Deceleration parameter

The deceleration parameter $q$ in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:

$q\ {\stackrel {{\mathrm {def}}}{=}}\ -{\frac {{\ddot {a}}a}{{\dot {a}}^{2}}}$

where $a$ is the scale factor of the universe and the dots indicate derivatives by proper time. The expansion of the universe is said to be "accelerating" if $\ddot{a}$ is positive (recent measurements suggest it is), and in this case the deceleration parameter will be negative.^[1] The minus sign and name "deceleration parameter" are historical; at the time of definition $\!q$ was thought to be positive, now it is believed to be negative.

The Friedmann acceleration equation can be written as

$3{\frac {{\ddot {a}}}{a}}=-4\pi G(\rho +3p)=-4\pi G(1+3w)\rho ,$

where $\!\rho$ is the energy density of the universe, $\!p$ is its pressure, and $\!w$ is the equation of state of the universe.

This can be rewritten as

$q={\frac {1}{2}}(1+3w)\left(1+K/(aH)^{2}\right)$

by using the first Friedmann equation, where $\!H$ is the Hubble parameter and $\!K=1,0$ or $\!-1$ depending on whether the universe is hyperspherical, flat or hyperbolic respectively.

The derivative of the Hubble parameter can be written in terms of the deceleration parameter:

${\frac {{\dot {H}}}{H^{2}}}=-(1+q).$

Except in the speculative case of phantom energy (which violates all the energy conditions), all postulated forms of matter yield a deceleration parameter $\!q\geq -1$. Thus, any expanding universe should have a decreasing Hubble parameter and the local expansion of space is always slowing (or, in the case of a cosmological constant, proceeds at a constant rate, as in de Sitter space).

Observations of the cosmic microwave background demonstrate that the universe is very nearly flat, so:

$q={\frac {1}{2}}(1+3w)$

This implies that the universe is decelerating for any cosmic fluid with equation of state $\!w$ greater than $\!-1/3$ (any fluid satisfying the strong energy condition does so, as does any form of matter present in the Standard Model, but excluding inflation). However observations of distant type Ia supernovae indicate that $\!q$ is negative; the expansion of the universe is accelerating. This is an indication that the gravitational attraction of matter, on the cosmological scale, is more than counteracted by the negative pressure of dark energy, in the form of either quintessence or a positive cosmological constant.

Before the first indications of an accelerating universe, in 1998, it was thought that the universe was dominated by dust with negligible equation of state, $\!w\approx 0$. This had suggested that the deceleration parameter was equal to one half; the experimental effort to confirm this prediction led to the discovery of possible acceleration.

References

1. Jones, Mark H.; Lambourne, Robert J. (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0-521-83738-5.