Is the Geometry of the Universe Determined by Its Density?

[Lapidus]

Just for clarity, the geometry of the universe is completely determined by the stuff that the universe contains. The parameter k in the R-W metric and in the Friedmann equation *is determined* by the density.

The curvature/ geometry of the universe is not independent of the density.

Correct or not?

Thank you!

 

[Mordred]

yes density determines geometry. Post #24 of this thread has an article containing further detail

https://www.physicsforums.com/showthread.php?t=735218#post4648216

 

[Lapidus]

Thanks!

But how is the sign of the curvature parameter k determined? As I understand, only the scale factor is dynamical in the Friedmann equation. So if our universe is sphere or a saddle, the curvature of the universe in our observable part could have been flatten out if the scale factor accelerated fast enough.

Also, there seems to be model universes in the textbooks (like in Ryden or in Carroll), where they assume a curvature of a universe (open/flat/closed) and then add energy components (matter/radiation/vaccum constant) to the model. Unfortunately, they nowhere say clearly if the curvature is a parameter outside of the model (the Friedmann equation) or not.So again my question: what physics determine if the universe is open, flat or closed?

Addendum: I just read in some notes that k never changes and that density, though it depends on time, always remains in whichever regime it starts with ( larger than one, equal to one, smaller than one).

So can we say then that k is given and can only be determined by measurement?

 

[bapowell]

The global curvature of the universe (whether k is positive, negative, or zero) depends on whether the density of the universe is greater than, less than, or equal to the critical density, $$\rho_c = \frac{3H^2}{8\pi G}$$.

 

[Mordred]

Ryden's Introduction in cosmology book has a variety of models. Single and multi-component
universes. She adds components to arrive at her benchmark model. If you work through each model it becomes clear how each component influences the universe. Bapowell has already answered how K is determined. That is also shown in Ryden's book.

 

[Lapidus]

Thanks again!

After reading some more I came to understand that the universe is "born" with a certain amount of matter, radiation, vacuum energy and with or without a certain curvature.

The density changes, but depending on the value of k, which is given when the universe comes into existence, the density remains larger than one, one or smaller than one.

So k is indeed not determined by any dynamics, but a given constant given at the start of a universe.