IGM Transmission Bias for z \geq 2.9 Lyman Continuum Detected Galaxies
R. Bassett, E.V. Ryan-Weber, J. Cooke, U. Meลกtri?, K. Kakiichi, L. Prichard, M. Rafelski
MMNRAS , 1โ20 (2021) Preprint 5 January 2021 Compiled using MNRAS L A TEX style ๏ฌle v3.0
IGM Transmission Bias for ๐ง โฅ R. Bassett , โ , E. V. Ryan-Weber , , J. Cooke , , U. Meลกtriฤ , , K. Kakiichi , ,L. Prichard , M. Rafelski , , Centre for Astrophysics and Supercomputing, Swinburne University of Technology, PO Box 218, Hawthorn VIC 3122, Australia ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia Department of Physics, University of California, Santa Barbara, CA 93106, USA Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218, USA Department of Physics & Astronomy, John Hopkins University, Baltimore, MD 21218, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Understanding the relationship between the underlying escape fraction of Lyman continuum(LyC) photons ( ๐ esc ) emitted by galaxies and measuring the distribution of observed ๐ esc values at high redshift is fundamental to the interpretation of the reionization process. Inthis paper we perform a statistical exploration of the attenuation of LyC photons by neutralhydrogen in the intergalactic medium using ensembles of simulated transmission functions.We show that LyC detected galaxies are more likely to be found in sightlines with higher-than-average transmission of LyC photons. This means that adopting a mean transmissionat a given redshift leads to an overestimate of the true ๐ esc for LyC detected galaxies. Wenote, however, that mean values are appropriate for ๐ esc estimates of larger parent samples thatinclude LyC non-detected galaxies. We quantify this IGM transmission bias for LyC detectionsin photometric and spectroscopic surveys in the literature and show that the bias is stronger forboth shallower observations and for fainter parent samples (i.e. Lyman ๐ผ emitters versus Lymanbreak galaxies). We also explore the e๏ฌects of varying the underlying probability distributionfunction (PDF) of ๐ esc on recovered values, showing that the underlying ๐ esc PDF may dependon sample selection by comparing with observational surveys. This work represents a ๏ฌrst stepin improved interpretation of LyC detections in the context of understanding ๐ esc from highredshift galaxies. Key words: intergalactic medium โ galaxies: ISM โ dark ages, reionization, ๏ฌrst stars
Understanding the details of cosmic reionization, the epoch at ๐ง (cid:39) โ
10 during which the hydrogen content of the intergalac-tic medium (IGM) transitioned from neutral to mostly ionized (e.g.Fan et al. 2006; Planck Collaboration et al. 2016; Greig & Mesinger2017; Mason et al. 2018), is a major goal of the international as-tronomical community. The general consensus currently favoursa picture in which ionizing, or Lyman continuum (LyC), photonsoriginating from young, massive stars and/or X-ray binaries andWolf-Rayet stars in star-forming galaxies are the primary driver.This picture is supported by extensive theoretical (e.g. Wise & Cen2009; Yajima et al. 2011; Paardekooper et al. 2015) and observa-tional (e.g. Inoue et al. 2006; Ouchi et al. 2009; Robertson et al.2015) e๏ฌorts. Active galactic nuclei (AGN), though pridigious pro-ducers of LyC emission, are expected to play only a minor role due โ E-mail: [email protected] (RB) to their low number density at ๐ง > ๐ esc ). The ๏ฌrst major challenge in using ๐ esc to un-derstand reionization is the fact that no LyC photons from galaxiesduring the Epoch of Reionization (EoR) will ever reach a telescopedue to absorption from intervening hydrogen. The second is theinherent faintness of LyC emission from galaxies (as demonstratedby pioneering works of Giallongo et al. 2002; Fernรกndez-Soto et al.2003; Inoue et al. 2005),which is driven largely by two key factors.The ๏ฌrst factor driving the faintness of LyC emission is that ๐ esc is typically found to be very low (or zero) as inferred from the lackof LyC detections in (e.g. Boutsia et al. 2011; Japelj et al. 2017; Bian& Fan 2020). This may, in part, be due to the fact that observationsof galaxies, and thus, their LyC emission, at high redshift ( ๐ง โฅ . ยฉ a r X i v : . [ a s t r o - ph . GA ] J a n R. Bassett et al. are limited to relatively high stellar mass ( ๐ โ โฅ ๐ (cid:12) ) galaxiesthat are likely to contain signi๏ฌcant quantities of neutral hydrogen(consistent with their high star-formation rates, SFRs, e.g. Steidelet al. 2001; Iwata et al. 2009; Nestor et al. 2011; Grazian et al.2016) that absorbs ionizing radiation before it can enter the IGMand drive reionization. Indeed, for the small sample of such knownLyC emitting galaxies at ๐ง (cid:38) .
8, the observed LyC ๏ฌux is relativelyfaint (e.g. Shapley et al. 2006; Micheva et al. 2017; Vanzella et al.2018). Even if ๐ esc is larger in lower mass galaxies, such galaxiesare inherently faint and their LyC emission will likely be at leastas di๏ฌcult to detect as their higher mass counterparts (apart fromthe rare cases of strong gravitational lensing, Bian et al. 2017;Rivera-Thorsen et al. 2019). The most straightforward way pastthis problem is to perform larger and deeper surveys targeting LyCemission across a range of redshifts. A variety such surveys arecurrently in progress.The second issue resulting in faint LyC emission is that theIGM itself contains large fractions of neutral hydrogen above ๐ง (cid:39) ๐ IGM ) of LyC photons for thatparticular sightline. This is troubling as observationally ๐ IGM and ๐ esc are degenerate meaning that, in order to estimate ๐ esc , a valueof ๐ IGM must be assumed that may or may not be appropriate for agiven IGM sightline. There is, however, hope of a way forward as thedi๏ฌerential column density distribution of HI absorption systems iswell constrained (e.g. Meiksin 2006; Becker et al. 2013; Rudie et al.2013), providing a statistical description of the probability that LyCphotons escaping galaxies will be absorbed by hydrogen in the IGMat a given redshift.Such a statistical approach to estimate ๐ IGM in a theoreticalcontext has been explored using Monte Carlo (MC) simulations foraround three decades (e.g. Mรธller & Jakobsen 1990; Bershady et al.1999; Inoue et al. 2014). Similarly, the application of such MC sim-ulations of ๐ IGM to detections (and non-detections) of LyC radiationhas a long history (e.g. Shapley et al. 2006; Siana et al. 2007; Steidelet al. 2018, S18 hereafter) In general, the most probable value of ๐ IGM at ๐ง > ๐ IGM > ๐ IGM (around ๐ ๐๐๐ ๐ก โผ
910 ร ) at ๐ง = 2.9-4.0 can be describedas bimodal with a sharp peak at ๐ IGM = . ๐ง โผ ๐ IGM sightlines is negligible. The resultis that LyC is unlikely to be observed from galaxies during the EoR.Using knowledge of the probability distribution of ๐ IGM ata given redshift, astronomers can put forward an estimate of ๐ esc for LyC detected galaxies. One method is to apply the full suiteof ๐ IGM models to a given observation (or set of observations),however this typically results in largely unconstrained ๐ esc valuesincluding a large number with the unphysical case of ๐ esc > ๐ IGM , (cid:104) ๐ IGM (cid:105) , among all simulatedsightlines thus providing a single ๐ esc value (S18, Bassett et al.2019; Fletcher et al. 2019; Meลกtriฤ et al. 2020, hereafter F19 andM20). The problem with this second method is that a single statisticbelies to complexities of the underlying ๐ IGM distribution. Indeed,the mean of a bimodal distribution will be found to lie betweenthe two peaks, and will not fall among the most likely values. Thisissue has been highlighted in the context of Ly ๐ผ transmission byByrohl & Gronke (2020) who ๏ฌnd that assuming a median or mean transmission curve โis misleading and should be interpreted withcautionโ.There exist, however, important observational priors that canprovide more realistic constraints on the most likely value of ๐ IGM for LyC detected galaxies. First and foremost, the fact that a galaxyhas been detected at LyC wavelengths means that ๐ IGM for thatgalaxy cannot be zero. This fact automatically reduces the under-lying bimodal ๐ IGM distribution for all sightlines to a unimodaldistribution for sightlines with LyC detections. In this case, stan-dard statistics such as the mean and median of ๐ IGM may be moreapplicable. Secondly, while the probability distribution function(PDF) of ๐ IGM is routinely considered, the underlying PDF of ๐ esc itself, which so far has been left out, may also be important. Aswe have stated, low or zero ๐ esc values seem to be preferred, whichis not re๏ฌected in current ๐ esc calculations. It is possible that thebroad behaviour of the ๐ esc PDF may be inferred through consid-eration of the detection rates in LyC surveys (this intriguing idea isexplored further in Section 4.3). It is likely that a full understandingof the underlying ๐ esc PDF of galaxies will require a theoreticalunderpinning through the careful analysis of high-resolution, hy-drodynamical simulations employing radiative transfer of ionizingphotons (e.g. Trebitsch et al. 2017; Rosdahl et al. 2018; Ma et al.2020).In this paper, we explore in detail the probability distributionsof both ๐ IGM and ๐ esc in the context of known LyC surveys athigh redshift. Our goal is to provide a statistically sound frameworkwithin which astronomers can calculate meaningful estimates of ๐ esc for both individual LyC detections as well as stacked samples. Inparticular, we show that both the assumption of the mean ๐ IGM valueand (to a lesser extent) a lack of consideration of the underlying ๐ esc PDF result in an overestimate of ๐ esc for LyC detected galaxies. Herewe quantify the IGM transmission bias, ๐ bias , as (cid:104) ๐ det (cid:105) โ (cid:104) ๐ IGM (cid:105) where (cid:104) ๐ det (cid:105) is the average IGM transmission for LyC detectedgalaxies for a given observational detection limit. We note that,although a transmission value is not inherently an additive quantity,our de๏ฌnition leads to a roughly redshift independent correctionto (cid:104) ๐ IGM (cid:105) as opposed to an alternative de๏ฌnition such as ๐ bias = (cid:104) ๐ det (cid:105)/(cid:104) ๐ IGM (cid:105) (see Section 3 for further discussion).This paper is organised as follows: in Section 2 we describeour method of generating simulated IGM sightlines and spectra ofmock LyC emitting galaxies, in Section 3 we describe the resultsof our various models, in Section 4 we explore the implications ofour results in the context of past and on-going LyC surveys, and inSection 5 we provide a brief summary of our ๏ฌndings.
In this Section we describe our method of producing mock obser-vations of LyC ๏ฌux from high redshift galaxies. There are threeprimary ingredients in creating an individual high redshift galaxyobservation for our simulation: an IGM transmission function, ๐ esc ,and the input SED model. Our method for producing an IGM trans-mission function is described in Section 2.1. Although the underly-ing PDF of ๐ esc for galaxies is largely unknown, we test two modelsdescribed in Section 2.2. Finally, we take our input SED model fromBPASSv2.1 (Eldridge et al. 2017, described further in Section 2.3),matching the assumed LyC to non-ionizing UV ๏ฌux ratio from pre-vious studies. In particular we compare with results from the KeckLyman Continuum Survey (KLCS, S18), the LymAn ContinuumEscape Survey (LACES, F19), and the ground based photometricwork of M20 based on deep ๐ข -band photometry from the Canada MNRAS000
In this Section we describe our method of producing mock obser-vations of LyC ๏ฌux from high redshift galaxies. There are threeprimary ingredients in creating an individual high redshift galaxyobservation for our simulation: an IGM transmission function, ๐ esc ,and the input SED model. Our method for producing an IGM trans-mission function is described in Section 2.1. Although the underly-ing PDF of ๐ esc for galaxies is largely unknown, we test two modelsdescribed in Section 2.2. Finally, we take our input SED model fromBPASSv2.1 (Eldridge et al. 2017, described further in Section 2.3),matching the assumed LyC to non-ionizing UV ๏ฌux ratio from pre-vious studies. In particular we compare with results from the KeckLyman Continuum Survey (KLCS, S18), the LymAn ContinuumEscape Survey (LACES, F19), and the ground based photometricwork of M20 based on deep ๐ข -band photometry from the Canada MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections Figure 1.
Left:
Example IGM transmission function for a galaxy at ๐ง = .
1. In gold and cyan are single transmission functions with highest and lowest ๐ IGM at880 < ๐ <
910 ร (range indicated by vertical, cyan, dotted lines) among our ensemble of 10,000 transmission functions at ๐ง = .
1. The black curve shows theaverage transmission for the entire ensemble.
Right:
The mean transmission of Ly ๐ผ (1210 < ๐ < < ๐ <
910 ร , gold) asa function of redshift for our simulated IGM transmission functions. Error bars indicate the range containing 68.1% of all values about the median in each bin.We note that mean and median values di๏ฌer given the complex, bimodal underlying distribution. Here we also compare to theoretical and observational workin the literature from Becker et al. (2013), Meiksin (2006), Inoue et al. (2014), and S18.
Figure 2.
A full statistical description of our 10,000 IGM transmissionfunctions at ๐ง = .
1. The shading represents the probability of a given ๐ IGM value at each wavelength with probability increasing from black to gold(note the colour scaling is logarithmic). Blueward of the Lyman limit (911.8ร , white dotted line) ๐ IGM is strongly peaked at ๐ IGM = 0. The behaviour at๏ฌxed ๐ shifts from unimodal at the shortest wavelengths to bimodal redwardof โผ
880 ร . For illustration we show the median and mean ๐ IGM functionsin black and cyan.
France Hawaii Telescope (CFHT) Large Area U-band Deep Survey(CLAUDS, Sawicki et al. 2019). ๐ IGM functions are produced following the method outlined in S18,Appendix B . We perform a Poisson sampling of the number of HIabsorbers in redshift intervals, ฮ ๐ง , from ๐ง = All code for producing IGM transmission curves is open source andavailable at https://github.com/robbassett/TAOIST_M C. ๐ง ๐๐ . Following Inoue et al. (2014) we select a value of ฮ ๐ง = ร โ ,noting however that deviations from this value would not a๏ฌect ourresults. The value of ๐ง ๐๐ for a given analysis is determined by theredshift of the galaxy, or sample of galaxies, being considered. Inthis work we create suites of 10,000 IGM transmission functions at10 discrete ๐ง ๐๐ values in the range 2.9-3.9 with ฮ ๐ง ๐๐ = . ๐ง = . ๐ง = . ๐ IGM function at a given ๐ง ๐๐ we must๏ฌrst produce a random sampling of hydrogen absorption systems inredshift bins of ฮ ๐ง = ร โ from ๐ง = ๐ง ๐๐ . This is achievedassuming a di๏ฌerential HI column density distribution, ๐ ( ๐ HI , ๐ ) ,following the prescriptions outlined for the โIGM+CGMโ model inS18 Appendix B. In each redshift interval we derive the expectednumber of absorption systems in each bin of ๐๐๐ ( ๐ HI ) (sampledfrom ๐๐๐ ( ๐ ๐ป ๐ผ ) = . โ . ฮ ๐๐๐ ( ๐ HI ) = .
1) as: ๐ abs = โซ ๐ HI , max ๐ HI , min โซ ๐ง + ฮ ๐ง๐ง ๐ โ ๐ฝ HI ๐ด ( + ๐ง ) ๐พ ๐๐ HI ๐๐ง (1)Where ๐ HI , min and ๐ HI , max are the lower and upper bounds, ๐ฝ isthe slope of ๐ ( ๐ HI , ๐ ) , ๐ด is a constant chosen to match observed ๐ abs , and ๐พ describes the redshift evolution of ๐ abs . Values for ๐ฝ , ๐ด , and ๐พ are taken directly from Table B1 of S18. We assume thepresence of absorption systems is a Poissonian process, thus foreach sightline the number of absorption systems at a given ๐ง and ๐ HI is calculated using numpy.random.poisson with ๐ set to ๐ abs .For each individual absorber in a given observed sightline, wethen apply the transmission function for LyC photons at ๐ rest โค . ๐ rest โฅ . not the LyCemitting galaxy. For LyC photons we apply the functional form: ๐ ๐ฟ๐ฆ๐ถ HI ( ๐ rest ) = ๐ HI ๐ HI ( ๐ rest ) (2)where ๐ rest is the photon frequency at the rest frame of agiven absorbtion system and ๐ HI ( ๐ rest ) is the frequency depen-dent interaction cross section of HI to ionising photons given by ๐ ๐ฟ ( ๐ rest / ๐ . ) โ . Here ๐ ๐ฟ is a constant with a value of 6 . ร โ cm (Osterbrock 1989). For Lyman series lines we use the following MNRAS , 1โ20 (2021)
R. Bassett et al. for each Lyman transition, ๐ (e.g. Inoue & Iwata 2008): ๐ ๐ ( ๐ rest ) = ๐ HI โ ๐๐ ๐ ๐ ๐ ๐ ๐๐ ๐ท ๐ ๐ ( ๐ rest ) (3)where ๐ ๐ and ๐ are the electron mass and charge, respectively, and ๐ is the speed of light. The parameter ๐ ๐ is the oscillator strengthof Lyman transition ๐ , which we take from tables provided with theVPFIT package (Carswell & Webb 2014). In our calculation weinclude the ๏ฌrst 32 Lyman series transitions. ๐ ๐ท = ๐ ๐ ( ๐ / ๐ ) is theDoppler broadening of the Lyman line at frequency ๐ ๐ where ๐ ,the Doppler parameter, is randomly sampled from (Hui & Rutledge1999): โ ( ๐ ) = ๐ ๐ ๐ ๐ โ ๐ ๐ / ๐ (4)with ๐ ๐ =
23 km s โ (e.g. Janknecht et al. 2006). Finally, ๐ ๐ ( ๐ ) ,the absorption pro๏ฌle, is taken as the analytic approximation ofthe Voigt pro๏ฌle given by Tepper-Garcรญa (2006). Here, as with ๐ ๐ ,we also sample ฮ ๐ , the damping constant for transition ๐ , from theVPFIT values. The total optical depth of an individual absorber isthen taken as ๐ ( ๐ ) = ๐ ๐ฟ๐ฆ๐ถ HI ( ๐ rest ) + ฮฃ ๐ ๐ ( ๐ rest ) , where ๐ refers to theobserved frame frequency, ๐ = ๐ rest /( + ๐ง ) . The total ๐ ( ๐ ) for agiven sightline is the sum of the ensemble of ๐ ( ๐ ) for all absorbersin that sightline.It is worth mentioning that our transmission curves are pro-duced as a function of wavelength, rather than frequency, and weemploy a ๏ฌxed resolution of ฮ ๐ = . ๐ IGM = ๐ โ ๐ , rather than considering ๐ computed asdescribed above. The reasons being ๏ฌrst that the value of ๐ IGM istypically included in calculations of ๐ esc and second that ๐ IGM hasa dynamic range between 0 and 1, which provides more intuitivecomparisons. We note that throughout this paper the symbol ๐ IGM may refer to a wavelength dependent transmission function or asingle value at some speci๏ฌed wavelength. We avoid introducing anexplicitly wavelength dependent symbol, i.e. ๐ IGM ( ๐ ) , as the usagehere is consistent with the conventions in the literature (e.g. Inoue& Iwata 2008).Example IGM transmission functions at ๐ง = . ๐ง = . ๐ rest =
910 ร transmission, respectively. At a given redshiftthe transmission of LyC in the IGM may vary from 0.0 to nearly1.0. In the right panel we show the redshift evolution of the meanLy ๐ผ and LyC transmission predicted by TAOIST-MC in comparisonwith observational and theoretical estimates from the literature. Inall cases, our model agrees, within errors, with previously reportedresults.We note that our measurements are systematically lower thansome previous results, which can be attributed to the inclusion ofthe circumgalactic medium component introduced in S18. Further-more, a single statistic (such as the mean) belies the complexity ofthe underlying ๐ IGM distribution as shown in Figure 2. Thus, we
Figure 3.
A comparison of the two ๐ esc PDFs used in this work. The โFlatโdistribution represents the case of no assumed prior when calculating ๐ esc and is representative of most studies in the literature. The alternative exploredhere is an exponentially declining models of the form PDF โ ๐ โ / ๐ , hereshown with ๐ = 0.50. do not place a large emphasis on di๏ฌerences between the averagevalues of ๐ IGM between di๏ฌerent studies. For theoretical ๐ IGM func-tions this behaviour may be, in part, attributed to the exact form ofthe di๏ฌerential ๐ HI distribution assumed and the details of the im-plementation. For example, Inoue & Iwata (2008) and S18 assumedi๏ฌerent behaviours for the exponent ๐ฝ of ๐ ( ๐ HI , ๐ ) producingdi๏ฌerent relative numbers of low and high ๐ HI systems. These dif-ferences will a๏ฌect the ๐ IGM of LyC and Ly ๐ผ di๏ฌerently and willappear as complex systematic o๏ฌsets between the mean ๐ IGM at agiven redshift between the two implementations. It should also bementioned that, to our knowledge, no study employing MC simula-tions of IGM transmission curves have accounted for the e๏ฌects ofHI clustering, which may further alter the mean ๐ IGM curve (see,however, Kakiichi & Dฤณkstra 2018, who demonstrate Ly ๐ผ may bemore attenuated from galaxies in high density environments).Di๏ฌerences in the behaviour between theoretical ๐ IGM imple-mentations are only apparent from the mean transmission curveswhile individual IGM transmission curves are likely indistinguish-able. The implications regarding the statistical behaviour of IGMsightline ensembles, however, is precisely the topic of this paper. Aswe will repeat, the absolute values of quantities calculated through-out will be imprinted with the assumptions regarding our ๐ HI dis-tribution sampling and may change slightly if di๏ฌerent implemen-tations are used. Thus, it is key to keep in mind that the absoluteresults are for our implementation only. Qualitatively, however ourresults are independent of the various input parameters. ๐ esc Distribution Functions
One of the key unknowns in this study is the distribution func-tion of ๐ esc for galaxies at ๐ง โฅ .
9. While quantifying ๐ esc fromgalaxies has been a long standing goal in the astrophysics of reion-ization, this parameter remains elusive. In a broad sense, a numberof studies have estimated the average ๐ esc required for all galax-ies in order to match the constraints on the timing of reionization, MNRAS000
9. While quantifying ๐ esc fromgalaxies has been a long standing goal in the astrophysics of reion-ization, this parameter remains elusive. In a broad sense, a numberof studies have estimated the average ๐ esc required for all galax-ies in order to match the constraints on the timing of reionization, MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections Figure 4.
Input 1500 ร absolute magnitude distributions for LBG (cyan)and LAE (gold) samples. Open histograms represent LRIS 1500 ร ๏ฌuxesfrom S18 and UV magnitudes from ground-based imaging reported in F19for LBGs and LAEs, respectively. Filled histograms represent one realisa-tion of 10,000 sampled values for our mock galaxy spectra produced usingcdf_sampler.py (see footnote 2) with the open histograms as inputs. Valueson the left y-axis refer to observed samples (open histograms) and on theright y-axis refer to mock samples (closed histograms), noting in the lattercase these values are based on an arbitrarily selected โparent sampleโ size. ๏ฌnding values in the range 0.05 < (cid:104) ๐ esc (cid:105) < ๐ esc over their lifetime seems van-ishingly small (Kimm & Cen 2014; Paardekooper et al. 2015, seealso Section 4.6 for a brief discussion of the 3D versus line-of-sight ๐ esc values).Given the lack of strong constraints on ๐ esc from the literature,for the ๏ฌducial model of our analysis, presented in Section 3.1,we simply uniformly apply values of ๐ esc between 0.0 and 1.0 toour mock spectra. This allows for mock spectra with the highestpossible LyC ๏ฌux for a given IGM sightline, representing the mostlikely galaxies to be detected in a LyC survey. As such, the resultsof our ๏ฌducial model should be interpreted as the minimum level of ๐ IGM bias expected for LyC detected galaxies.It seems most likely that allowing extremely high ๐ esc is un-realistic for the vast majority of real galaxies (e.g. Vanzella et al.2010; Siana et al. 2015; Japelj et al. 2017). In Section 3.2 we testthe e๏ฌects on our measured ๐ IGM bias of applying an additional,more realistic ๐ esc distribution, to our simulations. For this test, weassume an exponentially declining ๐ esc PDF, i.e. ๐ ( ๐ esc ) โ ๐ โ / ๐ ,resulting in a model with the most probable value of ๐ esc beingzero. For our exponentially declining ๐ esc PDF we choose a valueof ๐ = .
5, which is motivated by the observed detection rates ofKLCS (S18, see Section 4.3). We illustrate the relative PDF shapesof our ๏ฌducial and exponentially declining models in Figure 3 forclarity.
As mentioned above, the process of producing mock galaxy spectrafor our simulations requires three inputs: an underlying SED model,an IGM attenuation function, and a value for ๐ esc ( ๐ฟ๐ฆ๐ถ ) . We notethat in much of this work we ignore the e๏ฌects of dust attenuation(see, however, Section 4.4, simply noting that most LyC detectionsappear to originate from relatively dust free galaxies (e.g. S18). Similar to S18 we construct our SEDs from the BPASSv2.1 (El-dridge et al. 2017) models with ๐ โ = . ๐ผ = โ . ๐ (cid:12) . We employ a model with an expo-nentially declining SFR with an ๐ -folding time of 0.1 Gyr sampledat an age of โผ
200 Myr. This provides an input spectrum with anintrinsic LyC to UV ๏ฌux ratio, ( ๐ฟ / ๐ฟ ) ๐๐๐ก , of 0.18 (e.g. S18).Our SED model corresponds to a LyC photon production e๏ฌciency, ๐ ๐๐๐ , of log ( ๐ ๐๐๐ ) = 25.61 Hz erg โ , consistent with estimatesfor high redshift star-forming galaxies (e.g. Bouwens et al. 2016).We explore the e๏ฌect of altering ( ๐ฟ / ๐ฟ ) ๐๐๐ก on our results inSection 3.3.Each mock spectrum is scaled such that the non-ionizing UV๏ฌux matches a randomly sampled value characteristic of high red-shift, highly star-forming galaxies. The sampling of UV ๏ฌuxes isone key factor in our analysis as this ultimately determines the in-trinsic level of LyC ๏ฌux from galaxies in our mock samples. In thiswork we test samples taken two di๏ฌerent UV ๏ฌux distributions: onebased on the full sample of galaxies observed by the KLCS, whichis composed of a representative subsample of bright Lyman BreakGalaxies (LBGs) at 2.9 < ๐ง < ๐ง โผ ๐ผ emitters (LAEs) characteristic of galaxiestargeted by LACES (F19). For our LBG comparison UV valuesused in our work are sampled from measurements of LRIS spectraat ๐ rest = . We note, however, that some LBGs have beenshown to also exhibit Ly ๐ผ emission (e.g. Shapley et al. 2003), thusLBG and LAE classi๏ฌcations are based on selection methodology.Here, the important distinction is the relative non-ionizing UV ๏ฌuxwith LBGs being signi๏ฌcantly brighter.The sample of S18 covers a redshift range of ๐ง (cid:39) ฮ ๐ง = . ๐ง = . ๐ง = .
9. Thus, we must includea method to account for cosmological dimming of each of thesesamples when considering higher redshifts. In each case, we beginwith the absolute magnitude distributions shown in Figure 4 andassume that this distribution is roughly representative of a similarlyselected sample in each ฮ ๐ง = . ๐ esc , the latter following Section2.2. The current ๐ IGM function is applied to the input BPASSv2.1spectrum, then at all wavelengths shortward of 911.8 ร it is scaleduniformly by the randomly selected ๐ esc value. The resulting spec-trum is then scaled to match the randomly selected 1500 ร ๏ฌux. In both cases sampling of UV ๏ฌuxes is achieved usingthe histogram_oversampler class of the code cdf_sampler.py(https://github.com/robbassett/cdf_sampler) with spline ๏ฌtting enabled toremove sharp edges of the histogram binsMNRAS , 1โ20 (2021)
R. Bassett et al.
Figure 5.
Top:
Example BPASSv2.1 spectra used in this study. In black is shown the input spectrum and in gold and cyan we show the output spectra with thehigh and low IGM transmission curves shown in Figure 1. In this panel, both spectra are shown for the ๐ esc = 1.0 case. Bottom:
The e๏ฌect of our ๏ฌat treatmentof ๐ esc on the output spectra for the high IGM transmission spectrum shown in the top panel with ๐ esc varying from 0.0 to 1.0. For all spectra in both panels,we show the ๏ฌux in ๐น ๐ normalised to the ๏ฌux at a rest wavelength of 1500 ร . Thus, in each redshift bin we produce one million galaxy spec-tra ensuring that the 1500 ร ๏ฌux and ๐ esc distributions are wellsampled. Example spectra can be seen in Figure 5.We can summarise the construction of each individual mockspectrum with the following equation: ๐น ๐, ๐๐ ( ๐, ๐ง ) = ๐น ๐,๐๐๐ ( ๐ ) ๐น ๐ ,๐๐๐ ๐น ,๐๐๐ ๐ ๐ IGM ( ๐ ) ๐ ๐ esc ( ๐ ) (5)where ๐น ๐,๐๐๐ ( ๐, ๐ง ) is the input BPASS spectrum, ๐น ๐ ,๐๐๐ is the ๐ th randomly sampled 1500 ร ๏ฌux (noting again that here we haveincluded cosmological dimming), ๐น ,๐๐๐ is the 1500 ร ๏ฌux ofthe BPASS model (taken as the mean value at 1450 < ๐ < ๐ ๐ IGM is the current IGM transmission curve (we use the superscript ๐ to indicate that the same IGM transmission curve will be used 100times, thus it is not unique to mock spectrum ๐ ), and ๐ ๐ esc ( ๐ ) is astep function representation of the ๐ th randomly sampled ๐ esc valuegiven as: ๐ ๐ esc ( ๐ ) = (cid:40) ๐ ๐ esc if ๐ < .
81 if ๐ โฅ . The primary results of this paper concern quantifying the obser-vational bias in IGM transmission for samples of LyC detectedgalaxies. We reiterate that the initial results are based on tests per-formed on a ๏ฌducial dust-free, exponentially declining SFR SEDmodels at ๏ฌxed metallicity, IMF slope, and age (see Section 2.3for a full description). We have selected our ๏ฌducial model to have( ๐ฟ / ๐ฟ ) ๐๐๐ก โผ ๐ esc between 0 and 1.0. High ๐ esc will correspond to a bright LyC ๏ฌux,thus, we expect a preference towards detections at high ๐ esc in our๏ฌducial model. ๐ esc for real galaxies will be, on average, lower thanthe average of our ๏ฌducial model given the typically low value forobserved LyC emitters (e.g. S18). This means that the level of biasin ๐ IGM for detections seen in our ๏ฌducial model can be seen as alower limit to the true bias for observed galaxy samples.We explore the quantitative e๏ฌects of both altering the inputPDF of ๐ esc and changing the value of ( ๐ฟ / ๐ฟ ) ๐๐๐ก in Sections3.2 and 3.3, respectively. In the case of SED variations we test SEDswith ๐ ๐๐๐ values covering the range for exponentially declining SFRmodels using BPASSv2.1 spectra over available range of stellarpopulation ages provided. Here we quantify the bias in ๐ IGM a๏ฌecting samples of LyC detectedgalaxies when compared with the average ๐ IGM of all random sight-lines. Formally, we de๏ฌne this bias as: ๐ bias = (cid:104) ๐ det (cid:105) โ (cid:104) ๐ IGM (cid:105) (7)where (cid:104) ๐ det (cid:105) is the average ๐ IGM for galaxies with LyC detectedabove a speci๏ฌed detection limit and (cid:104) ๐ IGM (cid:105) is the average ๐ IGM for all sightlines. It is worth noting that transmission values are notinherently additive quantities and it could be argued that the def-inition ๐ bias = (cid:104) ๐ det (cid:105)/(cid:104) ๐ IGM (cid:105) is more sensible, and possibly morephysically motivated as it relates directly to a di๏ฌerence in opticaldepth/HI column density. Our choice of de๏ฌnition is motivated bythe fact that the resulting ๐ bias values are roughly redshift indepen-dent at ๏ฌxed observational detection limit (see, e.g., Section 3.1.3),providing a simpli๏ฌed framework for applying ๐ bias to a given set ofobservations. We also point out that, by de๏ฌnition, such a correctionwill never result in an unphysical transmission value for LyC detec-tions > ๐ bias with redshift whenassuming a fractional de๏ฌnition is primarily re๏ฌective of the red-shift evolution of (cid:104) ๐ IGM (cid:105) as one is dividing by a value increasingly
MNRAS000
MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections close to zero. Regardless, either ๐ bias de๏ฌnition mentioned herewill provide an equivalent correction, thus the choice is somewhatarbitrary.In this work, the calculation of ๐ bias is performed at 11 discreteredshifts in the range 2 . โค ๐ง โค . ฮ ๐ง = .
1. We also notethat, similar to ๐ IGM and (cid:104) ๐ IGM (cid:105) , ๐ bias can refer to a wavelengthdependent function, a single value at a speci๏ฌed wavelength, or anaverage value across a speci๏ฌed wavelength range. Due to tech-nical di๏ฌerences between LyC searches employing spectroscopy(e.g. S18) and photometry (F19, M20), we present the two casesseparately: spectroscopic biases are presented in Section 3.1.1 andphotometric biases are presented in Section 3.1.2. In all cases, wehave performed this experiment twice: once for an LBG-like sampleand once for a fainter, LAE-like sample (see Figure 4). Due to the in-herent faintness, our mock LAE samples are typically only detecteddeep HST F336W observations, which compare to F19 (particu-larly in our higher redshift bins) who achieve a depth of 30.24 mag.Thus, in most cases we only provide ๐ bias measurements for thiscomparison (as opposed to spectroscopy or CFHT ๐ข photometry).We provide a summary of our ๏ฌducial model in Section 3.1.3. Spectroscopic detection of LyC radiation provides a key advantageover photometric detections in terms of interpretation in the contextof estimating ๐ esc . The reason being that spectroscopy allows one toprobe the same rest frame wavelengths just shortward of the Lymanlimit, typically probed between 880-910 ร , independent of redshiftin theory. In practice, of course, the redshift range in which LyC canbe probed by a given set of spectroscopic observations is de๏ฌned bythe wavelength coverage of the instrument used. Furthermore, thedetection limits of a given instrument will be wavelength dependentdue to response variations of the detector. Thus, the experimentpresented here should be considered as a simulation of an idealisedspectroscopic instrument with uniform sensitivity to LyC radiationat 880-910 ร across the entire redshift interval from 2.9 < ๐ง < (cid:104) ๐ IGM (cid:105) functions in Figure 6 show that this wavelengthrange exhibits the largest (cid:104) ๐ IGM (cid:105) values at ๐ rest < . (cid:104) ๐ IGM (cid:105) and, thus, the highest probability of detection(at ๏ฌxed depth). This is simply due to the fact that the redshiftinterval of LyC absorption systems that a๏ฌect a given wavelengthincreases with decreasing wavelength. This means that at lowerwavelengths the probability of encountering a high column densitysystem in any individual sightline is higher.As we discuss later, this is not the case for photometric obser-vations which instead probe a ๏ฌxed ๐ obs range, thus a decreasing ๐ rest with increasing redshift. Another important and related pointis that ionizing radiation escaping from galaxies will be completelyabsorbed by intervening, high HI column density systems, result-ing in rapid drops in ๏ฌux based on the redshift of that interveningsystem (e.g. the drop at โผ
810 ร in Figure 1). This means that es-caping ionizing radiation from high redshift galaxies may only bevisible in a very small wavelength range and this behaviour willbe di๏ฌcult to capture and interpret from photometric observations,but will be seen clearly in spectroscopy. As a caveat, however, wenote that, without ancillary, high spatial resolution, space-basedphotometric data, it can be di๏ฌcult to rule out the possibility oflow redshift contamination from ground-based spectroscopic LyCdetections (Vanzella et al. 2010, 2012).The fact that spectroscopy probes the most transparent portionof the emitted spectrum from high redshift galaxies also suggests
Figure 6.
Left column: (cid:104) ๐ IGM (cid:105) for spectroscopically detected LyC emission.Shown are results at ๐ง = .
1, similar to the average redshift of S18 of โผ ๐ Jy ( โผ ๐ detection, dependent on individual targets, in the sample of S18. The meanand median (cid:104) ๐ IGM (cid:105) functions for detected LBG-like galaxies are shownwith cyan and gold lines with the gold shaded area enclosing 68% of ๐ IGM values for detected galaxies at a given wavelength. Thus, the lower bound ofthe gold shaded region is not representative of the transmission curve shapefor any individual sightline. (cid:104) ๐ IGM (cid:105) for LyC detected, LAE-like galaxies isshown in green, signi๏ฌcantly higher than for the LBG-like sample due totheir relative faintness. The increased dispersion seen for LAE samples is aresult of a smaller number of detections for such galaxies. The mean ๐ IGM forall sightlines is shown in black for comparison.
Right column:
Normalisedhistograms of ๐ IGM for all sightlines (black), detected, LBG-like galaxies(cyan), and detected, LAE-like galaxies (green). The mean for all galaxiesand detections are shown with corresponding vertical, dotted lines (matchedto corresponding open histograms), and the median for LBG-like detectionsis shown with a gold dotted line, noting that this line corresponds to the goldline of the left panel and has no matching histogram in the right panel. that spectroscopic detections of LyC may su๏ฌer from relativelylow ๐ bias at ๏ฌxed detection limits (noting however that photometricdetections are signi๏ฌcantly deeper for the same exposure time). Weshow this in Figure 6 where we show (cid:104) ๐ IGM (cid:105) for all 10,000 sightlinesin black and (cid:104) ๐ IGM (cid:105) for those where galaxies are detected with a ๏ฌuxabove 0.025 ๐ Jy, equivalent to โผ ๐ง = .
1, with colouredlines. It should be clari๏ฌed here that this detection limit is chosento be roughly matched to the faintest LyC detection reported in S18for the galaxy Westphal-MM37 (0.026 ๐ Jy). Considering the fullparent sample of S18, 0.025 ๐ Jy corresponds to a 1-5 ๐ detection asthe observational limits and noise characteristics exhibit complexdependencies on factors such as observational depth and sourceredshift (i.e. the observed wavelength of emitted LyC radiation).Thus, we reiterate that our results are representative of an idealisedversion of the S18 survey as we have not attempted to simulate thefull complexity of their spectroscopic observations.Returning to Figure 6, the cyan and gold lines indicate themean and median ๐ IGM curves for LyC detected, LBG-like galaxies(comparable to the S18 sample) while the green line shows themean ๐ IGM curve for LAE-like detections (comparable to the F19sample). Here we measure (cid:104) ๐ IGM (cid:105) in the rest frame wavelengthrange 880 โค ๐ rest โค
910 ร (indicated in Figure 6), also followingS18. We note that LAE-like galaxies are not representative of theS18 sample and are only detected at these spectroscopic limits in ourlowest redshift bins. In fact, overall detection rates at all redshiftsis lower for the more faint sample of LAEs, which accounts forthe increased dispersion seen in the (cid:104) ๐ det (cid:105) curve for LyC detected MNRAS , 1โ20 (2021)
R. Bassett et al.
LAEs. Given this comparison is to S18 who focus on LBGs, we donot place a large emphasis on this mock sample for spectroscopicobservations.Also shown in Figure 6 is the median and 68 percentile rangefor LyC detected LBGs in gold for comparison. The median valueof ๐ IGM for detections is seen to be larger in the Ly ๐ผ forest andlower beyond the Lyman limit, with a cross-over value around 880ร . The signi๏ฌcant di๏ฌerences between the mean and median IGMtransmission functions for detected galaxies is a re๏ฌection of thenon-Gaussian nature of the underlying ๐ IGM distribution (see Figure2). Regardless, the median and mean values of ๐ IGM for detectedgalaxies are similar and throughout the remainder of this work wefocus on the mean value.In the right column of Figure 6 we compare the histogramsof ๐ IGM at 880-910 ร between all sightlines (black) and thoseassociated with LBG-like galaxies detected above 0.025 ๐ Jy (cyan).We can see that the underlying distribution is bimodal with themost probable value of ๐ IGM being โผ ๐ IGM must not be zero. It is clear thatthe mean value of ๐ IGM for all sightlines falls between the peaks ofthe underlying ๐ IGM distributions and is thus not among the mostprobably values for detected galaxies.The fact that LyC detections can not occur at ๐ IGM = ๐ esc for LyC detectedgalaxies, and is key to the narrative of this work. Careful con-sideration of ๐ IGM variation in the estimate of ๐ esc for individualdetections is common practice (e.g. Shapley et al. 2006; Inoue et al.2011; Vanzella et al. 2016). Ultimately, the goal of this paper is toprovide a clear quanti๏ฌcation of this e๏ฌect. A primary applicationof our results will be for estimating (cid:104) ๐ esc (cid:105) for larger samples of LyCdetected galaxies that may be returned by future, extremely deepsurveys (see 5). It should also be mentioned that, when estimat-ing upper limits in ๐ esc for samples including LyC non-detections, (cid:104) ๐ IGM (cid:105) considering all simulated sightlines is appropriate (i.e. in-clusion of ๐ bias is unnecessary).The level of ๐ bias for LyC detections will also be sensitiveto the detection limits, ๐น ๐๐๐ , of a given set of observations. Weexplore the dependence between ๐ bias and spectroscopic detectionlimits in Figure 7. In the top panel of Figure 7 we show the value of (cid:104) ๐ IGM (cid:105) for galaxies with spectroscopically detected LyC emissionas a function of detection limit at redshifts in the range 2.9 โค ๐ง โค (cid:104) ๐ IGM (cid:105) for allsightlines with a dotted line of the same colour. The detection limitassumed in Figure 7 of 0.025 ๐ Jy is shown with a green, vertical,dotted line. We ๏ฌnd that at low detection limits the dependencebetween (cid:104) ๐ IGM (cid:105) and ๐น ๐๐๐ is similar in all redshift bins apart fromthe expected vertical o๏ฌsets due to the drop in (cid:104) ๐ IGM (cid:105) with redshift(reiterating, however, that the de๏ฌnition ๐ bias = (cid:104) ๐ det (cid:105)/(cid:104) ๐ IGM (cid:105) willresult in a clear redshift dependence). At each redshift the curve canbe well ๏ฌt by a power law of the form ๐ IGM โ ๐น ๐ฝ๐๐๐ with ๐ฝ in therange โผ ๐ฟ / ๐ฟ ) ๐๐๐ก or a di๏ฌerent input distribution of 1500 ร ๏ฌuxes). It is also worth reiterating that sensitivity variations acrossreal spectroscopic detectors will result in detection limit variationwith redshift at ๏ฌxed exposure time.In the bottom panel of Figure 7 we show ๐ bias as a function Figure 7.
The dependence of ๐ bias on the detection limit of spectroscopicobservations ( ๐น ๐๐๐ ) for LBG-like detections. Top: (cid:104) ๐ IGM (cid:105) as a functionof ๐น ๐๐๐ at redshifts between 2.9 and 3.9 (see lower panel for legend).Horizontal dotted lines show (cid:104) ๐ IGM (cid:105) of all sightlines at a given redshift, andthe vertical dotted line shows the ๐น ๐๐๐ assumed in Figure 6. At each redshiftwe ๏ฌt the curve of (cid:104) ๐ IGM (cid:105) for detected galaxies with a power law of the form (cid:104) ๐ IGM (cid:105)( ๐น ๐๐๐ ) = ๐๐น ๐๐๐๐ + ๐ . Bottom: ๐ bias as a function of ๐น ๐๐๐ for thesame redshift interval. We show a power-law ๏ฌt, ๐ bias ( ๐น ๐๐๐ ) = ๐๐น ๐๐๐๐ + ๐ ,to the combined data for all redshifts as a black dashed line. of ๐น ๐๐๐ at the same discrete ๐ง values between 2.9 and 3.9 with ฮ ๐ง = .
1. Overall we ๏ฌnd a very small scatter in ๐ bias with thedi๏ฌerence between the maximum and minimum ๐ bias at ๏ฌxed ๐น ๐๐๐ less that 0.01 at all redshifts in the range considered. Given thesmooth curves seen in Figure 7, it is tempting to provide the powerlaw ๏ฌts (of the form (cid:104) ๐ IGM (cid:105)( ๐น ๐๐๐ ) = ๐๐น ๐๐๐๐ + ๐ , dashed lines inFigure 7, top panel) at each redshift giving an analytical function forestimating ๐ bias as a function of ๐ง and ๐น ๐๐๐ , however we refrain fromdoing so as we would consider any application of such a function asan overinterpretation of Figure 7, which results from our particularimplementation for producing ๐ IGM functions as well as the variousinputs of our ๏ฌducial model (e.g. here we have only shown resultsfor LBG-like samples). For illustrative purposes we have ๏ฌt a powerlaw to the combined ๐ bias vs ๐น ๐๐๐ curves ๐ ๐๐๐๐ โ ๐น . ๐๐๐ . This ๏ฌt isshown in the bottom panel of Figure 7 with a black dashed line. Here,the choice of a power law is ad hoc, and no speci๏ฌc signi๏ฌcance isassigned to the ๏ฌt parameters.It is useful here take a step back and recall two importantpoints: ๏ฌrst there is signi๏ฌcant variation in ๐ IGM for individualsightlines at any redshift (see, e.g., Figure 1) and second the factthat we allow high ๐ esc values (up to 1.0) in our ๏ฌducial modelmeaning ๐ bias observed in our ๏ฌducial model represents the absoluteminimum ๐ bias for a given detection limit. Thus, we caution thereader from applying values of ๐ bias calculated using a similar model MNRAS000
1. Overall we ๏ฌnd a very small scatter in ๐ bias with thedi๏ฌerence between the maximum and minimum ๐ bias at ๏ฌxed ๐น ๐๐๐ less that 0.01 at all redshifts in the range considered. Given thesmooth curves seen in Figure 7, it is tempting to provide the powerlaw ๏ฌts (of the form (cid:104) ๐ IGM (cid:105)( ๐น ๐๐๐ ) = ๐๐น ๐๐๐๐ + ๐ , dashed lines inFigure 7, top panel) at each redshift giving an analytical function forestimating ๐ bias as a function of ๐ง and ๐น ๐๐๐ , however we refrain fromdoing so as we would consider any application of such a function asan overinterpretation of Figure 7, which results from our particularimplementation for producing ๐ IGM functions as well as the variousinputs of our ๏ฌducial model (e.g. here we have only shown resultsfor LBG-like samples). For illustrative purposes we have ๏ฌt a powerlaw to the combined ๐ bias vs ๐น ๐๐๐ curves ๐ ๐๐๐๐ โ ๐น . ๐๐๐ . This ๏ฌt isshown in the bottom panel of Figure 7 with a black dashed line. Here,the choice of a power law is ad hoc, and no speci๏ฌc signi๏ฌcance isassigned to the ๏ฌt parameters.It is useful here take a step back and recall two importantpoints: ๏ฌrst there is signi๏ฌcant variation in ๐ IGM for individualsightlines at any redshift (see, e.g., Figure 1) and second the factthat we allow high ๐ esc values (up to 1.0) in our ๏ฌducial modelmeaning ๐ bias observed in our ๏ฌducial model represents the absoluteminimum ๐ bias for a given detection limit. Thus, we caution thereader from applying values of ๐ bias calculated using a similar model MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections Figure 8. ๐ bias for photometrically detected LyC emission in the HST F336Wand CFHT ๐ข ๏ฌlters. Results are shown at ๐ง = . ๐ง = .
6. F336W andCFHT ๐ข transmission curves are shown as dashed gold and cyan lines, re-spectively. Detection limits are ๏ฌxed at 30.24 and 27.82 mag for F336Wand CFHT ๐ข , respectively (matched to F19 and M20). (cid:104) ๐ IGM (cid:105) for F336Wand CFHT ๐ข detected LBG-like galaxies are shown in gold and cyan, re-spectively, and (cid:104) ๐ IGM (cid:105) for all sightlines is shown in black. The green lineindicates (cid:104) ๐ IGM (cid:105) for LAE-like galaxies detected with the F336W ๏ฌlter, moresimilar to the sample of F19. The increased dispersion of the green line rel-ative to the gold line is driven by a decrease in the total number of detectedgalaxies. The Lyman limit is indicated in each panel by a vertical dottedline. to observations of individual galaxies when estimating ๐ esc withoutincluding these caveats. While spectroscopic detection of LyC radiation from galaxies pro-vides distinct advantages in terms of (cid:104) ๐ IGM (cid:105) , achieving this forlarge samples of galaxies is ine๏ฌcient. Photometric surveys havethe potential for detecting large samples of LyC emitting galaxiessimultaneously. Another important bene๏ฌt of photometric surveyswhen compared to spectroscopy is that photometry is signi๏ฌcantlymore sensitive (i.e. deeper) for the same exposure time. Further-more, in the case of space-based LyC detections, ancillary data isnot necessary to rule out the possibility of low redshift contamina-tion. Photometric LyC surveys must be performed in well studied๏ฌelds in which targeted galaxies already have accurate photometricredshift estimates (e.g. ZFOURGE ๏ฌelds Straatman et al. 2016) or,ideally, secure spectroscopic redshifts (e.g. 3DHST, DEIMOS10K,VANDELS, MUSE-wide, Momcheva et al. 2016; Hasinger et al.2018; Pentericci et al. 2018; Urrutia et al. 2019). In ๏ฌelds such asthese, speci๏ฌc redshift windows can be targeted using photomet-ric bands probing LyC emission such as HST F336W at ๐ง โผ . ๐ข at ๐ง โผ . calculation of the reported photometric ๏ฌux in- herently assumes a ๏ฌat ๏ฌux density across the ๏ฌlter. Of course, the interpretation of the photometric ๏ฌux can include more complexspectral behaviour, e.g. extreme [OIII]+H ๐ฝ emitters presented inForrest et al. (2017).The second drawback is that the observed wavelengths of pho-tometric bands are ๏ฌxed. This means that the ideal redshift for suchsurveys is at the point where the red cuto๏ฌ of the ๏ฌlter in questionfalls just below the Lyman limit (thus ๏ฌlter dependent). LyC radia-tion can be detected to higher redshifts (more likely for extremelydeep observations), however at high redshift the ๏ฌlter moves tobluer rest wavelengths where (cid:104) ๐ IGM (cid:105) is signi๏ฌcantly lower. Thisfact causes signi๏ฌcant complications when making comparisons ofLyC escape from photometric detections at di๏ฌerent redshifts. Wealso mention brie๏ฌy here that some photometric ๏ฌlters su๏ฌer fromso-called โred leakโ with a small amount of radiation at wavelengthslonger than the optimal cuto๏ฌ of the ๏ฌlter being transmitted (thoughthis is minimized for the new CFHT ๐ข ๏ฌlter used in M20, Sawickiet al. 2019). As such features will be included in the ๏ฌlter curvesused in our analysis, this e๏ฌect is implicitly accounted for.With these two drawbacks in mind we present the simulated ๐ bias for LyC detected galaxies for HST F336W and CFHT ๐ข de-tected galaxies in Figure 8. Here we use ๏ฌxed detection limits of30.24 and 27.82 mag for F336W and CFHT ๐ข , respectively (matchedto the limits of F19 and M20). The two panels in Figure 8 showthe mean ๐ IGM for all sightlines (black), for F336W LBG-like de-tections (gold), CFHT ๐ข LBG-like detections (cyan), and F336WLAE-like detections (green) at redshifts of 3.2 and 3.6 (LAE-likedetections for CFHT ๐ข are not shown as such detections are ex-tremely rare due to the relative shallowness of M20 photometry).We ๏ฌnd that ๐ bias for LBG-like galaxies is signi๏ฌcantly lower forF336W detections, however this is simply re๏ฌective of the greaterdepth of our F336W comparison rather than any intrinsic advantageof HST observations over ground-based for LyC detections. Com-paring F336W LAE-like versus LBG-like detections, we ๏ฌnd that ๐ bias for the former is โผ ๐ข ๏ฌlters are shown with dashedgold and cyan lines, highlighting the fact that the F336W and ๐ข ๏ฌlters exclusively probe LyC radiation at ๐ง > . ๐ง > . ๐ bias for the CFHT ๐ข ๏ฌlter at ๐ง = . ๐ esc is signi๏ฌcantly complicated (e.g. Bassettet al. 2019) and such cases should be avoided where possible. ๐ bias for photometry is also sensitive to observational detectionlimits. The variation in ๐ bias with detection limit (in magnitudes, ๐ ๐๐๐ ) is demonstrated in Figure 9 for the HST F336W ๏ฌlter. Solidlines show results for LBG-like detections and dashed lines forLAE-like detections. Similar to spectroscopic results presented inFigure 7, we ๏ฌnd that, at ๏ฌxed ๐ง , ๐ bias decreases linearly with anincreasing magnitude limit. When compared to the spectroscopicresults of Figure 7, with ฮ ๐ ๐๐๐๐ (cid:46) .
01 for all redshifts, we ๏ฌndmore variation with redshift. This is due to the changing rest-framewavelengths probed by the F336W ๏ฌlter with redshift. Again, amore signi๏ฌcant redshift evolution will be observed assuming thede๏ฌnition ๐ bias = (cid:104) ๐ det (cid:105)/(cid:104) ๐ IGM (cid:105) . For LAE-like detections, the factthat very few LyC ๏ฌuxes reach magnitudes brighter than 28.5 (andonly in the lowest redshift bins) means that detections occur in onlythose sightlines with the highest ๐ IGM (F336W). Thus the trendsshown for LAE samples in Figure 9 exhibit more scatter due to an
MNRAS , 1โ20 (2021) R. Bassett et al.
Figure 9.
The dependence of ๐ bias on the detection limit of F336W obser-vations ( ๐ ๐๐๐ ) at 3.2 โค ๐ง โค ๐ง = . ๐ bias is relatively constant at ๏ฌxed ๐ ๐๐๐ . Thelarger redshift variation when compared to Figure 7 and the divergent be-haviour for shallow observations at high redshift re๏ฌect the shifting restwavelengths probed by the F336W with increasing redshift. increased sensitivity to the stochasticity of our IGM transmissionfunctions. The dashed lines in Figure 9 also demonstrate why we๏ฌnd so few LyC detected LAEs for our mock spectroscopic andCFHT ๐ข observations given the depth of these two comparisons are๏ฌxed at โผ The results of our ๏ฌducial model for ๏ฌxed detection limits of 0.025 ๐ Jy ( โผ ๐ข , respectively, are summarised in Figure 10for mock observations of galaxies with 1500 ร ๏ฌux distributionscharacteristic of LBGs (F336W results for fainter, LAE-like galaxiesare also shown with dotted lines). As described in Sections 3.1.1 and3.1.2, spectroscopic detections at this depth (targeting a ๏ฌxed restwavelength window at 880 < ๐ rest <
910 ร ) experience a roughlyconstant ๐ bias of โผ โผ ๐ bias , which increases from0.157 at ๐ง = . ๐ง = . ๐ง = .
9. This change in ๐ bias of less than 2% is signi๏ฌcantlysmaller than the variance seen at any given redshift and is drivenentirely by our cosmological dimming (see Equation 6). Thus, weconclude that ๐ bias is e๏ฌectively constant at 3.0 < ๐ง < ๐ bias is found to be constant with redshift issomewhat counterintuitive. Instead, one may expect a monotonicincrease in ๐ bias with redshift due to the ๏ฌxed detection limit andlinear decrease in (cid:104) ๐ IGM (cid:105) . For our additive de๏ฌnition of ๐ bias , theconstant ๐ bias observed can be explained by a decrease in detectionrate with redshift where only the brightest galaxies contribute to ๐ bias at the high ๐ง end. This is illustrated in Figure 10 with thedetection percentages for spectroscopy at each redshift indicated inblack.Condsidering photometric detections, ๐ bias is seen to increasewhile the Lyman limit passes through the ๏ฌlter in question. At red- Figure 10.
A summary of ๐ bias for our ๏ฌducial model. Top: (cid:104) ๐ IGM (cid:105) asa function of redshift for all sightlines are shown with solid lines whiledashed lines show (cid:104) ๐ IGM (cid:105) for detected galaxies. Results for spectroscopy,F336W, and CFHT ๐ข are shown in black, gold, and cyan, respectively. Forour ๏ฌducial model we assume detection limits of 0.025 ๐ Jy ( โผ ๐ข . Bottom: ๐ bias as a function of redshift for each detection method. Error bars show the68 percentile range at each redshift. Values are calculated at ๏ฌxed redshiftsbetween 3.0 and 3.9 with ฮ ๐ง = 0.1, slight o๏ฌsets between methods are forclarity only. We also show the detection percentage for spectroscopy in black,which decreases signi๏ฌcantly with redshift, across the top of the bottompanel. In both panels, open symbols for photometric observations indicateredshifts at which a given ๏ฌlter probes (parially or entirely) wavelengthsredward of the Lyman limit (i.e. non-ionizing photons). shifts where a given ๏ฌlter has passed fully blueward of the Lymanlimit, the level of ๐ bias is seen to level o๏ฌ (within errors) at a valuedependent on the photometric depth. For our ๏ฌducial depths, thisplateau level is โผ โผ ๐ข ๏ฌlters (magnitude limit = 27.82),respectively. In the case of LAE-like 1500 ร ๏ฌux distributions, weshow results only for F336W as this comparison has signi๏ฌcantlydeeper ๏ฌux limits compared with spectroscopy and CFHT ๐ข (wheredetections of LAE-like samples are vanishingly rare). In the caseof LAEs, we ๏ฌnd that ๐ bias is roughly 0.1 higher than for LBGsat ๏ฌxed redshift, with values in the range โผ ๐ bias for photometric detectionsat the highest redshifts in the bottom panel of Figure 10. Unlikespectroscopic detections, by ๐ง โผ . ๐ rest where (cid:104) ๐ IGM (cid:105) is near zero. Furthermore, asseen in Figure 2, the ๐ IGM distribution at these wavelengths is askewed, unimodal distribution peaked at ๐ IGM = 0. This means thatthe probability of ๏ฌnding a sightline with ๐ IGM much higher thanzero is very low. This could explain why ๐ bias for photometry dipsat high ๐ง , as even those small number of detected galaxies will befound in sightlines approaching zero transmission at wavelengthsprobed by a given ๏ฌlter. This means that the level of ๐ bias seen atlower redshifts simply can not be maintained given the underlying MNRAS000
A summary of ๐ bias for our ๏ฌducial model. Top: (cid:104) ๐ IGM (cid:105) asa function of redshift for all sightlines are shown with solid lines whiledashed lines show (cid:104) ๐ IGM (cid:105) for detected galaxies. Results for spectroscopy,F336W, and CFHT ๐ข are shown in black, gold, and cyan, respectively. Forour ๏ฌducial model we assume detection limits of 0.025 ๐ Jy ( โผ ๐ข . Bottom: ๐ bias as a function of redshift for each detection method. Error bars show the68 percentile range at each redshift. Values are calculated at ๏ฌxed redshiftsbetween 3.0 and 3.9 with ฮ ๐ง = 0.1, slight o๏ฌsets between methods are forclarity only. We also show the detection percentage for spectroscopy in black,which decreases signi๏ฌcantly with redshift, across the top of the bottompanel. In both panels, open symbols for photometric observations indicateredshifts at which a given ๏ฌlter probes (parially or entirely) wavelengthsredward of the Lyman limit (i.e. non-ionizing photons). shifts where a given ๏ฌlter has passed fully blueward of the Lymanlimit, the level of ๐ bias is seen to level o๏ฌ (within errors) at a valuedependent on the photometric depth. For our ๏ฌducial depths, thisplateau level is โผ โผ ๐ข ๏ฌlters (magnitude limit = 27.82),respectively. In the case of LAE-like 1500 ร ๏ฌux distributions, weshow results only for F336W as this comparison has signi๏ฌcantlydeeper ๏ฌux limits compared with spectroscopy and CFHT ๐ข (wheredetections of LAE-like samples are vanishingly rare). In the caseof LAEs, we ๏ฌnd that ๐ bias is roughly 0.1 higher than for LBGsat ๏ฌxed redshift, with values in the range โผ ๐ bias for photometric detectionsat the highest redshifts in the bottom panel of Figure 10. Unlikespectroscopic detections, by ๐ง โผ . ๐ rest where (cid:104) ๐ IGM (cid:105) is near zero. Furthermore, asseen in Figure 2, the ๐ IGM distribution at these wavelengths is askewed, unimodal distribution peaked at ๐ IGM = 0. This means thatthe probability of ๏ฌnding a sightline with ๐ IGM much higher thanzero is very low. This could explain why ๐ bias for photometry dipsat high ๐ง , as even those small number of detected galaxies will befound in sightlines approaching zero transmission at wavelengthsprobed by a given ๏ฌlter. This means that the level of ๐ bias seen atlower redshifts simply can not be maintained given the underlying MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections ๐ IGM distribution for the wavelengths probed. At higher redshifts the ๐ IGM distribution becomes so strongly peaked at ๐ IGM = 0.0 that nodetections are expected, thus we do not expect the results presentedhere for 2.9 < ๐ง < ๐ IGM distribution at 880 < ๐ rest <
910 ร remains bimodal even athigh redshift, thus no obvious dip in ๐ bias is seen. Regardless, in allcases (cid:104) ๐ IGM (cid:105) is decreasing with redshift, thus detections becomerarer. This manifests as a decreasing 68 percentile range for ๐ bias ,a re๏ฌection of the drop in the numbers of detected galaxies withredshift.Finally, as mentioned at the start of Section 3.1, the de๏ฌnition ๐ bias = (cid:104) ๐ det (cid:105)/(cid:104) ๐ IGM (cid:105) is equally valid to the de๏ฌnition adopted inthis work. Under this alternative de๏ฌnition, a very clear trend be-tween ๐ bias and redshift is apparent increasing from โผ โผ โผ โผ
35 for F336W ob-servations for LAE samples. We reiterate that, although a fractionalde๏ฌnition may be more physical (in the sense that it relates directlyto a ratio of HI column densities), the redshift evolution of ๐ bias inthis case re๏ฌects primarily the fact that (cid:104) ๐ IGM (cid:105) moves increasinglyclose to zero with redshift while the actual di๏ฌerence in the meanIGM transmission between detections and all sightlines is roughlyconstant, as our chosen de๏ฌnition illustrates. Thus, our de๏ฌnitionprovides a simpli๏ฌed correction when calculating ๐ esc for LyC de-tected samples from an observational point of view. ๐ esc Distribution
It is expected that if LyC emission is detected from a given galaxy, itmust have a high ๐ esc and/or a high ๐ IGM . From current observationsof LyC emitters (particularly considering the large number of non-detections), it seems that ๐ esc values, i.e. 0.0-0.2, are most common(e.g. Boutsia et al. 2011; Grazian et al. 2016; Smith et al. 2018).The results presented for our ๏ฌducial model in Section 3.1, however,allow for ๐ esc values from 0.0 to 1.0 with no preference. This meansthat a large number of detections from our ๏ฌducial model exhibit alarge ๐ esc and are detected in sightlines with relatively low ๐ IGM . Ifwe instead choose an underlying ๐ esc distribution skewed towardslow ๐ esc , we might expect that the average ๐ IGM for detections willincrease, thus increasing ๐ bias .In this Section, we explore how altering the PDF of selected ๐ esc values a๏ฌects the level of ๐ bias and the distributions of ๐ esc forLyC detections. For comparison, the ๏ฌducial model can be treatedas a ๏ฌat PDF between 0 and 1. Here we test an alternative ๐ esc PDFmodel designed to give more weight to lower ๐ esc values. In thiscases we choose an exponentially declining ๐ esc PDF of the form:
๐๐ท๐น ( ๐ esc ) โ ๐ โ ๐ esc / ๐ (8)where ๐ represents an exponential cut o๏ฌ in ๐ esc . Here we test thevalue ๐ = 0.50 (see Figure 3) motivated by LyC detection ratesfrom S18 (see Section 4.3). For brevity our ๏ฌducial model will bedescribed as โ๏ฌatโ and our alternative model will be referred to as ๐ = 0.50. As with our ๏ฌducial model, for our ๐ = 0.50 model werecreate 100 mock spectra for each of our 10,000 IGM transmissionfunctions at each discrete redshift value as described in Section 2.3.We show example (cid:104) ๐ IGM (cid:105) curves at ๐ง = . ๐ esc PDF exhibits a lower ๐ bias than the ๐ =0.50 as expected. The increases in ๐ bias for both spectroscopic andphotometric detections are found to be only 0.01 and 0.02, respec-tively. These increases in ๐ bias are essentially negligible considering Figure 11. ๐ IGM curves for galaxies detected with F336W (top) and spectro-scopically, (bottom). Both panels show the results at ๐ง = . ๐ Jy ( โผ ๐ = 0.30 model exhibits only slightly higher ๐ IGM thanthe ๏ฌducial model. Spectroscopic (F336W) values of ๐ bias increase modestlyfrom 0.16 (0.10) for the ๏ฌducial model to 0.17 (0.13) for the ๐ = 0.50 model. the spread in (cid:104) ๐ IGM (cid:105) for LyC detections seen in Figure 10. Thus, inthe case of our ๐ = . ๐ bias when compared to the ๏ฌducial, ๏ฌat ๐ esc PDF and note that thisbehaviour is the same in all redshift bins. In cases where the un-derlying ๐ esc is more strongly skewed towards ๐ esc = 0 (i.e. smallervalues of ๐ ), the di๏ฌerence in ๐ bias when compared to a ๏ฌat PDFis certain to increase. Such a low ๐ model (or any other similarlyskewed ๐ esc PDF) may be appropriate for galaxy samples with se-lection biases di๏ฌerent from the LBG and LAE samples consideredhere if (cid:104) ๐ esc (cid:105) does indeed vary with galaxy properties (see Section4.3 for more discussion).The fact that ๐ bias for our ๐ = . ๐ esc PDF does not mean the two models areinterchangeable in regards to estimates of ๐ esc from observed sam-ples. To illustrate this, we show in Figure 12 the histograms of ๐ esc for detections only vs all trials at ๐ง = . ๐ esc dis-tributions and open histograms show the ๐ esc distribution for LyCdetections. We ๏ฌnd that the ๐ esc distributions of LyC detections (i.e.the posterior) for both observational methods is skewed towards ๐ esc MNRAS , 1โ20 (2021) R. Bassett et al.
Figure 12.
Histograms of ๐ esc for our two ๐ esc PDF models: ๏ฌat in the top rowand ๐ = .
50 on the bottom. The left column shows results for spectroscopyand the right for F336W. In each panel the underlying ๐ esc distribution isshown with a ๏ฌlled histogram, the ๐ esc distribution of detections with an openhistogram, and the mean value for detections is shown with a vertical dottedline and indicated in the top left of each panel. Note that each histogram hasbeen normalised by the maximum value for ease of comparison. = 1.0, inconsistent with the low values typically seen in observa-tions. The posterior for the ๐ = . ๐ = . ๐ esc value is lower by 0.07 and 0.12 for spectroscopyand F336W detections, respectively. Thus, although ๐ bias is roughlythe same between the two ๐ esc PDF models, the di๏ฌerences whenconsidering the inferred ๐ esc for galaxy samples is signi๏ฌcant.We note that the posterior distribution for the ๐ = . ๐ esc PDF model multiplied by the input ๐ = . ๐ esc PDF once the posterior for a ๏ฌatdistribution is determined for a given observational method anddetection limit without the need to run a separate analysis. We stressagain that the actual distribution of ๐ esc is essentially unknown,however we discuss possibilities for placing some constraints onthis in Sections 4.3 and 4.6. In this Section we brie๏ฌy explore the e๏ฌects that varying the SEDshape will have on our estimates of ๐ bias presented in Sections 3.1and 3.2. In regards to detecting LyC from a given galaxy above aspeci๏ฌed limit, the key di๏ฌerence resulting from a change in SEDshape will be a change in the ๏ฌux ratio of the LyC and UV ( ๐ rest โผ ๐น / ๐น ) ๐๐๐ , at a๏ฌxed ๐ IGM . The factor that will a๏ฌect ( ๐น / ๐น ) ๐๐๐ (in addition to ๐ IGM ) considered here is variation in the intrinsic ratio of LyCand UV emission, ( ๐ฟ / ๐ฟ ) ๐๐๐ก .To test the e๏ฌect of altering ( ๐ฟ / ๐ฟ ) ๐๐๐ก on our results wererun the analysis described in Section 2.3 for each age of our ex-ponentially declining BPASSv2.1 models (with ๐ -folding timescaleof 0.1 Gyr) in the range 6.0 < log(age) < ฮ log(age)= 0.1. The models produced exhibit ( ๐ฟ / ๐ฟ ) ๐๐๐ก in the range โผ ๐ ๐๐๐ from โผ < ๐ง < ฮ ๐ง = 0.1) with 1500 ร ๏ฌuxes sampled from an LBG-like distribution. We then repeat our measurements of LyC ๏ฌux asin previous sections and adopt the ๏ฌux limits of our ๏ฌducial model: ๐น ๐๐๐ (spectroscopy) = 0.025 ๐ Jy ( โผ ๐ ๐๐๐ (F336W) =30.24. The CFHT ๐ข comparison is not considered here as the rela-tively shallow nature of these observations results in prohibitivelyfew detections at low ( ๐ฟ / ๐ฟ ) ๐๐๐ก . For a similar reason, we alsodo not consider LAE-like samples in this section.We show the resulting ( ๐ฟ / ๐ฟ ) ๐๐๐ก versus ๐ bias for spectro-scopic, LBG-like LyC detections in the left panel of Figure 13. Sim-ilar to the results for our test on detection limits we ๏ฌnd only slightvariation in ๐ bias with redshift with a total spread in values of โผ ๐ฟ / ๐ฟ ) ๐๐๐ก above ( ๐ฟ / ๐ฟ ) ๐๐๐ก =0.15 (again, a fractional de๏ฌnition of ๐ bias will result in signi๏ฌcantredshift variation). For reference, we show the location of the ๏ฌdu-cial model presented in Section 3.1.1 with the dotted green line andshaded regions. Slight di๏ฌerences can be attributed to stochasticityas the analysis here represents and independent sample of 1500 ร ๏ฌuxes and ๐ esc values at the same ( ๐ฟ / ๐ฟ ) ๐๐๐ก value. Regard-less, the results of Figure 13 are consistent with those of 3.1.1 withinerrors.Results for F336W detections are shown similarly in the rightpanel of Figure 13. We show results at redshifts where F336Wpartially probes non-ionizing photons with dashed lines (i.e. ๐ง < . ๐ bias again attributed to the increased depthof the F336W observational comparison.Finally, we note that none of the models presented to this pointhave considered the e๏ฌects of dust attenuation on the observed LyC๏ฌux from mock galaxies. The e๏ฌect that dust will have on LyC willbe to further reduce the observed value of ๐น ( ๐ฟ๐ฆ๐ถ )/ ๐น ( ๐๐ ) relativeto ( ๐ฟ / ๐ฟ ) ๐๐๐ก . In this way, dust attenuation is a third levelof degeneracy between ๐ esc and ๐ IGM . Given the low attenuationfor LyC detections (e.g. S18), we ignore the e๏ฌects of dust simplynoting that detections should be biased towards galaxies with lowdust attenuation (or even none in the case of LAEs, e.g. Fletcheret al. 2019; Nakajima et al. 2020). ๐ IGM ( ๐ฟ๐ฆ๐ถ ) and ๐ IGM ( ๐ฟ๐ฆ๐ผ ) One major di๏ฌculty in accurately measuring ๐ esc from high red-shift galaxies is the unknown value of ๐ IGM . So far, there is noclear observational indicator of ๐ IGM ( LyC ) , which has necessitatedstatistical methods such as those explored in this paper. In the workof Inoue & Iwata (2008), however, it was argued that the ๐ IGM atLy ๐ผ wavelengths may correlate with ๐ IGM ( LyC ) (their Section 4.4,Figure 10). This claim is in direct contrast with previous results ofShapley et al. (2006) who found no such correlation at ๐ง = . MNRAS000
50 on the bottom. The left column shows results for spectroscopyand the right for F336W. In each panel the underlying ๐ esc distribution isshown with a ๏ฌlled histogram, the ๐ esc distribution of detections with an openhistogram, and the mean value for detections is shown with a vertical dottedline and indicated in the top left of each panel. Note that each histogram hasbeen normalised by the maximum value for ease of comparison. = 1.0, inconsistent with the low values typically seen in observa-tions. The posterior for the ๐ = . ๐ = . ๐ esc value is lower by 0.07 and 0.12 for spectroscopyand F336W detections, respectively. Thus, although ๐ bias is roughlythe same between the two ๐ esc PDF models, the di๏ฌerences whenconsidering the inferred ๐ esc for galaxy samples is signi๏ฌcant.We note that the posterior distribution for the ๐ = . ๐ esc PDF model multiplied by the input ๐ = . ๐ esc PDF once the posterior for a ๏ฌatdistribution is determined for a given observational method anddetection limit without the need to run a separate analysis. We stressagain that the actual distribution of ๐ esc is essentially unknown,however we discuss possibilities for placing some constraints onthis in Sections 4.3 and 4.6. In this Section we brie๏ฌy explore the e๏ฌects that varying the SEDshape will have on our estimates of ๐ bias presented in Sections 3.1and 3.2. In regards to detecting LyC from a given galaxy above aspeci๏ฌed limit, the key di๏ฌerence resulting from a change in SEDshape will be a change in the ๏ฌux ratio of the LyC and UV ( ๐ rest โผ ๐น / ๐น ) ๐๐๐ , at a๏ฌxed ๐ IGM . The factor that will a๏ฌect ( ๐น / ๐น ) ๐๐๐ (in addition to ๐ IGM ) considered here is variation in the intrinsic ratio of LyCand UV emission, ( ๐ฟ / ๐ฟ ) ๐๐๐ก .To test the e๏ฌect of altering ( ๐ฟ / ๐ฟ ) ๐๐๐ก on our results wererun the analysis described in Section 2.3 for each age of our ex-ponentially declining BPASSv2.1 models (with ๐ -folding timescaleof 0.1 Gyr) in the range 6.0 < log(age) < ฮ log(age)= 0.1. The models produced exhibit ( ๐ฟ / ๐ฟ ) ๐๐๐ก in the range โผ ๐ ๐๐๐ from โผ < ๐ง < ฮ ๐ง = 0.1) with 1500 ร ๏ฌuxes sampled from an LBG-like distribution. We then repeat our measurements of LyC ๏ฌux asin previous sections and adopt the ๏ฌux limits of our ๏ฌducial model: ๐น ๐๐๐ (spectroscopy) = 0.025 ๐ Jy ( โผ ๐ ๐๐๐ (F336W) =30.24. The CFHT ๐ข comparison is not considered here as the rela-tively shallow nature of these observations results in prohibitivelyfew detections at low ( ๐ฟ / ๐ฟ ) ๐๐๐ก . For a similar reason, we alsodo not consider LAE-like samples in this section.We show the resulting ( ๐ฟ / ๐ฟ ) ๐๐๐ก versus ๐ bias for spectro-scopic, LBG-like LyC detections in the left panel of Figure 13. Sim-ilar to the results for our test on detection limits we ๏ฌnd only slightvariation in ๐ bias with redshift with a total spread in values of โผ ๐ฟ / ๐ฟ ) ๐๐๐ก above ( ๐ฟ / ๐ฟ ) ๐๐๐ก =0.15 (again, a fractional de๏ฌnition of ๐ bias will result in signi๏ฌcantredshift variation). For reference, we show the location of the ๏ฌdu-cial model presented in Section 3.1.1 with the dotted green line andshaded regions. Slight di๏ฌerences can be attributed to stochasticityas the analysis here represents and independent sample of 1500 ร ๏ฌuxes and ๐ esc values at the same ( ๐ฟ / ๐ฟ ) ๐๐๐ก value. Regard-less, the results of Figure 13 are consistent with those of 3.1.1 withinerrors.Results for F336W detections are shown similarly in the rightpanel of Figure 13. We show results at redshifts where F336Wpartially probes non-ionizing photons with dashed lines (i.e. ๐ง < . ๐ bias again attributed to the increased depthof the F336W observational comparison.Finally, we note that none of the models presented to this pointhave considered the e๏ฌects of dust attenuation on the observed LyC๏ฌux from mock galaxies. The e๏ฌect that dust will have on LyC willbe to further reduce the observed value of ๐น ( ๐ฟ๐ฆ๐ถ )/ ๐น ( ๐๐ ) relativeto ( ๐ฟ / ๐ฟ ) ๐๐๐ก . In this way, dust attenuation is a third levelof degeneracy between ๐ esc and ๐ IGM . Given the low attenuationfor LyC detections (e.g. S18), we ignore the e๏ฌects of dust simplynoting that detections should be biased towards galaxies with lowdust attenuation (or even none in the case of LAEs, e.g. Fletcheret al. 2019; Nakajima et al. 2020). ๐ IGM ( ๐ฟ๐ฆ๐ถ ) and ๐ IGM ( ๐ฟ๐ฆ๐ผ ) One major di๏ฌculty in accurately measuring ๐ esc from high red-shift galaxies is the unknown value of ๐ IGM . So far, there is noclear observational indicator of ๐ IGM ( LyC ) , which has necessitatedstatistical methods such as those explored in this paper. In the workof Inoue & Iwata (2008), however, it was argued that the ๐ IGM atLy ๐ผ wavelengths may correlate with ๐ IGM ( LyC ) (their Section 4.4,Figure 10). This claim is in direct contrast with previous results ofShapley et al. (2006) who found no such correlation at ๐ง = . MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections Figure 13. ๐ bias as a function of intrinsic ratio of LyC (at 900ร ) to UV (at 1500ร ) luminosities for spectroscopy (left) and photometry (right). Solid lines showresults in each redshift bin as indicated in each legend while dotted lines in the right panel indicate redshifts at which the F336W ๏ฌlter contains contaminationfrom Ly ๐ผ forest photons as it has not passed fully into the LyC portion of the spectrum. The location corresponding to intrinsic ratios presented in Sections 3.1and 3.2 are indicated with dotted green lines and shaded regions. The top axis of both panels indicates ๐ ๐๐๐ for the BPASSv2.1 model at a given ( ๐ฟ / ๐ฟ ) ๐๐๐ก . Shapley et al. (2006) was due to those authors exploring ๐ IGM atonly one redshift. Here we test for a correlation between ๐ IGM ( LyC ) and ๐ IGM ( Ly ๐ผ ) for our simulated IGM transmission functions, not-ing that our simulations di๏ฌer from those of Shapley et al. (2006)and Inoue & Iwata (2008) in that we include a CGM component toour HI column density distributions following the work of S18 andRudie et al. (2013).To perform this test, we assess all one million IGM sightlineswe have produced in Section 2.1, measuring ๐ IGM for LyC at 880 < ๐ rest <
910 ร and for Ly ๐ผ at 1050 < ๐ rest < ๐ผ wavelength ranges used in these works arenot probing the same redshift range. Thus, we also measure analternative Ly ๐ผ wavelength range 1173 < ๐ rest < ๐ IGM ( LyC ) and ๐ IGM ( Ly ๐ผ ) in both wavelength ranges and also thecorrelation coe๏ฌcient of the combined data from all redshift bins.Figure 14 shows ๐ IGM ( LyC ) vs ๐ IGM ( Ly ๐ผ ) for all one millionsightlines colored by their redshift. Overall there appears to be acorrelation between the two values (albeit with large scatter), how-ever at any individual redshift such a correlation is less apparent.For ๐ IGM ( Ly ๐ผ ) at 1050-1170 ร we measure correlation coe๏ฌcientsat individual redshifts ๏ฌnding values in the range 0.05-0.08 indicat-ing no correlation with ๐ IGM ( LyC ) at ๏ฌxed redshift consistent withShapley et al. (2006). Considering all redshift bins together we ๏ฌnda drastic increase in the correlation coe๏ฌcient to 0.34. This is stilllower than the correlation of 0.86 quoted by Inoue & Iwata (2008),however, in this work the authors tested a much wider redshift rangefrom 0.2 to 6.0. Our results combined with those of Inoue & Iwata(2008) suggest that any apparent correlation between ๐ IGM at LyCand Ly ๐ผ wavelengths is driven only by the fact that both valuescorrelate similarly with redshift (e.g. Figure 1). For any individualgalaxy (or sample) at a given redshift, however, ๐ IGM ( Ly ๐ผ ) pro-vides no useful prediction for ๐ IGM ( LyC ) . Indeed, this is apparentfrom the contours shown in Inoue & Iwata (2008) Figure 10. Figure 14.
A comparison of ๐ IGM for LyC (880-910 ร ) and Ly ๐ผ (1070-1170 ร ) for all one million simulated sightlines. Points are coloured based ontheir source redshift. We ๏ฌnd that the apparent correlation seen between theIGM transmission of LyC and Ly ๐ผ radiation is driven by the fact that bothvalues exhibit individual redshift dependencies rather than any correlationbetween these two values. Indeed, there is no apparent correlation between ๐ IGM (LyC) and ๐ IGM (Ly ๐ผ ) at ๏ฌxed redshift. As we have pointed out, however, the Ly ๐ผ wavelength rangeconsidered in Shapley et al. (2006) and Inoue & Iwata (2008) isnot well matched to the LyC wavelength range they considered. Ifwe instead use our alternative Ly ๐ผ range, 1173-1213 ร , we ๏ฌnda signi๏ฌcant increase in the correlation coe๏ฌcient at ๏ฌxed redshiftrange to 0.32-0.37. Combining the values for all bins we ๏ฌnd amodest increase to 0.45. Thus, we ๏ฌnd a weak correlation between MNRAS , 1โ20 (2021) R. Bassett et al.
Figure 15.
Comparison of (cid:104) ๐ esc (cid:105) computed via Equation 9 (gold) and Equa-tion 10 (cyan) compared to the true (cid:104) ๐ esc (cid:105) at ๐ง = . ๐ง = . ๐น / ๐น ) ๐๐๐ among the 15galaxies and use this value to estimate (cid:104) ๐ esc (cid:105) using Equations 9 and 10.The 1-to-1 relation is shown with the thick dotted line while the thin dottedlines represent the average 68 percentile spread of the 15 galaxies selectedin individual trials (more description in text), which we ๏ฌnd to decreaseroughly linearly with increasing the mean ๐ esc for a given trial. ๐ IGM for LyC and Ly ๐ผ at ๏ฌxed redshift where the wavelength rangesfor these two are well matched. We note that a direct comparison tothe results of Inoue & Iwata (2008) and Shapley et al. (2006) maybe slightly tenuous as the IGM transmission curves produced theredo not include a CGM component while our models do. Indeed,this may be the reason that we ๏ฌnd such a large increase in thecorrelation coe๏ฌcient at ๏ฌxed redshift when the wavelength rangesof LyC and Ly ๐ผ are properly matched. Given the large scatter andthe fact that ๐ IGM ( LyC ) is found to be 0 for a range of ๐ IGM ( Ly ๐ผ ) at ๏ฌxed redshift, however, we would be hesitant to try and estimateone from the other regardless of the apparent weak correlation. ๐ bias on ๐ esc Estimates for Samples
The analysis presented in Section 3.1 was designed to predict theaverage bias for a sample of LyC detected galaxies, which in turn canbe used to estimate the average ๐ esc of the sample (e.g. S18, F19).Thus, it may not be appropriate to blindly apply values measuredhere to individual galaxies. Here we test the discrepancy between theaverage value of ๐ esc for a sample of LyC detections estimated withand without including ๐ bias when compared to the true average ๐ esc .This should be seen as a highly simpli๏ฌed test as all mock galaxiesrepresent dust-free BPASSv2.1 models with a ๏ฌxed ( ๐ฟ / ๐ฟ ) ๐๐๐ก of 0.18 (log ( ๐ ๐๐๐ /[ Hz erg โ ]) = . ๐ฟ / ๐ฟ ) ๐๐๐ก , and will thus decreasethe accuracy of ๐ esc estimates when compared to this test.The typical method of estimating ๐ esc is to employ an equationof the form (or similar to): ๐ esc = ( ๐น / ๐น ) ๐๐๐ ( ๐ฟ / ๐ฟ ) ๐๐๐ก ร (cid:104) ๐ IGM (cid:105) (9)noting that the e๏ฌects of dust attenuation are ignored here. In order toestimate ๐ esc for a given level of ๐ bias , Equation 9 must be modi๏ฌedin the following way: ๐ ๐๐๐๐ esc = ( ๐น / ๐น ) ๐๐๐ ( ๐ฟ / ๐ฟ ) ๐๐๐ก ร (cid:104) ๐ IGM (cid:105) + ๐ bias (10) Our test of the recovery of (cid:104) ๐ esc (cid:105) for a sample of spectroscopicallyLyC detected galaxies is performed on the mock observations de-scribed in Section 2.3. We ๏ฌrst select those mock galaxies withoutput 880 < ๐ rest <
910 ร ๏ฌuxes above the detection limit of 0.025 ๐ Jy. We then perform 5000 trials in which we randomly select 15mock LyC detections (matched to the number of detections in S18)and measure (cid:104) ( ๐น / ๐น ) ๐๐๐ (cid:105) of this subsample. For each trialwe calculate the average ๐ esc using Equations 9 and 10 and comparethis with the true (cid:104) ๐ esc (cid:105) for the 15 selected detections.The results of this test at ๐ง = . ๐ง = . ๐ = 0.5 ๐ esc PDF model are shown in Figure 15, though we ๏ฌnd similarresults for the ๏ฌat ๐ esc PDF of our ๏ฌducial model. Here we plotthe true (cid:104) ๐ esc (cid:105) versus two estimated values. Gold and cyan contoursshow the distribution for (cid:104) ๐ esc (cid:105) estimated using Equations 9 and 10,respectively. The thick dotted green line shows the 1-to-1 relation.The thin green lines are meant to be representative of the average68 percentile spread of the 15 galaxies from any individual trial. Toproduce these lines we measure the 68 percentile lower and upperbounds and the average values of ๐ esc for the 15 galaxies from eachof the 5000 trials. We ๏ฌnd that the upper and lower bounds for a giventrial decrease roughly linearly with increasing mean ๐ esc (albiet withsigni๏ฌcant scatter), thus we ๏ฌt each bound with a straight line as afunction of mean ๐ esc . In this way, we are attempting to illustrate,roughly, the expected speard in ๐ esc values for a random selectionof 15 LyC detected galaxies having a given mean ๐ esc value.At both redshifts there is good agreement between ๐ corresc and thetrue value, while failing to account for ๐ bias results in an overestimateof the average ๐ esc . The level of overestimation is lower at ๐ง = . ๐ esc in Equations 9 and 10 depends on thereciprocal of ๐ IGM , which is decreasing with redshift towards 0. At ๐ง = .
1, Only โผ (cid:104) ๐ esc (cid:105) values calculated usingEquation 9 fall within the range of typical โtrueโ ๐ esc values forour detected sample (noting this percentage is stochastic). At higherredshifts this falls to 0%. Considering ๐ corresc , we ๏ฌnd that, typically,less than 1% of trials fall outside of the rough con๏ฌdence intervalspresented in Figure 15. This test illustrates that not accounting for ๐ bias when estimating the stacked ๐ esc for detected galaxies canresult in a signi๏ฌcant overestimate of the true value.Of course, as has been repeated throughout this work, the ab-solute di๏ฌerences between ๐ esc and ๐ ๐๐๐๐ esc (as well as the fractionaldecrease) will have some dependence on the details of our methodfor producing IGM transmission curves (e.g. ๐ HI distributions), theassumed value(s) of ( ๐ฟ / ๐ฟ ) ๐๐๐ก , the assumed ๐ esc PDF, theinput distribution of 1500 ร ๏ฌuxes, etc. In addition, the inclusionof dust, choice of dust curve, and any assumed dependence be-tween E(B-V) and ๐ esc will further a๏ฌect these results. Althoughnot shown, we also performed the test presented here with dustattenuation included following the method outlined in Section 4.4(where E(B-V) values are sampled from a distribution characterisedby the observed values from S18) and ๏ฌnd similar results with asimilar level of scatter. This of course assumes that both the average ๐ธ ( ๐ต โ ๐ ) for detected galaxies as well as the exact form of theattenuation curve is precisely known. Inevitably, these values willbe highly uncertain for real observations resulting in a higher levelof scatter. Providing more realistic tests of the associated e๏ฌectson our stacking, while possible, would be highly model dependent,thus not particularly useful.The fact that ๐ IGM for LyC detected galaxies is expected tobe larger than (cid:104) ๐ IGM (cid:105) at a given redshift will be true regardlessof the exact implementations, however. Thus, the purpose of theillustration presented here is simply to highlight the fact that theassumption that (cid:104) ๐ IGM (cid:105) is representative of IGM sightlines towards
MNRAS000
MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections LyC detected galaxies will result in an overestimation of ๐ esc . Giventhat ๐ bias increases with decreasing observational depth, the over-estimation of ๐ esc will be the higher for shallower LyC surveys. ๐ esc PDF
As we have shown in Section 3.2, the value of (cid:104) ๐ esc (cid:105) inferred forstacked samples of LyC detections will depend on the PDF assumedfor ๐ esc . It has been repeated throughout this work that, observation-ally, there appears to be a preference for low (or zero) ๐ esc from highredshift galaxies (e.g. Japelj et al. 2017; Smith et al. 2018; Bian &Fan 2020). This creates a chain of circular reasoning, however, asaccurate measurements of ๐ esc thus requires knowledge of the PDFof ๐ esc , which seemingly requires accurate measurements of ๐ esc todetermine. The way forward is to determine an observational metricthat can help to determine the PDF ๐ esc that is independent of theindividual values of ๐ esc .In this Section we propose that the detection rates of LyCfrom dedicated surveys can be used to probe the parameters of agiven ๐ esc PDF. We construct two mock versions of the surveysof S18, F19, and M20, one with a ๏ฌat ๐ esc PDF and one withan exponentially declining ๐ esc PDF. In the latter case we tunethe value of ๐ (see Equation 8) to match the observed detectionrate of a given survey (more description to follow). In each case,we again use the same ๏ฌxed input BPASSv2.1 SED model as our๏ฌducial model with ( ๐ฟ / ๐ฟ ) ๐๐๐ก = 0.18. In all cases the input1500 ร ๏ฌuxes are sampled from a distribution matched to the ๏ฌuxesreported by each of those surveys. Thus, in this case, the F19 sample,which is made up of LAEs, have a characteristic 1500 ร ๏ฌux thatis lower than that of the LBG samle of S18 at the same redshiftresulting in a lower relative LyC ๏ฌux (see Figure 4). This is importantas the selection method of a given sample strongly in๏ฌuences thedistribution of galaxy properties included (e.g. typical 1500 ร ๏ฌux,( ๐ฟ / ๐ฟ ) ๐๐๐ก , among others), which ultimately determine theoutput LyC ๏ฌuxes, and thus the detectability of a given galaxy.Therefore, the toy model presented here is primarily for illustrativepurposes.The set of inputs described above is combined with our ๐ IGM sightlines to produce an output sample of LyC ๏ฌuxes. In the case ofS18 and F19 we simply use the ๐ IGM functions already produced,noting that this requires all of our mock galaxies to be at discreteredshifts with ฮ ๐ง = 0.1. For S18 we match the observed redshiftdistribution in each bin from that work and for F19 all galaxiesare simulated at ๐ง = .
1. As M20 explores signi๏ฌcantly higherredshifts we simply randomly sample values across the full redshiftrange (matched to the observed distribution from that work) andcreate new ๐ IGM functions each time. For each survey we produce10,000 mock observations. We then randomly draw subsamplesfrom these mock observations with sizes matched to the observedsample sizes in each paper and measure the detection rates of LyCfor each subsample with detection rates of 0.025 ๐ Jy ( โผ ๐ values for the exponential ๐ esc PDF, and coarsely tune the model such that the average detectionrate falls within the range quoted for each survey.The results of our test for KLCS and LACES are shown inFigure 16. In each panel the detection rate distribution for the tuned,exponentially declining model is shown in cyan (with the tuned ๐ value in the legend) and the distribution for the ๏ฌat model isshown in gold. For LACES the observed detection rate range isde๏ฌned by either only considering their โgoldโ sample (low) orconsidering the โgoldโ plus โsilverโ samples (high) and for KLCS Figure 16.
Detection rate distributions for our mock LACES (top) andKLCS (bottom) surveys. In each case we create 50 thousand mock spectraat the respective survey redshifts following Section 2.3, however we nowinclude the e๏ฌects of dust attenuation for the S18 comparison (see text).The observed detection rates are shown in gray while the detection ratesassuming a ๏ฌat and exponentially declining ๐ esc PDF are shown in gold andcyan, respectively. In each case, the value of ๐ for the exponentially decliningPDF is coarsely tuned to match the detection rate of a given survey. we take their detection rate of 15/124 โผ ๐ esc PDF model is seen to predict a detection rate thatis too large to reproduce the survey in question. While the ๐ =0.5 model is well matched to the observed detection rate. In thecase of LACES, however, though the average detection rate for the๏ฌat model is close to the upper limit for the detection rate of thatsuvey, it is di๏ฌcult to rule out a ๏ฌat PDF. The coarsely matchedexponential model requires a relatively high value of ๐ = 0.75.From Figure 12 we expect that the inferred ๐ esc values from thismodel will not di๏ฌer signi๏ฌcantly from a ๏ฌat distribution. In thecase of our mock M20 test, we were unable to reproduce the highdetection rate reported in the paper, which falls in the range 0.02-0.11 (1-5 out of 44) depending on the reliability cut for the LyCemitting galaxy candidates from that work. For our mocks we ๏ฌndan overall detection rate from the ๏ฌat ๐ esc PDF (which will givethe highest detection rate) of 0.005, thus only a small fraction ofrandom selections of 44 galaxies will even contain one detection.Taken together, this toy model test for our three comparisonsamples provides strong evidence that the underlying ๐ esc PDFs willbe sensitive to the selection bias of the galaxy sample in question. Inthe case of KLCS and LACES, the former probes the bright end ofthe UV luminosity function characterised by LBGs while the lattersigni๏ฌcantly fainter LAEs. The fact that the detection rate of KLCSrequires the ๐ esc PDF to be skewed towards 0 while LACES is notinconsistent with a ๏ฌat ๐ esc PDF points towards a scenario in whichfaint galaxies, on average, have a ๐ esc PDF less biased towards 0(similar to the results of Finkelstein et al. 2019). Our inability toreproduce the high detection rate of M20, even employing a ๏ฌat ๐ esc PDF, suggests that this sample may be biased towards high valuesof ๐ esc (though we have not tested such a model here). Interestingly,the goal of M20 was to provide a methodology for preferentiallyselecting high ๐ esc galaxies, consistent with the toy model presentedhere. The key point highlighted here is that we have shown thecalculation of ๐ esc to be sensitive to the underlying PDF (e.g. Figure12), which is in turn appears to depend on sample selection. Thus, MNRAS , 1โ20 (2021) R. Bassett et al. a consideration of the ๐ esc PDF should be considered in particularwhen comparing inferred ๐ esc values between disparate samples(e.g. LAEs vs LBGs).We reiterate that the exponentially declining ๐ esc PDF favouredhere is simply an ad-hoc solution chosen for its bias towards low ๐ esc values and a preference for ๐ esc = 0. This selection was moti-vated by the low detection rate of such emission and the, generally,low estimates of the average ๐ esc for large galaxy samples (e.g.Vanzella et al. 2010; Grazian et al. 2016; Smith et al. 2018). Thetrue functional form of the PDF of ๐ esc is very likely more complexand may include dependencies on galaxy properties such as, e.g.,stellar mass (Finkelstein et al. 2019; Naidu et al. 2020). Furtherclari๏ฌcation of this issue will require larger samples of LyC de-tections at ๐ง > ๐ข -band wavelengths. It is also likely that inputsfrom high-resolution, hydrodynamics simulations of high redshiftgalaxies that include full radiative transfer can help greatly withthe interpretation of detections (and non-detections), though run-ning such simulations is computationally expensive. Regardless, weshow here evidence that the most likely PDF for ๐ esc for LBG likegalaxies favours a model with a reasonable bias towards low ๐ esc . To this point, we have avoided one key topic in the study of opti-cal and UV radiation from star-forming galaxies: dust attenuation.In general, the level of attenuation at ๏ฌxed E(B-V) increases withdecreasing ๐ rest such that UV wavelengths experience the highestlevels of attenuation (i.e. lowest transmission) irrespective of thefunctional form of the assumed attenuation curve (e.g. Gordon &Clayton 1998; Calzetti et al. 2000; Reddy et al. 2016, etc). Thisstatement, of course, assumes that extending the chosen attenuationcurve to short wavelengths ( (cid:46) (cid:104) ๐ธ ( ๐ต โ ๐ )(cid:105) = 0.045(and 0.129 for full LBG parent sample, S18), and those of LACESall exhibit negligible attenuation ( ๐ธ ( ๐ต โ ๐ ) < (cid:104) ๐ธ ( ๐ต โ ๐ )(cid:105) (cid:39) ๐ esc calculations outlined in this paper is warranted. We advocatea methodology similar to that outlined in F19. First, a determina-tion of the stellar E(B-V) value should be computed based on theavailable photometric data for a given sample of objects. This canbe achieved through full SED ๏ฌtting or through calibrations suchas those based on the UV slope, ๐ฝ (e.g. Meurer et al. 1999). Inthe case of SED ๏ฌtting, we advocate a method only incorporatingbands redward of Ly ๐ผ , as shorter wavelengths are strongly a๏ฌectedby IGM attenuation (e.g. e๏ฌects not intrinsic to the galaxy) thatshould be treated independently to avoid added degeneracy in theSED model. The e๏ฌects of including or omitting ๏ฌux with wave-lengths shortward of Ly ๐ผ during the SED ๏ฌtting process will betested in future work (Bassett et al., in prep). The computed E(B-V)is combined with a choice of dust attenuation curve, ๐ ( ๐ ) , to correct the observed 1500 ร ๏ฌux to the โintrinsicโ, dust-free, value. Finally,the chosen input SED template for a given sample (either computedthrough SED ๏ฌtting or simply selecting a template with a reasonablevalue of ( ๐ฟ / ๐ฟ ) ๐๐๐ก ) is scaled to match the corrected 1500 ร ๏ฌux. Through this process, the intrinsic LyC ๏ฌux can be determined,noting this value will be dependent on the selection of the intrinsicSED.From the intrinsic LyC ๏ฌux calculated in this manner, one canthen determine the expected value of ๐ esc by comparing with theobserved value. In this way, any attenuation of LyC ๏ฌux due to dustis incorporated into the de๏ฌnition of ๐ esc , as pointed out by F19 (i.e.there is no distinction between dust attenuation and absorption ofLyC by neutral hydrogen). We also follow the methodology of F19who allow ๐ esc for a given value of ๐ธ ( ๐ต โ ๐ ) to only be as large asthe transmission allowed by the extrapolated dust attenuation curve.This is reasonable as, in the case of a galaxy with relatively large ๐ธ ( ๐ต โ ๐ ) , a value of ๐ esc = 1.0 would imply zero dust attenuationfor LyC and high attenuation at 1500 ร . We note, however, thatsuch a case is not entirely impossible given LyC emission is oftendominated by stellar populations with ages <
10 Myr while stellarpopulations as old as a few hundred Myr can provide signi๏ฌcant๏ฌux at 1500 ร (e.g. Eldridge et al. 2017), thus the emission at eachwavelength may originate from di๏ฌerent locations within a givengalaxy.We do note, however, that this maximum ๐ esc allowed by theassumed attenuation curve is highly model dependent. For examplea Small Magellanic Cloud attenuation curve (e.g. Gordon & Clayton1998) will have a much higher attenuation at LyC wavelengths whencompared to either a Calzetti et al. (2000) or Reddy et al. (2016)attenuation curve with the same 1500 ร attenuation. This resultsfrom the fact that the extension of the functional form of either aCalzetti et al. (2000) or Reddy et al. (2016) ๐ ( ๐ ) is signi๏ฌcantly๏ฌatter at ๐ < ๐ esc allowable for a given value of ๐ธ ( ๐ต โ ๐ ) . The study of LyC escape from galaxies in ground-based studies islimited to redshifts (cid:38) ๐ง โผ . ๐ง = .
42 by Sahaet al. (2020) with the Ultra-Violet-Imaging Telescope (UVIT) onboard AstroSat. We also note that there has been signi๏ฌcant activ-ity in spectroscopic detection of LyC from green pea galaxies at ๐ง โผ . โ ๐ bias .Here we focus on providing predictions for detection of LyCwithin the Ultraviolet Imaging of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey Fields (UVCANDELS;PI: Teplitz, PID 15647), a โผ
430 arcmin , 164-orbit Cycle 26UV HST program. UVCANDELS will provide 3-orbit depth ofWFC3/275W and parallel ACS/F435W in four CANDELS ๏ฌelds:GOODS-N, GOODS-S, EGS, and COSMOS.For our predictions we follow a similar procedure outlined inSection 4.3, however here we have produced 10,000 IGM trans-mission curves at both ๐ง = . ๐ง = . MNRAS000
430 arcmin , 164-orbit Cycle 26UV HST program. UVCANDELS will provide 3-orbit depth ofWFC3/275W and parallel ACS/F435W in four CANDELS ๏ฌelds:GOODS-N, GOODS-S, EGS, and COSMOS.For our predictions we follow a similar procedure outlined inSection 4.3, however here we have produced 10,000 IGM trans-mission curves at both ๐ง = . ๐ง = . MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections Figure 17.
UV magnitudes of mock star-forming galaxies at ๐ง = . ๐ง = . ๐ง โผ . ๐ง โผ . Top row:
LyC vs non-ionizing UVmagnitudes of mock observations.
Bottom row: histograms of magnitudes forLyC probing bands. In all panels the vertical dashed line indicates the depthsof UVCANDELS observations. Here mock galaxies are produced as dust-free, exponentially declining SFR BPASSv2.1 SEDs with ( ๐ฟ / ๐ฟ ) ๐๐๐ก = 0.184, and assuming a ๏ฌat ๐ esc PDF. We note that a larger value of( ๐ฟ / ๐ฟ ) ๐๐๐ก can produce a handful of individual LyC detected galaxies. ACS/F435W ๏ฌlters, respectively. To sample the input 1500 ร ๏ฌuxesor this comparison we sample from UV luminosity functions ofMoutard et al. (2020) derived from the CLAUDS survey for ๐ง = . ๐ง (cid:38) .
8) from Bouwens et al.(2015) for ๐ง = .
4. In both cases we use luminosity functions de-scribed by a Schechter function with ๐ผ = -1.4, ๐ โ = 2.708 ร โ ,and ๐ โ = โ .
623 at ๐ง = . ๐ผ = -1.64, ๐ โ = 1.97 ร โ , and ๐ โ = โ .
88 at ๐ง = .
4. For all mock galaxies we assume a valueof ( ๐ฟ / ๐ฟ ) ๐๐๐ก of 0.18.To sample the ๐ rest โผ ๐ง = . ๐ง = . ๐ depths of each ๏ฌlter are matchedto the observations at 27.0 mag for F275W, 28.0 for F435W, and28.4 for F814W. For the best chance of detecting LyC emissionwe produce roughly 100,000 mock observations of galaxies at eachredshift, signi๏ฌcantly more than should be expected in the UVCAN-DELS volume. We also assume a ๏ฌat ๐ esc PDF, to further increasethe possibility of producing galaxies with very bright LyC ๏ฌux.The results of this test are shown in Figure 17. The top rowshows the non-ionizing UV versus LyC magnitudes (observationalband is redshift dependent) and the bottom row show the his-tograms of F275W and F435W magnitudes in logscale. In bothpanels we show the magnitude limits of UVCANDELS for respec-tive LyC probing bands with a vertical dashed line. For galaxieswith ( ๐ฟ / ๐ฟ ) ๐๐๐ก = 0.18, the 5 ๐ limits of UVCANDELS aretoo shallow to detect individual galaxies within the UVCANDELSfootprint as the brightest. In the event that UV bright galaxies withsigni๏ฌcantly higher ( ๐ฟ / ๐ฟ ) ๐๐๐ก exist within the UVCANDELSfootprint, it may be possible that one or two individual detectionswill be found. We conclude that pushing observations to a depth of 30 mag and beyond in small, targeted ๏ฌelds (i.e. similar to the ๐ง = . ๐ esc for galaxy subsamples. In this sce-nario, prior information regarding the likelihood of escaping LyCemission (e.g. evidence of a hard ionizing spectrum or high Ly ๐ผ escape, if available) will be useful. This is due to the fact that,although high LyC ๏ฌux is more common for UV bright galaxies,galaxies with the same UV brightness are also commonly foundwith relatively low LyC ๏ฌux (as shown in Figure 17). Similarly, we๏ฌnd galaxies with relatively faint non-ionizing UV ๏ฌux with rela-tively high LyC ๏ฌux. Thus, simply stacking the galaxies with thehighest 1500 ร ๏ฌux does not guarantee that the galaxies with thehighest LyC ๏ฌux have been chosen.Finally, we note a few caveats to this analysis. First, this analy-sis has been performed using ๐ IGM curves produced independently,while UVCANDELS covers 4 individual ๏ฌelds. In the event thatthere is strong correlation in ๐ IGM across the ๏ฌeld, the resultingLyC ๏ฌuxes may be systematically higher (in the case of high ๐ IGM )or lower (in the case of low ๐ IGM ) for one particular ๏ฌeld. Consider-ation of the correlation of ๐ IGM between sources in individual ๏ฌeldsof a given area may require dedicated analysis of large scale simula-tions of the HI distributionns and is beyond the scope of this work.And second, this analysis has ignored variation in ( ๐ฟ / ๐ฟ ) ๐๐๐ก (as noted), assumed no dust attenuation, and employed a ๏ฌat ๐ esc PDF, all three of which will a๏ฌect our resulting LyC ๏ฌuxes. ๐ esc Ultimately the ongoing search for LyC emission from high redshiftgalaxies is closely connected with our understanding what types ofgalaxies are responsible for reionizing the universe. Characterisingthe population of strong LyC emitters will be key to informing ourpicture of the topological evolution of ionized regions during theEoR (Seiler et al. 2018). There exists, however, an inherent di๏ฌcultyregarding the interpretation of ๐ esc values measured observationallydue to the complex geometry of LyC escape from galaxies, inde-pendent of ๐ IGM . Indeed various models have been hypothesisedthat may provide slightly di๏ฌerent interpretations of the detectedLyC ๏ฌux in the context of measuring ๐ esc . A detailed discussion ofvarious LyC escape models can be found in Section 9.4 of S18.Crucially, it has been pointed out (e.g. Bassett et al. 2019;Barrow et al. 2020) that the detection of LyC from any individ-ual galaxy is re๏ฌective of only the fraction of LyC that is able toescape into our single line-of-sight. There is still no reliable wayof inferring if the observed ๐ esc value is re๏ฌective of ๐ esc in alldirections, i.e. the 3D ๐ esc (though intriguing indirect measurementtechniques for the 3D ๐ esc have been proposed, which warrant fur-ther exploration within an anisotropic LyC escape scenario, e.g.Zackrisson et al. 2013; Yamanaka et al. 2020). Similarly, the lackof LyC emission from any individual galaxy is not evidence of ๐ esc = 0.0 as large quantities of LyC photons could be escaping in di-rections other than our line-of-sight. As we have shown, the valueof ๐ esc is dependent on the assumed underlying PDF and currentdetection rates may disfavour a ๏ฌat distribution. One way to providea theoretically sound basis for our assumptions on the PDF of ๐ esc for galaxies or galaxy samples is through the careful considerationof high resolution hydrodynamical simulations.LyC escape can be measured in such simulations by applyingfull radiative transfer, then measuring ๐ esc from a large number of MNRAS , 1โ20 (2021) R. Bassett et al. sight lines towards the galaxy. This method provides the full threedimensional ๐ esc at a given time and has shown that even for indi-vidual galaxies ๐ esc is highly variable and can swing from 0 to 1within 100 Myr (Paardekooper et al. 2015; Trebitsch et al. 2017;Rosdahl et al. 2018) though there may be some mass dependenceon the 3D ๐ esc PDF. It has been shown, however, that for a galaxywith given 3D ๐ esc value the value of ๐ esc in any particular sight-line may vary from 0 to values larger than the true 3D value (e.g.Paardekooper et al. 2015, Figure 13). Thus, to construct the un-derlying ๐ esc PDF for a given sample of galaxies may require thecombination of the 3D ๐ esc of galaxies (with possible dependencieson mass or other properties) with the probability distribution of 2D ๐ esc (line-of-sight) for a given 3D ๐ esc value. Disentangling the var-ious dependencies on these underlying PDFs will require suites ofhigh resolution simulations with full radiative transfer, but is of theutmost importance in interpreting the 2D ๐ esc values from observa-tions with the true 3D ๐ esc distributions. Ultimately it is the full 3D ๐ esc values from galaxies that are of interest in the context of theEoR, which can only be connected to our 2D observational resultsthrough such a complex line of reasoning as is described here. In this paper we have explored the level of bias in the IGM trans-mission, ๐ IGM , for galaxies with LyC detections at ๐ง =3-4 under theobservational limits imposed by current instruments and surveys.Our tests were performed by simulating one million IGM transmis-sion functions in our redshift range of interest and applying theseto empirically motivated mock galaxy spectra constructed from theBPASSv2.1 models (Eldridge et al. 2017). We have also tested howthe level of IGM transmission bias, ๐ bias , depends on both the as-sumed probability distribution function, PDF, of ๐ esc and SED shape(which controls ( ๐ฟ / ๐ฟ ) ๐๐๐ก , a key value for measuring ๐ esc ).Our analysis has included modeling designed to approximate bothspectroscopic and photometric LyC detections from recent surveysof Steidel et al. (2018), Fletcher et al. (2019), and Meลกtriฤ et al.(2020).Broadly, we ๏ฌnd that, in all cases the average value of ๐ IGM at LyC wavelengths for galaxies with LyC detections is found tobe larger than the average ๐ IGM for all simulated sightlines at thesame redshift. This results from the fact that the underlying ๐ IGM distribution at 880 ร < ๐ rest <
910 ร is bimodal with the strongerpeak at ๐ IGM = 0, but the simple fact that the galaxy has beendetected means that ๐ IGM โ
0. Thus, the ๐ IGM distribution for LyC detected galaxies is unimodal with a peak at relatively high ๐ IGM ,while the mean for all sightlines falls below this due to the inclusionof the ๐ IGM = 0 peak. The result is that the assumption of a mean ๐ IGM for all sightlines when calculating (cid:104) ๐ esc (cid:105) for a sample of LyCdetected galaxies results in an overestimate of the true value. Thisresult is similar to the recent results of Byrohl & Gronke (2020) forLy ๐ผ transmission. Thus, it is becoming clear that, while tempting,using a single statistic (e.g. median or mean) when considering ๐ IGM for individual objects provides misleading results for LyC detectedsamples. Considering samples which include (are composed entirelyof) LyC non-detected galaxies, the use of (cid:104) ๐ IGM (cid:105) when calculatingupper limits on (cid:104) ๐ esc (cid:105) is appropriate, however. The remainder of ourconclusions can be summarised as follows: โข Assuming the an LBG-like UV ๏ฌux distribution and applyingdetection limits of Steidel et al. (2018), Fletcher et al. (2019), andMeลกtriฤ et al. (2020) we estimate minimum levels of ๐ bias to be โผ โผ โผ โข In the case of a UV ๏ฌux distribution more characteristic of LAEsample (e.g. those of Fletcher et al. 2019) a higher ๐ bias should beexpected. In this case, mock HST F336W observations similar toFletcher et al. (2019), the minimum ๐ bias increases to โผ โข We have shown in Section 3.2 that, although ๐ bias does notincrease signi๏ฌcantly assuming an ๐ esc PDF mildly biased towards0, there may be a slight decrease in the recovered ๐ esc value in sucha model. โข We have also demonstrated that the current detection rates ofLyC radiation from surveys may re๏ฌect information regarding theunderlying ๐ esc PDF. Our simpli๏ฌed model presented in Section 4.3,for example, appears to slightly disfavour a ๏ฌat ๐ esc PDF for LBGs(e.g. Steidel et al. 2018), though this may not be the case for LAEsamples (e.g. Fletcher et al. 2019).This ๏ฌnal point may suggest that fainter galaxies, representedby LAE samples, are more likely to exhibit a higher ๐ esc than brightgalaxies, represented by LBGs. Such a scenario is in agreement withother recent studies (e.g. Finkelstein et al. 2019). Our comparisonsin this context in Section 4.3 with the detection rates of Steidelet al. (2018) and Fletcher et al. (2019) are still in the realm oflow statistical signi๏ฌcance. Thus, con๏ฌrmation of these results willrequire larger samples of LyC detected galaxies on which to performa similar analysis.Of course, all of our results will depend on the various inputparameters of our models including the assumed distribution of1500 ร (rest-frame) ๏ฌuxes, our treatment (or lack thereof) of dustattenuation, our assumptions regarding the intrinsic luminosity ratio(( ๐ฟ / ๐ฟ ) ๐๐๐ก ) of galaxies, and even the details of our methodsfor producing ๐ IGM functions (e.g. HI distribution functions). Thus,we do not claim that the absolute values of ๐ bias from this work tobe in any way de๏ฌnitive. The purpose of this work is to highlightthe ways in which di๏ฌerent assumptions regarding the underlyingdistributions of ๐ IGM and ๐ esc a๏ฌect our attempts to estimate ๐ esc from galaxies. It is clear that signi๏ฌcant theoretical work is stillrequired to better understand these PDFs that are critical to ourinterpretation of LyC detections from observations.From an observational point of view, it is also clear that largersamples of LyC detections will be essential in disentangling the var-ious dependencies on ๐ esc (e.g. stellar mass, SFR, etc). It is possiblethat more e๏ฌcient searches can be conducted in the near future witha focus on increasing both depth and ๏ฌeld-of-view (FOV). Indeed,we ๏ฌnd the highest detection rates among our mock surveys for ourmock LACES survey (Fletcher et al. 2019, Section 4.3), primar-ily due to those observations reaching 30.24 mag. The drawbackis that this study is performed with WFC3, an instrument with arelatively small FOV. One possible future instrument that may pushLyC surveys to the next level is the Keck Wide Field Imager (KWFI,Gillingham et al. 2020) that is expected to achieve a signal to noiseof โผ ๐ข -band across a 1 degree diameterFOV in just under 8 hours of exposures (private communication).From our mock LACES survey, we estimate that โผ
50% of all simu-lated galaxies fall in the magnitude range between 30 and 32. Thus,the era of large samples of known LyC emitting galaxies may benear.
DATA AVAILABILITY
Simulated data used in this work is produced primarily using pub-licly available codes found at https://github.com/robbassett as wellas publicly available galaxy SED models from the BPASS collabora-
MNRAS000
MNRAS000 , 1โ20 (2021) he IGM Transmission of LyC Detections tion (Eldridge et al. 2017). Observational data used for comparisonis available from publications associated with those surveys. ACKNOWLEDGEMENTS
This research was conducted by the Australian Research CouncilCentre of Excellence for All Sky Astrophysics in 3 Dimensions(ASTRO 3D), through project number CE170100013. The authorswish to thank Chris Blake, Adam Batten, and Katinka Gerรฉb for use-ful and illuminating discussions. We also wish to thank our referee,Akio K. Inoue, for careful consideration of the manuscript, whichhas resulted in an improved focus within the context of currentstudies exploring LyC emission from galaxies at high redshift. Re-sults presented in this work have made extensive use of the python3programming language (Van Rossum & Drake 2009) and, in par-ticular, the authors wish to acknowledge the the numpy (Oliphant2006), matplotlib (Hunter 2007), and scipy (Virtanen et al. 2020)packages. MR and LP acknowledge support from HST programs15100 and 15647. Support for Program numbers 15100 and 15647were provided by NASA through a grant from the Space TelescopeScience Institute, which is operated by the Association of Universi-ties for Research in Astronomy, Incorporated, under NASA contractNAS5-26555.
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