The investigation of detectability of the relic gravitational waves based on the WMAP-9 and Planck

0.18 MB·

by  Basem Ghayour

The investigation of detectability of the relic gravitational waves
based on the WMAP-9 and Planck
Basem Ghayour1∗
1 School of Physics, University of Hyderabad, Hyderabad-500 046. India.
(Dated: January 27, 2017)
7 Abstract
1
0
Thegeneratedrelicgravitational waveswereunderwentseveralstagesofevolutionoftheuniverse
2

n such as inflation and reheating. These stages were affected on the shape of spectrum of the waves.
a
J As well known, at the end of inflation, the scalar field φ oscillates quickly around some point where

6
2 potential V(φ) = λφn has a minimum. The end of inflation stage played a crucial role on the

]
c further evolution stages of the universe because particles were created and collisions of the created
q
- particles were responsible for reheating the universe. There is a general range for the frequency
r
g
[ of the spectrum (0.3 10−18 0.6 1010)Hz. It is shown that the reheating temperature can
∼ × − ×

2
be affect on the frequency of the spectrum as well. There is constraint on the temperature from
v
6
cosmological observations based on WMAP-9 and Planck. Therefore it is interesting to estimate
9
8
allowed value of frequencies of the spectrum based on general range of reheating temperature
6
0
1. like few MeV . Trh . 1016 GeV, WMAP-9 and Planck data then compare the spectrum with
0
sensitivity of future detectors such as LISA, BBO and ultimate-DECIGIO. The obtained results
7
1
of this comparison give us some more chance for detection of the relic gravitational waves.
:
v
i
X
PACS numbers: 98.70.Vc,98.80.cq,04.30.-w
r
a
Keywords:
∗
[email protected]
1
I. INTRODUCTION
The relic gravitational waves generated in the early universe are very importance because
they provide useful information about physics of the early universe. The waves were un-
derwent several stages of evolution of the universe such as: inflation, reheating, radiation,
matter and acceleration. The evolution of the universe at various epoch is affected the shape
ofthespectrum ofthewaves. Therefore it isunavoidable theconsideration ofdifferent stages
of evolution of the universe on the study of spectrum of the waves that are to be observed
today. It is believed that the universe underwent a quasi exponential expansion in its early
stages of evolution known as inflation stage. This stage is very important on the evolution
of the universe because it provided the mechanism to formation of large scale structures in
the universe. At the end of inflation, the scalar field φ oscillates quickly around some point
where potential V(φ) = λφn has a minimum. The end of this stage played a crucial role on
the further evolution stages of the universe because particles were created and collisions of
the created particles were responsible for reheating the universe. The reheating was essen-
tial for the nucleosynthesis process since the inflation brought temperature of the universe
below for the requirement of thermo nuclear reactions. Towards the end of inflation, during
the reheating, the equation of state of energy for the universe is quite complicated and also
model-dependent [1]. Hence a new stage that called z-stage is introduced to allow a general
behaviour of reheating epoch [2].
Thereisageneralrangeforthefrequency ofthespectrum (0.3 10−18 0.6 1010)Hz. It
∼ × − ×
is shown that the reheating temperature (T ) canbe affect onthe frequency of the spectrum
rh
[3]. There is constraint on the T from cosmological observations based on WMAP-9 and
rh
Planck. However, the reheating temperature must be larger than a few MeV [4], for the
creation light elements, but less than the energy scale at the end of inflation, that is T .
rh
1016 GeV. Therefore it is interesting to estimate allowed value of frequencies of the spectrum
based on general range of T , WMAP-9 and Planck data then compare the spectrum with
rh
sensitivity of future detectors such as LISA [5], Big Bang Observer (BBO) [6] and ultimate
DECI-hertz Interferometer Gravitational-wave Observatory (ultimate -DECIGO) [7]. Hence
the main purpose of this work is investigation of this comparison. The obtained results of
this comparison give us some more chance for detection of the waves. We use the units
c = ~ = k = 1.
B
2
II. GRAVITATIONAL WAVES SPECTRUM IN EXPANDING UNIVERSE
The perturbed metric for a homogeneous isotropic flat Friedmann-Robertson-Walker uni-
verse can be written as
ds2 = S2(η)(dη2 (δ +h )dxidxj), (1)
ij ij
−
where S(η) is the cosmological scale factor, η is the conformal time and δ is the Kronecker
ij
delta symbol. The h are metric perturbations field contain only the pure gravitational
ij
waves and are transverse-traceless i.e; hij = 0,δijh = 0.
i ij
∇
The present study consider the shape of the spectrum of relic gravitational waves that
generatedbytheexpanding space timebackground. Thus theperturbedmattersourceisnot
taken into account. The gravitational waves are described with the linearized field equation
given by
√ g µh (x,η) = 0. (2)
µ ij
∇ − ∇
(cid:0) (cid:1)
The tensor perturbations have two independent physical degrees of freedom like h+ and h×
and called polarization modes. To compute the spectrum of gravitational waves h(x,η), we
express h+ and h× in terms of the creation (a†) and annihilation (a) operators,
√16πl d3k 1
h (x,η) = pl ǫp(k) [aphp(η)eik.x +a†ph∗p(η)e−ik.x], (3)
ij S(η) Z (2π)3/2 ij × √2k k k k k
Xp
where k is the comoving wave number, k = k , l = √G is the Planck’s length and
pl
| |
p = +, arepolarizationmodes. Thepolarizationtensorǫp(k)issymmetricandtransverse-
× ij
traceless kiǫpij(k) = 0,δijǫpij(k) = 0 and satisfy the conditions ǫijp(k)ǫpij′(k) = 2δpp′ and
ǫpij(−k) = ǫpij(k). The a and a† satisfy [apk,a†k′p′] = δpp′δ3(k − k′) and the initial vacuum
state is defined as ap 0 = 0 for each k and p.
k| i
For a fixed wave number k and a fixed polarization state p the linearized wave eq.(2)
gives
S′
h′′ +2 h′ +k2h = 0, (4)
k S k k
where prime means derivative with respect to the conformal time. Because the polarization
states are same, we consider h (η) without the polarization index.
k
Next, we rescale the filed h (η) by taking h (η) = f (η)/S(η), where the mode functions
k k k
f (η) obey the minimally coupled Klein-Gordon equation
k
S′′
f′′ + k2 f = 0. (5)
k − S k
(cid:16) (cid:17)
3
The general solution of the above equation is a linear combination of the Hankel function
with a generic power law for the scale factor S = ηq given by
(1) (2)
f (η) = A kηH (kη)+B kηH (kη). (6)
k k q−1 k q−1
2 2
p p
For a given model of the universe, consisting of a sequence of successive scale factors with
different q, we can obtain an exact solution f (η) by matching its value and derivative at
k
the joining points.
The approximate computation of the spectrum is calculated in two cases depending up
on the waves that are outside or within of the barrier. For the gravitational waves outside
barrier (k2 S′′/S) thecorresponding amplitude decrease ash 1/S(η)and forthe waves
k
≫ ∝
inside the barrier (k2 S′′/S), h = C simply a constant [8].
k k
≪
The history of expansion of the universe can be obtained as follows: The inflation stage
S(η) = l η 1+β, < η η , (7)
0 1
| | −∞ ≤
where 1+β < 0, η < 0 and l is a constant.
0
To make our analysis more general, we consider that the inflation stage was followed
by some interval of the z-stage (z from Zeldovich). In fact this stage is quite general that
considered by Zeldovich [9]. It can be governed by a softer than radiation matter, as well as
by a stiffer than radiation matter [11]. Towards the end of inflation, during the reheating,
the equation of state of energy in the universe can be quite complicated [1]. Hence this
z-stage is introduced to allow a general reheating stage. Therefore we define the reheating
stage in general form
S(η) = S (η η )1+βs, η < η η , (8)
z p 1 s
− ≤
where 1+β > 0 [8].
s
The radiation-dominated stage
S(η) = S (η η ), η η η , (9)
e e s 2
− ≤ ≤
and the matter-dominated stage
S(η) = S (η η )2, η η η , (10)
m m 2 E
− ≤ ≤
where η is the time when the dark energy density ρ is equal to the matter energy density
E Λ
ρ .
m
4
The value of redshift z at η is (1+z ) = S(η )/S(η ) 1.3 from Planck collaboration
E E E 0 E
∼
[12], where η is the present time.
0
The accelerating stage (up to the present)
S(η) = ℓ η η −γ, η η η , (11)
0 a E 0
| − | ≤ ≤
whereγ isΩ dependent parameter, andΩ istheenergy density contrast. Wetakeγ 1.97
Λ Λ
≃
[13] for Ω = 0.73 [14].
Λ
Except for β , there are ten constants in the expressions of S(η). By the continuity
s
conditions of S(η) and S′(η) at four given joining points η ,η ,η , and η , one can fix only
1 s 2 E
eight constants. The remaining two constants can be fixed by the normalization of S and
the observed Hubble constant. We put η η = 1 for the normalization of S, which fixes
0 a
| − |
the η , and the constant ℓ is fixed by the following calculation,
a 0
γ S2
= ℓ , (12)
H ≡ (cid:16)S′(cid:17)η0 0
where ℓ is the Hubble radius at present and H = 100 h km s−1 Mpc−1 with h 0.704 [14].
0
≃
The physical wavelength is related to the comoving wave number as λ 2πS(η)/k.
≡
Assuming that the wave mode crosses the horizon of the universe when λ/2π = 1/H [15],
then wave number k corresponding to the present Hubble radius is k = S(η )/ℓ = γ. Also
0 0 0 0
there is another wave number k = S(ηE) = k0 , whose corresponding wavelength at the
E 1/H 1+zE
time η is the Hubble radius 1/H.
E
By matching S and S′/S at the joint points, one gets
β−1 β−βs
l = ℓ bζ−(2+β)ζ 2 ζβζ1−βs, (13)
0 0 E 2 s 1
where b 1+β −(2+β), ζ S(η0), ζ S(ηE), ζ S(η2), and ζ S(ηs).
≡ | | E ≡ S(ηE) 2 ≡ S(η2) s ≡ S(ηs) 1 ≡ S(η1)
The power spectrum of gravitational waves is defined as
∞ dk
h2(k,η) = 0 hij(x,η)h (x,η) 0 . (14)
ij
Z k h | | i
0
Substituting eq.(3) in eq.(14) and taking the contribution from each polarization is same,
we get
4l
pl
h(k,η) = k h(η) . (15)
√π | |
Thus once the mode function h(η) is known, the spectrum h(k,η) follows.
5
The spectrum at the present time h(k,η ) can be obtained, provided the initial spectrum
0
is specified. The initial condition is taken to be during the inflation. The wave with wave
number k crossed over the horizon at a time η , when the wavelength λ /2π = 1/H(η ) =
i i i
S(η )/k [15]. Now we choose the initial condition of the mode function h as h (η ) =
i k k i
| |
1/S(η ). The initial amplitude of the power spectrum is
i
l
h(k,η ) = 8√π pl. (16)
i
λ
i
With λ /2π = 1/H(η ) it becomes
i i
S′(η )
i
= k. (17)
S(η )
i
Therefore initial amplitude of the spectrum is given by
2+β
k
h(k,η ) = A , (18)
i
(cid:18)k (cid:19)
0
where the constant A in eq.(18) can be determined by quantum normalization [8]:
l
A = 8√π pl. (19)
l
0
Thus the amplitude of the spectrum for different ranges are given as follows [8], [16], [17],
[18].
k 2+β
h(k,η ) = A , k k , (20)
0 E
k ≤
(cid:16) 0(cid:17)
k β−γ −2−γ
h(k,η0) = A (1+zE) γ , kE k k0, (21)
k ≤ ≤
(cid:16) 0(cid:17)
k β −2−γ
h(k,η0) = A (1+zE) γ , k0 k k2, (22)
k ≤ ≤
(cid:16) 0(cid:17)
k 1+β k0 −2−γ
h(k,η0) = A (1+zE) γ , k2 k ks, (23)
k k ≤ ≤
(cid:16) 0(cid:17) (cid:16) 2(cid:17)
k 1+β−βs ks βs k0 −2−γ
h(k,η ) = A (1+z ) γ , k k k . (24)
0 E s 1
k k k ≤ ≤
(cid:16) 0(cid:17) (cid:16) 0(cid:17) (cid:16) 2(cid:17)
The factor A in all the spectra is determined with the CMB data of WMAP-9 [19]. The
observed CMB anisotropies at lower multipoles is ∆T/T 0.44 10−5 at l 2 which
≃ × ∼
corresponds to the largest scale anisotropies that have observed so far. Thus one can gets
h(k0,η0) = A(1+zE)−2γ−γ 0.44 10−5r. (25)
≃ ×
6
where r is tensor to scalar ratio [20] (see the appendix.(A) for more details). The parameter
r taken 0.1 from Planck collaboration [12]. However, there is a point in the interpretation
∼
of δT/T at low multipoles. At present, the Hubble radius and Hubble diameter are ℓ ,
0
2ℓ respectively. The corresponding physical wave length of k at present is S(η )/k =
0 E 0 E
ℓ (1+z ) 1.32ℓ , which iswithin2ℓ andistheoretically observable. So, insteadofeq.(25),
0 E 0 0
≃
if δT/T 0.44 10−5 at l = 2 were taken as the amplitude of the spectrum at k , then we
E
≃ ×
have h(k ,η ) = 0.44 10−5 r1/2 yielding a smaller A than that in eq.(25) [21, 22]. Also
E 0
× ×
there is another normalization method that leads to decaying factor S(η )/S(η ) [23].
i 0
By taking ν as frequency, we can obtain ν = 0.30 10−18 Hz, ν = 0.36 10−18 Hz,
E 0
× ×
ν = 1.48 10−17 Hz, ν = 0.15 108 Hz.
2 s
× ×
The spectral energy density parameter Ω (ν) of gravitational waves is defined through
g
the relation ρ /ρ = Ω (ν)dν, where ρ is the energy density of the gravitational waves
g c g ν g
R
and ρ is the critical energy density. Therefore we have [8]
c
π2 ν 2
Ω (ν) = h2(k,η ) . (26)
g 0
3 ν
(cid:16) 0(cid:17)
We assume that the space time is spatially flat K = 0 with Ω = 1, then the fraction
density of relic gravitational waves must be less than unity, ρ /ρ < 1. In order to ρ /ρ
g c g c
dose not exceed the level of 10−5, the Ω (ν ) 10−6 in eq.(26) therefore we get ν 3 1010
g 1 1
≃ ≃ ×
Hz [8]. When the acceleration epoch is considered, the constraint becomes ν 4 1010 Hz
1
≃ ×
[3].
Sofarwedidnottaketheeffect ofreheating temperature onthespectrum ofgravitational
waves. Then we will consider this effect in the next section.
III. THE EFFECT OF REHEATING TEMPERATURE ON THE SPECTRUM
At the end of inflation, the scalar field φ oscillates quickly around some point where
potential V(φ) has a minimum. It is found that the scalar field oscillations behave like a
fluidwithp = ωρ, where theaverageequation ofstateω depends ontheformofthepotential
V(φ) [24]. For V(φ) = λφn, one has
n 2
ω = − . (27)
n+2
There is theoretical consideration during inflation and reheating stages for the equation of
state -1/3 < ω < 1 [3]. Due to eq.(27) the condition leads to n > 1. The upper bound
7
based on CMB observation for n gives n < 2.1 [25]. Therefore, we can write the range of n
as follows
1 < n < 2.1. (28)
There are two relations that connect the β and β with n [3]:
s
4 n
β = − , (29)
s
2(n 1)
−
and
n
β = 2 (1 n ), (30)
s
− − 2(n+2) −
where n is scalar spectral index. And also we can write the T as follows [26]
s rh
1 n 6
T = 3.36 10−68 − s exp( ), (31)
rh × r A 1 n
s s
−
where A is amplitude of the scalar perturbations. For taking in account the effect of the
s
T on the spectrum, we can consider the following relations [3, 8]:
rh
ν S(η ) S T g 1/3
s 2 rec rh 1
ζ = = = , (32)
s
ν S S(η ) T (1+z ) g
(cid:16) 2(cid:17) rec s CMB eq (cid:16) 2(cid:17)
ν1 (1+βs) S(ηs) mpl n 1/2TCMB g2 1/3 n+2
ζ = = = πA (1 n ) exp ,
1 ν S(η ) kp s − s 2(n+2) T g − 2(1 n )
(cid:16) s(cid:17) 1 0 h i rh (cid:16) 1(cid:17) h − s i
(33)
where S and T stand for the scale factor and the temperature at the recombination,
rec rec
respectively and kp = 0.002 Mpc−1 is pivot wavenumber. The g = 200 and g = 3.91 count
0 1 2
the effective number of relativistic species contributing to the entropy during the reheating
and that during recombination respectively [3]. Also we used T = T (1 + z ) with
rec CMB rec
T = 2.725 K = 2.348 10−13 GeV [14].
CMB
×
Using eq.(31), the obtained T for given n and A based on different types of
rh s s
the object for WMAP-9 (WMAP-9 +eCMB, WMAP-9 +eCMB+BAO, ...) and Planck
(Planck+lensing, Planck+WMAP-9, ...) are shown in tables.(III,III) respectively. There-
fore the acceptable range for T corresponds to the range few MeV . T . 1016 are
rh rh
obtained as follows
0.0108 T 7.8 108 GeV, WMAP 9, (34)
rh
≤ ≤ × −
8
The obtained T is based on WMAP-9. The n , A and m are taken from [27] and W stands for
rh s s
WMAP.
Object n A 109 m 1016(GeV) T (GeV)
s s rh
× ×
W 0.972 0.013 2.41 0.10 1.05 m 1.324 1029
pl
± ± ×
W +eCMB 0.9642 0.0098 2.43 0.084 1.35 m 7.8 108
pl
± ± ×
W +eCMB +BAO 0.9579+0.0081 2.484+0.073 1.61 m 0.0108
−0.0082 −0.072 pl
W +eCMB+H 0.9690+0.0091 2.396+0.079 1.16 m 1.378 1020
0 −0.0090 −0.078 pl ×
W +eCMB+BAO+H 0.9608 0.008 2.464 0.072 1.49 m 399.171
0 pl
± ±
0.090 T 3.4 107 GeV, Planck. (35)
rh
≤ ≤ ×
Thenwefindthefrequenciesν andν asfunctionoftheT withhelpofeqs.(32 35). The
s 1 rh
−
obtained frequencies are smaller than their initial amount (ν = 0.15 108 Hz, ν = 4 1010
s 1
× ×
Hz) due to T as follows
rh
0.78 10−9 ν 0.55 102 Hz, with n 1,
s
× ≤ ≤ × ∼
0.47 10−7 ν 0.44 103 Hz, with n 1, (36)
1
× ≤ ≤ × ∼
0.78 10−9 ν 0.55 102 Hz, with n 2.1,
s
× ≤ ≤ × ∼
0.28 103 ν 1.16 106 Hz, with n 2.1, (37)
1
× ≤ ≤ × ∼
for WMAP-9 and
0.65 10−8 ν 2.45 100 Hz, with n 1,
s
× ≤ ≤ × ∼
0.33 10−6 ν 0.23 102 Hz, with n 1, (38)
1
× ≤ ≤ × ∼
9
The obtained T is based on Planck. The n , A and m are taken from [12]. The P, le, W, and
rh s s
hl stand for Planck, lensing, WMAP and HighL respectively.
Object n ln(1010A ) m 1016(GeV) T (GeV)
s s rh
×
P 0.9616 0.0094 3.103 0.072 1.39 m 1.007 104
pl
± ± ×
P +le 0.9635 0.0094 3.085 0.057 1.30 m 3.4 107
pl
± ± ×
P +W 0.9603 0.0073 3.089+0.024 1.42 m 61.867
± −0.027 pl
P +W +hl 0.9585 0.007 3.090 0.025 1.49 m 0.090
pl
± ±
P +le+W +hl 0.9641 0.0063 3.087 0.024 1.38 m 4.531 103
pl
± ± ×
P +W +hl+BAO 0.9608 0.0054 3.091 0.025 1.41 m 422.196
pl
± ±
0.65 10−8 ν 2.45 100 Hz, with n 2.1,
s
× ≤ ≤ × ∼
0.57 103 ν 0.39 106 Hz, with n 2.1, (39)
1
× ≤ ≤ × ∼
for Planck. It is noted that the T does not change the frequencies less than ν and ν .
rh s 1
The obtained frequencies based on WMAP and Planck can affect on the shape of the
spectrumofthewaves intherangeν uptoν andgivesusinteresting resultsaboutdetection
s 1
ofthewaves. WeplotthespectruminFigs.[1, 2]comparedtothesensitivityofdetectorssuch
asLISA, BBO andultimate-DECIGO(black, redandgreencolorsrespectively). Theyellow,
pink and blue colors are based on the initial amount of (ν ,ν ), WMAP-9 and Planck data
s 1
respectively. Moreover the dashed (solid) of pink and blue colors are based on the minimum
(maximum) amount of (ν ,ν ) due to T from eqs.(34, 35) in both figures. It is noted that
s 1 rh
there are some overlap in dashed yellow and solid blue for both figures.
It is shown that all detectors can detect the waves corresponds to the amount of T in
rh
both figures. But, there is no chance for detection of the waves due to minimum amount of
T (dashed pink and blue colors) with all detectors while it exists for the maximum amount
rh
of T for the ultimate-DECIGO (solid pink and blue colors) in Figs.[1, 2]. The same thing
rh
10