Why Study Modified Gravity? Andrei Frolov (SFU) Unscreening - - PowerPoint PPT Presentation

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Why Study Modified Gravity? Andrei Frolov (SFU) Unscreening - - PowerPoint PPT Presentation

Jun 08, 2023 46 likes •596 views

U NSCREENING S CALARONS arXiv:1704.04114 , arXiv:1312.4625 Andrei Frolov Jun-Qi Guo (SFU) Daoyan Wang (UBC) Jos Toms Glvez Ghersi (SFU) Alex Zucca (SFU) Gravity and Cosmology 2018 Yukawa Institute for Theoretical Physics, Kyoto, Japan

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SLIDE 1

UNSCREENING SCALARONS

arXiv:1704.04114, arXiv:1312.4625

Andrei Frolov

Jun-Qi Guo (SFU) Daoyan Wang (UBC) José Tomás Gálvez Ghersi (SFU) Alex Zucca (SFU)

Gravity and Cosmology 2018 Yukawa Institute for Theoretical Physics, Kyoto, Japan 13 February 2018

Andrei Frolov (SFU) Unscreening Scalarons GC2018 1 / 33

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SLIDE 2

Why Study Modified Gravity?

Andrei Frolov (SFU) Unscreening Scalarons GC2018 2 / 33

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SLIDE 3

THE BEST ANSWER I FOUND SO FAR...

Andrei Frolov (SFU) Unscreening Scalarons GC2018 2 / 33

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SLIDE 4

MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G. S = f (R) 16πG + m

  • −g d 4x

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 3 / 33

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SLIDE 5

MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G. S = f (R) 16πG + m

  • −g d 4x

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 3 / 33

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SLIDE 6

MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G. S = f (R) 16πG + m

  • −g d 4x

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 3 / 33

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SLIDE 7

MAYBE IT’S GRAVITY WE DON’T UNDERSTAND

MODIFY EINSTEIN-HILBERT ACTION TO INCLUDE OTHER STUFF, E.G. S = f (R) 16πG + m

  • −g d 4x

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 3 / 33

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SLIDE 8

EXPECT DEVIATION FROM ΛCDM COSMOLOGY

Oyaizu, Lima & Hu (0807.2462)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 4 / 33

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SLIDE 9

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/r

100 1000 104

R/κ2 (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)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 5 / 33

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SLIDE 10

FIELD EQUATIONS IN F(R) GRAVITY

Vary the action with respect to the metric:

S = f (R) 16πG + m

  • −g d 4x

Einstein equations turn into a fourth-order equation:

f ′Rµν − f ′

;µν +

  • f ′ − 1

2 f

  • gµν = 8πG Tµν

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

f ′ = 1 3 (2f − f ′R) + 8πG 3 T

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 6 / 33

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SLIDE 11

A NEW SCALAR DEGREE OF FREEDOM

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

φ = V ′(φ) −

Matter directly drives the field φ by a force term:

= 8πG 3 (ρ − 3p)

Effective potential can be found by integrating

V ′(φ) ≡ d V d φ = 1 3 (2f − f ′R)

In practice, easier to obtain in parametric form:

d V d R ≡ d V d φ d φ d R = 1 3 (2f − f ′R)f ′′

Andrei Frolov (SFU) Unscreening Scalarons GC2018 7 / 33

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SLIDE 12

EXAMPLE I: (SAFE) UV MODIFICATION

0.5 1.0 1.5 2.0 2.5 –1 –0.5 0.5 1

V/M φ

2

in vacuum U (φ) = V (φ) + (φ∗ − φ) f (R) = R + R 2 M 2 φ = 2R M 2 V = 1 3 R 2 M 2 = M 2 12 φ2

massive scalar field!

scalar degree of freedom

φ is heavy and hard to

excite

Andrei Frolov (SFU) Unscreening Scalarons GC2018 8 / 33

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SLIDE 13

EXAMPLE I: (SAFE) UV MODIFICATION

0.5 1.0 1.5 2.0 2.5 –1 –0.5 0.5 1

in matter

V/M φ

2

in vacuum U (φ) = V (φ) + (φ∗ − φ) f (R) = R + R 2 M 2 φ = 2R M 2 V = 1 3 R 2 M 2 = M 2 12 φ2

massive scalar field!

scalar degree of freedom

φ is heavy and hard to

excite

Andrei Frolov (SFU) Unscreening Scalarons GC2018 8 / 33

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SLIDE 14

EXAMPLE II: (FAILED) IR MODIFICATION

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

V/µ2 φ

i n v a c u u m U (φ) = V (φ) + (φ∗ − φ) f (R) = R − µ4 R φ = µ4 R 2 V = 2 3

  • µ4

R − µ8 R 3

  • =

2 3 µ2 φ

1 2 − φ 3 2

  • field φ is unstable!

Dolgov & Kawasaki

(astro-ph/0307285)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 9 / 33

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SLIDE 15

EXAMPLE II: (FAILED) IR MODIFICATION

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

in matter

V/µ2 φ

i n v a c u u m U (φ) = V (φ) + (φ∗ − φ) f (R) = R − µ4 R φ = µ4 R 2 V = 2 3

  • µ4

R − µ8 R 3

  • =

2 3 µ2 φ

1 2 − φ 3 2

  • field φ is unstable!

Dolgov & Kawasaki

(astro-ph/0307285)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 9 / 33

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SLIDE 16

CAN WE COME UP WITH SOMETHING BETTER?

Hu and Sawicki [0705.1158]

f (R) = R − α(R/R0)n 1 + β (R/R0)n 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)/R0 R/R0 n=1 n=2 n=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)/R0 R/R0 n=1 n=2 n=4

... and many other models ...

Andrei Frolov (SFU) Unscreening Scalarons GC2018 10 / 33

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SLIDE 17

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

A B C D E F G

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 18

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B C D E F G A

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 19

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B C D E F G A

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 20

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B D E F G A C

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 21

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B D E F G A C

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 22

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B D E F G A C

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 23

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B D E F G A C

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 24

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B D E F G A C

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 25

DISAPPEARING COSMOLOGICAL CONSTANT

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

A B C D E F G

Starobinsky [0706.2041]

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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 11 / 33

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SLIDE 26

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

φ

δφ in vacuum φ∗

U (φ) = V (φ) + (φ∗ − φ)

in large R limit:

f (R) = R+Λ+ 1 R α

∞

  • n=0

µn R n φ ≡ f ′ − 1 ≃ − αµ0 R α+1 d V d R ≃ R f ′′ 3 = α(α + 1)µ0 3R α+1

weak power-law singularity:

V (φ) ≃ const−(α + 1)µ0 3|αµ0|γ |φ|γ γ = α α + 1

Andrei Frolov (SFU) Unscreening Scalarons GC2018 12 / 33

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SLIDE 27

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

φ

δφ in vacuum φ∗

U (φ) = V (φ) + (φ∗ − φ)

in large R limit:

f (R) = R+Λ+ 1 R α

∞

  • n=0

µn R n φ ≡ f ′ − 1 ≃ − αµ0 R α+1 d V d R ≃ R f ′′ 3 = α(α + 1)µ0 3R α+1

weak power-law singularity:

V (φ) ≃ const−(α + 1)µ0 3|αµ0|γ |φ|γ γ = α α + 1

Andrei Frolov (SFU) Unscreening Scalarons GC2018 12 / 33

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SLIDE 28

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

φ

δφ in vacuum in matter φ∗

U (φ) = V (φ) + (φ∗ − φ)

in large R limit:

f (R) = R+Λ+ 1 R α

∞

  • n=0

µn R n φ ≡ f ′ − 1 ≃ − αµ0 R α+1 d V d R ≃ R f ′′ 3 = α(α + 1)µ0 3R α+1

weak power-law singularity:

V (φ) ≃ const−(α + 1)µ0 3|αµ0|γ |φ|γ γ = α α + 1

Andrei Frolov (SFU) Unscreening Scalarons GC2018 12 / 33

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SLIDE 29

NEED UV COMPLETION! CAN IT SAVE THE DAY?

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

A B C D E F G

Starobinsky [0706.2041]

f (R) = R+λ

  • 1 + R 2−1 − 1
  • + R 2

M 2 φ = − 2λR (1 + R 2)2 + 2R M 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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 13 / 33

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SLIDE 30

NEED UV COMPLETION! CAN IT SAVE THE DAY?

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B C D E F

Starobinsky [0706.2041]

f (R) = R+λ

  • 1 + R 2−1 − 1
  • + R 2

M 2 φ = − 2λR (1 + R 2)2 + 2R M 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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 13 / 33

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SLIDE 31

NEED UV COMPLETION! CAN IT SAVE THE DAY?

V/R0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 –1.5 –1.0 –0.5 0.5 1.0 1.5

φ

B C D E F

Starobinsky [0706.2041]

f (R) = R+λ

  • 1 + R 2−1 − 1
  • + R 2

M 2 φ = − 2λR (1 + R 2)2 + 2R M 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 = −∞)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 13 / 33

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SLIDE 32

SPHERICAL SOLUTIONS IN F(R) GRAVITY

We need to solve a non-linear differential equation:

φ = −8π 3 G (ρ − 3p) + V ′(φ)

How do we understand its solutions?

“EQUILIBRIUM” REGIME: V ′(φ) = 8π 3 G (ρ − 3p)

chameleon mechanism

“BALLISTIC” REGIME: φ = −8π 3 G (ρ − 3p)

which one is realized depends on environment!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 14 / 33

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SLIDE 33

SPHERICAL SOLUTIONS IN F(R) GRAVITY

We need to solve a non-linear differential equation:

φ = −8π 3 G (ρ − 3p) + V ′(φ)

How do we understand its solutions?

“EQUILIBRIUM” REGIME: V ′(φ) = 8π 3 G (ρ − 3p)

chameleon mechanism

“BALLISTIC” REGIME: φ = −8π 3 G (ρ − 3p)

which one is realized depends on environment!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 14 / 33

slide-34
SLIDE 34

SPHERICAL SOLUTIONS IN F(R) GRAVITY

We need to solve a non-linear differential equation:

φ = −8π 3 G (ρ − 3p) + V ′(φ)

How do we understand its solutions?

“EQUILIBRIUM” REGIME: V ′(φ) = 8π 3 G (ρ − 3p)

chameleon mechanism

“BALLISTIC” REGIME: φ = −8π 3 G (ρ − 3p)

which one is realized depends on environment!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 14 / 33

slide-35
SLIDE 35

SPHERICAL SOLUTIONS IN F(R) GRAVITY

We need to solve a non-linear differential equation:

φ = −8π 3 G (ρ − 3p) + V ′(φ)

How do we understand its solutions?

“EQUILIBRIUM” REGIME: V ′(φ) = 8π 3 G (ρ − 3p)

chameleon mechanism

“BALLISTIC” REGIME: φ = −8π 3 G (ρ − 3p)

which one is realized depends on environment!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 14 / 33

slide-36
SLIDE 36

REGULAR STATIC SOLUTIONS EXIST, BUT...

Babichev & Langlois (0904.1382)

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 8.109 6.109 4.109 2.109

Φ Φmin

Andrei Frolov (SFU) Unscreening Scalarons GC2018 15 / 33

slide-37
SLIDE 37

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

0.01 0.2 0.4 0.6 0.8 1 φ r/(1+r)

Potential well of a compact object:

∆φ = −8π 3 G ρ + V ′(φ)

is this negligible?

∆Φ = 4πG ρ

For light scalaron, excitations of f(R) degree of freedom φ and Newtonian potential Φ are related:

φ ≈ φ∗ − 2 3 Φ

Effective Newton’s constant changes (non-linearly)!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 16 / 33

slide-38
SLIDE 38

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 )d Ω2

Make a black hole by collapsing a pulse of another scalar field ψ! 4 dynamical equations (in flat metric d γ2 = −d t 2 + d x 2):

r 2 = 2e −2σ(1 − r 2V (φ), σ = ... φ + 2 r ∇r · ∇φ = e −2σ

  • V ′(φ) + κ
  • 6

˜ T [ψ]

  • ψ + 2

r ∇r · ∇ψ =

  • 2

3 κ

  • ∇φ · ∇ψ
  • + 2 constraints (in ∂t x and ∂t t + ∂x x directions)...

Andrei Frolov (SFU) Unscreening Scalarons GC2018 17 / 33

slide-39
SLIDE 39

DYNAMICAL COLLAPSE IN HU-SAWICKI MODEL

2 4 6 8 2 4 6 8 10

x r

Initial value Final value

2 4 6 8 −2 −1 1 2

x σ

Initial value Final value

2 4 6 8 0.2 0.4 0.6 0.8 1

x

f ′

Initial value Final value

2 4 6 8 −0.5 0.5 1 1.5 2

x

ψ

Initial value Final value Andrei Frolov (SFU) Unscreening Scalarons GC2018 18 / 33

slide-40
SLIDE 40

DYNAMICAL COLLAPSE IN STAROBINSKY MODEL

−1.5 −1 −0.5 0.5 1 −4 −3 −2 −1 0 x 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 0.5 1 −4 −3 −2 −1 0 x 10

−7

χ U(χ) de Sitter point χ stops at χ = 0.

2 4 6 8 0.2 0.4 0.6 0.8 1

x

f ′

Initial value Final value

1 2 3 4 5 6 7 1 2 3 4 5 6 7

r = 0 Apparent horizon

x t

Andrei Frolov (SFU) Unscreening Scalarons GC2018 19 / 33

slide-41
SLIDE 41

INTERIOR SOLUTION IS KASNER! (+ 1 SCALAR)

10

−4

10

−3

10

−2

10

−7

10

−6

10

−5

10

−4

r τ

numerical results fitting

10

−4

10

−3

10

−2

10

2

10

3

10

4

r

exp(−2σ)

numerical results fitting

10

−4

10

−3

10

−2

10

−2

10

−1

r

f ′

numerical results fitting

10

−4

10

−3

10

−2

1.57 1.58 1.59 1.6 1.61

r

ψ

numerical results Andrei Frolov (SFU)

Unscreening Scalarons GC2018 20 / 33

slide-42
SLIDE 42

INTERIOR SOLUTION GOES TO VACUUM

10

−4

10

−3

10

−2

−10

11

−10

10

−10

9

−10

8

−10

7

rEF REF

numerical results fitting

10

−5

10

−4

10

−3

10

−2

10

−1

4.5 5 5.5 6 x 10

−7

R0( √ D − 1)

r

JF

RJF

numerical results fitting

10

−4

10

−3

10

−2

10

8

10

9

10

10

10

11

10

12

rEF CEF

numerical results fitting

10

−3

10

−2

10

−1

10

7

10

8

10

9

10

10

r

JF

CJF

numerical results fitting Andrei Frolov (SFU)

Unscreening Scalarons GC2018 21 / 33

slide-43
SLIDE 43

KASNER PARAMETERS VARY SPATIALLY

1 2 3 4 −0.5 0.5 1

C

Kasner parameters

p1 = p2 = p3 = 1/3 q =

  • 2/3

C =

  • 3/2

p1 p2 = p3 q

1 2 3 4 5 6 0.5 1 1.5

x

  • 2/3

C q

d s 2 = −d τ2 +

3

  • i=1

τ2pi d x 2

i ,

φ = q lnτ

  • pi = 1,
  • p 2

i = 1 − q 2

Andrei Frolov (SFU) Unscreening Scalarons GC2018 22 / 33

slide-44
SLIDE 44

QUICK SUMMARY Black holes in modified gravity are empty inside, but not Schwarzschild-de Sitter!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 23 / 33

slide-45
SLIDE 45

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), see arXiv:1704.04114...

They fall down!

Complete spectral accretion code publicly available:

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

(see movies attached)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 24 / 33

slide-46
SLIDE 46

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), see arXiv:1704.04114...

They fall down!

Complete spectral accretion code publicly available:

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

(see movies attached)

Andrei Frolov (SFU) Unscreening Scalarons GC2018 24 / 33

slide-47
SLIDE 47

FIELD EVOLUTION AT A GLANCE

Massive field Starobinsky

5 5 x 5 10 15 20 25 30 35 40 t 1.5 1.0 0.5 0.0 0.5 1.0 1.5 5 5 x 5 10 15 20 25 30 35 40 t 0.012 0.010 0.008 0.006 0.004 0.002

Andrei Frolov (SFU) Unscreening Scalarons GC2018 25 / 33

slide-48
SLIDE 48

TEST FIELD IN SCHWARZSCHILD BACKGROUND

Schwarzschild-like static background:

d s 2 = −g (r )d t 2 + d r 2 g (r ) + r 2 d Ω2

Scalar field φ = V ′(φ) − in tortoise coordinates d x = d r /g (r ):

−∂ 2

t φ + 1

r 2 ∂x

  • r 2 ∂xφ
  • = g
  • V ′(φ) −
  • Rewrite in a flux-conservative form with u ≡ ∂t φ and v ≡ r 2∂xφ:

− ∂t u + 1 r 2 ∂x v = g

  • V ′(φ) −
  • −∂t v + r 2 ∂x u

=

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

Andrei Frolov (SFU) Unscreening Scalarons GC2018 26 / 33

slide-49
SLIDE 49

SPECTRAL SOLVER IS ACCURATE AND EFFICIENT

Chebyshev basis φ(x) =

  • n

cnTn(θ) compactifying x/ℓ ≡ cotθ : Tn = cos(nθ), ∂xTn = n ℓ sin(nθ) sin2 θ, ∂ 2

x Tn

= n ℓ2

  • n cos(nθ) + 2cotθ sin(nθ)
  • sin4 θ,

Derivative operatorss become matrix multiplies on discretized field:

  • j

i jTn(x j) = ∂xTn(xi),

  • j

i jTn(x j) =

  • ∂x + 2g

r

  • ∂xTn(xi),

Andrei Frolov (SFU) Unscreening Scalarons GC2018 27 / 33

slide-50
SLIDE 50

BE CAREFUL WITH BOUNDARY CONDITIONS!!!

5 5 x 5 10 15 20 25 30 35 40 t 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Andrei Frolov (SFU) Unscreening Scalarons GC2018 28 / 33

slide-51
SLIDE 51

ABSORBING BOUNDARY CONDITIONS VIA PML

Deform real coordinate into a complex domain:

x → x + i f (x), ∂x → ∂x 1 + i f ′(x) ≡ ∂x 1 + γ(x)

∂t

Propagating wave e ik x−iωt becomes exponentially decaying!

∂t φ = u − w ∂t u = 1 r 2 ∂x v − γu ∂t v = r 2 ∂x(u − w) − γv ∂t w = g

  • V ′(φ) −
  • Reflection from absorption layer identically vanishes!

Andrei Frolov (SFU) Unscreening Scalarons GC2018 29 / 33

slide-52
SLIDE 52

ACCRETION OF SCALARON BY A BLACK HOLE

Starobinsky Hu-Sawicki

−4 −2 2 4 x 5 10 15 20 25 30 t −0.048 −0.042 −0.036 −0.030 −0.024 −0.018 −0.012 −0.006 0.000 φ/φ∗ −4 −2 2 4 x 5 10 15 20 25 30 t −0.064 −0.056 −0.048 −0.040 −0.032 −0.024 −0.016 −0.008 0.000 φ/φ∗

Andrei Frolov (SFU) Unscreening Scalarons GC2018 30 / 33

slide-53
SLIDE 53

SCALARON FLUX FALLS IN...

Starobinsky Hu-Sawicki

−4 −2 2 4 x 5 10 15 20 25 30 t −0.00032 −0.00024 −0.00016 −0.00008 0.00000 0.00008 0.00016 0.00024 0.00032 ˙ M[φ∗]2 −4 −2 2 4 x 5 10 15 20 25 30 t −0.0005 −0.0004 −0.0003 −0.0002 −0.0001 0.0000 0.0001 0.0002 0.0003 0.0004 ˙ M[φ∗]2

Andrei Frolov (SFU) Unscreening Scalarons GC2018 31 / 33

slide-54
SLIDE 54

... AND STATIC SCALARON PROFILE UNSCREENS

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

r/2M

10−4 10−3 10−2 10−1

|φ(r)static|/φ∗

r = 6M

λ = 1.0 λ = 3.0 λ = 5.0 λ = 7.0

Andrei Frolov (SFU) Unscreening Scalarons GC2018 32 / 33

slide-55
SLIDE 55

CURVATURE SINGULARITIES CAN FORM!

1.0 1.5 2.0 2.5 3.0 3.5 4.0

r/2M

1 2 3 4 5 6 7 8 9 10

R/R0

r = 6M

Andrei Frolov (SFU) Unscreening Scalarons GC2018 33 / 33