Simultaneously constraining the astrophysics of reionisation and the epoch of heating with 21CMMC
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 1 March 2018 (MN L A TEX style file v2.2)
Simultaneously constraining the astrophysics ofreionisation and the epoch of heating with 21CMMC
Bradley Greig (cid:63) & Andrei Mesinger Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
ABSTRACT
The cosmic 21 cm signal is set to revolutionise our understanding of the early Universe,allowing us to probe the 3D temperature and ionisation structure of the intergalacticmedium (IGM). It will open a window onto the unseen first galaxies, showing us howtheir UV and X-ray photons drove the cosmic milestones of the epoch of reionisation(EoR) and epoch of heating (EoH). To facilitate parameter inference from the 21 cmsignal, we previously developed : a Monte Carlo Markov Chain sampler of 3DEoR simulations. Here we extend to include simultaneous modelling of theEoH, resulting in a complete Bayesian inference framework for the astrophysics domi-nating the observable epochs of the cosmic 21 cm signal. We demonstrate that secondgeneration interferometers, the Hydrogen Epoch of Reionisation Array (HERA) andSquare Kilometre Array (SKA) will be able to constrain ionising and X-ray sourceproperties of the first galaxies with a fractional precision of order ∼ σ ). The ionisation history of the Universe can be constrained to within a few per-cent. Using our extended framework, we quantify the bias in EoR parameter recoveryincurred by the common simplification of a saturated spin temperature in the IGM.Depending on the extent of overlap between the EoR and EoH, the recovered astro-physical parameters can be biased by ∼ − σ . Key words: cosmology: theory – dark ages, reionisation, first stars – diffuse radiation– early Universe – galaxies: high-redshift – intergalactic medium
The 21 cm spin-flip transition of neutral hydrogen encodesa treasure trove of cosmological and astrophysical informa-tion (see e.g. Gnedin & Ostriker 1997; Madau et al. 1997;Shaver et al. 1999; Tozzi et al. 2000; Gnedin & Shaver 2004;Furlanetto et al. 2006; Morales & Wyithe 2010; Pritchard& Loeb 2012). The signal is expressed as the offset of the21 cm brightness temperature, δT b ( ν ), relative to the cos-mic microwave background (CMB) temperature, T CMB (e.g.Furlanetto et al. 2006): δT b ( ν ) ≈ x H I (1 + δ nl ) (cid:18) H d v r / d r + H (cid:19) (cid:18) − T CMB T S (cid:19) × (cid:18) z
10 0 . m h (cid:19) / (cid:18) Ω b h . (cid:19) mK , (1)where x H I is the neutral fraction, T S is the gas spin tem-perature, δ nl ≡ ρ/ ¯ ρ − H ( z ) is theHubble parameter, d v r / d r is the gradient of the line-of-sightcomponent of the velocity and all quantities are evaluatedat redshift z = ν /ν −
1, where ν is the 21 cm frequency. (cid:63) E-mail: [email protected]
Because equation (1) depends on the ionisation andtemperature as a function of space and time, the 21 cmsignal can provide insight into the formation, growth andevolution of structure in the Universe, the nature of the firststars and galaxies and their impact on the physics of the in-tergalactic medium (IGM; e.g. Barkana & Loeb 2007; Loeb& Furlanetto 2013; Zaroubi 2013). The most widely-studiedof these properties is the ionisation state: in the first billionyears, the Universe transitioned from being nearly fully neu-tral to being nearly fully ionised. This epoch of reionisation(EoR) was driven by the percolation of H II regions gener-ated by the ionising photons escaping from the first galaxies.Sourcing the x H I term in equation (1), the EoR should beevidenced by a rise and fall of large-scale fluctuations in the21 cm brightness temperature (e.g. Lidz et al. 2008).The second astrophysical term in equation (1) is theIGM spin temperature, T S . The spin temperature is thoughtto be regulated by the first sources of X-rays, which canheat the IGM from its post thermal decoupling adiabat totemperatures far above the CMB temperature. While theIGM is still adiabatically cooling, the (1 − T CMB /T S ) term inequation (1) can be of order − c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b B. Greig & A. Mesinger this epoch of heating (EoH) can provide the strongest 21cm signal, more than an order of magnitude larger than thatduring the EoR (e.g. Mesinger & Furlanetto 2007; Pritchard& Furlanetto 2007; Baek et al. 2010; Santos et al. 2010;McQuinn & O’Leary 2012; Mesinger et al. 2013). The 21cm signal can therefore be a powerful probe of high-energyprocesses in the first galaxies. The most likely of these X-raysources are high mass X-ray binaries (HMXBs; Power et al.2009; Mirabel et al. 2011; Fragos et al. 2013a; Power et al.2013) and/or the hot interstellar medium (ISM) within thefirst galaxies (e.g. Oh 2001; Pacucci et al. 2014). However,other alternative scenarios have been put forth includingmetal-free (Pop-III) stars (Xu et al. 2014), mini-QSOs (e.gMadau et al. 2004; Yue et al. 2013; Ghara et al. 2016), darkmatter annihilation (e.g. Cirelli et al. 2009; Evoli et al. 2014;Lopez-Honorez et al. 2016) or cosmic rays (Leite et al. 2017).Numerous 21 cm experiments are currently underway,attempting to detect the cosmic 21 cm signal. These fallinto two broad categories. The first are large-scale interfer-ometers, seeking to detect spatial 21 cm fluctuations. Theseinclude the Low Frequency Array (LOFAR; van Haarlemet al. 2013; Yatawatta et al. 2013), the Murchison WideField Array (MWA; Tingay et al. 2013), the Precision Ar-ray for Probing the Epoch of Reionisation (PAPER; Parsonset al. 2010), the Square Kilometre Array (SKA; Mellemaet al. 2013) and the Hydrogen Epoch of Reionisation Ar-ray (HERA; DeBoer et al. 2017). The second class are sin-gle dipole or small compact array global-sky experimentsseeking the volume averaged global 21 cm signal. Theseinclude the Experiment to Detect the Global EoR Signa-ture (EDGES; Bowman & Rogers 2010), the Sonda Cos-mol´ogica de las Islas para la Detecci´on de Hidr´ogeno Neutro(SCI-HI; Voytek et al. 2014), the Shaped Antenna Measure-ment of the Background Radio Spectrum (SARAS; Patraet al. 2015), Broadband Instrument for Global HydrOgenReioNisation Signal (BIGHORNS; Sokolowski et al. 2015),the Large Aperture Experiment to detect the Dark Ages(LEDA; Greenhill & Bernardi 2012; Bernardi et al. 2016),and the Dark Ages Radio Explorer (DARE; Burns et al.2012).As a first step in preparation for the wealth of data ex-pected from these 21 cm experiments, we developed a pub-licly available Monte Carlo Markov Chain (MCMC) EoRanalysis tool (Greig & Mesinger 2015). This isthe first EoR analysis tool to sample 3D reionisation sim-ulations (using ; Mesinger & Furlanetto 2007;Mesinger et al. 2011) within a fully Bayesian framework forastrophysical parameter space exploration and 21 cm exper-iment forecasting. In this introductory work, we adopted thecommon simplifying assumption of a saturated spin temper-ature: T S (cid:29) T CMB . However, the applicability of this satu-rated limit is dependent on the poorly-known strength andspectral shape of the X-ray background in the early Uni-verse. If the spin temperature is not fully saturated duringthe EoR, this could result in sizeable biases in the inferredEoR parameters (e.g. Watkinson & Pritchard 2015).Beyond its impact on EoR parameter recovery, the IGMspin temperature also encodes a wealth of information on thehigh-energy processes in the early Universe, as mentioned https://github.com/BradGreig/21CMMC above. Second generation interferometers, HERA and SKA,have the bandwidth and sensitivity to easily probe temper-ature fluctuations during the EoH.In this work, we extend to include a full treat-ment of the EoH, retaining our ability to perform on-the-fly sampling of 3D reionisation simulations. This updated is capable of jointly exploring the astrophysics ofboth the EoR and EoH, allowing us to maximise the scien-tific return of upcoming second-generation telescopes.We note that recent studies suggest that machine learn-ing can be a viable alternative to on-the-fly MCMC samplingof . Shimabukuro & Semelin (2017) used an ar-tificial neural network to predict astrophysical parameters,with an accuracy of ∼ tens of percent. This approach isfast, though producing Bayesian confidence limits becomesless straightforward. Alternately, Kern et al. (2017) bypassedthe on-the-fly sampling of 3D simulations by using an em-ulator trained on the 21 cm power spectrum. An emulatorcan be used in an MCMC framework, and is orders of mag-nitude faster at parameter forecasting compared to a directsampling of 3D simulations (such as ). This comesat the cost of ∼ < ten per cent in power spectrum accuracyover most of the parameter space, when the learning is per-formed on ∼ training samples (higher pre-cision can be obtained by increasing the size of the trainingset). Future work will test emulator accuracy on high-ordersummary statistics.The remainder of this paper is organised as follows. InSection 2 we summarise and the associated simulations used to generate 3D realisations of thecosmic 21 cm signal, outlining the updated astrophysicalparameterisation to model the EoH. In Section 3 we intro-duce our mock observations to be used in our 21 cm exper-iment forecasting, and present the forecasts and associateddiscussions in Section 4. We then explore the impact of as-suming the saturated IGM spin temperature limit in Sec-tion 5, before summarising the improvements to and finishing with our closing remarks in Section 6. Unlessstated otherwise, we quote all quantities in comoving unitsand adopt the cosmological parameters: (Ω Λ , Ω M , Ω b , n , σ , H ) = (0.69, 0.31, 0.048, 0.97, 0.81, 68 km s − Mpc − ), con-sistent with recent results from the Planck mission (PlanckCollaboration XIII 2016). In this section, we provide a short summary of the mainaspects of , before delving into the modelling of thecosmic 21 cm signal during the EoR and EoH in Section 2.1and developing an intuition about the full parameter set inSection 2.2. is a massively parallel MCMC sampler for ex-ploring the astrophysical parameter space of reionisation. Itwas built using a modified version of the easy to use pythonmodule cosmohammer (Akeret et al. 2013), which employsthe emcee python module developed by Foreman-Mackeyet al. (2013) based on the affine invariant ensemble sam-pler of Goodman & Weare (2010). At each proposed step inthe computation chain, performs a new, indepen-dent 3D reionisation of the 21 cm signal, using an optimisedversion of the publicly available simulation code c (cid:13)000
Because equation (1) depends on the ionisation andtemperature as a function of space and time, the 21 cmsignal can provide insight into the formation, growth andevolution of structure in the Universe, the nature of the firststars and galaxies and their impact on the physics of the in-tergalactic medium (IGM; e.g. Barkana & Loeb 2007; Loeb& Furlanetto 2013; Zaroubi 2013). The most widely-studiedof these properties is the ionisation state: in the first billionyears, the Universe transitioned from being nearly fully neu-tral to being nearly fully ionised. This epoch of reionisation(EoR) was driven by the percolation of H II regions gener-ated by the ionising photons escaping from the first galaxies.Sourcing the x H I term in equation (1), the EoR should beevidenced by a rise and fall of large-scale fluctuations in the21 cm brightness temperature (e.g. Lidz et al. 2008).The second astrophysical term in equation (1) is theIGM spin temperature, T S . The spin temperature is thoughtto be regulated by the first sources of X-rays, which canheat the IGM from its post thermal decoupling adiabat totemperatures far above the CMB temperature. While theIGM is still adiabatically cooling, the (1 − T CMB /T S ) term inequation (1) can be of order − c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b B. Greig & A. Mesinger this epoch of heating (EoH) can provide the strongest 21cm signal, more than an order of magnitude larger than thatduring the EoR (e.g. Mesinger & Furlanetto 2007; Pritchard& Furlanetto 2007; Baek et al. 2010; Santos et al. 2010;McQuinn & O’Leary 2012; Mesinger et al. 2013). The 21cm signal can therefore be a powerful probe of high-energyprocesses in the first galaxies. The most likely of these X-raysources are high mass X-ray binaries (HMXBs; Power et al.2009; Mirabel et al. 2011; Fragos et al. 2013a; Power et al.2013) and/or the hot interstellar medium (ISM) within thefirst galaxies (e.g. Oh 2001; Pacucci et al. 2014). However,other alternative scenarios have been put forth includingmetal-free (Pop-III) stars (Xu et al. 2014), mini-QSOs (e.gMadau et al. 2004; Yue et al. 2013; Ghara et al. 2016), darkmatter annihilation (e.g. Cirelli et al. 2009; Evoli et al. 2014;Lopez-Honorez et al. 2016) or cosmic rays (Leite et al. 2017).Numerous 21 cm experiments are currently underway,attempting to detect the cosmic 21 cm signal. These fallinto two broad categories. The first are large-scale interfer-ometers, seeking to detect spatial 21 cm fluctuations. Theseinclude the Low Frequency Array (LOFAR; van Haarlemet al. 2013; Yatawatta et al. 2013), the Murchison WideField Array (MWA; Tingay et al. 2013), the Precision Ar-ray for Probing the Epoch of Reionisation (PAPER; Parsonset al. 2010), the Square Kilometre Array (SKA; Mellemaet al. 2013) and the Hydrogen Epoch of Reionisation Ar-ray (HERA; DeBoer et al. 2017). The second class are sin-gle dipole or small compact array global-sky experimentsseeking the volume averaged global 21 cm signal. Theseinclude the Experiment to Detect the Global EoR Signa-ture (EDGES; Bowman & Rogers 2010), the Sonda Cos-mol´ogica de las Islas para la Detecci´on de Hidr´ogeno Neutro(SCI-HI; Voytek et al. 2014), the Shaped Antenna Measure-ment of the Background Radio Spectrum (SARAS; Patraet al. 2015), Broadband Instrument for Global HydrOgenReioNisation Signal (BIGHORNS; Sokolowski et al. 2015),the Large Aperture Experiment to detect the Dark Ages(LEDA; Greenhill & Bernardi 2012; Bernardi et al. 2016),and the Dark Ages Radio Explorer (DARE; Burns et al.2012).As a first step in preparation for the wealth of data ex-pected from these 21 cm experiments, we developed a pub-licly available Monte Carlo Markov Chain (MCMC) EoRanalysis tool (Greig & Mesinger 2015). This isthe first EoR analysis tool to sample 3D reionisation sim-ulations (using ; Mesinger & Furlanetto 2007;Mesinger et al. 2011) within a fully Bayesian framework forastrophysical parameter space exploration and 21 cm exper-iment forecasting. In this introductory work, we adopted thecommon simplifying assumption of a saturated spin temper-ature: T S (cid:29) T CMB . However, the applicability of this satu-rated limit is dependent on the poorly-known strength andspectral shape of the X-ray background in the early Uni-verse. If the spin temperature is not fully saturated duringthe EoR, this could result in sizeable biases in the inferredEoR parameters (e.g. Watkinson & Pritchard 2015).Beyond its impact on EoR parameter recovery, the IGMspin temperature also encodes a wealth of information on thehigh-energy processes in the early Universe, as mentioned https://github.com/BradGreig/21CMMC above. Second generation interferometers, HERA and SKA,have the bandwidth and sensitivity to easily probe temper-ature fluctuations during the EoH.In this work, we extend to include a full treat-ment of the EoH, retaining our ability to perform on-the-fly sampling of 3D reionisation simulations. This updated is capable of jointly exploring the astrophysics ofboth the EoR and EoH, allowing us to maximise the scien-tific return of upcoming second-generation telescopes.We note that recent studies suggest that machine learn-ing can be a viable alternative to on-the-fly MCMC samplingof . Shimabukuro & Semelin (2017) used an ar-tificial neural network to predict astrophysical parameters,with an accuracy of ∼ tens of percent. This approach isfast, though producing Bayesian confidence limits becomesless straightforward. Alternately, Kern et al. (2017) bypassedthe on-the-fly sampling of 3D simulations by using an em-ulator trained on the 21 cm power spectrum. An emulatorcan be used in an MCMC framework, and is orders of mag-nitude faster at parameter forecasting compared to a directsampling of 3D simulations (such as ). This comesat the cost of ∼ < ten per cent in power spectrum accuracyover most of the parameter space, when the learning is per-formed on ∼ training samples (higher pre-cision can be obtained by increasing the size of the trainingset). Future work will test emulator accuracy on high-ordersummary statistics.The remainder of this paper is organised as follows. InSection 2 we summarise and the associated simulations used to generate 3D realisations of thecosmic 21 cm signal, outlining the updated astrophysicalparameterisation to model the EoH. In Section 3 we intro-duce our mock observations to be used in our 21 cm exper-iment forecasting, and present the forecasts and associateddiscussions in Section 4. We then explore the impact of as-suming the saturated IGM spin temperature limit in Sec-tion 5, before summarising the improvements to and finishing with our closing remarks in Section 6. Unlessstated otherwise, we quote all quantities in comoving unitsand adopt the cosmological parameters: (Ω Λ , Ω M , Ω b , n , σ , H ) = (0.69, 0.31, 0.048, 0.97, 0.81, 68 km s − Mpc − ), con-sistent with recent results from the Planck mission (PlanckCollaboration XIII 2016). In this section, we provide a short summary of the mainaspects of , before delving into the modelling of thecosmic 21 cm signal during the EoR and EoH in Section 2.1and developing an intuition about the full parameter set inSection 2.2. is a massively parallel MCMC sampler for ex-ploring the astrophysical parameter space of reionisation. Itwas built using a modified version of the easy to use pythonmodule cosmohammer (Akeret et al. 2013), which employsthe emcee python module developed by Foreman-Mackeyet al. (2013) based on the affine invariant ensemble sam-pler of Goodman & Weare (2010). At each proposed step inthe computation chain, performs a new, indepen-dent 3D reionisation of the 21 cm signal, using an optimisedversion of the publicly available simulation code c (cid:13)000 , 000–000 (Mesinger & Furlanetto 2007; Mesinger et al. 2011) for com-putational efficiency. Using a likelihood statistic (fiduciallythe PS), it compares the model against a mock observationgenerated from a larger simulation with a different set ofinitial conditions. For further details, we refer the reader to(Greig & Mesinger 2015). In this work, we use an optimised version of the publicly-available version of v1.1 . employsapproximate but efficient modelling of the underlying astro-physics of the reionisation and heating epochs. The resulting21 cm PS during the EoR have been found to match thoseof computationally expensive radiative transfer simulationsto with tens of per cent on the scales relevant to 21 cm inter-ferometry, ∼ > produces a full, 3D realisation of the 21 cmbrightness temperature field, δT b (c.f equation 1) which isdependent on the ionisation, density, velocity and IGM spintemperature fields. The evolved IGM density field at anyredshift is obtained from an initial high resolution lineardensity field which is perturbed using the Zel’dovich ap-proximation (Zel’dovich 1970) before being smoothed ontoa lower resolution grid.The ionisation field is then estimated from this evolvedIGM density field using the excursion-set approach (Furlan-etto et al. 2004). The time-integrated number of ionisingphotons is compared to the number of neutral atoms withinregions of decreasing radius, R . These regions are computedfrom a maximum photon horizon, R mfp down to the individ-ual pixel resolution of a single voxel, R cell . A voxel at coor-dinates ( x , z ) within the simulation volume is then taggedas fully ionised if, ζf coll ( x , z, R, ¯ M min ) ≥ , (2)where f coll ( x , z, R, ¯ M min ) is the fraction of collapsed matterresiding within haloes more massive than ¯ M . min (Press &Schechter 1974; Bond et al. 1991; Lacey & Cole 1993; Sheth& Tormen 1999) and ζ is an ionising efficiency describing theconversion of mass into ionising photons (see Section 2.2.1).Partial ionisations are included for voxels not fully ionisedby setting their ionised fractions to ζf coll ( x , z, R cell , ¯ M min ).Since 21 cm observations use the CMB as a backgroundsource, the IGM spin temperature has to be coupled to thekinetic gas temperature for the signal to be detected. Thiscoupling is achieved through either collisional coupling orthe Ly α background from the first generation of stars (so-called Wouthuysen-Field (WF) coupling (Wouthuysen 1952; https://github.com/andreimesinger/21cmFAST This includes ionisations from both UV and X-ray sources.While X-rays can produce some level of pre-reionisation (e.g.Ricotti & Ostriker 2004; Dijkstra et al. 2012; McQuinn 2012;Mesinger et al. 2013) their predominant contribution is in pre-heating the IGM before reionisation (e.g. McQuinn & O’Leary2012).
Field 1958)). To compute the spin temperature, solves for the evolution of the ionisation, temperature andimpinging Ly α background in each voxel . These depend onthe angle-averaged specific intensity, J ( x , E, z ), (in erg s − keV − cm − sr − ), computed by integrating the comovingX-ray specific emissivity, (cid:15) X ( x , E e , z (cid:48) ) back along the light-cone: J ( x , E, z ) = (1 + z ) π (cid:90) ∞ z dz (cid:48) cdtdz (cid:48) (cid:15) X e − τ . (3)Here, e − τ corresponds to the probability that a photon emit-ted at an earlier time, z (cid:48) , survives until z owing to IGMattenuation (see Eq. 16 of Mesinger et al. 2011) and the co-moving specific emissivity is evaluated in the emitted frame, E e = E (1 + z (cid:48) ) / (1 + z ), with (cid:15) X ( x , E e , z (cid:48) ) = L X SFR (cid:20) ρ crit , Ω b f ∗ (1 + δ nl ) df coll ( z (cid:48) ) dt (cid:21) , (4)where the quantity in square brackets is the star-formationrate (SFR) density along the light-cone, with ρ crit , beingthe current critical density and f ∗ the fraction of galacticbaryons converted into stars.The quantity L X / SFR is the specific X-ray luminosityper unit star formation escaping the galaxies (in units of ergs − keV − M − (cid:12) yr). The specific luminosity is taken to bea power law in photon energy, L X ∝ E − α X , with photonsbelow some threshold energy, E , being absorbed inside thehost galaxy. Instead of the number of X-ray photons per stellarbaryon used in the default version of , here wenormalise the X-ray efficiency in terms of an integrated soft-band ( < − M − (cid:12) yr): L X < / SFR = (cid:90) E dE e L X / SFR . (5)This parametrisation is both physically-motivated (harderphotons have mean free paths longer than the Hubble lengthand so do not contribute to the EoH; e.g. McQuinn 2012;Das et al. 2017), and easier to directly compare with X-rayobservations of local star-forming galaxies. astrophysical parameter set In the previous section, we outlined the semi-numerical ap-proach, using , to obtain the 3D realisations of Note that within this work we do not vary the soft UV spectra ofthe first sources driving this epoch. This WF coupling epoch willbe extremely challenging to detect in comparison to the EoR andthe EoH. Nevertheless, we will return to this in future work. Formore specifics regarding the computation of the Ly α background,we refer the reader to Section 3.2 of (Mesinger et al. 2011) We note that in the computation of the heating and ionisa-tion rates in the default version of , these integralswere performed out to infinity, which could result in divergent be-haviour for α X ≤
0. Here we adopt an upper limit of 10 keV forcomputing the rate integrals. This choice is arbitrary and purelyfor numerical convenience, as the EoH only depends on the SEDbelow ∼ ∼
10 – 100 keV (e.g. Lehmer et al. 2013,2015).c (cid:13) , 000–000
B. Greig & A. Mesinger the 21 cm brightness temperature field. This parameteri-sation yields six free parameters to be sampled within ourBayesian framework. In this section, we provide more de-tailed descriptions for each of these parameters, providingphysical intuition for their impact on the IGM through X-ray heating or ionisation and defining their allowed ranges.To aid in this discussion, we provide Figure 1.For each parameter, denoted (i) - (vi), we providethe redshift evolution of the corresponding average neu-tral fraction ( top panel ), average 21 cm brightness tem-perature contrast ( middle panel ) and the amplitude ofthe 21 cm PS at k = 0 .
15 Mpc − ( bottom panel ) ( ¯ δT ∆ ( k, z ) ≡ k / (2 π V ) δ ¯ T ( z ) (cid:104)| δ ( k , z ) | (cid:105) k where δ ( x , z ) ≡ δT b ( x , z ) / ¯ δT b ( z ) − faint galaxies model (see Section 3 for furtherdetails). Before discussing the impact of each parameter, itis instructive to note the general features exhibited by thefiducial model. The EoR history in the top panel has a mono-tonic evolution with a midpoint around z ∼
8. The global21 cm signal in the middle panel shows a deep absorptiontrough, corresponding to when the X-ray heating rate sur-passes the adiabatic cooling rate (start of the EoH). This isfollowed by a small emission peak, corresponding to the on-set of the EoR (e.g. Furlanetto 2006). The height of the peakis determined by the relative overlap of the EoR and EoH;if the overlap is strong, reionisation can proceed in a coldIGM, with the global signal never switching to emission.The large-scale power in the bottom panel shows thecharacteristic three-peaked structure, driven by fluctuationsin the Ly α background (WF coupling); IGM gas tempera-ture (EoH), and the ionisation fraction (EoR), from rightto left, respectively (e.g. Pritchard & Furlanetto 2007; Baeket al. 2010). The redshift position of each peak thereforetraces the timing of each epoch, while the amplitude of the21 cm power traces the level of fluctuations determined bythe typical source bias and/or X-ray SED (e.g. Mesingeret al. 2013). Troughs on the other hand correspond to thetransitions between epochs, when the corresponding crossterms of the 21 cm PS cause the large-scale power to drop(e.g. Lidz et al. 2007; Pritchard & Furlanetto 2007; Mesingeret al. 2016). ζ The UV ionising efficiency of high- z galaxies (Equation 2)can be expressed as ζ = 30 (cid:18) f esc . (cid:19) (cid:18) f ∗ . (cid:19) (cid:18) N γ/b (cid:19) (cid:18) .
51 + n rec (cid:19) (6)where, f esc is the fraction of ionising photons escaping intothe IGM, f ∗ is the fraction of galactic gas in stars, N γ/b is Our choice of k -modes was motivated by the fact that the large-scale power in the 21 cm PS is the main discriminant between as-trophysical models (e.g. McQuinn et al. 2007; Greig & Mesinger2015), Moreover, these k -modes are detectable by upcoming ex-periments (i.e. they are sufficiently small to avoid being contam-inated by foregrounds yet large enough to obtain high signal tonoise). the number of ionising photons produced per baryon in starsand n rec is the typical number of times a hydrogen atom re-combines. While only the product of equation (6) is requiredfor generating the ionisation field, we provide some plausi-ble values for each of the terms. We adopt N γ ≈ f (cid:63) and f esc are ex-tremely uncertain, with observations and simulations plac-ing plausible values within the vicinity of ∼
10 per cent for f (cid:63) (e.g. Behroozi & Silk 2015; Sun & Furlanetto 2016) and f esc (e.g Paardekooper et al. 2015; Xu et al. 2016; Kimmet al. 2017). Finally, we adopt n rec ∼ .
5, similar to thosefound in the ’photon-starved’ reionisation models of (Sobac-chi & Mesinger 2014), which are consistent with emissivityestimates from the Ly α forest (e.g. Bolton & Haehnelt 2007;McQuinn et al. 2011). It is important to note that the f ∗ appearing here, is equivalent to the f ∗ in equation (4). Thatis, throughout all cosmic epochs, we adopt the same, con-stant value of 5 per cent for the fraction of galactic gas instars. In future work, we will relax this assumption.In this work, we adopt a flat prior over the fiducial rangeof ζ ∈ [10 , ζ = 100 chosen in Greig & Mesinger (2015).The extended range provides the flexibility to explore physi-cally plausible models where reionisation is driven by bright,rare galaxies (see Section 3). In panel (i) of Figure 1 we high-light the impact of varying ζ . As expected, ζ has a strongimpact on the EoR and an almost negligible impact on theEoH. As ζ is increased, the EoR peak shifts to earlier red-shifts and the width of the EoR peak reduces (i.e. shorterduration). For extremely large values of ζ , the EoR and EoHpeaks begin to merge, resulting in a larger amplitude EoRpeak, sourced by the contrast between the cold IGM patchespresent in the early EoH stages and the H II regions. R mfp The physical size of H II regions is regulated by the distanceionising photons can propagate into the IGM. This dependson the abundance of photon sinks (absorption systems suchas Lyman limit systems) and the corresponding recombina-tions of these systems. When the H II regions start approach-ing the typical separation of the photon sinks, an increasingfraction of ionising photons are used to balance recombina-tions, and the EoR can slow down (e.g. Furlanetto & Oh2005; Furlanetto & Mesinger 2009; Alvarez & Abel 2012).The details of this process can be complicated (e.g.Sobacchi & Mesinger 2014); however a common simplifi-cation in semi-numerical approaches is to adopt a maxi-mum horizon for the ionising photons within the ionisedIGM, R mfp7 , which is implemented as the maximum fil- For historical context, we adopt ‘ R mfp ’ to denote this effectivehorizon set by sub-grid recombinations. The sub-grid models ofSobacchi & Mesinger (2014) do not translate to a constant, uni-form value of R mfp , and further the mean free path is typicallylarger than R mfp (since the former is an instantaneous quantitywhile the latter depends on the cumulative contributions of sub-grid recombinations). However, these authors find that the ap-proximation of a constant R mfp can reproduce the 21 cm PS oftheir fiducial model to ∼
10 per cent. Future versions of c (cid:13)000
10 per cent. Future versions of c (cid:13)000 , 000–000 . . . . . . ¯ x H I Fiducial ( ↵ X = 1 . ↵ X = . ↵ X = 2 . T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc . . . . . . ¯ x H I Fiducial ( E = 0 . E = 0 . E = 0 . E = 1 . T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc . . . . . . ¯ x H I Fiducial (log ( L X < / SFR) = 40 . ( L X < / SFR) = 38 . ( L X < / SFR) = 42 . T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc . . . . . . ¯ x H I Fiducial ( R mfp = 15) R mfp = 5 R mfp = 25 T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc . . . . . . ¯ x H I Fiducial ( T minvir = 5 ⇥ ) T minvir = 10 T minvir = 10 T minvir = 5 ⇥ T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc . . . . . . ¯ x H I Fiducial ( ⇣ = 30) ⇣ = 10 ⇣ = 100 ⇣ = 250 T b [ m K ] Redshift , z T b [ m K ] k = 0 .
15 Mpc i) Vary ⇣ vi) Vary ↵ X ii) Vary R mfp [Mpc] iii) Vary T minvir [K] iv) Vary L X < / SFR [erg s M yr] v) Vary E [keV] Figure 1.
The impact of each of our six astrophysical model parameters on the evolution of the IGM neutral fraction (top panel), globalaveraged 21 cm brightness temperature contrast (middle panel) and the evolution of the PS amplitude at k = 0 .
15 Mpc − (bottompanel). We vary each model parameter across the full allowed prior range, holding the remaining five parameters fixed to the fiducial faint galaxies model values (see Section 3).c (cid:13) , 000–000 B. Greig & A. Mesinger tering scale in the excursion-set EoR modelling (see Sec-tion 2.1). Motivated by recent sub-grid recombination mod-els (Sobacchi & Mesinger 2014), we adopt a flat prior over R mfp ∈ [5 ,
25] cMpc.In panel (ii) of Figure 1, we show the impact of R mfp over cosmic history. As mentioned above, R mfp only becomesimportant in the advanced stages of reionization, when thetypical H II region scale approaches R mfp . The result is adelay of the late stages of the EoR for small values of R mfp . Limiting the photon horizon with a decreasing R mfp ( (cid:46)
15 Mpc) is also evidenced by a drop in the large-scale21 cm PS (e.g. McQuinn et al. 2007; Alvarez & Abel 2012;Mesinger et al. 2012; Greig & Mesinger 2015). Values of R mfp >
15 Mpc have little impact on the 21 cm PS (forthis combination of the other astrophysical parameters), asthe clustering of the ionising sources becomes the dominantsource of power (note the same PS amplitude for R mfp = 15and 25 Mpc). T minvir We define the minimum threshold for a halo hosting a star-forming galaxy to be its virial temperature, T minvir , which isrelated to the halo mass via, (e.g. Barkana & Loeb 2001) M minvir = 10 h − (cid:16) µ . (cid:17) − / (cid:18) Ω m Ω z m ∆ c π (cid:19) − / × (cid:18) T minvir . × K (cid:19) / (cid:18) z (cid:19) − / M (cid:12) , (7)where µ is the mean molecular weight, Ω z m = Ω m (1 + z ) / [Ω m (1 + z ) + Ω Λ ], and ∆ c = 18 π + 82 d − d where d = Ω z m −
1. The choice of T minvir acts as a step-functioncut-off to the UV luminosity function. Below T minvir , it is as-sumed that internal feedback mechanisms such as supernovaor photo-heating suppress the formation of stars. Above, ef-ficient star formation overcomes internal feedback, enablingthese haloes to produce ionising photons capable of con-tributing to the EoR and EoH. We shall consider a flat prioracross T minvir ∈ [10 , ] K within this work.Our lower limit, T minvir ≈ K is motivated by theminimum temperature for efficient atomic line cooling. Inprinciple, T minvir can be as low as ≈ K in the presence ofradiative cooling (Haiman et al. 1996; Tegmark et al. 1997;Abel et al. 2002; Bromm et al. 2002), however, star forma-tion within these haloes is likely inefficient (a few stars perhalo; e.g. Kimm et al. 2017) and can quickly ( z >
20) besuppressed by Lyman-Werner or other feedback processeswell before the EoR (Haiman et al. 2000; Ricotti et al. 2001;Haiman & Bryan 2006; Mesinger et al. 2006; Holzbauer &Furlanetto 2012; Fialkov et al. 2013). Our upper limit of T minvir ≈ K, is roughly consistent with the host halomasses of observed Lyman break galaxies at z ∼ framework, T minvir is will include the Sobacchi & Mesinger (2014) sub-grid recombina-tion model, eliminating R mfp as a free parameter. important both in the EoR and EoH, determining the ion-isation field ( f coll , equation 2) and the specific emissivityrequisite for the X-ray heating background ( df coll dt , equa-tion 4) respectively. This implies that the efficient star-forming galaxies responsible for reionisation are the samegalaxies which host the sources responsible for X-ray heat-ing (i.e. the physics of star formation drives both the X-rayheating and ionisation fields). As a result, T minvir affects thetiming of both the EoR and EoH.This is evident in panel (iii) of Figure 1. The EoR andEoH milestones are pushed to lower redshifts for an increas-ing T minvir . For a suitably large choice of T minvir (e.g. 5 × K),the timing of the EoH and WF-coupling peaks can be suffi-ciently delayed to result in their overlap. In addition to thetiming, T minvir , also impacts the amplitude of the fluctuations.Rarer (more biased) galaxies, corresponding to larger valuesof T minvir , are evidenced by more 21 cm power at a given epoch(e.g. McQuinn et al. 2007). L X < / SFRThe efficiency of X-ray heating is driven by the total inte-grated soft-band ( < ( L X < / SFR) ∈ [38 , to be one to two orders of magnitudebroader than the distribution seen in local populations ofstar forming galaxies (Mineo et al. 2012a), and their stacked Chandra observations (Lehmer et al. 2016). It also encom-passes values at high-redshifts predicted by population syn-thesis models (Fragos et al. 2013b).In Panel (iv) of Figure 1 we highlight the impact of L X < / SFR. For very high values of L X < / SFR, theEoH commences prior to the completion of WF-coupling. Asa result, no strong absorption feature in ¯ δT b is observed, andthe Ly α -EoH peaks in the 21 cm PS merge. In addition toheating, such high X-ray luminosities can also substantiallyionise the EoR (at the ∼ II regions and cold neutral IGM can yield an extremely large21 cm PS amplitude ( ∼ mK ). It is also roughly consistent with the limits proposed by Fialkovet al. (2017) using observations of the unresolved cosmic X-raybackground (e.g. Lehmer et al. 2012) and the upper limits on themeasured 21 cm PS from PAPER-64 (e.g. Ali et al. 2015; Poberet al. 2015). Though the specific model shown here exceeds the PAPER-64upper limits, for other choices of the EoR and EoH parameters,this same choice of L X < / SFR = 10 erg s − M − (cid:12) yr canresult in a 21 cm PS below these upper limits.c (cid:13)000
20) besuppressed by Lyman-Werner or other feedback processeswell before the EoR (Haiman et al. 2000; Ricotti et al. 2001;Haiman & Bryan 2006; Mesinger et al. 2006; Holzbauer &Furlanetto 2012; Fialkov et al. 2013). Our upper limit of T minvir ≈ K, is roughly consistent with the host halomasses of observed Lyman break galaxies at z ∼ framework, T minvir is will include the Sobacchi & Mesinger (2014) sub-grid recombina-tion model, eliminating R mfp as a free parameter. important both in the EoR and EoH, determining the ion-isation field ( f coll , equation 2) and the specific emissivityrequisite for the X-ray heating background ( df coll dt , equa-tion 4) respectively. This implies that the efficient star-forming galaxies responsible for reionisation are the samegalaxies which host the sources responsible for X-ray heat-ing (i.e. the physics of star formation drives both the X-rayheating and ionisation fields). As a result, T minvir affects thetiming of both the EoR and EoH.This is evident in panel (iii) of Figure 1. The EoR andEoH milestones are pushed to lower redshifts for an increas-ing T minvir . For a suitably large choice of T minvir (e.g. 5 × K),the timing of the EoH and WF-coupling peaks can be suffi-ciently delayed to result in their overlap. In addition to thetiming, T minvir , also impacts the amplitude of the fluctuations.Rarer (more biased) galaxies, corresponding to larger valuesof T minvir , are evidenced by more 21 cm power at a given epoch(e.g. McQuinn et al. 2007). L X < / SFRThe efficiency of X-ray heating is driven by the total inte-grated soft-band ( < ( L X < / SFR) ∈ [38 , to be one to two orders of magnitudebroader than the distribution seen in local populations ofstar forming galaxies (Mineo et al. 2012a), and their stacked Chandra observations (Lehmer et al. 2016). It also encom-passes values at high-redshifts predicted by population syn-thesis models (Fragos et al. 2013b).In Panel (iv) of Figure 1 we highlight the impact of L X < / SFR. For very high values of L X < / SFR, theEoH commences prior to the completion of WF-coupling. Asa result, no strong absorption feature in ¯ δT b is observed, andthe Ly α -EoH peaks in the 21 cm PS merge. In addition toheating, such high X-ray luminosities can also substantiallyionise the EoR (at the ∼ II regions and cold neutral IGM can yield an extremely large21 cm PS amplitude ( ∼ mK ). It is also roughly consistent with the limits proposed by Fialkovet al. (2017) using observations of the unresolved cosmic X-raybackground (e.g. Lehmer et al. 2012) and the upper limits on themeasured 21 cm PS from PAPER-64 (e.g. Ali et al. 2015; Poberet al. 2015). Though the specific model shown here exceeds the PAPER-64upper limits, for other choices of the EoR and EoH parameters,this same choice of L X < / SFR = 10 erg s − M − (cid:12) yr canresult in a 21 cm PS below these upper limits.c (cid:13)000 , 000–000 E The soft X-rays produced within galaxies can be absorbedby the intervening ISM, thus not being able to escape andcontribute to heating the IGM. The impact that this ISMabsorption has on the emergent X-ray SED depends on theISM density and metallicity. Early galaxies responsible forthe EoH are expected to be less polluted by metals thanlocal analogues. Indeed, Das et al. (2017) find that theemergent X-ray SED from simulated high-redshift galax-ies can be well approximated by a metal free ISM with atypical column density of log ( N H I / cm ) = 21 . +0 . − . . Asthe opacity of metal free gas is a steep function of energy,these authors find that the 21 cm PS from the emergingX-ray SED can be well approximated using the common as-sumption of a step function attenuation of the X-ray SEDbelow an energy threshold, E . In this work, we adopt aflat prior over E ∈ [0 . , .
5] keV which corresponds tolog ( N H I / cm ) ∈ [19 . , . .Panel (v) of Figure 1 highlights the impact of E . As L X < / SFR defines the total soft-band luminosity, as weincrease (decrease) E we effectively harden (soften) thespectrum of emergent X-ray photons. Since the absorptioncross section scales as ∝ E − , we would naively expectsmaller values of E to result in more efficient heating, shift-ing the minimum in the global signal to higher redshifts.However, this evolution is slightly reversed for very low val-ues, E ∼ < . E . Softer SEDs result in very inhomogeneous heating,with PS amplitudes larger by up to an order of magnitude(e.g. Pacucci et al. 2014). The amplitude of the EoH peakconsistently decreases for an increasing E , as the harderSEDs make the EoH more homogeneous. Eventually for E > . α X The spectral index, α X , describing the emergent spectrumfrom the first galaxies hosting X-ray sources depends onwhat is assumed to be the dominant process producing X-ray photons. In this work, we adopt a fiducial flat prior of α X ∈ [ − . , .
5] which should encompass the most relevanthigh energy X-ray SEDs that describe the first galaxies (e.g.HMXBs, host ISM, mini-quasars, SNe remnants etc.; see forexample McQuinn 2012; Pacucci et al. 2014).Finally, in panel (vi) of Figure 1, we illustrate the im-pact of α X . For an increasing α X , more soft X-ray photons The conversion to column densities is computed assuming aunity optical depth for a pristine, metal free, neutral ISM (i.e.the contribution from He II is negligible). are produced (as the soft-band luminosity is kept fixed) re-sulting in more efficient X-ray heating. This results in anincrease in the temperature fluctuations, driving a largeramplitude of the EoH peak in the 21 cm PS. Conversely, foran inverted spectral index, α X = − .
5, the emergent X-rayphotons are spectrally harder, producing inefficient uniformX-ray heating, and limited temperature fluctuations. Thus,the EoH peak in the 21 cm PS for α X = − . E , is much more potent inhardening/softening the SED than the spectral index, α X . In order to provide astrophysical parameter forecasting, wemust model the expected noise of the 21 cm experiments.Within this work, we focus solely on the 21 cm PS. To gen-erate the sensitivity curves for 21 cm interferometer experi-ments, we use the python module cmsense (Pober et al.2013, 2014). Below, we summarise the main aspects and as-sumptions required to produce telescope noise profiles.Firstly, the thermal noise PS is calculated at each grid-ded uv -cell according to the following (e.g. Morales 2005;McQuinn et al. 2006; Pober et al. 2014),∆ ( k ) ≈ X Y k π Ω (cid:48) t T , (8)where X Y is a cosmological conversion factor between theobserving bandwidth, frequency and comoving distance, Ω (cid:48) is a beam-dependent factor derived in Parsons et al. (2014), t is the total time spent by all baselines within a particular k -mode and T sys is the system temperature, the sum of thereceiver temperature, T rec , and the sky temperature T sky .We model T sky using the frequency dependent scaling T sky =60 (cid:0) ν
300 MHz (cid:1) − . K (Thompson et al. 2007).The sample variance of the cosmological 21 cm PS caneasily be combined with the thermal noise to produce thetotal noise PS using an inverse-weighted summation over allthe individual modes (Pober et al. 2013), δ ∆ ( k ) = (cid:32)(cid:88) i ,i ( k ) + ∆ ( k )) (cid:33) − , (9)where δ ∆ ( k ) is the total uncertainty from thermal noiseand sample variance in a given k -mode and ∆ ( k ) is thecosmological 21 cm PS (mock observation). Here we as-sume Gaussian errors for the cosmic-variance term, whichis a good approximation on large-scales.The largest primary uncertainty for 21 cm experimentsis dealing with the bright foreground contamination. How-ever, for the most part these bright foregrounds are spec-trally smooth and have been shown to reside within a con-fined region of cylindrical 2D k -space known as the ‘wedge’(Datta et al. 2010; Vedantham et al. 2012; Morales et al.2012; Parsons et al. 2012b; Trott et al. 2012; Hazelton et al.2013; Thyagarajan et al. 2013; Liu et al. 2014a,b; Thyagara-jan et al. 2015b,a; Pober et al. 2016). Outside of this ‘wedge’we are left with a relatively pristine observing window wherethe cosmic 21 cm signal is only affected by the instrumental https://github.com/jpober/21cmSensec (cid:13) , 000–000 B. Greig & A. Mesinger
Parameter HERA SKATelescope antennae 331 512Diameter (m) 14 35Collecting Area (m ) 50 953 492 602 T rec (K) 100 0.1T sky + 40Bandwidth (MHz) 8 8Integration time (hrs) 1000 (drift) 1000 (tracked) Table 1.
Summary of telescope parameters we use to computesensitivity profiles (see text for further details). thermal noise. At present, the location of the boundary sep-arating this observing window and the ‘wedge’-like featureis uncertain.Within cmsense three foreground removal strategiesare provided (Pober et al. 2014), “optimistic”, “moderate”and “pessimistic”. We defer the reader to this work for fur-ther details on these scenarios, highlighting here that wechoose to adopt the “moderate” scenario. This entails 21cm observations only within the pristine 21 cm window (i.e.avoiding the “wedge”), with the wedge location defined toextend ∆ k (cid:107) = 0 . h Mpc − beyond the horizon limit (Poberet al. 2014). Furthermore, this scenario includes the coher-ent summation over all redundant baselines within the arrayconfiguration allowing the reduction of thermal noise (Par-sons et al. 2012a).Within this work, we focus on the two second genera-tion 21 cm interferometer experiments capable of simulta-neously measuring the EoR and the EoH, namely the SKAand HERA. Below (and in Table 1) we summarise the spe-cific design features and assumptions required to model thetheoretical noise of both instruments:1. HERA:
We follow the design specifics outlined inBeardsley et al. (2014) with a core design consisting of 331dishes . Each dish is 14m in diameter closely packed intoa hexagonal configuration to maximise the total number ofredundant baselines (Parsons et al. 2012a). We model thetotal system temperature as T sys = 100 + T sky K. HERAwill operate in a drift-scanning mode, for which we assumea total 1080 hr observation, spread across 180 nights at 6hours per night.2.
SKA:
We use the latest available design for SKA-lowPhase 1, using the telescope positions provided in the mostrecent SKA System Baseline Design document . Specifi- Note, the final fully funded design for HERA consists of 350dishes, 320 in the core and 30 outriggers (DeBoer et al. 2017).For all intents and purpose for this work, the difference betweena 320 and 331 core layout is negligible. In Greig & Mesinger (2015) we used the original design priorto the re-baselining (the 50 per cent reduction in the number ofantennae dipoles), while in Greig et al. (2015) we investigatedseveral design layouts to maximise the sensitivity specifically forthe 21 cm PS following the re-baselining. The latest design forthe SKA results in reduced sensitivity when compared to bothour previous works, therefore we caution comparisons betweenthe SKA within this work and our previous studies. http://astronomers.skatelescope.org/wp-content/uploads/2016/09/SKA-TEL-SKO-0000422 02 SKA1 LowConfigurationCoordinates-1.pdf cally, SKA-low Phase 1 includes a total of 512 35m antennaestations randomly distributed within a 500m core radius.The total SKA system temperature is modelled as outlinedin the SKA System Baseline Design, T sys = 1 . T sky + 40 K.For the SKA we adopt a single, deep 1000 hr tracked scanof a cold patch on the sky.It is non trivial to perform a like-for-like comparisonbetween the two experiments, as HERA intends to per-form a rotational synthesis drift scan, whereas the SKA in-tends to conduct track scanned observations . These twostrategies result in considerably different noise PS. A sin-gle tracked field with the SKA will have considerably lowerthermal noise than HERA owing to the deeper integrationtime, therefore SKA will be superior on small scales (large k ) important for imaging. On the other hand, by observ-ing numerous patches of the sky rotating through the zenithpointing field of view of HERA per night, sample variancecan be better mitigated compared to the single tracked fieldof the SKA (i.e. HERA will have reduced noise on largescales, small k ). For the most part, the strongest constraintson the astrophysical parameters come from the large-scale(small k ) modes of the 21 cm PS, therefore using the PS asthe likelihood statistic will favour the approach of HERAover that of SKA. We quantify these claims further in theAppendix, showing the noise power spectra at different red-shifts.Further complicating a direct comparison is the factthat the SKA is planning a tiered survey. For simplicity,in this introductory work we only consider a single, deep1000 hr observation (though we will return to this in thefuture). However, we could have considered either the in-termediate 10 ×
100 hr or wide and shallow 100 ×
10 hrstrategies (e.g. Greig et al. 2015). These latter two surveysconcede thermal noise sensitivity (from the reduced per fieldintegration time) for an increased sample variance sensitiv-ity by surveying multiple fields. In terms of the simplifiedthree parameter EoR model considered in Greig & Mesinger(2015), the single, deep 1000 hr observation recovered thelargest uncertainties on the astrophysical parameters rela-tive to the intermediate or wide and shallow surveys (i.e. asingle, tracked field was the worst performed strategy).
Having outlined our astrophysical model to describe the EoRand EoH, we now introduce our mock observations of thecosmic 21 cm signal. It is impractical to vary all availableastrophysical parameters when creating mock observations,therefore, following Mesinger et al. (2016) we take two ex-treme choices for T minvir . This parameter characterises boththe timing of the epochs and the typical bias of the domi-nant galaxies, thus encoding the largest variation in the 21cm signal (e.g panel (iii) of Figure 1). Specifically, we adopt Additionally, HERA is a dedicated 21 cm experiment specifi-cally designed for a 21 cm PS measurement, while the SKA is amultidisciplinary experiment, with detection of the 21 cm signalonly one of many key science goals. Moreover, the instrument lay-out and design is tailored towards 3D tomographic imaging (e.g.Mellema et al. 2013) rather than the 21 cm PS.c (cid:13)000
Having outlined our astrophysical model to describe the EoRand EoH, we now introduce our mock observations of thecosmic 21 cm signal. It is impractical to vary all availableastrophysical parameters when creating mock observations,therefore, following Mesinger et al. (2016) we take two ex-treme choices for T minvir . This parameter characterises boththe timing of the epochs and the typical bias of the domi-nant galaxies, thus encoding the largest variation in the 21cm signal (e.g panel (iii) of Figure 1). Specifically, we adopt Additionally, HERA is a dedicated 21 cm experiment specifi-cally designed for a 21 cm PS measurement, while the SKA is amultidisciplinary experiment, with detection of the 21 cm signalonly one of many key science goals. Moreover, the instrument lay-out and design is tailored towards 3D tomographic imaging (e.g.Mellema et al. 2013) rather than the 21 cm PS.c (cid:13)000 , 000–000 T minvir = 5 × K ( faint galaxies ) and T minvir = 3 × K( bright galaxies ). These choices approximately match theevolution of the cosmic SFR density inferred from extrapo-lating the observed luminosity functions (Bouwens et al.2015) down to a UV magnitude of M UV = −
10 ( −
17) for the faint galaxies ( bright galaxies ) galaxies model (see fig-ure 5 of Das et al. 2017).In order to match the latest constraints on the electronscattering optical depth, τ e , from Planck ( τ e = 0 . ± . ζ = 30 for the faint galaxies model. This corresponds to the fiducial pa-rameter set outlined in equation (6). For the biased, rarergalaxies in the bright galaxies model, we adopt ζ = 200.We select a fiducial R mfp = 15 Mpc for both models. InTable 2 we summarise the adopted astrophysical parame-ter set for each of the two models, while also providing thecorresponding optical depth, τ e .We adopt the same X-ray source model for both the bright galaxies and faint galaxies models. We assumeHMXBs to be the dominant X-ray heating source within thefirst galaxies, and use the results of Das et al. (2017) to de-scribe the emergent X-ray SED . This corresponds to anenergy threshold, E = 0 . ( N H I / cm ) = 21 . α X = 1 .
0. Finally, we as-sume a HMXB soft band luminosity of L X < / SFR =10 erg s − M − (cid:12) yr, which is consistent with estimatesfrom the HMXB population synthesis models of Fragos et al.(2013b).In Figure 2 we provide the cosmic evolution of ourtwo EoR source models: faint galaxies (solid curve) and bright galaxies (dashed curve). The top, middle and bot-tom panels are the global reionisation history, global 21cm signal, and the 21 cm PS at k = 0 .
15 (black) and k = 0 . − (red) respectively. These show qualitativelythe same behaviour as we have discussed in the previoussection. These mock observations are generated from simu-lations with a volume of 600 Mpc , on a 400 grid smootheddown from the high-resolution initial conditions generatedon a 2400 grid.By construction, these two EoR models have a similarreionisation history, consistent with the latest observations.The faint galaxies model has a somewhat more extendedEoR, as the less biased DM halos which host the domi-nant galaxies form slower. Unlike for the observationally-constrained EoR history, we allow the EoH history to bedifferent in the two models. In other words, we take the sameX-ray luminosity per SFR. As a result, the faint galax-ies galaxy model has an earlier EoH (see the middle panel),governed by the formation of the first T minvir = 5 × Kstructures. In the future, we will provide a more generalised parameter-isation of the ionising source model by allowing each individualconstituent of ζ (equation 6) to vary with halo mass (Park et al.,in prep.). This will enable additional flexibility in the source mod-elling, while at the same time allowing high- z galaxy luminosityfunctions to be applied as priors to the source model prescription. Note that the typical star-forming haloes used by Das et al.(2017), e.g. (cid:46) × M (cid:12) , are smaller than the correspondingmasses set by our minimum on T minvir (e.g. 10 K). Therefore, wehave assumed that the intrinsic attenuation is independent of thehost galaxy mass. This, however, need not be the case. . . . . . . ¯ x H I − − − − δ T b [ m K ] Faint galaxiesBright galaxies
Redshift , z δ T b ∆ [ m K ] k = 0 .
15 Mpc − k = 0 . − Figure 2.
Cosmic evolution of the bright galaxies (dashed)and faint galaxies (solid) models used for the mock observationswithin this work.
Top: the evolution of the IGM neutral fraction.
Middle: global averaged 21 cm brightness temperature contrast.
Bottom: evolution of the PS amplitude at two different k -modes, k = 0 .
15 Mpc − (black) and k = 0 . − (red). Due to this, these two models will be interesting forour astrophysical parameter forecasting for both HERA andthe SKA. With a decreasing sensitivity with increasing red-shift, the location of the EoR and EoH peaks will impactthe resultant significance of a detection of the 21 cm signal.For example, in the bottom panel of Figure 2, it is evidentthat the bright galaxies model exhibits additional 21 cmpower relative to the faint galaxies model. Furthermore,this occurs at a lower redshift, where the instrumental noiseis expected to be lower. Naively, one would therefore ex-pect astrophysical parameters to be more tightly constrainedwith a mock bright galaxies signal.
We now quantify astrophysical parameter constraints foreach of these two models, for both second generation in-terferometers, HERA and the SKA. We first summarise our configuration, and then provide parameter con-straints while discussing their implications. setup
As in Greig & Mesinger (2015), we use the 21 cm PS as thelikelihood statistic to sample the astrophysical parameterspace in . We adopt a modelling uncertainty of 20per cent on the sampled 21 cm PS (not the mock observationof the 21 cm PS), which is added in quadrature with thetotal noise power spectrum (equation 8). As in Greig & A modelling uncertainty accounts for the inaccuracy of semi-numerical approaches such as . Here we simply takean uncorrelated fixed percentage error, with a value consistentwith the comparisons in Zahn et al. (2011). This constant errorpurely accounts for the observed fractional differences in the re-covered 21cm PS between the simulations. For example, it doesnot account for any potential errors arising from the differentalgorithms used to compute the IGM spin temperature. Futurec (cid:13) , 000–000 B. Greig & A. Mesinger ζ R mfp T minvir log ( T minvir ) log ( L X < / SFR) E log ( N H I ) α X τ e Source Model [Mpc] [K] [K] [erg s − M − (cid:12) yr] [keV] [cm − ] faint galaxies × bright galaxies × , 10 ] [4.0, 6.0] [38.0, 42.0] [0.1, 1.5] [19.3,23.0] [-0.5, 2.5] Table 2.
A summary of the astrophysical parameters used for our two mock observations, the corresponding electron scattering opticaldepth, τ e , as well as the adopted prior range for . See text for additional details. Mesinger (2015) we compare the 21 cm PS at each redshiftonly over the reduced k -space range, k = 0 .
15 – 1 . − ,which we deem to be free of cosmic variance and shot-noiseeffects arising within the simulations.We perform our forecasting using eight co-evolution redshifts ( z = 6 , , , , , ,
15 and 17). Note that ourchoice is relatively arbitrary, taken to span the EoH andEoR in both of our mock observations and is not driven byany computational or numerical reasons.For the MCMC sampling, we use smaller boxes thanused for the mock observations: 300 Mpc volume on a 200 grid smoothed down from a high-resolution 1200 grid. Boththe mock observations and the sampled 21 cm PS have thesame voxel resolution ( ∼ . . We note that the public version of is not optimised for our MCMC framework.Therefore, we heavily streamlined the computation for a sin-gle core implementation (since the advantage of isthat a realisation of is performed for each avail-able CPU core). For the 300 Mpc volume, 200 voxel setupabove, our streamlined version of is ∼ × fasteron a single core . For our astrophysical parameter forecast-ing, we perform ∼ × runs. As we are usingthe parallelised affine invariant ensemble sampler emcee , wecan achieve ∼ × samplings with the following setup:400 walkers, each performed for 200 iterations . Performed work will seek to characterise this uncertainty, potentially evenmediating it by comparing to suites of RT simula-tions. The observed 3D 21 cm signal is spatially dependent on the 2Dsky, with the third (line-of-sight) direction being frequency depen-dent. The line-of-sight axis encodes evolution along the light-cone,which can impact the observed 21 cm PS, when compared to pre-dictions from co-evolution cubes (e.g. Datta et al. 2012, 2014; LaPlante et al. 2014; Ghara et al. 2015). However, this effect is gen-erally only pronounced on much larger scales than considered here(note that our lowest k -modes, | k | = 0 .
15 Mpc − , correspond to amodest ∆ z ∼ . to operate directlyon the light-cone. This boost in computational efficiency arises owing to the in-clusion of additional interpolation tables to remove redundantcalculations. Additionally, is reduced to a single ex-ecutable removing file I/O and limiting memory overheads. We explored several combinations of the walker/iterations con-figuration to confirm our experimental setup was providing con-verged results for our MCMC. using 200 cores (i.e. 100 physical cores + 100 virtual cores)on a shared memory cluster, such a setup takes ∼ In Figure 3 we present the astrophysical parameter con-straints for our faint galaxies model for our assumed1000hr observation with both HERA (blue) and the SKA(red). Across the diagonals we provide the 1D marginalisedprobability distribution functions (PDFs) for each of the sixmodel astrophysical parameters. In the lower left half of thefigure, we provide the 2D marginalised joint likelihood con-tours, with crosses denoting the input fiducial values, whilethe thick and thin contours denote the 68 (1 σ ) and 95 (2 σ )percentiles, respectively. In the top right half of the figure,we provide the 2 σ marginalised constraints on the IGM neu-tral fraction, ¯ x H I , with respect to the reionisation historyof the mock observation (solid black curve). Note that togenerate this figure, we interpolate the reionisation historybetween the marginalised distributions for ¯ x H I at the eightco-evolution redshifts, sampling at a rate of ∆ z = 0 .
4. In Ta-ble 3 we provide the median and associated 16th and 84thpercentiles for each of our six astrophysical parameters forboth HERA and the SKA.From the relatively narrow 1D PDFs, it is clear thatboth HERA and the SKA can simultaneously constrain theEoR and EoH to high accuracy. The only parameter forwhich we do not achieve strong constraints is the X-rayspectral index, α X . However, this is not overly surprisinggiven the relatively small effect it has on the amplitude ofthe 21 cm PS over the entire allowed parameter range (seeFigure 1). Folding in the 20 per cent modelling uncertaintyand the instrumental noise, the relative difference in the 21cm PS amplitude across the full allowed parameter range isroughly consistent at about 2 σ . In terms of the reionisationhistory both HERA and the SKA recover comparably tightconstraints on the IGM neutral fraction. At 1 σ , we recoverconstraints on ¯ x H I of the order of ∼ ζ - T minvir , L X < / SFR- T minvir and E - α X . This is generally consis-tent with both Ewall-Wice et al. (2016) and Kern et al.(2017), whose mock observation most closely resembles our faint galaxies model. However, these authors find some-what stronger f X - E and f X - α X degeneracies (where f X cor-responds to the number of X-ray photons per stellar baryon,and can thus be related to our L X < / SFR for a given α X and E ). This discrepancy could arise due to (i) the inclu-sion of the 20 per cent modelling uncertainty which broadens c (cid:13)000
4. In Ta-ble 3 we provide the median and associated 16th and 84thpercentiles for each of our six astrophysical parameters forboth HERA and the SKA.From the relatively narrow 1D PDFs, it is clear thatboth HERA and the SKA can simultaneously constrain theEoR and EoH to high accuracy. The only parameter forwhich we do not achieve strong constraints is the X-rayspectral index, α X . However, this is not overly surprisinggiven the relatively small effect it has on the amplitude ofthe 21 cm PS over the entire allowed parameter range (seeFigure 1). Folding in the 20 per cent modelling uncertaintyand the instrumental noise, the relative difference in the 21cm PS amplitude across the full allowed parameter range isroughly consistent at about 2 σ . In terms of the reionisationhistory both HERA and the SKA recover comparably tightconstraints on the IGM neutral fraction. At 1 σ , we recoverconstraints on ¯ x H I of the order of ∼ ζ - T minvir , L X < / SFR- T minvir and E - α X . This is generally consis-tent with both Ewall-Wice et al. (2016) and Kern et al.(2017), whose mock observation most closely resembles our faint galaxies model. However, these authors find some-what stronger f X - E and f X - α X degeneracies (where f X cor-responds to the number of X-ray photons per stellar baryon,and can thus be related to our L X < / SFR for a given α X and E ). This discrepancy could arise due to (i) the inclu-sion of the 20 per cent modelling uncertainty which broadens c (cid:13)000 , 000–000 R m f p . . . l og ( T m i n v i r ) . . . l og ⇣ L X < k e V S F R ⌘ . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintyFaint Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp . . . log ( T minvir ) . . . log ⇣ L X < SFR ⌘ . . . . log ( N HI ) . . . . E . . . ↵ X [erg s M yr] [keV] [K] [Mpc] [cm ] R m f p . . . l og ( T m i n v i r ) . . . l og ⇣ L X < k e V S F R ⌘ . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintyFaint Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp . . . log ( T minvir ) . . . log ⇣ L X < SFR ⌘ . . . . log ( N HI ) . . . . E . . . ↵ X [Mpc] [K] [keV] [cm ] [erg s M yr] R m f p . . . l og ( T M i n v i r ) . . . l og ( L X / S F R ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintyFaint Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp [Mpc] . . . log ( T Minvir [K]) . . . log ( L X / SFR) . . . . log ( N HI / cm ) . . . . E [keV] . . . ↵ X . . . . . . . Redshift , z . . . . . . ¯ x H I Fiducial modelSKA (2 , interpolated)HERA331 (2 , interpolated) Figure 3.
Recovered 1 and 2D marginalised joint posterior distributions for our faint galaxies six parameter astrophysical model foran assumed 1000 hr on sky observation with HERA (blue) and SKA (red). Thick and thin contours correspond to the 68 (1 σ ) and 95(2 σ ) per cent 2D marginalised joint likelihood constraints, respectively, and crosses (black vertical dashed lines) denote the input modelparameters, defined to be ( ζ , R mfp , log ( T minvir ), log ( L X < / SFR), E , α X ) = (30, 15, 4.7, 40.0, 0.5, 1.0). Inset: The recoveredglobal evolution of the IGM neutral fraction. The solid black curve corresponds to the fiducial input evolution, whereas the error barscorrespond to the 2 σ limits on the recovered IGM neutral fraction. Note, these points are interpolated at ∆ z = 0 . faint galaxies ParameterModel/instrument ζ R mfp log ( T minvir ) log ( L X < / SFR) E α X [Mpc] [K] [erg s − M − (cid:12) yr] [keV]Full X-ray heatingHERA 331 34.69 +9 . − . +2 . − . +0 . − . +0 . − . +0 . − . +0 . − . SKA 33.99 +7 . − . +2 . − . +0 . − . +0 . − . +0 . − . +0 . − . Ignoring T S ( T S (cid:29) T CMB )HERA 331 18.98 +7 . − . +3 . − . +0 . − . - - -SKA 15.51 +4 . − . +3 . − . +0 . − . - - - Table 3.
Summary of the median recovered values (and associated 16th and 84th percentile errors) for the six parameter astrophysicalmodel describing the EoR and EoX, ζ , R mfp , log ( T minvir ), log ( L X < / SFR), E and α X . We assume a total 1000hr integrationtime with both the SKA and HERA. Our fiducial mock observation, corresponding to the faint galaxies model, assumes ( ζ , R mfp ,log ( T minvir ), log ( L X < / SFR), E , α X ) = (30, 15, 4.7, 40.0, 0.5, 1.0). We also provide the recovered biased constraints when weignore the IGM spin temperature fluctuations (i.e. T S (cid:29) T CMB ; see Section 5.1).c (cid:13) , 000–000 B. Greig & A. Mesinger our contours; (ii) the approximations made in those works[assumptions of Gaussian errors in Fisher matrices (Ewall-Wice et al. 2016) or modelling errors in an emulator methodKern et al. (2017)]; and/or (iii) our choice for the soft-band energy as a normalisation parameter (instead of the X-rayphoton number ), which can provide a more independent ba-sis vector for the EoH evolution .If we assume the 1D marginalised PDFs can be mod-elled by a normal distribution, which for the most part isreasonable at the 1 σ level (i.e. some tails begin to appearat ∼ σ ), we can provide some approximate fractional un-certainties for the astrophysical parameters. For the SKA(HERA), the 1 σ percentage errors are: ζ = 18 (24), R mfp = 16 (16), log ( T minvir ) = 1.4 (2.3), log ( L X < / SFR) =0.2 (0.3), E = 17 (14) and α X = 88 (73). The uncertaintyon the EoR parameters is comparable to what we obtainedin Greig & Mesinger (2015): ζ = 17 (22), R mfp = 18 (18),log ( T minvir ) = 2.4 (3.3). Therefore, despite increasing themodel complexity by including the EoH, the relative con-straints are comparable.Ewall-Wice et al. (2016) quote fractional precisions ontheir six parameter model, with 1-2 per cent accuracy on theEoR parameters and 6 per cent on their EoH parameters.Their constraints are smaller than ours by about an orderof magnitude for the EoR and factor of a few for the EoH.Approximately half of this difference can be attributed tothe inclusion of the modelling uncertainty (see e.g. Greig &Mesinger 2015). The remaining discrepancy can arise fromeither the fundamental assumptions in their Fisher matrixapproach, or their larger number of redshift samples (morethan a factor of two). In future, we will modify to directly work on the observed light cone, removing thenecessity of an ad-hoc sampling of co-evolution cubes.For our faint galaxies model, we find both HERAand the SKA will recover comparable parameter constraints.This is despite the significantly increased sensitivity achiev-able with the SKA, resulting in a larger total integratedsignal to noise (S/N). As pointed out in Greig & Mesinger(2015), the S/N is not a reliable metric for predicting aninstrument’s ability at parameter constraints, since modelconstraining power is biased towards large scales. This high-lights the importance of using parameter forecasting as afigure of merit, instead of just the total S/N. We cautionhowever that the SKA performance can be improved if onecan better mediate modelling uncertainties. Indeed, the in-creased thermal noise sensitivity on small-scales is washedout by our assumed 20 per cent modelling uncertainty. More- In selecting the soft-band X-ray luminosity instead of a harderX-ray band (e.g. 0.5 - 8 keV), we have preferentially minimised thedegeneracy between the X-ray luminosity and α X . Additionally,adopting a soft-band X-ray luminosity enables straightforwardcomparison with numerous observations of nearby galaxies (e.g.Tzanavaris & Georgantopoulos 2008; Mineo et al. 2012b; Fragoset al. 2013a; Lehmer et al. 2015, 2016). In the near future, weexpect observations of the intrinsic soft-band X-ray luminosityescaping the host galaxy to improve with the upcoming Athenatelescope (Barcons et al. 2012), which will provide a soft-band ef-fective area more than an order of magnitude larger that existingexperiments (T. Dauser, private communication). This will pro-vide stronger priors on the X-ray SED, even if 21cm observationsthemselves are less discriminatory. over, the SKA will be superior at tomography; using higher-order likelihood statistics should therefore favour the SKAover HERA. In Figure 4 we present our 1 and 2D joint marginalised poste-rior distributions for each of the six astrophysical parametersfor our bright galaxies model assuming a 1000hr observa-tion with HERA (blue) and the SKA (red). Table 4 providesthe median and associated 16th and 84th percentiles for eachof our astrophysical model parameters. As in the previoussection, we provide approximate fractional precisions on themodel parameters assuming normally distributed marginallikelihoods. For the SKA (HERA), the 1 σ percent errorsare: ζ = 17 (15), R mfp = 16 (12), log ( T minvir ) = 0.4 (0.6),log ( L X < / SFR) = 0.2 (0.2), E = 16 (17) and α X =80 (79).The constraints are comparable for most of the astro-physical parameters held fixed across the two models (i.e. R mfp , log ( L X < / SFR), E , α X ). The largest differenceis in R mfp , which shows tighter constraints relative to the faint galaxies model. At a given stage in the EoR, the H II regions in the bright galaxies model are larger and moreisolated, sourced by the brighter, rarer, more-biased sources.Recombinations play a larger role in limiting the growth ofsuch H II regions, whose characteristic sizes approach R mfp earlier in reionisation. Since the signal is more sensitive to R mfp , a degeneracy between ζ and R mfp emerges, with bothnow able to control the timing of the EoR: one can com-pensate for the slowing-down of the EoR due to recombina-tions (a smaller R mfp ) by increasing the ionising efficiency, ζ . This degeneracy leads to poorer overall constraints on ζ when compared to the faint galaxies model.These improvements in the constraints of R mfp are how-ever only available with HERA. The marginalised PDFs forthe SKA exhibit a noticeable tail towards increasing R mfp .The source of this tail can simply be attributed to the re-duced large-scale sensitivity of the SKA relative to HERAowing to our adopted observing strategies within this work(see Section 2.3 and Figure A2). The majority of studies that constrain the EoR with the cos-mic 21 cm signal assume that the IGM spin temperature issaturated (i.e. T S (cid:29) T CMB ). Since the corresponding term inthe brightness temperature saturates to unity in this regime(c.f. equation 1), assuming saturation greatly simplifies thecomputational load. However, the validity of such an ap-proximation is highly dependent on the poorly-constrainedrelative efficiencies of ionising and X-ray sources in the firstgalaxies. Not properly taking into account the IGM spintemperature can bias the recovered EoR parameter con-straints . Here, we quantify the impact of this using our In Ewall-Wice et al. (2016), these authors found that by in-correctly accounting for the EoH, their recovered uncertainties inc (cid:13)000
Summary of the median recovered values (and associated 16th and 84th percentile errors) for the six parameter astrophysicalmodel describing the EoR and EoX, ζ , R mfp , log ( T minvir ), log ( L X < / SFR), E and α X . We assume a total 1000hr integrationtime with both the SKA and HERA. Our fiducial mock observation, corresponding to the faint galaxies model, assumes ( ζ , R mfp ,log ( T minvir ), log ( L X < / SFR), E , α X ) = (30, 15, 4.7, 40.0, 0.5, 1.0). We also provide the recovered biased constraints when weignore the IGM spin temperature fluctuations (i.e. T S (cid:29) T CMB ; see Section 5.1).c (cid:13) , 000–000 B. Greig & A. Mesinger our contours; (ii) the approximations made in those works[assumptions of Gaussian errors in Fisher matrices (Ewall-Wice et al. 2016) or modelling errors in an emulator methodKern et al. (2017)]; and/or (iii) our choice for the soft-band energy as a normalisation parameter (instead of the X-rayphoton number ), which can provide a more independent ba-sis vector for the EoH evolution .If we assume the 1D marginalised PDFs can be mod-elled by a normal distribution, which for the most part isreasonable at the 1 σ level (i.e. some tails begin to appearat ∼ σ ), we can provide some approximate fractional un-certainties for the astrophysical parameters. For the SKA(HERA), the 1 σ percentage errors are: ζ = 18 (24), R mfp = 16 (16), log ( T minvir ) = 1.4 (2.3), log ( L X < / SFR) =0.2 (0.3), E = 17 (14) and α X = 88 (73). The uncertaintyon the EoR parameters is comparable to what we obtainedin Greig & Mesinger (2015): ζ = 17 (22), R mfp = 18 (18),log ( T minvir ) = 2.4 (3.3). Therefore, despite increasing themodel complexity by including the EoH, the relative con-straints are comparable.Ewall-Wice et al. (2016) quote fractional precisions ontheir six parameter model, with 1-2 per cent accuracy on theEoR parameters and 6 per cent on their EoH parameters.Their constraints are smaller than ours by about an orderof magnitude for the EoR and factor of a few for the EoH.Approximately half of this difference can be attributed tothe inclusion of the modelling uncertainty (see e.g. Greig &Mesinger 2015). The remaining discrepancy can arise fromeither the fundamental assumptions in their Fisher matrixapproach, or their larger number of redshift samples (morethan a factor of two). In future, we will modify to directly work on the observed light cone, removing thenecessity of an ad-hoc sampling of co-evolution cubes.For our faint galaxies model, we find both HERAand the SKA will recover comparable parameter constraints.This is despite the significantly increased sensitivity achiev-able with the SKA, resulting in a larger total integratedsignal to noise (S/N). As pointed out in Greig & Mesinger(2015), the S/N is not a reliable metric for predicting aninstrument’s ability at parameter constraints, since modelconstraining power is biased towards large scales. This high-lights the importance of using parameter forecasting as afigure of merit, instead of just the total S/N. We cautionhowever that the SKA performance can be improved if onecan better mediate modelling uncertainties. Indeed, the in-creased thermal noise sensitivity on small-scales is washedout by our assumed 20 per cent modelling uncertainty. More- In selecting the soft-band X-ray luminosity instead of a harderX-ray band (e.g. 0.5 - 8 keV), we have preferentially minimised thedegeneracy between the X-ray luminosity and α X . Additionally,adopting a soft-band X-ray luminosity enables straightforwardcomparison with numerous observations of nearby galaxies (e.g.Tzanavaris & Georgantopoulos 2008; Mineo et al. 2012b; Fragoset al. 2013a; Lehmer et al. 2015, 2016). In the near future, weexpect observations of the intrinsic soft-band X-ray luminosityescaping the host galaxy to improve with the upcoming Athenatelescope (Barcons et al. 2012), which will provide a soft-band ef-fective area more than an order of magnitude larger that existingexperiments (T. Dauser, private communication). This will pro-vide stronger priors on the X-ray SED, even if 21cm observationsthemselves are less discriminatory. over, the SKA will be superior at tomography; using higher-order likelihood statistics should therefore favour the SKAover HERA. In Figure 4 we present our 1 and 2D joint marginalised poste-rior distributions for each of the six astrophysical parametersfor our bright galaxies model assuming a 1000hr observa-tion with HERA (blue) and the SKA (red). Table 4 providesthe median and associated 16th and 84th percentiles for eachof our astrophysical model parameters. As in the previoussection, we provide approximate fractional precisions on themodel parameters assuming normally distributed marginallikelihoods. For the SKA (HERA), the 1 σ percent errorsare: ζ = 17 (15), R mfp = 16 (12), log ( T minvir ) = 0.4 (0.6),log ( L X < / SFR) = 0.2 (0.2), E = 16 (17) and α X =80 (79).The constraints are comparable for most of the astro-physical parameters held fixed across the two models (i.e. R mfp , log ( L X < / SFR), E , α X ). The largest differenceis in R mfp , which shows tighter constraints relative to the faint galaxies model. At a given stage in the EoR, the H II regions in the bright galaxies model are larger and moreisolated, sourced by the brighter, rarer, more-biased sources.Recombinations play a larger role in limiting the growth ofsuch H II regions, whose characteristic sizes approach R mfp earlier in reionisation. Since the signal is more sensitive to R mfp , a degeneracy between ζ and R mfp emerges, with bothnow able to control the timing of the EoR: one can com-pensate for the slowing-down of the EoR due to recombina-tions (a smaller R mfp ) by increasing the ionising efficiency, ζ . This degeneracy leads to poorer overall constraints on ζ when compared to the faint galaxies model.These improvements in the constraints of R mfp are how-ever only available with HERA. The marginalised PDFs forthe SKA exhibit a noticeable tail towards increasing R mfp .The source of this tail can simply be attributed to the re-duced large-scale sensitivity of the SKA relative to HERAowing to our adopted observing strategies within this work(see Section 2.3 and Figure A2). The majority of studies that constrain the EoR with the cos-mic 21 cm signal assume that the IGM spin temperature issaturated (i.e. T S (cid:29) T CMB ). Since the corresponding term inthe brightness temperature saturates to unity in this regime(c.f. equation 1), assuming saturation greatly simplifies thecomputational load. However, the validity of such an ap-proximation is highly dependent on the poorly-constrainedrelative efficiencies of ionising and X-ray sources in the firstgalaxies. Not properly taking into account the IGM spintemperature can bias the recovered EoR parameter con-straints . Here, we quantify the impact of this using our In Ewall-Wice et al. (2016), these authors found that by in-correctly accounting for the EoH, their recovered uncertainties inc (cid:13)000 , 000–000 R m f p . . . l og ( T m i n v i r ) . . . l og ⇣ L X < k e V S F R ⌘ . . . . E
50 100 150 200 ⇣ . . . ↵ X Bright Galaxies 1000hr20% modelling uncertaintyBright Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp . . . log ( T minvir ) . . . log ⇣ L X < SFR ⌘ . . . . log ( N HI ) . . . . E . . . ↵ X [erg s M yr] [keV] [K] [Mpc] [cm ] R m f p . . . l og ( T m i n v i r ) . . . l og ⇣ L X < k e V S F R ⌘ . . . . E
50 100 150 200 ⇣ . . . ↵ X Bright Galaxies 1000hr20% modelling uncertaintyBright Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp . . . log ( T minvir ) . . . log ⇣ L X < SFR ⌘ . . . . log ( N HI ) . . . . E . . . ↵ X [Mpc] [K] [keV] [cm ] [erg s M yr] R m f p . . . l og ( T M i n v i r ) . . . l og ( L X / S F R ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Bright Galaxies 1000hr20% modelling uncertaintyBright Galaxies 1000hr20% modelling uncertaintySKAHERA331
10 15 20 R mfp [Mpc] . . . log ( T Minvir [K]) . . . log ( L X / SFR) . . . . log ( N HI / cm ) . . . . E [keV] . . . ↵ X . . . . . . . Redshift , z . . . . . . ¯ x H I Fiducial modelSKA (2 , interpolated)HERA331 (2 , interpolated) Figure 4.
The same as Figure 3 except for the bright galaxies model. Crosses (black vertical dashed lines) denote the input modelparameters, defined to be ( ζ , R mfp , log ( T minvir ), log ( L X < / SFR), E , α X ) = (200, 15, 5.48, 40.0, 0.5, 1.0). bright galaxies ParameterModel/instrument ζ R mfp log ( T minvir ) log ( L X < / SFR) E α X [Mpc] [K] [erg s − M − (cid:12) yr] [keV]Full X-ray heatingHERA 331 188.59 +32 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . SKA 177.32 +33 . − . +2 . − . +0 . − . +0 . − . +0 . − . +0 . − . Ignoring T S ( T S (cid:29) T CMB )HERA 331 25.50 +12 . − . +1 . − . +0 . − . - - -SKA 15.65 +3 . − . +2 . − . +0 . − . - - - Table 4.
The same as Table 3 except now for the bright galaxies model. Our bright galaxies mock observation assumes ( ζ , R mfp ,log ( T minvir ), log ( L X < / SFR), E , α X ) = (200, 15, 5.48, 40.0, 0.5, 1.0). two mock 21 cm observations. The two adopted extrema in the EoR parameters could be biased. They explored this by fittingtheir EoR parameters by either (i) assuming a fixed EoH model;or (ii) properly fitting and marginalising over the EoH model pa-rameters. The fractional uncertainties from (i) were smaller than(ii) by almost a factor of two. This emphasises the importance of T minvir allow the exploration of vastly different levels of overlapbetween the EoR and EoH, enabling us to estimate the avail-able span in the corresponding bias. Here we use properly accounting for the EoH when performing astrophysicalparameter recovery from the 21 cm signal.c (cid:13) , 000–000 B. Greig & A. Mesinger to recover the fractional precision on the EoR parametersunder the simplification of T S (cid:29) T CMB , and compare theseto the constraints obtained by properly modelling T S . The 21 cm signal for the faint galaxies model transitionsfrom absorption to emission at z (cid:46)
12 (see e.g. the middlepanel of Figure 2). At lower redshifts, the IGM spin temper-ature continues to increase and the (1 − T CMB /T S ) factor ap-proaches unity. Assuming the saturated limit ( T S (cid:29) T CMB )results in a fractional error in the power spectrum less than10% when (1 − T CMB /T S ) (cid:38) .
9, corresponding to z ∼ . x H I ∼ . faint galaxies model considered in Mesinger et al. 2016).Therefore, the saturated spin temperature approximation isonly reasonable during the second half of reionisation in our faint galaxies model.In Figure 5, we present the 1 and 2D joint marginalisedposterior distributions for the EoR parameters, namely ζ , R mfp and log ( T minvir ). For this comparison, we only con-sider the SKA (the results for HERA are nearly identical asshown in Table 3), and run on the redshifts span-ning the EoR: z = 6 , , T minvir are offset by at least ∼ σ . For ζ , we recovermore modest offsets of ∼ . σ , and for R mfp we recover com-parable constraints. In all cases, the recovered astrophysicalconstraints return larger uncertainties, owing to the lowermarginalised likelihoods. For reference, the maximum like-lihood (ML: L = exp( − χ )) is a factor of four lower thanobtained with the full spin temperature modelling in Sec-tion 4.2.To understand this bias, in Figure 6 we show the 21 cmPS for the mock observation (black curve), the ML 21 cm PSassuming T S (cid:29) T CMB (red curve) and the 21 cm PS adopt-ing the ‘true’ EoR model parameters assuming T S (cid:29) T CMB (blue curve). Error bars denote the 20 per cent modellinguncertainty on the model 21 cm PS. The main impact of ig-noring the spin temperature can be seen by comparing theblue and black curves. Since both curves are generated fromthe same EoR model parameters, any discrepancies arisefrom the spin temperature. The largest discrepancy, as ex-pected, arises at the highest redshift, where the IGM is stillundergoing the final stages of X-ray heating. As discussedin Mesinger et al. (2016), during the EoR the cosmic H I patches effectively have the same, uniform IGM spin temper-ature, resulting in fairly negligible temperature fluctuationsfor the faint galaxies model (see also Pober et al. 2015).However, the amplitude of the 21 cm PS is decreased by afactor of (1 − T CMB /T S ) (compare the blue and black curves at z ∼ − T CMB /T S ) decreasein amplitude, a model which assumes T S (cid:29) T CMB will tendto prefer EoR models with intrinsically less power, i.e. thosein which the sources are less biased, having a lower T minvir . Alower T minvir implies more abundant ionising sources, whichmust be compensated for by decreasing ζ to attempt to re-cover the correct reionisation history. Therefore, the likeli-hood peaks at smaller virial temperatures and ionising effi-ciencies (c.f. the red curve, corresponding to T minvir = 10 . K and ζ = 15 . . Already for the faint galaxies model we find significantbiases in the recovered EoR parameters under the assump-tion that the spin temperature is saturated. In the brightgalaxies model, the EoH and EoR overlap even morestrongly, in effect maximising this bias in parameter recov-ery. We quantify this in Figure 7, in which we provide themarginalised distributions, and in the lower half of Table 4where we provide the median, 16th and 84th percentiles forour recovered EoR model parameters.For both ζ and T minvir we recover marginalised constraintsdiscrepant at > σ . As in the faint galaxies model, R mfp remains consistent to within 1 σ , but it instead prefersmarginally lower values, R mfp ∼
10 Mpc. In the inset ofFigure 7, the recovered reionisation history is discrepant atthe > σ level, beyond z ∼ .
5. In Figure 8 we present the21 cm PS from the mock bright galaxies model, the MLestimate and the ‘true’ EoR model parameters assuming thesaturated spin temperature limit. It is immediately evidentthat all the constraining power for the ML arises from the21 cm PS at z = 6 and 7. At z >
7, the saturated limitcannot reproduce the reduced amplitude of the mock 21 cmPS .This is not surprising. As discussed in Section 4.3,the global averaged 21 cm brightness temperature contrastis still in absorption at z > T S (cid:46) T CMB for a significant fraction of the simulation vol-ume. At the same time, the IGM is 35 per cent ionised(¯ x H I = 0 .
65) by z = 8, indicating reionisation is well under-way. This significant overlap of the EoR and EoH breaks thefundamental assumption of the saturated spin temperaturelimit. Furthermore, at z = 8 the bright galaxies modelis transitioning closely to the T S ≡ T CMB limit, producing aprecipitous drop in the 21 cm PS amplitude. Relative to the bright galaxies model in the saturated limit (blue curve)this is a factor of ∼
40 difference in the 21 cm PS amplitude.Under the saturated limit assumption, there is no avenue tomimic such behaviour, resulting in hugely discrepant astro-physical parameter constraints.This highlights the importance of properly including the Note, that the ML from our six parameter model was com-puted from eight different redshifts, whereas in the saturated limitwe only considered four redshifts. As additional redshift data de-creases the unnormalised ML value of a model, the relative dif-ferences here are even larger. For reference, the ML is ∼ times lower when we assume T S (cid:29) T CMB . c (cid:13)000
40 difference in the 21 cm PS amplitude.Under the saturated limit assumption, there is no avenue tomimic such behaviour, resulting in hugely discrepant astro-physical parameter constraints.This highlights the importance of properly including the Note, that the ML from our six parameter model was com-puted from eight different redshifts, whereas in the saturated limitwe only considered four redshifts. As additional redshift data de-creases the unnormalised ML value of a model, the relative dif-ferences here are even larger. For reference, the ML is ∼ times lower when we assume T S (cid:29) T CMB . c (cid:13)000 , 000–000 R m f p . . . l og ( T m i n v i r ) . . . l og ( L X ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintySKASKA (T S >> T CMB )
10 15 20 R mfp [Mpc] . . . log ( T minvir [K]) . . . log ( L X ) . . . . log ( N HI ) . . . . E [keV] . . . ↵ X R m f p . . . l og ( T m i n v i r ) . . . l og ( L X ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintySKASKA (T S >> T CMB )
10 15 20 R mfp [Mpc] . . . log ( T minvir [K]) . . . log ( L X ) . . . . log ( N HI ) . . . . E [keV] . . . ↵ X . . . . . . . Redshift , z . . . . . . ¯ x H I Fiducial modelwith T S (2 , interpolated)T S >> T CMB (2 , interpolated) Figure 5.
The impact of the common saturated spin temperature approximation ( T S (cid:29) T CMB ) on EoR parameter inference. We showthe recovered 1 and 2D joint marginalised posterior distributions for the faint galaxies model assuming a 1000 hr on sky observationwith the SKA. Red curves correspond to our fiducial constraints which include the spin temperature modelling (marginalising over theEoH parameter) accounting for the IGM spin temperature fluctuations (Figure 3), whereas the purple curves are the recovered constraintswhen ignoring the IGM spin temperature fluctuations (i.e. T S (cid:29) T CMB ). Thick and thin contours correspond to the 68 (1 σ ) and 95 (2 σ )per cent marginalised joint likelihood contours, respectively, and crosses (black vertical dashed lines) denote the input model parameters,defined to be ( ζ , R mfp , log ( T minvir )) = (30, 15, 4.7). Inset: The recovered global evolution of the IGM neutral fraction. The solid blackcurve corresponds to the fiducial input evolution, whereas the error bars correspond to the 2 σ limits on the recovered IGM neutralfraction. Note, all points are interpolated at ∆ z = 0 . δ T b ∆ [ m K ] z = 6 . z = 7 . − k (Mpc − )10 δ T b ∆ [ m K ] z = 8 . − k (Mpc − ) z = 9 . Mock Obs . (Faint galaxies)ML (T S >> T CMB ) ‘ True ’ EoR params . (T S >> T CMB ) Figure 6.
The 21 cm PS corresponding to the maximum likelihood model assuming a saturated spin temperature ( T S (cid:29) T CMB ; redcurve), compared to the fiducial mock 21 cm PS of the faint galaxies model (black curve). The blue curve corresponds to the “true”EoR model parameters, ( ζ , R mfp , log ( T minvir )) = (30, 15, 4.7), but is computed assuming a saturated spin temperature. The grey shadedregion corresponds to the 1 σ observational uncertainty for an assumed 1000hr observation with the SKA, while the error bars denoteour assumed 20 per cent modelling uncertainty on the 21 cm PS. Hatched regions denote k -modes outside of our nominal fitting range.c (cid:13) , 000–000 B. Greig & A. Mesinger R m f p . . . l og ( T m i n v i r ) . . . l og ( L X ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Faint Galaxies 1000hr20% modelling uncertaintySKASKA (T S >> T CMB )
10 15 20 R mfp [Mpc] . . . log ( T minvir [K]) . . . log ( L X ) . . . . log ( N HI ) . . . . E [keV] . . . ↵ X . . . . . . . Redshift , z . . . . . . ¯ x H I Fiducial modelwith T S (2 , interpolated)T S >> T CMB (2 , interpolated) R m f p . . . l og ( T m i n v i r ) . . . l og ( L X ) . . . . E
50 100 150 200 ⇣ . . . ↵ X Bright Galaxies 1000hr20% modelling uncertaintySKASKA (T S >> T CMB )
10 15 20 R mfp [Mpc] . . . log ( T minvir [K]) . . . log ( L X ) . . . . log ( N HI ) . . . . E [keV] . . . ↵ X Figure 7.
The same as Figure 5, except for our bright galaxies model. Crosses (black vertical dashed lines) denote the input modelparameters, defined to be ( ζ , R mfp , log ( T minvir )) = (200, 15, 5.48). δ T b ∆ [ m K ] z = 6 . z = 7 . − k (Mpc − )10 δ T b ∆ [ m K ] z = 8 . − k (Mpc − ) z = 9 . Mock Obs . (Bright galaxies)ML (T S >> T CMB ) ‘ True ’ EoR params . (T S >> T CMB ) Figure 8.
The same as Figure 6, except for our bright galaxies model. The fiducial bright galaxies
EoR parameter set correspondsto, ( ζ , R mfp , log ( T minvir )) = (200, 15, 5.48). EoH. Incorrectly ignoring the EoH and associated IGM spintemperature fluctuations when interpreting a realistic ob-servation could significantly bias the inferred EoR sourcemodel.
Detecting the cosmic 21 cm signal during the EoR and theEoH stands to reveal insights into the formation, growth and evolution of structure in the Universe. However, howdo we interpret the underlying astrophysics once we have adetection? To aid this, we developed , a massivelyparallel Bayesian MCMC analysis tool, which performs full3D reionisation simulations (using ) on the fly, forrecovering EoR astrophysical parameter constraints. Secondgeneration experiments such as HERA and the SKA, alongwith global 21 cm experiments, will be able to measure boththe EoR and the pre-heating of the IGM by X-rays. There- c (cid:13)000
Detecting the cosmic 21 cm signal during the EoR and theEoH stands to reveal insights into the formation, growth and evolution of structure in the Universe. However, howdo we interpret the underlying astrophysics once we have adetection? To aid this, we developed , a massivelyparallel Bayesian MCMC analysis tool, which performs full3D reionisation simulations (using ) on the fly, forrecovering EoR astrophysical parameter constraints. Secondgeneration experiments such as HERA and the SKA, alongwith global 21 cm experiments, will be able to measure boththe EoR and the pre-heating of the IGM by X-rays. There- c (cid:13)000 , 000–000 fore, in order to facilitate simultaneous astrophysical fore-casting of the EoR and EoH and consequently aid in the de-velopment and construction of the data reduction and signalextraction pipelines, in this work we extend intothe epoch of X-ray heating.We demonstrate that both HERA and the SKA will beable to simultaneously constrain the astrophysics of reioni-sation and the X-ray heating. We consider two models de-scribing the mock observation: faint galaxies and brightgalaxies . These are intended to encompass the physicallyplausible region of parameter space provided by extrapolat-ing the faint end of the observed UV luminosity function.Assuming a 1000 hr observation of the 21 cm PS at eightco-evolution redshifts we recover the fractional precision onour six parameter model describing the sources responsiblefor the EoR and the EoH. These parameters include an ion-ising source efficiency ( ζ ), effective photon horizon ( R mfp ),minimum halo mass of ionising sources ( T minvir ), soft-band lu-minosity ( < L X < / SFR), min-imum energy threshold for the attenuation of the X-rays bythe host galaxy ISM ( E ) and the spectral index of the X-ray source SED ( α X ). Additionally assuming a 20 per centmodelling uncertainty in the power spectrum, we recover theparameters of the mock signals with the following percent-age error (1 σ ): • faint galaxies: ζ = 18 (24), R mfp = 16 (16),log ( T minvir ) = 1.4 (2.3), log ( L X < / SFR) = 0.2 (0.3), E = 17 (14) and α X = 88 (73), respectively for the SKA(HERA). • bright galaxies: ζ = 17 (15), R mfp = 16 (12),log ( T minvir ) = 0.4 (0.6), log ( L X < / SFR) = 0.2 (0.2), E = 16 (17) and α X = 80 (79), respectively for the SKA(HERA).Both the SKA and HERA perform equally as well at simul-taneously constraining the astrophysics of reionisation andthe epoch of X-ray heating due to their comparable sensitivi-ties on the large scales (under our assumptions regarding thesurvey strategy) which most strongly discriminate betweenthe astrophysical models.With our expanded framework, we also quantify theimpact of the common assumption of a saturated spin tem-perature, T S (cid:29) T CMB , during the EoR. Our faint galaxies model has a relatively distinct EoH and EoR, for which wewould typically expect T S (cid:29) T CMB to be a reasonable ap-proximation during the bulk of reionisation. Nevertheless,even this modest overlap of epochs leads to biases in ζ and T minvir of up to ∼ σ , and a reduction in the ML by a factorof >
4. The bright galaxies model on the other hand rep-resents the extreme case of EoR and EoH overlap. For thismodel, the recovered constraints are discrepant at > σ ,with the ML under in the saturated limit being a factor of ∼ lower than in the full model. Therefore, adopting thesaturated spin temperature approximation can significantlybias inferences on EoR parameters. ACKNOWLEDGEMENTS
We thank Adrian Liu and Nicholas Kern for comments ona draft version of this work. This work was supported bythe European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 638809 – AIDA – PI: Mesinger).
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In Figures A1 and A2 we present the mock 21 cm PS (blackcurves) for both of our EoR source models faint galaxies and bright galaxies , respectively. We show the 21 cm PSfor each of the eight co-evolution redshifts we use in thiswork to achieve our astrophysical constraints, covering boththe EoR and the EoH. For each, we provide the correspond-ing total noise PS for HERA (blue) and SKA (red) for ourassumed 1000hr observational strategy (see Section 2.3 formore details). Finally, the black dashed curve is the 20 percent modelling uncertainty, which we have applied to themock 21 cm PS (it is applied to the sampled 21 cm PS notthe mock observation in ) to provide reference tothe dominate source of error at any k -mode.It is interesting to note that while most of the EoH pa-rameters are tightly constrained for both mock observations,these constraints appear to be arise solely from a relativelynarrow range in redshifts, which pick up the rise and fallof the large scale power. This emphasises the importance ofobserving the cosmic 21 cm signal in an extended frequencyrange, allowing us to pick up the major milestones in the sig-nal which drive strong astrophysical parameter constraints. c (cid:13) , 000–000 B. Greig & A. Mesinger − δ T b ∆ [ m K ] z = 6 . z = 7 . z = 8 . − δ T b ∆ [ m K ] z = 9 . z = 10 . − k (Mpc − ) z = 13 . − k (Mpc − )10 − δ T b ∆ [ m K ] z = 15 . − k (Mpc − ) z = 17 .
11 Mock Obs . (Faint galaxies)SKA 1000hrHERA 1000hr20% of Mock Obs . (modelling uncertainty) Figure A1.
The 21 cm PS (black, solid curve) of the faint galaxies (Section 3) mock observation at all eight co-eval redshifts usedin this work. Red and blue solid curves correspond to the noise curves (thermal + sample variance) for an assumed 1000 hr observationwith the SKA and HERA, respectively (see Section 2.3). The black dashed curve is 20 per cent of the mock 21 cm PS, highlighting thescale of the assumed modelling uncertainty used in this work. Representing it in this format highlights the dominant source of error atany redshift, or k -mode. Hatched regions correspond to k -modes beyond the 21 cm PS fitting region used in . − δ T b ∆ [ m K ] z = 6 . z = 7 . z = 8 . − δ T b ∆ [ m K ] z = 9 . z = 10 . − k (Mpc − ) z = 13 . − k (Mpc − )10 − δ T b ∆ [ m K ] z = 15 . − k (Mpc − ) z = 17 .
11 Mock Obs . (Bright galaxies)SKA 1000hrHERA 1000hr20% of Mock Obs . (modelling uncertainty) Figure A2.
Same as Figure A1 except now for the bright galaxies model (see Section 3).c (cid:13)000