Cosmology with the Wide-Field Infrared Survey Telescope - Multi-Probe Strategies

Designing a multi-probe analysis for the galaxies observed with the WFIRST reference survey can be broadly split into the following steps:

1. 1.

   Choose broad categories of cosmological probes that are to be combined: For our WFIRST reference survey these are weak lensing, galaxy clusters, galaxy clustering (photometric and spectroscopic).

2. 2.

   Define specific probe combinations and summary statistics that make up the data points of the data vector, which in our case are one-point functions and two-point functions that represent the corresponding probes. We do not consider higher-order correlation functions.

3. 3.

   Define the galaxy samples that are associated with the aforementioned probes. We use the WFIRST exposure time calculator (ETC) (Hirata et al., 2012) to compute realistic survey scenarios for WFIRST’s coverage of area and depth in a given band. We fix the time per exposure and vary the number of exposures to build up depth over the survey area of a given scenario. For the HLS Reference Survey this area is 2,000 deg². The total survey time for a given number of exposures includes a simple prescription for overheads and is correct to approximately 10$\%$.

   In order to obtain accurate redshift distributions we closely follow Hemmati et al. (2019) in applying the ETC results to the CANDELS data set (Grogin et al., 2011; Koekemoer et al., 2011), which is the only data set available that is sufficiently deep in the near-infrared to model WFIRST observations. The ETC has a built-in option to obtain a weak lensing catalog based on an input catalog of detected sources. The criteria for galaxies to be considered suitable for weak lensing are $S/N$>18 (J+H band combined, matched filter), ellipticity dispersion $\sigma_{\epsilon}<0.2$, and resolution factor R>0.4, where we used the Bernstein & Jarvis (2002) convention (i.e. $\epsilon=(a^{2}-b^{2})/(a^{2}+b^{2})$ instead of $(a-b)/(a+b)$).

   We apply these selections to the CANDELS catalog and obtain our source sample for the WFIRST HLS 4 NIR band survey. For the lens sample we select CANDELS galaxies with $S/N>10$ in each of the 4 WFIRST bands. Our WFIRST analysis assumes LSST photometry from the ground, hence we further down-select both samples by imposing a $S/N>5$ cut in each LSST band except for u-band (we note that 50% of our galaxy sample has $S/N>5$ in the u-band as well).

   The resulting number densities for the HLS are

   	$\displaystyle\bar{n}_{\mathrm{source}}=N_{\mathrm{source}}/\Omega_{\mathrm{s}}$	$\displaystyle=$	$\displaystyle 51\;\mathrm{galaxies/arcmin^{2}}$		(1)
   	$\displaystyle\,\bar{n}_{\mathrm{lens}}=N_{\mathrm{lens}}/\Omega_{\mathrm{s}}$	$\displaystyle=$	$\displaystyle 66\;\mathrm{galaxies/arcmin^{2}}\,.$		(2)

   where $\Omega_{\mathrm{s}}$ is the WFIRST survey area. We impose a $z_{\mathrm{min}}=0.25$ for the lens sample and define 10 tomographic bins for each sample such that $\bar{n}^{i}_{x}=\bar{n}_{x}/10$. These tomographic bins are then convolved with a Gaussian distribution, which is further described in Sect. 3.3.

   We consider two different Gaussian photo-z scenarios: an optimistic variation with mean zero and narrow width of $\sigma_{z}=0.01$ and a more pessimistic scenario with broader kernel of $\sigma_{z}=0.05$. The resulting redshift distributions are depicted in Fig. 2.

   Multi-probe FoM summary
   Probe	Individual	Cumulative
   Cosmic shear	9.8	9.8
   3x2	23.46	23.46
   Clusters	3.86	31.56
   RSD+BAO	8.19	89.54
   SNIa	24.62	300.11

   • •

     Source galaxy sample, for which we require position, photometric redshift, and galaxy shape measurements.

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     Lens galaxy sample, for which we require position and photometric redshift measurements.

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     Galaxy clusters, for which we require position, photometric redshift and optical richness estimates for galaxy clusters that are identified in the overall galaxy catalog.

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     Spectroscopic galaxy sample, which requires measurements of positions and spectroscopic redshifts.

4. 4.

   Define exact analysis choices: Given that we are looking at 2-point functions as summary statistics, we need to decide on the exact auto and cross-galaxy samples that constitute a cosmological probe. Further, we need to define the exact binning within each probe, in particular which angular scales and tomographic redshift binning are considered. The decision tree for these choices is complex and takes into account our ability to accurately model physics and systematics at specific angular scales and redshifts, and in particular our ability to model the correlations across all data points in the covariance matrix. For the WFIRST data vector that we use to simulate the HLS Reference Survey, we choose:

   • •

     Source galaxies – cosmic shear: In terms of angular binning we universally choose 25 logarithmically spaced Fourier mode bins ranging from $l_{\rm{min}}=30$ to $l_{\rm{max}}=15000$ for all two-point functions in our data vector, however we impose different scale cuts for the different probes. The idea of universal binning across probes is driven by the desire to avoid computing cross-covariances of probes with different $l$-binning. For the cosmic shear part of the data vector we impose a scale cut of $l_{\rm{max}}\rm{(cosmic\,shear)}=4000$, which leaves 20 bins that carry information. The ten tomographic bins translate into 55 auto-and cross power spectra.

   • •

     Lens galaxies – photometric clustering: The redshift distribution for the lens sample is further detailed in Sect. 3.3 and divided into 10 tomographic bins. We exclude $l-$bins, if scales below $R_{\mathrm{\min}}=2\pi/k_{\mathrm{max}}=21\mathrm{\;Mpc/h}$ contribute to the Limber integral (see Eq. (5)), which imposes a redshift dependent scale cut in the $l$-binning.

   • •

     Lens $\times$ source galaxies – photometric galaxy-galaxy lensing: The galaxy-galaxy lensing part of the data vector assumes the lens galaxy sample as foreground and the source galaxy sample as background galaxies; we only consider source-lens combinations where the source bin is fully behind the lens bin in redshift. We again impose a cut-off at $R_{\mathrm{\min}}=21\mathrm{Mpc/h}$.

   • •

     Galaxy cluster number counts: This is the one one-point function we include in our data vector. We split our cluster sample into four cluster redshift bins (0.4-0.6,0.6-0.8,0.8-1.0,1.0-1.2) and 4 cluster richness bins between $\lambda_{\mathrm{min}}=40$ and $\lambda_{\mathrm{max}}=220$ in each redshift bin.

   • •

     Galaxy clusters $\times$ source galaxies – cluster weak lensing: In order to calibrate the cluster mass–richness relation (Eq. 26), we consider the stacked weak lensing signal from all combinations of cluster redshift and richness bins with source galaxies, with the restriction that source galaxies are located at higher redshift than the galaxy clusters. Specifically, we use the cluster lensing power spectrum in the angular range $4000<l<15000$, which corresponds mostly to the 1-halo cluster lensing signal.

   • •

     Spectroscopic $\times$ spectroscopic – spectroscopic galaxy clustering: While our analysis considers all cross-covariance terms for the 5 cosmological probes above, WFIRST’s spectroscopic clustering is treated as an independent probe whose cosmological information is determined separately and added a posteriori. This is an approximation, however the derivation of a 2D+3D joint covariance is beyond the scope of this paper and deferred to future work. Our spectroscopic clustering data vector is comprised of 3D power spectrum fourier modes $P(k,\mu)$ and we select 100 logarithmic bins ranging from $k_{\mathrm{min}}=0.001$ to $k_{\mathrm{max}}=0.3$ h/Mpc, 10 linearly spaced $\mu$ bins from $0-1.0$, and 7 density weighted redshift bins that start at $0.83$ and range out to 3.7. This data vector captures both the BAO and RSD information.

Given the data vector $\mathbf{D}$, we sample the joint parameter space of cosmological $\mathbf{p}_{\mathrm{c}}$ and nuisance parameters $\mathbf{p}_{\mathrm{n}}$ using the emcee¹⁵¹⁵15https://emcee.readthedocs.io/en/stable/ (Foreman-Mackey et al., 2013), which is based on the affine-invariant sampler of Goodman & Weare (2010). At each step we compute the posterior using Bayes’ theorem

$P_{r}(\mathbf{p}_{\mathrm{c}},\mathbf{p}_{\mathrm{n}})$ denotes the prior probability which in our case is based on the WFIRST SNIa survey forecast from (Hounsell et al., 2018). Specifically, we reran the “Imaging: Allz (optimistic)” scenario (c.f. Sect. 5.4 and Table 13 in Hounsell et al., 2018) centered it on the fiducial cosmology of our analysis (see Table 2). We did not include any information from CMB or BAO experiments, which explains the different contours compared to (Hounsell et al., 2018).

The cosmological information from the HLS enters our simulations through the second term in Eq. (3), i.e. the likelihood, $L(\mathbf{D}|\mathbf{p}_{\mathrm{c}},\mathbf{p}_{\mathrm{n}})=N\,\times\,\exp(-\frac{1}{2}\chi^{2}(\mathbf{p}_{\mathrm{c}},\mathbf{p}_{\mathrm{n}}))$. We assume that the errors of this data vector are distributed as a multi-variate Gaussian

which is composed of two $\chi^{2}=(\mathbf{D}-\mathbf{M})^{t}\,\mathbf{C}^{-1}\,(\mathbf{D}-\mathbf{M})$ terms reflecting our approximation that the cosmological information from HLSS and HLIS is independent. We note that future work should explore correlations between HLIS and HLSS and develop a joint covariance matrix for these measurements. $N$ is a normalization constant.

Based on the analysis choices (probes, redshifts, scales) described in Sect. 2.1 we compute the data vectors and covariance matrices for HLIS and HLSS at the fiducial cosmology and systematics parameters (see Tables 2, 5, 7, for the different probes). In case of the HLSS survey the covariance matrix is diagonal and further described in Sect. 5, in case of the HLIS the matrix has significant off-diagonal terms.

Figure 3 illustrates the structure of the matrix with the auto-probe matrices denoted as numbers 1-5 corresponding to cosmic shear (1), galaxy-galaxy lensing (2), galaxy clustering (3), cluster number counts(4), and cluster weak lensing (5). Calculation of the individual terms of the covariance can be found in the Appendix (Eqs: A2-A14) of Krause & Eifler (2017).

Since this covariance matrix is calculated analytically and not estimated from either simulations or data, it can be considered noise-free and is easily invertible. It does not inherently limit the number of data points that can enter our analysis, which would be the case if the covariance were computed from a limited set of realizations (see e.g., Taylor et al., 2013; Dodelson & Schneider, 2013, for details on these constraints).

We calculate the angular power spectrum between redshift bin $i$ of observable $A$ and redshift bin $j$ of observables $B$ at projected Fourier mode $l$, $C_{AB}^{ij}(l)$, using the Limber and flat sky approximations (we refer to e.g. Fang et al., 2019, for the potential impact when analyzing data):

where $\chi$ is the comoving distance, $q_{A}^{i}(\chi)$ are weight functions of the different observables given in Eqs. (6-7), and $P_{AB}(k,z)$ the three dimensional, probe-specific power spectra detailed below. The weight function for the projected galaxy density in redshift bin $i$,$q_{\delta_{\mathrm{g}}}^{i}(\chi)$, is given the normalized comoving distance probability of galaxies in this redshift bin

with $n_{\mathrm{lens}}^{i}(z)$ the redshift distribution of galaxies in (photometric) galaxy redshift bin $i$ (c.f. Eq. 17), and $\bar{n}_{\mathrm{lens}}^{i}$ the angular number densities of galaxies in this redshift bin (c.f. Eq. 1). For the convergence field, the weight function $q_{\kappa}^{i}(\chi)$ is the lens efficiency,

with $n_{\mathrm{source}}^{i}(z)$ the the redshift distribution of source galaxies in (photometric) source redshift bin $i$ (Eq. 17), $\bar{n}_{\mathrm{source}}^{i}$ the angular number densities of source galaxies in this redshift bin (Eq. 1), and $a(\chi)$ the scale factor.

The three-dimensional power spectra $P_{AB}(k,z)$ can be expressed through the matter density power spectrum $P_{mm}(k,z)$. For the purpose of this section $P_{mm}(k,z)$ corresponds to the density power spectrum $P_{\delta\delta}(k,z)$, where we use the Takahashi et al. (2012) fitting formula to model nonlinear evolution. Noting that $P_{AB}=P_{BA}$, we describe the different cases in Eqs. (8,9,23). For $A=\kappa$, this is trivial,

For quantities related to the galaxy density, we note that we only consider the large-scale galaxy distribution, where it is valid to assume that the galaxy density contrast on these scales can be approximated as the non-linear matter density contrast times an effective galaxy bias parameter $b_{g}(z)$

Since there is no compelling model of modified gravity, we adopt phenomenological modified gravity parameters $(\mu_{0},\Sigma_{0})$ which we define similar as e.g., Simpson et al. (2013).

In this parameterization the expressions for the Newtonian potential $\Psi$ and the curvature potential $\Phi$ that govern the perturbed Friedmann-Robertson-Walker metric

are altered. Within general relativity $\Psi=\Phi$ holds. The $(\mu,\Sigma)$ parameters give additional freedom to the Newtonian gravitational potential $\Psi$ experienced by non-relativistic particles and the lensing potential $(\Phi+\Psi)$ experienced by relativistic particles, specifically

We assume that $\mu(a)$ and $\Sigma(a)$ are both scale independent. Furthermore, since their motivation was to explain the dark energy phenomenon, we assume that the modified gravity parameters scale with the dark energy density, i.e.,

where $\Omega_{\Lambda}$ is the present day dark energy density. Note that in the case of general relativity, $\mu_{0}=\Sigma_{0}=0$.

The $\mu_{0}$ parameter modifies the growth of linear density perturbation such that

which changes the growth function, and consequently the density-density power spectrum $P_{\delta\delta}$ and all projected power spectra described in Eq. (5).

The $\Sigma_{0}$ parameter only affects lensing related quantities, which in a 3x2pt analysis means the galaxy-shear and shear-shear power spectrum. Specifically, Eq. (5) is modified as

where the exponent $k=2$ if $A=B=\kappa$, $k=0$ if $A=B=\delta_{\mathrm{g}}$, and $k=1$ if either $A=\kappa$ or $B=\kappa$.

We parameterize uncertainties arising from systematics through nuisance parameters, which are summarized with their fiducial values and priors in Table 2. Our default likelihood analysis includes the following systematics:

We use the WFIRST exposure time calculator (ETC) version 16 of Hirata et al. (2012) to compute galaxy densities and redshift distributions for our baseline scenario (c.f. Table 6) and then consider doubling (halving) the survey area, doubling (halving) the galaxy number density, and decreasing the minimum scale which we include in our analysis (see Fig. 8).

Following (Seo & Eisenstein, 2003; Wang et al., 2013) we model the cosmological information from redshift space distortions (RSD) and Baryon Acoustic Oscillations (BAO) through features in the observed power spectrum

where we assume that the 3D Fourier mode $\bf k$ can be decomposed into a line-of-sight $k_{\parallel}$ and a transverse $k_{\perp}$ component with $\mu=k_{\parallel}/|\bf k|$ as the cosine of the angle between the 3D vector and the line-of sight. The arguments for the observed power spectrum $k^{\rm ref}_{\perp}$ and $k^{\rm ref}_{\parallel}$ are computed at a reference cosmology, indicated through the superscript ^ref. In order to relate the observed power spectrum to the true underlying power spectrum a correction factor $\left([D_{A}^{\rm ref}(z)]^{2}H(z)\right)/\left([D_{A}(z)]^{2}H^{\rm ref}(z)\right)$ which accounts for the volume difference between the two cosmologies is introduced. The $P_{\mathrm{shot}}$ term describes residual uncertainties that remain after subtracting the shot noise term computed from the inverse number density of galaxies. These residuals occur, e.g., because of galaxy clustering bias (Seljak, 2000). Equation (5.1) accounts for residual redshift uncertainty in our measurement, e.g. from fitting emission lines, through the damping factor $e^{-k^{2}\mu^{2}\sigma^{2}_{r,z}}$. Following Wang et al. (2013) we consider the dewiggeled power spectrum

where $P_{0}$ defines the normalization of the linear power spectrum at redshift zero, $n_{s}$ is the spectral index, and the (dewiggeled) transfer function $T^{2}_{\rm dw}(k,z)$ is given by

where $g_{\mu}(k,z)=1-\mu^{2}+\mu^{2}[(1+f_{g}(z))^{2}-1]$ (Eisenstein et al., 2007, c.f.) and $f_{g}(z)$ being the linear growth factor.

The BAO transfer function is defined as the difference between the linear matter transfer functions with and without baryons, and the exponential damping due to nonlinear effects is only applied to the transfer function associated with BAO. The uncertainty in nonlinear effects that are still present in the power spectrum even after reconstruction (Seo & Eisenstein, 2007; Padmanabhan et al., 2012) is paramterized through

In case no reconstruction algorithm is applied nonlinear effects in structure growth, galaxy bias, and redshift space distortions are fully present and $p_{\mathrm{NL}}=1.0$. We assume an optimistic reconstruction algorithm in line with Wang et al. (2013) of $p_{\mathrm{NL}}=0.5$, which corresponds to $k_{*}=0.24h/\mathrm{Mpc}$. We allow for uncertainty in the reconstruction algorithm through varying $k_{*}$ and marginalize over a Gaussian prior with 10% uncertainty in the fiducial value.

The dewiggled model characterized through Eq. 30 will break down on small scales where RSD couples with the damping factor but has been shown to work well on quasi-linear scales (Angulo et al., 2008).

We bin the observable power spectrum linearly in $k$ (100 bins between $k_{\mathrm{min}}=0.001$ and $k_{\mathrm{max}}=0.3$) and $\mu$ (10 bins between 0 and 1) and assume 7 bins in redshift (c.f. Table 6). We model the fractional error of said power spectrum as detailed in Seo & Eisenstein (2003)

where $n$ refers to the galaxy number density within a given redshift bin, which again are specified in Table 6.

Figure 8 shows the variation of the WFIRST and BAO and RSD measurements on $w_{0}$ and $w_{a}$. We again use the emcee sampler to cover the parameter space; each chain is $>3$M steps and, in addition to the cosmological parameters mentioned in Table 2, we sample the 11 systematics parameters specified in Table 7. Specifically, we account for uncertainties in the level of shot noise $P_{\mathrm{shot}}$ (1 parameter), uncertainties in galaxy bias modeling parameterized through one free parameter $b_{i}$ in each redshift bin (7 parameters), uncertainties in redshift measurements $\sigma^{2}_{r,z}$ (1 parameter), uncertainties in modeling peculiar velocities $\sigma_{p}$ in each redshift bin (7 parameters), uncertainty in residual nonlinear effects $k_{*}$ (1 parameter).

Figure 8 shows the change in constraining power when increasing/decreasing the survey area (left), increasing/decreasing the number density of galaxies (middle panel) and when changing our fiducial $k_{\mathrm{max}}$ from 0.3 to 0.25 and 0.2. Note that the observing time is not held fixed in the left and middle panel (as opposed to the calculations in Sect. 5.2), which means that when considering twice the area in the left panel this implies doubling the observing time compared to reference HLSS survey. We summarize the FoMs in Table 8 and find that the difference for different $k_{\mathrm{max}}$ is negligible, and that there is a slight preference for going deeper compared to going wider.

We note that including an absolute measurement of the BAO scale imprinted in the CMB would notably increase the information compared to the HLSS survey alone. In Fig. 9 we include information from (Planck Collaboration et al., 2018) on the acoustic angular scale $\theta_{*}=r_{*}/(1+z)D_{a}$, where $r_{∗}$ is the comoving sound horizon at recombination and $D_{a}$ is the comoving angular diameter distance to the CMB. The combined likelihood of Planck TT, TE, EE, low-E measurements gives $\theta_{*}=0.0104109\pm 0.0000030$, which we re-center to our fiducial cosmolgy and use as a prior in Fig. 9.

In addition to the MCMC analysis in the previous subsection we explore the science return of the HLSS using a Fisher analysis on constraining the BAO scale $s$ and RSD parameter combination $f\sigma_{8}$ as a function of redshift.

For this analysis we run the ETC in BAO survey mode, using either galaxies observed in H$\alpha$ and [N_II] (compilation option -NII) or in [O_III] (-DOIII_GAL) as tracers. For the H$\alpha$ and [N_II] detections, we use model option 992, an average of the three galaxy luminosity functions given in Pozzetti et al. (2016), which were derived specifically for Euclid and WFIRST; in all cases, the [N_II] luminosity function (used to enhance the signal-to-noise of detected galaxies) is assumed to be $0.37$ times the H$\alpha$ luminosity function. For the [O_III] detections we use model 1992, an average of three luminosity functions: Mehta et al. (2015) and Colbert et al. (2013), two different analyses of the WFC3 grism, and Khostovan et al. (2015), based on ground-based narrow-band surveys. In both the H$\alpha\,$+[ N_II] and [O_III] scenarios, we use an updated galaxy size distribution from a mock catalog based on COSMOS data originally based on Jouvel et al. (2009).

The resulting redshift distributions are shown in Fig. 10, which are used as input for the trade-off studies of the forecast results in this section. In each panel, we vary the survey depth from 0.5x, 1x to 2x the fiducial depth (shown in thick, normal and thin lines respectively), while the different panels show the distributions obtained with different $S/N$ cutoffs (6.5, 5 and 3.5). As expected, the number densities increase significantly when lower $S/N$ cutoffs are chosen. Note that in this section, we explore the impact of survey depth at fixed observation time, so the area of the survey is scaled proportionally in what follows.

Using each of the above-mentioned distributions, we compute the fractional error $\sigma_{p_{i}}/p_{i}=[F^{-1}]_{ii}$ on parameter $p_{i}$, where the Fisher information matrix for parameters $p_{i}$ and $p_{j}$ is given by

assuming spatially constant galaxy density $n$, we have

There are two separate Fisher matrices, one for the RSD cosntraints on $f\sigma_{8}$, and another for the BAO constraints on $s$. For the RSD constraint, we follow McDonald & Seljak (2009) (using only one tracer) and model the observed galaxy power spectrum as in Eq. (5.1) but without the distance ratios for changing cosmology as we fix the background cosmology

and we marginalize over $\sigma_{r,z}=\sigma_{r,v}(1+z)/H(z)$. We adopt the fiducial value of $\sigma_{r,v}=0.001$ which is dominated by the observational redshift uncertainty of the grism. Furthermore, for the RSD forecast, we assume perfect reconstruction with $k_{*}=\infty$.

For the BAO constraints, we calculate errors for the Hubble parameter $H$ and the angular diameter distance $D$ and report their best constrained combination $s$. Again we use Eq. (33) but this time, modeling the galaxy power spectrum as defined in Eq. 5.2 with the following differences: First, the fractional reconstruction capability $p_{\rm NL}$ is set by how well the displacements can be determined given the level of shot noise in the data in linear theory. Second, the nonlinear damping of the BAO wiggles is modelled with a slightly different $k_{*}^{-1}=8.0\,\mathrm{Mpc}/h\times(\sigma_{8}/0.8)\,p_{\rm NL}$. Finally $\sigma_{r,z}$ is not marginalized for the BAO forecast but is fixed at the same fiducial value mentioned above.

For both BAO and RSD forecasts, we use the inverse galaxy number density for the galaxy shot noise, and the same linear bias model as in DESI Collaboration et al. (2016) for emission line galaxies (ELG) as is appropriate for the WFIRST GRS: $b_{\mathrm{ELG}}(z)D(z)=0.84$, where $D(z)$ is the growth factor normalized at $z=0$.

The Fisher matrices are computed at a fixed flat cosmology consistent with Planck 2015 best-fit (baseline model 2.6) Ade et al. (2016) and we separately evaluate fractional errors on parameters for the H_α+ [N_II] and [O_III] samples before inverse-variance combining them. In Fig. 11 we show the combined fractional error on the BAO scale $s$ (left) and RSD parameter $f\sigma_{8}$ (right). Note that the H_α is the dominant sample up to $z\approx 1.9$, beyond which the [O_III] sample becomes the only available sample.

We consider different survey strategies varying depth (top row of Fig. 11) and $S/N$ (middle row) starting from a pilot survey with default area $A=2000$ deg² and $S/N$ cutoff 5. We fix the total HLSS observation time to 0.6 years in all cases. In the top panels, we show results for a deeper (twice deeper, half the area) and a wider survey (twice the area, half the depth) compared to the pilot survey. For both $s$ and $f\sigma_{8}$, the wide survey would improve the low-$z$ constraints, whereas the deep survey is more powerful at higher $z$, as expected.

Since the aggregate constraint (shown in text beside each curve) is dominated by better errors at low-$z$, the wide survey would improve on the total constraint on parameters compared to the deep survey (e.g. 0.3% vs 0.4% for $s$ and 0.7% vs 1.1% for $f\sigma_{8}$). On the other hand, if dark energy behaviour at higher $z$ becomes an important science case, the deep survey improves constraining power by almost a factor of $2-3$ over the wide option.

In the middle row of Fig. 11, we also show the impact of different $S/N$ cutoffs for galaxy detections at fixed area and depth. We compare our default case of $S/N=5$ with a conservative $S/N=6.5$, and a more optimistic $S/N$ cutoff of 3.5. As expected, a lower $S/N$ cutoff yields better constraints everywhere in $z$, with more improvement at higher $z$ as fainter and distant galaxies are more affected by the cut. There is factor of 2 improvement at high $z$ between the curves at $S/N=6.5$ and 5. The same is true for 5 and 3.5, we however note that $S/N=3.5$ is not likely going to be a realistic value for reliable detections.

We perform a similar analysis but for an extended HLSS survey that lasts 2 years instead of 0.6 years and at only half the depth of the pilot survey, which allows us to survey 13,559 deg² (see bottom row of Fig. 11). We show results for 3 different $S/N$ cut and again find unsurprisingly that a $S/N$ cut of 3.5 improves constraining power substantially compared to the more realistic $S/N=5$ and the conservative $S/N=6.5$ cuts.