Conformal time analytical expression

[anuradha]

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...

Anuradha

 

[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:

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