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For a flat background dominated by one constant-$w$ barotropic equation of state, assume $\delta P=w\delta \rho$ and no scalar anisotropic stress. The Einstein field equations then reduce the common Newtonian-gauge potential to the stated Fourier transform equation, with $\mathcal{H}=2/[(1+3w)\tau ]$. For radiation the regular solution is $3{\Phi }_0(\sin x-x\cos x)/x^3$, $x=k\tau /\sqrt{3}$, and oscillates with a decaying envelope after sound-horizon entry. For pressureless matter the two solutions are a constant and ${\tau }^{-5}$. A non-adiabatic pressure perturbation would supply an additional source, so constant background $w$ alone does not justify the homogeneous equation.

A general scalar cosmological perturbation has four scalar metric functions: a lapse perturbation, a scalar shift, a spatial curvature perturbation and a scalar spatial shear. The two scalar functions in a cosmological gauge transformation can set the shift and shear to zero. For nondegenerate spatial modes this gives conformal Newtonian gauge, leaving the two potentials $\psi$ and $\varphi$. The scalar traceless spatial Einstein equation makes their difference proportional to scalar anisotropic stress. In general relativity, with no scalar anisotropic stress and the usual boundary conditions, $\psi =\varphi$. Pressure alone does not spoil equality; free-streaming relativistic species can do so through their anisotropic stress.

Write $\Psi =(\varphi +\psi )/2$, the Weyl lensing potential. A conformal rescaling leaves the unparameterized null geodesics unchanged, although the affine parameter changes. To first order, the conformal optical metric for scalar perturbations hasThe only nonzero connection coefficients at this order areHere spatial indices are raised with ${\delta }^{ij}$. An affine parameter $\lambda$ for this rescaled metric therefore obeysThe first line is also ${\eta }_{,\lambda \lambda }+2{\eta }_{,\lambda }d\Psi /d\lambda +2x_{,\lambda }^i{\eta }_{,\lambda }{\partial }_i\Psi =0$, since the derivative of the potential is along the ray.

Put $v^i=d{x}^i/d\eta$. The chain rule gives $x_{,\lambda \lambda }^i={\eta }_{,\lambda }^{2}d{v}^i/d\eta +v^i{\eta }_{,\lambda \lambda }$. Eliminating the affine acceleration givesThe extra spatial-gradient term inside the parentheses does not replace the spatial term already contained in the total derivative.

At reception, the null condition is $|v{|}^2=1+4{\Psi }_0$. With the specified unit propagation vector $e$, this impliesThe zero-order trajectory is $x^i=e^i(\eta -{\eta }_0)$. In every term already containing a first-order potential, we may substitute $v=e$. Defining the transverse projector $P_{\perp }^{ij}={\delta }^{ij}-e^i{e}^j$, the first-order geodesic equation becomesIntegrating with the reception position and velocity gives the first-order photon trajectory in scalar perturbations:Its first derivative at reception is $e^i+2e^i{\Psi }_0$, and differentiating twice gives the previous equation. For a first-order calculation, evaluating the integrals on the unperturbed ray is sufficient; shifting their evaluation points by the first-order displacement changes them only at second order. This is the Born approximation in cosmological lensing.

To fix the angular-deflection sign, define ${\chi }_s={\eta }_0-{\eta }_s>0$, $n_{\mathrm{obs}}=-e$, and $n_{\mathrm{src}}=x_s/|x_s|$. The latter is the unlensed coordinate direction to the source, while the former is its apparent direction measured by the observer. The longitudinal integral changes the radial distance, not the direction at this order. The transverse displacement isDividing by ${\chi }_s$ yieldsThis is the angular deflection kernel for cosmological lensing. It depends on the sum of the two metric potentials; setting them equal reduces it to the familiar twice-the-Newtonian-potential expression. With angular derivatives, the lensing potential is $2{\int }_0^{{\chi }_s}[({\chi }_s-\chi )/({\chi }_s\chi )]\Psi d\chi$, and its angular gradient is $\alpha$.

Gravitational lensing displaces apparent source positions and produces magnification and shear through angular derivatives of the deflection. Weak gravitational lensing of galaxy shapes probes the foreground mass distribution, including dark matter. Lensing also remaps the cosmic microwave background, smoothing its acoustic features and converting part of its E-mode polarization into B modes. Strong deflections can produce multiple images, though a single first-order ray calculation does not describe that regime completely.

For a single barotropic perfect fluid in general relativity, the perturbations obey $\delta P=w\delta \rho =w\bar{\rho }\delta$. This adiabatic closure is needed: constant background $P/\rho$ alone would not eliminate an independent entropy perturbation. The absence of scalar anisotropic stress allows the common potential used in Newtonian gauge in cosmology.

Substituting the density constraint into the pressure equation gives the gravitational potential evolution of a barotropic fluidSince $a\propto {\tau }^{2/(1+3w)}$, the conformal Hubble parameter is $\mathcal{H}=2/[(1+3w)\tau ]$. The bracket cancels identically, leavingFor a Fourier transform mode, ${\nabla }^2$ becomes $-k^2$.

During radiation domination, $w=1/3$ and ${\Phi }_k^{\prime \prime }+4{\Phi }_k^{\prime }/\tau +(k^2/3){\Phi }_k=0$. Set $x=k\tau /\sqrt{3}$ and write $\Phi =u/x$. The resulting equation is $u_{xx}+2u_x/x+(1-2/x^2)u=0$, so the two Spherical Bessel functions in the hint giveAt $x\ll 1$, the two solutions approach a constant and a mode proportional to ${\tau }^{-3}$. The regular adiabatic mode, normalized to its primordial potential, isAfter entry into the sound horizon, $x\gg 1$, the potential oscillates at cosmological sound speed $1/\sqrt{3}$ with envelope $x^{-2}\propto a^{-2}$. The Hubble radius and sound horizon differ by the sound-speed factor; outside the Hubble radius the regular potential is constant, while well inside it radiation supports acoustic oscillations.

During matter domination, $w=0$ gives ${\Phi }_k^{\prime \prime }+6{\Phi }_k^{\prime }/\tau =0$ at every wavenumber. HenceThe growing density mode has a constant potential both outside and inside the Hubble radius; the other potential mode decays. Pressureless matter has zero cosmological sound speed, so horizon entry does not produce the radiation acoustic decay. These formulas cover both independent solutions, while the subsequent sketches select the regular adiabatic growing mode.

Here primes mean conformal time derivatives, $d\tau =dt/a$, and spatial derivatives use comoving coordinates in Newtonian gauge in cosmology. Set ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$. The linear cosmological Euler equation gives ${\theta }_m^{\prime }=-\mathcal{H}{\theta }_m-{\nabla }^2\Phi$, while the continuity equation gives ${\theta }_m=-{\delta }_m^{\prime }+3{\Phi }^{\prime }$.

Differentiate the continuity equation and substitute:Thus the linear matter density equation in Newtonian gauge isThe common potential in these supplied equations assumes that the relevant scalar anisotropic stress is neglected. No Fourier-sign convention is needed for the derivation in position space.

Relativistic pressure gives the radiation fluid a sound speed $c_s^2=1/3$. Well inside the sound horizon, $k\gg \mathcal{H}$, radiation linear cosmological density perturbations undergo rapid pressure-supported oscillations instead of sustained gravitational growth. Their potential decays. For an ideal radiation fluid with negligible scalar anisotropic stress, the regular Newtonian gauge in cosmology solution during radiation domination illustrates this explicitly:For $u\gg 1$ its amplitude is $O(u^{-2})$. Thus radiation in cosmology remains essential to the background Friedmann equation, while its rapidly oscillating gravitational forcing can be smoothed out when following the slowly varying cold dark matter density mode. This is the approximation behind the Mészáros equation.

The important limitation is that “subhorizon” alone is not a pointwise theorem that $|{\Phi }_r|\ll |{\Phi }_m|$. Radiation can dominate the instantaneous potential during early radiation domination, and the forcing near horizon entry seeds the later logarithmic matter mode. We neglect that transient and rapidly oscillating forcing in the subsequent coarse-grained growth equation, not radiation's contribution to the expansion rate. Close to horizon entry, or for accurate oscillatory evolution, the coupled radiation and matter perturbation equations must be retained. For the slow matter potential, the metric time derivatives in part (a) are also suppressed relative to its spatial Laplacian by $O((\mathcal{H}/k{)}^2)$. The approximation smooths radiation perturbations while retaining radiation in cosmology in the background cosmic expansion.

For one Fourier mode, the trace-free differential operator in the Einstein field equations is $-k^2{Q}_{ij}$, with $Q_{ij}={\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3$. Part (ii) gives ${\Pi }_{ij}=(4/5){\bar{\rho }}_{\nu }{\Theta }_2{Q}_{ij}$. The two minus signs in the field equation therefore implyInsert ${\Theta }_2={\psi }_0{k}^2{\eta }^2/6$. During radiation domination, $a\propto \eta$, $\mathcal{H}=1/\eta$, and the Friedmann equation gives $8\pi G{a}^2\bar{\rho }=3{\mathcal{H}}^2$. ThusThese are the radiation-era adiabatic potentials with free-streaming neutrinos; photons are taken to have negligible scalar anisotropic stress.

To normalize the potentials, use $\bar{P}=\bar{\rho }/3$. The denominator in the comoving curvature perturbation becomesWith $\dot{\varphi }=0$, the conserved curvature therefore obeys $\mathcal{R}=-{\varphi }_0-{\psi }_0/2$. Combining it with the potential ratio givesAs a sign and normalization check, $f_{\nu }=0$ gives ${\varphi }_0={\psi }_0=-2\mathcal{R}/3$. Increasing the free-streaming radiation fraction increases ${\varphi }_0/{\psi }_0$ while preserving the supplied curvature convention. The TeX's malformed definition of $\mathcal{H}$ is restored by the PDF: $\mathcal{H}=\dot{a}/a$.

In the absence of scalar anisotropic stress, the traceless spatial Einstein field equations giveThe photon mass-shell condition $g_{\mu \nu }{P}^{\mu }{P}^{\nu }=0$, together with $p^2=g_{ij}{P}^i{P}^j$ and $ϵ=ap$, gives to first orderUse the time component of the geodesic equation, divide it by $d\eta /d\lambda =P^0$, and substitute the stated Christoffel symbols. Keeping the perturbed ratios $P^i/P^0=(1+\Phi +\Psi ){\hat{p}}^i$ where they multiply the background $\mathcal{H}$ terms makes those terms cancel. Comparing the result with the total derivative of $P^0=ϵa^{-2}(1-\Psi )$ yieldsThe first term is the time-dependent gravitational redshift and the second is the gravitational frequency shift along the photon direction.

In synchronous gauge in cosmology, $h_{0\mu }=0$, so the explicit metric terms vanish. Symmetry of the covariant energy-momentum tensor and the background metric implyHence the two expansion terms in part (c) combine to $3H\delta T^0{}_j$. Substituting the scalar velocity potential and scalar anisotropic stress definitions gives

Use the Fourier convention $f(\mathbf{x})={\int }_{\mathbf{k}}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$ with ${\int }_{\mathbf{k}}=\int d^{3}k/(2\pi {)}^3$. Neglect anisotropic stress and assume the pressureless velocity is irrotational, soFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the nonlinear continuity equation giveswhere the alpha mode-coupling kernel and its symmetrization areTaking the divergence of the Euler equation giveswith the symmetric beta mode-coupling kernel

In the Einstein-de Sitter universe, $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. At linear order the ansatz gives ${\tilde{\theta }}^{(1)}={\tilde{\delta }}^{(1)}$. At second order, the continuity and Euler equations becomewhere each integral includes the momentum-conserving measure above. Eliminating ${\tilde{\theta }}^{(2)}$ yieldswith the standard perturbation theory density kernel

For regular adiabatic initial conditions on superhorizon scales during radiation domination, the unweighted neutrino Boltzmann hierarchy gives ${\Theta }_1=\psi k\eta /2$ and ${\Theta }_2=\psi (k\eta {)}^2/6$. With $f_{\nu }={\bar{\rho }}_{\nu }/({\bar{\rho }}_{\nu }+{\bar{\rho }}_{\gamma })$ and neutrinos the only source of scalar anisotropic stress, the trace-free Einstein field equations gives $\varphi =(1+2f_{\nu }/5)\psi$. For the sign convention $\mathcal{R}=-\varphi -\psi /2$,

Using the unweighted neutrino Boltzmann hierarchy convention and ${\Pi }_{ij}=-(4{\bar{\rho }}_{\nu }/3){\Pi }_{\nu }({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$, the scalar scalar anisotropic stress is ${\Pi }_{\nu }=-3{\Theta }_2/5$. Only the quadrupole contributes, since the trace-free tensor $e_i{e}_j-{\delta }_{ij}/3$ has angular degree two. Different stress and multipole sign conventions change this displayed coefficient.

The average of the two conformal Newtonian gauge potentials controls scalar gravitational lensing. A conformal rescaling makes the spatial optical metric flat and its time component $1+4\Psi$ to first order. Vanishing scalar anisotropic stress makes $\varphi =\psi$ in general relativity, but lensing depends on their sum even when they differ.