# How many parameters are needed to fix the scale factor?

## Main Question or Discussion Point

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?