Download Homework 2 Problem 1 Determine: 1) the relationship of the energy

Homework 2
Problem 1
Determine:
1) the relationship of the energy density of radiation and the temperature in
the expanding universe.
2) Calculate how the number density of photons changes with temperature
and calculate todayʼs value.
3) What is the temperature that corresponds to p-anti-p collisions at the
LHC collider? To what era in the cosmological history does this
correspond? Ignore g, the effective number degrees of freedom of
particles. [Hint: You can assume that the kinetic energy of particles is Ek =
3/2 kT].
Hint: remember that photons are bosons and the following relationships for the
black body radiation apply: the emitted power per unit area of per unit solid angle
and per unit frequency
Sometimes it is also written as a spectral energy density:
which has units of energy per unit volume per unit frequency. Rewrite this as a
function of wavelength. The total energy density in the black body radiation is the
integral between 0 and infinity over all frequencies/wavelengths and this energy
density for radiation is
When you will solve the integral you will end up with the Stefan-Boltzmann but
4
with the constant of proportionality with T expressed as a function of the Planck
constant, c and the Boltzmann constant. You have solved 1).
Then calculate from the spectral energy density above the number density of
photons in a narrow range of wavelengths and integrate over all wavelengths.
You may find useful the following integrals between 0 and infinity:
Solution
1)
So the total energy density is:
u=
8π 5 (kT )4
15(hc)3
= 7.56 × 10−15 (T /K)4 erg/cm3
2) The number density is the integral over all frequencies of the energy
density/energy of each photon:
That gives for T = 2.7K about 400 cm-3.
3) LHC is 14 TeV in the CM and each proton-antiproton beam has kinetic energy
of 7 TeV. We can use the kinetic energy-temperature relationship
We also know that for relativistic matter or radiation:
That comes from the Friedmann equation and the fact that
Hence:
And it was in the hadron era.
Problem 2
What is the maximum kinetic energy that a WIMP of mass 100 GeV and 1 GeV of
kinetic energy can transfer to an electron at rest? Why if the incident particle
would be a neutrino the recoil energy is low?
Some possible Solutions
The electron mass is: me = 0.51 MeV << mW = 100 GeV => μ ≈ m. Since Ek = 1
GeV.
From lecture notes and invariance of sum of 4-momenta squared in the CM and
LAB:
equal and opposite in the CM frame:
�Momenta p are
�1/2
p2
p2
2
(m + M) + 2MEK = (m + 2m ) + (M + 2M ) = m + M + p2 ( 2m1 + 2M1 )
and the recoil energy of the nucleus is maximum for head on collision when the
electron in the lab takes all the momentum 2p. Also me << mw => μ≈me
and we neglect terms in p^4 because the speed of the WIMPs << c:
For M = 100 GeV and Ek = 1 GeV=> Emax = 20 keV.
Another possible solution is using energy and momentum conservation equations
(in the rest frame of the electron) and head-on collision.
With
For the neutrino case see slides in the lecture section of the web page.
Practical problem on background discrimination in Xenon 10
The figure shows the ratio of the 2 scintillation signals in the LXe TPC Xenon2 10
obtained from a calibration to reproduce typical ER (electron recoil) and NR
(nuclear recoil) signatures. ER is the background of the experiment and NR the
signal aside from a potential background of neutrons that we assume completely
eliminated by the shielding.
1) Generate randomly calibration samples for NR and ER similar to the ones in
the first figure below. [Hint: once you fix an energy value, generate randomly the
log10(S2/S1) using a Gaussian with mean decreasing exponentially with energy.
The exponential should be similar to the red line for ER and to the blue line for
NR. If you want to simulate the effect of a decreasing spread with energy for the
NR points use again an exponentially decreasing value of the sigma of the
Gaussian].
2) Transform the ER data in a flat band as shown in the second figure. [Hint: if
you have doubts on how to do this read the Xenon paper in the Handouts section
of the course page.]
3) Define your cut as a band between the Mean of the NR calibration data and
Mean - 3 sigma of the NR acceptance region (see blue lines in the second plot
below).
After the calibrations, the experiment starts the data taking. Data resemble the
ER points since dark matter signals are very rare. You know that 5000 ER events
correspond to 1 month of data taking.
4) How many events fall in the signal region for a year of data?
5) If you would make 100 experiments equivalent to this, what would be the
average number of events in a year?
6) What do you think are the weak points of this experiment? What can the
experiment do to improve and propose an improved detector using this
technique?
Solution
1) analysis.C and C++ code dmgen3.c
2) is done by subtracting the red line (mean of ER) to their data and obtain a plot
similar to the plot on the right below.