Extensions of two-ﬁeld mimetic gravity Yunlong Zhengabch 1and Haomin Raodefg aCenter for Gravitation and Cosmology College of Physical Science and Technology Yangzhou

2025-05-06 0 0 500.43KB 18 页 10玖币

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Extensions of two-ﬁeld mimetic gravity

Yunlong Zheng,a,b,c,h,1and Haomin Raod,e,f,g

aCenter for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou

University, Yangzhou 225009, China

bCAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, Uni-

versity of Science and Technology of China, Hefei, Anhui 230026, China

cSchool of Astronomy and Space Science, University of Science and Technology of China, Hefei,

Anhui 230026, China

dSchool of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced

Study, UCAS, Hangzhou 310024, China

eUniversity of Chinese Academy of Sciences, 100190 Beijing, China

fInterdisciplinary Center for Theoretical Study, University of Science and Technology of China,

Hefei, Anhui 230026, China

gPeng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

hICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy

E-mail: zhyunl@ustc.edu.cn,raohaomin@ucas.ac.cn

Abstract: Two-ﬁeld mimetic gravity was recently realized by looking at the singular

limit of the conformal transformation between the auxiliary metric and the physical metric

with two scalar ﬁelds involved. In this paper, we reanalyze the singular conformal limit and

ﬁnd a more general solution for the conformal factor A, which greatly broadens the form

of two-ﬁeld mimetic constraint and thus extends the two-ﬁeld mimetic gravity. We ﬁnd

the general setup still mimics the role of dark matter at the cosmological background level.

Moreover, we extend the action by introducing extra possible term for phenomenological

interests. Surprisingly, some special cases are found to be equivalent to general relativity,

k-essence theory and Galileon theory. Finally, we further extend the theory by allowing the

expression of mimetic constraint to be arbitrary without imposed condition, and show that

the dark matter-like behavior is unaﬀected even in this extension.

1Corresponding author

arXiv:2210.10499v2 [gr-qc] 12 Apr 2023

Contents

1 Introduction 1

2 Singular conformal limit and two-ﬁeld mimetic constraint 3

2.1 Singular conformal limit 3

2.2 Two-ﬁeld mimetic constraint 4

2.3 Speciﬁc forms 4

3 General two-ﬁeld mimetic gravity 6

3.1 Equations of motion 7

3.2 Cosmological implications 7

4 Extension with introduced extra term and its relations with other modi-

ﬁed theories of gravity 8

4.1 Case where ψcan be eliminated 9

4.1.1 Subcase without introducing the term Q9

4.1.2 Subcase with Q=Q(φ, ψ, X)10

4.1.3 subcase with Q=G2(φ, X)−Z10

4.1.4 subcase with Q=Q(Y)10

4.2 Case where neither of the ﬁelds can be eliminated 11

4.3 Degrees of freedom and Ostrogradski ghost 11

5 Further extension 12

6 Conclusion and discussions 12

A The Condition on A in the singular limit 13

1 Introduction

Theories of modiﬁed gravity [1–4] have attracted considerable attention in the last few

years as possible solutions for the dark energy, dark mater, the singularity problem [5–8]

and so on. Mimetic scenario was ﬁrst proposed by Chamseddine and Mukhanov [9] as

a modiﬁcation of Einstein’s general relativity where the scalar ﬁeld play the role of dark

matter. The idea is to express the physical metric gµν in the Einstein-Hilbert action in

terms of an auxiliary metric ˜gµν and a scalar ﬁeld φ, as follows

gµν =˜gαβφαφβ˜gµν ,(1.1)

– 1 –

in which φα=∇αφis the covariant derivative of the scalar ﬁeld with respect to spacetime,

so that the physical metric is invariant under Weyl rescalings of ˜gµν . The above relation

then leads to a constraint equation

gµν φµφν= 1 .(1.2)

The gravitational equations by varying the Einstein-Hilbert action, which is constructed

from the physical metric gµν , contains an extra scalar mode which can mimic the cold dark

matter. Alternatively, one can impose the above constraint by employing a Lagrange mul-

tiplier [10] in the action. The mimetic gravity written in the Lagrange multiplier formalism

takes the following form

S=Zd4x√−g1

2R+λ

2(gµν φµφν−1) + Lm,(1.3)

where we have used the unit reduced Planck mass M2

p= 1/(8πG)=1and the most negative

signature for the metric.

For phenomenological applications, the mimetic model was generalized in Ref. [11,

12] by introducing a potential term V(φ). With appropriate choice of the potential, the

generalized model can provide inﬂation, bounce, dark energy, and so on. This has stimulated

extensive cosmological and astrophysical interests [13–30], and mimetic scenario has also

been applied in various modiﬁed gravity theories [31–44]. The Hamiltonian analysis of

diﬀerent kinds of mimetic models were investigated in Refs. [45–52]. There are also some

other theoretic developments [53–57]. See Ref. [58] for a review.

Although the original mimetic gravity is free of pathologies, it is shown that the scalar

perturbation is non-propagating due to the mimetic constraint even being oﬀered a po-

tential, thus can’t be quantized in a usual way. Besides, this may also lead to caustic

singularities. To remedy these issues, higher derivative terms of the mimetic ﬁeld (2φ)2are

introduced [11] to promote the scalar mode to be propagating. The equation of motion for

the scalar perturbation indeed has the wave-like form by choosing appropriate coeﬃcient.

However, the detailed analysis in the action formalism indicates that the mimetic model

with higher derivatives always suﬀer from ghost or gradient instability at the level of linear

perturbations [59,60]. Then it was suggested [61–63] to circumvent this diﬃculty by con-

sidering the direct couplings of the higher derivatives of the mimetic ﬁeld to the spacetime

curvature.

Another interesting aspect of mimetic gravity is its close relation with non-invertible

disformal transformation. It is pointed out in Refs. [64–66] that mimetic gravity can be

realized through a non-invertible disformal transformation [67] where the number of degrees

of freedom (DOFs) is no longer preserved between the two frames [68] and the additional

DOF plays the role of dark matter. In this regard, the two-ﬁeld extension of the mimetic

scenario [69,70] was recently proposed by looking at the singular limit in the special case

of conformal transformation

gµν =A(φ, ψ, ˜

X, ˜

Y , ˜

Z)˜gµν ,(1.4)

– 2 –

where ˜

X≡˜gµν φµφν,˜

Y≡˜gµν ψµψν,˜

Z≡˜gµν φµψν, and the conformal factor Ais the

function of φ, ψ, ˜

X, ˜

Y, and ˜

Z. Note that the notations used in this paper are slightly

diﬀerent from those of Ref. [69]. The condition for singular limit is derived to be

A(φ, ψ, ˜

X, ˜

Y , ˜

Z) = ˜

XA,˜

X+˜

Y A,˜

Y+˜

ZA,˜

Z,(1.5)

where a comma denotes a partial derivative with respect to the argument. Then the authors

of [69] claimed that the nontrivial solution for Ais

A=α˜

X+β˜

Y+ 2γ˜

Z , (1.6)

where αand βare arbitrary functions of the scalar ﬁelds φand ψ. After imposing shift

symmetries on both φand ψ, it was found [69,70] that this setup still mimics the dark

matter at the cosmological background level. Recently, mimetic gravity is extended to the

multi-ﬁeld setup in [71] with a curved ﬁeld space manifold.

In this paper, we will reanalyze the singular limit of conformal transformation with

two scalar ﬁelds, and then extend the two-ﬁeld mimetic theory. This paper is organized

as follows. In section 2, we obtain the more general solution for the condition on the

conformal factor Ain singular limit, and give several explicit forms of function Aas examples

to illustrate new possibilities. In section 3, the generalized two-ﬁeld mimetic gravity is

proposed and the dark matter-like ﬂuid is conﬁrmed. In section 4, we extend the theory by

introducing extra possible terms and some special cases are found to be equivalent to general

relativity, k-essence theory and Galileon theory. In section 5, further possible extensions

are proposed. Section 6is our conclusion.

2 Singular conformal limit and two-ﬁeld mimetic constraint

As pointed out by Refs. [64,65], one can ﬁnd the original single ﬁeld mimetic gravity

by performing a singular conformal transformation to the pure Einstein-Hilbert action.

Similarly, two-ﬁeld extension of mimetic gravity can be found by looking at the singular

limit of the conformal transformation (1.4) which involves two scalar ﬁelds [69]. Requiring

the eigenvalue for the Jacobian of the transformation ∂gµν

∂˜gαβ to be zero, the condition on A

in the singular limit (1.4) is obtained as A(φ, ψ, ˜

X, ˜

Y , ˜

Z) = ˜

XA,˜

X+˜

Y A,˜

Y+˜

ZA,˜

Z.In this

section, we will reanalyze the condition on the conformal factor A, and then we derive the

corresponding two-ﬁeld mimetic constraint and show several forms of Ato illustrate new

possibilities.

2.1 Singular conformal limit

One can write the condition (1.5) on Ain a simpler way

A(φ, ψ, c ˜

X, c ˜

Y , c ˜

Z) = c A(φ, ψ, ˜

X, ˜

Y , ˜

Z),(2.1)

which holds for any number or scalar ﬁeld c. We will prove the equivalence between eq. (1.5)

and eq. (2.1) in the appendix A. This means A(φ, ψ, ˜

X, ˜

Y , ˜

Z)is a homogeneous function

of degree one with respect to ˜

X, ˜

Yand ˜

Z. Writing the kinetic term ˜

X, ˜

Y , ˜

Zin term of

– 3 –

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摘要：

Extensionsoftwo-eldmimeticgravityYunlongZheng,a;b;c;h;1andHaominRaod;e;f;gaCenterforGravitationandCosmology,CollegeofPhysicalScienceandTechnology,YangzhouUniversity,Yangzhou225009,ChinabCASKeyLaboratoryforResearchesinGalaxiesandCosmology,DepartmentofAstronomy,Uni-versityofScienceandTechnologyofChin...

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Extensions of two-ﬁeld mimetic gravity Yunlong Zhengabch 1and Haomin Raodefg aCenter for Gravitation and Cosmology College of Physical Science and Technology Yangzhou

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