radiation-dominated early universe
TRANSCRIPT
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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