Radiation-Dominated Early Universe

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The Early UniverseCody Arceneaux

May 7, 2013


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Timeline

Time (s) Temperature (K)

0.01 1 x 1011

0.1 3x1010

1.0 1 x 1010

12.7 3 x 109

168 1 x 109

1980 3 x 108

1.78 x 104 1 x 108

1.20 x 1013

30004.34 x 1017 2.725

Table 9.1 from Pathria et. al.


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Relativistic ParticlesFor relativistic fermions (+) or bosons (-) of nearly

zero chemical potential P(T) = kT a() ln(1 e-) d

n(T) = a()1

e 1d

u(T) = a() e 1d

s(T) =

where the density of states is

a() =

2()

where is the spin degeneracy


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Relativistic Particles

For photons, the pressure (P), number density (n),energy density (u), and entropy density (s) are

P(T) =()

45()

n(T) =2 3 ()

()

u(T) =()

15()

s(T) =4 ()

45()


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Relativistic Particles

Ptot(T) = (2 + 21/4 + 7/2) P(T)/2 = (43/8)P(T)

ntot(T) = (2 + 9/2 + 7/2) = (19/4)n(T)

utot(T) = (2 + 21/4 + 7/2) u(T)/2 = (43/8)u(T)

stot(T) = (2 + 21/4 + 7/2) s(T)/2 = (43/8)s(T)

Particles Fermi/Bose

Factor

Spin

Degeneracy

Number of

Species

Photons 1 2 1Neutrinos 7/8 1 3

Anti-Neutrinos 7/8 2 3

Electrons 7/8 2 1

Positrons 7/8 2 1

Table 9.2 from Pathria et. al.


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Scale Factor

Useful to define a scale factor R. The scale

factor is the size of the universe for a time being

observed divided by the size of the current

observable universe. The scale factor can be

related to the redshift z by

R = (1 + z)-1

where

z = /rest


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Matter- vs. Radiation-Dominated

For the radiation-dominated universe, the relationship betweendensity and volume needs to be found via

R3(1+w) = 0

where w is constant that for pressureless dust is zero but is 1/3 forradiation

R-4 R t1/2

For the matter-dominated universe, the relationship between densityand volume should be

R3m= m,0 R-3

R t2/3


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Transition

To find when the transition between the radiation-dominated universe occurred and the matter-dominateduniverse, it is just necessary to find when rel= matter.Another way to find this is to use the density parameter

, which is a ratio between the observed density and thecritical density needed for a flat universe. In that case, allthat is needed to find when rel= matter. These are givenby

matter

=8 G matter

3

where H is the Hubble parameter (time dependent) and

rel =4 G rel

3


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Transition

Setting these to be equal and using data from theWilkinson Microwave Anisotropy Probe (WMAP)gives scale factor for the transition of Rr,m= 3.05 x

10-4

. The temperature of the universe after the BigBang went as

R T = T0where T is the temperature for a particular R value

and T0is the current temperature. This would implythat the temperature at the transition was Tr,m=8920K with redshift zr,m= 3270.


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Cosmic Microwave Background


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RecombinationThe chemical potential for protons, electrons, and hydrogen atoms are

p= mpc2+ kT ln(npp

3)kT ln2

e= mec2+ kT ln(nee

3)kT ln2

H= mHc2+ kT ln(nHH

3)kT ln4

The number densities are related by

nH= npnee3e Ry

where Ry is the ionization energy of a Rydberg atom (1 Ry = 13.6 ev)and

np=ne

due to charge neutrality.

Since the protons after nucleosynthesis are free or combined into hydrogenatoms

np+ nH= (12q) n

where is the baryon-to-photon ratio.


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RecombinationCombining the previous equations gives the Saha equation for

neutral hydrogen fraction

fH=

+= (1+ fH)

2s

where s is

s = 4 (3) 2(1 - 2q) ( )3 2 e Ry

which leads to

fH=1+2 1+4

2

Small values of the baryon-to-photon ratio and Ry/mec2leads

to Recombination occurring at

kT Ry

ln(

(

) )

3000K


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Acoustic Peak


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Acoustic Peak Peak occurs at ~0.6 deg., which corresponds to 480 Mly

This means the preferred separation for galaxies will be480 Mly

Dark Matter: Lower dark matter density increases

peaks while increased normal matter densityincreases odd peaks

Combining the acoustic peak of 480 Mly with redshiftand optical measurements of a baryon acoustic peakfor nearby galaxies allows for a more accuratemeasurement of distance on cosmological scales

Can then be used to constrain some of the propertiesof dark energy


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References

Carroll, Bradley W., and Ostile, Dale A. (2007). AnIntroduction to Modern Astrophysics.

Pathria, R. K., and Beale, Paul D. (2011). StatisticalMechanics.

Other WMAP Images. http://map.gsfc.nasa.gov/resources/otheri mages.html. November 23,2012

Radiation-Dominated Early Universe

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