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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
Quantum Correlations and BH
Cosmology
The end
Gonzalo J. Olmo July 5th, 2005 - p. 2/26
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
Quantum Correlations and BH
Cosmology
The end
Gonzalo J. Olmo July 5th, 2005 - p. 2/26
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 3/26
Part I:Quantum correlations
andblack holes
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 4/26
Gravitational collapse and quantization
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 4/26
Gravitational collapse and quantization
Consider a scalar field
φ(x) = ∑[aiui(x)+a†i u∗i (x)]
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 4/26
Gravitational collapse and quantization
Consider a scalar field
“IN” expansion:
φ(x) = ∑i[aini uin
i (x)+aini
†uini∗(x)]
Vacuum state:aini |0〉in = 0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 4/26
Gravitational collapse and quantization
Consider a scalar field
“IN” expansion:
φ(x) = ∑i[aini uin
i (x)+aini
†uini∗(x)]
Vacuum state:aini |0〉in = 0
“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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 4/26
Gravitational collapse and quantization
Consider a scalar field
“IN” expansion:
φ(x) = ∑i[aini uin
i (x)+aini
†uini∗(x)]
Vacuum state:aini |0〉in = 0
“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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficientsFUTURE = "OUT" region
I−
I+
Incomingmatter
Flat geometry
BH geometry
Singularity
Event H
orizo
n
Outside the BH
= "IN" regionPAST
Future
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficients
FUTURE = "OUT" region
I−
I+
Incomingmatter
Flat geometry
BH geometry
Singularity
Event H
orizo
n
Outside the BH
= "IN" regionPAST
Future
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficients
FUTURE = "OUT" region
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficients
FUTURE = "OUT" region
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 )
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficients
FUTURE = "OUT" region
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)
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 )
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 5/26
Bogolubov coefficients
FUTURE = "OUT" region
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)
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 )
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 )
Number of particles:
in〈0|Nouti |0〉in = ∑k |βik|2
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 6/26
Black Holes evaporate
Number of particles detected atI+ (Hawking 1974):
in〈0|Nouti |0〉in = ∑k |βik|2 = 1
e8πMωi−1
⇒ Planckian spectrum atT = ~
8πκBM
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 6/26
Black Holes evaporate
Number of particles detected atI+ (Hawking 1974):
in〈0|Nouti |0〉in = ∑k |βik|2 = 1
e8πMωi−1
⇒ Planckian spectrum atT = ~
8πκBM
Uncorrelated outgoing radiation→ THERMAL state
(Parker 1975),(Wald 1975)
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 6/26
Black Holes evaporate
Number of particles detected atI+ (Hawking 1974):
in〈0|Nouti |0〉in = ∑k |βik|2 = 1
e8πMωi−1
⇒ Planckian spectrum atT = ~
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 7/26
Evaporation with backreaction
The outgoing radiation modifies the geometry. Thiseffect (backreaction) could restore the correlations.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 7/26
Evaporation with backreaction
The outgoing radiation modifies the geometry. Thiseffect (backreaction) could restore the correlations.
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 7/26
Evaporation with backreaction
The outgoing radiation modifies the geometry. Thiseffect (backreaction) could restore the correlations.
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
Involved computation ofα andβ:
Unknownin〈0|Nouti |0〉in = ?
Unknown density matrix,|0〉in→ ?
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 7/26
Evaporation with backreaction
The outgoing radiation modifies the geometry. Thiseffect (backreaction) could restore the correlations.
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
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 8/26
Bogolubov -Vs- Correlator
The Bogolubov coefficientsα andβ allow toconstruct magnitudes such asin〈0|Nout
i |0〉in .
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 8/26
Bogolubov -Vs- Correlator
The Bogolubov coefficientsα andβ allow toconstruct magnitudes such asin〈0|Nout
i |0〉in .
The two-point correlator allows to "see" thecorrelations among the outgoing particles.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 8/26
Bogolubov -Vs- Correlator
The Bogolubov coefficientsα andβ allow toconstruct magnitudes such asin〈0|Nout
i |0〉in .
The two-point correlator allows to "see" thecorrelations among the outgoing particles.
Can we determinein〈0|Nouti |0〉in directly from
in〈0|φ(x1)φ(x2)|0〉in ?
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 8/26
Bogolubov -Vs- Correlator
The Bogolubov coefficientsα andβ allow toconstruct magnitudes such asin〈0|Nout
i |0〉in .
The two-point correlator allows to "see" thecorrelations among the outgoing particles.
Can we determinein〈0|Nouti |0〉in directly from
in〈0|φ(x1)φ(x2)|0〉in ?
YES!
⇒We can bypass the computation ofα andβ !!!
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 9/26
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 9/26
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
Using the scalar product( f1| f2) =−i dΣµ f1←→∂ µ f ∗2
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 9/26
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
Using the scalar product( f1| f2) =−i dΣµ f1←→∂ µ f ∗2
Product of operators:
aout†i aout
j = (uouti (x1)|(uout∗
j (x2)|: φ(x1)φ(x2) :))
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 9/26
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
Using the scalar product( f1| f2) =−i dΣµ f1←→∂ µ f ∗2
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 10/26
Example: Conformal Invariance
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 10/26
Example: Conformal Invariance
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
Normal-ordered two-point function: φ(x1)φ(x2) : ≡ φ(x1)φ(x2)− out〈0|φ(x1)φ(x2)|0〉out
in〈0|: φ(x1)φ(x2) :|0〉in ≡∣
∣
∣
∂y∂x
∣
∣
∣
∆/d
x1
∣
∣
∣
∂y∂x
∣
∣
∣
∆/d
2
C|y(x1)−y(x2)|2∆ − C
|x1−x2|2∆
It vanishes for Conformal Transf.⇒ in〈0|Nouti |0〉in = 0.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 10/26
Example: Conformal Invariance
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
Normal-ordered two-point function: φ(x1)φ(x2) : ≡ φ(x1)φ(x2)− out〈0|φ(x1)φ(x2)|0〉out
in〈0|: φ(x1)φ(x2) :|0〉in ≡∣
∣
∣
∂y∂x
∣
∣
∣
∆/d
x1
∣
∣
∣
∂y∂x
∣
∣
∣
∆/d
2
C|y(x1)−y(x2)|2∆ − C
|x1−x2|2∆
It vanishes for Conformal Transf.⇒ in〈0|Nouti |0〉in = 0.
βi j should vanish for all Conformal Transf.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
x
R+I L
+
I L−
I R
−
y0
MirrorTrajectory
y(x)
y
I
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
x
R+I L
+
I L−
I R
−
y0
MirrorTrajectory
y(x)
y
I
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Hyperbolic trajectory (WBR):
y(x) =
x x≤ 0x
1+a2xx≥ 0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Hyperbolic trajectory (WBR):
y(x) =
x x≤ 0x
1+a2xx≥ 0
Birrell-Davies, [Cambridge Univ.Press(1982)]
⇒ steady flux of particles alongx > 0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Hyperbolic trajectory (WBR):
y(x) =
x x≤ 0x
1+a2xx≥ 0
Birrell-Davies, [Cambridge Univ.Press(1982)]
⇒ steady flux of particles alongx > 0
No particles forx1×x2 > 0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Hyperbolic trajectory (WBR):
y(x) =
x x≤ 0x
1+a2xx≥ 0
Birrell-Davies, [Cambridge Univ.Press(1982)]
⇒ steady flux of particles alongx > 0
No particles forx1×x2 > 0
Particles localized aboutx≈ 0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 11/26
Moving-mirrors and black holes
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]
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
]
Exponential trajectory (NBR):y(x) = y0− 1
Ce−Cx ⇒ THERMAL
Hyperbolic trajectory (WBR):
y(x) =
x x≤ 0x
1+a2xx≥ 0
Birrell-Davies, [Cambridge Univ.Press(1982)]
⇒ steady flux of particles alongx > 0
No particles forx1×x2 > 0
Particles localized aboutx≈ 0
Thunderbolt localized aty = y0
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 12/26
Beyond the number of particles
The coefficientsαik,β jk never appear alone:
in〈0|aouti aout
j |0〉in =−~(β∗α†)i j in 〈0|aouti
†aoutj |0〉in = +~(ββ†)i j
in〈0|aouti aout
j†|0〉in = +~(αα†) ji in 〈0|aout
i†aout
j†|0〉in =−~(αβT)i j
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 12/26
Beyond the number of particles
The coefficientsαik,β jk never appear alone:
in〈0|aouti aout
j |0〉in =−~(β∗α†)i j in 〈0|aouti
†aoutj |0〉in = +~(ββ†)i j
in〈0|aouti aout
j†|0〉in = +~(αα†) ji in 〈0|aout
i†aout
j†|0〉in =−~(αβT)i j
In expectation values only the "OUT" indices arefree. The "IN" indices are always summed over.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 12/26
Beyond the number of particles
The coefficientsαik,β jk never appear alone:
in〈0|aouti aout
j |0〉in =−~(β∗α†)i j in 〈0|aouti
†aoutj |0〉in = +~(ββ†)i j
in〈0|aouti aout
j†|0〉in = +~(αα†) ji in 〈0|aout
i†aout
j†|0〉in =−~(αβT)i j
In expectation values only the "OUT" indices arefree. The "IN" indices are always summed over.
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
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 12/26
Beyond the number of particles
The coefficientsαik,β jk never appear alone:
in〈0|aouti aout
j |0〉in =−~(β∗α†)i j in 〈0|aouti
†aoutj |0〉in = +~(ββ†)i j
in〈0|aouti aout
j†|0〉in = +~(αα†) ji in 〈0|aout
i†aout
j†|0〉in =−~(αβT)i j
In expectation values only the "OUT" indices arefree. The "IN" indices are always summed over.
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
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 13/26
Summary and Conclusions
Alternative approach to study radiation problems:
α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 13/26
Summary and Conclusions
Alternative approach to study radiation problems:
α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in
Clear visualization of particle production:particles are produced when the correlator deviates from its
vacuum value.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 13/26
Summary and Conclusions
Alternative approach to study radiation problems:
α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in
Clear visualization of particle production:particles are produced when the correlator deviates from its
vacuum value. Technically more accessible and intuitive.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 13/26
Summary and Conclusions
Alternative approach to study radiation problems:
α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in
Clear visualization of particle production:particles are produced when the correlator deviates from its
vacuum value. Technically more accessible and intuitive.
Clarifies an apparent tension between particlecreation and energy fluxes in curved space.
Outline
Quantum Correlations and BH
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
Gonzalo J. Olmo July 5th, 2005 - p. 13/26
Summary and Conclusions
Alternative approach to study radiation problems:
α,β - Vs - N,Cin terms of correlation functions:in〈0|: φ(x1)φ(x2) :|0〉in
Clear visualization of particle production:particles are produced when the correlator deviates from its
vacuum value. Technically more accessible and intuitive.
Clarifies an apparent tension between particlecreation and energy fluxes in curved space.
Allows to detect localized thunderbolts.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 14/26
Part II:Cosmology
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 15/26
Standard cosmologies
Two basic assumptions:
Cosmological principle:isotropy and homogeneity.
Large scale dynamics governed by gravity.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 15/26
Standard cosmologies
Two basic assumptions:
Cosmological principle:isotropy and homogeneity.
Large scale dynamics governed by gravity.
First assumption⇒ kinematics:
ds2 =−dt2 +a2(t)[
dr2
1−kr2 + r2dΩ2]
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 15/26
Standard cosmologies
Two basic assumptions:
Cosmological principle:isotropy and homogeneity.
Large scale dynamics governed by gravity.
First assumption⇒ kinematics:
ds2 =−dt2 +a2(t)[
dr2
1−kr2 + r2dΩ2]
Second assumption⇒ dynamics ofa(t) .
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 15/26
Standard cosmologies
Two basic assumptions:
Cosmological principle:isotropy and homogeneity.
Large scale dynamics governed by gravity.
First assumption⇒ kinematics:
ds2 =−dt2 +a2(t)[
dr2
1−kr2 + r2dΩ2]
Second assumption⇒ dynamics ofa(t) .
Observations tell us about the geometry of theuniverse:k≈ 0, a(t0) > 0.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 15/26
Standard cosmologies
Two basic assumptions:
Cosmological principle:isotropy and homogeneity.
Large scale dynamics governed by gravity.
First assumption⇒ kinematics:
ds2 =−dt2 +a2(t)[
dr2
1−kr2 + r2dΩ2]
Second assumption⇒ dynamics ofa(t) .
Observations tell us about the geometry of theuniverse:k≈ 0, a(t0) > 0.
a(t0) > 0 was unexpected only 10 years ago.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 16/26
Accelerating Universe
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 17/26
Mechanism for the acceleration
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 17/26
Mechanism for the acceleration
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
Modified dynamics:aa =??
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 17/26
Mechanism for the acceleration
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
Modified dynamics:aa =??
Others
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 18/26
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 18/26
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
Starobinsky model(1980): f (R) = R+ R2
M⇒ Leads to early-time inflation
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 18/26
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
Starobinsky model(1980): f (R) = R+ R2
M⇒ Leads to early-time inflation
Carroll et al. model(2004): f (R) = R− µ4
R⇒ Leads to late-time acceleration
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 18/26
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
Starobinsky model(1980): f (R) = R+ R2
M⇒ Leads to early-time inflation
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 19/26
Metric and Palatini formalisms
Scalar curvature and Ricci tensor:
R = gµνRµν
Rµν = −∂µΓλλν +∂λΓλ
µν +ΓλµνΓρ
ρλ−ΓλνρΓρ
µλ
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 19/26
Metric and Palatini formalisms
Scalar curvature and Ricci tensor:
R = gµνRµν
Rµν = −∂µΓλλν +∂λΓλ
µν +ΓλµνΓρ
ρλ−ΓλνρΓρ
µλ
In Metric formalism:Γαβγ = gαλ
2
(
∂gλγ∂xα +
∂gλβ∂xγ − ∂gβγ
∂xλ
)
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 19/26
Metric and Palatini formalisms
Scalar curvature and Ricci tensor:
R = gµνRµν
Rµν = −∂µΓλλν +∂λΓλ
µν +ΓλµνΓρ
ρλ−ΓλνρΓρ
µλ
In Metric formalism:Γαβγ = gαλ
2
(
∂gλγ∂xα +
∂gλβ∂xγ − ∂gβγ
∂xλ
)
In Palatiniformalism:Γαβγ is independent ofgµν.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 19/26
Metric and Palatini formalisms
Scalar curvature and Ricci tensor:
R = gµνRµν
Rµν = −∂µΓλλν +∂λΓλ
µν +ΓλµνΓρ
ρλ−ΓλνρΓρ
µλ
In Metric formalism:Γαβγ = gαλ
2
(
∂gλγ∂xα +
∂gλβ∂xγ − ∂gβγ
∂xλ
)
In Palatiniformalism:Γαβγ is independent ofgµν.
Only for f (R) = a+bR the two formalism lead tothe same equations of motion.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 19/26
Metric and Palatini formalisms
Scalar curvature and Ricci tensor:
R = gµνRµν
Rµν = −∂µΓλλν +∂λΓλ
µν +ΓλµνΓρ
ρλ−ΓλνρΓρ
µλ
In Metric formalism:Γαβγ = gαλ
2
(
∂gλγ∂xα +
∂gλβ∂xγ − ∂gβγ
∂xλ
)
In Palatiniformalism:Γαβγ is independent ofgµν.
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 20/26
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 20/26
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
As part of effective actions:(more involved method)
From quantum effects in curved space From low-energy limits of string/M theory
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 20/26
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
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 21/26
Constraining the gravity lagrangian
Take a clean scenario to test gravity.
Cosmology is not a clean laboratory. The solar system is more appropriate.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 21/26
Constraining the gravity lagrangian
Take a clean scenario to test gravity.
Cosmology is not a clean laboratory. The solar system is more appropriate.
Compute the predictions of the theory in thatregime: post-Newtonian limit .
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 21/26
Constraining the gravity lagrangian
Take a clean scenario to test gravity.
Cosmology is not a clean laboratory. The solar system is more appropriate.
Compute the predictions of the theory in thatregime: post-Newtonian limit .
Confront predictions with experimental data.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 21/26
Constraining the gravity lagrangian
Take a clean scenario to test gravity.
Cosmology is not a clean laboratory. The solar system is more appropriate.
Compute the predictions of the theory in thatregime: post-Newtonian limit .
Confront predictions with experimental data.
Determine the observational constraints onf (R).
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 22/26
PN limit I: Scalar-Tensor representation
The original actionS= 12k2 d4x
√−g f(R)+Sm(gµν,ψ)
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 22/26
PN limit I: Scalar-Tensor representation
The original actionS= 12k2 d4x
√−g f(R)+Sm(gµν,ψ)
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.
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 22/26
PN limit I: Scalar-Tensor representation
The original actionS= 12k2 d4x
√−g f(R)+Sm(gµν,ψ)
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.
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 23/26
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 23/26
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
]
.
Fundamental constraint:R0
[
f ′(R0)R0 f ′′(R0)
−1]
L2S≫ 1
LS is a relatively short lengthscale.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 23/26
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
]
.
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 24/26
PN limit III: Palatini formalism or ω =−3/2
The metricgµν ≈ ηµν +hµν:
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 24/26
PN limit III: Palatini formalism or ω =−3/2
The metricgµν ≈ ηµν +hµν:
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].
Fundamental constraint:R f′(R)∣
∣
∣
f ′(R)R f′′(R) −1
∣
∣
∣L2(ρ)≫ 1
whereL2(ρ)≡ (k2ρc/φ0)−1
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 24/26
PN limit III: Palatini formalism or ω =−3/2
The metricgµν ≈ ηµν +hµν:
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].
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
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 25/26
Summary and Conclusions
Is the cosmic speed-up driven byf (R) gravities?
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 25/26
Summary and Conclusions
Is the cosmic speed-up driven byf (R) gravities?
Solar system experiments impose severeconstraints on the lagrangianf (R).
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 25/26
Summary and Conclusions
Is the cosmic speed-up driven byf (R) gravities?
Solar system experiments impose severeconstraints on the lagrangianf (R).
Non-linear contributions dominant at lowcurvatures(1/R, logR, . . . ) are ruled out byobservations.
Outline
Quantum Correlations and BH
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
Summary and Conclusions
The end
Gonzalo J. Olmo July 5th, 2005 - p. 25/26
Summary and Conclusions
Is the cosmic speed-up driven byf (R) gravities?
Solar system experiments impose severeconstraints on the lagrangianf (R).
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
Quantum Correlations and BH
Cosmology
The end
Gonzalo J. Olmo July 5th, 2005 - p. 26/26
Thanks!¡Gracias!
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