Introduction to Cosmology

154 Cosmic Inflation

σeh

σph

5 here and now

structures in place

reionization

recombination

Big Bang

Last Scattering

decoupling

20

Redshift z

11001089890 0

∞

Figure 7.1 A co-moving space/conformal time diagram of the Big Bang. The observer (here

and now) is at the center. The Big Bang singularity has receded to the outermost dashed circle,

and the horizon scale is schematically indicated at last scattering. It corresponds to an arc of

angle휃today. Reproduced from [1] by permission of J. Silk and Macmillan Magazines Ltd.

represents the extent of the LSS that could have come into causal contact from푡= 0

to푡LSS. If the event horizon is larger than the particle horizon, then all the Universe we

now see (in particular the relic CMB) could not have been in causal contact by푡LSS.

The event horizon휎eh, is obtained by substituting푎(푡)∝(푡LSS∕푡)−^2 ∕^3 from Equa-

tion (5.39) into Equation (2.50) and integrating over the full epoch of matter dom-

ination from푡LSSto푡max=푡 0. Assuming flat space,푘=0, we have

휎eh∝

∫

푡 0

푡LSS

d푡

(

푡LSS

푡

) 2 ∕ 3

= 3 푡LSS

[(

푡 0

푡LSS

) 1 ∕ 3

− 1

]

. (7.11)

Let us take푡LSS= 0 .35Myr and푡 0 =15Gyr. Then the LSS particle horizon휎phis seen

today as an arc on the periphery of our particle horizon, subtending an angle

180

휋

[휎

ph

휎eh

]

LSS

≃ 1. 12 ∘. (7.12)

This is illustrated in Figure 7.1, which, needless to say, is not drawn to scale. It follows

that the temperature of the CMB radiation coming from any 1∘arc could not have been

causally connected to the temperature on a neighboring arc, so there is no reason why

they should be equal. Yet the Universe is homogeneous and isotropic over the full 360

∘

.

This problem can be avoided, as one sees from Equations (7.6)–(7.9), when the net
pressure is negative, for example, when a cosmological constant dominates. In such
acase,푎(푡)∝econst.⋅푡[the case푤=−1 in Equation (5.38)]. If a cosmological constant
dominates for a finite period, say between푡 1 and푡 2 <푡LSS, then a term econst.⋅(푡^2 −푡^1 )
enters into Equation (7.10). This term can be large, allowing a reordering of horizons
to give휎ph>휎eh.
The age of the Universe at temperature 20MeV was푡=2ms and the distance scale
2 ct. The amount of matter inside that horizon was only about 10−^5 푀⊙, which is very