Relic Gravitational Waves in the Accelerating Universe

0.21 MB·English

by  Yang Zhang

Relic Gravitational Waves in the Accelerating Universe
Yang Zhang, Yefei Yuan, Wen Zhao, and Ying-Tian Chen
Astrophysics Center
University of Science and Technology of China
5
Hefei, Anhui, China
0
0
2

n
Abstract
a
J

7
1 Observations have recently indicated that the Universe at the present

1 stage is in an accelerating expansion, a process that has great implica-
v
9 tions. We evaluate the spectrum of relic gravitationalwaves in the current
2
accelerating Universe and find that there are new features appearing in
3
1
the resulting spectrum as compared to the decelerating models. In the
0
5
low frequency range the peak of spectrum is now located at a frequency
0
h/ ν (Ωm)1/3ν , where ν is the Hubble frequency, and there appears
E Ω H H
p ≃ Λ
a new segment of spectrum between ν and ν . In all other intervals
- E H
o
Ω
r of frequencies ν , the spectral amplitude acquires an extra factor m,
t ≥ H ΩΛ
s
a due to the current acceleration, otherwise the shape of spectrum is simi-
:
v lar to that in the decelerating models. The recent WMAP result of CMB
i
X
anisotropies is used to normalize the amplitude for gravitational waves.
r
a
The slope of the power spectrum depends sensitively on the scale factor
a(τ) τ 1+β during the inflationary stage with β = 2 for the exact de
∝ | | −
Sitter space. With the increasing of β, the resulting spectrum is tilted
to be flatter with more power on high frequencies, and the sensitivity of
the second science run of the LIGO detectors puts a restriction on the
parameter β 1.8. We also give a numerical solution which confirms
≤ −
these features.
PACS numbers: 98.80.-k, 98.80.Es, 04.30.-w, 04.62+v,
Key words: gravitational waves, accelerating universe, dark energy
e-mail: [email protected]
1
1. Introduction.
The inflationary expansion of the early Universe can create a stochastic background of relic gravita-
tional waves, which is important in cosmology and has been extensively studied in the past [1] [2] [3]
[4] [5]. The spectrum of relic gravitational waves, as is to be observed today, depends not only on the
details of the early stage of inflationary expansion, but also on the expansion behavior of the subsequent
stages, including the current one. The calculations of spectrum so far [6] [7] [8] [9] [10] [11] have been
done for the case that the current stage is in a decelerating expansion. The resulting spectrum has
been put among the candidate list of sources for the gravitational wave detectors either in operation
[12][13], or under construction [14] [15]. The astronomical observations on SN Ia [16] [17] indicate that
the Universe is currently under accelerating expansion, which may be driven by the cosmic dark energy
(Ω 0.7) plus the dark matter (Ω 0.3) [18]. This is further supported by the recent WMAPresults
Λ m
∼ ∼
on CMB anisotropies [19] with Ω = 1. By the wave equation, the evolution of relic gravitational waves
depends on the expanding spacetime background, and the wave amplitudes depend on whether the wave
lengths are inside or outside the Hubble radius. The Universe under accelerating expansion has a Hubble
radius as a function of time that differs from that in the conventional models of decelerating expansion.
Therefore, onecanexpect that, if anearlier matter-dominated stage ofdecelerating expansionisfollowed
by the current accelerating expansion, the outcome for the spectrum of relic gravitational waves will be
altered.
In this paper we study the impact of the current accelerating expansion on the relic gravitational
waves. We will first sketch, as a setup, the well-known formulations of the gravitational waves in an
expanding spacetime [20], and give the explicit scale factor a(τ) for a sequence of successive expanding
epochs, including the current epoch of accelerating expansion. We then evaluate the power spectrum
of relic gravitational waves and find that the current accelerating expansion does change the spectrum,
including its shape and amplitude. Throughout the paper we shall work with units with c = h¯ = 1,
otherwise it will be pointed out. We also use notations similar to that of Grishchuk [8] for convenience
for comparison.
2. The Gravitational Wave Equation.
Incorporating the perturbations to the spatially flat Robertson-Walker space-time, the metric is
ds2 = a2(τ)[dτ2 (δ +h )dxidxj], (1)
ij ij
−
where where τ is the conformal time, the perturbations of spacetime h is a 3 3 symmetric matrix
ij
×
containing generally the scalar, vector, and tensor parts. The gravitational wave field is the tensorial
portion of h , which is transverse-traceless
ij
∂ hij = 0, δijh = 0. (2)
i ij
2
We are interested only in the creation of the relic gravitational waves by the expanding spacetime
background, the perturbed matter source is therefore not taken into account. Moreover, as the relic
gravitational waves are very weak, in the sense that h 1, so one need just study the linearized field
ij
≪
equation:
∂ (√ g∂µh (x,τ)) = 0. (3)
µ ij
−
In quantum theory of gravitational waves, the field h is a field operator, which is written as a sum of
ij
the plane wave Fourier modes
√16πl 2 ∞ 1
h (x,τ) Pl d3kǫ(λ)(k) [a(λ)h(λ)(τ)eik·x +a†(λ)h(λ)∗(τ)e−ik·x], (4)
ij (2π)23 λ=1Z−∞ ij √2k k k k k
X
where l = √G is the Planck’s length, the two polarizations ǫ(λ)(k), λ = 1,2, are symmetric and
Pl ij
transverse-traceless kiǫi(jλ)(k) = 0, δijǫi(jλ)(k) = 0, and satisfy the conditions ǫ(λ)ij(k)ǫ(ijλ′)(k) = 2δλλ′,
and ǫi(jλ)(−k) = ǫi(jλ)(k), the creation and annihilation operators satisfy [ak(λ),a†k(′λ′)] = δλλ′δ3(k − k′),
and the initial vacuum state is defined as
(λ)
a 0 >= 0 (5)
k
|
for each k and λ. As a matter of fact, this definition depends on the choice of the mode function
(λ) (λ)
h (τ); different h (τ) define different vacuum states, a point that will be further explained later.
k k
In the vacuum state the energy density of gravitational waves t = 1 h hij gives the zero-point
00 32πG ij,0 ,0
energy
1 ∞ 2
< 0 t d3x 0 >= h¯ d3k ω < 0 a(λ)a(λ)† 0 > .
00 k k k
| | 2 −∞ | |
Z Z λ=1
X
For a fixed wave number k and a fixed polarization state λ, the Eq.(3) reduces to the second-order
ordinary differential equation [20]
h(λ)′′(τ)+2a′h(λ)′(τ)+k2h(λ) = 0, (6)
k a k k
(λ)
where the prime denotes d/dτ. Since the equation of h (τ) for each polarization is the same, we
k
(λ)
denote hk (τ) by hk(τ). One rescales the field hk(τ) as
µk(τ)
hk(τ) , (7)
a(τ)
and the equation for µk is
a′′
µ′′ +(k2 )µ = 0. (8)
k − a k
This equationcan beregarded asthe equation for one-dimensional oscillator in a giveneffective potential
barrier a′′/a. For a given spacetime background with a generic power-law form of the scale factor
a(τ) τα, (9)
∝
3
where α isareal number, positiveornegative, the general solution isalinear combinationof theHankel’s
functions:
µ (τ)A √kτH(1) (kτ)+B √kτH(2) (kτ). (10)
k k (α−1) k (α−1)
2 2
Given a model of the expansion of Universe, consisting of a sequence of successive a(τ) as in Eq.(9) with
different α, one can construct an exact solution µ (τ) by matching its values and its derivatives at the
k
joining points. One may may also numerically solve Eq.(6) and give the corresponding power spectrum
as we will present later in the paper. The vacuum state 0 >, determined by the mode function h (τ),
k
|
is therefore fixed by a choice of coefficients A and B . In the limit k , or τ ,
k k
→ ∞ → ±∞
2
√kτH(1) (kτ) i−αe−ikτ,
(α−12) → sπ
2
√kτH(2) (kτ) iαeikτ,
(α−12) → sπ
approaching the positive and negative frequency modes, respectively. For instance, if we choose B = 0,
k
then in the limit k , or τ ,
→ ∞ → ±∞
hk(τ) e−ikτ,
∝
giving the positive frequency mode. This choice yields the so-called adiabatic vacuum [21] through (4)
and (5). The amplitude of the power spectrum of the gravitational waves depends only on the initial
value of hk(τ) at the horizon-crossing.
| |
The following two limiting cases are useful for an approximate evaluation of the spectrum. Outside
the barrier k2 a′′ (equivalent to k a′) the gravitational wave field reduces to
≫ a ≫ a
eikτ e−ikτ
h (τ) A +B , (11)
k k k
→ a(τ) a(τ)
having a decreasing amplitude
h (τ) 1/a(τ). (12)
k
∝
Inside the barrier k2 a′′ (equivalent to k a′) the gravitational wave field
≪ a ≪ a
τ dτ′
h (τ) C +D . (13)
k → k k a2(τ′)
Z
The second term D τ dτ′ is small for the models that we shall study in the following, so the long
a2(τ′)
wave-length limit of hR is simply a constant:
k
h (τ) = C . (14)
k k
Thus, as a function of τ, h (τ) has simple approximate behaviors in the two limiting cases, and we will
k
use these to estimate the spectrum at present stage.
3. Epochs of Expanding Universe
4
The history of overall expansion of the Universe can be modelled as the following sequence of
successive epochs of power-law expansion.
The initial stage (inflationary)
a(τ) = l τ 1+β, < τ τ , (15)
0 1
| | −∞ ≤
where 1+β < 0, and τ < 0. The special case of β = 2 is the de Sitter expansion.
1
−
The z-stage
a(τ) = a (τ τ )1+βs, τ τ τ , (16)
z p 1 s
− ≤ ≤
where β +1 > 0. Towards the end of inflation, during the reheating, the equation of state of energy in
s
the Universe can be quite complicated and is rather model-dependent [22]. So this z-stage is introduced
to allow a general reheating epoch, as has been advocated by Grishchuk [8].
The radiation-dominated stage
a(τ) = a (τ τ ), τ τ τ . (17)
e e s 2
− ≤ ≤
The matter-dominated stage
a(τ) = a (τ τ )2, τ τ τ , (18)
m m 2 E
− ≤ ≤
where τ is the time when the dark energy density ρ is equal to the matter energy density ρ . Before
E Λ m
the discovery of the accelerating expansion of the Universe, the current expansion was usually taken to
be in this matter-dominated stage, which is a decelerating one. Now, following the matter-dominated
stage, we add an epoch of accelerating stage . The value of the redshift z at the time τ is given by
E E
(1+z ) = a(τ )/a(τ ), where τ is the present time. Since ρ is constant and ρ (τ) a−3(τ), one
E H E H Λ m
∝
has
ρ ρ
Λ Λ
1 = = .
ρ (τ ) ρ (τ )(1+z )3
m E m H E
If we take the current values Ω 0.7 and Ω 0.3, then it follows that
Λ m
∼ ∼
Ω
1+z = ( Λ)1/3 1.33.
E
Ω ∼
m
The accelerating stage (up to the present)
a(τ) = l τ τ −1, τ τ τ . (19)
H a E H
| − | ≤ ≤
ThisstagedescribestheacceleratingexpansionoftheUniverse, whichisthenewfeatureinourmodeland
will induce some modifications to the spectrum of the relic gravitational waves. It should be mentioned
that the actual scale factor function a(τ) differs from Eq.(19), since the matter component exists in
the current Universe. However, the dark energy is dominant, so (19) is an approximation to the current
expansion behavior.
5
Given a(τ) for the various epochs, the derivative a′ = da/dτ and the ratio a′/a follow immediately.
Except for the β that is imposed upon as the model parameter, there are ten constants in the above
s
expressions of a(τ). By the continuity conditions of a(τ) and a(τ)′ at the four given joining points
τ , τ , τ , and τ , one can fix only eight constants. The remaining two constants can be fixed by the
1 s 2 E
overall normalization of a and by the observed Hubble constant as the expansion rate. Specifically, we
put τ τ = 1 as the normalization of a, which fixes the constant τ , and the constant l is fixed
H a a H
| − |
by the following calculation
1 a2
= l , (20)
H ≡ a′! H
τH
so l is just the Hubble radius at present. Then everything is fixed up. In the expanding Robertson-
H
Walker spacetime the physical wave length is related to the comoving wave number by
2πa(τ)
λ , (21)
≡ k
and the wave number k corresponding to the present Hubble radius is
H
2πa(τ )
H
k = = 2π. (22)
H
l
H
There is another wave number,
2πa(τ ) k
E H
k = ,
E
≡ 1/H 1+z
E
whose corresponding wave length at the time τ is the the Hubble radius 1/H.
E
By matching a and a′/a at the joint points, we have derived, for example,
β−1 β−βs
l = l bζ−(2+β)ζ 2 ζβζ1+βs, (23)
0 H E 2 s 1
where b 1 + β −(1+β), which is defined differently from Grishchuk’s [8], ζ a(τH), ζ a(τE),
≡ | | E ≡ a(τE) 2 ≡ a(τ2)
ζ a(τ2), and ζ a(τs). With these specifications, the functions a(τ) and a′(τ)/a(τ) are fully
s ≡ a(τs) 1 ≡ a(τ1)
determined. In particular, a′(τ)/a(τ) rises up during the accelerating stage , instead of decreasing as in
the matter-dominated stage. This causes the modifications to the spectrum of relic gravitational waves.
4. The Spectrum of Gravitational Waves.
The power spectrum h(k,τ) of relic gravitational waves is defined by the following equation
∞ dk
h2(k,τ) < 0 hij(x,τ)h (x,τ) 0 >, (24)
ij
k ≡ | |
Z0
where the right-hand-side is the vacuum expectation value of the operator hijh . Substituting Eq.(4)
ij
into the above, and taking the contribution from each polarization to be the same, one reads the power
spectrum
4l
Pl
h(k,τ) = k h (τ) (25)
k
√π | |
6
Once the mode function h (τ) is given, the spectrum h(k,τ) follows.
k
The initial condition is taken to be during the inflationary stage. For a given wave number k, its
wave crossed over the horizon at a time τ , i.e. when the wave length λ = 2πa(τ )/k is equal to
i i i
1/H(τ ), the Hubble radius at time τ . Eq.(15) yields H(τ ) = l τ 2+β/ 1+β , and for the exact de
i i i 0 i
| | | |
Sitter expansion with β = 2 one has H(τ ) = l . Note that a different k corresponds to a different
i 0
−
time τ . Now choose the initial condition of the mode function h (τ) as
i k
1
h (τ ) = . (26)
k i
| | a(τ )
i
Then the initial amplitude of the power spectrum is
l
h(k,τ ) = 8√π Pl. (27)
i
λ
i
From λ = 1/H(τ ) it follows that a′(τi) = k . So the initial amplitude of the power spectrum is
i i a(τi) 2π
k
h(k,τ ) = A( )2+β, (28)
i
k
H
where the constant
l
A = 8√πb Pl. (29)
l
0
For the case of β = 2 the initial spectrum is independent of k. The power spectrum for the primordial
−
perturbations ofenergydensity isP(k) h(k,τ ) 2, anditsspectral indexnisdefinedasP(k) kn−1.
H
∝ | | ∝
Thus one reads off the relation n = 2β +5. The exact de Sitter expansion with β = 2 will yield the
−
so-called scale-invariant spectral index n = 1.
Once the initial spectrum is specified, we can derive the spectrum h(k,τ ) at the present time τ .
H H
For k k , the wave lengths 2πa(τH) in this range are even greater than the present Hubble radius
≤ E k
l , one has k < a′/a throughout the whole expansion up to the present, so the amplitude remains the
H
same constant as the initial one in Eq.(28):
k
h(k,τ ) = A( )2+β, k k . (30)
H E
k ≤
H
During the whole period inside the barrier, the spectral amplitude remains h(k,τ ) approximately
i
until the wave leaves the barrier and begins to decrease as 1/a(τ). Let a (k) be the scale factor at
∗∗
this moment. For those very long wavelength modes with k k k , during the current epoch of
E H
≤ ≤
accelerating expansion, h (τ) stops decreasing as soon as the barrier a′/a is higher than k at a time τ(k)
k
earlier than τ , so h (τ) has decreased by a factor a∗∗(k), and the amplitude of the present spectrum
H k a(τ(k))
is given by
k a (k)
h(k,τ ) = A( )2+β ∗∗ . (31)
H
k a(τ(k))
H
The decreasing factor is written as
a (k) a (k) a(τ )
∗∗ ∗∗ E
= .
a(τ(k)) a(τ ) a(τ(k))
E
7
During the matter-dominated stage one has a (k) τ2 k−2 and a(τ ) τ2 k−2, and during the
∗∗ ∝ ∝ E ∝ E ∝ E
accelerating stage one has a(τ ) τ−1 k and a(τ(k)) τ−1 k, so the decreasing factor is
E ∝ E ∝ E ∝ ∝
a (k) k
∗∗ = ( E)3.
a(τ(k)) k
Thus one gets
k k k 1
h(k,τ ) = A( )2+β( E)3A( )β−1 , k k k . (32)
H k k k (1+z )3 E ≤ ≤ H
H H E
where the relation kE = 1 has been used. The spectrum has a rather stiff slope with the power-law
kH 1+zE
index β 1. The occurrence of this segment of power spectrum is a new feature of the model of
−
accelerating expansion that is absent in the decelerating model. The wave lengths corresponding to this
(k , k ) segment are very long, comparable to the present Hubble radius, and can only possibly be
E H
observed through the CMB anisotropies at low multipoles.
For all the wave number k > k , as soon as the waves leave the barrier at a (k), the modes h (τ)
H ∗∗ k
have been decreasing all the way up to the present time τ . So it has been reduced by a factor a∗∗(k),
H a(τH)
and the amplitude of present spectrum is given by
k a (k)
h(k,τ ) = A( )2+β ∗∗ . (33)
H
k a(τ )
H H
We use this formula to obtain the following result for the spectrum of the gravitational waves in the
remaining range of wave numbers.
For k k k , the wave number does not hit the barrier, so
H 2
≤ ≤
a (k)a (k)a(τ ) k 1
∗∗ ∗∗ E = ( E)2 ,
a(τ ) a(τ ) a(τ ) k (1+z )
H E H E
one obtains
k 1
h(k,τ ) = A( )β , k k k . (34)
H k (1+z )3 H ≤ ≤ 2
H E
The spectrum in this interval differs from that of the matter-dominated model by an extra factor
1 = Ωm 0.43. The wave-lengths in this range are very long, but are still shorter than l . The
(1+zE)3 ΩΛ ∼ H
spectrum in this interval may contribute to, and, therefore, have its imprints in CMB anisotropies. Let
us estimate the value k . Assuming that the equality of radiation and matter occurred at the redshift
2
z 3454, as indicated by the WMAP observation [19], one has
2
≃
k a(τ ) a(τ ) a(τ ) 1
2 = ( E )12 = ( H )21( E )21 √3454 51.
k a(τ ) a(τ ) a(τ ) ≃ √1.33 ≃
E 2 2 H
For k k k , the calculation is similar to the previous case with the result
2 s
≤ ≤
k k 1
h(k,τ ) = A( )1+β( H) , k k k . (35)
H k k (1+z )3 2 ≤ ≤ s
H 2 E
8
Again the extra factor 1/(1+z )3 appears. Notice that this range of frequency covers the one that the
E
detectors of LIGO and LISA are sensitive.
For k k k , the calculation yields
s 1
≤ ≤
k k k 1
h(k,τ ) = A( )1+β−βs( s )βs H , k k k . (36)
H k k k (1+z )3 s ≤ ≤ 1
H H 2 E
It also contains an extra factor 1 . This is the high frequency range which is still beyond the current
(1+zE)3
detection.
5. Determining the Parameters.
Tocompletelydeterminethespectrum, wealsoneedtospecifythevaluesofν ,A, ν ,β ,thatappear
1 s s
in the expressions for the spectrum h(k,τ ). Since the wave number is proportional to the frequency,
H
k ν, theratiosofthewavenumberscanbereplacebythatofthefrequencies, e.g., k/k = ν/ν ,etc.,
H H
∝
in the above formulae of the spectrum h(k,τ ). The Hubble frequency ν = 1/l = H 2 10−18Hz.
H H H
≃ ×
An estimate of the highest frequency ν can be made as given in [8]. From the expression (36) one
1
has
l ν l
h(k ,τ ) = 8√π Pl( 1 ) = 8√π Pl, (37)
1 H
l ν l
H H 1
wheretheexpressions (23)forb/l and(29)forAhavebeenused. Thespectralenergy densityparameter
0
Ω (ν) of the gravitational waves is defined through the relation ρ /ρ = Ω (ν)dν, where ρ is the
g g c g ν g
energy density of the gravitational waves and ρ is the critical energy densitRy. One reads
c
π2 ν
Ω (ν) = h2(k,τ )( )2.
g H
3 ν
H
If it isimposed that atthe highest frequency ν the valueΩ (ν ) not exceed the level of 10−6, asrequired
1 g 1
by the rate of the primordial nucleogenesis, then one gets ν = 3 1010Hz.
1
×
Next let us estimate the overall factor A in the spectrum h(k,τ ). If the CMB anisotropies at low
H
multipolesareinducedbythegravitationalwaves,or,ifthecontributionsfromthegravitationalwavesand
from the density perturbations are of the same order of magnitude, we may assume ∆T/T h(k,τ ).
H
≃
This will determine A. The observed CMB anisotropies [19] at lower multipoles is ∆T/T 0.37 10−5
≃ ×
at l 2, which corresponds to the largest scale anisotropies that have observed so far. Taking this to
∼
be the perturbations at the Hubble radius 1/H yields
1
h(k ,τ ) = A = 0.37 10−5. (38)
H H (1+z )3 ×
E
Actually the observed ∆T/T varies with l, for instance, ∆T/T 1.04 10−5 at l 10, which would
≃ × ∼
give an outcome for A greater than that in Eq.(38) by a factor 2.8.
∼
However, there is a subtlety here on the interpretation of ∆T/T at low multipoles, whose corre-
sponding scale is very large l . At present the Hubble radius is l , and the Hubble diameter is 2l .
H H H
∼
9
On the other hand, the smallest characteristic wave number is k , whose corresponding physical wave
E
length at present is 2πa(τH) = l (1 + z ) 1.32l , which is within the Hubble diameter 2l , and
kE H E ≃ H H
is theoretically observable. So, instead of (38), if ∆T/T 0.37 10−5 at l 2 were taken as the
≃ × ∼
amplitude of spectrum at ν , one would have h(k ,τ ) = A 1 = 0.37 10−5, yielding a smaller
E E H (1+zE)2+β ×
A than that in Eq.(38) by a factor (1+z )1−β 2.2.
E
∼
We now check the range of β allowed. During the inflationary expansion when the k-mode wave
entered the barrier with λ = 1/H(τ ), from which it follows that λ = l0(kH)2+β. For the classical
i i i b k
treatment of the background gravitational field to be valid, this wave length should be greater than the
Planck length, λ > l , so
i Pl
ν 8√π
( )2+β < .
ν A
H
At the highest frequency ν = ν , this yields a constraint
1
8√π ν
1
β < 2+ln( )/ln( ), (39)
− A ν
H
which depends on A. Thus, for A given in (38), one obtains the upper limit β < 1.78. We remark
−
that this range is larger than that in the decelerating model [8], which allows for only β < 1.9.
−
Finally we give an estimate of the allowed values of β and ν . Plugging b/l given by (23) into
s s 0
A = 8√πblPl, using ν /ν = 58.8 and l /l = 1.238 1061, one has
l0 2 H H Pl ×
A ν ν
1.484 1058 = ( 1 )−β( 1)βs. (40)
× (1+z )3 ν ν
E H s
Given a set values of A and β, one can take ν and β to satisfy this relation. From Eq.(36) it is seen
s s
that, for a fixed ν , a smaller β tends to increase slightly the amplitude h(ν,τ ). For definiteness, we
s s H
take ν = 108 Hz as an example in the following. For A given in (38), taking β = 1.8, 1.9, then
s
− −
β = 0.598, 0.552, respectively.
s
−
With these results we plot the spectrum h(k,τ ) vs the frequency ν for two values of parameter
H
β = 1.9, 1.8 in Figure 1. Besides, for convenience of comparison, the spectrum for the decelerating
− −
model is also plotted for β = 1.9. It is clearly seen that, for the same parameter β = 1.9 in the
− −
whole range ν ν , the accelerating model has an amplitude lower by a factor of 2.2 than that
H
≥ ∼
of the decelerating model. This is due to the extra factor 1 = Ωm, as has been demonstrated in
(1+zE)3 ΩΛ
the previous expressions (34), (35), and (36). Interestingly, the larger value β = 1.8 gives a flatter
−
spectrum h(ν,τ ) with an overall higher amplitude in the range ν ν . In particular, in the higher
H 2
≥
frequency range (10−4 0) Hz covered by LISA, the amplitude is h 10−19 10−22 for the β = 1.8
∼ ≃ ∼ −
case, about 10 to 102 higher than the β = 1.9 case. In the even higher frequency range (10 104)
− ∼
Hz covered by LIGO, the amplitude is h 10−23 10−25 for the β = 1.8 case, also 10 to 102 higher
≃ ∼ −
than the β = 1.9 case. Thus an inflationary model of larger index n = 2β +5 will predict a stronger
−
signal of relic gravitational waves in higher frequencies.
10