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

7 Vacuum energy and Inflation
In Section 5 we introduced Einstein’s field equations, Eq. 26, including the
comsological constant (Λ) term. Since g µν represents flat spacetime, this term
is uniformly distributed. For the Friedmann equation, Eq. 27, this term gets
included as Λ/3 on the RHS. Since this term has no R dependence, then for
sufficiently large R, a non-zero Λ will eventually dominate the RHS of Eq.
27. In this case, the Friedmann equation becomes
Ṙ = (Λ/3)R ⇒ dR R = √Λ/3 dt ⇒ R(t) ∝ e αt , (33)
√
where α ≡ Λ/3, as in Table 1.
Now, we make the connection between this funny Λ term in Einstein’s
equations and the idea of vacuum energy from quantum field theory (QFT).
In quantum mechanics, the uncertainty principle states that there can be a
non-conservation of energy ∆E over a time ∆t such that
∆E∆t ≥ ¯h/2 .
In QFT, this must occur with particle pairs such as e + e − appearing and
disappearing, consistent with the uncertainty relation above. On average,
there is a net contribution which is non-zero. This represents the vacuum
energy. So the vacuum is not a state of nothing, it is simply the lowest energy
level, or ground state. Hence, Einstein’s general relativity will couple to this
vacuum energy. In addition, in QFT we expect the vacuum to have a uniform
energy density, just like the Λ field. Hence, we identify vacuum energy with a
non-zero Λ. Hence, this allows us to possibly connect episodes of spacetime
evolution influenced by Λ with fundamental elementary particles and fields.
Such is the case for inflation, discussed below.
We complete the identification of Λ with vacuum energy by including it
as a contribution to ρ in the Friedmann equation:
ρ v =
Λ
8πG .
We used ρ v in this form in the previous section. We will not get into specific
particle physics manifestations for inflation. We will motivate it using the
observations of the CMBR and then discuss the properties the inflationary
era must possess.
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