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Problem 3: Radiation Universe OUR UNIVERSE ALSO contains a homogenous, isotropic distribution of radiation called the Cosmic Microwave Background. Radiation refers to particles with zero rest mass like photons7. The pressure of radiation is $p = \epsilon/3$ so $w = 1/3$. 3a) Consider a flat universe ($\Omega_0 = 1$) with only radiation ($w = 1/3$) within it. Solve for the time-dependent scale factor $a(t)$. 3b) Find an expression for the current age of the Universe for a flat, radiation dominated Universe. Is it greater than or less than $t_0$ for the flat matter-dominated case? 3c) The flat universe ($\Omega_0 = 1$) will keep expanding forever, but a closed ($\Omega_0 > 1$) radiation dominated Universe will at some point stop expanding and turn around, and then start shrinking. For a closed radiation dominated Universe find an expression for the maximum size that the scale factor ever reaches (as a function of $\Omega_0$, the scaled radiation density at the present time). 3d) Optional; not graded The solution for $a(t)$ for the case $\Omega_0 \neq 1$ is harder to work out, but can be done in a straightforward fashion and put in a relatively simple form. Show that for an open ($\Omega_0 < 1$) radiation dominated Universe $a(t) = \left[ (H_0 t + \sqrt{\Omega_0^2 - \Omega_0} )^2 - \Omega_0 \right]^{1/2}$ (11) Neutrinos have a rest mass but it is very small, so they too act as a background of radiation. *You do not need to integrate to solve for $a(t)$ here; you can use the dynamical equation before integration to find the turnaround point of the Universe. Think about the speed at which $a(t)$ is changing at this turnaround point.

          Problem 3: Radiation Universe
OUR UNIVERSE ALSO contains a homogenous, isotropic distribution
of radiation called the Cosmic Microwave Background. Radiation
refers to particles with zero rest mass like photons7. The pressure of
radiation is $p = \epsilon/3$ so $w = 1/3$.
3a) Consider a flat universe ($\Omega_0 = 1$) with only radiation ($w = 1/3$)
within it. Solve for the time-dependent scale factor $a(t)$.
3b) Find an expression for the current age of the Universe for a flat,
radiation dominated Universe. Is it greater than or less than $t_0$ for the
flat matter-dominated case?
3c) The flat universe ($\Omega_0 = 1$) will keep expanding forever, but a
closed ($\Omega_0 > 1$) radiation dominated Universe will at some point stop
expanding and turn around, and then start shrinking. For a closed
radiation dominated Universe find an expression for the maximum
size that the scale factor ever reaches (as a function of $\Omega_0$, the scaled
radiation density at the present time).
3d) Optional; not graded The solution for $a(t)$ for the case $\Omega_0 \neq 1$
is harder to work out, but can be done in a straightforward fashion
and put in a relatively simple form. Show that for an open ($\Omega_0 < 1$)
radiation dominated Universe
$a(t) = \left[ (H_0 t + \sqrt{\Omega_0^2 - \Omega_0} )^2 - \Omega_0 \right]^{1/2}$
(11)
Neutrinos have a rest mass but it
is very small, so they too act as a
background of radiation.
*You do not need to integrate to solve
for $a(t)$ here; you can use the dynamical
equation before integration to find
the turnaround point of the Universe.
Think about the speed at which $a(t)$ is
changing at this turnaround point.

Problem 3: Radiation Universe
OUR UNIVERSE ALSO contains a homogenous, isotropic distribution
of radiation called the Cosmic Microwave Background. Radiation
refers to particles with zero rest mass like photons7. The pressure of
radiation is p = ϵ/3 so w = 1/3.
3a) Consider a flat universe (Ω0 = 1) with only radiation (w = 1/3)
within it. Solve for the time-dependent scale factor a(t).
3b) Find an expression for the current age of the Universe for a flat,
radiation dominated Universe. Is it greater than or less than t0 for the
flat matter-dominated case?
3c) The flat universe (Ω0 = 1) will keep expanding forever, but a
closed (Ω0 > 1) radiation dominated Universe will at some point stop
expanding and turn around, and then start shrinking. For a closed
radiation dominated Universe find an expression for the maximum
size that the scale factor ever reaches (as a function of Ω0, the scaled
radiation density at the present time).
3d) Optional; not graded The solution for a(t) for the case Ω0 ≠ 1
is harder to work out, but can be done in a straightforward fashion
and put in a relatively simple form. Show that for an open (Ω0 < 1)
radiation dominated Universe
a(t) = [ (H0 t + √(Ω0^2 - Ω0) )^2 - Ω0 ]^1/2
(11)
Neutrinos have a rest mass but it
is very small, so they too act as a
background of radiation.
*You do not need to integrate to solve
for a(t) here; you can use the dynamical
equation before integration to find
the turnaround point of the Universe.
Think about the speed at which a(t) is
changing at this turnaround point.

Added by Jose Ignacio M.

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