Physical Foundations of Cosmology

9.6 Delayed recombination and the finite thickness effect 371

ηrdetermined by the condition

μ′′=μ′^2. (9.50)
Since recombination is really spread over an interval of time, we will useηr
henceforth to represent the conformal time when the visibility function takes its
maximum value. This maximum is located within the narrow redshift range 1200>
z>900. During this short time interval, the scale factor and the total number density
ntdo not vary substantially, so we can use their values atη=ηr. On the other hand,
the ionization fractionXchanges by several orders of magnitude over this same
interval of time, so its variation cannot be ignored. Substituting (9.48) in (9.50), we
can re-express the condition determiningηras

X′r−(σTnta)rX^2 r, (9.51)

where the subscriptrrefers to the value atηr. For redshifts 1200>z>900,Xis
well described by (3.202). The time variation ofXis mainly due to the exponential
factor, and hence

X′−

1. 44 × 104

z

HX, (9.52)

whereH≡a′/a.Substituting this relation in (9.51) we obtain

XrHrκ(σTnta)−r^1 , (9.53)

whereκ≡ 14400 /zr.Together with (3.202), this equation determines that the vis-
ibility function reaches its maximum atzr1050, irrespective of the values of
cosmological parameters. At this time, the ionization fractionXrisκ 13 .7 times
larger than the ionization fraction at decoupling, as defined by the conditiont∼τγ
(see (3.206)). Near its maximum, the visibility function can be well approximated
as a Gaussian:

μ′exp(−μ)∝exp

(

−

1

2

(μ−lnμ′)′r(ηL−ηr)^2

)

. (9.54)

Calculating the derivatives with the help of (9.52) and (9.53), we obtain

μ′exp(−μ)

(κHη)r

√

2 πηr

exp

(

−

1

2

(κHη)^2 r

(

ηL

ηr

− 1

) 2 )

, (9.55)

where the prefactor has been chosen to satisfy the normalization condition
∫
μ′exp(−μ)dηL= 1.

The expression inside the square brackets in (9.49) does not change as much
as the oscillating exponent and we obtain a good estimate just taking its value at