Hubble parameter

The Hubble parameter is the fractional expansion rateand satisfies $\dot{H}+H^2=\ddot{a}/a$.

For the normal flow with zero shift vector, $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ obeys $H_{\mathrm{loc}}=N^{-1}{\partial }_t\log a_{\mathrm{loc}}=-K/3$ in the negative extrinsic curvature of a spatial hypersurface convention. The expansion scalar is $\theta =-K=3H_{\mathrm{loc}}$.

For conformal time $d\tau =dt/a$, the conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH=\dot{a}$, where primes and dots denote conformal and cosmic time derivatives respectively. Its inverse, in units $c=1$, is the comoving Hubble radius. It differs from the ordinary Hubble parameter by one factor of the scale factor.

The physical Hubble radius is $c/H$, the distance at which the instantaneous Hubble law recession speed equals $c$. Its comoving counterpart is $c/(aH)$. Neither radius is generally the same as a particle horizon or a causal boundary: those depend on integrals of the past or future expansion history.

The Hubble constant is the present value $H_0$ of the Hubble parameter. It sets the local proportionality between recession velocity and distance in the Hubble law.

Differentiating $H=\dot{a}/a$ gives

The comoving Hubble radius is $(aH{)}^{-1}$. It grows during ordinary decelerating expansion and shrinks during sufficiently accelerated inflation, allowing modes to cross from sub-Hubble to super-Hubble scales.

With $\mathcal{H}=aH=\dot{a}$, the Friedmann acceleration equation givesThe coefficient is positive when $a>0$, $\rho >0$ and $H\ne 0$. Thus growth of the comoving Hubble radius and linear growth of departures from flatness have the same sign criterion in an expanding universe. This local criterion is distinct from a global particle horizon, which depends on the whole past history.

In a spatially flat universe containing components with constant $w_i$,

For pressureless matter and dark energy with equation of state $w(z)$,

Hubble friction is the damping term proportional to the Hubble parameter in a field or fluid equation in an expanding universe. A homogeneous scalar field, for example, has the term $3H\dot{\varphi }$ in its equation of motion.

The homogeneous expansion velocity is proportional to physical position: