Download Autumn semester 2013-14 - The University of Sheffield

PHY323
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DEPARTMENT OF PHYSICS AND ASTRONOMY
Autumn Semester (2013-2014)
DARK MATTER AND THE UNIVERSE
2 HOURS
Answer question ONE (Compulsory) and TWO other questions.
All questions are marked out of ten. The breakdown on the right-hand side of
the paper is meant as a guide to the marks that can be obtained from each
part.
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COMPULSORY
1
(a) Give two reasons why neutrinos are excluded from dominating dark matter in the
Universe.
[2]
(b) A basic model for dark matter in galaxies assumes a density distribution of the
form known as an isothermal sphere. Briefly describe the meaning of this term and
also state why we expect dark matter in galaxies not to co-rotate with the baryonic
component.
[2]
(c) Define the term critical density in cosmology and derive an expression for it in
terms of the Hubble constant squared and the gravitational constant G.
[2]
(d) With the aid of a suitable graph explain how measurements of the abundance of
light elements in the Universe can be used to determine the number density of
baryons in the Universe. What does this tell us about the fraction of non-baryonic
matter in the Universe?
[2]
(e) Axions are a potential dark matter candidate. Briefly explain how axions may
arise in particle physics and state what characteristics they have that may allow them
to be detected in the laboratory.
[2]
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Relic neutrinos from the Big Bang do form a small component of the dark matter. In
addition neutrinos are important in searches for neutralino WIMP dark matter,
because they may be produced when WIMPs gravitationally collect inside massive
bodies such as the Sun or Earth and annihilate with each other there.
(a) Briefly explain what fundamental property of neutralinos is important for this
annihilation process. Outline how the neutrinos produced may be detected on Earth
as a means to so-called indirect detection of neutralino dark matter. State why the
best sensitivity for this indirect detection technique is for spin-coupled neutralinos.
Include a brief description of one experiment currently in operation which could
perform such an indirect search for neutralinos using neutrino detection.
[3]
(b) The capture rate of WIMPs in the Sun C is dependent on the flux of WIMPs
impinging on the Sun φΧ , the solar mass M, the proton mass mp, and the WIMPproton interaction cross section σΧp . Deduce a simple expression for C using these
parameters. Hence, using typical assumed values for the WIMP flux φΧ = 105 cm-2s-1
21 -1
-41
2
and cross section
€ σΧp = 10 cm show that the capture rate would be ~ 1.2 × 10 s . [2]
(c) Inside the Sun, the€WIMPs will occasionally encounter each other and annihilate
€
such that at some point during the Sun’s 4.6 billion year evolution
we might expect
€
the in-Sun
WIMP
annihilation
rate
to
become
equal
to
the
capture
rate. Assuming
€
this to be the case, and stating two important assumptions you make, estimate the
flux of neutrinos to be expected on Earth in cm-2s-1.
[2]
(d) The number density of relic neutrinos is expected to be equal to 9/11 times the
number density of photons in the cosmic microwave background radiation (CMBR).
Write an expression for the energy density of CBMR photons u(T) in terms of a
constant a = 7.6 x 10-15 erg cm-3 K-4 and hence, taking the temperature of the CMBR
to be 2.7K, show that the number density of neutrinos in the Universe
is nn ~ 900 cm-3.
[2]
(e) Briefly comment on the relative experimental challenge of detecting the relic
neutrinos compared to the neutrinos from WIMP annihilations in the Sun.
[1]
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3
Supersymmetry (SUSY) theory offers a potential candidate particle for dark matter
we call the neutralino, that may be detectable either through creation in an
accelerator such as the Large Hadron Collider (LHC), or directly in the laboratory.
In both cases two important issues to consider in the design of experiments are the
observable energy expected and any backgrounds that may mask the signal.
(a) Briefly describe the motivation in particle physics for introducing SUSY and
write an equation for the neutralino in terms of a combination of SUSY particle
states. Define the term LSP in this context and state its importance for dark matter.
[2]
(b) Briefly explain in general terms how we would expect a neutralino to manifest
itself in an experiment such as ATLAS at the LHC. State what typical event energies
are involved in such an observation and what type of events constitute the
background. Outline why it is difficult to be sure that a population of SUSY particle
candidates discovered by this technique is actually responsible for the dark matter.
[2]
(c) So-called two-phase liquid xenon technology provides a means of detecting
neutralinos directly. Briefly describe how this technology is able to reject electron
background.
[1]
(d) The radioactive isotope 85Kr, which undergoes β-decay with a 10.8 year half-life,
is a major source of background in liquid xenon technology. Consider a dark matter
detector with 200 kg of liquid xenon that contains one 85Kr nucleus for every 106 Xe
nuclei (atomic mass 131). The detector has an electron background rejection
efficiency of 107. Estimate the background rate in the detector from the 85Kr decays.
Hence determine by what factor the level of 85Kr contamination must be reduced by
purification of the xenon to ensure no more than one background event from this
isotope is seen in 1 year of operation. State any assumptions you make.
[4]
(e) Apart from electron and gamma background, state the form and origin of one
other main background that plagues direct dark matter experiments. Explain why
such backgrounds are not important in searches for neutralinos at the LHC.
[1]
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The concept of microlensing is a useful tool in the search for baryonic dark matter,
both in the form of MACHOs in the vicinity of our Galaxy and of more distant black
holes in the Universe.
(a) Give an estimate of the fraction of dark matter in the Universe likely to be in
baryonic form and state two techniques, other than microlensing, useful in searches
for the existence of black holes as dark matter candidates.
[2]
(b) The use of microlensing relies on measuring changes in the observed luminosity
of an object behind the lensing mass. Outline, with the aid of suitable sketches, the
expected typical form of this luminosity change with time, firstly for the case of a
MACHO and secondly for an intergalactic black hole where the background object is
a Quasar. For both cases show the typical timescales involved and state what
phenomena might lead to a false detection.
[2]
(c) In microlensing, we can write that the impact parameter b for a MACHO lens
object of mass M situated half way between an observer on Earth and a distant
luminous star being lensed by the passing MACHO, is given by
b=
2MGD
c2
where D is the distance from the observer to the background object. Define the term
impact parameter in this situation and give, in solar mass units, a typical value
expected for the maximum
€ mass of a MACHO. Taking this value for M and the
equation above, calculate in milliarcsec the angular extent of a microlensing event
caused by passage of a MACHO between the Earth and a star in the LMC, assuming
the distance between the Earth and the MACHO is 25 kpc. Take the distance from
the centre of the Galaxy to the LMC to be 50 kpc. State any assumptions made.
[2]
(d) Consider now the case where a black hole at distance 10 Mpc from Earth causes
lensing of a background luminous object 20 Mpc from Earth. Calculate the mass of
the black hole in solar masses assuming that it results in the same angular extent as
calculated in (c). State any assumptions.
[2]
(e) Re-write the expression for the impact parameter in (c) for the case where the lens
is not equidistant between the Earth and background luminous object. Hence, state
how the angular diameter of lensing events is changed if the position of the lens is
shifted along the line of sight, assuming the distance between the background object
and observer is kept constant.
[2]
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5
Observations of the rotation curves of galaxies lead to a simplified conclusion that
the matter density has form ρ(r) ∝ r −2 where r is the radial distance from the centre
of the galaxy in question.
(a) Although the equation above is a good basic model, briefly outline why it is
unrealistic for extreme
values of r. Outline what sort of astrophysical processes
€
might result in more general deviations from this model.
[2]
(b) Consider an unusual galaxy for which the rotation velocity v is given by:
v(r) = v 0 + Xr
where r is the radial distance from the centre of the galaxy, and the parameters v0 and
X are constants. Show that for this form of rotation curve the total mass M(r) inside
radius r is given by:
€
Ar 3 + Br 2 + Cr
M(r) =
G
where G is the gravitational constant and A, B and C are constant functions of
parameters v0 and X.
[2]
€
(c) Given that measurements
of the rotation curve for the galaxy above indicate that
-1
v0 = 100 km s and X = 38 km s-1 kpc-1, calculate values for the constants A, B and C.
State any assumptions you make.
[2]
(d) Now deduce an equation for the density distribution of dark matter in the same
galaxy, and hence show that one component is independent of radius with value:
ρ(r) const
3X 2
=
.
4 πG
[2]
(e) Calculate a value for this constant density component in GeV/cm3 and comment
on the result relative to the density of dark matter thought to be present in our own
€
Galaxy in the vicinity of the Earth.
[2]
END OF EXAMINATION
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