Photon temperature multipole (${\Theta }_{\ell }$)

For Fourier wavevector $\mathbf{k}$ and photon direction $\hat{\mathbf{p}}$, the temperature perturbation is expanded as

For a fixed angular order $m$, write $\Theta ={\sum }_{\ell }{C}_{\ell }{q}_{\ell }{Y}_{\ell m}$. Streaming by $ik\mu$ contributes $ik(C_{\ell +1}/C_{\ell })a_{\ell +1,m}{q}_{\ell +1}+ik(C_{\ell -1}/C_{\ell })a_{\ell ,m}{q}_{\ell -1}$. Thus rescaling the expansion coefficients also rescales the hierarchy couplings and source amplitudes. Replacing $C_{\ell }=(-i{)}^{\ell }\sqrt{(2\ell +1)/(8\pi )}$ by ${\tilde{C}}_{\ell }=(-i{)}^{\ell }\sqrt{2\pi /(2\ell +1)}$ requires ${\tilde{q}}_{\ell }=(2\ell +1)q_{\ell }/(4\pi )$; the temperature field is unchanged. This prevents mixing incompatible photon temperature multipole conventions.

Let $⟨F⟩=(4\pi {)}^{-1}\int Fd\Omega$ and $f=\bar{f}(ϵ)-ϵ{\bar{f}}^{\prime }(ϵ)\Theta (\mathbf{e})$ with comoving energy $ϵ=aE$. If ${ϵ}^4\bar{f}$ vanishes at both integration endpoints, integration by parts gives $-\int {ϵ}^4{\bar{f}}^{\prime }dϵ=4\int {ϵ}^3\bar{f}dϵ$. The kinetic stress-energy tensor then gives ${\delta }_{\gamma }=4⟨\Theta ⟩$, $v_{\gamma }^i=3⟨\Theta e^i⟩$, $\delta P=\bar{\rho }{\delta }_{\gamma }/3$, and ${\pi }^{ij}=4\bar{\rho }⟨\Theta (e^i{e}^j-{\delta }^{ij}/3)⟩$. If $\Pi =-\pi$ is used, the last expression changes sign. Frequency independence of $\Theta$ is essential to this common temperature description.

A direction-dependent fractional photon temperature change. For a thermal spectrum, $f(ϵ)=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ is its linear expansion at fixed comoving energy. For a nonthermal spectrum the same ansatz is a brightness dilation, rather than necessarily a thermodynamic temperature. In unweighted Legendre polynomial multipoles, $\Theta ={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }$, the photon density, velocity and anisotropic stress are ${\delta }_{\gamma }=4{\Theta }_0$, $v_{\gamma }=-{\Theta }_1$ and ${\Pi }_{\gamma }=-3{\Theta }_2/5$.

The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination ${\Theta }_0+\Psi$.

In Newtonian gauge, ${\Theta }_0+\Psi$ combines the intrinsic photon temperature perturbation with the gravitational redshift at emission. With constant potentials and negligible baryon loading it oscillates as $\cos (k{r}_s)$ for adiabatic initial conditions.

For a constant growing adiabatic mode with curvature convention $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/[4\pi G{a}^2(\bar{\rho }+\bar{P})]$, ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. The superhorizon photon continuity equation conserves ${\Theta }_0-\varphi$. With radiation initial value ${\Theta }_0=\mathcal{R}/3$, it gives ${\Theta }_{0,\mathrm{mat}}=2\mathcal{R}/5$ and $({\Theta }_0+\psi {)}_{\mathrm{mat}}=-\mathcal{R}/5$ when $\psi =\varphi$.

The Sachs-Wolfe effect is the large-angle CMB temperature anisotropy produced by intrinsic photon-temperature perturbations and gravitational redshift at last scattering. For an adiabatic mode during matter domination, ${\Theta }_0+\Psi =\Psi /3$ on superhorizon scales.

The Doppler CMB anisotropy is the line-of-sight velocity contribution from the last-scattering plasma. With the convention used here it contributes $-\hat{\mathbf{n}}\cdot {\mathbf{v}}_e$ to the observed fractional temperature perturbation.

The integrated Sachs-Wolfe effect is the CMB temperature shift produced when the gravitational potentials traversed by a photon evolve with time. Its line-of-sight source is proportional to ${\Phi }^{\prime }+{\Psi }^{\prime }$.

The Rees-Sciama effect is the microwave temperature shift produced by time-dependent gravitational potentials in nonlinear structure. A photon gains energy entering a potential well and loses it leaving; the shifts do not cancel when the well changes during transit. To leading order in its effect on the photon spectrum, this is a frequency-independent thermodynamic temperature shift, unlike the thermal Sunyaev-Zeldovich effect. A nonlinear potential can evolve even during matter domination, when linear growing-mode potentials are constant.

The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, $v_b=-3{\Theta }_1$ under the convention used here.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.