Formula for the dark energy density (dark energy formula)

[Alexroma]

I propose a formula for the dark energy density (dark energy formula):

Ω_Λ = Ω_bA_eh/t²

In this equation:
Ω_Λ is the dark energy density, with the estimated value from 0.712 to 0.743, according to WMAP seven-year data release [1];
Ω_b is the baryon density, 0.0456 ±0.0016, according to WMAP seven-year data release [1];
A_eh is the area of a sphere represented by the event horizon of the Universe, calculated as 4πr², where r is the distance to the event horizon (currently estimated at about 16 billion light years [2]);
t is the age of the Universe, currently 13.75 ±0.11 billion years, according to WMAP seven-year data release [1].

This formula adequately describes the state of the Universe, as the calculated value of the dark energy density is in conformity with its actual value within the established margins (from 0.712 to 0.743):
for the minimum calculated value, Ω_Λ = 0.044×4π×16²/13.86² = 0.737
for the maximum calculated value, Ω_Λ = 0.0472×4π×16²/13.64² = 0.816

References
[1] Table 8 on p. 39 of Jarosik, N., et.al. (WMAP Collaboration). Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results (PDF). nasa.gov (2010). (http://lambda.gsfc.nasa.gov/product...year/basic_results/wmap_7yr_basic_results.pdf, retrieved 2011-10-18).
[2] Tamara M. Davis, Charles H. Lineweaver. Misconceptions about the Big Bang, Scientific American (2005), and Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, Publications of the Astronomical Society of Australia, 2004, 21, 97-109.

 

[BillSaltLake]

To get good agreement with observation, we need Ω_Λ≈0 for early times. Your formula would have Ω_Λ> Ω_b then.

 

[Alexroma]

You are right, my formula doesn't work for "early times", but there is a good reason for that: the acceleration began only about 5 billion years ago. My formula makes no sense for the times when there was no acceleration, because there wasn't cosmological event horizon either. I mean, there is no "event horizon" in the models of universes dominated by matter or by radiation.

 

[Chalnoth]

You are right, my formula doesn't work for "early times", but there is a good reason for that: the acceleration began only about 5 billion years ago. My formula makes no sense for the times when there was no acceleration, because there wasn't cosmological event horizon either. I mean, there is no "event horizon" in the models of universes dominated by matter or by radiation.

Then why would you think this is an explanation?

 

[Alexroma]

Then why would you think this is an explanation?

Good point. May be, this formula works only for the present time. May be, it's not complete. May be, it's just a coincidence that it works. I'll be working on it further.

 

[Alexroma]

Reality check: how it works with 2013 Planck Collaboration results

I checked if this dark energy formula works with the cosmological parameter values (Planck+WP+highL+BAO, Best fit) produced in 2013 by the Planck Collaboration [1].
It turned out that it perfectly works with the addition of one previously missed element, i.e. density fluctuations at 8h⁻¹ Mpc (σ₈).

The dark energy formula now looks like this:
Ω_Λ = σ₈Ω_bA_eh/t²

In this equation:
Ω_Λ is the dark energy density, 0.6914, according to Planck 2013 results [1];
σ₈ is the density fluctuations at 8h⁻¹ Mpc, 0.8288, according to Planck 2013 results [1];
Ω_b is the baryon density, 0.05, according to Planck 2013 results [2];
A_eh is the area of a sphere represented by the event horizon of the Universe, calculated as 4πr², where r is the distance to the event horizon (currently estimated at about 16 billion light years [3]);
t is the age of the Universe, 13.7965 billion years, according to Planck 2013 results [1].

With all this cosmological parameter values, the formula adequately describes the state of the Universe, as the calculated value of the dark energy density (0.7004) is in accordance with the empirical evidence (0.6914).

References
[1] Planck 2013 results. I. Overview of products and scientific results. Astronomy & Astrophysics manuscript no. Planck Mission 2013. March 22, 2013. http://arxiv.org/pdf/1303.5062v1.pdf, page 36, Table 9.
[2] Planck 2013 results. I. Overview of products and scientific results. Astronomy & Astrophysics manuscript no. Planck Mission 2013. March 22, 2013. http://arxiv.org/pdf/1303.5062v1.pdf, page 34.
[3] Tamara M. Davis, Charles H. Lineweaver. Misconceptions about the Big Bang, Scientific American (2005), and Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, Publications of the Astronomical Society of Australia, 2004, 21, 97-109.