Holographic Gas and Dark Energy

Miao Li

Institute of Theoretical Physics　

2009.03.19

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1. Holographic dark energy

The basic postulate of HDE is

where L is some infrared cut-off.

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This formula is motivatde by the energybound

thus, this model of dark energy is calledholographic dark energy model.

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The model was proposed more than 4years ago, and has been studied extensively both theoretically and numerically.

But, there has been no concretely proposal for the nature of this energy

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We made one such concrete proposal:

The energy may be that of a unusualgas.

(Miao Li, Xiao-Dong Li, Chunshan Lin andYi Wang, Holographic Gas as Dark Energy, arXiv:0811.3332, Commun.Theor.Phys.51:181-186,2009 )

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Since the gas emulates a system withGibbons-Hawking entropy, we call it aholographic gas.

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2. The gas

We start with the single particle partitionfunction:

where w is the degeneracy of particle withenergy k.

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Let

then the single particle partition is

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For a Maxwell-Boltzmann gas, the totalpartition is

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For other kinds of gas (Bose-Einstienand Fermi-Dirac):

where

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When the chemical potential vanishes, wehave simple relations

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Since the free energy thus the entropyscales as

and if we use the Hawking temperature

we wish the entropy to scale linearly inthe area, then

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For all three kinds of gas, we have

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We deduce

Or more generally for a modified dispersion relation

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Finally, for a gas in a ball of radius R, take the Hawking entropy and Gibbons-Hawking temperture

we obtain

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Compare with the hoolographic dark energy model, we deduce

To have a finite partition function, a>-3,thus c<1!

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What does c<1 mean? If we take R to bethe radius of the future event horizon, we have a phantom dark energy model(see below).

For a more general dispersion relation:

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3. Stability of the gas

The gas is stable. Three features guarantee this.

(a) Positive specific heat

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(b) Positive mechanical pressure

(c) Smallness of fluctuations

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4. Cosmology

The evolution of holographic gas is dictated by the first law

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If R is a co-moving scale, we usuallyassume dS=0, then the effective pressureis equal to the mechanical one.

But we SHOULD take R to be the radiusof some kind of horizon, in general dS>0thus the effective pressue can be negative!

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For instance, if

then

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and

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5. Infinite stastics

Another kind of gas consists of all distinguishable particles

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Thus

and

c can be arbitrarily close to 1.

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Conclusion:

A holographic gas is a viable model for darkenergy, and gives rise to the holographic darkenergy formula automatically.

The gas is very unusual in that there is a hugedegeneracy for low energy particles.

This property is currently under close study.

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