Effect of redshift bin mismatch on the cross correlation between the DESI Legacy Imaging Survey and the Planck CMB lensing potential

The standard model of cosmology or the $\Lambda$ cold dark matter ($\Lambda$CDM) model, well established based on a vast array of observations, provides us with a comprehensive framework to understand the structure, origin, and evolution of the Universe. The parameters governing the $\Lambda$CDM model have been constrained with unparalleled accuracy by precise measurements of the cosmic microwave background (CMB) (Ade et al. 2021; Dutcher et al. 2021; Planck Collaboration et al. 2020b; Aiola et al. 2020; Adachi et al. 2020). However, despite the remarkable success of the $\Lambda$CDM model in providing a satisfactory description of the astrophysical and cosmological probes, it cannot explain some of the key ingredients in our understanding of the Universe, namely dark energy (Perlmutter et al. 1999; Riess et al. 1998), dark matter (Trimble, 1987) and inflation (Guth, 1981).

The gravitational lensing of the CMB photons carries information about the growth of structure at redshifts of $1-3$. Cross correlations between tracers of the large-scale structure and CMB lensing are excellent probes of the evolution of the large-scale gravitational potential and can be used to test the validity of the $\Lambda$CDM model. Several works over the past decade have established the importance of such cross-correlation measurements in testing the cosmological model (Saraf et al. 2022; Miyatake et al. 2022; Robertson et al. 2021; Krolewski et al. 2021; Darwish et al. 2021 Abbott et al. 2019; Bianchini & Reichardt 2018; Singh et al. 2017; Bianchini et al. 2016; Bianchini et al. 2015). On the other hand, observations of the probes of large-scale structure, such as weak gravitational lensing and galaxy clustering (Abbott et al. 2022; Philcox & Ivanov 2022; Secco et al. 2022; Amon et al. 2022; Asgari et al. 2021; Skara & Perivolaropoulos 2020; Joudaki et al. 2017; Macaulay et al. 2013) have consistently pointed out a $2-3\,\sigma$ difference in the strength of matter clustering — quantified by the $S_{8}\equiv\sigma_{8}\sqrt{\frac{\Omega_{m}}{0.3}}$ parameter — compared to the conclusions drawn from the CMB-only analysis (Planck Collaboration et al., 2020a).

A large number of studies that implemented the tomographic cross-correlation measurements — by dividing the galaxy samples into narrow redshift bins — reported differences in the values of the $\sigma_{8},\Omega_{m}$, or $S_{8}$ parameters compared to the values derived from CMB measurements (see Alonso et al. 2023; Wang et al. 2023; Yu et al. 2022; White et al. 2022; Pandey et al. 2022; Chang et al. 2023; Sun et al. 2022; Krolewski et al. 2021; Hang et al. 2021; Marques & Bernui 2020; Peacock & Bilicki 2018; Giannantonio et al. 2016). Other works like Bianchini & Reichardt (2018), Amon et al. (2018), Blake et al. (2016), Giannantonio et al. (2016), and Pullen et al. (2016) find consistent deviations in the values of $D_{g}$ and $E_{g}$ statistics when testing the $\Lambda$CDM model with different galaxy surveys. Tomographic measurements with upcoming large-scale structure surveys, such as Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al. 2019a; LSST Science Collaboration et al. 2009), Euclid (Laureijs et al., 2011), Nancy Grace Roman Space Telescope (Spergel et al., 2013), Dark Energy Spectroscopic Instrument (DESI; Dey et al. 2019), and Spectro-Photometer for the History of the Universe, Epoch of Reionization, and Ices Explorer (SPHEREx; Doré et al. 2014) will help us explore the $\sigma_{8}-\Omega_{m}$ or $S_{8}$ tension in unprecedented detail.

The tomographic cross-correlation measurements with photometric galaxy surveys allow us to study the evolution of $\sigma_{8}$ or $S_{8}$ to high redshifts but are plagued by photometric redshift errors. The errors on redshift lead to the placement of a fractions of galaxies in wrong redshift bins, referred to here as ‘the redshift bin mismatch’ of galaxies. The effects of redshift bin mismatch on the estimation of parameters have been explored in a few ways (Hang et al. 2021 (hereafter H21); Stölzner et al. 2021; Balaguera-Antolínez et al. 2018). Another attempt to mitigate the bin mismatch of galaxy redshifts and self-calibrate the redshift distributions was proposed by Zhang et al. (2010), who used the scattering matrix formalism. Zhang et al. (2017) proposed an algorithm based on a non-negative matrix factorisation method to compute the scattering matrix and correct the angular power spectra for the redshift scatter. However, the non-negative matrix factorisation method becomes computationally expensive with increasing redshift bins and data points in the angular power spectra. We proposed an alternative method in Shekhar Saraf & Bielewicz (2024) for fast and efficient computations of the scattering matrix validated with Monte Carlo simulations of the LSST survey.

In the present paper, we apply the proof of concept presented in Shekhar Saraf & Bielewicz (2024) to the cross correlation between the Planck CMB lensing potential map and the photometric galaxy catalogues from Data Release 8 of the Dark Energy Spectroscopic Instrument Legacy Imaging Survey prepared by H21. We highlight the importance of precise modelling of the photometric redshift error distributions and quantify the impact of redshift bin mismatch correction on estimation of the $\sigma_{8}$ parameter. We show that mitigation of redshift bin mismatch is necessary as a second-stage correction in order to obtain unbiased estimates of parameters. We provide a fast and accurate method for redshift bin mismatch correction. We describe the data used in our analysis in section 2 and the modelling of the redshift error distributions in section 3. In section 4, we describe the procedure for estimating the redshift distribution of galaxies taking into account the photometric redshift errors. We expand on the parameter estimation procedure using maximum likelihood estimation in section 5. The mock catalogues used to validate our analysis pipeline are presented in section 6, and various methods for computing the covariance matrix are summarised in section 7. We present the key results from our analysis and comparisons with H21 in section 8. Finally, in section 9, we summarise the findings from our work presented in this paper.

Precise modelling of the distribution of photometric redshift error is required to propagate redshift uncertainties in the estimation of parameters. H21 adopted a modified Lorentzian function (Eq. 3) to model the redshift error distribution of $\Delta z=z_{s}-z_{p}$ ($s\equiv$ spectroscopic; $p\equiv$ photometric) as a function of $z_{p}$, $p(z_{s}-z_{p}|z_{p})$.

where $\mathcal{N}$ is the normalisation factor and $x_{0},\sigma$ and $a$ are the parameters to be constrained for every tomographic bin. The best fit-values of $x_{0},\sigma$ and $a$ from H21 are quoted in Table 1.

Fig. 2 shows the photometric error distributions compared to the best-fit modified Lorentzian functions. The modified Lorentzian functions provide a good estimate near the peak of the error distributions but fail to capture the broader tails of the error distribution. We attempted to model the broad wings of the error distributions with a sum of three Gaussians. As shown in Fig. 2, our sum of Gaussians model better captures the tails of the error distributions, at the marginal expense of increasing the number of free parameters in the fitting function. An important point to note is we only seek to fit the broad wings that immediately follow the peak of the error distributions and not the extremities, which is to avoid over-fitting. In section 8.2.1, we detail the impact of modelling the wings of the error distributions on the estimation of parameters.

An estimate of the true redshift distribution can be computed using either the convolution method or the deconvolution method.

We focused on estimating two parameters, the linear galaxy bias $b$ and amplitude of cross correlation $A$, from measurements of the galaxy auto-power spectrum and the cross-power spectrum between CMB lensing and galaxy density, using the maximum likelihood estimation method. The galaxy bias accounts for the fact that galaxies are biased tracers of the underlying total matter distribution. The amplitude of cross correlation is a phenomenological parameter rescaling the amplitude of the cross-power spectrum. The cross correlation amplitude can be used to test the validity of the underlying cosmological model, which in our study is the $\Lambda$CDM model (Saraf et al. 2022; Marques & Bernui 2020; Bianchini et al. 2015). The galaxy auto-power spectrum scales as $b^{2}$, whereas the cross-power spectrum depends on the product of the parameters $b\times A$ and induces a degeneracy in the estimation of parameters. We break this degeneracy using a joint likelihood function of the form

where $N_{L}$ is the number of data points and $\hat{C}_{L}$ is the measured joint power spectrum, $\hat{C}_{L}=(\hat{C}_{L}^{\kappa g},\hat{C}_{L}^{gg})$. Also, $C_{L}(b,A)$ is the joint theoretical power spectrum template defined as $C_{L}(b,A)=(AC_{L}^{\kappa g}(b),C_{L}^{gg}(b))$, and the covariance matrix is given as

In section 7, we outline different ways to estimate the covariance matrix $\text{Cov}_{LL^{\prime}}$.

We employed the publicly available EMCEE (Foreman-Mackey et al., 2013) package to effectively sample the parameter space, and estimated the galaxy bias $b$ and the cross-correlation amplitude $A$. We used flat priors for the parameters with ${b\in[0,10]}$ and ${A\in[-5,5]}$. The other cosmological parameters entering the likelihood estimation through theoretical power spectrum templates were set to their constant values for the fiducial background cosmology described in Planck Collaboration et al. (2020a) (i.e. the flat $\Lambda$CDM cosmology with the best-fit Planck + WP + highL + lensing parameters, where WP stands for the WMAP polarisation data at low multipoles, highL is the high-resolution CMB data from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), and lensing refers to the inclusion of Planck CMB lensing data in the parameter likelihood).

To gain insight into how the observed redshift errors impact the estimation of parameters, we created mock catalogues following the same approach to simulations as in our previous paper, Shekhar Saraf & Bielewicz (2024). We used the publicly available code FLASK (Xavier et al., 2016) and created $300$ Monte Carlo simulations of correlated log-normal galaxy density field with observed DESI-LIS physical properties (quoted in Table 1) and Planck CMB lensing convergence field. The photometric redshifts were generated from the observed photometric redshift error distributions $p(z_{p}-z_{s}|z_{s})$. We divided the simulated galaxy catalogue into five redshift bins with intervals of $(0.0,0.3,0.45,0.6,0.8,1.0]$. We introduced an additional redshift bin of $0.8\leq z<1.0$ in simulations to account for the scatter of galaxies outside the redshift range considered in the analysis, which is $0.0\leq z<0.8$. The theoretical power spectra for galaxy auto-correlation, CMB convergence auto-correlation and their cross correlation were computed from the Limber approximation (Limber, 1953) using the relation

where $\{x,y\}\in\{\kappa,g\}$, $\kappa\equiv$ convergence and $g\equiv$ galaxy over-density. Here, $\chi$ is the comoving distance, and $\chi_{*}$ is the comoving distance to the surface of the last scattering at redshift $z_{*}\simeq 1100$. Also, $P\bigg{(}k=\frac{\ell+1/2}{\chi},z(\chi)\bigg{)}$ is the matter power spectrum which is generated using the publicly available cosmology code CAMB⁶⁶6https://camb.info/(Lewis et al., 2000) with the HALOFIT prescription to take into account the non-linear distribution of matter. $W^{\kappa}$ and $W^{g}$ are lensing and galaxy kernels given by

We used a redshift-dependent galaxy bias (Solarz et al. 2015; Moscardini et al. 1998; Fry 1996)

where $b_{0}=1.3$ and $D(z)$ is the normalised growth function

where $\gamma=0.55$ is the growth index for the general relativity (Linder, 2005).

We used the CMB convergence noise power spectrum $N_{\ell}^{\kappa\kappa}$ from Planck PDR3 to add noise to the simulated CMB lensing convergence maps. To introduce noise in the galaxy density field, we let FLASK Poisson sample the number of galaxies in mock catalogues.

In this section, we explore three different ways to estimate the covariance matrix $\text{Cov}_{LL^{\prime}}$ (Eq. 8).

We followed the analysis choice of H21 and binned the measured galaxy auto-power spectra and cross-power spectra between the DESI-LIS photometric galaxy catalogue and the Planck minimum-variance CMB lensing potential map with a bin width of $\Delta\ell=10$ in the multipole range $10<\ell<500$. In this section, we present our estimates of galaxy bias and cross-correlation amplitude, taking into account the effects of redshift bin mismatch of objects across tomographic bins.

We performed a tomographic cross-correlation analysis between the minimum-variance CMB lensing convergence map from Planck Collaboration et al. (2020b) and photometric galaxy catalogues from the Data Release 8 of Dark Energy Spectroscopic Instrument Legacy Imaging Survey prepared by H21. We divided the galaxy density field into four redshift slices covering $0<z<0.8$, with a photometric redshift precision of $\frac{\sigma_{z}}{1+z}$ in the range $0.012-0.015$.

H21 used a modified Lorentzian function to approximate the photometric redshift error distribution. Here, we show that the modified Lorentzian function performs well only near the peaks of the error distribution but fails to capture the tails of the error distribution. Although the number of objects in the tails of the error distribution will be smaller than in the peak, we show that broad tails significantly affect the estimation of parameters by adopting a better fitted sum of Gaussians model for the redshift error distributions. We find that our ‘sum of Gaussians’ model gives $\sim 1-1.5\,\sigma$ smaller values for the cross-correlation amplitude and significantly higher estimates of the galaxy linear bias than H21. As the sum of Gaussians model better captures the peculiarities of the error distributions, we treat the parameters estimated under this model as our baseline results.

We implemented the scattering matrix formalism presented in Shekhar Saraf & Bielewicz (2024) to correct for the scatter of objects across redshift bins causing biased estimation of parameters. For our baseline analysis with the sum of Gaussians fit to the redshift error distributions, we find a $\sim 1-2\,\sigma$ higher values of the amplitude of cross correlation from three out of four tomographic bins after correction for leakage, whereas no significant change was observed in the second redshift bin. This may be caused by a small proportion of objects with redshifts between $0.2<z<0.4$, that have assigned photometric redshifts between $0.4<z<0.6$, as mentioned by H21 in section 3.2 of their paper. Due to this systematic error in the redshift calibration, we may underestimate the scattering matrix elements for the second redshift bin. Our estimates of cross-correlation amplitude are consistent with the $\Lambda$CDM expectations, except for the last redshift bin where we get a value that is higher by $\sim 2.5\,\sigma$, making it completely consistent with $\Lambda$CDM expectations. The difference in the last bin could be the result of taking into account objects with redshift of higher than 0.8 in our analysis, which were not considered by H21. However, the galaxy bias differs significantly from H21, primarily due to the different models of galaxy bias.

As shown by Saraf et al. (2022), different CMB lensing maps produce significantly different cross-correlation amplitudes. To test this dependence, we estimated galaxy linear bias and cross-correlation amplitude using Sunyaev-Zeldovich deprojected (SZ-deproj) and temperature-only (TT) reconstructions of the CMB lensing convergence maps. We found the results from SZ-deproj and TT maps to be consistent with the MV map overall. The SZ-deproj map yields an $\sim 1\,\sigma$ higher value for cross-correlation amplitude in the second tomographic bin and the TT map completely resolves the deviation on the amplitude with respect to the $\Lambda$CDM model for the last two tomographic bins, but with poorer $\chi^{2}$ values. Finally, we estimated the impact of the redshift mismatch correction on the $\sigma_{8}$ parameter. We find that the corrected power spectra yields estimates that are consistent with the standard cosmological model for the fourth bin, smaller by around $\sim 1\,\sigma$ for the first and third bins, and smaller by $\sim 3\,\sigma$ for the second bin.

In this study, we demonstrate the importance of precise modelling of the photometric redshift error distributions when estimating parameters such as $\sigma_{8}$ from cross-correlation measurements and present a working example of scattering matrix formalism to correct for the redshift bin mismatch of objects in tomographic cross-correlation analysis. The $\sigma_{8}-\Omega_{m}$ tension is one of the pressing challenges of modern cosmology. Upcoming large-scale structure surveys such as Vera C. Rubin Observatory Legacy Survey of Space and Time (Ivezić et al. 2019b; LSST Science Collaboration et al. 2009), are expected to have smaller redshift errors. This will enable us to put tighter constraints on the $\sigma_{8}-\Omega_{m}$ tension with future tomographic cross-correlation measurements. However, the statistical errors of these surveys will also be small and the net outcome of the bin mismatch systematic error will depend on the interplay between redshift errors and survey statistical errors. From the results for simulations of the LSST survey presented in our previous publication Shekhar Saraf & Bielewicz (2024), we can expect the redshift bin mismatch to remain a significant systematic error even for future photometric datasets. Here we show that the method developed in Shekhar Saraf & Bielewicz (2024) based on the use of the scattering matrix offers a robust way to mitigate bin mismatch, which should enable unbiased inference of cosmological parameters from future tomographic measurements.