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Re: [Phys-l] cosmology question



Regarding Mark's question:
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.
They relate according to:

k*c^2/(H_0*R)^2 = [Omega]_m + [Omega]_lambda - 1

Where k = 0 for a spatially flat universe, k = +1 for a spatial 3-sphere, and k = -1 for a hyperbolic spatial isotropic 3-Lobachevskyoid universe. Also, c = local speed limit of causation, H_0 = current value of the Hubble parameter, R = current spatial scale factor size of the universe (radius of the 3-sphere if k = 1, or the 'imaginary radius' of the hyperbolic space if k = -1), [Omega}_m = the dimensionless relative mean density of matter, and [Omega]_lambda = the dimensionless relative density of dark energy (AKA, cosmological Constant).

The WMAP and other data suggest that k = 0, [Omega]_m = 0.272, and [Omega]_lambda = 0.728. Note the universe is spatially flat as long as [Omega]_m & [Omega]_lambda add up to precisely unity. This seems to be the case within observational error.

The dimensionless so-called deceleration parameter is given by:

q_0 = [Omega]_m/2 - [Omega]_lambda.

Plugging in these values gives a deceleration parameter of about q_0 = - 0.592. A negative deceleration parameter corresponds to a positively accelerating cosmic scale factor for the universe. If there was no dark energy q_0 would necessarily be positive and the universe would have to decelerate its expansion. But it appears that [Omega]_lambda = 0.728, instead, and the universe is expanding at an accelerating rate.

BTW, [Omega]_m = [Omega]_b + [Omega]_c where [Omega]_b = 0.0455 = the dimensionless mean density of ordinary baryonic matter, and [Omega]_c = 0.227 = the dimensionless mean density of (cold) dark matter.

These dimensionless densities are relative to critical values. The critical value of the cosmological constant is 3*(H_0/c)^2 and the critical mean matter density is 3*(H_0)^2/(8*[pi]*G), where G is Newton's universal gravitational constant. The *meaning* of the critical cosmological constant is it is the value the CC would have to have in a hypothetical universe that expanded at the current rate our universe does but was *both* spatially flat and devoid of all matter. The *meaning* of the critical mean matter density is that it is the mean matter density that a hypothetical universe would have to have if it expanded at the same rate ours does and was *both* spatially flat and had no nonzero cosmological constant.

BTW, the current relative spatial expansion rate of the universe is given the Hubble parameter, i.e. H_0 = 70.3 km/s/Mpc = 1/(13.91 Gyr).

David Bowman