Scale factor

The cosmological scale factor $a(t)$ converts comoving separations into physical separations in an FLRW universe.

Cosmic expansion is the increase of physical distances between comoving observers as the scale factor grows. Its instantaneous fractional rate is the 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:

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is $n_p=a^{-3}{n}_c$. A conserved population moving with homogeneous expansion has constant $n_c$ despite declining proper density. Formation, mergers or changing sample selection can make $n_c$ evolve.

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,where $\mathbf{u}$ is physical velocity and $\mathbf{v}$ is peculiar velocity.

For a fluid in an expanding universe, the peculiar-velocity equation is the Euler equations for an inviscid fluid after decomposing physical velocity into Hubble flow and peculiar velocity. It contains the Hubble-drag term $H\mathbf{v}$ and force $-\nabla \Phi /a$ from the peculiar gravitational potential.

The peculiar-velocity divergence is $\theta =\nabla \cdot \mathbf{v}$. For irrotational linear perturbations it captures the scalar part of the peculiar velocity field.

Light emitted at $t_e$ and observed at $t_0$ has $1+z=a(t_0)/a(t_e)$.

Cosmic expansion stretches an emitted or absorbed spectral feature's wavelength by $1+z$. A small observed wavelength interval at fixed transition wavelength gives a redshift interval by division by the rest wavelength. Cosmological radial distances then require the Hubble parameter and, for proper distance at that epoch, an extra factor $1/(1+z)$.

A local redshift spacing between absorbers corresponds to proper path length $c\Delta z/[H(z)(1+z)]$. Using the same rest-frame transition for all lines gives $\Delta z=\Delta \lambda /{\lambda }_{\mathrm{rest}}$. This conversion assumes cosmological redshift dominates the line spacing and uses a local Hubble-rate approximation.

Redshift drift is the change in a comoving source's cosmological redshift with observer time. It directly probes $\dot{z}=(1+z)H_0-H(z)$.

Differentiating $1+z=a_0/a(t)$ gives

Cosmic chronometers infer the Hubble parameter from differential ages of passively evolving galaxies:They probe $H(z)$ locally in redshift rather than through a distance integral.

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,