Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA -...
Transcript of Gravity and scalar fields - NTUA · Gravity and scalar fields Thomas P. Sotiriou SISSA -...
Gravity and scalar fields
Thomas P. SotiriouSISSA - International School for Advanced Studies, Trieste
(...soon at the University of Nottingham)
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Thomas P. Sotiriou
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15Mpc
7th Aegean Summer School, Paros, Sep 23rd 2013
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Thomas P. Sotiriou
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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
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
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
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
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
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
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
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...
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
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
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
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
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
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
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
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
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
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
[m]
Thomas P. Sotiriou
10−5 100 105 1010
GR
[m−2]
10−40
10−30
10−20
10−10
Cur
vatu
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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
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
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
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
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
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
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