Linear bias forecasts for emission line cosmological surveys

TL;DR

This paper forecasts the linear galaxy bias for high-redshift H-alpha emitters in Euclid-like and WFIRST-like surveys using halo occupation models, finding a consistent bias evolution with redshift and agreement with existing measurements.

Contribution

It introduces a physically motivated HOD-based simulation approach to forecast galaxy bias for upcoming emission line surveys, incorporating dust attenuation calibration.

Findings

Bias increases with redshift, following approximately b(z) = 0.7z + 0.7.

Forecasted biases are consistent with measurements from the HiZELS survey.

Euclid-like and WFIRST-like surveys produce broadly similar bias estimates.

Abstract

We forecast the linear bias for H${\rm \alpha}$-emitting galaxies at high redshift. To simulate a Euclid-like and a WFIRST-like survey, we place galaxies into a large-volume dark matter halo lightcone by sampling a library of luminosity-dependent halo occupation distributions (HODs), which is constructed using a physically motivated galaxy formation model. We calibrate the dust attenuation in the lightcones such that they are able to reproduce the H{\alpha} luminosity function or the H{\alpha} cumulative number counts. The angle-averaged galaxy correlation function is computed for each survey in redshift slices of width $\Delta z = 0.2$. In each redshift bin the linear bias can be fitted with a single, scale-independent value that increases with increasing redshift. Fitting for the evolution of linear bias with redshift, we find that our Euclid-like and WFIRST-like surveys are both consistent within error with the relation $b(z) = 0.7z + 0.7$. Our bias forecasts are consistent with bias measurements from the HiZELS survey. We find that the Euclid-like and WFIRST-like surveys yield linear biases that are broadly consistent within error, most likely due to the HOD for the WFIRST-like survey having a steeper power-law slope towards larger halo masses.

Tables

Table 1. Table 1: Specifications for the lightcone catalogues considered in this work. Note that the Euclid-like lightcone is constructed by applying a completeness limit to the MXXL-15K. The WFIRST-like lightcone is derived from the MXXL-2K lightcone in a similar manner. The sky areas for the MXXL-15K and the Euclid-like lightcones have been rounded to the nearest integer – in reality these lightcones each have an area of ( 3 / 8 ) × 4 ​ π 3 8 4 𝜋 (3/8)\times 4\pi steradians.

Catalogue	Flux Limit	Right	Declination	Area	Redshift	Completeness
Name	$\left[\mathrm{erg}{s}^{-1}{\mathrm{cm}}^{-2}\right]$	Ascension		$\left[{\mathrm{deg}}^2\right]$	Range
MXXL-15K	$2\times {10}^{-16}$	$0^{\circ }⩽\alpha ⩽{270}^{\circ }$	$0^{\circ }⩽\delta ⩽{90}^{\circ }$	15470		1.0
Euclid-like	$2\times {10}^{-16}$	$0^{\circ }⩽\alpha ⩽{270}^{\circ }$	$0^{\circ }⩽\delta ⩽{90}^{\circ }$	15470		0.45
MXXL-2K	$1\times {10}^{-16}$	${50}^{\circ }⩽\alpha ⩽{100}^{\circ }$	$-{20}^{\circ }⩽\delta ⩽{20}^{\circ }$	2000		1.0
WFIRST-like	$1\times {10}^{-16}$	${50}^{\circ }⩽\alpha ⩽{100}^{\circ }$	$-{20}^{\circ }⩽\delta ⩽{20}^{\circ }$	2000		0.7

Table 2. Table 2: Catalogue specifications and linear bias fits for our Euclid-like catalogues, measured in redshift-space (upper table) and in real-space (lower table). The properties shown are: the effective redshift of the galaxies in the redshift slice, z eff subscript 𝑧 eff z_{\rm eff} ; the mean number of galaxies used to compute the correlation function, N ¯ gal subscript ¯ 𝑁 gal \bar{N}_{\rm gal} ; the mean value for the ratio of randoms to galaxies, N ¯ ran / N ¯ gal subscript ¯ 𝑁 ran subscript ¯ 𝑁 gal \bar{N}_{\rm ran}/\bar{N}_{\rm gal} ; the mean and standard deviation of the halo mass, M h subscript 𝑀 ℎ M_{h} , for that galaxy sample; and the linear bias fit. The columns show the results for each redshift slice considered. Each table shows the equivalent results when adopting a WISP-calibrated lightcone or a HiZELS-calibrated lightcone.

Calibration	Property
Euclid-like (redshift-space)
WISP	$z_{\mathrm{eff}}$	0.991	1.19	1.38	1.59	1.79
	${\overline{N}}_{\mathrm{gal}}$	13175397	8688180	5045749	2912673	1748220
	${\overline{N}}_{\mathrm{ran}}/{\overline{N}}_{\mathrm{gal}}$	10	10	14	25	30
	${\log }_{10}\left(M_h/h^{-1}{M}_{\odot }\right)$	$11.8\pm \mathrm{\hspace{0.17em}0.4}$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$
	$b_{\mathrm{lin}}\pm \delta b_{\mathrm{lin}}$	$1.40\pm \mathrm{\hspace{0.17em}0.03}$	$1.51\pm \mathrm{\hspace{0.17em}0.05}$	$1.70\pm \mathrm{\hspace{0.17em}0.04}$	$1.82\pm \mathrm{\hspace{0.17em}0.02}$	$1.96\pm \mathrm{\hspace{0.17em}0.09}$
HiZELS	$z_{\mathrm{eff}}$	0.988	1.19	1.39	1.59	1.79
	${\overline{N}}_{\mathrm{gal}}$	7785058	4642807	2635798	1687841	1215424
	${\overline{N}}_{\mathrm{ran}}/{\overline{N}}_{\mathrm{gal}}$	10	10	15	25	29
	${\log }_{10}\left(M_h/h^{-1}{M}_{\odot }\right)$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$
	$b_{\mathrm{lin}}\pm \delta b_{\mathrm{lin}}$	$1.42\pm \mathrm{\hspace{0.17em}0.03}$	$1.55\pm \mathrm{\hspace{0.17em}0.05}$	$1.69\pm \mathrm{\hspace{0.17em}0.04}$	$1.88\pm \mathrm{\hspace{0.17em}0.05}$	$1.9\pm \mathrm{\hspace{0.17em}0.1}$

Table 3. Table 6: Catalogue specifications and linear bias fits for the MXXL-15K lightcone, measured in redshift-space (upper table) and in real-space (lower table). In each instance, the properties shown are: the effective redshift of the slice, z eff subscript 𝑧 eff z_{\rm eff} ; the mean number of galaxies used to compute the correlation function, N ¯ gal subscript ¯ 𝑁 gal \bar{N}_{\rm gal} ; the mean value for the ratio of randoms to galaxies, N ¯ ran / N ¯ gal subscript ¯ 𝑁 ran subscript ¯ 𝑁 gal \bar{N}_{\rm ran}/\bar{N}_{\rm gal} ; the mean and standard deviation of the halo mass, M h subscript 𝑀 ℎ M_{h} , for that galaxy sample; and the linear bias fit. The columns show the results for each redshift slice considered. Each table shows the bias fits when adopting a WISP-calibrated version of the lightcone and a HiZELS-calibrated version of the lightcone.

Calibration	Property
MXXL-15K (redshift-space)
WISP	$z_{\mathrm{eff}}$	0.991	1.19	1.38	1.59	1.79
	${\overline{N}}_{\mathrm{gal}}$	29281758	19305864	11214261	6472414	3886909
	${\overline{N}}_{\mathrm{ran}}/{\overline{N}}_{\mathrm{gal}}$	10	10	10	15	25
	${\log }_{10}\left(M_h/h^{-1}{M}_{\odot }\right)$	$11.8\pm \mathrm{\hspace{0.17em}0.4}$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$
	$b_{\mathrm{lin}}\pm \delta b_{\mathrm{lin}}$	$1.40\pm \mathrm{\hspace{0.17em}0.02}$	$1.52\pm \mathrm{\hspace{0.17em}0.05}$	$1.70\pm \mathrm{\hspace{0.17em}0.03}$	$1.83\pm \mathrm{\hspace{0.17em}0.02}$	$1.95\pm \mathrm{\hspace{0.17em}0.09}$
HiZELS	$z_{\mathrm{eff}}$	0.988	1.19	1.39	1.59	1.79
	${\overline{N}}_{\mathrm{gal}}$	7785058	4642807	2635798	1687841	1215424
	${\overline{N}}_{\mathrm{ran}}/{\overline{N}}_{\mathrm{gal}}$	10	10	15	25	29
	${\log }_{10}\left(M_h/h^{-1}{M}_{\odot }\right)$	$11.8\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$	$11.9\pm \mathrm{\hspace{0.17em}0.3}$
	$b_{\mathrm{lin}}\pm \delta b_{\mathrm{lin}}$	$1.42\pm \mathrm{\hspace{0.17em}0.03}$	$1.55\pm \mathrm{\hspace{0.17em}0.05}$	$1.69\pm \mathrm{\hspace{0.17em}0.04}$	$1.88\pm \mathrm{\hspace{0.17em}0.05}$	$1.9\pm \mathrm{\hspace{0.17em}0.1}$

Equations

$${\xi }_{\mathrm{gal}}(r)=b^2(r){\xi }_{\mathrm{DM}}(r),$$

$$\frac{⟨N_{\mathrm{cen}}\left(>L_{\mathrm{cen}}|M_{200},z\right)⟩}{⟨N_{\mathrm{cen}}\left(>L_{\min }|M_{200},z\right)⟩}=x_2,$$

$${\sigma }^2\left(M_{200}\right)=\frac{G{M}_{200}}{2R_{200}},$$

$${\log }_{10}\left(L_{\mathrm{H}\alpha }^{\mathrm{att}}\right)={\log }_{10}\left(L_{\mathrm{H}\alpha }^0\right)-0.4A_{\mathrm{H}\alpha }.$$

$${\xi }_{\mathrm{LS}}(r)=\frac{DD(r)-2DR(r)+RR(r)}{RR(r)},$$

$$\delta b_{\mathrm{lin}}=\sqrt{\frac{1}{N}\sum _{r⩽r_{\mathrm{cut}}}{\left(b(r)-b_{\mathrm{lin}}\right)}^2},$$

$${\log }_{10}\left(-{\log }_{10}\left(N_{\mathrm{cen}}\right)\right)\propto m_{\mathrm{fit}}{\log }_{10}\left(M_{200}\right)$$

Full text

††pagerange: Linear bias forecasts for emission line cosmological surveys–LABEL:lastpage††pubyear: 2024

Linear bias forecasts

1 Introduction

Probing the nature of the driving force behind the observed accelerated expansion of the Universe continues to be one of the major goals of modern cosmology. Most of the observational evidence gathered to date is consistent with the theory that the expansion is the result of a mysterious phenomenon known as dark energy, although existing observations lack the statistical precision to allow alternative theories to be differentiated and conclusively ruled out. Up and coming cosmological missions aim to make very precise measurements of different cosmological probes of the expansion history of the Universe and the growth rate of cosmic large-scale structure in order to distinguish between the competing theories (e.g. Albrecht et al., 2006; Guzzo et al., 2008; Wang, 2008a, b).

For missions such as the ESA-led Euclid mission (Laureijs et al., 2011) and the NASA-led Wide Field Infrared Survey Telescope mission (WFIRST, Dressler et al., 2012; Green et al., 2012; Spergel et al., 2015), dark energy will be probed using a spectroscopic galaxy redshift survey, which will measure the galaxy clustering and redshift-space distortions of the galaxy distribution, and a photometric survey, which will be used to measure weak gravitational lensing shear of the galaxies. Specifically for the galaxy clustering measurements, these missions will measure the baryon acoustic oscillations (BAO) in the galaxy population, which can be used as a standard ruler to probe the expansion history of the Universe (Blake & Glazebrook, 2003; Seo & Eisenstein, 2003), and redshift-space distortions, which are sensitive to the growth rate of the large-scale structure (Kaiser, 1987; Song & Percival, 2009).

The spectroscopic redshift surveys of the Euclid and WFIRST missions will target many tens of millions of emission line galaxies (ELGs), which will be identified using near-IR grism spectroscopy. The primary targets for the spectroscopic galaxy redshift surveys are ${\rm H\alpha}$-emitting galaxies between redshifts $0.9\lesssim z\lesssim 2$. The Euclid wide-area survey will cover 15,000 square degrees to an ${\rm H\alpha}$ line flux limit of $2\times 10^{-16}\,{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$ and is expected to detect ${\rm H\alpha}$-emitting galaxies in the redshift range $0.9\lesssim z\lesssim 1.8$. In a similar fashion, the WFIRST High Latitude Survey (HLS) will cover $\sim 2200$ square degrees to a fainter ${\rm H\alpha}$ line flux limit of $1\times 10^{-16}\,{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$ and is expected to detect ${\rm H\alpha}$-emitting galaxies in a redshift range of $1\lesssim z\lesssim 2$. Due to their observing strategy, both missions are expected to detect additional emission lines belonging to galaxies outside of these redshift ranges. For instance, both Euclid and WFIRST expect to detect [OIII] emission from galaxies at $z\gtrsim 2$. In this work we focus on the redshift surveys of ${\rm H\alpha}$-emitting galaxies. The Euclid and WFIRST dark energy missions are designed to be highly complementary: the large area and shallower depth of Euclid is ideal for optimising statistical precision in cosmological measurements, whilst the smaller area and greater depth of WFIRST is ideal for understanding the systematics that are expected to dominate over measurement uncertainties.

One of the systematics that must be taken into account is galaxy bias, which describes how galaxies trace the underlying dark matter distribution. Galaxy formation does not occur uniformly in space, but occurs primarily in the peaks of the matter density field. Galaxies are therefore biased tracers of the density field, sampling only the over-dense regions (e.g. Kaiser 1984; Bardeen et al. 1986; see Desjacques et al. 2018 for a recent review). The bias, $b$, of a population of galaxies is defined according to,

[TABLE]

where $\xi_{\rm gal}(r)$ is the galaxy correlation function and $\xi_{\rm DM}(r)$ is the dark matter correlation function, expressed as a function of spatial separation $r$. Observational evidence indicates that different galaxy populations, identified for example by luminosity, stellar mass or morphological type, display different clustering amplitudes (e.g. Davis & Geller, 1976; Dressler, 1980; Guzzo et al., 1997; Norberg et al., 2001, 2002; Zehavi et al., 2002, 2011; Guo et al., 2013; Skibba et al., 2014; Kim et al., 2015; McCracken et al., 2015; Hatfield et al., 2016; Favole et al., 2017; Law-Smith & Eisenstein, 2017; Cochrane et al., 2017, 2018; Durkalec et al., 2018). This has been further confirmed with simulations of galaxy formation (e.g. Kauffmann et al., 1997; Benson et al., 2000a; Orsi et al., 2010; Contreras et al., 2013). As such, different galaxy populations are expected to display different bias values. This can be understood in terms of the halo model, which predicts that different galaxy populations commonly reside in dark matter haloes of different mass. The hierarchical nature of structure formation means that the bias of galaxies will additionally vary with redshift (e.g. Fry, 1996; Tegmark & Peebles, 1998; Hui & Parfrey, 2008; Basilakos et al., 2012; Mirbabayi et al., 2015). In the non-linear regime the bias is typically scale-dependent, but in the linear regime, at scales $r\gtrsim 50\,h^{-1}{\rm Mpc}$, the bias approaches a constant value (Mann et al., 1998; Peacock & Smith, 2000; Seljak, 2000; Benson et al., 2000b; Blanton et al., 2000; Berlind & Weinberg, 2002; Verde et al., 2002). We refer to the constant bias on large-scales as linear bias, $b(r)\sim b_{\rm lin}$.

To date, many of the clustering analyses of emission line galaxies at high redshift have been carried out using data from the High-z Emission Line Survey (HiZELS, Geach et al., 2008; Sobral et al., 2009, 2013) – a ground-based, narrow band survey capable of detecting emission line galaxies up to $z\sim 9$. Using a sample of $\sim 700$ ${\rm H\alpha}$-emitters at $z=0.84$, Sobral et al. (2010) observed that the clustering strength of ${\rm H\alpha}$-emitters is strongly dependent on the ${\rm H\alpha}$ luminosity. Geach et al. (2012) examined the clustering of 370 ${\rm H\alpha}$-emitters at $z=2.23$ to measure a bias of $2.4^{+0.1}_{-0.2}$ and estimate that the dark matter haloes hosting ${\rm H\alpha}$-emitters have a typical mass of $\log_{10}\left(M_{h}/h^{-1}{\rm M_{\odot}}\right)=11.7\pm 0.1$. Geach et al. (2012) additionally construct a parametric model for the halo occupation distribution (HOD) of their ${\rm H\alpha}$-emitters and attempt to fit this model to their data. Due to the size of their galaxy sample, Geach et al. (2012) were unable to place strong constraints on the HOD parameters. However, despite this, they were able to constrain the effective halo mass of the ${\rm H\alpha}$-emitters to be $\log_{10}\left(M_{\rm eff}/h^{-1}{\rm M_{\odot}}\right)=12.1^{+0.1}_{-0.2}$. Recently Cochrane et al. (2017) built upon the previous HiZELS results by carrying out an extensive clustering analysis of $\sim 3000$ ${\rm H\alpha}$-emitters at $z=0.8$, $\sim 450$ emitters at $z=1.47$ and $\sim 730$ emitters at $z=2.23$, split into both differential and cumulative luminosity bins. Using Markov Chain Monte Carlo sampling, Cochrane et al. (2017) fit a parametrised HOD to their clustering results to estimate the bias and halo mass of their ${\rm H\alpha}$-emitters. They find that the bias of the emitters increases both with redshift and with luminosity, and that the emitters are hosted by haloes with mass typically $\sim 10^{12}h^{-1}{\rm M_{\odot}}$.

In this work we use simulated galaxy catalogues to forecast the linear bias as a function of redshift for ${\rm H\alpha}$-emitting galaxies in a Euclid-like redshift survey and a WFIRST-like redshift survey. Forecasts of the galaxy bias are important for two reasons. First, understanding the bias of the ${\rm H\alpha}$-emitting galaxies will be crucial for the Euclid and WFIRST missions if we are to correctly infer the dark matter clustering from the galaxy clustering measurements, and thereby accurately estimate cosmological parameters (Gaztañaga et al., 2012; Salvador et al., 2019). Clerkin et al. (2015) provide a comparison of several bias evolution prescriptions and the impact that choice of prescription has on cosmological parameter constraints. Second, the figure-of-merit of a cosmological mission, which indicates the ability of a mission to successfully measure dark energy, is sensitive to the understanding of galaxy bias. As such forecasting of the bias is vital for helping optimise the observational strategy of a cosmological mission. For example, Orsi et al. (2010) used galaxy mock catalogues built using the Galform semi-analytical galaxy formation model to compare the effective volumes that could be probed by an ${\rm H\alpha}$-selected galaxy survey and an H-band selected galaxy survey. They determine that for an ${\rm H\alpha}$-selected galaxy survey to probe an effective volume comparable to that of an ${\rm H_{AB}}=22$ slit-based survey, the ${\rm H\alpha}$-selected survey would need to probe down to a flux depth of $10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$. Orsi et al. (2010) additionally provide some forecasts for the luminosity dependence of the bias of ${\rm H\alpha}$-emitters between $z=0$ and $z=2$, though due to the limited volume of the underlying N-body simulation they are unable to estimate the bias well into the linear regime ($r\gtrsim 50\,h^{-1}{\rm Mpc}$).

The structure of the paper is as follows: in Section 2 we give details of how we construct our lightcone galaxy catalogues; in Section 3 we discuss how we calibrate our lightcone catalogues to match the luminosity function and cumulative number counts of ${\rm H\alpha}$-emitters; in Section 4 we present our linear bias forecasts; in Section 5 we compare the linear bias values for the Euclid-like and WFIRST-like surveys; and in Section 6 we summarise and conclude.

2 Mock Catalogue Construction

For this work we use a pair of mock catalogues built using the methodology of Smith et al. (2017), who populate the dark matter haloes of the Millennium XXL simulation (MXXL, Angulo et al., 2012) using a set of luminosity-dependent HODs. In this section we provide further details on the construction of these mock catalogues.

2.1 The Millennium XXL Simulation

In this work we use the Millennium XXL simulation (MXXL, Angulo et al., 2012), which uses $6720^{3}=303,464,448,000$ particles of mass $6.17\times 10^{9}h^{-1}{\rm M_{\odot}}$ to follow the non-linear, hierarchical growth of dark matter structure within a cubic volume of $3\,\,h^{-1}{\rm Gpc}$. The cosmology adopted in the MXXL simulation is a ${\rm\Lambda}$CDM cosmology with parameters identical to the cosmological parameters adopted for the Millennium Simulation (Springel et al., 2005). These parameters are: a baryon matter density $\Omega_{{\rm b}}=0.045$; a total matter density $\Omega_{{\rm m}}=\Omega_{{\rm b}}+\Omega_{{\rm CDM}}=0.25$, where $\Omega_{{\rm CDM}}$ is the density in cold dark matter; a dark energy density $\Omega_{{\rm\Lambda}}=0.75$; a Hubble constant $H_{0}=100h\,{\rm km\,s}^{-1}{\,\rm Mpc}^{-1}$, where $h=0.73$; a primordial scalar spectral index $n_{\rm s}=1$ and a fluctuation amplitude $\sigma_{8}=0.9$. This cosmological parameter set is consistent with the first year results from the Wilkinson Microwave Anisotropy Probe (Spergel et al., 2003). We note that this cosmology is discrepant with the latest cosmological parameters measured from the Planck mission (Planck Collaboration et al., 2018), but argue that this should not be of major concern since our ignorance in the modelling of galaxy formation will dominate over any discrepancies in cosmology. In Fig. 1 we show, for reference, the non-linear matter power spectrum and corresponding real-space correlation function as computed for the MXXL cosmology using the open source Nbodykit111https://github.com/bccp/nbodykit python package (Hand et al., 2018). The BAO peak is clearly visible and, given the paremeters of the simulation, is located at approximately $110\,h^{-1}{\rm Mpc}$.

Particle positions and velocities are stored in 63 snapshots at fixed epochs spaced approximately logarithmically in expansion factor between $z=20$ and $z=0$, as was done for the Millennium Simulation (see Lemson & Virgo Consortium, 2006). Halo merger trees have also been constructed using the Subfind structure finding algorithm (Springel et al., 2001). Smith et al. (2017) additionally constructed friends-of-friends (FOF) merger trees using the most massive subhalo in each FOF group. Further details regarding the MXXL simulation can be found in Angulo et al. (2012).

Smith et al. (2017) constructed a lightcone catalogue of dark matter haloes from the MXXL simulation using the methodology described in Merson et al. (2013). An observer is first placed in the simulation box at a randomly chosen position and the simulation box is replicated about the observer so as to generate a sufficiently large volume reaching out to the user-specified redshift. Using the positions of haloes and their associated descendants, one can determine the pair of snapshots between which any particular merger tree intercepted the past lightcone of the observer (if it intercepts the lightcone at all). Once the pair of snapshots has been identified, a binary search algorithm is used to interpolate along the trajectory of the halo to compute the exact location at which the halo crosses the lightcone. Interpolation of the halo trajectories between pairs of snapshots is done using cubic interpolation, with the initial position and velocity of a halo and its corresponding descendant used to set the boundary conditions. The lightcone that Smith et al. (2017) built is all-sky and extends out to redshift $z=2.2$.

2.2 Populating the MXXL dark matter haloes

Ideally we would like to populate the haloes of the MXXL simulation with galaxies by running a physically motivated galaxy formation model directly on the MXXL halo merger trees. Unfortunately the mass resolution of the MXXL simulation is too poor, with a minimum resolved halo mass of $1.22\times 10^{11}h^{-1}{\rm M_{\odot}}$, such that running a semi-analytical galaxy formation model on the MXXL trees directly would yield an unrealistic galaxy population (e.g. see Angulo et al., 2014). We note, however, that this minimum halo mass is just under an order of magnitude smaller than the typical halo mass for ${\rm H\alpha}$-emitters estimated by Geach et al. (2012).

To populate the MXXL haloes we therefore use the pipeline presented in Smith et al. (2017). This pipeline follows the methodology of Skibba et al. (2006), whereby haloes are populated with galaxies of different luminosities by random sampling of probability distribution functions created from sets of luminosity-dependent halo occupation distributions (HODs) that evolve with redshift.

Smith et al. (2017) originally developed this pipeline for the purpose of constructing an $r$-band selected galaxy catalogue to mimic the Dark Energy Spectroscopic Instrument Bright Galaxy Survey (DESI BGS, DESI Collaboration et al., 2016). For broad-band photometrically-selected samples such as this, the HOD is well understood and can be parametrised easily, (e.g. Zehavi et al., 2011; Zhai et al., 2017). However, the HOD of ${\rm H\alpha}$-emitting galaxies is much less well understood, although some parametrisation has been attempted based upon clustering results from HiZELS (Geach et al., 2012; Cochrane et al., 2017). Some attempts have also been made to fit the HOD parameters for simulated star-forming galaxies and simulated [OII]-emitters (Contreras et al., 2013; Gonzalez-Perez et al., 2018; Cochrane & Best, 2018).

Rather than adopting any parametrisation for the HODs of ${\rm H\alpha}$-emitting galaxies, we again consider using a semi-analytical galaxy formation model, but in this case using a model to generate a library of luminosity-dependent HODs that we can interpolate over as a function of luminosity and redshift. We choose to use the open source galaxy formation model Galacticus (Benson, 2012).

2.2.1 The Galacticus galaxy formation model

The Galacticus222Here we use version 0.9.4 of Galacticus, which is publicly available from: bitbucket.org/galacticusdev/galacticus. The Mercurial hash I.D. for the particular revision used is: 4787d94cd86e. semi-analytical galaxy formation model is designed to follow the formation and evolution of a galaxy population within a merging hierarchical distribution of dark matter haloes. The astrophysics governing the baryonic processes occurring within the dark matter haloes is described using sets of coupled ordinary differential equations (ODEs). These processes include the rate of radiative gas cooling, the quiescent star formation rate, the chemical enrichment of the stellar and gaseous components, as well as the regulation of feedback processes from supernovae and active galactic nuclei. By calling the ODE solver within Galacticus at various epochs, one can compute the star formation histories of a population of galaxies from high redshift to the present day. Given a stellar initial mass function (IMF), these histories can be convolved with a single stellar population synthesis model to generate a spectral energy distribution for each galaxy, with which we can compute photometric luminosities for a specified set of filter transmission curves. By default Galacticus adopts the Flexible Stellar Population Synthesis code of Conroy et al. (2010), with a Chabrier (2003) IMF.

Emission line luminosites are computed for the Galacticus galaxies by interpolating over tabulated libraries generated from the Cloudy photo-ionisation code (Ferland et al., 2013). These libraries store emission line luminosities as a function of (i) the ionizing continuua luminosities for various species (Hi, Hei and Oii), (ii) the hydrogen gas density, (iii) the metallicity of the interstellar medium and (iv) the volume filling factors of Hii regions. All of these properties can be computed for the galaxies from Galacticus and so by interpolating over the Cloudy tables we can obtain emission line luminosities that are consistent with the other galaxy properties. Further details regarding the calculation of the emission line luminosities are provided in Merson et al. (2018).

Typically the Galacticus model parameters are calibrated to reproduce numerous observational statistics of the galaxy population, in particular those statistics of the local Universe that are the most tightly constrained. The version of Galacticus that we use here is the same as the version used in Merson et al. (2018), which employs the standard parameter set. This version of Galacticus has been calibrated to reproduce a range of galaxy statistics, particularly those in the local Universe. Emphasis has been placed on reproducing the $z=0$ galaxy stellar mass function from the Sloan Digital Sky Survey, as measured by Li & White (2009). Full details of the calibration procedure for Galacticus can be found in Benson (2014) and Knebe et al. (2015).

2.2.2 Generating a library of luminosity-dependent HODs

As we have previously mentioned, the poor mass resolution of the MXXL simulation means that we are unable to run Galacticus directly on the MXXL merger trees. Instead we run the model on the halo merger trees of the Millennium Simulation, which we remind the reader has a cosmology identical to that of the MXXL simulation. Populations of galaxies are output at 31 snapshots with redshifts in the range $0.7<z<2.2$ (i.e. completely spanning the redshift ranges of Euclid and WFIRST). For each snapshot we split the galaxies into 31 cumulative luminosity-limited samples, with blended luminosity limits evenly separated in log-space between $L_{{\rm H\alpha}+[NII]}=10^{38}h^{-2}{\rm erg}\,{\rm s}^{-1}{}$ and $L_{{\rm H\alpha}+[NII]}=10^{43}h^{-2}{\rm erg}\,{\rm s}^{-1}{}$. We select galaxies by their blended ${\rm H\alpha}{}+\left[{\rm N_{II}}\right]$ luminosity as the resolution of the grisms proposed for Euclid will be too poor to deblend the ${\rm H\alpha}$ line and the $\left[{\rm N_{II}}\right]$ doublet. See Faisst et al. (2018) and Martens et al. (2019) for discussions on the impact that ${\rm H\alpha}$ and $\left[{\rm N_{II}}\right]$ line blending can have on Euclid and WFIRST cosmological measurements. To introduce $\left[{\rm N_{II}}\right]$ contamination into the ${\rm H\alpha}$ luminosities from Galacticus, we follow the methodology adopted by Merson et al. (2018). The blended luminosities have no dust attenuation applied. For each luminosity-limited sample we construct the HOD by computing the mean number of central and satellite galaxies in haloes binned by halo mass, $M_{200}$, which we define as the mass within an over-density with an average density corresponding to 200 times the mean density of the Universe. Each HOD is measured in 26 mass bins evenly separated in log-space between $M_{200}=10^{9.7}h^{-1}{\rm M_{\odot}}$ and $M_{200}=10^{14.7}h^{-1}{\rm M_{\odot}}$. At this stage we now have a tabulated library of luminosity-dependent HODs over which we can interpolate as a function of halo mass, luminosity and redshift. Note that in the library we store three sets of HODs: the HODs for the central galaxies only, the HOD for the satellite galaxies only, and the combined HOD.

In Fig. 2 we show the combined HODs for our 31 blended luminosity-limited samples as measured from running Galacticus on three of the redshift snapshots of the Millennium Simulation. The shapes of these HODs are consistent with the shapes of emission line luminosity-selected HODs presented elsewhere in the literature (e.g. Geach et al., 2012; Contreras et al., 2013; Gonzalez-Perez et al., 2018; Cochrane & Best, 2018). For the faintest luminosity limits considered, the HODs show a smooth step-like function, similar to the HODs for galaxy samples selected using broad-band photometry. For masses below $10^{11}\,h^{-1}{\rm M_{\odot}}$ there is evidence for a secondary step feature, particularly towards lower redshift. However, we note that at such low masses we are approaching the resolution limit of the Millennium Simulation and so the extent to which this secondary step is real is uncertain.

Towards brighter luminosity limits the amplitude of the HOD decreases with the step-like shape of the HOD disappearing as the number of galaxies with bright ${\rm H\alpha}$ luminosity rapidly declines, particularly for very massive haloes. For the brightest luminosity limits considered, the distribution of central galaxies is much more peaked, with a maximum value at a halo mass of approximately $10^{12}\,h^{-1}{\rm M_{\odot}}$. This is understandable as ${\rm H\alpha}$ emission occurs in regions of ongoing star-formation and so ${\rm H\alpha}$ luminosity is correlated with galaxy star-formation rate, which, due to feedback from active galactic nuclei, does not increase monotonically with increasing halo mass. The quenching of satellite galaxies in massive haloes also leads to a decline in the number of satellite galaxies with bright ${\rm H\alpha}$ emission. In haloes with mass above $10^{12}h^{-1}{\rm M_{\odot}}$, for $L_{{\rm H\alpha}+\left[{\rm N_{II}}\right]}\gtrsim 10^{41}{\rm erg}\,{\rm s}^{-1}$ the satellite fraction is below approximately 5 per cent. Indeed the occupation number for satellite galaxies still appears to follow a power law relation, but with a slope that decreases with increasing luminosity. The impact of these quenching mechanisms is that for bright luminosities the HOD turns over for haloes with mass $M_{200}\gtrsim 10^{12}\,h^{-1}{\rm M_{\odot}}$. We can see from Fig. 2 that the turn-over is more significant towards lower redshifts. In addition we can see that the amplitude of the HOD decreases with increasing luminosity limit, suggesting that galaxies that are bright in ${\rm H\alpha}$-emission are becoming increasingly rare. This change in the amplitude of the HOD, along with the change in the strength of the turnover, is consistent with the observed decline in the global star-formation rate density towards the present day (e.g. Madau et al., 1998; Arnouts et al., 2005; Ly et al., 2007; Shim et al., 2009; Brammer et al., 2011; Madau & Dickinson, 2014).

2.2.3 Placing galaxies into the MXXL haloes

Having generated a library of HODs we now randomly sample the HODs to populate the haloes in the MXXL halo lightcone. Note that by doing so, we lose any environmental-dependence (i.e., assembly bias). This does not matter for our present purposes, since we are only interested in measuring the linear bias on large scales.

In order to sample the HODs we will need to interpolate over the library as a function of halo mass, luminosity and redshift. At low masses we can interpolate linearly in log-space between the tabulated values. However, for the most massive haloes in the MXXL lightcone, approaching $10^{15}h^{-1}{\rm M_{\odot}}$, we will need to extrapolate beyond the halo mass range of the tabulated HODs. The limited volume of the Millennium Simulation means that our library of tabulated HODs from Galacticus extends only to $M_{200}=10^{14.7}h^{-1}{\rm M_{\odot}}$. In order to extrapolate to the higher halo masses we therefore decide to fit smoothed functional forms to the tabulated occupation distributions for the central galaxies and the satellite galaxies. Details of this fitting procedure are provided in Appendix A.

The functional fits to the tabulated HODs are shown in Fig. 3. In the majority of cases the fits provide a reasonably good description of the tabulated HODs. In some cases, particularly at lower redshifts, the noise in the tabulated HODs for the brightest luminosity limits leads to overlap between the HODs of different luminosity limits. In these instances fits to the HODs will cross such that interpolation between the HODs as a function of luminosity would yield a constant HOD with increasing luminosity, which can be a problem in the random sampling of the HODs. This issue can be resolved by applying small vertical shifts of less than 0.3 dex to the amplitude of the HODs of the brightest luminosity limits. Given the rarity of bright ${\rm H\alpha}$-emitting galaxies in massive haloes at lower redshift, applying these offsets leads to only negligible differences in the number density of galaxies in our lightcone catalogues (Smith, 2018).

Galaxies are then placed into each halo using a methodology presented in Smith et al. (2017) and Smith (2018). The first step in the process is to select a minimum luminosity, $L_{\rm min}$, for the lightcone catalogue. Here we adopt $L_{\rm min}=10^{40}h^{-2}{\rm erg}\,{\rm s}^{-1}{}$, which, at the lowest redshift in the mock, is about an order of magnitude fainter than the luminosity limit for a WFIRST-like selection. Following this we identify all of the haloes that could host a central galaxy with $L_{{\rm H\alpha}+\left[{\rm N_{II}}\right]}\geqslant L_{\rm min}$ by interpolating over the HOD with the corresponding luminosity threshold. We assign a central galaxy to each of these haloes for which $x_{1}<\left<N_{\rm cen}\left(>L_{\rm min}|M_{200},z\right)\right>$, where $\left<N_{\rm cen}\left(>L_{\rm min}|M_{200},z\right)\right>$ is the mean number of central galaxies brighter than $L_{\rm lim}$ that could be found in a halo of mass $M_{200}$ at redshift $z$, and $x_{1}$ is a random number drawn from a uniform distribution $0\leqslant x_{1}\leqslant 1$. Central galaxies are placed at the centre-of-mass of the haloes, with a velocity equal to the velocity of their host halo. The luminosity of the galaxy, $L_{\rm cen}$, is determined by interpolating over the library of HODs to find the luminosity that satisfies,

[TABLE]

where $x_{2}$ is a second random number, again drawn from a $0\leqslant x_{2}\leqslant 1$ uniform distribution.

The number of satellite galaxies assigned to a halo is determined by drawing from a Poisson distribution with a mean equal to $\left<N_{\rm sat}\left(>L_{\rm min}|M_{200},z\right)\right>$, where $\left<N_{\rm sat}\left(>L_{\rm min}|M_{200},z\right)\right>$ is the mean number of satellite galaxies brighter than $L_{\rm lim}$ that could be found in a halo of mass $M_{200}$ at redshift $z$. Satellite galaxies are placed randomly within the halo following a Navarro et al. (1997) profile and are given random velocities relative to the velocity of the halo. These velocities are drawn from an isotropic Maxwell-Boltzmann distribution with a line-of-sight velocity dispersion, $\sigma^{2}\left(M_{200}\right)$, given by,

[TABLE]

where $R_{200}$ is the radius of a sphere, centred on the halo centre-of-mass, within which the enclosed density is 200 times the mean density of the Universe. The luminosities of the satellite galaxies are assigned using random number generation in the same manner as was done for assigning the luminosities of the central galaxies.

2.3 Catalogue specifications

We employ the methodology laid out above to construct two lightcone catalogues: one covering an area of $15\,470\,{\rm deg}^{2}$ (MXXL-15K), and another covering a smaller area of $2\,000\,{\rm deg}^{2}$ (MXXL-2K). Note that in reality the MXXL-15K lightcone covers $3/8^{\rm ths}$ of the sky. The specifications for these two lightcones are shown in Table 1. With these two lightcones we are able to simulate a Euclid-like survey and a WFIRST-like survey (see $\S$ 4.1).

At this point we note that due to the wide-angle area of the lightcones and their redshift extent, as well as the finite volume of the MXXL simulation, we expect these lightcones to contain repeated structures. Given the cosmology of the simulation, Smith et al. (2017) determined that an observer placed at the very centre of the box would not observe any repeated structures for $z\lesssim 0.5$. Therefore we expect to observe repeated structures, although we do not expect this to significantly affect our clustering measurements. However, we should expect this feature to lead to under-estimates in the uncertainties on our clustering measurements, and therefore the uncertainties on the bias estimates. Note that those effects should be negligible, since the mass density variance is very small on scales larger than $3\,h^{-1}{\rm Gpc}$.

3 Mock Catalogue Calibration

Before we can compute the galaxy bias it is necessary to calibrate the lightcone catalogues so that they reproduce, as closely as possible, existing observations of ${\rm H\alpha}$-emitting galaxies. To calibrate the mock catalogue we apply dust attenuation to the galaxy luminosities, using $\chi^{2}$ minimisation to determine the dust attenuation, $A_{\rm{\rm H\alpha}}$, that leads to the best agreement with a particular observational dataset. The dust attenuation $A_{\rm{\rm H\alpha}}$ relates the un-attenuated luminosity, $L^{0}_{\rm{\rm H\alpha}}$, and attenuated luminosity, $L^{\rm att}_{\rm{\rm H\alpha}}$, according to,

[TABLE]

Here we choose to calibrate the catalogues to match the luminosity functions and cumulative number counts of ${\rm H\alpha}$-emitters. Being able to reproduce the luminosity function helps ensure that we have the correct number density, whilst reproducing the cumulative number counts confirms that we have the correct total number of galaxies (e.g. Pozzetti et al., 2016; Valentino et al., 2017; Merson et al., 2018).

3.1 Observational datasets

For calibration of the MXXL-15K and MXXL-2K lightcones we adopt (i) the cumulative number counts of ${\rm H\alpha}$-emitters from the Wide Field Camera 3 (WFC3) Infrared Spectroscopic Parallels Survey (WISP, Atek et al., 2010, 2011) as measured by Mehta et al. (2015), and (ii) the luminosity function of ${\rm H\alpha}$-emitters from HiZELS as measured by Sobral et al. (2013).

The WISP survey is a pure-parallel near-infrared grism spectroscopic survey that was carried out using the G141 ($1.2\,-\,1.7\mu{\rm m}$, $R\sim 130$) and G102 ($0.8\,-\,1.2\mu{\rm m}$, $R\sim 210$) grisms on the Hubble Space Telescope WFC3. Given the wavelength ranges of these grisms, WISP is able to directly detect the ${\rm H\alpha}$ emission line in galaxies out to $z\lesssim 1.5$. Note that the Euclid and WFIRST observing strategies will be comparable to the WISP observing strategy. The survey targeted emission line galaxies in several hundred high-latitude fields. Estimates for the stellar mass of a subset of the WISP galaxies analysed by Domínguez et al. (2013), suggest these galaxies have stellar masses in the approximate range $10^{7}h^{-1}{\rm M_{\odot}}\lesssim M_{\star}\lesssim 10^{11}h^{-1}{\rm M_{\odot}}$, assuming a Chabrier (2003) initial mass function. For calibration we use the most recent published cumulative number counts measured by Mehta et al. (2015) who in their analysis used galaxies from 52 separate fields covering an area of $182\,{\rm arcmin}^{2}$ with a quality flag $<16$ down to a flux limit of $(3-5)\times 10^{-17}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$. Specifically, the counts that we use correspond to the cumulative counts of ${\rm H\alpha}$-emitters without any correction for $\left[{\rm N_{II}}\right]$ contamination, i.e. counting the galaxies according to their ${\rm H\alpha}+\left[{\rm N_{II}}\right]$ blended flux (see Table 5 of Mehta et al. 2015). We consider the blended flux counts in the redshift ranges $0.7<z<1.5$ and $0.8<z<1.2$.

In contrast to the WISP survey, HiZELS is a ground-based panoramic survey of emission line galaxies covering several square degrees that was carried out on the United Kingdom Infrared Telescope, the Subaru Telescope and the Very Large Telescope. Custom-made ${\rm NB}_{J}$, ${\rm NB}_{H}$, ${\rm NB}_{K}$ and ${\rm NB921}$ narrow band filters were employed to target emission line galaxies out to $z\sim 9$. The specifications of the narrow band filters allowed ${\rm H\alpha}$-emitters to be detected in thin redshift slices centred approximately at $z=0.4$, $0.84$, $1.47$ and $2.23$. Further details regarding the survey design can be found in Geach et al. (2008), Sobral et al. (2009), Sobral et al. (2012) and Sobral et al. (2013). Estimates by Cochrane et al. (2018) of the stellar masses of the subset of HiZELS galaxies used in their clustering analysis suggest that these galaxies have stellar masses in the approximate range $10^{8}h^{-1}{\rm M_{\odot}}\lesssim M_{\star}\lesssim 10^{11}h^{-1}{\rm M_{\odot}}$, assuming a Chabrier (2003) initial mass function. For our calibration we use the completeness corrected luminosity functions of ${\rm H\alpha}$-emitters presented in Table 4 of Sobral et al. (2013). For their analysis, Sobral et al. (2013) selected ${\rm H\alpha}$-emitters using a combination of colour-colour selections and photometric redshifts when available.

3.2 Number counts calibration

As mentioned in the previous subsection, for calibrating the number counts of the lightcone we use the WISP cumulative number counts from Mehta et al. (2015), who measured the number counts in two broad redshift bins, $0.7<z<1.5$ and $0.8<z<1.2$. For each of these redshift bins we compute the number counts using the dust-free luminosities from the lightcone catalogue and use $\chi^{2}$ minimisation to determine the optimum value for $A_{\rm{\rm H\alpha}}$ needed to shift these counts into agreement with the observations. Note that this approach assumes that $A_{\rm{\rm H\alpha}}$ has a single fixed value for each redshift bin. We note that Domínguez et al. (2013) observe a dust attenuation that varies with observed ${\rm H\alpha}$ luminosity, though their attenuation estimates are consistent with a fixed value attenuation that is independent of luminosity (see also Merson et al. 2018).

The results of our calibration procedure are shown in Fig. 4 where we compare the number counts for the MXXL-2K lightcone to the observed cumulative number counts from Mehta et al. (2015) in the redshift ranges $0.7<z<1.5$ (left-hand panel) and $0.8<z<1.2$ (right-hand panel). In each panel the dotted blue line shows the number counts assuming the dust-free luminosities from the lightcone, and the solid blue line shows the number counts for the value of $A_{\rm{\rm H\alpha}}$ that leads to the best match to the observed counts in that particular redshift bin, i.e. the value that minimises the $\chi^{2}$ statistic. The blue shaded region shows the range of allowed number counts given the uncertainty on $A_{\rm{\rm H\alpha}}$, which we define as the range of $A_{\rm{\rm H\alpha}}$ values whose $\chi^{2}$ value is within one standard deviation of the minimum $\chi^{2}$ value. We see that for both redshift bins the calibrated counts from the lightcone are in excellent agreement with the observed counts, particularly at faint flux limits. Towards brighter flux limits, however, the lightcone under-predicts the number of galaxies. Since the number of bright ${\rm H\alpha}$-emitters is only a very small fraction of the samples for Euclid and WFIRST, we expect this deficit to have negligible effect on our results. Thus, for a Euclid-like flux limit of $2\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$, or a WFIRST-like flux limit of $1\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$, the lightcone shows a deficit of bright galaxies but predicts the correct overall number of galaxies. We do not show the counts for the MXXL-15K lightcone but confirm that they are in equally good agreement with the WISP observations as the MXXL-2K lightcone.

The WISP-calibrated values for $A_{\rm{\rm H\alpha}}$ are shown in Fig. 5, where the diamond symbols indicate the calibration results for the $0.7<z<1.5$ redshift bin and square symbols indicate the calibration results for the $0.8<z<1.2$ redshift bin. Filled symbols correspond to the results for the MXXL-2K lightcone and empty symbols correspond to the results for the MXXL-15K lightcone. For both redshift bins the calibration results for the MXXL-2K lightcone are in excellent agreement with the results for the MXXL-15K lightcone. Furthermore, for both lightcones the value for $A_{\rm{\rm H\alpha}}$ obtained by calibrating to the $0.7<z<1.5$ counts is consistent within error with the value obtained by calibrating to the $0.8<z<1.2$ counts. Therefore, adopting the calibration result for either redshift bin has negligible impact on the counts. This can be seen in the left-hand panel of Fig. 4 where the black dot-dashed line shows the counts over the redshift range $0.7<z<1.5$ computed assuming the value of $A_{\rm{\rm H\alpha}}$ obtained by calibrating to the counts in the $0.8<z<1.2$ bin. Going forward, we therefore adopt the results from the calibration to the counts in the $0.8<z<1.2$ bin, which we note in the next section are marginally closer in value to the dust attenuation values obtained from calibrating the lightcone luminosity function. The calibration using the luminosity function, which yields the HiZELS attenuation factors, is discussed in the next subsection.

3.3 Luminosity function calibration

For calibrating the lightcone luminosity function we use the HiZELS luminosity functions, as presented by Sobral et al. (2013), measured at $z=0.84$, $z=1.47$ and $z=2.23$. For each redshift, we compute the dust-free luminosity function from the lightcone using a thin redshift slice centred on the appropriate HiZELS redshift. Note that since $z\simeq 2.23$ is beyond the upper redshift extent of our lightcone, in this case we use a thin redshift slice with $z=2.2$ as the upper bound. As with the cumulative counts, we use $\chi^{2}$ minimisation to determine the optimum value for $A_{\rm{\rm H\alpha}}$ needed to shift these luminosity functions into agreement with the observational estimates. Note that prior to our calibration we have adjusted the HiZELS luminosity functions in two ways. First, Sobral et al. (2013) originally corrected for dust attenuation by shifting the ${\rm H\alpha}$ luminosity bright-wards by $A_{\rm{\rm H\alpha}}=1\,{\rm mag}$ (0.4 dex). Therefore, to ensure that we are comparing to the dust-attenuated HiZELS luminosity function we have removed this shift and made the luminosities fainter by 0.4 dex. Second, the HiZELS luminosity functions assumed ${\rm H\alpha}$ luminosities that had been corrected to remove $\left[{\rm N_{II}}\right]$ contamination and so for our calibration we re-introduce the $\left[{\rm N_{II}}\right]$ contamination. We do this by boosting the HiZELs luminosity function bright-wards by a factor of $4/3$. This factor was chosen to match the median correction of $\left[{\rm N_{II}}\right]/\left({\rm H\alpha}+\left[{\rm N_{II}}\right]\right)\approx 0.25$ reported by Sobral et al. (2013). Note that this correction factor shows some variation with redshift (with the correction being slightly smaller at higher redshift), though this variation is negligible in size.

The calibrated luminosity functions for the MXXL-2K lightcone are shown in Fig. 6. In each panel the blue dotted line shows the luminosity function computed using the dust-free luminosities and the symbols correspond to the observed HiZELS luminosity functions, with the open symbols corresponding to luminosity bins that were excluded in our calibration procedure. These bins were left out following a preliminary calibration that indicated that the MXXL-2K lightcone is incomplete at these faint luminosities and so including these luminosity bins would unfairly bias our calibration results. The two vertical dotted lines indicate the luminosities at each redshift that correspond to the Euclid and WFIRST flux limits. For reference we also show the Schechter (1976) functional fits presented by Cochrane et al. (2017) and the luminosity functions predicted by the third of the three empirical models presented by Pozzetti et al. (2016), both which we have similarly corrected by a factor of $4/3$ in order to introduce $\left[{\rm N_{II}}\right]$ contamination and applied an attenuation of $A_{\rm{\rm H\alpha}}=1\,{\rm mag}$.

In each panel the solid blue lines show the MXXL-2K luminosity function for the value of $A_{\rm{\rm H\alpha}}$ that minimises the $\chi^{2}$ statistic and leads to the best match to the HiZELS luminosity function at that particular redshift. The blue shaded region shows the range of allowed luminosity functions given the uncertainty on $A_{\rm{\rm H\alpha}}$, which again is defined as the range of $A_{\rm{\rm H\alpha}}$ values whose $\chi^{2}$ value is within one standard deviation of the minimum $\chi^{2}$ value. We can see that at each redshift the MXXL-2K lightcone is able to reproduce very well the HiZELS luminosity function brightwards of the Euclid flux limit. For $z\simeq 1.47$ and $z\simeq 2.2$ the lightcone provides a good match to the HiZELS luminosity function down to the WFIRST flux limit. At $z\simeq 0.84$ we see the impact of incompleteness in the lightcone luminosity function as it falls below the observations for luminosities faintwards of the Euclid flux limit. We do not expect this to have a significant impact our clustering analysis however as we note that $z\simeq 0.84$ is the below the lower redshift limit of WFIRST. We do not show the luminosity functions for the MXXL-15K lightcone but can confirm that they are in equally good agreement to the HiZELS observations as the MXXL-2K lightcone.

The dust attenuation values for our luminosity function calibration are also shown with circular symbols in Fig. 5. As with the number counts calibration, the attenuations obtained for the MXXL-15K and the MXXL-2K lightcones are in excellent agreement. The values for $A_{\rm{\rm H\alpha}}$ from our luminosity function calibration are much larger than the values obtained from the WISP number counts calibration, particularly at low redshift. However, we note that at $z\simeq 0.84$ and $z\simeq 2.2$ the $A_{{\rm H\alpha}}$ values that we have obtained are consistent with the $A_{{\rm H\alpha}}=1\,{\rm mag}$ that was originally assumed by Sobral et al. (2013). Using the median $\left[{\rm O_{II}}\right]/{\rm H\alpha}$ line ratio as a dust attenuation indicator (see Sobral et al. 2012 for details), Sobral et al. (2013) estimated that their $z\simeq 1.47$ galaxy sample should have an attenuation of $A_{{\rm H\alpha}}=0.8\,{\rm mag}$, which, as seen in Fig. 5, is consistent within error with our $z\simeq 1.47$ calibration result. Additionally, Garn et al. (2010) determined that $A_{{\rm H\alpha}}\approx 1.2$ at $z\simeq 0.84$, which is also consistent within the errors with our calibration result at $z\simeq 0.84$. At $z\simeq 2.2$ our luminosity function calibrated attenuation is consistent with the attenuation obtained from our calibration to the WISP number counts.

We note that Cochrane et al. (2017) presented updated Schechter (1976) functional fits to the measured HiZELS luminosity functions, which at $z=0.84$ differ from the functional fits presented in Sobral et al. (2013). It is clear from Fig. 6, where the Cochrane et al. (2017) fits are shown as thick dotted lines, that calibrating the MXXL lightcones to reproduce the Cochrane et al. (2017) fits would require a dust attenuation that is equal to or stronger than the attenuation required to reproduce the Sobral et al. (2013) measurements. Adopting the Cochrane et al. (2017) fits, instead of the Sobral et al. (2013) measurements, may therefore widen the discrepancy between the dust attenuation needed to reproduce the WISP number counts and the HiZELS luminosity functions.

To see how much of an impact the discrepancy between the WISP-calibrated attenuation and the HiZELS-calibrated attenuation has on the luminosity function, we show in Fig. 6 the luminosity function obtained when assuming the dust attenuation $A_{\rm{\rm H\alpha}}$ calibrated to reproduce the WISP counts in the redshift range $0.8<z<1.2$. Using the WISP-calibrated attenuation leads to the lightcone over-predicting the luminosity function, particularly for bright luminosities at low redshift. At $z\simeq 2.2$ the luminosity function assuming the HiZELS-calibrated attenuation and the luminosity function assuming the WISP-calibrated attenuation are virtually identical. For $z\simeq 0.84$ and $z\simeq 1.47$ the difference between the HiZELS-calibrated luminosity function and the WISP-calibrated luminosity function is negligible for luminosities corresponding to the Euclid and WFIRST flux limits. We additionally show in Fig. 4 the cumulative number counts obtained when assuming the HiZELS-calibrated dust attenuation. As expected, the HiZELS-calibrated counts are lower than the WISP-calibrated counts, but the discrepancy decreases towards fainter flux limits, such that at the Euclid and WFIRST flux limits the discrepancy is less than a factor of two.

3.4 Discussion of calibration results

Why the calibration to the WISP number counts and the calibration to the HiZELS luminosity functions lead to different values for the dust attenuation is not immediately clear. One possibility is that the WISP number counts and the HiZELS luminosity functions are probing different luminosity ranges. As shown in Fig. 4, the WISP cumulative number counts from Mehta et al. (2015) have been measured at flux limits spanning $1\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$ to $1\times 10^{-15}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}$. Note that the fainter limit corresponds to the WFIRST flux limit. These two flux limits have been plotted as vertical dotted and dashed lines in the panels of Fig. 6. We can see that at $z\simeq 0.84$ and $z\simeq 1.47$ this flux range covers most of the luminosity range measured by the Sobral et al. (2013) luminosity functions, with only the faintest few luminosity bins outside of the flux range. This would suggest that the small difference in luminosity ranges probed by the cumulative counts and the luminosity functions is less likely to be the cause of the discrepancy. At $z\simeq 2.2$ the flux range is restricted to brighter luminosities, though at this redshift the WISP-calibrated attenuations and HiZELS-calibrated attenuations are consistent.

The most likely cause of the discrepancy is the different observing strategies and selection functions employed by the WISP and HiZELS surveys. For example, narrow band surveys, such as HiZELS, rely on continuum detection to identify emission lines which means that these surveys typically miss the lowest mass galaxies that spectroscopic grism surveys, such as WISP, are more sensitive to (Mehta et al., 2015). As such, for a given ${\rm H\alpha}$ flux limit the galaxies detected by WISP will typically have a lower stellar mass compared to those detected by HiZELS (J. Colbert, private communication). Therefore, given the relations observed between stellar mass and dust attenuation $A_{{\rm H\alpha}}$ (Garn & Best, 2010; Domínguez et al., 2013), we expect the WISP dataset to typically have a weaker dust attenuation, as we can see in Fig. 5. Additionally, the thin-shell volumes probed by narrow band surveys mean that volume densities estimated by such surveys are sensitive to cosmic variance introduced by the presence of large-scale structure (Mehta et al., 2015). As such, cosmic variance may also be contributing to the discrepancy in dust attenuation that our calibration procedure has returned.

Using the HiZELS-calibrated attenuation results in Fig. 5 we can construct a toy model of how the attenuation evolves with redshift, such that we can apply a HiZELS-calibrated attenuation to any galaxy in the mock catalogue by interpolating over these results as a function of redshift. Finally, we note that in our toy model we assume that $A_{\rm{\rm H\alpha}}$ depends only on redshift and not additionally on the ${\rm H\alpha}$ luminosity. However, for our purposes we argue that adding a luminosity-depenence is not necessary given that we are able to reproduce the HiZELS luminosity functions very well down to the luminosity limits of relevance for Euclid and WFIRST.

In summary, given the difference in the dust attenuation required to reproduce the WISP number counts and the HiZELS luminosity functions, we choose to carry out our bias forecasts using both a WISP-calibrated version of the lightcones (adopting the values for calibration to the counts in the $0.8<z<1.2$ bin) and a HiZELS-calibrated version of the lightcones.

3.5 Galaxy number densities

Before progressing to examine the clustering of the galaxies in the lightcones, we examine the number densities of the galaxies as a check that our calibration yields number densities that are consistent with previous estimates.

In Fig. 7 we show in the upper panel the number densities for the galaxies in the MXXL-15K lightcone, and in the lower panel the number densities for the galaxies in the MXXL-2K lightcone. We have applied a flux limit of $2\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$ to the dust-attenuated luminosities in the MXXL-15K lightcone and a flux limit of $1\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$ to the dust-attenuated luminosities in the MXXL-2K lightcone. In both panels the solid line shows the number densities when adopting the WISP-calibrated dust attenuations and the dashed line shows the number densities when adopting the HiZELS-calibrated dust attenuations. The dotted line corresponds to applying no dust attenuation. The symbols correspond to the number densities presented in Merson et al. (2018) obtained by applying three different dust attenuation methods to a Galacticus lightcone. The dust attenuation methods considered in Merson et al. (2018) were: (i) interpolation over the library of dust curves from Ferrara et al. (1999), (ii) the Charlot & Fall (2000) model, where the dust attenuation follows a power law with wavelength, and (iii) the Calzetti et al. (2000) dust screen law.

Considering first the MXXL-15K lightcone we see that when adopting a WISP-calibrated dust attenuation, the galaxy number density is consistent with the results from Merson et al. (2018). Adopting a HiZELS-calibrated dust attenuation leads to a number density that is slightly below the Merson et al. (2018) results, but typically remains within approximately a factor of two. The fact that the number densities for the WISP-calibrated attenuations are in better agreement than the HiZELS-calibrated attenuations is not surprising as the Galacticus lightcone in Merson et al. (2018) was calibrated to reproduce the WISP number counts from Mehta et al. (2015). Turning to the MXXL-2K lightcone we see that the number densities for the WISP-calibrated attenuations and the HiZELS-calibrated attenuations are both consistent with the number densities from Merson et al. (2018), with the difference between the two being consistent in size with the scatter between the number densities predicted for the different dust methods used in Merson et al. (2018). The redshift distributions for the MXXL-15K and MXXL-2K lightcones are provided in Appendix B.

Overall, when we apply the WISP-calibrated and HiZELS-calibrated dust attenuations to the MXXL-15K and MXXL-2K lightcones, the number densities that we obtain are entirely consistent with previous number density estimates.

4 Linear bias forecasts

Having calibrated our lightcone catalogues we now proceed to make forecasts for the linear bias of ${\rm H\alpha}$-emitters as a function of redshift for a Euclid-like survey and a WFIRST-like survey. To calculate the linear bias we will first need to compute the correlation functions for the galaxies in our catalogues.

4.1 Catalogue preparation

The first step is to build catalogues for our Euclid-like and WFIRST-like surveys. We firstly apply a flux limit to the dust attenuated luminosities from the appropriate lightcone. For our Euclid-like surveys we apply a flux limit of $2\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$ to select galaxies from the MXXL-15K lightcones and for our WFIRST-like surveys we apply a flux limit of $1\times 10^{-16}{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}{}$ to select galaxies from the MXXL-2K lightcones.

Secondly, we introduce incompleteness into the surveys by applying the appropriate completeness expected for each redshift survey: 45 per cent for Euclid (Laureijs et al., 2011) and 70 per cent for WFIRST (Spergel et al., 2015). Both surveys have completeness significantly below 100 per cent due to slitless effects. This incompleteness will arise in the Euclid and WFIRST galaxy redshift surveys due to the inability to measure redshifts for faint galaxies with particularly noisy spectra, or for galaxies in high density regions where slitless spectra will overlap and could be confused.

Incompleteness is introduced by using random number generation to discard the appropriate fraction of galaxies. Note that we simply draw random numbers from a uniform distribution and make no attempt to introduce a density-dependent or luminosity-dependent incompleteness. For our Euclid-like surveys we apply a completeness limit of 0.45, consistent with the completeness limit estimated in Laureijs et al. (2011), and reject a random 55 per cent of the galaxies. For our WFIRST-like surveys we apply a completeness limit of 0.7, consistent with the completeness limit provided in Spergel et al. (2015), and reject a random 30 per cent of the galaxies. The specifications for our Euclid-like and WFIRST-like surveys are listed in Table 1 alongside the specifications of the MXXL-15K and MXXL-2K lightcones. In Fig. 8 we show the redshift distributions for our Euclid-like and WFIRST-like surveys, as well as for the MXXL-15K and the MXXL-2K lightcones.

In our clustering analysis we consider two versions of a Euclid-like survey and WFIRST-like survey: one calibrated to reproduce the WISP cumulative number counts, and one calibrated to reproduce the HiZELS luminosity functions. Additionally, for each version we repeat the analysis twice. In the first instance we take the observed redshifts of the galaxies (i.e. with peculiar velocities included), which we shall refer to as redshift-space, and in the second instance we take the cosmological redshifts of the galaxies (i.e. with peculiar velocities removed), which we will refer to as real-space. Note that galaxies are not observed in real-space, but we include this analysis to show the impact of redshift-space distortions.

4.2 Clustering analysis

For our clustering analysis we split each of our Euclid-like and WFIRST-like surveys into 5 equal redshift bins of width $\Delta z=0.2$ and compute the two-point galaxy correlation function in each bin. For the Euclid-like surveys these bins cover the redshift range $0.9\leqslant z<1.9$, and for the WFIRST-like surveys these bins cover the redshift range $1\leqslant z<2$.

In order to compute the galaxy correlation functions we generate individual catalogues of randoms for each redshift bin. In each instance we generate a sufficiently large number of randoms (at least ten times the number of galaxies in the redshift bin), with the number chosen arbitrarily such that adding any additional randoms does not improve our recovery of the galaxy correlation function. The number of randoms required increases with redshift as the number density of galaxies decreases. For the lower redshifts the number of randoms was also limited by available computing resources. The right ascension and declination of the randoms were found by sampling uniform distributions bounded by the right ascension limits and the cosine of the declination limits of the particular lightcone. To assign redshifts to the randoms we do not sample a uniform distribution as this would not reproduce the correct shape of the galaxy redshift distribution across the bin. Instead we assign redshifts by first replicating the redshifts of the galaxies, until we match the required number of randoms, and then randomly shuffling the galaxy redshifts.

We compute the angle-averaged galaxy two-point correlation function, $\xi(r)$, using the open-source TreeCorr package (Jarvis et al., 2004), and the Landy & Szalay (1993) estimator,

[TABLE]

where $DD(r)$ are the weighted galaxy-galaxy pair counts, $DR(r)$ are the weighted galaxy-random pair counts and $RR(r)$ are the weighted random-random pair counts as a function of separation $r$ in units of $\,h^{-1}{\rm Mpc}$. For each redshift bin the correlation function is calculated 5 times, each time using a different random seed when applying the survey incompleteness (see $\S$ 4.1) and using a different catalogue of randomly generated positions. The correlation function is calculated between $50\,h^{-1}{\rm Mpc}$ and $150\,h^{-1}{\rm Mpc}$ using 48 linearly spaced bins.

In Table 4.2 we report for our Euclid-like surveys the average number of galaxies in each redshift bin after introducing incompleteness and the effective redshift of these galaxies. We also report the average ratio of the number of galaxies to the number of random positions. The number of randoms was increased with increasing redshift to account for the decreasing number densities of the galaxy samples. Memory limitations of available computing resources forced us to limit the number of randoms, but in each case we ensured that a sufficient number was used to adequately recover the clustering signal of the galaxies. We report equivalent numbers for our WFIRST-like surveys in Table 4.2.