[SOLVED] Physics 396 homework set 11

1. In Homework Set 10, we showed that
a(t) = a0 t
2/3(1+w) (1)
is a solution to the flat Friedman equation, for a constant equation of state parameter w,
given by
a
2 8
3
a2 = 0. (2)It is again noted that w = 0 for pressureless matter, w = 1/3 for radiation, and w = 1
for a constant vacuum energy density. It is further noted that the Hubble constant H0 is
defined by the expression
H0
a(t0)
a(t0)
, (3)where a dot indicates a time-derivative. The Hubble time, tH, is defined by
tH
1
H0
, (4)
which gives an upper bound and a rough estimate for the age of a decelerating Universe.a) Given a Hubble constant of H0 = 72.0 (km/s)/Mpc, calculate the Hubble time in
years to three significant figures.
b) Inserting Eq. (1) into Eq. (3), find an expression for t0, the current age of the
Universe in terms of the Hubble time.c) For a Universe containing only radiation, calculate the current age of this Universe.
d) For a Universe containing only pressureless-matter, calculate the current age of this
Universe.2. In Homework Set 6, we arrived at a 3D non-Euclidean line element of a three-sphere of
radius R from a fictitious flat 4D Euclidean space via the coordinate transformation
x = R sin sin cos
y = R sin sin sin
z = R sin cos
w = R cos , (5)where 0 < , 0 < , and 0 < 2. Here we wish to arrive at a three-surface
in flat 4D spacetime that is the analog of the three-surface of a sphere in flat 4D Euclidean
space. This geometry describes that of a Lorentz hyperboloid.a) Consider the coordinate transformation
x = R sinh sin cos
y = R sinh sin sin
z = R sinh cos
t = R cosh , (6)
where 0 < 1, 0 < , and 0 < 2.Setting R = 1, show that the above
transformation obeys the equation of constraint
t
2 + x2 + y2 + z2 = 1. (7)
b) Calculate dx, dy, dz, dt.
c) Calculate dx2 + dy2.
d) Calculate dx2 + dy2 + dz2.e) Show that the 4D Minkowski spacetime line element
ds2 = dt2 + dx2 + dy2 + dz2 (8)
becomes that of a 3D non-Euclidean line element of a three-surface known as the
Lorentz hyperboloid of the form
dS2 = d2 + sinh2 (d2 + sin2 d2
) (9)
under the above coordinate transformation.f) By defining r sinh , show that the above line element takes the form
dS2 = dr2
1 + r2 + r2
(d2 + sin2 d2
). (10)Notice that the line element of Eqs. (9) and (10) equate to a t = const. slice of a
FRW homogeneous open universe3. In Homework Set 6, we found the embedding diagram for a 2D equatorial slice
(t = const., = /2) of a FRW homogeneous closed universe. This equated to finding a
curved 2D surface in 3D Euclidean space with the same intrinsic geometry as a 2D
equatorial slice (t = const., = /2) of the 4D homogeneous closed universe.a) Consider the t = const. slice of a FRW homogeneous open universe given by Eq. (9).
Construct the corresponding 2D equatorial slice ( = /2) for this homogenous
opene universe, analogous to Eq. (7.41) of your text.b) Following a procedure similar to that of Section 7.7 of your text, show that the curved
2D surface obeys the dierential equation
dz
d = R
q
1 cosh2
(). (11)c) Show that this whole 2D equatorial slice of the FRW homogeneous open universe
cant be embedded as an axisymmetric surface in flat 3D Euclidean space.
Extra Credit: Create your own Universe4. In class, we arrived at an equation of motion describing the scale factor of the form
1
2
da
dt

2
+ Uef f (a) = c
2 , (12)
where
Uef f (a)
1
2

va2 +
m
a +
r
a2

. (13)Eq. (12) is the Friedman equation for a flat, open, or closed Universe and is of the form
of a 1D conservation of energy expression for a particle in Newtonian mechanics with
either zero, negative, or positive total energy, respectively. This expression connects the
time evolution of the Universe to its spatial geometry and energy density.It is noted that the cosmological parameters obey the constraint equation
v + m + r + c = 1, (14)
where 0 m, r 1, however, c, v can be positive, negative, or zero. In this problem,
your task is to find values for the cosmological parameters m, v, r and c, subjected
to the constraint given by Eq. (14), that will generate a Universe that eitheri. expands until the scale factor reaches a maximum value and then contracts to a
Big Crunch or
ii. contracts until the scale factor reaches a minimum value before bouncing to an
expanding Universe.Using your favorite plotting software (i.e. Maple, Matlab, Wolframalpha, etc.)
a) plot Uef f (a) vs a for your choice of cosmological parameters with 1 c 1.
b) plot c/2 with 1 c 1.c) Explicitly state your choice of parameters that satisfy i. and ii. These two curves
should be positioned on the same plot.