% EU5Q3.TEX, 8.286 QUIZ 3, Spring 1998, corrected copy:
%   Corrections:
%     Black-body formula: p=-1/3 u -> p = 1/3 u
%     Number of points that various problems were worth
\input gmacro.tex
\input euemacro.tex

\headline{\smallsl 8.286 QUIZ 3, Spring 1998 \hfill p.~\folio}
\nopagenumbers

\def\rhat{\hat r}
\def\ihat{\hat \imath}
\def\jhat{\skew{1}\hat \jmath}
\def\khat{\hat k}
\def\points#1{{\it (#1 points)}}
\def\footfont{\small}

{\baselineskip=10.5pt \parskip=0pt
\centerline{MASSACHUSETTS INSTITUTE OF TECHNOLOGY}
\centerline{Physics Department}
\smallskip
\line{Physics 8.286: The Early Universe
\hfill April 28, 1998}
\line{Prof.~Alan Guth \hfill Corrected Version}
\medskip
\centerline{\large\bf QUIZ 3}
}

\head{USEFUL INFORMATION:}

{\narrow{30pt} \parindent=0pt \parskip=\medskipamount
\def\dsp{\displaystyle}
{\bf COSMOLOGICAL EVOLUTION:}
  $$ \left({\dot R \over R}\right)^2 = {8\pi \over 3}G\rho - {k
     c^2 \over R^2} $$
  $$ \ddot R = - {4\pi \over 3}G \left(\rho + {3p \over
     c^2}\right) R $$
\medskip

{\bf EVOLUTION OF A FLAT ({\setmath $\Omega \equiv \rho / \rho_c =
1$}) UNIVERSE:} 
  $$\eqalign{R(t) &\propto t^{2/3} \qquad
     \hbox{(matter-dominated)}  \cr
   R(t) &\propto t^{1/2} \qquad \hbox{(radiation-dominated)} \cr}
     $$
\medskip

{\bf EVOLUTION OF A MATTER-DOMINATED}\par \vskip -\parskip
{\bf UNIVERSE:}\par
  $$ \left({\dot R \over R}\right)^2 = {8\pi \over 3}G\rho - {k
     c^2 \over R^2} $$
  $$ \ddot R = - {4\pi \over 3}G \rho  R $$
  $$ \rho(t) = {R^3(t_i) \over R^3(t)}\, \rho(t_i) $$
\medskip
\table \spacea=20pt \baselineskip=18pt
Closed ($\Omega > 1$): &$\dsp ct = \alpha(\theta-\sin \theta) \ ,$
                &$\dsp {R \over \sqrt{k}} = \alpha( 1 - \cos\theta) \ ,$
                &where $\dsp \alpha\equiv {4\pi\over 3} {G \rho %
                  R^3 \over k^{3/2} c^2}$ 
Open ($\Omega < 1$): &$\dsp ct = \alpha \left(\sinh\theta-\theta\right)$ 
                &$\dsp {R \over \sqrt{\kappa}}= \alpha\left( %
                  \cosh\theta-1\right) \ , $
%                &where $\dsp \alpha\equiv {4\pi\over 3} {G \rho %
%                  R^3 \over \kappa^{3/2} c^2} \ .$
                &$\eqalign{\hbox{where\ } \dsp \alpha &\equiv %
                  {4\pi\over 3} {G \rho R^3 \over \kappa^{3/2} %
                  c^2} \ , \cr \kappa &\equiv - k \ .\cr}$

\medskip

{\bf COSMOLOGICAL REDSHIFT:}
  $$1 + Z \equiv {\lambda_{\rm observed} \over \lambda_{\rm
     emitted}} = {R(t_{\rm observed}) \over R(t_{\rm emitted})}
     $$ 

\medskip

{\bf ROBERTSON-WALKER METRIC:}
  $$ds^2 = -c^2 \, d \tau^2 = -c^2 \, dt^2 + R^2(t) \left\{{dr^2
     \over 1 - kr^2} + r^2 \left(d\theta^2 + \sin^2 \theta \,
     d\phi^2\right)\right\} $$

{\bf SCHWARZSCHILD METRIC:} 
  $$\eqalign{ds^2 = - c^2 d \tau^2 &= - \left(1 - {2 G M \over r
     c^2}\right) c^2 dt^2 + \left(1 - {2 G M \over r
     c^2}\right)^{-1} dr^2 \cr
   &\qquad + r^2 d \theta^2 + r^2 \sin^2 \theta \, d \phi^2 \ ,
     \cr} $$
\medskip

{\bf GEODESIC EQUATION:}
  $$\eqalign{{d\over d\lambda} \left\{ g_{ij} {dx^j\over
     d\lambda}\right\} &= {1\over 2} \left( \partial_i
     g_{k\ell}\right) {dx^k\over d\lambda} \ {dx^\ell\over
     d\lambda} \cr 
   \hbox{or:} \qquad {d \over d\tau} \left\{g_{\mu \nu} {dx^\nu
     \over d\tau} \right\} &= {1 \over 2} \left(\partial_\mu
     g_{\lambda \sigma} \right) {d x^\lambda \over d\tau} {d
     x^\sigma \over d\tau} \cr}$$

\medskip
{\bf COSMOLOGICAL CONSTANT:} 
  $$p_{\rm vac} = - \rho_{\rm vac} c^2 \qquad \rho_{\rm vac} =
     {\Lambda c^2 \over 8 \pi G} $$
\qquad where $\Lambda$ is the cosmological constant.

\medskip
{\bf PHYSICAL CONSTANTS:}
  $$\eqalignno{
     &\eqalign{k = \hbox{Boltzmann's constant} &= 1.381 \times
          10^{-16}\,\hbox{erg}/^\circ \hbox{K} \cr 
        &= 8.617 \times 10^{-5}\,\hbox{eV}/^\circ \hbox{K}\ ,\cr}
        \cr
     \noalign{\medskip} 
     &\eqalign{\hbar = {h\over 2\pi} &= 1.055 \times
          10^{-27}\;\hbox{erg-sec} \cr
        &= 6.582 \times 10^{-16}\;\hbox{eV-sec}\ ,\cr}\cr
     \noalign{\medskip}
     &c = 2.998 \times 10^{10}\ \hbox{cm/sec} \cr
     \noalign{\medskip}
     &1\ \hbox{eV} = 1.602 \times 10^{-12}\ \hbox{erg}\ .\cr}$$

\medskip
{\bf BLACK-BODY RADIATION:} 
  \medskip
  $$\eqalign{  
   u &= g {\pi^2\over 30}\ {(kT)^4\over (\hbar c)^3} \qquad
   p = {1 \over 3} u \qquad \rho = u/c^2 \cr
   \noalign{\medskip}
   n &= g^* {\zeta (3)\over \pi^2}\ {(kT)^3\over (\hbar c)^3}\qquad
   s = g {2\pi^2\over 45}\ {k^4 T^3\over (\hbar c)^3}\ , \cr}$$
\qquad where
  $$\eqalign{
   g &\equiv \left\{\vcenter{
     \hbox{1 per spin state for bosons (integer spin)}
     \hbox{7/8 per spin state for fermions (half-integer spin)}}
          \right. \cr          
     % \}  (added to balance parentheses for screen editor)
   \noalign{\medskip}
   g^* &\equiv \left\{\vcenter{
     \hbox{1 per spin state for bosons}
     \hbox{3/4 per spin state for fermions ,}} \right.
     % \}  (added to balance parentheses for screen editor)
     \cr}$$
\qquad and
  $$\zeta(3) = {1\over 1^3} + {1\over 2^3} + {1\over 3^3} +
     \cdots \approx 1.202 \ .$$

\medskip
{\bf EVOLUTION OF A FLAT RADIATION-DOMINATED UNIVERSE:}
  $$kT= \left( {45\hbar^3 c^5\over 16\pi^3 gG}\right)^{1/4}\
     {1\over \sqrt{t}}$$
\indent \hangindent 30pt For $m_\mu = 106 \hbox{ MeV} \gg kT \gg m_e = 0.511
     \hbox{ MeV}$, $g=10.75$ and then
  $$kT = {0.860\;\hbox{MeV}\over \sqrt{t\ (\hbox{in
     sec})}}$$

% \medskip
% {\bf CHEMICAL EQUILIBRIUM:}
%  $$ n_i = g_i {(2 \pi m_i k T)^{3/2} \over (2 \pi \hbar)^3} \,
%      e^{(\mu_i - m_i c^2)/kT} \ .$$
% \qquad $\vtop{\hbox{$\eqalign{\hbox{where } n_i &= \hbox{number
%      density of particle}\cr
%              g_i &= \hbox{number of spin states of particle}\cr
%              m_i &= \hbox{mass of particle}\cr
%              \mu_i &= \hbox{chemical potential}\cr}$}}$
% 
% \hang For any reaction, the sum of the $\mu_i$ on the left-hand
%      side of the reaction equation must equal the sum of the
%      $\mu_i$ on the right-hand side.  Formula assumes gas is
%      nonrelativistic ($kT \ll m_i c^2$) and dilute ($n_i \ll {(2
%      \pi m_i k T)^{3/2} / (2 \pi \hbar)^3}$).
 
}

\vfill\eject

\head{PROBLEM 1: DID YOU DO THE READING? \points{20}}

In Silk's chapter on ``The First Millisecond,'' he discusses a
number of types of particles that may have been produced.  One of
those particles is a hypothetical entity called a {\it photino}. 

\item{(a)} \points{5} The photino is related to the photon by a
symmetry.  It is plausible that the underlying laws of physics
exhibit this symmetry, but at the present time there is no firm
evidence that this is true.  What is the name of this symmetry?

\item{(b)} \points{5} If photinos do make up a significant amount
of the dark matter, they would occasionally undergo annihilation. 
Silk mentions two possible products of such annihilations that
might be detectable.  Name one of them.

\noindent In Weinberg's chapter titled ``A Historical
Diversion,'' he discusses the history of nucleosynthesis.  

\item{(c)} \points{5} In 1964 an important paper was published,
concluding that the large amount of helium in the universe could
not have been synthesized in ordinary stars, but could have been
produced in the early stages of a ``big bang.''  Surprisingly,
one of the authors of this paper is better known for being one of
the strongest advocates of the steady-state theory.  Name either
of the two authors of this paper. 

%The following problem was taken from Problem 1(b), 1998 Review
%   Problems 3.
\noindent Finally, please answer the following question from the
Review Problems for Quiz 3:

\item{(d)} \points{5} In Chapter 6 of {\bf The First Three
Minutes}, Steven Weinberg discusses three reasons why the
importance of a search for a $3^\circ\,$K microwave radiation
background was not generally appreciated in the 1950s and early
1960s.  Choose those three reasons from the following list:

\itemitem{(i)} The earliest calculations erroneously predicted a
cosmic background temperature of only about $0.1^\circ\,$K, and
such a background would be too weak to detect.

{\parskip=\smallskipamount
\itemitem{(ii)} There was a breakdown in communication between
theorists and experimentalists.

\itemitem{(iii)} It was not technologically possible to detect a
signal as weak as a $3^\circ\,$K microwave background until about
1965.

\itemitem{(iv)} Since almost all physicists at the time were
persuaded by the steady state model, the predictions of the big
bang model were not taken seriously.

\itemitem{(v)} It was extraordinarily difficult for physicists
to take seriously {\it any} theory of the early universe.

\itemitem{(vi)} The early work on nucleosynthesis by Gamow, Alpher,
Herman, and Follin, et al., had attempted to explain the origin
of all complex nuclei by reactions in the early universe.  This
program was never very successful, and its credibility was
further undermined as improvements were made in the alternative
theory, that elements are synthesized in stars.
\par}


\vfill\eject

\head{PROBLEM 2: A REVISED THERMAL HISTORY OF THE UNIVERSE
\points{35}} 

\noindent {\it As promised there is one question verbatim from
the homework or review problems--- specifically, from Problem Set
4.  Part (a) of this question calls for a numerical answer, but
since you were not told to bring calculators, you need not carry
out the arithmetic.  Your answer should be expressed, however, in
``calculator-ready'' form--- that is, it should be an expression
involving pure numbers only (no units), with any necessary
conversion factors included, and with the units of the final
answer specified at the end.  (For example, if you were asked how
far a light pulse in vacuum travels in 5 minutes, you could
express the answer as $2.998 \times 10^8 \times 5 \times 60$
meters.)  Similarly, in part (d) you may express your answer in
terms of a trigonometric or exponential function, which you need
not evaluate.}

Suppose a New Theory of the Weak Interactions (NTWI) was
proposed, in which the weak interactions are somewhat weaker than
in the standard model.  This problem will deal with the
cosmological consequences of such a theory.

The NTWI will predict that the neutrinos in the early universe
will decouple at a higher temperature than in the standard model.
Suppose that this decoupling takes place at $kT \approx 200$ MeV\.
This means that when the neutrinos cease to be thermally coupled
to the rest of matter, the hot soup of particles would contain
not only photons, neutrinos, and $e^+$-$e^-$ pairs, but also
$\mu^+$, $\mu^-$, $\pi^+$, $\pi^-$, and $\pi^0$ particles.  (The
muon, you will recall, is a particle that behaves almost
identically to an electron, except that its rest energy is 106
MeV.  The pions are the lightest of the mesons, with zero angular
momentum and rest energies of 135 MeV and 140 MeV for the neutral
and charged pions, respectively.)

\item{(a)} \points{10} According to the standard particle physics      
model, what is the mass density $\rho $  of the universe when   
$kT \approx  200$ MeV?  What is the value of $\rho $  at this
temperature, according to NTWI?

\item{(b)} \points{10} According to the standard model, the
temperature today of the thermal neutrino background should be
$(4/11)^{1/3}T_\gamma $, where $T_\gamma \approx 2.7^\circ\,$K is
the temperature of the thermal photon background.  What does the
NTWI predict for the temperature of the thermal neutrino
background?
 
\item{(c)} \points{5} According to the standard model, what is
the ratio today of the number density of thermal neutrinos to the
number density of thermal photons?  What is this ratio according
to NTWI?

\item{(d)} \points{10} Since the reactions that interchange
protons and neutrons involve neutrinos, these reactions "freeze
out" at roughly the same time as the neutrinos decouple.  At
later times the only reaction which effectively converts neutrons
to protons is the free decay of the neutron.  Despite the fact
that neutron decay is a weak interaction, we will assume that it
occurs with the usual 15 minute mean lifetime.  Would the helium
abundance predicted by the NTWI be higher or lower than the
prediction of the standard model?  To within 5 or 10\%, what
would the NTWI predict for the percent abundance (by weight) of
helium in the universe? 

\vfill\eject

\head{PROBLEM 3: THE EFFECT OF PRESSURE ON COSMOLOGICAL EVOLUTION
\points{20}}

A radiation-dominated universe behaves differently from a
matter-dominated universe because the pressure of the radiation
is significant.  In this problem we explore the role of pressure
for several fictitious forms of matter.

\item{(a)} \points{10} For the first fictitious form of matter,
the mass density $\rho$ decreases as the scale factor $R(t)$
grows, with the relation
  $$\rho(t) \propto {1 \over R^5(t)} \ .$$
What is the pressure of this form of matter?  {\it [Hint: the
answer is proportional to the mass density.]}

\item{(b)} \points{5} Find the behavior of the scale factor
$R(t)$ for a flat universe dominated by the form of matter
described in part (a).  You should be able to determine the
function $R(t)$ up to a constant factor.

\item{(c)} \points{5} Now consider a universe dominated by a
different form of fictitious matter, with a pressure given by
  $$p={1 \over 6} \rho c^2 \ .$$
As the universe expands, the mass density of this form of matter
behaves as
  $$\rho(t) \propto {1 \over R^n(t)} \ .$$
Find the power $n$.

\vfill\eject

\head{PROBLEM 4: PROPERTIES OF BLACK-BODY RADIATION \points{25}}

In answering the following questions, remember that you can refer
to the formulas at the front of the exam.  Since you were not
asked to bring calculators, you may leave your
answers in the form of algebraic expressions, such as
$\pi^{32}/\sqrt{5 \zeta(3)}$.

\item{(a)} \points{5} For the black-body radiation (also called
thermal radiation) of photons at temperature $T$, what is the
average energy per photon?

\item{(b)} \points{5} For the same radiation, what is the average
entropy per photon?

\item{(c)} \points{5} Now consider the black-body radiation of a
massless boson which has spin zero, so there is only one spin
state.  Would the average energy per particle and entropy per
particle be different from the answers you gave in parts (a) and
(b)?  If so, how would they change?

\item{(d)} \points{5} Now consider the black-body radiation of
electron neutrinos.  These particles are fermions with spin 1/2,
and we will assume that they are massless and have only one
possible spin state.  What is the average energy per particle for
this case?

\item{(e)} \points{5} What is the average entropy per particle
for the black-body radiation of neutrinos, as described in part
(d)? 

% \centerline{\ugraphbx{}{harri10a}{1213\pix}{1381\pix}}

%\vfill\eject
%\vbox{} \vfill
%\hbox{\hskip-0.74truein \ugraphbx{}{non2a}{2418\pix}{1266\pix}}

%\centerline{\ugraphbx{}{non2b}{1518\pix}{1250\pix}}
%\vskip 0pt plus 1.2 fill
\bye

