Graviton mass constraint from CMB

0.3 MB·

by  N. Malsawmtluangi

Graviton mass constraint from CMB
N. Malsawmtluangi ∗and P. K. Suresh †
School of Physics, University of Hyderabad.
P. O. Central University, Hyderabad 500046. India.
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Abstract

n
a TheeffectofprimordialmassivegravitationalwavesontheBB-modecorrelationangular
J
powerspectrumofCMBisstudiedforseveralinflationmodels. Theangularpowerspectrum

2 withtheBICEP2/KeckArrayandPlanckjointdatasuggestsfurtherconstraintonthelower
1 and upperboundson themass of primordial gravitons

]
c
q 1 Introduction
-
r
g
[ The force of gravity is believed to be mediated by spin-2 particles called gravitons which are

commonlyconsideredasmassless. However,initiatingwiththeideaofaspin-2particlewithnon-
1
zerorestmass,severalapproacheshavebeentakentointroducemasstograviton[1,2,3,4,5,6].
v
6 Endowinggravitonwithmassleadstoextradegreesoffreedomwhichdonotdecoupleasgraviton
1 mass approaches to zero implying that the general relativistic (GR) case cannot be recovered
3 [2, 3]. Some of the approaches to massive gravity suffer from pathologies like the presence of
3
ghost mode [4], discontinuity when the mass approaches to zero limiting case and so on [5],
0
and several theories have been proposed to fix these problems and also attempted to formulate
.
1
a consistent theory of massive gravity [6, 7, 8, 9, 10, 11]. At the same time, there have been
0
several attempts to estimate the mass of graviton from astrophysical sources and primordial
7
gravitational waves (GWs) [12, 13, 14, 15, 16, 17, 18]. It is proposed that if graviton mass
1
: is comparable to the Hubble parameter, then it can provide a repulsive effect at cosmological
v
distancesandhenceleadtothelatetimecosmicacceleration,therebysuggestingthatthemassive
i
X gravitons responsible for the current accelerating phase of the universe instead of dark energy.
r
a Inthis paper,we considerthe Lorentz-violatingmassivegravitytheory inwhichthe Lorentz
invariance is spontaneously broken by a convenient choice of the vacuum for the Goldstone
fields, and the mass parameters are chosen in such a way that the pathologies are absent, and
the scalar and vector modes behave exactly like those in the general relativistic case. Hence,
the modification ofthe gravitycomes only fromthe tensormodes andthe dispersionrelationof
gravitationalwavesacquiresaneffective massandis relativistic[7, 8]. Accordingto this theory,
the bound on the primordial graviton mass comes from the exponential decay in the Yukawa
potential, putting the upper bound for the graviton mass to be 10−30 eV [8, 19]. The lower
≤
bound for graviton mass has been proposed to be 1.239 10−32 eV [20]. The small mass of
×
graviton is expected to have an effect on the temperature anisotropy and polarization spectra
∗e-mail: [email protected]
†e-mail: [email protected]
1
of the cosmic microwave background (CMB) [19, 21]. The imprint of primordial gravitational
waves on CMB anisotropy can be observed through the angular power spectrum of CMB in
the form of B-mode polarization [22, 23, 24, 25]. The observations of B-mode polarization on
CMB would not only verify the theory of inflation itself but also help in constraining many
inflation models [26, 27, 28]. The detection of B-mode polarization of CMB or the primordial
GWitselfwouldprovideaclearboundonthemassofprimordialgraviton. Hence,inthispaper,
we study the effect of the primordial massive GWs on the BB mode correlation angular power
spectrum of CMB for various inflation models and the results are compared with the recent
BICEP2/Keck Array and Planck collaboration data [29] and thereby to obtain constraint on
the mass of primordial gravitons.
2 Massive gravitational waves
For massive gravity, the action can be written in terms of the Einstein-Hilbert action and the
Goldstone action as [7, 8],
S = S +S ,
EH G
= d4x√ g[ m2 R+Λ4F(Zij)], (1)
− − pl
Z
where Λ characterizes the cutoff energy scale for low energy effective theory. F is a function
of the Goldstone field, metric components and its derivatives. The second term in the above
action leads to violation of the Lorentz symmetry. It is assumed that ordinary matter field is
minimally coupled to the metric.
The argument Zij can be obtained with the help of the following expressions
Zij = XγWij,
X = Λ−4gµν∂ ζ0∂ Φ0,
µ ν
ViVj
Wij = Λ−4gµν∂ Φi∂ Φj ,
µ ν
− X
Vi = Λ−4gµν∂ Φ0∂ Φi, (2)
µ ν
where Φ0(x), Φi(x), (i=1,2,3) are the four scalar fields and γ is considered as a constant free
parameter.
For Eq.(1), the vacuum solutions corresponding to the flat Friedmann-Lemaitre-Robertson-
Walker (FLRW) metric can be written as
g = a2η ,
µν µν
Φ0 = Λ2t, (3)
Φi = Λ2xi.
where a is the scale factor for the FLRW metric and η is the flat space metric.
µν
The metric g with perturbations can be written as
µν
g =a2η +δg , (4)
µν µν µν
wherethemetricperturbationsδg aretakenafterthespontaneousLorentzsymmetrybreaking.
µν
2
The components of the metric perturbation are given by
δg = 2a2ϕ,
00
δg = a2(N ∂ A), (5)
0i i i
−
δg = a2[ h ∂ Q ∂ Q +2(ψδ ∂ ∂ E)],
ij ij i j j i ij i j
− − − −
where ϕ, ψ, A and E are scalar fields, N and Q are transverse vector fields and h is the
i i ij
transverse-tracelesstensor perturbation.
By expanding √ g+δg, X(g+δg), Vi(g+δg), Wij(g+δg) in Eq.(3) and using Eq.(1) we
−
get the Lagrangianas
m2
L = pl(m2h h +2m2h h m2h h +m2h h 2m2h h ), (6)
m 2 0 00 00 1 0i 0i− 2 ij ij 3 ii jj − 4 00 ii
where the mass parameters are given by [30],
Λ4
m2 = [XF +2X2F ],
0 m2 X XX
pl
2Λ4 1
m2 = [ XF WF + XWF ],
1 m2 − X − W 2 VV
pl
2Λ4
m2 = [WF 2W2F ], (7)
2 m2 W − WW2
pl
Λ4
m2 = [WF +2W2F ],
3 m2 W WW1
pl
Λ4
m2 = [XF +2XWF ],
4 −m2 X XW
pl
where
W = 1/3δ Wij,
ij
−
∂F
= F ,
X
∂X
∂2F
= F ,
∂X2 XX
∂F
= F δ , (8)
∂Wij W ij
∂2F
= F δ ,
∂Vi∂Vj VV ij
∂2F
= F δ δ +F (δ δ +δ δ ),
∂Wij∂Wkl WW1 ij kl WW2 ik jl ij jk
∂2F
= F δ .
∂X∂Wij XW ij
There is a number of different regions in the mass parameter space where massive gravityis
described by a consistent low-energy effective theory with strong coupling scale Λ (mm )1/2
pl
∼
whichimplies aghost-freescenario. Eachofthese regionsischaracterizedbycertainfine-tuning
relations between the mass parameters. In the vector sector, provided m = 0, the vector field
2
6
behaves in the same way as in the Einstein theory in the gauge F = 0; hence there are no
i
3
propagating vector perturbations and gravity is not modified in this sector unless one takes
into account the non-linear effects or higher derivative terms [7, 8]. In the scalar sector, the
scalar field has massless limit which coincides with the GR expression; hence there is no vDVZ
discontinuity. In the tensor sector, only the transverse-traceless perturbations h are present
ij
and their field equation is that of a massive field with the mass m with helicity-2; hence there
2
are two massive spin-2 propagating degrees of freedom.
The perturbed metric for a flat FLRW universe can be written as
ds2 =a2(τ)[ dτ2+(δ +h )dxidxj], (9)
ij ij
−
here δ is the flat space metric and τ is the conformal time defined by dτ = dt.
ij a
The dynamical equation of motion for massive gravitational waves can be written as
h(m)′′(τ)+2 h(m)′(τ)+k2h(m)(τ)+a2m2 h(m)(τ)=0, (10)
ij H ij ij gw ij
where m m is the mass of the gravitonand = a′ is the Hubble parameter.
gw ≡ 2 H a
The massive tensor perturbation h(m) can be expanded in the Fourier space as
ij
D ∞ d3k
h(m)(x,τ) = [h(m)(p)(τ)c(m)(p)ε(m)(p)(k)eik.x
ij (2π)23 Z−∞ √2Ek ij ij ij
+h(m)(p)∗(τ)c(m)(p)†ε(m)(p)∗(k)e−ik.x], (11)
ij ij ij
where D = √16πl is the normalization constant, l is the Planck length, E is the energy of
pl pl k
the mode, (p) is the polarization index and the superscript (m) stands for the massive tensor
perturbation.
The two polarization states ε(p), p=1,2 are symmetric and transverse-tracelessand satisfy
ij
the conditions
ε(ijp)δij =0, ε(ijp)ki =0, ε(ijp)ε(p′)ij =2δpp′, ε(ijp)(-k)=ε(ijp)(k).
Thesepolarizationsarelinearandarecalledtheplus(+)polarizationandcross( )polarization.
×
The creation and annihilation operators c(p)† and c(p) satisfy the following relations
k k
c(kp),ck(p′′)† = δpp′δ3(k−k′), (12)
h c(p),c(p′)i = c(p)†,c(p′)† =0. (13)
k k′ k k′
h i h i
Using Eq.(11) in Eq.(10), we get
h(m)′′(τ)+2 h(m)′(τ)+(k2+a2m2 )h(m)(τ)=0. (14)
k H k gw k
Here after we drop the polarization index (p) and the index (m) for notational convenience.
The mode function can be taken in the following form
µ (τ)=a(τ)h (τ). (15)
k k
Using Eq.(15) in Eq.(14) we get
a′′
µ′′+ k2+a2m2 µ =0. (16)
k gw− a k
(cid:18) (cid:19)
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The dispersion relation can be written as [31]
k2
+m2 =w2, (17)
a2 gw
where w is known as the effective frequency.
For the adiabatic vacuum, Eq.(14) has the solution
h (τ) e−iwaτ. (18)
k
∝
For the frequency lower than the rate of cosmic expansion, w2 2, the mode is termed
≪ H
super-horizon mode. The tensor amplitudes are frozen and the mode stays outside the horizon
and is constant, and its absolute value is
h = (k), τ <τ , (19)
k ex k
| | A
where (k)= Hex , is the amplitude of the mode at the time of its generation and is
Aex mplk3/2 Hex
the expansionrateatthe time ofhorizonexitduring inflation, τ is the time ofhorizonre-entry
k
and m is the reduced Planck mass.
pl
Whenw is comparableto the rate ofcosmicexpansion, w2 2, for amode with comoving
≃H
momentumk,thecorrespondingtimeiscalledhorizoncrossingtime. Assumingthatthehorizon
re-entry takes place sufficiently rapidly, i.e., τ τ , then Eq.(18) can be rewritten as
k
≃
(k)
h (τ)= C e−iwaτ, τ τ , (20)
k k
w a3 ≃
k k
p
where w w(τ )= indicates horizon re-entry and (k) is a constant of integration.
k k k
≡ H C
With the evolutionof the universe, the modes re-enter the horizonand their amplitudes are
nolongerconstant. Thefrequencybecomeshigherthantherateofcosmicexpansion,w2 2,
≫H
calledsub-horizonmode. Oncethe modere-entersthehorizon,itoscillates. Itssolutionisgiven
by Eq.(18)
(k)
h (τ)= C e−iwaτ, τ >τ . (21)
k k
w(τ)a3(τ)
p
Using Eq.(19), Eq.(20) and Eq.(21), we get
h (τ) w a3
| k | = k k , τ >τ . (22)
ex(k) sw(τ)a3(τ) k
A
Replacing w by k/a and τ by τGR, GR indicating the massless case, we get the corresponding
k k
solution in the massless case as
hGR(τ) aGR
| k | = k , τ >τGR. (23)
(k) a(τ) k
ex
A
The two-point correlation function for the massive gravitational waves can be written as
d
P(w ) 0h hij 0 , (24)
0 ij
≡ dlnw h | | i
0
5
where
D2 ∞ dk
0h (x,τ)hij(x,τ)0 = k2 h (τ)2 . (25)
h | ij | i 2π2 | k | k
Z0
Therefore one gets
w2 2k3
P(w )= 0 h (τ )2, (26)
0 w2 m2 π2 | k 0 |
0 − gw,0
where
k =a w2 m2 ,
0 0− gw,0
d dk q w2
= 0 .
dlnw k w2 m2
0 (cid:18) (cid:19) 0− gw,0
Using Eq.(22), the power spectrum for the massive gravitationalwaves is obtained as
2k3 k′a 2 w a
P(w ) = 2(k) k k k
0 π2 A ka w a
(cid:18) 0 (cid:19) 0 0
k′a 2 w a
k k k
= P(k), (27)
ka w a
(cid:18) 0 (cid:19) 0 0
where k′ =a w and P(k)= 2k3 2(k) is known as the primordial power spectrum.
0 0 π2 A
Using Eq.(23), the power spectrum for the massless case can be written as
aGR 2
P (w )= k′ P(k′). (28)
GR 0
a
(cid:18) 0 (cid:19)
By taking the ratio of Eq.(27) to Eq.(28), we obtain
P(w ) P(k) k′a 2 w a
0 k k k
=
P (w ) P(k′) kaGR w a
GR 0 (cid:18) k′ (cid:19) 0 0
P(k)
= S2(w ), (29)
P(k′) 0
where the enhancement factor S(w ) can be written as
0
k′a w a
k k k
S(w )= . (30)
0 kaGR w a
k′ r 0 0
The dispersion relation at the time of horizon re-entry is
w m (τ ). (31)
k gw k
≃
The cosmic expansion rate is comparable to the effective mass of the gravitational waves
when all modes re-enter the horizon simultaneously, then
(τ ) m (τ ).
k gw k
H ≃
Therefore, we have η η , a a , and w m (τ )= khc.
k ≃ hc k ≃ hc Hk ≃H hc ≃ gw hc ahc
6
Byconsideringthemasstermwhichdominatesthefrequencymodestillpresenttime,weget
k
0
w m = ,
0 gw,0
≃ a
0
k′ k .
0
≃
For long wavelength modes, the enhancement factor becomes [31]
−1
a k w2 2
S(w ) hc hc 0 1 . (32)
0 ≃ aGk0Rr k0 m2gw,0 − !
The massive short wavelength modes behave almost similar to their massless counterparts
and hence, are not considered here.
3 Inflation
In the simplest inflationary scenario, the sudden expansion of early universe is driven by a
canonicalsinglescalarfieldcalledtheinflaton. Intheslowrollinflationaryscenario,theinflaton
slowly rolls down its potential which is almost flat.
The equation of motion for the inflaton with effective potential V can be written as
φ¨+3Hφ˙ +V′(φ)=0, (33)
where the Hubble parameter H is determined by the energy density of the inflaton field,
φ˙2
ρ = +V,
φ
2
so that the Friedmann equation can be written as
1 1
H2 = φ˙2+V(φ) . (34)
3m2 2
pl (cid:18) (cid:19)
In the slow-rolllimit, the Hubble parameter takes the following form
V
H2 . (35)
≃ 3m2
pl
The slow-rollcondition is characterizedin terms of the slow-rollparameters defined in terms of
the inflaton potential and its derivatives as follows
m2 V′ 2
pl
ǫ 1,
≡ 2 V ≪
(cid:18) (cid:19)
V′′
η m2 1, (36)
≡ pl V ≪
(cid:18) (cid:19)
these arethe sufficientbut notnecessaryconditions. As longasthe slow-rollconditions aresat-
isfiedtheinflationcontinues. The slow-rollapproximationcanbe usedto studythe fluctuations
generated during inflation.
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The strength of the tensor fluctuations can be measured with respect to that of the scalar
fluctuations in terms of the tensor-to-scalarratio r given by
P (k)
T
r 16ǫ. (37)
≡ P (k) ≃
S
The tensor spectral index can be given by the parameter ǫ as
n = 2ǫ. (38)
T
−
It can be seen that both r and n are determined by the equation of state during inflation.
T
They are useful inunderstanding the dynamics ofthe earlyuniverse as wellas indistinguishing
various inflationary models.
3.1 Inflation models
In this work, we consider the single field slow-roll inflation models for which the corresponding
tensor-toscalarratiolieswithin (10−3)andr <0.07[32]. Thescalarpowerspectrumforeach
O
model is taken to be P =2.43 10−9.
S
×
R2 Inflation model (Starobinsky model)
This model is based on the higher order gravitationalterms with the action [33]
m2 R2
S = d4x√ g pl R+ , (39)
− 2 6m2
Z (cid:18) (cid:19)
where R is the Ricci scalar and m is the inflaton mass.
ThemodelcanberepresentedintheformofEinsteingravitywithanormalizedinflatonfield
with effective potential,
V(φ)=M4(1 e−√2/3φ/mpl)2. (40)
−
Thetensor-to-scalarratioforthismodelisobtainedasr =3.25 10−3. Theslow-rollparameters
×
obtained for the model are
ǫ = 2.03 10−4,
×
η = 1.63 10−2. (41)
− ×
The calculated tensor power spectrum with the tensor spectral index n = 4.06 10−4 is
T
− ×
P =7.9 10−12.
T
×
Arctan Inflation model
This model is consideredas a large field inflation where the inflaton field starts at a large value
and then evolves to the minimum potential [34, 35]. The effective potential for this model is
given by
φ
V(φ)=M4 1 arctan , (42)
− µ
(cid:20) (cid:18) (cid:19)(cid:21)
8
where µ/m = 10−2 is a free parameter which characterizes the typical vacuum expectation
pl
value at which inflation takes place, M/m =10−3.
pl
Thetensor-to-scalarratioforthismodelisfoundasr =1.38 10−2. Thecalculatedslow-roll
×
parameters are,
ǫ = 8.62 10−4,
×
η = 3.0 10−2. (43)
×
TheobtainedtensorpowerspectrumisP =3.35 10−11forwhichthetensorspectralindex
T
×
has the value n = 1.72 10−3.
T
− ×
Higgs Inflation model
Inthismodel,theHiggsfieldisconsideredtoplaytheroleoftheinflaton. Thefieldisconsidered
to be non-minimally coupled to gravity [36]. The effective potential for this model is
V(φ)=M4(1+e−√2/3φ/mpl)−2. (44)
The tensor-to-scalar ratio for this model is r = 2.83 10−3. The corresponding slow-roll pa-
×
rameters are,
ǫ = 1.77 10−4,
×
η = 1.48 10−2. (45)
− ×
The tensor power spectrum is obtained as P =6.87 10−12 with n = 3.53 10−4.
T T
× − ×
Inverse Monomial Inflation model
This model is considered in the context of quintessential inflation where the inflaton need not
necessarily decay and hence, may survive through the present epoch. Since the inflaton does
not decay, radiation is created via gravitational particle production [37, 38, 39]. The effective
potential for this model is
φ −p
V(φ)=M4 , (46)
m
(cid:18) pl(cid:19)
where p is a positive parameter, M/m =10−1.
pl
The calculated tensor-to-scalar ratio for this model is r = 2.0 10−3 and the slow-roll
×
parameters are,
ǫ = 1.25 10−4,
×
η = 3.33 10−4. (47)
×
The tensor power spectrum is found as P =4.86 10−12 with n = 2.50 10−4.
T T
× − ×
9
Loop Inflation model
Thismodelisstudiedinthecontextofspontaneoussymmetrybreakingwhichalterstheflatness
ofthe potentialandtakesthe formoflogarithmicfunction for one loopordercorrection[40,41,
42]. The effective potential for this model is
φ
V(φ)=M4 1+αln , (48)
m
(cid:20) (cid:18) pl(cid:19)(cid:21)
where α=g2/16π2 tunes the strength of radiative effects, M =1016 GeV.
The tensor-to-scalar ratio for this model is obtained as r = 4.34 10−2. The calculated
×
slow-rollparameters are,
ǫ = 3.09 10−3,
×
η = 2.06 10−2. (49)
− ×
The calculated tensor power spectrum with the tensor spectral index n = 6.18 10−3 is
T
− ×
P =1.2 10−10.
T
×
4 The BB-mode angular power spectrum of CMB
The expression for computing the BB-mode correlation angular power spectrum of CMB is
given by [43, 44]
CBB = (4π)2 dkk2P (k)
l T
Z
τH j (x) 2
dηg(τ)h (τ) (8x+2x2∂ ) l , (50)
×(cid:12)(cid:12)Z0 k n x x2 ox=k(τ0−τ)(cid:12)(cid:12)
where x=k(τ0 τ), g(τ)=κ(cid:12)(cid:12)e−κ is the probability distribution of the last sc(cid:12)(cid:12)attering with κ as
−
the differential optical depth and j (x) is the spherical Bessel function.
l
The CMB angularspectrumfor the BB mode correlationwith the slow-rollinflationmodels
are obtainedby using the CAMB code with κ=0.08andk =0.002Mpc−1 as the tensor pivot
0
scale. The obtained results are presented in Figs.1, 2, 3, 4 and 5. The limit (BK BK
× −
αBK P)/(1 α) is taken from the BKP joint data after subtraction of dust contribution of
× −
the BICEP2/KeckArray band which gives the fiducial value α=0.04 [29].
5 Discussion and conclusion
The BB mode correlation angular power spectrum of CMB for the primordial massive gravi-
tational waves for the Starobinsky (R2), Arctan, Higgs, Inverse monomial and Loop inflation
models is studied in the context of Lorentz violating massive gravity model. It is observed for
each inflation model that, for gravitational waves with mass m & 1.4 10−16 Hz, there is
gw
×
enhancement in the power level compared to the massless gravitational waves case while there
isdecreaseinthepowerlevelinthecaseofm <1.4 10−16 Hz. TheBBmodeangularpower
gw
×
spectrum of CMB for gravitational waves with mass m 1.4 10−16 Hz ( 5.79 10−31
gw
≃ × ≡ ×
10