Observational Cosmology

Problem Set 1

Due Thursday 18 Feb 2002

- Derive the age of a flat (Ω
_{m}= 1) universe in terms of the Hubble constant H_{0}.

What is this age if H_{0}= 72 km/s/Mpc?

How does this compare to the minimum age of globular clusters,*T*> 11 Gyr?_{GC} - An important cosmological parameter is the deceleration parameter,
defined as

q = - R (d ^{2}R/dt^{2})(dR/dt)^{-2}.- a) For Λ = 0, derive an expression for q in terms of
Ω
_{m}.

Take a moment to reflect on this expression.

Does it make sense to you that q and Ω_{m}should be connected this way? - b) Derive q for non-zero Λ in terms of
Ω
_{m}and Ω_{Λ}. - c) Suppose one observation indicates that the geometry of the
universe is flat, and another measures Ω
_{m}= 0.3.

Can these be reconciled? By what value of q_{0}& Ω_{Λ}?

- a) For Λ = 0, derive an expression for q in terms of
Ω
- Using the Friedmann equation (with Λ = 0), show that
- a) H
^{2}= H_{0}^{2}(1+z)^{2}(1+z Ω_{m0}) - b) Ω
_{m}= Ω_{m0}(1+z)/(1+z Ω_{m0}) - c) Plot Ω
_{m}(z) vs. log(1+z) to at least z=1000 for several choices of Ω_{m0}.

- Does it appear fair to ignore the curvature term in the early universe?
- Does this evolution pose a philosphical problem?

- a) H
- Repeat problem 3 including Λ and assuming a flat geometry.
- a) Find the expression for H
^{2}. - b) Find the expression for Ω
_{m}. - c) Plot Ω
_{m}(z).- Suppose Ω
_{m0}= 0.3.

What was the redshift of matter-cosmological constant equality? - Is the coincidence problem solved by any flat model?

- Suppose Ω

- a) Find the expression for H
- 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?