Constraining Gravity at Large Scales with the 2MASS Photometric Redshift Catalog and Planck Lensing

The 2MASS Photometric Redshift catalog¹ (Bilicki et al. 2014) is an almost all-sky galaxy sample that includes photometric information for approximately 935,000 sources. The catalog has been built by cross-matching the near-IR 2MASS Extended Source Catalog (Jarrett et al. 2000, XSC) with optical SuperCOSMOS (Peacock et al. 2016, SCOS) and mid-IR Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) data. This results in a flux-limited catalog to K_S ≤ 13.9 mag (Vega), roughly corresponding to the 2MASS XSC full-sky completeness limit. The multi-wavelength information allows to estimate accurate photo-zs for the sources by employing neural network algorithms trained on subsamples drawn from spectroscopic redshift surveys overlapping with 2MASS. Inferred photo-zs are virtually unbiased (δz ≃ 0) and their random errors are ∼15% (rms normalized scatter of σ_z ≃ 0.015). 2MPZ sources lie in the redshift range of z ≲ 0.4 with the median being z_med ≃ 0.08 and 95% of the sources below z ≲ 0.17. We define our baseline sample by retaining all of the galaxies in the photometric redshift range 0 ≤ z ≤ 0.24, but do not apply any magnitude cut. Finally, we construct an overdensity map ${\delta }_{g}(\hat{{\boldsymbol{n}}})=n(\hat{{\boldsymbol{n}}})/\bar{n}-1$, where $n(\hat{{\boldsymbol{n}}})$ is the number of objects in a given pixel and $\bar{n}$ is the mean number of objects per pixel in the unmasked area, in the HEALPix² (Gorski et al. 2005) format with a resolution parameter N_side = 256 (approximately 137 pixel size).

We use the publicly available³ CMB lensing convergence map reconstructed by the Planck team (Planck Collaboration et al. 2016b). The lensing convergence map has been extracted by applying the quadratic lensing estimator developed by Okamoto & Hu (2003) to SMICA foreground-cleaned CMB temperature (T) and polarization (P) maps. Different quadratic combinations of the T and P maps are then combined to form a minimum-variance (MV) estimate of the lensing convergence κ bandpass filtered between 8 ≤ ℓ ≤ 2048. The MV lensing reconstruction is provided in the HEALPix format at a resolution of N_side = 2048, corresponding to 17 pixel size.

Even though the 2MPZ catalog and Planck have almost full-sky coverage, there are regions unsuitable for cosmological studies due to observational effects. The main obstacle is the obstruction of view by our own Galaxy, creating the so-called Zone of Avoidance. On top of this, we have to exclude regions contaminated by Galactic foregrounds, such as dust, stars, bad seeing, as well as areas with incomplete coverage.

We construct a 2MPZ fiducial mask over which we carry out our cosmological analysis following Alonso et al. (2015). Briefly, we assume the reddening E(B − V) map from Schlegel et al. (1998) to trace the Galactic extinction in the K-band as A_K = 0.367 E(B − V), and consider the star density n_star at each 2MPZ galaxy position. We then create HEALPix maps of A_K and n_star at a resolution N_side = 64, and discard all pixels for which A_K > 0.06 and ${\mathrm{log}}_{10}{n}_{\mathrm{star}}\gt 3.5$. This eliminates regions near the Galactic plane and the Magellanic Clouds, and covers f_sky = 0.69.

We also have a mask for the Planck CMB lensing data set. The Planck mask is the combination of (i) a 70% Galactic mask; (ii) a point source mask at 143 and 217 GHz; (iii) a mask that removes Sunyaev–Zel'dovich (SZ) clusters with signal-to-noise S/N > 5 in the Planck SZ catalog PSZ1; (iv) as well as the SMICA T and P confidence masks. The sky fraction left for the lensing reconstruction is approximately f_sky = 0.67.

We then multiply the 2MPZ and Planck masks to construct our fiducial mask. With this fiducial mask, the main galaxy sample has 639,673 sources over f_sky = 0.62. This corresponds to a galaxy number density of $\bar{n}=8.2\times {10}^{5}$ sr⁻¹, equivalent to 24.5 deg⁻² or 1.3 pix⁻¹.

The basic idea behind combining CMB lensing and galaxy clustering measurements is that the two observables respond to the underlying dark matter field in complementary ways. Whereas lensing measurements are sensitive to the integrated matter distribution along the LOS, galaxy surveys provide a biased sparse sampling of the dark matter field. More quantitatively under general relativity, the CMB lensing convergence κ and galaxy overdensity δ_g fields can be written as LOS projections of the 3D dark matter density contrast δ:

$$\begin{eqnarray}&&X(\hat{{\boldsymbol{n}}})={\int }_{0}^{\infty }{dz}\,{W}^{X}(z)\delta (\chi (z)\hat{{\boldsymbol{n}}},z).\end{eqnarray} \tag{ 1 }$$

In the above equation, X = {κ, g}. The kernels W^X(z) encode each observable's response to the underlying dark matter distribution:

$$\begin{eqnarray}&&{W}^{\kappa }(z)=\displaystyle \frac{3{{\rm{\Omega }}}_{m}}{2c}\displaystyle \frac{{H}_{0}^{2}}{H(z)}(1+z)\chi (z)\displaystyle \frac{{\chi }_{* }-\chi (z)}{{\chi }_{* }},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{W}^{g}(z)=b(z)\displaystyle \frac{{dN}}{{dz}}.\end{eqnarray} \tag{ 3 }$$

Here, H(z) is the Hubble factor at redshift z, χ(z) and χ_* are the comoving distances to redshift z and to the last scattering surface. Ω_m and H₀ are the present-day values of matter density and Hubble parameter, respectively. In Equation (3), we assumed a linear, local, and deterministic galaxy bias b(z) to relate the galaxy overdensity δ_g to the matter overdensity δ (Fry & Gaztanaga 1993), while the galaxy sample unit-normalized redshift distribution is denoted as dN/dz. Since we are selecting galaxies based on photo-z's, we must also account for the photo-z uncertainties. We do this by convolving the sample's photometric redshift distribution with the catalog's photo-z error function (Sheth & Rossi 2010):

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dz}}={\int }_{0}^{\infty }{dz}\,W({z}_{\mathrm{ph}})\displaystyle \frac{{dN}}{{dz}}({z}_{\mathrm{ph}})p(z| {z}_{\mathrm{ph}}).\end{eqnarray} \tag{ 4 }$$

In this equation, W(z_ph) defines the redshift bin—usually a top-hat function in photo-z space—while the photo-z error function is modeled as a Gaussian of redshift-dependent width, $p(z| {z}_{\mathrm{ph}})\sim { \mathcal G }(0,{\sigma }_{z}(1+z))$, where σ_z = 0.015 (Bilicki et al. 2014; Balaguera-Antolínez et al. 2017). The resulting redshift distribution for our galaxy sample is shown as the solid blue line in Figure 1.

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The knowledge of the galaxy redshift distribution is a key ingredient needed to translate the observed power spectra into a growth factor estimate. We quantify the impact of the assumed redshift distribution on the results of the analysis in Section 5.3.

For scales smaller than ℓ ≳ 10, we can adopt the so-called Limber approximation (Limber 1953) and evaluate the theoretical angular auto- and cross-power spectra as

$$\begin{eqnarray}&&{C}_{{\ell }}^{{XY}}={\int }_{0}^{\infty }\displaystyle \frac{{dz}}{c}\displaystyle \frac{H(z)}{{\chi }^{2}(z)}{W}^{X}(z){W}^{Y}(z)P\left(k=\displaystyle \frac{{\ell }+\tfrac{1}{2}}{\chi (z)},z\right).\end{eqnarray} \tag{ 5 }$$

Equation (5) can be thought of as a weighted integral of the matter-power spectrum P(k, z) = P(k, 0)D²(z), where D(z) is the linear growth function normalized to unity at z = 0:

$$\begin{eqnarray}&&D(z)=\exp \left\{-{\int }_{0}^{z}\displaystyle \frac{{[{{\rm{\Omega }}}_{{\rm{m}}}(z^{\prime} )]}^{\gamma }}{1+z^{\prime} }{dz}^{\prime} \right\},\end{eqnarray} \tag{ 6 }$$

and γ ≈ 0.55 is the growth index in the case of general relativity (Linder 2005). We compute the nonlinear P(k, z) using the CAMB⁴ code with the Halofit prescription (Lewis et al. 2000; Smith et al. 2003).⁵ As shown in Balaguera-Antolínez et al. (2017), the impact of RSD and the Limber approximation is ≲5% and confined to scales ℓ ≲ 10. Given that this level of theoretical uncertainty is smaller than the statistical errors and that we limit the analysis to ℓ > 10, we ignore these effects here. In Equation (3) we also neglect the effect of the lensing magnification bias (see Bianchini et al. 2016 for the expression including these effects).

Examining Equation (5), one notices that the auto-power spectrum scales as ${C}_{{\ell }}^{{gg}}\propto {b}^{2}(z){D}^{2}(z)$, while the cross-spectrum scales as ${C}_{{\ell }}^{\kappa g}\propto b(z){D}^{2}(z)$. Thus, an appropriate combination of the two can eliminate the bias and break the degeneracy between galaxy bias and cosmic growth. Giannantonio et al. (2016) devised a bias-independent estimator for photo-z surveys to recover the cosmic growth information:

$$\begin{eqnarray}&&{\hat{D}}_{G}={\left\langle \displaystyle \frac{{\hat{C}}_{{\ell }}^{\kappa g}}{{{/}\!\!\!{C}}_{{\ell }}^{\kappa g}}\sqrt{\displaystyle \frac{{{/}\!\!\!{C}}_{{\ell }}^{{gg}}}{{\hat{C}}_{{\ell }}^{{gg}}}}\right\rangle }_{{\ell }}.\end{eqnarray} \tag{ 7 }$$

Here, the hat represents measured power spectra while slashed quantities denote theoretical power spectra calculated by removing the growth function from the Limber integration, i.e., the matter-power spectrum in Equation (5) is evaluated at z = 0. We emphasize that Equation (7) is averaged over the range of multipoles included in the analysis.

In the limit of no galaxy bias evolution over a bin (narrow redshift bins), the D_G estimator has the advantage of being bias independent: the true bias shows up in both denominator and numerator and thus cancels out, as does the assumed bias. Its expectation value is $\langle {D}_{G}\rangle =D$ on linear scales, although one might be concerned about the impact of nonlinearities. We test for the dependence of the growth factor constraint on the choice of angular scale cuts below. Note that the D_G estimator will scale as ${\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2}$ due to its dependence on the matter-power spectrum and CMB lensing kernel.

In this work, we measure the D_G statistic in the harmonic domain by combining the observed ${\hat{C}}_{{\ell }}^{\kappa g}$ and ${\hat{C}}_{{\ell }}^{{gg}}$ spectra. We work with maps at an HEALPix resolution of N_side = 256 and convert between different resolutions using the HEALPix built-in ud_grade routine. Power spectra are extracted using a pseudo-C_ℓ method based on MASTER algorithm (Hivon et al. 2002) that deconvolves for the mask induced mode-coupling and pixelization effects.

Operating with cross-power spectra as for ${\hat{C}}_{{\ell }}^{\kappa g}$ has a number of advantages. A cross-spectrum is free of noise bias and it is less prone to systematics as the systematics and noise rarely correlate between different experiments and observables.

The analysis also uses galaxy–galaxy auto-spectrum ${\hat{C}}_{{\ell }}^{{gg}}$ which has to be noise subtracted. Here, we do not debias for the shot-noise term, ${N}_{{\ell }}^{{gg}}=1/\bar{n}$, but rely on a jackknifing approach instead (see, for example, Ando et al. 2018). We randomly split the galaxy catalog in two and create two galaxy overdensity maps ${\delta }_{g}^{1}$ and ${\delta }_{g}^{2}$. From these, we form a pair of half-sum and half-difference maps, ${\delta }_{g}^{\pm }=({\delta }_{g}^{1}\pm {\delta }_{g}^{2})/2$. The former map will contain both signal and noise, while the latter will be noise only. We then extract their auto-power spectra and evaluate the total galaxy auto-power spectrum as ${\hat{C}}_{{\ell }}^{{gg}}={\hat{C}}_{{\ell }}^{++}-{\hat{C}}_{{\ell }}^{--}$.

We estimate both angular power spectra in linearly spaced band powers of width Δℓ = 10 between 10 ≤ ℓ ≤ 250, where the lower limit is imposed by the filtering applied to the Planck lensing map. The maximum multipole is not limited by the data (as long as ℓ_max ≲ 2N_nside). Instead, the choice of ℓ_max is motivated by the desire to avoid strongly nonlinear scales. To reduce potential contaminations from nonlinearities, in our baseline analysis we set ℓ_max to the angular scale subtended by the density modes that are entering the nonlinear regime at z ≃ 0.08, i.e., ${{\rm{\Delta }}}^{2}({k}_{\mathrm{NL}})={k}_{\mathrm{NL}}^{3}{P}^{\mathrm{lin}}$ $({k}_{\mathrm{NL}})/(2{\pi }^{2})\approx 1$. Therefore, we set ℓ_max = 70 and explore below the robustness of the results against different choices of ℓ_max. We have also checked that adopting a finer bin width of Δℓ = 5 has a negligible impact on the results of the analysis.

Assuming that both fields behave as Gaussian random distributed variables on the scales of interest, we evaluate the error bars as

$$\begin{eqnarray}&&{({\rm{\Delta }}{\hat{C}}_{L}^{{XY}})}^{2}=\displaystyle \frac{1}{(2L+1){\rm{\Delta }}{\ell }{f}_{\mathrm{sky}}}[{({\hat{C}}_{L}^{{XY}})}^{2}+{\hat{C}}_{L}^{{XX}}{\hat{C}}_{L}^{{YY}}],\end{eqnarray} \tag{ 8 }$$

where Δℓ is the bin width of a band power centered at L and ${\hat{C}}_{L}^{{XX}}({\hat{C}}_{L}^{{XY}}$) is the auto- (cross-)spectrum comprehensive of noise bias. By setting X = Y in Equation (8), one finds the expression for the auto-power spectra uncertainties. The validity of this assumption has been tested by Balaguera-Antolínez et al. (2017), who have compared the Gaussian error bars with uncertainties estimated trough jackknife re-sampling and galaxy mocks methods.

As we mentioned in Section 3.1, the D_G estimator involves averaging over some angular scales. To this end, we make use of the uncertainties information given by Equation (8) and apply an inverse-variance weighting when averaging across multipoles ℓ's. The details concerning the weighting scheme can be found in Appendix.

For the survey specifications discussed in Section 2, and assuming a galaxy bias b ≃ 1.2 (Alonso et al. 2015), we forecast an overall signal-to-noise (S/N) in the multipole range ℓ ∈ [10, 70] of ≈4.3 and ≈37 for the cross- and auto-correlation respectively. As will be seen in the next section, this forecast is consistent with the real D_G measurement uncertainty.

The extracted cross-power spectrum between the Planck CMB lensing map and the 2MPZ catalog is shown in Figure 3 (left panel), together with the galaxy auto-power spectrum (right panel). A clear cross-correlation signal can be seen, even though the CMB lensing kernel and 2MPZ redshift distribution are not perfectly matched (see Figure 1).

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In Figure 3, we show the best-fit curves and their 1σ uncertainties as the solid gray lines and contours, respectively. A popular parametrization of the galaxy auto-power spectrum is the power law C_ℓ = Aℓ^−γ. We have checked that the auto-power spectrum shape is well approximated by a power-law function with A = 0.027 ± 0.007 and γ = 1.35 ± 0.06. This is broadly consistent with Balaguera-Antolínez et al. (2017) findings.

As a consistency check, we individually estimate the best-fit galaxy bias by comparing the observed ${\hat{C}}_{{\ell }}^{\kappa g}$ and ${\hat{C}}_{{\ell }}^{{gg}}$ to the theoretical predictions in the multipole range 10 ≤ ℓ ≤ 70.⁷ To this end, we assume Gaussian likelihoods as $-2\mathrm{ln}{ \mathcal L }({\boldsymbol{d}}| {\boldsymbol{\theta }})\propto {\chi }^{2}$, where ${\chi }^{2}=[{\boldsymbol{d}}-{\boldsymbol{t}}$ $({\boldsymbol{\theta }}){]}^{T}{{\rm{C}}}^{-1}[{\boldsymbol{d}}-{\boldsymbol{t}}({\boldsymbol{\theta }})]$, ${\boldsymbol{d}}$ is the data vector (measured band powers), ${\boldsymbol{t}}({\boldsymbol{\theta }})$ is the theory vector predicted by model parameters ${\boldsymbol{\theta }}$, and C⁻¹ is the covariance matrix. The posterior space $p({\boldsymbol{\theta }}| {\boldsymbol{d}})\propto { \mathcal L }({\boldsymbol{d}}| {\boldsymbol{\theta }})\pi ({\boldsymbol{\theta }})$ is then sampled via a Markov chain Monte Carlo (MCMC) method implemented in the public emcee code (Foreman-Mackey et al. 2013). The covariance matrix is assumed to be diagonal with elements given by Equation (8), while priors $\pi ({\boldsymbol{\theta }})$ are taken to be flat. The cross-power spectrum analysis reveals a galaxy bias of ${b}_{\kappa g}=1.33\pm 0.35$ and a χ² = 2.3 for ν = 6−1 = 5 degrees of freedom (DOF), corresponding to a PTE of ∼80%, while for the auto-power spectrum we find ${b}_{{gg}}=1.13\pm 0.02$ and a χ² = 5.2 for ν = 5, translating to a PTE of approximately 39%. Both these galaxy biases are consistent with each other and in good agreement with those found by Stölzner et al. (2017), Balaguera-Antolínez et al. (2017). For both the auto- and cross-correlation, we can estimate the significance of the detection by calculating $S/N=\sqrt{{\chi }_{\mathrm{null}}^{2}-{\chi }_{\mathrm{bf}}^{2}}$, where ${\chi }_{\mathrm{null}}^{2}={\chi }^{2}(b=0)$ and ${\chi }_{\mathrm{bf}}^{2}$ is the χ² evaluated at the best-fit. We find S/N ≈ 3.7 and ≈36 for the cross- and auto-power spectrum respectively, in good agreement with the values estimated in Section 3.2 albeit slightly lower in the case of cross-correlation. This might due to an overestimation of the cross-spectrum uncertainties based on Equation (8) or to a simple statistical fluke.

After extracting the power spectra, we apply the D_G estimator to the Planck and 2MPZ data sets and obtain ${\hat{D}}_{G}=1.03\pm 0.19$ as shown in Figure 4. The error bars are estimated in a Monte Carlo approach. To this end, we generate N_sims = 500 correlated Gaussian realizations of the CMB convergence and galaxy fields with statistical properties that match the observed data (a thorough description of these steps can be found in Bianchini et al. 2015). Then, we use the extracted auto- and cross-spectra from the simulation ensemble to obtain N_sims estimates of D_G. We quote the scatter across these estimates as our 1σ uncertainty on the measured ${\hat{D}}_{G}$.

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Following Giannantonio et al. (2016), we assume the cosmic growth history shape D_G(z) to be fixed at high redshift by the fiducial cosmology and then fit for its amplitude A_D as ${\hat{D}}_{G}({z}_{\mathrm{med}})={A}_{D}{D}_{G}^{\mathrm{th}}({z}_{\mathrm{med}})$. The result of the fit is A_D = 1.06 ± 0.20, in good agreement with ΛCDM. Note that the magnitude of the uncertainties ΔA_D is comparable to what found by Giannantonio et al. (2016) for the DES sample, although their result hints to a lower amplitude value, being 1.7σ away from A_D = 1. We also stress that the two analyses are complementary in their redshift ranges: 0 ≤ z ≤ 0.24 in this work versus 0.2 ≤ z ≤ 1.2 for the DES analysis.

Of course, the expected signal does depend on cosmology, specifically the parameter combination, ${\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2}$. The high value observed for D_G could be explained by increasing the ${\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2}$ combination by about ≈6% over the fiducial cosmology. The dark and light gray regions in Figure 4 show the 1σ and 2σ variations in the predicted D_G across a Planck chain. Specifically, we randomly draw 5000 points from Planck chains and calculate the linear growth functions D(z) for each of the 5000 cosmologies. We normalize the curve for model i by multiplying by ${({\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2})}_{i}/{({\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2})}_{\mathrm{fid}.}$ (Giannantonio et al. 2016). The uncertainty in cosmological parameters induces a scatter of approximately ${\sigma }_{{D}_{G}}\approx 0.03$.

While the scatter is small compared with the overall uncertainty, we choose to include the cosmological uncertainty into the A_D estimate. We do this by stacking the A_D likelihoods across 5000 randomly selected points, ${{\boldsymbol{\theta }}}_{\mathrm{cosmo}}^{i}$, from the stationary Planck chains. For each parameter set, we calculate the likelihood

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }({\boldsymbol{d}}| {A}_{D},{{\boldsymbol{\theta }}}_{\mathrm{cosmo}})\propto \displaystyle \frac{{[{\hat{D}}_{G}({{\boldsymbol{\theta }}}_{\mathrm{cosmo}})-{A}_{D}{D}_{G}^{\mathrm{th}}({{\boldsymbol{\theta }}}_{\mathrm{cosmo}})]}^{2}}{{\rm{\Delta }}{\hat{D}}_{G}^{2}},\end{eqnarray} \tag{ 9 }$$

at each step i. Then, we marginalize over the cosmological parameters by stacking the posteriors evaluated at each step. The resulting posterior distribution for the growth factor amplitude A_D is shown in Figure 5. When we allow the underlying cosmological parameters to vary, we find ${A}_{D}=1.00\pm 0.21$, in excellent agreement with the ΛCDM expectations and in line with the constraint for the fiducial cosmology.

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One of the main goals of structure probes like this D_G measurement is to inform us about dark energy and modified gravity models. In Figure 4, we explore the parameter space associated to more exotic scenarios in which we allow for departures from the standard ΛCDM. For the sake of simplicity, we consider modifications of the standard model where there is no induced scale-dependent growth, such as the Linder γ parametrization (Linder 2005) of the growth rate f(z) = [Ω_m(z)]^γ, with γ(z) = γ₀ + γ_az/(1 + z), or the dynamical dark energy model with time-dependent equation of state as w(z) = w₀ + w_az/(1 + z) (Chevallier & Polarski 2001). Note that to highlight the effect of changing the single dark energy/modified gravity parameters in Figure 4, we fix γ₀ (w₀) to its standard value γ₀ = 0.55 (w₀ = −1) when varying γ_a (w_a) and vice versa. As done in the previous section, these predictions have been normalized such that D_Gⁱ matches the ΛCDM one at high-z and apply the ${({\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2})}_{i}/{({\sigma }_{8}{{\rm{\Omega }}}_{m}{H}_{0}^{2})}_{\mathrm{Planck}}$ rescaling (Giannantonio et al. 2016). Although the data is not yet sufficiently sensitive to allow for an interesting test of models, the result seems to favor values of the dark energy equation of state w₀ < −1, or γ₀ values smaller than the predicted value in GR, γ₀ ≈ 0.55. As we will show in Section 5.4, future experiments will be able to discriminate between these models. Note also that combining measurements of f(z)σ₈(z) from RSD with that of the linear growth factor D(z) at a given redshift can potentially break the degeneracy between the cosmic matter density Ω_m and the growth index γ.

We now turn to investigating the impact of different analysis choices on the results. The first choice we consider is the range of angular scales used in the D_G estimator. Effectively, the question is to whether extending the range into the nonlinear regime leads to a shift in the observed D_G value. To zeroth order, we would expect nonlinear structure growth to cancel out for similar reasons as the bias cancellation. However, if nonlinear structure growth is biasing the result, we should see a monotonic shift in D_G as the maximum multipole is increased. In Figure 6, we show the estimated ${\hat{D}}_{G}$ value as function of the maximum multipole ℓ_max included in Equation (7). The shaded regions reflect the 1σ scatter in the growth factor estimates across the N_sims simulations presented in Section 5.2. The dotted vertical line reflects the angular multipole of modes that are entering the nonlinear regime at z = 0.08. There is no significant shift in the D_G value with ℓ_max, and we conclude that is unlikely that the inferred amplitude of the growth factor is affected by the improper inclusion of nonlinear scales.

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In the baseline analysis, we compute the theoretical power spectra including nonlinear corrections from Halofit. We test the impact of this decision in Figure 7, where we show the D_G values estimated with and without the inclusion of such corrections. As can be seen, the impact is rather mild, with the D_G value in the linear case drifting toward higher values, but still within ∼0.4σ of the baseline result. This can be understood as follows. The nonlinear impact at low redshift is more pronounced for the galaxy auto-power spectrum rather than the cross-spectrum, meaning that $\sqrt{{{/}\!\!\!{C}}_{{\ell }}^{{gg},\mathrm{lin}}/{{/}\!\!\!{C}}_{{\ell }}^{{gg},\mathrm{nl}}}$ > $({{/}\!\!\!{C}}_{{\ell }}^{\kappa g,\mathrm{lin}}/{{/}\!\!\!{C}}_{{\ell }}^{\kappa g,\mathrm{nl}})$. Then, it follows that ${D}_{G}^{\mathrm{lin}}\gt {D}_{G}^{\mathrm{nl}}$.

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In Figure 6, we also show the impact of using uniform weighting (dot–dash blue) instead of the baseline inverse-variance weighting (solid orange) of modes entering the D_G estimator. The inverse-variance weighting de-weights the higher noise high angular modes, whereas these noisy modes can pull the estimate around in the uniform weight case. As a result, the uniform weighting case is somewhat more dependent on the choice of ℓ_max. When the weights are applied, the values are more stable and almost independent on ℓ_max.

We also test the dependence of the results on the limiting lower magnitude in the K-band. By raising the threshold ${K}_{S}^{\min }$ we progressively select intrinsically brighter, therefore more biased, objects at higher redshifts. We show the results in Figure 7 and conclude that results are stable against magnitude cuts.

The knowledge of how galaxies are statistically distributed as a function of redshift is a key ingredient of the present analysis as it allows us to predict the theoretical power spectra entering Equation (7). Any mismatch between the true and assumed redshift distribution could potentially bias the inferred value of the growth factor. The first test that we conduct is to assume the redshift distribution of the spectroscopic 2MPZ subsample (orange histogram in Figure 1) to be representative of the whole sample. By doing so, we are effectively testing the robustness of the analysis with respect to systematic shifts in the assumed galaxy redshift distribution, as the spectroscopic dN/dz peaks toward lower redshifts than the fiducial one, because most of the sources with spec-z comes from the shallow subsamples of 2MPZ, 2MRS and 6dFGS (see Figure 1). In this case, we find best-fit linear galaxy biases of b_κg = 1.44 ± 0.40 and b_gg = 1.06 ± 0.03 for the cross- and auto-power spectrum respectively. While the bias constraint from ${C}_{{\ell }}^{\kappa g}$ is almost unaffected by changes in the dN/dz, the best-fit galaxy bias from ${C}_{{\ell }}^{{gg}}$ decreases by ≈9% to compensate the shift in the redshift distribution. In turn, this translates into a growth factor of about ${\hat{D}}_{G}=1.17\pm 0.20$, approximately 14% higher than our baseline value but still within 0.7σ (see Figure 7). We stress that the result of this test does not imply the presence of any systematic effect, as we have purposely input an erroneous redshift distribution (the spec dN/dz), while photo-zs are known to be virtually unbiased. We then test for the smearing effect of photo-z error by performing the analysis considering the full photometric dN/dz and taking σ_z = 0 and 0.03. For the two cases, respectively, we find ${\hat{D}}_{G}=1.03\pm 0.19$ and ${\hat{D}}_{G}=1.02\pm 0.19$, meaning that a broadening or narrowing of the dN/dz distribution has a negligible impact on the result.

As a final remark, we note that care has to be taken when interpreting results based on the D_G as well as the E_G estimators. The main reason is that there is a mismatch between the lensing and clustering kernels, meaning that these two measurements probe structures at different effective redshifts. A further complication to this picture is represented by the scale and redshift dependence of the clustering bias. To account for these effects, correction factors have been devised in the context of E_G measurements (Reyes et al. 2010). For example, Pullen et al. (2016) found that corrections to the E_G estimator are ≲6% out to scales of about ℓ ∼ 400 in the case of the CMASS-Planck lensing cross-correlation. Given the 20% statistical error characterizing the D_G measurement presented in this work, we neglect these systematics corrections but caution the reader that they could affect the analysis of forthcoming CMB and LSS data sets, hence requiring a careful modeling using simulations.

In this subsection, we look to the future and predict the capability of future LSS and CMB surveys to measure D_G. For a fixed area of overlapping sky, the key factors that determine the overall S/N are the lensing reconstruction noise and the galaxy number density.

On the optical side, we consider a Large Synoptic Survey Telescope (LSST)-like photometric redshift survey (LSST Science Collaboration et al. 2009). LSST is expected to image few billions of galaxies over a large range of redshifts, optimally overlapping with the CMB lensing kernel. We assume the LSST "gold" sample, defined by a magnitude limit of 25.3 in the i band, to be our representative galaxy sample. This is expected to include ∼40 galaxies arcmin⁻² with the photometric redshift distribution being well approximated by the following functional form

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dz}}\propto {\left(\displaystyle \frac{z}{{z}_{0}}\right)}^{2}\exp \left(-\displaystyle \frac{z}{{z}_{0}}\right).\end{eqnarray} \tag{ 10 }$$

Here, z₀ = 0.0417i−0.744, resulting in a median of about 0.8 given our magnitude cut choice. We model the linear galaxy bias evolution as b(z) = 1 + 0.84z, accordingly to the LSST science book (LSST Science Collaboration et al. 2009). The photo-z errors requirement for the LSST gold sample is σ_z/(1 + z) < 0.05 with a goal of 0.02. Here, we take a conservative approach and assume an intermediate value of σ_z(z) = 0.03(1 + z) and split the main galaxy sample in photo-z bins of width given by $3\times {\sigma }_{z}(\bar{z})$, where $\bar{z}$ is the bin center (Alonso et al. 2017). This choice results in 15 redshift bins.

On the CMB side, we consider two upcoming ground-based surveys: the Simons Observatory⁸ (SO) and the CMB Stage-4 (CMB-S4; Abazajian et al. 2016). These surveys will provide multi-frequency measurements of the microwave sky and deliver high S/N maps of the CMB lensing convergence. To estimate the CMB lensing noise curves, we consider a θ_FWHM = [1.4, 2] arcmin beam and Δ_T = [6, 1] μK-arcmin noise level for SO and CMB-S4, respectively. We also assume the lensing reconstruction to be performed with CMB modes from about ℓ_min = 30 up to ${{\ell }}_{\max }^{E,B}=5000$ for polarization and up to ${{\ell }}_{\max }^{T}=3000$ for temperature, to reflect the impact of foregrounds in the intensity maps.

In our forecasting setup, we consider the LSS and CMB surveys to overlap over an area of about 18,000 deg². The expected uncertainties are calculated by assuming that the auto- and cross-power spectra will be measured from ${{\ell }}_{\min }=10$, reflecting the difficulty to recover the largest scales from ground observations, up to a redshift-dependent cutoff multipole given by ${{\ell }}_{\max }(z)={k}_{\mathrm{NL}}(z)\chi (z)$ to avoid the inclusion of nonlinear scales. This cutoff scale goes from ${{\ell }}_{\max }\approx 30$ at low redshift, up to more than 3000 for higher redshift. We show the forecasted D_G, along with the current measurements, in Figure 8. To give a rough estimate of how the sensitivity to D_G varies across the experimental landscape, we calculate the total S/N integrated over angular scales and redshift bins z_i as

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=\sqrt{\displaystyle \sum _{{z}_{i}}{\left(\displaystyle \frac{{D}_{G}({z}_{i})}{{\rm{\Delta }}{D}_{G}({z}_{i})}\right)}^{2}}.\end{eqnarray} \tag{ 11 }$$

We can also predict at what significance level a certain data sets combination can differentiate between standard ΛCDM and a given alternative model. To this end, we calculate

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{{z}_{i}}{\left(\displaystyle \frac{{D}_{G}^{\mathrm{DE}/\mathrm{MG}}({z}_{i})-{D}_{G}({z}_{i})}{{\rm{\Delta }}{D}_{G}({z}_{i})}\right)}^{2},\end{eqnarray} \tag{ 12 }$$

where D_G and ${D}_{G}^{\mathrm{DE}/\mathrm{MG}}$ are the growth factor calculated for ΛCDM and a certain dark energy/modified gravity model respectively. Then, we can quote $\sqrt{{\chi }^{2}}$ as the significance of the discrimination between two scenarios (Pullen et al. 2015). As can be seen in Table 1, LSST high galaxy number density will allow for high S/N measurements of D_G, making possible the discrimination between different exotic models at high significance. Specifically, the lower lensing reconstruction noise that characterizes the forthcoming CMB surveys will improve the overall S/N by a factor 3.4 and 5 with respect to Planck for SO and CMB-S4, respectively.

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Table 1.  Total S/N of the D_G Measurement for the LSST Photo-z Survey Combined with Current and Upcoming CMB Surveys

Survey	S/N	$\sqrt{{\chi }^{2}}\quad [w=-1.2]$	$\sqrt{{\chi }^{2}}\quad [{\gamma }_{0}=0.25]$	$\sqrt{{\chi }^{2}}\quad [{\gamma }_{a}=0.5]$
LSST x Planck	92	2.1	8.7	3.5
LSST x SO	312	8.2	31.3	13.4
LSST x CMB-S4	468	13.1	48.3	21.3

Note. We also report the significance of discrimination between ΛCDM and a CPL model with w₀ = −1.2, and two Linder models with γ₀ = 0.25 and one with γ_a = 0.5.

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