# Conformal time analytical expression

1. Aug 5, 2010

Hi all,
Anybody pls help me to convert cosmic time to conformal time (numerically)...
is there any analytical expression for conformal time, \tau , except d\tau=dt/a(t) ?
can we approximate \tau \approx -1/aH ??

pls help...

2. Aug 5, 2010

### bapowell

You can obtain an analytic expression for \tau whenever you know the time dependence of the scale factor, a(t). During de Sitter expansion, one has

$$a(t) \propto e^{Ht}$$

giving a conformal time of $$\tau = 1/aH$$. So that approximation you give is only good for near de Sitter expansion, when the rate of change of the Hubble parameter is small:

$$\frac{\dot{H}}{H^2} \ll 1$$.