Equation of state (cosmology)

Cosmology

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The main stages of the Universe
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Expansion of the Universe
- Cosmological redshift
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Theoretical studies

Gravitational Instability
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Cosmological models
- Cosmological singularity
- Big explosion
- Model de sitter
- Hot Universe Model
- Cosmic inflation
- Friedman Universe
  - Friedmann equation
  - Attendant distance
  - Lambda CDM Model
  - Cosmological equation of state
  - Critical density

Equation of state of cosmological model - dependence $p(\varepsilon )$ ${\ displaystyle p (\ varepsilon)}$ ${\ displaystyle p (\ varepsilon)}$ pressure from the mass energy density of the medium in this model.

In Friedmann's theory , not only is the density of matter created, but also the pressure of the medium: the density of the effective gravitating energy $\varepsilon _{G}=\varepsilon +3\cdot p,$ ${\ displaystyle \ varepsilon _ {G} = \ varepsilon +3 \ cdot p,}$ ${\ displaystyle \ varepsilon _ {G} = \ varepsilon +3 \ cdot p,}$ Where $p$ ${\ displaystyle p}$ $p$ - pressure of the medium, and $\varepsilon$ ${\ displaystyle \ varepsilon}$ $\ varepsilon$ - the energy density of the medium $\varepsilon =c^{2}\cdot \rho ,$ ${\ displaystyle \ varepsilon = c ^ {2} \ cdot \ rho,}$ ${\ displaystyle \ varepsilon = c ^ {2} \ cdot \ rho,}$ Where $\rho$ ${\ displaystyle \ rho}$ $\ rho$ - mass energy density of the medium, $c$ ${\ displaystyle c}$ $c$ - the speed of light.

Pressure is expressed through the equation of state $p(\varepsilon ),$ ${\ displaystyle p (\ varepsilon),}$ ${\ displaystyle p (\ varepsilon),}$ or use the dimensionless parameter - the ratio of pressure to energy density $w={\frac {p}{\varepsilon }},$ ${\ displaystyle w = {\ frac {p} {\ varepsilon}},}$ ${\ displaystyle w = {\ frac {p} {\ varepsilon}},}$ then the equation of state:

p=w\cdot \varepsilon .

{\ displaystyle p = w \ cdot \ varepsilon.}

{\ displaystyle p = w \ cdot \ varepsilon.}

For different environments $w$ ${\ displaystyle w}$ $w$ has a different meaning. Below we assume that the density of the medium is above zero. The following 9 options are possible:

1. Phantom energy (phantom energy) (see phantom cosmology ) is a medium with negative gravity greater (in magnitude ) than that of a vacuum.

w<-1.

{\ displaystyle w <-1.}

{\ displaystyle w <-1.}

With such an equation of state, the density of the medium increases with time, the negative gravity increases and after a finite time becomes infinite, and a Big Gap will occur in the Universe. Another feature of such an environment is that the speed of sound in it is higher than the speed of light. $c .$ ${\ displaystyle c.}$ ${\ displaystyle c.}$

2. Vacuum - a medium with negative gravity.

w=-1.

{\ displaystyle w = -1.}

{\ displaystyle w = -1.}

Respectively:

p_{V}=-\varepsilon _{V}

{\ displaystyle p_ {V} = - \ varepsilon _ {V}}

{\ displaystyle p_ {V} = - \ varepsilon _ {V}}

(only such an equation of state is compatible with the definition of vacuum as a form of energy with a constant density everywhere and always, regardless of the reference system).

\varepsilon _{G}=-2\cdot \varepsilon _{V}.

{\ displaystyle \ varepsilon _ {G} = - 2 \ cdot \ varepsilon _ {V}.}

{\ displaystyle \ varepsilon _ {G} = - 2 \ cdot \ varepsilon _ {V}.}

In the Einstein equations, the vacuum energy is described by the cosmological constant $\Lambda ={\frac {8\cdot \pi \cdot G}{c^{4}}}\cdot \varepsilon _{V}.$ ${\ displaystyle \ Lambda = {\ frac {8 \ cdot \ pi \ cdot G} {c ^ {4}}} \ cdot \ varepsilon _ {V}.}$ ${\ displaystyle \ Lambda = {\ frac {8 \ cdot \ pi \ cdot G} {c ^ {4}}} \ cdot \ varepsilon _ {V}.}$

According to the latest data ^[1], the vacuum energy density in the Universe is $\Omega _{\Lambda }=0{,}728_{-0{,}016}^{+0{,}015}$ ${\ displaystyle \ Omega _ {\ Lambda} = 0 {,} 728 _ {- 0 {,} 016} ^ {+ 0 {,} 015}}$ ${\ displaystyle \ Omega _ {\ Lambda} = 0 {,} 728 _ {- 0 {,} 016} ^ {+ 0 {,} 015}}$ from critical density .

3. Quintessence - a medium with negative gravity is lower than that of a vacuum.

-1<w<-{\frac {1}{3}}.

{\ displaystyle -1 <w <- {\ frac {1} {3}}.}

{\ displaystyle -1 <w <- {\ frac {1} {3}}.}

Only when $w<-{\frac {1}{3}}$ ${\ displaystyle w <- {\ frac {1} {3}}}$ ${\ displaystyle w <- {\ frac {1} {3}}}$ there is negative gravity, therefore, only under this condition the expansion of the universe accelerates, that is, the nature of dark energy is either a vacuum, or phantom energy, or quintessence.

4. The environment in which there is no positive and negative gravity.

w=-{\frac {1}{3}}.

{\ displaystyle w = - {\ frac {1} {3}}.}

{\ displaystyle w = - {\ frac {1} {3}}.}

5. The environment in which gravity is lower than that of dust.

-{\frac {1}{3}}<w<0.

{\ displaystyle - {\ frac {1} {3}} <w <0.}

{\ displaystyle - {\ frac {1} {3}} <w <0.}

6. Dust cloud, ordinary baryonic matter and cold dark matter (no environmental pressure, $p=0$ ${\ displaystyle p = 0}$ $p = 0$ ).

w=0.

{\ displaystyle w = 0.}

{\ displaystyle w = 0.}

Respectively:

p_{M}=0;

{\ displaystyle p_ {M} = 0;}

{\ displaystyle p_ {M} = 0;}

\varepsilon _{G}=\varepsilon _{M}.

{\ displaystyle \ varepsilon _ {G} = \ varepsilon _ {M}.}

{\ displaystyle \ varepsilon _ {G} = \ varepsilon _ {M}.}

According to the latest data ^[1], the energy density of ordinary cold baryonic matter in the Universe is $\Omega _{b}=0{,}0456\pm 0{,}0016$ ${\ displaystyle \ Omega _ {b} = 0 {,} 0456 \ pm 0 {,} 0016}$ ${\ displaystyle \ Omega _ {b} = 0 {,} 0456 \ pm 0 {,} 0016}$ from the critical density , and the density of cold dark matter is $\Omega _{c}=0{,}227\pm 0{,}014$ ${\ displaystyle \ Omega _ {c} = 0 {,} 227 \ pm 0 {,} 014}$ ${\ displaystyle \ Omega _ {c} = 0 {,} 227 \ pm 0 {,} 014}$ from the critical density, which in total gives $\Omega _{m}=0{,}272_{-0{,}015}^{+0{,}016}$ ${\ displaystyle \ Omega _ {m} = 0 {,} 272 _ {- 0 {,} 015} ^ {+ 0 {,} 016}}$ ${\ displaystyle \ Omega _ {m} = 0 {,} 272 _ {- 0 {,} 015} ^ {+ 0 {,} 016}}$ from critical density.

7. The environment in which gravity is higher than that of dust, but lower than that of radiation.

0<w<{\frac {1}{3}}.

{\ displaystyle 0 <w <{\ frac {1} {3}}.}

{\ displaystyle 0 <w <{\ frac {1} {3}}.}

8. Ultrarelativistic environment (radiation, photons, and other ultrarelativistic particles), including relic radiation ; also massive particles in the early Universe, when the temperature (expressed in energy units) far exceeds the masses of the particles:

w={\frac {1}{3}}.

{\ displaystyle w = {\ frac {1} {3}}.}

{\ displaystyle w = {\ frac {1} {3}}.}

The behavior of the Universe was determined by a close to this equation of state in the time interval from the Planck era to the recombination era.

Respectively:

p_{R}={\frac {1}{3}}\cdot \varepsilon _{R};

{\ displaystyle p_ {R} = {\ frac {1} {3}} \ cdot \ varepsilon _ {R};}

{\ displaystyle p_ {R} = {\ frac {1} {3}} \ cdot \ varepsilon _ {R};}

\varepsilon _{G}=2\cdot \varepsilon _{R}.

{\ displaystyle \ varepsilon _ {G} = 2 \ cdot \ varepsilon _ {R}.}

{\ displaystyle \ varepsilon _ {G} = 2 \ cdot \ varepsilon _ {R}.}

9. The environment in which gravity is higher than that of radiation.

{\frac {1}{3}}<w.

{\ displaystyle {\ frac {1} {3}} <w.}

{\ displaystyle {\ frac {1} {3}} <w.}

Similarly, when $w>1$ ${\ displaystyle w> 1}$ ${\ displaystyle w> 1}$ the speed of sound in such an environment is higher than the speed of light $c$ ${\ displaystyle c}$ $c$ .

Links

modcos.com - Website about modern cosmology

Notes

↑ ¹ ² Parameters of the ΛCDM model according to the WMAP (WMAP Seven-year Cosmological Parameter Summary) (English)

[WMAP_Recommended_Parameter_Values-1] ¹ ² Parameters of the ΛCDM model according to the WMAP (WMAP Seven-year Cosmological Parameter Summary) (English)

Equation of state (cosmology)

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Notes

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