A derivation of the cosmological constant
v. 7 n. 6
NOTICE
The means for calculating the present day cosmological constant was outlined in recent Letters, but the focus was a time varying cosmological parameter to purportedly update general relativity by incorporating the recent results of the dark energy survey, DESI; the other focus was the possible absence of stellar singularities because of this cosmological parameter. According to the first year survey results, it appears dark energy was more prevalent in the early universe, implying a variable cosmological parameter instead of a constant, and dark energy as an active participant in a repulsive aspect of gravitation itself rather than being a passive background constant for space, so that dark energy is perhaps recognized as gravitation, such being fundamentally repulsive and only phenomenally attractive at smaller scales. [1][2][3] This has also been discussed in Newtonian terms. [4]
It was determined that the proposed time varying cosmological parameter is
Λ(t) = (3/c^2) [ a2/a + (a1/a)^2 ], ..................................................... (1)
where a is the scale factor, a1 is the first derivative of the scale factor and a2 the second, so that the parameter is determined exclusively in terms of the scale factor, which describes the relative expansion of the universe and was discussed in detail. [2] This arrangement of the scale factor in Equation (1) represents key parameters in the Friedmann equations, such as the energy density of the universe at large throughout much of its evolution. Again, the Friedmann equations are a solution of the general relativity relations for conventional simplifying assumptions.
At about 30,000 years after the beginning according to standard cosmology, the universe transitioned from a radiation to a matter dominated era where the energy density, rho, decreased at a slower rate, from
a or a(t) ∝ t^1/2 to
a(t) ∝ t^2/3,
where t is time; that is, expansion slowed down in the era when the universe cooled and matter began clumping into recognizable systems. Listing a(t) and its derivatives,
a(t) ∝ t^2/3
a1(t) ∝ 2/3 t^-1/3
a2(t) ∝ -2/9 t^-4/3;
substituting and simplifying these into Equation (1),
Λ(t) = 4 / (3t^2 c^2). ............................................................................... (2)
Given the current age of the universe as t_o ≈ 13.8 billion years and substituting into Equation (2), the proposed cosmological parameter at present might be,
Λ(t_o) ≈ 7.84 x 10^-53 m^-2.
There doesn't seem to be any other theoretical estimates, but the conventional figure from observation is,
Λ = 1.09 ± 0.04 x 10^-52 m^-2
so that the proposed theoretical figure is well within an order of magnitude of the measured value. The radiation and inflation eras have not been considered in the theoretical estimate given the age of the universe to only one decimal place. This suggests that the age might be refined by calculating the cosmological constant to greater precision considering the other eras, then adjusting the age with the resulting refinement of Equation (2).
[4] i.e., An explanation of dark matter and dark energy from unmodified Newtonian gravity* | LinkedIn