Lecture 6: The CMBR Flashcards
Launched in 1989 the Cosmic Background Explorer (COBE) satellite confirmed that
the radiation had an almost perfect black body spectrum, with T=2.725K
To produce a black body spectrum requires that particles
interact frequently
with one another so that they reach a thermal equilibrium (i.e. in which the interactions between the particles proceed equally frequently in both the forward and backward reactions)
starting with energy density, to get the total energy we
integrate over all frequencies from zero to infinity
first change variables
total energy expression simplification
define alpha (radiation constant) which groups together all constants to give E_rad=alphaT^4
the present-day value of the dimensionless radiation density is negligible compared with
matter and dark energy terms
how to write down expression for how radiation density scales with redshift
from including the contribution of radiation density in the expression for the Hubble parameter as a function of redshift
key results needed to explain the CMBR in terms of radiation: the mean energy per unit volume is
u= alpha T^4
key results needed to explain the CMBR in terms of radiation:the mean number of photons per unit volume
n is proportional to T^3
key results needed to explain the CMBR in terms of radiation: each photon has energy
kBT/2 per dimension and there are three spatial dimensions
key results needed to explain the CMBR in terms of radiation: the entropy per unit volume is
s = 4/3 alpha T^3
key results needed to explain the CMBR in terms of radiation: the equation of state for radiation
p=1/3pc^2 = 1/3 U_r
For other (effectively) massless
particles such as neutrinos or fermions, the physics stays the same but
the leading coefficients change
the equation of state for radiation combined with the fluid equation confirms that
pr is indeed inversely proportional to the fourth power of the scale factor
pra^4 = pr0 a0^4
first law of thermodynamics
dU=TdS - pdV
we investigate the temperature evolution of the CMBR by starting with
the first law of thermodynamics and assuming that the photon gas expands isentropically (reversibly and adiabatically)
aT=constant is an important equation because
the inverse relationship between scale factor and temperature means that the Big Bang (when a approached 0) really was hot (T approaching infinity)
if the universe cools as it expands, it must have been
much hotter in the past
For a boson gas, the mean number of photons in thermal equilibrium at temperature T is given, with respect to frequency, by
the Bose-Einstein distribution
g_eff represents
the number of spin states (two for photons)
If the total entropy remains constant as the universe expands then
the total number of photons, nV , must also stay the same.
V nf df = V’ nf’ df’
V’=
v (a’/a)^3
f’=
f(a/a’)
subbing in T’=T(a/a’) and f’=f(a/a’) cancels out
the factor of (a/a’)^3 so the frequency reduces in proportion to 1/a and blackbody form is preserved
