**7.4 Results**

**7.4.3 Measuring the lensing potential**

It is well known that the measurement of the growth rate requires an analysis of galaxy surveys which are sensitive to redshift-space distortions [237, 232, 86, 94]. Isolating other effects can lead to an analysis which is more sensitive to other parameters.

In this section we study especially how one can measure the lensing potential with galaxy surveys. The lensing potential is especially sensitive to theories of modified gravity which often have a different lensing potential than General Relativity, see,

7.4. RESULTS 123 e.g. [21, 238, 113]. The lensing potential out to some redshift z is defined by [110]

Ψκ(n, z) = ∫

rs(z)

0 drrs−r

rsr (Ψ(rn, t) +Φ(rn, t)). (7.16)
Denoting its power spectrum by C_{`}^{Ψ}(z, z^{′}), we can relate it to the lensing contribution
C_{`}^{lens}(z, z^{′}) [106] to the angular matter power spectrum C`(z, z^{′})by

C_{`}^{lens}(z, z^{′}) =`^{2}(`+1)^{2}C_{`}^{Ψ}(z, z^{′}). (7.17)
We shall see, that this lensing power spectrum can be measured from redshift integrated
angular power spectra of galaxy surveys.

To study this possibility, we first introduce the signal-to-noise for the different terms which contribute to the galaxy power spectrum as defined in Ref. [106]. For completeness we list these terms in Appendix7.6. The signal-to-noise for a given term is given by

where ˜C` is calculated neglecting the term under consideration (e.g., lensing), and the r.m.s. variance is given by

It is also useful to introduce a cumulative signal-to-noise that decides whether a term is observable within a given multipole band. We define the cumulative signal-to-noise by

Note that C`−C˜` contains not only the auto-correlation of a given term, but also its cross-correlations with other terms so that it can be negative. Especially, for small

`’s the lensing term is dominated by its anti-correlation with the density term and is therefore negative.

Eq. (7.18) estimates the contribution of each term to the total signal. If its signal-to-noise is larger than 1, in principle it is possible measure this term and therefore to constrain cosmological variables determined by it. To evaluate the signal-to-noise of the total C`’s, which is the truly observed quantity, we set ˜C`=0.

In Figure7.9 we show the signal-to-noise for different width of the redshift window function. We consider a tophat window for the narrowest case, ∆z =0.01 and Gaussian window functions with standard deviations ∆z ≳ 0.05(1+z), which corresponds to Euclid photometric errors [162] for the panels on the second line. The sky fraction,

5 10 50 100 500
10^{-6}

10^{-4}
0.01
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{ HSNL{

*z=1,*Dz=0.01

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{ HSNL{

*z=1,*Dz=0.1

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{ HSNL{

*z=1,*Dz=0.5

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{ HSNL{

*z=1, Dirac*

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{ HSNL{

0.5<z<2.5

Figure 7.9: Signal-to-noise for different terms: total (solid, black), redshift-space dis-tortions (dashed, red), lensing (dotted, magenta), potential terms (dot-dashed, blue).

The plot in the top line is computed with a tophat window function with half-width

∆z =0.01. The cases in the second line correspond to Gaussian window functions of half-width ∆z = 0.1,0.5 around ¯z = 1. These three plots are compatible with Euclid specifications. In the third line, we show two extreme situations. The Dirac z-window function corresponds to an infinitesimal z-bin (left) and a very wide redshift range, 0.5<z <2.5. This last case shows a large lensing signal. Notice the zero-crossing of the lensing term which is indicated by a downward spike in this log-plot. The shadowed regions correspond to nonlinearity scale estimated at the mean redshift of the bin.

7.4. RESULTS 125
fsky, and the galaxy distribution dN/dz are compatible with Euclid specifications (see
black lines in Fig. (7.2)). In particular, shot noise turns out to be negligible in this
analysis. The shadowed regions in Figure 7.9 show the nonlinearity scales, estimated
as `max=2πr(¯z)/λmin, where ¯z is the mean redshift of the bin and λmin=68h^{−1} Mpc.

As expected [64], redshift-space distortions and purely relativistic terms are mainly important at large scales, while lensing has a weaker scale dependence. For small ∆z, apart from the usual density term, redshift-space distortions are the main contribution.

Their signal-to-noise is larger than one, which allows to constrain the growth factor.

As ∆z increases, redshift-space distortions are washed out, and their signal decreases significantly. On the other hand, lensing and potential terms increase. This is due to the fact that these terms depend on integrals over z that coherently grow as the width of the z-window function increases. While potential terms always remain sub-dominant, the lensing signal-to-noise becomes larger than 1 for the value of ∆z =0.5 already at `≈60.

As a reference, we also show the signal-to-noise for an infinitesimal bin width (Dirac z-window function). This corresponds to the largest possible C` amplitude. As in the case ∆z = 0.01, redshift-space distortion is of the same order as the density term.

Note , however, that in reality for very narrow bins, shot noise becomes important and
decreases (S/N)_{`}, especially for large multipoles.

The case of a uniform galaxy distribution between 0.5<z <2.5 is also shown. We assumefsky =1 and neglect shot noise. In this configuration the lensing term has a very large signal-to-noise. This can be used to constrain the lensing potential by comparing the observable (total) C`’s to the theoretical models. In practice one may adapt this study to catalogs of radio galaxies, which usually cover wide z ranges but with poor redshift determination which is not needed for this case, for previous studies see, e.g., [58, 204, 266].

In Figure 7.10 the cumulative signal-to-noise, Eq. (7.20), is shown as function of the maximum multipole considered in the sum. Contrary to all the other terms, the cumulative signal-to-noise of the potential terms never exceeds 1. We therefore conclude that the considered experiment is not able to measure the potential terms. It is not clear, whether another feasible configuration would be sensitive to them. Notice also how the lensing term really ’kicks in’ after the zero-crossing, when it is not longer dominated by its anti-correlation with the density term but by the contribution from the autocorrelation.