Re: [Phys-l] cosmology question

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding Mark's response:

> Thanks for the clear and detailed answer.
>
> My question arises from an IB examination question "Explain,
> with reference to the possible fate of the universe, the
> significance of the critical density of matter in the universe".
> It's a question that has come up several times over the past 15
> years or so, and the answer expected has not changed. I was
> wanting to be clear about what modification dark energy brings
> about. 
>
> Mark.

I'm surprised that this question is still on the test with the same expected answer.  Before the discovery of dark energy there was a fairly simple relationship between the fate, the geometric spatial structure, and the mean mass density in the universe.  Once dark energy is included into the mix the relationships between these things have become much less straightforward, so much so, that asking the question about them on the IB examination is, I think, asking a much too difficult question.  This is because the correct answer involves the tricky relationships between the roots of a cubic polynomial as functions of changes in the coefficients of the equation.  The cubic polynomial in question comes from the RHS of the Friedmann equation (i.e. the 1st order differential equation whose solution gives the history of the spatial scale factor for the universe).

The Friedmann equation is:

x*(dx/dt)^2/H_0^2 = [Omega]_m + (1 - [Omega]_m - [Omega]_lambda)*x + [Omega]_lambda*x^3 .

Here the function x(t) is the spatial cosmic scale factor of the universe at time t in units of its current value, i.e. x(t) is the ratio of the spatial 'size' of the universe at time t to the 'size' it has now.  The boundary condition on x(t) is: x(now) = 1.  The quantities H_0, [Omega]_m, and [Omega]_lambda are the same here as they were in my previous post.
The universe will expand forever if the RHS of this equation has no positive root in x greater than 1.  If it *does* have a root bigger than x = 1 then the universe will stop expanding when the cosmic scale factor x reaches this value, and then it will begin to re-contract.

If the RHS polynomial has a positive root that is less than x = 1 then the universe will have grown to its current state from a previous 'big bounce' from a prior state when the universe's scale factor was that size.

If the RHS polynomial has both a positive root less than unity *and* a root bigger than unity then the universe not only grew to its present 'size' from a previous bounce, but it will re-contract to that bounce size again after it reaches its maximal size at the upper root.  In essense in such a situation the model has the universe's 'size' trapped between a maximal and minimal 'size' and oscillates between them.
If there is *no* positive root on the RHS polynomial less than unity then the universe grew to its current 'size' from a big bang singularity (or nearly a singularity where the model breaks down) .  If there is still a root bigger than unity then after growing to a future maximal 'size' is will recontract to a big crunch state of a similar singular condition  from which it started out in the initial Bang.

The recent parameter values of [Omega]_m = 0.272, and [Omega]_lambda =  0.728 make the RHS boil down to the polymomial:

0.272 + 0.728*x^3 .

This polynomial has only one real root at about x = -0.720.  Since there are no positive roots at all the universe expanded from a big bang singularity, and will expand forever.

Before the discovery of dark energy (or a non-zero value for the cosmological constant) the above Friedmann equation was a linear polynomial because the coefficient of the cubic term dropped out (because [Omega]_lambda = 0 when there is no dark energy).  In that prior situation the Friedmann equation was:

x*(dx/dt)^2/H_0^2 = [Omega]_m + (1 - [Omega]_m)*x

In this case the RHS has a root at x = [Omega]_m/([Omega]_m - 1).  Since [Omega]_m is necessarily positive the model predicted a Big Bang singularity in the past because there then can't be a root in the range: 0 < x < 1.  Note that the model *does* have a future halted expansion and a future Big Crunch whenever [Omega]_m  > 1.  OTOH, it would expand forever if [Omega]_m <= 1.  Now, in that prior model, the spatial geometry of the universe is also determined by the value of the mass density.  If [Omega]_m > 1 then k = +1 and the universe is the 3-dimensional hyperspherical hypersurface of a 4-dimensional hyper-ball.  If [Omega]_m = 1 then at this critical mass density the universe is spatially flat.  And if [Omega]_m < 1 then k = -1 and the spatial universe is a 3-dimensional hyperbolic space.

We thus see that when [Omega]_lambda = 0 (no dark energy) the relationship between the mass density, the geometric structure, and the history/fate of the universe are all simply related.  But when [Omega]_lambda <> 0 then the relationship between these things becomes more complicated.  Another simplification that can be considered when dark energy is included is to consider the special case of a spatially flat universe.  This is relevant because the universe really *does* appear to be spatially flat.  In this special case

1 = [Omega]_m + [Omega]_lambda

and the RHS of the Friedmann equation becomes:

1 -  [Omega]_lambda + [Omega]_lambda*x^3

This expression has a single real root at x = (1 - 1/Omega]_lambda)^(1/3).  This root is negative, and the universe expands forever from a Big Bang singularity, whenever 0 < [Omega]_lambda < 1 (as the actual case seems to be).  OTOH, this root is positive and bigger than 1, and the universe later recontracts to a Big Crunch in the future, whenever [Omega]_lambda < 0 (i.e. whenever the dark energy density is negative).  OTTH, if [Omega]_lambda > 1 then the universe expands forever after starting out from an initial state that was a big bounce of finite non-zero scale factor 'size' from a previous contraction.

Dave Bowman