On Gauge Invariant Cosmological Perturbations in UV-modified Horava Gravity

0.36 MB·

by  Sunyoung Shin

arXiv:1701.xxxx [hep-th]
January 17, 2017
On Gauge Invariant Cosmological Perturbations in UV-modified
Hoˇrava Gravity
Sunyoung Shin1 and Mu-In Park∗2
1208, Nokwon A. Sangga, 30, Dunsannam-ro, Seo-gu, Daejeon, 35235, Korea
2Research Institute for Basic Science,
7 Sogang University, Seoul, 121-742, Korea
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0 Abstract
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We consider gauge invariant cosmological perturbations in UV-modified, z = 3 Hoˇrava gravity
n
withonescalarmatterfield,whichhasbeenproposedasarenormalizablegravitytheorywithoutthe
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J ghostprobleminfourdimensions. Inordertoexhibititsdynamicaldegrees offreedom,weconsider

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theHamiltonianreductionmethodandfindthat,bysolvingalltheconstraintequations,thedegrees
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of freedom are the same as those of Einstein gravity: One scalar and two tensor (graviton) modes

] when a scalar matter field presents. However, we confirm that there is no extra graviton modes
h
t and general relativity is recovered in IR, which achieves the consistency of the model. From the
-
p UV-modification termswhichbreakthedetailed balanceconditioninUV,weobtainscale-invariant
e
h powerspectrumsfornon-inflationarybackgrounds,likethepower-lawexpansions,withoutknowing
[ the details of early expansion history of Universe. This could provide a new framework for the Big

1 Bang cosmology. Moreover, we find that tensor and scalar fluctuations travel differently in UV,
v
generally. We present also some clarifying remarks about confusing points in the literatures.
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4
8 PACS numbers: 04.50.Kd,04.60.-m, 04.62.+v,04.80.Cc
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0
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1
0
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:
v
i
X
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a
∗ E-mail address: [email protected], Corresponding author
1
Several years ago, Hoˇrava proposed a renormalizable, higher-derivative gravity theory,
without ghost problems, by considering different scaling dimensions for space and time [1].
Fromthelackoffull diffemorphism (Diff)beyondgeneral relativity (GR)limit, itsconstraint
structure is quite complicated and not completely understood yet [2–4].
As a closely related problem, there have been confusions regarding the extra graviton
mode, known as the ‘scalar graviton’, and the recovery of GR in IR limit [5–10]. However, in
the non-projectable case, which allows arbitrary space-time dependent linear perturbations
in the lapse function N so that there exists the “local” Hamiltonian constraint as well as
the momentum constraints, it has been shown that the scalar mode can be consistently
decoupled from the usual tensor graviton modes in the spatially flat (k = 0) cosmological
background as well as the flat (vanishing cosmological constant, Λ = 0) Minkowski vacuum
for some appropriate gauge choices [11–14]. Later, a more complete analysis has been done
by considering the most general expressions of the cosmological perturbations for metric as
well as that of a canonical scalar matter field and it has been shown that, without choosing
any gauge, there are one scalar and two tensor modes as in GR, by solving all the constraint
equations in the Hamiltonian reduction method [15] 1.
However, there is a serious physical problem in those works since scale-invariant power
spectrums for the quantized cosmic scalar fluctuations can not be obtained, in disagreement
with observational data. This is basically due to the exact cancelation of the sixth-order-
spatial derivative terms for scalar fluctuations, from the “detailed balance” condition, which
was originally introduced to construct the four-dimensional, power-counting renormalizable
gravity action with some limited number of independent coupling constants, motivated by
reminiscent methods in condensed matter systems [1].
On the other hand, it is already known that the detailed balance condition is too restric-
tive to get a viable model in IR limit since the usual Schwarzschild black hole solution as
well as the Newtonian potential in the weak field approximation for vanishing cosmological
constant can not be obtained. Similar to the IR problem, one can also cure the above UV
problem by considering some “UV-breaking” of the detailed balance condition so that the
sixth-spatial derivatives for the scalar fluctuations do not cancel.
In this regard, there have been several works already for the flat Minkowski vacuum
[14, 17, 18], FRW cosmology [17, 19, 20], cosmological perturbations [21] ( [22, 23] for the
projectable case, [24–26] for the extended case), but still a through analysis, similar to [15]
is lacking. One of the purpose of this paper is to fill the gap by considering the gauge
invariant cosmological perturbations in the Hamiltonian reduction approach, without the
above mentioned UV problem.
In this paper, we consider gauge invariant cosmological perturbations in UV-modified
Hoˇrava gravity with one scalar matter field so that the scalar cosmological fluctuation can
be also (power-counting) renormalizable with the dynamical critical exponent z = 3 in
four dimensions. By solving all the constraints using the Hamiltonian reduction method,
we find that only one scalar and two tensor (graviton) degrees of freedom are left when
a scalar matter field presents, as in Einstein gravity. This confirms that there is no extra
scalar graviton mode and GR is recovered in IR, which provides the consistency of the
1 In the projectable case [15], where the lapse function is a function of time only, or in the extended model
with the dynamical lapse function [8, 16], there exists one extra scalar gravitonmode. But in this paper,
we will not consider those cases since it not clear whether it could be a viable model even in our solar
system, not to mention pathological ghost behaviors [7, 9].
2
model. Furthermore, we obtain the scale-invariant power spectrums for non-inflationary
backgrounds, like the power-law expansions with the power of 1/3 < p < 1, without knowing
the details of early expansion history of Universe. Moreover, we find that tensor and scalar
fluctuations travel differently in UV, generally. This could provide a new framework for
the Big Bang cosmology. In addition, we revisit several debating issues which have been
discussed earlier within the Lagrangian approach with the appropriate gauge choices in [12]
and confirm their resolutions in the Hamiltonian approach in a gauge independent way. We
have also clarified several confusing points which have not been discussed clearly in [15].
To this ends, we start by considering the ADM decomposition of the metric,
ds2 = N2c2dη2 +g dxi +Nidη dxj +Njdη (1)
ij
−
(cid:16) (cid:17)(cid:16) (cid:17)
andtheUV-modifiedHoˇravagravityactionwithz = 3, a ‘laHoˇrava,whichispower-counting
renormalizable [1], is given by
2
S = dηd3x√gN K Kij λK2 + , (2)
g κ2 ij − V
Z (cid:20) (cid:16) (cid:17) (cid:21)
ǫijk
= σ +ξR+α R2 +α R Rij +α R Rl
1 2 ij 3 il j k
−V √g ∇
+ α R iRjk +α R jRik +α R iR, (3)
4 i jk 5 i jk 6 i
∇ ∇ ∇ ∇ ∇ ∇
where
1
′
K = (g N N ) (4)
ij ij i j j i
2N −∇ −∇
is the extrinsic curvature (the prime (′) denotes the derivative with respect to η), ǫijk is
the Levi-Civita symbol, R and R are the Ricci tensor and scalar of the three-dimensional
ij
(Euclidean) spacial geometry, respectively, and κ,λ,ξ,α are coupling constants 2. From the
i
prescription of thedetailed balancecondition, the number ofindependent coupling constants
can be reduced to six, i.e., κ,λ,µ,ν,Λ ,ω for a viable model in IR [13, 14],
W
3κ2µ2Λ2 κ2µ2Λ2 κ2µ2(4λ 1) κ2µ2 κ2µ
σ = W , ξ = W , α = − , α = , α = ,
8(3λ 1) −8(3λ 1) 1 32(3λ 1) 2 − 8 3 2ν2
− − −
κ2
α = = α = 8α , (5)
4 −82ν4 − 5 − 6
in contrast to four fundamental constants in GR, i.e., Newton constant G, speed of light
c, cosmological constant Λ, and the fixed coupling constant, λ = 1. However, in the below
we do not restrict to this case only, at least for the UV couplings α ,α ,α so that the
4 5 6
power-counting renormalizable and scale-invariant cosmological scalar fluctuations can be
obtained.
Forthepower-counting renormalizablematteraction, weconsider z = 3scalar fieldaction
[27, 28],
S = dηd3x√gN 1 φ′ Ni∂ φ 2 V(φ) Z(∂ φ) , (6)
m 2N2 − i − − i
Z (cid:20) (cid:16) (cid:17) (cid:21)
2 The notations differ from those of [15] as κ2 = 2κ2 ,µ = ξ ,α = α ,α = α ,α =
(Here) (Here) 1 2(Here) 2 1(Here) 4
α = 8α =α .
5 6 4(Here)
− −
3
where
3
Z(∂ φ) = ξ ∂(n)φ∂i(n)φ, (7)
i n i
n=1
X
with the superscript (n) denoting n-th spatial derivatives, and V(φ) is the matter’s potential
without derivatives.
The actions (2) and (6) are invariant under the foliation preserving Diff [1]
δxi = ζi(η,x), δη = f(η),
− −
δg = ∂ ζkg +∂ ζkg +ζk∂ g +fg′ ,
ij i jk j ik k ij ij
δN = ∂ ζjN +ζj∂ N +ζ′jg +fN′ +f′N ,
i i j j i ij i i
δN = ζj∂ N +fN′ +f′N,
j
δφ = ζj∂ φ+fφ′. (8)
j
In order to study the cosmological perturbations around the homogeneous and isotropic
backgrounds, we expand the metric and the scalar field as,
N = a(η)[1+ (η,x)], N = a2(η) (η,x) , g = a2(η)[δ +h (η,x)], (9)
A i B i ij ij ij
φ = φ (η)+δφ(η,x),
0
by considering spatially flat (k = 0) backgrounds and the conformal (or comoving) time η,
for simplicity 3. By substituting the metric and scalar field of (9) into the actions one can
obtain the linear-order perturbation part of the total action S = S +S as follows (up to
g m
some boundary terms)
6(1 3λ) 1
δ S = dηd3x a2 − 2 φ′2 +a2(V σ)
1 Z ("− κ2 H −(cid:18)2 0 0 − (cid:19)#A
1 2(1 3λ) 1
+ − 2 +2 ′ + φ′2 a2(V σ) h
2 "− κ2 (cid:16)H H(cid:17) (cid:18)2 0 − 0 − (cid:19)#
− φ′0′ +2Hφ′0 +a2Vφ′0 δφ , (10)
(cid:16) (cid:17) o
which results the background equations, known as the Friedman’s equations,
κ2 1
2 = φ′2 +a2(V σ) , (11)
H −6(1 3λ) 2 0 0 −
− (cid:18) (cid:19)
κ2 1
2 +2 ′ = φ′2 a2(V σ) , (12)
H H 2(1 3λ) 2 0 − 0 −
− (cid:18) (cid:19)
φ′′ +2 φ′ +a2V = 0, (13)
0 H 0 φ0
with the comoving Hubble parameter a′/a, V V(φ ),V (∂V/∂φ) ,h hi,
H ≡ 0 ≡ 0 φ0 ≡ φ0 ≡ i
and indices raised and lowered by δ . Here, it important to note that there is no higher-
ij
derivative corrections to the background equations of (11) and (12) for spatially flat case
and so the background equations are the same as those of GR [13]. However, even in this
3 Forthe metricwith the physicaltime dt=adη, ds2 = 2c2dη2+g (dxi+ idη)(dxj+ jdη), onecan
ij
−N N N
obtain =N/a=(1+ ), =N /a=a .
i i i
N A N B
4
case, the higher-derivative effects can appear in the perturbed parts. Moreover, we note
that the spatial curvature k and cosmological constant Λ can be independent parameters
only when either the space-time is time-dependent or matter exists, which has been confused
sometimes in the literatures. 4
The quadratic part of the total perturbed action is given by
2a2 1
δ S = dηd3x (1 3λ) 3 2 + (2∂ i h′) +(1 λ)(∂ i)2 + ∂ ∂i j
2 Z ( κ2 (cid:20) − H(cid:16) HA A B − (cid:17) − iB 2 iBj B
∂ hij′ + 1h ′hij′ +λ ∂ ih′ 1h′2 +a2ξ + 1h ∂ ∂ hij ∆h
i j ij i i j
− B 4 B − 4 A 2 −
(cid:18) (cid:19)(cid:21) (cid:18) (cid:19)(cid:16) (cid:17)
1 1 a2 1
+a2 δφ′2 φ′δφ′ + 2φ′2 +∂ iφ′δφ V δφ2 a2V δφ φ′δφh′
"2 −A 0 2A 0 iB 0 − 2 φ0φ0 − φ0 A− 2 0 #
a4 (2) +δZ , (14)
− V
(cid:16) (cid:17)o
where ∆ δij∂ ∂ is the spatial Laplacian, δZ = 3 ξ ∂(n)δφ∂i(n)δφ, and (2) is the
≡ i j n=1 n i V
quadratic part of the potential in (3).
V P
Now, in order to separate the scalar, vector, and tensor contributions, we consider the
most general decompositions,
= ∂ +S ,
i i i
B B
h = 2 δ +∂ ∂ +∂ F +H˜ , (15)
ij ij i j (i j) ij
R E
where S and F are transverse vectors, and H˜ is a transverse-traceless tensor, i.e.,
i i ij
∂ Si = ∂ Fi = H˜ = ∂ H˜i = 0. (16)
i i i j
Then, the pure tensor, vector, and scalar parts of the total action is given by, respectively,
2 α α
δ S(t) = dηd3x a2 H˜′ H˜ij′ +ξH˜ ∆H˜ij + 2∆H˜ ∆H˜ij + 3ǫijk∆H˜ ∆∂ H˜l
2 κ2 ij ij a2 ij a3 il j k
Z (cid:20)
α
4∆H˜ ∆2H˜ij , (17)
−a4 ij
(cid:21)
δ S(v) = 1 dηd3x a2∂ Sj Fj′ ∂ S F′ , (18)
2 κ2 i − i j − j
Z (cid:16) (cid:17) (cid:16) (cid:17)
2(1 3λ)
δ S(s) = dηd3x a2 − 3 ′2 6 ′ +3 2 2 2( ′ )∆( ′)
2
( κ2 R − HAR H A − R −HA B−E
Z h i
2(1 λ) 2
+ − [∆( ′)]2 2ξ( +2 )∆ + (8α +3α )(∆ )2
κ2 B −E − R A R a2 1 2 R
2 1
(3α +2α +8α )∆ ∆2 a2V δφ V δφ2 δZ
−a4 4 5 6 R R− φ0A − 2a2 φ0φ0 −
1 1
+ δφ′2 φ′δφ′ + φ′2 2 +[∆( ′) 3 ′]φ′δφ . (19)
2 − 0 A 2 0 A B −E − R 0
(cid:27)
4 In [1], the metric perturbations were considered around (spatially flat and static) Minkowski vacuum
(k = 0,Λ = 0) but the vacuum solution can not be the solution of the gravity with detailed balance !
[13, 29, 30]
5
Here, it is important to note that sixth-order-derivative terms in the gravity action (2),
which are required in the power-counting renormalizability in four dimensions, contribute to
scalar as well as tensor perturbations, through the specific combination of ‘3α +2α +8α ’
4 5 6
for the former but through only ‘α ’ for the latter. This implies that
4
tensor and scalar perturbations travel differently in UV, generally.
On the other hand, unlike the tensor and scalar parts, the vector perturbation is not
dynamical, from the lack of kinetic terms.
Now, in order to exhibit the true dynamical degrees of freedom we consider the Hamil-
tonian reduction method [31], for the cosmologically perturbed actions (17)-(19) [15, 32].
For the tensor part, the reduction is rather trivial since it is already in the unconstrained
form with two physical modes which can be interpreted as two polarizations of the (primor-
dial) gravitational waves as in GR [13, 14, 33–35]. In other words, the perturbed action (17)
can be written as the first-order form,
δ S(t) = dηd3x ΠijH˜′ (t) , (20)
2 ij −H
Z (cid:16) (cid:17)
κ2 α
(t) = ΠijΠ a2ξH˜ij∆H˜ α ∆H˜ij∆H˜ 3ǫijk∆H˜ ∆∂ H˜l
H 8a2 ij − ij − 2 ij − a il j k
α
+ 4∆H˜ij∆2H˜ , (21)
a2 ij
with the conjugate momentum,
δ(δ S(t)) 4a2
Πij 2 = H˜ij′, (22)
≡ δH˜′ κ2
ij
but without any constraint terms.
Next, for the vector part, the action (18) can written as the first-order form,
δ S(v) = dηd3x Πi∆F′ (v) , (23)
2 i −H
Z (cid:16) (cid:17)
κ2
(v) = Πi∆Π +Πi∆S , (24)
H −4a2 i i
with the conjugate momentum,
Πi δ(δ2S(v)) = 2a2 Si Fi′ . (25)
≡ δ∆F′ κ2 −
i (cid:16) (cid:17)
But, from the equations of motion for ∆S ,
i
δ (v)
0 = H = Πi, (26)
−δ∆S
i
which is a constraint equation, one obtains the vanishing action (23) and its Hamiltonian
(24), which show the non-dynamical nature of the vector perturbation.
6
Finally, for the scalar part, which is the most non-trivial one, the action (19) can be
written as the first-order form, after some computations,
δ S(s) = dηd3x Π δφ′ +Π ′ +Π ∆ ′ (s) Π ∆ , (27)
2 δφ R ∆E A ∆E
R E −H −AC − B
Z (cid:16) (cid:17)
κ2 3 4 1 λ κ2
H(s) = 8a2 "−ΠRΠ∆E + 2Π2∆E + κ2Πδφ2 + 2(1−3λ)ΠR2#+ 4(1 3λ)φ′0ΠRδφ
− −
3κ2a2 a2
+ φ′2δφ2 +a2 δZ + V δφ2 +2a2ξ ∆ 2(8α +3α )(∆ )2
8(1 3λ) 0 2 φ0φ0 ! R R− 1 2 R
−
2
+ (3α +2α +8α )∆ ∆2 , (28)
a2 4 5 6 R R
= Π φ′ + Π +4a2ξ∆ +a2 3 φ′ +a2V δφ, (29)
CA δφ 0 H R R H 0 φ0
(cid:16) (cid:17)
with the conjugate momenta
δ(δ S(s))
Π 2 = a2(δφ′ φ′ ) , (30)
δφ ≡ δδφ′ − 0A
δ(δ S(s)) 4(1 3λ)
ΠR ≡ δ2 ′ = a2( κ−2 [3(R′ −HA)−∆(B −E′)]−3φ′0δφ) , (31)
R
δ(δ S(s)) 4(1 3λ) 4(1 λ)
Π∆E ≡ δ∆2 ′ = a2( κ−2 (R′ −HA)− κ−2 ∆(B −E′)−φ′0δφ) . (32)
E
Here, we note that 2 terms in (19) are canceled without using the background equation
A
(11) 5 and only the linearly-dependent terms remain in (27) so that , as well as , be the
A B
Lagrange multiplier.
Then, from the equations of motion for and ∆ , which give the following constraint
A B
equations, respectively,
= 0, Π = 0, (33)
A ∆E
C
one obtains the reduced action, by eliminating Π in (27),
R
δ S(s) [δφ,Π ; ] = dηd3x Π δφ′ (s) , (34)
2 red δφ R δφ −Hred
Z (cid:16) (cid:17)
1 κ2(1 λ)φ′2 κ2(1 λ)
Hr(sed) = 2a2 1+ 8(1 −3λ) 02!Πδφ2 + 8(1 −3λ)φ′0(3φ′0H+a2Vφ0) δφΠR
− H −
a4 κ2(1 λ) a2V 2 3κ2
+ − 3φ′ + φ0 + φ′2 +V δφ2 +a4δZ
2 8(1 3λ) 0 ! 4(1 3λ) 0 φ0φ0
− H −
κ2(1 λ) a2(1 λ) 
+ − ξφ′Π ∆ + − ξ(3φ′ +a2V ) δφ∆ +2a2ξ ∆
2(1 3λ) 0 δφ R 2(1 3λ) 0H φ0 R R R
− −
κ2(1 λ)a2ξ2 2
+ − 2(8α +3α ) (∆ )2 + (3α +2α +8α )∆ ∆2 , (35)
" (1 3λ) 2 − 1 2 # R a2 4 5 6 R R
− H
5 This is in contrast to the on-shell result in [15].
7
which shows one dynamical scalar degree of freedom for the matter δφ and one non-
dynamical field from the metric.
R
Inordertoobtainagauge-invariantdescriptionforthetruedynamical degreesoffreedom,
we consider a new variable,
φ′
ζ δφ 0 , (36)
≡ − R
H
which is gauge invariant since transforms as δ = f under the foliation preserving Diff
R R H
(8), and its conjugate momentum,
a3 φ′ ′
Π Πδφ 0 . (37)
ζ
≡ − a ! R
H
Then, after some analysis, we find that the reduced action (34) can be written as
δ Sˆ(s) [ζ,Π ; ] = dηd3x Π ζ′ ˆ(s)(ζ,Π ; ) , (38)
2 red ζ R ζ −Hred ζ R
Z h i
ˆ(s)(ζ,Π ; ) = A(ζ,Π ) B(ζ,Π )∆ +∆ Θ , (39)
Hred ζ R ζ − ζ R R R
where
A(ζ,Π ) = A Π 2 +A Π η +ζA ζ, (40)
ζ 1 ζ 2 ζ 3
B(ζ,Π ) = B Π +B ζ, (41)
ζ 1 ζ 2
Θ(ζ,Π ) = Θ +Θ ∆+Θ ∆2, (42)
ζ 1 2 3
whose explicit forms are given in Appendix A.
From the equations of motion for ∆ ,
R
δ (s)
0 = H = 2Θ B ∆R, (43)
δ∆ R− ≡ C
R
one can eliminate also and finally obtain the physical action, with only the physical
R
variables ζ and Π ,
ζ
δ S(s) [ζ,Π ] = dηd3x Π η′ (s)(ζ,Π ) , (44)
2 ⋆ ζ ζ −H⋆ ζ
Z h1 i
(s)(ζ,Π ) = A(ζ,Π ) B(ζ,Π )∆Θ−1B(ζ,Π ), (45)
H⋆ ζ ζ − 4 ζ ζ
where the non-local operator Θ−1 is defined as ΘΘ−1 = 1, which may satisfy Θ−1Θ = 1 when
the zero mode can be ignored. This completes the Hamiltonian reduction of the system,
which ends up with only the gauge invariant, physical degrees of freedom.
The usual second-order action form is given by
1 1 ′ 1
δ S(s) = dηd3x ζ′ −1ζ′+ζ −1 + ( )2 −1 ζ , (46)
2 ⋆ 4 G1 4 G2G1 4 G2 G1 −G3
Z (cid:26) (cid:20) (cid:16) (cid:17) (cid:21) (cid:27)
where
1 1 1
A (B )2∆Θ−1, A B B ∆Θ−1, A (B )2∆Θ−1, (47)
1 1 1 2 2 1 2 3 3 3
G ≡ − 4 G ≡ − 2 G ≡ − 4
8
using the Hamilton’s equation,
ζ′ = ζ, d3x (s) = 2 Π + ζ, (48)
{ H⋆ } G1 ζ G2
Z
with the Poisson bracket,
ζ(η,x),Π (η,y) = δ3(x y), (49)
ζ
{ } −
which can be read from the symplectic structure in the action (46) [36].
By introducing a new canonical variable,
u (2 )−1/2ζ, (50)
1
≡ G
the action (46) can be written as
δ S(s) = dηd3x1 u′2 u 1 ′ −1 ′ 1 ′ −1 2 −1 ′ 2 +4 u .
2 ⋆ 2 − 2 G1G1 − 4 G1G1 −G1 G2G1 −G2 G1G3
Z (cid:26) (cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) (cid:27)(51)
with its equations of motion as
u′′ = ω2u, (52)
− u
ω2 = 1 ′ −1 ′ 1 ′ −1 2 −1 ′ 2 +4 . (53)
u 2 G1G1 − 4 G1G1 −G1 G2G1 −G2 G1G3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Now, in order to be compared with some observational data about the early Universe, let
us consider UV limit 6 of our cosmological perturbations. First, for the scalar perturbations,
the field equation of the canonical scalar field u (52) reduces to, in UV limit7,
u′′ = ω2 u, (54)
− u(UV)
6ξ α˜ κ2(1 λ)z2
ω2 = − 3 4 2+ − ∆3, (55)
u(UV) a2z2 " 4(1 3λ)a2#
−
where 3α˜ 3α + 2α + 8α , z aφ′/ . Here, it is important to note that there are
4 ≡ 4 5 6 ≡ 0 H
sixth-spatial derivatives, as required by the scale invariance of the observed power spectrum
[20, 38] as well as the (power-counting) renormalizability [1]. This occurs only when there
are sixth-derivative terms in the starting scalar action (6), (7) (i.e., ξ = 0 in (7)) as well
3
6
as some breaking of the detailed balance condition in sixth-derivative terms for the gravity
action (2) (i.e., α˜ = 0) 8.
4
6
Regarding the scale invariance of the power spectrums, it has been noted that Hoˇrava
gravity could provide an alternative mechanism for the early Universe without introducing
the hyphothetical inflationary epoch [20]: The basic reason of the alternative mechanism
comes from the momentum-dependent speeds of gravitational perturbations which could
6 For IR limit, the usual Mukhanov equation u′′ ∆u (z′′/z)u is obtained [15].
− −
7 The result(55)canbe alsocheckedinthe Lagrangianapproachof[12]: Forthe UV-modifiedtermsinthe
potentialpartof(3),thecorrectedcoefficientsinUVlimitared˜ =g ϕ˙2/H2 6α˜ ,Ω= (g ϕ˙2/H2)[1
Ψ 3 4 3
− − −
(1 6α˜ H2/g ϕ˙2)−1]∆3 and then one can obtain the same UV limit as in (55) [37].
4 3
−
8 The UV limit of (55) differs from that of the scalar graviton, ω2 (1 λ)α˜ ∆3/(1 3λ), which vanishes
4
∼ − −
in the GR limit of λ 1 [14, 22].
→
9
be much larger than the current, low energy (i.e., IR) speed c so that the exponentially
expanding early spacetime could be mimicked. 9 Especially, it has been argued that even
the power-law expansions [41] (t is the physical time, defined by dt = adη),
a = a tp, (1/3 < p < 1) (56)
0
could produce the scale-invariant power spectrums. In this case, from the relations,
a = a1/(1−p)[(1 p)η]p/(1−p), (57)
0 −
(1 3λ) a2
z2 = − ( ′ 2)
κ2 2 H −H
H 2/(1−p)
(1 3λ)a
= − 0 [(1 p)η]2p/(1−p), (58)
− κ2 p −
[ we have used the background equation (12) in the first line of (58) ] one finds that the UV
frequency in (55) can be written as
6ξ α˜ κ2 1 λ
ω2 = − 3 4 2p − ∆3. (59)
u(UV) (1 3λ)a4 − 4 !
−
Moreover, for some analytic computations, we consider only one interesting case of p =
1/2, which corresponds to the radiation-dominated era at the early Universe. Then, it is
easy to see that the normalized mode function is given by 10[20, 33]
h¯ 2 ik3 η dη
u = M a exp , (60)
k s 2k3 − 2 a2!
M Z
(1 3λ)
2 = − − ,
M v6κ2ξ α˜ [1 (1 λ)/4]
u 3 4
u − −
t
from the standard normalization condition,
i
∗ ′ ∗′
u ,u (u u u u ) = 1 (61)
h k ki ≡ h¯ k k − k k
with the Fourier expansion,
dk3
u(η,x) = uk(η)eik·x. (62)
(2π)3
Z
Then, the (dimensionless) power spectrum 11 ∆2(k) for the quantum field ζˆ of the ζ
ζ
perturbation,
2π2
0 ζˆk(η)ζˆk′(η) 0 = (2π)3δ(k+k′) ∆2(k) (63)
| | k3 ζ
D E
9 This idea is reminiscent of, so-called, the “varying speed of light (VSL)” model [39, 40]. This fact seems
to be another justification for the direction of Hoˇrava gravity, other than the original motivation for
renormalizability [1].
10 This is the only case where a′′/a and z′′/z vanish.
11 For some introductory materials about the power spectrum, see [42], for example.
10