Linear bias and halo occupation distribution of emission line galaxies from Nancy Grace Roman Space Telescope

The synthetic galaxy catalog used in this paper has been constructed using the Galacticus galaxy formation model (Benson 2012). Similar to the other SAMs, Galacticus parametrizes the astrophysical processes and performs the evolution of galaxy populations within a distribution of dark matter halos and their merger trees. The processes governing galaxy formation and evolution include gas cooling, star formation, feedback from supernovae, black hole formation and so on. By parametrizing these processes as ordinary differential equations (ODE) and calling the ODE solver, Galacticus can perform a simulation of galaxies within a sufficiently large volume in a timely manner and output details for the galaxy populations, including the star formation history, galaxy morphology, spectral energy distribution (SED), photometric luminosities for a set of filter transmission curves and emission line luminosities.

Before using Galacticus to produce the galaxy catalog, we need to determine the free parameters in the model due to the poor prior knowledge of the astrophysical processes. This can be non-trivial since the typical number of free parameters is 15 or more (Wechsler & Tinker 2018). In our work, we do not limit ourselves to local galaxies to calibrate the model, but compare the model prediction with galaxy populations at higher redshifts relevant to Roman. The parameters of this model have been calibrated in Zhai et al. (2019), including the parameters for the physics of galaxy formation and the dust-attenuation model. In particular, the dust model is calibrated to produce consistent prediction of H$\alpha$ luminosity function compared with observations from the ground-based narrow-band High-z Emission Line Survey (HiZELS, Geach et al. 2008; Sobral et al. 2009; Sobral et al. 2013), or the number counts data collected from Wide Field Camera 3 (WFC3) Infrared Spectroscopic Parallels survey (WISP; Atek et al. 2010, 2011; Mehta et al. 2015). The dust model applied is the Calzetti et al. (2000) model with parameter $A_{V}$ to describe the strength of dust attenuation. The result is $A_{V}=1.652$ for calibration based on WISP, and $A_{V}=1.916$ for HiZELS. Note that higher value of $A_{V}$ means stronger dust attenuation, thus the HiZELS-based calibration results a lower number density of the galaxy sample compared with the WISP-based calibration, with a preference of selecting brighter galaxies. In this paper, we present the bias measurement for ELGs with both dust models and investigate the impact on the large scale structure analysis.

The galaxy population in this analysis is selected by the emission line luminosity. Galacticus can output the number of ionizing photons for various species (HI, He I and [OII]), the metallicity of the interstellar medium (ISM), the hydrogen gas density and the volume filling factor of HII regions. We use these parameters to interpolate the tabulated libraries from the CLOUDY photo-ionization code (Ferland et al. 2013) and compute the emission line luminosity for each galaxy. More details of the method can be found in Merson et al. 2018.

The key ingredient for SAMs like Galacticus is the set of merger trees of dark matter halos, which can be approximately constructed using the Press-Schechter formalism (Press & Schechter 1974), or come from a cosmological N-body simulation. Here we have chosen the later for high fidelity, and use the merger trees extracted from the UNIT simulation ¹¹1https://unitsims.ft.uam.es (Chuang et al. 2019) which assumes a spatially flat $\Lambda$CDM model with parameters consistent with Planck 2016 measurement (Planck Collaboration et al. 2016). The simulation contains 4096³ particles with a box-size of $1h^{-1}$Gpc. This simulation has a mass resolution of $10^{9}h^{-1}M_{\odot}$ with data product covering redshift range of $0<z<99$. The large volume and high resolution makes this simulation sufficient for the next generation galaxy surveys, including DESI, and those planned for Roman and Euclid. We refer the readers to Chuang et al. (2019) for more details of the UNIT simulation and its data products. The merger trees of the dark matter halos are constructed using the Consistent Trees software (Behroozi et al. 2013) and the final product contains more than 160 million merger trees. Applying Galacticus to this simulation allows us to build a light cone catalog of galaxies. In particular, we use the method from Kitzbichler & White (2007) to determine where the dark matter halos enter the lightcone of the observer. The resulting catalog has an area of $\sim 2000\text{deg}^{2}$, consistent with the current baseline design of Roman HLSS.

In the top row of Figure 1, we present the luminosity function (LF) of H$\alpha$ and [OIII] emission line galaxies at $1<z<3$, with both dust free and dust attenuated results. The evolution with redshift indicates the star forming history of the galaxies. In order to validate the simulation, we compare the LF of H$\alpha$ and [OIII] emission lines with current observations at selected redshifts. The middle row shows the LF of H$\alpha$ galaxies compared with HiZELS measurements, which was used to calibrate the SAM model. In particular, the dust model with $A_{V}=1.916$ is chosen to match the LF at $z=1.47$ from HiZELS. Our model underestimates the LF at higher redshift, indicating either the need for improvement in the Galacticus model, or that a redshift dependent dust model is required. The bottom panel shows the comparison of [OIII] emission line galaxies with WISP measurements. Our model shows mild deviation compared with WISP measurements, however the amplitude is roughly consistent. The performance can be improved by calibrating the SAM with more observational data sets.

In this paper, we focus on the forecast of galaxy linear bias for Roman HLSS, but the results can also be applicable to surveys like the one planned for Euclid. The observing strategies can impact the galaxy selection and thus linear bias measurements. Roman grism has a wavelength range of $1.0-1.93$ microns, which determines the redshift range for the emission lines of interest, as shown In Figure 2, which includes the three primary lines H$\alpha$, [OII] and [OIII]. We also shows the [NII] and H$\beta$ lines as they are the main contaminants to H$\alpha$ and [OIII] respectively, due to the closeness of the emission line wavelength. Since the current Galacticus SAM model significantly underestimates the strength of [OII] emission, we will only consider H$\alpha$ and [OIII] lines throughout this paper, as they define the expected Roman galaxy samples.

One of the key characteristics of a survey is its depth, or sensitivity, i.e. the emission line flux limit for an ELG survey. We consider three different line flux limits in our analysis: $0.5\times 10^{-17}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ as a reference case, $1.0\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ as the $6.5\sigma$ nominal depth for Roman HLSS, and $2.0\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ as the $3.5\sigma$ depth of a Euclid-like GRS. The faint limit of $0.5\times 10^{-17}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ is included to facilitate a depth versus area optimization study of the Roman HLSS. As the primary target, H$\alpha$ emission line is detectable at $z<2.0$. Therefore we split the analysis into two subsamples with $z>2.0$ and $z<2.0$.

At $z<2.0$, Roman science requirements specify that at least two emission lines are used in measuring a redshift, with the strong line above the line flux limit. The detection of the second line may not require its strength to be above the line flux limit at 6.5$\sigma$, thus we allow for different thresholds for the second emission line. At $z>2.0$, we only consider the [OIII] line, required to be observed above the flux limit at 6.5$\sigma$, since we are not including [OII] in this study due to the current limits of Galacticus. The impact of the line flux threshold of the [OII] line will be studied in future work.

For the galaxies with $z<2.0$, we first compute the emission line flux for H$\alpha$ and [OIII], then we set the flux limits by two variables. The first is $f_{\mathrm{lim,1}}$, this is the lower limit of the stronger line (either H$\alpha$ or [OIII]), as chosen above. The second is $f_{\mathrm{lim,2R}}$ which sets the lower limit of the weak line in units of $f_{\mathrm{lim,1}}$, therefore $f_{\mathrm{lim,2R}}$ is dimensionless. We choose three values and investigate the impact on the large scale structure analysis: $f_{\mathrm{lim,2R}}=0.25,0.5,1.0$. For instance when $f_{\mathrm{lim,1}}=1.0\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ and $f_{\mathrm{lim,2R}}=0.5$, we select galaxies with the strongest emission line brighter than $f_{\mathrm{lim,1}}$, and the second emission line brighter than $f_{\mathrm{lim,1}}*f_{\mathrm{lim,2R}}=0.5\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$. At $z>2.0$, we just apply $f_{\mathrm{lim,1}}$ to select [OIII] emitting galaxies.

In Figure 3, we show the number densities of the selected galaxies with different flux limits. The curves of the same color merge at $z\sim 2$ due to the selection algorithm since there is only one flux cut on [OIII]. The result shows a monotonic decrease as redshift increases. The flux limit parameter $f_{\mathrm{lim,2R}}$ for the second emission line has weaker impact than $f_{\mathrm{lim,1}}$, but can be important when brighter flux cut and stronger dust-attenuation are assumed.

For galaxies with $z<2.0$, both H$\alpha$ and [OIII] lines can be observed, and their relative strength may not be a constant. In Figure 4, we plot the fraction of galaxies with H$\alpha$ as the stronger line, as a function of redshift. The left and middle panels show that as we go to higher redshifts, the H$\alpha$ dominance decreases with redshift regardless of the dust model, for sufficiently faint H$\alpha$ line flux cut. The right panel shows that this is not true for brighter H$\alpha$ line flux cut, for which the H$\alpha$ dominance flattens at higher redshifts. This indicates that there are more bright H$\alpha$ emitters than bright [OIII] emitters at high redshifts. In addition, Figure 4 shows the significant impact of the dust model. The intrinsic result (dust free with $A_{\mathrm{V}}=0$) shows that the H$\alpha$ dominance is around 30 to 70% in this redshift range. However, when dust model is applied, almost all galaxies have stronger H$\alpha$ than [OIII] emission (>80%), indicating that the [OIII] emission line experiences more dust attenuation as predicted by the Calzetti model. Since the H$\alpha$ emission is only present at $z<2$, we will refer to H$\alpha$ galaxies and galaxies at $z<2$ interchangeably in the following section, and similarly [OIII] galaxies refer to galaxies with $z>2$.

Because galaxies have peculiar velocities, the observed redshift is different from the cosmological redshift due to cosmic expansion. This RSD effect can change the measured galaxy distribution and the resultant clustering signal. In our simulation, we add this effect into the galaxy catalog by perturbing the cosmological redshift with $v_{p}/(ac)$, where $v_{p}$ is the line-of-sight component of the velocity, $a$ is the scale factor and $c$ is the speed of light. We will present clustering measurements in both real and redshift space in the following sections.

Galaxies do not perfectly trace the underlying matter distribution. They preferentially live in the peaks of the matter density field. This makes the galaxies biased tracers of large scale structure, which preferentially sample the over-dense regions (Kaiser 1984; Bardeen et al. 1986). In addition, the processes of galaxy formation can introduce additional deviations of the galaxy distribution from matter distribution. These factors result a relationship between the spatial distribution of galaxies and the dark matter density field: the galaxy bias. Neglecting the stochasticity and non-locality, the galaxy density contrast can be written as a function of the underlying dark matter density contrast on some scale (Coil 2013) $\delta_{g}=f(\delta_{m})$, where $\delta\equiv\rho/\bar{\rho}-1$ and $\bar{\rho}$ is the mean density.

On large scales (also known as linear scales), where the density fluctuations are small and evolve linearly, we can expand the function $f$ and define the linear galaxy bias through $\delta_{g}=b\delta_{m}$. In terms of the 2-point correlation function, we can measure the galaxy bias by comparing the clustering amplitudes of galaxies and matter

where $\xi_{gg}$ and $\xi_{mm}$ are galaxy and matter correlation function as a function of spatial separation, respectively.

With the simulated galaxy catalog from Galacticus, we compute the galaxy correlation function using the Landy & Szalay (1993) estimator,

where $DD,DR$ and $RR$ are suitably normalized numbers of (weighted) data-data, data-random, and random-random pairs in each galaxy separation bin. The random catalogs are first generated with uniform distribution on a sphere and then truncated to have the same right ascension and declination boundary as the galaxy catalog. The redshifts of the random catalog are randomly drawn from the galaxy catalog to have the same radial distribution. The total number of randoms is 10 times larger than galaxy catalog to assure stable measurement of clustering. Following the same strategy as Merson et al. (2019), we measure the correlation function for each galaxy sample 5 times with different random catalogs. We measure the correlation function at spatial scales up to 150 Mpc$h^{-1}$. In Figure 5, we present the clustering measurement at a few redshift bins for galaxies with $1<z<2$ for both real and redshift space. We find that given our sample selection and dust model, the BAO peak can be recovered successfully. The worst case corresponds to high redshift galaxies with the brightest flux cut for the emission lines, which results in a low galaxy number density and thus the clustering signal is impacted by noise significantly. The attenuation from two dust models give similar impacts on the clustering amplitude, although their predictions for the number densities of ELGs are different (Zhai et al. 2019). The clustering amplitude in redshift space is higher than real space, consistent with expectation of the enhancement due to RSD effect.

In order to measure the linear bias, we compute the non-linear correlation function of dark matter $\xi_{mm}$ using the CLASS and Halofit functionality in the code Nbodykit (Hand et al. 2018) with Planck 2016 cosmology, to be consistent with the dark matter simulation used in our SAM simulation. The result is shown as the cyan curve in each panel of Figure 5. The galaxy bias is computed by taking the ratio between the correlation functions of galaxies and dark matter. The lower row of each panel shows the resultant $b^{2}$. Same as for the galaxy correlation function, the bias is also obtained by using the mean of five repeat measurements. The error bars are omitted to clearly present the result.

The figure shows that the bias is close to a constant at scales from 10 to 50 or 60 $h^{-1}$Mpc. At scales below 10 $h^{-1}$Mpc, the non-linearity of the dark matter dynamics comes into play and can induce scale dependent bias. On the other hand, at scales above 50 or 60 $h^{-1}$Mpc, the galaxy bias deviates from a constant value and presents complicated behavior, especially around the BAO scale. This feature is more significant in redshift space than in real space. This result is also pointed out in the earlier attempt presented in Merson et al. (2019), where the galaxies are H$\alpha$ only galaxies rather than the samples defined using two emission lines as in this paper. However, the cause for this distortion on the largest scales are similar, and a combination of several factors, such as the RSD effect, sample variance, mode coupling of the cosmic density perturbation and so on.

With the measured galaxy correlation function, we can estimate the constant value of bias $b$ by fitting the $b^{2}$ as shown in Figure 5 with a constant. Based on the measurement, we fit the data within $10<s<50$ $h^{-1}$Mpc. The uncertainty of the bias estimate adopts the same strategy as in Merson et al. (2019) based on the root-mean-square (RMS) of the difference between $b$ and the fitted mean $b_{\text{lin}}$

where $N$ is the number of bins for the galaxy correlation function within $10<s<50$ $h^{-1}$Mpc. As an example, Figure 6 displays the resulting measurement of linear bias for galaxies selected with $f_{\text{lin},1}=1.0\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ and $f_{\text{lin, 2R}}=1.0$, i.e. only galaxies with both H$\alpha$ and [OIII] flux brighter than $1.0\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$. It shows a close to linear relation for galaxy bias as a function of redshift, for both real and redshift space. The two dust models calibrated to match the WISP number counts and HiZELS H$\alpha$ LF respectively give quite consistent results. Note that the huge errorbar in the redshift space measurement for the galaxy subsample at highest redshift bin is due to the noisy measurement of the correlation function.

The measured galaxy bias as a function of redshift can be simply described as $b_{\text{lin}}(z)=az+b$, where $a$ and $b$ are gradient and intercept respectively. Combining this model with the bias measurements, we construct a simple $\chi^{2}$ and perform a Monte Carlo Markov Chain (MCMC) test with the python code emcee (Foreman-Mackey et al. 2013) to obtain the constraints on $a$ and $b$. We then sample from the posterior to estimate the 16 and 84 percentile as the uncertainty. In Figure 7, we show the fitting results for different dust models and sample selections. We can see that the dust model removes fainter and less massive galaxies. This can increase the average halo mass of the galaxy sample to increase the bias, as expected. The difference between the two dust models is not as significant as we find in the clustering measurements. The fitting result of the linear model is summarized in Table 1 for real space and Table 2 for redshift space with different sample selections and dust model.

The flux limit of the strong line, i.e. $f_{\text{lin},1}$ has a direct impact on the linear bias. Increasing its value selects galaxies with brighter H$\alpha$ or [OIII] emissions. This is consistent with the relationship between H$\alpha$ or [OIII] luminosity and host halo mass (see e.g. Zhai et al. 2019). However, at higher redshifts, the impact is less significant, which is partially due to the larger fraction of [OIII] dominated galaxies and thus reduces the effect. The dependence of linear bias on $f_{\text{lin, 2R}}$, the flux limit of the secondary emission line is more complicated. The result doesn’t present a monotonic relation. The reason is partially due to the flux ratio of H$\alpha$/[OIII], which has a clear dependence on the H$\alpha$ luminosity. However this dependence decreases with higher redshift (see for example Fig 7 in Zhai et al. 2019). Thus the scatter of flux ratio H$\alpha$/[OIII] at a given H$\alpha$ luminosity indicates that a galaxy with bright H$\alpha$ emission doesn’t necessarily have bright [OIII] emission and vice versa. In general, we find that the dust models as well as the flux limit for the emission lines can affect our estimate of the linear bias at a few to $\sim$ten percent level.

Halo Occupation Distribution (HOD) is a statistical approach to describe the connection between galaxies and dark matter halos. It has been used to interpret the observations of galaxy clustering over a wide range of redshifts and luminosities, see e.g. Zheng et al. (2005, 2007); Zehavi et al. (2011); White et al. (2011); Zhai et al. (2017). In practice, it splits the galaxies into centrals and satellites. The investigations of massive galaxies have built simple parametrizations to describe the functional forms for the centrals and satellites. However, our understanding of the HOD of the ELGs is relatively poor, although some pioneering work has been done based on simulated or observed ELGs, for instance Geach et al. (2012); Contreras et al. (2013); Cochrane & Best (2018); Gonzalez-Perez et al. (2018); Avila et al. (2020) and references therein.

The SAM simulation in our work provides a reasonable framework for the measurement of HOD of the ELGs. In Figure 8, we present the measured HOD using different selections and dust models for galaxies in a few redshift slices. The first prominent feature is that the HOD for centrals has a double peak shape as a function of halo mass. The valley is around $10^{12.4}M_{\sun}h^{-1}$ and this position has no significant dependence on sample selection. This double peak HOD for centrals differs from the results of studies based on luminous red galaxies (LRGs), which are consistent with a monotonic function for central occupation. We note that this double peak behavior in the HOD can be a combination of different factors, including sample selection, dust model and galaxy formation physics. In the earlier attempt of Merson et al. (2019), the HOD for H$\alpha$ galaxies is measured based on Millennium simulation. Regardless of the different sample selections and parameter sets for the SAM, they also find similar, but weaker, behavior as ours. This feature becomes less pronounced when a higher threshold for luminosity is adopted, consistent with the tendency shown in Figure 8. When we increase the flux limit for the emission line or dust attenuation to reduce the galaxy number density, the double peak feature becomes less significant. This indicates that the occupancy of the second peak is dominated by less mass galaxies. Previous work based on observational data or simulation shows that the occupation of central galaxies only peaks at low mass range and quickly declines at high mass end (Contreras et al. 2013; Gonzalez-Perez et al. 2018; Avila et al. 2020; Hadzhiyska et al. 2021).

The double-peaked nature present in the predicted HODs can be traced back to the presence of two distinct sequences in the plane of cenral galaxy star formation rate and halo mass in the Galactics models. We have examined the origin of these two sequences and find that they are the result of the fact that, in the UNIT simulation merger trees, some significant fraction of dark matter halos undergo periods of mass loss (i.e. their total mass decreases with time).

Galacticus assumes that halos accrete baryons from their surroundings at a rate proportional to their halo mass growth rate. During periods of mass loss in a halo, Galacticus instead holds the baryonic content of the halo fixed, and let it begin to increase again when the halo has grown beyond its previous greatest mass.

As a consequence, halos undergoing mass loss have no new gas supply, and so the star formation rate of their central galaxies quickly declines, leading to the formation of a second sequence of galaxies in the plane of star formation rate and halo mass.

These periods of mass loss from halos may be physical (driven by merging events which cause mass to be ejected), or may be purely numerical in origin (due to the choice of halo mass definition, or to failings in the halo finder and merger tree builder to link halos together over time). A detailed examination of the origins of these periods of mass loss, and how best to model their effect on the baryonic content of halos, is beyond the scope of this paper, but will be explored in a future work.

The satellite occupation from Galacticus is consistent with expectations, and can be represented by a functional form close to a power law, which is similar to massive galaxies at lower redshift. However, we note that this power low can break for high redshift galaxies.

In this section we present the linear bias measurements for galaxies at $z>2$. Note that for this paper, the sample selection for these high redshift galaxies ($z>2$) is different from the lower redshift sample ($z<2$); only the [OIII] line flux is used, as $f_{\text{lim, 1}}$, to define the flux-selected samples. The [OII] line can be used as the second emission line for robust redshift determination for the Roman $z>2$ ELG sample, which will be implemented in future work pending improvement of [OII] line flux predictions from Galacticus.

In the top rows of Figure 9, we present the correlation function of the galaxies with a few example redshift slices. The bias measurement $b^{2}$ is shown in the bottom row. We find that the BAO peak can be recovered with these sparser samples, compared with the $z<2$ galaxies. However, due to the lower number density, the clustering measurement becomes noisy and the BAO signal is erased to certain extent. This can be improved by enlarging the redshift bins in the analysis to include more galaxies in the measurement of the two point correlation function. The ratio of the galaxy and the dark matter correlation functions also shows a close to constant behavior at scales $10<s<50$ $h^{-1}$Mpc, similar to the H$\alpha$ galaxies and thus enables an estimate of a constant bias on linear scales.

In Figure 10, we present the bias measurement for [OIII] ELGs using the same method as in previous section. The galaxies are selected if the [OIII] line flux is higher than $0.5\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$. The stronger dust attenuation (higher $A_{V}$) selects brighter and more massive galaxies, which increases the galaxy bias as expected. Given the estimated uncertainty, the two dust models give consistent result within $1\sigma$. Compared with the H$\alpha$ ELGs, the two dust models increase the bias estimates significantly for the [OIII] ELGs, due to the fact that dust imposes more attenuation on [OIII] emission than H$\alpha$. The distribution of the measurements in the figure also presents a linear relation of galaxy bias with redshift. We fit with a linear relation as introduced in Section 3.2 and present the result in Figure 11. We note that the flux limit has a stronger effect on the bias of the $z>2$ galaxies than the $z<2$ galaxies; the bias can change by roughly 20%. The nominal depth for Roman galaxy survey of flux above $1\times 10^{-16}\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}$ is able to observe a significant number of [OIII] emitting and highly biased galaxies. This can provide robust measurement for the clustering signal to infer cosmological information. Compared with the galaxies at $z<2$, these [OIII] galaxies are more biased due to the early phases of the dark matter evolution and the redshift dependence of dark matter halo bias. The fitting result of the linear bias model is summarized in Table 3.

We measure the HOD of these [OIII] galaxies to better understand their distribution within dark matter halos and present the result in Figure 12 for a few redshift slices. The prominent feature is similar to that of the H$\alpha$ galaxies at $z<2$. The central occupation shows a clear double-peak behavior as a function of halo mass. Either it is caused by physical reasons of mass loss of dark matter halos, or numerical issues in the simulation, we will investigate this in future work. Similar to the $z<2$ galaxies, the satellite occupation is also close to a power-law form, but with some break at high redshift.

In order to further investigate whether our Galacticus simulation can make reasonable predictions for HODs, we compare the HOD results with the latest eBOSS ELG measurement (Avila et al. 2020). The eBOSS ELG program creates a catalog of thousands of galaxies within the redshift range of $0.6<z<1.1$, selected using the DECaLS photometric survey. The finalized sample has an average redshift $z=0.865$ with a number density of $n_{\mathrm{eBOSS}}=2.187\times 10^{-4}(\mathrm{Mpc}/h)^{-3}$. At this redshift, the Roman HLSS can only observe H$\alpha$ emission due to the wavelength range of its grism (see Figure 2). Therefore we use the H$\alpha$ flux to define our galaxy mock. We apply flux limit $f_{\mathrm{lim}}=5\times 10^{-16}\mathrm{ergs}^{-1}\mathrm{cm}^{-2}$ and $10\times 10^{-16}\mathrm{ergs}^{-1}\mathrm{cm}^{-2}$ with two dust models $A_{v}=1.6523$ and $A_{V}=1.9156$. This give us four different galaxy samples with number densities 1.65, 1.08, 0.38 and 0.27 times that of eBOSS ELG. We measure their HODs and compare with the eBOSS measurement in Figure 13. Although our galaxy sample has different target selection than eBOSS which uses [OII] doublet to identify galaxy redshift, our prediction of HOD has similar amplitude when the number density is close to that of eBOSS. The satellite occupancy shows excellent agreement in terms of the shape and amplitude. The halo mass dependence can be described by a power-law at high mass end. The central occupancy in both Galacticus and eBOSS shows a similar shape with a peak at intermediate mass scale, with eBOSS peaking at slightly lower mass scale. In addition, the overall amplitude of the central occupancy of Galacticus is higher and its shape flattens at high mass end instead of dropping quickly. This discrepancy can be caused by a combination of factors: the difference in the selection algorithms of Roman GRS and eBOSS, the calibration of Galacticus for the parameters governing star formation history and galaxy formation, the dust models used in the analysis and so on.

In Figure 14, we present the bias measurement of emission line galaxies for the entire redshift range of Roman HLSS. We apply $f_{\mathrm{lim}}=1\times 10^{-16}\mathrm{ergs}^{-1}\mathrm{cm}^{-2}$ and $f_{\text{lin, 2R}}=1.0$ to choose H$\alpha$ galaxies, and $f_{\mathrm{lim}}=1\times 10^{-16}\mathrm{ergs}^{-1}\mathrm{cm}^{-2}$ for [OIII] galaxies. Using the measurement with dust model $A_{\text{V}}=1.652$, we perform a linear fit of the bias measurement for H$\alpha$ and [OIII] galaxies respectively, shown as the red line with shaded area. The fitting result can also be found from Table 1, 2 and 3. In redshift space, we summarize the results as

Our previous tests show that the practical choices for the dust model and flux limits for the emission lines won’t have significant impact on the estimate of the linear bias. Therefore the result quoted above is a reasonable description for future analysis, especially for the investigation of the science forecast of Roman HLSS.