How many parameters are needed to fix the scale factor?

[kdv]

Never mind!

I realized the answer to my question.





Consider solving Einstein's equations in a FLRW universe and assume that there is no other force acting on matter/energy beside gravity.

Let's say I want to find the full time evolution of the scale factor a(t) (and knowing this will give me the value of Hubble's constant, obviously).

What is the minimum information required to answer this?

One one hand, there is the matter/energy content of the universe at the present time, so that's $$\rho_{now}, p_{\now},$$ and $$\Lambda$$

Is that all that is required to determine how the scale factor a(t) varies with time and to determine the curvature constant "k" of the universe?

In addition, to find the full a(t) I guess that we need some boundary condition for example $$a(t_{now}) = 1$$. And finally, to fix the choice of origin of the time axis one imposes in addition a ``Big Bang" type condition a(t=0) = 0.

Is that all that is required to fix uniquely a(t) as well as the curvature k?

I thought so...but now I am doing the calculation for one of the Friedmann models. I am using d'Inverno.


Consider the (almost) simplest model.

Consider a "dust" distribution of matter. So P=0. And let's say I fix the density now to be some value $$\rho_{now}$$. Consider the simplest case: no cosmological constant.

In addition, let's set $$a_{now} = 1$$ and $$a(t=0) = 0$$.

It seems that this is all that should be required to fix completely the universe.

However, the way d'Inverno does it is that he uses all those conditions but in addition imposes the value of k. So this seems to be required as an input. I thought this would come out as an output.

So does that mean that imposing the density, pressure and value of the cosmological constant does not determine the curvature of the universe? Or am I missing something?

Thanks in advance

 

[AndyW]

for your help!



Hello,

Thank you for sharing your question and thought process. It appears that you have a good understanding of the necessary inputs to determine the time evolution of the scale factor in a FLRW universe. To answer your question, yes, the matter/energy content, including the density, pressure, and cosmological constant, are the minimum information required to determine the scale factor and curvature of the universe. However, as you mentioned, a boundary condition (such as the present value of the scale factor) and a choice of origin for the time axis are also necessary to uniquely determine the scale factor.

In terms of the curvature, it is not an input, but rather an output of the calculations. The value of k is determined by the other inputs and the boundary conditions. It is possible that the source you are using may have a slightly different approach or notation, leading to confusion about the role of k in the equations. I would recommend double-checking your calculations and consulting other sources for confirmation.

I hope this helps clarify your understanding. Keep up the good work in exploring Einstein's equations and the FLRW universe!