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Cosmology Problem Set 2

Due Friday April 29 at 5pm Central Time

作业代写成功案例

物理代写|宇宙学代写Cosmology Problem Set 2
Problem 1.

Consider a flat universe containing only matter and cosmological constant.
(a). Suppose a signal is sent from some comoving coordinate $r$ at a time between $t_{i}$ and $t_{f}$ to a cosmological observer at $r=0$. Show that the largest possible value of $\mathrm{r}, r_{\max }$, is given by the formula
$$
r_{\max }=\int_{t_{i}}^{t_{f}} \frac{d t}{a(t)} .
$$
(a). Using this result, prove that if there is no dark energy, all events in spacetime will eventually become visible to the observer at $r=0$.
(c). Next show that if $\Omega_{\Lambda}>0$ that there are events which we can never see. The boundary of the future events we can see is the cosmological event horizon.
(d). In terms of the parameters of the universe now $\left(\Omega_{M}, \Omega_{R}, H_{0}\right)$, find an equation determining the maximum redshift $z_{\max }$ defined such that if an object at $z \leq z_{\max }$ emits a signal now we can see it at some point in the future. For the parameters describing our universe, what is the numerical value of $z_{\max }$ ?

Problem 2.

Consider a relativistic gas of particles of mass $m$ in thermal equilibrium at temperature $T$. The number density of particles with momentum of magnitude $k$ is:
$$
n(k)=\frac{4 \pi g k^{2}}{(2 \pi \hbar)^{3}}\left(\frac{1}{\exp \left(\sqrt{k^{2}+m^{2}} / k T\right)-\sigma}\right),
$$
where $g$ is the number of spin states of the particle species and $\sigma=+1$ if the particles are bosons (Bosé-Einstein distribution) and $\sigma=-1$ if the particles are fermions (Fermi-Dirac distribution). (If the above formula for $n(k)$ is unfamiliar to you, review it!).
(a). Use the second law of thermodynamics (the formula for a differential change in entropy) to find $\rho(T)$ and $p(T)$ in terms of integrals involving the distribution $n(k)$. (You do not need to do the integrals.)
(b). Show that the results obtained for $\rho(T)$ and $p(T)$ imply that such particles obey an equation of state of the form
$$
p=w_{\sigma}(k T / m) \rho .
$$
In other words the equation of state is a linear relationship between pressure and energy density, where the coefficient depends only on the statistics of the particles $(\sigma=\pm 1)$ and the ratio of the temperature to the mass.
(c). By computing the integrals defining pressure and energy density, show that for cold matter, $k T \ll m, w_{\sigma}$ vanishes (for both bosons and fermions).
(d). By computing the integrals defining pressure and energy density, show that for hot matter, $k T \gg m, w_{\sigma}=1 / 3$ (for both bosons and fermions).
(e). Make a plot of $w_{\sigma}(T / m)$ and verify that it interpolates between the answers for cold and hot matter above.

Problem 3.

Consider a gas of helium-4. Let the temperature $T$ be sufficiently small that the gas is nonrelativistic, but sufficiently hot that the helium may be ionized either singly, to He ${ }^{+}$, or doubly, to $\mathrm{He}^{++}$. We assume that the plasma is in a state of chemical and thermal equilibrium, and denote by $n$ the total number density of all helium nuclei (ionized or not).
(a). Let $X_{k}$ for $k=1,2$ be the fraction of helium atoms in the plasma that are $k$-ionized, and $X_{e}$ be the fraction of total electrons that are unbound. Derive the coupled Saha equations:
$\frac{1-X_{1}-X_{2}}{X_{1}\left(X_{1}+2 X_{2}\right)}=\frac{n}{4}\left(\frac{m_{e} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left(\frac{B_{1}}{k T}\right), \quad \frac{X_{1}}{X_{2}\left(X_{1}+2 X_{2}\right)}=n\left(\frac{m_{e} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left(\frac{B_{2}}{k T}\right)$ where $B_{1}$ and $B_{2}$ are the first and second binding energy of helium.
(b). In our universe, helium makes up about $24 \%$ of the baryonic matter density. Taking this as input, and using the results of part a) make a plot of the ionization fractions $X_{e}, X_{1}, X_{2}$ for temperatures between $5000^{\circ}$ Kelvin and $20000^{\circ}$ Kelvin, and briefly describe your findings. At about what temperature is $99 \%$ of the helium in the universe plasma in electrically neutral atoms? At what temperature is it $99 \%$ doubly ionized?
(c). Suppose the laws of physics were modified so that there was no such thing as hydrogen and all the baryonic matter in the universe was found in helium with negligible amounts of heavier elements. In this hypothetical universe, determine the temperature at last scattering of photons by charged matter. (Assume in this hypothetical universe that the total helium density is again $24 \%$ of the baryon density in our universe. Assume also that $\Omega_{M}$ takes the same value as in our universe, so the hypothetical universe contains a bit more dark matter to compensate for the missing hydrogen.)

物理代写|宇宙学代写Cosmology Problem Set 2

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FALL 2021

 NumberCourse InformationInstructor
Lower-Division Courses
Unique number links to course detail (UT EID required)
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48250 · TTH 9:30-11AM · WEL 3.502 (HYBRID/BLENDED)
 ENDL, M
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48255 · MWF 10-11AM · WEL 3.502
 FINKELSTEIN, K
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48260 · TTH 11-12:30PM · WEL 3.502
 OFFNER, S
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48265 · TTH 12:30-2PM · WEL 3.502
 FINKELSTEIN, S
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48275 · MWF 2-3PM · WEL 3.502
 FINKELSTEIN, K
AST 301
[syllabus]
INTRODUCTION TO ASTRONOMY
48279 · MWF 1-2PM · WEL 3.502
 RIES, J
AST 307
[syllabus]
INTRODUCTORY ASTRONOMY
48280 · TTH 11-12:30PM · WEL 2.246
 CASEY, C
AST 309L
[syllabus]
SEARCH FOR EXTRATERRESTRIAL INTELLIGENCE
48285 · TTH 12:30-2PM · WEL 2.110
 MORLEY, C
AST 309R
[syllabus]
GALAXIES, QUASARS, AND THE UNIVERSE
48295 · MWF 11-12PM · WEL 3.502
 BERG, D
AST 110CCONFERENCE COURSE IN ASTRONOMY
48300
 
AST 210CCONFERENCE COURSE IN ASTRONOMY
48305
 
AST 310CCONFERENCE COURSE IN ASTRONOMY
48310
 
Upper-Division Courses
Unique number links to course detail (UT EID required)
NumberCourse InformationInstructor(s)
AST 352K
[syllabus]
STELLAR ASTRONOMY
48315 · TTH 9:30-11AM · PMA 15.216B (HYBRID/BLENDED)
 DINERSTEIN, H
AST 353
[syllabus]
ASTROPHYSICS
48320 · TTH 12:30-2PM · WEL 2.246 (HYBRID/BLENDED)
 KUMAR, P
AST 275
[syllabus]
TOPICS IN ASTRONOMY RESEARCH – FRI
48325 · F  2-3PM · PMA 15.201
 MONTGOMERY, M
AST 375
[syllabus]
WHITE DWARF STARS – FRI
48330 · F  2-3PM · PMA 15.201 (HYBRID/BLENDED)
 MONTGOMERY, M
AST 376C
[syllabus]
COSMOLOGY
48350 · TTH 11-12:30PM · PMA 5.104 (HYBRID/BLENDED)
 SHAPIRO, P
AST 376R
[syllabus]
PRACTICAL INTRO TO RESEARCH
48355 · TTH 3:30-5PM · PMA 15.201 (HYBRID/BLENDED)
 JOGEE, S
AST 175CCONFERENCE COURSE IN ASTRONOMY
48335
 
AST 275CCONFERENCE COURSE IN ASTRONOMY
48340
 
AST 375CCONFERENCE COURSE IN ASTRONOMY
48345
 
AST 379HHONORS TUTORIAL COURSE
48360
 
Graduate Courses
Unique number links to course detail (UT EID required)
NumberCourse InformationInstructor(s)
AST 380ERADIATIVE PROCESSES
48365 · MW 1:30-3PM · PMA 15.216B (HYBRID/BLENDED)
 WINGET, D
AST 382D
[syllabus]
ASTRONOMICAL DATA ANALYSIS
48370 · MW 10-11:30AM · PMA 15.216B (HYBRID/BLENDED)
 BOWLER, B
AST 383D
[syllabus]
STELLAR STRUCTURE AND EVOLUTION
48375 ·TTH 11-12:30PM · PMA 5.112 (HYBRID/BLENDED)
 HAWKINS, K
AST 396C
[syllabus]
ELEMENTS OF COSMOLOGY
48395 ·TTH 2-3:30PM · PMA 15.216B (HYBRID/BLENDED)
 BOYLAN-KOLCHIN, M
AST 390FSTARS, PLANETS & ISM SEMINAR
48380 · W 12-1PM · PMA 15.216B (HYBRID/BLENDED)
 
AST 390GGALAXIES AND COSMOLOGY SEMINAR
48385 · M 12-1PM · PMA 15.216B (HYBRID/BLENDED)
 
AST 391GRADUATE RESEARCH IN ASTRONOMY
48390
 
AST 698ATHESIS
48400
 
AST 698BTHESIS
48405
 
AST 398RMASTER’S REPORT
48410
 
AST 399WDISSERTATION
48415
 
AST 699WDISSERTATION
48420
 
AST 999WDISSERTATION
48425