When does the warping of space time peak during the very early universe?

If gravity is a separate force and has taken its current form by 1 Plank time, then the very small 1 Plank length space-time must be warped beyond description according to General Relativity, or is it? Is it possible that space-time was strained by the General Relativity warping to beyond the point where it could return to a strain-free configuration, if the mass / energy source disappeared, and that this is what drove the expansion of space-time?

As space-time expands during the Grand Unification epoch, does the warping of space-time increase, or perhaps the warping of space time per unit volume, as the strain field has "room" (i.e., both the space and time) to fully develop? If so, when does the peak space-time strain density occur? Could this explain the inflationary expansion?

To answer this question, one has to choose how to quantify "the warping of space-time". In general, it takes 20 numbers (independent components of the Riemann curvature tensor) at every point to specify how spacetime is warped!

A reasonable approach is to look at the Ricci scalar curvature $$R$$, which boils down this complexity into a single number. In a homogeneous and isotropic spacetime such as ours, it is just a function of cosmological time, $$R(t)$$, because it is the same at all points in space. Using the usual spatially-flat Friedmann metric for describing our universe, it is given by

$$R(t)=6\left(\frac{\ddot{a}(t)}{a(t)}+\frac{\dot{a}(t)^2}{a(t)^2}\right)$$

where $$a(t)$$ is the Friedmann scale factor. So if we know how the scale factor evolves with time, we know how the scalar curvature evolves with time.

The evolution of the scale factor is complicated because in different eras different types of energy dominate in the universe. Let's simplify by looking at three cases where only one type of energy dominates.

If the universe is filled with only matter, then

$$a\propto t^{2/3}.$$

Plugging this into the equation above gives

$$R=\frac{4}{3t^2},$$

If the universe is filled with only radiation, then

$$a\propto t^{1/2}.$$

which gives

$$R=0,$$

so the curvature vanishes!

If the universe is filled with an inflaton field or dark energy, then

$$a\propto e^{Ht}$$

which gives

$$R=12H^2$$

which is a constant.

In none of these cases does the curvature ever increase with time.

So, without doing a more detailed analysis combining matter, radiation, and inflation or dark energy, we find that the warping is maximum at the Big Bang (it was infinite, according to classical General Relativity, when the universe was a point) and has been decreasing ever since as the universe has expanded.

• I knew that quantification of warping of space-time was crucial. (I was thinking along the line of entropy.) I can see how there would be a single number for a specific space-time configuration / history, but does not space time warping vary locally (e.g., near a large mass concentration versus far away from any mass concentration)? Dec 1 '19 at 21:34
• Yes, but cosmologists ignore those local variations in curvature in order to be able to have a reasonable model of the whole universe. Since on large scales there seem to be about the same number of galaxies in each cube, say, a billion light-years across, they treat the universe as if it were smooth and homogeneous rather than lumpy. This is sufficient to understand the overall large-scale dynamics. Then within that they look at how lumps evolve, such as galaxy formation or variations in the CMB. Dec 1 '19 at 21:39
• This is somewhat similar to understanding a gas, where one can talk about density, temperature, and pressure without having to track individual molecules. One averages over volumes containing many molecules. Dec 1 '19 at 21:44
• In lumps like galaxies the curvature can increase. At the singularity inside a black hole, it has become infinite again, just like it was at the Big Bang. But the overall average curvature of the universe as a whole decreases as it expands. Dec 1 '19 at 21:48
• Is it possible that for T > 0K, the space-time needs a certain amount of curvature, analogous to entropy, and that if curvature decreased below, then the space-time expansion slows, stops, reverses? Dec 1 '19 at 21:58