The linear power spectrum of observed source number counts

TL;DR

The work addresses how the observed angular counts of astrophysical sources relate to the underlying cosmological perturbations within linear general relativity. It develops a self-consistent, gauge-invariant framework starting from a non-perturbative Jacobi-map formulation and specializes to linear scalar perturbations in a flat FRW universe, deriving a comprehensive expression for the perturbation to counts that includes density fluctuations, redshift-space distortions, lensing, radial displacements, magnification bias, and ISW-related effects. The authors extend the formalism to flux- and magnitude-limited samples and provide a practical numerical implementation (CAMB sources) to compute auto- and cross-spectra with CMB temperature and polarization, quantifying the relative importance of each term across redshift and scale. They also analyze selection-function versus total-count interpretations and discuss implications for cross-correlations with CMB and weak lensing, highlighting the role of bias modeling and non-linearities on small scales. Overall, the paper delivers a robust, widely applicable tool for interpreting counts and their correlations in current and future large-scale structure surveys, reducing biases from incomplete or inconsistent treatments of relativistic effects.

Abstract

We relate the observable number of sources per solid angle and redshift to the underlying proper source density and velocity, background evolution and line-of-sight potentials. We give an exact result in the case of linearized perturbations assuming general relativity. This consistently includes contributions of the source density perturbations and redshift distortions, magnification, radial displacement, and various additional linear terms that are small on sub-horizon scales. In addition we calculate the effect on observed luminosities, and hence the result for sources observed as a function of flux, including magnification bias and radial-displacement effects. We give the corresponding linear result for a magnitude-limited survey at low redshift, and discuss the angular power spectrum of the total count distribution. We also calculate the cross-correlation with the CMB polarization and temperature including Doppler source terms, magnification, redshift distortions and other velocity effects for the sources, and discuss why the contribution of redshift distortions is generally small. Finally we relate the result for source number counts to that for the brightness of line radiation, for example 21-cm radiation, from the sources.

Figures (8)

• Figure 1: Three-dimensional power spectrum of the Newtonian-gauge counts source $\delta_n$ (solid) and the synchronous gauge source $b \delta_m^\text{syn}$ (dashed) at $z=0$ and $z=3$ with bias $b=1.5$ (relative to the matter perturbation $\delta_m^\text{syn}$) assuming constant comoving source density. The dash-dotted line shows the spectrum for $\delta_n$ re-scaled (incorrectly) from $z=3$ to $z=0$ with the scale-independent growth factor $D(\eta)$ appropriate to $\delta_m^\text{syn}$. On large scales, $\delta_n$ grows more slowly in time than $\delta_m^\text{syn}$ due to the velocity term in Eq. (\ref{['biaseq']}) which goes as $\mathcal{H} \dot{D}$. Note that neither power spectrum is directly observable, but the difference between them illustrates the importance of using a full analysis on scales $k\ll 0.01\text{Mpc}^{-1}$.
• Figure 2: Angular power spectra of total number counts, based on the approximation of Eq. (\ref{['deltaN_approx']}), for Gaussian window functions $W(z)$ at $z=0.45$ (left; $\sigma_z=0.03$ and constant bias $b=1.5$) and $z=0.6$ (centre; $\sigma_z = 0.05$ and $b=1.95$), and their cross-correlation (right) using non-linear corrections from Halofit Smith:2002dz. Solid lines have no lensing ($2-5s=0$), dashed lines have lensing but no magnification bias ($s=0$), and dot-dashed lines have $s=0.6$. Note that these are barely distinguishable in the auto-power spectra. The window functions are similar to those actually measured using LRG surveys Blake:2006kvPadmanabhan:2006ku. Lensing is a significant source of correlation when the cross-correlation is otherwise small, c.f. Ref. LoVerde:2007ke. The effect is well below cosmic variance on an individual $l$, but above cosmic variance over a range $\Delta_l \sim 100$. For $s=0$ the effect of magnification is negative since a magnified area has less sources per solid angle.
• Figure 3: Angular power spectra for all sources in a Gaussian window function at $z=0.1$ ($\sigma_z = 0.01$, $b=1$), with (thick solid) and without (dashed) the ${\cal O}({\hat{\mathbf{n}}}\cdot{\mathbf{v}}/\mathcal{H}\chi)$ radial-displacement term. The dot-dashed line shows the equivalent result for a magnitude-limited survey with constant $2-5s = 1$, and the bottom panel shows the fractional differences compared to the result with no radial-displacement terms.
• Figure 4: Fractional error compared to the full result for the counts angular power spectrum $C_l$ with broad Gaussian window functions peaking at various redshifts with $\sigma_z=0.3 z$. The error shown is that obtained when various types of term in the full result of Eq. (\ref{['eq:countsfin_magbias']}) are individually neglected. The 'redshift' term is for redshift-distortions (radial derivative of velocity), 'velocity' terms are proportional to ${\hat{\mathbf{n}}}\cdot {\mathbf{v}}$ and include the radial displacement effect, 'lensing' is the convergence term, and 'potentials' includes the effects of gravitational potentials at the source, time delay and the ISW. Solid and dashed lines indicate terms whose neglect reduces and increases the power spectrum respectively. The 'standard' result is the approximation given by Eq. \ref{['standard_delta']} and for this the fractional error compared to the full result is shown (with solid/dashed lines denoting an excess/deficit). There is no source evolution and (unrealistically) $b=1$ and $s=0$.
• Figure 5: The fractional difference in the counts angular power spectrum (left) and CMB temperature cross-correlation (right), compared with the 'standard' result calculated using Eq. \ref{['standard_delta']} for the distribution given in Eq. \ref{['sampleP']}. Thick lines are our limiting full results for the case of no source evolution (solid), in which case the source selection function is non-trivial, and the case when all sources are observed (dashed), so that $P(z)=\bar{n}(z)$ implying source evolution. The thin red line shows the change in the standard result from dropping the redshift-distortion term; this modification to the 'standard' result brings it close to the full result when all sources are observed as expected from the arguments in Sec. \ref{['sec:totalcounts']}. The blue dashed-dot line shows the change in the standard result when $\delta_N^\text{syn}$ is replaced with the Newtonian gauge $\delta_N$ (no source evolution), which is significantly larger than the difference between the standard and full results.