#### A covariant Lagrangian for stable nonsingular bounce

HJE
A covariant Lagrangian for stable nonsingular bounce
Yong Cai 0 1 3
Yun-Song Piao 0 1 2 3
0 P. O. Box 2735, Beijing 100190 , China
1 Beijing 100049 , China
2 Institute of Theoretical Physics, Chinese Academy of Sciences
3 School of Physics, University of Chinese Academy of Sciences
The nonsingular bounce models usually suffer from the ghost or gradient instabilities, as has been proved recently. In this paper, we propose a covariant effective theory for stable nonsingular bounce, which has the quadratic order of the second order derivative of the field φ but the background set only by P (φ, X). With it, we explicitly construct a fully stable nonsingular bounce model for the ekpyrotic scenario.
Classical Theories of Gravity; Spacetime Singularities
1 Introduction 2 3 4
15] (see also [16, 17]), along the road beyond the cubic Galileon (even the Horndeski
theory [18–20]). Moreover, the developments of scalar-tensor theory (the GLPV [21] and
DHOST theory [22–24], the mimetic gravity [25, 26]) might also be able to provide us with
some chances to implement stable nonsingular cosmologies. However, due to the complexity
of relevant theories, which component is required for a stable bounce is not clear. Thus so
far building a realistic and stable model is still difficult.
In refs. [8, 9], with the EFT of nonsingular cosmologies, it has been found that the
operator R(3)δg00 is significant for the stability of nonsingular bounce. Actually, in unitary
gauge, without getting involved in the specific theories,
Ladd−oper ∼
M24(t)
2
(δg00)2 +
2
m˜24(t) R(3)δg00
(1.1)
– 1 –
∼ (∂ζ)2 at quadratic order.
might be the least set of operators added to GR to cure the instabilities, since (δg00)2 ∼ ζ
˙2
In this paper, based on the covariant description of the R(3)δg00 operator, we propose a
covariant theory for stable nonsingular bounce, which has the quadratic order of the second
order derivative of the field φ but the background set only by P (φ, X). We illuminate
its application by constructing a fully stable nonsingular bounce model for the ekpyrotic
scenario [27, 28].
Note added. Several days after our paper appeared in arXiv, the preprint [29] appeared,
in which somewhat similar analysis is done in beyond Horndeski model with sort of similar
where X = φμφμ, φμ = ∇μφ and φμ = ∇μφ.
Codazzi relation, it is straightforward (though tedious) to find
R(3) is the Ricci scalar on the 3-dimensional spacelike hypersurface. Using the
GaussR(3) = R −
φμν φμν
− ( φ)2
+
2(φννμφμ − φνμμφν )
+
,
X
X
2φμφμν φνσφσ
X2
−
2φμφμν φν φ
X2
with φμν = ∇ν ∇μφ and φννμ = ∇μ∇ν ∇ν φ. It is simple to check that the right hand side
of eq. (2.2) is 0 at the background level.
We define Sδg00R(3) = R d4x√
−gLδg00R(3) , and have
Lδg00R(3) =
f1(φ)
2
f
2
+
R − 2
f
2X
δg00R(3)
X Z
=
fφφd ln X −
fφ +
Z fφ d ln X
2
φ
X2
φμν φμν
− ( φ)2
−
f − 2XfX [φμφμρφρν φν − ( φ)φμφμν φν ]
after integration by parts, where f (φ, X) = f1 1 + fX2
has the dimension of mass squared,
f2(φ) is defined in (2.1), and the total derivative terms have been discarded. One useful
formula for obtaining eq. (2.3) is
2B(φ, X)φμφμν φν = ∇μ φ
BdX
− X
μ Z
Z ∂B dX −
∂φ
φ
Z
BdX .
– 2 –
S =
Z
d x
4 √
−g
2
Mp2 R + P (φ, X)
!
+ Sδg00R(3) ,
which is a covariant theory equivalent to GR plus the set of operators in (1.1), since
M24(t) = φ˙4PXX and m˜24(t) = f1(φ).
The covariant action (3.1) actually belongs to a subclass of the DHOST theory [22, 23]
(see appendix A for details), which could avoid the Ostrogradski instability, up to quadratic
order of the second order derivative of φ. Ijjas and Steinhardt used the quartic Horndeski
action in [13]. In (2.3), though the nonminimal coupling f (φ, X)R is similar to that in [13],
terms ∼
φ, φμν φμν , ( φ)2, ( φ)φμφμν φν and φμφμρφρν φν also appear simultaneously
with the coefficients set by δg00R(3), so that the effect of Sδg00R(3) on background is canceled
accurately. Here, the background is set only by P (φ, X). In [14], ( φ)2 is used, which shows
itself the Ostrogradski ghost, see also earlier [30], how to remove it requires argumentation.
The quadratic action of scalar perturbation for (3.1) is in which
ζ
S(2) =
Z
˙2
a3Qs ζ − cs a2
2 (∂ζ)2
d4x ,
Qs =
2φ˙4PXX − Mp2H˙
H2
, cs2Qs = Mp2
c˙3
a − 1
3.1
The covariant theory
Here, the EFT proposed is
2f1(φ) =
H Z
a
a Qscs2 + Mp2 dt − Mp2.
– 3 –
and c3 = a(1 + M2fp12 )/H. We can see that the sound speed of scalar perturbation can be
directly modified by f1(φ), namely, the function before δg00R(3) operator. Therefore, the
gradient instability of scalar perturbation could be cured by proper choice of f1(φ), while
that of tensor perturbation is unaffected by Sδg00R(3) , hence is same with that of GR.
A fully stable nonsingular bounce (Qs > 0 and cs2 = 1) can be designed with (3.1). In
the bounce phase, H˙ > 0. However, Qs > 0 can be obtained, since P (φ, X) contributes
φ˙4PXX in Qs. While around the bounce point H ≃ 0,
2
cs ∼ −H
˙
1 +
2f1
Mp2
.
Thus we will have cs2 > 0 for 2f1 < −Mp2, as has been clarified in refs. [8, 10]. It should be
mentioned that if f1 = 0, we have cs2 ∼ −H˙ < 0 around the bounce point, thus Sδg00R(3)
is needed to contribute f1. Here, we always could set cs2 ∼ O(1) with a suitable f1(φ) (see
also [10]) which satisfies
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
HJEP09(217)
With (3.1), building a nonsingular bounce model is simple. The ghost-free nonsingular
2
bounce is set by P (φ, X), while cs ≃ 1 is set by using suitable f1 and f2 in (2.1).
As a specific model, we set P (φ, X) in (3.1) as
k0
P (φ, X) =
Around φ ≃ 0, we have
where the potential is ekpyrotic-like
V (φ) = − 2
V0 eφ/M1 1 − tanh
φ
M2
,
with constant M1, M2, V0, and k0, κ1 responsible for the switching of the sign before X/2
around φ ≃ 0, and q0, κ2 for the appearance of X2 around φ ≃ 0, see [31] for a similar
P (φ, X), which might allow for a supersymmetric counterpart [32].
The background equations are
Initially φ ≪ −M2, −1/√κ1, −1/√κ2, we have P (φ, X) = −X/2+V0eφ/M1 , the Universe
is in the ekpyrotic phase with the equation of state parameter
3Mp2H2 = −2φ˙2PX − P ,
Mp2H˙ = φ˙2PX .
Mp2
3M1
ωekpy =
given by (3.5), and f2(φ) = φ˙(t(φ)).
can be obtained, while cs2 = 1 can be obtained by setting suitable f1(φ) in (2.3), which is
The background evolution is numerically plotted in figure 1. We show the behaviors of
f1(φ) and f2(φ) with respect to φ in figure 2 while we require cs2 = 1 throughout. In both
figures 1 and 2, we set k0 = 1.2, κ1 = 30, q0 = 1.25, κ2 = 20, V0 = 2 × 10−7, M1 = 0.22
and M2 = 0.1. We set the initial condition of φ as φini = −0.54 and φ˙ini = 2.24 × 10−4,
while the initial value of t is tini = −2000. We see that with f1 and f2 plotted in figure 2,
the Lagrangian (3.1) with P (φ, X) in (3.6) will bring a fully stable nonsingular bounce
(Qs > 0 and cs2 = 1).
(3.6)
(3.7)
0.6
0.4
1.5
1.0
-0.2
-0.4
-0.6
(a) f1(φ) for cs2 ≡ 1.
(b) f2(φ) for cs2 ≡ 1.
φ [6–9]. Actually, in [10, 38], it is observed that the Galileon
HJEP09(217)
The exploration of stable nonsingular bounce has been still a significant issue. Recently, it
has been found in refs. [8, 9] that the operator R(3)δg00 in EFT of nonsingular cosmologies
is significant for the stability of bounce. Here, based on the covariant description of the
R(3)δg00 operator, we propose a covariant theory (3.1) for stable nonsingular bounce.
Our (3.1) is actually a subclass of the DHOST theory [22, 23], but the cosmological
background is set only by P (φ, X). The P (φ, X) nonsingular bounce model could be
ghostfree [31, 37], but suffers the problem of cs2 < 0, which can not be dispelled by using the
Galileon interaction ∼
interaction only moves the period of cs2 < 0 to the outside of the bounce phase, but can
not remove it, see also earlier [39]. Thus it could be imagined that the quadratic order of
the second order derivative of φ, i.e., φμν φμν , ( φ)2, φμφμρφρν φν and ( φ)φμφμν φν , might
play crucial roles in stable nonsingular bounce model. However, due to the complexity of
relevant theories, what kind of combination of these components is required for a stable
cosmological bounce is unclear. Here, the corresponding combination (2.3) is just what
told by the covariant description of the R(3)δg00 operator.
With (3.1), the design of stable nonsingular bounce model is simple, as illuminated
for the ekpyrotic scenario. Our work actually offers a concise way to the fully stable
nonsingular cosmologies. See also [40–48] for other interesting studies.
Here, the importance of the EFT of nonsingular cosmologies is obvious. Actually, the
role of R(3)δK in EFT [8] is similar to that of R(3)δg00, where Kμν is the extrinsic curvature
on the 3-dimensional spacelike hypersurfaces. The covariant description of R(3)δK involves
the term ∼ ( φ)R, which might have the Ostrogradski ghost unless certain constraint is
imposed. This issue will be revisited. In mimetic gravity [25, 26] (see e.g. [
49
] for review),
since the mimetic constraint suggests δg00 = 0 (which is the source of instabilities [
50–
53
]), one might apply the operator R(3)δK to make the (possibly-built) nonsingular bounce
stable,1 instead of R(3)δg00. The mimetic gravity with the couple ( φ)R has been proposed
in ref. [54]. We will back to the relevant issues.
Acknowledgments
We thank Mingzhe Li, Taotao Qiu and Youping Wan for helpful discussions. This work is
supported by NSFC, No. 11575188, 11690021, and also supported by the Strategic Priority
Research Program of CAS, No. XDA04075000, XDB23010100.
A
Correspondence with a subclass of DHOST theory
Up to cubic order of φμν , the covariant action of DHOST can be written as (see e.g., [24])
SDHOST =
Z
d x
4 √
−g hp(φ, X) + q(φ, X) φ + g2(φ, X)R + C(μ2ν)ρσφμν φρσ
+g3(φ, X)Gμν φμν + C(μ3ν)ρσαβφμν φρσφαβi ,
(A.1)
1Communication with Mingzhe Li.
– 6 –
with
and
with
where R and Gμν denote the usual 4-dimensional Ricci scalar and Einstein tensor associated
with the metric gμν , respectively;
C(μ2ν)ρσφμν φρσ = X aA(φ, X) L(A2) ,
L(2) = φμν φμν ,
1
2
L(2) = ( φ)2 ,
L(2) = φμφμρφρν φν , L(2) = (φμφμν φν )2 ,
4 5
L(2) = ( φ)φμφμν φν ,
3
C(μ3ν)ρσαβφμν φρσφαβ = X bA(φ, X)L(A3) ,
5
A=1
10
A=1
1
L(3) = ( φ)3 ,
L(3) = ( φ)φμν φμν ,
2
L(3) = ( φ)2 φμφμν φν ,
4
L(3) =
5
φ φμφμν φνρφρ ,
L(3) = φμφμν φνρφρσφσ , L(3) = φμφμν φνρφρ φσφσλφλ ,
7 8
L(3) =
9
φ (φμφμν φν )2 , L(130) = (φμφμν φν )3 ;
L(3) = φμν φνρφρμ ,
3
L(3) = φμν φμν φρφρσφσ ,
6
extra conditions on the functions aA and bA need to be satisfied so that there is no extra
propagating degree of freedom, see [24] and references therein for further discussions.
Comparing with (A.1), we find our model (3.1) corresponds to the covariant form of
DHOST theory with
p(φ, X) = P (φ, X) − 2
X Z
g2(φ, X) =
Mp2 + f
2
a1 = −a2 =
,
f
2X
,
and bA = 0.
fφφd ln X ,
q(φ, X) = −fφ −
g3(φ, X) = 0 ,
a3 = −a4 =
Z fφ d ln X ,
2
f − 2XfX ,
X2
(A.6)
a5 = 0 ,
In the EFT formalism, the quadratic action for DHOST theory can be written as
SD(2H)OST =Z d3xdta3 M 2
2
δKμν δKμν
− 1+ 3 αL δK2 +(1+αT ) R(3) δ h
2
a3 +δ2R(3) (A.7)
√
!
+H2αK δN 2 +4HαBδKδN +(1+αH )R(3)δN +4β1δKδN˙ +β2δN˙ 2 + βa23 (∂iδN )2 ,
where δN = δg00/2, δ2R(3) stands for the second order term in the perturbative expansion
of R(3), the dimensionless time-dependent functions αL, αT , αK , αB, αH , β1, β2 and β3
satisfy certain conditions so that there is no extra propagating degree of freedom, see [24]
for details.
– 7 –
(A.2)
(A.3)
(A.4)
(A.5)
HJEP09(217)
Comparing with (A.7), we find our model (3.1) corresponds to
M = Mp ,
αK =
4M24
Mp2H2 =
4XM2p2PHX2X ,
αH =
2Mm˜p224 =
2f1(φ)
Mp2
.
αL = αT = αB = 0 ,
β1 = β2 = β3 = 0 ,
(A.8)
Note that the results in eqs. (A.8) should be evaluated at background level in the quadratic
action if we derive them from eqs. (A.6) by using formulae given in eqs. (2.14) of [24].
According to the above results, our model (3.1) belongs to a subclass of the DHOST
theory with αL = 0 and αT = 0. As has been pointed out in ref. [24], in such a DHOST
HJEP09(217)
positive times.
theory, in the linear regime for a Minkowski background (namely, at the limit a = 1 and
1 1
H = 0) the Newton’s constant is GN = 8πMp2 (1+αH)2 . In our specific numerical example,
the Universe is nearly slowly expanding Friedmann Universe at large positive times, which
is nearly Minkowskian. However, in that limit, the contribution from Lδg00R(3) (or the value
1
of f1(φ)) is already vanishing, i.e., αH = 0, hence GR is retrieved and GN = 8πMp2 at large
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