First Constraints on Growth Rate from Redshift-Space Ellipticity Correlations of SDSS Galaxies at $0.16<z<0.70$

Cosmological parameters have been precisely determined via various observations: cosmic microwave background (Planck Collaboration et al., 2020), large-scale structure of the universe (Alam et al., 2017), and gravitational lensing (Hikage et al., 2019). However, the origin of the accelerating expansion of the universe, namely, dark energy or/and modification of Einstein’s gravity theory, is still a complete mystery in fundamental physics. Thus, deeper and wider galaxy surveys are ongoing to better understand the expansion and growth history of the universe (Takada et al., 2014; DESI Collaboration et al., 2016).

In parallel, we need to keep exploring methods that maximize the use of cosmological information encoded in given observations. There is a growing interest in using intrinsic alignment (IA) of galaxy shapes (Croft & Metzler, 2000; Heavens et al., 2000; Hirata & Seljak, 2004) as a geometric and dynamical probe of cosmology complimentary to galaxy clustering. Although there are various observational studies of IA, they mainly focused on the contamination to weak gravitational-lensing measurements (e.g., Mandelbaum et al., 2006; Okumura et al., 2009; Joachimi et al., 2011; Li et al., 2013; Singh et al., 2015; Tonegawa & Okumura, 2022). The anisotropy of three-dimensional IA statistics has been detected by Singh & Mandelbaum (2016). The full cosmological information of IA, however, had not been investigated at that time.

To fully exploit cosmological information encoded in anisotropic IA, theoretical modeling of the three-dimensional IA correlations has been developed (Okumura & Taruya, 2020; Okumura et al., 2020; Kurita et al., 2021). A series of our papers (Taruya & Okumura, 2020; Chuang et al., 2022; Okumura & Taruya, 2022) has also shown that the three-dimensional IA statistics in redshift space provide additional constraints on the linear growth rate of the universe, $f=d\ln{\delta_{m}}/d\ln{a}$ ($a$ and $\delta_{m}$ being the scale factor and matter density perturbation), which is used to test modified gravity models. Furthermore, recent studies showed that IA can be used as probes of not only modified gravity models but also other effects such as primordial non-Gaussianity, neutrino masses, and gravitational redshifts (Schmidt et al., 2015; Lee et al., 2023; Zwetsloot & Chisari, 2022; Saga et al., 2023).

In this paper, besides conventional galaxy density correlation functions, we measure intrinsic ellipticity correlation functions from various galaxy samples in the Sloan Digital Sky Survey (SDSS) and SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). We then present the first joint constraints on the growth rate from the galaxy IA and clustering. Otherwise stated, we assume a flat $\Lambda$CDM model determined by Planck Collaboration et al. (2020) as our fiducial cosmology throughout this paper.

We analyze the galaxy distribution over $0.16\leq z\leq 0.70$ from the SDSS-II (Eisenstein et al., 2001) and SDSS-III BOSS (Reid et al., 2016). First, we use the luminous red galaxy (LRG) sample ($0.16\leq z\leq 0.47$) from the SDSS Data Release 7 (DR7). Galaxies in the sample have rest-frame $g$-band absolute magnitudes, $-23.2<M_{g}<-21.2$ ($H_{0}=100\,{\rm km}~{}{\rm s}^{-1}~{}{\rm Mpc}^{-1}$) with $K+E$ corrections of passively evolved galaxies to a fiducial redshift of 0.3. The components of the ellipticity are defined as

where $q$ is the minor-to-major-axis ratio ($0\leq q\leq 1$) and $\beta_{x}$ is the position angle of the ellipticity from the north celestial pole to east. We use the ellipticity of LRG defined by the $25~{}{\rm mag}~{}{\rm arcsec}^{-2}$ isophote in the $r$ band. This LRG sample is similar to that used in Okumura et al. (2009) and Okumura & Jing (2009) but slightly extended from DR6 to DR7, with the total number of the LRG used being $105,334$.

We also use $353,804$ LOWZ ($0.16\leq z\leq 0.43$) and $761,567$ CMASS ($0.43\leq z\leq 0.70$) galaxy samples from the BOSS DR12. For these samples, we adopt the ellipticity defined by the adaptive moment (Bernstein & Jarvis, 2002). While this method optimally corrects for the point-spread function (PSF) in the determined ellipticity, it is found to result in a small bias (Hirata & Seljak, 2003). The residual PSF remains in the shape autocorrelation function at large scales (Singh & Mandelbaum, 2016). As we show below, the correlation functions of these samples are very noisy, and they do not contribute to cosmological constraints below.

As in our earlier studies, we set the axis ratio in Equation (5) to $q=0$ (Okumura & Jing, 2009; Okumura et al., 2009, 2019, 2020). We are not interested in the amplitude of IA and marginalize it over. This simplification will not affect results below.

In this section, we measure the redshift-space correlation functions of galaxy density and IA from the SDSS samples, and estimate their covariance matrix.

As a conventional clustering analysis, we use a galaxy autocorrelation (GG) function in redshift space, $\xi_{gg}^{s}(\boldsymbol{r})=\left\langle\delta_{g}^{s}(\boldsymbol{x}_{1})\delta_{g}^{s}(\boldsymbol{x}_{2})\right\rangle$, where superscript $s$ denotes the quantity defined in redshift space, $\boldsymbol{r}=\boldsymbol{x}_{2}-\boldsymbol{x}_{1}$, and $\delta_{g}^{s}(\boldsymbol{x})$ is the galaxy number density fluctuation. We adopt the Landy & Szalay (1993) estimator to measure it,

where $DD$, $RR$, and $DR$ are the normalized counts of galaxy–galaxy, random–random, and galaxy–random pairs, respectively. We then obtain the multipole moments,

where $r=|\boldsymbol{r}|$, $\mu_{\boldsymbol{r}}$ is the direction cosine between the line of sight and $\boldsymbol{r}$, and ${\cal L}_{\ell}$ is the $\ell$th-order Legendre polynomials. To obtain the multipoles via Eq. (7), we estimate $\xi_{gg}^{s}(r,\mu_{\boldsymbol{r}})$ with the angular bin size of $\Delta\mu_{\boldsymbol{r}}=0.1$ in Eq. (6) and take the sum over $\mu_{\boldsymbol{r}}$.

The first row of Figure 1 shows multipole moments of the GG correlation functions. The first, second, and third columns show the results from the LRG, LOWZ, and CMASS samples, respectively. Since the hexadecapole is noisy, we analyze only the monopole and quadrupole moments (Kaiser, 1987). These correlation functions have been measured in various previous works (e.g., Samushia et al., 2012; Alam et al., 2017) and our measurements are consistent with theirs.

Next, we introduce intrinsic alignment statistics, which are density-weighted quantities. The galaxy position–intrinsic ellipticity (GI) correlation, $\xi_{g+}^{s}$, and intrinsic ellipticity–ellipticity (II) correlations, $\xi_{+}^{s}$ and $\xi_{-}^{s}$, are defined by

where $W_{g+}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})=\gamma_{+}(\boldsymbol{x}_{2})$ and $W_{\pm}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})=\gamma_{+}(\boldsymbol{x}_{1})\gamma_{+}(\boldsymbol{x}_{2})\pm\gamma_{\times}(\boldsymbol{x}_{1})\gamma_{\times}(\boldsymbol{x}_{2})$. For the II correlations, we label $\xi_{+}^{s}$ and $\xi_{-}^{s}$ individually as II$(+)$ and II($-$) correlations, respectively. The GI correlation function is estimated as (Mandelbaum et al., 2006),

where $S_{+}D$ is the sum over all pairs with separation $\boldsymbol{r}$ of the $+$ component of the ellipticity, $S_{+}D=\sum_{i\neq j|\boldsymbol{r}}{\gamma_{+}(j|i)}$, with $\gamma_{+}(j|i)$ being the ellipticity of galaxy $j$ measured relative to the direction to galaxy $i$, and $S_{+}R$ is defined similarly. The II correlation functions are estimated as

where $S_{+}S_{+}=\sum_{i\neq j|\boldsymbol{r}}{\gamma_{+}(j|i)\gamma_{+}(i|j)}$ and similarly for $S_{\times}S_{\times}$. Finally, multipole moments for the IA correlations, $\xi_{g+,\ell}^{s}$ and $\xi_{\pm,\ell}^{s}$, are obtained via the same equation as Equation (7). Again, since the hexadecapole is noisy, we analyze only the $\ell=0$ and $\ell=2$ moments.

The second, third, and bottom rows of Figure 1 respectively present redshift-space multipole moments of the GI, II($+$) and II($-$) correlation functions. Both the monopole and quadrupole of the GI correlation are clearly detected in all the three samples. Particularly, LRG are the brightest galaxy sample and shows the strongest signal because IA has a strong luminosity dependence. Though LOWZ has a redshift range similar with LRG, it targets fainter galaxies and thus has higher number density. Therefore, the LOWZ sample shows lower GI amplitude, confirming the earlier detection by Singh & Mandelbaum (2016). We find even a lower GI signal in the CMASS sample. The monopole of the II correlation is clearly detected for the LRG sample, as in Okumura et al. (2009), while the newly measured quadrupole is noisier and consistent with zero. Those for the LOWZ and CMASS samples have much lower amplitude, and are somewhat consistent with zero. Furthermore, their shapes are determined by the adaptive moment and have nonzero correlation due to the PSF at $r>30\,h^{-1}\,{\rm Mpc}$ (Singh & Mandelbaum, 2016).

We estimate the covariance matrix for the measured correlation functions, ${\rm C}_{ij}^{X_{\ell}X^{\prime}_{\ell^{\prime}}}\equiv{\rm C}\left[\xi_{X,\ell}(r_{i}),\xi_{X^{\prime},\ell^{\prime}}(r_{j})\right]$, with $X=\{gg,g+,+,-\}$ and $\ell=\{0,2\}$, using the jackknife resampling method. While jackknife is not an unbiased error estimator, it provides reliable error bars for the statistics whose error is dominated by the shape noise (Mandelbaum et al., 2006). The error bars shown in Figure 1 are the square root of the diagonal components of the covariance matrix.

Here we present theoretical models to interpret the measured correlation functions. Since theoretical models are naturally provided in Fourier space, we first present models for the power spectra, $P_{X}^{s}$, perform the Fourier transform,

where $X=\{gg,g+,+,-\}$, and obtain the multipole moments $\xi_{X,\ell}^{s}$ via Equation (7).

We perform the likelihood analysis and constrain the growth rate parameter $f\sigma_{8}$ from the three SDSS galaxy samples. Particularly, we show how well the constraints are improved by combining IA statistics with the conventional galaxy clustering statistics. We compare the measured statistics, $\xi_{X,\ell}^{s}$, where $X=\{gg,g+,+,-\}$ and $\ell=\{0,2\}$, to the corresponding predictions. The $\chi^{2}$ statistic is given by

where $\Delta_{i}^{X_{\ell}}=\xi_{X,\ell}^{s,{\rm obs}}(r_{i})-\xi_{X,\ell}^{s,{\rm th}}(r_{i};{\boldsymbol{\theta}})$ is the difference between the observed correlation function and theoretical prediction with ${\boldsymbol{\theta}}$ being a parameter set to be constrained. The analysis is performed over the scales adopted, $r_{\rm min}\leq r_{i}\leq r_{\rm max}$. Since the jackknife method underestimates the covariance at large scales, we set the maximum separation $r_{\rm max}=100\,h^{-1}\,{\rm Mpc}$. Moreover, as described in Sec. 2, the II correlation functions of LOWZ and CMASS are affected by the residual PSF at $r>30\,h^{-1}\,{\rm Mpc}$ (Singh & Mandelbaum, 2016). We thus set $r_{\rm max}=25\,h^{-1}\,{\rm Mpc}$ for the II correlations of these samples. In Appendix A, we investigate how our constraints change with $r_{\rm min}$, and we adopt $r_{\rm min}=10\,h^{-1}\,{\rm Mpc}$. In Appendix B, we provide further argument that our cosmological constraints are not biased by the effect of the uncorrected PSF. For the clustering-only analysis, the covariance is a $20\times 20$ matrix, while for the full analysis of clustering and IA, it is a $60\times 60$ matrix for LRG and $48\times 48$ for LOWZ and CMASS. The data points used for the analysis are enclosed by the vertical lines in Figure 1.

Figure 2 shows the parameter constraints obtained from the LRG sample. The blue and orange contours are results with the clustering-only analysis and its combination with IA statistics, respectively. For the clustering-only analysis, after marginalizing over $b\sigma_{8}$ and $\sigma_{v}$, we obtain the constraint as $f\sigma_{8}=0.5196^{+0.0352}_{-0.0354}$ ($68\%$ confidence level). For the combined analysis of clustering and IA, we obtain $f\sigma_{8}=0.5322^{+0.0293}_{-0.0291}$ by further marginalizing over the shape bias parameter $b_{K}$. Namely, the constraint on $f\sigma_{8}$ is improved by $19\%$ by adding the IA statistics. Note that, as we set $q=0$ in Equation (5), the definition of $b_{K}$ here is different from literature and one cannot directly compare the values.

The left and right panels of Figure 3 show results similar to Figure 2 but for LOWZ and CMASS, respectively. Using LOWZ, we obtain $f\sigma_{8}=0.5043^{+0.0226}_{-0.0229}$ (GG only), and $f\sigma_{8}=0.4937^{+0.0201}_{-0.0201}$ (GG$+$IA). The LOWZ is a denser sample than the LRG by targeting fainter galaxies, and thus, even the galaxy clustering alone puts tighter constraints. However, combining the IA statistics, LRG provides almost as a strong constraint as LOWZ. CMASS is also a fainter population at higher redshift, $0.43<z<0.70$. With the GG-only analysis, we obtain $f\sigma_{8}=0.4614^{+0.0156}_{-0.0154}$, and with the GG+IA analysis, $f\sigma_{8}=0.4628^{+0.0149}_{-0.0151}$. Our analysis of these three galaxy samples demonstrates that the contribution of IA to cosmological constraints can be enhanced by adopting an optimal weighting to brighter galaxies (Seljak et al., 2009). Exploring such an optimization is our future work.

The best-fit nonlinear models jointly fitted for the clustering and IA statistics are shown by the solid curves in Figure 1. Reduced $\chi^{2}$ values obtained for LRG, LOWZ, and CMASS samples are $\chi^{2}/\nu=1.85,1.14$, and $2.42$, respectively, where $\nu$ is the degree of freedom, $\nu=56$ for LRG and $\nu=44$ for LOWZ and CMASS. The large $\chi^{2}$ value for the CMASS sample is due to small error bars in the GG correlation. If we adopt $r_{\rm min}=15\,h^{-1}\,{\rm Mpc}$, the minimum $\chi^{2}$ is reduced to $\chi^{2}/\nu=1.68$. Accordingly, the best-fitting value of $f\sigma_{8}$ is shifted (see Figure 5 in Appendix A).

Finally, Figure 4 summarizes the constraints on $f\sigma_{8}$ from the three galaxy samples we considered. As shown in the lower panel, the constraint gets tighter by adding IA statistics to the galaxy clustering statistics. Overall, the derived results are consistent with the prediction of $\Lambda$CDM determined from the Planck satellite experiment (Planck Collaboration et al., 2020). It indicates that combining IA and clustering statistics enables us to obtain robust and tight constraints.

We have presented the first cosmological constraints using IA of the SDSS galaxies. We have measured the redshift-space GI and II correlation functions of LRG, LOWZ, and CMASS galaxy samples. By comparing them with the models of nonlinear alignment and RSD effects, we have constrained the growth rate of the density perturbation, $f(z)\sigma_{8}(z)$. We found that combining IA with clustering enhances the growth rate constraint by $\sim 19\%$ compared to the clustering-only analysis for the LRG sample. This improved constraint on $f\sigma_{8}$ is only slightly worse than that obtained from the LOWZ, which is a much denser sample by targeting fainter galaxies. This indicates a potential that the contribution of the IA statistics can be further enhanced by adopting an optimal weighting to brighter galaxies.

In this work we considered only the dynamical constraint via RSD. However, baryon acoustic oscillations (BAOs) observed in the galaxy distributions (Eisenstein et al., 2005) were shown to be also encoded in galaxy IA statistics and thus useful to tighten geometric constraints (Chisari & Dvorkin, 2013; Okumura et al., 2019). The cosmological analysis of IA simultaneously using RSD and BAO will be shown in our future work.

The benefits of using IA can be further enhanced by improving the model. In this paper we worked with a simple extension of the NLA model to include partly the FoG effect (T. Okumura et al. 2023, in preparation). However, more sophisticated nonlinear models of IA statistics have been proposed recently (Blazek et al., 2019; Vlah et al., 2020; Akitsu et al., 2021; Matsubara, 2022). These models enable us to use the measured IA correlation functions down to smaller scales, which will enhance the science return from IA of galaxies.