An Introduction to Modern Astrophysics

Bradley W. Carroll, Dale A. Ostlie

Chapter 29

Cosmology - all with Video Answers

Educators

Chapter Questions

Problem 1

It might be argued that the inverse square law for light, shown below, would provide a solution to Olbers's paradox. To see that this is not so, consider a uniform distribution of stars with $n$ stars per unit volume, each of luminosity $L$. Imagine that two thin, spherical shells of stars with radii $r_{1}$ and $r_{2}$ are centered on Earth; let the thickness of each shell be $\Delta r .$ Show that the same energy flux reaches Earth from each shell.
$$F=\frac{L}{4 \pi r^{2}}$$

Farhanul Hasan

Farhanul Hasan

Numerade Educator

Problem 2

Suppose that all of the matter in the universe were converted into energy in the form of blackbody radiation. Take the average density of matter to be the WMAP value for the density of baryonic matter, $\rho_{b, 0} .$ Use the below equation for the energy density of blackbody radiation to find the temperature of the universe in this situation. At what wavelength would the blackbody spectrum peak? In what region of the electromagnetic spectrum is this wavelength found? Explain how your result may be applied to Olbers's paradox.
\[
u=\frac{4 \pi}{c} \int_{0}^{\infty} B_{\lambda}(T) d \lambda=\frac{4 \sigma T^{4}}{c}=a T^{4}
\]

Farhanul Hasan

Numerade Educator

Problem 3

Show by substitution that Eqs. ( 32 ) and (34) are solutions to Eq. (11) for a closed universe $(k>0).$
$$\begin{aligned}
&\left(\frac{d R}{d t}\right)^{2}-\frac{8 \pi G \rho_{0}}{3 R}=-k c^{2}\\
&\begin{aligned}
R_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k c^{2}}[1-\cos (x)] \\
t_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k^{3 / 2} c^{3}}[x-\sin (x)]
\end{aligned}
\end{aligned}$$

Farhanul Hasan

Numerade Educator

Problem 4

Show by substitution that Eqs. ( 36 ) and ( 38 ) are solutions to Eq. ( 11 ) for an open universe $(k<0).$
$$\begin{aligned}
R_{\text {open }} &=\frac{4 \pi G \rho_{0}}{3|k| c^{2}}[\cosh (x)-1] \\
t_{\text {open }} &=\frac{4 \pi G \rho_{0}}{3|k|^{3 / 2} c^{3}}[\sinh (x)-x]
\end{aligned}$$

Farhanul Hasan

Numerade Educator

Problem 5

Derive Eqs. (33) and (35) from Eqs. (32) and $(34),$ respectively
\[
\begin{aligned}
R_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k c^{2}}[1-\cos (x)] \\
&=\frac{1}{2} \frac{\Omega_{0}}{\Omega_{0}-1}[1-\cos (x)] \\
t_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k^{3 / 2} c^{3}}[x-\sin (x)] \\
&=\frac{1}{2 H_{0}} \frac{\Omega_{0}}{\left(\Omega_{0}-1\right)^{3 / 2}}[x-\sin (x)]
\end{aligned}
\]

Farhanul Hasan

Numerade Educator

Problem 6

Derive Eqs. (37) and (39) from Eqs. (36) and (38), respectively
\[
\begin{aligned}
R_{\text {open }} &=\frac{4 \pi G \rho_{0}}{3|k| c^{2}}[\cosh (x)-1] \\
&=\frac{1}{2} \frac{\Omega_{0}}{1-\Omega_{0}}[\cosh (x)-1] \\
t_{\text {open }} &=\frac{4 \pi G \rho_{0}}{3|k|^{3 / 2} c^{3}}[\sinh (x)-x] \\
&=\frac{1}{2 H_{0}} \frac{\Omega_{0}}{\left(1-\Omega_{0}\right)^{3 / 2}}[\sinh (x)-x]
\end{aligned}
\]

Farhanul Hasan

Numerade Educator

Problem 7

(a) Use Eq. (11) to find an expression for the maximum scale factor $R$ in a closed universe. Does your answer agree with Eq. (33)?
$$\begin{aligned}
&\left(\frac{d R}{d t}\right)^{2}-\frac{8 \pi G \rho_{0}}{3 R}=-k c^{2}\\
&=\frac{1}{2} \frac{\Omega_{0}}{\Omega_{0}-1}[1-\cos (x)]
\end{aligned}$$
(b) Find the lifetime of a closed universe (expressed as a multiple of the Hubble time, $t_{H}$ ) as a function of the density parameter, $\Omega_{0}.$

Farhanul Hasan

Numerade Educator

Problem 8

Derive Eqs. ( $40-$ 42) for the age of the universe using Eqs. ( 31 ), ( 33 ), ( 35 ), $(37),(39),$ and (4).
$$\begin{aligned}
&R=\frac{1}{1+z}\\
&=\left(\frac{3}{2}\right)^{2 / 3}\left(\frac{t}{t_{H}}\right)^{2 / 3} \quad\left(\text { for } \Omega_{0}=1\right)
\end{aligned}$$
$$\begin{array}{l}
=\frac{1}{2} \frac{\Omega_{0}}{\Omega_{0}-1}[1-\cos (x)] \\
=\frac{1}{2 H_{0}} \frac{\Omega_{0}}{\left(\Omega_{0}-1\right)^{3 / 2}}[x-\sin (x)] \\
=\frac{1}{2} \frac{\Omega_{0}}{1-\Omega_{0}}[\cosh (x)-1] \\
=\frac{1}{2 H_{0}} \frac{\Omega_{0}}{\left(1-\Omega_{0}\right)^{3 / 2}}[\sinh (x)-x] \\
\frac{t_{\mathrm{flat}}(z)}{t_{H}}=\frac{2}{3} \frac{1}{(1+z)^{3 / 2}} \quad\left(\text { for } \Omega_{0}=1\right)
\end{array}$$
$$\begin{array}{c}
\frac{t_{\mathrm{closed}}(z)}{t_{H}}=\frac{\Omega_{0}}{2\left(\Omega_{0}-1\right)^{3 / 2}}\left[\cos ^{-1}\left(\frac{\Omega_{0} z-\Omega_{0}+2}{\Omega_{0} z+\Omega_{0}}\right)-\frac{2 \sqrt{\left(\Omega_{0}-1\right)\left(\Omega_{0} z+1\right)}}{\Omega_{0}(1+z)}\right] \\
\left(\operatorname{for} \Omega_{0}>1\right) \\
\frac{t_{\mathrm{open}}(z)}{t_{H}}=\frac{\Omega_{0}}{2\left(1-\Omega_{0}\right)^{3 / 2}}\left[-\cosh ^{-1}\left(\frac{\Omega_{0} z-\Omega_{0}+2}{\Omega_{0} z+\Omega_{0}}\right)+\frac{2 \sqrt{\left(1-\Omega_{0}\right)\left(\Omega_{0} z+1\right)}}{\Omega_{0}(1+z)}\right] \\
\left(\operatorname{for} \Omega_{0}<1\right)
\end{array}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Consider a one-component universe of pressureless dust.
(a) Show that
\[
\Omega(t)=\frac{\rho(t)}{\rho_{c}(t)}=1+\frac{k c^{2}}{(d R / d t)^{2}}
\]
which describes how $\Omega$ varies with time. What does this have to say about the nature of the early universe?
(b) Show that $d R / d t \rightarrow \infty$ as $t \rightarrow 0 .$ What does this say about the difference between a closed, a flat, and an open universe at very early times?

Farhanul Hasan

Numerade Educator

Problem 10

For a one-component universe of pressureless dust, show that
\[
\frac{1}{\Omega}-1=\left(\frac{1}{\Omega_{0}}-1\right)(1+z)^{-1}
\]
What happens as $z$ increases?

Farhanul Hasan

Numerade Educator

Problem 11

Show that in the limit $(1+z) \gg 1 / \Omega_{0},$ Eq. $(\quad 41)$ reduces to Eq. $(\quad 43) .$ Hint: First write Eq. $(\quad 41)$ in terms of a variable $u \equiv 1 /\left[\Omega_{0}(1+z)\right],$ and then expand the equation in a Taylor series about $u=0 .$ You may find
\[
\cos ^{-1}(1-x)=\sqrt{2} x^{1 / 2}+\frac{\sqrt{2}}{12} x^{3 / 2}+\cdots
\]
for $x \ll 1$ to be useful
$\frac{t_{\text {closed }}(z)}{t_{H}}=\frac{\Omega_{0}}{2\left(\Omega_{0}-1\right)^{3 / 2}}\left[\cos ^{-1}\left(\frac{\Omega_{0} z-\Omega_{0}+2}{\Omega_{0} z+\Omega_{0}}\right)-\frac{2 \sqrt{\left(\Omega_{0}-1\right)\left(\Omega_{0} z+1\right)}}{\Omega_{0}(1+z)}\right]$
$$\frac{t(z)}{t_{H}}=\frac{2}{3} \frac{1}{(1+z)^{3 / 2} \Omega_{0}^{1 / 2}}$$

Stephen Hobbs

Numerade Educator

Problem 12

Derive the acceleration equation, Eq. ( 51 ).
\[
\frac{d^{2} R}{d t^{2}}=-\frac{4}{3} \pi G\left(\rho+\frac{3 P}{c^{2}}\right) R
\]

Farhanul Hasan

Numerade Educator

Problem 13

Assuming that $P=0$, show that Eq. $(\quad 51$ ) for the acceleration of a mass shell can be found from Newton's second law by considering the gravitational force on an expanding shell.
$$\frac{d^{2} R}{d t^{2}}=-\frac{4}{3} \pi G\left(\rho+\frac{3 P}{c^{2}}\right) R$$

Farhanul Hasan

Numerade Educator

Problem 14

Consider a model of the universe consisting of neutral hydrogen atoms for which the average $(\mathrm{rms})$ speed of the atoms is $600 \mathrm{km} \mathrm{s}^{-1}($ approximately the speed of the Local Group relative to the Hubble Flow ). Show that $\rho \gg P / c^{2}$ for the gas. For an adiabatically expanding universe, for what value of $R$ and $z$ will $\rho=P / c^{2} ?$

Manish Jain

Numerade Educator

Problem 15

By inserting the equation of state $P=w \rho c^{2}$ into the fluid equation, Eq. $(\quad 50),$ show that $R^{3(1+w)} \rho=$ constant $=\rho_{0},$ where $\rho_{0}$ is the present value of $\rho.$
$$\frac{d\left(R^{3} \rho\right)}{d t}=-\frac{P}{c^{2}} \frac{d\left(R^{3}\right)}{d t}$$

Farhanul Hasan

Numerade Educator

Problem 16

Show that for a pressureless dust universe, $q(t)=\frac{1}{2} \Omega(t),$ which is Eq. (55)
\[
q(t)=\frac{1}{2} \Omega(t)
\]

Farhanul Hasan

Numerade Educator

Problem 17

The deuterium $\left(_{1}^{2} \mathrm{H}\right)$ nucleus is not very tightly bound.
(a) Calculate the binding energy of the deuterium nucleus, using values of $m_{H}=1.007825 \mathrm{u}$ $m_{n}=1.008665 \mathrm{u},$ and $m_{D}=2.014102 \mathrm{u}.$
(b) What is the wavelength of a photon with this energy?
(c) From Wien's law, at what temperature is this the characteristic energy of a blackbody photon?

Farhanul Hasan

Numerade Educator

Problem 19

The carbon absorption lines that are formed when the light from a distant quasar, $\mathrm{Q} 1331+70$ passes through an intergalactic cloud have been measured by Antoinette Songaila and her colleagues. The relative strengths of the lines indicate that the temperature of the cloud is $7.4 \pm 0.8 \mathrm{K},$ and the lines show a redshift of $z=1.776 .$ How does the temperature of the cloud compare with the temperature of the CMB at that redshift? (If there are sources of heating for the cloud in addition to the $\mathrm{CMB}$, then its temperature must be considered as an upper limit to the temperature of the CMB.

Farhanul Hasan

Numerade Educator

Problem 19

In $1941,$ microwave observations detected absorption lines due to cyanogen molecules (CN) in molecular clouds. A cyanogen molecule has three first excited rotational states, each of which is degenerate and has an energy that is $4.8 \times 10^{-4} \mathrm{eV}$ above the ground state. An analysis of the absorption lines shows that for every 100 molecules in the ground state, there are 27 others that are in one of the three first excited states. Assuming that the molecular clouds are in thermal equilibrium with the $\mathrm{CMB}$, use the Boltzmann equation shown below to estimate the temperature of the CMB.
$$\frac{N_{b}}{N_{a}}=\frac{g_{b} e^{-E_{b} / k T}}{g_{a} e^{-E_{a} / k T}}=\frac{g_{b}}{g_{a}} e^{-\left(E_{b}-E_{a}\right) / k T}$$

Farhanul Hasan

Numerade Educator

Problem 20

Channel 6 on your television consists of radio waves with wavelengths between $3.41 \mathrm{m}$ and $3.66 \mathrm{m} .$ Consider a 25,000 -watt television station located $70 \mathrm{km}$ from your home. Use the below equation for the energy density of blackbody radiation to estimate the ratio of the number of channel 6 photons to the number of $\mathrm{CMB}$ photons that your television antenna picks up in this wavelength band. (Hint: For the television broadcast, recall that the energy density of an electromagnetic wave is related to the time-averaged Poynting vector by $u=\langle S\rangle / c .$

Zachary Warner

Numerade Educator

Problem 21

Use the below equation for the relativistic Doppler shift to derive Eq. (61). Show that Eq. (61) reduces to Eq. (62) when $v \ll c.$
$$v_{\mathrm{obs}}=\frac{v_{\mathrm{rest}} \sqrt{1-u^{2} / c^{2}}}{1+(u / c) \cos \theta}=\frac{v_{\mathrm{rest}} \sqrt{1-u^{2} / c^{2}}}{1+v_{r} / c}$$
$$\begin{aligned}
&T_{\text {moving }}=\frac{T_{\text {rest }} \sqrt{1-v^{2} / c^{2}}}{1-(v / c) \cos \theta}\\
&T_{\text {moving }} \simeq T_{\text {rest }}\left(1+\frac{v}{c} \cos \theta\right)
\end{aligned}$$

Zachary Warner

Numerade Educator

Problem 22

Calculate the magnitude of the variation in the temperature of the CMB due to the Sun's peculiar velocity.

Farhanul Hasan

Numerade Educator

Problem 23

In this problem you will approximate the physics that produces the Sunyaev-Zel'dovich effect.
(a) First, estimate the shift in the frequency of a low-energy CMB photon as it is scattered by a high-energy electron (inverse Compton scattering) of speed $v_{e}$ in the hot intracluster gas of a rich cluster of galaxies. Although photons can be scattered from any direction into any direction, we will consider the four equally likely situations shown in Fig. $31 .$ Show that the average shift in the frequency of these four photons is
\[
\frac{\Delta v}{v}=\frac{v_{e}^{2}}{c^{2}}=3 \frac{k T_{e}}{m_{e} c^{2}}
\]
where $T_{e}$ is the temperature of the electron gas. Evaluate this expression using $T_{e}=10^{8} \mathrm{K}$ and explain why you could assume that in the rest frame of the electron, the change in the wavelength $\left(\sim \lambda_{C}\right)$ of the photon could be neglected.
(b) About what fraction of the CMB photons will be scattered as they pass through the intracluster gas? Assume an electron number density of $n_{e}=10^{4} \mathrm{m}^{-3}$ and a cluster radius of 3 Mpc.
(c) Use the increase $\Delta v$ of the peak frequency with Wien's law (Eq. 59 ) to obtain an approximate expression for the effective decrease in the temperature of the $\mathrm{CMB}, \Delta T / T_{0}$ (the Sunyaev-Zel'dovich effect)
\[
\frac{v_{\max }}{T}=5.88 \times 10^{10} \mathrm{Hz} \mathrm{K}^{-1}
\]

Chai Santi

Numerade Educator

Problem 24

Show that in the general equation of state $P=w u$ (Eq. $\quad 52$ ), $w=1 / 3$ for relativistic particles $\left(E \gg m c^{2}\right) .$ Hint: The pressure integral, may prove useful.
\[
\begin{array}{c}
P=w u=w \rho c^{2} \\
P=\frac{1}{3} \int_{0}^{\infty} n_{p} p v d p
\end{array}
\]

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 25

Consider a comoving sphere whose surface expands with the universe. Let it be centered at the origin and filled with CMB photons. Show that Eq. ( 81 ), $R^{4} \rho_{\text {rel }}=\rho_{\text {rel. } 0}$, is consistent with the conservation of energy within the sphere.
\[
R^{4} \rho_{\mathrm{rel}}=\rho_{\mathrm{rel}, 0}
\]

Farhanul Hasan

Numerade Educator

Problem 26

Some quantities obey an exponential time-behavior of the form $f(t)=f_{0} e^{t / \tau},$ where $\tau$ is the characteristic time for the system under consideration.
(a) Show that \[ \tau=\left(\frac{1}{f} \frac{d f}{d t}\right)^{-1}\]
This expression can be used to define a characteristic time for any function, regardless of whether its behavior is exponential.
(b) Use the scale factor, $R(t)$, to show that the characteristic time for the expansion of the universe is $\tau_{\exp }(t)=1 / H(t)$.
(c) Assuming a flat universe containing only matter and radiation, find an expression (valid in both the radiation era and the matter era) for the characteristic expansion time $\tau_{\mathrm{cxp}}$ as a function of the scale factor $R$.

Linh Vu

Numerade Educator

Problem 27

(a) Show that deep in the radiation era when $R \ll R_{r, m},$ Eq. (84) is well approximated by Eq. (86).
(b) Solve the Friedmann equation for a flat, one-component universe that contains only relativistic particles, and compare your result with Eq. (86).
$$\begin{aligned}
&t(R)=\frac{2}{3} \frac{R_{r, m}^{3 / 2}}{H_{0} \sqrt{\Omega_{m, 0}}}\left[2+\left(\frac{R}{R_{r, m}}-2\right) \sqrt{\frac{R}{R_{r, m}}+1}\right]\\
&R(t)=\left(\frac{16 \pi G g_{*} a}{3 c^{2}}\right)^{1 / 4} T_{0} t^{1 / 2}
\end{aligned}$$

Manik Pulyani

Numerade Educator

Problem 28

Use a procedure similar to that used to obtain Eq. ( 28 ) to show that a one-component universe of relativistic particles is flat in the limit $z \rightarrow \infty$.
$$\Omega=\left(\frac{1+z}{1+\Omega_{0} z}\right) \Omega_{0}=1+\frac{\Omega_{0}-1}{1+\Omega_{0} z}$$.

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 29

Assuming that the present density of baryonic matter is given by Eq. ( 17 ), what was the density of matter at the time of Big Bang nucleosynthesis, when $T \sim 10^{10} \mathrm{K} ?$
$$\rho_{b, 0}=4.17 \times 10^{-28} \mathrm{kg} \mathrm{m}^{-3} \quad(\text { for } h=0.71)$$,

Farhanul Hasan

Numerade Educator

Problem 30

One factor that contributed to the cessation of the reactions that formed neutrons at roughly $10^{10} \mathrm{K}[\mathrm{Eqs} .(\quad 93-\quad 95)]$ was the annihilation of electron-positron pairs that occurred at that time. When the temperature became too low, the electron-positron pairs could not be replaced by pair production. (This removed the supply of electrons that could combine with protons to form neutrons.) By setting the characteristic thermal energy of a photon, $k T,$ equal to the rest energy of an electron-positron pair, estimate the temperature below which an annihilated pair will not readily be replaced.
$$\begin{aligned}
n & \rightleftharpoons p^{+}+e^{-}+\bar{v}_{e} \\
n+e^{+} & \rightleftharpoons p^{+}+\bar{v}_{e} \\
n+v_{e} & \rightleftharpoons p^{+}+e^{-}.
\end{aligned}$$

Farhanul Hasan

Numerade Educator

Problem 31

In this problem, you will show that when the temperature of the universe was about $10^{9} \mathrm{K}$, all of the neutrons would have combined with protons to form helium nuclei.
(a) Using arguments similar to those leading up to the equation below show that the number of collisions between a neutron and a proton that occur within a time $\Delta t$ is $n_{p} \sigma v \Delta t,$ where $n_{p}$ is the number density of protons, $\sigma$ is the neutron's collision cross section, and $v$ is the speed of the neutron.
$$\ell=\frac{v t}{n \sigma v t}=\frac{1}{n \sigma}$$.
(b) Evaluate $n_{p} \sigma v \Delta t$. If the result is $\gg 1$, then each neutron had ample opportunity to combine with a proton. Let $\Delta t$ be the characteristic timescale of the universe at the time of helium formation, and use $\sigma=\pi(2 r)^{2}$, where $r \simeq 10^{-15} \mathrm{m}$ is the radius of a neutron. The number density of protons can be estimated from the baryonic mass density when $T=10^{9} \mathrm{K}$.

Farhanul Hasan

Numerade Educator

Problem 32

(a) Use the cross section for electron scattering shown below to find an expression for the average time between scatterings of a photon by free electrons.
$$\sigma_{T}=\frac{1}{6 \pi \epsilon_{0}^{2}}\left(\frac{e^{2}}{m_{e} c^{2}}\right)^{2}=6.65 \times 10^{-29} \mathrm{m}^{2}$$.
(b) Assuming that the electrons remain free, at what value of $R$ and $z$ will the average time between scatterings equal the characteristic expansion time $\tau_{\text {exp }}$ of Problem $26 ?$ (Use WMAP values.) What is the age of the universe when this occurs? This is when a flat universe of matter and radiation would have become transparent due solely to its expansion (no recombination).

David Collins

Numerade Educator

Problem 33

Solve Eq. (101) for a composition of pure hydrogen to find the temperature when half of the electrons and protons have combined to form neutral atoms.
$$\frac{f}{1-f}=\frac{m_{H} R^{3}}{f \rho_{b, 0}}\left(\frac{2 \pi m_{e} k T_{0}}{h^{2} R}\right)^{3 / 2} e^{-\chi_{1} R / k T_{0}}$$.

Keshav Singh

Numerade Educator

Problem 34

Calculate the time of decoupling, $t_{\text {dec }}$, for a universe of matter and radiation using the WMAP values for $z_{\text {dec }}=1089$ and other quantities. Compare your answer with the WMAP result of $379_{-7}^{+8}$ kyr.

Ajay Singhal

Numerade Educator

Problem 35

Using WMAP values for a universe of matter and radiation, estimate the time interval $\Delta t$ between when recombination began (say, when $99 \%$ of the hydrogen atoms were ionized) and when recombination ended (say, when $1 \%$ of the hydrogen atoms were ionized). What is the difference $\Delta z$ between the values of the redshift $z$ at these two times? This is the thickness (in terms of $z$ ) of the "surface" of last scattering. Compare your answers with the WMAP results of $\Delta t=118_{-2}^{+3}$ kyr and $\Delta z=195 \pm 2 .$ Assume a composition of pure hydrogen.

Suzanne W.

Numerade Educator

Problem 36

Suppose that Earth were a perfectly smooth sphere. If you drew a circle of radius $D=100 \mathrm{me}$ ters on Earth's surface, what discrepancy would you find between the expected and measured values of the circle's circumference?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 37

Follow a procedure similar to that used to obtain Eq. ( 28 ) to derive an expression for the total density parameter, $\Omega(z),$ as a function of $z .$ Verify that your expression reduces to Eq. $(\quad 28)$ if $\Omega_{\mathrm{rcl}, 0}=\Omega_{\Lambda, 0}=0 .$ What does your expression say about the geometry of an early, three-component universe?
$$\Omega=\left(\frac{1+z}{1+\Omega_{0} z}\right) \Omega_{0}=1+\frac{\Omega_{0}-1}{1+\Omega_{0} z}$$.

Raj Bala

Numerade Educator

Problem 38

Use the Robertson-Walker metric, Eq. ( 106 ), to show that the proper area $(d t=0$ ) of a spherical surface, centered at the origin and passing through comoving coordinate $\sigma$, is $4 \pi[R(t) \varpi]^{2}$.
$$(d s)^{2}=(c d t)^{2}-R^{2}(t)\left[\left(\frac{d \varpi}{\sqrt{1-k w^{2}}}\right)^{2}+(\varpi d \theta)^{2}+(\varpi \sin \theta d \phi)^{2}\right]$$.

Narayan Hari

Numerade Educator

Problem 39

Einstein originally introduced the cosmological constant $\Lambda$ to stabilize his model of a pressureless dust universe against expansion or contraction.
(a) Find an expression for $\Lambda$ in terms of the density $\rho_{m}$ of a static model of a pressureless dust universe with a cosmological constant.
(b) Find an expression for the curvature $k$ for this static model. Is this model universe closed, open, or flat?
(c) Explain why Einstein's static model is in an unstable equilibrium, so any departure from equilibrium (expansion or contraction) will tend to increase.

Sheh Lit Chang

University of Washington

Problem 40

Evaluate $\Omega_{m}, \Omega_{\mathrm{rel}},$ and $\Omega_{\wedge}$ at the time of decoupling $(z=1089)$ using WMAP values.

Farhanul Hasan

Numerade Educator

Problem 41

Show that Eq. (122) may be written as
\[H=H_{0}\left[\sum_{i} \Omega_{i, 0}(1+z)^{3\left(1+w_{i}\right)}+\left(1-\Omega_{0}\right)(1+z)^{2}\right]^{1 / 2}.\]
where $w$ is the coefficient from the equation of state $P_{i}=w_{i} \rho_{i} c^{2}$ and the $" i "$ subscripts identify one of the components of the universe (i.e., pressure-less dust, relativistic particles, or dark energy).
$$H=H_{0}(1+z)\left[\Omega_{m, 0}(1+z)+\Omega_{\mathrm{rcl}, 0}(1+z)^{2}+\frac{\Omega_{\Lambda, 0}}{(1+z)^{2}}+1-\Omega_{0}\right]^{1 / 2}$$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 42

Derive Eq. (123) for a general expression of the deceleration parameter,
$$\begin{aligned}
&q(t)=\frac{1}{2} \sum_{i}\left(1+3 w_{i}\right) \Omega_{i}(t).\\
&q(t)=\frac{1}{2} \sum_{i}\left(1+3 w_{i}\right) \Omega_{i}(t).
\end{aligned}$$

R M

Numerade Educator

Problem 43

Use the acceleration equation to show that the acceleration of the universe changed sign (from negative to positive) when the scale factor was
\[
R_{\text {sceel }}=\left(\frac{\Omega_{m, 0}}{2 \Omega_{\Lambda, 0}}\right)^{1 / 3}.
\]
Evaluate the value of $R_{\text {accel }}$ and $z_{\text {accel }}$ at this time with WMAP values.

Manik Pulyani

Numerade Educator

Problem 44

(a) Use Eq. ( 129 ) to find an expression for the lookback time, $t_{L},$ as a function of the redshift $z$.
$$t(R)=\frac{2}{3} \frac{1}{H_{0} \sqrt{\Omega_{\Lambda, 0}}} \ln [\sqrt{\left(\frac{\Omega_{\Lambda, 0}}{\Omega_{m, 0}}\right) R^{3}}+\sqrt{1+\left(\frac{\Omega_{\Lambda .0}}{\Omega_{m, 0}}\right) R^{3}}]$$.
(b) The below Figure shows the comoving space density of active galactic nuclei (AGN) as a function of redshift. Using your expression for the lookback time with WMAP values, replot the "ChoMP + CDF + ROSAT" data (marked with filled circles) with $t_{L} / t_{H}$ on the horizontal axis (the lookback time as a fraction of the Hubble time). How would you characterize the decline in the space density of AGN with increasing lookback time?

Ren Jie Tuieng

Numerade Educator

Problem 45

The cosmological constant becomes dominant as the scale factor $R$ becomes increasingly larger in the $\Lambda$ era.
(a) Show that the Hubble parameter is a constant in a flat universe deep in the $\Lambda$ era.
(b) Suppose that, starting today $\left(t=t_{0}, \text { when } R=1\right),$ only the cosmological constant contributes to the Friedmann equation." Solve the Friedmann equation and show that for $A>0,$ the scale factor will increase exponentially.
(c) Use WMAP values to evaluate the characteristic time for the exponential expansion (cf. Problem 26 ).

Eduard Sanchez

Numerade Educator

Problem 46

In the matter era, the distance to the particle horizon for a closed, one-component universe of pressureless dust is given by $$d_{h}(z)=\frac{c}{H_{0}(1+z) \sqrt{\Omega_{0}-1}} \cos ^{-1}\left[1-\frac{2\left(\Omega_{0}-1\right)}{\Omega_{0}(1+z)}\right]$$.
In this problem, you will derive this expression for $d_{h} .$ First change variables in Eq. to obtain
\[d_{h}(t)=R(t) \int_{0}^{\frac{1}{1+2}} \frac{c d R}{R(d R / d t)},\]
where the limits of integration range from $R=0$ (at $t=0$ ) to $R=1 /(1+z)$ (at time $t$ ). Then show that
\[
\left(\frac{d R}{d t}\right)^{2}=H_{0}^{2}\left(\frac{\Omega_{0}}{R}-\Omega_{0}+1\right)
\].
and make this substitution into the denominator of the integral. You may find
\[
\int \frac{d x}{\sqrt{b x-a x^{2}}}=\frac{1}{\sqrt{a}}\left[\cos ^{-1}\left(1-\frac{2 a x}{b}\right)-\frac{\pi}{2}\right]
\]
to be useful.

Dominador Tan

Numerade Educator

Problem 47

Use the results of Problem 46 to find the ratio of the distance to the particle horizon to the circumference of a closed, matter-dominated universe. What happens at very early times, as $z \rightarrow \infty ?$ Show that at the time of maximum expansion (just before the closed universe begins to collapse), this ratio is equal to one-half. This means that at the moment of the Big Crunch ending the collapse, the particle horizon encompasses the entire universe.

Manish Jain

Numerade Educator

Problem 48

Using Eq. (163) for the comoving coordinate, $w,$ of a photon just now arriving from the present particle horizon in a flat universe, find the maximum proper distance of the photon during its journey. Express your answer as a fraction of the model's particle horizon, $d_{h, 0}=$ $3 c t_{0} .$ At what time $\left(t / t_{0}\right)$ is the photon at this distance? Carefully explain the meaning of the phrase "just now arriving from the present particle horizon."
$$\sigma=\sigma_{e}-3 c t_{0}\left(\frac{t}{t_{0}}\right)^{1 / 3}$$.

Penny Riley

Numerade Educator

Problem 49

Consider the (unrealistic) model of a flat, one-component universe of pressureless dust, as described in Section 1 of "cosmology".
(a) Show that for this model,
$$\varpi=\frac{2 c}{H_{0}}\left(1-\frac{1}{\sqrt{1+z}}\right)$$.
(b) Find an expression for the proper distance to an object with redshift $z$ for this model.
(c) Find an expression for the horizon distance in this model. Evaluate this using WMAP values, and compare your result with the more accurate value of Eq. (159).
$$d_{h, 0}=4.50 \times 10^{26} \mathrm{m}=14,600 \mathrm{Mpc}=14.6 \mathrm{Gpc}$$

Sarah Mccrumb

Numerade Educator

Problem 50

Derive Mattig's relation for a one-component universe of pressureless dust,
\[
\varpi=\frac{2 c}{H_{0}} \frac{1}{1+z} \frac{1}{\Omega_{0}^{2}}\left[\Omega_{0} z-\left(2-\Omega_{0}\right)(\sqrt{1+\Omega_{0} z}-1)\right],
\]
or in terms of the deceleration parameter $q_{0}=\Omega_{0} / 2$ (valid for this one-component model),
$$\varpi=\frac{c}{H_{0}} \frac{1}{1+z} \frac{1}{q_{0}^{2}}\left[q_{0} z-\left(1-q_{0}\right)(\sqrt{1+2 q_{0} z}-1)\right]$$.
Show that this is valid for a flat, an open, and a closed one-component universe of pressureless dust. Note that Eq. ( 198 ) in the previous problem and Eq. ( 199 ) are in agreement when $\Omega_{0}=1$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

Show that Eq. (178) is the Taylor-series expansion of Eq. (168) about $z=0$.
$$I(z) \equiv \int_{0}^{z} \frac{d z^{\prime}}{\sqrt{\Omega_{m, 0}\left(1+z^{\prime}\right)^{3}+\Omega_{\mathrm{rel}, 0}\left(1+z^{\prime}\right)^{4}+\Omega_{\Lambda, 0}+\left(1-\Omega_{0}\right)\left(1+z^{\prime}\right)^{2}}}.$$
$$I(z)=\int_{0}^{z}\left\{1-\left(1+q_{0}\right) z^{\prime}+\left[\frac{1}{2}+2 q_{0}+\frac{3}{2} q_{0}^{2}+\frac{1}{2}\left(1-\Omega_{0}\right)\right] z^{\prime 2}+\cdots\right\} d z^{\prime}$$

Eduard Sanchez

Numerade Educator

Problem 52

In this problem, you will carry out a general derivation of Eq. ( 180 ) without reference to any specific model of the universe.
$$\varpi \simeq \frac{c z}{H_{0}}\left[1-\frac{1}{2}\left(1+q_{0}\right) z\right] \quad(\text { for } z \ll 1)$$.
(a) Expand the scale factor, $R(t),$ in a general Taylor series about the present time, $t_{0},$ and obtain
$$=1-H_{0}\left(t_{0}-t\right)-\frac{1}{2} H_{0}^{2} q_{0}\left(t_{0}-t\right)^{2}+\cdots$$
(b) Show that $1 / R(t)$ is then given by
$$\frac{1}{R(t)}=1+H_{0}\left(t_{0}-t\right)+H_{0}^{2}\left(1+\frac{1}{2} q_{0}\right)\left(t_{0}-t\right)^{2}+\cdots$$.
[Hint: Use long division to divide 1 by $R(t)$.]
(c) Use Eq. $(\quad 142), 1 / R(t)=1+z,$ with the result of part (b) to write the expansion for $z$ about the present time. Now solve for $t_{0}-t$ and express the result as a series in $z$ to get
$$\begin{aligned}
&t_{0}-t=\frac{z}{H_{0}}-\left(1+\frac{1}{2} q_{0}\right) \frac{z^{2}}{H_{0}}+\cdots\\
&\frac{1}{R\left(t_{e}\right)}=\frac{\lambda_{0}}{\lambda_{e}}=1+z.
\end{aligned}$$
(d) Consider a photon that is emitted at comoving coordinate $w$ at time $t$ and received on Earth at the present time $t_{0} .$ Use an equation similar to Eq. (138) to find an approximate expression for $w .$ You need only use the first two terms in the expansion in part (b) for $1 / R(t)$ in the left-hand integral. For the right-hand integral, use a two-term Taylor series for $1 / \sqrt{1-k \varpi^{2}} .$ You should find that
$$\begin{aligned}
&\varpi=c\left(t_{0}-t\right)\left[1+\frac{1}{2} H_{0}\left(t_{0}-t\right)\right]+\cdots\\
&\int_{t_{e}}^{t_{0}} \frac{c d t}{R(t)}=\int_{0}^{\varpi_{e}} \frac{d \varpi}{\sqrt{1-k w^{2}}}
\end{aligned}$$
(e) By substituting the expression for $t_{0}-t$ from part (c) into this equation for $w,$ show that, to second order in $z$,
$$\varpi=\frac{c z}{H_{0}}\left[1-\frac{1}{2}\left(1+q_{0}\right) z\right]+\cdots$$.
This result is very important because it does not rely on any particular model of the universe. It is valid even if the cosmological constant, $\Lambda,$ is not equal to zero.

Sana Riaz

Numerade Educator

Problem 53

Use Eq. ( 196 ) from Problem 41 to show that the luminosity distance may be written as
$d_{L}=\frac{c(1+z)}{H_{0} \sqrt{\left|1-\Omega_{0}\right|}} \operatorname{sinn}\left\{\sqrt{\left|1-\Omega_{0}\right|} \int_{0}^{z} \frac{d z^{\prime}}{\left[\sum_{i} \Omega_{i, 0}\left(1+z^{\prime}\right)^{3\left(1+w_{i}\right)}+\left(1-\Omega_{0}\right)\left(1+z^{\prime}\right)^{2}\right]^{1 / 2}}\right\},$
cosmology: Problem Set
where
\[
\begin{aligned}
\operatorname{sinn}(x) & \equiv \sinh (x) & & \text { if } & \Omega_{0}<1, \\
&=x & & \text { if } & \Omega_{0}=1, \\
& \equiv \sin (x) & & \text { if } & \Omega_{0}>1.
\end{aligned}\]
This is Eq. (1) of Garnavich et al. (1998), which explains how this equation may be used with the distance modulus, Eq. ( 186 ), to place limits on the value of $w_{\Lambda}$ for the equation of state of dark energy. The WMAP result is that $w_{\Lambda}<-0.78$.
$$m-M=5 \log _{10}\left(d_{L} / 10 \mathrm{pc}\right)$$

Andrew Eddins

Emory University

Problem 54

Assume a flat, one-component universe of pressureless dust for this problem.
(a) Show that the angular diameter observed for an extended object of linear diameter $D$ at redshift $z$ is
$$\theta=\frac{H_{0} D}{2 c} \frac{(1+z)^{3 / 2}}{\sqrt{1+z}-1}$$.
(b) Find the value of the redshift for which $\theta$ is a minimum. Compare your result with Fig. 30.
(c) What is the smallest value of $\theta$ when observing a cluster of galaxies with a diameter of 1 Mpc? Use $h=0.71$.

Mayukh Banik

Numerade Educator

Problem 55

Model the hot intragalactic gas in a rich cluster of galaxies as a homogeneous sphere of radius $R$ and temperature $T_{e}$. Let $F_{X}$ be the $X$ -ray flux observed for the gas, and let $\theta$ be the angular diameter of the gas, as observed from Earth.
(a) Show that if the Sunyaev-Zel'dovich effect, $\Delta T / T_{0},$ is measured for the cluster, then the Hubble constant may be calculated as
cosmology: Problem Set
\[H_{0}=C f(z) \frac{F_{X} T_{e}^{3 / 2}}{\theta\left(\Delta T / T_{0}\right)^{2}}.\]
where $f(z)$ is a function of the redshift $z$ of the cluster and $C$ is a constant factor you must determine.
(b) Use the data in Fig. $\quad 12$ for the clusters, together with $\theta=46^{\prime \prime}$ and $k T_{e}=8.0 \mathrm{keV}$ for Abell $697,$ and $\theta=69^{\prime \prime}$ and $k T_{e}=7.2 \mathrm{keV}$ for Abell 2218 (from Jones et al., 2005 ) to evaluate $h$, the Hubble parameter. To crudely compensate for the gas not being isothermal, take $T_{e}$ to be half the value obtained from the data. How do your answers compare with the WMAP value of $h=0.71 ?$
FIGURE $12 \quad$ Radio contours showing the Sunyaev-Zel'dovich effect superimposed on ROSAT images of the clusters Abell $697(\Delta T=1047 \mu \mathrm{K}, z=0.282)$ and Abell $2218(\Delta T=797 \mu \mathrm{K}$ $z=0.171$ ). The contour interval is $60 \mu \mathrm{Jy}$ (left) and $80 \mu \mathrm{Jy}$ (right). The dashed contours indicate a decrease in the received radio flux density. (Figure adapted from Jones et al., $M N R A S, 357,518$ $2005 .)$

Hubert Agamasu

Numerade Educator

Problem 56

In Eq. $(166),$ the limits of integration on the left-hand side span a time interval that lies entirely in the past. Use Eq. (132) to approximate the scale factor on the left-hand side. On the right-hand side, consider the case of $t_{\max }>t_{0},$ so the limits of integration span a time interval that lies entirely in the future. Use Eq. ( 133 ) for the scale factor on the right-hand side. [Note that very distant objects have already emitted the last photon that will ever reach us $\left.\left(t_{\max }<t_{0}\right), \text { so we must restrict our attention to nearer objects to ensure that } t_{\max }>t_{0} \cdot\right]$ Show that
\[
t_{\mathrm{mva}} \simeq \frac{t_{n}}{\sqrt{\Omega_{\Lambda, 0}}} \ln \left[\left(\frac{\sqrt{\Omega_{m, 0}}}{2 \sqrt{\Omega_{\Lambda, 0}}}\right)^{1 / 3}\left(\frac{\sqrt{1+z}}{\sqrt{1+z}-1}\right)\right].
\]
Using WMAP values, what is the largest redshift for which $t_{\text {rma }}>t_{H}$ ? Find the maximum visible age, in units of $t_{H},$ for sources at values of $z$ of $0.1,0.5,1,$ and 1.5.
cosmology: Problem Set
\[
\begin{array}{c}
R(t) \simeq\left(\frac{3}{2} H_{0} t \sqrt{\Omega_{m, 0}}\right)^{2 / 3}=\left(\frac{3 \sqrt{\Omega_{m, 0}}}{2}\right)^{2 / 3}\left(\frac{t}{t_{H}}\right)^{2 / 3}, \\
R(t) \simeq\left(\frac{\Omega_{m, 0}}{4 \Omega_{\Lambda, 0}}\right)^{1 / 3} e^{H_{0} t \sqrt{\Omega_{\Lambda, 0}}}. \\
\int_{t_{e}}^{t_{0}} \frac{d t}{R(t)}=\int_{t_{1}}^{t_{f}} \frac{d t}{R(t)}.
\end{array}
\]

Jacquelinne S. Mejia Sandoval

Jacquelinne S. Mejia Sandoval

Numerade Educator

Problem 57

Question is missing.

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Elizabeth Edge

Numerade Educator

Problem 58

Use WMAP values for this problem.
(a) $\mathrm{On}$ a single graph, plot the luminosity distance, the present proper distance, the angular diameter distance, the relativistic Hubble law distance estimate the below equation, and the non-relativistic Hubble law distance shown below for values of $z$ between 0 and 4 Express these distances in units of $c / H_{0}$.
$$d \simeq \frac{c}{H_{0}} \frac{(z+1)^{2}-1}{(z+1)^{2}+1},$$
$$d=\frac{c z}{H_{0}}$$.
(b) From your results, determine the value of $z$ when the relativistic Hubble law distance estimate differs from the proper distance by more than $10 \%$.

Natalie Anderson

Natalie Anderson

Numerade Educator

Problem 59

A distant radio galaxy, $8 \mathrm{C} 1435+63,$ has a redshift of $z=4.25 .$ Assume WMAP values for this problem.
(a) How old was the universe at this redshift? Express your answer both in terms of years and as a fraction of the present age of the universe.
(b) What is the present proper distance (in $\mathrm{Mpc}$ ) to $8 \mathrm{C} 1435+63 ?$
(c) What was the proper distance (in Mpc) to $8 \mathrm{C} 1435+63$ when its light was emitted?
(d) What is the luminosity distance to $8 \mathrm{C} 1435+63 ?$
(e) What is the angular diameter distance to $8 \mathrm{C} 1435+63 ?$
(f) The angular diameter of the nucleus of $8 \mathrm{C} 1435+63$ is about $5 "$. What is the linear diameter of the galaxy (in units of $\mathrm{kpc}$ )?
(g) Suppose the galaxy's redshift were $z=1 .$ What would its linear diameter be [using the same angular diameter as in part (d)]? Further information about $8 \mathrm{C} 1435+63,$ which may be the progenitor of a cD elliptical galaxy, can be found in Spinrad, Dey, and Graham (1995).

Farhanul Hasan

Numerade Educator