Conformal time doubt My professor for study FRLW metric respect to conformal time started from
$$
ds^2=c^2dt^2-a^2(t) f(r,\Omega)
$$
and, after that, define the conformal time $\eta$ as
$$
cdt=a(\eta) d\eta
$$
but Wikipedia, for example, define conformal time as
$$
cdt=a(t) d \eta
$$
can you help me?
 A: $\eta$ can be considered a function of $t$, and $t$ can be considered a function of $\eta$, so $a$ can be considered a function of either one. Which parameterization is more useful depends on what you are doing.
As a simple analogy, sometimes in 1D classical mechanics one is interested in $v(t)$ and sometimes (such as when using an energy-conservation argument) in $v(x)$.
Mathematicians tend not to do this kind of thing, since $v(t)$ and $v(x)$ are different functions of their arguments! Physicists, on the other hand, are comfortable naming the two different functions “$v$” because they are both velocity functions; they seldom even write the argument.
Cosmologists only care that $a$ is the Friedmann scale factor. Whether it is expressed as a function of cosmological time, conformal time, or something else like temperature is less important.
A: One of the main advantages of using the conformal time is that the action of the scalar and non-massive vector fields are reduced to a form similar to that which occurs in a plane-Minkowski spacetime.
The scale factor a (t) has the meaning of the radius of curvature of space at time t (either in the closed space / 3-sphere / model or in the open / 3-hyperboloid model.
In the flat space model a (t) has no physical meaning.
In the flat space model, the relationship of the scale factor at different instants of time a (t1) / a (t2) has physical significance.
A: usually you use this transformation
$$dt=a(\eta)\,d\eta\tag 1$$
Example
$$a(\eta)=\eta$$
$\Rightarrow$
$$t=\int \eta\,d\eta=\frac{\eta^2}{2}+c~\,\text{with}~c=0$$
thus
$$\eta=\sqrt{2}\,t$$
is $~dt=a(t)\,d\eta\,?$
$$dt=a(\eta)\,d\eta=\eta\,d\eta=\sqrt{2}\,t\,\sqrt{2}\,dt\ne dt$$
thus
$$dt=a(t)\,d\eta$$
is only fulfill  if $\eta=t$, but  not for a general case
