[SOLVED] Physics 396 homework set 10

1. A distant galaxy has a redshift of z = 0.2. According to Hubbles law, how far away was
the galaxy, in light years, when the light was emitted if the Hubble parameter is
72 km/s/Mpc?2. When the flat Robertson-Walker line element, given by Eq. (18.1) in your text, and the
perfect fluid energy-momentum tensor are inserted into the Einstein equations of GR, one
arrives at a set of non-linear, ordinary dierential equations of the form
8 = 3
a 2
a2 (1)
8p =

2
a
a
+
a 2
a2

(2)
0= + 3
a
a
( + p) (3)
where and p are the density and pressure, respectively.It is noted that a dot represents
a time derivative, a double-dot represents two time derivatives. Also notice that
Eq. (3) is not a field equation, but rather a conservation statement. These dierential
equations represent the Standard Model of Cosmology.a) Show that Eq. (3) is a redundant equation by inserting Eqs. (1) and (2) for and p
into Eq. (3). Hint: when performing derivatives, notice that a 2
a2 = a
a
2
.Having 2 equations and 3 unknowns (, p, and a), we have an underdetermined system
of equations. At this stage one usually employs an equation of state (EoS) of the form
p = w (4)
where w is called the equation of state parameter.1 In general, w can depend on time,
however, here we will treat it as a constant. We now have 3 equations and 3 unknowns.At this point, Eq. (3) will yield an expression relating the density to the scale factor.
1Notice: w = 0 corresponds to pressureless matter, w = 1/3 corresponds to radiation, and w = 1 corresponds
to vacuum energy.8Tp
This will allow us to decouple Eqs. (1) and (2), thus yielding two dierential equations
involving only the scale factor.b) Using Eqs. (4) and (3), find = (a). Hint: eliminate the pressure from Eq. (3) in
favor of the density, separate variables, and integrate.
Answer:
(t) = 0a(t)
3(1+w) (5)c) Using Eqs.(1)-(3) and (4), obtain an expression for the acceleration of the Universe,
a/a , in terms of the density of the Universe.
d) For what values of w does one obtain an accelerated expansion of the Universe?3. Reconsider the flat Friedman-Robertson-Walker field equations of cosmology, given by
Eqs. (1) (3).
a) Show that a(t) = a0 t
2/3(1+w) is a solution to Eq. (1).
b) Show that a(t) = a0 t
2/3(1+w) is a solution to Eq. (2).c) Solve Eqs. (1) and (2) for when w = 1.
Hint: Use the fact that
d
dt a
a

= a
a a 2
a2 . (6)4. In class, we arrived at an equation of motion describing the scale factor of the form
1
2H2
0
da
dt 2
+ Uef f (a)=0, (7)
where
Uef f (a)
1
2

va2 +
m
a
+
r
a2

(a(t0) = 1). (8)It is noted that Eq. (7) is of the form of a 1D conservation of energy expression for a
particle with zero energy in Newtonian mechanics.Using your favorite plotting software
(i.e. Maple, Matlab, Wolframalpha, etc.), plot Uef f (a) vs a for
a) Matter-dominated flat FRW spacetime, m = 1, r = 0, v = 0,
b) Radiation-dominated flat FRW spacetime, m = 0, r = 1, v = 0,c) Vacuum-dominated flat FRW spacetime, m = 0, r = 0, v = 1,
d) Our flat FRW spacetime, m = 0.3, r = 5 105, v = 0.7.Your a-axis should have a range of 0.5 <a< 3.5 to properly show the curves. These four
curves should be positioned on the same plot.e) Using the 1D conservation of energy expression given by Eq. (7) and your four plots,
discuss qualitatively the behavior of the scale factor a(t) for all time for each of
these cases.Does one get accelerated or decelerated expansion? Do we get expansion
and then contraction? Do we get contraction and then expansion?