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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]
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
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
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
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
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
)
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
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
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
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
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
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
)]
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
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
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∞)
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
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
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-
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
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
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
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
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]
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-
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
<
∞
>
∞
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-
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-
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-
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]
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.
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
)
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
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]
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
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)
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
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
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)
]
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
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
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
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]
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)
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
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
)
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)
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)
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
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
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).
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
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)]
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
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
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
Next steps
Introduction
The McVittie metric
Dynamic accretion
Conclusion
D. C. Guariento Moriond Cosmology 2014 – 17
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
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
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
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]
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!
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
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
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
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
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
)
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)
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)