ESTIMATING LUMINOSITY FUNCTION CONSTRAINTS FROM HIGH-REDSHIFT GALAXY SURVEYS

Recent studies using Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) observations have discovered tens of candidate galaxies at redshifts z ≳ 7 (Bouwens et al. 2009, 2010a, 2010b; Oesch et al. 2010a, 2010b; Bunker et al. 2010; McLure et al. 2010; Yan et al. 2009; Wilkins et al. 2010; Labbé et al. 2009, 2010; Finkelstein et al. 2009). The new WFC3 observations have broadened our knowledge of the highest redshift galaxies yet found, complementing z ≳ 7 galaxy searches with the Near-Infrared Camera and Multi-Object Spectrometer (Kneib et al. 2004; Bouwens et al. 2004, 2005; Egami et al. 2005; Henry et al. 2007, 2008, 2009; Richard et al. 2008; Bradley et al. 2008; Bouwens et al. 2008; Zheng et al. 2009; Oesch et al. 2009; Gonzalez et al. 2010), ground-based dropout selections (Richard et al. 2006; Stanway et al. 2008; Ouchi et al. 2009a; Hickey et al. 2010; Castellano et al. 2010), and narrowband Lyα emission surveys (Parkes et al. 1994; Kodaira et al. 2003; Santos et al. 2004; Willis & Courbin 2005; Willis et al. 2008; Taniguchi et al. 2005; Stark & Ellis 2006; Iye et al. 2006; Kashikawa et al. 2006; Stark et al. 2007a; Cuby et al. 2007; Ota et al. 2008; Ouchi et al. 2009b; Hibon et al. 2010; Sobral et al. 2009). The existence of star-forming galaxies at z ≳ 7 has been well established by these studies, and the importance of these high-redshift galaxies for reionization and subsequent galaxy formation at lower redshifts will likely motivate the dedication of large telescope allocations to detailing their abundance. The purpose of this paper is to develop a method to rapidly compare possible photometric survey strategies for detecting large numbers of z ∼ 7 galaxies and to forecast constraints on the z ∼ 7 galaxy luminosity function (LF) achieved by different HST survey designs.

Determination of the constraining power a survey can obtain requires an estimate of the uncertainty in the abundance of galaxies as a function of luminosity. In addition to the Poisson uncertainty inherent in galaxy counts, cosmic sample variance induced by density fluctuations and galaxy clustering must be accounted for (see, e.g., Newman & Davis 2002; Somerville et al. 2004; Stark et al. 2007b). A particularly powerful approach for estimating these uncertainties and determining the resulting potential constraints on the LF of galaxies in the Ultra Deep Field (UDF) was presented by Trenti & Stiavelli (2008). These authors used cosmological simulations to determine the abundance and spatial distribution of dark matter halos and then applied a model for the halo mass-to-light ratio to determine the abundance of galaxies of a given luminosity. Poisson and sample variance uncertainties were estimated by drawing pencil beam realizations of the survey from the cosmological volume (see also Kitzbichler & White 2007). A maximum-likelihood approach was then used to study constraints on the LF for various dropout selections.

The maximum-likelihood estimation of LF parameters based on mock catalogs can account for detailed selection effects and spatial correlations in addition to the Poisson and sample variance uncertainties. Such simulations have clear advantages for estimating the completeness of magnitude-limited surveys or understanding systematic effects introduced by dropout color selections. However, the need for halo catalogs from cosmological simulations introduces two limitations. First, the volume of the simulation should probe many independent realizations of the modeled survey. However, future high-redshift galaxy surveys with WFC3 or other instruments may take the form of deeper versions of surveys such as the Spitzer Extended Deep Survey (SEDS; Fazio et al. 2008), Exploration Science program, or the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007b, 2007a). These surveys are each ∼1 deg² or larger, and large volume (L ≳ 200 h⁻¹ Mpc) cosmological simulations are required to probe multiple independent samples of the surveys' high-redshift galaxy populations. For instance, a ∼1 deg² survey at 6.5 ≲ z ≲ 7.5 has a comoving volume of V ≈ 3 × 10⁶ h⁻³ Mpc³. The largest simulation used by Trenti & Stiavelli (2008) would provide less than two independent samples of such a volume, and even the Millennium Simulation (Springel et al. 2005) with a comoving box size L = 500 h⁻¹ Mpc would only provide ∼40 independent samples of such a wide high-redshift survey. Second, the method requires access to and manipulation of cosmological simulation results. This requirement may pose an unwanted computational overhead for those interested in rapid estimates and comparisons of potential constraints from a wide range of survey designs.

A simpler methodology for estimating survey constraints on the abundance of high-redshift galaxies that does not directly require halo catalogs from cosmological simulations is therefore desirable for performing rapid comparisons of survey designs. Hence, we seek an approximate method for forecasting LF parameter constraints that relies on descriptions of galaxy and dark matter halo abundance and clustering, Poisson and cosmic sample variance, and parameter covariances that are analytical or easily calculable through numerical methods. We utilize a simple model for describing the clustering of z ∼ 7 galaxies based on fiducial empirical estimates of the high-redshift galaxy LF (Oesch et al. 2010b) and abundance matching between galaxies and dark matter halos (e.g., Conroy et al. 2006; Conroy & Wechsler 2009). We then adopt a common approach to translate galaxy clustering and matter fluctuations into an estimate of the cosmic sample variance (see the various calculations in, e.g., Newman & Davis 2002; Somerville et al. 2004; Stark et al. 2007b; Trenti & Stiavelli 2008). With this estimate of the sample variance and Poisson uncertainty from an assumed fiducial model for the abundance of galaxies, we use a Fisher matrix formalism to characterize the likelihood function and estimate z ∼ 7 LF parameter constraints. The presented methodology is fast and flexible, and can be used with appropriate extensions, to estimate constraints on galaxy abundance for other survey sample selections and redshifts.

Motivated by the exciting recent HST WFC3 results, we focus on modeling z ∼ 7 dropout survey designs. While we choose to study broadband searches for high-redshift galaxies, narrowband surveys for high-redshift Lyα emission present another interesting class of survey designs. The rapid progress in detecting increasing numbers of high-redshift Lyα emitters using narrowband surveys has motivated theoretical efforts both to understand and predict the abundance of Lyα emitters. The observable properties of the high-redshift Lyα emitter population are particularly difficult to model owing to the uncertain escape fraction, intergalactic medium absorption, and other radiative transfer effects, as well as uncertainties in our knowledge of the galaxy formation process (e.g., Haiman 2002; Santos 2004; Barton et al. 2004; Wyithe & Loeb 2005; Le Delliou et al. 2006; Hansen & Oh 2006; Davé et al. 2006; Furlanetto et al. 2006; Tasitsiomi 2006; McQuinn et al. 2007; Nilsson et al. 2007; Mao et al. 2007; Kobayashi et al. 2007, 2010; Stark et al. 2007b; Mesinger & Furlanetto 2008; Fernandez & Komatsu 2008; Tilvi et al. 2009; Dayal et al. 2009, 2010; Samui et al. 2009). While the astrophysics involved in these studies are tremendously interesting and provide another route to probe high-redshift galaxies, we will only examine the statistical constraining power of various broadband survey designs and will not attempt to model the Lyα emitter population.

This paper is organized as follows. Forecasting constraints on z ∼ 7 LF function parameters requires a model of the sources of error and covariances in the data. In Section 2, we discuss sample variance uncertainties owing to cosmic density fluctuations. In Section 3, we review the Fisher matrix formalism and show how to apply the formalism to forecast LF parameter uncertainties accounting for cosmic sample and Poisson variances. To perform actual forecasts for the z ∼ 7 LF, we review existing HST WFC3 survey data in Section 5 and define fiducial model surveys in Section 6. In Section 7, we combine the expected constraints from existing surveys with forecasts of LF constraints from model surveys. We discuss our results and possible caveats in Section 8, and summarize and conclude in Section 9.

Throughout, we work in the context of a ΛCDM cosmology consistent with joint constraints from the 5 year Wilkinson Microwave Anisotropy Probe, Type Ia supernovae, Baryon Acoustic Oscillation, and Hubble Key Project data (Freedman et al. 2001; Percival et al. 2007; Kowalski et al. 2008). Specifically, we adopt a Hubble parameter h = 0.705, matter density Ω_m = 0.274, dark energy density Ω_Λ = 0.726, baryon density Ω_b = 0.0456, relativistic species density Ω_r = 4.15 × 10⁻⁵, spectral index n_s = 0.96, and rms density fluctuations in 8 h⁻¹Mpc radius spheres of σ₈ = 0.812 (Komatsu et al. 2009). We report all magnitudes in the AB system (Oke & Gunn 1983).

We wish to evaluate the relative merits of various galaxy survey designs in terms of their ability to constrain the galaxy LF. To perform this evaluation, we must determine the quality of each design in terms of the number of galaxies of a given luminosity the survey will discover (the Poisson variance) and the intrinsic scatter expected for the survey volume given variations in the cosmological density field (the cosmic sample variance). This section of the paper formally defines each source of uncertainty and describes how these variances are calculated.

We define cosmic sample variance as the fluctuations in a volume-averaged quantity owing to density inhomogeneities seeded by the matter power spectrum. We will use the terms "cosmic variance" and "sample variance" interchangeably, but elsewhere in the literature cosmic variance is taken to equal the sample variance only in the limit of the entire volume of the universe (e.g., Hu & Kravtsov 2003).

2.1. Dark Matter Density Variance

Density fluctuations, or differences between the local matter density ρ_m(x) and the mean matter density $\bar{\rho }_{m}$, can be described in terms of a local matter overdensity $\delta (\mathbf {x}) \equiv [\rho _{m}(\mathbf {x}) - \bar{\rho }_{m}]/\bar{\rho }_{m}$. Consider a survey of comoving volume V at redshift z. For "unbiased" quantities measured within V that spatially cluster like the dark matter, such that the two-point correlation function ξ(r) is identical to the dark matter correlation function ξ_m(r) = 〈δ(x)δ(x + r)〉, the cosmic variance is simply the dark matter variance

$$\begin{eqnarray} &&D^{2}(z)\sigma _{\mathrm{DM}}^{2} \equiv \langle \delta ^{2}(\mathbf {x}) \rangle = D^{2}(z) \int \frac{{d}^{3} k}{(2\pi)^{3}} P(k) |\hat{W}(\mathbf {k},V)|^{2},\quad \end{eqnarray} \tag{ 1 }$$

where $\hat{W}(\mathbf {k},V)$ is the Fourier transform of the survey volume window function W(x) (whose geometry may introduce a dependence on the direction of the wavenumber k), D(z) is the linear growth function, and P(k) is the isotropic linear ΛCDM power spectrum. To calculate P(k), we use the transfer function of Eisenstein & Hu (1998) that includes the effects of baryons. We ignore possible nonlinear corrections to the power spectrum (e.g., Peacock & Dodds 1996; Smith et al. 2003). The window function is normalized such that ∫d³xW(x) = 1. For a spherical volume of comoving radius R = 8 h⁻¹ Mpc, Equation (1) would provide σ_DM = σ₈ at redshift z = 0. The linear growth function

$$\begin{equation} D(z) = D_{0} H(z) \int _{z}^{\infty } \frac{(1+z^{\prime }) {d}z^{\prime }}{H^{3}(z^{\prime })} \end{equation} \tag{ 2 }$$

has a normalization constant D₀ chosen such that D(z = 0) = 1. The Hubble parameter

$$\begin{eqnarray} H(z) &=& H_{0}[\Omega _{\mathrm{r}}(1+z)^{4} + \Omega _{\mathrm{m}}(1+z)^{3} \nonumber \\ &&+ (1-\Omega _{\mathrm{m}}-\Omega _{\Lambda }-\Omega _{\mathrm{r}})(1+z)^{2} + \Omega _{\Lambda }]^{1/2} \end{eqnarray} \tag{ 3 }$$

describes the rate of change of the universal scale factor $H\equiv \dot{a}/a$ as a function of the matter density Ω_m, relativistic species density Ω_r, and dark energy density Ω_Λ (taken to be a cosmological constant).

2.2. Dark Matter Halo Abundance and Clustering

For galaxy surveys, where quantities of interest depend on the abundance and clustering of galaxies, the sample variance will depend on the bias of dark matter halos hosting the observed systems. We can define the bias in terms of the correlation function as b² = ξ_h/ξ_m, where ξ_h is the correlation function of dark matter halos. Given a halo mass function, the bias of halos with mass m can be estimated using the peak-background split formalism (e.g., Kaiser 1984; Mo & White 1996; Sheth & Tormen 1999) or measured directly from the simulations via the halo correlation function or the halo power spectrum. We adopt the latter approach.

We use the dark matter halo mass function measured by Tinker et al. (2008) from a large suite of cosmological simulations (Kravtsov et al. 2004; Warren et al. 2006; Crocce et al. 2006; Gottlöber & Yepes 2007; Yepes et al. 2007). The Tinker et al. (2008) mass function can be written as a function of the dark matter halo mass m in terms of the "peak height,"

$$\begin{equation} \nu = \frac{\delta _{c}}{ D(z) \sigma (m)}, \end{equation} \tag{ 4 }$$

where δ_c = 1.686 is the spherical collapse barrier (see, e.g., Gunn & Gott 1972; Bond & Myers 1996), D(z)σ(m) is the square root of the dark matter variance (Equation (1)) evaluated in a spherical volume of comoving radius $R = (3m/4\pi \bar{\rho }_{m})^{1/3}$. Here, $\bar{\rho }_{m}$ is the background matter density. The linear growth function D(z) is given by Equation (2).

With the definition of peak height ν in Equation (4), the Tinker et al. (2008) halo mass function can be written as

$$\begin{equation} \frac{{d}n_{\mathrm{h}}}{{d}m} = \frac{\bar{\rho }_{m}}{m} f(\nu) \frac{{d}\nu }{{d}m}, \end{equation} \tag{ 5 }$$

where the function

$$\begin{equation} f(\nu) = \alpha [ 1 + (\beta \nu)^{-2\phi }]\nu ^{2\eta }e^{-\gamma \nu ^{2}/2} \end{equation} \tag{ 6 }$$

is often called the "first crossing distribution." Tinker et al. (2008) find that the parameter values α = 0.245, β = 0.757, γ = 0.853, ϕ = −0.659, and η = −0.341 fit well the z = 2.5 simulated mass function measured for halos defined with a spherical overdensity Δ = 200 relative to the background matter density (see also Section 4 of Tinker et al. 2010). The abundance of dark matter halos more massive than m is then just n_h(>m) = ∫^∞_m(dn_h/dm)dm.

We choose the Tinker et al. (2008) mass function because it is accurate to ≲5% for halos in the mass range 10¹¹ h⁻¹ M_☉ ⩽ m ⩽ 10¹⁵ h⁻¹ M_☉ at redshift z = 0 and improves on previous approximations by 10%–20% (see, Sheth & Tormen 1999). Tinker et al. (2008) demonstrate that the halo mass function does not have a redshift-independent, universal form and that the normalization of the first crossing distribution evolves at the 20%–50% level between z = 0 and z = 2.5. However, the halo mass function has not been calibrated at the redshifts of interest (z ≫ 2.5), and following the advice in Section 4 of Tinker et al. (2008) we will use the z = 2.5 first crossing distribution as the best available approximation². We note that using any other previously published mass function from the literature (e.g., Sheth & Tormen 1999) will therefore introduce an unknown error in the abundance of halos at high redshifts. Tinker et al. (2008) estimate this error could be as large as ∼20%–50% for galaxy-sized halos.³

For the bias b relating halo and dark matter clustering, we will use the results of Tinker et al. (2010) who measure the halo bias as a function of peak height ν in a manner consistent with the halo mass function of Tinker et al. (2008). The bias function b(ν) is constrained by the halo first crossing distribution f(ν) by requiring that dark matter is not biased against itself, i.e.,

$$\begin{equation} \int b(\nu) f(\nu) {d}\nu = 1. \end{equation} \tag{ 7 }$$

Under this constraint, Tinker et al. (2010) find that the fitting function

$$\begin{equation} b(\nu) = 1 - A\frac{\nu ^{a}}{\nu ^{a} + \delta _{c}^{a}} + B\nu ^{b} + C\nu ^{c} \end{equation} \tag{ 8 }$$

with parameters A = 1.0, a = 0.1325, B = 0.183, b = 1.5, C = 0.265, and c = 2.4 provides an accurate match to the bias of dark matter halos defined with a spherical overdensity of Δ = 200 relative to the background matter density. As demonstrated by Tinker et al. (2010), Equation (8) reproduces the simulated halo clustering better than the analytical formulae of Mo & White (1996) or Sheth et al. (2001) calculated using the peak-background split formalism. Tinker et al. (2010) find that the bias b(ν) as a function of peak height ν is nearly redshift independent, and we will adopt Equation (8) for b(ν) at all redshifts.

2.3. Galaxy Abundance and Clustering

Our main premise is to use the Fisher matrix approach to estimate the constraints on a model for the abundance of galaxies that reproduces well the observed source counts. The definition of the model for galaxy abundance is therefore important. Further, the clustering of galaxies directly influences the covariances of the data, and knowledge of the galaxy spatial distribution is therefore also quite important. Since we are interested in the characteristics of the high-redshift galaxy population for which little clustering information is known, we will estimate the galaxy clustering bias by associating luminous galaxies with dark matter halos of similar comoving abundance and assigning those galaxies the bias of their associated halos (as provided by Equation (8)). When more detailed clustering information is available, as is the case at lower redshifts, additional constraints on the connection between galaxy and halo populations are attainable (see, e.g., Lee et al. 2009).

We will adopt the commonly used Schechter (1976) model for the abundance of galaxies. Specifically, the expected number density $\bar{n}_{i}$ of galaxies in the ith luminosity bin of width ΔM about magnitude M_i can be written as

$$\begin{equation} \bar{n}_{i} = \int _{M_{i}-\Delta M/2}^{M_{i}+\Delta M/2} \Phi (M){\it dM}, \end{equation} \tag{ 9 }$$

where the Schechter (1976) function

$$\begin{eqnarray} &&\Phi (M) = \case{2}{5} \ln (10) \phi _{\star }\big[10^{\frac{2}{5}(M_{\star }- M)}\big]^{\alpha +1} \exp \big[ - 10^{\frac{2}{5}(M_{\star }-M)}\big]\quad \ \ \ \end{eqnarray} \tag{ 10 }$$

describes the distribution of galaxy luminosities, ϕ_⋆ is the LF normalization in comoving Mpc⁻³ mag⁻¹, M_⋆ is the characteristic galaxy luminosity in AB magnitudes, and α is the faint-end slope. We will often refer to the parameters of this Schechter (1976) model in terms of the vector p = [log₁₀ϕ_⋆, M_⋆, α], and it is these parameters for which we will forecast constraints. The fiducial values for the parameters p used in the Fisher matrix calculation will be selected in Section 5.

In a manner similar to Equation (9), we can also define the comoving abundance $\bar{n}_{L}$ of galaxies more luminous than magnitude M as $\bar{n}_{L}(<M) = \int _{-\infty }^{M} \Phi (M){\it dM}$, where the negative lower limit follows from the definition of magnitudes.

With a model for the abundance of galaxies, we will associate galaxies with dark matter halos of similar abundance to estimate the galaxies' spatial clustering. The abundance $\bar{n}_{i}$ of galaxies in the range M_i ± ΔM/2 can be written as $\bar{n}_{i} \equiv \bar{n}_{L}(<M_{i} + \Delta M/2) - \bar{n}_{L}(<M_{i} - \Delta M/2)$. We match the abundance of galaxies and halos as

$$\begin{equation} n_{\mathrm{h}}(> m) \simeq \bar{n}_{L}(<M) \end{equation} \tag{ 11 }$$

at the minimum and the maximum luminosity of galaxies in each magnitude bin (e.g., M = M_i + ΔM/2 and M = M_i − ΔM/2), which provides a mass range m_i ± Δm/2 of halos with a similar abundance (e.g., Conroy et al. 2006; Conroy & Wechsler 2009). The comoving number density of these halos is simply n_h,i ≡ n_h(>m_i − Δm/2) − n_h(>m_i + Δm/2), with $n_{\mathrm{h},i}= \bar{n}_{i}$. The resulting connection between galaxy luminosity and halo mass is simplistic, but more sophisticated stellar mass–halo mass relations could be incorporated into our approach when warranted by the constraining power of the available data (see, e.g., Behroozi et al. 2010). We adopt ΔM = 0.25 mag throughout, but we have checked that our conclusions also hold for ΔM = 0.5 or ΔM = 0.1.

The bias b_i of galaxies in the range M_i ± ΔM/2 is then approximated as the number-weighted average clustering of halos of mass m_i ± Δm/2. We can express b_i as

$$\begin{eqnarray} b_{i} &=& \left[\int _{m_{i} - \Delta m/2}^{m_{i} + \Delta m/2} b(m) \frac{{d}n_{\mathrm{h}}}{{d}m} {d}m\right] \,{\times}\, \left[\int _{m_{i} - \Delta m/2}^{m_{i} + \Delta m/2} \frac{{d}n_{\mathrm{h}}}{{d}m} {d}m\right]^{-1},\nonumber\\ \end{eqnarray} \tag{ 12 }$$

where b(m) ≡ b[ν(m)] is as defined in Equation (8).

2.4. Sample Covariance from Galaxy Abundance and Clustering

Equation (9) provides the average expected abundance of galaxies in a survey, given our fiducial LF model Φ(M). Owing to spatial density fluctuations on large scales, the actual measured number density of galaxies in the magnitude range M ± ΔM/2 at location x will be

$$\begin{equation} n_{i}(\mathbf {x},z) = \bar{n}_{i}[1 + b_{i} \delta (\mathbf {x},z)], \end{equation} \tag{ 13 }$$

where δ(x, z) is the local linear overdensity and the bias b_i is determined in Equation (12). The large-scale structure of the matter density field will cause the galaxy counts to covary. The sample covariance S_ij between galaxies in the ith and jth magnitude bins is simply the average squared difference between the measured galaxy density n and the expected average galaxy density $\bar{n}$ for each bin. We can then write the sample covariance as

$$\begin{equation} S_{ij}\equiv \langle (n_{i} - \bar{n}_{i})(n_{j} - \bar{n}_{j})\rangle, \end{equation} \tag{ 14 }$$

where the average is taken over all N_fields fields of the survey. Given Equations (1), (8), (9), (12), and (13), we can evaluate the elements of the sample covariance matrix S as

$$\begin{equation} S_{ij}= \frac{b_{i} b_{j} \bar{n}_{i} \bar{n}_{j}}{N_{\mathrm{fields}}} D^{2}(z)\int \frac{{d}^{3} k}{(2\pi)^{3}} \hat{W}_{i}(\mathbf {k}) \hat{W}^{\star }_{j}(\mathbf {k}) P(k), \end{equation} \tag{ 15 }$$

where $\hat{W}_{i}(\mathbf {k})$ is the k-space window function for the survey field volume of the ith magnitude bin. Depending on, e.g., the redshift distribution of sources with different magnitudes, or some luminosity-dependent completeness, we could have $\hat{W}_{i}(\mathbf {k})\ne \hat{W}_{j}(\mathbf {k})$ in general. However, throughout the rest of the paper we will consider only galaxy densities and variances within the entire effective survey volume, such that the elements of the sample covariance matrix S refer to luminosity bins within the same volume of each field. We will therefore write $\hat{W}_{i}(\mathbf {k})\hat{W}^{\star }_{j}(\mathbf {k}) = |\hat{W}_{V}(\mathbf {k})|^{2}$, where

$$\begin{equation} \hat{W}_{V}(\mathbf {k}) = \mathrm{sinc}\left(\frac{k_{x}r\Theta _{x}}{2}\right)\mathrm{sinc}\left(\frac{k_{y}r\Theta _{y}}{2}\right)\mathrm{sinc}\left(\frac{k_{z} \delta r}{2}\right) \end{equation} \tag{ 16 }$$

is an approximate k-space window function for the effective volume V of a survey at comoving radial distance r, comoving radial width δr, and rectangular area Θ_x × Θ_y in square radians.⁴ The function sinc(x/2) = 2sin(x/2)/x is the Fourier transform of the Heaviside Π(x) unit box. Similar window functions were adopted by Newman & Davis (2002) and Stark et al. (2007b) in their estimates of cosmic variance.

Unless otherwise specified, when discussing the cosmic sample variance uncertainty or error we will refer to the averaged quantity

$$\begin{equation} \sigma _{\mathrm{CV}}\equiv \langle b\rangle D(z)\sigma _{\mathrm{DM}}/\sqrt{N_{\mathrm{fields}}}, \end{equation} \tag{ 17 }$$

where 〈b〉 is the average bias of all galaxies in a survey field. This quantity σ_CV is the cosmic variance uncertainty that is often reported for surveys, but is distinct from the elements sample covariance matrix S_ij since the latter involves the bias of galaxies in individual luminosity bins.

2.4.1. Multiple Fields and Sample Covariance

Splitting a survey into N_fields multiple fields can reduce the sample covariance in the combined data by probing statistically independent regions in space. The sample variance will scale roughly as S ∝ 1/N_fields (e.g., Newman & Davis 2002); however, the actual gain depends on the shape of the ΛCDM power spectrum through the survey volume and geometry. For a fixed amount of observing time, splitting a survey into N_fields = 2 fields will rescale the volume of each field by V ∝ 1/N_fields and result in a corresponding increase in the typical dark matter density fluctuations in each field. For very large surveys, the decrease in the volume per field can (at least partially) offset the gains achieved by probing multiple independent samples (Muñoz et al. 2009).

The left panel of Figure 1 shows the rms density fluctuations $D(z)\sigma _{\mathrm{DM}}/\sqrt{N_{\mathrm{fields}}}$ in a survey at redshifts 6.5 ⩽ z ⩽ 7.5 as a function of total area for multiple fields (N_fields = 1, 2, 4). For a galaxy survey, the sample variance in each luminosity bin will be increased by a factor of the galaxy bias (see Equation (15)). While the uncertainty from rms density fluctuations will improve with the addition of statistically independent samples, the fractional improvement is less than $1-1/\sqrt{N_{\mathrm{fields}}}$ for large volumes. The right panel of Figure 1 shows the fractional improvement gained by multiple fields. For a flat power spectrum, the improvement would be $1-1/\sqrt{2}=0.293$ for N_fields = 2 and $1-1/\sqrt{4}=0.5$ for N_fields = 4.

Zoom In Zoom Out Reset image size

2.5. Poisson Variance from Galaxy Abundance

Number-counting statistics will naturally introduce a Poisson variance into the galaxy number count statistics. The diagonal Poisson variance matrix P will only add to the total covariance for counts within a single magnitude bin (i.e., only when i = j). For definiteness, we will express the Poisson covariance as

$$\begin{equation} P_{ij} = \frac{\delta _{ij} \bar{n}_{i}}{V_{i}}, \end{equation} \tag{ 18 }$$

where the Kronecker δ_ij = 1 for i = j and δ_ij = 0 for i ≠ j.

Fisher (1935) illustrated how to infer inductively the properties of statistical populations from data samples. By approximating the likelihood function as a Gaussian near its maximum and assuming a parameterized model, the uncertainties in the model parameters allowed by a future data set can be estimated directly from the data covariances. Interested readers should refer to the excellent and detailed discussion of the Fisher matrix formalism provided in Section 2 of Tegmark et al. (1997), but we outline the general approach below.

We aim to estimate the uncertainties on model parameters p_μ (the "parameter covariance matrix" C) achieved by the data produced by some fiducial survey. The quality of the future data for each fiducial survey will be characterized by the "data covariance matrix" D. The elements D_ij of the total data covariance matrix are simply the sum of the sample covariance and Poisson uncertainties described in Sections 2.4 and 2.5, which we can write as

$$\begin{equation} D_{ij}= S_{ij}+ P_{ij}, \end{equation} \tag{ 19 }$$

where P_ij only contribute when i = j (see Equation (18)). Following Lima & Hu (2004), who applied the Fisher matrix approach to the parameter estimation of the mass-observable relation in galaxy cluster surveys, we will express our approximate Fisher matrix as

$$\begin{equation} F_{\mu \nu }= \sum _{ij} \frac{\partial \bar{n}_{i} }{\partial p_{\mu }} (\mathbf {D}^{-1})_{ij} \frac{\partial \bar{n}_{j} }{\partial p_{\nu }} + \frac{1}{2} \mathrm{Tr}\left[\mathbf {D}^{-1}\frac{\partial \mathbf {S}}{\partial p_{\mu }}\mathbf {D}^{-1}\frac{\partial \mathbf {S}}{\partial p_{\nu }}\right] \end{equation} \tag{ 20 }$$

(see, e.g., Holder et al. 2001; Hu & Kravtsov 2003; Lima & Hu 2005; Hu & Cohn 2006; Cunha & Evrard 2009; Wu et al. 2009, and especially the discussion in Section III of Lima & Hu 2004). Here, Tr(A) = ∑^m_i=1A_ii for an m × m matrix. The vector elements p_μ reflect the parameters of the data model $\bar{n}$. The first term of Equation (20) models second derivatives of the likelihood function in the Poisson error-dominated regime (Holder et al. 2001), while the second term models the sample covariance-dominated regime (Tegmark et al. 1997, see also Appendix A of Vogeley & Szalay 1996). The derivatives $\partial \bar{n}/ \partial p_{\mu }$ of the LF model are computed directly by differentiating Equation (9). The derivatives ∂S/∂p_μ of the sample covariance matrix are evaluated numerically since changes to the model LF alter the galaxy bias b in Equation (15) for a given luminosity bin in a nontrivial way.

Once the Fisher matrix F is calculated, estimating the parameter covariance matrix C becomes straightforward. The elements of the parameter covariance matrix are approximated as

$$\begin{equation} C_{\mu \nu }\approx (\mathbf {F}^{-1})_{\mu \nu }. \end{equation} \tag{ 21 }$$

The marginalized uncertainty on parameter p_μ is then

$$\begin{equation} \sigma _{\mu } \equiv C_{\mu \mu }^{1/2} = (\mathbf {F}^{-1})_{\mu \mu }^{1/2}. \end{equation} \tag{ 22 }$$

Similarly, we can estimate the unmarginalized error on each parameter as σ^u_μ = F^−1/2_μμ. However, in what follows when we discuss the "error" or "uncertainty" on LF parameters we mean the marginalized error unless otherwise stated.

We will apply the above formalism to estimate the relative constraining power of possible galaxy surveys, but we will focus on evaluating such surveys in the context of existing and forthcoming data from observational programs already underway (i.e., the WFC3 Ultradeep Field Guest Observation and Great Observatories Origins Deep Survey Early Release Science (GOODS ERS) data). Our statistical formalism provides a simple way to incorporate constraints from prior data. The combined constraints C_combo of a prior observation C_prior supplemented by the forecasted constraints of a future survey C can be estimated as

$$\begin{equation} \mathbf {C}_{\mathrm{combo}}= \big(\mathbf {C}^{-1} + \mathbf {C}_{\mathrm{prior}}^{-1} \big)^{-1}, \end{equation} \tag{ 23 }$$

or, in other words, the combined parameter covariance matrix is the inverse of the sum of the Fisher matrices of the prior and future surveys. Depending on the magnitude of the off diagonal elements of C and C_prior, the combined parameter covariance matrix C_combo can provide a substantially different correlation between parameters than either the prior or future surveys produce individually. As a result, the marginalized uncertainty on parameters can benefit substantially by combining surveys with different characteristics. These ramifications of Equation (23) will become more apparent in Section 7.

Our statistical formalism for forecasting constraints on properties of the galaxy population requires a model for the survey data. Given the approach outlined in Section 2, the relevant characteristics of each survey include the total area A_tot, the number of fields N_fields, and the minimum and maximum redshifts of the survey, z_min and z_max (that, in combination with A_tot, determine the survey volume V). The limiting magnitude depth of the survey M_max strongly influences source statistics of the survey by determining the faintest luminosity bin calculated via Equation (9). The completeness of the survey f_comp and halo occupation fraction f_occ change the cosmic sample variance by altering the halo–galaxy correspondence in Equation (11).

Of these survey characteristics, we will keep z_min, z_max, f_comp, and f_occ fixed between surveys. We will assume that the surveys are effectively volume-limited (f_comp = 1) over the redshift range of interest. Given a complete volume-limited survey, the choice of minimum and maximum redshifts roughly corresponds to the filter choice defining a dropout selection. We will adopt z_min = 6.5 and z_max = 7.5, which roughly approximates the redshift selection of the (z₈₅₀−Y₁₀₅) versus (Y₁₀₅−J₁₂₅) color selection of Oesch et al. (2010b, see their Figure 1). Similar selections can be defined for I₈₁₄ dropouts. Our calculations can easily be extended to different redshift selections, but we adopt this redshift range since the fiducial abundance of z ∼ 7 WFC3 UDF GO galaxy candidates appears increasingly robust (see, e.g., the discussion in Section 2 of McLure et al. 2010), the characteristic ultraviolet (UV) luminosity of galaxies is decreasing with redshift (e.g., Bouwens et al. 2008), and the abundance of dark matter halos hosting galaxies is rapidly declining at earlier epochs.

Our model surveys will consist of WFC3 H-band coverage with equal coverage in an additional, bluer WFC3 filter. The existing and ongoing UDF GO and GOODS ERS surveys will use the F160W band (see Section 5), but using the F140W band buys ≈0.3–0.5 mag in sensitivity for the same exposure time, depending on the source luminosity (see below). While the dropout color selection is perhaps better for F160W, we will assume in our forecasts that future surveys will utilize F140W. Our results would be similar for F160W surveys to similar limiting depths. We will characterize the abundance of high-redshift galaxies in terms of a rest-frame UV LF. We must therefore adopt a color conversion between H_AB magnitude and rest-frame luminosity appropriate for z ∼ 7. In an approximation to the conversion used by Oesch et al. (2010b), we estimate that H_AB ≈ 29.0 translates into M_UV ≈ −18.2. We have checked that our general conclusions about the relative constraining power of survey designs are insensitive to changes in this conversion (e.g., ±0.4 mag in M_UV).

Lastly, HST observations are conducted using some number N_orbits per pointing that effectively determines M_max. During each orbit, the field visibility depends on the field declination, and the available on-source integration time also depends on observatory and instrument overheads such as guide star acquisition, filter changes, dithering, and readout. The UDF GO and GOODS ERS surveys are at a declination of δ ≈ −27 deg, and for ease of comparison we will assume all future surveys have |δ| < 30 deg. This roughly equatorial declination range provides a visibility of 54 minutes/orbit.⁵ Given the additional observatory and instrument overheads, we will calculate all sensitivities using a 46 minutes/orbit exposure time. Given the compact character of the observed WFC3 z ∼ 7 galaxy candidates (e.g., Oesch et al. 2010a), we will report optimum 5σ point source sensitivities. Table 1 lists these sensitivities for a flat F_ν spectrum source, as a function of N_orbits for both the F140W and F160W filters.⁶

The discussion in Section 2 makes clear that the combination of different survey designs can potentially provide increased constraints beyond that achieved by individual data sets. Even duplicate surveys will reduce the Poisson variance and potentially the sample variance (especially if the fields are widely separated on the sky). In the absence of significant systematic biases, using prior data will generally improve the expected parameter uncertainty obtained by future experiments. We will therefore rely on the expected constraints achieved by the ongoing WFC3 UDF GO (PI Illingworth, Program ID 11563) and GOODS ERS (PI O'Connell, Program ID 11359) programs to augment the fiducial survey designs evaluated in Sections 6 and 7. In this section, we will calculate the expected constraints provided by the UDF GO and GOODS ERS data.

Table 2 describes the field geometry, number of fields N_fields, total area, and expected H-band 5σ point source depth for the UDF GO and GOODS ERS survey designs. Numerous analyses of the initial UDF GO data release have already been performed (e.g., Bouwens et al. 2010a; Oesch et al. 2010b, 2010a; McLure et al. 2010; Bunker et al. 2010; McLure et al. 2010; Yan et al. 2009; Finkelstein et al. 2009), but we will consider the expected constraints provided by the entire 192 orbit program. The GOODS ERS data have not yet been released (but for initial analyses on unreleased ERS data see Wilkins et al. 2010; Labbé et al. 2009), and we will also use the Fisher matrix approach to estimate the constraints provided by that survey.

Table 2. Existing Surveys

Survey	Field Geometry	Nfields	Total Area	Norbitsa	H-band Depth	Ref.
Name			(arcmin2)		(AB Mag.)
UDF GO	205 × 227	2	9.3	19	28.76	1
	205 × 227	1	4.7	38	29.14
GOODS ERS	52 × 103b	1	53.3	3	27.73c	2

Notes. ^aThe number of orbits per pointing. ^bThe GOODS ERS survey is a 2 × 4 WFC3 IR mosaic dithered to match the UV/visible channel field of view. ^cFor details, see the discussion in Section 5. References. (1) http://www.stsci.edu/observing/phase2-public/11563.pdf; (2) http://www.stsci.edu/hst/proposing/old-proposing-files/goods-cdfs.pdf.

Download table as:  ASCIITypeset image

The UDF GO program is comprised of three WFC3 pointings. Two of the WFC3 pointings use the F160W filter with 19 orbits. Using the WFC3 ETC, we estimate these observations will reach H_AB ≈ 28.76. The UDF GO observations also will have 38 F160W orbits in the HUDF that will reach H_AB ≈ 29.14.⁷ The remaining 116 orbits in the program will be used for observing in bluer filters.

The GOODS ERS survey will have three-orbit depth in F160W, using 24 orbits (out of a total 104) for H-band observations. We estimate that these observations will reach a sensitivity of H_AB ≈ 27.73.⁸ The remaining 80 orbits in the program will be used for observations with other filters and grisms.

5.1. Forecasted Constraints for Existing Surveys

The forecasted constraints achieved by the UDF GO and GOODS ERS surveys are plotted in Figure 2. Each panel shows the constraints for the UDF GO (blue region) and GOODS ERS (light blue region) surveys separately, and in combination (dark blue region). The constraints are plotted for the M_⋆ − ϕ_⋆ (left panel), M_⋆ − α (middle panel), and ϕ_⋆ − α (right panel) projections. In Figure 2 (and in similar figures throughout the paper), the shaded regions correspond to standard Gaussian contours.

Zoom In Zoom Out Reset image size

Figure 2 highlights some general properties of the performance of different kinds of surveys for providing LF parameter constraints, as well as specific features of the UDF GO and GOODS ERS surveys.

• 1.  
  The covariances between LF parameters are significant and positive. In terms of the Pearson's correlation coefficient ρ for parameters x and y, defined as

  $$\begin{equation} \rho = \frac{C_{xy}}{\sigma _{x}\sigma _{y}}, \end{equation} \tag{ 24 }$$

  the forecasts calculate typical correlation coefficients of ρ ≳ 0.9. For a given M_⋆ a narrow range of log₁₀ϕ_⋆ or α are permitted by the data, even as the marginalized uncertainties can be as large as ∼30%–40% fractionally for log₁₀ϕ_⋆ and α.
• 2.  
  The orientation of the constraint ellipse forecasted for each survey can differ significantly depending on the parameter uncertainties, even if the parameter correlation coefficients for the separate surveys are similar. The orientation of the constraint ellipse major axis with respect to the x-axis in the x–y parameter plane can be characterized by the angle

  $$\begin{equation} \Theta \equiv \frac{1}{2}\arctan \left[\frac{2\rho \sigma _{x}\sigma _{y}}{\sigma _{x}^{2}-\sigma _{y}^{2}}\right] = \frac{1}{2}\arctan \left[\frac{2C_{xy}}{C_{xx}-C_{yy}}\right], \end{equation} \tag{ 25 }$$

  which depends on the correlation ρ and the parameter uncertainties. If the angle Θ differs between separate surveys, then the constraints achieved by combining the surveys can improve dramatically.

For reference, the calculated marginalized and unmarginalized errors for p as well as the Pearson's correlation coefficient ρ and the angle Θ for each pair of parameters are listed for the existing surveys in Table 3.

Table 3. Forecasted Constraints for Existing Surveys

Survey	Marg. Uncert.	Unmarg. Uncert.	Marg. Uncert.	Unmarg. Uncert.	Marg. Uncert.	Unmarg. Uncert.
Name	M⋆	M⋆	log10ϕ⋆	log10ϕ⋆	α	α
UDF GO	0.837	0.160	0.655	0.093	0.566	0.162
GOODS ERS	1.338	0.171	1.026	0.145	1.852	0.520
Combined	0.524	0.117	0.425	0.078	0.415	0.154
Survey	Pearson ρ	Pearson ρ	Pearson ρ	Θ(M⋆ − ϕ⋆)	Θ(M⋆ − α)	Θ(ϕ⋆ − α)
Name	M⋆ − log10ϕ⋆	M⋆ − α	log10ϕ⋆ − α	(deg)	(deg)	(deg)
UDF GO	0.98	0.91	0.95	38	33	41
GOODS ERS	0.99	0.96	0.95	37	55	62
Combined	0.97	0.89	0.93	39	38	44

Download table as:  ASCIITypeset image

As Figure 2 and Table 3 show, the UDF GO and GOODS ERS surveys will already provide interesting constraints on the abundance of z ∼ 7 galaxies. When combined, the unmarginalized uncertainties on the LF parameters will be ΔM_⋆ ∼ 0.1 mag, Δlog₁₀ϕ_⋆ ≲ 0.1, and Δα ∼ 0.15. For the full UDF GO survey, we find that the unmarginalized uncertainty for the faint-end slope is Δα ∼ 0.16. Using a single UDF GO field and a limiting depth of F160W ∼ 29 AB for a single pointing, Oesch et al. (2010b) report a faint-end slope uncertainty of Δα ∼ 0.33 when ϕ_⋆ and M_⋆ are fixed (i.e., the unmarginalized uncertainty on α). If we use the same single pointing area and depth, and the same cosmology, our estimate of the unmarginalized uncertainty would increase to Δα ∼ 0.26. The GOODS ERS and UDF GO surveys are complementary in that the depth of the UDF GO survey provides a beneficial constraint on the faint-end slope α. This UDF GO constraint on α rotates the UDF GO error ellipse relative to the GOODS ERS constraint in the M_⋆ − α and ϕ_⋆ − α projections, thereby reducing the corresponding parameter uncertainties. Individually, the GOODS ERS uncertainties will be considerably larger than those obtained by the UDF GO survey, since the GOODS ERS survey lacks sufficient depth to tightly constrain the LF faint-end slope and is not wide enough to tightly constrain M_⋆ or ϕ_⋆.

While the UDF GO and GOODS ERS surveys achieve appreciable unmarginalized constraints, the covariances between the LF parameters are large. The marginalized uncertainties calculated for the LF parameters are ΔM_⋆ ∼ 0.5 mag, Δϕ_⋆ ∼ 0.4, and Δα ∼ 0.4. Accounting for covariances, these marginalized parameter uncertainties correspond to a fractional uncertainty in the total number of galaxies with M_UV < −18 of ≈25%, increasing to a factor of ≈2 uncertainty in the total number of galaxies with M_UV < M_⋆. To improve the constraints on the number of galaxies with M_UV < −18 (M_UV < M_⋆) to ≈5% (≈50%) would require marginalized parameter uncertainties of approximately ΔM_⋆ ∼ 0.2 mag, Δϕ_⋆ ∼ 0.2, and Δα ∼ 0.2 depending on their covariances. To reach such constraints, these UDF GO and GOODS ERS surveys would need to be complemented by either wider area or deeper surveys. We now consider some fiducial model surveys that could achieve these constraints in combination with the UDF GO and GOODS ERS data.

The complete UDF GO and GOODS ERS surveys will provide extremely interesting initial data on the abundance of z ∼ 7 galaxies, but the marginalized uncertainties on the LF parameters achieved by those surveys will still permit uncertainties of ∼25% in the total number of galaxies at M_UV ≲ −18. We can repeat the calculations from Section 5 for fiducial model surveys to illustrate what constraints wider or deeper surveys can achieve when combined with the UDF GO and GOODS ERS data.

The model surveys are designed to be appropriate for an HST Multi-Cycle Treasury Program,⁹ which can receive up to 750 orbits per HST cycle.¹⁰ We consider six possible model surveys that we estimate would require 450–900 total orbits to acquire filter coverage with the WFC IR channel.

As discussed in Section 4, we will assume the model surveys will use the F140W filter owing to its increased throughput relative to F160W. The sensitivity of each survey is determined by selecting a number N_orbits of orbits per pointing, assuming 46 minutes/orbit exposure time, and using the WFC3 IR channel ETC. The total number of orbits for each survey was then determined by selecting the number of fields N_fields, a mosaic geometry per field, and multiplying the number of pointings in each mosaic by N_orbits (and then doubling to account for comparable coverage in a bluer WFC3 filter).

The survey models are designed to cover a large range in total area (A_tot ≈ 14–3600 arcmin²), field numbers (N_fields = 1–4), orbits per pointing (N_orbits = 0.5–125), limiting depth (H_AB ≈ 27–30), and total number of orbits (450–900). We design each survey to approximate possible HST WFC3 tilings of existing surveys; as such, these model surveys represent realistic extensions of existing HST and Spitzer surveys to hundreds of orbits of WFC3 coverage. Summaries of the model surveys can be found in Table 4, and are ordered by decreasing limiting depth and increasing total area. Brief descriptions of the models follow.

Table 4. Model Surveys

Model	Field Mosaic	Field Geometry	Nfields	Total Area	Norbitsa	H-band Depth	Total Orbitsb
Name	(Point × Point)			(arcmin2)		(AB Mag.)
Survey A	1 × 1	205 × 227	3	13.96	125	30.12	750
Survey B1	5 × 7	103 × 159	2	325.7	8	28.62	896
Survey B2	5 × 7	103 × 159	2	325.7	6	28.46	672
Survey B3	5 × 7	103 × 159	2	325.7	4	28.23	448
Survey C	4 × 13	82 × 295	4	967.6	2	27.84	832
Survey D	26 × 30	590 × 615	1	3628.5	0.5	27.00	780

Notes. ^aThe number of orbits per pointing. ^bWe assume each survey will require comparable coverage in two WFC3 filters, which doubles the required number of total orbits.

Download table as:  ASCIITypeset image

6.1. Survey A

6.2. Survey B1

6.3. Survey B2

6.4. Survey B3

6.5. Survey C

6.6. Survey D

6.7. Field Size Comparison

We show an illustrative comparison of the existing and model survey areas in Figure 3. The UDF GO, GOODS ERS, and Surveys A, B1, B2, B3, C, and D areas are shown as white boxes overlaid on a thin 10 h⁻¹ Mpc slice through a ΛCDM cosmological simulation of comoving size L = 250 h⁻¹ Mpc at z ∼ 7 (Tinker et al. 2008). The blue scale image shows the projected dark matter surface density calculated from the dark matter particle distribution of the simulation. The comoving length scale corresponding to an angle of θ = 1 deg at z = 7 is 108.7 h⁻¹ Mpc for the adopted WMAP5 cosmology. This comparison illustrates the characteristic angular size of large-scale structures at z ∼ 7, as well as the survey areas required to probe representative samples of the high-redshift dark matter density distribution. The separation between fields is not to scale, and model surveys incorporating different fields would likely be more widely spaced to probe statistically independent regions on the sky.

Zoom In Zoom Out Reset image size

The forecasted constraints calculated for the model Surveys A, B1, B2, B3, C, and D are summarized in Table 5 and presented in Figures 4–9. In each figure, the shaded areas show the projected constraints for each model survey in the M_⋆ − ϕ_⋆ (left panel), M_⋆ − α (middle panel), and ϕ_⋆ − α (right panel) LF parameter planes. The axes ranges in Figures 4–9 are identical (and much smaller than in Figure 2), and the plotted constraints are directly comparable. A description of the forecasted constraints for each model survey is as follows.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

7.1. Survey A

7.2. Surveys B1, B2, and B3

7.3. Survey C

7.4. Survey D

We have considered the problem of forecasting constraints on parameters of the z ∼ 7 LF given the characteristics of on-going surveys and models for potential future survey designs. The purview of our calculation was purposefully narrow since a more comprehensive evaluation of galaxy surveys could involve many additional questions we have not addressed. We now turn to a variety of possible caveats that stem from considering photometric galaxy survey designs more generally.

We have focused on forecasting constraints for the LF. Our approach was modeled after Fisher matrix calculations that used the abundance of galaxy clusters to forecast cosmological parameters constraints (Hu & Kravtsov 2003; Lima & Hu 2004, 2005; Cunha & Evrard 2009; Wu et al. 2009), but other previous calculations have forecasted cosmological parameter constraints from galaxy clustering (e.g., Vogeley & Szalay 1996; Matsubara & Szalay 2001, 2003; Linder 2003; Albrecht et al. 2009). The incorporation of galaxy clustering data can circumvent some assumptions made in Section 2 when using simple abundance matching to assign galaxy bias by replacing the sample variance estimates in Equations (15) and (17) by an integral over the galaxy correlation function. Other estimates of how cosmic variance uncertainty is influenced by galaxy bias have taken a similar approach (e.g., Newman & Davis 2002; Somerville et al. 2004; Stark et al. 2007b; Trenti & Stiavelli 2008).

Our calculations have regarded a limited but interesting redshift regime near z ∼ 7. While we have found that the combination of existing deep/narrow surveys with a future wide/shallow survey or a future ultradeep/narrow survey would provide tight constraints on the z ∼ 7 LF, studies of the galaxy population at higher and lower redshifts could require substantially different surveys. For instance, the z ≳ 8 dropout candidates identified in the UDF GO data are all fainter than H_AB = 27.7 (Bouwens et al. 2009; Bunker et al. 2010; McLure et al. 2010; Yan et al. 2009). The decreasing abundance of relatively bright galaxies with increasing redshift will tend to favor deeper and narrow surveys. Our calculations can easily be extended to estimate the constraining power of various surveys designs for higher-redshift galaxy populations, but we will save such estimates for future work when better fiducial estimates of the z ≳ 8 LF are available.

Our Fisher matrix approach requires the use of a fiducial model for the abundance of z ∼ 7. We adopt the Oesch et al. (2010b) estimate of the galaxy LF, which was determined by scaling the characteristic magnitude M_⋆ and normalization ϕ_⋆ from lower redshift data and then fitting for the faint-end slope α. If the z ∼ 7 galaxy LF differs substantially from the Oesch et al. (2010b) estimate, then our forecasted constraints could be similarly inaccurate. For instance, if the normalization ϕ_⋆ was considerably lower or the characteristic magnitude M_⋆ much fainter than that estimate by Oesch et al. (2010b), then the relative benefit of combining the UDF GO and GOODS ERS data with a wide/shallow survey over a narrow/ultradeep design could be reduced.

The calculations in Sections 2.4.1 and 7 suggest that splitting wide surveys into multiple fields to probe statistically independent regions of the universe may not dramatically improve constraints on the galaxy LF. While this conclusion depends strongly on the total volume of the survey, other considerations such as scheduling, field observability, or sky backgrounds could make multiple fields advantageous compared with a single contiguous field of the same total area.

The abundance matching calculation also requires either knowledge or assumption about the completeness of the survey and the fraction of dark matter halos occupied by galaxies. We have assumed that the surveys are essentially volume-limited and that there is a one-to-one correspondence between galaxies and dark matter halos. Both of these assumptions are likely imperfect, and some estimates of the high-redshift occupation fraction are as low as 20% (Stark et al. 2007b). The influence of these assumptions over the forecasted parameter constraints depends on the character of the survey. For a given observed LF, reducing the halo occupation fraction or the survey completeness acts to reduce the effective galaxy bias in the survey by either increasing the number of halos per galaxy in the survey or increasing the number of undetected galaxies. In either case, the Poisson uncertainty is based on the observed number of galaxies and is unaffected. For purposes of constraining the observed LF, wide surveys are fairly insensitive to either assumption, since Poisson uncertainty in the abundance of bright galaxies plays a large role in their error budget. The change in sample variance for narrow surveys can lead to a degradation of the marginalized parameter constraints (while the unmarginalized constraints can improve) by increasing the parameter correlations. However, the relative effect is small and the degradation is only 2× for simultaneously low incompleteness (f_comp = 0.1) and small occupation fraction (f_occ = 0.1).

Motivated by the exciting initial galaxy survey data obtained by newly installed WFC3 on the HST, we have attempted to quantify how well on-going and possible future infrared surveys with WFC3 will constrain the abundance of galaxies at z ∼ 7. Our primary methods and results include the following.

• 1.  
  We perform a Fisher matrix calculation to forecast constraints on the galaxy LF achievable by a survey with a given depth, area, and number of fields. In our approach, the constraints on the LF normalization ϕ_⋆, characteristic magnitude M_⋆, and faint-end slope α that a survey can achieve directly relate to the sample cosmic variance and Poisson uncertainty on the observed galaxy abundance through the Fisher matrix. For a fiducial LF model, the abundance of observed galaxies and dark matter halos are matched (e.g., Conroy et al. 2006; Conroy & Wechsler 2009) to estimate the bias of galaxies of a given luminosity. The galaxy bias is combined with the rms density fluctuations within the survey volume to calculate the sample cosmic variance (e.g., Newman & Davis 2002; Somerville et al. 2004; Stark et al. 2007b; Trenti & Stiavelli 2008), while the Poisson variance simply scales with the square root of the number of observed galaxies. The constraining power of each survey is then calculated from the Fisher matrix using the Schechter (1976) model of the LF, its derivatives, the sample cosmic variance and Poisson uncertainties, and any data covariance. The combined constraints from multiple surveys can be estimated easily by summing their Fisher matrices. Similar calculations should prove useful for designing future photometric surveys and estimating their constraining power for the galaxy LF.
• 2.  
  Using the Fisher matrix calculations, we estimate the constraints on the abundance of z ∼ 7 galaxies that will be achieved with the entire forthcoming UDF GO and GOODS ERS HST WFC3 IR channel data. Using the z ∼ 7 galaxy LF estimated by Oesch et al. (2010b) as a fiducial model, we calculate that the combined UDF GO and GOODS ERS data will achieve marginalized (unmarginalized) LF parameter constraints of ΔM_⋆ ≈ 0.5 mag (0.1 mag), Δlog₁₀ϕ_⋆ ≈ 0.4 (0.1), and Δα ≈ 0.4 (0.15) when the surveys are fully completed. These marginalized constraints correspond to uncertainties in the total number of z ∼ 7 galaxies with magnitudes M_UV < −18 (M_UV < M_⋆ ≈ −19.8) of 25% (200%), after accounting for covariances between the LF parameters. These surveys will provide the first detailed information on z ∼ 7 galaxy populations, but abundance of the bright end of the z ∼ 7 LF will remain uncertain without further data.
• 3.  
  We also forecast z ∼ 7 LF constraints provided by a variety of model WFC3 surveys that would each require ∼450–900 HST orbits. The six model surveys considered cover a large range of areas (14–3600 arcmin²) and depths (H_AB = 27–30 in F140W) to study the relative value area and depth for constraining the abundance of z ∼ 7 galaxies. When combined with the forthcoming UDF GO and GOODS ERS data, all the surveys considered produce interesting LF constraints (see Table 5). We find that a ∼1 deg² survey to H_AB ≈ 27 in F140W provides the tightest combined marginalized constraints (ΔM_⋆ ≈ 0.14, Δϕ_⋆ ≈ 0.11, Δα ≈ 0.20) on the abundance of z ∼ 7 galaxies of all survey designs we consider, but only by a small margin. This survey would require 780 total orbits, including equal coverage in a bluer WFC3 filter to define a drop out color selection. In contrast, the abundance of faint galaxies would be best constrained by increasing depth of the HST ultradeep fields to ∼125 orbits per pointing (H_AB ≈ 30.1 in F140W), which provides marginalized LF constraints of ΔM_⋆ ≈ 0.22, Δϕ_⋆ ≈ 0.15, and Δα ≈ 0.10 for 750 total orbits (including equal coverage in a bluer WFC3 filter).
• 4.  
  We also consider the usefulness of splitting surveys into N_fields multiple fields to probe independent samples and reduce cosmic variance uncertainties (e.g., Newman & Davis 2002). We show that the shape of the ΛCDM power spectrum limits the statistical gain of splitting a high-redshift survey into multiple fields to ≲10% (for N_fields = 2) when the survey area is large (≳0.5 deg²). We suggest that this statistical gain should be weighed against any scientific gains achieved by probing large contiguous areas.

Initial analyses of the UDF GO data have already demonstrated that the installation of WFC3 on HST will transform our knowledge of high-redshift galaxy populations at z ∼ 7 and beyond (e.g., Bouwens et al. 2009, 2010a, 2010b; Oesch et al. 2010a, 2010b; Bunker et al. 2010; McLure et al. 2010; Yan et al. 2009; Wilkins et al. 2010; Labbé et al. 2009, 2010; Finkelstein et al. 2009). Our work has attempted to quantify expectations for the constraining power of the UDF GO and GOODS ERS surveys and forecast constraints achievable with more extensive future surveys using WFC3 or other instruments. These calculations illustrate how truly powerful the refurbished HST is for exploring high-redshift galaxy populations and emphasize how exciting near-term gains in our knowledge of z ≳ 7 galaxies will be.

I thank Peter Capak and Nick Scoville for useful advice on modeling the survey data and WFC3 observations, Richard Ellis for helpful discussions, and Anatoly Klypin for permission to use his cosmological simulation. I am supported by a Hubble Fellowship grant, program number HST-HF-51262.01-A provided by NASA from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. I also thank the Caltech Astronomy Department for hosting me during my Hubble Fellowship.