lack holes in modified gravity models...black holes in modified gravity models arxiv:1312.4625+ work...

BLACK HOLES IN MODIFIED GRAVITY MODELSarXiv:1312.4625 + WORK IN PROGRESS

Andrei Frolov

Jun-Qi Guo (SFU)Daoyan Wang (UBC)

José Tomás Gálvez Ghersi (SFU)Alex Zucca (SFU)

16th Canadian Conference on General Relativityand Relativistic Astrophysics

SFU Segal Building, Vancouver, BC, Canada7 July 2016

Andrei Frolov (SFU) Black Holes in Modified Gravity Models CCGRRA-16 1 / 26

http://arxiv.org/abs/1312.4625

Why Study Modified Gravity?


THE BEST ANSWER I FOUND SO FAR...


MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G.

S =

∫

§

f (R )16πG

+L m

ª

p

−g d 4 x

UV MODIFICATION:

f (R ) =R +R 2

M 2

Starobinsky (1980)

IR MODIFICATION:

f (R ) =R −µ4

R

Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]

FOR F(R) THEORY TO MAKE SENSE WE NEED:

f ′ > 0 – otherwise gravity is a ghost

f ′′ > 0 – otherwise gravity is a tachyon


http://arxiv.org/abs/astro-ph/0303041


MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G.

S =

∫

§

f (R )16πG

+Lm

ª

p

−g d 4 x

UV MODIFICATION:

f (R ) =R +R 2

M 2

Starobinsky (1980)

IR MODIFICATION:

f (R ) =R −µ4

R

Capozziello et. al. [astro-ph/0303041]Carroll et. al. [astro-ph/0306438]

FOR F(R) THEORY TO MAKE SENSE WE NEED:

f ′ > 0 – otherwise gravity is a ghost

f ′′ > 0 – otherwise gravity is a tachyon




EXPECT DEVIATION FROM ΛCDM COSMOLOGYO

yaizu,Lima

&H

u(0807.2462

)



INVOKE CHAMELEON TO SATISFY LOCAL TESTS

r/r

ρ (g

cm

-3)

0.1 1 10 100 1000

10-20

10-10

1

R/κ2 (n=4, | fR0|=0.1)

ρ

r/r100 1000 104

R/κ

2 (g

cm

-3)

10-24

10-23

|fR0|=0.001|fR0|=0.01|fR0|=0.05|fR0|=0.1

ρ

n=4

Hu & Sawicki (0705.1158)



FIELD EQUATIONS IN F(R) GRAVITY

Vary the action with respect to the metric:

S =

∫

§

f (R )16πG

+Lm

ª

p

−g d 4 x

Einstein equations turn into a fourth-order equation:

f ′Rµν− f ′;µν+

f ′−1

2f

gµν = 8πG Tµν

A new scalar degree of freedom φ ≡ f ′−1 appears:

f ′ =1

3(2 f − f ′R ) +

8πG

3T

Can rewrite fourth-order field equation as two second order ones!


A NEW SCALAR DEGREE OF FREEDOM

Equation for φ ≡ f ′−1 is just a scalar wave equation:

φ =V ′(φ)−F

Matter directly drives the field φ by a force term:

F =8πG

3(ρ−3p )

Effective potential can be found by integrating

V ′(φ)≡d V

dφ=

1

3(2 f − f ′R )

In practice, easier to obtain in parametric form:

d V

d R≡

d V

dφ

dφ

d R=

1

3(2 f − f ′R ) f ′′


EXAMPLE I: (SAFE) UV MODIFICATION

0

0.5

1.0

1.5

2.0

2.5

–1 –0.5 0 0.5 1

V/M

φ

2

in vacuum

U (φ) =V (φ) +F (φ∗−φ)

f (R ) =R +R 2

M 2

φ =2R

M 2

V =1

3

R 2

M 2=

M 2

12φ2

massive scalar field!

scalar degree of freedomφ is heavy and hard toexcite


EXAMPLE I: (SAFE) UV MODIFICATION

0

0.5

1.0

1.5

2.0

2.5

–1 –0.5 0 0.5 1

in matter

V/M

φ

2

in vacuum

U (φ) =V (φ) +F (φ∗−φ)

f (R ) =R +R 2

M 2

φ =2R

M 2

V =1

3

R 2

M 2=

M 2

12φ2

massive scalar field!

scalar degree of freedomφ is heavy and hard toexcite


EXAMPLE II: (FAILED) IR MODIFICATION

0

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6 0.8 1

V/µ2

φ

in vacuum

U (φ) =V (φ) +F (φ∗−φ)

f (R ) =R −µ4

R

φ =µ4

R 2

V =2

3

µ4

R−µ8

R 3

=2

3µ2

φ12 −φ

32

field φ is unstable!Dolgov & Kawasaki

(astro-ph/0307285)



EXAMPLE II: (FAILED) IR MODIFICATION

0

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6 0.8 1

in matter

V/µ2

φ

in vacuum

U (φ) =V (φ) +F (φ∗−φ)

f (R ) =R −µ4

R

φ =µ4

R 2

V =2

3

µ4

R−µ8

R 3

=2

3µ2

φ12 −φ

32

field φ is unstable!Dolgov & Kawasaki

(astro-ph/0307285)



CAN WE COME UP WITH SOMETHING BETTER?

Hu and Sawicki [0705.1158]

f (R ) =R −α (R/R0)n

1+β (R/R0)nR0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

10-2 10-1 100 101 102

(R-F

)/R

0

R/R0

n=1n=2n=4

Starobinsky [0706.2041]

f (R ) =R+λ

1

(1+ (R/R0)2)n −1

R0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

10-2 10-1 100 101 102

(R-F

)/R

0

R/R0

n=1n=2n=4

... and many other models ...




DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φABCD

E

F

G


f (R ) =R +λ

1+R 2−1−1

φ =−2λR

(1+R 2)2

A singularity (R =+∞)B stable dS min ( f ′ = 0 )C unstable dS max ( f ′ = 0 )D critical point ( f ′′ = 0 )E flat spacetime ( f ′ = 0 )F critical point ( f ′′ = 0 )G singularity (R =−∞)




V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φBCD

E

F

G

A


f (R ) =R +λ

1+R 2−1−1

φ =−2λR

(1+R 2)2





V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φB

D

E

F

G

AC


f (R ) =R +λ

1+R 2−1−1

φ =−2λR

(1+R 2)2





V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φABCD

E

F

G


f (R ) =R +λ

1+R 2−1−1

φ =−2λR

(1+R 2)2




SINGULARITY IS FINITE DISTANCE AWAY!

V/R0

–0.22

–0.20

–0.18

–0.16

–0.14

–0.12

–0.10

–0.18 –0.14 –0.10 –0.06 –0.02 0

curvature singularity

B

A

φδφ

in vacuum

φ∗

U (φ) =V (φ) +F (φ∗−φ)

in large R limit:

f (R ) =R+Λ+1

Rα

∞∑

n=0

µn

R n

φ ≡ f ′−1'−αµ0

Rα+1

d V

d R'

R f ′′

3=α(α+1)µ0

3 Rα+1

weak power-law singularity:

V (φ)' const−(α+1)µ0

3 |αµ0|γ|φ|γ

γ=α

α+1


SINGULARITY IS FINITE DISTANCE AWAY!

V/R0

–0.22

–0.20

–0.18

–0.16

–0.14

–0.12

–0.10

–0.18 –0.14 –0.10 –0.06 –0.02 0

curvature singularity

B

A

X

φδφ

in vacuum

in matter

φ∗

U (φ) =V (φ) +F (φ∗−φ)

in large R limit:

f (R ) =R+Λ+1

Rα

∞∑

n=0

µn

R n

φ ≡ f ′−1'−αµ0

Rα+1

d V

d R'

R f ′′

3=α(α+1)µ0

3 Rα+1

weak power-law singularity:

V (φ)' const−(α+1)µ0

3 |αµ0|γ|φ|γ

γ=α

α+1


NEED UV COMPLETION! CAN IT SAVE THE DAY?

V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φABCD

E

F

G


f (R ) =R+λ

1+R 2−1−1

+R 2

M 2

φ =−2λR

(1+R 2)2+

2R

M 2




NEED UV COMPLETION! CAN IT SAVE THE DAY?

V/R0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

φBCD

E

F


f (R ) =R+λ

1+R 2−1−1

+R 2

M 2

φ =−2λR

(1+R 2)2+

2R

M 2




SPHERICAL SOLUTIONS IN F(R) GRAVITY

We need to solve a non-linear differential equation:

φ =−8π

3G (ρ−3p ) +V ′(φ)

How do we understand its solutions?

“EQUILIBRIUM” REGIME:

V ′(φ) =8π

3G (ρ−3p )

chameleon mechanism

“BALLISTIC” REGIME:

φ =−8π

3G (ρ−3p )

which one is realized depends on environment!


REGULAR STATIC SOLUTIONS EXIST, BUT...

Babichev & Langlois (0904.1382)

0 2 4 6 8 10-0.10

-0.08

-0.06

-0.04

-0.02

0.00

r

0.0 0.5 1.0 1.5 2.0 2.5-8.´10-9

-6.´10-9

-4.´10-9

-2.´10-9

0

Φ

Φmin



QUASI-STATIC BALL COLLAPSE IN F(R) GRAVITY

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

r/(1+r)

Potential well ofa compact object:

∆φ = −8π

3Gρ + V ′(φ)

︸︷︷︸

is this negligible?

∆Φ = 4πGρ

For light scalaron, excitations of f(R)

degree of freedom φ and Newtonian

potential Φ are related:

φ ≈φ∗−2

3Φ

Effective Newton’s constantchanges (non-linearly)!


DYNAMICAL COLLAPSE IN F(R) GRAVITY

General spherical symmetric spacetime metric (in Einstein frame):

d s 2 = e −2σ(x ,t )(−d t 2+d x 2) + r 2(x , t )dΩ2

Make a black hole by collapsing a pulse of another scalar field ψ!

4 dynamical equations (in flat metric dγ2 =−d t 2+d x 2):

r 2 = 2e −2σ(1− r 2V (φ), σ= ...

φ+2

r∇r ·∇φ = e −2σ

V ′(φ) +κp

6T [ψ]

ψ+2

r∇r ·∇ψ=

√

√2

3κ

∇φ ·∇ψ

+ 2 constraints (in ∂t x and ∂t t + ∂x x directions)...


DYNAMICAL COLLAPSE IN HU-SAWICKI MODEL

0 2 4 6 80

2

4

6

8

10

x

r

Initial value

Final

value

0 2 4 6 8−2

−1

0

1

2

x

σ

Initial value

Final

value

0 2 4 6 80

0.2

0.4

0.6

0.8

1

x

f′

Initial value

Final value

0 2 4 6 8−0.5

0

0.5

1

1.5

2

x

ψ

Initial value

Final value


DYNAMICAL COLLAPSE IN STAROBINSKY MODEL

−1.5 −1 −0.5 0 0.5 1−4

−3

−2

−1

0x 10

−6

χ

U(χ)

χ stops at χ = 0.

0.96 0.97 0.98 0.99 1

−3.58

−3.57

−3.56x 10

−6

−0.5 0 0.5 1−4

−3

−2

−1

0x 10

−7

χ

U(χ)

de Sitter point

χ stops at χ = 0.

0 2 4 6 80

0.2

0.4

0.6

0.8

1

x

f′

Initial value

Final value

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

r = 0

Apparent horizon

x

t


INTERIOR SOLUTION IS KASNER! (+ 1 SCALAR)

10−4

10−3

10−2

10−7

10−6

10−5

10−4

r

τnumerical resultsfitting

10−4

10−3

10−2

102

103

104

r

exp(−

2σ)

numerical resultsfitting

10−4

10−3

10−2

10−2

10−1

r

f′


10−4

10−3

10−2

1.57

1.58

1.59

1.6

1.61

r

ψ

numerical results


INTERIOR SOLUTION GOES TO VACUUM

10−4

10−3

10−2

−1011

−1010

−109

−108

−107

rEF

REF


10−5

10−4

10−3

10−2

10−1

4.5

5

5.5

6x 10

−7

R0(√

D − 1)

rJF

RJF


10−4

10−3

10−2

108

109

1010

1011

1012

rEF

CEF


10−3

10−2

10−1

107

108

109

1010

rJF

CJF



KASNER PARAMETERS VARY SPATIALLY

0 1 2 3 4−0.5

0

0.5

1

C

Kasner

parameters

p1 = p2 = p3 = 1/3

q =√

2/3

C =√

3/2 p1p2 = p3q

0 1 2 3 4 5 60

0.5

1

1.5

x

√

2/3

Cq

d s 2 =−dτ2+3∑

i=1

τ2pi d x 2i , φ = q lnτ

∑

pi = 1,∑

p 2i = 1−q 2


QUICK SUMMARY

Black holes in modified gravity are emptyinside, but not Schwarzschild-de Sitter!


BIG ASTROPHYSICAL QUESTIONS

Where do scalaron hair go after collapse?

How much scalaron radiation is produced?

Back-reaction on matter inbinary black hole collisions?


TEST FIELD IN SCHWARZSCHILD BACKGROUND

Schwarzschild-like static background:

d s 2 =−g (r )d t 2+d r 2

g (r )+ r 2 dΩ2

Redefine field and switch to tortoise coordinate:

φ(r, t ) =u (r, t )

r, d r∗ =

d r

g (r )

Then equation of motion becomes flat wave equation with effective mass:

φ =−∂ 2

t + ∂2

r∗−M

r g (r )u (r, t ), M =

g ′g

r

Still self-coupled, but much easier to understand than full collapse...


ACCRETION OF SCALARON BY A BLACK HOLE

Where do scalaron hair go after collapse?

Worked it out with José Tomás Gálvez Ghersi& Alex Zucca (SFU), stay tuned for the paper...

They fall down!

Complete spectral accretion code publicly available:https://github.com/andrei-v-frolov/accretion

(see movies attached)


https://github.com/andrei-v-frolov/accretion

lack holes in modified gravity models...black holes in modified gravity models arxiv:1312.4625+ work...

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