Does Positive Curvature Dominate FRW Cosmological Models in Inflation Theory?

[TrickyDicky]

The FRW solutions are the basis of the current LCDM cosmological model. They include three possible spatial curvatures, positive, flat and negative. The last two possibilities imply infinite extension for any t>0, while positive curvature would give finite space for finite time by definition.
Would these considerations lead to conclude that only the positive curvature case of the FRW cosmological model is used in the current inflationary LCDM cosmology? I mention it because it seems that only the topologically compact case with positive curvature is susceptible to be flattened by inflation, the flat case isn't for obvious reasons and the negative curvature because being infinite for any t>0 cannot be flattened any further by exponential inflation and in any case its original curvature should have been as small as the current measures error bars allow which is quite small.

 

[Orodruin]

because being infinite for any t>0 cannot be flattened any further by exponential inflation

This is not true. Inflation can still suppress the curvature of a hyperbolic space. In the case of no curvature, space is already flat and will be so also after inflation.

 

[George Jones]

Adding to what Oroduin wrote:

Inflation does (spatially) flatten open universes.

Let ##\Omega## be density with respect to critical density, so ##\Omega = 1## if the universe is spatially flat. If ##\Omega_r## is the density of radiation, ##\Omega_m## is the density of matter, and ##\Omega_\Lambda## is the density of the cosmology constant term, then

$$\Omega = \Omega_r + \Omega_m + \Omega_\Lambda,$$
and ##\Omega = 1## if the universe is spatially flat.

Define the curvature parameter ##\Omega_k## by

$$\Omega_k = 1 - \Omega = 1 - \left( \Omega_r + \Omega_m + \Omega_\Lambda \right),$$
so that ##\Omega_k = 0## when the universe is spatially flat.

Suppose the universe is not spatially flat, so ##\Omega_k \ne 0##. The equation of evolution for the curvature parameter is

$$\frac{\Omega_k}{dt} = \Omega_k H \left( \Omega_m -2\Omega_r - 2\Omega_\Lambda \right).$$
During inflation, ##\Omega_\Lambda## dominates and ##H## and ##\Omega_\Lambda## are both (essentially) constant, so that

$$\frac{\Omega_k}{dt} = -2\Omega_k H \Omega_\Lambda,$$
which drives ##\Omega_k## towards zero, i.e., towards spatial flatness.

 

[TrickyDicky]

Yes, that's right. It's no use trying to visualize the curvature evolution of an Infinite hypersurface, it leads one to misleading conclusions.
So in the case that's flat to start with, what would one need inflation for? And if so why not pick it as the general case and even bother with inflation at all?

 

[Chalnoth]

Yes, that's right. It's no use trying to visualize the curvature evolution of an Infinite hypersurface, it leads one to misleading conclusions.
So in the case that's flat to start with, what would one need inflation for? And if so why not pick it as the general case and even bother with inflation at all?

There's still the horizon problem. If you don't have inflation, then spots on the sky more than about a degree apart or so would never have been in causal contact before the CMB was emitted. How did the CMB know to be the same temperature if different parts of the sky could never possibly have any causal links between them?

Inflation, by modifying the expansion history of the very early universe, gives everything in the observable universe more than enough time to come to equilibrium before inflation ends.

 

[bapowell]

So in the case that's flat to start with, what would one need inflation for? And if so why not pick it as the general case and even bother with inflation at all?

That's like coming upon a pencil standing perfectly on its tip and not seeking an explanation for it. One can propose that the pencil just started out that way, but this would certainly be considered a very special rather than "general case".

 

[timmdeeg]

It seems there are no theoretical predictions regarding the spatial curvature of the universe before it underwent inflation. What would determine the curvature in this pre-inflationary era? The comparison of actual and critical energy density as the FRW model tells? However is this model applicable at all before negative pressure starts to play its role?

 

[bapowell]

It seems there are no theoretical predictions regarding the spatial curvature of the universe before it underwent inflation. What would determine the curvature in this pre-inflationary era? The comparison of actual and critical energy density as the FRW model tells? However is this model applicable at all before negative pressure starts to play its role?

Yes, the early curvature depends on the density ratio if we extrapolate the FRW solution back to pre-inflationary times. While we currently have no observational data from the pre-inflationary epoch, FRW should be applicable because the initial inflationary patch needed to be sufficiently isotopric to support inflation.

 

[Chalnoth]

It seems there are no theoretical predictions regarding the spatial curvature of the universe before it underwent inflation.

That's not entirely true. The amount of spatial curvature when inflation began is model-dependent. The most naive estimate would suggest that the curvature should be of order 1, which inflation would then rapidly dilute away to flatness. In some rather simple models, however, it is expected that the universe would start out flat (that is, most of the universes in the ensemble from the model are flat).

 

[timmdeeg]

Thank you, bapowell and Chalnoth.

That's not entirely true. The amount of spatial curvature when inflation began is model-dependent. The most naive estimate would suggest that the curvature should be of order 1, which inflation would then rapidly dilute away to flatness. In some rather simple models, however, it is expected that the universe would start out flat (that is, most of the universes in the ensemble from the model are flat).

Hmm, interesting. So, it seems that these models are based on certain assumptions regarding the energy density at that times. Naively thinking I would expect dominating attractive gravity (k=1 model) and dominating repelling gravity resp. (flat model). However out of what does the universe consist in the pre-inflationary epoch? Presumably there exists neither (not yet) a cosmological constant nor matter/radiation.

 

[bapowell]

However out of what does the universe consist in the pre-inflationary epoch? Presumably there exists neither (not yet) a cosmological constant nor matter/radiation.

That really depends. There are models related to the Hawking no boundary proposal in which the universe pops from nothing into existence in an inflationary state (Alex Vilenkin did much early work on this idea in the 80's). There are also models where the inflaton is simply another quantum field (along with the Higgses, matter fields, etc) that comes to dominate the energy density at some time, kicking off the inflationary expansion. Prior to this, you presumably have radiation dominated expansion just like the classical big bang cosmology, though it is unclear whether the universe would be in thermal equilibrium at this time (that's an important question that has implications for how inflation might get underway).

 

[timmdeeg]

Thanks for this helpful comment. From this it seems a rather challenging job to select the true model.

 

[Chalnoth]

Thanks for this helpful comment. From this it seems a rather challenging job to select the true model.

Yes. Very. So much so that it's very possible that we may never know the correct model.