Jeans wavenumber ($k_J$)

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.