Physical Foundations of Cosmology

380 Cosmic microwave background anisotropies

Problem 9.9Show thatηx/η 0 Iz−x^1 /^2 ,where

I≡ 3

(

(^) 
(^) m

) 1 / 6 ⎡

⎣

∫y

0

dx

(sinhx)^2 /^3

⎤

⎦

− 1

(9.84)

andy≡sinh−^1 ( (^) /
m)^1 /^2 .In a flat universe (^) = 1 − (^) m,and the numerical
fitting formula
I −m^0.^09 (9.85)
approximates (9.84) to an accuracy better than 1% over the interval 0. 1 <
m< 1.
Verify thatηx/ηris equal to
ηr
ηx



(

zx

zeq

) 1 / 2 [(

1 +

zeq

zr

) 1 / 2

− 1

]

. (9.86)

Combining the relations from Problem 9.9 we obtain

ηr

η 0

=

1

√

zr

[(

1 +

zr

zeq

) 1 / 2

−

(

zr

zeq

) 1 / 2 ]

I. (9.87)

Using this result together with (9.60) forσ, (9.78) becomes

lf 1530

(

1 +

zr

zeq

) 1 / 2

I−^1 , (9.88)

where we recall from (9.61) that

zr

zeq

 7. 8 × 10 −^2

(

(^) mh^275

)− 1

(9.89)

for three neutrino species.

The result is that the finite thickness damping coefficientlfdepends only weakly

on the cosmological term and (^) mh^275 .For (^) mh^275  0 .3 and (^) h^275  0 .7, we have
lf  1580 ,whereas for (^) mh^275 1 and (^) h^275 0, we findlf  1600.
The scalelSdescribing the combination of finite thickness and Silk damping
effects can be calculated in a similar way.
Problem 9.10Using the estimate (9.62) for the Silk dissipation scale, show that
lS 0. 7 lf

⎧

⎪⎨

⎪⎩

1 + 0. 56 ξ

1 +ξ

+

0. 8

ξ( 1 +ξ)

(

(^) mh^275

) 1 / 2

[

1 +

(

1 +zeq/zr

)− 1 / 2 ]^2

⎫

⎪⎬

⎪⎭

− 1 / 2

. (9.90)