Cosmological black holes and self-gravitating fields from exact...
Transcript of Cosmological black holes and self-gravitating fields from exact...
![Page 1: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/1.jpg)
D. C. Guariento Moriond Cosmology 2014 – 1
Cosmological black holes and self-gravitating
fields from exact solutions
Daniel C. Guariento
in collaboration with N. Afshordi, A. M. da Silva, M. Fontanini,
E. Papantonopoulos and E. Abdalla
based on [1207.1086], [1212.0155], [1312.3682], [1404.xxxx]
![Page 2: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/2.jpg)
Motivation
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
![Page 3: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/3.jpg)
Motivation
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects
Expanding universe
![Page 4: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/4.jpg)
Motivation
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects
Expanding universe
Coupling between local effects and cosmological evolution
Causal structure
Interaction through Einstein equations
![Page 5: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/5.jpg)
Motivation
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 2
Exact solutions of Einstein equations
Black holes in the presence of self-gravitating matter
Two competing effects:
Gravitationally bound objects
Expanding universe
Coupling between local effects and cosmological evolution
Causal structure
Interaction through Einstein equations
Field source:
Consistency analysis
Evolution and interaction from equations of motion
![Page 6: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/6.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 3
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −
(
1− m2a(t)r
)2
(
1 + m2a(t)r
)2dt2+a2(t)
(
1 +m
2a(t)r
)4(dr2 + r2dΩ2
)
![Page 7: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/7.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 3
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −(1− m
2r
)2
(1 + m
2r
)2dt2 +
(
1 +m
2r
)4 (dr2 + r2dΩ2
)
a(t) constant: Schwarzschild metric
![Page 8: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/8.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 3
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −dt2 + a2(t)(dr2 + r2dΩ2
)
a(t) constant: Schwarzschild metric
m = 0: FLRW metric
![Page 9: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/9.jpg)
The McVittie metric
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 3
Cosmological black holes: McVittie solution [McVittie, MNRAS 93,325 (1933)]
ds2 = −
(
1− m2a(t)r
)2
(
1 + m2a(t)r
)2dt2+a2(t)
(
1 +m
2a(t)r
)4(dr2 + r2dΩ2
)
a(t) constant: Schwarzschild metric
m = 0: FLRW metric
Unique solution that satisfies
Spherical symmetry
Perfect fluid
Shear-free
Asymptotically FLRW
Singularity at the center
![Page 10: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/10.jpg)
Metric features
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 4
Past spacelike singularity at a = m2r
Ricci scalar
R = 12H2 + 6H
(m+ 2ar
m− 2ar
)
where H(t) ≡ a/a
![Page 11: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/11.jpg)
Metric features
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 4
Past spacelike singularity at a = m2r
Ricci scalar
R = 12H2 + 6H
(m+ 2ar
m− 2ar
)
where H(t) ≡ a/a
Fluid has homogeneous density
ρ(t) =3
8πH2
Expansion is homogeneous (Hubble flow) and shear-free
![Page 12: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/12.jpg)
Metric features
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 4
Past spacelike singularity at a = m2r
Ricci scalar
R = 12H2 + 6H
(m+ 2ar
m− 2ar
)
where H(t) ≡ a/a
Fluid has homogeneous density
ρ(t) =3
8πH2
Expansion is homogeneous (Hubble flow) and shear-free
Pressure is inhomogeneous
p(r, t) =1
8π
[
−3H2 + 2H
(m+ 2ar
m− 2ar
)]
![Page 13: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/13.jpg)
Areal radius coordinates
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 5
Causal structure is more easily seen on areal radius coordinates
[N. Kaloper, M. Kleban, D. Martin, 1003.4777]
ds2 = −(
1− 2m
r
)2
dt2 +
dr
√
1− 2mr
−Hrdt
2
+ r2dΩ2
![Page 14: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/14.jpg)
Areal radius coordinates
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 5
Causal structure is more easily seen on areal radius coordinates
[N. Kaloper, M. Kleban, D. Martin, 1003.4777]
ds2 = −(
1− 2m
r
)2
dt2 +
dr
√
1− 2mr
−Hrdt
2
+ r2dΩ2
Apparent horizons: zero expansion of null radial geodesics
(dr
dt
)
±= R (rH ±R) = 0
+ outgoing
− ingoing
![Page 15: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/15.jpg)
Areal radius coordinates
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 5
Causal structure is more easily seen on areal radius coordinates
[N. Kaloper, M. Kleban, D. Martin, 1003.4777]
ds2 = −(
1− 2m
r
)2
dt2 +
dr
√
1− 2mr
−Hrdt
2
+ r2dΩ2
Apparent horizons: zero expansion of null radial geodesics
(dr
dt
)
±= R (rH ±R) = 0
+ outgoing
− ingoing
Only ingoing geodesics have a solution: 1− 2mr
−H(t) r2 = 0
r+ Outer (cosmological) horizon
r− Inner horizon
If H(t) → H0 for t → ∞ apparent horizons become
Schwarszchild-de Sitter event horizons (r− → r∞)
![Page 16: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/16.jpg)
Light cones and apparent horizons
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 6
0
tmin
t
2m r
rout
![Page 17: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/17.jpg)
Light cones and apparent horizons
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 6
0
tmin
t
2m r
rout
rin
![Page 18: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/18.jpg)
Light cones and apparent horizons
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 6
0
tmin
t
2m r
routrinr+r-
![Page 19: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/19.jpg)
Light cones and apparent horizons
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 6
0
tmin
t
2m r
r+r-rinrout
![Page 20: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/20.jpg)
Asymptotic behavior
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 7
Inner horizon is an anti-trapping surface for finite coordinate times
![Page 21: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/21.jpg)
Asymptotic behavior
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 7
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)d
dt(r − r∗) = RrH +O
(R2)> 0
![Page 22: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/22.jpg)
Asymptotic behavior
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 7
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)d
dt(r − r∗) = RrH +O
(R2)> 0
If H0 > 0 the inner horizon r∞ is traversable in finite proper time
Black hole at Shwarzschild-de Sitter limit
![Page 23: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/23.jpg)
Asymptotic behavior
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 7
Inner horizon is an anti-trapping surface for finite coordinate times
Singular surface r∗ = 2m lies in the past of all events (McVittie big
bang)d
dt(r − r∗) = RrH +O
(R2)> 0
If H0 > 0 the inner horizon r∞ is traversable in finite proper time
Black hole at Shwarzschild-de Sitter limit
If H0 = 0, some curvature scalars become singular at r∞
Possibly no BH interpretation at Schwarzschild limit
[K. Lake, M. Abdelqader, 1106.3666]
![Page 24: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/24.jpg)
Penrose diagrams
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 8
Geodesic completion with Schwarzschild-de Sitter
i+
b +i0
+
r = 0
r = 2m
( -)
r+r-
![Page 25: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/25.jpg)
Penrose diagrams
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 8
Geodesic completion with Schwarzschild-de Sitter
<
∞
>
∞
![Page 26: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/26.jpg)
Penrose diagrams
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 8
Geodesic completion with Schwarzschild-de Sitter
i+
b +i0
+
-
+
-
r = 0
r = 0
…
r = 2m
( -)
r+r-
![Page 27: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/27.jpg)
McVittie with compact Φ
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 9
i+
b +i0
+
r = 0
r = 2m
( -)
r+r-
![Page 28: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/28.jpg)
McVittie with compact Φ
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 9
i+
-i0 +i0
+
+
-
r = 0
…
r = 2m
te
b
( -)
r+r-
![Page 29: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/29.jpg)
Convergence to H0
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 10
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)[A. M. da Silva, M. Fontanini, DCG, 1212.0155]
![Page 30: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/30.jpg)
Convergence to H0
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 10
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)[A. M. da Silva, M. Fontanini, DCG, 1212.0155]
Causal structure of McVittie depends on how fast H → H0.
![Page 31: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/31.jpg)
Convergence to H0
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 10
Causal structure depends on cosmological history
Horizon behavior at t → ∞ depends on the set Φ(H−)[A. M. da Silva, M. Fontanini, DCG, 1212.0155]
Causal structure of McVittie depends on how fast H → H0.
Example: cold dark matter (“dust”) and cosmological constant
H(t) = H0 coth
(3
2H0t
)
Convergence depends on the sign of η
η ≡ R(r∞)
3
[R′(r∞)
H0− 1
]
− 1
η depends only on the product λ ≡ mH0, (0 < λ < 13√3
)
![Page 32: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/32.jpg)
Cosmological history
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 11
-1
-0.5
0
0.5
1
1.5
0 0.0755 13 3
= m H0
< 0
(-) compact
> 0
(-) non-compact
= 0
![Page 33: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/33.jpg)
Scalar fields
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 12
McVittie is also a solution of a scalar field with a modified kinetic term
minimally coupled to GR
[E. Abdalla, N. Afshordi, M. Fontanini, DCG, E. Papantonopoulos, 1312.3682]
![Page 34: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/34.jpg)
Scalar fields
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 12
McVittie is also a solution of a scalar field with a modified kinetic term
minimally coupled to GR
[E. Abdalla, N. Afshordi, M. Fontanini, DCG, E. Papantonopoulos, 1312.3682]
Gravity and Cuscuton field
S =
∫
d4x√−g
[1
2R+ µ2
√X − V (φ)
]
with X = −1
2φ;αφ;α
![Page 35: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/35.jpg)
Scalar fields
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 12
McVittie is also a solution of a scalar field with a modified kinetic term
minimally coupled to GR
[E. Abdalla, N. Afshordi, M. Fontanini, DCG, E. Papantonopoulos, 1312.3682]
Gravity and Cuscuton field
S =
∫
d4x√−g
[1
2R+ µ2
√X − V (φ)
]
with X = −1
2φ;αφ;α
Field has constant Kαα on homogeneous surfaces
Kαα =
1
µ2
dV
dφ= 3H(t)
![Page 36: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/36.jpg)
Scalar fields
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 12
McVittie is also a solution of a scalar field with a modified kinetic term
minimally coupled to GR
[E. Abdalla, N. Afshordi, M. Fontanini, DCG, E. Papantonopoulos, 1312.3682]
Gravity and Cuscuton field
S =
∫
d4x√−g
[1
2R+ µ2
√X − V (φ)
]
with X = −1
2φ;αφ;α
Field has constant Kαα on homogeneous surfaces
Kαα =
1
µ2
dV
dφ= 3H(t)
Einstein equations and equations of motion give consistent results
V (φ) = −6πµ4 (φ+ C)2 =3
8πH2
![Page 37: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/37.jpg)
Scalar fields
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 12
McVittie is also a solution of a scalar field with a modified kinetic term
minimally coupled to GR
[E. Abdalla, N. Afshordi, M. Fontanini, DCG, E. Papantonopoulos, 1312.3682]
Gravity and Cuscuton field
S =
∫
d4x√−g
[1
2R+ µ2
√X − V (φ)
]
with X = −1
2φ;αφ;α
Field has constant Kαα on homogeneous surfaces
Kαα =
1
µ2
dV
dφ= 3H(t)
Einstein equations and equations of motion give consistent results
V (φ) = −6πµ4 (φ+ C)2 =3
8πH2
Uniform expansion, shear-free solutions are unique to this type of field
![Page 38: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/38.jpg)
Uniqueness of the cuscuton solution
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 13
K-essence action: S =∫d4x
√−g[12R+K(X,ϕ)
]
![Page 39: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/39.jpg)
Uniqueness of the cuscuton solution
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 13
K-essence action: S =∫d4x
√−g[12R+K(X,ϕ)
]
Einstein equations in McVittie
−2XK,X +K = − 3
8πH2
2XK,XX +K,X = 0
![Page 40: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/40.jpg)
Uniqueness of the cuscuton solution
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 13
K-essence action: S =∫d4x
√−g[12R+K(X,ϕ)
]
Einstein equations in McVittie
−2XK,X +K = − 3
8πH2
2XK,XX +K,X = 0
Unique solution: cuscuton
K(X,ϕ) = A(ϕ) +B(ϕ)√X
A(ϕ) =3
8πH2 , B(ϕ) = constant
Dual to Horava-Lifshitz gravity with anisotropic Weyl symmetry
![Page 41: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/41.jpg)
Uniqueness of the cuscuton solution
Introduction
The McVittie metric
Properties
Penrose diagrams
Dependence with
cosmological history
Scalar fields
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 13
K-essence action: S =∫d4x
√−g[12R+K(X,ϕ)
]
Einstein equations in McVittie
−2XK,X +K = − 3
8πH2
2XK,XX +K,X = 0
Unique solution: cuscuton
K(X,ϕ) = A(ϕ) +B(ϕ)√X
A(ϕ) =3
8πH2 , B(ϕ) = constant
Dual to Horava-Lifshitz gravity with anisotropic Weyl symmetry
McVittie is also a vacuum solution of modified gravity
![Page 42: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/42.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
![Page 43: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/43.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
![Page 44: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/44.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
McVittie class is shear-free
![Page 45: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/45.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
McVittie class is shear-free
Bulk viscosity reabsorbed into pressure
T rr = T θ
θ = T φφ = p− 3ζ
(
H +2m
2ar −m
)
![Page 46: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/46.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
McVittie class is shear-free
Bulk viscosity reabsorbed into pressure
T rr = T θ
θ = T φφ = p− 3ζ
(
H +2m
2ar −m
)
Landau-Eckart hydrodynamical model
Fluid temperature has an extra term
T√−gtt = T∞(t) +
m
4πχmln(√
−gtt)
![Page 47: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/47.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
McVittie class is shear-free
Bulk viscosity reabsorbed into pressure
T rr = T θ
θ = T φφ = p− 3ζ
(
H +2m
2ar −m
)
Landau-Eckart hydrodynamical model
Fluid temperature has an extra term
T√−gtt = T∞(t)
︸ ︷︷ ︸
equilibrium
+m
4πχmln(√
−gtt)
![Page 48: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/48.jpg)
Accretion of general fluids
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 14
Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]
Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)
McVittie class is shear-free
Bulk viscosity reabsorbed into pressure
T rr = T θ
θ = T φφ = p− 3ζ
(
H +2m
2ar −m
)
Landau-Eckart hydrodynamical model
Fluid temperature has an extra term
T√−gtt = T∞(t) +
m
4πχmln(√
−gtt)
︸ ︷︷ ︸
heat flow
![Page 49: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/49.jpg)
Accretion through heat flow
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 15
An imperfect fluid gives an exact solution to Einstein equations
![Page 50: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/50.jpg)
Accretion through heat flow
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 15
An imperfect fluid gives an exact solution to Einstein equations
Two arbitrary functions: a(t) and m(t).
![Page 51: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/51.jpg)
Accretion through heat flow
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 15
An imperfect fluid gives an exact solution to Einstein equations
Two arbitrary functions: a(t) and m(t). Expansion scalar and fluid density are related via (t, t)
component of field equations
ρ(r, t) =3
8π
[
H +2m
2ar −m
]2
=1
8π
Θ2
3
![Page 52: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/52.jpg)
Accretion through heat flow
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 15
An imperfect fluid gives an exact solution to Einstein equations
Two arbitrary functions: a(t) and m(t). Expansion scalar and fluid density are related via (t, t)
component of field equations
ρ(r, t) =3
8π
[
H +2m
2ar −m
]2
=1
8π
Θ2
3
Energy flow through heat transfer; temperature gradient gives m
T =1√−gtt
[
T∞(t) +m
4πχmln(√
−gtt)]
![Page 53: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/53.jpg)
Null geodesics
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 16
^ r
t
^r+static
^r-static
^r+dyn
^r-dyn
![Page 54: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/54.jpg)
Null geodesics
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 16
^ r
t
^r+s
^r-s
^r+d
^r-d
![Page 55: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/55.jpg)
Null geodesics
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 16
^ r
t
^r+s
^r-s
^r-d
^r = 2 m
![Page 56: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/56.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
![Page 57: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/57.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
Causal structure of the generalized McVittie metric
Apparent horizons
Cauchy horizons on the past singularity
![Page 58: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/58.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
Causal structure of the generalized McVittie metric
Apparent horizons
Cauchy horizons on the past singularity
Thermodynamics: entropy; temperature; first and second laws
Reabsorb heat current into rest frame of the fluid: accretion
![Page 59: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/59.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
Causal structure of the generalized McVittie metric
Apparent horizons
Cauchy horizons on the past singularity
Thermodynamics: entropy; temperature; first and second laws
Reabsorb heat current into rest frame of the fluid: accretion
Other shear-free members of the family of solutions:
Kustaanheimo-Qvist class
![Page 60: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/60.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
Causal structure of the generalized McVittie metric
Apparent horizons
Cauchy horizons on the past singularity
Thermodynamics: entropy; temperature; first and second laws
Reabsorb heat current into rest frame of the fluid: accretion
Other shear-free members of the family of solutions:
Kustaanheimo-Qvist class
Heat flow and anisotropic stress in higher orders of L: Stay tuned
[1404.xxxx]
![Page 61: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/61.jpg)
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
Causal structure of the generalized McVittie metric
Apparent horizons
Cauchy horizons on the past singularity
Thermodynamics: entropy; temperature; first and second laws
Reabsorb heat current into rest frame of the fluid: accretion
Other shear-free members of the family of solutions:
Kustaanheimo-Qvist class
Heat flow and anisotropic stress in higher orders of L: Stay tuned
[1404.xxxx]
Thank you!
![Page 62: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/62.jpg)
Horndeski action
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 18
Can we have a field as source for generalized McVittie?
Additional terms in the action must look like heat flow
Most general scalar action: Horndeski
First term added to the k-essence action: kinetic gravity braiding
[C. Deffayet, O. Pujolàs, I. Sawicki, A. Vikman, 1008.0048]
Sϕ =
∫
d4x√−g [K(X,ϕ) +G(X,ϕ)ϕ]
with ϕ = gαβϕ;αβ
Up to total derivatives, the Lagrangian may be rewritten as
L = K +Gϕ
= K −G;αϕ;α
= K + 2XG,ϕ −G,Xϕ;αX;α
![Page 63: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/63.jpg)
Full Horndeski action
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 19
S =
∫
d4x√−g
(
1
2R+
3∑
n=0
L(n)
)
where
L(0) =K(X,ϕ)
L(1) =G(X,ϕ)ϕ
L(2) =G(2)(X,ϕ),X
[
(ϕ)2 − ϕ;αβϕ;αβ]
+RG(2)(X,ϕ)
L(3) =G(3)(X,ϕ),X
[
(ϕ)3 − 3ϕϕ;αβϕ;αβ + 2ϕ;αβϕ
;αρϕ β;ρ
]
− 6Gµνϕ;µνG(3)(X,ϕ)
L(2) and L(3) components of Tµν have non-vanishing anisotropic
stress
![Page 64: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/64.jpg)
Energy-momentum tensor
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 20
Energy-momentum tensor of the KGB term
Tµν = (K −G;αϕ;α) gµν + (K,X +ϕG,X)ϕ;µϕ;ν +2G(;µϕ;ν)
Equivalent fluid four-velocity uµ
uµ =ϕ;µ
√2X
Energy-momentum tensor of the equivalent fluid
ρ ≡ 2X (K,X +ϕG,X)−√2Xϕ;αG
;α
p ≡ K −G;αϕ;α
qµ ≡√2XhµαG
;α
![Page 65: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/65.jpg)
Einstein equations
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 21
tr and tt Einstein equations
− m
mϕ= 8πXG,X
−1
3Θ2 = 8π
K − 2X[
G,ϕ +K,X + 3√2XG,XΘ
]
solution of the tr equation
G(X,ϕ) = g0(ϕ) lnX + g1(ϕ)
with
g0(ϕ) = − 1
8π
m
mϕ
Inserting in the tt equation, taking derivative with respect to r and
using the definition of X
K,X + 2XK,XX + 2[(1 + lnX)g′0 + g′1 + 24πg20
]= 0
![Page 66: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/66.jpg)
Einstein equations
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 22
Solution of tt equation
K = f1(ϕ) + f2(ϕ)√X + 2X
[(2− lnX)g′0 − g′1 − 24πg20
]
Plugging the solution into rr Einstein equation and solving for f1 and
f2
f1 = − 3
8π(H −M)2
f2 =
√2
4πϕ
[
H −M + 3M(
H − M)]
(m
m≡ M and
a
a≡ H
)
![Page 67: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/67.jpg)
Equation of motion
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 23
Consistency check: equation of motion of the scalar
K,ϕ +G,ϕϕ+ [(K,X +G,Xϕ)ϕ;µ +G;µ];µ = 0
Inserting the solutions for G and K
G = g0(ϕ) lnX + g1(ϕ)
K = f1(ϕ) + f2(ϕ)√X + 2X
[(2− lnX)g′
0− g′
1− 24πg2
0
]
where
g0 = − 1
8πϕM
f1 = − 3
8π(H −M)
2
f2 =√2
4πϕ
[
H −M + 3M(
H − M)]
the equation of motion is identically satisfied
Reduces to McVittie/Cuscuton when G = 0 (m = 0)
![Page 68: Cosmological black holes and self-gravitating fields from exact …moriond.in2p3.fr/J14/transparencies/Friday/guariento.pdf · 2014. 3. 28. · D. C. Guariento Moriond Cosmology](https://reader035.fdocuments.in/reader035/viewer/2022071411/61072ffa2eb2ea54bd33600e/html5/thumbnails/68.jpg)
Action coefficients
Introduction
The McVittie metric
Dynamic accretion
Conclusion
Extra slides
D. C. Guariento Moriond Cosmology 2014 – 24
Functions g0, f1 and f2 can be written in terms of ϕ
g0 = − 1
8π
d (lnm)
dϕ⇒ m(ϕ) = e−8π
∫g0dϕ
f2 = − 1√3π
(
8πg0 +f ′1√−f1
)
H =
√
−8πf13
− 8πg0ϕ
Two functions necessary, plus ϕ(t)
Compare with cuscuton case: K(X,ϕ) = A(ϕ) +B(ϕ)√X
A(ϕ) =3
8πH2 , B(ϕ) = constant
One function necessary, plus ϕ(t)