Linear perturbations in Eddington-inspired Born...

Linear perturbations in Eddington-inspired

Born-Infeld gravity

Yu-Xiao Liu (4��)

Institute of Theoretical Physics, Lanzhou University

Based on: Ke Yang, Xiao-Long Du, YXL, arXiv:1307.2969.

ICTS, USTC Sep 5, 2013

ITP,LZU Linear perturbations of EiBI Gravity

Outline

1. Introduction to EiBI gravity

2. Linear Perturbations

3. The stability of the perturbations

4. Conclusion and discussion



1915: General Relativity

It provides precise descriptions to a variety of phenomena in

our Universe for almost a century.

It also suffers various troublesome theoretical problems: dark

matter/energy, nonrenormalization, singularity...

The the Einstein-Hilbert action is

SEH[g] =

∫d4x√−g [R(g)− 2Λ] . (1)



Modified Gravity

Scalar-tensor (Brans-Dicke) gravity

Einstein-Aether gravity

F(R) gravity and general higher-order theories,

Horava-Lifschitz gravity

Galileons

Ghost Condensates

Models of extra dimensions: KK, ADD, RS, DGP

Born-Infeld Gravity

Bimetric theories

· · ·ITP,LZU Linear perturbations of EiBI Gravity

1924, Eddington proposed a purely affine gravity.

Eddington gravity

SEdd(Γ) =1

8πG

2

κ

∫d4x√−|κRµν(Γ)|, (2)

⇓ (gµν = κRµν)

gµν;λ = 0, i.e., Γλµν =1

2gλρ(gρµ,ν + gρν,µ − gµν,ρ)

⇓ (Λ = 1κ)

Rµν = Λgµν (3)

Eddington’s theory is totally equivalent to the GR with Λ.

HOWEVER, it is incomplete because matter is not included.


Consider the Palatini action for gravity with Λ

SP[g,Γ] =

∫d4x

(√−g gµνRµν(Γ)− 2Λ

), (4)

Eliminating the connection using its own EoM gives

SEH[g] =

∫d4x√−g [R(g)− 2Λ] , (5)

If Λ 6= 0, eliminating the metric yields [Annals Phys. 162(1985)31]

SEdd[Γ] =2

Λ

∫d4x√−|Rµν(Γ)|. (6)

SP[g,Γ] is called the Parent action, while the Einstein-Hilbert

action and Eddington’s action are its daughters.

SEH[g] and SEdd[Γ] are said to be dual to each other, and in

many respects they are equivalent.


The vector Born-Infeld theory (1934 Born, Infeld)

SVBI =

∫d4x√−|gµν + Fµν |. (7)

The Born-Infeld gravity [Deser and Gibbons, CQG 15, L35 (1998)]

SBI[gµν ] =

∫d4x√−|gµν − l2Rµν +Xµν(R)|. (8)

Xµν(R) must be chosen such that the action is free of ghost.

For the vector BI theory, the EoM is of second order.

For the spin two theory this is not automatic and requires the

addition of Xµν(R).


Inspired by the Eddington and BI gravity theories, a new

theory was put forward by Banados and Ferreira [PRL 105 (2010)

011101].

Eddington Inspired Born-Infeld (EiBI) gravity

SEiBI[g,Γ,Φ] =2

κ

∫d4x

(√−|gµν + κRµν(Γ)| − λ

√−|gµν |

)+SM (g,Γ,Φ), (9)

where the dimensionless parameter λ must be nonvanishing.


By varying the action with respect to the metric simply gives√−|gµν + κRµν |

[(gµν + κRµν)−1

]αβ=√−|gµν |

(λgαβ − κTαβ

),

(10)

By introducing an auxiliary metric

qµν = gµν + κRµν , (11)

the variation to the connection simply gives us qµν;σ = 0, i.e., Γ is

just the Christoffel symbol of the auxiliary metric.

Then Eq. (10) is rewritten as√−|qµν | qαβ = λ

√−|gµν | gαβ − κ

√−|gµν | Tαβ, (12)

Eqs. (11) and (12) and matter field equations form a

complete set of equations of the theory.ITP,LZU Linear perturbations of EiBI Gravity

Some properties of the EiBI gravity:

When κR� g, SEiBI → SEdd.

When κR� g, the EiBI action reproduces the GR with Λeff in

lowest order approximation:

SEiBI ≈1

2κ

∫dnx

√−|gµν |

(R− 2Λeff +

κ

4RR− κ

2RµνR

νµ

)+SM [g,Γ,Φ], (13)

where Λeff ≡ (λ− 1)/κ.

In the nonrelativistic limit, the EiBI theory gives the modified

Poisson equation

∇2Φ = −1

2ρ− κ

4∇2ρ. (14)

So, it reproduces Einstein gravity precisely within the vacuum

but deviates from it in the presence of source.


Homogeneous and isotropic Universe

ds2 = −dt2 + a2(t)d~x · d~x (15)

With coupling to an ideal fluid Tµν = (p+ ρ)uµuν + pgµν , the

Friedmann equation is given by

H2 =2

3

G

F 2, (16)

F = 1− 3κ(ρT + pT )(1− ω − bκρT − κpT )

(1 + κρT )(1− κpT ),

G =1

κ[1 + 2U − 3

U

V],

U = (1 + κρT )−1/2(1− κpT )3/2,

V = a2(1 + κρT )1/2(1− κpT )1/2, (17)

with ρT = ρ+ Λκ and pT = p− Λ

κ .


Focus on the evolution of the scale factor at early times.

Assume radiation domination: ρT = ρ, pt = p = 13ρ.

Define ρ = κρ, the Friedmann equation becomes

3H2(ρ) =

[ρ− 1 +

√(1 + ρ)(3− ρ)3

3√

3

](1 + ρ)(3− ρ)2

κ(3 + ρ2)2.

For small ρ we recover the conventional Friedmann universe,

H2 ≈ ρ/3.

But at high density there is a stationary point H2 = 0

at ρ = 3 (ρ = 3/κ) for κ > 0, and

at ρ = −1 (ρ = −1/κ) for κ < 0.


The Hubble rate H2 as a function of energy density ρ

The new stationary points correspond to a maximum

density ρB (ρB = 3/κ for κ > 0 and ρB = −1/κ for κ < 0)


The scale factor a as a function of time t

The maximum density ρB corresponds to a minimum

length aB in cosmology.

Thus the universe may be entirely singularity free.


arXiv:0801.4103.


arXiv:1204.1691.


arXiv:1210.1521.

Motivation:

Do the linear scalar and vector perturbations stable in

Eddington regime?

So, we study the full linear perturbations of a

homogeneous and isotropic spacetime in the EiBI gravity.


2. Linear Perturbations of EiBI cosmology

2.1 The perturbed metrics

The background space-time metric is

ds2 = gµνdxµdxν

= −dt2 + a2(t)δijdxidxj . (18)

The background auxiliary metric is

ds′2 = qµνdxµdxν

= −X2(t)dt2 + a2(t)Y 2(t)δijdxidxj . (19)


The perturbed space-time metric is

ds2 = gµνdxµdxν = (gµν +Hµν)dxµdxν

= (−1 + h00(x))dt2 + a2(t)(δij + hij(x))dxidxj

+2h0i(x)dtdxi. (20)

The perturbed auxiliary metric is

ds′2 = qµνdxµdxν = (qµν + Πµν)dxµdxν

= X2(t)(−1 + γ00(x))dt2 + a2(t)Y 2(t)(δij + γij(x))dxidxj

+2Y 2(t)γ0i(x)dtdxi. (21)

h ≡ ηµνhµν , γ ≡ ηµνhµν . (22)


2.2 The background field equation

The 1st field equation reads√−|qµν |qµν = λ

√−|gµν |gµν − κ

√−|gµν |Tµν , (23)

Y 3

X= λ+ κρ, (24)

XY = λ− κP. (25)

The 2nd field equation

qµν = gµν + κRµν , (26)

X2 = 1 + 3κ[a

a+Y

Y− a

a

X

X+ 2

a

a

Y

Y− X

X

Y

Y], (27)

Y 2 = 1 + κY 2

X2(a

a+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y− X

X

Y

Y+Y

Y+ 2

Y 2

Y 2).


2.3 The energy-momentum tensor

Here the matter is the perfect fluid,

Tµν = Pgµν + (P + ρ)uµuν , (28)

and the static observer is uµ = (1, 0, 0, 0) with gµνuµuν = −1.

The 1st order perturbation of the energy-momentum tensor is

δT 00 = δρ+ ρh00, (29)

δT i0 = −a−2ρh0i + a−2(P + ρ)δui, (30)

δT ij = a−2δPδij − a−2Phij . (31)


2.4 The first perturbed field equation

Perturbing the first field equation√−|qµν |qµν = λ

√−|gµν |gµν − κ

√−|gµν |Tµν , (32)

we can get

γ00 = h00 +κδρ

2(λ+ κρ)+

3κδP

2(λ− κP ),

γ0i = h0i −(P + ρ)

(λ+ κρ)κδui,

γij = hij + [κδρ

2(λ+ κρ)− κδP

2(λ− κP )]δij . (33)


2.5 The perturbed Ricci tensor

Perturbing the Ricci tensor

Rµν = ∂λΓλµν − ∂νΓλλµ + ΓλλαΓαµν − ΓαµλΓλαν . (34)

we have

δR00 = −1

2

X2

Y 2a−2∂i∂iγ00 −

3

2(a

a+Y

Y)∂0γ00 −

1

2∂0∂0γii

−(a

a− 1

2

X

X+Y

Y)∂0γii + a−2∂0∂iγi0

+a−2(2Y

Y− X

X)∂iγi0,

δR0i =Y 2

X2[a

a+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+Y

Y+ 2

Y 2

Y 2− X

X

Y

Y]γi0

−[a

a+Y

Y]∂iγ00 +

1

2∂0∂jγij −

1

2∂0∂iγjj −

1

2a−2∂j∂jγi0

+1

2a−2∂i∂jγj0,


δRij =a2Y 2

X2[a

a+Y

Y+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+ 2

Y 2

Y 2− X

X

Y

Y]

×[γ00δij + γij ] +1

2

a2Y 2

X2[a

a+Y

Y]∂0γ00δij

+1

2

a2Y 2

X2[3a

a− X

X+ 3

Y

Y]∂0γij −

1

2

Y 2

X2[a

a− X

X+ 3

Y

Y]∂iγj0

−1

2

Y 2

X2[a

a− X

X+ 3

Y

Y]∂jγi0 +

1

2

a2Y 2

X2[a

a+Y

Y]∂0γkkδij

−Y2

X2[a

a+Y

Y]∂kγk0δij +

1

2∂i∂jγ00 +

1

2

a2Y 2

X2∂0∂0γij

+1

2∂k∂jγki +

1

2∂k∂iγkj −

1

2∂k∂kγij −

1

2∂i∂jγkk

−1

2

Y 2

X2∂0∂iγj0 −

1

2

Y 2

X2∂0∂jγi0. (35)


2.6 The perturbed field equation

With the scalar-vector-tensor decomposition of the

perturbed metric hµν and δui

h00 = −E, hi0 = ∂iF +Gi,

hij = Aδij + ∂i∂jB + ∂jCi + ∂iCj +Dij ,

δui = ∂iδu+ δUi,

where ∂iCi = ∂iGi = ∂iδUi = 0, ∂iDij = 0, and Dii = 0,

the 2nd perturbed field equation

δqµν = δgµν + κδRµν (36)

can be calculated as


00-component:

1

2

X2

Y 2a−2∇2E + 3(

a

a+Y

Y− a

a

X

X+ 2

a

a

Y

Y− X

X

Y

Y)E

+3

2(a

a+Y

Y)E − 1

2[3A+∇2B]− (

a

a− 1

2

X

X+Y

Y)[3A+∇2B]

+a−2∇2F + a−2(2Y

Y− X

X)∇2F − κ

4a−2X

2

Y 2

∇2δρ

λ+ κρ

−3κ

4a−2X

2

Y 2

∇2δP

λ− κP −3κ

4∂0∂0[

δρ

λ+ κρ] +

3κ

4∂0∂0[

δP

λ− κP ]

−3κ

4(3a

a+ 3

Y

Y− X

X)∂0[

δρ

λ+ κρ]− 3κ

4(a

a+Y

Y+X

X)∂0[

δP

λ− κP ]

−1

2[1 + 3κ(

a

a+Y

Y− a

a

X

X+ 2

a

a

Y

Y− X

X

Y

Y)][

δρ

λ+ κρ+

3δP

λ− κP ]

−κa−2∂0[P + ρ

λ+ κρ∇2δu]− κa−2(2

Y

Y− X

X)P + ρ

λ+ κρ∇2δu = 0. (37)


i0-component:

∂i

{[a

a+Y

Y]E − ∂iA−

κ

2∂0[

∂iδρ

λ+ κρ] +

κ

2∂0[

∂iδP

λ− κP]

−κ2

[a

a+Y

Y](

∂iδρ

λ+ κρ+

3∂iδP

λ− κP) +

P + ρ

λ+ κρ∂iδu

}+

1

2∇2Ci −

1

2a−2∇2Gi +

κ

2a−2 P + ρ

λ+ κρ∇2δUi +

P + ρ

λ+ κρδUi

= 0. (38)


ij-component:{− a2Y 2

X2[a

a+Y

Y+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+ 2

Y 2

Y 2− X

X

Y

Y]E

−1

2

a2Y 2

X2[a

a+Y

Y]E +

1

2

a2Y 2

X2A− 1

2∇2A

− Y2

X2[a

a+Y

Y]∇2F +

1

2

a2Y 2

X2[3a

a+ 3

Y

Y− X

X]A

+1

2

a2Y 2

X2[a

a+Y

Y][3A+∇2B] +

κ

4

a2Y 2

X2∂0∂0[

δρ

λ+ κρ]

−κ4

a2Y 2

X2∂0∂0[

δP

(λ− κP )]− κ

4[∇2δρ

λ+ κρ− ∇

2δP

λ− κP ]

+κ

2

a2Y 2

X2[a

a+Y

Y+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+ 2

Y 2

Y 2− X

X

Y

Y]

×[ δρ

λ+ κρ+

3δP

λ− κP ] +κ

4

a2Y 2

X2[7a

a+ 7

Y

Y− X

X]∂0[

δρ

λ+ κρ]

−1

2a2[

δρ

λ+ κρ− δP

λ− κP ] + κY 2

X2[a

a+Y

Y]P + ρ

λ+ κρ∇2δu

−κ4

a2Y 2

X2[3a

a+ 3

Y

Y− X

X]∂0[

δP

λ− κP ]

}δij


∂i∂j

{− 1

2E − 1

2A+

1

2

a2Y 2

X2B − Y 2

X2F + κ

δP

λ− κP

+1

2

a2Y 2

X2[3a

a+ 3

Y

Y− X

X]B − Y 2

X2[a

a− X

X+ 3

Y

Y]F

+κY 2

X2∂0[

P + ρ

λ+ κρδu] + κ

Y 2

X2[a

a− X

X+ 3

Y

Y]P + ρ

λ+ κρδu

}+1

2

a2Y 2

X2[3a

a+ 3

Y

Y− X

X][∂iCj + ∂jCi]

+1

2

a2Y 2

X2[∂iCj + ∂jCi]−

1

2

Y 2

X2[∂iGj + ∂jGi]

−1

2

Y 2

X2[a

a− X

X+ 3

Y

Y][∂iGj + ∂jGi] +

κ

2

Y 2

X2

P + ρ

λ+ κρ[∂iδUj + ∂jδUi]

+κ

2

Y 2

X2

P + ρ

λ+ κρ[∂iδUj + ∂jδUi]−

κ2

2

Y 2

X2

ρ(P + ρ)

(λ+ κρ)2[∂iδUj + ∂jδUi]

+κ

2

Y 2

X2[a

a− X

X+ 3

Y

Y]P + ρ

λ+ κρ[∂iδUj + ∂jδUi]

−1

2∇2Dij +

1

2

a2Y 2

X2Dij +

1

2

a2Y 2

X2[3a

a+ 3

Y

Y− X

X]Dij = 0.


A. Scalar modes:

The 00-component of the perturbed equation (36) gives

1

2

X2

Y 2a−2∇2E + 3(

a

a+Y

Y− a

a

X

X+ 2

a

a

Y

Y− X

X

Y

Y)E +

3

2(a

a+Y

Y)E

−1

2(3A+∇2B)− (

a

a− 1

2

X

X+Y

Y)(3A+∇2B) + a−2∇2F

+a−2(2Y

Y− X

X)∇2F − κ

4a−2X

2

Y 2

∇2δρ

λ+ κρ− 3κ

4a−2X

2

Y 2

∇2δP

λ− κP

−3κ

4∂0∂0

δρ

λ+ κρ− 3κ

4(3a

a+ 3

Y

Y− X

X)∂0

δρ

λ+ κρ+

3κ

4∂0∂0

δP

λ− κP

−1

2[1 + 3κ(

a

a+Y

Y− a

a

X

X+ 2

a

a

Y

Y− X

X

Y

Y)](

δρ

λ+ κρ+

3δP

λ− κP )

−3κ

4(a

a+Y

Y+X

X)∂0

δP

λ− κP − κa−2∂0[

P + ρ

λ+ κρ∇2δu]

−κa−2(2Y

Y− X

X)P + ρ

λ+ κρ∇2δu = 0. (39)


The part of ij-component of (36) proportional to δij gives

−a2Y 2

X2(a

a+Y

Y+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+ 2

Y 2

Y 2− X

X

Y

Y)E +

1

2

a2Y 2

X2A

−1

2

a2Y 2

X2(a

a+Y

Y)E − 1

2∇2A− Y 2

X2(a

a+Y

Y)∇2F

+1

2

a2Y 2

X2(3a

a+ 3

Y

Y− X

X)A+

1

2

a2Y 2

X2(a

a+Y

Y)(3A+∇2B)

+κ

4

a2Y 2

X2∂0∂0

δρ

λ+ κρ− κ

4

a2Y 2

X2∂0∂0

δP

λ− κP

−κ4(∇2δρ

λ+ κρ− ∇

2δP

λ− κP )− 1

2a2(

δρ

λ+ κρ− δP

λ− κP )

+κ

2

a2Y 2

X2(a

a+Y

Y+ 2

a2

a2− a

a

X

X+ 6

a

a

Y

Y+ 2

Y 2

Y 2− X

X

Y

Y)

×( δρ

λ+ κρ+

3δP

λ− κP ) +κ

4

a2Y 2

X2(7a

a+ 7

Y

Y− X

X)∂0

δρ

λ+ κρ

−κ4

a2Y 2

X2(3a

a+ 3

Y

Y− X

X)∂0

δP

λ− κP + κY 2

X2(a

a+Y

Y)P + ρ

λ+ κρ∇2δu

= 0. (40)


The part of i0-component of (36) with the form ∂iS (where

S is any scalar) gives

(a

a+Y

Y)E − A− κ

2∂0

δρ

λ+ κρ+κ

2∂0

δP

λ− κP+P + ρ

λ+ κρδu

−κ2

(a

a+Y

Y)(

δρ

λ+ κρ+

3δP

λ− κP) = 0. (41)

The part of ij-component of (36) of the form ∂i∂jS gives

−1

2E − 1

2A+

1

2

a2Y 2

X2B +

1

2

a2Y 2

X2(3a

a+ 3

Y

Y− X

X)B − Y 2

X2F

−Y2

X2(a

a− X

X+ 3

Y

Y)F + κ

δP

λ− κP+ κ

Y 2

X2∂0(

P + ρ

λ+ κρδu)

+κY 2

X2(a

a− X

X+ 3

Y

Y)P + ρ

λ+ κρδu = 0. (42)


The 0-component of the perturbed conservation equation

gives

δρ+ 3a

a(δρ+ δP ) +

1

2(P + ρ)(3A+∇2B)

−a−2(P + ρ)∇2(F − δu) = 0. (43)

The part of i-component of the perturbed conservation equation of

the form ∂iS (where S is any scalar) gives

δP +1

2(P + ρ)E + (P + ρ)δu+ (P + ρ)δu+ 3

a

a(P + ρ)δu = 0.(44)


B. Vector modes:

The part of i0-component of the perturbed equation (36)

with the form Vi (where Vi is any vector satisfying ∂iVi = 0) gives

1

2∇2Ci −

1

2a−2∇2Gi +

κ

2a−2 P + ρ

λ+ κρ∇2δUi +

P + ρ

λ+ κρδUi = 0.(45)

The part of i-component of the perturbed conservation equation

with the form Vi (where Vi is any vector satisfying ∂iVi = 0) gives

(P + ρ)δUi + (P + ρ)δUi + 3a

a(P + ρ)δUi = 0. (46)


The part of ij-component of (36) with the form ∂iVj (where

Vj is any vector satisfying ∂jVj = 0) gives

1

2

a2Y 2

X2Cj +

1

2

a2Y 2

X2(3a

a+ 3

Y

Y− X

X)Cj −

1

2

Y 2

X2Gj

−1

2

Y 2

X2(a

a− X

X+ 3

Y

Y)Gj +

κ

2

Y 2

X2∂0(

P + ρ

λ+ κρδUj)

+κ

2

Y 2

X2(a

a− X

X+ 3

Y

Y)P + ρ

λ+ κρδUj = 0. (47)


B. Tensor modes:

The part of ij-component of (36) with the form of a

transverse-traceless tensor is

−∇2Dij +a2Y 2

X2Dij +

a2Y 2

X2(3a

a+ 3

Y

Y− X

X)Dij = 0. (48)


3. The stability of the perturbations

The perturbed equations involve 7 scalars modes

E,F,A,B, δρ, δP, δu, 3 transverse vectors modes Ci, Gi, δUi,

and 1 transverse-traceless tensor mode Dij .

For scalar perturbed modes we work in the Newtonian gauge,

i.e., we set B = F = 0 in the linear perturbed equations.

For vector mode, we fix the gauge freedom to eliminate Ci.

Finally, with the state equation P = ωρ, then δP = ωδρ, all

the remaining perturbed modes are solvable.

The dominant component in Eddington regime is the highly

relativistic ideal gas (ω = 1/3, δP = 13δρ) and the

cosmological constant can be neglected (λ = 1).


3.1 The case κ > 0

For κ > 0, the approximate background solution near the

maximum density (t→ −∞) is given by

[Escamilla-Rivera, Banados, and Ferreira, PRD 85(2012)087302],

[Scargill, Banados, and Ferreira, PRD 86 (2012) 103533]

a = aB[1 + e

√83κ

(t−t0)],

X = U12 = 2e

34

√83κ

(t−t0),

Y = V12 = 2e

14

√83κ

(t−t0). (49)


3.1 The case κ > 0

A. Scalar perturbations

For scalar perturbations, the solution (t→ −∞) is given by

A ' 2c1(xi)e74b(t−t0), (50)

E ' −2c1(xi)e114b(t−t0), (51)

δρ ' −12

κc1(xi)e

74b(t−t0), (52)

δu ' 4

7bc1(xi)e

74b(t−t0) + c2(xi). (53)

Therefore, the scalar perturbations vanish when the universe

approaches the maximum density and we conclude that the scalar

perturbations are stable in the Eddington regime.


3.1 The case κ > 0

B. Transverse vector modes

For transverse vector modes, from Eqs. (46) and (47), we

have

Gi ' c3(xi), (54)

δUi ' c4(xi). (55)

The vector perturbations are also stable in the Eddington regime.


3.1 The case κ > 0

C. Transverse-traceless tensor mode

The transverse-traceless tensor mode is given by Eq. (48),

with the approximate solution (49), the dominant part simply gives

Dij = 0. (56)

So it gives that

Dij ∝ m(x)t+ n(x). (57)

When the universe approaches the maximum density (t→ −∞),

the tensor perturbation is divergent, it causes an instability in the

Eddington regime.


3.2 The case κ < 0

For κ < 0, the approximate background solution near the

maximum density (t→ 0) is given by

[Escamilla-Rivera, Banados, and Ferreira, PRD 85(2012)087302],

[Scargill, Banados, and Ferreira, PRD 86 (2012) 103533]

a = aB

(1− 2

3κ|t|2), (58)

X =2√3

(−κ/2)1/4 |t|−12 , (59)

Y =2√3

(−2/κ)1/4 |t|12 . (60)


3.2 The case κ < 0

The perturbations are approximately given by

A ' κ

2C1(xi) |t|

32 , E ' κ2

16C1(xi)|t|−

12 , (61)

δρ ' C1(xi) |t|32 , δu ' −κ

2

16C1(xi) |t|

12 , (62)

δU ' C2(xi), Gi ' C3(xi)|t|−2 +2

3C2(xi), (63)

Dij ' C4(xi)|t|−1 + C5(xi). (64)

Therefore, scalar, vector and tensor modes will all cause

instabilities in the Eddington regime in this case.


Conclusion and discussion

We studied the full linear perturbations in the radiation era of

a homogeneous and isotropic spacetime in EiBI theory.

For κ > 0, the scalar and transverse vector modes are stable,

while the tensor mode is unstable in Eddington regime.

For κ < 0, all the scalar, vector and tensor modes cause

instabilities in Eddington regime.

It may necessary to consider the nonlinear perturbations.



In Einstein theory, in the radiation era, ( aa)2 = ρ3 , a = a0

√t.

The linear perturbed modes are

E = −A ' d1t− 3

2 + d2, (65)

δu ' d1t− 1

2 − d2

2t, (66)

δρ ' 3

2d1t− 7

2 − 3

4d2t−2, (67)

Gi ' d3t− 1

2 , δUi ' d4t12 , (68)

Dij ' d5t− 1

2 + d6. (69)

All modes are all unstable in early universe.



Thus the EiBI cosmology with κ > 0 presents as an interesting

theory with stable scalar and vector modes.

This was also demonstrated in the following references:

Pani, Cardoso, Delsate, Compact stars in eddington inspired gravity,

PRL 107 (2011) 031101;

P. Avelino and R. Ferreira, Bouncing eddington-inspired born-infeld

cosmologies: an alternative to inflation?, PRD 86 (2012) 041501;

P. Pani, T. Delsate, and V. Cardoso, Eddington-inspired born-infeld

gravity. phenomenology of non-linear gravity-matter coupling, PRD

85 (2012) 084020,

where the EiBI theory with positive κ shows more well

properties than negative one.



For a 5D braneworld model with

SEiBI =1

κ

∫d5x

(√−|gPQ + κRPQ(Γ)| − λ

√−|gPQ|

)−∫d5x√−|gPQ|

(1

2gMN∂Mφ∂Nφ+ V (φ)

),

ds2 = a2(y)ηµνdxµdxν + dy2,

We found in [PRD 85(2012)124053] a brane solution for κ > 0

and the TT tensor perturbation is stable.


Thank you !


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