General Relativistic Description of the Observed Galaxy Power Spectrum: Do We Understand What We Measure?

TL;DR

This work presents a fully general-relativistic framework for the observed galaxy power spectrum, showing that the real-space matter fluctuation defined on the hypersurface of the observed redshift, ${m_{\delta z}}$, induces anisotropy in the power spectrum $P_{\hat{m}_{\delta z}}(k,\mu_k)$ and evolves with redshift, in contrast to the standard Newtonian description. The authors derive the full observed galaxy power spectrum $P_g(k,\mu_k)$ within GR, expressing it as $P_g(k,\mu_k)=P_\phi(k)\,T_g({\bf k},z)\,T_g^*({\bf k},z)$ with a transfer function that mixes bias, velocity, and gravitational potentials, and they compare it to the standard $P_{\rm std}$ to quantify large-scale deviations. They provide explicit forms for the GR transfer with coefficients $\mathcal{N}$ and $\mathcal{M}$ and show that the GR corrections become significant at high redshift ($z\gtrsim 3$), where they cannot be neglected for interpreting large-scale modes. The analysis also evaluates the detectability of these GR effects under a cosmic-variance-limited survey, finding that while the deviations are small at $z\leq3$, they should be measurable at higher redshifts, underscoring the necessity of the GR framework for future surveys and offering a pathway to test gravity on cosmological scales.

Abstract

We extend the general relativistic description of galaxy clustering developed in Yoo, Fitzpatrick, and Zaldarriaga (2009). For the first time we provide a fully general relativistic description of the observed matter power spectrum and the observed galaxy power spectrum with the linear bias ansatz. It is significantly different from the standard Newtonian description on large scales and especially its measurements on large scales can be misinterpreted as the detection of the primordial non-Gaussianity even in the absence thereof. The key difference in the observed galaxy power spectrum arises from the real-space matter fluctuation defined as the matter fluctuation at the hypersurface of the observed redshift. As opposed to the standard description, the shape of the observed galaxy power spectrum evolves in redshift, providing additional cosmological information. While the systematic errors in the standard Newtonian description are negligible in the current galaxy surveys at low redshift, correct general relativistic description is essential for understanding the galaxy power spectrum measurements on large scales in future surveys with redshift depth z>3. We discuss ways to improve the detection significance in the current galaxy surveys and comment on applications of our general relativistic formalism in future surveys.

Figures (3)

• Figure 1: Matter power spectrum $P_{{\hat{m}}_{\delta z}}(k,\mu_k)$ at various redshifts. Upper panel: Solid lines represent the matter power spectrum at $z=0$, computed by using Eq. (\ref{['eq:realP']}) along the line-of-sight direction ($\mu_k=1$; thick) and along the transverse direction ($\mu_k=0$; thin). For reference, various lines indicated in the legend show power spectra of perturbation variables in the conformal Newtonian gauge and the synchronous gauge. Bottom panel: matter power spectrum $P_{{\hat{m}}_{\delta z}}(k,\mu_k)$ at $z>0$, but with its amplitude normalized to match $P_{{\hat{m}}_{\delta z}}(k,\mu_k)$ at $z=0$. Solid and dashed lines represent $P_{{\hat{m}}_{\delta z}}(k,\mu_k)$ with $\mu_k=1$ and $\mu_k=0$, respectively. The horizon scale at $z=0$ is shown as a vertical line.
• Figure 2: Observed galaxy power spectrum $P_g(k,\mu_k)$ at $z=1$. Thick lines represent the observed galaxy power spectrum, computed by using the full general relativistic description in Eq. (\ref{['eq:fullpobs']}) along the line-of-sight direction ($\mu_k=1$; solid) and along the transverse direction ($\mu_k=0$; dashed). Dotted lines show the observed galaxy power spectrum using the standard method in Eq. (\ref{['eq:stdpok']}). For comparison, the matter power spectrum $b^2P_{{\hat{m}}_{\delta z}}(k,\mu_k)$ is shown as thin lines with the galaxy bias factor $b=2$.
• Figure 3: Systematic errors in theoretical modeling of the observed galaxy power spectrum. Upper panel: fractional difference of $P_g(k,\mu_k)$ at various redshift slices, compared to the standard description $P_{\rm std}(k,\mu_k)$ in Eq. (\ref{['eq:stdpok']}) along the line-of-sight direction (solid) and along the transverse direction (dashed). Bottom panel: detection significance of the departure from the standard description in a cosmic-variance limited survey as a function of maximum wavenumber. The survey volume is divided into four spherical shells with redshift range $z=0\sim1$ (bottom solid), $1\sim3$, $3\sim6$, and $6\sim10$ (top solid), and the signal is computed at the mean redshift of each shell.