CONSTRAINTS ON GRAVITY AND DARK ENERGY FROM THE PAIRWISE KINEMATIC SUNYAEV–ZEL'DOVICH EFFECT

Even though on large scales the universe appears homogenous and isotropic, initial local matter overdensities form galaxies and galaxy clusters and evolve into the large scale structure of the universe. The growth of these structures depends on the underlying physical theory and can therefore be used to constrain cosmological models.

According to linear theory, the matter over-density, ${\delta }_{m}$, is related to the velocity of dark matter particles, ${\dot{\delta }}_{m}\propto {v}_{m}$, which connects the time evolution of the perturbations to the dark matter velocity. Any tracer of the underlying dark matter velocity distribution can be used to constrain cosmology and in particular modified gravity models. In a variety of modified gravity scenarios the evolution of the density perturbations can be quite different from standard gravity even though the background expansion of the universe is undistinguishable from a ΛらむだCDM universe (for example Linder & Cahn 2007; Amendola 2008; Laszlo 2008). The linear perturbation equations have a solution of the form ${\delta }_{m}({\boldsymbol{x}},t)={D}_{a}(t)\delta (x)$, factorizing the spatial and temporal dependency, with D_a being the growth factor. We can define the growth rate at a given scale factor, a, as

$$\begin{eqnarray}&&{f}_{g}(a)\equiv \displaystyle \frac{d\mathrm{ln}{D}_{a}}{d\mathrm{ln}a}\end{eqnarray} \tag{ 1 }$$

to parametrize the growth of structure. The growth rate is well approximated by ${f}_{g}(a)\approx {{\rm{\Omega }}}_{m}{(a)}^{\gamma }$ with the growth index $\gamma \simeq 0.55$ for standard gravity (Wang & Steinhardt 1998; Linder 2005). Pairwise velocity statistics can be used to constrain the cosmological model of the universe and the underlying theory of gravity (Davis & Peebles 1977; Ferreira et al. 1999; Juszkiewicz et al. 1999).

We analytically model the expected large scale motion of clusters under cosmological gravitational interactions by considering the properties of dark matter particles, in linear theory, and then using a halo model to infer the velocity statistics of gravitationally bound halos, which we use as proxies for galaxy clusters.

Following the formalism outlined in (Sheth et al. 2001), we assume linear theory to describe the mean pairwise streaming velocity, v, between two dark matter particles, at positions ${{\boldsymbol{r}}}_{{\boldsymbol{i}}}$ and ${{\boldsymbol{r}}}_{{\boldsymbol{j}}}$, in terms of their comoving separation $r=| {{\boldsymbol{r}}}_{{\boldsymbol{i}}}-{{\boldsymbol{r}}}_{{\boldsymbol{j}}}|$,

$$\begin{eqnarray}&&v(r,a)=-\displaystyle \frac{2}{3}{f}_{g}(a)H(a){ar}\displaystyle \frac{\bar{\xi }(r,a)}{1+\xi (r,a)}\end{eqnarray} \tag{ 2 }$$

where ξくしー is the dark matter two-point correlation function and $\bar{\xi }$ the volume averaged correlation function, respectively defined as,

$$\begin{eqnarray}&&\xi (r,a)=\displaystyle \frac{1}{2{\pi }^{2}}\int {{dkk}}^{2}{j}_{0}({kr})P(k,a),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\bar{\xi }(r,a)=\displaystyle \frac{3}{{r}^{3}}{\displaystyle \int }_{0}^{r}{dr}\prime r{\prime }^{2}\xi (r\prime ,a),\end{eqnarray} \tag{ 4 }$$

with $P(k,a)$ being the dark matter power spectrum and ${j}_{0}(x)=\mathrm{sin}(x)/x$ is the zeroth order spherical Bessel function.

The properties of dark matter halos of mass M, relative to the dark matter distribution, can be modeled using a halo bias

$$\begin{eqnarray}&&b(M,a)=1+\displaystyle \frac{{\delta }_{\mathrm{crit}}^{2}-{\sigma }_{0}^{2}(M,a=1)}{{\sigma }_{0}^{2}(M,a=1){\delta }_{\mathrm{crit}}{D}_{a}},\end{eqnarray} \tag{ 5 }$$

where $M(R)=4\pi {R}^{3}\bar{\rho }/3$, $\bar{\rho }$ is the average cosmological matter density, the critical overdensity is taken to have the standard ΛらむだCDM value of ${\delta }_{\mathrm{crit}}\approx 1.686$, and the zeroth order moment of the mass distribution squared is

$$\begin{eqnarray}&&{\sigma }_{0}^{2}(m,a)=\displaystyle \frac{1}{2{\pi }^{2}}{\displaystyle \int }_{0}^{\infty }{{dkk}}^{2}P(k,a){W}^{2}({kR}(M))\end{eqnarray} \tag{ 6 }$$

with a top-hat window function $W(x)=3(\mathrm{sin}x-x\mathrm{cos}x)/{x}^{3}$.

Surveys will generally include cluster halos over a range of masses above some limiting mass threshold, ${M}_{\mathrm{min}}$. To analyze the mass statistics we consider a mass averaged cluster pairwise velocity statistic, $V$, for pairs of clusters separated by a comoving distance r

$$\begin{eqnarray}&&V(r,a)=-\displaystyle \frac{2}{3}H(a){{af}}_{g}(a)\displaystyle \frac{r{\bar{\xi }}_{h}(r,a)}{1+{\xi }_{h}(r,a)},\end{eqnarray} \tag{ 7 }$$

which has an analogous expression to that in (2) (Sheth et al. 2001; Bhattacharya & Kosowsky 2008), with

$$\begin{eqnarray}&&{\xi }_{h}(r,a)=\displaystyle \frac{1}{2{\pi }^{2}}\int {{dkk}}^{2}{j}_{0}({kr}){P}_{\mathrm{lin}}(k,a){b}_{h}^{(2)}(k),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\bar{\xi }}_{h}(r,a)=\displaystyle \frac{3}{{r}^{3}}{\displaystyle \int }_{0}^{r}{dr}\prime r{\prime }^{2}\xi (r\prime ,a){b}_{h}^{(1)}(k).\end{eqnarray} \tag{ 9 }$$

The mass-averaged halo bias moments, ${b}_{h}^{(q)}$, are given by

$$\begin{eqnarray}&&{b}_{h}^{(q)}(k,a)=\displaystyle \frac{{\displaystyle \int }_{{M}_{\mathrm{min}}}^{{M}_{\mathrm{max}}}{dM}\;M\;n(M,a){b}^{q}(M,a){W}^{2}[{kR}(M)]}{{\displaystyle \int }_{{M}_{\mathrm{min}}}^{{M}_{\mathrm{max}}}{dM}\ M\ n(M,a){W}^{2}[{kR}(M)]}\end{eqnarray} \tag{ 10 }$$

where $n(M,a)$ is the number density of halos of mass M, given by the Jenkins mass function, and the Gaussian window function, $W(x)=\mathrm{exp}(-{x}^{2}/2)$, with $M(R)={(2\pi )}^{3/2}\bar{\rho }{R}^{3}$.

In Figure 1 we show the mean pairwise velocity, $V$ as a function of cluster separation r for a number of cosmological models at z = 0.15 for a survey with limiting mass ${M}_{\mathrm{min}}={10}^{14}\ {M}_{\odot }$, assuming all other survey specifications are fixed (left panel) and for various assumptions on the limiting mass (right panel). The figure suggests that, as with other linear growth rate related statistics, the Equation of state, w₀, and growth exponent, γがんま, have degenerate effects on the pairwise velocity amplitude, through their effects on the growth factor, and do not alter the shape of the function. However, as indicated in Section 3.4, the redshift dependence of these parameters helps to break the degeneracy. To be more specific, the amplitude of V as a function of z is different for variations in γがんま compared to w₀. Increasing the minimum cluster mass shifts the peak of the pairwise velocity function to larger scales (on scales below 60 Mpc h⁻¹) and boosts the overall amplitude on scales larger than this, because the larger clusters have a larger streaming velocity.

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Figure 2 shows the effect of different window function when calculating the mass-averaged halo moments ${b}_{h}^{(q)}(k,a)$. For large minimum mass, a top-hat filter induces sharp features in $V$ due to edge effects of the window function at small scales, whereas a Gaussian filter leads to smoother results. Therefore, in the following we assume a Gaussian window function.

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Measurements of cluster velocities are subject to a number of statistical and systematic uncertainties. First, discreteness effects need to be taken into consideration; a smooth continuous field is typically assumed to underly a discrete distribution of local objects, which leads to shot noise. For a large sample size the shot noise should be approximately Gaussian resulting in an error proportional to $1/N$ (Eisenstein & Zaldarriaga 1999), where N is the number of objects in the sample. If the number of objects (e.g., clusters) in the sample is not sufficiently large, the Gaussian limit breaks down, and an additional non-Gaussian contribution to the shot noise can become relevant (Cohn 2006).

Second, as in any cosmological survey, the measurement will be subject to cosmic variance due to the finite size of the sample. Third, in addition to the statistical errors we include a velocity measurement error (Bhattacharya & Kosowsky 2008) to account for the accuracy of the measurements and the uncertainty in the optical depth of the clusters. The total covariance for the mean pairwise velocity is therefore a combination of cosmic variance, shot noise, and the velocity measurement error:

$$\begin{eqnarray}{C}_{V}^{\mathrm{total}}(r,r\prime ) & = & {C}_{V}^{\mathrm{cosmic}}(r,r\prime )+{C}_{V}^{\mathrm{shot}}(r,r\prime )\\ & & +{C}_{V}^{\mathrm{measurement}}(r,r\prime ).\end{eqnarray} \tag{ 11 }$$

A detailed derivation of the covariance terms can be found in the Appendix. We summarize the results here.

Defining an estimator for the mean pairwise velocity, $\hat{V}$, enables the covariance matrix to be calculated using

$$\begin{eqnarray}&&{C}_{V}(r,r\prime )=\langle \hat{V}(r)\hat{V}(r\prime )\rangle -\langle \hat{V}(r)\rangle \langle \hat{V}(r\prime )\rangle \end{eqnarray} \tag{ 12 }$$

where $\langle \ldots \rangle$ is the volume average. For analyzing a survey we include binning, as observations will be combined not at just one radius r but in bins of width ${\rm{\Delta }}r$,

$$\begin{eqnarray}&&\hat{V}(r)\to {V}_{{\rm{\Delta }}}(r)=\displaystyle \frac{1}{{V}_{\mathrm{bin}}}{\displaystyle \int }_{r-{\rm{\Delta }}r/2}^{r+{\rm{\Delta }}r/2}{\tilde{r}}^{2}d\tilde{r}\int d{\rm{\Omega }}\hat{V}(\tilde{r}),\end{eqnarray} \tag{ 13 }$$

assuming spherical symmetry and where a Δでるた subscript indicates binned quantities over bins of size ${\rm{\Delta }}r$.

The covariance between the mean pairwise velocities of two cluster pairs, with the two pairs separated by r and r' and using bin width ${\rm{\Delta }}r$, can be expressed as

$$\begin{eqnarray}&&{C}_{V}(r,r\prime )\\ &&=\displaystyle \frac{4}{{\pi }^{2}{V}_{s}(a)}{\left(\displaystyle \frac{H(a)a}{1+{\xi }_{h}(r,a)}\right)}^{2}{f}_{g}{(a)}^{2}\\ &&\times \left[\displaystyle \int {dk}{\left({P}_{\mathrm{lin}}(k,a){b}_{h}^{(1)}(k)+\displaystyle \frac{1}{{n}_{\mathrm{cl}}(a)}\right)}^{2}{W}_{{\rm{\Delta }}}({kr}){W}_{{\rm{\Delta }}}({kr}\prime )\right.\\ &&+\left.\displaystyle \frac{{\rm{\Delta }}r}{{V}_{{\rm{\Delta }}}(r\prime )}\displaystyle \int {dkk}\displaystyle \frac{{P}_{\mathrm{lin}}(k,a){b}_{h}^{(1)}(k)}{{n}_{\mathrm{cl}}{(a)}^{2}}{W}_{{\rm{\Delta }}}({kr})\right],\end{eqnarray} \tag{ 14 }$$

where V_s(a) is the survey volume, ${n}_{\mathrm{cl}}(a)$ is the number density of clusters, and

$$\begin{eqnarray}&&{W}_{{\rm{\Delta }}}({kr})=3\displaystyle \frac{{R}_{\mathrm{min}}^{3}\tilde{W}({{kR}}_{\mathrm{min}})-{R}_{\mathrm{max}}^{3}\tilde{W}({{kR}}_{\mathrm{max}})}{{R}_{\mathrm{max}}^{3}-{R}_{\mathrm{min}}^{3}}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\tilde{W}(x)=\displaystyle \frac{2\mathrm{cos}(x)+x\mathrm{sin}(x)}{{x}^{3}}.\end{eqnarray} \tag{ 16 }$$

The first term in (14) is the Gaussian contribution to the covariance, which includes both cosmic variance ($\propto P$) and shot noise ($\propto 1/{n}_{\mathrm{cl}}$). The second term is an additional contribution that is often neglected, which arises if the Gaussian limit breaks down; we refer to this term as "Poisson" shot noise as in (Cohn 2006). While we find it is subdominant in comparison to the Gaussian terms for a mass cut-off $M\leqslant 1\times {10}^{14}\ {M}_{\odot }$ (see Figure 4), it can be important for surveys with smaller cluster number densities. The purely Gaussian shot noise contribution on the other hand is not insignificant and should be included.

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We include a contribution to the covariance due to the uncertainty in measuring the velocity given by (Bhattacharya & Kosowsky 2008),

$$\begin{eqnarray}&&{C}_{V}^{\mathrm{measurement}}(r,r\prime )=\displaystyle \frac{2{\sigma }_{v}^{2}}{{N}_{\mathrm{pair}}}{\delta }_{r,r\prime }\end{eqnarray} \tag{ 17 }$$

where ${\sigma }_{v}$ is the measurement error discussed in more detail in Section 3.4, and ${N}_{\mathrm{pair}}$ is the number of pairs in each separation bin given by

$$\begin{eqnarray}{N}_{\mathrm{pair}}(r,a) & = & \displaystyle \frac{{n}_{\mathrm{cl}}(a){V}_{s}(a)}{2}\\ & & \times \left({V}_{{\rm{\Delta }}}(r){n}_{\mathrm{cl}}(a)+4\pi {r}^{2}{n}_{\mathrm{cl}}(a){\xi }_{h}(r,a){\rm{\Delta }}r\right).\end{eqnarray} \tag{ 18 }$$

As shown in Figure 3, the number of cluster pairs increases rapidly with decreased minimum mass. The measurement error term in the covaraince is proportional to $1/{N}_{\mathrm{pairs}}$ and will increase quickly with an increasing number of bins since the number of cluster pairs directly depends on the size of the r-bin.

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Figure 4 shows the diagonal elements of the covariance matrix for the different covariance components, for a bin width of ${\rm{\Delta }}r=2\ \mathrm{Mpc}\;{{\rm{h}}}^{-1}$. As cluster separation increases, the covariance becomes dominated by cosmic variance, while at smaller separations $\lesssim 40\ \mathrm{Mpc}/{\rm{h}}$, the contributions from each of the terms becomes comparable. As a result of the multiple contributions to the covariance matrix, and their respective sensitivities to bin size and cluster separation, the total covariance matrix slightly depends upon the number of bins. The measurement error and shot noise can be reduced by choosing a coarser binning with the trade-off of decreased resolution and loss of information. On the other hand, the fractional contribution of the cosmic variance will increase as the size of the bins increases. Once the cosmic variance dominates nothing can be gained from a coarser binning. A very coarse binning marginally reduces the constraints, e.g., using ${\rm{\Delta }}r=20\ \mathrm{Mpc}/{\rm{h}}$ lowers the figures of Merit (FoM) by 30% compared to ${\rm{\Delta }}r=2\ \mathrm{Mpc}/{\rm{h}}$, however, any bin size smaller than ${\rm{\Delta }}r=5\ \mathrm{Mpc}/{\rm{h}}$ leads to equivalent results. Throughout the analysis we assume a bin size of ${\rm{\Delta }}r=2\ \mathrm{Mpc}/{\rm{h}}$.

Off-diagonal covariances between cluster pairs of different separations are important. Figure 5 shows the covariance contributions from cosmic variance and shot noise and indicates the comparative importance of off-diagonal terms. The off-diagonal contributions have a notable effect on the Fisher matrix amplitudes as a function of separation, giving rise to the differences between the left and right panels in Figure 4. The right panel shows the effect on the Fisher matrix of changing key model assumptions, the minimum detectable cluster mass and the mean pairwise velocity uncertainty. Altering the mass limit has a larger effect than comparable changes to the the measurement error because the number of clusters and cluster pairs strongly depend on the limiting mass (see Figure 3), changing shot noise as well as the measurement error contribution to the covariance significantly.

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For our analysis we consider constraints on 9 cosmological parameters:

$$\begin{eqnarray}&&{\boldsymbol{p}}=\{{{\rm{\Omega }}}_{b}{h}^{2},{{\rm{\Omega }}}_{m}{h}^{2},{{\rm{\Omega }}}_{k},{{\rm{\Omega }}}_{{\rm{\Lambda }}},{w}_{0},{w}_{a},{n}_{s},\mathrm{ln}{A}_{s},\gamma \}\end{eqnarray} \tag{ 19 }$$

where ${{\rm{\Omega }}}_{b}$, ${{\rm{\Omega }}}_{m}$, ${{\rm{\Omega }}}_{k}$ and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ are the dimensionless baryon, matter, curvature and dark energy densities respectively, h is the Hubble constant in units of 100 km s⁻¹ Mpc⁻¹, w₀ and w_a are the dark energy equation of state parameters, such that the equation of state is $w(a)={w}_{0}+(1-a){w}_{a}$, γがんま is the growth rate exponent, such that ${f}_{g}={{\rm{\Omega }}}_{m}{(a)}^{\gamma }$, and n_s and A_s are the spectral index and normalization of the primordial spectrum of curvature perturbations.

Throughout this paper we assume a fiducial model that is a ΛらむだCDM cosmological model with parameters consistent with those adopted in (Laureijs et al. 2011): ${{\rm{\Omega }}}_{b}{h}^{2}$ $=\;0.021805,{{\rm{\Omega }}}_{m}{h}^{2}$ $=\;0.1225,{{\rm{\Omega }}}_{k}$ $=\;0,{{\rm{\Omega }}}_{{\rm{\Lambda }}}$ $=\;0.75,{w}_{0}$ $=\;-0.95,\;$ ${w}_{a}$ $=\;0,{n}_{s}=1,$ $\;\mathrm{ln}({10}^{10}{A}_{s})=3.1954$.

We calculate constraints on cosmological parameters using the Fisher Matrix formalism. The covariance between two parameters ${p}_{\mu }$ and ${p}_{\nu }$, from (19), is given by

$$\begin{eqnarray}&&{F}_{\mu \nu }=\displaystyle \sum _{i}^{{N}_{z}}\displaystyle \sum _{p,q}^{{N}_{r}}\displaystyle \frac{\partial V({r}_{p},{z}_{i})}{\partial {p}_{\mu }}{\mathrm{Cov}}_{i}^{-1}({r}_{p},{r}_{q},{z}_{i})\displaystyle \frac{\partial V({r}_{p},{z}_{i})}{\partial {p}_{\nu }},\end{eqnarray} \tag{ 20 }$$

where ${Cov}({r}_{p},{r}_{q},{z}_{i})$ is the covariance matrix between two clusters pairs as defined in 2.3, including a redshift bin with mid-point z_i and the clusters in each pair having comoving separations of r_p and r_q. N_z and N_r are the number of redshift and spatial separation bins, respectively.

We quote results in terms of the Dark Energy FoM (Albrecht et al. 2006) defined as

$$\begin{eqnarray}&&\mathrm{FoM}=\mathrm{det}{\left[({F}^{-1})\right]}_{{w}_{0},{w}_{a}}^{-1/2}\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{\mathrm{FoM}}_{\mathrm{GR}}=\mathrm{det}{\left[({F}_{\mathrm{GR}}^{-1})\right]}_{{w}_{0},{w}_{a}}^{-1/2}.\end{eqnarray} \tag{ 22 }$$

${({F}_{\mathrm{GR}}^{-1})}_{{w}_{0},{w}_{a}}$ is the 2 × 2 submatrix of the inverted Fisher matrix excluding the modified gravity parameter γがんま. This procedure is equivalent to marginalizing over the 7 parameters (for MG) or 6 parameters (for GR) of the model considered.

Throughout this paper we consider results in combination with either simply a Planck-like CMB prior or a Dark Energy Taskforce (DETF; Albrecht et al. 2006) prior that includes CMB, SN, and non-kSZ related LSS constraints on the background cosmological and dark energy parameters. We do not include a prior on the modified gravity parameters unless stated otherwise. For the Planck-like CMB survey, we consider complementary constraints on the cosmological parameters from the temperature (T) and polarization (E) measurements up to l = 3000 as summarized in Table 1.⁴

Table 1.  CMB Survey Specifications

Parameter	Frequency (GHz)
	100	143	217
${f}_{\mathrm{sky}}$	⋯	0.8	⋯
${\theta }_{\mathrm{FWHM}}$(arcmin)	10.7	8.0	5.5
${\sigma }_{T}$(μみゅーK)	5.4	6.0	13.1
${\sigma }_{E}$(μみゅーK)	⋯	11.4	26.7

Note. For the sky coverage, ${f}_{\mathrm{sky}}$, beam size, ${\theta }_{\mathrm{FWHM}}$, and noise levels per pixel for the temperature and polarization detections at 3 frequencies, for a Planck-like survey.

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We forecast cosmological constraints for three different combinations of surveys: (1) a current (Stage II) CMB survey, such as ACTPol (Niemack et al. 2010), combined with a galaxy sample that includes spectroscopic redshifts, such as SDSS BOSS (Comparat et al. 2013), (2) a near-term (Stage III) survey, such as Advanced ACTPol (Calabrese et al. 2014), also combined with SDSS BOSS, and (3) a longer-term (Stage IV) survey, such as CMB-S4 (Abazajian et al. 2013), combined with a next generation spectroscopic survey, such as DESI (Levi et al. 2013).

The mean cluster pairwise velocity can be measured by cross-correlating the kSZ signal with cluster positions and redshifts. For the cluster sample, we assume that a spectroscopic survey provides redshifts to LRGs over an overlapping area with the CMB survey. Recent studies show that the kSZ signal can be extracted from the CMB maps using LRGs of the BOSS survey as a proxy for clusters (Hand et al. 2011). Using LRGs creates a large, precise positioned sample of tracers to extract the kSZ correlation.

However, there are several factors that need to be considered in using LRGs as cluster tracers. LRGs are not perfect tracers of a cluster's center, with perhaps 40% of bright LRGs and 70% of faint LRGs off-centered, satellite galaxies (Hikage et al. 2012) that may be related to cluster mergers (Martel et al. 2014). The imprecise match between LRGs and clusters could lead to detrimental misalignments, such as trying to extract the kSZ signal from positions that are not associated with clusters or an incomplete cluster catalog if spectroscopic measurements of an LRG near the cluster center were not obtained. The theoretical mean cluster pairwise velocity is an observable averaged over all cluster pairs assuming a complete sample above a limiting minimum mass. While (Hand et al. 2011) optimize the angular size of the CMB sub-map used in the stacking approach to minimize the overall covariance, this does not ensure that the cluster sample obtained from the LRGs is complete. Further studies are needed to quantify the effects of using LRGs as cluster tracers and ensure that no bias is introduced in the analysis before this approach can be used for cosmological constraints. Additional clustering information, such as the halo occupation distribution of the galaxy sample, must be used to quantify the uncertainty of LRGs as accurate cluster tracers and ensure that no bias is introduced in the cosmological constraints. Modeling the number and distribution of galaxies within the dark matter halos could include both full simulations and constraints from large scale structure data (White et al. 2011).

Another issue is that the uncertainty in the minimum mass of the cluster sample associated with the LRGs is difficult to estimate, although, the minimum mass uncertainty could be treated as an additional nuisance parameter in the analysis.

To acknowledge these issues in our forecasts, we assume a scenario that aims to maximize cluster completeness and purity, with a well defined cluster mass cut-off, rather than cluster number density. We select a survey area that has photometric and spectroscopic galaxy catalogs and overlapping CMB kSZ data. Specifically, we consider BOSS and a DESI-like survey, for which we expect photometric catalogs to exist over the survey area. We note that Euclid spectroscopic and imaging surveys, and LSST imaging with overlapping WFIRST imaging and spectroscopy would also provide future valuable data sets at higher redshifts. The uncertainties in the cosmological parameters evolve as the square root of the sky coverage. Requiring spectroscopic redshifts, e.g., from BOSS, limits the survey area, but provides confidence that the comoving cluster separation can be accurately calculated as in (Hand et al. 2012). Photometric information allows cluster detection, and mass estimates, using algorithms, such as the friends-of-friends, as used in redMaPPer (Rykoff et al. 2014), to maximize the completeness and purity of the cluster sample, with the drawback of a limited number of clusters and a volume-limited catalog. A study of using only photometric information to extract the kSZ signal can be found in (Keisler & Schmidt 2013).

The survey specification assumed in our analysis for the CMB and large scale structure Stage II, III, and IV surveys are given in Table 2. We assume a BOSS-like spectroscopic survey for Stages II and III and a DESI-like Stage IV survey with redshift ranges that are determined by the redshift coverage of the LRG sample and assume joint photometric survey data. We assume Stages II and III have access to the same or comparable LRG surveys so retain the same limiting mass, but do slightly increase CMB overlap with these data due to the larger survey area planned for Advanced ACTPol (Calabrese et al. 2014). For Stage IV we assume a deeper LRG survey that provides lower minimum mass, higher z, and larger overlap. Our minimum mass assumptions are conservative, and will likely be improved upon at each respective stage. As an example, the LSST survey projects that the minimum detectable cluster mass at $z\sim 0.6$ will be lower than $\sim 5\times {10}^{13}\ {M}_{\odot }$ after a single visit image in all bands, and be better than ${10}^{13}\ {M}_{\odot }$ in all bands in the complete 10 years survey (Abell et al. 2009). Similarly the SDSS-derived MaxBCG Catalog already achieves 90% purity and >85% completeness for clusters of masses exceeding ${10}^{14}\ {M}_{\odot }$ (Koester et al. 2007).

Table 2.  Reference Survey Specifications Used to Model Stages II, III, and IV kSZ Cluster Surveys

		Survey Stage
Survey	Parameters	II	III	IV
CMB	${\rm{\Delta }}{T}_{\mathrm{instr}}$ ($\mu K\mathrm{arcmin}$)	20	7	1
Galaxy	${z}_{\mathrm{min}}$	0.1	0.1	0.1
	${z}_{\mathrm{max}}$	0.4	0.4	0.6
	No. of z bins, Nz	3	3	5
	${M}_{\mathrm{min}}$ (${10}^{14}{M}_{\odot }$)	1	1	0.6
Overlap	Area (1000 sq. deg.)	4	6	10

Note. The expected instrument sensitivity of the CMB survey, ${\rm{\Delta }}{T}_{\mathrm{inst}}$, along with the assumed optical large scale structure survey redshift range ${z}_{\mathrm{min}}\lt z\lt {z}_{\mathrm{max}}$, redshift binning, and minimum detectable cluster mass, ${M}_{\mathrm{min}}$ are shown. We consider an effective sky coverage by estimating the degree of overlap between the respective cmb and optically selected cluster data sets.

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The measurement error for the radial peculiar velocity, v, of a cluster is a combination of the instrumental sensitivity as well as the uncertainty in the optical depth, τたう, for each cluster as the kSZ signal is proportional to τたう as follows (Sunyaev & Zeldovich 1980),

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{T}_{\mathrm{kSZ}}}{{T}_{\mathrm{CMB}}}=-\displaystyle \frac{v}{c}\tau ,\end{eqnarray} \tag{ 23 }$$

where ${T}_{\mathrm{CMB}}$ is the temperature of the CMB. We estimate the total measurement error by adding those two sources of uncertainty in quadrature as

$$\begin{eqnarray}&&{\sigma }_{v}=\sqrt{{\sigma }_{\mathrm{instr}}^{2}+{\sigma }_{\tau }^{2}.}\end{eqnarray} \tag{ 24 }$$

The accuracy of the instrument is given by

$$\begin{eqnarray}&&{\sigma }_{\mathrm{instr}}=\displaystyle \frac{{\rm{\Delta }}{T}_{\mathrm{instr}}}{{\rm{\Delta }}{T}_{\mathrm{kSZ}}}\times v=\displaystyle \frac{{\rm{\Delta }}{T}_{\mathrm{pixel}}/\sqrt{{N}_{\mathrm{pixel}}}}{\tau v/{{cT}}_{\mathrm{CMB}}}\times v\end{eqnarray} \tag{ 25 }$$

where ${\rm{\Delta }}{T}_{\mathrm{pixel}}$ is the sensitivity of the instrument per pixel and ${N}_{\mathrm{pixel}}$ being the number of pixels of a cluster. We assume that an average size cluster will have ${N}_{\mathrm{pixel}}\approx 4$ and an instrument sensitivity as summarized in Table 2. The uncertainty in the optical depth is given by

$$\begin{eqnarray}&&{\sigma }_{\tau }=\displaystyle \frac{{\rm{\Delta }}\tau }{\tau }\times v.\end{eqnarray} \tag{ 26 }$$

Assumed uncertainties contributing to the measurement error are summarized in Table 3. We use the scatter in the optical depth, $| {\rm{\Delta }}\tau /\bar{\tau }|$, and the mean value of τたう from simulations (Battaglia et al. 2010),⁵ to obtain an indicative estimate for the intrinsic dispersion in τたう averaged over all cluster masses. For the fiducial analysis we do not include any further dispersion arising from potential additional measurement accuracy in determining τたう. Section 3.4 includes a discussion of the impact of additional factors affecting the measurement error on the cosmological constraints.

Table 3.  The Assumed Individual Contribution from Instrument Sensitivity, ${\sigma }_{\mathrm{instr}}$, and Uncertainty in τたう, ${\sigma }_{\tau }$

Parameter	Survey Stage	Redshift bin
		0.15	0.25	0.35	0.45	0.55
${10}^{3}\tau$		3.45	2.27	1.84	1.45	1.20
${({{\rm{\Delta }}}_{\tau }/\bar{\tau })}^{2}$		⋯	⋯	0.15	⋯	⋯
${\sigma }_{\tau }$ (km s−1)		⋯	⋯	120	⋯	⋯
${\sigma }_{\mathrm{instr}}$	Stage II	290	440	540	⋯	⋯
(km s−1)	Stage III	100	150	190	⋯	⋯
	Stage IV	15	22	27	34	42
${\sigma }_{v}$	Stage II	310	460	560	⋯	⋯
(km s−1)	Stage III	160	200	230	⋯	⋯
	Stage IV	120	120	120	120	130

Note. The values of τたう and fractional uncertainty in τたう, ${({\rm{\Delta }}\tau /\bar{\tau })}^{2}$, are estimated from simulations assuming a convolution over a 13 beam. ${\sigma }_{v}$ is the total measurement uncertainty for the reference case.

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In this section we discuss the potential of upcoming kSZ surveys to constrain dark energy and modified gravity parameters. Figure 1 shows both the equation of state, driving the expansion history, and γがんま, that modifies the growth history of density perturbations, have qualitatively similar effects on the pairwise velocity function through their effect on the linear growth factor. For cluster measurements in each individual redshift bin this creates a degeneracy between the equation of state and γがんま parameters. As shown in Figure 6, the use of multiple redshift bins allows the differences in the evolution of the growth rate for the dark energy and modified gravity parameters to be distinguished. The constraints on w₀ and γがんま from the low and high redshift bins are markedly orthogonal; in combination this complementarity tightens the constraints, in particular on the growth factor. In Figure 7 we present the 2D marginalized constraints in the ${w}_{0}-\gamma$ parameter plane for kSZ in combination with a Planck-like CMB and DETF priors on all parameters excluding γがんま, including CMB, BAO, weak lensing and SNe measurements for a combination of Stage III-like surveys (Albrecht et al. 2006). The suite of Stage III DETF-motivated observables would provide stronger constraints on the equation of state, through the addition of geometric measurements that constrain the expansion history. These break the degeneracy between w₀ and γがんま from kSZ and CMB measurements alone. With the addition of a CMB prior on the data, however, the data can constrain γがんま to 9%, 8% and 5% respective in the Stage II through IV survey specifications. The kSZ is a less powerful tool for constraining the dark energy equation of state. A Stage IV-like survey can achieve FoMs of ${\mathrm{FoM}}_{\mathrm{GR}}=61$ with a CMB prior (which has ${\mathrm{FoM}}_{\mathrm{GR}}=1.15$ alone), and $\mathrm{FoM}=292$ with DETF Stage III data included (${\mathrm{FoM}}_{\mathrm{GR}}=116$).

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Complementary, contemporaneous constraints from baryonic acoustic oscillations and SNe Ia will provide significantly tighter constraints on the background expansion history and the equation of state. If we include the impact of a DETF Stage III prior on all parameters, excluding the growth factor, the degeneracy between the equation of state and growth factor is significantly reduced and the projected constraints on γがんま are improved, with fractional errors of 5% and 3% for Stage III and IV surveys. Table 4 summarizes the dark energy FoM, assuming modified gravity (9 parameters, marginalizing over γがんま), GR (8 parameters, fixing γがんま) and a flat, GR cosmology (7 parameters, fixing γがんま and ${{\rm{\Omega }}}_{k}$), as well as the $1\sigma$ constraints of w₀, w_a (marginalizing over γがんま) and γがんま for Stage II-, III-, and IV- like surveys, as specified in Table 2.

Table 4.  [Left Columns] Results for the Reference Survey Assumptions as Summarized in Table 2 Including [Top Rows] Planck Priors and [Lower Rows] Constraints on the Background Cosmological Parameters (Excluding γがんま) from the DETF Stage III Survey

Data Set	Parameter	Fiducial assumptions			+ Uncertainty in ${M}_{\mathrm{min}}$			+ Lower ${M}_{\mathrm{min}}$
		Stage II	Stage III	Stage IV	Stage II	Stage III	Stage IV	Stage II	Stage III	Stage IV
+CMB	${\mathrm{FoM}}_{\mathrm{MG}}$	5	6	26	4	9	19	7	9	154
	${\mathrm{FoM}}_{\mathrm{GR}}$	7	10	47	6	12	29	10	15	195
	${\mathrm{FoM}}_{\mathrm{flat}}$	33	43	102	28	40	84	50	63	289
	$\sigma ({w}_{0})$	0.73	0.70	0.35	0.62	0.39	0.41	0.67	0.63	0.09
	$\sigma ({w}_{a})$	2.6	2.5	1.2	2.2	1.4	1.5	2.5	2.3	0.3
	${\rm{\Delta }}\gamma /\gamma$	0.09	0.08	0.05	0.10	0.07	0.05	0.07	0.06	0.02
+DETF	${\mathrm{FoM}}_{\mathrm{MG}}$	119	130	208	⋯	⋯	⋯	⋯	⋯	⋯
	${\mathrm{FoM}}_{\mathrm{GR}}$	128	139	229	⋯	⋯	⋯	⋯	⋯	⋯
	${\mathrm{FoM}}_{\mathrm{flat}}$	176	191	321	⋯	⋯	⋯	⋯	⋯	⋯
	$\sigma ({w}_{0})$	0.10	0.09	0.07	⋯	⋯	⋯	⋯	⋯	⋯
	$\sigma ({w}_{a})$	0.30	0.29	0.24	⋯	⋯	⋯	⋯	⋯	⋯
	${\rm{\Delta }}\gamma /\gamma$	0.07	0.05	0.03	⋯	⋯	⋯	⋯	⋯	⋯

Note. For reference, the Planck-like Fisher matrix alone has ${\mathrm{FoM}}_{\mathrm{GR}}=1.15$ and DETF has ${\mathrm{FoM}}_{\mathrm{GR}}=116$. [Central columns] Results in which the impact of an uncertainty in the exact minimum mass of the cluster sample, ${M}_{\mathrm{min}}$, is included by marginalizing over ${M}_{\mathrm{min}}$ as a nuisance parameter with a 15% prior imposed. [Right columns] Results for a more optimistic mass cut-off of ${M}_{\mathrm{min}}=4\times {10}^{13}\ {M}_{\odot }$ for Stages II and III and ${M}_{\mathrm{min}}=1\times {10}^{13}\ {M}_{\odot }$ for Stage IV with marginalization over ${M}_{\mathrm{min}}$ with a 15% prior imposed as well. Constraints as a function of ${M}_{\mathrm{min}}$ are also shown at the top of Figure 8.

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These results could provide valuable complementary constraints to those on γがんま from spectroscopic galaxy clustering surveys. Projections include constraints of ${\rm{\Delta }}\gamma /\gamma \simeq 5\%$ from measurements at $z\gt 0.65$ using OII for a DESI-like survey (Font-Ribera et al. 2014), and comparable from a Euclid-like ${\rm{H}}\alpha$ survey, for which (Amendola et al. 2012) projected ${\rm{\Delta }}\gamma /\gamma =4\%$ (assuming a luminosity function Geach et al. 2009) that has since been revised downwards to lower ${\rm{H}}\alpha$ number counts (Wang et al. 2012; Colbert et al. 2013).

For cluster abundance measurements knowing the precision with which the minimum mass is known is important. To assess the degree of precision required for the pairwise measurements we consider the impact on the cosmological constraints of marginalizing over the minimum mass, with a 15% prior on ${M}_{\mathrm{min}}$. The middle panel of Table 4 shows the effects of this marginalization: the constraints are loosened only slightly compared to the fiducial case, that has no marginalization over the minimum mass. This implies that a precise knowledge of the minimum mass is not crucial to achieve cosmological constraints. An explanation for the comparative insensitivity of the dark energy constraints to uncertainties in M_min, can be understood with reference to Figure 1. While varying dark energy parameters and M_min both change the large scale pairwise velocity amplitude the minimum mass also changes the shape of the pairwise velocity function. This means that uncertainties in the minimum mass can be discerned from those in dark energy, and do not translate into a comparable degradation of constraints on w or γがんま.

The measurement uncertainty on the mean pairwise velocity decreases with the number of clusters used for the cross-correlation. The upper panels of Figure 8 presents the dependence of the FoM and ${\rm{\Delta }}\gamma /\gamma$ constraints on the assumed minimum observed mass. The increased number density of clusters and cluster pairs arising from a lower mass bound, below $\sim {10}^{14}\ {M}_{\odot }$, significantly improves the statistical uncertainties in the pairwise velocity. For our analysis we integrated over a Jenkins mass function using the minimum observed mass as our lower limit. As the number density of clusters drops off quickly for higher masses the constraints deteriorate strongly for a minimum mass above $M\gt 2\times {10}^{14}\ {M}_{\odot }$.

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Assuming that the complications discussed in Section 2.5, in determining LRG centrality and cluster mass estimates, can be controlled, in principle one could achieve much higher number densities and a smaller minimum mass. This would increase the number of pairs in the cluster sample. The window functions for lower mass halos would include additional information in the mass averaged statistics from the power spectrum at smaller scales that would lead to tighter constraints on the cosmological parameters. The right columns of Table 4 show the results assuming a more optimistic mass cut-off than the reference case, ${M}_{\mathrm{min}}=4\times {10}^{13}\ {M}_{\odot }$ for Stages II and III and ${M}_{\mathrm{min}}=1\times {10}^{13}\ {M}_{\odot }$ for Stage IV. To account for the uncertainty in mass we marginalize over the minimum mass assuming a 15% prior. The GR FoM, with a CMB prior, are improved from ${\mathrm{FoM}}_{\mathrm{GR}}=6$ and ${\mathrm{FoM}}_{\mathrm{GR}}=10$ for Stages II and III to ${\mathrm{FoM}}_{\mathrm{GR}}=12$ and ${\mathrm{FoM}}_{\mathrm{GR}}=15$, compared to the reference scenarios, and by a factor of 1.8 for Stage IV. The uncertainty in γがんま reduces to ${\rm{\Delta }}\gamma /\gamma =0.07$, 0.06, and 0.02 for Stages II, III, and IV.

As shown in Figure 4, the inverse covariance rises at lower cluster separations so that the inclusion of cluster pairs at small separation can have a potentially significant effect on improving the dark energy constraints. Simulation show a deviation from the predicted theoretical mean pairwise velocity, however, starting at separations of $r\lt 45\ \mathrm{Mpc}/{\rm{h}}$ (Bhattacharya & Kosowsky 2008) so that the nonlinear corrections to the cluster motion needs to be considered. Equation (2) has two major deficiencies: It relies on linear theory to model the underlying dark matter distribution (Juszkiewicz et al. 1999; Sheth et al. 2001) and it assumes a linear, scale-independent bias (Sheth et al. 2001). The former leads to a discrepancy of the dark matter pairwise velocity with linear theory at nonlinear scales around $r\leqslant 10\ \mathrm{Mpc}/{\rm{h}}$, the latter introduces deviations at even larger scales. It is worthwhile, therefore, to assess how accurately the mean pairwise velocity of clusters can be modeled in the transition to the nonlinear regime and how the cosmological constraints depend upon the assumed limiting minimum mass.

In Figure 8 we highlight the sensitivity of the FoM and uncertainty in the modified gravity parameter γがんま to the assumptions about the smallest cluster separations to be included in the analysis, parametrized here by ${r}_{\mathrm{min}}$. For a Stage IV-like survey including all scales up to $r=5\ \mathrm{Mpc}/{\rm{h}}$ increases the FoM by 50% compared to an analysis with separations above $50\ \mathrm{Mpc}/{\rm{h}}$ excluded and halves the uncertainty on γがんま.

In this work we chose a moderate approach cutting off our analysis in the mildly nonlinear regime using a minimum separation of ${r}_{\mathrm{min}}=20$ Mpc/h. On-going work on using an perturbative approach to model nonlinearities (Okumura et al. 2014) and improved N-body simulations suggests that the formalism will be improved in the near future to fully exploit the mildly nonlinear regime.

Central to utilizing the kSZ for cosmology, is the ability to measure the pairwise momentum accurately, and then in turn extract the pairwise velocity, from the momentum, through being able to determine the cluster optical depths. In this section we investigate in more detail the sensitivity of the constraints to these important effects.

As described in Section 2.5, the measurement error of a given cluster is given by the combination in quadrature of the instrument noise and the uncertainty in the optical depth of the cluster. In the fiducial analysis we include an uncertainty in the measurement of τたう based on the intrinsic dispersion in the optical depth observed in cluster simulations, averaged over all masses. While this does not include the measurement error in estimating the optical depth, it also does not include additional information in the mass dependence of the optical depth that could reduce the intrinsic dispersion estimator through the creation of a fitting function. Possible ways to estimate τたう beyond the scope of this paper include combining thermal SZ and X-ray observations to break the electron temperature-optical depth degeneracy that will partially affect even multi-frequency arcminute resolution observations (Sehgal et al. 2005). This technique relies on theoretical assumptions and modeling to connect the electron temperature to the X-ray temperature, which need more detailed testing against simulations. A polarization sensitive stage IV CMB survey may be able to measure τたう by stacking clusters to extract the polarization signal introduced by the scattering, which depends directly on the optical depth (see, e.g., Sazonov & Sunyaev 1999).

To understand the impact of greater uncertainty in the determination of τたう on the cosmological constraints, we consider two potential forms of uncertainties, shown in Figure 9. The first is the effect of increased statistical dispersion, ${\sigma }_{\tau }$ in the optical depths of the cluster sample and the second is a systematic offset in the τたう. For the latter, we introduce a nuisance parameter, ${b}_{\tau }(z)$, in each redshift bin that scales the amplitude of the mean pairwise velocity, $\hat{V}(z)={b}_{\tau }(z)V(z)$, and consider its effect on cosmological constraints when marginalizing over ${b}_{\tau }(z)$. Additionally we consider a constant, redshift independent nuisance parameter, ${b}_{\tau }$, that scales the amplitude across all clusters. For clarity, when studying the impact of ${b}_{\tau }$ we remove the ${\sigma }_{\tau }^{2}$ contribution to the covariance and purely parameterize the uncertainty in τたう through a prior on ${b}_{\tau }$.

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Table 2 shows that for a near-term Stage II survey the noise will be dominated by the instrument accuracy, for a more sensitive Stage III both components become comparable, and for a Stage IV survey the velocity accuracy may be limited by the accuracy of τたう. This is reflected in the top panels of Figure 9 in which varying the amplitude of ${\sigma }_{\tau }$ between 0 and 1000 km s⁻¹ only minimally changes the constraints on the dark energy FoM and the constraints on γがんま for Stages II and III.

For Stage II and III surveys, conclusions for the effect of ${b}_{\tau }$ on w₀ and w_a are similar to those for ${\sigma }_{\tau }$. The constraints on these dark energy parameters are principally determined by the Planck-like prior, independent of the kSZ constraints, and uncorrelated with ${b}_{\tau }$. For the Stage IV survey the kSZ constraints provide additional constraints on the equation of state, increasing their correlation with ${b}_{\tau }$, and the prior has a more pronounced effect on improving the FoM once below ${\rm{\Delta }}{b}_{\tau }\lesssim {10}^{-1}$. Equivalently, Stages II and III are not affected by the assumptions on the τたう bias model; marginalizing over the amplitude in each redshift bin yields similar results to introducing a constant bias factor across all redshifts. For Stage IV slightly larger FoM are achieved for a redshift independent ${b}_{\tau }$ model without imposing any prior.

For the growth parameter, which is predominantly constrained by the kSZ data, the model assumptions on the τたう nuisance parameter are more important. Marginalizing over the amplitude in each redshift bin, ${b}_{\tau }(z)$, without imposing any prior doubles the uncertainty in γがんま compared to a constant, redshift independent nuisance parameter ${b}_{\tau }$. The difference between this behavior and the FoM constraints (shown in Figure 9 lower panels) indicates that the redshift dependence of the FoM versus γがんま helps to break the degeneracy between them. A prior on the bias ${\rm{\Delta }}{b}_{\tau }\lesssim {10}^{-1}$ leads to a factor of 5–10 improvement in the parameter constraints for the redshift dependent ${b}_{\tau }(z)$ model and a factor of 3–4 improvement for a constant ${b}_{\tau }$. For the Stage IV survey we find that the multiple-redshift bins and improved covariance reduce the degeneracy between the τたう bias parameter and γがんま, so that the systematic bias and the growth parameter can be constrained simultaneously by the data, and the prior has less effect.

Beyond uncertainties in τたう, the measurement uncertainty also depends on the peculiar velocity of the cluster, see (24). Even though the peculiar velocities of clusters are in principle distributed over a range of velocities, here for simplicity we assume a rms velocity of v = 300 km s⁻¹ that corresponds to the peak velocity of the distribution found in simulations (Sheth & Diaferio 2001) for all clusters to calculate the total measurement error. Fortunately the peak velocity does not strongly depend on the mass of the cluster (Sheth & Diaferio 2001). While future observations will reduce the velocity measurement uncertainty, there is an irreducible error of around ${\sigma }_{v}=(50-100)$ km s⁻¹ on the cluster peculiar motion due to internal motion within the cluster (Nagai et al. 2003; Holder 2004) that will ultimately limit the CMB observations.

In the previous sections we parametrized modified gravity models using one extra parameter γがんま that is assumed constant across all redshifts. Not all modified gravity models are well represented by such a simple parameterization. Some models are better fit by a more general parametrization that allows for a monotonic redshift dependence in γがんま, $\gamma (a)={\gamma }_{0}+(1-a){\gamma }_{a}$ (Wu et al. 2009), equivalent to the dark energy ${w}_{0}-{w}_{a}$ model. Table 5 summarizes the FOM as well as the $1\sigma$ constraints on {w₀,w_a,${\gamma }_{0}$,${\gamma }_{a}$}. Introducing an additional extra parameter loosens the constraints on the parameters with the advantage of imposing a smaller theoretical prior on modified gravity. Figure 10 shows the $1\sigma$ and $2\sigma$ constraints on ${\gamma }_{0}-{\gamma }_{a}$ for Stages II, III, and IV.

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Table 5.  Marginalized Constraints on the Dark Energy and Modified Gravity Parameters in the ${\gamma }_{0}$ and ${\gamma }_{a}$ Parameterization

Paramerer	$\gamma ={\gamma }_{0}+{\gamma }_{a}(1-a)$
	Stage II	Stage III	Stage IV
${\mathrm{FoM}}_{\mathrm{MG}}$	127	137	211
$\sigma ({w}_{0})$	0.10	0.09	0.07
$\sigma ({w}_{a})$	0.29	0.28	0.23
${\rm{\Delta }}{\gamma }_{0}/{\gamma }_{0}$	0.31	0.23	0.08
$\sigma ({\gamma }_{a})$	0.69	0.51	0.14

Note. A summary of the dark energy FoM and 1σしぐま marginalized constraints on the dark energy parameters in the ${\gamma }_{0}-{\gamma }_{a}$ parametrization for Stages II, III and IV scenarios in combination with a DETF prior on all parameters except ${\gamma }_{0}$ and ${\gamma }_{a}$.

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An even more general approach is to directly constrain the growth rate in redshift bins as a "model-independent" way. This approach is particularly applicable for spectroscopic galaxy surveys which can isolate peculiar velocity data, and hence the growth rate, in precise redshift bins. In Figure 11 we present the forecasts for this parametrization. We find, in combination with the Stage III DETF constraints on the equation of state, an uncertainty in f_g of around 2% at $z\sim 0.3-0.6$ for Stage IV surveys. These constraints provide a complementary measurement to the growth rate measurements from redshift space clustering of emission line galaxies and LRGs with a DESI-like survey. On its own, without the additional DETF stage III data, forecasts for DESI project 1.4%–1.6% errors on the growth factor multiplied by ${\sigma }_{8}$ over a comparable redshift range to that considered in our analysis (Levi et al. 2013). DESI, along with Euclid (Amendola et al. 2012) and WFIRST (Spergel et al. 2013) will also provide complementary spectroscopic constraints on the growth rate at higher redshifts, $1\lt z\lt 3$.

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