A tale of two sites -- II: Inferring the properties of minihalo-hosted galaxies with upcoming 21-cm interferometers
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 25 September 2020 (MN L A TEX style file v2.2)
A tale of two sites - II: Inferring the properties ofminihalo-hosted galaxies with upcoming 21-cminterferometers
Yuxiang Qin (cid:63) , Andrei Mesinger , Bradley Greig , and Jaehong Park , Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy School of Physics, University of Melbourne, Parkville, VIC 3010, Australia ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) School of Physics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
25 September 2020
ABSTRACT
The first generation of galaxies is expected to form in minihalos, accreting gas throughH cooling, and possessing unique properties. Although unlikely to be directly detectedin UV/infrared surveys, the radiation from these molecular-cooling galaxies (MCGs)could leave an imprint in the 21-cm signal from the Cosmic Dawn. Here we quantifytheir detectability with upcoming radio interferometers. We generate mock 21-cmpower spectra using a model for both MCGs as well as more massive, atomic-coolinggalaxies (AGCs), allowing both populations to have different properties and scalingrelations. The galaxy parameters are chosen so as to be consistent with: (i) high-redshift UV luminosity functions; (ii) the upper limit on the neutral fraction from QSOspectra; (iii) the Thomson scattering optical depth to the CMB; and (iv) the timing ofthe recent putative EDGES detection. The latter implies a significant contribution ofMCGs to the Cosmic Dawn, if confirmed to be cosmological. We then perform Bayesianinference on two models including and ignoring MCG contributions. Comparing theirBayesian evidences, we find a strong preference for the model including MCGs, despitethe fact that it has more free parameters. This suggests that if MCGs indeed playa significant role in the Cosmic Dawn, it should be possible to infer their propertiesfrom upcoming 21-cm power spectra. Our study illustrates how these observations candiscriminate among uncertain galaxy formation models with varying complexities, bymaximizing the Bayesian evidence. Key words: cosmology: theory dark ages, reionization, first stars diffuse radiationearly Universe galaxies: high-redshift intergalactic medium
The first galaxies of our Universe are expected to form outof pristine gas, cooling inside so-called “minihalos” (withmass M h ∼ – 10 M (cid:12) ) via rotational-vibrational tran-sitions of H (e.g. Haiman et al. 1996, 1997; Yoshida et al.2003, 2006). The first episodes of star formation, evolution,and feedback inside these first-generation, molecular-coolinggalaxies (MCGs) can be very different from later generationsthat were mostly built-up out of pre-enriched material insidedeeper potential wells (e.g. Haiman et al. 1999; Tumlinson &Shull 2000; Abel et al. 2002; Schaerer 2002; Bromm & Lar-son 2004; Yoshida et al. 2006; McKee & Tan 2008; Whalenet al. 2008; Turk et al. 2009; Heger & Woosley 2010; Wise (cid:63) E-mail: [email protected] et al. 2012; Xu et al. 2016b; Kimm et al. 2016). Moreover,star formation inside MCGs is expected to be transient, ta-pering off as a growing Lyman-Werner (LW) backgroundstarts to effectively photodissociate H (e.g. Johnson et al.2007; Ahn et al. 2009; Holzbauer & Furlanetto 2012; Fialkovet al. 2013; Jaacks et al. 2018; Schauer et al. 2019).Unfortunately, MCGs are likely too faint to observe di-rectly using UV or infrared telescopes in the foreseeablefuture (e.g. O’Shea et al. 2015; Xu et al. 2016b). Theirtransient nature also makes low-redshift detection or evensearching for stellar relics in the nearby Universe very chal-lenging (Beers & Christlieb 2005; Tornatore et al. 2007; Na-gao et al. 2005, 2008; Lai et al. 2008; Suda et al. 2008; Roed-erer et al. 2014; Liu & Bromm 2020).A promising alternative is to study MCGs throughthe imprint their radiation fields leave in the intergalactic c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Qin et al. medium (IGM; Fialkov et al. 2013; Mirocha et al. 2017;Muoz 2019). In the standard hierarchical structure forma-tion paradigm, there should have existed a period at thestart of the Cosmic Dawn in which the radiation back-grounds were dominated by MCGs. If we can observe IGMproperties at a high enough redshift, we could indirectlystudy the properties of MCGs (e.g. Ciardi et al. 2006; Mc-Quinn et al. 2007; Ahn et al. 2012; Visbal et al. 2015; Mi-randa et al. 2017; Mesinger et al. 2012; Koh & Wise 2018).Luckily, the cosmic 21-cm signal is set to revolution-ize our understanding of the early Universe (for a recentreview, see Mesinger 2019). Sourced by the spin-flip transi-tion of neutral hydrogen, the cosmic 21-cm signal is sensi-tive to the ionization and thermal state of the IGM. Theseare in turn determined by the ionizing, soft UV and X-rayemission from the first galaxies. Therefore, the high-redshift21-cm signal should encode information about the birth, dis-appearance, spatial distribution, and typical spectral energydistributions (SEDs) of MCGs.Many experiments are striving to measure the signal.These can be broadly divided into global signal experimentsand interferometers measuring 21-cm fluctuations. The for-mer includes the Shaped Antenna measurement of the back-ground RAdio Spectrum (SARAS; Singh et al. 2018), theLarge-aperture Experiment to Detect the Dark Age (LEDA;Price et al. 2018), Probing Radio Intensity at high-Z fromMarion (PRI Z M; Philip et al. 2019), and the Experiment toDetect the Global EoR (i.e. Epoch of Reionization) Signa-ture (EDGES; Bowman et al. 2018). The latter has recentlyclaimed a detection with an absorption feature at z ∼
17, in-citing much debate as to its cosmological origin (e.g. Hillset al. 2018; Bradley et al. 2019; Sims & Pober 2019; Mirocha& Furlanetto 2019; Mu˜noz & Loeb 2018; Fialkov et al. 2018;Ewall-Wice et al. 2018; Mebane et al. 2019; Qin et al. 2020).Existing interferometers, such as the Low-Frequency Array(LOFAR ; van Haarlem et al. 2013; Patil et al. 2017), theMurchison Widefield Array (MWA ; Tingay et al. 2013;Beardsley et al. 2016) and the Precision Array for Prob-ing Epoch of Reionisation (PAPER ; Parsons et al. 2010),are focusing on measuring the 21-cm power spectrum, gen-erally at z <
10. These instruments are serving as precursorsand pathfinders for the next-generation radio telescopes: theHydrogen Epoch of Reionization Arrays (HERA ; DeBoeret al. 2017) and the Square Kilometre Array (SKA ), whichpromise to deliver 3-dimensional imaging and a high S/Nmeasurement of the 21-cm power spectrum (PS) out to z ∼ < Wouldwe be able to confidently extract the imprint of MCGs fromthe signal, and distinguish them from more evolved, second-generation galaxies?
Bayesian inference provides us with aclean framework to answer such a question. Specifically,Bayesian evidence allows us to perform model selection, http://eor.berkeley.edu http://reionization.org/ quantifying if data prefers one theoretical model over an-other. It has a built-in Occam’s razor factor, penalizing ad-ditional model complexity unless explicitly required by thedata (for a recent review of Bayesian inference in astronomy,see Trotta 2017).In this work, we quantify the detectability of MCGsfrom a mock measurement of the cosmic 21-cm PS, expectedfrom a 1000h integration with SKA1-low. Our mock signalis generated by self-consistently following the evolution ofboth MCGs and more massive atomically-cooled galaxies(ACGs), as described in Qin et al. (2020, hereafter Paper-I). From this mock observation, we infer the properties ofthe underlying galaxies using a model having only a popu-lation of ACGs, and a model allowing for both populations:ACGs and MCGs. We compute the Bayes factor of thesetwo models, quantifying if the mock observation providessufficient evidence for an additional population of MCGs.This paper is organized as follows. We briefly summa-rize our model in Section 2. In Section 3, we present ourmock observation, chosen so that the timing of the globalsignal is consistent with the putative EDGES detection. InSection 4, we perform parameter inference using two galaxymodels, presenting the corresponding Bayesian evidence. Fi-nally, we conclude in Section 5. In this work, we adopt thefollowing cosmological parameters: (Ω m , Ω b , Ω Λ , h, σ , n s =0.31, 0.048, 0.69, 0.68, 0.81, 0.97), consistent with Planck (Planck Collaboration et al. 2016b, 2018)).
To model the 21-cm signal we use the public code 21cm fast (Mesinger & Furlanetto 2007; Mesinger et al. 2011) withthe latest update from Paper-I. In Paper-I, we extended thegalaxy models of 21cm fast to include a separate populationof MCGs, with properties independent to those of ACGs.Here we briefly summarize our procedure for characterizingthese galaxies and their corresponding emissivities; for moredetails, please see Paper-I.We define two distinct galaxy populations on the basisof the cooling channel through which they obtained the bulkof their gas – ACGs and MCGs. These two populations aredefined via exponential “window functions” over the halomass ( M h ) function, d n/ d M h (for an in-depth discussion ofthis choice, see Paper-I). Specifically, the number density ofactively star-forming galaxies is φ = d n d M h × exp (cid:18) − M atomcrit M h (cid:19) exp (cid:18) − M molcrit M h (cid:19) exp (cid:18) − M h M coolcrit (cid:19) , (1)where the superscripts “atom” and “mol” are used to dis-tinguish ACGs and MCGs, respectively, as they are allowedto have different properties and scaling relations.We see from equation (1) that the occupancy fractionof ACGs starts dropping below a characteristic mass scaleof M atomcrit = max (cid:104) M coolcrit , M ioncrit , M SNcrit (cid:105) . (2) https://github.com/21cmfast/21cmFAST c (cid:13) , 000–000 inihalos in 21-cm PS Here we account for three physical processes that can sup-press star formation: (i) inefficient cooling, M coolcrit (corre-sponding to a virial temperature of ∼ K; Barkana &Loeb 2001); (ii) photoheating feedback from inhomogeneousreionization, M ioncrit (e.g. Sobacchi & Mesinger 2014; see alsoEfstathiou 1992; Shapiro et al. 1994; Thoul & Weinberg1996; Hui & Gnedin 1997); and (iii) supernova feedback , M SNcrit (Haiman et al. 1999; Wise & Abel 2007; Dalla Vecchia& Schaye 2008, 2012; Hopkins et al. 2014; Keller et al. 2014;Kimm et al. 2016; Hopkins et al. 2017).On the other hand, the occupancy fraction of MCGspicks up below the atomic cooling threshold, M coolcrit , andextends down to M molcrit = max (cid:104) M disscrit , M ioncrit , M SNcrit (cid:105) , (3)where the additional term, M disscrit , accounts for the coolingefficiency of H in the presence of an inhomogeneous LWbackground (e.g. Machacek et al. 2001; Draine & Bertoldi1996; Johnson et al. 2007; Ahn et al. 2009; Wolcott-Greenet al. 2011; Holzbauer & Furlanetto 2012; Visbal et al. 2015).We adopt power-law relations for the stellar ( M ∗ ) tohalo mass ratio (Moster et al. 2013; Sun & Furlanetto 2016;Mutch et al. 2016; Ma et al. 2018; Tacchella et al. 2018;Behroozi et al. 2019; Yung et al. 2019) M ∗ M h = Ω b Ω m × min , f atom ∗ , (cid:16) M h M (cid:12) (cid:17) α ∗ f mol ∗ , (cid:16) M h M (cid:12) (cid:17) α ∗ , (4)where three free parameters ( f atom ∗ , , f mol ∗ , and α ∗ ) set thenormalizations and scaling index.We assume that the stellar mass is on average built-upover some fraction of the Hubble time, t ∗ H ( z ) − , resultingin a star formation rate of SFR = M ∗ H ( z ) /t ∗ . Here, forcomputational convenience, we fix t ∗ = 0 . ∼ few times the halo dynamical time), noting that thereis a strong degeneracy between f ∗ and t ∗ , and the priordistribution over these two parameters results in a relativeinsensitivity of the results to t ∗ (Park et al. 2019; Paper-I).To compare with observed UV LFs (e.g. Finkelstein et al.2015; Bouwens et al. 2015a, 2016; Livermore et al. 2017;Atek et al. 2018; Oesch et al. 2018; Bhatawdekar et al. 2019),we also compute the corresponding 1500˚A luminosity witha conversion factor L / SFR = 8 . × erg s − Hz − yr(Madau & Dickinson 2014).We allow ACGs and MCGs to have different UV ioniz-ing escape fractions, also with power law scalings with halomass. However, for computational convenience, here we as-sume no evolution with halo mass or redshift resulting injust two additional free parameters, f atomesc and f molesc ).The dominant sources of X-rays in the very early Uni-verse are expected to be high mass X-ray binaries (HMXBs;Sanderbeck et al. 2018). Motivated by models and observa-tions of HMXBs (e.g. Mineo et al. 2012; Fragos et al. 2013;Pacucci et al. 2014), we assume their population-averaged Following Paper-I, here we also assume M SNcrit is smaller than theother relevant mass scales, so that we can maximize the impor-tance of MCG and thus match the timing of the putative EDGESdetection. Note that SNe feedback could still be responsible forthe power-law scaling of the stellar to halo mass relation if starformation is feedback-limited (e.g. Wyithe & Loeb 2013). specific X-ray luminosity scales linearly with the SFR of hostgalaxies, and has a power-law SED with an energy spectralindex of -1. We assume that only X-rays with energy greaterthan E = 500 eV can escape the host galaxy and interactwith the IGM. This value is motivated by high resolutionhydrodynamic simulations of the ISM in the first galaxies(Das et al. 2017). Moreover, we characterize the X-ray lumi-nosity of early galaxies with their soft-band ( < L X / ˙ (cid:12) d E = E − (cid:82) d EE − × (cid:40) L atomX < / ˙ (cid:12) L molX < / ˙ (cid:12) , (5)where we include two more free parameters (i.e. L atomX < / ˙ (cid:12) and L molX < / ˙ (cid:12) ) as the total soft-band luminosity per SFRfor ACGs and MCGs.Based on these galaxy properties, we can calculate 1)the ionization and heating rates by X-rays; 2) the Lyman- α coupling coefficient between the IGM spin and kinetic tem-peratures; 3) the LW radiation intensity and the critical halomass characterising the radiative feedback from LW sup-pression; as well as 4) the UV ionizing photon budget andthe critical mass for photoheating feedback (see equation1). It is worth noting that we also include inhomogeneousrecombinations (Sobacchi & Mesinger 2014), adopting a sub-grid density distribution from Miralda-Escud´e et al. (2000)but adjusted for the mean density in each cell, and accountfor density-dependent attenuation of the local ionizing back-ground according to (Rahmati et al. 2013). These quantitiesare then used to follow the temperature and ionization stateof each gas element in our simulation, which are in turn usedto compute the 21-cm signal. For more details see Mesingeret al. (2011) and Paper-I. We create a mock 21-cm observation from a simulation boxwith a comoving volume of (500Mpc) and a 256 grid.While the full parameter space of our model is very large(17 dimensional; see Table 1 in Paper-I), in this proof-of-concept work, we limit it to the 7 parameters that drive thelargest signal variation and are most relevant for the earlyCosmic Dawn signal. These include f atom ∗ , , f mol ∗ , , α ∗ , f atomesc , f molesc , L atomX < / ˙ (cid:12) and L molX < / ˙ (cid:12) . Table 1 summarizes theirphysical meaning, and shows the fiducial values we use tomake our mock observation. These fiducial values are chosenin order for the mock observation to be consistent with thefollowing observations:(i) the galaxy UV LFs at z ∼ −
10 (Bouwens et al. 2015a,2016; Oesch et al. 2018);(ii) the upper limit on the neutral fraction at z ∼ . x H i < . . , σ ; Mc-Greer et al. 2015);(iii) the CMB Thomson scattering optical depth from Planck ( τ e = 0 . ± . , σ ; Planck Collaboration et al.2016a); and c (cid:13) , 000–000 Qin et al.
Table 1.
A list of the free parameters varied in this work, together with their descriptions, fiducial values used for the mock observation,and the recovered values (median with [14, 86] percentiles) obtained from the two 21 cmmc runs. Other model parameters are heldconstant, using the values discussed in the text and in Paper-I.Parameters Description Mock 21 cmmc results log f atom ∗ , Stellar to halo mass ratio at M vir = 10 M (cid:12) M (cid:12) for ACGsMCGs − . − . +0 . − . − . +0 . − . log f mol ∗ , − . − . +0 . − . - α ∗ Stellar to halo mass power-law index 0 . . +0 . − . . +0 . − . log f atomesc Escape fraction of ionizing photons for ACGsMCGs − . − . +0 . − . − . +0 . − . log f molesc − . − . +0 . − . -log L atomX < / ˙ (cid:12) Soft-band X-ray luminosity per SFR (erg s − M − (cid:12) yr) for ACGsMCGs 40.5 40 . +1 . − . . +1 . − . log L molX < / ˙ (cid:12) . +1 . − . - (iv) the timing of the recent putative detection of anabsorption profile centered at 78 ± f atom ∗ , ∼
6% and α ∗ =0 . f atomesc =6% ensures reionization of the fiducialmodel finishes by z ∼ .
9, with the inferred τ e consis-tent with results from Planck . On the other hand,log (cid:104) L atomX < / ˙ (cid:12) / erg s − M (cid:12)− yr (cid:105) =40 . f mol ∗ , ∼ .
2% and f molesc =6%. Finally,an enhanced X-ray luminosity of MCGs, here we takelog (cid:104) L atomX < / ˙ (cid:12) / erg s − M (cid:12)− yr (cid:105) =41 .
7, is needed to re-produce an 21-cm absorption trough centred at ∼ We only consider the timing from EDGES that is expected to bedriven by minihalos (see Paper-I and also Mirocha & Furlanetto2019). This allows us to select an optimistic model for our proof-of-concept study, in which minihalos play an important role. Theamplitude of the reported signal cannot be explained by standardphysics (e.g. Mu˜noz & Loeb 2018; Fialkov et al. 2018; Ewall-Wice et al. 2018; Mebane et al. 2019), and some exotic expla-nations could have a large impact also on the power spectrum.However, partial degeneracy with unidentified systematics and/orforegrounds (e.g. Hill & Baxter 2018; Spinelli et al. 2018; Bradleyet al. 2019; Sims & Pober 2019) could bring the amplitude in linewith standard models, without evoking exotic physics. Our mockPS corresponds to such a scenario. This is based on the strong degeneracy between f mol ∗ , and theX-ray luminosity per SFR found in Paper-I. Note that althoughACGs and MCGs are assumed to share the same specific X-rayluminosity in Paper-I, a smaller L atomX < / ˙ (cid:12) used in this workdoes not have a significant impact to the timing of the absorp-tion trough because the contribution from ACGs at high redshifts( z (cid:38)
15) is small. in the upper panel of Fig. 1 and show its globally averaged21-cm brightness temperature evolution, EoR history as wellas the Thomson scattering optical depth ( τ e = 0 . α background, coupling the spin( T s ) and kinetic temperatures ( T k ). The brightness temper-ature ( δT b ) is negative (i.e. the IGM is seen in absorptionagainst the CMB) and decreases as the IGM adiabaticallycools faster than the CMB. For our choice of galaxy pa-rameters, δT b reaches its minimum at z ∼
17, before X-rayheating becomes significant, eventually heating the IGM totemperatures above the CMB by z ∼ −
14. As reionizationprogresses, the signal starts fading until z ∼ . i ). Following Greig & Mesinger (2018), we compress the cosmic21-cm lightcone into 3D averaged power spectra. The PSfrom the mock observation (generated from a unique initialseed) and the forward-modelled simulations (in Sec. 4) arecalculated from the same comoving volume of the lightcone.For computing efficiency, forward-modeled simulations havea factor of 2 smaller volume than the mock while keepingthe same resolution, i.e. (250Mpc) and 128 cells. There-fore, we compute the PS from 12 independent sub volumesof the lightcone between z = 5 . ∼ z = 30( ∼ . We usethe public package (Pober et al. 2013, 2014). Note that HERA is expected to provide comparable astrophys-ical parameter recovery as SKA1-low, using the power spectrumsummary statistic and the fiducial galaxy models from Park et al.(2019) and Greig et al. (2020). However, our fiducial model here ischosen to have a significant contribution of MCGs driving a muchearlier epoch of heating, as motivated by the putative EDGESc (cid:13) , 000–000 inihalos in 21-cm PS z = 5.77 z = 6.37 z = 7.04 z = 7.81 Mock2pop1pop z = 8.70 z = 9.73 z = 10.94 z = 12.35 z = 14.04 z = 16.06 z = 18.52 z = 21.56 k [Mpc ] T b / m K ] Figure 1.
Top panel : A slice of the 21-cm lightcone from our mock observation. The central redshifts of the 12 independent box samples,which are used to calculate the 21-cm PS, are indicated by the vertical dashed lines. Note that the spatial range of the vertical axis isfrom 250 to 500cMpc, half the entire lightcone length (500cMpc).
Lower panels: evolution of the 21-cm PS. Solid black curves correspondto the mock observation, with gray shaded regions indicating the 1 σ noise from a 1000h observation with SKA1-low. Only power withinthe range of k = 0 . − − is considered when performing the Bayesian inference. The [14, 86] percentiles of the recovered posteriorsfrom Fig. 2 are bracketed by the colored lines ( using red dash-dotted lines, using blue dashed lines). We adopt the “moderate” foreground removal configura-tion, which excises foreground contaminated modes from thecylindrical k -space “wedge” (which is assumed to extend at k (cid:107) ≈ . h Mpc − k mode is (Morales 2005; McQuinn et al. 2006;Parsons et al. 2012)∆ ( k ) ≈ k (1 + z ) π (cid:112) Ω m (1 + z ) + Ω Λ Ω BT , (6) detection. As thermal noise dominates at the corresponding lowfrequencies and SKA1-low should have smaller thermal noise thanHERA for a fixed integration time, for this proof-of-concept studywe make the slightly more optimistic choice of SKA1-low whencomputing the noise. where Ω and B correspond to the solid angle of the primarybeam size (e.g. ∼ . z = 17) and observing bandwidth(8MHz), respectively. We use the SKA1-low antennae config-uration described in Greig et al. (2020). The system temper-ature is taken to be T N = ( T sky + T rec )(2 Bt ) − . where T sky and T rec represent the sky and receiver temperatures whilethe factor, √ Bt , reflects the number of independent mea-surements during the integration time, t . Following Thomp-son et al. (2017), the sky is modelled as being dominatedby Galactic synchrotron emission and scales with frequency( ν ) as T sky = 60K( ν/ − . . On the other hand, thereceiver is assumed to be kept at 40K with an addition of0 . T sky reflecting its response to the sky (Pober et al. 2014).The total uncertainty on the 21-cm PS ( σ ∆ ) is ob-tained by summing over the individual modes, i , (Poberet al. 2013), and adding the cosmic variance of the mockobservation (reasonably assuming it is Gaussian distributedat the relevant scales; Mondal et al. 2016: (cid:20) σ ∆ ( k ) (cid:21) = (cid:88) i (cid:32) , i + ∆ (cid:33) . (7) c (cid:13) , 000–000 Qin et al.
The gray shaded regions in Fig. 1 show the resulting1 σ uncertainty on the mock cosmic signal. We note largeuncertainties at high redshifts while most constraints fromthe 21-cm PS come from large scales and z <
15. When per-forming inference, we additionally exclude the modes outsidethe range 0 . (cid:54) k/ Mpc − (cid:54) In this section, we use 21 cmmc (Greig & Mesinger 2015)to constrain astrophysical parameters and perform modelselection using the following observations: • the mock 21-cm power spectra discussed in Sec. 3.1; • the observed galaxy LFs at z ∼ −
10 (Bouwens et al.2015a, 2016; Oesch et al. 2018); • the upper limits on the neutral fraction at z ∼ . • the Thomson scattering optical depth of the CMB(Planck Collaboration et al. 2016a).We perform inference using the following two models:(i) : the “full” model, including both MCGs andACGs, used to generate the mock observation; and(ii) : a single population model consisting only ofACGs. is characterized with the 7 free parameters listed inTable 1, while only has the four parameters relevant forACGs (i.e. excluding the ones labelled “mol”). It is clear that cannot fully reproduce the mock observation: ACGs aretoo biased at early times and are not sensitive to the buildup of the inhomogeneous LW background. However, giventhe limited sensitivity of even SKA1-low during the CosmicDawn, will we be able to say with certainty that the model is incorrect?This question can be readily answered with Bayesian in-ference. Using the built-in Occam’s razor in the Bayes factor,we can quantify whether the unique properties of MCGs areneeded to explain the “observation”, or whether the sim-pler, single-population model can adequately mimic the sig-nal. Is the additional model complexity of justified bythe data? If not, we might not be able to detect minihalogalaxies even indirectly with upcoming interferometers.Before presenting our results in Sec. 4.3, we briefly re-view the basics of Bayesian model selection (Sec. 4.1), aswell as the MultiNest sampler we use inside 21 cmmc (Sec.4.2).
Bayes’ law states that the posterior probability distribution[ P ( θ | O , M )] of model ( M ) characterized by parameters ( θ )when constrained by observations ( O ) is equal to the prod-uct of our prior knowledge [ P ( θ | M )] and the likelihood func-tion [ P ( O | θ, M )] divided by the evidence [ P ( O | M )] P ( θ | O , M ) = P ( θ | M ) P ( O | θ, M ) P ( O | M ) . (8)While the posterior represents our belief about themodel after taking the observation into account, the prior re-flects our knowledge before . The likelihood measures howwell a parameter combination θ can reproduce the observeddata O .The Bayesian evidence, also known as the marginal like-lihood, is central to model selection. It can be computed byintegrating the likelihood, weighted by the prior, over theentire parameter space: P ( O | M ) = (cid:73) θ d θP ( O | θ, M ) P ( θ | M ) ∼ δθP ( O | θ max , M ) P ( θ max | M ) (9)The last step in equation (9) approximates the in-tegral trapezoidally around the maximum likeli-hood, P ( O | θ max , M ) (e.g. Trotta 2017). Here, δθ and∆ θ ≡ P − ( θ max | M ) characterize widths of the likelihoodand prior, respectively. The factor δθ/ ∆ θ is commonlyreferred to as Occam’s factor, as it penalizes models whichhave a prior volume that is larger than the likelihood.There are many model selection criteria (Liddle 2004,2007) to answer whether the increased complexity (seemore in Kunz et al. 2006) of a model involving a higher-dimensional parameter space is justifiable by the observa-tion – in our case, whether upcoming 21-cm PS measure-ments can be used to detect minihalo-hosted galaxies. Here.we use an empirical scale (Jeffreys 1939) based on the ra-tio of the evidences of the two models, the so-called Bayesfactor. Specifically, the probability of the model beingpreferred over is 75.0% (weak), 92.3% (moderate) and99.3% (strong) if ln B ≡ ln [ P ( O | M ) /P ( O | M )] is 1,2.5 and 5, respectively. cmmc (Greig & Mesinger 2015) is a Bayesian sampler of21-cm lightcones, allowing for cosmological and astrophys-ical parameter inference from the 21-cm signal (Greig &Mesinger 2017, 2018). In its default configuration, 21 cmmc employs an ensemble sampler ( emcee ; Goodman & Weare2010; Foreman-Mackey et al. 2013; Akeret et al. 2013) toexplore the parameter space, which does not require the ev-idence to generate a proposal distribution. This makes the Here we use a flat prior over the following ranges: f atom ∗ , ∈ [10 − ,
1] in logarithmic space; f mol ∗ , ∈ [10 − , − ] in logarithmicspace; α ∗ ∈ [ − . , f atom(mol)esc ∈ [10 − ,
1] in logarithmic space;and L atom(mol)X < / ˙ (cid:12) ∈ [10 , ] erg s − M − (cid:12) yr in logarithmic space. https://github.com/21cmfast/21CMMC c (cid:13) , 000–000 inihalos in 21-cm PS log f atom*,10 =1.25 +0.050.19 ( 1.25)1.28 +0.060.24 ( 1.25) . . . . . * * =0.50 +0.160.11 (0.49)0.43 +0.270.09 (0.41) . . . . . l o g f a t o m e s c log f atomesc =1.22 +0.140.21 ( 1.19)1.14 +0.120.14 ( 1.14) .
04 0 .
54 2 .
04 3 . l o g L a t o m X < k e V / e r g s M y r log L atomX<2keV/ erg s M yr =40.50 +1.281.88 (39.69)41.21 +1.170.22 (41.05) . . . . . l o g f m o l * , log f mol*,7 =2.75 +0.890.61 ( 2.42) . . . . . l o g f m o l e s c log f molesc =1.22 +0.931.05 ( 1.70) . . . . . log f atom*, 10 .
04 0 .
54 2 .
04 3 . l o g L m o l X < k e V / e r g s M y r . . . . . * . . . . . log f atomesc . . . . log L atomX<2keV/ erg s M yr . . . . . log f mol*, 7 . . . . . log f molesc . . . . log L molX<2keV/ erg s M yr log L molX<2keV/ erg s M yr =41.70 +1.072.01 (41.42) x H I Mock
Frequency/MHz T b / m K Mock: 79.8MHzEDGES: 78.0±1.0(1 )±19.0(FWHM)2pop: 78.6 +7.013.1
MHz1pop: 85.0 +5.59.1
MHz e d n / d e Mock0.062Planck0.058±0.012 2pop0.060 +0.0050.005 +0.0020.003 x HI ( z = 5.9) d n / d x H I ( z = . ) McGreer+15<0.06+0.05
Information Criterialn[ P ( | )] =ln [ P ( | max , ) ] =9.18 ± 0.32-7.9233.01 ± 0.25-20.44 redshift Figure 2.
Marginalized posterior distributions from our two astrophysical models: (i) in red / dash-dotted lines; (ii) in blue /dashed lines. While the model only considers 4 parameters describing ACGs, includes additional 3 parameters representing theproperties of MCGs. Both results use the following observations when computing the likelihood: (i) the observed galaxy LFs at z ∼ − z ∼ . x H i ) andbrightness temperature ( δT b ) as well as the PDF of τ e and x H i at z =5 . (cid:13) , 000–000 Qin et al. evaluation of the Bayesian evidence computationally chal-lenging in a high-dimensional parameter space (see the firstpart of equation 9).In this work, we include the
MultiNest sampler(Feroz & Hobson 2008; Feroz et al. 2009, 2019; Buchneret al. 2014) in 21 cmmc , which implements nested sampling– converting the variable of integration in equation (9) fromthe high-dimensional parameter space to the 1D prior space(see more in Skilling 2004) P ( O | M ) = (cid:90) d P ( θ | M ) P [ O | P ( θ | M )] , (10)where d P ( θ | M ) ≡ dθP ( θ | M ) represents the differential ofprior volume. By reducing the prior volume around higherprobability regions at each step when new sampling pointsare drawn, MultiNest computes the posterior and calcu-lates the Bayesian evidence as a “by-product”. The currentpublic version of 21 cmmc allows the user to choose between emcee and
MultiNest samplers.It is worth noting that the recent development of by Binnie & Pritchard (2019) also introduced
MultiNest into 21 cmmc . They found the posterior of a 3-parameter 21-cm model inferred from mock observations tobe consistent between and the original 21 cmmc .This encourages us to apply it to our updated 21-cm simu-lations using more sophisticated galaxy models.
In Fig. 2, we present the marginalized posteriors from ourtwo models ( / in blue/red), including model pa-rameters, global 21-cm signals, EoR histories, and the op-tical depths. The corresponding 21-cm power spectra areshown in Fig. 1.For both models, the properties of ACGs are tightlyconstrained, including f atom ∗ , , α ∗ and f atomesc . We caution how-ever that these parameters, especially f atomesc , are overcon-strained (e.g. compared to Park et al. 2019) due to the factthat several ACG parameters are kept fixed in this demon-strative study (most importantly M SNcrit , α esc , t ∗ ).In the absence of MCGs, we see that the model dramatically overestimates the X-ray luminosi-ties of ACGs, with the 1D PDF peaking sharply atlog ( L atomX < / ˙ (cid:12) / erg s − M − (cid:12) yr) ∼ . − .
4: a factor of ∼ −
75 times higher than the “true” value of the mocksignal. Moreover, the model prefers a lower α ∗ (i.e. asteeper stellar mass function), despite the fact that the UVLFs already constrain this parameter (e.g. Park et al. 2019).Thus, the posterior prefers galaxy models with moreefficient star formation in lower mass halos (i.e. smaller α ∗ )and with higher X-ray emissivities (i.e. larger L atomX < / ˙ (cid:12) ),in order to (partially) compensate for the missing populationof MCGs.From the global evolution of the neutral fraction andbrightness temperatures, as well as the power spectra, wesee that the model does indeed perform a reasonablejob at capturing the mock observation. Differences emerge atthe highest redshifts, when the radiation fields have a higher https://github.com/rjw57/MultiNesthttps://github.com/JohannesBuchner/PyMultiNest relative contribution from MCGs. Even with a higher X-ray emissivity and steeper stellar mass functions, the ACG-only model cannot fully capture the evolution of the ACG +MCG mock observation. ACGs are more biased galaxies, andare insensitive to LW feedback which can prolong the earlyevolution of IGM properties in feedback-dominated MCGmodels (e.g. Ahn et al. 2009; Holzbauer & Furlanetto 2012;Fialkov et al. 2013). Thus, compared to the mock signal, the model has: (i) a more rapid evolution of cosmic mile-stones; and (ii) a higher 21-cm PS during the epochs whena single field (i.e. temperature or Ly α coupling) sources thefluctuations, thus making cross terms negligible and allow-ing the 21-cm PS to be roughly estimated analogously to thehalo model with a bias term for the galaxies (e.g. Pritchard& Furlanetto 2007; McQuinn & DAloisio 2018). Indeed, wesee that the model has a more rapid evolution of theearly stages of reionization (see also Ahn et al. 2009). More-over, during the epoch of heating when the 21-cm signal issourced by temperature fluctuations (12 ∼ < z ∼ < prefers power spectra that are too high, and results in a toorapid evolution during the transition to the earlier, Ly α -dominated epoch ( z ∼ > model recovers thefiducial parameters of the mock observation quite well. Theinferred global evolution of the neutral fraction and bright-ness temperature, as well as the power spectra, are all con-sistent with the mock observation, without any notable bias.The X-ray luminosity of MCGs is well constrained, to within ∼ L atomX < / ˙ (cid:12) – L molX < / ˙ (cid:12) marginalized posterior, we see thatthis is due to a partial degeneracy allowing ACGs to domi-nate the epoch of heating for those models in which MCGsdo not emit significant soft X-rays.The ionizing escape fraction of MCGs is poorly con-strained, as they do not have a significant contribution toreionization. However, models with both high f mol ∗ , and f molesc are excluded as they would result in a Thomson scatteringoptical depth that is too high (see Paper-I and Visbal et al.2015).For completeness, we also present the marginalized UVLFs of ACGs, MCGs (only in ) and all galaxies in Fig.3. We see that the ACGs and total LFs are tightly con-strained at the bright end by currently available observations(Bouwens et al. 2015a, 2016; Oesch et al. 2018). Comparedto the mock observation, both and results areconsistent at M < −
8. At fainter magnitudes, only the model recovers the UV LFs, since MCGs dominate inthis regime.Finally, we come to the main question of this work:can we quantitatively claim that is a better fit tothe data, given that it has more free parameters comparedto ? We quantify this using the Bayesian evidence:ln [ P ( O | M )] = − . ± .
32 and − . ± .
25 for and , respectively. These result in a Bayes factor ofln B ≡ ln [ P ( O | M ) /P ( O | M )] ∼
24, suggesting theprobability of being preferred over by the data (i.e.the mock 21-cm PS) is > .
3% (Jeffreys 1939). We thereforeconclude that the (mock) data require the additional param-eters characterizing MCG (i.e. f mol ∗ , , f molesc and L molX < / ˙ (cid:12) ).This means that it might be possible to indirectly detect thefootprint of MCGs in upcoming 21-cm power spectra mea- c (cid:13) , 000–000 inihalos in 21-cm PS z=6z=6 MCGs ACGs MCGs+ACGs z=8z=8 z=10z=10 z=15z=15 Mock2pop1pop M / M p c d e x Figure 3.
UV luminosity functions of MCGs, ACGs and all galaxies from the model posteriors: ( in red, in blue). Lines andshaded regions represent the median and [16, 84] percentiles. Observational estimates used in the inference (Bouwens et al. 2015a, 2017;Oesch et al. 2018) are shown in grey at the bright end. surements. We caution that this conclusion is based on theassumption that minihalo-hosted galaxies truly play a sig-nificant role in the IGM evolution during the cosmic dawn(as would be the case if, for example, the EDGES detectionis genuinely cosmological).
In this work, we quantify the detectability of minihalosfor upcoming 21-cm interferometers. We compute a mock21-cm signal, motivated by the timing of the putativeEDGES detection, which would be driven by X-ray lumi-nous, molecularly-cooled galaxies (Paper-I). The result addi-tionally agrees with the observed high-redshift galaxy UV lu-minosity functions (Bouwens et al. 2015a, 2016; Oesch et al.2018), the upper limit on the neutral hydrogen fraction at z ∼ . Planck satellite (Planck Collaboration et al. 2016a).We calculate the 21-cm power spectra (PS) from this model,including telescope noise corresponding to a 1000-hour inte-gration with SKA1-low and moderate foreground avoidance.These mock observations are then fed to the 21 cmmc driver(Greig & Mesinger 2015), upgraded to allow for nested sam-pling (Feroz & Hobson 2008), and used to constrain twomodels: (i) , including both MCGs and their massiveatomic-cooling galaxy (ACG) counterparts; and (ii) ,considering only ACGs.We note that the model is able to partially com-pensate for the missing population of MCGs by preferring asteeper stellar mass function (smaller α ∗ ) and a more X-ray luminous population of HMXBs (higher L atomX < / ˙ (cid:12) ). How-ever, without a transient population of MCGs, the more bi-ased galaxies in the model result in a somewhat morerapid evolution of cosmic milestones, with a higher PS dur-ing the epoch of heating.We quantify the preference of the mock observation forthe more sophisticated galaxy model using the Bayesian ev-idence. We obtain ln [ P ( O | M )] = − . ± .
32 and a max-imum likelihood of ln [ P ( O | θ max , M )] = − .
92 for .These, compared to the result (i.e. − . ± .
25 and − . > .
3% probability of being pre-ferred over by the data (i.e. the mock 21-cm PS) accord-ing to the Jeffreys’ scale (Jeffreys 1939). Thus if minihalo-hosted galaxies indeed have a significant impact on high-redshift IGM properties (as would be the case if the timingof the EDGES signal is proven to be cosmological Bowmanet al. 2018), we should be able to indirectly infer their ex-istence and their properties from upcoming 21-cm observa-tions.More generally, our study showcases how upcoming 21-cm measurements can be used to discriminate against un-certain galaxy formation models, of varying complexity (seealso Binnie & Pritchard 2019). Although we used two sim-plified, nested models here, the same analysis can be appliedto even more sophisticated galaxy models (e.g. Moster et al.2013; Sun & Furlanetto 2016; Mutch et al. 2016; Ma et al.2018; Tacchella et al. 2018; Behroozi et al. 2019; Yung et al.2019). The need for additional complexity can be directlytested via the Occam’s razor factor of the Bayesian evidence,by adding additional model parameters until the evidence ismaximized. c (cid:13) , 000–000 Qin et al.
ACKNOWLEDGEMENTS
This work was supported by the European Research Coun-cil (ERC) under the European Unions Horizon 2020 researchand innovation programme (AIDA –
DATA AVAILABILITY
The data underlying this article will be shared on reasonablerequest to the corresponding author.
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