Cosmic expansion

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 rate

H = \frac{a ˙}{a},

(1)

and satisfies

\dot{H} + H^{2} = \overset{a}{¨} / a

Hubble constant ( $H_{0}$ )

 0  0

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.

Hubble parameter identity

 0  0

Differentiating

H = \overset{a}{˙} / a

gives

\dot{H} = \frac{a ¨}{a} - H^{2} .

(1)

Comoving Hubble radius ( $(a H)^{- 1}$ )

 0  0

The comoving Hubble radius is

(a H)^{- 1}

. It grows during ordinary decelerating expansion and shrinks during sufficiently accelerated inflation, allowing modes to cross from sub-Hubble to super-Hubble scales.

Hubble parameter for constant-equation-of-state components

 0  0

In a spatially flat universe containing components with constant

w_{i}

H (z) = H_{0} [\sum_{i} Ω_{i, 0} (1 + z)^{3 (1 + w_{i})}]^{1/2} .

(1)

Hubble parameter for matter and dynamical dark energy

 0  0

For pressureless matter and dark energy with equation of state

w (z)

H (z) = H_{0} [Ω_{m, 0} (1 + z)^{3} + Ω_{DE, 0} (1 + z)^{3} exp (3 \int_{0}^{z} \frac{w ( z ^{'} )}{1 + z ^{'}} d z^{'})]^{1/2} .

(1)

Hubble friction

 0  0

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