Physical Foundations of Cosmology

116 The hot universe

to be a constant. Equation (3.160) is then readily integrated:

(

1 +

2 R 2

XD(T)

)

=

(

1 +

2 R 2

XD(T∗)

)

exp

⎛

⎝ 4 R 2 η 10

∫T∗

T

α(T)dT

⎞

⎠, (3.161)

where the temperature is expressed in MeV.As the temperature decreases, the
deuterium concentration freezes out atXDf≡XD(T→ 0 ).Taking into account
thatXD(T∗)∼R 1 R 2 ,we obtain

XDf

2 R 2

exp(Aη 10 )− 1

, (3.162)

where

A≡ 4 R 2

∫T∗

T

α(T)dT∼ 4 R 2 α

(

T∗

)

TMeV∗. (3.163)

The coefficientAdepends only weakly onη 10 ; it increases by a factor of 2 asη 10
goes from 1 to 10^2 .Taking as an estimateA 0. 1 ,we find good agreement with
the results of the numerical calculations shown in Figure 3.8.
Forη 10 < 1 /A∼ 10 ,(3.162) simplifies to

XDf

2 R 2

Aη 10

∼ 4 × 10 −^4 η− 101. (3.164)

For this range ofη 10 the deuterium freeze-out abundance decreases in inverse
proportion toη 10 .This dependence onη 10 can easily be understood. Forη 10 < 10 ,
the freeze-out concentrationXDf is larger thanR 2  2 × 10 −^5 and, according to
(3.160), DD reactions dominate in destroying deuterium. The deuterium freeze-out
is then determined by the conditionX ̇D/XD∼λDDXDf∼t−^1 .SinceλDD∝nN∝
η 10 ,we find that, to leading order,XDf∝η 10 −^1.
Forη 10 > 10 ,(3.162) becomes

XDf 2 R 2 exp(−Aη 10 ). (3.165)

In this case the deuterium abundance decays exponentially withη 10 and decreases
by five orders of magnitude, from 10−^5 to 10−^10 ,whenη 10 changes from 10 to
100 (Figure 3.8). In a universe with high baryon density, the reaction Dp→^3 Heγ
dominates in the destruction of deuterium whenXD<R 2  2 × 10 −^5 .Hence, the
freeze-out concentration is determined by the term linear inXDin (3.160).
Thus, deuterium turns out to be an extremely sensitive indicator of the baryon
density in the universe. The observational data certainly rule out the possibility of
a flat universe composed only of baryonic matter.