Linear cosmological perturbation theory

This is the first-order sector of cosmological perturbation theory.

Linear cosmological perturbation theory expands the metric and matter fields to first order about a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric.

Mass conservation in a Lagrangian dust displacement gives $1+\delta =\det (I+{\nabla }_q\psi {)}^{-1}$, hence $\delta =-{\nabla }_q\cdot \psi$ to first order. The perturbing gravitational potential determines the longitudinal part of the force, so the displayed equation holds for the curl-free displacement under the usual homogeneous boundary convention. A divergence-free transverse displacement instead satisfies ${\ddot{\psi }}_T+2H{\dot{\psi }}_T=0$. Taking the divergence gives $\ddot{\delta }+2H\dot{\delta }-4\pi G\bar{\rho }\delta =0$ without assuming the full displacement is longitudinal. The background Poisson equation with positive source requires $\nabla {\Phi }_0=+4\pi G\bar{\rho }r/3$.

For $h_{ij}=\mathrm{diag}(L,0,0)$, the determinant is zero for every $L$, even when one perturbation component is arbitrarily large. Linearized metric theory instead requires the eigenvalues of $h$ relative to the background spatial metric to be small, equivalently all its components in a background orthonormal frame to be small. The determinant of $1+h$ then expands as $1+\mathrm{tr}h+O(h^2)$, and the volume element as $1+\mathrm{tr}h/2+O(h^2)$.

A cosmological gauge transformation changes the identification of a perturbed spacetime with its homogeneous background. Individual field perturbations can change without changing physical observables. At first order the metric change is the linear metric gauge-transformation law; matter scalars obey the corresponding background-shift rule.

For a nonzero Fourier mode in synchronous gauge, write $h_{ij}=h{\delta }_{ij}/3+h_s({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$. Under the passive displacement ${\xi }^{\mu }=(T,i{\hat{k}}_{i}L)e^{i\mathbf{k}\cdot \mathbf{x}}$, cancellation of spatial scalar shear requires $h_s+2kL=0$, and cancellation of the shift requires $kT-L^{\prime }=0$. The resulting cosmological Newtonian potential is $\Phi =(h_s^{\prime \prime }+\mathcal{H}{h}_s^{\prime })/(2k^2)$ and the spatial potential is $\Psi =-(h-h_s)/6-\mathcal{H}{h}_s^{\prime }/(2k^2)$. Both remain unchanged under residual synchronous-gauge freedom, for which $T^{\prime }+\mathcal{H}T=0$ and $L^{\prime }=kT$.

For the conformal metric with $g_{0i}=-a^2{\partial }_{i}B$ and trace-free scalar spatial shear $E_{ij}=({\partial }_i{\partial }_j-{\delta }_{ij}{\nabla }^2/3)E$, the passive coordinate change $\tilde{\eta }=\eta +T$, ${\tilde{x}}^i=x^i+{\partial }^{i}L$ gives the displayed rules and $\tilde{\varphi }=\varphi +\mathcal{H}T+{\nabla }^{2}L/3$. The combination $C=\varphi +{\nabla }^{2}E/3$ changes by $\mathcal{H}T$, while $B+v$ changes by $T$. Hence $\mathcal{R}=-C+\mathcal{H}(B+v)$ is the comoving curvature perturbation in this sign convention.

Density is a scalar field. Under a first-order time shift $\tilde{\tau }=\tau +T$, its background changes by ${\bar{\rho }}^{\prime }T$, so its perturbation changes by the negative of that amount. Spatial shifts act only at second order on a homogeneous background density.

For the passive convention $\tilde{x}=x+\xi$, expanding the tensor coordinate-transformation law around a fixed background gives this Lie derivative formula. The background time shift and Jacobian terms must both be retained. On an FLRW metric, the scalar lapse, shift and spatial potentials transform under two scalar coordinate functions.

A vector cosmological perturbation is the transverse-vector sector of linear metric and matter perturbations. For a Fourier mode with nonzero wavevector $\mathbf{k}$, transversality removes the component parallel to $\mathbf{k}$, leaving two helicity components. In a gauge with $d{s}^2=a^2[-d{\eta }^2+2B_{i}d\eta d{x}^i+{\delta }_{ij}d{x}^{i}d{x}^j]$, ${\partial }_i{B}_i=0$ describes the vector metric perturbation.

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.

An adiabatic mode is a long-wavelength physical perturbation whose leading local field configuration is generated by a residual gauge transformation. Its action on shorter modes produces a cosmological soft theorem or consistency relation.

An adiabatic perturbation obeys ${\Delta }_r=(4/3){\Delta }_c$. In a radiation background this makes the cold-matter density equation ${\Delta }_c^{\prime \prime }+{\Delta }_c^{\prime }/\tau =4{\Delta }_c/{\tau }^2$, whose growing solution is ${\tau }^2$. In a matter background it instead becomes ${\Delta }_c^{\prime \prime }+2{\Delta }_c^{\prime }/\tau =6{\Delta }_c/{\tau }^2$, with the same growing conformal-time exponent and a different scale-factor dependence.

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaître-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.

If the tensor metric perturbation is $h_{ij}=2E_{ij}$ and the polarization tensors have contraction $M_{ij}^{(p)}{M}^{(q)\ast ij}={\delta }_{pq}$, the quadratic action for each Fourier amplitude is $(M_{\mathrm{Pl}}^2/2)\int a^2(|{\psi }_{(p)}^{\prime }{|}^2-k^2|{\psi }_{(p)}{|}^2)$. Thus each helicity is normalized like a real massless scalar by the displayed rescaling. Its power per polarization is $(H_k/(2\pi M_{\mathrm{Pl}}){)}^2$. Factors of two in other tensor spectra depend on the definitions of the metric amplitude, the polarization norm and whether both helicities are summed.

A complex helicity-$+2$ plane wave along the $z$ axis has polarization contraction ${\sin }^2\theta e^{2i\phi }$. Its remaining tensor CMB line-of-sight source depends only on $\cos \theta$. Orthogonality of the azimuthal factors of spherical harmonics therefore restricts its temperature coefficients to $m=2$. A real perturbation also includes the conjugate mode.

For a matter-era superhorizon tensor with $h=h_{\mathrm{prim}}[1-(k\eta {)}^2/10+\cdots ]$, the tensor CMB line-of-sight source gives ${\Theta }_{22}\propto h_{\mathrm{prim}}(k{\eta }_0{)}^2$ and ${\Theta }_{32}/{\Theta }_{22}=-ik{\eta }_0/(3\sqrt{7})$ when ${\eta }_{\ast }\ll {\eta }_0$. The extra spatial phase makes the octupole one order smaller.

For a tensor cosmological perturbation $h_{ij}$, the time geodesic equation in a local orthonormal tetrad gives $d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$. The photon comoving energy removes the background redshift; the dot is a partial time derivative, whereas the energy derivative follows the ray.

For massless neutrinos with a temperature perturbation $\Theta$ and background energy density ${\bar{\rho }}_{\nu }$, the physical trace-free spatial stress-energy tensor isSome conventions define ${\Pi }^{ij}=-{\pi }_{\nu }^{ij}$ and correspondingly reverse the sign of the stress source in the gravitational-wave equation.

Projecting free-streaming neutrinos onto the tensor spherical harmonic givesHere $j_2$ is a Spherical Bessel function. The continuous value at zero follows by integrating $(1-{\mu }^2{)}^2$, whose integral is $16/15$.

For either tensor helicity, neutrino free streaming produces the memory equationThe neutrino tensor free-streaming kernel encodes the angular propagation of the perturbation. Energy transferred to neutrino directional anisotropy damps the oscillation amplitude of a gravitational wave after horizon entry. This effect was calculated by Weinberg.

A scalar cosmological perturbation transforms as a scalar under spatial rotations and can be represented using scalar metric potentials, density and pressure perturbations, and the longitudinal part of the fluid velocity.

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.

For the Newtonian gauge in cosmology with potentials $\psi ,\varphi$, photon comoving momentum $ϵ$ obeys $d\ln ϵ/d\eta ={\varphi }^{\prime }-\mathbf{e}\cdot \nabla \psi =-d\psi /d\eta +{\varphi }^{\prime }+{\psi }^{\prime }$. The homogeneous physical-energy redshift is separated by $E=ϵ/a$. The total derivative follows the unperturbed photon path at linear order.

Let $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$ and $\bar{\Pi }=\mathcal{D}\bar{\varphi }$. For negative-expansion extrinsic curvature of a spatial hypersurface, its trace perturbation is $\kappa$. With $\delta \Pi =\mathcal{D}\delta \varphi -\bar{\Pi }\Psi$, scalar evolution is $\mathcal{D}\delta \Pi -\Psi \mathcal{D}\bar{\Pi }+3H\delta \Pi -\kappa \bar{\Pi }-\Delta \delta \varphi +V_{,\varphi \varphi }\delta \varphi =0$. Combining background acceleration terms produces the standard lapse source $-\bar{\Pi }(\kappa +\mathcal{D}\Psi -3H\Psi )+2\Psi V_{,\varphi }$.

A comoving gauge in cosmology chooses hypersurfaces with zero scalar momentum density for the specified fluid or total system. For cold pressureless matter it can be implemented by taking the matter velocity potential to vanish. The choice of component must be stated in a multi-fluid system; density contrasts on this slicing can differ from those in a Newtonian or spatially flat gauge.

The uniform-density curvature perturbation is the spatial scalar curvature measured on density-defined hypersurfaces, with an explicitly chosen sign convention. If the spatial scalar potentials are $C,E$, one convention is $\zeta =-C+{\nabla }^{2}E/3+\mathcal{H}\delta \rho /{\bar{\rho }}^{\prime }$. Its two time-shift terms cancel under a cosmological gauge transformation. The density derivative must be nonzero. This is a different slicing from the comoving curvature perturbation, although the variables are related for regular adiabatic long-wavelength modes, with signs depending on convention.

For spatial metric $a^2(1-2\Phi ){\delta }_{ij}$, the combination $\zeta =-\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is invariant under a time displacement: $\Phi$ shifts by $\bar{N}HT$ and $\delta \rho$ by $3\bar{N}H(\bar{\rho }+\bar{P})T$. The opposite overall sign is also valid. The sum $\Phi +\delta \rho /[3(\bar{\rho }+\bar{P})]$ is not invariant. This is a useful convention check before applying superhorizon conservation of uniform-density curvature.

With the convention $\zeta =\Phi -\delta \rho /[3(\bar{\rho }+\bar{P})]$, energy conservation gives ${\zeta }^{\prime }=\mathcal{H}\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})+(\nabla \cdot \mathbf{v})/3$. The non-adiabatic pressure perturbation vanishes for adiabatic cosmological perturbations, and regular velocity gradients are suppressed on superhorizon scales. Curvature is then conserved to leading gradient order, allowing primordial initial conditions to be propagated between cosmological eras.

For $N=\bar{N}(1+\Psi )$, $g_{0i}=a^2{\partial }_{i}B$ and spatial scalar convention $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}-2{\partial }_i{\partial }_{j}E]$, a passive displacement $({\xi }^0,{\partial }^i\lambda )$ gives $\tilde{\Psi }=\Psi -{\dot{\xi }}^0-(\dot{\bar{N}}/\bar{N}){\xi }^0$, $\tilde{B}=B+{\bar{N}}^2{\xi }^0/a^2-\dot{\lambda }$, $\tilde{\Phi }=\Phi +(\dot{a}/a){\xi }^0$ and $\tilde{E}=E+\lambda$. These follow from subtracting the background metric Lie derivative.

For spatial scalar convention $g_{ij}=a^2[(1-2\Psi ){\delta }_{ij}-2E_{,ij}]$ and $g_{0i}=a^2{B}_{,i}$, a passive gauge displacement gives $\tilde{E}=E+\lambda$ and $\tilde{B}=B-\dot{\lambda }+{\bar{N}}^2{\xi }^0/a^2$. Moving from synchronous gauge to Newtonian gauge therefore requires $\lambda =-E_S$ and ${\xi }^0=-a^2{\dot{E}}_S/{\bar{N}}^2$. A scalar density transforms by $\delta \tilde{\rho }=\delta \rho -\dot{\bar{\rho }}{\xi }^0$, giving the displayed density contrast relation. Keeping the background lapse function explicit prevents cosmic-time and conformal-time factors from being confused.

For the negative extrinsic curvature convention and a shift vector entering the line element as $d{x}^i-N^{i}dt$, the trace-free scalar curvature perturbation is $\delta K^i{}_j{}_{\mathrm{TF}}=-({\partial }^i{\partial }_j-{\delta }_j^i\Delta /3)\chi$. Here spatial raised derivatives use $a^{-2}{\delta }^{ij}$, so $\Delta =a^{-2}{\nabla }^2$. Changing the sign of the metric shift vector changes the corresponding definition of $\chi$.

With negative extrinsic curvature convention, the background trace is $\bar{K}=-3H$. The scalar expansion perturbation is $\kappa =K+3H$. For lapse function $N=\bar{N}(1+\Psi )$, induced metric $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and shift vector $N_i=-a^2{B}_{,i}$, it is $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )-\Delta \chi$, where $\chi$ is the scalar shear potential. The expansion of the future normal congruence is $-K$, so its perturbation has the opposite sign.

For $N_i=a^2{B}_{,i}$, ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$, put ${\chi }_{+}=a^2(B-\dot{E})/\bar{N}$. Then $K^i{}_j=-H{\delta }^i{}_j+(\dot{\Phi }/\bar{N}+H\Psi ){\delta }^i{}_j+{\partial }^i{\partial }_j{\chi }_{+}$ and $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )+\Delta {\chi }_{+}$. Defining $\chi =-{\chi }_{+}$ instead makes the trace-free tensor coefficient negative and the trace gradient term $-\Delta \chi$. A sign change must affect both pieces.

Synchronous gauge sets the lapse and shift perturbations to zero, $h_{0\mu }=0$. Its coordinates follow freely falling observers, but residual coordinate freedom produces gauge modes that must be distinguished from physical perturbations.

Define $\delta g_{ij}=a^2{h}_{ij}$ and $h_{ij}=h{\delta }_{ij}/3+({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)h_S$. Under the passive coordinate shift $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$, removing the traceless spatial perturbation gives $L=h_S/(2k)$. Removing the mixed time-space metric gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$. The remaining diagonal metric components are the displayed Newtonian potentials. This sign convention is opposite to defining the synchronous perturbation by $g_{ij}=-a^2({\delta }_{ij}+h_{ij})$.

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.

For the FRW metric in conformal time, a displacement ${\xi }^{\mu }=(\alpha ,{\xi }^i)$ changes a raw metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of synchronous gauge in cosmology. Their homogeneous solutions leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.

For constant $P=w\rho$, the perfect fluid stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar{\rho }[(1+\delta ){\delta }^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. Stress-energy conservation yields the synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge; the pressure projection includes $v_i{\bar{P}}^{\prime }$, which changes the velocity damping from $\mathcal{H}$ to $(1-3w)\mathcal{H}$.

// Target: cosmology.bigb

For constant $P=w\rho$ and velocity potential $v_i=i{k}_i\theta$, stress-energy conservation gives the displayed density equation in synchronous gauge in cosmology. Here $h$ is the trace of the spatial metric perturbation, and primes denote conformal time. In particular cold baryons obey ${\delta }_b^{\prime }=k^2{\theta }_b-h^{\prime }/2$. A positive sign in front of $k^2{\theta }_b$ on the left contradicts this potential convention.

Preserving $\Psi =B=0$ under the scalar gauge transformations with a background lapse requires $\bar{N}{\xi }^0=C(\mathbf{x})$ and $\dot{\lambda }=C(\mathbf{x})\bar{N}/a^2$. Thus arbitrary spatial functions $C,D$ remain after imposing synchronous gauge in cosmology. This freedom can generate changes in the remaining scalar metric perturbations.

A residual synchronous time displacement ${\xi }^0=C(\mathbf{x})/a$ changes a fluid density contrast by $-{\bar{\rho }}^{\prime }{\xi }^0/\bar{\rho }$. For constant equation of state this gives the displayed mode: it scales as ${\tau }^{-2}$ during radiation domination and as ${\tau }^{-3}$ during matter domination. Such density changes alone do not indicate physical structure growth. Choosing the cold-matter rest frame fixes the residual time displacement, while a spatial relabeling fixes the remaining integration convention for the metric trace.

For scalar perturbations of a generic fluid in cosmic time, momentum conservation in synchronous gauge in cosmology gives

Newtonian gauge removes scalar shear and writes scalar metric perturbations in terms of gravitational potentials. The fluid Euler equation then contains an explicit gradient of the Newtonian potential.

The potential $\Phi$ describes the lapse perturbation in conformal Newtonian gauge. The spatial curvature perturbation is conventionally denoted $\Psi$; the two are equal when the linear scalar anisotropic stress vanishes. On scales where the Newtonian limit applies, $\Phi$ agrees with the peculiar gravitational potential. Its Fourier-mode variance and the power spectrum of a comoving curvature perturbation are different observables.

Rescale the perturbed cosmological metric by $a^{-2}(1+2\varphi )$. To first order, its spatial part is Euclidean and only its lapse is perturbed. The unparameterized null geodesics are unchanged. The nonzero linear connection coefficients are ${\hat{\Gamma }}_{00}^0=2{\partial }_{\eta }\Psi$, ${\hat{\Gamma }}_{0i}^0=2{\partial }_i\Psi$ and ${\hat{\Gamma }}_{00}^i=2{\partial }^i\Psi$. The affine parameter is that of the rescaled metric, not necessarily that of the original spacetime.

For reception at the origin with unit propagation direction $e$, the null geodesic condition fixes $v({\eta }_0)=(1+2{\Psi }_0)e$. The perturbative spatial acceleration is $x^{i\prime \prime }=2e^{i}d\Psi /d\eta -2({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi$. Its integrated trajectory is $x^i=e^i(\eta -{\eta }_0)+2e^i{\int }_{{\eta }_0}^{\eta }\Psi d{\eta }^{\prime }-2{\int }_{{\eta }_0}^{\eta }(\eta -{\eta }^{\prime })({\delta }^{ij}-e^i{e}^j){\partial }_j\Psi d{\eta }^{\prime }$. Potentials in these already first-order integrals can be evaluated on the straight background ray.

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.

For pressureless matter in conformal time and Newtonian gauge,Its linear curl implies that vorticity decays as $a^{-1}$.

For pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.

On subhorizon scales, the Newtonian potential sourced by the matter density contrast obeys

The longitudinal velocity perturbation of a cosmological fluid can be represented by a scalar velocity potential through $\delta T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_i\delta u$.

Scalar anisotropic stress is the trace-free scalar part of the spatial stress perturbation. Under a common convention it enters as $\delta T^i{}_j=\delta P{\delta }_j^i+{\partial }^i{\partial }_j{\pi }^S$.

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.

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 $\mathcal{R}=-\varphi -\psi /2$,

Flat gauge sets the scalar perturbation of the spatial metric to zero. Scalar fluctuations then reside in the lapse, shift, and matter fields, making the field fluctuation directly visible while the lapse and shift remain constrained variables.

A cosmological transfer function relates a late perturbation to a specified primordial perturbation mode by mode. It records the scale-dependent evolution caused by horizon entry, pressure, free streaming, and changes in the dominant cosmic component.

During matter domination with nearly constant gravitational potential, a radiation density perturbation obeys a forced acoustic equation. The observable Sachs-Wolfe combination is the homogeneous oscillatory part.

Pressure-supported radiation perturbations well inside the horizon oscillate rather than exhibit secular gravitational growth. In the ideal relativistic fluid, $c_s^2=1/3$ and the leading conformal frequency is $k/\sqrt{3}$. Their rapidly changing source has a small late-time response in the slowly varying cold-matter density mode.

Let ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

In deep radiation domination, an oscillatory radiation density source in the matter equation has order ${\Delta }_r/{\tau }^2$ and frequency of order $k$. Its rapidly varying particular response is therefore of order ${\Delta }_r/(k\tau {)}^2$. For $k\tau \gg 1$ it does not control the secular matter growth, though forcing near horizon entry sets the constant and logarithmic homogeneous amplitudes.

At fixed time after equality, the comoving cold-dark-matter density contrast per unit primordial curvature grows as $k^2$ for $k\ll k_{\mathrm{eq}}$ and only logarithmically for $k\gg k_{\mathrm{eq}}$. Small-scale modes entered during radiation domination and could grow only logarithmically until equality.

Assume all relevant modes begin outside the horizon at common conformal time ${\tau }_i$, grow as ${\tau }^2$ until radiation-era horizon entry, then freeze until matter-radiation equality, and again grow as ${\tau }^2$ in matter domination. With horizon entry ${\tau }_h=2\pi /k$, $k_{\mathrm{eq}}=2\pi /{\tau }_{\mathrm{eq}}$ and final time ${\tau }_0$, direct multiplication of the growth factors gives the displayed transfer with $G=({\tau }_0/{\tau }_i{)}^2$. A primordial density power spectrum $P_i=Ak$ becomes $P_0=A{G}^{2}k$ below equality wavenumber and $A{G}^2{k}_{\mathrm{eq}}^4{k}^{-3}$ above it. This approximation neglects the true logarithmic radiation-era matter growth, baryonic acoustic structure and late acceleration; a conventionally normalized transfer divides out $G$.

The matter equation in the barotropic matter-string model contains no wavenumber. Growing initial data therefore evolve as ${\delta }_c(k,\eta )=A_{k}D(\eta )$ with the same $D$ for every mode. For regular adiabatic modes initially outside the horizon during matter domination, $A_k$ is proportional to $k^2$ times the primordial curvature amplitude. The density-per-curvature transfer is consequently proportional to $k^{2}D(\eta )$ for modes entering after equality. Factoring out both $k^2$ and $D$ gives a constant shape transfer on those scales. Comparing with continued matter-era growth instead introduces the common suppression $D(\eta )/\eta$, not a new wavenumber dependence.

In the idealization that subhorizon cold matter freezes during radiation domination, early-entering modes have an entry density of order the primordial curvature, retain it until equality, and then share the same matter-era growth factor. Their density-per-curvature transfer becomes approximately independent of $k$. Matching this plateau to the matter-era $k^2$ transfer gives the displayed sharp-transition model, with its high-$k$ coefficient dependent on the matching convention. A scale-invariant curvature spectrum then gives density power proportional to $k$ below equality and $k^{-3}$ above it. Actual cold matter grows logarithmically in the radiation era, adding the familiar high-$k$ logarithms of the Mészáros effect.

For the regular growing mode in a purely matter-dominated flat background, $h=-2{\delta }_c$ and ${\delta }_c=C{\tau }^2$. The synchronous Einstein constraint $\mathcal{H}{h}^{\prime }+k^2{h}^{-}/3=3{\mathcal{H}}^2{\delta }_c$ with $\mathcal{H}=2/\tau$ gives $h^{-}=60C/k^2$. Hence $\zeta =h^{-}/6+{\delta }_c/3=10C/k^2+C{\tau }^2/3$, conserved to leading order on superhorizon scales. Identifying its early constant value gives the displayed density transfer; it includes a $k^2$ factor that is factored out of the conventionally normalized matter transfer function.

The dimensional matter power spectrum contains a factor $k^{n_s}$, rather than $k^{n_s+3}$. The cosmological transfer function tends to one below the matter-radiation equality scale and to $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ above it. The smooth spectrum therefore scales as $k^{n_s}$ at small wavenumber and $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$ at large wavenumber. The logarithmic factor records slow cold-matter growth during radiation domination. Multiplying by $k^3$ instead produces dimensionless matter power.