Both our BOSS×BOSS and FIRAS×BOSS models require the dark matter angular power spectrum of the overdensity field, $C_{\ell }^{\delta }(z,z^{\prime })$. We calculate this angular power spectrum with the Boltzmann code CLASS (Di Dio et al. 2013, 2014), using cosmological parameters inferred from the Planck 2015 (Ade et al. 2016) temperature and low-ℓ polarization maps (TT + LowP). The Halofit routine (Smith et al. 2003) provides nonlinear corrections to the power spectrum. However, on the several-degree scales of this analysis, the fluctuations are well-described by linear perturbation theory ($k_{\rm max} \sim \frac{\ell _{\rm max}}{\chi (z)_{\rm min}} \sim \frac{50}{880}$ h/Mpc ∼0.06 h/Mpc), and nonlinear corrections are small. CLASS computes the angular power spectrum from the 3D power spectrum, P(k), according to the equation
(11)$$\begin{eqnarray*}
C_{\ell }^{A\times B}(z,z^{\prime }) = \frac{2}{\pi } \int k^2 P^{\delta }(k,z{=}0) W_A^{\rm tot}(k,z)W_B^{\rm tot}(k,z^{\prime }) \mathrm{d}k,
\end{eqnarray*}$$where Pδ(k, z = 0) is the dark matter power spectrum at the current epoch, and, if there are no redshift space distortions (RSDs),
(12)$$\begin{eqnarray*}
W_A(k, z) = b_A \int \phi _z(z^{\prime \prime }) G(z^{\prime \prime },k) j_{\ell }[k \chi (z^{\prime \prime })] \mathrm{d}z^{\prime \prime },
\end{eqnarray*}$$where bA is the bias for dark matter tracer A, ϕz(z″) is a tophat redshift selection function that is non-zero only over the range of the redshift slice centered at redshift z and normalized to integrate to 1, jℓ is a spherical Bessel function of the first kind with parameter ℓ, G(z″, k) is the growth factor, and χ(z″) is the radial comoving distance to the shell at redshift z″. Linear RSDs can be included (Fisher, Scharf & Lahav 1994; Padmanabhan et al. 2007) by replacing WA(k, z) with $W_A^{\rm tot}(k,z)$, where
(13)$$\begin{eqnarray*}
W_A^{\rm tot}(k, z) &amp;= &amp;W_A(k, z) + W_A^{\rm RSD}(k,z)\nonumber \\
W_A^{\rm RSD}(k,z) &amp;=&amp;b_A \int \beta _A(z^{\prime \prime }) \phi _z(z^{\prime \prime }) \times \nonumber \\
&amp;&amp; \biggl \lbrace \frac{2\ell ^2+2\ell -1}{(2\ell +3)(2\ell -1)}j_{\ell }[k \chi (z^{\prime \prime })]\nonumber \\
&amp; &amp;- \frac{\ell (\ell -1)}{(2\ell -1)(2\ell +1)}j_{\ell -2}[k \chi (z^{\prime \prime })]\nonumber \\
&amp; &amp;- \frac{(\ell +1)(\ell +2)}{(2\ell +1)(2\ell +3)}j_{\ell +2 }[k \chi (z^{\prime \prime })] \biggr \rbrace \mathrm{d}z^{\prime \prime },
\end{eqnarray*}$$where βA(z) = f(z)/bA, with f(z) being the logarithmic growth rate of linear perturbations in the matter power spectrum. The $C_{\ell }^{A\times B}(z,z^{\prime })$ that results from using $W_A^{\rm tot}(k, z)$ and $W_B^{\rm tot}(k, z)$ contains cosmological terms proportional to bAbB, proportional to (bA + bB)/2, and independent of both bA and bB. We label these terms $C_{\ell }^{(2)}(z,z^{\prime })$, $C_{\ell }^{(1)}(z,z^{\prime })$, and $C_{\ell }^{(0)}(z,z^{\prime })$ respectively. They are calculated from CLASS via linear combinations of Cℓ(z, z′) computations without RSD and bias 1, with RSD and bias 1, and with RSD and bias 0. With this formalism, the cross-power spectrum of two biased matter tracers is given by
(14)$$\begin{eqnarray*}
C_{\ell }^{A\times B}(z,z^{\prime }) = b_A b_B C_{\ell }^{(2)}(z,z^{\prime }) + \frac{b_A {+} b_B}{2}C_{\ell }^{(1)}(z,z^{\prime }) + C_{\ell }^{(0)}(z,z^{\prime }). \nonumber \\
\end{eqnarray*}$$The bias dependence of these terms is reminiscent of the Kaiser correction in power spectrum space. Indeed, these equations can be derived by including the Kaiser enhancement term in a plane-wave expansion of the power spectrum and integrating along the line-of-sight (Padmanabhan et al. 2007).
Citation: Padmanabhan et al. 2007
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Citation: Padmanabhan et al. 2007
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