Constraining gravity with a new precision $E_G$ estimator using Planck $+$ SDSS BOSS data

The $E_G$ statistic is a discriminating probe of gravity developed to test the prediction of general relativity (GR) for the relation between gravitational potential and clustering on the largest scales in the observable Universe. We present a novel high-precision estimator for the $E_G$ statistic using CMB lensing and galaxy clustering correlations that carefully matches the effective redshifts across the different measurement components to minimize corrections. A suite of detailed tests is performed to characterize the estimator’s accuracy, its sensitivity to assumptions and analysis choices, and the non-Gaussianity of the estimator’s uncertainty is characterized. After finalization of the estimator, it is applied to Planck CMB lensing and SDSS CMASS and LOWZ galaxy data. We report the first harmonic space measurement of $E_G$ using the LOWZ sample and CMB lensing and also updated constraints using the final CMASS sample and the latest Planck CMB lensing map. We find ${\widehat{E}}_G^{\mathrm{Planck}+\mathrm{CMASS}}=0.36_{-0.05}^{+0.06}(68.27\%)$ and ${\widehat{E}}_G^{\mathrm{Planck}+\mathrm{LOWZ}}=0.40_{-0.09}^{+0.11}(68.27\%)$, with additional subdominant systematic error budget estimates of 2% and 3%, respectively. Using ${\mathrm{\Omega }}_{\mathrm{m},0}$ constraints from Planck and SDSS BAO observations, $\mathrm{\Lambda }\mathrm{CDM}$-GR predicts $E_G^{\mathrm{GR}}(z=0.555)=0.401\pm 0.005$ and $E_G^{\mathrm{GR}}(z=0.316)=0.452\pm 0.005$ at the effective redshifts of the CMASS and LOWZ based measurements. We report the measurement to be in good statistical agreement with the $\mathrm{\Lambda }\mathrm{CDM}$-GR prediction and report that the measurement is also consistent with the more general GR prediction of scale independence for $E_G$. This work provides a carefully constructed and calibrated statistic with which $E_G$ measurements can be confidently and accurately obtained with upcoming survey data.

Figure 1

Summary of the $E_G$ statistic measurements presented in this work. Shown is $E_G$ as a function of redshift (lower) and for each of the two measurements at their effective redshift as a function of angular scale $\ell$ (upper). We present a measurement using Planck CMB lensing combined with SDSS CMASS galaxies (orange circles) and combined with SDSS LOWZ galaxies (blue pentagons) with 68.27% confidence ranges for the error. The $\mathrm{\Lambda }\mathrm{CDM}$-GR expectation for the fiducial cosmology (assuming ${\mathrm{\Omega }}_{\mathrm{m},0}=0.3111\pm 0.0056$) is shown as a black line with a gray one sigma uncertainty band.

Figure 2

Accuracy of the $E_G$ estimator presented in this work for CMB lensing combined with the CMASS (orange, solid line) and LOWZ (blue, dashed line) galaxy samples with one indicating consistency between estimator and input. (Upper panel) The overall systematic error of the estimator, $S_{\mathrm{\Gamma }}^{\ell }$, before considering astrophysical systematics. (Middle) Our estimate of the systematic error from bias redshift evolution within the galaxy samples of CMASS and LOWZ, $S_b^{\ell }$. (Lower) The overall impact of magnification bias, $C_{\alpha }^{\ell }$, which we correct for with the binned correction shown as dots.

Figure 3

Visualizing the off-diagonal correlation matrix between $C_{\ell }^{\kappa g*}$ and $C_{\ell }^{gg}$ for Planck and CMASS. Shown is the correlation matrix defined as $\mathrm{Cov}(C_{\ell }^{XY},C_{\ell }^{AB})/({\sigma }_{\ell }^{XY}{\sigma }_{\ell }^{AB})$ for the bins in the cosmological range. The upper triangle shows the simulated covariance which is the baseline for the results presented in this work and the lower triangle shows the analytic covariance. An arrow indicates the direction of increasing $\ell$ for each axis. Correlations between $C_{\ell }^{gg}$ and $C_{\ell }^{\kappa g*}$ are up to around 0.35 for our largest-scale bin. The off-diagonals of the simulated covariance match the analytic covariance well within statistical uncertainty. For a comparison of the diagonals of the covariances see Fig. 6. For Planck $+$ LOWZ see Appendix pp3.

Figure 4

Demonstration of pipeline accuracy on simulated data. The ratio of the mean estimated cross-correlation, ${\widehat{C}}_{\ell }^{\kappa g}$, (upper) and autocorrelation, ${\widehat{C}}_{\ell }^{gg}$, (lower) with the input theory spectrum is shown for the cases involving CMASS (orange) and LOWZ (blue, slightly shifted to the right for readability) with the statistical uncertainty of a single measurement, $\sigma$, and of the mean for $N_{\mathrm{sims}}=480$ simulations, $\sigma /\sqrt{N_{\mathrm{sims}}}$ (red).

Figure 5

Comparing our analytic estimate of the shot noise in the galaxy sample [solid black line for CMASS, dashed black line for LOWZ] with an estimate from data for both CMASS [orange circles] and LOWZ (blue pentagons). The data estimate has large statistical uncertainty on large scales and gets more accurate for small scales (high $\ell$). The analytic estimate matches the constraint from the data well.

Figure 6

Measurement of (Left) the angular cross-power spectrum, $C_{\ell }^{\kappa g*}$, for the CMB lensing map based on Planck PR4 and the reweighted CMASS galaxy sample and (Right) the angular auto-CMASS galaxy power spectrum. (Top) The angular power spectrum with simulation-based errors as well as our theory curve. (Middle) Comparison of the analytic (solid blue line) and jackknife error estimates (dashed green) to the baseline marginalized errors from the simulations. (Lower) The signal-to-noise ratio, calculated as the ratio of measurement and measurement uncertainty is shown (filled orange). For reference, the ratio of the theory expectation and the measurement uncertainty is also shown [dashed black]. For Planck $+$ LOWZ see Appendix pp3.

Figure 7

RSD fit to the clustering signal of the CMASS data set. We show the corner-plot of the fitting parameters $f{\sigma }_8,b{\sigma }_8$ as well as the derived parameter $\beta =f{\sigma }_8/(b{\sigma }_8)$ that we use as part of our $E_G$ estimate. For each parameter, we also show the mean value and 68.27% error. For LOWZ see Appendix pp3.

Figure 8

Characterization of the non-Gaussian PDF for the ${\widehat{E}}_G$ estimator for the fiducial cosmology and Planck $+$ CMASS data covariance (orange curve). The fiducial value for $E_G$ [vertical black line] is shown along with the mean of the PDF (vertical orange full line) and median (vertical orange dotted line). The distribution of 100,000 samples from the theory model of the data vector and covariance are also shown (blue shading). The top panel shows the case for combined measurement using all bins and the middle panel the case of only using the first bin. In the bottom panel, we show for comparison the result when assuming Gaussianity (green dashed) for the combined measurement.

Figure 9

Characterizing the statistical consistency test between the $E_G$ measurement and the $\mathrm{\Lambda }\mathrm{CDM}$-GR prediction by applying it to simulations. For each of the set of 480 simulated Planck $+$ CMASS-like data vectors drawn around the fiducial cosmology, we calculate the ${\mathrm{PTE}}_{\mathrm{GR}}$ for consistency with the GR value. Results are shown when combining all multipole bins in the cosmological range [orange solid] and for two single band powers at the largest ($\ell =53$) [blue line] and smallest scale ($\ell =382$) [green dashed] of the cosmological analysis range. The ${\mathrm{PTE}}_{\mathrm{GR}}$ distributions are consistent with the expected flat distribution for a valid test.

Figure 10

Characterizing the statistical consistency test of the $E_G$ measurement for scale independence for the case of Planck $+$ CMASS. (Upper) The minimum value for the ${\chi }^2$ distance measure. The ${\chi }^2$ distribution for the degrees of freedom (gray dashed curve) is shown to be consistent with the results from the 480 simulations [orange] and the distribution for 100,000 samples around the fiducial cosmology and baseline (simulation-derived) covariance. The value of significance for the ${\chi }^2$ distribution (gray dashed vertical) and for the 100,000 samples (95% of sampled distance measures fall below) (blue full vertical) are shown to be in good agreement. (Lower) The resulting ${\mathrm{PTE}}_{\mathrm{scale}-\mathrm{indep}}$ values for the 480 simulations. The ${\mathrm{PTE}}_{\mathrm{scale}-\mathrm{indep}}$ distribution is consistent with the expected flat distribution for a valid test. Our result for the Planck $+$ CMASS ${\widehat{E}}_G$ estimate, ${\mathrm{PTE}}_{\mathrm{scale}-\mathrm{indep}}=0.52$, obtained in Sec. 8 is shown as a vertical red line. For Planck $+$ LOWZ see Appendix pp3.

Figure 11

Measurements of ${\widehat{E}}_G^{\ell }$ for (Upper) Planck $+$ CMASS and (Lower) Planck + LOWZ for each multipole bin. The median value is indicated as dots and the errors show 68.27% confidence ranges. The marginalized errors for all three of the covariance estimate approaches are given; simulations (black), analytic (blue), and jackknife (green) with the latter two shifted to the right for visibility. The GR prediction is shown (solid line) for reference with the one sigma uncertainty (gray shaded).

Figure 12

The PDF of the measurement ${\widehat{E}}_G$ marginalized over the cosmological range of scales for Planck $+$ CMASS (upper panel) and Planck $+$ LOWZ (lower panel). Shown is the PDF of the ${\widehat{E}}_G$ measurement using the baseline simulation-based covariance (solid orange line) with the median [vertical orange] and central 68.27% confidence range (shaded orange) highlighted. The resulting PDF for the measurement when using the analytic covariance is also shown (dashed blue line). The $\mathrm{\Lambda }\mathrm{CDM}$-GR prediction is shown as a vertical black line with the one sigma uncertainty shaded. Our measurement is in good agreement with the GR prediction with ${\mathrm{PTE}}_{\mathrm{GR}}=0.46$ for Planck $+$ CMASS and ${\mathrm{PTE}}_{\mathrm{GR}}=0.64$ for Planck $+$ LOWZ.

Figure 13

The sensitivity of the baseline ${\widehat{E}}_G$ results presented in this work to analysis choices and corrections applied. The $\mathrm{\Lambda }\mathrm{CDM}$-GR predicted value and one sigma error based on the fiducial cosmology (gray band) and the 68.27% confidence range for the baseline measurements from this work (orange band) are used as references. These are compared with the results using an analytic covariance instead of simulation-based, restricting to smaller scales with ${\ell }_{\min }=59$ and ${\ell }_{\min }=106$, analyzing the angular power spectra for the BOSS North and South patches only and restricting the autocorrelation to the same footprint as the cross-correlation. Additionally, below the gray dashed line, the impact of removing well-motivated analysis steps is shown; removing the magnification bias correction (negligible for Planck $+$ LOWZ), not correcting the CMB lensing normalization, not reweighting the galaxy sample for the cross-correlation and not accounting for the non-Gaussianity of the estimator uncertainty.

Figure 14

Overview of the cosmological constraints based on the ${\widehat{E}}_G$ estimator presented in this work applied to the Planck PR4 CMB lensing map and the SDSS BOSS DR12 CMASS and LOWZ galaxy samples. Shown are the measurements of $E_G$ at their respective effective redshift for Planck $+$ CMASS (orange dots) and Planck $+$ LOWZ [blue pentagons]. The $\mathrm{\Lambda }\mathrm{CDM}$-GR expectation for the fiducial cosmology (assuming ${\mathrm{\Omega }}_{\mathrm{m},0}=0.3111\pm 0.0056$) is shown as a black line with a gray one sigma uncertainty band. Literature results that use earlier versions of the Planck and SDSS data and different approaches to estimate the $E_G$ statistic are shown in gray. Note that the redshifts of the different measurements are not shifted for visibility, other results using Planck and SDSS BOSS data quote different effective redshifts. The statistical 68.27% confidence ranges for each measurement are shown, systematic error budgets are not plotted, they are reported as subdominant for all cases.

Figure 15

Observational masks of the CMASS and LOWZ samples in equatorial coordinates and Mollweide projection. The values show the fractional coverage of each pixel combining survey footprint, local completeness, and accounting for veto regions. Shown are the pixels with coverage fraction above 60% which is the lower cutoff used in the analysis.

Figure 16

A summary of how the improvements to the ${\widehat{E}}_G$ estimator proposed in this work impact the overall accuracy of the estimator for our fiducial cosmology. Shown is the case of CMB lensing combined with a CMASS-like galaxy sample without considering FKP weights with $z_{\mathrm{eff}}=0.531$. Shown are the analytic $E_G$ value for the fiducial cosmology (black dashed), ${\widehat{E}}_G$ using the estimator in this work [orange, full], our estimator but using $C_{\ell }^{\kappa g}$ rather than the reweighted $C_{\ell }^{\kappa g*}$ (blue, dot-dashed) and the estimator used in Pullen et al. [34] for both the redshift assumed in that work, $z=0.57$, (yellow, dotted) and at the effective redshift of the sample, $z=0.531$ [green, full]. See Appendix pp2 for details.

Figure 17

RSD fit to the clustering signal of the LOWZ dataset. The corner plot shows the fitting parameters $f{\sigma }_8,b{\sigma }_8$ as well as the derived parameter $\beta =f{\sigma }_8/(b{\sigma }_8)$ used as part of our $E_G$ estimate. For each parameter, we also show the mean value and one sigma error.

Figure 18

Visualizing the off-diagonal correlation matrix between $C_{\ell }^{gg}$ and $C_{\ell }^{\kappa g*}$ for Planck and LOWZ. Shown is the correlation matrix defined as $\mathrm{Cov}(C_{\ell }^{XY},C_{\ell }^{AB})/({\sigma }_{\ell }^{XY}{\sigma }_{\ell }^{AB})$ with the diagonal removed for readability. The upper triangle shows the simulated covariance and the lower triangle the analytic covariance. Correlations between $C_{\ell }^{gg}$ and $C_{\ell }^{\kappa g*}$ are up to around 0.35 for our narrowest large-scale bins. The off-diagonals of the simulated covariance match the analytic covariance well within statistical uncertainty. For a comparison of the diagonals of the covariances see Fig. 19.

Figure 19

Measurement of (Left) the angular cross-power spectrum, $C_{\ell }^{\kappa g*}$, for CMB lensing map based on Planck PR4 and the reweighted LOWZ galaxy sample and [Right] the angular auto-LOWZ galaxy power power spectrum (equivalent figure for CMASS shown in Fig. 6). (Top) The angular power spectrum with simulation-based errors as well as our theory curve. (Middle) Comparison of the analytic (solid blue line) and jackknife error estimates [dashed green] to the baseline marginalized errors from the simulations. (Lower) The signal-to-noise ratio, calculated as the ratio of measurement and measurement uncertainty is shown (filled orange). For reference, the ratio of the theory expectation and the measurement uncertainty is also shown (dashed black).

Figure 20

Characterizing the statistical test for consistency of our measurement with scale independence for Planck $+$ LOWZ. Content as described for CMASS in Fig. 10.