How did inflation affect time-like space-time? Alan Guth gave an overview of inflation in a series of videos for World Science U. The emphasis is on how inflation expanded a patch of space to the size of a marble, with no mention of how time-like space-time was affected.
 A: During the brief period of cosmic inflation, spacetime had, to a reasonable approximation, the geometry of deSitter space. See
https://en.m.wikipedia.org/wiki/De_Sitter_universe#Modelling_cosmic_inflation
In the distant future, the geometry will again approach that of de Sitter space as dark energy dominates more and more over matter.
It is not meaningful to talk about “time-like spacetime”. It is intervals and paths in spacetime that can be time-like, space-like, or light-like.
The reason that Guth focused on what was happening to space and not time is that the metric of a homogeneous and isotropic universe can be written in the simple form
$$ds^2=a(t)^2ds_3^2-c^2dt^2$$
where $ds_3^2$ is the 3-d metric for space... either flat Euclidean space or a space that has uniform positive or negative curvature. (The positive case, for example, is a 3-sphere.)
As you can see from this metric, nothing is happening to the size of spacetime in the time direction. All that happens is that the size of spatial directions grows or shrinks with time. In the case of inflation and deSitter space, the scale factor $a(t)$ actually grows exponentially with time!
