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The galaxy density contrast measures a fractional departure of galaxy number density from its ensemble or survey mean. Linear galaxy bias relates its large-scale three-dimensional component to the matter density contrast by ${\delta }_g=b_g{\delta }_m$. An angular sample averages this field with its normalized redshift selection; a probability density $p(z)$ per unit cosmological redshift corresponds to a distance kernel $H(z)p(z)/c$ in a spatially flat background.

The four contributions have distinct physical origins. The intrinsic photon temperature perturbation is ${\delta }_{\gamma }/4$: a local blackbody radiation energy density satisfies ${\rho }_{\gamma }\propto T^4$, so its first-order fractional perturbation is four times the fractional temperature perturbation. The endpoint cosmological Newtonian potential term ${\psi }_{\ast }$ is the gravitational redshift between the emitting plasma and the observer; escaping a negative potential well lowers the measured photon energy. Together with the intrinsic contribution it produces the ordinary Sachs-Wolfe effect. The term $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_{\ast }$ is the Doppler CMB anisotropy, caused by the plasma's line-of-sight peculiar velocity at last scattering, with the source's direction convention.

Finally, $2\int d\chi {\psi }^{\prime }$ is the Integrated Sachs-Wolfe effect. A photon passing through a static potential well gains energy on entry and loses it on exit, with no net shift. When the potential evolves while the photon crosses, the two shifts fail to cancel. The factor two is the sum of the temporal and spatial scalar-potential contributions when anisotropic stress is negligible, so the two potentials are equal. The prime is a partial derivative at fixed comoving coordinate; it is not the total derivative along the light ray.

For a low-cosmological redshift galaxy sample, the intrinsic, endpoint-potential and last-scattering Doppler CMB anisotropy terms are localized near $z\simeq 1100$, whereas the galaxy density contrast is measured much nearer us. In the Limber approximation their radial kernels have no overlap, so their leading cross correlations vanish. The late Integrated Sachs-Wolfe effect samples the same nearby evolving potentials as the galaxies, giving the desired late-time CMB-matter cross correlation. The early Integrated Sachs-Wolfe effect near matter-radiation equality is likewise radially separated.

This is the leading linear, separated-kernel description intended here. The primary Cosmic microwave background anisotropy and late density contrast share primordial initial conditions, so their exact large-angle correlation need not be identically zero outside the Limber approximation. Additional secondary effects, such as the Sunyaev-Zeldovich effect, can also correlate with galaxies in real maps; they are absent from the stipulated linear expression.

Let $p(z)$ be the normalized galaxy probability density per unit cosmological redshift, $\int p(z)dz=1$, and let $b_g$ be its linear galaxy bias. In a spatially flat background with $c=1$, $d\chi =dz/H(z)$. Thus the projected galaxy density contrast isso its radial kernel is $q_g=Hbp$. The cosmological Poisson equation in Fourier transform variables givesHere ${\Omega }_m$ denotes today's matter density parameter, not its time-dependent value. For a growing linear cosmological density perturbation, write $\delta (\mathbf{k},a)=D(a){\delta }_{\mathrm{ini}}(\mathbf{k})$, where $D$ is the linear growth factor, and define the logarithmic linear growth rate $f=d\ln D/d\ln a$. At fixed comoving coordinate,ConsequentlyIn particular, $f=1$ during matter domination makes the linear potential constant and eliminates this Integrated Sachs-Wolfe effect.

To use the supplied Limber approximation without hiding its scale dependence in a radial kernel, take $U=\delta$, $V={\psi }^{\prime }$, $q_g=Hbp$, and $q_T=2$. The corresponding three-dimensional cross-power spectrum iswhere $P$ is the matter power spectrum. Substitution givesThe Limber approximation relation $k\chi =\ell +1/2$ cancels ${\chi }^{-2}{k}^{-2}$, yieldingBoth powers of $H$ are necessary: one converts the redshift selection to a distance kernel, and the other differentiates the potential in conformal time. The factor two in the Integrated Sachs-Wolfe effect cancels the factor $1/2$ from the cosmological Poisson equation. For positive galaxy bias and $f<1$, the predicted correlation is positive: an overdense region whose negative potential becomes shallower gives a positive integrated temperature shift. The result assumes linear, scale-independent growth and the Newtonian cosmological Poisson equation; the Limber approximation becomes less accurate at the very lowest angular multipoles.