Unequal-time correlators for cosmology

Measurements of the power spectrum from large-scale structure surveys have, to date, assumed an equal-time approximation, where the full cross-correlation power spectrum of the matter density field evaluated at different times (or distances) has been approximated either by the power spectrum at a fixed time or in an improved fashion, by a geometric mean $P(k;r_1,r_2)=[P(k;r_1)P(k;r_2){]}^{1/2}$. In this paper we investigate the expected impact of the geometric mean ansatz and present an application in assessing the impact on weak-gravitational-lensing cosmological parameter inference, using a perturbative unequal time correlator. As one might expect, we find that the impact of this assumption is greatest at large separations in redshift $\mathrm{\Delta }z\gtrsim 0.3$ where the change in the amplitude of the matter power spectrum can be as much as 10 percent for $k\gtrsim 5h{\mathrm{Mpc}}^{-1}$. However, of more concern is that the corrections for small separations, where the clustering is not close to zero, may not be negligibly small. In particular, we find that for a Euclid- or LSST-like weak lensing experiment, the assumption of equal-time correlators may result in biased predictions of the cosmic shear power spectrum, and that the impact is strongly dependent on the amplitude of the intrinsic alignment signal. To compute unequal-time correlations to sufficient accuracy will require advances in either perturbation theory to high $k$ modes or extensive use of simulations.

Figure 1

Left panel: The fractional difference between $2D(z)D(z^{\prime })$ and $D^2(z)+D^2(z^{\prime })$, where $D(z)$ is the linear growth factor, for a $\mathrm{\Lambda }\mathrm{CDM}$ cosmology with parameters set at the Planck [6] maximum likelihood values. Right panel: A cross section through the left panel plot for $z^{\prime }=1$.

Figure 2

The fractional difference between $P^{\mathrm{UETC}}(k;r,r^{\prime })$ and $[P^{\mathrm{ETC}}(k;r)P^{\mathrm{ETC}}(k;r^{\prime }){]}^{1/2}$ and for a $\mathrm{\Lambda }\mathrm{CDM}$ cosmology with parameters set at the Planck [6] maximum likelihood values. In the left panel we set $z=1$ and show the variation as a function of $z^{\prime }$ for several $k$ modes. In the right panel we set $z=1$ and show the variation as a function of $k$ modes for several $z^{\prime }$ values.

Figure 3

The functions $C_{\ell }^{\delta \delta ,\mathrm{UETC}}(r_1,r_2)$ (blue) and $C_{\ell }^{\delta \delta ,\mathrm{ETC}}(r_1,r_2)$ (red) for several different $\ell$ modes and redshift values defined in Eq. (14). The title of each plot gives the $\ell$ mode and $z$ value used.

Figure 4

Left panel: Equation (16) as a function of $r^{\prime }$ for $r=3000$. The red line is calculated using numerical integration, and the blue line is the analytic solution multiplied by ten so that the functional form can be distinguished. The black line shows the absolute difference between the two. Right panel: The same as the left except $r=1000$. Both plots are for $\ell =1000$ and $K=10h{\mathrm{Mpc}}^{-1}$.

Figure 5

Left panel: The fractional change in a projected power spectrum for a Gaussian projection kernel $F_A(r,r_1)\propto \exp [-(r-r_1{)}^2/2{\sigma }_r^2]$, fixing $F_B(r,r_2)={\delta }^D(r-r_2)$. We set $r_B=r(z=1)$ and $r_A=r(z^{\prime }=0.9)$. The lines correspond to different widths of the projection kernel. Right panel: The same as the left panel except with $r_A=r(z^{\prime }=0.5)$.

Figure 6

The difference ${\ell }^2|C_{\ell }^{\mathrm{UETC}}(z,z^{\prime })-C_{\ell }^{\mathrm{ETC}}(z,z^{\prime })|$ between the unequal-time projected power spectra and the equal-time projected power spectra as a function of the $\ell$ mode for each redshift bin combination for the $GG$, $GI$, and $II$ cases.

Figure 7

The change in the integrated systematic power spectrum, Eq. (23), caused by the equal-time assumption, as a function of the intrinsic alignment amplitude $A_{\mathrm{IA}}$. In the left axes, we show this normalized by the required value of ${10}^{-7}$, and the right axes show the unnormalized value.