F. K. Lamb Fall 2006
PHYSICS 598AST: ASTROPHYSICS
Sixth Homework Assignment
1. This problem is designed to help you review the standard (hot big-bang) cosmological model.
(a) In Robertson-Walker spacetimes, unlike in the Minkowski spacetime of special relativity, there is a
universal time and a local standard of rest. Assuming that our universe is described by a Robertson-
Walker spacetime, describe two ways observers everywhere can synchronize their watches and two ways
they can determine their velocities relative to the local standard of rest.
(b) What is meant by “the horizon problem” in the standard cosmological model and how is it resolved
by inflation?
(c) What is meant by “the flatness problem” in the standard cosmological model and how is it resolved
by inflation?
(d) The age of the universe is measured to be about 13 Gyr. Neutrons have a half-life of only 614 seconds
but are still plentiful in the universe today. Why haven’t they decayed by now?
(e) Which of the following chemical elements were produced during big-bang nucleosynthesis: H, D, 3He,
4He, 7Li, 12C, 16O, 56 Fe.
2. In the inflationary model of the very early universe, quantum fluctuations produce small initial variations
(fluctuations) in the density of mass-energy and the curvature that eventually grow in amplitude to
produce the nonlinear large-scale structures we see today.
(a) Let PΦ(k) be the power density of the fluctuations in the field Φ at wavenumber k. A spectrum of
the form PΦ(k) = const.is sometimes called scale-free or scale-invariant. Why?
(b) A spectrum of the form PΦ(k) = k−3is also sometimes called scale-free or scale-invariant. Why?
(c) How does the physical wavelength of a given density fluctuation δ(~x)≡δρ(~x)/¯ρevolve as the universe
expands?
(d) How is the comoving wavenumber related to the physical wavenumber? Why is it more convenient
to use the comoving wavenumber rather than the physical wavenumber to analyze the evolution of
perturbations?
(e) Measurements by the COBE and WMAP satellites have shown that the temperature of the cosmic
microwave background (CMB) varies on angular scales of about 7◦and hence that the universe is not
isotropic, yet for most purposes we still use the isotropic Robertson-Walker spacetimes to describe it.
Why? What is thought to be the reason for the temperature variation of the CMB?
3. Two currently open questions in modern cosmology are the nature of the dark matter and the dark
energy.
(a) Explain what is meant by “dark matter” and cite two measurements that support its existence.
(b) Explain what is meant by “dark energy” and cite two measurements that support its existence.
(c) What fraction of the closure density of the universe is thought to be contributed by the following
constituents at the present epoch: baryons, dark matter, dark energy, radiation.
(d) In traditional relativistic cosmology, the fate of the universe is uniquely related to its topologyy (i.e.,
whether it is open or closed). Explain this statement. Is this still thought to be true? Explain.
