#### ASTR 622 Observational Cosmology Problem Set 1Due Thursday 18 Feb 2002

1. 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?

2. An important cosmological parameter is the deceleration parameter, defined as
q = - R (d2R/dt2)(dR/dt)-2.
• a) For Λ = 0, derive an expression for q in terms of Ωm.
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 & ΩΛ?

3. Using the Friedmann equation (with Λ = 0), show that
• a) H2 = H02 (1+z)2 (1+z Ωm0)
• b) Ωm = Ωm0 (1+z)/(1+z 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?

4. 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?

5. 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?