Gravity and scalar fields

Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham) µm 1AU 15Mpc Quantum Gravity General Relativity unknown corrections? General Relativity plus General Relativity plus Dark Matter and Dark Energy General Relativity plus Dark Matter

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7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou GR + ?? GR GR + DM + DE

10 5 0 5 10 15 20 25 10− 10− 10 10 10 10 10 10 [m]

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 Thomas P. Sotiriou General Relativity

The action for general relativity is 1 S = d4x√ g (R 2Λ) + S (gµν ,ψ) 16πG − − m ￿ S m 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: 1. Diffeomorphism invariance 2. Requirement to have second order equations 3. Requirement to have 4 dimensions 4. Requirement to have no other fields

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

To go beyond GR one can Add extra fields Allow for more dimensions EFT equivalent: adding extra fields Allow for higher order equations

Classically equivalent to adding extra fields Give up diffeomorphism invariance

Classically equivalent to adding extra fields

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

The action of the theory is ω S = d4x√ g ϕR 0 µϕ ϕ V (ϕ)+L (g ,ψ) BD − − ϕ ∇ ∇µ − m µν ￿ ￿ ￿ and the corresponding field equations are

ω 1 G = 0 ϕ ϕ g λϕ ϕ µν ϕ2 ∇µ ∇ν − 2 µν ∇ ∇λ ￿ ￿ 1 V (ϕ) + ( ϕ g ϕ) g ϕ ∇µ∇ν − µν ￿ − 2ϕ µν

(2ω + 3) ϕ = ϕV￿ 2V 0 ￿ − Solutions with constant ϕ are admissible and are GR solutions.

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

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

V = m2 (ϕ ϕ )2 − 0 around static, spherically symmetric stars a nontrivial configuration is necessary and

2ϕ0 2ω0 +3 exp[ mr] h 2ω0+3 γ ii|i=j = − − ≡ h00 ￿ 2ϕ0 2ω0 +3+exp[ mr] − 2ω0+3 ￿ So, hiding the scalar requires, either a very large mass (short range) or a very large Brans-Dicke parameter

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

The action for scalar-tensor theory is

ω(ϕ) S = d4x√ g ϕR µϕ ϕ V (ϕ)+L (g , ψ) ST − − ϕ ∇ ∇µ − m µν ￿ ￿ ￿ Makes sense as an EFT for a scalar coupled nonminimally to gravity

The generalization ω 0 ω ( ϕ ) 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.

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

Assume you have a theory without a potential which admits solutions such that ω(φ ) 0 →∞ 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

T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70, 2220 (1993) Celebrated demonstration that strong field effects can be very important...

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

The scalar field satisfies the following equation

2 (2ω + 3) ϕ = ω￿( ϕ) + ϕV ￿ 2V + T ￿ − ∇ −

ϕ is sourced by matter

The mass is affected by the presence of matter Sun Potentials can be designed to have large mass locally

Strong influence by the boundary conditions

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

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

Under a conformal transformation and a scalar field redefinition

2 gµν = A (φ)˜gµν φ = φ[ϕ] the action of scalar tensor theory becomes 1 S = d4x g˜ R˜ ˜ µφ ˜ φ U(φ)+L (g , ψ) ST − − 2∇ ∇µ − m µν ￿ ￿ ￿ and the equation￿ of motion for φ is

3 ˜ φ U ￿(φ)+A (φ) A￿(φ)T =0 ￿ − so there is an effective potential

U U(φ) A4(φ)T/4 eff ≡ −

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

In spherical symmetry 2 d 2 d 3 φ + φ = U ￿ + A A￿ρ dr2 r dr Assume that 1 1 1 U(φ)= µ2φ2 + λφ4 A(φ)=1+ φ2 −2 4 2M 2 Then the effective potential is 1 ρA3 1 U = µ2 φ2 + λφ4 eff 2 M 2 − 4 ￿ ￿ Spontaneous symmetry breaking when ρ =0 Vanishing VEV when ρA3 >M2µ2 K. Hinterbichler and J. Khoury, Phys. Rev. Lett. 104, 231301 (2010)

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

Usual shortcomings: They usually don’t solve the old CC problem. Powerful no-go theorem by Weinberg! (modulo exceptions) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989) 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 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 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∂ν φ)(∂ ∂ φ) ( φ)2]δφ ￿ µ ν − ￿ Inspired by galileons: scalar that enjoy galilean symmetry Most general action given by Horndeski in 1974! It includes well-know terms, such as

µν αβµν µν 2 ( µφ)( ν φ)G φ R R 4R R + R ∇ ∇ αβµν − µν The new terms are highly￿ non-linear and can lead to ￿ Vainshtein screening

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 Ostrogradksi instability

∂L d ∂L d2 ∂L L = L(q, q,˙ q¨) + =0 → ∂q − dt ∂q˙ dt2 ∂q¨ Four pieces of initial data, four canonical coordinates ∂L d ∂L ∂L Q = q, P = ; Q =˙q, P = 1 1 ∂q˙ − dt ∂q¨ 2 2 ∂q¨ The corresponding hamiltonian is

H = P1Q2 + P2q¨ + L(Q1,Q2, q¨)¨q =¨q(Q1,Q2,P2)

Linear in P1 Unbound from below

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 f(R) gravity as an exception

S = d4x√ gf(R) − ￿ can be written as

S = d4x√ g [f(φ)+ϕ(R φ)] − − ￿ Variation with respect to φ yields

ϕ = f ￿(φ) One can then write

4 S = d x√ g [ϕR V (ϕ)] V (ϕ)=f(φ) f ￿(φ)φ − − − ￿ Brans-Dicke theory with ω 0 =0 and a potential

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

Consider the equation (in flat spacetime)

flat ￿ φ =0 and the set of equations

￿φ =0 Rµνκλ =0 The first equation is invariant only for inertial observers The second set is observer independent However, R =0 g = η µνκλ ⇒ µν µν All solutions for the new field break the symmetry!

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

The action of the theory is 1 S = d4x√ g( R M αβµν u u ) æ 16πG − − − ∇α µ∇β ν æ ￿ where

αβµν αβ µν αµ βν αν βµ α β M = c1g g + c2g g + c3g g + c4u u gµν and the aether is implicitly assumed to satisfy the constraint

µ u uµ =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 Hypersurface orthogonality ∂ T Now assume u = α α µν g ∂µT ∂ν T and choose T as the time coordinate￿ TT 1/2 uα = δαT (g )− = NδαT Replacing in the action and defining one gets

ho 1 3 √ ij 2 (3) i Sæ = dT d xN h Kij K λK +ξ R + ηa ai 16πGH − ￿ ￿ ￿ with a i = ∂ i ln N and the parameter correspondence G 1 1+c c H = ξ = λ = 2 η = 14 G 1 c 1 c 1 c æ − 13 − 13 − 13 T. Jacobson, Phys. Rev. D 81, 101502 (2010)

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

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10 10 stellar and − intermediate mass neutron stars black holes

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Gravity Probe B double pulsar Cavendish Curvature 30 experiment 10−

lunar ranging

Mercury’s precession 40 10−

2 [m− ]

7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou Neutron Stars as Nuclear Physics labs 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 ...

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Black holes as gravity labs

Black holes are fully described by just gravity! No matter No microphysics Simple enough yet full GR solutions: perfect!

Additional motivation They contain horizons: causal structure probes They contain singularities: this is exactly were GR is supposed to be breaking down!

Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Are black holes different?

Not necessarily! Consider scalar-tensor theory (or f(R) gravity) ω(ϕ) S = d4x√ g ϕR µϕ ϕ V (ϕ)+L (g ,ψ) st − − ϕ ∇ ∇µ − m µν ￿ ￿ ￿ 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 No difference from GR?

Actually there is...

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

They even lead to new effects, e.g. floating orbits V. Cardoso et al., Phys. Rev. Lett. 107, 241101 (2011)

Cosmic evolution could also lead to scalar “hair” T. Jacobson, Phys. Rev. Lett. 83, 2699 (1999) M. W. Horbatsch and C. P. Burgess, JCAP 010, 1205 (2012) More general couplings lead to “hairy” solutions P. Kanti et al., Phys. Rev. D 54, 5049 (1996)

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

What if there is matter around a black hole?

3 ˜ φ U ￿(φ)+A (φ) A￿(φ)T =0 ￿ − In general no φ = constant solutions when T =0 unless ￿ A￿(φ0)=0 U ￿(φ0)=0 Hint that a small amount of matter might perhaps drastically change the solutions 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 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