Problem Set 1
Due Thursday 18 Feb 2002
- Derive the age of a flat (Ωm = 1) universe in terms of the
Hubble constant H0.
What is this age if H0 = 72 km/s/Mpc?
How does this compare to the minimum age of globular clusters,
TGC > 11 Gyr?
- An important cosmological parameter is the deceleration parameter,
q = - R (d2R/dt2)(dR/dt)-2.
- a) For Λ = 0, derive an expression for q in terms of
Take a moment to reflect on this expression.
Does it make sense to you that q and Ωm should be
connected this way?
- b) Derive q for non-zero Λ in terms of
Ωm and ΩΛ.
- c) Suppose one observation indicates that the geometry of the
universe is flat, and another measures Ωm = 0.3.
Can these be reconciled? By what value of q0 &
- Using the Friedmann equation (with Λ = 0), show that
- a) H2 = H02
(1+z)2 (1+z Ωm0)
- b) Ωm = Ωm0
- c) Plot Ωm(z) vs. log(1+z) to at least z=1000
for several choices of Ωm0.
- Does it appear fair to ignore the curvature term in the early universe?
- Does this evolution pose a philosphical problem?
- Repeat problem 3 including Λ and assuming a flat geometry.
- a) Find the expression for H2.
- b) Find the expression for Ωm.
- c) Plot Ωm(z).
- Suppose Ωm0 = 0.3.
What was the redshift of matter-cosmological constant equality?
- Is the coincidence problem solved by any flat model?
- Consider two observers Kang and Kodos at fixed comoving coordinates
in an expanding matter dominated FRW universe. A pulse emitted
from the location of Kang is observed by Kodos to have redshift z=3.
Kodos instantly emits a return pulse in reply.
What is the redshift of the return pulse observed by Kang?