Jeans instability

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 a matter-dominated fluid with constant $c_s^2=w\ll 1$, combine stress-energy conservation in synchronous gauge with the scalar trace Einstein equation. Discard pressure corrections to the background gravitational and velocity-damping terms while retaining the potentially large pressure-gradient term $w{k}^2\delta$. This produces the displayed leading nonrelativistic equation. It is not an exact equation at finite $w$. At negligible pressure the growing and decaying matter-era solutions are ${\tau }^2$ and ${\tau }^{-3}$.

// Target: cosmology.bigb

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

During radiation domination, $a\propto \tau$ and $8\pi G{\rho }_R{a}^2=3/{\tau }^2$. The adiabatic initial conditions reduce the CDM density equation in a matter-radiation universe to ${\delta }_C^{\prime \prime }+{\delta }_C^{\prime }/\tau -4{\delta }_C/{\tau }^2=0$. Its Euler-Cauchy equation powers are two and minus two. The growing solution is the displayed one, in the stated CDM-comoving synchronous gauge.

In the leading radiation domination limit, neglect $k^2{\delta }_r$ and the matter contribution to gravity. The radiation equation gives $S^{\prime \prime }=0$. Substitution into ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau =3{\delta }_r/{\tau }^2$ givesThus adiabatic initial conditions select both $S_0=0$ and $S_1=0$, leaving the growing and decaying power-law modes. The four constants describe the leading supplied coupled system; regularity and additional initial Einstein field equations constraints can further restrict their physical interpretation.

With physical wavelength $2\pi /k$, the Jeans length is $2\pi /k_J$, where the Jeans wavenumber is $\sqrt{4\pi G\rho }/c_s$. Linear pressure support dominates gravity below this length, whereas longer wavelengths can grow. In an expanding background the growth also depends on the Hubble damping term, so the static threshold does not by itself give the time dependence.

When baryons are a small fraction of the gravitating matter, their perturbation is forced by the cold dark matter potential. Comparing pressure frequency $c_s^2{k}^2$ with $4\pi G{a}^2{\bar{\rho }}_C$ yields the scale ${\lambda }_J=c_s\sqrt{\pi /(G{\bar{\rho }}_C)}$ for following the growing matter mode. It is not an absolute statement that every sub-Jeans baryon perturbation is zero or cannot be forced: a slowly varying response is ${\delta }_B\simeq (k_J^2/k^2){\delta }_C$. In the pressureless limit the difference from CDM is a constant plus a decaying mode during matter domination, so baryons catch up in relative amplitude.

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.

Suppose a collisionless particle has momentum $p=T_r\propto a^{-1}$ during pure radiation domination. Since $a\propto t^{1/2}$, integrating $\chi ={\int }_0^{t}v(t^{\prime })d{t}^{\prime }/a(t^{\prime })$ gives the displayed comoving distance. The physical distance at observation is $a\chi =H^{-1}\mathrm{arsinh}(y)/y$, $y=m/T_r$. The printed expression without $a^{-1}$ is consequently a physical distance, or a comoving distance in coordinates normalized to $a=1$ at observation. Its ratio tends to one in the relativistic limit and to $\log (2y)/y$ in the nonrelativistic limit. The accumulated scale differs from an instantaneous velocity-over-expansion-rate estimate.

Spatial translation symmetry in a flat FRW metric conserves the covariant canonical momentum $q_i=m{a}^{2}d{x}^i/ds$ of a freely falling massive particle. An orthonormal spatial frame gives $p_{\mathrm{phys},i}=q_i/a$, so physical momentum redshifts as $a^{-1}$ even while the particle changes from relativistic to nonrelativistic motion. The local energy is $E=\sqrt{m^2+q^2/a^2}$ and the coordinate velocity is $d\mathbf{x}/dt=\mathbf{q}/[a\sqrt{q^2+m^2{a}^2}]$.

A freely falling particle's physical momentum is measured in the local orthonormal frame of a comoving observer. With proper peculiar velocity $\mathbf{v}=ad\mathbf{x}/dt$ and Lorentz factor $\gamma =(1-v^2{)}^{-1/2}$, it is $m\gamma \mathbf{v}$. It differs by one factor of the scale factor from the conserved comoving momentum. For a nonrelativistic particle, its peculiar velocity therefore decays as $a^{-1}$.