4 Relative strength of the B-mode

Whereas the presence of a B-mode, and an additional contribution to the E-mode due to source clustering must occur, one needs to estimate the relative amplitude of this effect as compared to the "usual'' cosmic shear strength described by $P_\kappa$. This estimate requires a model for the source clustering, i.e., a model for the function $g(w;\theta)$. g can be related to the three-dimensional correlation function $\xi_{\rm gg}$ of galaxies,

$$% g(w;\theta)=\int_{-\infty}^\infty {\rm d}(\Delta w)\; \xi_{\rm gg}\left( \sqrt{ (\Delta w)^2+f^2(w)\theta^2} \right).$$ (65)

If we assume that the correlation function behaves like a power-law, $\xi_{\rm gg}(r)=[r/r_0(w)]^{-\gamma}$, where r₀(w) is the comoving correlation length, and $\gamma\simeq 1.7$, then

$$% g(w;\theta)=C~r_0(w)\left( f(w)\theta\over r_0(w) \right)^{1-\gamma},$$ (66)

where $C={\sqrt{\pi}~\Gamma([\gamma-1]/2)/\Gamma(\gamma/2)}$, and $\Gamma(x)$ is the Gamma function. This yields for the angular two-point correlation function

$$% \omega(\theta)=\theta^{(1-\gamma)} C\int_0^{w_{\rm H}}{\rm ... ... p^2(w)~r_0(w)~\left( f(w)\over r_0(w) \right)^{1-\gamma}\cdot$$ (67)

A useful parameterization of $\omega(\theta)$ is $\omega=A(1') (\theta/1')^{(1-\gamma)}$. Fixing $\gamma=1.7$, one obtains a relation between the correlation length r₀(w) and the redshift distribution $\bar p(w)$,

 

$$% \int_0^{w_{\rm H}}{\rm d} w\;\bar p^2(w)~r_0(w)~\left( f(w)\over r_0(w) \right)^{-0.7} {=}1.65\times 10^{-5}~{A(1')\over 0.02} ,$$ (68)

where the fiducial value of A(1') was taken from McCracken et al. (2001). Note that in McCracken et al. (2001), essentially the same data set has been used as in the cosmic shear analysis of van Waerbeke et al. (2001); in particular, the depth of the data are the same. The above-quoted value for the angular clustering strength at¹1'corresponds to the faintest flux threshold considered in McCracken et al., which is very similar to the flux limit employed in the cosmic shear analysis. It must be mentioned, however, that the galaxies used in the cosmic shear analysis do not form a truly flux-limited sample, since additional cuts are used, e.g. a size cut. Hence, a precise estimate of the angular correlation function of those galaxies which were used for the cosmic shear analysis cannot be given.

For this reason, we shall assume the power-law dependence of $\omega(\theta)$ as given above; in addition, we will make the simplifying assumption that the comoving clustering length r₀(w) is independent of distance w; this assumption is not too critical, since the function $\bar p^2(w)$ is relatively well peaked and therefore large w-variations of the correlation length are not probed. Then, (69) determines this constant comoving correlation length r₀. We obtain in this case

$$g(w;\theta)=\omega(\theta){[f(w)]^{1-\gamma} \over \int{\rm d} w'~\bar p^2 (w')~[f(w)]^{1-\gamma} }\cdot$$

The power-law dependence of g on $\theta$ implies that $P_{\rm c}(\ell;\theta)$ also behaves like $\theta^{1-\gamma}$. Since the angular correlation function is small compared to unity, even on scales of a few arcseconds, we shall neglect $\omega(\theta)$ in the denominator of the integrand in (61); this greatly simplifies the calculation of the power spectra due to source clustering, since the $\theta$-integration can then be carried out first, making use of Eqs. (11.4.33, 34) of AS.

In order to make further progress, we need to assume a redshift distribution for the sources from which the shear is measured. We employ the form (Brainerd et al. 1996)

$$% \bar p_z(z)\propto z^2 \exp\left\lbrack -\left( z\over z_0 \right)^\beta \right\rbrack,$$ (69)

and shall consider $\beta=3/2$ in the following, yielding a mean redshift of $\left\langle z \right\rangle\approx 1.5 ~z_0$.

Figure 2: Dimensionless power spectra $\ell ^2 P(\ell )$, as a function of of wavenumber $\ell$. The solid curve corresponds to the power spectrum $\ell ^2 P_\kappa (\ell )$ that is the "standard'' power spectrum of the projected mass density. The dotted curve displays $\ell ^2 P_{\rm c}(\ell , 1')$, and the two dashed curves correspond to the E- and B-mode power caused by the source clustering. Here, a $\Lambda$CDM model was used, with shape parameter $\Gamma =0.21$, normalization $\sigma _8=1$, and the source redshift distribution is characterized by z0=1, yielding $\left\langle z \right\rangle\approx 1.5$. Other parameters for the model used here are mentioned in the text.

In Fig. 2 we show an example of the various power spectra considered here; all power spectra are multiplied by $\ell^2$ to obtain dimensionless quantities. For this figure, we employed a standard $\Lambda$CDM model with $\Omega_\Lambda=0.7$ and $\Omega_0=0.3$ and normalization $\sigma _8=1$. Sources are distributed in redshift according to the foregoing prescription, with z₀=1. The amplitude of the angular correlation function of galaxies was chosen to be A(1')=0.02, and the slope of $\gamma=1.7$ for the three-dimensional correlation function was used; the corresponding correlation length in this case is $r_0\approx 4.7~h^{-1}~{\rm Mpc}$. To calculate the three-dimensional power spectrum and its redshift evolution, we used the Peacock & Dodds (1996) prescription for the non-linear evolution of $P_\delta(k,w)$.

The power spectrum of the projected mass density, $P_\kappa(\ell)$, is the same as that in SvWJK, except for a slightly different choice of the cosmological parameters. The spectrum $P_{\rm c}(\ell;1')$ is very much smaller than $P_\kappa(\ell)$, as expected from the smallness of the amplitude of the angular correlation function; in fact, the ratio $P_{\rm c}(\ell;1')/ P_\kappa(\ell)$ is nearly constant at a value of approximately (1+B)A(1'), with $B\approx 1.2$ for this choice of the parameters.

The behavior of the power spectra which arise from source clustering, $P_{\rm cE}$ and $P_{\rm B}\equiv P_{\rm cB}$, as a function of $\ell$is quite different. First, both of these spectra are very similar, which is due to the fact that the J₄-term in (61) is much smaller than the J₀-term. Second, although both of these spectra are small on large angular scales, i.e. at small $\ell$, their relative value increases strongly for smaller angular scales. Hence, as expected, the relative importance of source clustering increases for larger $\ell$. What is surprising, though, is that these power spectra have the same amplitude as $P_\kappa$ at a value of $\ell\sim 6.7\times 10^5$, corresponding to an angular scale of $\theta = 2\pi/\ell\sim 2''$, and the relative contribution of the B-mode amounts to about 2% at an angular scale of 1'. It should be noted here that cosmic shear has already been measured on scales below 1'; therefore, source clustering gives rise to a B-mode component in cosmic shear which is observable.

We shall now consider the behavior of $P_{\rm B}(\ell)$ for large values of $\ell$. The aforementioned properties of $P_{\rm c}(\ell;\theta)$ can be summarized as

$$P_{\rm c}(\ell;\theta)\approx (1+B) A(1')~P_\kappa(\ell) \left( \theta\over 1' \right)^{1-\gamma}\cdot$$

Inserting this result into (61), neglecting the J₄-terms and considering the limit $\ell\to \infty$ yields

$$\ell^2P_{\rm B}(\ell)\approx a_1~ B~ A(1')~ (\ell 1')^{\gamma-1} \int{\rm d}\ell~\ell~P_\kappa(\ell),$$

where $a_1=2^{1-\gamma}\Gamma[(3-\gamma)/2]/\Gamma[(\gamma-1)/2] \approx 0.335$ for $\gamma=1.7$. Hence, $P_{\rm B}\propto \ell^{\gamma-3}$ at large $\ell$. To obtain an approximate value for the integral in the preceding equation, we shall describe the power spectrum $P_\kappa$ by a simple function,

$$\ell^2 P_\kappa(\ell)\sim B_{\rm m}{ (\ell/\ell_{\rm m})^\alp... ...over \left\lbrack 1+(\ell/\ell_{\rm m})^2 \right\rbrack^\beta}$$

with $B_{\rm m}\sim 3.7\times 10^{-3}$, $\alpha\sim 0.7$, $\beta\sim 0.6$, and $\ell_{\rm m}\sim 7\times 10^3$. Using A(1')=0.02, this then yields

$$\ell^2 P_{\rm B}(\ell)\approx 1.5\times 10^{-4} \left( \ell\over 10^5 \right)^{0.7},$$

which is a reasonably good description of the result in Fig. 2 for large $\ell$. Furthermore, we can obtain the ratio $P_{\rm B}/P_\kappa$ in the limit of large $\ell\gg \ell_{\rm m}$, which yields

$${P_{\rm B}(\ell)\over P_\kappa(\ell)} \approx \left( \ell\over 4.9\times 10^5 \right)^{1.2},$$

and roughly predicts the correct crossing point between these two power spectra seen in Fig. 2.

Figure 3: For the same model as in Fig. 2, several correlation functions are plotted. The solid line shows $\xi _+(\theta )$; in fact, the correlation function $\xi _{\rm E+}$ cannot be distinguished from $\xi _+$ on the scale of this figure; their fractional difference is less than 1%, even on the smallest scale shown. The two B-mode correlation functions are shown as well as $\xi _-$ and $\xi _{\rm E-}$. Note that the difference between the latter two is larger than that of the corresponding "+''-correlation functions.

In Fig. 3 we have plotted several correlation functions; they have been calculated from the power spectra plotted in Fig. 2 by using (20). The first point to note is that $\xi _{\rm E+}$differs from $\xi _+$ by less than 1% for angular scales larger than 3''; hence, the relative contribution caused by the source correlation is even smaller than that seen in the power spectra. This is due to the fact that the correlation function is a filtered version of the power spectra, however with a very broad filter. This implies that even at small $\theta$ scales, the correlation function is not dominated by large values of $\ell$, where the contribution from source clustering is largest, but low values of $\ell$ contribute significantly. The influence of source clustering on the "-'' modes is larger, since the filtering function for those are narrower (i.e., J₄(x) is a more localized function that J₀(x)), and $\xi _{\rm E-}$ differs from $\xi _-$ appreciably on scales below about 1'.

Finally, in Fig. 4 we have plotted the aperture measures. On scales below about 1', the dispersion of $M_\perp$ is larger than about 1% of that of $M_{\rm ap}$. Hence, the ratio of these E- and B-mode aperture measures are very similar to that of the corresponding power spectra.

Figure 4: Aperture measures, for the same model as used in Fig. 2. Shown here is the dispersion of the aperture mass, $\left\langle M_{\rm ap}^2 \right\rangle$, the corresponding function in the absence of source correlations (noted by the subscript "0'') and $\left\langle M_\perp^2 \right\rangle$, which is the aperture measure for the B-mode. As expected from the power spectra shown in Fig. 2, and the fact that the aperture measures are a filtered version of the power spectra with a very narrow filter function, the B-mode aperture measure is considerably smaller than $M_{\rm ap}$ itself.

The fact that $P_{\rm cE}$ and $P_{\rm B}$ are very similar in amplitude means that by measuring $P_{\rm B}$, one can make an approximate correction of $P_{\rm E}$, obtaining a value close to $P_\kappa$ by subtracting $P_{\rm B}$ from $P_{\rm E}$. Owing to the relative amplitude of these correlation-induced powers, such a correction may be needed in future high-precision measurements of the cosmic shear.