Testing cosmological structure formation using redshift-space distortions

TL;DR

The paper develops and tests a bias-free estimator for cosmological structure growth from redshift-space distortions by exploiting anisotropies in the galaxy power spectrum. It introduces a Legendre-m multipole framework and a Gaussian damping extension to recover the matter power spectrum scaled by $f^2$ on large scales, enabling measurement of $f\sigma_{8,\mathrm{mass}}$ independent of galaxy bias. Using a high-resolution $\Lambda$CDM N-body simulation, the authors demonstrate that on large scales $P_{\theta\theta} \approx f^2 P_{\mathrm{mass}}$ and $P_{g\theta}^2/(P_{gg} P_{\mathrm{mass}}) \approx 1$, while characterizing quasi-linear deviations via SSVD with a damping scale $\sigma$ that grows with $k$; the proposed estimator $\hat{P} = (P_{g\theta})^2 / P_{gg}$ can robustly reconstruct the matter spectrum’s shape with amplitude $f^2$. The work highlights the complementarity with weak lensing, extends redshift-space analyses into quasi-linear regimes, and lays groundwork for extracting cosmological information from future spectroscopic surveys.

Abstract

Observations of redshift-space distortions in spectroscopic galaxy surveys offer an attractive method for observing the build-up of cosmological structure. In this paper we develop and test a new statistic based on anisotropies in the measured galaxy power spectrum, which is independent of galaxy bias and matches the matter power spectrum shape on large scales. The amplitude provides a constraint on the derivative of the linear growth rate through f.sigma_8. This demonstrates that spectroscopic galaxy surveys offer many of the same advantages as weak lensing surveys, in that they both use galaxies as test particles to probe all matter in the Universe. They are complementary as redshift-space distortions probe non-relativistic velocities and therefore the temporal metric perturbations, while weak lensing tests the sum of the temporal and spatial metric perturbations. The degree to which our estimator can be pushed into the non-linear regime is considered and we show that a simple Gaussian damping model, similar to that previously used to model the behaviour of the power spectrum on very small scales, can also model the quasi-linear behaviour of our estimator. This enhances the information that can be extracted from surveys for LCDM models.

Figures (7)

• Figure 1: Mass density and velocity power spectra recovered from three different simulations. Solid lines correspond to the primary simulation used in this paper, and for this simulation we show power spectra of the overdensity (solid squares), velocity divergence (solid circles) and the curl of the velocity (solid triangles). For comparison, the dashed line is the corresponding power spectra from a simulation with the same initial conditions and cosmology, run with a PM code (so lower force resolution), and the dotted lines corresponds to a simulation with a box half as big (so twice the force resolution, 8 times the mass resolution but 1/8 the volume).
• Figure 2: Recovered power spectra from our simulation, with redshift-space distortions along one axis of the simulation box (solid circles). Power spectra are shown for all mass (lower data), and for the three halo catalogues described in the text (upper data, with the more biased power spectra corresponding to larger mass thresholds). These data were fitted for $k<0.05\,h\,{\rm Mpc}^{-1}$ using the model given by Eq. (\ref{['eq:pgs_comb_lin']}). Ratios between model and data are shown in the lower panels, and panels from left to right are for different SSVD models. See Section \ref{['sec:sph_av_nl']} for details.
• Figure 3: $P_{1\,{\rm axis}}^s(k)$ calculated from the simulation are shown by the solid circles - these are the same data in the upper panels of Fig. \ref{['fig:pk']}. The lowest amplitude power spectrum corresponds to the mass, while the other three are from halo catalogues, as described in the text, where the amplitude is an increasing function of mass limit. For comparison we also plot the model given by Eqns. \ref{['eq:pk_one']}, where the free parameters fitted are $P_{gg}$, $P_{g\theta}$, $P_{\theta\theta}$, and $\sigma$ where a SSVD model is included (solid lines). To determine these models, we have fitted $P_{1\,{\rm axis}}^s$, $P_{2\,{\rm axes}}^s$, and $P_{3\,{\rm axes}}^s$ for $k<0.7\,h\,{\rm Mpc}^{-1}$, where the data were equally weighted in $\log k$. In order to highlight differences, we plot the model for $P_{1\,{\rm axis}}^s$ divided by the data in the lower panels (solid lines). Similar lines are also plotted for $P_{2\,{\rm axes}}^s$ (dashed), and $P_{3\,{\rm axes}}^s$ (dotted).
• Figure 4: Upper panels: $P_{gg}$ (dotted lines), $P_{g\theta}/f$ (dashed lines), $P_{\theta\theta}/f^2$ (solid lines) recovered by fitting $P_{0\,{\rm axes}}^s$, $P_{1\,{\rm axis}}^s$, $P_{2\,{\rm axes}}^s$, and $P_{3\,{\rm axes}}^s$ for $k<0.7\,h\,{\rm Mpc}^{-1}$. Four power spectra are plotted, corresponding to the mass and the halo catalogues. The amplitude of $P_{gg}$ and $P_{g\theta}$ is lowest for the mass, and is an increasing function of the halo mass limit. For $P_{\theta\theta}$, the deviation from the shape of $P_{gg}$ is weakest for the mass, and is an increasing function of halo mass limit. The lower panels show these power spectra divided by $P_{gg}$ for the mass to highlight deviations between the shapes of the power spectra.
• Figure 5: For $P_{g\theta}$, we can remove a deterministic linear galaxy density bias dependence by dividing by $\sqrt{P_{gg}}$. If the galaxy density bias is well described by a local linear model and there is no velocity bias, then we expect $P^2_{g\theta}/P_{gg}=P_{\rm mass}$, so we divide $P_{g\theta}$ by a further factor of $\sqrt{P_{\rm mass}}$ to highlight deviations from this model. The solid line was calculated from a fit with no small-scale dispersion correction, the dotted line assumed an Exponential model, and the dashed line a Gaussian model, both with single scale-independent variance.