Table Of ContentFully Loop-Quantum-Cosmology-corrected propagation of gravitational waves during
slow-roll inflation
J. Grain∗
Institut d’Astrophysique Spatiale, Universit´e Paris-Sud 11, CNRS
Bˆatiments 120-121, 91405 Orsay Cedex, France
T. Cailleteau,† A. Barrau,‡ and A. Gorecki§
Laboratoire de Physique Subatomique et de Cosmologie, UJF, INPG, CNRS, IN2P3
53, avenue des Martyrs, 38026 Grenoble cedex, France
(Dated: January 29, 2010)
Thecosmologicalprimordialpowerspectrumisknowntobeoneofthemostpromisingobservable
to probe quantum gravity effects. In this article, we investigate how the tensor power spectrum is
0 modifiedbyloop quantumgravity corrections. Thetwomost important quantumterms,holonomy
1 and inverse-volume, are explicitly taken into account in a unified framework. The equation of
0 propagation of gravitational waves is derived and solved for one set of parameters.
2
PACSnumbers: 04.60.Pp,04.60.Bc,98.80.Cq,98.80.Qc
n
a
J
9
2
I. INTRODUCTION
]
c
The inflationary scenario is currently the favored paradigm to describe the first stages of the evolution of the
q
Universe (see, e.g., [1] for a recent review). Although still debated, it has received many experimental confirmations,
-
r including from the WMAP 5-year results [2], and solves most cosmologicalparadoxes.
g
On the other hand, a fully quantum theory of gravityis necessaryto investigate situations where generalrelativity
[
(GR) breaks down. The big bang is an example of such a situation where the backward evolution of a classical
2 space-timecomestoanendafterafiniteamountoftime. AmongthetheorieswillingtoreconciletheEinsteingravity
v with quantum mechanics, loop quantum gravity(LQG) is appealing as it is based on a nonperturbative quantization
2
of3-spacegeometry(see,e.g., [3]foranintroduction). Loopquantumcosmology(LQC)isafinite, symmetryreduced
9
model of LQG suitable for the study of the whole Universe as a physical system (see, e.g., [4]).
8
2 Inthisarticle,weconsidertheinfluenceofLQCcorrectionstogeneralrelativityontheproductionandpropagation
. of gravitational waves during inflation. We first derive the equation of propagation of gravity waves with both
0
holonomy and inverse-volume corrections. This equation is then reexpressed with the commonly used cosmological
1
variables. It is finally solved for a specific set of parameters and the primordial power spectrum is derived. The aim
9
0 ofthisworkistoconcludeourpreviousstudies[5]and[6]where,respectively,onlyholonomyandonlyinverse-volume
: corrections were considered. By combining both terms, we show that the inverse-volume correction dominates over
v
the holonomy one and dictates the overall shape of the tensor spectrum.
i
X Quite a lot of work has already been devoted to gravitational waves in LQC [7]. Our approach assumes the
r background to be described by the standard slow-roll inflationary scenario whereas LQC corrections are taken into
a
accounttocomputethepropagationoftensormodes. Thisapproachisheuristicallyjustified(todecouplethephysical
effects) and intrinsically plausible (as, on the one hand, the LQC-driven superinflation can only be used to set the
proper initial conditions to a standard inflationary stage if the horizon and flatness problems are both to be solved
[8] andas, on the other hand, it seems that the quantum bounce cantrigger ona standardinflationaryphase [9]). In
addition,veryfew studies sofarhavetakeninto accountboth the holonomyandthe inverse-volumecorrections. This
latter term is somehow more speculative than the former one as it was shown to exhibit a fiducial cell dependence
(see,e.g.,[10]). Forthesakeofcompletenessitishoweverobviouslyworthconsideringthefullycorrectedpropagation
of gravitationalwaves.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
2
II. EQUATION OF PROPAGATION FOR THE GRAVITON
The derivation of the equation of propagation of gravitational waves with both holonomy and inverse-volume
correctionsextensivelyusesthe materialdevelopedin[11]: notations,conventionsandframeworkofthisworkarethe
same and will not be explicitly restated. We begin by considering a Friedmann-Lemaˆıtre-Robertson-Walkeruniverse
with a spatial metric q which will be perturbed to account for gravitational waves. Hereafter, N and Na are
ab
respectively the lapse function and the shift function. The metric components read as follows:
g = N2+q NaNb = a2(η), (1)
00 ab
− −
g = q Nb =0, (2)
0a ab
g = q =a2(η)(δ +h ). (3)
ab ab ab ab
As usual in the formalism of LQC, we use the Ashtekar variables for an homogeneous and isotropic background:
the connection A¯i, and the triad density E¯a. They can be written as a function of two other variables (k¯,p¯) as
a i
E¯a = p¯δa,
i i
A¯i = K¯i +Γ¯i
a a a
K¯i = k¯δi, Γ¯i =0,
a a a
N¯a = 0, N¯ =√p¯. (4)
Hamilton-Jacobi equations will be used to determine the perturbed part of the Ashtekar variables. The Hamiltonian
constraint reads as
1
H[N] = 2κ d3xN|detE|−21EjaEkb(ǫijkFaib−2(1+γ2)K[ibKaj]), (5)
ZΣ
where Fi =∂ Ai ∂ Ai +ǫijkAjAk is the field strength. The Hamiltonian for a matter field Φ is given by
ab a b− b a a b
1p2 +EaEb∂ Φ∂ Φ
H = d3x Φ i i a b + detEc V(Φ) . (6)
matter Z 2 |detEjc| q| j|
q
With Eq.(4) and these Hamiltonians, the backgroundis described by
1
Hfond[N¯]= d3xN¯ 6√p¯k2 , (7)
G 2κ −
ZΣ
(cid:2) (cid:3)
and
1p2
Hmatter[N¯]=ZΣd3x(cid:18)2p¯Φ32 +p¯32V(Φ)(cid:19). (8)
Perturbing the canonical variables (and going through the appropriate Poissonbracket) leads to:
1 k¯2
H [N¯]= d3xN¯ 6√p¯k¯2 (δEcδEdδkδj)+√p¯(δKjδKkδcδd)
G 2κ − − 2p¯3/2 j k c d c d k j
ZΣ (cid:20)
2k¯ 1
(δEcδKj) (δ δjkEcδef∂ ∂ Ed) , (9)
−√p¯ j c − p¯3/2 cd j e f k
(cid:21)
where only the tensor perturbations (i.e. gravitationalwaves) are considered in δEa.
i
This classical Hamiltonian is to be modified by quantum corrections. Because loop quantization is based on
holonomies, i.e. exponentials of the connection rather than direct connection components, one needs to substitute in
the gravitationalsector
sin(mµ¯γk¯)
k¯ (10)
→ mµ¯γ
3
where µ¯ is a new parameter related to the action of the fundamental Hamiltonian on a lattice state. In addition,
becauseofinversepowersofthedensitizedtriadwhich,whenquantized,becomesanoperatorwithzerointhediscrete
part of its spectrum, the matter and gravitationalHamiltonians must be modified by introducing the function
l2 n
α(p¯,δEa)=1+λqn =1+λ PL . (11)
i p¯
(cid:18) (cid:19)
At a semiclassical level, i.e. q 1, the same parametric form of α can be used in both the matter Hamiltonian
≪
and the gravitationalHamiltonian. However,the two positive and real valued constants λ and n may differ from one
sector to another. In the following, (S, λ , s) and (D, λ , d) will therefore denote (α, λ, n) for the gravitational
s d
sector and the matter sector respectively. With these two corrections, the Hamiltonians read
1 sinµ¯γk¯ 2
Heff[N¯] = d3xN¯S(p¯) 6√p¯ , (12)
G 2κZΣ "− (cid:18) µ¯γ (cid:19) #
1 p2
Hmatter[N¯] = ZΣd3x(cid:18)2D(q)p¯Φ23 +p¯23V(Φ)(cid:19), (13)
with Heff the effective gravitationalHamiltonian describing the homogeneous background. The equations of motion
G
for (k¯,p¯), i.e. the background equations, can be obtained in the Hamiltonian formalism
p¯˙ = p¯,Heff[N¯]+H [N¯] ; k¯˙ = k¯,Heff[N¯]+H [N¯] , (14)
{ G matter } { G matter }
leading to:
sin(2µ¯γk¯)
p¯˙ = 2 p¯ S(p¯,δE) , (15)
· · · 2µ¯γ
(cid:18) (cid:19)
k¯˙ = κ ∂Hmatter p¯∂S sin(µ¯γk¯) 2 S sin(µ¯γk¯) 2+2p¯∂ sin(µ¯γk¯) 2 . (16)
3V0 ∂p¯ − ∂p¯ ·(cid:18) µ¯γ (cid:19) − 2 "(cid:18) µ¯γ (cid:19) ∂p¯(cid:18) µ¯γ (cid:19) #
The same modification is applied to the perturbed gravitational Hamiltonian. Denoting HPhen the effective per-
G
turbed quantum-corrected gravitationalHamiltonian, it reads with both holonomy and inverse-volume corrections
1 sinµ¯γk¯ 2 1 sinµ¯γk¯ 2
HPhen[N]= d3xN¯S(p¯,δEa) 6√p¯ (δEcδEdδkδj)
G 2κZΣ i "− (cid:18) µ¯γ (cid:19) − 2p¯3/2 (cid:18) µ¯γ (cid:19) j k c d
2 sin2µ¯γk¯ 1
+√p¯(δKjδKkδcδd) (δEcδKj) (δ δjkEcδef∂ ∂ Ed) . (17)
c d k j − √p¯ 2µ¯γ j c − p¯3/2 cd j e f k
(cid:18) (cid:19) (cid:21)
We now turn to the equation of motion of the graviton. The perturbed densitized triad is
1
δEa = p¯ha. (18)
i −2 i
As has been done for the homogeneous canonical variables, it is possible to define the equation of motion for the
perturbations:
δE˙a = δEa,HPhen[N¯]+H [N¯]
i { i G matter }
δ
= δKj(x),δEa(y) (HPhen[N¯]+H [N¯]),
−{ b i }δ(δKj) G matter
b
δK˙i = δKi,HPhen[N¯]+H [N¯]
a { a G matter }
δ
= δKi(x),δEb(y) (HPhen[N¯]+H [N¯]).
{ a j }δ(δEb) G matter
j
This leads to:
1
δE˙a = (p¯˙ha+p¯h˙a) (19)
i −2 i i
sin(2µ¯γk¯)
= S(p¯,δE) p¯ δKl δc δb δEa . (20)
− · · c· a· i − 2µ¯γ · i
(cid:20) (cid:18) (cid:19) (cid:21)
4
By combining those equations and using the expression of p¯˙, one obtains the expression of δKi as a function of hi
a a
and of h˙i. The expression of δKi is:
a a
1 1 sin(2µ¯γk¯)
δKi = h˙i + hi. (21)
a 2S a 2 2µ¯γ a
(cid:18) (cid:19)
The equation of motion will lead to another derivative with respect to η. The Hamilton-Jacobi equation for the
perturbed connection can now be used to find the final equation of propagation for gravitationalwaves:
1 h¨i 1 ∂S sin(2µ¯γk¯) ∂ sin(2µ¯γk¯)
δK˙i = a h˙i + h˙i +hi
a 2" S − S2 ∂η · a (cid:18) 2µ¯γ (cid:19) a a· ∂η (cid:18) 2µ¯γ (cid:19)#
δ
= δKi(x),δEb(y) (HPhen[N¯]+H [N¯]).
{ a j }δ(δEb) G matter
j
As
δHPhen 1 δS 1 2 sin(µ¯γk¯) 2
G = d3(x) N¯ [...]+ d3(x)N¯S (δEc δl δj)
δ(δEjb) 2κZΣ · · δ(δEjb) 2κZΣ "−2p¯23 (cid:18) µ¯γ (cid:19) l · b· c
2 sin(2µ¯γk¯) 2
δKj (δ δjk δ ∂ ∂ (δEd)) ,
− √p¯(cid:18) 2µ¯γ (cid:19) b − p¯23 bd· · ef e f k (cid:21)
where [...] stands for the term beginning with[ 6√p¯ sinµ¯γk¯ 2 ...] in (17), one obtains (with δef∂ ∂ (δEd) =
− µ¯γ − e f k
2(δEd)= 1p¯ 2hd) (cid:16) (cid:17)
∇ k −2 ·∇ k
δHPhen 1 δS
δKi,δEb G = √p¯ [...]
{ a j} δ(δEb) 2 δ(δEb)
j j
1 1 sin(µ¯γk¯) 2 sin(2µ¯γk¯) h˙i sin(2µ¯γk¯)
+ S hi a + hi + 2hi (22)
2 "2(cid:18) µ¯γ (cid:19) a−(cid:18) 2µ¯γ (cid:19) S (cid:18) 2µ¯γ (cid:19) a! ∇ a#
1 h¨i 1 ∂S sin(2µ¯γk¯) ∂ sin(2µ¯γk¯) δH [N¯]
= a h˙i + h˙i +hi κ matter . (23)
2" S − S2 ∂η a (cid:18) 2µ¯γ (cid:19) a a∂η (cid:18) 2µ¯γ (cid:19)#− δ(δEjb)
After quite a lot of algebra, the equation of motion of the graviton can be derived:
1 sin(2µ¯γk¯) p¯∂S
h¨i +2S h˙i 1 S2 2hi +S2T hi +S i =κSΠi , (24)
2 a 2µ¯γ a − S ∂p¯ − ∇ a Q a Aa Qa
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
where
p¯∂µ¯ sin(µ¯γk¯) 4
T = 2 (µ¯γ)2 ,
Q
− µ¯ ∂p¯ µ¯γ
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∂H δEcδjδi δH
Πi = matter j a c cos(2µ¯γk¯)+ matter,
Qa 3V ∂p¯ p¯ δ(δEa)
0 ! i
1 δS ∂S sin(µ¯γk¯) 2
i = √p¯ [...] p¯ cos(2µ¯γk¯) hi.
Aa 2 δ(δEa) − ∂p¯ µ¯γ a
i (cid:18) (cid:19)
As usual, requiring an anomaly-free constraint algebra in the presence of quantum corrections requires i to vanish.
Aa
Itshouldbenoticedthattheinverse-volumecorrectionisinvolvedineachterm,throughtheS andD factors,whereas
the holonomy correction is only involved in the h˙i term, in T and in Πi .
a Q Qa
5
It is worth studying a bit more into the details of this Πi source term as it seems to have been misunderstood in
Qa
severalworks. Inparticular,ithasoftenbeeneitherneglectedormiscomputed. Withoutholonomyandinverse-volume
correction, this term reads as
1 ∂H δEcδjδi δH
Πi = matter j a c + matter , (25)
a 3V ∂p¯ p¯ δ(δEa)
" 0 ! i #
with, in this case,
1 1
Ea =p¯δa , δEa = p¯ha , detE = ǫ ǫijkEaEbEc. (26)
i i i −2 i 3! abc i j k
At the zeroth order in gravitationalperturbation, one can show that
1p2
H¯matter =ZΣd3xN¯ 2p¯φ32 +p¯23V(φ)!, (27)
and the nonlinear H is given by
matter
1 1p2
H =H¯ + d3xN¯ δEaδEbδjδi φ V(φ) , (28)
matter matter ZΣ 4√p¯ i j a b 2p¯3 − !
thus leading to
δH N¯ δEb 1p2
matter = jδjδi φ V(φ) . (29)
δ(δEia) 2 √p¯ a b 2p¯3 − !
Restrictingtothefirstorderinperturbation,thederivativewithrespecttop¯canbeevaluatedandonefinallyobtains
1 ∂H δEcδjδi δH
matter j a c = matter. (30)
3V ∂p¯ p¯ − δ(δEa)
0 i
This easily establishes that classically Πi = 0. However, when LQC corrections are taken into account the source
a
term may not vanishanymore (because of the derivative of D with respect to p¯for the inverse-volumecorrectionand
because of the cosine term for the holonomy one).
When only inverse-volume corrections are considered, the source term is still given by Eq. (25) but the matter
Hamiltonian now reads
H = H¯ +H(δ) (31)
matter matter matter
1p2 1 1p2
= ZΣd3xN¯" D(p¯,δEia)2p¯φ32 +p¯32V(φ)!+ 4√p¯δEiaδEjbδajδbi D(p¯,δEia)2p¯φ3 −V(φ)!#,
which leads, at the leading order, to
δH δEb 1p2 p2 δD
matter =N¯ j δjδi φ V(φ) + φ , (32)
δ(δEia) "2√p¯ a b 2p¯3 − ! 2p¯23 δ(δEia)#
and
1 ∂H δEcδjδi 1 δEcδjδi 3 D 3 ∂Dp2 1
matter j a c =N¯ j a c p2 + √p¯V(φ)+ φ . (33)
3V0 ∂p¯ p¯ 3 p¯ !"−4p¯52 φ 2 ∂p¯ 2 p¯23#
We finally obtain
1 δEcδjδi ∂H δH p2 1 δEcδjδi ∂D δD
Πi,(IV) = j a c matter + matter = φ j a c + . (34)
Qa 3V0 p¯ ! ∂p¯ δ(δEia) 2p¯23 "3 p¯ ! ∂p¯ δ(δEia)#
6
However,becauseof the anomaly-freecondition(see Eq. (27)of[11]), this termis vanishing. This means that, atthe
leading order, Πi,(IV) =0.
Qa
Considering now the holonomy correction alone, one can expand the cosine term in Πi and show that
Qa
1 ∂H¯ δEcδjδi
Πi,(holo) = 2µ¯γsin2 µ¯γk¯ matter j a c .
Qa − 3V0 ∂p¯ p¯ !
(cid:0) (cid:1)
ConsideringsimultaneouslythetwotypesofcorrectionsandusingtheexplicitexpressionofthematterHamiltonian,
one obtains the full LQC source term:
1 ∂H¯ δEcδjδi
Πi,(LQC) = 2µ¯γsin2 µ¯γk¯ matter j a c , (35)
Qa − 3V0 ∂p¯ p¯ !
(cid:0) (cid:1)
as expected from the vanishing inverse-volume source term.
III. SCHRO¨DINGER EQUATION FOR THE FOURIER MODES
The energy density and the pressure of the cosmologicalfluid can be written as
1 δH 1 δH [N¯]
matter matter
ρ= , p= . (36)
V0p¯23 δN¯ −N¯V0 δ( detE )
| |
With the Hamiltonian constraint, one obtains p
1 sin(µ¯γk¯) 2 δH
0= d3xS 6√p¯ + matter, (37)
2κZΣ "− (cid:18) µ¯γ (cid:19) # δN¯
which finally leads to
3S sin(µ¯γk¯) 2
ρ= . (38)
κ p¯ µ¯γ
(cid:18) (cid:19)
Defining astheHubble parameterwithrespecttothe conformaltime ( =a−1da(η)/dη)), weobtainthequantum
H H
Friedmann equations:
sin(2µ¯γk¯) 2
2 = S2
H 2µ¯γ
(cid:18) (cid:19)
p¯κ κ p¯
= S2 ρ 1 µ¯2γ2 ρ , (39)
S 3 − 3S
(cid:16) (cid:17)
which lead, with ρ =3/(κµ¯2γ2p¯), to
c
κ ρ
2 =a2 ρ S . (40)
H 3 − ρ
(cid:18) c(cid:19)
This equation, which has already been found in [12], includes all the LQC corrections and shows that the holonomy
term, leading to the bounce, is the most important one as far as the background in concerned. This conclusion will
be radically modified for perturbations.
The equation of motion for the graviton can now be reexpressed in terms of the commonly-used cosmological
variables. By taking into account Eq. (38) and µ¯2p¯=l2 , one obtains
PL
p¯∂µ¯ sin(µ¯γk¯) 4 κa2
S2T = 2 (Sµ¯γ)2 = ρ2. (41)
Q
− µ¯ ∂p¯ µ¯γ 3 ρ
(cid:18) (cid:19) (cid:18) (cid:19) c
The multiplicative factor of h˙i in Eq. (24) can be reexpressed as a function of the Hubble parameter
a
sin(2µ¯γk¯) p¯∂S 1aS˙
2S 1 =2 1 . (42)
(cid:18) 2µ¯γ (cid:19)(cid:18) − S ∂p¯(cid:19) H − 2a˙ S!
7
Finally, the source term can be explicitly computed
hi ρ p¯ Φ˙2 1D˙ a
Πi = a ρ 1 .
Qa S ρc2" − D(q)a2 − 6Da˙!#
As in [11], we use the effective parametrizationS =1+λs(q)−2s with q =(a/lPL)2. The equationof propagationcan
now be written as
a˙ 1∂ln(S)
h¨+2 1 h˙ S2 2+M2(a) h=0, (43)
a − 2 ∂ln(a) − ∇
(cid:18) (cid:19)
(cid:0) (cid:1)
with
ρ 2 Φ˙2 1D˙ a
M2(a)=κ a2 ρ 1 . (44)
ρc 3 − D(q)a2 − 6Da˙!!
This can be usefully expressed as an equation for the spatial Fourier transform h of h
k
a˙ 1aS˙
h¨ +2 1 h˙ +(S2k2 M2(a))h =0. (45)
k k k
a − 2a˙ S! −
The variables are changed according to φ =h a/√S, leading to a Schr¨odinger-like equation
k k
2
a¨ a˙ S˙ 3 S˙ 1S¨
φ¨ + S2k2 +M2(a) + φ =0. (46)
k −a − aS 4 S! − 2S k
IV. POWER SPECTRUM
The mainquestionto addressis to investigateif one correction,either holonomyorinverse-volume,dominates over
the other as far as the production of gravitational waves during inflation is concerned. The system describing the
dynamics is
κ ρ
2 = a2 ρ S ,
H 3 − ρ
(cid:18) c(cid:19)
a˙ 1aD˙
0 = Φ¨ +2 1 Φ˙ +a2DV (Φ),
k k ,Φ
a − 2a˙ D!
2
a¨ a˙ S˙ 3 S˙ 1S¨
0 = φ¨ + S2k2 +M2(a) + φ ,
k −a − aS 4 S! − 2S k
which is unfortunately much too difficult to be analytically solved. We therefore turnto the approach developed in
[5, 6]. The backgroundevolution is assumed to be classical(D 1) with the scale factor given by the usual slow-roll
≈
approximation a(η) = l η −1−ǫ. In this case, the effective Schr¨odinger equation d2 +E (η) V(η) φ (η) = 0,
0| | dη2 k − k
reads, to first order in λs, as h i
s
l
E (η) = S2k2 = 1+2λ PL η s(1+ǫ) k2, (47)
k s
l | |
(cid:20) (cid:18) 0 (cid:19) (cid:21)
2+3ǫ 6 1 (1+4ǫ)
V(η) = + η −2(1−ǫ)
η2 κρ l2 | |
c 0
l s 12 1 (1+4ǫ) 1
+ λ PL η s−2+ǫ(s+2)+s(1+2ǫ)η s(1+ǫ)−2 s(s 1+ǫ(2s 1))η s(1+ǫ)−2 .
s l − κ ρ l2 | | | | − 2 − − | |
(cid:18) 0 (cid:19) (cid:20) c 0 (cid:21)
(48)
8
To implement initial conditions, we consider the limit η where the adiabatic vacuum holds. Of course, if
→ −∞
higherordertermsinλ weretobe included, the vacuumwouldnotbe the sameanymore. However,wehavechecked
s
that the adiabaticity condition would still be fulfilled in the relevant wavenumber range.
It is possible to solve analytically this equation, at least for one set of parameters: s=2 and ǫ=0. It becomes
d2φ l 2 2 3 1 1 l 2 12 1 1
k + 1+2λ PL η2 k2 1 λ PL +1 φ =0. (49)
dη2 " s(cid:18) l0 (cid:19) ! − η2 (cid:18) − κρcl02(cid:19)− s(cid:18) l0 (cid:19) (cid:20)− κ ρcl02 (cid:21)# k
By some appropriate changes of variables, this equation can be turned into a Whittaker equation. The solution can
be expressed with Kummer functions and the Wronskian condition φ ∂ φ+ φ+∂ φ = 16iπ/M2 allows one to
k η k − k η k PL
normalize the modes. The field is then given at the end of inflation by
φk(c)= MPL2(√k√2π2Z)14e2iπae−2icc41+µU(cid:18)12 +µ−v,1+2µ,ic(cid:19), (50)
and the resulting primordial tensor power spectrum is
2
PT(k)= M126 k3−2µH02(√2Z)−2µ Γ(Γb(−a)1)e−2iπv , (51)
PL (cid:12) (cid:12)
(cid:12) (cid:12)
with (cid:12) (cid:12)
(cid:12) (cid:12)
1 1 3 8γ2l2 i γ2l2
a = +µ v = + 1+ PL + k2 Z 1 4 PL , (52)
2 − 2 4s 9l02 √32Zk2 (cid:18) − (cid:18) − l02 (cid:19)(cid:19)
3 8γ2l2
b = 1+2µ=1+ 1+ PL, (53)
2s 9l02
i γ2l2
v = k2 Z 1 4 PL , (54)
−√32Zk2 − − l2
(cid:18) (cid:18) 0 (cid:19)(cid:19)
where Z = (l /l )2λ and γ2 = 3/(κρ l2 ). The ultraviolet limit of this spectrum can be easily derived and leads
PL 0 s c PL
to
2
l 3 Z
PTUV(k)=16π3 Pl2L 1+ 2k2(1−4ǫ) k−34ω, (55)
(cid:18) 0 (cid:19) (cid:18) (cid:19)
with ω =γ2l2 /l2. On the other hand, the infrared limit is given by
PL 0
2
PTIR(k)=16π3 lPlL (Z(1−4ω))−23k3eπ√Z8 (1−k4ω). (56)
(cid:18) 0 (cid:19)
Those results show that the k + limit of the power spectrum is in agreement with the general relativistic
→ ∞
behavior with the addition of a slight tilt. The ultraviolet spectrum is nearly asymptotically scale invariant. This is
notsurprisingasboththeholonomycorrection(encodedinthek−43ω term)andtheinverse-volumecorrection(encoded
in the 1+ 3 Z(1 4ǫ) term), taken individually, lead to this behavior. The infrared limit is more interesting as,
2k2 −
in this case, the holonomy and inverse-volume corrections lead to very different spectra. The result obtained here
(cid:0) (cid:1)
shows that the power spectrum is exponentially divergent, in exact agreement with the limit obtained with the
inverse-volume correction alone. This proves that, under the standard inflationary background evolution hypothesis,
the inverse-volume term strongly dominates over the holonomy one. This is to be contrasted with the background
evolution in the very remote past where the holonomy term alone leads to the replacement of the singularity by a
bounce.
V. CONCLUSION
This work derives the fully LQC-corrected equation of motion for gravitational waves. This equation is expressed
in terms of cosmological variables and is explicitly solved for a given set of parameters in a standard inflationary
background. It is shown that the spectrum remains exponentially infrared divergent, as for a pure inverse-volume
correction. This reinforces the use of primordial gravitational waves as a strong probe of loop quantum gravity
effects. The next step is naturally to build a fully consistent model which includes all the corrections for both the
perturbations and the background .
9
Acknowledgments
This work was supported by the Hublot - Gen`eve company.
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