Linear cosmological density perturbation

For pressureless subhorizon matter perturbations, a Fourier mode in conformal time obeyswhere $\mathcal{H}=a^{\prime }/a$.

In a flat matter-dominated universe with $a\propto t^{2/3}$ and $P\propto {\rho }^{4/3}$, the squared sound speed scales as $t^{-2/3}$. Each comoving Fourier mode then satisfies an Euler-Cauchy equation, with gravitational growth at long wavelengths and logarithmic-time acoustic oscillations at short wavelengths.

When the characteristic exponents of the density-mode Euler-Cauchy equation coincide, the second independent solution gains a logarithm. In the polytropic matter-era example the complete solution at this threshold is $t^{-1/6}(A+B\log t)$.

The peculiar gravitational potential is the Newtonian potential after subtracting the potential of the homogeneous expanding background. In comoving coordinates it obeys a Poisson equation sourced by the density contrast, so its gradient accelerates matter relative to the Hubble flow.

The density contrast is the fractional perturbation $\delta =(\rho -\bar{\rho })/\bar{\rho }$.

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as $a$, because ${\mathcal{H}}^2\propto a^{-1}$. Outside the Hubble radius, the constant $-2\Phi$ term dominates and is gauge dependent. The comoving density combination ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ cancels it for ${\theta }_c=2k^2\Phi /(3\mathcal{H})$, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.

The smoothed matter density variance measures the root-mean-square linear density contrast after applying a window corresponding to a Lagrangian mass scale. For a homogeneous isotropic field it is ${\sigma }^2=(1/(2{\pi }^2))\int k^{2}P(k)|W(kR){|}^{2}dk$. The variance convention, growth epoch and smoothing mass-density relation must agree with the collapse barrier used in a halo abundance calculation.

A spherical top-hat window function averages a field uniformly inside a sphere of radius $R$ and gives zero weight outside it. Its normalized real-space kernel is $W_R(\mathbf{x})=3/(4\pi R^3)$ for $|\mathbf{x}|\le R$, and its Fourier transform is

The linear growth factor multiplies the growing mode of a linear cosmological density perturbation, so $\delta (\mathbf{x},t)=D(t)\delta (\mathbf{x},t_0)/D(t_0)$ when scale-independent growth applies.

With physical matter and radiation densities, primordial amplitudes and spatial flatness fixed, extra smooth late dark energy increases $H_0$ but keeps ${\Omega }_m{H}_0^2$ fixed. Provided the dark energy is negligible at equality, the early cosmological transfer function and equality wavenumber in physical units are unchanged. Earlier accelerated expansion suppresses the linear growth factor, lowering present dimensional matter power by its squared growth ratio. Units involving $h$ can additionally relabel the horizontal and vertical axes.

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.

For pressureless matter on subhorizon scales, the linear growth factor obeysThe second term is dilution by cosmic expansion, while the final term drives Jeans instability.

Replacing the self-gravity term in the matter growth equation by $(1-f)4\pi G{\bar{\rho }}_{m}D$, with $0<1-f<1$, weakens clustering while leaving the background expansion unchanged.

During matter domination, the growing mode under a reduced clustering strength $1-f$ isIt grows more slowly than the standard $t^{2/3}$ mode whenever $0<f<1$.

In a matter-dominated universe, the long-wavelength pressureless growth equation has modes

When matter self-gravity is negligible during radiation domination, $a\propto t^{1/2}$ and $\ddot{\delta }+2H\dot{\delta }=0$ gives $\delta =C_1+C_2\log a$.

In a radiation-dominated background with matter self-gravity retained, ${\delta }_{aa}+{\delta }_a/a-3\delta /(2a{a}_{\mathrm{eq}})=0$. A change to $x=\sqrt{6a/a_{\mathrm{eq}}}$ gives the Modified Bessel differential equation of order zero. The Modified Bessel function of the first kind increases as $1+3a/(2a_{\mathrm{eq}})+\cdots$; the Modified Bessel function of the second kind decreases with a leading negative logarithm. The early solution space has the constant/logarithmic behaviors, not the $a$ growth of matter domination. This background approximation must not be extrapolated through equality.

During curvature domination $a\propto t$, so a pressureless perturbation without a self-gravity term has modes $1$ and $a^{-1}$. The surviving mode is constant.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys

The Jeans wavenumber separates pressure-supported modes from gravitationally unstable density modes.

Jeans instability is the gravitational growth of density perturbations whose self-gravity overcomes pressure support. For physical sound speed $c_s$, the comoving threshold isModes with $k\ll k_J$ grow gravitationally, whereas modes with $k\gg k_J$ undergo pressure-supported acoustic oscillations.

For the screened attraction $-Gm{e}^{-\alpha r}/r$ with $\alpha \ge 0$, $({\nabla }^2-{\alpha }^2)\varphi =4\pi G\rho$. A static homogeneous barotropic medium with sound speed $v_s>0$ has ${\omega }^2=k^2[v_s^2-4\pi G{\rho }_0/(k^2+{\alpha }^2)]$. A growing mode exists iff $4\pi G{\rho }_0>v_s^2{\alpha }^2$, with $0<k<\sqrt{4\pi G{\rho }_0/v_s^2-{\alpha }^2}$. For nonzero screening the growth rate tends to zero as $k\to 0$; the homogeneous mode is not an exponentially growing density fluctuation. Finite interaction range can therefore remove the instability entirely, while all sufficiently small nonzero wavenumbers remain unstable when the threshold is exceeded.

The Jeans swindle is the removal of the unperturbed uniform density's gravitational field when treating an infinite homogeneous Newtonian medium as static. Without such subtraction or an external supporting prescription, a nonzero constant density and zero background gravitational acceleration cannot satisfy Poisson's equation. Linear perturbations then obey ${\nabla }^2\delta \varphi =4\pi G\delta \rho$. This is a specified idealization, not an exact isolated infinite equilibrium.

In a static homogeneous universe, a density mode obeysModes with $k>k_J$ oscillate as sound waves, while modes with $k<k_J$ have one exponentially growing and one exponentially decaying solution.

Up to a convention-dependent numerical factor, a collisional fluid has comoving Jeans length ${\lambda }_{J,\mathrm{com}}=(c_s/a)\sqrt{\pi /(G\bar{\rho })}$.

Before cosmological recombination, photon-baryon sound speed controls the baryonic comoving Jeans length. The tightly coupled photon-baryon fluid has $c_s^2=c^2/[3(1+R_b)]$, where $R_b=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })\propto a$. Using total background density in the instantaneous dynamical Jeans estimate, the scale grows as $a$ during radiation domination, approaches a plateau in strongly baryon-loaded matter domination, and drops when photon decoupling removes photon pressure support. Subsequently, adiabatic monatomic-gas cooling gives $T_b\propto a^{-2}$ and a comoving Jeans length proportional to $a^{-1/2}$. Residual Compton scattering can delay that cooling law.

For collisionless matter, velocity dispersion replaces sound speed in the Jeans estimate. After free decoupling, a nonrelativistic thermal velocity dispersion redshifts as $a^{-1}$.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

Collisionless free streaming is the ballistic motion of particles out of an overdensity. It erases perturbations below the comoving distance travelled by particles while their thermal velocities are appreciable.

For a barotropic fluid, the squared sound speed is $c_s^2=\partial P/\partial \rho$. A Fourier density mode obeysin linear Newtonian perturbation theory.

The squared adiabatic sound speed is the ratio of background pressure and energy-density time derivatives. For a barotropic equation of state it is $dP/d\rho$. The non-adiabatic pressure perturbation is the difference $\delta P-c_a^2\delta \rho$; in a general field or multicomponent model this background quantity need not equal the rest-frame propagation speed.

If a smooth component dominates the expansion but does not cluster, pressureless matter still satisfiesAs ${\bar{\rho }}_m$ becomes negligible, Hubble friction leaves a constant mode and a decaying mode, so structure growth freezes.

When $H$ is constant and the matter source is negligible,so ${\delta }_m=C_1+C_2{a}^{-2}$. The growing matter-era mode therefore approaches a constant after cosmological-constant domination begins.

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives

The growing solution of the matter perturbation equation in a spatially flat matter-plus-cosmological-constant universe isnormalized so that $D_{+}(a)\sim a$ during early matter domination.

When $H\to H_{\Lambda }>0$, the growth integral has an integrable $a^{-3}$ tail. Its product with $H$ tends to a constant, with leading correction proportional to $a^{-2}$ in a matter-plus-Lambda background. Both displayed basis functions can tend to constants; a linear combination cancels that constant and isolates the truly decaying late-time $a^{-2}$ behavior.

Changing the lower limit in $H(a){\int }_{a_i}^{a}d\tilde{a}/(\tilde{a}H{)}^3$ adds a constant multiple of $H(a)$, the independent matter-era decaying solution. Thus a finite lower limit defines a useful independent solution but can include a decaying admixture. A pure growing normalization is selected by an appropriate linear combination.

If a perturbation has amplitude $A\ll 1$ at decoupling, it needs a growth factor of at least $A^{-1}$ before cosmological-constant domination freezes its growth. The order-of-magnitude requirement isup to the collapse threshold and the gradual transition encoded by the exact growth factor.

During radiation domination in a flat expanding universe, $a(t)\propto t^{1/2}$ and $H=1/(2t)$.

The Mészáros effect is the suppression of subhorizon cold-dark-matter growth during radiation domination. The growing solution is only logarithmic until matter-radiation equality.

In a flat universe containing pressureless matter and smooth radiation in cosmology, put $y=a/a_{\mathrm{eq}}$ and $C=H_0^2{\Omega }_{m,0}^2/{\Omega }_{r,0}$. The Friedmann equation gives ${\mathcal{H}}^2=C(1+y)/y^2$, so $y^{\prime 2}=C(1+y)$ and $y^{\prime \prime }=C/2$. The subhorizon cosmological Poisson equation gives ${\delta }^{\prime \prime }+\mathcal{H}{\delta }^{\prime }=3C\delta /(2y)$. The chain rule produces the displayed equation. Smoothing radiation neglects its rapid acoustic forcing after horizon entry; it is not an exact treatment of all radiation perturbations.

A basis of the solution space of a homogeneous linear differential equation for the Mészáros equation on $y>0$ isSubstitution verifies both solutions; their Wronskian is $-1/[y\sqrt{1+y}]$, so they are independent. At small $y$, $D_{+}=1+O(y)$ and $D_{-}=\log (4/y)-3+O(y\log y)$. At large $y$, $D_{+}\sim 3y/2$ and $D_{-}\sim 4/(15y^{3/2})$. Matching the actual early-time radiation forcing determines which constant and logarithmic combination is excited.

During matter domination in a spatially flat universe, pressureless matter controls the expansion, $a(t)\propto t^{2/3}$, and the growing linear density mode is proportional to $a$.

In a flat matter-dominated era with $a\propto {\tau }^2$, one has $\mathcal{H}=2/\tau$ andThe growing mode is proportional to the scale factor.

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeysIts growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

After neglecting the decaying mode, the linear growth between conformal times ${\tau }_1$ and ${\tau }_2$ is

A mode of comoving wavenumber $k$ and physical wavelength $2\pi a/k$ crosses a conformal horizon of physical size $ac\tau$ when

With $a(t_0)=1$, $k_0=2\pi /(c{t}_0)$, and $1+z_{\mathrm{eq}}=(t_0/t_{\mathrm{eq}}{)}^{2/3}$,

A mode crossing during matter domination acquires the growth factor $({\tau }_0/{\tau }_H{)}^2$. An initial amplitude proportional to ${\tau }_H^2{k}^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.

The density power spectrum records the squared Fourier-mode amplitude, up to the chosen statistical normalization. Scale dependence acquired from primordial amplitudes and subsequent growth determines its spectral shape.

If primordial horizon-crossing perturbations satisfy $V⟨|{\delta }_k{|}^2⟩=C/k^3$, remain frozen during radiation domination, and grow as $a$ during matter domination, then

Neutrino free streaming transports neutrinos out of small-scale overdensities. Massive neutrinos therefore contribute to the homogeneous matter density while clustering inefficiently below their free-streaming scale, slowing the growth of cold-matter perturbations.

For massless collisionless neutrinos in Newtonian gauge in cosmology, expand their temperature fluctuation as $\Theta (\mu )={\sum }_{\ell \ge 0}(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$, without a $(2\ell +1)$ factor. The Legendre polynomial recurrence relation givesThe lower-neighbour term is absent at $\ell =0$. Multipoles with the more usual $(2\ell +1)$ weighting equal ${\Theta }_{\ell }/(2\ell +1)$.