In addition, we ascribe to the present epoch the condition that the density of matter determined by the use of the present moment Hubble parameter
and from the Friedmann-Einstein equation must be equal.
With solving this equation in terms of H, the obtained value (H) is achieved corresponding to the Hubble parameter
in the inflation time .
Therefore, measurements of the Hubble parameter
from the CMB spectrum (r [right arrow] [R.sub.U]) will give a value different from and larger than H = 1/T; we find:
H = [??]/a is the Hubble parameter
and the dot represents the derivative with respect to time.
The Hubble parameter
in Figure 2 rose sharply from 1.0 x [10.sup.-37] to 2.0 x [10.sup.-37] s before undergoing an oscillatory behavior.
The generalized mean Hubble parameter
H can be expressed as H = [??]/R = (1/3)([H.sub.x] + [H.sub.y] + [H.sub.z]), where [H.sub.x] = [[??].sub.1]/[b.sub.1], [H.sub.y] = [[??].sub.2]/[b.sub.2], and [H.sub.z] = [[??].sub.3]/[b.sub.3] are the directional Hubble parameters
in the directions of x, y, and z, respectively.
Now we try to explain how the universe exploded and expanded, we start from our assumptions we made before and find the Hubble parameter
and try to find the dark energy and matter.
The Hubble parameter
is the ratio of a galaxy's recession velocity to its distance and describes the rate at which the universe is expanding.
where H [equivalent to] [??]/a is the Hubble parameter
. For the Hubble volume,
where [beta] = 0.5804 and H0 is the so called Hubble constant, the value of the Hubble parameter
H(t) at t = [T.sub.0], the current age of the Universe.
* The Hubble parameter
, the rate of the universe's expansion today, is 70.1 [+ or -] 1.3 km per second per megaparsec.
where H = [??]/a is the Hubble parameter
, [r.sub.c] = [m.sup.2.sub.pl]/(2[m.sup.3.sub.5])  is the crossover length scale reflecting the competition between 4D and 5D effects of gravity, and [epsilon] = [+ or -]1 corresponds to the two branches of solutions of the DGP model.
where H(t) is time dependent Hubble parameter
, and that pressure [p.sub.m] = 0 (matter is treated as dust), one has
Then, by using independently measured numbers like the Hubble parameter
, they can infer (1) how far away the galaxy was when it emitted the light we see now, (2) how far it now lies from Earth, and (3) how far the light traveled in the interim.
Recently, a large class of flat nonsingular FRW type cosmologies, where the vacuum energy density evolves like a truncated power-series in the Hubble parameter
H, has been discussed in the literature [19-22] (its dominant term behaves like [[rho].sub.[LAMBDA]](H) [varies] [H.sup.n+2], n > 0).