Scalar cosmological perturbation

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.