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Eddington inspired Born-Infeld gravityprospects, problems, and extensions

Terence Delsate Jan Steinhoff (speaker)

Centro Multidisciplinar de Astrofısica (CENTRA)Instituto Superior Tecnico (IST)

Lisbon

April 8th, 2013, Mons, Belgium

Supported by through STE 2017/1-1

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 1 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 2 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 3 / 22

Motivation

Born-Infeld (BI) nonlinear electrodynamics:

S =1κ2

∫d4x

[√−det(gab + κFab)−

√−det(gab)

][M. Born and L. Infeld, Proc. R. Soc. A 144 (1934) 425–451]

Arises as low-energy effective theory from certain string theories.[E. Fradkin and A. A. Tseytlin, Phys. Lett. B 163 (1985) 123]

Born-Infeld-Einstein gravity actions, e.g.:

S =2γκ

∫d4x

[√−det(gab + κRab)− λ

√−det(gab)

][S. Deser and G. W. Gibbons, Class. Quant. Grav. 15 (1998) L35–L39]

Eddington inspired Born-Infeld (EiBI) gravity: Palatini variation[M. Banados and P. G. Ferreira, Phys. Rev. Lett. 105 (2010) 011101]

Many similarities to Palatini f (R)![T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82 (2010) 451–497]

Metric formalism for BI-gravity actions inconsistent? (ghosts, instabilities)

Palatini variation: more fundamental than metric formalism? (next slide)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 4 / 22

Classification of gauge theories of gravityfrom book: M. Blagojevic, F.W. Hehl, ”Gauge Theories of Gravitation”, arXiv:1210.3775

[see Figure in arXiv:1210.3775]

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 5 / 22

LiteratureD. N. Vollick, Phys. Rev. D 69 (2004) 064030M. Banados and P. G. Ferreira, Phys. Rev. Lett. 105 (2010) 011101P. Pani, V. Cardoso, and T. Delsate, Phys. Rev. Lett. 107 (2011) 031101J. Casanellas, P. Pani, I. Lopes, and V. Cardoso, Astrophys. J. 745 (2012) 15P. P. Avelino, Phys. Rev. D 85 (2012) 104053P. Pani, T. Delsate, and V. Cardoso, Phys. Rev. D 85 (2012) 084020T. Delsate and J. Steinhoff, Phys. Rev. Lett. 109 (2012) 021101Y.-X. Liu, K. Yang, H. Guo, and Y. Zhong, Phys. Rev. D 85 (2012) 124053C. Escamilla-Rivera, M. Banados, and P. G. Ferreira, Phys. Rev. D 85 (2012) 087302A. De Felice, B. Gumjudpai, and S. Jhingan, Phys. Rev. D 86 (2012) 043525P. Avelino and R. Ferreira, Phys. Rev. D 86 (2012) 041501P. Avelino, JCAP 1211 (2012) 022I. Cho, H.-C. Kim, and T. Moon, Phys. Rev. D 86 (2012) 084018P. Pani and T. P. Sotiriou, Phys. Rev. Lett. 109 (2012) 251102J. H. Scargill, M. Banados, and P. G. Ferreira, Phys. Rev. D 86 (2012) 103533Y.-H. Sham, P. T. Leung, and L.-M. Lin, Phys. Rev. D 87 (2013) 061503(R)S. Jana and S. Kar, arXiv:1302.2697 [gr-qc]

I. Cho and H.-C. Kim, arXiv:1302.3341 [gr-qc]

M. Bouhmadi-Lopez, C.-Y. Chen, and P. Chen, arXiv:1302.5013 [gr-qc]

S. Rajagopal and A. Kumar, arXiv:1303.6026 [gr-qc]

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 6 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 7 / 22

Field EquationsD. N. Vollick, Phys. Rev. D 69 (2004) 064030

Action of EiBI coupled to matter Ψ: Λ =λ− 1κ

, g = det(gab)

S[g, Γ, Ψ] =2γκ

∫d4x

[√−det(gab + κRab[Γ])− λ

√−g]

+ SM [g, Γ, Ψ]

Define auxiliary metric qab such that:

Γcab =

12

qcd (∂aqbd + ∂bqad − ∂dqab)

Algebraic field equation:

τ(

gab − γκ

λT ab

)= qab

τ :=

√gq

, q = det(qab), T ab :=1√−g

δSM

δgab

Differential field equation:

gab = λqab − κRab

In the following, we set λ = 1 (i.e. Λ = 0) and γ = 8πGJan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 8 / 22

CosmologyM. Banados and P. G. Ferreira, Phys. Rev. Lett. 105 (2010) 011101

For κ > 0:no big bang singularity!loitering phase at early timessimilar to Einstein universe

For κ < 0:no singularity!bounce

-4 -2 0 2 40

2

4

6

8

10

12

t

a�

a B

Κ < 0

Κ > 0

Maximal density ρB, depends on sign(κ) and EOSLeads to minimal scale factor aB

How generic is the singularity avoidance in this theory?Similar singularity avoidance in Palatini f (R)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 9 / 22

Compact StarsP. Pani, V. Cardoso, and T. Delsate, Phys. Rev. Lett. 107 (2011) 031101

For κ > 0:repulsive effectmaximal mass increasesmay save excluded EOS

For κ < 0:attractive effectmaximal mass decreases

For κ > 0 the theory admitsdust stars with EOS P = 0!

8 10 12 14 16

0.5

1.0

1.5

2.0

2.5

3.0

R Hkm L

M�M�

GRΚΡ0=0.1ΚΡ0=0.2ΚΡ0=0.3ΚΡ0=0.4ΚΡ0=-0.01ΚΡ0=-0.05ΚΡ0=-0.1

Κ<0

Κ>0

Still a maximum mass exists: collapse to black hole not avoidedAt the surface the auxiliary metric qab is smoothBut the ”true” metric gab is maybe not even continuous!

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 10 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 11 / 22

Apparent source

Manipulate field equations:

τ(gab − 8πGκT ab) = qab gab = qab − κRab

Einstein equation for auxiliary metric

Rab = 8πG

[T a

b −12δa

bT cc

]with the ”apparent” stress tensor

T ab = τT a

b +δa

b

8πG[τ − 1− 4πGκτT ]

τ =

√gq

=1√

det(δab − 8πGκT a

b)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 12 / 22

Apparent source

Manipulate field equations:

τ(δab − 8πGκT a

b) = qacgcb

=

qacgcb = δab − κRa

b

Einstein equation for auxiliary metric

Rab = 8πG

[T a

b −12δa

bT cc

]with the ”apparent” stress tensor

T ab = τT a

b +δa

b

8πG[τ − 1− 4πGκτT ]

τ =

√gq

=1√

det(δab − 8πGκT a

b)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 12 / 22

Apparent source

Manipulate field equations:

τ(δab − 8πGκT a

b) = qacgcb = qacgcb = δab − κRa

b

Einstein equation for auxiliary metric

Rab = 8πG

[T a

b −12δa

bT cc

]with the ”apparent” stress tensor

T ab = τT a

b +δa

b

8πG[τ − 1− 4πGκτT ]

τ =

√gq

=1√

det(δab − 8πGκT a

b)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 12 / 22

Apparent source

Manipulate field equations:

τ(δab − 8πGκT a

b) = qacgcb = qacgcb = δab − κRa

b

Einstein equation for auxiliary metric

Rab = 8πG

[T a

b −12δa

bT cc

]with the ”apparent” stress tensor

T ab = τT a

b +δa

b

8πG[τ − 1− 4πGκτT ]

τ =

√gq

=1√

det(δab − 8πGκT a

b)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 12 / 22

Fluid: Modified EOS

T ab = τT a

b +δa

b

8πG[τ − 1− 4πGκτT ]

τ =

√gq

=1√

det(δab − 8πGκT a

b)

Interesting result for ideal fluid:

T ab = (ρ+ P) uaub + Pδa

b, uaubgab = −1

⇒ T ab = (ρq + Pq) vavb + Pqδ

ab , vavbqab = −1

Pq =τ

2(ρ− P) +

τ − 18πGκ

, ρq =τ

2(ρ+ 3P)− τ − 1

8πGκ

τ =[(1 + 8πGκρ)(1− 8πGκP)3]− 1

2 , ρq + Pq = τ(ρ+ P)

Similar: baryon number density, entropy, temperatureFor dust P = 0:

Pq = πGκρ2q +O(ρ3

q)

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 13 / 22

Viability, Phenomenology, and Constraints

Coupling between gravity and matter less explored→ less constrainedTheory is equivalent to general relativity in vacuumIn vacuum EiBI can not be distinguished from GR with source T a

b

Phenomenologically qab, T ab, ρq , and Pq can be qualified as ”apparent”

Interesting: constraint on κ from observations of the sun(acoustic oscillation modes, neutrinos)

|κ| < 3 · 105 m5

s2kg

[J. Casanellas, P. Pani, I. Lopes, and V. Cardoso, Astrophys. J. 745 (2012) 15]

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 14 / 22

Energy Conditions (EC)

EC used in the literature:Null EC: ρ+ P ≥ 0Weak EC: ρ+ P ≥ 0, ρ ≥ 0Strong EC, SEC: ρ+ P ≥ 0, ρ+ 3P ≥ 0Dominant EC: ρ ≥ |P|Causal EC: |ρ| ≥ |P|

Most important: Null ECNull EC violation associated with pathologieslike traversable worm holes, warp drives, etc.Is Null EC fulfilled in apparent sectorif it holds for the real EOS?

ρq + Pq = τ(ρ+ P), τ ≥ 0

⇒ Yes, it is!

Plots: illustrate Strong EC (γ = 8πG)

- 2 -1 0 1 2

- 2

-1

0

1

2

Ρ� ΓΚ

P

ΓΚ

Apparen t SEC violation for real SEC, ΓΚ > 0

- 2 -1 0 1 2

- 2

-1

0

1

2

Ρ� ΓΚ

Apparen t SEC violation for real SEC, ΓΚ < 0

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 15 / 22

Analysis of Singularity Avoidance

τ has poles for finite values of ρ and P:

τ =

√gq

=1√

(1 + 8πGκρ)(1− 8πGκP)3

Pole for 8πGκP → 1:Can happen for κ > 0Maximal pressure Pmax = 1/8πGκ(corresponding to singular ρq ∝ τρ ... )EOS is considerably softened

Pole for 8πGκρ→ −1:Can happen for κ < 0Max. energy density ρmax = 1/8πG|κ|EOS is considerably hardened

Counterexample: dust universe (plots)”Usual” EOS should avoid singularities!

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

t

ΓΚ = 1

Ρ

Pq

KHgL

KHqL

-2 -1 0 1 2

-0.5

0.0

0.5

1.0

t

ΓΚ = -1

Ρ

Pq �10

KHgL�15

KHqL�15

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 16 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 17 / 22

Problems and ExtensionsP. Pani and T. P. Sotiriou, Phys. Rev. Lett. 109 (2012) 251102

Theory is well behaved in qab sectorBut Ricci scalar R[g] at surface of stars is singular if (near the surface)

P = KρΓ0 with Γ > 3/2

Generic problem, differential structure of theory in terms of gab:higher order derivatives of matter in sourceThe metric gab is too sensitive to sharp matter profilesSame problem appears for Palatini f (R)

But implications are discussed controversially in Palatini f (R) gravityCan probably be cured by adding further degrees of freedom:torsion, nonmetricity, . . .

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 18 / 22

Extensions from bimetric action approach?

A bimetric linearization of the action reads:

SG =1

8πG

∫d4x√−q[R[q]− 2

λ

κ+

1κ

(qabgab − 2

√gq

)]+ SM [g]

Stringy anlogon: from Nambu-Goto to Polyakov actionNo ghosts!Only the metric coupling to matter is measurable”Cutoff” 1/8πGκ appears as coupling parameterStarting point to modify the gravity-matter coupling?Similar bimetric action used in asymptotic safety scenarioE. Manrique, M. Reuter, and F. Saueressig, Annals Phys. 326 (2011) 440–462

More complicated bimetric actions appear in New Massive GravityS. Hassan and R. A. Rosen, JHEP 1107 (2011) 009S. Hassan and R. A. Rosen, JHEP 1202 (2012) 126

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 19 / 22

Outline

1 Motivation: Born-Infeld structures and Palatini variation

2 EiBI and some of its properties

3 EiBI as realization of modified coupling

4 Problems and Extensions

5 Conclusions

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 20 / 22

Conclusions

Exiting features of EiBI:BI structure may originate from quantized gravityPalatini variation is very naturalSome singularities are avoidedDust stars (pressureless) for κ > 0Coupling between gravity and matter less explored/constrainedInteresting phenomenology, as it deviates from GR only inside matter

Problems:Maximum NS mass vs. singularity avoidanceSimilar to Palatini f (R), same problems, e.g.:Problems at surface/phase transitions, like singular curvature

Possible extensions:Modification using bimetric actionModification by relaxing conditions on connection: torsion, nonmetricity

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 21 / 22

Thank you for your attention

and for support by the German Research Foundation

Jan Steinhoff (CENTRA, IST) EiBI gravity: prospects, problems, and extensions Mons, April 8th, 2013 22 / 22