Here are the HW questions that we suggest for AST 322 this semester. This 
list may change as we go through the semester.

Barbara Ryden: ``Intro to Cosmology -- 2nd edition'' -- Spring 2020

Unless indicated in class otherwise, H/W sets are due as following [In all
cases, H/W may be handed in early for +1 extra credit point, or it may be
handed in up to a week late for -2 debit points. H/W's in a given set may not 
be handed in after they are graded and handed back to the students or their
answers are discussed in class]: 

H/W 1: Th Jan. 23

H/W 2: Th Feb. 06

H/W 3: Th Feb. 13

H/W 4.1-4.3: Th Feb. 27

H/W 4.4a+4.5 [plus 4.4b for extra credit]: Th Mar 06

H/W 5.1-5.4: Th March 20

H/W 5.5:     Th March 27

H/W 6.7-6.9: Th April 03 

[H/W 5.6 is a term project topic that you may chose. Or you may do it for honors
credit in addition to your term project that we will discuss after spring break].

All term projects are due Apr. 18 (early bird extra credit) or Apr 27 midnight 
(hard deadline).

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H/W 1.0: Verify (NOT prove from first principles) that Ry Eq. 1.1-1.5 are
correct. Then given the values of G, c, h_bar, verify that the listed values of
l_p (m), M_P (kg), t_P (s), E_p (eV) and T_P (K) are correct. What do these
extremely small or large values mean physically? (HINT: What do these values
imply for the early Universe and/or how are these units defined)

H/W 1.1: Derive from the order-of-magnitude equations for the energy density 
during the Planck time: rho_P = c^7 / ( h_bar . G^2 ). What is rho_P in physical 
units? Compare this to the rho_o value today of 2.7 x 10^-27 kg/m^3 (see Ry 
2.2 pg. 11). What does this imply? 

H/W 1.2a: The Schwarzschild radius of a black hole is r_s = 2 G M_BH / c^2.
Derive the Schwarzschild radius of the Universe during the Planck time, r_s,P. 

H/W 1.2b: How does r_s,P compare to l_P? Describe the fallacy here. 

H/W 1.2c: Briefly suggest possible ways out of this dilemma. 

H/W 1.2d (EXTRA CREDIT): Later on this semester, we will ask you to compute
r_s,U for the universe today, and compare it to the Hubble radius r_H today.
Describe the same issue here as in 1.2a--1.2c. 

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H/W 2.1: Confirm the Delta_rho/rho values in Fig. 2 (a), (b), (c), (d), (e). 
Fig. 2e refers to Dr. Windhorst's handwritten notes in Ch. 2 referring to the 
mass density of a black hole, where the radius is the Schwarzschild radius r_S
(see Chapter 1 for the definition of r_S). For (a)-(d) you can use the rho_0
and rho values given in Ryden, but for (d) just choose the approximate rho
value that yields the correct overdensity.

H/W 2.2: Given a value of H_o, calculate R_o in Gpc and t_o in years as
accurately as you currently can (after Ch 6, you will be given the exact
formulae or methods to do so).

H/W 2.3: With what you are given in Ry Chapter 2.4,

a) Prove Wien's law starting from hf=4.97kT: lambda_peak = (0.29 cmK) / T. 
What is the ``color'' (and peak wavelength) of the Sun really?

b) Prove that the mean energy in Planck curve of temp T: E_mean = 2.7 k T (K). 
What is the mean energy of a photon coming from the Sun? And what is it for a
cosmic microwave background photon? 

(EXTRA CREDIT) Derive Wien's Law [lambda_peak = (0.29 cm K)/T ] starting from 
Planck's Law (Eq. 2.27). (Hint: you’ll have to make a change of
variables to write Planck's law in terms of wavelength)

(EXTRA CREDIT): If I told you, as we will in Ch 8, that stars half the
temperature of the Sun can begin to ionize Hydrogen, AND that both the Sun 
and the cosmic microwave background are subject to the same Hydrogen
(ionization) physics, what does that tell you about the redshift of the cosmic
microwave background? Discuss briefly. 

H/W 2.4: Like at the end of Ch. 2, prove that indeed V(z) = (r_0)^3 / (1+z)^3, 
T(z) = T_0 (1+z), and eps_gamma (z) = alpha (T_0)^4 (1+z)^4. For T(z) you must
start with Eq 2.37, following Ryden's derivation but showing intermediate steps.

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H/W 3.1: Show that, or derive how Ry 3.20 and 3.21 follow from the previous
equations. We will use these metrics a lot in what follows. [Note the typo in
Ry Eq. 3.20: The two square exponents INSIDE the first set of square brackets
shouldn't be there ...]

H/W 3.2: See Ry Fig. 3.3. Show that the actual downwards deflection of a
light ray --- horizontally emitted in this closed elevator or box --- is
~2x10^-16 m in the presence of Earth's gravity (or an equivalent acceleration of
the elevator!) across the box length of 2 m. [Note typo in Fig. 3.2's caption:
2x10^-14 should read 2x10^-16 !] 

H/W 3.3: Show that the metric as written in 3.36 is mathematically equivalent to
that in 3.33+3.34 for all three values of k (=+1, 0 -1). We will use both
notations of these metrics interchangeably in what follows. 

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H/W 4.1: Derive from Equation 4.10, combined with Newton’s 2nd Law, that 
Equation 4.12 is true, where U is a constant of integration.

H/W 4.3: Show that Equation 4.28a is true (from 4.20). What is the special 
meaning of the case kappa = 0? What is the actual value of the critical density 
of the universe rho_0 that you derive in that case? i.e., show that Equation 
4.28b is true. Assuming the constants given in the text, calculate the numerical 
value of rho_0 from 4.28b. [Do 4.3 before 4.2]

H/W 4.2: Derive Equation 4.17 from the equations that come before it. Then 
derive Equation 4.18 from that. Solve Equation 4.18 for a(t) in the case of 
kappa = 0, i.e., U = 0. Then sketch this solution. Discuss and sketch the 
solutions for the case of kappa = +1 (U < 0) and kappa = −1 (U > 0).

(EXTRA CREDIT) What is the end behavior of your a(t) for the case of kappa = 0? 
What does this mean physically? Hint: calculate lim(t->inf)da/dt. Derive from 
Equation 4.18 that a(t) has critical points (minima or maxima) determined by 
Equation 4.19. Discuss what this means for the case of kappa > 0 and kappa < 0.

H/W 4.4a Show that (4.35) and (4.36) are true. Discuss the critical
implications of the equation that follows from this: 
[1 - 1/Omega(t)] = [1 - 1/Omega_0(t)]/(1+z) [link your finding to Inflation]. 

4.4b This part for extra credit only: [Harder; TAs may give you further
hints!]: Use (4.26)+(4.33) to derive Omega(t) as a function of (1+z), and then
prove that: [1 - 1/Omega(t)] = [1 - 1/Omega_0(t)]/(1+z) . This equation may be
assumed, but is needed to give the discussion in 4.4a. 

H/W 4.5: Re. (4.58) and following eqs., discuss the physical meaning of the
following cases of the expanding universe's content, all for w<1:

w~=0

w=1/3 

w=-1/3 or smaller

w=-1

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H/W 5.1: Show from the previous Eqs. that (5.9) is a valid solution of (5.8).
Then use this result to answer: What w-component dominates the expansion of the
universe for:

a) a ---> 0 ( or z ---> infinity); and 

b) a ---> infinity (or t ---> infinity). 

c) Discuss the cases w=0 and w=1/3.

H/W 5.2: For what value of z_Lambda_m is eps_Lambda = eps_m? Discuss what this
means for the growth of galaxies with time (no more than 1-2 paragraphs). 

Hint: In preparation for a possible term project, do a little literature search
(use ADS abstract link on Links) on the cosmic star-formation rate vs. redshift
SFR(z). (e.g., Madau, P. & Dickinson, M. 2014, Ann. Rev. A&Ap, 52, p. 415). The
PDF is available here: https://arxiv.org/abs/1403.0007 . Based on their Fig. 9,
discuss what impact the expansion and Lambda (> 0) may have had on the growth of
galaxies with cosmic time, and whether this is visible in the data. 

[For extra credit: You could also discuss how the galaxy merger rate has changed
as a function of redshift, by comparing the mix of spiral and elliptical galaxies
in high redshift clusters to those at low redshifts. (Hint: You need to know that
galaxy mergers tend to transform spiral galaxies into ellipticals; Do a
literature search of how the mix of spiral and elliptical galaxies has changed in
galaxy clusters with redshift, as observed e.g. with Hubble).] 

H/W 5.3: Show that (5.38), (5.39), (5.41) and (5.42) follow from the previous 
math (for w .neq. -1). Discuss the meaning of each of these equations for
simple cases like w=0 and w=1/3. E.g., calculate t_o (in units of 1/H_o) for
the current value of eps_o if w=0 or w=1/3.

H/W 5.4: Show that (5.48), (5.49), and (5.50) follow from the previous math.
Plot d_p(t_o) and t(z) for w=0 and w=1/3, and discuss their meaning. (As before,
you may omit the cases --- for now --- where a w-value causes a zero-divide). 

H/W 5.5: By differentiating the appropriate equations in section 5.3, derive for
what redshift z_max the proper distance d_p(t_e) reaches a maximum value
d_p(t_e)_max (See Fig. 5.3). What is this maximum value d_p(t_e)_max in units of
R_o = c/H_o for each of the w-values that we discussed (w=0, 1/3, -1, etc)? For
one w-value of your choice, plot the resulting d_p(t_e) vs. z [using your
favorite plotting program], and verify that your derived d_p(t_e)_max value is
indeed the maximum of that curve.

For suggestions how to proceed for each w-value, see the equations in the Ryden
5.3 lecture notes and look for the breakdown of this problem into H/W 5.5a, H/W
5.5b, or H/W 5.5c. You don't have to do all suggested H/W 5.5* (although you may
do so for extra credit), and the TAs may suggest which one you need to solve.
Also look ahead to Ch 6.1-6.3 for Taylor series approaches to these solutions.

Now discuss the following: If Congress gave you a fixed large budget to build a
the best telescope you can build within that budget, but passed a law that you
could move --- for ONE time --- to any universe of your liking (using a
Hawking-Penrose wormhole that the DoD had been developing), discuss which
universe would you move to to make optimal use of that telescope? Motivate your
answer well, since you may only move once! (Hint: Consider that you wish to
observe galaxies with a telescope of finite resolution in that universe at
redshifts that are as large as possible; Ponder Fig. 5.3 and its consequences
carefully!).

H/W 5.6 is a possible Term Project or Honors Project!: Evaluate a(t) from the
integral equation (5.83) for a reasonable grid of (Omega_r, Omega_m,
Omega_Lambda) values, and plot these a(t) [following Fig. 5.7] for a grid of
(Omega_m, Omega_Lambda) values as suggested in Fig. 5.6. Sample Fig. 5.6 with a
dense enough grid of (Omega_m, Omega_Lambda) values to provide examples of a(t)
for all reasonable universes (see the *** in Fig. 5.6). 

Remainder H/W based on Ry Ch 5 are numbered 6.7, 6.8. 6.9 (since chapters 5+6
of Ryden's 1st edition were combined into Chapter 5 of Ryden's 2nd edition).

[We ask you only do 6.7a, 6.8b, and 6.9 --- the TA's will verify this]:

H/W 6.7a: For Omega_0>1, show that (5.90) and (5.91) are a solution of (5.89).
Then sketch or plot a(t) for this universe, and show that it results in a big
crunch after a time given in (5.91). At what time is the maximum expansion
reached?

H/W 6.7b (EXTRA CREDIT): For Omega_0<1, show that (5.93) and (5.94) are a
solution of (5.89). Then sketch or plot a(t) for this universe, and verify that
it expands forever.

H/W 6.8a (EXTRA CREDIT): For Omega_Lambda,0<0, show that (5.97), (5.99), (5.99)
are a valid solution of the Friedmann equation (5.96) in the presence of Lambda.
Then sketch or plot a(t) for this universe, and show that it results in a big
crunch after a time given in (5.98).

H/W 6.8b: For Omega_Lambda,0>0, show that (5.100)--(5.103) are a solution of
(5.96) in the presence of Lambda. Then sketch or plot a(t) for this universe,
and verify that it expands forever. Verify that the resulting (5.105) and
(5.106) are true for this universe (this is the Universe we live in!).

H/W 6.9: Show that (5.109)--(5.113) are valid solutions for the Friedmann
equation (5.108) in the case of a Radiation+Matter only universe. (This is the
universe we lived in before Lambda took over at 3.5 Gyr ago (at
z_Lambda,m=0.29). Using (5.111)--(5.113), verify the value of z_r,m and t_rm
that we will use a lot later on.

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Ry Ch  6: H/W 6.1, 6.2, 6.3, 6.9

Ry Ch  7: H/W 7.3, 7.4, 7.6

Ry Ch  8: H/W 8.1, 8.2, 8.4

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