Download LAST YEAR`S EXAM

determine the redshift at which there was equality between radiation and baryons
(ignore neutrinos).
Cosmology Problem Sheet 3
Q8 Describe how measurements of individual spiral galaxies and of clusters of
galaxies lead to the conclusion that the majority of matter in these systems must be
dark matter.
Q1 Describe the two main problems with Big Bang cosmology that are solved by
cosmological inflation. Describe without equations the concept of inflation and how it
solves these two problems
Q9 (a) Solve the fluid equation for non-relativistic matter (p=0) and radiation (p=
"c2/3) to determine how the density of each varies with the scale factor.
Q2 Show using the acceleration equation that an accelerating rate of Universal
expansion requires
P<-!c2/3
(b)Then use the results in the Friedman equation to determine how the scale factor of
the Universe varies with time for a matter dominated and a radiation dominated flat
Universe with no cosmological constant.
Q3 Using the Friedmann equation, show that a Universe where the dynamics are
dominated by a cosmological constant will expand exponentially with time as
a(t) " exp(!t)
and determine !.
Q4 The Friedmann equation can be written as
| "TOT ! 1 |=
|k|
a2H 2
(c)Show that a flat Universe with no cosmological constant that is currently matter
dominated will have previously been radiation dominated. Given that the energy
density in radiation is given by
#="radc2=$T4
and that the current ratio of radiation to matter density is approximately 2.8x10-4,
determine the scale factor and temperature at the time of matter-radiation equality. If
the current temperature of the CMB is 2.725K and the Universe is 4x1017 seconds old,
determine the age of the Universe at the time of matter-radiation equality.
Assuming that inflation occurred between 10-36 and 10-34 seconds after the Big Bang,
and that during this time the Hubble parameter is constant, determine the value of
#TOT after inflation if #TOT=100 immediately before.
LAST YEAR’S EXAM
A1
(a)
(10 marks)
Assuming that the Moon orbits the Earth in a circular orbit of radius 384,000
km, and that the mass of the moon is much less than that of the Earth, use the
Virial theorem to estimate the mass of the Earth, stating any additional
assumptions.
(b)
(10 marks)
State the cosmological principle and explain why this implies that the Universe
is both homogeneous and isotropic.
(c)
(10 marks)
Using the Friedmann and fluid equations, and assuming no cosmological
constant, derive the acceleration equation.
(d)
(10 marks)
Briefly describe how measurements of individual spiral galaxies and of
clusters of galaxies lead to the conclusion that the majority of matter in these
systems must be dark matter. Why does a comparison of the level of
fluctuations in the Cosmic Microwave Background to the range and amount of
structure seen in the current-day Universe indicate that much dark matter must
be non-baryonic?
Q5 Seconds after the Big Bang, the distribution of proton and neutron energies are
described by Maxwell-Boltzmann distributions. Show that by the time the
temperature of the Universe is T=9.3x109K, the neutron to proton ratio is 0.2
He4 production only begins when the temperature is T=1.2 x109K, 400 seconds after
the Big Bang. Given that the half-life of free neutrons is 614 seconds, determine the
same ratio this time. Then show that the fraction of total baryonic mass in He4 is
approximately 0.22.
Q6 Estimate the temperature of the Universe at the time of decoupling if it assumed
that decoupling occurs when the energy of a typical photon is the same as the
ionization energy of hydrogen (13.6eV). Explain why this does not give the correct
answer. Given that the microwave background was formed at a redshift z~1000, what
was the approximate temperature of the Universe when decoupling occurred?
Q7 Using the temperature of the CMB today, determine the current energy density in
the CMB in terms of the critical density (i.e. determine !rad). Determine the number
density of CMB photons. Given that the current baryon density is !B=0.02h-2 ,
B2
(a)
(b)
(c)
B3
(a)
(b)
(c)
B4
(15 marks)
Describe two of the main problems with Big Bang cosmology that are
solved by cosmological inflation. Describe the concept of inflation and
how it solves these two problems.
(5 marks)
Assuming that the field or particle which gives rise to inflation can be
expressed mathematically as an effective cosmological constant term in the
Friedmann equation, determine how the scale factor varies with time during
inflation.
(10 marks)
Assuming that inflation occurred between 10-36 and 10-34 seconds after the
Big Bang, during which time the Hubble parameter is constant, determine
the value of #TOT after inflation if #TOT=100 immediately before it.
Comment on this answer in relation to the problems mentioned in part (a)
of this question.
(12 marks)
Given that the temperature of the CMB today is 2.725K, estimate the
temperature of the Universe at the time of decoupling if it is assumed that
decoupling occurs when the energy of a typical photon is the same as the
ionization energy of hydrogen (13.6eV). Explain why this does not give the
correct answer. Given that the microwave background was formed at a redshift
z~1000, what was the approximate temperature of the Universe when
decoupling occurred?
(10 marks)
Using determine the current energy density in the CMB in terms of the
critical density (i.e. determine !rad). Given that the current baryon density
is !B=0.02h-2 , determine the ratio between between the number densities
of photons and baryons.
(8 marks)
If the Universe is 4x1017 seconds old, determine the age of the Universe at the
time of matter-radiation equality. Assume a flat universe with no cosmological
constant.
Then show that !(t) " t-2
(d) (8 marks)
Assuming the same equation of state, but for a Universe with k<0, what
value of $ gives a Friedmann equation where the density term varies as the
curvature term (assume no cosmological constant). Show that this gives a
scale factor that increases linearly with time.
Formula Sheet
Virial Theorem
2K+V=0
Friedmann equation:
Alternative to above:
Universe
!
Effective dimensionless density of
Cosmological constant
Fluid equation
Redshift
1+z= a(t0)/a(t), t0=today
Equation of state for:
Non-relativistic matter
Energy density in radiation
(8 marks)
Then use the Friedmann equation to show that a(t) " t2/3 .
(6 marks)]
#="radc2=$T4
Expansion rate in a flat Universe that is :
Radiation dominated
a(t)&t1/2
a(t)2H2&t-1
$
(c)
!
Radiation
(a)
(b)
a˙˙
4 #G %
3p ( +
="
'$ + 2 * +
a
3 &
c ) 3
Acceleration equation
Maxwell Boltzmann distribution
$
| kc 2 |
, where !TOT=!+!
a2H 2
Dimensionless density of component of
For a flat Universe with no cosmological constant and an equation of state
P=($-1)!c2 with 0<$<2:
(8 marks)
Use the fluid equation to show that !(a) "a-3 , where a is the scale factor
of the Universe.
| "TOT #1 |=
Matter dominated
a(t)&t2/3
a(t)2H2&t-2/3
%