Correcciones cuánticas de agujeros negros y cosmología · 2010. 12. 29. · Gonzalo J. Olmo...

Gonzalo J. Olmo

Correcciones cuánticas de agujeros negrosy

cosmología

(or Quantum correlations, black holes and cosmology)

Gonzalo J. Olmo Alba

Thesis advisor: José Navarro Salas

Universidad de Valencia

Outline

Quantum Correlations and BH

Cosmology

The end

Gonzalo J. Olmo July 5th, 2005 - p. 2/26

Outline

Outline


Cosmology

The end


Outline

Part I:Quantum correlations and black holes Subject: New approach for radiation problems in curved space. Structure:

Black hole evaporation following the standard formalism. Difficult application of the standard approach when

backreaction effects are considered. New approach to solve the problems: correlation functions.

Outline


Cosmology

The end


Outline

Part I:Quantum correlations and black holes Subject: New approach for radiation problems in curved space. Structure:

Black hole evaporation following the standard formalism. Difficult application of the standard approach when

backreaction effects are considered. New approach to solve the problems: correlation functions.

Part II:Cosmology Subject: Cosmic speed-up due to new gravitational dynamics? Structure:

Observational evidence for the cosmic accelerated expansion. Possible explanations: dark energy, modified dynamics, . . . Modified dynamics:f (R) gravities. Analyze the solar system constraints on these theories.

Outline


Gravitational collapse

Bogolubov coefficients

Black Holes evaporate

Evaporation with backreaction

Bogolubov -Vs- Correlator

New approach

Example: Conformal Invariance

Moving-mirrors and BH

Beyond N

Summary and Conclusions

Cosmology

The end


Part I:Quantum correlations

andblack holes

Outline







New approach



Beyond N


Cosmology

The end


Gravitational collapse and quantization

Outline







New approach



Beyond N


Cosmology

The end



Consider a scalar field

φ(x) = ∑[aiui(x)+a†i u∗i (x)]

Outline







New approach



Beyond N


Cosmology

The end




“IN” expansion:

φ(x) = ∑i[aini uin

i (x)+aini

†uini∗(x)]

Vacuum state:aini |0〉in = 0

Outline







New approach



Beyond N


Cosmology

The end




“IN” expansion:


i (x)+aini

†uini∗(x)]


“OUT” expansion:

φ(x) = φH +φI+

φH = ∑ j [aHj uH

j (x)+aHj

†uH

j∗(x)]

φI+ = ∑ j [aoutj uout

j (x)+aoutj

†uoutj∗(x)]

Vacuum state:aouti |0〉out = 0

Outline







New approach



Beyond N


Cosmology

The end




“IN” expansion:


i (x)+aini

†uini∗(x)]


“OUT” expansion:

φ(x) = φH +φI+

φH = ∑ j [aHj uH

j (x)+aHj

†uH

j∗(x)]

φI+ = ∑ j [aoutj uout

j (x)+aoutj

†uoutj∗(x)]

Vacuum state:aouti |0〉out = 0

In general|0〉in 6= |0〉out

Outline







New approach



Beyond N


Cosmology

The end


Bogolubov coefficientsFUTURE = "OUT" region

I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future

Outline







New approach



Beyond N


Cosmology

The end



FUTURE = "OUT" region

I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future

Outline







New approach



Beyond N


Cosmology

The end




I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future

Outside the Black Hole:

uoutj = ∑i [α ji uin

i +β ji uin∗i ]

aouti = ∑ j [α∗i j ain

j −β∗i j ainj

†]

α ji = (uoutj ,uin

i ) , β ji =−(uoutj ,uin∗

i )

Outline







New approach



Beyond N


Cosmology

The end




I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future



i +β ji uin∗i ]


j −β∗i j ainj

†]

α ji = (uoutj ,uin


i )

At the horizon:

uHj = ∑i [γ ji uin

i +η ji uin∗i ]

aHi = ∑ j [γ∗i j ain

j −η∗i j ainj

†]

γ ji = (uHj ,uin

i ) , η ji =−(uHj ,uin∗

i )

Outline







New approach



Beyond N


Cosmology

The end




I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future

Vacuum state:|0〉in = S|0〉out

S= S(α,β,γ,η)



i +β ji uin∗i ]


j −β∗i j ainj

†]

α ji = (uoutj ,uin


i )

At the horizon:


i +η ji uin∗i ]


j −η∗i j ainj

†]

γ ji = (uHj ,uin


i )

Outline







New approach



Beyond N


Cosmology

The end




I−

I+

Incomingmatter

Flat geometry

BH geometry

Singularity

Event H

orizo

n

Outside the BH

= "IN" regionPAST

Future

Vacuum state:|0〉in = S|0〉out

S= S(α,β,γ,η)



i +β ji uin∗i ]


j −β∗i j ainj

†]

α ji = (uoutj ,uin


i )

At the horizon:


i +η ji uin∗i ]


j −η∗i j ainj

†]

γ ji = (uHj ,uin


i )

Number of particles:

in〈0|Nouti |0〉in = ∑k |βik|2

Outline







New approach



Beyond N


Cosmology

The end



Number of particles detected atI+ (Hawking 1974):

in〈0|Nouti |0〉in = ∑k |βik|2 = 1

e8πMωi−1

⇒ Planckian spectrum atT = ~

8πκBM

Outline







New approach



Beyond N


Cosmology

The end





e8πMωi−1


8πκBM

Uncorrelated outgoing radiation→ THERMAL state

(Parker 1975),(Wald 1975)

Outline







New approach



Beyond N


Cosmology

The end





e8πMωi−1


8πκBM

Uncorrelated outgoing radiation→ THERMAL state

(Parker 1975),(Wald 1975)

BIG PROBLEM : quantum information not radiated (Hawking 1976)

Apparent conflict betweenQM andGR:

Non-unitary evolution of quantum states

(Information Loss Problem)

Outline







New approach



Beyond N


Cosmology

The end



The outgoing radiation modifies the geometry. Thiseffect (backreaction) could restore the correlations.

Outline







New approach



Beyond N


Cosmology

The end




Charged black holes represent good toy models.A.Fabbri, D.Navarro, J.Navarro-Salas and G.J.O. , Phys.Rev.D (2003)

Strong correlations appear in the outgoing radiation:

Crel =Cwbr(x1,x2)Cnbr(x1,x2)

∼ e2κ|x1−x2|

|x1−x2|4 whereC(x1,x2)≡ in〈0|φ(x1)φ(x2)|0〉in

Outline







New approach



Beyond N


Cosmology

The end







∼ e2κ|x1−x2|


Involved computation ofα andβ:

Unknownin〈0|Nouti |0〉in = ?

Unknown density matrix,|0〉in→ ?

Moving-mirror model physically equivalent but . . . Unclear computation ofα andβ. Unclear relation between particles and energy fluxes.

Outline







New approach



Beyond N


Cosmology

The end



The Bogolubov coefficientsα andβ allow toconstruct magnitudes such asin〈0|Nout

i |0〉in .

Outline







New approach



Beyond N


Cosmology

The end




i |0〉in .

The two-point correlator allows to "see" thecorrelations among the outgoing particles.

Outline







New approach



Beyond N


Cosmology

The end


New approach

With the decomposition

φI+ = ∑[aoutj uout

j (x)+aoutj

†uoutj∗(x)]

We construct the normal-ordered operator

: φ(x1)φ(x2) : ≡ φ(x1)φ(x2)− out〈0|φ(x1)φ(x2)|0〉out

Outline







New approach



Beyond N


Cosmology

The end


New approach



j (x)+aoutj

†uoutj∗(x)]



Using the scalar product( f1| f2) =−i dΣµ f1←→∂ µ f ∗2

Outline







New approach



Beyond N


Cosmology

The end


New approach



j (x)+aoutj

†uoutj∗(x)]




Product of operators:

aout†i aout

j = (uouti (x1)|(uout∗

j (x2)|: φ(x1)φ(x2) :))

Outline







New approach



Beyond N


Cosmology

The end


New approach



j (x)+aoutj

†uoutj∗(x)]




Product of operators:

aout†i aout

j = (uouti (x1)|(uout∗

j (x2)|: φ(x1)φ(x2) :))

Number of particles:

in〈0|Nouti |0〉in = 1

~dΣµ

1dΣν2[u

outi (x1)

←→∂ µ][uout∗

i (x2)←→∂ ν] in〈0|: φ(x1)φ(x2) :|0〉in

Outline







New approach



Beyond N


Cosmology

The end



In d-dimensional Minkowski space, a conformallyinvariant field theory satisfies:

in〈0|φ(y1)φ(y2)|0〉in =C

|y1−y2|2∆

in〈0|φ(y1)φ(y2)|0〉in =

∣

∣

∣

∣

∂x∂y

∣

∣

∣

∣

∆/d

x1

∣

∣

∣

∣

∂x∂y

∣

∣

∣

∣

∆/d

x2

in〈0|φ(x1)φ(x2)|0〉in

Outline







New approach



Beyond N


Cosmology

The end


Moving-mirrors and black holes

x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Number of particles:in〈0|Nout

k |0〉in =− 1π

∞−∞ dx1dx2uk(x1)u∗k(x2)×

×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]

Exponential trajectory (NBR):y(x) = y0− 1

Ce−Cx ⇒ THERMAL

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :

in〈0|Toutxx (x)|0〉in =

=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :


=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL

Hyperbolic trajectory (WBR):

y(x) =

x x≤ 0x

1+a2xx≥ 0

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :


=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL


y(x) =

x x≤ 0x

1+a2xx≥ 0

Birrell-Davies, [Cambridge Univ.Press(1982)]

⇒ steady flux of particles alongx > 0

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :


=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL


y(x) =

x x≤ 0x

1+a2xx≥ 0



No particles forx1×x2 > 0

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :


=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL


y(x) =

x x≤ 0x

1+a2xx≥ 0




Particles localized aboutx≈ 0

Outline







New approach



Beyond N


Cosmology

The end



x

R+I L

+

I L−

I R

−

y0

MirrorTrajectory

y(x)

y

I

Flux to I+R :


=− ~

24π

[

y′′′(x)y′(x) −

32

(

y′′(x)y′(x)

)2]


k |0〉in =− 1π


×[

y′(x1)y′(x2)[y(x1)−y(x2)]2

− 1(x1−x2)2

]


Ce−Cx ⇒ THERMAL


y(x) =

x x≤ 0x

1+a2xx≥ 0




Particles localized aboutx≈ 0

Thunderbolt localized aty = y0

Outline







New approach



Beyond N


Cosmology

The end




in〈0|aouti aout



in〈0|aouti aout


i†aout


In expectation values only the "OUT" indices arefree. The "IN" indices are always summed over.

Outline







New approach



Beyond N


Cosmology

The end




in〈0|aouti aout



in〈0|aouti aout


i†aout



Instead ofαik,β jk we can use

Ci j = ~−1

in〈0|aouti aout

j |0〉in = (β∗α†)i j

Ni j = ~−1

in〈0|aouti

†aoutj |0〉in = (ββ†)i j

We are free to choose between two representations:α,β N,C

"IN" and "OUT" indices "OUT" indices

Outline







New approach



Beyond N


Cosmology

The end



Alternative approach to study radiation problems:

α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in

Outline







New approach



Beyond N


Cosmology

The end





Clear visualization of particle production:particles are produced when the correlator deviates from its

vacuum value.

Outline







New approach



Beyond N


Cosmology

The end






vacuum value. Technically more accessible and intuitive.

Outline







New approach



Beyond N


Cosmology

The end







Clarifies an apparent tension between particlecreation and energy fluxes in curved space.

Outline







New approach



Beyond N


Cosmology

The end







Clarifies an apparent tension between particlecreation and energy fluxes in curved space.

Allows to detect localized thunderbolts.

Outline


Cosmology

Standard cosmologies

Accelerating Universe

Mechanism for the acceleration

f(R) gravities

Metric and Palatini formalisms

Constraining the lagrangian

PN limit I: Scalar-Tensor

PN limit II: Metric

PN limit III: Palatini


The end


Part II:Cosmology

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end



Two basic assumptions:

Cosmological principle:isotropy and homogeneity.

Large scale dynamics governed by gravity.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end






First assumption⇒ kinematics:

ds2 =−dt2 +a2(t)[

dr2

1−kr2 + r2dΩ2]

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end







ds2 =−dt2 +a2(t)[

dr2

1−kr2 + r2dΩ2]

Second assumption⇒ dynamics ofa(t) .

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end







ds2 =−dt2 +a2(t)[

dr2

1−kr2 + r2dΩ2]


Observations tell us about the geometry of theuniverse:k≈ 0, a(t0) > 0.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end







ds2 =−dt2 +a2(t)[

dr2

1−kr2 + r2dΩ2]


Observations tell us about the geometry of theuniverse:k≈ 0, a(t0) > 0.

a(t0) > 0 was unexpected only 10 years ago.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end



Empty ModelFlat Dark Energy ModelClosed Dark Energy ModelDecelerating ModelDusty Decelerating Model

Binned Data

Closed Matter Only Modelde Sitter ModelEvolving Supernovae

0.0 0.5 1.0 1.5 2.0-1

0

1

Redshift

∆DM

fain

tbr

ight

Ned Wright - 8 Jul 2003

Big dots represent type-Ia supernovae.

The expansion began to accelerate some 5000 million years ago.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end



Dark energy: some stuff with negative pressure.

In GR⇒ aa =−4πG

3 (ρtot +3Ptot)

Matter ρm∼ 1/a3 , Pm = 0

Radiation ρr ∼ 1/a4 , Pr = ρr/3

Cosmological constantρΛ = constant=−PΛ

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end





3 (ρtot +3Ptot)




Modified dynamics:aa =??

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end





3 (ρtot +3Ptot)




Modified dynamics:aa =??

Others

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


f(R) gravities: motivation and examples

" f (R) gravities" stands for:

S= 12k2 d4x

√−g f(R)+Sm(gµν,ψ)

GR is the casef (R) = R GR + cosmological constant isf (R) = R−2Λ

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




S= 12k2 d4x



Starobinsky model(1980): f (R) = R+ R2

M⇒ Leads to early-time inflation

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




S= 12k2 d4x





Carroll et al. model(2004): f (R) = R− µ4

R⇒ Leads to late-time acceleration

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




S= 12k2 d4x





Carroll et al. model(2004): f (R) = R− µ4

R⇒ Leads to late-time acceleration

GR could just be a good approximation atintermediate curvatures.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end



Scalar curvature and Ricci tensor:

R = gµνRµν

Rµν = −∂µΓλλν +∂λΓλ

µν +ΓλµνΓρ

ρλ−ΓλνρΓρ

µλ

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R = gµνRµν


µν +ΓλµνΓρ

ρλ−ΓλνρΓρ

µλ

In Metric formalism:Γαβγ = gαλ

2

(

∂gλγ∂xα +

∂gλβ∂xγ − ∂gβγ

∂xλ

)

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R = gµνRµν


µν +ΓλµνΓρ

ρλ−ΓλνρΓρ

µλ


2

(

∂gλγ∂xα +


∂xλ

)

In Palatiniformalism:Γαβγ is independent ofgµν.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R = gµνRµν


µν +ΓλµνΓρ

ρλ−ΓλνρΓρ

µλ


2

(

∂gλγ∂xα +


∂xλ

)


Only for f (R) = a+bR the two formalism lead tothe same equations of motion.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R = gµνRµν


µν +ΓλµνΓρ

ρλ−ΓλνρΓρ

µλ


2

(

∂gλγ∂xα +


∂xλ

)


Only for f (R) = a+bR the two formalism lead tothe same equations of motion.

Observations should help to determine bothf (R)and the right formalism.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


Search for a suitable f (R)

By trial and error:(very common method)

R− µ4

R R− µ4

R +bR2 R−alogR cRn R− 6asinhR

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R− µ4

R R− µ4


As part of effective actions:(more involved method)

From quantum effects in curved space From low-energy limits of string/M theory

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




R− µ4

R R− µ4


As part of effective actions:(more involved method)

From quantum effects in curved space From low-energy limits of string/M theory

Ask Nature about the admissiblef (R) functions.(Method of this Thesis)

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


Constraining the gravity lagrangian

Take a clean scenario to test gravity.

Cosmology is not a clean laboratory. The solar system is more appropriate.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end





Compute the predictions of the theory in thatregime: post-Newtonian limit .

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end






Confront predictions with experimental data.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end






Confront predictions with experimental data.

Determine the observational constraints onf (R).

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


PN limit I: Scalar-Tensor representation

The original actionS= 12k2 d4x


can be rewritten as follows:

S= 12κ2 d4x

√−g[

φR(g)− ωφ (∂µφ∂µφ)−V(φ)

]

+Sm

where φ≡ d fdR and V(φ) = R f′(R)− f (R)

E.O.M. (3+2ω)φ+2V−φ dVdφ = k2T

Metric ⇒ ω = 0 ⇒ dynamical. Palatini⇒ ω =−3/2 ⇒ non-dynamical.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end






S= 12κ2 d4x

√−g[


]

+Sm




This is more than aBrans-Dicketheory. In B-D V(φ) = 0 (or near an extremum) and

ω is determined by observations(ωobs> 40.000).

Now ω is fixed andV(φ) is to be determined.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end






S= 12κ2 d4x

√−g[


]

+Sm




This is more than aBrans-Dicketheory. In B-D V(φ) = 0 (or near an extremum) and

ω is determined by observations(ωobs> 40.000).

Now ω is fixed andV(φ) is to be determined. We want to constraint the form ofV(φ)⇔ f (R).

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


PN limit II: Metric formalism or ω = 0

The metricgµν ≈ ηµν +hµν:

h(2)00 ≈ 2GM⊙

r + V06φ0

r2 G = k2

8πφ0

[

1+ e−mϕr

3

]

Gexp= const.

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2]

δi j γ = 3−e−mϕr

3+e−mϕr γexp≈ 1

with mϕ2≡ φ0V ′′0 −V ′0

3 = R0

[

f ′(R0)R0 f ′′(R0) −1

]

.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




h(2)00 ≈ 2GM⊙

r + V06φ0

r2 G = k2

8πφ0

[

1+ e−mϕr

3

]

Gexp= const.

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2]



with mϕ2≡ φ0V ′′0 −V ′0

3 = R0

[

f ′(R0)R0 f ′′(R0) −1

]

.

Fundamental constraint:R0

[

f ′(R0)R0 f ′′(R0)

−1]

L2S≫ 1

LS is a relatively short lengthscale.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




h(2)00 ≈ 2GM⊙

r + V06φ0

r2 G = k2

8πφ0

[

1+ e−mϕr

3

]

Gexp= const.

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2]



with mϕ2≡ φ0V ′′0 −V ′0

3 = R0

[

f ′(R0)R0 f ′′(R0) −1

]

.

Fundamental constraint:R0

[

f ′(R0)R0 f ′′(R0)

−1]

L2S≫ 1

LS is a relatively short lengthscale.

Conclusion:

−2Λ≤ f (R)≤ R−2Λ+ l2R2

2

l2≪ L2S is a bound to the current range of the scalar interaction

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end


PN limit III: Palatini formalism or ω =−3/2


h(2)00 ≈ 2GM⊙

r + V06φ0

r2 + log(

φ(ρ)φ0

)

G = κ2

8πφ0

(

1+ MVM⊙

)

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2− log(

φ(ρ)φ0

)]

δi j γ = M⊙−MVM⊙+MV

with M⊙ ≡ d3x′ρ(t,x′)/φ , MV ≡ k−2 d3x′[V0−V(φ)/φ].

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




h(2)00 ≈ 2GM⊙

r + V06φ0

r2 + log(

φ(ρ)φ0

)

G = κ2

8πφ0

(

1+ MVM⊙

)

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2− log(

φ(ρ)φ0

)]



Fundamental constraint:R f′(R)∣

∣

∣

f ′(R)R f′′(R) −1

∣

∣

∣L2(ρ)≫ 1

whereL2(ρ)≡ (k2ρc/φ0)−1

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




h(2)00 ≈ 2GM⊙

r + V06φ0

r2 + log(

φ(ρ)φ0

)

G = κ2

8πφ0

(

1+ MVM⊙

)

h(2)i j ≈

[

2γGM⊙r −

V06φ0

r2− log(

φ(ρ)φ0

)]



Fundamental constraint:R f′(R)∣

∣

∣

f ′(R)R f′′(R) −1

∣

∣

∣L2(ρ)≫ 1

whereL2(ρ)≡ (k2ρc/φ0)−1

Conclusion:

f (R)≤ α+ l2R2

2 + R2

√

1+(l2R)2 + 12l2

log[l2R+√

1+(l2R)2]

f (R)≥ α− l2R2

2 + R2

√

1+(l2R)2 + 12l2

log[l2R+√

1+(l2R)2]

Expanding inl2Rwe get:

α+R− l2R2

2 ≤ f (R)≤ α+R+ l2R2

2

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end



Is the cosmic speed-up driven byf (R) gravities?

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end




Solar system experiments impose severeconstraints on the lagrangianf (R).

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end





Non-linear contributions dominant at lowcurvatures(1/R, logR, . . . ) are ruled out byobservations.

Outline


Cosmology




f(R) gravities




PN limit II: Metric



The end





Non-linear contributions dominant at lowcurvatures(1/R, logR, . . . ) are ruled out byobservations.

Viable models are almost equivalent toR−2Λ intheir late-time cosmological predictions.

Outline


Cosmology

The end


Thanks!¡Gracias!

Correcciones cuánticas de agujeros negros y cosmología · 2010. 12. 29. · Gonzalo J. Olmo...

Documents

Transcript of Correcciones cuánticas de agujeros negros y cosmología · 2010. 12. 29. · Gonzalo J. Olmo...