Physical Foundations of Cosmology

394 Cosmic microwave background anisotropies

(9.128), we can calculate the integral with the help of (9.43). The result, valid for
η 0 /ηrl1, reads

l(l+ 1 )ClT

2

15 π

l(l+ 1 )

(l+ 3 )(l− 2 )

Bgw≈ 4. 2 × 10 −^2 Bgw. (9.130)

(For example,η 0 /ηr55 for (^) mh^275  0. 3 .) This estimate fails when applied to
the low multipoles and in particular to the quadrupole (l=2).The quadrupole can
be calculated numerically and the result,
l(l+ 1 )ClT

∣

∣

l= 2 ^4.^4 ×^10

− (^2) Bgw, (9.131)
does not differ greatly from the expression in the right hand side of (9.130).
Problem 9.16Verify that the relative contribution of the tensor and scalar pertur-
bations generated during inflation to the quadrupole is
ClT= 2
ClS= 2
 10. 4 cs

(

1 +

p

ε

)

, (9.132)

where the expression on the right hand side must be estimated at the moment
when the perturbations responsible for the quadrupole cross the Hubble scale on
inflation. Taking 1+p/ε∼ 10 −^2 andcs=1 we find that the gravitational waves
should contribute about 10% to the quadrupole component. Inkinflation where
cs1 their contribution can be negligible.

As we found in Section 7.3.2, the amplitude of the gravitational waves which
entered the horizon is inversely proportional to the scale factor and, fork>η−r^1 ,
their spectrum atη=ηris already significantly modified. For instance, fork
ηeq−^1 ,

∣∣

h^2 k(ηr)k^3

∣∣

O( 1 )Bgw

(

1

kηeq

) 2 (

zr

zeq

) 2

. (9.133)

Substituting this expression into (9.128), we obtain

l(l+ 1 )ClT∝Bgw

(

leq

l

) 2

forlleq,whereleq≡η 0 /ηeq(leq150 for (^) mh^275  0 .3). In the intermediate
region 55<l< 150 ,the amplitudel(l+ 1 )ClTalso decays. Note that all results
in this section were derived in the approximation of instantaneous recombination,
which is valid in the most interesting range of the multipoles.
Problem 9.17Assuming thatηeqηrdetermine howl(l+ 1 )ClTdepends onl
forη 0 /ηeqlη 0 /ηr.