- The Ωm-H0 Diagram
Two papers independently claim to measure the age of the universe and
the Hubble constant:
Plot these as constraints in
a diagram of Ωm0 vs. H0 assuming
- No cosmological constant (ΩΛ0 = 0), and
- A flat universe (Ωm0+ΩΛ0 = 1).
Use the intersection of the constraint on age and H0 to
estimate the mass density Ωm0 that is implied in each case,
including an estimate of the uncertainty.
- How does the result differ with and without Λ?
- Matter-Radiation Equality
Find the redshift of matter-radiation equality for
several conceivable mass densities:
- Ωm0 = 0.045 (normal matter only)
- Ωm0 = 0.27 (ΛCDM)
- Ωm0 = 1 (SCDM).
In each case, note whether matter-radiation equality happens
before or after recombination (z = 1090).
The current radiation density
is Ωr0 = 9 x 10-5
(this includes both photons and neutrinos).
- Expansion during Radiation Domination
Use the Friedmann equation to derive the expansion rate a(t) when
radiation dominates [i.e., ignore the other terms].
Keep the terms for H0 and
Ωr0 to obtain a proper equality.
- Big Bang Nucleosynthesis
Estimate the mass fraction of primordial helium YP assuming
that all available neutrons are incorporated into 4He nuclei
after 5 minutes for the following situations:
- a) Standard BBN: start from the freez-out neutron-to-proton ratio
of nn/np = 0.2, and make use of the decay time
of the free neutron (τn = 880 s)*
to obtain the standard BBN prediction for YP.
- b) Fast neutron decay: what would the helium mass fraction be if
the neutron decay time were an order of magnitude faster
(τn = 88 s)?
[Think: which way should this drive YP?]
- c) Small splitting: what would the helium mass fraction be if
the proton-neutron mass difference were an order of magnitude smaller?
(0.129 MeV instead of 1.29 MeV. For this part, the decay time is normal,
τn = 880 s, as is the freeze-out temperature, 0.8 MeV)
[Think: what does this affect? Which way does it drive YP?]
*Note that the e-folding time τ differs from the half-life t1/2
by the factor ln(2).