cosmological black holes and self-gravitating ﬁelds from exact...

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


Exact solutions of Einstein equations

Black holes in the presence of self-gravitating matter

Motivation

Introduction

The McVittie metric

Dynamic accretion

Conclusion




Two competing effects:

Gravitationally bound objects

Expanding universe

Motivation

Introduction

The McVittie metric

Dynamic accretion

Conclusion






Expanding universe

Coupling between local effects and cosmological evolution

Causal structure

Interaction through Einstein equations

Motivation

Introduction

The McVittie metric

Dynamic accretion

Conclusion






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


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


Scalar fields

Dynamic accretion

Conclusion



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


Scalar fields

Dynamic accretion

Conclusion



ds2 = −dt2 + a2(t)(dr2 + r2dΩ2

)


m = 0: FLRW metric

The McVittie metric

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



ds2 = −

(

1− m2a(t)r

)2

(

1 + m2a(t)r

)2dt2+a2(t)

(

1 +m

2a(t)r

)4(dr2 + r2dΩ2

)


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


Scalar fields

Dynamic accretion

Conclusion


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


Scalar fields

Dynamic accretion

Conclusion



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


Scalar fields

Dynamic accretion

Conclusion



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


Scalar fields

Dynamic accretion

Conclusion


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


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion




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


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion




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


Scalar fields

Dynamic accretion

Conclusion


0

tmin

t

2m r

rout


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


0

tmin

t

2m r

rout

rin


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


0

tmin

t

2m r

routrinr+r-


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


0

tmin

t

2m r

r+r-rinrout

Asymptotic behavior

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


Inner horizon is an anti-trapping surface for finite coordinate times

Asymptotic behavior

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



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


Scalar fields

Dynamic accretion

Conclusion




bang)d


(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


Scalar fields

Dynamic accretion

Conclusion




bang)d


(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


Scalar fields

Dynamic accretion

Conclusion


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


Scalar fields

Dynamic accretion

Conclusion



<

∞

>

∞

Penrose diagrams

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



i+

b +i0

+

-

+

-

r = 0

r = 0

…

r = 2m

( -)

r+r-

McVittie with compact Φ

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


i+

b +i0

+

r = 0

r = 2m

( -)

r+r-

McVittie with compact Φ

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


i+

-i0 +i0

+

+

-

r = 0

…

r = 2m

te

b

( -)

r+r-

Convergence to H0

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion


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


Scalar fields

Dynamic accretion

Conclusion




Causal structure of McVittie depends on how fast H → H0.

Convergence to H0

Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion




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


Scalar fields

Dynamic accretion

Conclusion


-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


Scalar fields

Dynamic accretion

Conclusion


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


Scalar fields

Dynamic accretion

Conclusion





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


Scalar fields

Dynamic accretion

Conclusion






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


Scalar fields

Dynamic accretion

Conclusion






S =

∫

d4x√−g

[1

2R+ µ2

√X − V (φ)

]

with X = −1

2φ;αφ;α


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


Scalar fields

Dynamic accretion

Conclusion






S =

∫

d4x√−g

[1

2R+ µ2

√X − V (φ)

]

with X = −1

2φ;αφ;α


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


Scalar fields

Dynamic accretion

Conclusion


K-essence action: S =∫d4x

√−g[12R+K(X,ϕ)

]


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



√−g[12R+K(X,ϕ)

]

Einstein equations in McVittie

−2XK,X +K = − 3

8πH2

2XK,XX +K,X = 0


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



√−g[12R+K(X,ϕ)

]


−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


Introduction

The McVittie metric

Properties

Penrose diagrams

Dependence with


Scalar fields

Dynamic accretion

Conclusion



√−g[12R+K(X,ϕ)

]


−2XK,X +K = − 3

8πH2

2XK,XX +K,X = 0

Unique solution: cuscuton

K(X,ϕ) = A(ϕ) +B(ϕ)√X

A(ϕ) =3


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


Generalized McVittie metric: m → m(t) [V. Faraoni and A. Jacques, 0707.1350]


Introduction

The McVittie metric

Dynamic accretion

Conclusion



Imperfect fluid (heat conductivity χ, bulk viscosity ζ , shear viscosity η)


Introduction

The McVittie metric

Dynamic accretion

Conclusion




McVittie class is shear-free


Introduction

The McVittie metric

Dynamic accretion

Conclusion





Bulk viscosity reabsorbed into pressure

T rr = T θ

θ = T φφ = p− 3ζ

(

H +2m

2ar −m

)


Introduction

The McVittie metric

Dynamic accretion

Conclusion






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)


Introduction

The McVittie metric

Dynamic accretion

Conclusion






T rr = T θ

θ = T φφ = p− 3ζ

(

H +2m

2ar −m

)



T√−gtt = T∞(t)

︸︷︷︸

equilibrium

+m

4πχmln(√

−gtt)


Introduction

The McVittie metric

Dynamic accretion

Conclusion






T rr = T θ

θ = T φφ = p− 3ζ

(

H +2m

2ar −m

)



T√−gtt = T∞(t) +

m

4πχmln(√

−gtt)

︸︷︷︸

heat flow

Accretion through heat flow

Introduction

The McVittie metric

Dynamic accretion

Conclusion


An imperfect fluid gives an exact solution to Einstein equations


Introduction

The McVittie metric

Dynamic accretion

Conclusion



Two arbitrary functions: a(t) and m(t).


Introduction

The McVittie metric

Dynamic accretion

Conclusion



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


Introduction

The McVittie metric

Dynamic accretion

Conclusion



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


^ r

t

^r+static

^r-static

^r+dyn

^r-dyn

Null geodesics

Introduction

The McVittie metric

Dynamic accretion

Conclusion


^ r

t

^r+s

^r-s

^r+d

^r-d

Null geodesics

Introduction

The McVittie metric

Dynamic accretion

Conclusion


^ r

t

^r+s

^r-s

^r-d

^r = 2 m

Next steps

Introduction

The McVittie metric

Dynamic accretion

Conclusion


Next steps

Introduction

The McVittie metric

Dynamic accretion

Conclusion


Causal structure of the generalized McVittie metric

Apparent horizons

Cauchy horizons on the past singularity

Next steps

Introduction

The McVittie metric

Dynamic accretion

Conclusion



Apparent horizons


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



Apparent horizons




Other shear-free members of the family of solutions:

Kustaanheimo-Qvist class

Next steps

Introduction

The McVittie metric

Dynamic accretion

Conclusion



Apparent horizons






Heat flow and anisotropic stress in higher orders of L: Stay tuned

[1404.xxxx]

Next steps

Introduction

The McVittie metric

Dynamic accretion

Conclusion



Apparent horizons






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


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


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


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


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


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


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


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


One function necessary, plus ϕ(t)

cosmological black holes and self-gravitating ﬁelds from exact...

Documents