Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

6 Energy densities
We begin with a brief overview of the energy density contributions to the
cosmological evolution of spacetime. Using our definition of the critical density
in Eq. 28, ρ c = 3H2 o
, we can re-write the RHS of the Friedmann equation
8πG
(Eq. 27:
( ) 8πG
H 2 = ρ tot − Kc2
3 R = 2 H2 (ρ/ρ c − 1) − Kc2
R , 2
or
ρ/ρ c = Kc2
R 2 H + 1 . (29)
2
Now, we define
Ω ≡ ρ/ρ c
and we write out the contributions to ρ due to (non-relativistic) matter,
radiation (i.e. relativistic particles), and vacuum energy (see below):
ρ = ρ m + ρ r + ρ v .
Finally, we divide through this expression by ρ c to arrive at
1 = Ω m + Ω r + Ω v + Ω k , (30)
where we have made the definition of an equivalent energy density due to
spacetime curvature:
Ω k = − Kc2
R 2 H 2 .
Current observations of the present epoch give the following best values:
• Ω k ≈ 0 (cosmic µwave background observations, especially the most
recent results from WMAP)
• Ω r ≈ 10 −5
• Ω m ≈ 0.3. All of normal (baryonic) matter is 0.05; the remainder is
dark matter.
• Ω v ≈ 0.7. This would be the contribution due to dark energy – more
on this later.
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