Codex @codex  0

In a measuring observer's orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor. Using the positive Euclidean spatial metric ${\delta }_{ij}$, defineThe trace here is the spatial Euclidean trace. Writing a Lorentzian mixed trace $T^{\hat{k}}{}_{\hat{k}}$ without its signature convention can reverse its sign in signature $(+---)$.

An isotropic perfect fluid in general relativity has $T^{\hat{i}\hat{j}}=P{\delta }^{ij}$ in its rest frame and therefore zero anisotropic stress. Different directional pressures or an angular quadrupole of particle momenta can give nonzero ${\pi }^{ij}$. In particular, the photon quadrupole determines the linear photon contribution.

Some linear cosmological perturbation theory conventions use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{\hat{i}\hat{j}}=P{\delta }^{ij}-{\Pi }^{ij}$. This sign must accompany a photon Euler equation; trace removal and the choice of sign are separate conventions.

In a spatially flat universe containing separately conserved pressureless matter and a coasting fluid with $w=-1/3$, the ratio $\eta ={\rho }_s/{\rho }_m$ is proportional to the scale factor. If the coasting component is modeled by the barotropic fluid perturbation equations with no anisotropic stress, its contribution to the trace gravitational source vanishes because $1+3w=0$. The matter growing mode isA decaying solution is $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$. The reduction of order formula with integrating factor ${\eta }^{3/2}\sqrt{1+\eta }$ produces the integral solution. It has $D=\eta [1-4\eta /7+O({\eta }^2)]$ during matter domination and tends to $5/2$ during coasting-fluid domination. Thus growth freezes rather than continuing in proportion to the scale factor. The background geometry is flat despite the same background expansion law as an open dust universe.

For scalar perturbations in cosmic time with $d{s}^2=-d{t}^2+a^2({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j$, define the covariant fluid velocity potential $u$ by $\delta u_i={\partial }_{i}u$. The linear momentum density is $T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_{i}u$. Normalize scalar anisotropic stress bywhere derivatives inside these parentheses are Cartesian comoving derivatives. Then ${\partial }_j{\pi }^j{}_i=2{\partial }_i{\nabla }^2\Pi /(3a^2)$. Spatial stress-energy conservation gives, for nonzero scalar Fourier modes,Writing ${\pi }^S=2\Pi /(3a^2)$ reproduces the form with ${\nabla }^2{\pi }^S$; the coefficient depends on that stress normalization. The time derivative acts on the entire momentum-density product, not only on $u$.

For pressureless matter with zero scalar stress, background $\dot{\bar{\rho }}+3H\bar{\rho }=0$ reduces the equation to $\dot{u}=0$. The physical peculiar velocity is $\mathbf{v}=\nabla u/a$ and therefore decays as $a^{-1}$. This checks the difference between covariant velocity potential and physical velocity and the expansion factor in the conservative equation.

For a barotropic perfect fluid with constant $w$, its cosmological adiabatic sound speed satisfies $c_s^2=w$. If $w<0$, the short-wavelength pressure term has the wrong sign for restoring oscillations: a mode grows as $\exp (\sqrt{-w}k\tau )$ when expansion can be neglected. For $w=-1/3$ in a coasting background with constant conformal expansion rate $K$, the exponents are $-K\pm \sqrt{K^2+k^2/3}$. This is a gradient instability, not ordinary positive-pressure Jeans instability. A network with tension or elastic anisotropic stress needs different constitutive perturbation equations; a negative background equation of state alone does not establish its perturbation sound speed.

The Newtonian limit of general relativity describes slowly moving matter in a weak, approximately static gravitational field. With metric signature $(-,+,+,+)$ and $x^0=ct$, identify the Newtonian gravitational potential $\Phi$ byNeglecting time derivatives and terms of higher order in particle velocity divided by the speed of light, the spatial geodesic equation becomesThis recovers Newtonian gravitational acceleration rather than merely matching a metric coefficient.

For nonrelativistic matter, $T_{00}\simeq \rho c^2$ dominates the stress-energy tensor. With negligible anisotropic stress, the weak-field Einstein tensor gives $G_{00}\simeq 2{\nabla }^2\Phi /c^2$. Matching $G_{\mu \nu }=\kappa T_{\mu \nu }$ to the Poisson equationtherefore fixes $\kappa =8\pi G/c^4$. Setting $c=1$ gives the commonly written $h_{00}=-2\Phi$ and $\kappa =8\pi G$. Relativistic pressure, rapid time variation or strong gravitational fields requires terms omitted in this limit.

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.

The Friedmann-Lemaître-Robertson-Walker metric implements the cosmological principle: on the scales being modelled, the spatial slices have spatial homogeneity and cosmological isotropy. There is a family of comoving observers whose orthogonal slices share one cosmic time and one scale factor. Each spatial slice is a locally maximally symmetric space, with time-independent normalized $k$.

The matter is represented by a perfect fluid in general relativity, at rest relative to the comoving observers, with energy density $\varrho (t)$ and pressure $p(t)$ depending only on cosmic time. Its stress-energy tensor isThere is no preferred spatial direction, so there is no anisotropic stress or spatial heat flux in that rest frame. Gravity obeys the Einstein field equations; the equations in this paper contain no separate cosmological constant, although a component with $p=-\varrho$ can represent one. An equation of state relating $p$ and $\varrho$ closes the evolution equations. Spatial homogeneity and cosmological isotropy are idealizations of the cosmological background; they do not assert that individual local structures are uniform.

There is a sign error in the printed synchronous line-of-sight integrand. Direct contraction of the stated decomposition gives $h_{ij}^{\prime }{n}^i{n}^j=(h^{\prime }-h_S^{\prime })/3+(\hat{\mathbf{k}}\cdot \mathbf{n}{)}^2{h}_S^{\prime }$, with a plus sign in its last term. In the positive spatial-perturbation convention of part a, the redshift term is half this contraction. The printed minus sign cannot yield the requested Newtonian expression with the supplied transformations.

The consistent identity can be derived explicitly. Work mode by mode with $q=\mathbf{k}\cdot \mathbf{n}$, $E(\tau )=e^{iq(\tau -{\tau }_0)}$ along the unperturbed ray; the shift of the phase origin is immaterial. Put $A=(h-h_S)/6$. Part a gives $A=\Psi -\mathcal{H}T$, while $\Phi =-\mathcal{H}T-T^{\prime }$. Therefore $A^{\prime }={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. The corrected synchronous integral issince $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$. This is the photon redshift identity with positive spatial perturbations.

The supplied fluid shifts are ${\delta }_{\gamma }^N={\delta }_{\gamma }+4\mathcal{H}T$ and ${\mathbf{v}}_{\gamma }^N={\mathbf{v}}_{\gamma }+i\mathbf{k}T$. At emission, the synchronous intrinsic and Doppler terms are consequently $E_d[{\delta }_{\gamma }^N/4+{\mathbf{v}}_{\gamma }^N\cdot \mathbf{n}-\mathcal{H}T-iqT{]}_d$. Adding the lower integral endpoint cancels the velocity shift and leaves $-\mathcal{H}T-T^{\prime }=\Phi$. The upper endpoint is the local observer monopole/dipole $[T^{\prime }-iqT{]}_0$, removed by temperature calibration and the observer-rest-frame convention. With zero anisotropic stress, $\Phi =\Psi$, the observable higher-multipole result isAll fields are evaluated along the ray; the integral contains the time derivative at fixed spatial position. Retaining the printed negative angular term would leave the additional $-2q^2\int ETd\tau$, generally a gauge-dependent, nonzero term. It is therefore not repaired merely by discarding observer monopole and dipole.

The first term is the intrinsic photon temperature perturbation, important in acoustic compressions and rarefactions. The velocity term is the Doppler shift and is prominent near acoustic, roughly degree and subdegree, scales; it is suppressed on very large scales. The potential at emission is the ordinary Sachs-Wolfe effect; combined with the intrinsic term it gives the familiar large-angle adiabatic matter-era result $\Phi /3$. The final term is the Integrated Sachs-Wolfe effect, caused by changing potentials. It vanishes for constant matter-era potentials, receives an early contribution near radiation-matter transition, and has an important late large-angle contribution when curvature or dark energy causes potentials to evolve. The sign of the Doppler term follows the photon-propagation direction convention used in the ray; reversing that direction reverses it consistently.

A superhorizon density contrast is gauge dependent. The stated growth laws describe the growing adiabatic mode in the comoving-gauge density contrast (and the corresponding usual synchronous growing mode), not an arbitrary coordinate density perturbation. Work with a flat, single-fluid background with constant $w=p/\rho$, negligible anisotropic stress, $c=1$, and a comoving Fourier wavenumber $K$.

Here is a derivation from the linear gravitational constraints and evolution equation. For an adiabatic mode, the scalar gravitational potential $\Phi$ obeysPrimes denote conformal time derivatives. The Friedmann equation gives $a\propto {\tau }^{2/(1+3w)}$ and $\mathcal{H}=2/[(1+3w)\tau ]$, so the bracket vanishes. On superhorizon scales, $K\tau \ll 1$, the leading solutions are a constant potential and a decaying solution proportional to ${\tau }^{-(5+3w)/(1+3w)}$. Keeping the growing adiabatic branch therefore makes $\Phi$ time independent to leading order.

Combining the time-time and time-space linear Einstein constraints eliminates the coordinate velocity and gives the comoving density constraintThis is the relativistic comoving constraint, not a subhorizon Newtonian approximation. Since $\rho \propto a^{-3(1+w)}$, it implies $\Delta \propto (a^2{H}^2{)}^{-1}\propto a^{1+3w}$. Also $a\propto t^{2/[3(1+w)]}$ in cosmic time, whenceBoth epochs here mean their respective leading growing solution; matching through matter-radiation equality requires the full two-component evolution. The constant superhorizon curvature or potential is consistent with this growth because the comoving density perturbation carries the additional factor $(K/aH{)}^2$.

Write $\mathcal{H}=a^{\prime }/a$ for the conformal Hubble parameter and ${\theta }_N=i\mathbf{k}\cdot {\mathbf{v}}_N$ for the velocity divergence of a Fourier mode. Work in the synchronous gauge comoving with the cold dark matter, so ${\mathbf{v}}_C=0$. This remains consistent with its pressureless momentum equation ${\mathbf{v}}_C^{\prime }+\mathcal{H}{\mathbf{v}}_C=0$. Its density equation then gives ${\delta }_C^{\prime }=-h^{\prime }/2$.

For the string component, $1+3w_S=0$. Thus it makes no contribution to the trace gravitational source in the supplied perfect-fluid model. Substitute $h^{\prime }=-2{\delta }_C^{\prime }$ and $h^{\prime \prime }=-2{\delta }_C^{\prime \prime }$ in that trace equation to obtainThe absence of a pressure-gradient term reflects the zero cosmological sound speed of the cold matter.

For the $w_S=-1/3$ coasting fluid, the continuity and momentum equations becomeThe second equation follows by taking $i\mathbf{k}\cdot$ the vector equation; in particular, $i\mathbf{k}\cdot i\mathbf{k}=-k^2$, which fixes the pressure sign. Differentiate the first equation and use ${\theta }_S={\delta }_C^{\prime }-3{\delta }_S^{\prime }/2$:Eliminating ${\delta }_C^{\prime \prime }$ with the cold-matter equation givesThese are equations for the specified idealized barotropic string fluid. A cosmic string network with additional anisotropic stress need not obey the same pressure response.

The Jeans length is the physical wavelength at which the restoring pressure term and self-gravity balance. In the elementary static-fluid approximation,Wavelengths longer than ${\lambda }_J$ are unstable to self-gravitating collapse, whereas shorter ones have restoring sound oscillations. Expansion and relativistic effects modify this elementary criterion. Ideal cold dark matter has $c_s=0$, hence no positive pressure-supported Jeans length in these equations. For the specified negative-pressure barotropic string fluid, $c_s^2=-1/3$; there is no real restoring-pressure Jeans threshold.

For $\eta \gg 1$, the matter growth equation has leading termsThe gravitational term is one inverse power smaller than the derivative coefficients. Dropping it at leading order gives $({\eta }^2{D}_{\eta }{)}_{\eta }=0$, hence a constant mode and a decaying ${\eta }^{-1}$ mode. A more controlled check is possible: direct substitution gives $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$ as an exact solution. Reduction of order then gives the growing mode, normalized by $D_{+}/\eta \to 1$ at early times,The integral grows as $\eta +O(\log \eta )$, while the prefactor is ${\eta }^{-1}[1+O({\eta }^{-1})]$. Thuswith corrections of order $\log \eta /\eta$. The perturbation freezes because the matter fraction tends to zero while the coasting fluid controls the expansion.

To describe the requested cold-dark-matter transfer function, its normalization must be specified. The early matter-era Hamiltonian constraint, in the negative-expansion convention for $K$, gives $-4k^2\Psi /a^2-4H\kappa =16\pi G\delta \rho$. For the growing comoving dust mode, the continuity equation gives $\kappa ={\dot{\delta }}_C=H{\delta }_C$ in proper time. Since $16\pi G\bar{\rho }=6H^2$, this yields ${\delta }_C=-2k^2\Psi /[5(aH{)}^2]$. On early superhorizon scales $\Psi$ is the conserved curvature amplitude, up to the chosen overall sign, and $(aH{)}^2\propto a^{-1}$. This establishes the $k^{2}a$ initial growing behavior rather than assuming identical horizon-entry amplitudes at all epochs. For regular adiabatic modes outside the horizon during early matter domination, $A_k=\mathcal{A}{k}^2{\mathcal{R}}_{\ast }(k)$, with a $k$-independent constant $\mathcal{A}$ and primordial curvature amplitude ${\mathcal{R}}_{\ast }$. The matter equation above has no $k$, so at fixed late timeConsequently the density-per-curvature transfer is proportional to $k^2{D}_{+}$ on scales entering after equality. The conventional shape transfer, which factors out $k^2$ and the common growth factor, is a plateau, $T(k)\simeq 1$ on these scales. Comparing with a matter-only universe that continues growing as $\eta$ instead gives a common amplitude suppression $D_{+}/\eta \simeq 5/(2\eta )$, still independent of $k$. A decline caused solely by later horizon entry would contradict the supplied scale-independent matter equation: the regular synchronous growing solution already evolves outside the Hubble radius. This is the scale-independent matter suppression by a coasting fluid.

There is also a limiting entry scale. Since $a^2{\rho }_S$ is constant, define $k_{\ast }^2=(8\pi G/3)a^2{\rho }_S$. ThenA mode enters the Hubble radius only if $k>k_{\ast }$; $k\gg k_{\ast }$ enters during matter domination, whereas $k\downarrow k_{\ast }$ enters arbitrarily late. Modes $k\le k_{\ast }$ never enter in this ideal coasting future. The requested post-equality entrants occupy $k_{\ast }<k<k_{\mathrm{eq}}$. The following original diagram distinguishes the two transfer conventions and marks this restricted interval; it does not attempt to model modes entering during radiation domination.

The adiabatic initial conditions for separately conserved components have equal ${\delta }_N/(1+w_N)$. Therefore the radiation relation is replaced here byTo check dynamically, put $S={\delta }_S-2{\delta }_C/3$. At $k=0$, subtracting two-thirds of the matter equation from the string equation gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$. Selecting $S=0$ and no independent relative-velocity mode preserves the relation. Thus the growing adiabatic string density contrast follows the matter mode and is also approximately frozen at late times to leading order in gradients. This is the adiabatic matter-string density relation.

At late times $\mathcal{H}\to k_{\ast }$, while ${\Omega }_C$ and ${\delta }_C^{\prime }$ vanish. The homogeneous string equation becomesFor $k\gg k_{\ast }$, one solution grows rapidly with exponent approximately $k/\sqrt{3}-k_{\ast }$, instead of undergoing radiation-like acoustic oscillations. This is a gradient instability of a negative-pressure perfect fluid. For $k\ll k_{\ast }$, the growing exponent is only $k^2/(6k_{\ast })$, while the other exponent is approximately $-2k_{\ast }$. The leading zero-gradient approximation is frozen, but at fixed nonzero superhorizon $k$ a slow instability can accumulate over sufficiently long conformal time. The strict gradient limit and infinite-future limit therefore need not commute.

These are the apparent behaviors of the equations given. They do not establish rapid collapse of a realistic cosmic string network: tension, anisotropic stress and the constitutive sound response were excluded in replacing the network by this barotropic perfect fluid. Once the modeled string perturbation becomes nonlinear, the linear calculation itself ceases to apply.

A scalar cosmological perturbation has two scalar gauge freedoms. In Newtonian gauge in cosmology, these remove the scalar shift and spatial shear, leaving the lapse potential $\Phi$ and spatial-curvature potential $\Psi$:A perfect fluid has no scalar anisotropic stress. The traceless spatial Einstein field equations therefore set $\Phi =\Psi =\varphi$ for nonzero spatial Fourier modes, producing the line element in the question. Homogeneous perturbations include background and gauge changes and require separate treatment. The single potential describes the physical scalar perturbation in this gauge.

Retain zeroth and first spatial-gradient orders in the long-wavelength approximation in cosmology. Spatial Ricci curvature, $D_i{D}_{j}N$, $q^2$ and scalar anisotropic stress are of second gradient order. Put $A^i{}_j={\tilde{K}}^i{}_j$ and $A^2=A^i{}_j{A}^j{}_i$. The supplied Einstein field equations then reduce toThe scalar-field stress projections show that the momentum constraint is a first-gradient equation and must be retained. The square $A^2$ is not a spatial derivative and cannot yet be dropped merely by invoking the long-wavelength approximation in cosmology. The time-derivative dot on the trace-free evolution equation is visible in the PDF and missing from the TeX aid.

Using $NH=\dot{a}/a$, the last equation becomes ${\partial }_t(a^3{A}^i{}_j)\simeq 0$. HenceThe time-independent spatial tensor must also satisfy the momentum constraint and the symmetry inherited from $K_{ij}$. This is inflationary shear damping: expanding regions lose anisotropic expansion, without requiring the frozen spatial metric shape to vanish. The matter evolution, useful below, is $\dot{\varphi }=N\Pi$ and $\dot{\Pi }\simeq -N(3H\Pi +V_{,\varphi })$, the leading Klein-Gordon equation in each local region.

In the decoupled perfect fluid in general relativity limit, photons have $w=1/3$ and no anisotropic stress, while cold baryons have $w\simeq 0$ with a small pressure correction $c_b^2{\delta }_b$. The density equations express particle and energy conservation; Thomson scattering transfers momentum rather than leading-order rest-frame energy. The photon Euler equation consequently gains a drag toward the baryon velocity potential, and the baryon Euler equation gains the equal-and-opposite drag weighted by their inertias.

Let ${\Gamma }_T=a{n}_e{\sigma }_T>0$. With the velocity potential used in part (i), these forces are $-{\Gamma }_T({\theta }_{\gamma }-{\theta }_b)$ and $-R{\Gamma }_T({\theta }_b-{\theta }_{\gamma })$, with $R=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$. Multiplication by the photon and baryon enthalpies, $4{\bar{\rho }}_{\gamma }/3$ and ${\bar{\rho }}_b$, makes their sum zero. Note that this paper's $R$ is the reciprocal of the usual baryon loading parameter. Thomson scattering also relaxes the photon quadrupole, giving the $-{\Gamma }_T{\sigma }_{\gamma }$ term in the stipulated simplified hierarchy. This quadrupole is an anisotropic stress, omitted by a perfect-fluid model; finite relaxation creates Silk damping.

The printed baryon density sign and collision normalization are inconsistent with part (i). The density equation must be ${\delta }_b^{\prime }-k^2{\theta }_b=-h^{\prime }/2$. As a direct check, with $h^{\prime }=0$, nonzero common velocity, and initially ${\delta }_{\gamma }=4{\delta }_b/3$, the printed plus sign gives $({\delta }_{\gamma }-4{\delta }_b/3{)}^{\prime }=8k^2\theta /3$, violating the tight-coupling adiabatic condition immediately. Likewise division of the collision forces by $k^2$ belongs to neither this potential convention nor the usual common velocity-divergence convention; the physical relaxation rate is ${\Gamma }_T$, independent of $k$. Both printed drag terms still cancel in the weighted sum used below, so that particular algebraic combination is unaffected by this normalization error.

For a nonrelativistic distribution, $T^{ij}=\int d^{3}p{p}^i{p}^{j}f/E$ is down by $v^2$ relative to the rest-mass energy density, and the corresponding anisotropic stress is also suppressed. The streaming terms coupling successive angular moments are proportional to $kv$, and stresses from the velocity distribution disappear in the cold limit. Thus, on scales where velocity dispersion is negligible, density and bulk velocity suffice; the photon Boltzmann hierarchy has no analogous small-speed suppression. This argument needs a cold distribution or a controlled small-dispersion approximation. Collisionless nonrelativistic matter with appreciable dispersion or multistreaming need not be an exact perfect fluid, and near photon decoupling the higher photon moments cannot all remain negligible.

A small direction-dependent temperature change $T=\bar{T}(1+\Theta )$ shifts a thermal spectrum according to $\bar{f}(ϵ/(1+\Theta ))=\bar{f}(ϵ)-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$ at first order. Hence $\boxed{\Theta =\delta T/\bar{T}}$ is the angular photon temperature perturbation. For a general spectrum the same ansatz describes an energy-independent brightness dilation; a literal thermodynamic temperature interpretation additionally assumes a thermal shape without spectral distortions.

Use $E=ϵ/a$ in the tetrad stress-energy tensor. The background photon energy density isIntegration by parts gives $-\int dϵ{ϵ}^4{\bar{f}}^{\prime }=4\int dϵ{ϵ}^3\bar{f}$, assuming the boundary term vanishes. Thus each directional energy perturbation is four times its temperature perturbation. Writing $⟨\cdot {⟩}_{\Omega }=(4\pi {)}^{-1}\int d\Omega$, we haveIn the paper's convention for photon angular temperature moments, there is no $(2\ell +1)$ factor multiplying ${\Theta }_{\ell }$. Legendre polynomial orthogonality gives $⟨\Theta {⟩}_{\Omega }={\Theta }_0$ and $⟨e^i\Theta {⟩}_{\Omega }=-i{\hat{k}}^i{\Theta }_1/3$. Comparing with the stated density and velocity conventions yieldsFor completeness, $⟨e^i{e}^j{P}_2(\hat{\mathbf{k}}\cdot \mathbf{e}){⟩}_{\Omega }=({\hat{k}}^i{\hat{k}}^j-{\delta }^{ij}/3)/5$. The $(-i{)}^2$ quadrupole phase then gives $\boxed{{\Pi }_{\gamma }=-3{\Theta }_2/5}$. These moment relations are photon angular temperature moments and keep the anisotropic stress convention consistent with the subsequent hierarchy.

The collisionless Boltzmann equation is conservation of $f$ along the photon trajectory. To linear order, use the unperturbed trajectory $d\mathbf{x}/d\eta =\mathbf{e}$ when differentiating $\Theta$. Deflection of $\mathbf{e}$ is first order and multiplies the first-order angular dependence of $\delta f$, while the background has no angular dependence; it therefore contributes only at second order. Similarly, the energy derivative of $\delta f$ times the perturbed energy change is second order. ThusFor a nontrivial spectrum this identifies the linear Free-streaming photon Boltzmann equation:In Newtonian gauge in cosmology, substitute the specified scalar gravitational-redshift source to writeMultiply its angular average by four. Since $⟨e^j{⟩}_{\Omega }=0$, the photon angular temperature moments imply the photon continuity equationFor the photon Euler equation, multiply by $3e^i$ and average. The source is $-3⟨e^i{e}^j{⟩}_{\Omega }{\partial }_j\psi =-{\partial }^i\psi$, while the second angular moment isIt follows thatThe background energy density is spatially homogeneous, so no spatial derivative acts on $\bar{\rho }$. The minus sign of its anisotropic stress term follows from the PDF's $\Pi =-\pi$ convention. Switching to the conventional trace-free stress $\pi$ changes that term to a plus sign.

Use a local orthonormal tetrad and units in which the speed of light is one. A particle has four-momentum $p^{\hat{a}}=E(1,\mathbf{e})$ and moves with velocity $\mathbf{e}$. Thus its energy density contributes $E$, its momentum density contributes $E{e}^i$, and its momentum flux contributes $E{e}^i{e}^j$. Integrating the phase-space distribution function gives the kinetic stress-energy tensor. The measure $d^{3}p/E$ is the future mass-shell Lorentz-invariant phase-space measure, so the same expression transforms as a spacetime stress-energy tensor; fixed state-counting or polarization factors are understood to be included in $f$.

Define $⟨F{⟩}_{\Omega }=(4\pi {)}^{-1}\int Fd\Omega$. The photon momentum measure is $d^{3}p=E^{2}dEd\Omega$, so every component of the kinetic stress-energy tensor has a radial factor $E^{3}dE=a^{-4}{ϵ}^{3}dϵ$. An isotropic background has $⟨e^i{⟩}_{\Omega }=0$ and $⟨e^i{e}^j{⟩}_{\Omega }={\delta }^{ij}/3$. HenceThis radiation pressure equation of state requires isotropy and masslessness, not a thermal spectrum.

The distribution perturbation is $\delta f=-ϵ{\bar{f}}^{\prime }(ϵ)\Theta$. Assume finite energy density and endpoint behavior ${ϵ}^4\bar{f}(ϵ)\to 0$ at zero and infinity, as for the Planck photon distribution. Integration by parts then givesConsequently the perturbed energy density and momentum density areSince $q^i=(\bar{\rho }+\bar{P})v^i=(4\bar{\rho }/3)v^i$, the photon angular temperature moments give $\delta =4⟨\Theta {⟩}_{\Omega }$ and $v^i=3⟨\Theta e^i{⟩}_{\Omega }$.

The sign of anisotropic stress must be specified. Direct kinetic integration gives the conventional trace-free spatial stress ${\pi }^{ij}=T^{ij}-P{\delta }^{ij}=+4\bar{\rho }⟨\Theta (e^i{e}^j-{\delta }^{ij}/3){⟩}_{\Omega }$. The PDF uses the opposite sign, ${\Pi }^{ij}=-{\pi }^{ij}$, equivalently $T^{ij}=P{\delta }^{ij}-{\Pi }^{ij}$. With that convention, all requested moments areTaking the spatial trace also givesThe angular temperature description assumes $\Theta$ is independent of photon energy; independent spectral distortions would require additional energy-dependent moments.

A perfect fluid in general relativity has isotropic rest-frame pressure and no rest-frame heat flux or anisotropic stress. In units $c=1$ and signature $(-,+,+,+)$, let $u^a{u}_a=-1$, let $\rho$ be rest-frame energy density and $p$ rest-frame pressure. Its stress-energy tensor isThe energy measured with the fluid is $T_{ab}{u}^a{u}^b=\rho$. The orthogonal projection $h^a{}_b={\delta }^a{}_b+u^a{u}_b$ isolates spatial directions and gives isotropic projected stress $p{h}_{ab}$. A fluid moving relative to another observer can have nonzero momentum density even though its own rest-frame stress is isotropic.

Projecting stress-energy conservation along and perpendicular to the four-velocity givesThese are the comoving energy balance and relativistic Euler equation. The inertial factor is enthalpy density $\rho +p$. An equation of state is needed to close the evolution; relativistic dust sets $p=0$, while an isotropic radiation fluid has $p=\rho /3$. Perfect-fluid behavior does not require irrotational motion: the potential-flow action below describes a restricted subset.

In Newtonian gauge in cosmology, the Free-streaming photon Boltzmann equation is $\dot{\Theta }+\mathbf{e}\cdot \nabla \Theta =\dot{\varphi }-\mathbf{e}\cdot \nabla \psi$. Angular averaging and the photon angular temperature moments give the displayed continuity equation. Its dipole yields the photon Euler equation, with the sign of the anisotropic stress term determined by the stress convention.

Let $q=\mathbf{k}\cdot \mathbf{n}$, $E=e^{iq(\tau -{\tau }_0)}$ and $T=h_S^{\prime }/(2k^2)$. The synchronous-to-Newtonian transformation with positive spatial perturbations gives $(h^{\prime }-h_S^{\prime })/6={\Phi }^{\prime }+{\Psi }^{\prime }+T^{\prime \prime }$. Since $[E(T^{\prime }-iqT){]}^{\prime }=E(T^{\prime \prime }+q^{2}T)$, the displayed identity follows by integration. The emission endpoint combines intrinsic and velocity shifts into the ordinary Sachs-Wolfe effect; the local observer endpoint is a monopole plus a dipole. Without anisotropic stress the remaining integral is the Integrated Sachs-Wolfe effect, $2\int E{\Phi }^{\prime }d\tau$. The angular term in the synchronous contraction is positive for this metric convention.