quantum avoidance of the big rip singularity in eddington-inspired-born-infeld theory ·...

Quantum avoidance of the big rip singularityin Eddington-inspired-Born-Infeld theory

Che-Yu Chen

Institute of physics, National Taiwan UniversityLeCosPA, National Taiwan University

Work done under the supervision of Dr. M. Bouhmadi-Lopez (Universidade da BeiraInterior (Portugal)) and Prof. Pisin Chen (NTU and Standford University)

5th Dec. 2016

Che-Yu Chen (National Taiwan University) 05-12-2016 1 / 33

References

Eur. Phys. J. C 74, 2802 (2014), arXiv:1302.5013

M. Bouhmadi-Lopez, C-Y- Chen and P. Chen

Eur. Phys. J. C 75, 90 (2015), arXiv: 1406.6157

M. Bouhmadi-Lopez, C-Y- Chen and P. Chen

JCAP 1611, no. 11, 023 (2016), arXiv:1609.00700

M. Bouhmadi-Lopez and C-Y- Chen


Outline

1 Introduction

2 Big Rip in the EiBI theory

3 Quantization of the EiBI theoryStandard approach: Start from the actionA “phenomenological” approach: Start from the Friedmann equations

4 Conclusion


Outline

1 Introduction



4 Conclusion


Introduction-1-: motivation

Einstein general relativity (GR) is an extremely successful theory for

more than a century

There are some motivations for looking for possible extension of GR:

singularity at high energies, late time acceleration...

To explain the acceleration, some kinds of Dark Energy or

modifications are needed: C.C, quintessence, Generalized Chaplygin

Gas(GCG), phantom...

More singularities?


Big Rip

Caldwell, Kamionkowski, Weinberg (2003)


Introduction-2-: motivation

Born-Infeld action for electrodynamics in 1934:√|gµν + κFµν |

Theories of gravity with Born-Infeld determinantal action have been

widely investigated:√|gµν + κGµν |

S. Deser and G. W. Gibbons (1998)

BUT most of them suffer from ghost instability (pure metric

formalism)

An alternative Born-Infeld determinantal theory within Palatini

formalism has been proposed - Eddington-inspired-Born-Infeld gravity

Banados, Ferreira (2010)


The theory-1-:

SEiBI(g , Γ,Ψ) = 2κ

∫d4x

[√|gµν + κRµν(Γ)| − λ

√−g]

+ Sm(g ,Ψ)

Palatini formalism: g , Γ, Sm(g ,Ψ)

The compatibility: Γ⇔ qµν ≡ gµν + κRµν

No torsion T = 0

κ is the only parameter characterizing the theory (8πG = c = 1)

λ is related to the cosmological constant Λ

Recover GR+C.C at low curvature limit

Sg ≈ 12

∫d4x√−g[R − 2(λ−1)

κ

]


The theory-2-:

SEiBI(g , Γ,Ψ) = 2κ

∫d4x


√−g]

+ Sm(g ,Ψ)

A radiation dominated universe: Big Bang is avoided through aLoitering effect if κ > 0

a→ al when t → −∞The Big Bang is substituted by a bounce if κ < 0

5 10 15 20t

1

2

3

4

5a�al

-2 -1 1 2 3 4ΚΡ

-0.3

0.3ΚH2

Banados, Ferreira (2010)


Outline

1 Introduction



4 Conclusion


The EiBI scenario and the Big Rip

We consider an EiBI universe (κ > 0) filled with radiation, matter

and, a phantom dark energy component with a constant equation of

state w < −1

The physical metric gµν : a→∞, ρ→∞

H2 ≈ 4√|w |3

3(3w+1)2 ρ→∞, and dHdt ≈

2√|w |3

(3w+1)2 |1 + w |ρ→∞ at a finite t

(Big Rip)

For the same w , the singular event is slightly postponed in the EiBI

theory compared with GR

M. Bouhmadi-Lopez, C. Y. Chen, P. Chen (2014),(2015)


The evolution of a bound structure near Big Rip

The evolution of a bound system can be estimated by

r =a

ar − GM

r2+

L2

r3, r2φ = L

S. Nesseris, L. Perivolaropoulos (2004)

Near the singularity:

r(t) = A1(tmax − t)2

3(1+w) + A2(tmax − t)1− 23(1+w) ,

φ(t) ≈ 3(w+1)1−3w

LA1

2 (tmax − t)3w−1

3(w+1) + φ0

M. Bouhmadi-Lopez, C. Y. Chen, P. Chen (2015)


The EiBI scenario and the Big Rip

The geometry corresponding to the connection (or qµν)

The auxiliary metric qµν (t → t, a→ a, so that qµν = [−1, a2δij ]):

Hq2 ≡ ( 1

adadt

)2 → 13κ , and a ∝ eHq t when t →∞ (de-Sitter)

Does not couple to matter (observers)

M. Bouhmadi-Lopez, C. Y. Chen, P. Chen (2015)


The full analysis of EiBI phantom model

Singularity in GR EiBI physical metric EiBI auxiliary metric

Big Rip Big Rip expanding de-Sitter

past Sudden past Type IV (0 < α ≤ 2) contracting de-Sitter

(α > 0) past Sudden (α > 2)

future Big Freeze future Big Freeze (−3 < α < −1) expanding de-Sitter

(α < −1) future Type IV (α = −3)

future Sudden (α < −3)

past Type IV past Sudden (−2/3 < α < −1/3) past Type IV

(−1 < α < 0) (1)past Type IV

(α 6= −n/(n + 1)) (2)finite past without singularity finite past without singularity

past loitering effect (ab > amin) Big Bang

finite past without singularity finite past without singularity finite past without singularity

(α = −n/(n + 1))

(−1 < α < 0)

past loitering effect (ab > amin) Big Bang

Little Rip Little Rip expanding de-Sitter


Outline

1 Introduction



4 Conclusion


Quantizing EiBI theory? Why?

The big rip singularity is still unavoidable in the EiBI theory→ The theory is still NOT complete

Some quantum effects are expected near the classical big ripsingularity

Quantum avoidance of the DE singularities in GR

The quantization of theories constructed upon Palatini formalism:(g , Γ)

Some results which are model-independent? Further step to quantumgravity?


Quantizing EiBI theory? How?

FLRW ansatz → Quantum cosmology

Quantum Geometrodynamics → Wheeler DeWitt (WDW) equationHΨ = 0

The DeWitt condition: Ψ→ 0singularity avoidance

Does it sufficiently imply a vanishing probability? NO

A potential hint for singularity avoidance

B. S. DeWitt (1967)


Outline

1 Introduction



4 Conclusion


How to start? The alternative action

SEiBI = 2κ

∫d4x


√−g]

+ Sm(g ,Ψ)

The problematic square root√...

It may be possible to use Lagrangian multipliers → too many variables

An alternative action which is dynamically equivalent!!

T. Delsate, J. Steinhoff (2012)

Sa = λ∫d4x√−q[R(q)− 2λ

κ + 1κ

(qαβgαβ − 2

√gq

)]+ Sm(g ,Ψ)

{g ,Γ}→{g ,q}Linear on R(q) ≡ qαβRβα(q) (like SEH)

q → dynamical g → non-dynamical

The field equations are exactly the SAME!!

T. Delsate, J. Steinhoff (2012)


FLRW universe

The FLRW ansatz

gµνdxµdxν = −N(t)2dt2 + a(t)2d~x2,

qµνdxµdxν = −M(t)2dt2 + b(t)2d~x2.

Perfect fluid assumption→

L = λMb3[− 6b2

M2b2− 2λ

κ+

1

κ

(N2

M2+ 3

a2

b2− 2

Na3

Mb3

)]− 2ρ(a)Na3

E.O.M

N: gauge fixing M: Firedmann eq. of b

a: relating a and b b: Rauchaudhuri eq. of b


Constructing the Hamiltonian H

Conjugate momenta:

pb =∂L∂b

= −12λb

Mb

The Hamiltonian

H = M[−

p2b

24λb+

2λ2

κb3 +

1

κλ(λ+ κρ(a))2 a

6

b3− 3λ

κba2]

Canonical quantization: {a, b, pb} → Operators → Quantum level


The first factor ordering: The WDW equation

b3H = 0→[∂2

∂x2 + V1(a, x)]Ψ(a, x) = 0, x = ln(

√λb)

V1(a, x) =24

κ~2

[2e6x − 3a2e4x + (λ+ κρ(a))2a6

]WKB approximation: If V1 → +∞ when a→∞, then Ψ→ 0

Ψ(a, x) ≈ V1(a, x)−14 exp

{± i

∫ x√V1(a, x ′)dx ′

}Goal: To prove V1 → +∞ when a→∞ for all x

M. Bouhmadi-Lopez, C. Y. Chen (2016)


First factor ordering: Ψ→ 0 when a→∞

V1(a, x) = 24κ~2 e

6x [2− 3δ + (λ+ κρ(a))2δ3] δ ≡ a2e−2x ∝ a2/b2

If when a→∞....

δ → 0 (a� b →∞): the −3δ term can be neglectedδ →∞ (a� b):

V1(a, x) ≈ 24

κ~2(λ+ κρ(a))2a6

δ → finite constant (a ≈ b):

V1(a, x) ≈ 24

κ~2(λ+ κρ(a))2a6

V1 → +∞ when a→∞ for all x!! Ψ→ 0



First factor ordering: Ψ→ 0 when a→∞

A guiding example: If a and b are related through E.O.Mand p = wρ (w < −1)

Without WKB approximation:

Ψ(x) = C1J0(A1e3x) + C2Y0(A1e

3x) A1 ≡4√

3κ~2

When x →∞ (as well as a)

Ψ(x) ≈√

2

πA1e−3x/2

[C1 cos

(A1e

3x − π

4

)+ C2 sin

(A1e

3x − π

4

)]M. Bouhmadi-Lopez, C. Y. Chen (2016)


Second factor ordering: The WDW equation

H = 0→[∂2

∂y2 + V2(a, y)]Ψ(a, y) = 0, y ≡ (

√λb)3/2

Similar strategy...

V2(a, y) =32

3κ~2y2[2−3η+(λ+κρ(a))2η3

]η ≡ a2y−4/3 ∝ a2/b2

WKB approximation: If V2 → +∞ when a→∞, then Ψ→ 0

Ψ(a, y) ≈ V2(a, y)−14 exp

{± i

∫ y √V2(a, y ′)dy ′

}M. Bouhmadi-Lopez, C. Y. Chen (2016)


Second factor ordering: Ψ→ 0 when a→∞

V2(a, y) = 323κ~2 y

2[2− 3η + (λ+ κρ(a))2η3

]η ≡ a2y−4/3 ∝ a2/b2

If when a→∞....

η → 0 (a� b →∞):

η →∞ (a� b):

η → finite constant (a ≈ b):

V2 → +∞ when a→∞ for all y !! Ψ→ 0



Second factor ordering: Ψ→ 0 when a→∞

A guiding example: If a and b are related through E.O.Mand p = wρ (w < −1)

Without WKB approximation:

Ψ(y) = C1√yJ1/4(A1y

2) + C2√yY1/4(A1y

2)

When y →∞ (as well as a)

Ψ(y) ≈

√2

πA1y

[C1 cos

(A1y

2 − 3π

8

)+ C2 sin

(A1y

2 − 3π

8

)]M. Bouhmadi-Lopez, C. Y. Chen (2016)


Outline

1 Introduction



4 Conclusion


The “phenomenological” Wheeler DeWitt equation

A phenomenological WDW equation from the Friedmann equations

κH2 ≈4√|w |3

3(3w + 1)2a−3(1+w) κH2

q = κ(1

b

db

dt

)2≈ 1

3

t = rescaled time

HtΨ(a, b) = HgHqΨ(a, b) = 0 Ψ(a, b) = ψ1(a)ψ2(b)

c.f. The direct product of two independent spaces( d2

da2+

48√|w3|

κ~2(3w + 1)2a4+ε

)ψ1(a)

( d2

db2+

12

κ~2b4)ψ2(b) = 0



The vanishing of the wave function(d2

da2 +48√|w3|

κ~2(3w+1)2 a4+ε)ψ1(a)

(d2

db2 + 12κ~2 b

4)ψ2(b) = 0

Two possibilities:(d2

da2 +48√|w3|

κ~2(3w+1)2 a4+ε)ψ1(a) = 0

(d2

db2 + 12κ~2 b

4)ψ2(b) = K1(b)

or...(d2

da2 +48√|w3|

κ~2(3w+1)2 a4+ε)ψ1(a) = K2(a)

(d2

db2 + 12κ~2 b

4)ψ2(b) = 0

K1 and K2 are bounded functions

In both cases, ψ1(a)→ 0 when a→∞ and ψ2(b) is bounded for all b

Ψ(a, b) = ψ1(a)ψ2(b)→ 0 when a→∞ for all b



Outline

1 Introduction



4 Conclusion


Conclusion

The big rip singularity is unavoidable in the EiBI theory, even though

most singularities can be alleviated in it

The bound structures would be ripped apart before the singularity

Quantum geometrodynamics within EiBI framework - quantum

cosmology

Standard approach: An alternative action → Lagrangian → H→WDW eq. (two factor ordering)

Phenomenological approach: Two Friedmann eqs. → Ht → WDW eq.

DeWitt condition is satisfied → big rip is expected to be avoided in

quantum level


Conclusion

Thank you for your attention:)


quantum avoidance of the big rip singularity in eddington-inspired-born-infeld theory ·...

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