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198	5 Cosmology and General Relativity

5.9.4 Calculation of the Power Spectrum in the Two Regimes

Let us now consider the power spectrum for short and long wave-lengths respectively.

5.9.4.1 Short Wave-Lengths

According to our previous discussion in the short wave-length regime, which can be deﬁned as

we just have \|uκ (η)\|2	1		κη & 1							(5.9.74)
	1	so that we ﬁnd:
	π κ3	so that we ﬁnd:
		PΦ (κ)	κη	&	1	ϕ			1	2
		PΦ (κ)		&		ϕ		2 a

			=			−	m2		˙
			=			−	π		˙
								P	H	(5.9.75)
									H	(5.9.75)

The last line of the above equation follows from use of the exact result (5.9.8) and further transformation of the η-derivatives into t -derivatives.

5.9.4.2 Long Wave-Lengths

The method to obtain information on the wave-function and hence on the power spectrum for long wave-lengths:

κη ' 1

(5.9.76)

relies on solving once again the propagation equation in the approximation κ2 → 0. This means that in (5.9.56) we forget the term in κ2 and we are left with the equation:

uk −	θ
uk −	θ uk = 0	(5.9.77)

A basis of two independent solutions of the above ordinary differential equation of the second order is immediately found as follows:

u1	= θ		(5.9.78)
	η dη
u2	= θ η0		(5.9.79)
		θ 2
Indeed one can easily verify that the Wronskian of these two solutions is:
u1u2 − u2u1 = 1			(5.9.80)

5.9 Primordial Perturbations of the Cosmological Metric and of the Inﬂaton

199

Correspondingly we can write the generic solution of (5.9.77) as follows:

uk(η) = c1θ + c2θ

η dη

θ 2

η0

= Akθ

η dη

(5.9.81)

θ 2

Indeed the integral

η0

dη

is just some number so that the contribution from the

θ 2

η0

ﬁrst solution can

always be reabsorbed into a redeﬁnition of the initial point of inte-

gration. The integration constant Ak has instead to be ﬁxed by means of boundary conditions. Using the exact result (5.9.8) in the deﬁnition (5.9.27) of the function θ we can rewrite it as it follows:

1 1				1	H	−1/2
θ =	√				− H 2	(5.9.82)

		2π G	a

Using this expression and the deﬁnition of the conformal Hubble function H we get:


	dη	= 2π G		dη a2 1 −					H
	θ 2								H 2
		= 2π G		a2				a2 dη
						−				(5.9.83)
			H
Using this result and multiplying by θ =					H					ϕ
							and by the factor			a necessary to con-
					aϕ

vert a u-mode into a mode of the gravitational potential we obtain the following

long wave-length result:	1 − a2	a2 dη
σk A k	1 − a2	a2 dη
	H				a
= A k	1 − a	a dt	= dt		a	a dt	(5.9.84)
	H			d	1

The last line follows from conversion of the η-derivatives into t -ones; furthermore we have reabsorbed the factor 2π G into the integration constant A k

Apart from the initial approximation consisting in neglecting the κ2 term for large wave-lengths the above result is exact. No approximation about cosmic evolution has been introduced so far. When the propagation of perturbations takes place in a slow-rolling universe we are in an approximately exponential phase where:

a(t) exp[H t]

a dt

a(t)

(5.9.85)

In this regime from (5.9.84) we obtain:

slow-roll

d 1

σk

A k

= −A k H 2

(5.9.86)

200	5 Cosmology and General Relativity

On the other hand if we consider the same Fourier component σk during the postinﬂationary radiation-dominated era we are in an approximately square root phase where:

a(t) m√t

a dt

3 mt 2

a dt

(5.9.87)

and we get:

radiation era 2

σk

A k

(5.9.88)

We conclude that, as observed, in the post-inﬂationary age, the power spectrum of the gravitational potential has the following form:

for κ η

| &

PΦ (κ)

κ > aH

π ˙

(5.9.89)

94 |

k|2κ3

for κ|η| '

κ < aH

The last column yielding the separating condition between the short and long wave-length regimes, follows from the approximate behavior of the scale factor in the almost exponential phase of inﬂation. There, in conformal time, we have: a −1/(H η) and therefore η (aH )−1. The physical interpretation of (5.9.89) is quite clear. At every cosmic time t ,

λκ (t) ≡	a(t)	(5.9.90)
	κ

is the effective wave-length of the Fourier mode κ, which is constantly stretched by the expansion of the Universe. Short wave-lengths are those that are shorter than the Hubble radius at the same time:

λκ (t) < H (t)−1

(5.9.91)

Long wave lengths are those larger than the Hubble radius. In an exponential phase of expansion the Hubble radius is also the event horizon (see (5.6.38)) which remains approximately constant while the scale factor and hence all the wave-lengths rapidly grow. Hence if the exponential phase lasts long enough the wave-lengths of almost all modes κ exit the Hubble radius and becomes frozen. Indeed no physical process can inﬂuence a mode whose characteristic scale is larger than the event horizon.

Quite different is the evolution of wave-lengths in radiation and matter dominated universes. In both these cases the Hubble radius grows linearly in time:

H (t)−1 t

(5.9.92)

while the wave-lengths grow either the Hubble radius (particle horizon

1	2	. Hence no mode which is inside
as t 2	or as t 3	. Hence no mode which is inside

in these cases) at some time t will exit it in the

5.9 Primordial Perturbations of the Cosmological Metric and of the Inﬂaton

201

future. On the contrary modes which were out of the Hubble horizon at the end of inﬂation can reenter it in the subsequent radiation dominated or matter dominated era.

5.9.4.3 Gluing the Long and Short Wave-Length Solutions Together

In view of these considerations the formula (5.9.89) for the power spectrum is quite exhaustive provided we can ﬁx the integration constant A k which encodes all the information. This step can be achieved by equating the long and short wave-length form of the mode σk at the transition time κ|η| = 1, namely by setting:

ϕ (

−

1)3/4e±i(ηκ− iπ2ν )

| =

κ√π κ

−A k H 2

at κ η

(5.9.93)

From (5.9.93) we obtain:

iψ mP2

H 2

A k = e

κ3√

(5.9.94)

where we used the exact result (5.8.23) and where eiψ is an η dependent phase factor whose explicit form is irrelevant since we are interested in the square modulus of A k.

In this way the long wave-length form of the power spectrum becomes:

	4 m2		H 4
PΦ (κ) = −		P			(5.9.95)
PΦ (κ) = −	9	π		H	(5.9.95)
				˙	κ=aH

The value of the above expression resides in the following. In the post inﬂationary age we can use (5.9.95) for all those modes whose wave-length was inside the Hubble radius at the beginning of inﬂation but exited it before the end of inﬂation. This condition gives the range:

(H a)f > κ > (H a)i

(5.9.96)

where the sufﬁx i/f means that we have to evaluate the speciﬁed quantity at the beginning and at the end of inﬂation, respectively. If inﬂation lasts long enough and produces 60 or 70 e-foldings the range described in (5.9.96) goes over as many order of magnitudes and encompasses the whole observable universe.

The power spectrum is observed today but the Hubble function and its derivative appearing in it refer to the inﬂation-age, when they were almost constant.

5.9.4.4 The Spectral Index

It is customary to characterize the behavior of the power spectrum by means of a so called spectral index, deﬁned as follows:

	d ln PΦ (κ)	1			d
nS = 1 +		= 1 +		κ		PΦ	(5.9.97)
	d ln κ		PΦ		dκ

202	5 Cosmology and General Relativity

Imagine that the power spectrum has a power-like behavior:

PΦ (κ) κα

(5.9.98)

then the spectral index would just be:

nS = 1 + α

(5.9.99)

In case of scale invariant spectra, namely PΦ (λκ) = PΦ (κ), the spectral index is exactly nS = 1. It is very interesting that, by implementing the slow-roll approximation, the spectral index can be calculated and related to the slow-roll parameters of the potential. To this effect let us observe that, by deﬁnition, we have d ln κ = d ln(aH ). On the other hand in the slow roll approximation d ln(aH ) d ln a and we have:

d ln a	=	d ln a dt	=	H	−	3H 2		2W	(5.9.100)
dφ	=	dt dφ	=	φ	−	W	− W		(5.9.100)
				˙

In the above equations φ and sionless potential, introduced tions:

W are the dimensionless scalar ﬁeld and the dimenin (5.8.19) and we have used the two slow-roll equa-

H		3		;	˙	−	3H
	2	2	W (φ)		φ		W	(5.9.101)
			W (φ)		φ			(5.9.101)

Using these tools we can rewrite:

d ln PΦ (κ)	= −	W (φ) d			(5.9.102)
				PΦ (κ)
d ln κ		2W	dφ

Substituting in (5.9.102) the expression (5.9.95) of the power-spectrum and using once again the slow-roll approximation (5.9.101) we ﬁnally obtain:

	d ln κ	= −	2	3 W		− 2	W	(5.9.103)
	d ln PΦ (κ)		1	W	2		W
	d ln PΦ (κ)		1	W			W
This result immediately yields:
	nS = 1 + 6ηW − 9εW							(5.9.104)
where we used the slow-roll parameters of the						potential deﬁned		in (5.8.27)
and (5.8.30).

Thus the shape of the primeval inﬂationary potential deﬁnes the behavior of the power spectrum occurring in the two-point function of the quantized scalar ﬁeld. The power spectrum, on its turn, is an experimentally accessible datum since it is directly related to the anisotropies of the Cosmic Microwave Background. How this can happen is outlined in the next section.

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