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问题 1.

Problem 4. (30 points)
A bit about inflation. Let’s assume for the moment that inflation began when the universe was $10^{-36} \mathrm{~s}$ old, and ended when it was $10^{-33} \mathrm{~s}$. I want you to suppose that before inflation, the density of the universe was critical, with all of the contents being made up of a fluid with $w=1 / 3$.
a) ( 3 points) Write down the Friedman equation for the period before inflation.
b) ( 6 points) What was the Hubble constant as a function of time?
c) ( 3 points) What was the Hubble constant at the beginning of inflation?



a) $w=1 / 3$ describes radiation, meaning that the Friedmann equation will only have one term corresponding to $\Omega_r$.
$$
\frac{1}{H_0^2}\left(\frac{\dot{a}}{a}\right)^2=\frac{1}{a^4}
$$
3
b) Simplifying the Friedmann equation, we have.
$$
H_0=a \dot{a}
$$
Integrating this by separation of variables, we find that
$$
H_0 \Delta t=\frac{1}{2} \Delta\left(a^2\right)
$$
c) Let inflation begin when $a=1$, and integrate from $a=0$ and $t=0$. This implies that
$$
\begin{gathered}
H_0=\frac{1}{2} \frac{\Delta a^2}{\Delta t} \
H_0=\frac{1}{2} \frac{1}{\left(10^{-36} \mathrm{~s}\right)} \
H_0=5 \times 10^{35} \mathrm{~s}^{-1}
\end{gathered}
$$

approx 0.8
$$

问题 2.

d) ( 3 points) If the density is critical, what was the effective mass density of the field?
e) (3 points) What was the equivalent temperature? Use $a_B T^4=\rho_c c^2$.
f) (3 points) What energy (in GeV) does that correspond to?

d) Equation 4.31 relates $H_0$ to the critical density.
$$
\begin{gathered}
\rho_{c, 0}=\frac{3 H_0^2}{8 \pi G} \
\rho_{c, 0} \approx \frac{1}{8} \frac{\left(5 \times 10^{35} \mathrm{~s}^{-1}\right)^2}{\left(6.11 \times 10^{-11} \mathrm{~s}^{-2} \cdot\left(\mathrm{kg} / \mathrm{m}^3\right)^{-1}\right)} \
\rho_{c, 0} \approx 5 \times 10^{80} \frac{\mathrm{kg}}{\mathrm{m}^3}
\end{gathered}
$$
e) Equation 2.29 gives us a value for $a_B$, which relates the blackbody temperature of the universe to the energy density:
$$
\begin{gathered}
a_B T^4=\rho_{c, 0} c^2 \
T=\left(\frac{\rho_{c, 0} c^2}{a_B}\right)^{1 / 4} \
T=\left(\frac{\left(5 \times 10^{80} \frac{\mathrm{kg}}{\mathrm{m}^3}\right)\left(3 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}\right)^2}{7.566 \times 10^{-16} \frac{\mathrm{J}}{\mathrm{m}^3} \cdot \mathrm{K}^{-4}}\right)^{1 / 4} \
T \approx 2 \times 10^{28} \mathrm{~K}
\end{gathered}
$$
f) Using the temperature we find,
$$
E=k_B T
$$
$$
E=\left(8.617 \times 10^{-14} \mathrm{GeV} \cdot \mathrm{K}^{-1}\right)\left(2 \times 10^{28} \mathrm{~K}\right)
$$
$$
E \approx 2 \times 10^{15} \mathrm{GeV}
$$

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