On the sensitivity of weak gravitational lensing to the cosmic expansion function

The power spectrum of cosmological weak gravitational lensing, specifically the expectation for the power spectrum of the weak gravitational shear $\gamma$ caused by cosmic structures, can be expressed as (Bartelmann & Schneider, 2001; Bartelmann, 2010; Schneider, 2006).

The shear power spectrum $C_{\ell}^{\gamma}$ at an angular wave number $\ell$ is determined by a line-of-sight integral of the density-fluctuation power spectrum $P_{\delta}(k)$ at wave number $k=\ell/w$, geometrically weighed with the squared distance ratio between the matter fluctuations acting as gravitational lenses and the sources. In (1), $w$ is the comoving radial distance running from the source at $w=w_{\mathrm{s}}$ to the observer at $w=0$. The parameter of the present matter density is $\Omega_{\mathrm{m0}}$, and $cH_{0}^{-1}$ is the Hubble radius. Since density fluctuations grow with time, and thus also grow along the line-of-sight, variations of the growth factor change the density-fluctuation power spectrum $P_{\delta}$ and thus also affect our expectation for the shear power spectrum. In addition to the indirect variation mediated by the growth factor, the shear power spectrum is also directly affected by variations in the expansion function because the cosmic expansion affects the geometry of space and thereby all distance measures.

We substitute the comoving distance $w$ in (1) by the scale factor $a$ as an integration variable and find

Aiming at analyzing the impact of variations in the expansion function $E(a)$ and the growth factor $D_{+}(a)$ on this shear power spectrum, we can restrict ourselves to the integral in (2) because the prefactors are constants. Thus, our main object of interest is the integral

containing the geometrical weight function

Our uncertainty analysis proceeds in the three steps enumerated below. Within each step, we consider different angular wave numbers $\ell$ and compare results for linearly and non-linearly evolving density-fluctuation power spectra $P_{\delta}$.

1. 1.

   As motivated in the introduction, we wish to answer the question: If we vary the cosmic expansion function by $\Delta E(x)$ at a certain scale factor $x$, which we henceforth call the perturbation scale factor, how does the integral $I(a_{\mathrm{s}},\ell)$ change relatively? Moving the perturbation scale factor $x$ from $a_{\mathrm{s}}$ to unity then results in a relative uncertainty distribution for fixed $a_{\mathrm{s}}$ and $\ell$, depending on $x$, which can be calculated using Gaussian error progression,

   $$R(x,a_{\mathrm{s}},\ell):=\frac{\Delta I_{x}(a_{\mathrm{s}},\ell)}{I(a_{\mathrm{s}},\ell)}=\left|\frac{\delta\ln I(a_{\mathrm{s}},\ell)}{\delta E(x)}\right|\cdot\Delta E(x)\;.$$

   (5)

   We shall refer to $R(x,a_{\mathrm{s}},\ell)$ as the two-dimensional uncertainty distribution since we shall later fix $\ell$ and vary $x$ and $a_{\mathrm{s}}$. The function $R(x,a_{\mathrm{s}},\ell)$ answers the question: What is the relative change of the weak-lensing power spectrum $C_{\ell}^{\gamma}$ in response to a change in the expansion function somewhere between the source and the observer?

   Variations of the expansion function $E(a)$ also cause variations of the growth factor $D_{+}(a)$. Possible additional systematic uncertainties in $D_{+}$ are negligible in comparison. Thus, the uncertainties $\Delta D_{+}$ are dominated by the uncertainties $\Delta E$, propagated through the differential equation (19) for linear density fluctuations. Thus, $\Delta D_{+}$ depends linearly on $\Delta E$ with the functional derivative $\delta D_{+}/\delta E$ as the slope, which will be calculated in Sect. 2.2.3. Thus, the uncertainties in $D_{+}$ are implicitly taken into account in (5), as we shall also see in Sect. 2.2.

2. 2.

   We additionally vary the scale factor $a_{\mathrm{s}}$ of the source and thus arrive at the two-dimensional relative uncertainty distribution $R(x,a_{\mathrm{s}},\ell)$ at fixed $\ell$. We restrict our analysis to sources at scale factors $a_{\mathrm{s}}\in[1/3,2/3]$, corresponding to source redshifts $z_{\mathrm{s}}\in[0.5,2]$.

3. 3.

   Accounting for all perturbations along the line-of-sight, achieved by integrating the relative uncertainty distribution over all perturbation scale factors $x$, finally gives the total relative uncertainty for a given source at scale factor $a_{\mathrm{s}}$ as a function of the angular wave number $\ell$,

   $$\bar{R}(a_{\mathrm{s}},\ell)=\int_{a_{\mathrm{s}}}^{1}\text{d}x\,R(x,a_{\mathrm{s}},\ell)=\int_{a_{\mathrm{s}}}^{1}\text{d}x\,\frac{\Delta I_{x}(a_{\mathrm{s}},\ell)}{I(a_{\mathrm{s}},\ell)}\;.$$

   (6)

   We shall refer to this function as one-dimensional since we vary $a_{\mathrm{s}}$ at fixed angular scale $\ell$.

This numerical evaluation needs the following inputs:

• •

  the expansion function $E(a)$ and its derivative as well as the growth factor $D_{+}$;

• •

  the power spectrum $P_{\delta}$ of matter-density fluctuations;

• •

  the matter-density parameter today, $\Omega_{\mathrm{m0}}$; and

• •

  the exponent $\gamma$ of the logarithmic derivative of the growth factor.

With these ingredients, we can evaluate the functional derivative $\delta D_{+}(a)/\delta E(x)$ solving (25) with standard integration routines, in our case taken from the GNU Science Library. The derivatives $\alpha(k,a)$ and $\kappa(k,a)$ of the power spectrum with respect to the scale factor $a$ and the wave number $k$, defined in (15) and (17), also need to be calculated numerically except for the derivative of the linear power spectrum with respect to $a$.

It is interesting to consider an Einstein-de Sitter universe as a reference cosmology. It contains only matter with critical density. Since all the input functions are then known analytically, we can compare the analytic solution for the functional derivative in such a model universe with the numerical results. The characteristic functions in this universe are

Using these, we find from (26)

for the functions $g(x),h(x)$ and $\Gamma(x,a)$ and, from (28), the functional derivative of the normalised growth factor becomes

This analytic result is shown in the top panel of Fig. 4. It is opposed there to the numerical result for an Einstein-de Sitter universe in the centre panel for comparison, and to the functional derivative of $D_{+}$ obtained by evaluating (28) numerically as described in the following subsection, shown in the bottom panel. The analytical and the numerical results for the Einstein-de Sitter universe are identical, thus confirming our numerical implementation. Despite the difference in the cosmological background evolution between the left and central panels compared to the right panel, the results are quite similar.

With the functional derivative $\delta D_{+}(a)/\delta E(x)$ of the growth factor with respect to the empirically reconstructed expansion function at hand, we can now numerically evaluate the functional derivative of the integral $I(a_{\mathrm{s}},\ell)$ from (3) with respect to the expansion function $E(a)$ and the uncertainty distributions $R(x,a_{\mathrm{s}},\ell)$ and $\bar{R}(a_{\mathrm{s}},\ell)$ defined in (5) and (6) based on the empirical expansion function $E(a)$ and its uncertainty $\Delta E(a)$. Note that $R(x,a_{\mathrm{s}},\ell)$ differs from the logarithmic derivative $\delta\ln I(a_{\mathrm{s}},\ell)/\delta E(x)$ by the factor $\Delta E(x)$, i.e. by a specific function quantifying the uncertainty of the expansion function. Thus, before we proceed to the uncertainty distributions $R$ and $\bar{R}$, we first show in Fig. 6 the logarithmic derivative $\delta\ln I(a_{\mathrm{s}},\ell)/\delta E(x)$ of the weak-lensing power spectrum with respect to the expansion function for sources at $a_{\mathrm{s}}=0.5$ or $z=1$ and for the three different angular wave numbers $\ell\in[200,1100,2000]$, corresponding to angular scales of $\theta\approx 180^{\circ}/\ell\in[54^{\prime},9.8^{\prime},5.4^{\prime}]$.

The figure illustrates the two interesting aspects that the logarithmic derivative of the shear power spectrum is larger for the linearly than for the non-linearly growing power spectrum, and that it flattens at perturbation scale factors $x\approx 0.7-0.8$ (i.e. at perturbation redshifts $z\approx 0.45-0.25$) as the angular scale shrinks, i.e. as $\ell$ increases. Both aspects are due to the non-linear growth, with substantially enhances the shear power spectrum and introduces a peak in the logarithmic derivative $\kappa(a,k)$ of the density-fluctuation power spectrum with respect to wave number, as shown in Fig. 1. For non-linearly growing structures, the functional derivatives vary little with both the perturbation scale factor and the angular wave number. On intermediate and small angular scales, it is $\approx 2-6$, showing that a relative change in the expansion function of, say, $1\,\%$ changes the shear power spectrum by $\approx 2-6\,\%$.

We then proceed to the uncertainty distribution $R(x,a_{\mathrm{s}},\ell)$ for $\ell=2000\mathop{\hat{\approx}}5.4^{\prime}$, which we show in Fig. 7. We focus on small angular scales because non-linear effects are important there (Bartelmann & Schneider, 2001; Schneider, 2006).

The two panels of Fig. 7 show results assuming a linearly (top) and a non-linearly (bottom) evolving power spectrum. For $x<a_{\mathrm{s}}$, the distributions vanish identically because then the perturbation of the expansion function precedes the emission by the source.

For linear structure growth, the distribution has a broad and shallow minimum at intermediate perturbation scale factors $x$ and grows for later-time perturbations, $x\to 1$, and with decreasing source distance, $a_{\mathrm{s}}\to 2/3$. Late-time perturbations affect cosmic distances and structure growth, and for less distant sources, the shear signal decreases. Both effects together cause the increase of $R$ for increasing source and perturbation scale factors. Non-linear structure growth reduces $R$ because it enhances the signal. The distributions for other angular scales look qualitatively very similar.

The relative uncertainty integrated over perturbation scale factors $x$, i.e. the one-dimensional relative uncertainty $\bar{R}(a_{\mathrm{s}},\ell)$, is shown in Fig. 8. Again, we show results there for the linearly and the non-linearly evolving power spectra, and for large and intermediate angular scales ($\ell=200$ and $\ell=1100$) in addition to small angular scales ($\ell=2000$). As before, the source scale factor $a_{\mathrm{s}}$ varies between $1/3$ and $2/3$, corresponding to source redshifts between $z_{s}=2$ and $z_{s}=0.5$, respectively.

For linear structure growth and intermediate to small angular scales, the total relative uncertainty increases from $\approx 10\,\%$ for sources at redshift $z_{s}\approx 2$ to $\approx 20\,\%$ for more nearby sources at redshift $z_{s}\approx 0.5$. The main reason for this increase is that the cosmological weak-lensing signal is getting weaker as the source redshift decreases. With the non-linearly evolving power spectrum, the uncertainty is somewhat smaller, increasing from $\approx 9\,\%$ for distant to $\approx 14\,\%$ for more nearby sources. This again reflects that the signal increases for non-linearly growing density fluctuations, reducing its relative uncertainty. The different shape of the curve for $\ell=200$ and non-linear structure growth is due to the transition of the spectrum from linear to nonlinear behaviour, quantified by the function $\kappa(k,a)$ defined in (17) and shown in Fig. 1. The solid curves in Fig. 8 show that the expansion function is empirically known well enough already to predict the cosmological weak-lensing power spectrum with a relative uncertainty of $\approx 10\,\%$, without assuming any specific cosmological parameters apart from $\Omega_{\mathrm{m0}}$.