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Re: [Phys-l] cosmology question



I finally got enough time to work out a more complete answer to Mark S.'s original cosmology question from 05/24/2011.

In the old days before dark energy we used to say that a flat
universe, i.e. critical density, would expand at a decreasing
rate with the rate of expansion tending to zero.
Now we know about dark energy. The rate of expansion is
accelerating. But CMBR observations tell us that the universe
is pretty much flat. How do flatness, critical density and fate
of the universe relate to each other in a universe with dark
energy?

Mark.

Previously when addressing this question I referred the reader to the various roots of the RHS polynomial in the Friedmann equation as determining the behavior of the universe. But since making those posts I have worked out and classified those roots and their values so a structure/origin/destiny diagram of the universe can be drawn as functions of the two parameters [Omega]_m and [Omega]_lambda for the (currently adequate) Lambda-CDM cosmology model (i.e. big bang model that has cold dark matter and dark energy). A description of the results is given here. The model for what follows zeros out the background pressure of the matter, the energy and pressure of any radiation that may be present, and it treats the dark energy as an effective cosmological constant (making the w = -1 for the dark energy equation of state). These assumptions are good for a universe in its current low density & low temperature era long after the first moments of the big bang when radiation and pressure effects would be significant and when the dark energy might be dynamically varying.

Recall the notation:

[Omega]_m = Capital Omega with a lower right subscript m = total mean dimensionless mass density of the universe in units of the critical mass density (includes both the mass densities of the ordinary baryonic matter and the dark matter).

[Omega]_lambda = Capital Omega with a lower right subscript of lambda = dimensionless dark energy density (or cosmological constant) of the universe in units of its critical value.

The critical mass density has the value 3*(H_0)^2/(8*[pi]*G), where H_0 is the current Hubble parameter, and G is Newton's universal gravitational constant. It is the value the mean mass density would have to be in a hypothetical universe that was spatially flat, and was expanding as fast as ours is, but for which the dark energy/cosmological constant exactly vanished. The critical cosmological constant has the value 3*(H_0/c)^2, where c is the local speed limit of causation. It is the value the cosmological constant would have to be in a hypothetical universe that was spatially flat, and was expanding as fast as ours is, but for which there was no matter at all.

Also recall if [Omega]_m + [Omega]_lambda > 1 then the spatial structure of the universe is a closed, bounded and positively curved 3-sphere (i.e. the 3-d hyperspherical surface of a ball in 4-dimensions).

If [Omega]_m + [Omega]_lambda = 1 then the universe is spatially flat.

If [Omega]_m + [Omega]_lambda < 1 then the universe is an open, unbounded and negatively curved hyperbolic 3-space.

In what follows I will presume that the physically allowed ranges of [Omega]_m & [Omega]_lambda are given by:

0 <= [Omega]_m and

-[infinity] < [Omega]_lambda < +[infinity] .

The reason [Omega]_m is taken as nonnegative is that there is no such thing, AFAWK, as a negative physical mass density. This makes our allowed parameter space in ([Omega]_lambda , [Omega]_m) a half-plane. Each point in that parameter space can be considered as a potentially allowable universe. From the data from the WMAP project and other data, such as from type Ia supernovae, it seems that the our particular universe exists at the point (0.728, 0.272) with some uncertainty in the last sig fig. Here we imagine the parameter space with [Omega]_lambda plotted horizontally and [Omega]_m plotted vertically. The physically allowable region is the upper half-plane consisting of the 1st & 2nd quadrants.

Before assigning the attributes to each point in the upper half-plane of the parameter space it is convenient to define two useful real-valued functions of a nonnegative real argument, x:

A(x) = { 1 - x + 3*x*sin((1/3)*arcsin(1 - 1/x)), when x >= 1} &
{ 0, when 0 <=x <= 1}

B(x) = { 1 - x + 3*x*cos((1/3)*arccos(1/x - 1)), when x >= 1/2} &
{ 1 - x + 3*x*cosh((1/3)*arccosh(1/x - 1)), when 0 <= x <= 1/2}

These functions define boundary curves in the ([Omega]_lambda , [Omega]_m) half-plane separating regions of qualitatively different behavior in the past or future. The A(x) function describes the boundary separating a region having a Big Crunch in the future from a region having a Big Freeze in the future. The B(x) function describes the boundary separating a region having a Big Bang singularity in the past from a region having a Big Bounce in the past from which the universe subsequently expanded.

Each point in the half-plane has 3 relevant attributes: 1) a structure attribute labeling the universe's spatial structure and topology (i.e. spherical, flat, or hyperbolic), 2) a history attribute describing the origin of our expanding universe as having started from either a big bang singularity of (nearly) infinite density *or* from a finite density bounce from a previous contracting phase of the universe, and 3) a destiny attribute describing the fate of our universe as either expanding forever in an unbounded way, or eventually halting its expansion and then recontracting to a big crunch singularity. In what follows I will label each of these three attributes as:

1) structure: S = closed 3-spherical, F = spatially flat, & H = open 3-hyperbolic.

2) history: BBg = Big Bang from an initial singularity in the finite past, Bnc = universe expanded from a finite density earlier "bounce" state that reversed an previous collapse in the finite past, IOi = infinitely old universe that has always been expanding forever from an asymptotic past state of infinite density, IOf = infinitely old universe that has always been expanding forever from an asymptotic past state of finite density.

3) destiny: BF = Big Freeze as the universe expand forever in an unbounded way approaching a zero density state in the infinite future, BC = Big Crunch as the universe halts its expansion in a finite time in the future and reverses itself resulting in an infinite density future singularity in the finite future, EH = expansion of the universe gradually halts at a finite density, taking an infinite time to finish halting.

There is one possibility omitted from this classification scheme, i.e. the situation for which the expanding universe starts expanding from a previous finite density 'Bounce' state but later halts its expansion, and then recollapses to another future finite density bounce causing the universe to oscillate. This possibility is omitted because it can be shown that it only occurs when [Omega]_m is negative (i.e. in the unphysical negative half-plane of our parameter space).

In describing the attributes of the universes in our half-plane we consider, piecewise, strips of various ranges of values of [Omega]_m, each with a set of ranges of [Omega]_lambda.

For [Omega]_m > 1 we have universe attributes (structure, history, destiny):
(H, BBg, BC) when [Omega]_lambda < 1 - [Omega]_m
(F, BBg, BC) when [Omega]_lambda = 1 - [Omega]_m
(S, BBg, BC) when 1 - [Omega]_m < [Omega]_lambda < A([Omega]_m)
(S, BBg, EH) when [Omega]_lambda = A([Omega]_m)
(S, BBg, BF) when A([Omega]_m) < [Omega]_lambda <= B([Omega]_m)
(S, IOf, BF) when [Omega]_lambda = B([Omega]_m)
(S, Bnc, BF) when B([Omega]_m) < [Omega]_lambda

For [Omega]_m = 1 we have the universe attributes:
(H, BBg, BC) when [Omega]_lambda < 0
(F, BBg, BF) when [Omega]_lambda = 0
(S, BBg, BF) when 0 < [Omega]_lambda < B(1)
(S, IOf, BF) when [Omega]_lambda = B(1)
(S, Bnc, BF) when B(1) < [Omega]_lambda

For 0 < [Omega]_m < 1 we have the universe attributes:
(H, BBg, BC) when [Omega]_lambda < A([Omega]_m) = 0
(H, BBg, BF) when 0 = A([Omega]_m) <= [Omega]_lambda < 1 - [Omega]_m
(F, BBg, BF) when [Omega]_lambda = 1 - [Omega]_m (Our universe seems to be in this case.)
(S, BBg, BF) when 1 - [Omega]_m < [Omega]_lambda < B([Omega]_m)
(S, IOf, BF) when [Omega]_lambda = B([Omega]_m)
(S, Bnc, BF) when B([Omega]_m) < [Omega]_lambda

For 0 = [Omega]_m we have the universe attributes:
(H, BBg, BC) when [Omega]_lambda < = 0
(H, BBg, BF) when 0 <= [Omega]_lambda < 1
(F, IOi, BF) when [Omega]_lambda = 1 (de Sitter Universe)
(S, Bnc, BF) when 1 < [Omega]_lambda

These attributes describe the spatial topological structure and qualitative any Friedmann-Robertson-Walker model universe having any non-negative matter density, any real-valued Cosmological Constant, negligible background pressure, insignificant radiation energy and pressure, and a simple cosmological constant model for the Dark Energy.

David Bowman