gravity and scalar fields - ntua · gravity and scalar fields thomas p. sotiriou sissa -...

27
Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Upload: others

Post on 26-Jun-2020

4 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Gravity and scalar fields

Thomas P. SotiriouSISSA - International School for Advanced Studies, Trieste

(...soon at the University of Nottingham)

Page 2: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

10−3010−35 10−25 10−20 10−15 10−10 10−5 100 105 1010 1015 1020 1025 [m]

Thomas P. Sotiriou

1AUµm

Qua

ntum

Gra

vity

Gen

eral

Rel

ativ

ity

Gen

eral

Rel

ativ

ity p

lus

Dar

k M

atte

r

Gen

eral

Rel

ativ

ity p

lus

Dar

k M

atte

r an

d D

ark

Ener

gy

Gen

eral

Rel

ativ

ity

plus

un

know

n co

rrec

tions

?

15Mpc

7th Aegean Summer School, Paros, Sep 23rd 2013

Page 3: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

10−10 [m]

Thomas P. Sotiriou

10−5 100 105 1010 1015 1020 1025

GRGR + ?? GR + DM + DE

We should not just wait for the right effective action to come from fundamental theory!

Instead we should:

“Interpret” experiments Combine them with theory (technical naturalness, effective field theory, ...)Construct interesting toy theories (classical alternatives to GR)Use them to “re-interpret” experiments and give feedback to QG model building

There are also large scale puzzles that call for an answer! Can it be gravity?

7th Aegean Summer School, Paros, Sep 23rd 2013

Page 4: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

General Relativity

S =1

16πG

�d4x

√−g (R− 2Λ) + Sm(gµν ,ψ)

The action for general relativity is

has to match the standard model in the local frame and minimal coupling with the metric is required by the equivalence principle.

Gravitational action is uniquely determined thanks to:

Sm

1. Diffeomorphism invariance

2. Requirement to have second order equations3. Requirement to have 4 dimensions4. Requirement to have no other fields

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 5: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Beyond GR

Add extra fields

Allow for more dimensions

To go beyond GR one can

EFT equivalent: adding extra fields

Classically equivalent to adding extra fields

Classically equivalent to adding extra fields

Allow for higher order equations

Give up diffeomorphism invariance

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 6: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Taming the extra fields

Extra fields can lead to problems

Classical instabilities

Quantum mechanical instabilities (negative energy)

These are particularly hard problems when the fields are nonminimally coupled to gravity.

Supposing that these have been tamed, the major problem becomes to

hide the extra fields in regimes where GR works well

make them re-appear when GR seems to fail

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 7: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Brans-Dicke theory

SBD =

�d4x

√−g

�ϕR− ω0

ϕ∇µϕ∇µϕ− V (ϕ) + Lm(gµν ,ψ)

Gµν =ω0

ϕ2

�∇µϕ∇νϕ− 1

2gµν ∇λϕ∇λϕ

+1

ϕ(∇µ∇νϕ− gµν�ϕ)− V (ϕ)

2ϕgµν

(2ω0 + 3)�ϕ = ϕV � − 2V

Solutions with constant are admissible and are GR solutions.ϕ

The action of the theory is

and the corresponding field equations are

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 8: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Brans-Dicke theory

However, they are not the only ones. E.g. for

around static, spherically symmetric stars a nontrivial configuration is necessary and

V = m2 (ϕ− ϕ0)2

So, hiding the scalar requires, either a very large mass (short range) or a very large Brans-Dicke parameter

γ ≡ hii|i=j

h00=

2ω0 + 3− exp[−�

2ϕ0

2ω0+3mr]

2ω0 + 3 + exp[−�

2ϕ0

2ω0+3mr]

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 9: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Scalar-tensor theory

The action for scalar-tensor theory is

Makes sense as an EFT for a scalar coupled nonminimally to gravity

The generalization does not really solve the problem with the range of the force.

If the mass of the scalar is large then no large scale phenomenology

There are interesting theories with large or zero mass though.

SST =

�d4x

√−g

�ϕR− ω(ϕ)

ϕ∇µϕ∇µϕ− V (ϕ) + Lm(gµν ,ψ)

ω0 → ω(ϕ)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 10: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Spontaneous scalarization

Assume you have a theory without a potential which admits solutions such that

Then the theory will admit GR solutions around matter!

However they will not necessarily be the only ones...

The non-GR configuration is preferred for sufficiently large central density

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

ω(φ0) → ∞

T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70, 2220 (1993)

Celebrated demonstration that strong field effects can be very important...

Page 11: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Chameleons

The scalar field satisfies the following equation

(2ω + 3)�ϕ = −ω�(∇ϕ)2 + ϕV � − 2V + T

is sourced by matterϕ

Sun

Strong influence by the boundary conditions

The mass is affected by the presence of matter

Potentials can be designed to have large mass locally

J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 12: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Einstein frame

Under a conformal transformation and a scalar field redefinition

gµν = A2(φ)g̃µν φ = φ[ϕ]

the action of scalar tensor theory becomes

and the equation of motion for is

SST =

�d4x

�−g̃

�R̃− 1

2∇̃µφ∇̃µφ− U(φ) + Lm(gµν ,ψ)

φ

�̃φ− U �(φ) +A3(φ)A�(φ)T = 0

so there is an effective potential

Ueff ≡ U(φ)−A4(φ)T/4

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 13: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Symmetrons

In spherical symmetry

U(φ) = −1

2µ2φ2 +

1

4λφ4 A(φ) = 1 +

1

2M2φ2

d2

dr2φ+

2

r

d

drφ = U � +A3A�ρ

Assume that

Then the effective potential is

Ueff =1

2

�ρA3

M2− µ2

�φ2 +

1

4λφ4

Spontaneous symmetry breaking when

Vanishing VEV when

ρ = 0

ρA3 > M2µ2

K. Hinterbichler and J. Khoury, Phys. Rev. Lett. 104, 231301 (2010)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 14: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Dynamical DE models

Usual shortcomings:

They usually don’t solve the old CC problem. Powerful no-go theorem by Weinberg! (modulo exceptions)

They usually do not solve the new CC problem either! Turning a tiny cosmological constant to a tiny mass of a field, a tiny coupling for some extra terms, a strange potential, etc. is not really a solution

S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)

It is only meaningful to “rename” the scale of the CC if this makes it (technically) natural!

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 15: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Generalized Galileons

One can actually have terms in the action with more than 2 derivatives and still have second order equations:

δ((∂φ)2�φ) = 2[(∂µ∂νφ)(∂µ∂νφ)− (�φ)2]δφ

Inspired by galileons: scalar that enjoy galilean symmetry

Most general action given by Horndeski in 1974!

It includes well-know terms, such as

The new terms are highly non-linear and can lead to Vainshtein screening

(∇µφ)(∇νφ)Gµν φ

�RαβµνRαβµν − 4RµνRµν +R2

A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79, 064036 (2009)G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 16: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Ostrogradksi instability

L = L(q, q̇, q̈) → ∂L

∂q− d

dt

∂L

∂q̇+

d2

dt2∂L

∂q̈= 0

Four pieces of initial data, four canonical coordinates

Q1 = q, P1 =∂L

∂q̇− d

dt

∂L

∂q̈; Q2 = q̇, P2 =

∂L

∂q̈

The corresponding hamiltonian is

H = P1Q2 + P2q̈ + L(Q1, Q2, q̈) q̈ = q̈(Q1, Q2, P2)

Linear in

Unbound from below

P1

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 17: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

f(R) gravity as an exception

S =

�d4x

√−gf(R)

S =

�d4x

√−g [f(φ) + ϕ(R− φ)]

ϕ = f �(φ)

S =

�d4x

√−g [ϕR− V (ϕ)] V (ϕ) = f(φ)− f �(φ)φ

can be written as

Variation with respect to yieldsφ

One can then write

Brans-Dicke theory with and a potentialω0 = 0

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 18: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Symmetries and fields

Consider the equation (in flat spacetime)

�flatφ = 0

and the set of equations

The first equation is invariant only for inertial observers

The second set is observer independent

However,

�φ = 0 Rµνκλ = 0

Rµνκλ = 0 ⇒ gµν = ηµν

All solutions for the new field break the symmetry!

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 19: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Einstein-aether theory

Sæ =1

16πGæ

�d4x

√−g(−R−Mαβµν∇αuµ∇βuν)

Mαβµν = c1gαβgµν + c2g

αµgβν + c3gανgβµ + c4u

αuβgµν

The action of the theory is

where

and the aether is implicitly assumed to satisfy the constraint

uµuµ = 1

Most general theory with a unit timelike vector field which is second order in derivatives

T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 20: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Hypersurface orthogonality

Now assume uα =∂αT�

gµν∂µT∂νT

and choose as the time coordinate

uα = δαT (gTT )−1/2 = NδαT

Replacing in the action and defining one gets

with and the parameter correspondence

GH

Gæ= ξ =

1

1− c13λ =

1 + c21− c13

η =c14

1− c13

ai = ∂i lnN

Sho

æ =1

16πGH

�dTd3xN

√h�KijK

ij−λK2+ξ(3)R+ ηaiai�

T

T. Jacobson, Phys. Rev. D 81, 101502 (2010)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 21: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

[m]

Thomas P. Sotiriou

10−5 100 105 1010

GR

[m−2]

10−40

10−30

10−20

10−10

Cur

vatu

re

lunar ranging

Mercury’s precession

double pulsarCavendish experiment

Gravity Probe B

stellar and intermediate mass

black holesneutron stars

7th Aegean Summer School, Paros, Sep 23rd 2013

Page 22: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Neutron star physics description requires

1. gravity (Einstein’s equations)

2. Magnetohydrodynamics (MHD equations)

3. Microphysics (equation of state)

There is too much uncertainty in 3. to start meddling with 1.

Better strategy: Assume GR and try to get insight into the microphysics, i.e.

Nuclear physics

Superfluidity

...

Neutron Stars as NuclearPhysics labs

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 23: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Black holes as gravity labs

Black holes are fully described by just gravity!

Simple enough yet full GR solutions: perfect!

They contain horizons: causal structure probes

They contain singularities: this is exactly were GR is supposed to be breaking down!

No matter

No microphysics

Additional motivation

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 24: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Are black holes different?

Not necessarily! Consider scalar-tensor theory (or f(R) gravity)

Sst =

�d4x

√−g

�ϕR− ω(ϕ)

ϕ∇µϕ∇µϕ− V (ϕ) + Lm(gµν ,ψ)

Black holes that are

Endpoints of collapse, i.e. stationary

isolated, i.e. asymptotically flat

are GR black holes!

T. P. S. and V. Faraoni, Phys. Rev. Lett. 108, 081103 (2012)

(generalizes the 1972 proof for Brans-Dicke theory)

S.W. Hawking, Comm. Math. Phys. 25, 167 (1972)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 25: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

No difference from GR?

Actually there is...

Perturbations are different!

They even lead to new effects, e.g. floating orbits

Cosmic evolution could also lead to scalar “hair”

More general couplings lead to “hairy” solutions

E. Barausse and T.P.S., Phys. Rev. Lett. 101, 099001 (2008)

V. Cardoso et al., Phys. Rev. Lett. 107, 241101 (2011)

P. Kanti et al., Phys. Rev. D 54, 5049 (1996)

T. Jacobson, Phys. Rev. Lett. 83, 2699 (1999)M. W. Horbatsch and C. P. Burgess, JCAP 010, 1205 (2012)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 26: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Black holes with matter

What if there is matter around a black hole?

�̃φ− U �(φ) +A3(φ)A�(φ)T = 0

In general no constant solutions when unless

Hint that a small amount of matter might perhaps drastically change the solutions

T �= 0

A�(φ0) = 0

φ =

U �(φ0) = 0

Even when these conditions are satisfied

Spontaneous scalarization

Superradiant instability

V. Cardoso, I. P. Carucci, P. Pani and T. P. S., Phys. Rev. Lett. 111, 111101 (2013)

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013

Page 27: Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Conclusions and Perspectives

Scalar-tensor theories are simple, interesting examples of modified gravity

A scalar field can be traded of for more derivatives or lack of symmetry

There are mechanisms to hide the scalar field from local test, yet have it dominate cosmological behaviour

Naturalness is still a problem...

Strong gravity systems might be able to reveal the existence of scalar fields

Even in theories whose black hole solutions are the same as in GR there is new black hole phenomenology!

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013