Taking into account the cosmological parameters recently produced by the Planck Collaboration [1], I propose dark energy formula which is based on E = MC^{2 }and shows correlation between dark energy density, baryon density, cosmological event horizon and density fluctuations:

.

In this equation:

.is the dark energy density, 0.6914, according to.Planck 2013 results[1];

is the density fluctuations at 8h^{−1}Mpc, 0.8288, according toPlanck 2013 results[1];

is the baryon density, 0.05, according toPlanck 2013 results[2];

is the area of a sphere represented by the event horizon of the Universe, calculated as , where is the distance to the event horizon (currently estimated at about 16 billion light years [3]);

is the age of the Universe, 13.7965 billion years, according toPlanck 2013 results[1].

This formula proves to adequately describe the state of the Universe, as the calculated value of the dark energy density (0.7004) corresponds with the parameter based on the Plank Satellite measurements (0.6914).

Also, this formula can be transformed to be denominated in joules (kg m^{2}/s^{2})

For that purpose, the parameters (dark energy density) and (baryon density) should be replaced for (dark energy in joules) and (baryon mass in kg), and parameter should be transformed as follows:

..

Where:

.is an area of a sphere with radius , which represents the distance to cosmic event horizon, denominated in meters.

is a distance to cosmic event horizon in light seconds (1 light second (ls) = 299 792 458 m)

is the speed of light in vacuum (C = 299 792 458 m/s)

After a small additional transformation, i.e. , the final equation has the following form, resembling the mass-energy equivalence formula :

..

Where:..

is dark energy value, in joules

is density fluctuations at 8h^{−1}Mpc, 0.8288, according toPlanck 2013 results[1]

is baryon mass, in kg (as calculated below)

is speed of light in vacuum

is a distance to cosmic event horizon in light seconds, at present time about 5x10^{17}ls (about 16 billion light years [2])

is an age of Universe in seconds, now 4.37x10^{17}s (13.8 billion years [1])

Baryon mass of the universe region delimited by the cosmic event horizon (conveniently called “local universe”) is calculated based on the baryon density and on the premises that the total mass of the local universe equals its critical density times its volume, with the following formula:...

Where:...

is critical density of the universe

is baryon density, 0.048, according toPlanck 2013 results[1]

is the total volume of the local universe, calculated as , where is the distance to cosmic event horizon in meters (1.51x10^{26}m);

is the Hubble constant, 67.77 km/Mpcs, according toPlanck 2013 results[1], or 2.17x10^{-18}s^{-1}

is the gravitational constant, 6.67x10^{-11}Nm^{2}kg^{-2 }

After all the values are applied, we have:.and.

Finally, after all the values are applied to , the following value for the dark energy is produced:..

How this calculated dark energy value corresponds with model? According to that model, the dark energy value for 1 km^{3}is half a joule (citation needed). Thus, the total dark energy value for the local universe with volume 1.44x10^{79}m^{3}, or 1.44x10^{70}km^{3}, is 7.20x10^{69}joules.

As it was demonstrated, calculation of the dark energy value on the base of dark energy formula produces the result that is only slightly (less than 1 per cent) different from the value established according to model.

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), andExpanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, Publications of the Astronomical Society of Australia, 2004, 21, 97-109.