Astroparticle Physics

12.6 Solution to the Flatness Problem 257a final time t f , and suppose that during this time the energydensity is dominated by vacuum energy ϱ v , which could resultfrom some inflaton field. The expansion rate H duringthis period is given by√8πGH =3 ϱ v . (12.24)So from t i to t f , the scale factor increases by a ratioR(t f )R(t i ) = eH(t f−t i ) ≡ e N , (12.25)where N = H(t f − t i ) represents the number of e foldingsof expansion during inflation.Referring back to (12.3), it was shown that the Friedmannequation could be written asnumber of e foldingsof the scale factorΩ − 1 = k Ṙ 2 . (12.26)Now, during inflation, one has R ∼ e Ht , with H = Ṙ/Rconstant. Therefore, one findsΩ during inflationΩ − 1 ∼ e −2Ht (12.27)during the inflationary phase. That is, during inflation, Ω isdriven exponentially towards unity.What one would like is to have this exponential decreaseof |Ω − 1| to offset the divergence of Ω from unity duringthe radiation- and matter-dominated phases. To do this onecan work out how far one expects Ω to differ from unityfor a given number of e foldings of inflation. It has beenshown that during periods of matter domination one has|Ω − 1| ∼t 2/3 , see (12.4), and during radiation domination|Ω − 1| ∼t, see (12.6). Let us assume inflation starts at t i ,ends at t f , then radiation dominates until t mr ≈ 50 000 a, followedby matter domination to the present, t 0 ≈ 14 × 10 9 a.The current difference between Ω and unity is thereforeΩ is driven to unityduring inflation( )( ) 2/3|Ω(t 0 ) − 1| =|Ω(t i ) − 1| e −2H(t f−t i ) tmr t0.t f t mr(12.28)In Chap. 11 it was seen that the WMAP data indicate|Ω(t 0 ) − 1| < 0.04. If, for example, one assumes that theinflaton field is connected to the physics of a Grand UnifiedTheory, then one expects inflation to be taking place