This question is a continuation of
https://physics.stackexchange.com/questions/62034/how-to-calculate-a-scalar-curvature-fast .

Let's have Lorentz-Fock spacetime with an interval
$$
d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \frac{\hat {t}^{2}}{R^{2}}d\hat {l}^{2}\right), \qquad (1)
$$
and
$$
d\hat {l}^{2} = \frac{d \hat {r}^{2}}{1 + \frac{\hat {r}^{2}}{R^{2}}} + \hat {r}^{2}\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) , \quad t_{0} = \frac{R}{c} .
$$
Recently, I realized that expression $(1)$ is looks like interval for Frieedman-Robertson-Walker model. Let's have 
$$
r = R \sinh(\psi ).
$$
So $(1)$ can be transformed as
$$
d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat {t}^{2} - \hat {t}^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right]\right) \qquad (2).
$$
This expression is almost corresponding Frieedman-Robertson-Walker metric. To fully comply I need to **change variables**:
$$
\tau = \frac{t_{0}^{2}}{\hat {t}}, d \tau = -\frac{t_{0}^{2}d \hat {t}}{\hat t^{2}},
$$
and $(2)$ can be rewrite as
$$
d\hat {s}^{2} = c^{2}d \tau^{2} - c^{2}\tau^{2}\left[d \psi^{2} + \sinh^{2}(\psi)\left( d\theta^{2} + \sin^{2}(\theta )d\varphi^{2}\right) \right],
$$
which is equal to FRW metric.

So, my questions are following:

 1. Is it possible to make the change of variables specified?

 2. What can I do with singularity of $\tau $ for $\hat {t} -> 0$?