Probing Dark Energy with the Kunlun Dark Universe Survey Telescope

Antarctic Plateau, especially the Kunlun Station under construction by the Polar Research Institute of China (PRIC), provides a unique opportunity for wide-field astronomical surveys targeting cosmological studies. Astronomical site survey of Dome A, Antarctica was enabled by the International Polar Year endorsed PANDA program (Yang et al. 2009) led by the PRIC and the Chinese Center for Antarctica Astronomy (CCAA). Several international teams contributed to this effort. In particular, the power and on-site laboratory system built by the University of New South Wales (Lawrence et al. 2009) has provided the platform for all the site survey instruments.

In its two years of operation, the site survey effort proves practically all aspects of the theoretical expectations of the Dome A site for astronomical observations. Preliminary analyses show the boundary layer of atmospheric turbulence to be around 10–20 m during the Antarctic winter (Ashley et al. 2010). Similar to the relatively better studied neighboring Dome C site (Fossat et al. 2010), the Dome A site may enjoy free-atmospheric seeing conditions of about 0.3" seeing above this boundary layer, thus making it an ideal site for high angular resolution wide-area surveys.

The temperature at Dome A is around -60 to -70°C, making it the coldest spot on the surface of the Earth. This implies a very low thermal background emission in the thermal infrared. The Dome A site is thus also the best site for astronomical observations in the near-infrared wavelength region.

One other especially exciting property of the site is the lack of water vapors, due to the high altitude and the low temperature. This implies that the site is also ideal for terahertz observations, which have been impossible from any temperate sites on Earth.

While quantitative site properties are still under analysis and longer-term monitoring is still needed to firmly establish the astronomical potential of the Dome A site, there is no doubt that a survey project at Dome A can be highly complementary to programs such as the Large Synoptic Survey Telescope⁸ (LSST; LSST Science Collaborations 2009) and Euclid.⁹ For example, a survey at Dome A can provide near-infrared data that complement the deep optical band survey of the LSST; a deep survey in the near-infrared combined with the optical data from LSST can reveal high-redshift objects at z ∼ 10, which are not detectable in the optical. The Kunlun Dark Universe Survey Telescope (KDUST) is a 6–8 m wide-area survey telescope being designed by the CCAA. The preliminary design includes a 3 × 3 deg² optical camera with 0.15'' pixel^-1, and an infrared camera of 1 × 1 deg² at 0.1'' pixel^-1 optimized for 1–3.5 μm surveys.

One of the key science missions of KDUST is to investigate the mystery of the accelerated cosmic expansion (Riess et al. 1998; Perlmutter et al. 1999) using multiple techniques. In this article, we estimate how well an ideal 10,000 deg² ugrizyJH survey can constrain the dark energy equation of state (EOS) with weak lensing (WL), baryon acoustic oscillations (BAOs), and Type Ia supernova (SNe) luminosity distances. These dark energy probes have different sensitivities to the cosmic expansion and structure growth, as well as various systematic uncertainties in the observations, and hence are highly complementary to each other for constraining dark energy properties (e.g., Knox et al. 2005; Zhan 2006; Zhan et al. 2009).

The SNe technique relies on the standardizable candle of the SNe intrinsic luminosity (Phillips 1993) to measure the luminosity distance, D_L(z). Dark energy properties can then be inferred from the distance–redshift relation. The BAO technique utilizes the standard ruler of the baryon imprint on the matter (and hence galaxy) power spectrum (Peebles & Yu 1970; Bond & Efstathiou 1984) to measure the angular diameter distance, D_A(z), and, if the redshifts are sufficiently accurate, the Hubble parameter, H(z) (Eisenstein et al. 1998; Cooray et al. 2001; Blake & Glazebrook 2003; Hu & Haiman 2003; Linder 2003a; Seo & Eisenstein 2003). The WL technique has the advantage that it can measure both D_A(z) from the lensing kernel and the growth factor of the large-scale structure G(z) (Hu & Tegmark 1999; Huterer 2002; Refregier 2003; Takada & Jain 2004; Knox et al. 2006a; Zhan et al. 2009).

Because of the excellent seeing condition and infrared accessibility at Dome A, KDUST has a number of advantages for the commonly used cosmological probes. For example, the signal-to-noise ratio for point sources is inversely proportional to the seeing. Thus, a 6 m telescope at Dome A (∼0.3'' median seeing in the optical) would be equivalent to a 14 m telescope at a temperate site (∼0.7'' seeing) for point-source observations with the same sky background level. An 8 m KDUST could detect SNe out to redshift 3 in the K_dark (2.27–2.45 μm, redward of K) band (Kim et al. 2010). Although high- z distances are not sensitive to conventional dark energy, they can be used to determine the mean curvature accurately, which, in turn, helps constrain dark energy EOS (Linder 2005; Knox et al. 2006b). Moreover, even though dark energy is thought to be subdominant at high redshift, there is no direct evidence to prove one way or another. Measurements of SNe at z > 2 will provide crucial data for tests of early dark energy.

Small and stable seeing is particularly helpful for WL. One could resolve more galaxies at the same surface brightness limit, which reduces the shape noise for shear measurements. Fine resolution helps measure the shape accurately and reduce the shear measurement systematic errors. In addition, deep JH photometry can track the 4000 Å break of an elliptical galaxy to z ∼ 3 and improve photometric redshifts (photo- z), as well as systematic uncertainties in the photo- z error distribution (Abdalla et al. 2008), which has a large impact on WL constraints on the dark energy EOS (Huterer et al. 2006; Ma et al. 2006; Zhan 2006).

Adding K or K_dark band will certainly improve galaxy photo- z, especially at z ≳ 3. However, currently planned multiband dark energy surveys use galaxies at z ≲ 3, and their concern is the confusion between z ≲ 0.5 ellipticals and 2 ≲ z ≲ 3.5 star-forming galaxies, which is greatly mitigated by u and JH bands (Abdalla et al. 2008). Another consideration is that Dome A is far more advantageous at the K_dark band than at the K band because of the low thermal background there, so K_dark is likely to be chosen over K. This would leave a considerable gap in the wavelength coverage and reduce the already-small gain on photo- z in the useful redshift range for dark energy investigations. Therefore, we do not discuss utilities of wavebands beyond H in this article. Nevertheless, the K_dark band is crucial for a broad range of other sciences and will be an important aspect of the KDUST survey.

This article is organized as follows. Section 2 discusses the survey plan of KDUST, including that of its pathfinder in context of the LSST survey. We then consider a joint KDUST and LSST survey in § 3 for constraining the dark energy EOS with BAO, WL, and Type Ia SN techniques. The results are presented in both two-parameter space, where the dark energy EOS is parameterized using the Chevallier-Polarski-Linder parameterization w(z) = w₀ + w_a z(1 + z)^-1 (Chevallier & Polarski 2001; Linder 2003b), and in model-independent principle component space. The conclusion is drawn in § 4.

Dome A has a great potential for cosmology, but, given other ambitious projects that will be concurrent with KDUST, one must carefully plan the KDUST survey to make the best use of the Dome A site. As we discuss subsequently, a potentially efficient strategy for KDUST is to focus on the near-infrared (NIR) bands and incorporate optical data from LSST or other surveys.

In the optical bands, a 6 m KDUST is about twice as fast as LSST for surveying sky-dominated point sources at the same sky level, in which case the survey speed is proportional to the aperture and field of view and inversely proportional to the seeing disk area. In reality, aurorae increase the sky brightness in short wavelengths. It is estimated that the sky brightness at Dome A would be twice as bright as that at the best temperate site in B and 20%–30% brighter in V (Saunders et al. 2009); measurements have shown that the median i -band sky brightness at Dome A in 2008¹⁰ was 19.81 mag arcsec^-2 and 20.46 mag arcsec^-2 during dark time, better than that at other good sites (Zou et al. 2010). With the preceding considerations, the 6 m KDUST could survey 10,000 deg² to LSST depths in griz and much deeper in y (∼26 mag, 5σ point sources) in 2.5 yr. The NIR camera of KDUST would have a much smaller field of view. It could reach J = 25 mag and H = 24.6 mag over the same area in 3.5 yr.

Since LSST plans to survey the southern sky in ugrizy, it is not absolutely necessary for KDUST to survey in the optical again, except in the y band. LSST would spend 20% of its time in the y band to achieve a 5-σ limiting magnitude of 24.4 for point sources, which is 2.8 mag shallower than its r -band limit. From photo- z consideration, it is desirable to have the y -band limit not too much shallower than the limits in shorter wavebands. KDUST could improve the situation in its 10,000 deg² survey area, which would be covered by LSST as well. A joint effort of KDUST in yJH and LSST in ugrizy would save both projects time while achieving better performance in measuring the dark energy EOS.

Here, we use SNAP and LSST as references to model an ideal 10,0000 deg² survey (named "ideal 10k") in ugrizyJH bands to a depth comparable with that of SNAP. We assume that its galaxy redshift distribution follows

with z_∗ = 0.5. The projected galaxy number density is 70 arcmin^-2, and the distribution peaks at 2z_∗. For LSST, we adopt z_∗ = 0.3 and (LSST Science Collaborations 2009). The model parameters of the survey data are summarized in Table 1.

The galaxy distribution n_i(z) in the i th bin is sampled from n(z) by Ma et al. (2006) and Zhan (2006):

where the subscript p denotes photo- z space, and define the extent of bin i, and is the probability of assigning a galaxy that is at true-redshift z to the photo- z bin between z_p = a and b. We approximate the photo- z error to be Gaussian with bias δz and rms σ_z = σ_z0(1 + z), and the probability becomes

The normalization I(0,∞; z) implies that galaxies with a negative photo- z have been excluded from n(z). We use 40 photo- z bias δz and 40 photo- z rms σ_z parameters evenly spaced between z = 0 and five to model the photo- z error distribution in the z - z_p space; the photo- z bias and rms at any redshift are linearly interpolated from these 80 parameters. Note that the photo- z parameters are assigned independetly of galaxy bins. We assume that σ_z0 = 0.03 for the ideal survey and σ_z0 = 0.05 for LSST.

Uncertainties of the photo- z parameters (or the photo- z error distribution in general) have a large impact on the dark energy constraints from WL and are referred to as photo- z systematics. Therefore, the prior on the photo- z error distribution is an important quantity to specify when reporting WL constraints on dark energy. To reduce the dimension of the investigation, we set a simple function for all the priors on the photo- z bias parameters, σ_P(δz) = 0.2σ_z, and peg the priors on photo- z rms parameters to those on the bias parameters: . For Gaussian photo- z errors, these priors correspond to a calibration sample of 25 spectra per redshift interval of the photo- z parameters. We set σ_P(δz) = 0.3σ_z, reflecting larger uncertainties with fewer filter bands. However, such a difference in the photo- z priors is not important if one performs a joint analysis of the BAO and WL techniques, which takes advantage of the self-calibration of photo- z error distribution by galaxy power spectra (referenced in Schneider et al. 2006; Zhan 2006; Zhang et al. 2009).

LSST will benefit from KDUST yJH data in a number of ways. In the 10,000 deg² overlap region, KDUST data will (1) improve photo- z directly; (2) increase the galaxy sample with those that would not meet the LSST optical photometry selection criteria, or whose photo- z could not be reliably determined in the absence of the deep yJH data, or whose shape could not be measured well because of the larger seeing at the LSST site; and (3) provide calibration of various systematic errors such as those in shear measurements. Even in the nonoverlap region, LSST could still improve photo- z and shear measurements because of the calibration within the overlap region. This is reflected in the last column of Table 1, where we construct a joint survey combining LSST and KDUST.

For the SN data, we assume that KDUST would obtain a SNAP-like sample of 2000 SNe reaching redshift 1.7, as well as a sample of 1000 local and nearby SNe. Details of the assumptions for the SNAP experiments and the calculation of the Fisher matrices can be found in Pogosian et al. (2005) and in Zhao et al. (2009).

In this section, we estimate the constraints on the dark energy EOS from the joint survey of KDUST and LSST using BAO, WL, and SN techniques.

3.1. Baryon Acoustic Oscillations and Weak Lensing

The angular power spectra of the galaxy number density n(θ) and (E-mode) shear γ(θ) can be written as (Hu & Jain 2004; Zhan 2006)

where lowercase subscripts correspond to the tomographic bins, uppercase superscripts label the observables, e.g., X = g for galaxies or γ for shear, is the dimensionless power spectrum of the density field, and k = ℓ/D_A(z). The window functions are

where b(z) is the linear galaxy clustering bias, and Ω_m and H₀ are, respectively, the matter fraction at z = 0 and Hubble constant. The galaxy redshift distribution n_i(z) in the i th tomographic bin is an average of the underlying three-dimensional galaxy distribution over angles, and the mean surface density is the total number of galaxies per steradian in bin i. For WL, we use 10 bins evenly spaced between z = 0 to 3.5, and for BAO, we use 30 bins from z = 0.15 to 3.5 with bin width proportional to 1 + z.

The 1σ statistical error of the mean power spectrum in the multipole range (ℓ,ℓ + Δℓ) is given by

where is the Kronecker delta function, f_sky is the sky coverage, g_rms ≡ 1, and γ_rms = 0.18 + 0.042z.

Figure 1 shows several examples of the galaxy auto and cross power spectra, and Figure 2 shows shear auto and cross power spectra. Cross power spectra between foreground galaxy bins and background shear bins, , are not shown, but they have characteristic shapes similar to those in Figure 1 and Figure 2. The amplitude of the galaxy power spectra and the strength of the BAO feature decreases with increasing photo- z rms error, because a larger photo- z rms means more smoothing in the radial direction. The amplitude of the shear power spectrum increases as redshift increases, because higher-redshift galaxies are lensed by more intervening matter and, hence, have stronger shear signal fluctuations. The amplitude of the galaxy cross power spectrum between two bins is very sensitive to the separation between the two bins in true-redshift space. Hence, galaxy cross power spectra can be used to calibrate the photo- z error distribution.

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3.2. Constraints on w₀ and w_a

We use the Fisher information matrix (Tegmark 1997) to estimate the errors of the parameters of interest. In summary, the Fisher matrix of the parameter set {q_α} is given by

with for galaxies and shear. The minimum marginalized error of q_α is . Independent Fisher matrices are additive; a prior on q_α, σ_P(q_α), can be introduced via .

We extend the additive and multiplicative shear power spectrum errors in Huterer et al. (2006) to include the galaxy power spectrum errors:

where ρ^X determines how strongly the additive errors of two different bins are correlated, and η^X and account for the scale dependence of the additive errors. Note that the multiplicative error of galaxy number density is degenerate with the galaxy clustering bias and is hence absorbed by b_i. At the levels of systematics that future surveys aim to achieve, the most important aspect of the (shear) additive error is its amplitude (Huterer et al. 2006), so we simply fix ρ^X = 1 and η^X = 0. For more comprehensive accounts of the preceding systematic uncertainties, see Huterer et al. (2006), Jain et al. (2006), Ma et al. (2006), and Zhan (2006).

Forecasts on dark energy constraints are sensitive to the priors on the shear multiplicative errors () and the amplitudes of the shear additive errors (), so we list them in Table 1. See Wittman (2005), Massey et al. (2007), and Paulin-Henriksson et al. (2008) for detailed work on these systematic uncertainties and Zhan et al. (2009) for a discussion on and for LSST. It has been demonstrated that a small and stable point-spread function in space leads to better shear measurements (Kasliwal et al. 2008). In the absence of a detailed investigation, we simply choose values or priors of these systematic error parameters that are somewhat arbitrarily between what might be achieved by space projects and what have been used in LSST forecasts. We infer from the Sloan Digital Sky Survey galaxy angular power spectrum (Tegmark et al. 2002) that the additive galaxy power spectrum error due to extinction, photometry calibration, and seeing will be at the level. Since the amplitude of is fairly low compared with the galaxy power spectra, its value does not affect the forecasts much.

In summary, the parameter set includes 11 cosmological parameters and 170 nuisance parameters. The cosmological parameters are w₀, w_a, the matter density ω_m, the baryon density ω_b, the angular size of the sound horizon at the last scattering surface θ_s, the curvature parameter Ω_k, the scalar spectral index n_s, the running of the spectral index α_s, the primordial Helium fraction Y_p, the election optical depth τ, and the normalization of the primordial curvature power spectrum . Note that Y_p and τ are solely constrained by the cosmic microwave background (CMB), which is introduced as priors. The nuisance parameters include 40 photo- z bias parameters, 40 photo- z rms parameters, 40 galaxy clustering bias parameters, 30 galaxy additive noise parameters, 10 shear additive noise parameters, and 10 parameters for shear calibration errors. We use multipoles 40 ≤ ℓ ≤ 2000 for WL and 40 ≤ ℓ ≤ 3000 for BAO. In addition, we require for BAO to reduce the influence of nonlinear evolution. The lower cut in ℓ is to minimize the dependence of the forecasts on particular models of dark energy perturbation and the integrated Sachs-Wolfe effect, which affect only very large scales.

Figure 3 presents the forecasts of 1σ error contours of the dark energy EOS parameters w₀ and w_a for the ideal 10k survey WL (solid line), BAO (dotted line), SNe (dot-dashed line), joint BAO and WL (dashed line), and the three combined (shaded area). Since different probes have different parameter degeneracy directions in the full parameter space, including both cosmological parameters and nuisance parameters (such as the photo- z bias and rms parameters), a joint analysis can reduce the error significantly. One example is that the BAO technique can determine the curvature parameter and the matter density far better than the SNe technique, whereas the latter needs strong priors on Ω_K and ω_m in order to place tight constraints on w₀ and w_a (Linder 2005; Knox et al. 2006b). Another example is that the WL technique is sensitive to the priors on the photo- z parameters, which can be calibrated by the BAO technique. By comparing the results in Figure 3 with the LSST BAO + WL result in Figure 4, one sees that the ideal 10,000 deg² ugrizyJH survey can place comparable constraints on the dark energy EOS parameters to the LSST survey.

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We show the dark energy constraints from combinations of LSST with the KDUST 10,000 deg² survey and LSST with half of the KDUST survey (labeled as KDUST5k) in Figure 4. The KDUST5k survey could be carried out by a 2.5 m KDUST pathfinder in 12 yr. We assume that 5000 deg² high-resolution yJH imaging is enough to reach the systematic calibration floor, so the improvement to the LSST survey outside the KDUST survey area is kept the same for both proposals. Under this condition, the WL + BAO constraints on w₀ and w_a from LSST + KDUST5k are nearly the same as those from LSST + KDUST ; they improve the LSST-alone dark energy task force (DETF, Albrecht et al. 2006) figure of merit (FOM) by 30%. Adding SNAP-like SN data from KDUST can also significantly increase the DETF FOM; the result is as good as LSST or LSST + KDUST joint BAO and WL constraints on w₀ and w_a in the absence of systematics.

The current constraint on {w₀,w_a} is (Zhao & Zhang 2010)

derived from a joint Markov chain Monte Carlo analysis of the recently released SNe "Constitution" sample (Hicken et al.2009), the WMAP 5 yr data (Komatsu et al.2009),¹¹ and the Sloan Digital Sky Survey luminous red galaxy sample (Tegmark et al.2006). In this analysis, the dark energy perturbation, which is important in the parameter estimation (Zhao et al. 2005; Fang et al. 2008), was consistently included based on the treatment developed in Zhao et al. (2005). The central values indicate that the "quintom" scenario (Feng et al. 2005) is mildly favored; namely, the EOS today is w(z)|_z=0 = w₀ > -1, while EOS in the far past w(z)|_z=∞ = w₀ + w_a < -1. This is consistent with the recent published result using the Constitution SNe sample (Shafieloo et al. 2009; Biswas & Wandelt 2009; Qi et al.2009; Wei 2009; Huang et al. 2009). The error bars in equation 4 can be used to estimate the FOM of current data, and we find that KDUST + LSST can improve the current FOM by 2 orders of magnitude.

3.3. Principal Component Analysis on w(z)

To investigate the constraints on dark energy equation of state w(z) from the surveys in a model-independent way, we follow Huterer & Starkman (2002) and Crittenden et al. (2009) and employ a principle component analysis (PCA) approach.

We choose 40 uniform redshift bins, stretching to a maximum redshift of z = 3, and allow the high-redshift (z > 3) equation of state to vary. To avoid w(z) with infinite derivatives, each bin rises and falls following a hyperbolic tangent function with a typical transition width dz of the order of 10% of the width of a bin. We choose a constant w = -1.0 as the fiducial model, which is consistent with all the present data. Since dark energy perturbations play a crucial role in the parameter estimation, we use a modified version of CAMB, which allows us to calculate dark energy perturbation for an arbitrary w(z) consistently (Zhao et al. 2005).

For the PCA, we calculate the Fisher matrices based on four kinds of observables: supernovae, CMB anisotropies, galaxy number counts correlation functions, and WL observations. We also include all the possible cross-correlations among these, including CMB × galaxy and CMB × WL, which are sensitive to the integrated Sachs-Wolfe effect, as well as galaxy weak-lensing correlations.

We first calculate the Fisher matrices for each of the observables, , where the indices i,j run over the parameters of the theory, in our case, the binned w_i(z). We then find the normalized eigenvectors and eigenvalues of this matrix {e_i(z),λ_i} and write

where the rows of W are the eigenvectors and Λ is a diagonal matrix with elements λ_i. The Fisher matrix is an estimate of the inverse covariance matrix that we expect the data to give us and the eigenvalues reflect how well the amplitude of each eigenvector can be measured. The true behavior of the equation of state can be expanded in the eigenvectors as

and the expected error in the recovered amplitudes is given by .

For our forecasts, we assume three surveys: the ideal 10k survey, LSST, and KDUST + LSST as listed in Table 1. We also combine the Planck survey for CMB and SNAP SNe. We assume a flat universe and marginalize over the intrinsic SN magnitude M and the galaxy bias parameters.

In the upper panel of Figure 5, we show the spectra of eigenvalues σ^-2(α_i) of the Fisher matrices for these three tomographic surveys, and in the lower panel, we show the ratio of the eigenvalues for LSST + KDUST to that for LSST. As we can see, the ideal 10k survey is competitive to LSST on the dark energy constraints, and adding KDUST to LSST can improve the constraints on the first few eigenmodes by as much as 85%. The first five best determined eigenvectors are shown in Figure 6. As we can see, the N th eigenmode has N - 1 nodes in z, and the "sweet spot"—the redshift where the uncertainty of w(z) gets minimized—is at z ∼ 0.2. This is consistent with the analysis done in Huterer & Starkman (2002) and in Crittenden et al. (2009). We also find that adding KDUST to LSST makes the eigenmodes stretch to slightly higher z because of the higher redshift reach with the addition of KDUST data.

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As stated in Crittenden et al. (2009), all of the eigenvectors are informative, no matter how large the error bars are, if we have no prior knowledge of the possible w(z) behaviors. However, even without a physical model for w(z), we would still be surprised if w(z) were much too positive (w≫1/3) or much too negative (w ≪ -1). This motivates us to choose some theoretical priors to roughly separate the eigenmodes into those that are informative relative to the priors and those that are not. To apply the theoretical prior, we follow Crittenden et al. (2009) to choose a correlation function describing fluctuations of w(z) away from the fiducial model. This smoothness prior can filter out the high-frequency modes, while the low-frequency modes remain unaffected. Also, as long as there are sufficient bins compared with the correlation length, the prior largely wipes out dependence on the precise choice of binning.

As elaborated in Crittenden et al.(2009), the deviations of the equation of state from its fiducial model can be encapsulated in a correlation function,

and the covariance matrix of the binned equation of state is then

where the i th bin is from z_i to z_i + Δ, and we assume that all bins have the same width Δ = z_i+1 - z_i. The variance of the mean equation of state over all the bins is

And we assume that ξ_w(z) = ξ_w(0)/(1 + (z/z_c)²), where ξ_w(0) is the variance of w at any given point and z_c is the correlation length.

To apply the prior, we need to specify the correlation length z_c and tune ξ_w(0) so that the error in the mean, σ_m, is in the range of [0.2,0.5] in order to be consistent with the observational uncertainty. We tried correlation lengths in the range of 0.1 ≤ z_c ≤ 0.4, where the upper limit was beginning to be strong enough to impact the observed modes for the SN. Then the resulting prior takes the form

where C_ij ≡ 〈δw_iδw_j〉. This prior naturally constrains the high-frequency modes without over constraining the lower-frequency modes that are typically probed by experiments.

Note that assuming no correlations among the bins is equivalent to using a delta function for the correlation prior, e.g., ξ(z) = ξ₀δ(z). In such a case, one finds 〈δw_iδw_j〉 = ξ₀δ_ij/Δ and the mean variance is . Thus, assuming a fixed total range, the bin variance should grow with the number of bins to keep the mean variance unchanged.

In order to evaluate the value added by a given survey, we use the mean squared error (MSE) as a FOM. In the absence of any priors, this is expected to be

Taking into account priors, F in this expression is replaced by C^-1 + F. The MSE arising from the prior alone is MSE = TrC, and it is independent of the shape of the correlation function. Adding more experimental data reduces the MSE, and the amount by which the MSE is reduced can be seen as a measure of how informative the experiment is.

In Table 2 we vary ξ_w(0) and z_c while holding the prior constraint on the mean fixed, σ_m = 0.3. For the case of a diagonal prior, the MSE for the prior alone is given by . Adding the data, one could significantly improve the constraints on some of the eigenmodes, thus reducing the MSE. As the prior is diagonal, the new eigenmodes are also eigenmodes of the prior with the same eigenvalue, given by (shaded region in the upper panels of Fig. 5). The reduction of the MSE thus roughly tells us how many modes can be constrained compared with the prior. For example, adding LSST reduces the MSE from 144 to 112, meaning that there are (144 - 112)/3.6 ≃ 9 modes that can be constrained by LSST, and similarly, the ideal 10k survey is able to constrain seven modes. With KDUST + LSST, one can actually constrain 10 eigenmodes. These numbers can also be counted in the upper panel of Figure 5 above the prior threshold shown in shade.

For a correlated prior, we can analogously put a lower bound on the number of modes constrained by data using the MSE shown in Table 2. For instance, for the case of z_c = 0.1 and z_c = 0.4, LSST combined with KDUST can at least constrain (35 - 18.1)/3.6 ≃ 4 and (11.5 - 3.7)/3.6 ≃ 2 modes, respectively. As we see, the number of new modes estimated in this way is reduced as the prior correlation length is increased. This is as expected—as z_c increases, more high-frequency modes will be filtered out by the smoothness prior.

Dome A offers a very competitive site for studying dark energy. Given the amount of resources required to build a large telescope and run a massive survey there, one must give the highest priority to programs that cannot be easily carried out elsewhere. Thus, a reasonable strategy is to focus on NIR imaging and collaborate with other surveys for optical data. Using LSST as an example, we show that a high-resolution 5000–10,000 deg² KDUST survey in yJH bands could improve LSST BAO + WL constraints on the dark energy EOS parameters w₀ and w_a by reducing the photo- z and shear measurement systematics. A SNAP-like SN sample plus a large local and nearby SN sample from KDUST would further boost the DETF FOM by more than a factor of 2.

In addition to forecasts for the w₀ - w_a parametrization, we also apply a PCA approach to investigate the constraints on the dark energy EOS w(z) in a model-independent way. We find that regarding the number of the constrained eigenmodes of w(z), an ideal 10,000 deg² ugrizyJH survey, combined with Planck, can constrain seven eigenmodes, while KDUST+LSST can allow us to constrain three more modes.

We have not discussed dark energy probes such as strong lensing, cluster counting, and higher-order statistics of the same galaxy and shear data, which could further tighten the constraints on the dark energy EOS. Strong lensing constrains dark energy through the time delay effect, as well as the counting of strong lenses. It is also an excellent probe of dark matter halo structures and, hence, can be used to measure dark matter particle properties. With high-resolution imaging, one could extract more cosmological information from strong lensing observations. Therefore, Dome A could be particularly advantageous for strong lensing studies.

Dark energy forecasts depend crucially on the assumed properties of the survey data, including all the systematics. Dome A has many advantages over other ground sites and has an environment close to that in space. Hence, we use well-studied LSST and SNAP as references to make crude estimates of the data for this investigation. Further work and detailed modeling are needed to give a more realistic assessment of the Dome A site for studying dark energy.

We thank David Bacon, Rob Crittenden, Kazuya Koyama, Bob Nichol and Levon Pogosian for useful discussions. G. Z. is supported by an Engineering Research Center grant. H. Z. is supported by the Bairen program from the Chinese Academy of Sciences and the National Basic Research Program of China grant no. 2010CB833000. The work of L. Wang is partially supported by National Science Foundation (NSF) grant AST-0708873. X. Z. is supported in part by the NSF of China.

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