ASTR/PHYS 328/428
Cosmology
Problem Set 2
Due Thursday 19 Sep 2024

    The energy density of radiation may be neglected in this problem set (Ωr = 0 & P = 0).

  1. The Age of the Universe
    The locally measured Hubble constant is H0 ≈ 75 km/s/Mpc (e.g., Tully et al. 2016).
    1. What is the Hubble time for this value of the Hubble constant?
      [Hint: this requires a straightforward unit conversion.]
    2. How does this Hubble time compare to the age of the oldest stars, TGC ≈ 13 Gyr?
      (GC stands for globular clusters, which are the oldest stellar systems known. See Valcin et al. 2020).
    3. Now use the Friedmann equation to derive the age of a critical density (Ωm = 1) universe.
      What is the age of the universe now?
      How does it compare to the ages of globular clusters?
    4. Turning the question around, if we are sure that Ωm = 1, what limit does the age of these oldest stars impose on the value of the Hubble constant?

  2. Deceleration Parameter
    An important cosmological parameter is the deceleration parameter, defined as
    q = - a (d2a/dt2)(da/dt)-2.
    1. 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?
    2. Derive q for non-zero Λ in terms of Ωm and ΩΛ.
    3. 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. Friedmann Equation
    Using the Friedmann equation (with Λ = 0), show that
    1. H2 = H02 (1+z)2 (1+z Ωm0)
    2. Ωm = Ωm0 (1+z)/(1+z Ωm0)
    3. Plot Ωm(z) vs. log(1+z) to at least z=1000 for several choices of Ωm0.
      1. Does it appear fair to ignore the curvature term in the early universe?
      2. Does it seem like an odd coincidence if we live at a time when we observe Ωm0 close to, but not equal to, one? Why?

  4. Friedmann Equation with Λ
    Repeat problem 3 including Λ and assuming a flat geometry.
    1. Find the expression for H2.
    2. Find the expression for Ωm.
    3. Plot Ωm(z).
      1. Suppose Ωm0 = 0.3.
        What was the redshift of matter-cosmological constant equality?
      2. Is the coincidence problem solved by every flat model? any flat model?
    4. Plot H/(1+z) vs. z out to z = 2.5 for Ωm0 = 0.04, 0.3, and 1 for both flat cosmologies (with appropriate Λ) and open cosmologies (with Λ = 0 from problem 3). Also plot these data for H(z).
      1. How do you choose H0?
      2. Does it matter which formula you use when Ωm0 = 1?
      3. What does the curvature of these lines tell you about the deceleration history q(z) of the models?
      4. Are there models that are clearly preferred? Excluded?

    Extra credit; required for ASTR/PHYS 428

  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?