor Damped Lyman- α absorbers at z > . is statistically different(at 4 σ significance) to those at lower redshifts. However it is ob-servationally challenging and expensive to draw a sufficiently largesample of nearby H i galaxies and distant Lyman- α absorbers in or-der to provide strong constraints on the redshift of evolution of thecold phase gas. An alternative approach is to infer the spin temper-ature indirectly from much larger 21-cm absorption line surveys bycomparing the outcomes with the expected detection yield (Darlinget al. 2011; Allison et al. 2016, 2020; Grasha et al. 2020). Recently,Curran (2017, 2019) showed that the 21-cm absorber population,and hence spin temperature, may evolve with the star formationhistory Universe. However, they use an evolutionary model for thecovering fraction of radio sources that would mimic any perceivedevolution in the spin temperature. Model-independent methods aretherefore required to verify such a claim.In this paper we present a statistical measurement of the N HI -weighted harmonic mean spin temperature in DLAs at intermediatecosmological redshifts ( . < z < . ). This uses a Bayesiantechnique first proposed by Allison et al. (2016) to infer the H i spintemperature by comparing the expected and actual detection yieldsin 21-cm absorption line surveys. The results presented here arederived from a recent commissioning project with the AustralianSquare Kilometre Array Pathfinder (ASKAP; DeBoer et al. 2009;Hotan et al. 2021), which carried out a survey for 21-cm absorptionlines in a sample of 53 bright compact radio sources (Sadler et al.2020; hereafter S20). This is the first demonstration of this methodto measure the spin temperature at cosmological distances. In futurework we plan to use data from the ASKAP First Large AbsorptionSurvey in HI (FLASH; e.g. Allison et al. 2016, 2020) to measurethe harmonic mean spin temperature to greater precision and enablemeasurements as a function of redshift.Throughout this paper we use a flat Λ CDM cosmology with Ω m = 0 . , Ω Λ = 0 . and H = 70 km s − Mpc − (e.g. Spergelet al. 2007). We use data from the 21-cm absorption line survey carried outby S20 during commissioning of the ASKAP telescope. S20searched 21-cm redshifts between z = 0 . and . towards asample of 53 radio sources selected from the Australia Telescope20 GHz (AT20G) Survey catalogue (Murphy et al. 2010). They de-tected four intervening H i 21-cm absorbers towards PKS 0834 − −
02, PKS 1610 −
77 and PKS 1830 − − − −
02 andPKS 1830 −
211 were already known, the sample was selected basedon source flux density and declination and so is not biased by known detections. We refer the reader to S20 for further details of the sam-ple selection, observations and individual detections (see also Alli-son et al. 2017, 2019 for further details of the ASKAP detectionstowards PKS 1830 −
211 and PKS 1740 − − and the spatial resolution, using natural weighting, isapproximately 1.8 arcmin for the 6-antenna ASKAP Boolardy En-gineering Test Array (BETA; Hotan et al. 2014; McConnell et al.2016) and 50 arcsec for the ASKAP-12 array (Hotan et al. 2021).The spectral baseline is flat for the dynamic range requirements ofthis survey, although as noted by S20 there is an additional non-Gaussian noise component for the ASKAP-12 spectra, which isdiscussed further in subsubsection 2.3.1. We use these data to de-termine the sensitivity to the 21-cm optical depth for each spectralinterval, thereby allowing us to estimate the expected number of in-tervening 21-cm absorber detections as a function of the propertiesof the foreground H i and background sources. We automate line detection using a bespoke software tool calledFLASHfinder (Allison et al. 2012), which uses the PyMulti-Nest (Buchner et al. 2014) implementation of MultiNest (Feroz& Hobson 2008; Feroz et al. 2009, 2013). The multi-modal capa-bility of MultiNest enables more than one line to be detected in agiven spectrum. Detection significance is given by the Bayes factor B (e.g. Kass & Raftery 1995), a statistic that is equal to the ratioof Bayesian evidences for a line model to a null model. Since noprior preference is given for either the line or null models, which isreasonable if we are testing for the unknown incidence of 21-cm ab-sorption lines, B is equal to the odds in favour of the line model. Tocalculate the Bayesian evidence we use the likelihood function for anormal distribution, with standard deviation equal to the measuredrms noise in each spectral channel.For the purpose of 21-cm absorption line detection we usea single Gaussian profile to model the optical depth in velocity,which is then converted into the observed absorption line. We usenon-informative priors for the model parameters within ranges setby the data and physically realistic limits; for the line position weuse a uniform prior over the full range of the spectrum, for thefull width at half maximum (FWHM) we use a loguniform priorbetween 0.1 and 2000 km s − , and for the peak optical depth we usea loguniform prior between 1 per cent of the median rms noise per18.5 kHz channel and a maximum value of 100.A Gaussian profile is appropriate given that broadening isexpected to be dominated by Doppler shift due to line-of-sightthermal, turbulent and bulk motion of the gas. A typical 21-cmabsorption line will have a more complex velocity profile due tothe relative kinematic and spatial distributions of the foregroundabsorber and background radio source. However, the effect of morecomplex model choices on detecting high signal-to-noise lines isnot expected to be significant. MNRAS000
211 and PKS 1740 − − and the spatial resolution, using natural weighting, isapproximately 1.8 arcmin for the 6-antenna ASKAP Boolardy En-gineering Test Array (BETA; Hotan et al. 2014; McConnell et al.2016) and 50 arcsec for the ASKAP-12 array (Hotan et al. 2021).The spectral baseline is flat for the dynamic range requirements ofthis survey, although as noted by S20 there is an additional non-Gaussian noise component for the ASKAP-12 spectra, which isdiscussed further in subsubsection 2.3.1. We use these data to de-termine the sensitivity to the 21-cm optical depth for each spectralinterval, thereby allowing us to estimate the expected number of in-tervening 21-cm absorber detections as a function of the propertiesof the foreground H i and background sources. We automate line detection using a bespoke software tool calledFLASHfinder (Allison et al. 2012), which uses the PyMulti-Nest (Buchner et al. 2014) implementation of MultiNest (Feroz& Hobson 2008; Feroz et al. 2009, 2013). The multi-modal capa-bility of MultiNest enables more than one line to be detected in agiven spectrum. Detection significance is given by the Bayes factor B (e.g. Kass & Raftery 1995), a statistic that is equal to the ratioof Bayesian evidences for a line model to a null model. Since noprior preference is given for either the line or null models, which isreasonable if we are testing for the unknown incidence of 21-cm ab-sorption lines, B is equal to the odds in favour of the line model. Tocalculate the Bayesian evidence we use the likelihood function for anormal distribution, with standard deviation equal to the measuredrms noise in each spectral channel.For the purpose of 21-cm absorption line detection we usea single Gaussian profile to model the optical depth in velocity,which is then converted into the observed absorption line. We usenon-informative priors for the model parameters within ranges setby the data and physically realistic limits; for the line position weuse a uniform prior over the full range of the spectrum, for thefull width at half maximum (FWHM) we use a loguniform priorbetween 0.1 and 2000 km s − , and for the peak optical depth we usea loguniform prior between 1 per cent of the median rms noise per18.5 kHz channel and a maximum value of 100.A Gaussian profile is appropriate given that broadening isexpected to be dominated by Doppler shift due to line-of-sightthermal, turbulent and bulk motion of the gas. A typical 21-cmabsorption line will have a more complex velocity profile due tothe relative kinematic and spatial distributions of the foregroundabsorber and background radio source. However, the effect of morecomplex model choices on detecting high signal-to-noise lines isnot expected to be significant. MNRAS000 , 1–12 (2020)
I spin temperature at cosmological distances − − − ln B N o . D e t ec t i o n s Negative featuresPositive featuresNoise model
Figure 1.
The distribution of positive and negative features detected inthe data as a function of the detection statistic, the Bayes factor B . Solidhistograms denote results for the S20 spectra. For comparison, the emptyhatched histograms denote the larger data set of the ASKAP wide-fieldsurvey of the GAMA 23 field, undertaken during FLASH early science(A20). Regions of overlapping negative and positive features are colouredmagenta. The solid black line denotes a model fit to the distribution ofpositive features detected in the A20 spectra. There are six negative featuresthat are clearly reliable, of which four are intervening 21-cm absorbers, oneis 21-cm absorption associated with the radio galaxy PKS B1740 −
517 andone is OH 18-cm absorption towards PKS 1830 −
211 (see S20 for furtherdetails).
Our detection statistic, B , is the Bayesian odds in favour of a featurerepresented by the absorption-line model, with features at ln B (cid:38) strongly favoured. Artefacts that remain in the data after process-ing, and that are not accounted for by the null model, will reducethe detection reliability (also known as the purity or fidelity). Thisissue is particularly problematic for absorption-line surveys, whereany bandpass error not corrected by calibration is amplified in thesource spectrum. Possible solutions include further data processingtechniques to remove or mask artefacts, adapting the null model toaccount for them, or distinguishing them from true lines in param-eter space, a posteriori . Here we use the latter method, focusingspecifically on the detection statistic B , but will explore other ap-proaches in future work. We characterise the distribution of false absorption-like detectionsby running our detection method on the inverted spectra, and theninspect the distribution of positive (emission-like) features detectedas a function of B . This is analogous to the procedure in blindsurveys for H i 21-cm emission in the nearby Universe (e.g. Serraet al. 2012, 2015) and CO emission at higher redshifts (e.g Walteret al. 2016; Pavesi et al. 2018; Lenkić et al. 2020) that use negative(absorption-like) features to determine reliability. The implicit as-sumption is that the incidence of true lines amongst the referencedistribution is negligible compared with the incidence of featuresdue to noise and/or artefacts. We discuss the validity of this assump-tion for our data in subsubsection 2.3.2 below.In Figure 1 we show the distribution of detected negative https://github.com/drjamesallison/flashfinder and positive features as a function of B . For comparison, we in-clude results from the larger ASKAP wide-field blind survey of theGAMA 23 field by Allison et al. (2020) (hereafter A20), which wasundertaken during FLASH early science. The reliability of bothsurveys is limited by a non-Gaussian contribution to the spectralnoise, at the level of ∼ per cent of the signal, which is causedby incorrect firmware weights used for correcting the coarse chan-nelisation in the ASKAP-12 array . Because the additional noiseis multiplicative, it is preferentially detected towards sources withhigher signal-to-noise continuum. This behaviour is evident in theresults shown Figure 1; the distribution of positive features detectedin S20 is skewed to higher ln( B ) values than A20, which is con-sistent with the proportion of higher signal-to-noise spectra in thatsample.We note that in blind surveys for CO emission the reliabilityof low signal-to-noise lines is typically determined by modellingthe distribution of false detections assuming negative/positive sym-metry, thereby enabling fainter sources to be included in the COluminosity function. However, for the data considered here it isnot certain that the additional positive and negative features gener-ated by the channelisation error would be distributed symmetrically.Without further characterisation of the additional noise we cannotdetermine the reliability of excess negative features within the lowsignal-to-noise region. We therefore take a conservative approachby selecting a ln( B ) threshold above which we are confident thatany absorption-like features are reliable.We capture the behaviour of the well-sampled distribution ofpositive features in the A20 spectra by fitting a skew normal distri-bution in the logarithm of ln B , with parameter λ = − (Azzalini& Valle 1996). We stress that there was no basis for this choiceother than the model provides a reasonably accurate analytical rep-resentation of the observed distribution. We extrapolate from thismodel that there is less than 0.003 probability for positive featuresabove ln B = 16 , corresponding to an expectation of less than onedetection. This is the reliability threshold adopted by A20 to deter-mine the completeness of those data. In the case of the S20 spectra,the distributions are relatively under-sampled and harder to model;we also note the excess of negative features in the low signal-to-noise regime that could either be caused by real absorption lines orsimply bias in the behaviour of the noise. Although it is not possi-ble to determine if some of these excess negative features are trueintervening 21-cm absorbers, none are detected at the 21-cm lineposition in the spectrum corresponding to the source redshift andso we can rule out associated absorption. Given that the distributionappears to be skewed to higher ln B than that of A20, we choose athreshold of ln B = 20 , above which no detections of negative orpositive features are apparent in the low signal-to-noise distribution. As previously mentioned, our reliability analysis assumes that thereference distribution of positive (emission-like) features is domi-nated by noise and not contaminated with a significant fraction oftrue spectral lines. We discuss the validity of that assumption here.First we consider the possibility that the positive features maycontain detections of H i 21-cm emission. We note that none ofthe positive features are detected at the 21-cm line position in thespectrum corresponding to the source redshift, and so we can rule This error is not present in the earlier 6-antenna BETA data, and has beencorrected for observations with the full ASKAP-36 array.MNRAS , 1–12 (2020)
J. R. Allison
H i emission associated with the host galaxy. However, it is alsopossible that we may detect H i emission from other galaxies in ourline of sight. In the case of the S20 spectra, the lowest redshift is z = 0 . with a typical spectral rms noise of ∼ mJy beam − per 18.5 kHz, which for an H i line width of ∼ km s − givesa 5-sigma detection limit of M HI ∼ . × M (cid:12) (e.g. Meyeret al. 2017). Likewise, for the A20 spectra, the lowest H i redshiftis z = 0 . with a median rms noise of 3.2 mJy beam − , giving acorresponding mass limit of M HI ∼ . × M (cid:12) . Both limitsare well beyond the high-mass end of the local H i mass function(e.g. Jones et al. 2018) and so we conclude that the serendipitousdetection of 21-cm emission in these spectra is very unlikely.Of the other emission lines that we could possibly detect atthese frequencies, the most likely are luminous masing emissionfrom the main 1665 and 1667 MHz lines of the hydroxyl radical(OH). The S20 spectra cover OH redshifts between z = 0 . and . , which for a maser line width of ∼ km s − corresponds toa 5-sigma detection limit between L = 0 . and . × L (cid:12) .In the case of the A20 spectra, they cover OH redshifts between z = 0 . and . with a corresponding 5-sigma detection limitbetween . × and . × L (cid:12) . First we consider OH emission(and absorption) associated with host galaxies of both the radiosources and the reliably identified 21-cm absorbers. Only the knownOH absorption line towards PKS1830 −
211 is detected. Secondly,we consider the possibility of serendipitously detecting line-of-sightmegamaser emission. We can use the low-redshift ( z < . ) OHmegamaser luminosity function measured by Darling & Giovanelli(2002) to estimate the expected number of detections. Assuming atotal sky area covered by the S20 spectra ∼ .
05 deg (based onthe angular resolution of each spectrum) the expected number ofserendipitously detected OH megamasers is negligible (i.e. N OH (cid:28) ), even if the luminosity function evolves by an order of magnitudeat higher redshifts. Likewise, for the 1253 spectra of A20 the totalsky area covered is ∼ .
76 deg , again giving a negligible numberof expected OH megamasers.Clearly contamination from other emission and absorptionlines is not an issue with these data. However, future large 21-cm absorption surveys with the SKA and pathfinder telescopes willbe sensitive to a greater volume and will need to take into consider-ation these other emission and absorption lines when determiningthe reliability of their detections. For these future surveys, in addi-tion to OH, we will need to consider that other molecular species athigh redshift (for example the H CO 6-cm line) could give rise toconfusing absorption lines that would affect 21-cm absorption reli-ability. Furthermore, there are other masing transitions that lead topositive features that would contaminate the reference distribution,such as the radio-frequency recombination lines and the conjugateOH satellite lines at 1612 and 1720 MHz.
We define the completeness as the probability that an absorption linewith a given peak signal-to-noise ratio (S/N) and width is recoveredfrom the data using our detection method and reliability threshold.This is determined by randomly populating our sample spectra with1000 Gaussian-profile absorption lines in bins of peak S/N andFWHM and measuring the fraction that are recovered. The resultsare shown in Figure 2 and define the completeness as a bivariatefunction of the peak S/N and FWHM.We now consider how these are used to determine the com-pleteness in terms of the physical properties of absorption lines.For a line with peak optical depth τ , the peak S/N in i ’th spectral Peak S / N in 18 . C o m p l e t e n e ss [ p e r ce n t ] ∆ v FWHM [km s − ]510204080160 Figure 2.
The expected completeness of reliable detections ( ln( B ) > )in the S20 spectra, as a function of the peak S/N in a single 18.5-kHz (5.3– 7.8 km s − ) channel. The lines denote smoothing of the simulated data(circles) using a Savitzky-Golay filter (Savitzky & Golay 1964). channel towards the j ’th source is given by (S / N) i,j = [1 − exp( − τ )] c f S c i,j σ i,j , (1)where c f is the source covering factor, S c i,j is the unabsorbed con-tinuum flux density, and σ i,j is the rms noise. If we consider aGaussian profile, then the peak optical depth is given in terms ofthe physical properties of the absorber by τ = 0 . (cid:20) N HI × cm − (cid:21) (cid:20) T s
100 K (cid:21) − (cid:20) ∆ v FWHM
30 km s − (cid:21) − , (2)where N HI is the H i column density, T s is the spin temperature,and ∆ v FWHM the FWHM. We can therefore use Equation 1 andEquation 2 to calculate the expected peak S/N in a given spectralchannel as a function of the physical properties of the absorber, andthen use the results of Figure 2 to determine the completeness C i,j . To infer the spin temperature, we need to determine how manyintervening absorbers we would have expected to detect in the dataas a function of T s . In the discrete limit, the expected number ofdetected intervening absorbers is evaluated by the following sumover J sight-lines, each with I j spectral channels, µ abs ( θ ) = J (cid:88) j =1 I j (cid:88) i =1 δX i,j (cid:90) N max N min f ( N HI , z i,j ) C i,j ( N HI , θ ) w j ( z i,j + ∆ z asc i,j ) d N HI , (3)where θ is the set of parameters at the absorber that determinedetection, { T s , c f , ∆ v FWHM } , δX i,j is the absorption path lengthspanned by the i, j ’th spectral channel, and f ( N HI , z ) is the columndensity distribution function, equal to the frequency of absorbersintersecting a given sight-line with a column density between N HI and N HI + d N HI per unit column density per unit comoving ab-sorption path. The completeness, C i,j ( N HI , θ ) , is as defined in sub-section 2.4. We integrate over column densities between N min and MNRAS000
30 km s − (cid:21) − , (2)where N HI is the H i column density, T s is the spin temperature,and ∆ v FWHM the FWHM. We can therefore use Equation 1 andEquation 2 to calculate the expected peak S/N in a given spectralchannel as a function of the physical properties of the absorber, andthen use the results of Figure 2 to determine the completeness C i,j . To infer the spin temperature, we need to determine how manyintervening absorbers we would have expected to detect in the dataas a function of T s . In the discrete limit, the expected number ofdetected intervening absorbers is evaluated by the following sumover J sight-lines, each with I j spectral channels, µ abs ( θ ) = J (cid:88) j =1 I j (cid:88) i =1 δX i,j (cid:90) N max N min f ( N HI , z i,j ) C i,j ( N HI , θ ) w j ( z i,j + ∆ z asc i,j ) d N HI , (3)where θ is the set of parameters at the absorber that determinedetection, { T s , c f , ∆ v FWHM } , δX i,j is the absorption path lengthspanned by the i, j ’th spectral channel, and f ( N HI , z ) is the columndensity distribution function, equal to the frequency of absorbersintersecting a given sight-line with a column density between N HI and N HI + d N HI per unit column density per unit comoving ab-sorption path. The completeness, C i,j ( N HI , θ ) , is as defined in sub-section 2.4. We integrate over column densities between N min and MNRAS000 , 1–12 (2020)
I spin temperature at cosmological distances N max , which are selected based on the range of column densities forwhich we expect the data to be sensitive to 21-cm absorption. Thesource redshift weighting function, w j ( z + ∆ z asc ) , is the probabil-ity that the j ’th source is located at a redshift greater than z +∆ z asc .The offset in redshift is used to exclude absorption associated withthe host galaxy of the source, and is given by ∆ z asc = (1 + z )∆ v asc /c, (4)where ∆ v asc = 3000 km s − .To determine µ abs as a function of spin temperature, wemarginalise over the velocity width and covering factor µ abs ( T s ) = (cid:90) (cid:90) µ abs ( θ ) p ( c f , ∆ v FWHM | T s ) d( c f ) d(∆ v FWHM ) , (5)where p ( c f , ∆ v FWHM | T s ) is the joint conditional probability dis-tribution for c f and ∆ v FWHM . We assume that the covering factorand velocity width are independent, so this can be expressed as aproduct of the individual conditional probability distributions forthese two quantities.In the remainder of this section we discuss in further detail thefactors and assumptions that are used in Equation 3 and Equation 5.
For the 50 sources in the S20 sample with reliable redshifts, theweighting w j ( z ) is simply given by w j ( z ) = (cid:40) z < z src j otherwise , (6)where z src j is the redshift of the j ’th source. Typical uncertainties inthe optical spectroscopic redshifts of about 50 km s − correspond toa fractional uncertainty in each sight-line of less than 0.05 per cent.For the remaining 3 sources without spectroscopic redshifts,we use the statistical redshift distribution given by De Zotti et al.(2010) for bright radio sources ( S . > mJy) in the CombinedEIS-NVSS survey of Radio Sources (CENSORS; Brookes et al.2008), w j ( z ) = (cid:90) ∞ z N src ( z (cid:48) ) d z (cid:48) (cid:30)(cid:90) ∞ N src ( z (cid:48) ) d z (cid:48) , (7)where N src ( z ) = 1 .
29 + 32 . z − . z + 11 . z − . z . (8)Measurement uncertainty in the CENSORS redshift distributioncontributes a fractional uncertainty in each sight-line that increaseswith redshift from approximately 4 per cent at z = 0 . to 8 per centat z = 1 . . Since these sight-lines comprise only 5 per cent of thesample, the uncertainty in the total path length due to the sourceredshifts is only about 0.3 per cent. An absorption line measured from a spectrum is an average overthe spatial extent of the unresolved background continuum source.For the interstellar atomic gas in galaxies, distinct structures areevident on linear sizes as small as 100 pc (Braun 2012), and theircolumn density distribution has been measured in both the nearbyUniverse, using resolved studies of 21-cm emission (e.g. Zwaanet al. 2005b; Braun 2012), and at cosmological distances usingLyman- α absorption towards compact UV emission from AGN (e.g. Noterdaeme et al. 2012; Neeleman et al. 2016; Rao et al. 2017;Bird et al. 2017).In the case of H i 21-cm absorption the background radio sourcecan extend well beyond the AGN, so that at the intervening galaxya large fraction of the flux density is distributed on scales largerthan that probed by Lyman- α absorption. The effect of averagingover extended radio sources is to distribute the 21-cm absorbers tolower optical depths than would be expected given the distributionmeasured at higher spatial resolution (Braun 2012). The impact onour model is to over predict how many 21-cm absorbers we mightexpect to detect. It is therefore useful to define a source coveringfactor, c f , that corrects for the unknown areal fraction of the sourceflux density that is subtended by a foreground absorbing structureof putative constant optical depth (see Equation 1).Ideally we need information about the mas-structure of eachsource at sub-GHz frequencies, in order to fully understand the ex-pected effect of source size for 21-cm absorption (e.g. Braun 2012;K14). In the absence of this information for our sample, we insteadextrapolate from what we understand about the source morphologyat other wavelengths. S20 selected sources from the AT20G cat-alogue (Murphy et al. 2010) that have flux densities greater than500 mJy at 20 GHz and 1.5 Jy at 1.4 GHz/843 MHz, thereby includ-ing a high fraction of radio-loud QSOs with compact radio emission.Their analysis of very long baseline interferometric (VLBI) imagingat 5 – 15 GHz in the literature indicated that these sources typicallyhave 20 – 60 per cent of their flux density in components smallerthan ∼ mas, corresponding to projected sizes between 50 and80 pc for the redshift range used in this survey. They note that this isconsistent with the results of Horiuchi et al. (2004), who found thatfor a larger complete and flux-density limited sample of 303 radiosources at 5 GHz, about 50 per cent of the flux density is typicallycontained in a 10 mas component, with 20 per cent from a radio coreof average size 0.2 mas.If we assume that the ratio of the 10-mas flux density to thetotal flux density is a reasonable proxy for the covering factor, thenbased on the above results we might expect that a typical value is (cid:104) c f (cid:105) ≈ . . We note that this also assumes that the source structureat 5 GHz is representative of the structure at sub-GHz frequencies.However, Kanekar et al. (2009a) (see also K14) carried out a VLBIimaging study at sub-GHz frequencies of the radio-loud quasars intheir DLA sample, using the ratio of core-to-total flux density asa proxy for the covering factor. Allison et al. (2016) used a two-tailed Kolmogorov-Smirnov (KS) test to show that the hypothesisthat covering factors obtained by Kanekar et al. are drawn from a − uniform distribution is true at the level of p = 0 . , but not for p = 0 . . Visual inspection of the distribution shows that there maybe a paucity of quasars in the Kanekar et al. sample for coveringfactors less than c f ∼ . which is consistent with the Horiuchiet al. (2004) result that 20 per cent of the flux density in radio-loudAGN selected at 5 GHz is within the sub-mas radio core.Based on these results, we marginalise the expected numberof absorbers over the covering factor by drawing randomly froma uniform prior between c f = 0 and 1. We further assume thatthe covering factor is not conditional on the spin temperature. It ispossible that the true distribution of covering factors may be skewedto lower values than uniform, in which case we would underestimatethe expected detection yield and therefore overestimate the inferredspin temperature. MNRAS , 1–12 (2020)
J. R. Allison ∆ v FWHM [km s − ] . . . . . . p ( ∆ v F W H M ) [ p e r k m s − ] N abs Figure 3.
The sample distribution of velocity widths in the literature, forintervening 21-cm absorbers at z > . (blue histogram). Errorbars denotethe standard deviation given by √N abs . The red line is a log-normal fit tothe data, from which we draw our prior distribution of widths. Sensitivity to a resolved 21-cm absorption line of fixed equiva-lent width is inversely related to the square-root of its width. Wemarginalise the expected number of detections over the distributionof velocity widths for a sample of detected intervening 21-cm ab-sorbers in the literature . We assume that this sample distribution isdrawn from a sufficiently large range of observations with differentsensitivities and spectral resolutions to be representative of the pop-ulation and not censored by the underlying sensitivity to velocitywidth. This is shown in Figure 3, along with a log-normal fit to thedata, from which we draw random line widths.By using this single distribution for the velocity width, we haveassumed that it is not conditional on the spin temperature. This istrue for absorption lines where the dominant broadening mechanismis from bulk rotational or turbulent motion of the gas, but not whenthermal broadening is important. Since the spin temperature is ei-ther equal to or less than the gas kinetic temperature, depending onthe dominant mechanism for excitation of the 21-cm line (e.g. Pur-cell & Field 1956; Field 1958, 1959; Bahcall & Ekers 1969; Liszt2001), this places a lower limit constraint on the velocity width. Wetherefore apply the following additional constraint on the allowedrange of velocity widths ∆ v FWHM ≥ (cid:114) k B m H T s , (9)where k B is the Boltzmann constant and m H is the mass of ahydrogen atom. Note that for typical H i spin temperatures in therange 100 – 1000 K, this corresponds to lower bounds on the FWHM References for the literature sample of velocity widths shown in Figure 3:Briggs, de Bruyn & Vermeulen (2001); Carilli, Rupen & Yanny (1993);Chengalur, de Bruyn & Narasimha (1999); Chengalur & Kanekar (2000);Curran et al. (2007); Davis & May (1978); Ellison et al. (2012); Guptaet al. (2009, 2012, 2013); Kanekar & Chengalur (2001, 2003); Kanekaret al. (2001, 2006, 2009b, 2013, 2014); Kanekar & Briggs (2003), Kanekar,Chengalur & Lane (2007); Kanekar (2014); Lane & Briggs (2001); Lovellet al. (1996); York et al. (2007); Zwaan et al. (2015). between 2 and 7 km s − , which is consistent with the literaturedistribution. The column density distribution function, f ( N HI ) , gives the fre-quency of absorbers intercepting a given sight-line with a col-umn density between N HI and N HI + d N HI , per unit columndensity per unit comoving absorption path. In Equation 3 we in-tegrate over column densities that span equivalent DLA systems( N min = 2 × cm − , N max = ∞ ), which are thought to havesufficient warm neutral gas to temperature-shield and produce CNM(Kanekar et al. 2011). The four intervening absorbers detected inthis sample all have integrated optical depths that correspond tocolumn densities in the equivalent DLA range for T s > K and c f < (S20), suggesting that this is the range of column densities forwhich our data sensitive. The expected number of 21-cm absorbersshould therefore be considered as those with column densities inthe equivalent DLA range.Precise measurements of f ( N HI ) have been obtained using21-cm emission line surveys in the local Universe (e.g. Zwaan et al.2005b; Braun 2012) and DLAs at cosmological distances (e.g. No-terdaeme et al. 2012; Neeleman et al. 2016; Rao et al. 2017; Birdet al. 2017). Searches for DLAs at UV-wavelengths using archival HST data by Neeleman et al. (2016) and Rao et al. (2017) are mostclosely matched in redshift to our data, but the sample sizes areinsufficient to provide precise measurements of f ( N HI ) across thefull range of equivalent DLA column densities. We therefore lin-early interpolate between the results of Zwaan et al. (2005b) andBraun (2012) at z = 0 , and Bird et al. (2017) at z = 2 , who obtainedthe most precise DLA measurement yet using a catalogue generatedby a Gaussian Process (GP) method from the Sloan Digital SkySurvey III Data Release 12 (Garnett et al. 2017). z f ( N HI ) : spatial resolution and self-absorption In the low- z local Universe, the column density distribution func-tions measured by Zwaan et al. (2005b) and Braun (2012) disagreesignificantly, the former being steeper than expected at high col-umn densities for a random oriented gas disc. The measurementby Zwaan et al. (2005b) was carried out using HI emission imagesfrom a sample of 355 galaxies observed in the Westerbork HI Sur-vey of Irregular and Spiral Galaxies (WHISP; van der Hulst et al.2001) with a maximum linear resolution of about 1.3 kpc. In con-trast, Braun (2012) only looked at the H i images of three galaxiesin the Local Group (M31, M33 and the Large Magellanic Cloud),but in much greater detail at 100 pc resolution and thousands ofindependent sight lines.There are several possible reasons for the differences seen inthese two measurements. First, the lower resolution of the studyby Zwaan et al. (2005b) will re-distribute small-scale structures tolower column densities, and thus steepen f ( N HI ) . Braun (2012) sug-gested that this is broadly consistent with their data once smoothed toa similar resolution, but not reproduced in detail. Secondly, by mod-elling each emission line profile in detail, Braun (2012) showed that21-cm self-absorption can have a significant effect in reducing highcolumn densities in the range < log ( N HI ) < . However,we note that despite the exquisite spatial detail and self-consistencywithin the Local Group sample, it is not yet clear how representativeit is of the H i distribution in the larger galaxy population at z = 0 .Given the discrepancy between these measurements, we cal- MNRAS000
The sample distribution of velocity widths in the literature, forintervening 21-cm absorbers at z > . (blue histogram). Errorbars denotethe standard deviation given by √N abs . The red line is a log-normal fit tothe data, from which we draw our prior distribution of widths. Sensitivity to a resolved 21-cm absorption line of fixed equiva-lent width is inversely related to the square-root of its width. Wemarginalise the expected number of detections over the distributionof velocity widths for a sample of detected intervening 21-cm ab-sorbers in the literature . We assume that this sample distribution isdrawn from a sufficiently large range of observations with differentsensitivities and spectral resolutions to be representative of the pop-ulation and not censored by the underlying sensitivity to velocitywidth. This is shown in Figure 3, along with a log-normal fit to thedata, from which we draw random line widths.By using this single distribution for the velocity width, we haveassumed that it is not conditional on the spin temperature. This istrue for absorption lines where the dominant broadening mechanismis from bulk rotational or turbulent motion of the gas, but not whenthermal broadening is important. Since the spin temperature is ei-ther equal to or less than the gas kinetic temperature, depending onthe dominant mechanism for excitation of the 21-cm line (e.g. Pur-cell & Field 1956; Field 1958, 1959; Bahcall & Ekers 1969; Liszt2001), this places a lower limit constraint on the velocity width. Wetherefore apply the following additional constraint on the allowedrange of velocity widths ∆ v FWHM ≥ (cid:114) k B m H T s , (9)where k B is the Boltzmann constant and m H is the mass of ahydrogen atom. Note that for typical H i spin temperatures in therange 100 – 1000 K, this corresponds to lower bounds on the FWHM References for the literature sample of velocity widths shown in Figure 3:Briggs, de Bruyn & Vermeulen (2001); Carilli, Rupen & Yanny (1993);Chengalur, de Bruyn & Narasimha (1999); Chengalur & Kanekar (2000);Curran et al. (2007); Davis & May (1978); Ellison et al. (2012); Guptaet al. (2009, 2012, 2013); Kanekar & Chengalur (2001, 2003); Kanekaret al. (2001, 2006, 2009b, 2013, 2014); Kanekar & Briggs (2003), Kanekar,Chengalur & Lane (2007); Kanekar (2014); Lane & Briggs (2001); Lovellet al. (1996); York et al. (2007); Zwaan et al. (2015). between 2 and 7 km s − , which is consistent with the literaturedistribution. The column density distribution function, f ( N HI ) , gives the fre-quency of absorbers intercepting a given sight-line with a col-umn density between N HI and N HI + d N HI , per unit columndensity per unit comoving absorption path. In Equation 3 we in-tegrate over column densities that span equivalent DLA systems( N min = 2 × cm − , N max = ∞ ), which are thought to havesufficient warm neutral gas to temperature-shield and produce CNM(Kanekar et al. 2011). The four intervening absorbers detected inthis sample all have integrated optical depths that correspond tocolumn densities in the equivalent DLA range for T s > K and c f < (S20), suggesting that this is the range of column densities forwhich our data sensitive. The expected number of 21-cm absorbersshould therefore be considered as those with column densities inthe equivalent DLA range.Precise measurements of f ( N HI ) have been obtained using21-cm emission line surveys in the local Universe (e.g. Zwaan et al.2005b; Braun 2012) and DLAs at cosmological distances (e.g. No-terdaeme et al. 2012; Neeleman et al. 2016; Rao et al. 2017; Birdet al. 2017). Searches for DLAs at UV-wavelengths using archival HST data by Neeleman et al. (2016) and Rao et al. (2017) are mostclosely matched in redshift to our data, but the sample sizes areinsufficient to provide precise measurements of f ( N HI ) across thefull range of equivalent DLA column densities. We therefore lin-early interpolate between the results of Zwaan et al. (2005b) andBraun (2012) at z = 0 , and Bird et al. (2017) at z = 2 , who obtainedthe most precise DLA measurement yet using a catalogue generatedby a Gaussian Process (GP) method from the Sloan Digital SkySurvey III Data Release 12 (Garnett et al. 2017). z f ( N HI ) : spatial resolution and self-absorption In the low- z local Universe, the column density distribution func-tions measured by Zwaan et al. (2005b) and Braun (2012) disagreesignificantly, the former being steeper than expected at high col-umn densities for a random oriented gas disc. The measurementby Zwaan et al. (2005b) was carried out using HI emission imagesfrom a sample of 355 galaxies observed in the Westerbork HI Sur-vey of Irregular and Spiral Galaxies (WHISP; van der Hulst et al.2001) with a maximum linear resolution of about 1.3 kpc. In con-trast, Braun (2012) only looked at the H i images of three galaxiesin the Local Group (M31, M33 and the Large Magellanic Cloud),but in much greater detail at 100 pc resolution and thousands ofindependent sight lines.There are several possible reasons for the differences seen inthese two measurements. First, the lower resolution of the studyby Zwaan et al. (2005b) will re-distribute small-scale structures tolower column densities, and thus steepen f ( N HI ) . Braun (2012) sug-gested that this is broadly consistent with their data once smoothed toa similar resolution, but not reproduced in detail. Secondly, by mod-elling each emission line profile in detail, Braun (2012) showed that21-cm self-absorption can have a significant effect in reducing highcolumn densities in the range < log ( N HI ) < . However,we note that despite the exquisite spatial detail and self-consistencywithin the Local Group sample, it is not yet clear how representativeit is of the H i distribution in the larger galaxy population at z = 0 .Given the discrepancy between these measurements, we cal- MNRAS000 , 1–12 (2020)
I spin temperature at cosmological distances . . . . . . . z − − d ( ∆ X ) / d z Total N HI = 2 × cm − N HI = 2 × cm − T s = 100 K T s = 1000 K Figure 4.
The change in absorption path length with redshift. Lines areshown for the total interval spanned by the survey (regardless of spin temper-ature), and for intervals sensitive to DLAs ( × cm − ) and super-DLAs( × cm − ), for spin temperatures T s = 100 and 1000 K. culate the expected number of 21-cm absorbers using both distri-butions and use the resulting difference as the standard error (witha median of 27 per cent). The possibility that a significant fraction of quasars with dustyintervening absorbers are missing from flux-limited optical sampleshas long been the subject of vigorous investigation. If true, then theseoptical surveys would underestimate the number density of high- N HI DLAs at z > and there would be a corresponding steepeningof the column density distribution function that we use here.Until recently there was no strong evidence of a large missingfraction of optical DLAs; in a combined Bayesian analysis of theoptical and radio constraints Pontzen & Pettini (2009) found thatonly 7 per cent of DLAs are expected to be missing from optically-selected samples due to dust obscuration, with at most 19 per centmissing at 95 per cent confidence. However, Krogager et al. (2019)have recently considered the colour selection used in SDSS-II DataRelease 7, in addition to the magnitude limit, and found that thefraction of missing DLAs could be as much as 28 per cent at z ≈ and 42 per cent at z ≈ , for the lowest dust-to-metal ratio of log κ Z = − . .The column density distribution function we use here wasmeasured by Bird et al. (2017) using spectra from SDSS-III DR12,for which a similar analysis has yet to be carried out. Krogager et al.(2019) state that although the selection criteria used for SDSS-IIIis more complex, it may still be susceptible to the same biases inSDSS-II. In the absence of further information, we calculate theexpected number of 21-cm absorbers in our data assuming that thecolumn density distribution function of Bird et al. (2017) may besystematically low by 20 per cent, using the resulting difference asthe standard error (with a median of 9.7 per cent). The absorption path length defines the comoving interval of thesurvey that is sensitive to intervening absorbers. The infinitesimalpath length element d X spanned by a redshift element d z is given analytically by d X = d z (1 + z ) E ( z ) − , (10)where d z = (1 + z ) d νν , (11) z = ν HI ν − , (12) E ( z ) = (cid:112) (1 + z ) Ω m + (1 + z ) (1 − Ω m − Ω Λ ) + Ω Λ , (13) ν is the corrected observed frequency, and ν HI is the rest frequencyof the H i 21-cm line, equal to 1420.40575177 MHz (Hellwig et al.1970).In the discrete limit, We can estimate the expected total absorp-tion path length sensitive to a given column density by evaluatingthe following sum over J sight lines, each with I j channels ∆ X ( N HI ) = J (cid:88) j =1 I j (cid:88) i =1 C i,j ( N HI , θ ) w j ( z i,j + ∆ z asc i,j ) δX i,j , (14)where the completeness and source redshift weighting are as de-fined above. Again marginalising over the above distributions for ∆ v FWHM and c f , and assuming a fiducial T s = 100 K, theexpected total path length sensitive to DLA column densities( N HI > × cm − ) is ∆ X = 8 . z = 4 . and forsuper-DLAs ( N HI > × cm − ) is ∆ X = 35 (∆ z = 19) .For a higher spin temperature of T s = 1000 K, these intervals de-crease to ∆ X = 0 .
17 (∆ z = 0 . and ∆ X = 7 . z = 4 . ,respectively. In Figure 4 we show the differential absorption pathlength, which gives the sensitivity to 21-cm absorbers as a functionof redshift. Since observations with the more sensitive ASKAP-12telescope were undertaken at higher frequencies, this sensitivityfunction is skewed to lower redshifts. We show the expected number of detected intervening 21-cm ab-sorbers as a function of spin temperature in Figure 5. The 68 per centuncertainty is also given, equal to the quadrature sum of the standarderror due to measurement uncertainty (assumed in total to be about10 per cent), systematic differences between the 21-cm emissionline surveys, and the dust obscuration in DLA surveys. The medianuncertainty in µ abs is about 30 per cent, but increases significantlyat high spin temperatures due to uncertainty in the number of highcolumn density systems that dominate at these sensitivities. The ex-pected number of detections at the mass-weighted harmonic meanspin temperature for the Milky Way ISM ( ≈ K; Murray et al.2018) is . ± . , which is notably less than the four detectionsobtained by S20 and indicates possible tension with the data.We derive the posterior probability density function for thespin temperature using Bayes’ Theorem, p ( T s |N abs ) = Pr( N abs | T s ) p ( T s )Pr( N abs ) , (15) In the notation used here, p ( x ) is the probability density function for acontinuous variable x , Pr( < x ) is the probability mass that a continuousvariable has a value less than x , and Pr( y ) is the probability mass functionfor a discrete variable y.MNRAS , 1–12 (2020) J. R. Allison T s [K] − − − − µ a b s µ abs Pr( < T s |N abs = 4)MW ,
300 K . . . . . . P r( < T s | N a b s = ) Figure 5.
The expected number of detected intervening 21-cm absorbers(blue line) and 68 per cent uncertainty (shaded region), and the correspond-ing cumulative posterior probability for the spin temperature, conditional ondetecting 4 absorbers (red line). The two red lines correspond to differentchoices of prior: p ( µ abs ) ∝ / √ µ abs (solid) and /µ abs (dot-dashed). where Pr( N abs | T s ) is the likelihood function, p ( T s ) is the priorprobability density, and Pr( N abs ) is the normalising constantknown as the marginal likelihood or model evidence. In section 3we determined the probability density for the detection yield ( µ abs )conditional on the spin temperature, p ( µ abs | T s ) . We model this asa normal distribution with mean and standard deviation as shownin Figure 5. These results are then used to determine the probabilityof detecting N abs as a function of spin temperature by evaluatingthe following integral, Pr( N abs | T s ) = (cid:90) Pr( N abs | µ abs ) p ( µ abs | T s ) d µ abs , (16)where the likelihood for µ abs is modelled as a Poisson distributiongiven by Pr( N abs | µ abs ) = µ abs N abs N abs ! e − µ abs . (17)To determine the posterior probability density given in Equa-tion 15 we also need to choose an appropriate prior, p ( T s ) . Forsmaller surveys, where the number of detections are fewer and hencethe likelihood is less constraining, this choice can have a significantaffect on the inferred spin temperature. Since we have no strongprior information on the value of the spin temperature, we choosea non-informative prior. In the context of the Poisson process beingconsidered here, the spin temperature prior is given in terms of thedetection yield by p ( T s ) = (cid:90) p ( T s | µ abs ) p ( µ abs ) d µ abs , (18)where p ( T s | µ abs ) is just the inverse of the above-mentioned normaldistribution, and p ( µ abs ) is the prior for the detection yield. Wechoose the non-informative Jeffreys prior, p ( µ abs ) ∝ / √ µ abs ,which corresponds to invariance of the detection yield under achange of parameterisation (Jeffreys 1946). This is an improperprior, but valid for a finite range of spin temperatures. We choose . < T s < K , covering all reasonable values of the spintemperature expected from our understanding of the physical con-ditions in the interstellar medium of galaxies (e.g. Wolfire et al. 2003). An equally suitable choice of non-informative prior wouldbe p ( µ abs ) ∝ /µ abs (e.g. Jeffreys 1961; Novick & Hall 1965;Villegas 1977), which we also consider here in our results.We can now estimate the posterior probability for the spintemperature given that four intervening 21-cm absorbers were de-tected in the S20 data. In Figure 5 we plot the resulting cumulativeprobability function, Pr( < T s |N abs ) , calculated by integrating theposterior probability density given by Equation 15. We find that T s < K with 95 per cent probability, increasing to 387 K for theabove alternative choice of non-informative prior.
Since the 21-cm line optical depth is inversely dependent on thespin temperature, the value inferred from such a measurement is acolumn-density-weighted harmonic mean over the line-of-sight H ipresent in each thermally distinct phase, T s = N HI (cid:34)(cid:88) i N i T i (cid:35) − , (19)where N HI = (cid:80) i N i . In the case of a simple two-phase model,where the spin temperatures of the individual phases are known,then the measurement of T s can be inverted to infer their relativefraction. In practice the coldest phase dominates the measurementand this expression reduces to a simple ratio of the cold fractionand temperature. However, in general any predictive model of themulti-phased ISM can be tested by measuring T s .In the Milky Way ISM there are three observed thermally dis-tinct phases: the cold (CNM; T CNM ∼ K), unstable (UNM; T UNM ∼ K) and warm neutral medium (WNM; T WNM ∼
10 000
K), with total mass fractions of approximately 28, 20 and52 per cent respectively (Heiles & Troland 2003; Murray et al.2018). The mass-weighted harmonic mean spin temperature over theMilky Way is therefore approximately 300 K, and is seen to be con-stant with galactocentric radius outside of the solar circle (Dickeyet al. 2009). This suggests that the phases are sufficiently mixed thatrandom DLA-like sight-lines observed through our Galaxy wouldgive on average a measurement of T s ∼ K.Given that the relative fraction of neutral phases depends onthe gas-phase metallicity, dust abundance and ambient UV fieldof individual galaxies (Wolfire et al. 2003), we expect T s to varywithin the population. Such variation is seen within the Local Group,where the Small Magellanic Cloud has a much smaller fraction ofCNM than other members, although this is somewhat mitigated bya correspondingly lower CNM temperature (Dickey et al. 2000).Likewise we expect to see variation in the T s inferred from DLAssimply due to variance created by sampled sight-lines; indeed a highintrinsic scatter is seen in the spin temperature of individual DLAswithin a given redshift range (see Figure 8 and K14).Despite this variance in the T s for individual objects, we cantest for systematic changes in the physical conditions of H i gas withredshift. For the individual DLAs in the sample of K14, an estimateof the population T s can be obtained by taking the sample N HI -weighted harmonic mean. Likewise, the value of T s we infer fromEquation 15 is the N HI -weighted harmonic mean over the DLApopulation within the redshift range spanned by the survey, and cantherefore be usefully compared with direct measurements at otherredshifts. MNRAS000
K), with total mass fractions of approximately 28, 20 and52 per cent respectively (Heiles & Troland 2003; Murray et al.2018). The mass-weighted harmonic mean spin temperature over theMilky Way is therefore approximately 300 K, and is seen to be con-stant with galactocentric radius outside of the solar circle (Dickeyet al. 2009). This suggests that the phases are sufficiently mixed thatrandom DLA-like sight-lines observed through our Galaxy wouldgive on average a measurement of T s ∼ K.Given that the relative fraction of neutral phases depends onthe gas-phase metallicity, dust abundance and ambient UV fieldof individual galaxies (Wolfire et al. 2003), we expect T s to varywithin the population. Such variation is seen within the Local Group,where the Small Magellanic Cloud has a much smaller fraction ofCNM than other members, although this is somewhat mitigated bya correspondingly lower CNM temperature (Dickey et al. 2000).Likewise we expect to see variation in the T s inferred from DLAssimply due to variance created by sampled sight-lines; indeed a highintrinsic scatter is seen in the spin temperature of individual DLAswithin a given redshift range (see Figure 8 and K14).Despite this variance in the T s for individual objects, we cantest for systematic changes in the physical conditions of H i gas withredshift. For the individual DLAs in the sample of K14, an estimateof the population T s can be obtained by taking the sample N HI -weighted harmonic mean. Likewise, the value of T s we infer fromEquation 15 is the N HI -weighted harmonic mean over the DLApopulation within the redshift range spanned by the survey, and cantherefore be usefully compared with direct measurements at otherredshifts. MNRAS000 , 1–12 (2020)
I spin temperature at cosmological distances . . . . . . . z − − d ( ∆ X ) / d z Total N HI = 2 × cm − N HI = 2 × cm − T s = 100 K T s = 1000 K Figure 6.
As Figure 4, but including data from the wide field survey of theGAMA23 field, undertaken as a FLASH early science projct (A20).
As previously mentioned in subsubsection 2.3.1, A20 (Allison et al.2020) recently reported results from a wide-field H i 21-cm absorp-tion survey of the GAMA 23 field (Liske et al. 2015), undertakenas a FLASH early science project with ASKAP-12. They searchedfor H i absorption towards 1253 radio sources at a median rmsnoise level of 3.2 mJy beam − per 18.5kHz, covering redshifts be-tween z = 0 . and 0.79 over a sky area of approximately 50 deg .In a purely blind search of the data Allison et al. did not detectany absorption lines, but by cross-matching radio sight lines withknown optically-selected galaxies they did find an absorber associ-ated with gas in the outer regions of an early type galaxy. Allisonet al. calculated the expected detection rate as a function of the spintemperature to covering factor, but unlike the method describedhere, assumed a constant line width ∆ v FWHM = 30 km s − anda fixed completeness threshold corresponding to a S/N of 5.5 in asingle channel. They reported that if the typical spin temperature tocovering factor ratio at these redshifts is equal to 300 K, then theprobability of detecting no absorbers is about 64 per cent.It is straight forward to update our statistical measurement ofthe spin temperature to include this additional information, giventhat the linear combination of independent Poisson processes isitself a Poisson process with mean equal to the sum of individualmeans. We re-analyse the data using the method described in thispaper, and the completeness simulations carried out by A20 toa reliability threshold of ln B > . The total absorption pathlength in the combined S20 and A20 surveys can be estimatedusing Equation 14, which for T s = 100 K we find that ∆ X = 15 ( ∆ z = 8 . ) for DLA column densities, and ∆ X = 120 ( ∆ z = 69 )for super-DLA column densities. Likewise, for T s = 1000 K theintervals are ∆ X = 0 . ( ∆ z = 0 . ) and ∆ X = 13 ( ∆ z =7 . ), sensitive to DLAs and super-DLAs, respectively. In Figure 6we show the differential absorption path length as a function ofredshift, giving the sensitivity function to 21-cm absorbers overthe interval spanned by the combined surveys. Given that the A20survey spanned redshifts between z = 0 . and . , there is acorresponding increase in the path length over this range.In Figure 7 we show the combined number of 21-cm absorbersexpected to be detected in both surveys as a function of spin temper-ature, and the updated spin temperature probability conditional ondetecting 4 absorbers. For the Jeffreys prior we find that T s < Kwith 95 per cent probability, increasing to 567 K for the alternative T s [K] − − − − µ a b s µ abs Pr( < T s |N abs = 4)MW ,
300 K . . . . . . P r( < T s | N a b s = ) Figure 7.
As Figure 5, but including data from the wide field survey of theGAMA23 field, undertaken as a FLASH early science project (A20). prior. Given that the blind survey of the GAMA 23 field did notyield any further detections of intervening 21-cm absorbers, ourupper limit on the harmonic mean spin temperature does increase,as one would expect. However, we caution that the sources in theGAMA 23 survey were selected purely based on their total fluxdensity, and so the distribution of covering factors for those sourcescould be significantly skewed to lower values than the uniform priorassumed here. It is therefore likely that the spin temperature is lowerthan this limit and closer to that determined from just the survey ofS20. T s in individual DLAs There have been several 21-cm absorption line surveys of knownindividual DLAs with the goal of directly measuring the spin tem-perature, although it should be noted that these too suffer from un-certainty in source covering factor. These targeted surveys have beencompiled and summarised into a single study by K14, who foundthat DLAs at z > . have a statistically (4 σ significance) differentdistribution of spin temperatures to those at lower redshifts, and areon average higher. They also found evidence (3.5 σ significance)of an anti-correlation between the spin temperature and metallic-ity in DLAs. These results are expected if the DLAs observed inthe earlier Universe are on average more relatively metal poor (e.g.Rafelski et al. 2012; Cooke et al. 2015; De Cia et al. 2018) andtherefore lacked sufficient coolants in the gas to form a significantfraction of CNM via fine structure cooling.In Figure 8 we plot as a function of redshift the 95 per centupper limit on the N HI -weighted harmonic mean spin temperaturefrom our statistical measurement, and the direct measurements fromthe DLA sample of K14. Using the Kaplan-Meier estimator of thesurvival function to include lower limits , we calculate the medianand 95 per cent confidence interval for the N HI -weighted harmonic We make use of the LifeLines survival analysis package ( https://github.com/CamDavidsonPilon/lifelines ).MNRAS , 1–12 (2020) J. R. Allison ρ m o l [ M (cid:12) M p c − ] z abs T s [ K ] This workKanekar et al . (2014) MW ,
300 K ρ mol Figure 8.
The N HI -weighted harmonic mean spin temperature versus ab-sorber redshift for this work (red point; 95 per cent upper limit) and individ-ual DLAs in the literature (blue points; K14). The blue dashed line (median)and shaded region (95 per cent confidence interval) are an estimate of the N HI -weighted harmonic mean in ∆ z = 1 bins from the K14 sample (usingsurvival analysis to include limits). Also shown are the evolution of themolecular gas density predicted by the best fitting curve to observationaldata by Walter et al. (2020) (solid magenta curve) and the depletion-timemodel of Tacconi et al. (2020) (dot-dashed magenta curve). mean in redshift bins of ∆ z = 1 . The results show an evolutionwith redshift towards larger spin temperatures that is consistent withthe conclusions of K14, and in the lowest redshift bin is consistentwith our upper limit. In a recent Green Bank Telescope (GBT) survey for intervening 21-cm absorbers towards 252 radio sources, spanning redshifts between z = 0 and . , Grasha et al. (2020) reported ten detections in a totalcomoving absorption path length sensitive to DLAs of ∆ X ∼ .By comparing their measurement of the cosmological H i massdensity from 21-cm absorbers with prior measurements at the sameredshifts, they obtained a mean spin temperature to covering factorratio of T s /c f ∼ K. For all values of c f ≤ their estimate of themean spin temperature is consistent with our 95 per cent upper limiton the harmonic mean. However, since Grasha et al. did not providea confidence interval for this measurement we cannot yet draw arobust statistical comparison. We note that the total absorption pathlength covered by the GBT survey is a factor ∼ greater thanour data (and is more sensitive to lower column densities), butachieved only a factor ∼ greater detection yield. This apparentinconsistency in outcomes between the surveys, and the inferredspin temperature, can be resolved by applying our method to thesedata in future work. It is now well established that the cosmic star formation rate den-sity underwent a rapid acceleration in the early Universe, peakedat z ≈ . − . , and then declined by a factor of − tothe present epoch (Madau & Dickinson 2014). Since stars form in self-gravitating clouds of dense molecular gas imbedded within theISM (McKee & Ostriker 2007), it is expected that a similarly strongevolution should be observed in the cosmological mass density ofthe molecular gas in galaxies. Observations of the bulk tracers ofmolecular gas in galaxies – CO emission (e.g. Decarli et al. 2019,2020; Lenkić et al. 2020; Riechers et al. 2020a,b; Fletcher et al.2021), supplemented by far-infrared and mm-wavelength observa-tions of the dust continuum (e.g. Berta et al. 2013; Scoville et al.2017; Magnelli et al. 2020) – support this expected strong evolution.In Figure 8 we show the best fitting parametric curve to theobservational data by Walter et al. (2020), alongside that expectedfrom the depletion-time model of Tacconi et al. (2020); both agreewithin the uncertainties of the data and show a peaked-evolutionthat mirrors that of the SFR density (albeit slightly delayed). Bybalancing the flow rates between phases of the baryonic matter,Walter et al. (2020) show that the observed changes in SFR andmulti-phase gas densities can be simply modelled by changes in thenet conversion rates of ionised to atomic gas, and atomic to molec-ular gas. Both appear to have declined by an order of magnitudesince z ≈ .However, when examined in detail the atomic gas in galaxiesis far more complex; it is multi-phased and spans two orders ofmagnitude in density and temperature (Wolfire et al. 2003). Asmentioned already in this paper, recent observations of the MilkWay ISM show the H i to exist in two stable phases in pressurebalance, the denser CNM ( T s ∼ K) and the more diffuse WNM( T s ∼
10 000
K), as well as a substantial fraction of unstable UNM( T s ∼ K) that is generated by dynamical processes such asturbulence and supernova shocks (Heiles & Troland 2003; Murrayet al. 2018). If the denser and cooler CNM is a pre-requisite formolecular cloud and star formation (e.g. Krumholz et al. 2009),then evolution of the conversion rate of H i to H could be matchedby a similar evolution in the relative fraction of the warm to coldphase atomic gas.Evolution in the physical state of the atomic gas would betraced by the H i spin temperature inferred from 21-cm absorptionline surveys. As yet no strong constraints on the detailed evolution of T s are possible from the existing data shown in Figure 8. However,we note that at intermediate cosmological redshifts ( z ∼ . − )both our upper limit on T s and the DLAs in the sample of K14have values that are consistent with being lower than that of theMilky Way ISM at z = 0 . This could indicate evolution of theISM towards higher CNM fractions at intermediate cosmologicalredshifts, as would be expected for a higher molecular gas density. We use a Bayesian technique to infer the harmonic mean spintemperature from a recent 21-cm absorption line survey with theASKAP telescope towards 53 compact radio sources (S20). Con-ditional on the outcome of four detections, we obtain a 95 per centupper limit on the harmonic mean spin temperature of T s ≤ Kin DLAs at . < z < . . This measurement is consistentwith 21-cm absorption line surveys of known individual DLAs atthe same redshift and is possibly indicative of higher fractions ofcold phase gas ( T ∼ K) at z ∼ than is typical of the localISM. Such a result would be expected if the cold fraction in galax-ies evolves similarly to the molecular and star forming densities,resulting from changes in the physical conditions of the interstellarmedium. However, larger 21-cm line surveys are required to verifysuch a claim. MNRAS000
K), as well as a substantial fraction of unstable UNM( T s ∼ K) that is generated by dynamical processes such asturbulence and supernova shocks (Heiles & Troland 2003; Murrayet al. 2018). If the denser and cooler CNM is a pre-requisite formolecular cloud and star formation (e.g. Krumholz et al. 2009),then evolution of the conversion rate of H i to H could be matchedby a similar evolution in the relative fraction of the warm to coldphase atomic gas.Evolution in the physical state of the atomic gas would betraced by the H i spin temperature inferred from 21-cm absorptionline surveys. As yet no strong constraints on the detailed evolution of T s are possible from the existing data shown in Figure 8. However,we note that at intermediate cosmological redshifts ( z ∼ . − )both our upper limit on T s and the DLAs in the sample of K14have values that are consistent with being lower than that of theMilky Way ISM at z = 0 . This could indicate evolution of theISM towards higher CNM fractions at intermediate cosmologicalredshifts, as would be expected for a higher molecular gas density. We use a Bayesian technique to infer the harmonic mean spintemperature from a recent 21-cm absorption line survey with theASKAP telescope towards 53 compact radio sources (S20). Con-ditional on the outcome of four detections, we obtain a 95 per centupper limit on the harmonic mean spin temperature of T s ≤ Kin DLAs at . < z < . . This measurement is consistentwith 21-cm absorption line surveys of known individual DLAs atthe same redshift and is possibly indicative of higher fractions ofcold phase gas ( T ∼ K) at z ∼ than is typical of the localISM. Such a result would be expected if the cold fraction in galax-ies evolves similarly to the molecular and star forming densities,resulting from changes in the physical conditions of the interstellarmedium. However, larger 21-cm line surveys are required to verifysuch a claim. MNRAS000 , 1–12 (2020)
I spin temperature at cosmological distances For the data considered in this work, the total absorption pathlength sensitive to 21-cm absorption is only ∆ X ∼ − .However, future large-scale surveys with the pathfinder telescopes tothe Square Kilometre Array, including ASKAP FLASH (e.g. A20,Allison et al. in preparation) and the MeerKAT Absorption LineSurvey (MALS; Gupta et al. 2016), are expected to span intervalsthat are three orders of magnitude larger ( ∆ X ∼ − ) andwill provide correspondingly stronger constraints on the evolutionof cold gas in galaxies. Applying the method presented in this paperto these larger surveys is expected to constrain the harmonic meanspin temperature to a precision of ∼
10 per cent (Allison et al. 2016)and will allow self-consistent direct measurements of evolution ofthe cold phase gas in several redshift bins.
ACKNOWLEDGEMENTS
We are grateful to Robert Allison and Hengxing Pan for usefuldiscussions on Bayesian priors. We also thank Elaine Sadler forhelpful comments on a previous version of the manuscript. Finally,we thank the reviewer, Jeremy Darling, for comments that helpedimprove the presentation and clarity of the paper.JRA acknowledges support from a Christ Church Career De-velopment Fellowship. Parts of this research were conducted by theAustralian Research Council Centre of Excellence for All-sky Astro-physics in 3D (ASTRO 3D) through project number CE170100013.This work was supported by resources provided by the PawseySupercomputing Centre with funding from the Australian Govern-ment and the Government of Western Australia, including compu-tational resources provided by the Australian Government under theNational Computational Merit Allocation Scheme (project JA3).The Australian SKA Pathfinder is part of the Australia Tele-scope National Facility which is managed by CSIRO. Operationof ASKAP is funded by the Australian Government with supportfrom the National Collaborative Research Infrastructure Strategy.ASKAP uses the resources of the Pawsey Supercomputing Centre.Establishment of ASKAP, the Murchison Radio-astronomy Obser-vatory and the Pawsey Supercomputing Centre are initiatives ofthe Australian Government, with support from the Government ofWestern Australia and the Science and Industry Endowment Fund.We have made use of Astropy, a community-developed corePYTHON package for astronomy (Astropy Collaboration et al.2013), and NASA’s Astrophysics Data System Bibliographic Ser-vices.
DATA AVAILABILITY
The data underlying this article were accessed from the AustraliaTelescope National Facility (ATNF). The derived data generated inthis research were originally published by Sadler et al. (2020) andAllison et al. (2020), and will be shared on reasonable request tothe corresponding author.
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