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Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi, SM, Gleyzes, Langlois, JCAP06(2015) 008

Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

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Page 1: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Modified Gravity inside

Astrophysical Bodies

Shuntaro Mizuno (Waseda)

2nd APCTP-TUS workshop on dark energy

2015. 8. 5

Saito, Yamauchi, SM, Gleyzes, Langlois, JCAP06(2015) 008

Page 2: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Contents

1. Introduction

4. Summary and discussions

2. Breaking of Vainshtein mechanism in beyond Horndeski

3. Impact of the breaking on the stellar structure

Page 3: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

1. Introduction

Page 4: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Cosmic acceleration

・ Expansion of the current Universe is accelerating.

Dark energy or Modified gravity ?

・ Successful modified gravity should have the deviation from GR

only on cosmological (IR) scales and must not spoil the success of it

in the solar-system experiments.

redshift z

ap

pa

ren

t lu

min

osity

accelerating

decelerating

Page 5: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Modified gravity theories

・ Scalar-tensor theory, f(R) gravity, Galileon, ,,,,

・ GR: massless spin 2 particle (graviton)

In general, to modify gravity on large scales, a new d.o.f is introduced.

light scalar d.o.f universally coupled to matter

It also mediates a new long-range force (fifth force) on small scales

・ Massive gravity

a mass term for the graviton

massless multiplet (2 d.o.f) massive multiplet (5 d.o.f)

The helicity 0 mode works in a similar way

Page 6: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Fifth force

・ Prototype Brans-Dicke model as a concrete example

Constraints from solar-system experiment

the scalar field couples with matter !!

with

But for relativistic objects, this coupling does not work.

Page 7: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Screening mechanisms

interaction terms are important to screen the fifth force

around non-relativistic objects

・ Chameleon mechanism

・ Vainshtein mechanism

f(R) theory

Galileon, Massive gravity

The scalar field becomes massive

The scalar field strongly coupled with derivative interaction

・ Poisson equation for the scalar field (non-relativistic objects)

Page 8: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

2. Breaking of Vainshtein mechanism

in beyond Horndeski

Page 9: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Behavior of metric perturbations

・ Metric perturbation in Newtonian gauge

・ General Relativity

・ Scalar-tensor theories

・ Desirable behavior

0 GR non-GR

Page 10: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Horndeski theory Horndeski, `74,

Kobayashi, Yamaguchi, Yokoyama, `11

Deffayet,Gao, Steer, Zahariade, `11

・The most general scalar-tensor theory with second-order field equations

・Having 4 arbitrary functions of and

・Because of the nonlinear derivative interaction terms,

Vainshtein mechanism works around non-relativistic objects

Covariant Galileon

Page 11: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Beyond Horndeski theory Gleyzes, Langlois, Piazza, Vernizzi, `14

・This generalization gives higher order field equations.

・But # of initial conditions remain same for the time hypersurfaces

that coincide with the uniform scalar field hypersurfaces.

No new d.o.f. is introduced !!

Page 12: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Strategy for analyzing Vainshtein mechanism

・ Constructing an effective theory on small scales

Kimura, Kobayashi, Yamamoto, PRD85 (2012), 024023

Narikawa, Kobayashi, Yamauchi, Saito, PRD87 (2013), 124006

Koyama, Niz, Tasinato, PRD88 (2013), 021502(R)

De Fellice, Kase, Tsujikawa, PRD85 (2012), 044059

for Horndeski theory

・ Finding a spherically symmetric solution

Algebraic eq. for ,

Kobayashi, Watanabe, Yamauchi, PRD91 (2015)6, 064013

Page 13: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

New terms from beyond Horndeski theory

・ also generate the same Horndeski terms in ,

which can be absorbed in the redefinition of the coefficients.

・ Only when , these new terms appear.

・ From now on, we concentrate on the third term.

Page 14: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Impact of new term on spherical overdensity

Page 15: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,
Page 16: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,
Page 17: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,
Page 18: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,
Page 19: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

3. Impact of the breaking

on the stellar structure

Saito, Yamauchi, SM, Gleyzes, Langlois, JCAP06(2015) 008

Page 20: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Model for stellar interiors

Static, spherically symmetric, polytropic model

・ Euler equation (hydrostatic equilibrium)

・ Poisson equation

・ Equation of state

with

only the gravitational law is modified !!

polytropic index

1 for neutron stars

3 for main sequence stars like our Sun

Page 21: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Modified Lane-Emden (MLE) equation

・ Intro. dimensionless variable and function

・ Closed equation for the density

This reduces to the standard Lane-Emden equation for

: energy density at the center of a star

・ Boundary condition s for at the center of a star

Page 22: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Numerical solutions for MLE equation

• For , the density blows up and never approaches to zero.

• For , the force is always attractive.

For smaller , gravity becomes stronger in the inner region

but weaker in the outer region

Page 23: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Analytic solution for MLE equation

・ MLE equation with n=1

It is well known that there is an analytic solution for

We find that analytic solutions exist for arbitrary

: hypergeometric function

holds identically !!

Page 24: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Universal bound on the modification (1)

・ Expansion of and near the center

inserting these expansions into MLE eq. , at lowest order in

pressure increases for

・ Modified Poisson equation

gravity becomes repulsive for

Page 25: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Universal bound on the modification (2)

• For a physically reasonable EoS, one expects increases with

The density is increasing near the center when

• We will show that this property continues for higher radii.

Page 26: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Universal bound on the modification (3)

This contradicts the assumption

The density continues to increase for higher radii

No physically sensible profile for any reasonable EoS for

Page 27: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Modifications in Radius and Mass for

More sensitive to for larger polytropic index

Radius Mass

Once the solution is computed, the radius and mass of the object are

determined by first zero of the function

Page 28: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

4. Summary and discussions

Page 29: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Summary

• Behavior of the scalar d.o.f. in Horndeski theory and

beyond Horndeski was studied based on effective theory.

• In beyond Horndeski, a new term partially breaks

Vainshtein mechanism is even inside the Vainshtein radius.

• We show that the stellar radial density profile is modified

by this effect in the nonrelativistic limit.

• We obtain a universal bound on the amplitude of the

modification, independently of the details of the EoS.

Page 30: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Discussions

• Calculations in more realistic situations

• Other interesting effects from beyond Horndeski

• The disformal coupling plays an important role

Realistic EoS, dynamics, rotation, relativistic corrections

Koyama, Sakstein, `15

Structure formation, Local gravity test

Kase’s talk Tsujikawa `15

Page 31: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Appendix

Page 32: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Spherical overdensity in beyond Horndeski

・ Basic equations

with

Page 33: Modified Gravity inside Astrophysical Bodies · Modified Gravity inside Astrophysical Bodies Shuntaro Mizuno (Waseda) 2nd APCTP-TUS workshop on dark energy 2015. 8. 5 Saito, Yamauchi,

Expression of the parameters

・ Concrete model (quartic Lagrangian of beyond Horndeski)

with

When and , is simplified as

cannot be large with positive value