The first direct measurement of gravitational potential decay rate at cosmological scales and improved dark energy constraint

The integrated Sachs-Wolfe (ISW) effect probes the decay rate ($DR$) of large scale gravitational potential and therefore provides unique constraint on dark energy (DE). However its constraining power is degraded by the ISW measurement, which relies on cross-correlating with the large scale structure (LSS) and suffers from uncertainties in galaxy bias and matter clustering. In combination with lensing-LSS cross-correlation, $DR$ can be isolated in a way free of uncertainties in galaxy bias and matter clustering. We applied this proposal to the combination of the DR8 galaxy catalogue of DESI imaging surveys and Planck cosmic microwave background (CMB) maps. We achieved the first $DR$ measurement, with a total significance of $3.2\sigma$. We verified the measurements at three redshift bins ($[0.2,0.4)$, $[0.4, 0.6)$, $[0.6,0.8]$), with two LSS tracers (the"low-density points"and the conventional galaxy positions). Despite its relatively low S/N, the addition of $DR$ significantly improves dark energy constraints, over SDSS baryon acoustic oscillation (BAO) data alone or Pantheon supernovae (SN) compilation alone. For flat $w$CDM cosmology, the improvement in the precision of $\Omega_m$ is a factor of 1.8 over BAO and 1.5 over SN. For the DE equation of state $w$, the improvement factor is 1.3 over BAO and 1.4 over SN. These improvements demonstrate $DR$ as a useful cosmological probe, and therefore we advocate its usage in future cosmological analysis.

This degeneracy can be broken in a model independent way combining lensing-LSS cross-correlation measurement (Zhang 2006). Essentially, the ratio of two cross-correlations measuresφ/φ, 1 up to a prefactor depending on the geometry of the universe but free of galaxy bias and matter clustering. Furthermore, it is less sensitive to the survey masks, whose impacts on the cross-correlation measurements are largely canceled out 1 For more general cases such as the case of modified gravity or dark energy with significant anisotropic stress, φ should be replaced with the lensing potential Φ L ≡ (φ − ψ)/2. φ and ψ are defined through dτ 2 = (1 + 2ψ)dt 2 − (1 + 2φ)γ ij dx i dx j in the Conformal Newtonian Gauge. in the ratio. In Dong et al. (2021a,b), we have measured both cross-correlations combining DESI imaging surveys and Planck maps. We explored two LSS tracers, namely the traditional galaxy positions, and low-density points (LDP) recently proposed by Dong et al. (2019). The ISW effect was detected at 3.2σ and the CMB lensing was detected at 56σ. Here we combine these measurements to determine the decay rate. Due to measurement errors, we can not simply take the ratio, otherwise the measured ratio will be biased. We have developed a unbiased estimator of measuring the ratio of two data sets. (Sun et al., in preparation) and will apply it here. There are also other studies that using the Planck CMB lensing map to help calibrate the bias of the sources in the ISW measurements, but in different ways (Ferraro et al. 2015;Planck Collaboration et al. 2016a).
This paper is organized as follows. §2 introduces the methodology and data sets used in our analysis. §3 presents the measured decay rates at three redshifts. They significantly improve the dark energy constraint over existing baryon acoustic oscillation (BAO) or type Ia supernovae measurements. We discuss and conclude in §4. The estimator adopted to measure DR is briefly introduced in the Appendix A. The results for H(z) are plotted for comparison. For identical fractional measurement error, the constraint of Ωm from DR will be a factor of 4 better than that from H on the median, and constraint of w from DR will be a factor of 7 better than that from H for 0 < z < 1. A cosmology of Ωm = 0.3 and w = −1 is used for the plots.
∆z/2. The coefficient DR is the decay rate that we can measure, This relations holds for both flat and curved universes. For brevity we focus on a flat universe hereafter. H(z), a and c are the Hubble parameter at redshift z, the scale factor and the speed of light respectively. D φ is the linear growth factor of gravitational potential φ (Linder 2005). Along with the onset of dark energy, φ decays with time. So in Eq. 2 we include explicitly a negative sign to make DR positive. Notice that DR defined above differs from the desired decay rate d ln D φ /d ln a by factors in the last parenthesis, since lensing has an extra geometry dependence W L (z) = [1 − χ(z)/χ(z s )]/χ(z). χ(z) = cdz/H(z) is the comoving radial coordinate. For CMB lensing, the source redshift z s = 1100. Fortunately, these extra factors do not depend on H 0 and thus avoid uncertainties in H 0 . The above relation is expected to be valid for z m 0.2 and z m > ∆z (Zhang 2006). We numerically verify that it is accurate to 5% over the multipole range < 300. 2 This is sufficiently precise given the ∼ 30% measurement error in DR.
At redshifts where the ISW effect can be detected (z 1), DR is determined by the matter density Ω m and the dark energy equation of state w. Due to the intrinsic dependence of decay rate onφ/φ, its sensitivity to w is a factor of 3/4/5 higher at z = 0.3/0.5/0.7 than that of H (Fig. 1). This superior sensitivity also holds for Ω m (Fig. 1). Later on we will find that such superior sensitivity partly compensates its significantly weaker measurement S/N, and leads to significant improvement in DE constraint over BAO and SNe Ia.

CMB Temperature Map and Lensing Map
In the following, we use C Il and C φl to represent the ISW-LDP cross-power spectrum and lensing-LDP crosspower spectrum. We use the Planck SMICA temperature product ( For both products, the thermal Sunyaev-Zeldovich (tSZ) effect has been deprojected. The Planck temperature product is provided as a full-sky map, while the lensing product is provided as spherical harmonic coefficients a lm of the lensing convergence in HEALPix FITS format (Górski et al. 2005) with max = 4096, for which we multiply them by a factor of 2/ /( + 1) to produce the lensing potential.
To match the resolution of the LSS tracer density map, we downgrade the above CMB products to a lower resolution with Nside = 512. During the analysis, we adopt the Planck mask to remove pixels contaminated by Galactic dust or known points sources. Notice that to avoid systematic biases to the lensing data introduced by the mask effect, aliasing effect and reconstruction noise, we apply a tophat cut in multipole space for generating the lensing potential for which the details are introduced in Appendix B. The same filter is applied to the CMB temperature to eliminate its effect on the DR measurement.

Galaxy Catalogue and LDP Identification
For LSS tracers, we base on the photo-z galaxy catalog from the DESI Legacy Imaging Surveys. The DESI instrument is designed to measure the redshifts of galaxies and quasars from the northern hemisphere in a 14,000 deg 2 survey, for which the imaging data are provided by three projects: the Beijing-Arizona Sky Survey (BASS), the Mayall z-band Legacy Survey (MzLS), the DECam Legacy Survey (DECaLS). In combination with the Dark Energy Survey (DES), the joint sky coverage of the Data Release 8 (DR8) of the Legacy Surveys approaches ∼ 20000 deg 2 , which is a key to reduce the statistical error in the ISW measurement. We select galaxies from the DR8 photometric galaxy catalog 3 and form volumelimited galaxy samples. The DR8 photometric catalog (Zou et al. 2019) is selected upon the three optical bands (g, r, z) and mid-infrared bands observed by the Widefield Infrared Survey Explorer satellite (Zou et al. 2019;Silva et al. 2016;Flaugher 2005;Blum et al. 2016;Abbott et al. 2018). The photo-z of each galaxy is estimated by a local linear regression algorithm (Beck et al. 2016;Gao et al. 2018). The catalogue also provides apparent magnitudes in g, r, z bands, and stellar masses of galaxies.
LDPs depend on the galaxy sample selected. Following Dong et al. (2021a), we select galaxies with r-band absolute magnitudes 4 M r < −21.5 and 0.2 <≤ z P < 0.4. This results into 1.33 × 10 6 galaxies. We exclude all regions within R s = 3 radius 5 of any galaxies in this sample and define the remaining sky positions as LDPs. Statistically speaking, LDPs correspond to underdense regions (Dong et al. 2019(Dong et al. , 2021a. We apply the same operations on galaxies at 0.4 < z P < 0.6 and 0.6 < z P < 0.8 respectively (Table 1).
We sample LDPs on equal-area HEALPix grids at N side = 4096, which corresponds to an angular resolution of 0.859 . We then follow Dong et al. (2021a) to define the pixelized LDP over-density field δ LDP at Nside=512 resolution, 3 http://batc.bao.ac.cn/ ∼ zouhu/doku.php?id=projects: desi photoz:; 4 The absolute magnitudes used have not been K-corrected. Since galaxies within the same photo-z bin have similar Kcorrection and the LDP generation is sensitive only to relative brightness between these galaxies, this lack of K-correction is not an issue for our purpose.
5 Larger Rs or more galaxies locates denser low density areas but less number of LDPs. The adopted Rs = 3 is a balance between the two.
LDP galaxy Figure 3. The gravitational potential decay rate DR measured at 0.2 < z < 0.4, 0.4 < z < 0.6 and 0.6 < z < 0.8. Red(green) data points show the results obtained with LDPs(galaxies). Theoretical curves of five wCDM cosmologies are also shown for demonstration.
Here f LDP is the fraction of each N side = 512 pixel occupied by LDPs, andf LDP is the mean quantity averaged over the survey area. δ LDP is tightly correlated with the matter overdensity δ m and galaxy number overdensity δ g , as we have verified in N-body simulations. The adopted magnitude cut, ∆z, R s and the definition of δ LDP have yielded significant detection of the ISW-LDP cross-correlation (Dong et al. 2021a). A combined mask from both galaxy catalogue 6 and the Planck survey is applied to the LDP over-density field. Furthermore, to avoid possible foreground contaminations from the Galactic plane, we put an additional galactic mask with |DEC| > 30 • to both the CMB maps and the LDP (galaxy) maps. We perform all the analysis in the Galactic coordinate system.  Table 1). We then follow the same procedure in Dong et al. (2021a,b) to measure the ISW-LDP cross-power spectrum C Il and the CMB lensing-LDP cross-power spectrum C φl . Considering that C Il mainly arises from large angular scale ( < 50) while the measurement of C φl is noisy at < 10, we choose ( min , max ) ∼ (9, 117) and six angular bins equally spaced in ln .
Instead of directly taking the ratio of these two crosscorrelations, we use the likelihood method for estimating DR and the corresponding error bar (Appendix A). It directly evaluates P (DR) using the exact analytical expression (Eq. A1). Since it involves no multi-parameter fitting, it is computationally fast. Furthermore, the resulting P (DR) is unbiased and relies on no cosmological assumptions other than the proportionality relation. The obtained P (DR) of three redshift bins are shown in Fig. 2. P (DR) is very close to Gaussian. Therefore later we will adopt a Gaussian P (DR) in cosmological parameter fitting.
The results of DR are shown in Table 1 & Fig.3. The detection significances of three redshifts are 1.62σ, 2.66σ and 0.82σ, respectively. The total significance is 3.2σ. For comparison, we also measure DR with the same set of galaxies used for generating LDPs, and obtain a total ∼ 3.2σ measurement. The measurements including the S/N are fully consistent with each other (Table.1). Because the sign of the cross-correlations of C Il and C φl is flipped for LDP, the sign of DR is identical to that measured with galaxies.

Constraint on a Flat wCDM Model
For the dark energy constraint, we restrict the analysis to the flat wCDM cosmology. To demonstrate its constraining power on DE, we show DR(z) of various flat wCDM cosmology in Fig. 3. DR first increases with redshift (DR ∝ azH) until z ∼ 1, and then begins to decrease due to vanishingφ. A lower Ω m or a less negative w leads to a higher DR. In particular, since DR = 0 if Ω m = 1, a non-zero measurement provides a smoking gun evidence of dark energy.
Nevertheless, the ∼ 3σ measurement of DR seems disappointing in constraining power in w. However, there are three points that boost its DE constraining power to be much stronger than than that implied by the measurement S/N. We emphasize two here and postpone the third after the likelihood analysis. Firstly, as shown in Fig. 1, the sensitivity of DR to w and Ω m is a factor of ∼ 4 higher than that of H. Secondly, DR relies on fewer parameters than other data. For flat ΛCDM, it only relies on Ω m . For flat wCDM, it relies on Ω m and w. As comparisons, both BAO and SNe Ia depend on extra cosmological parameters such as H 0 (or r s ), and extra nuisance parameters in the data analysis.
Since p(DR) is nearly Gaussian, we estimate the posterior distribution of (Ω m , w) by L ∝ exp(−χ 2 /2) and Here σ DR (z i ) is the corresponding error bar of the i-th DR measurement.  Partly due to the Ωm-w degeneracy direction and partly due to its higher sensitivity on Ωm and w, the inclusion of DR significantly improves over BAO or SN alone. . Improvement in wCMD constraints by including the DR measurement. "peak" refers to the best-fit value of the parameter and "mean" refers to the average value of the parameter. The improvement in parameter uncertainty is 1.4-2. |DEC > 30 • | is an additional mask adopted to all maps used for deriving the main results in this paper (solid lines). We also show the results obtained without adopting this mask for comparison (dotted lines), which are consistent with the solid lines. Fig.4 shows the constraints from DR measured by LDPs. 7 Due to limited S/N, DR measurements alone suffer from a strong degeneracy between Ω m and w. Since for a flat universe Ω DE = 1 − Ω m , the degeneracy direction just restates that higher dark energy density or less negative w (and therefore longer duration of dark energy dominance) results into higher DR. This behav-ior is consistent with behaviors of theoretical curves in Fig. 3.
Nevertheless, Ω m -w constraints from DR are still useful, since they are highly complementary to constraints from BAO and SNe Ia. Fig.4 shows the contours from SDSS BAO data (Alam et al. 2021), 8 which includes observations of galaxy and quasar samples from the SDSS, BOSS, and the eBOSS surveys at z < 2.2 and Lyα forest observations over 2 < z < 3.5 (Neveux et al. 2020;Hou et al. 2021;de Mattia et al. 2021;Tamone et al. 2020;Raichoor et al. 2021;Gil-Marín et al. 2020;Bautista et al. 2021). It also shows the contours from the Pantheon SNe Ia data (Scolnic et al. 2018). Clearly, the Ω m -w degeneracy direction of DR is largely orthogonal to those of BAO and SN. The reason is that, for a flat universe both BAO and SN constrain Ω m and w through their impacts on the same H(z), although with different redshift weights. So the degeneacy direction is δw . z * is the redshift where most constraining power locates. Therefore for both BAO and SN, c BAO < 0 and c SN < 0. In contrast, c DR > 0 (Fig.1).
Therefore the inclusion of DR measurement significantly improves over BAO/SN constraints of Ω m and w (  -Columns for Ωm, Ωm and σ(Ωm) refers to the best-fit value of the matter density, the average value and the scatter with respect to Ωm . Columns of w have the same meaning. We show the results obtained by adopting an additional galactic mask of |DEC| > 30 • for measuring DR or not. The former result is more safe to use, while the latter shows smaller σw (especially for DR+SN). We also show the results obtained from a joint analysis of DR of three redshift slices for consistency check, labelled as 'joint 3z'. These results are consistent with each other.
between w constrained form BAO/SN. Comparing to BAO, DR+BAO shift the bestfit w closer to −1. As a cross-check, we also perform the analysis with DR measured from galaxy positions. The obtained constraints are consistent with LDP (Table 2).

DISCUSSION
We detect the gravitational potential decay rate (DR) in a model-independent way by combining the ISW-LSS cross-correlation with CMB lensing-LSS crosscorrelation. The large overlap in survey region between DESI DR8 photo-z catalogue and Planck 2018 CMB data release enables us to detect DR at a total significance of 3.2σ within the redshift range of 0.2 < z < 0.8. We demonstrate that DR highly complements BAO/SN, and therefore significantly improves their wCDM constraints.
In the above, we measure DR independently for each redshift. One may wonder whether a joint analysis of the cross-power spectrum from three redshift slices could improve the detection of DR, since non-zero crosscorrelations are found between different redshifts, possibly caused by the same adopted mask, mixture of galaxies due to the photo-z error, as well as the wide ISW and CMB-lensing kernels. To do so, we consider the full covariance by including autocorrelations, C Il (z i ) − C Il (z i ) (C φl (z i ) − C φl (z i )), and cross-correlations, C Il (z i ) − C Il (z j ) (C φl (z i ) − C φl (z j ), i = j). We then fit values of DR of three redshifts simultaneously (Table 3). The cross-correlation between different redshifts is weak, so

Galaxy redshift
Ng DR(LDP) DR(gal) 0.2 < z P < 0.4 1.33 × 10 6 0.111 ± 0.062 0.097 ± 0.056 0.4 < z P < 0.6 1.66 × 10 6 0.177 ± 0.070 0.172 ± 0.067 0.6 < z P < 0.8 1.54 × 10 6 0.068 ± 0.073 0.046 ± 0.070 As the ISW effect peaks around z ∼ 0.5, our current measurements of DR focus on the low redshift (0.2 < z < 0.8). Considering that the optimistic S/N of the full-sky ISW signal for a standard ΛCDM model is around 7σ (Giannantonio et al. 2008), our detections of DR are already close to the statistical limit set by limited sky coverage and redshift range. However, it is necessary to include higher redshift in the future for the following reasons. First, more independent volumes are available for higher z, meaning lower statistical error. Second, the measurement of DR at high z, even a null detection, is useful for distinguishing between different dark energy models. DESI, with over 30 million spectroscopic galaxy and quasar redshifts out to z = 3.5 and 14000 deg 2 sky coverage, will further improve the DR S/N, while reducing possible systematic errors in the current measurement (e.g. photometric redshift bias).
In addition to the DESI surveys in the northern sky, the Large Synoptic Survey Telescope (LSST, LSST Dark Energy Science Collaboration (2012)) under construction will cover about 18,000 deg 2 of the southern sky in the main survey for the next decade, with deeper image depths (r ∼ 24.5). This will double the sky coverage of ISW/lensing-LSS cross-correlation measurement and improve DR measurement by another ∼ 40%.
At last, there are still other possibilities to further explore. For example, DR could also be detected through ISW-LSS cross-correlation + galaxy shear-LSS crosscorrelation. Therefore, it can serve as a cross-check for , the ratio R can be estimated by Bayesian analysis. To be model independent, we take the parameters to be marginalized as λ ≡ d theory 1 and adopt a flat prior P (λ). The marginalization can be done analytically and the the posterior distribution of R is given by Here Q = i A T i C −1 i A i , A 1 = I, A 2 = RI, and C i is the covariance matrix of d i . To measure DR by this estimator, we replace d 1 and d 2 with the CMB lensing-LSS cross-correlation C Φl ( ) and the ISW-LSS cross-correlation C Il ( ). We use simulations to determine the covariance matrix. Noise Planck Full Focal Plane (FFP106) simulations (Planck Collaboration et al. 2020) are used to determine the mean-field bias and the covariance matrix estimation of C Φl ( ) (Dong et al. 2021b). While for C Il , we simulate CMB temperature maps with the Planck cosmology to estimate the covariance matrix (Dong et al. 2021a). Fig.2 shows P (DR). We then find the bestfit DR and σ DR .
The estimator can be extended to more general cases of the theory-data mapping matrixes A 1,2 and we refer readers to Sun et al. 2022 (in preparation) for details. B. FILTERING THE CMB DATA In this work, we produce the CMB lensing potential based on the Planck convergence product κ m : Φ m = 2/ /( + 1)κ m . However, the construction of κ m in observation is contaminated with noise, which propagates onto Φ m and is amplified by a factor of 2/ /( + 1). Therefore, there is a risk for directly using Φ m , especially in the existence of mask. We find that our adopted mask is highly correlated with the lowest multipoles of Φ m ( ≤ 5). Consequently, the contaminated cross power spectra at ≤ 5 are leaked onto higher and severely interferes our measurement of C φl at all scales. To avoid such effect, we adopt a tophat filter with 5 < < 1024 for generating the lensing potential map in this study. The result is shown in Fig.6, in which we have performed the scale cut test by varying the range of the tophat filter in multipole space: min < < max . We find that min = 5 is a safe choice as the cross-power spectra of higher multipoles is almost identical to the one when adopting a larger min . The choice of max is to prevent the aliasing of high-noise power spectrum 9 .