Astroparticle Physics

12.10 Outlook on Inflation 263years, the universe has been undergoing a renewed quasiexponentialexpansion compatible with a value of Ω Λ,0 =ϱ v,0 /ϱ c,0 ≈ 0.7. (The subscript Λ refers to the relation betweenvacuum energy and a cosmological constant; the subscript0 denotes as usual present-day values.) The criticaldensity is ϱ c,0 = 3H0 2/8πG,whereH 0 ≈ 70 km s −1 Mpc −1is the Hubble constant. This gives a current vacuum energydensity of around (¯hc ≈ 0.2GeVfm)current vacuum energydensityϱ v,0 ≈ 10 −46 GeV 4 . (12.39)In the example of inflation considered above, however, themagnitude of the vacuum energy density is related to the expansionrate during inflation by (12.15). Furthermore, it wasargued that the expansion rate is approximately related to thestart time of inflation by H ≈ 1/t i . So, if one believes thatinflation occurred at the GUT scale with, say, t i ≈ 10 −38 s,then one needs a vacuum energy density of, see (9.40),dark energy at inflationϱ v = 3m2 Pl32πt 2 i≈ 10 64 GeV 4 . (12.40)The vacuum energies now and those which existed duringan earlier inflationary period may well have a common explanation,but given their vast difference, it is by no meansobvious what their relationship is.One could try to argue that inflation took place at a latertime, and therefore had a smaller expansion rate and correspondinglylower ϱ v . The latest possible time for inflationwould be just before the Big Bang Nucleosynthesis era,around t ≈ 1 s. With inflation any later than this, the predictionsof BBN for abundances of light nuclei would be alteredand would no longer stand in such good agreement with observation.Even at an inflation time of t i = 1 s, one wouldneed a vacuum energy density of ϱ v ≈ 10 −12 GeV 4 , still 34orders of magnitude greater than what is observed today.In addition to those observable features mentioned, inflationpredicts the existence of gravitational waves. Thesewould present a fossilized record of the first moments oftime. In particular, the energy density of gravity waves isexpected to beϱ gravity waves = h2 ω 2which gives32πG(12.41)inflation timegravitational wavesfrom inflation