High- 𝑧 LAEs

PROBING COSMIC REIONIZATION AND MOLECULAR GAS GROWTH WITH TIME

2.3 Models

2.3.1 Tracers of Large-Scale Structure

2.3.1.4 High- 𝑧 LAEs

metallicity 𝑍 ∼ 0.05𝑍_⊙ during the EoR following Sun & Furlanetto (2016). The SFRD informed by UV data can then be expressed as

¤ 𝜌_∗(𝑧) =

∫ 𝑀_max 𝑀_min

d𝑀 d𝑛 d𝑀

𝑀¤_∗(𝑀 , 𝑧), (2.17) where we choose𝑀_min =10⁸𝑀_⊙, corresponding to the minimum halo mass for star formation implied by the atomic cooling threshold, and 𝑀_max =10¹⁵𝑀_⊙. As will be discussed in Section 2.3.3, the SFR, 𝑀¤_∗, as a function of halo mass and redshift can be specified by the star formation efficiency (SFE) and the rate at which halo mass grows. The shapes of both [C ii] luminosity function and power spectrum are therefore affected by the halo mass dependence of these factors. Since the reionization history is irrelevant to star formation after reionization was complete, we do not match the SFRD inferred from UV observations to that obtained by extrapolating the CIB model to𝑧 ≳ 5, which is itself highly uncertain.

The spatial fluctuations of [C ii] emission can be described by the [C ii] auto- correlation power spectrum

𝑃_{C II}(𝑘 , 𝑧) = 𝐼¯²

C II(𝑧)𝑏¯²

C II(𝑧)𝑃_𝛿𝛿(𝑘 , 𝑧) +𝑃^shot

C II(𝑧). (2.18) The mean [C ii] intensity is

¯

𝐼_{C II}(𝑧) =

∫ 𝑀_max 𝑀_min

d𝑀 d𝑛 d𝑀

𝐿_{C II}[𝐿_UV(𝑀 , 𝑧)]

4𝜋 𝐷²

𝐿

𝑦(𝑧)𝐷²

𝐴 , (2.19)

and ¯𝑏_{C II}(𝑧) is the [C ii] luminosity-averaged halo bias factor defined as

¯

𝑏_{C II}(𝑧) =

∫^𝑀_max

𝑀_min d𝑀(d𝑛/d𝑀)𝑏(𝑀 , 𝑧)𝐿_{C II}[𝐿_UV(𝑀)]

∫ ^𝑀_max

𝑀_min d𝑀(d𝑛/d𝑀)𝐿_{C II}[𝐿_UV(𝑀)]

. (2.20)

The shot-noise term is 𝑃^shot

C II(𝑧) =

∫ 𝑀_max 𝑀_min

d𝑀 d𝑛 d𝑀

(

𝐿_{C II}[𝐿_UV(𝑀 , 𝑧)]

4𝜋 𝐷²

𝐿

𝑦(𝑧)𝐷²

𝐴

)2

. (2.21) Similar to the CO case, we use the scaling factors given in Sun et al. (2019) to account for the effects of𝜎_{C II}on the [C ii] power spectrum.

Ly𝛼emission onto the halo catalogs from the Simulated Infrared Dusty Extragalactic Sky (SIDES, Béthermin et al. 2017) simulation. Analytic models have been widely used to investigate physical properties of high-redshift LAEs (e.g., Samui et al. 2009;

Jose et al. 2013; Mas-Ribas & Dijkstra 2016; Mas-Ribas et al. 2017a,b; Sarkar &

Samui 2019). Here, to model Ly𝛼luminosity of LAEs, we assume that Ly𝛼photons are solely produced by recombinations under ionization equilibrium. As a result, for a given halo mass and redshift, it can be approximately related to the SFR by

𝐿_Ly𝛼 =

𝑓_𝛾𝑀¤_∗(𝑀 , 𝑧)/𝜂

𝑚_p/(1−𝑌) (1− 𝑓_esc)𝑓^Ly

𝛼

esc 𝑓_Ly𝛼𝐸_Ly𝛼 , (2.22) where𝑚_pis the mass of hydrogen atom. The ionizing photon produced per stellar baryon 𝑓_𝛾, the escape fraction of ionizing photons 𝑓_esc, the fraction of recombinations ending up as Ly𝛼 emission 𝑓_Ly𝛼 and the helium mass fraction𝑌 are taken to be 𝑓_𝛾 = 4000 (typical for low-metallicity Pop II stars with a Salpeter initial mass function), 𝑓_esc =0.1, 𝑓_Ly𝛼=0.67 and𝑌 =0.24, respectively. The factors(1− 𝑓_esc) and 𝑓^Ly

𝛼

esc account for the fraction of ionizing photons failing to escape (and thus leading to recombinations) and the fraction of Ly𝛼photons emitted that eventually reach the observer. Because the production of Ly𝛼emission is also subject to local dust extinction, a scale factor log𝜂 = ⟨𝐴_UV⟩/2.5, whose value is specified by the dust correction formalism described in Appendix 2.9, is included here to obtain the obscured star formation rate. As in cases of [C ii] and CO emission, we consider a log-normal scatter 𝜎_Ly𝛼 around the mean 𝐿_Ly𝛼–𝑀 relation above, which makes the observed LAE luminosity function a convolution of the intrinsic function with the log-normal distribution. In our model, we take 𝑓^Ly

𝛼

esc =0.6 and𝜎_Ly𝛼 =0.3 dex, consistent with the observationally determined Ly𝛼escape fraction (Jose et al. 2013) and the dispersion about the luminosity–halo mass relation (More et al. 2009), to obtain reasonably good fits to the luminosity functions measured by Konno et al.

(2018), as shown in Figure 2.5. The luminosity–halo mass relation is then used to paint both [C ii] and Ly𝛼emission onto dark matter halos catalogued to obtain maps of LAE spatial distribution and [C ii] intensity fluctuations.

The limiting magnitude𝑚^AB

lim of LAE surveys can be related to the line luminosity 𝐿_Ly𝛼of LAEs by𝐿_Ly𝛼 =4𝜋 𝐷²

𝐿𝐹_Ly𝛼and

𝐹_Ly𝛼 =3×10⁻⁵× 10^{(8.90−𝑚ABlim)/2.5}Δ𝜆 𝜆²

erg s⁻¹cm⁻²,

where we take Δ𝜆 = 131 Å and 𝜆 = 8170 Å for 𝑧 = 5.7 and Δ𝜆 = 120 Å and 𝜆=9210 Å for𝑧 =6.6 as specified in Konno et al. (2018). Meanwhile, to generate

42.6 42.8 43.0 43.2 43.4 43.6 43.8 44.0

log(L Ly /[ergs ¹ ])

7 6 5 4 3

LA E [M pc 3 de x 1 ] ^{z = 5.7}

z = 6.6

Konno+18 This work

Figure 2.5: A comparison between modeled and observed LAE luminosity func- tions. The luminosity functions at𝑧 =5.7 and 𝑧 = 6.6 predicted by our analytical model (solid curves) are compared against the observed ones taken from Konno et al. (2018) (data points and dotted curves).

mock LAE catalogs we consider limiting magnitudes of the planned, ultra-deep (UD) survey of the HSC, namely 𝑚^AB

lim = 26.5 and 26.2 at 𝑧 = 5.7 and 6.6, respectively, which correspond to minimum Ly𝛼 luminosities of log(𝐿_Ly𝛼/erg s⁻¹) = 42.3 and 42.4. For such survey depths, we predict the comoving number density of LAEs to be 𝑛^𝑧=5.7

LAE = 1.4 ×10⁻³Mpc⁻³ and 𝑛^𝑧=6.6

LAE = 5.7× 10⁻⁴Mpc⁻³ by integrating the LAE luminosity functions our model implies. As a result, no more than a few LAEs are expected to exist in the survey volume of TIME due to its limited survey area of about 0.01 deg². One caveat to our LAE model is that we ignore the impact of patchy reionization on the spatial distribution of LAEs through the Ly𝛼 transmission fraction, which is affected by, and thus informs, the growth of ionized bubbles around LAEs (e.g., Santos et al. 2016). We note, though, that for estimating the [C ii]–LAE cross-correlation TIME will measure, our simple model calibrated against the LAE luminosity functions from the SILVERRUSH survey should suffice. In fact, thanks to the large survey areas covered (14 and 21 deg²at 𝑧 = 5.7 and 6.6, respectively), the patchiness effect is already captured, at least in part, by the observed LAE statistics. To fully address the suppression on the LAE number density due to patchy reionization, both numerical (e.g., McQuinn et al.

2007) and semi-analytical (e.g., Dayal et al. 2008) methods can be applied. We

will explore how such effects may be probed by the [C ii]–LAE cross-correlation in future work.

Therefore, we consider the measurement of two-point correlation function, instead of power spectrum, to maximally extract the information about large-scale correlation between distributions of LAEs and [C ii] intensity. In general, for a given normalized selection function N (𝑧), the angular correlation function is related to the spatial correlation function by the Limber equation

𝜔(𝜃 , 𝑧) =

∫

d𝑧^′N (𝑧^′)

∫

d𝑧^′′N (𝑧^′′)𝜉[𝑟(𝜃 , 𝑧^′, 𝑧^′′), 𝑧] , (2.23) where we approximateN (𝑧)by top-hat functions over𝑧 =5.67–5.77 and𝑧=6.52–

6.63 corresponding to the bandwidths of narrow-band filters used in the SILVER- RUSH survey (Ouchi et al. 2018; Konno et al. 2018). Specifically, the angular cross-correlation function between the [C ii] intensity map measured by TIME and the LAE distribution is (in units of Jy/sr)

𝜔_{C II×LAE}(𝜃) ≡

𝑁(𝜃)

Í

𝑖

Δ𝐼^𝑖

C II(𝜃)

𝑁(𝜃) ≈ 𝑏_LAE𝑏¯_{C II}𝐼¯_{C II}𝜔_DM(𝜃), (2.24) where for the bin𝜃,Δ𝐼^𝑖

C II(𝜃) = 𝐼^𝑖

C II(𝜃) −𝐼¯_{C II}denotes the [C ii] intensity fluctuation at pixel𝑖, whereas 𝑁(𝜃) denotes the total number of LAE-pixel pairs. Determined from the LAE distributions generated with our semi-analytical approach, the LAE bias 𝑏_LAE ≈ 6 at both𝑧 = 5.7 and 6.6 is consistent with the upper limits on𝑏_LAE estimated from the SILVERRUSH survey. The approximation is valid on large scales where the clustering of LAEs and [C ii] emission are linearly biased tracers of the dark matter density field. The dark matter angular correlation function𝜔_DM is derived using Equation (2.23) from the spatial correlation function

𝜉_DM(𝑟 , 𝑧) = 1 2𝜋²

∫

d𝑘 𝑘²𝑃_𝛿𝛿(𝑘 , 𝑧)sin(𝑘 𝑟) 𝑘 𝑟

. (2.25)