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Astrophysical consequences of Lorentz violations in gravity

Enrico Barausse

(Institut d'Astrophysique de Paris/CNRS)

in collaboration with

Diego Blas, Ted Jacobson, Thomas Sotiriou, Kent Yag & Nico Yunes

Testing GR with Present and Future Astrophysical Observations,

January 8, 2014

Testing GR @ Ole Miss, Jan 8, 2014

Outline

● Lorentz violation in gravity: why and how

Einstein aether and khronometric theory (= low-energy Horava gravity).

Based on Jacobson, Mattingly, Eling, Foster, Horava ...

● Lorentz violation implies violations of the strong equivalence principle:

The motion of neutron stars (the “sensitivities” and dipolar gravitational-wave emission) and constraints from pulsars. Based on

Yagi, Blas, Yunes, EB arXiv:1307.6219; Yagi, Blas, EB, Yunes arXiv:1311.7144

● Black hole solutions and universal horizons. Based on

EB, Jacobson & Sotiriou PRD 83, 124043 (2011); EB and Sotiriou CQG 30 244010 (2013)

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Lorentz violation in gravity: why?

● LV may give better UV behavior (Horava), QG completions generally lead to LV

● LV allows MOND-like (Bekenstein, Blanchet & Marsat) or DE-like phenomenology

● Strong constraints in matter sector, weaker ones in gravity sector (caveat: constraints expected to percolate from gravity to matter sector)

● Solar system/isolated & binary pulsar experiments historically used to constrains LV in weak field (1 PN) regimes (“preferred-frame parameters”: Nordvedt, Kramer, Wex, Damour, Esposito Farese), but surprises may happen in stronger-field regimes

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Einstein-aether theory

● We want to specify a (local) preferred time “direction” timelike aether field with unit norm

● Most generic action (in 4D) that's quadratic in derivatives is given (up to total derivatives) by

● To satisfy weak equivalence principle, matter fields couple minimally to metric (and not directly to aether)

U μ

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Khronometric gravity

● To specify a global time, U must be hypersurface orthogonal (“khronometric” theory)

● Because U is timelike, T can be used to as time coordinate

● 3 free parameters vs 4 of AE theory (because aether is hypersurface orthogonal)

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Khronometric vs Horava gravity

● L4 and L

6 contain 4th- and 6th-order terms in the spatial derivatives

● Lower bound on M* depends on details of percolation of Lorentz violations from gravity to matter: from Lorentz violations in gravity alone, , but precise bounds depend on percolation

● Theory remains perturbative at all scales if

● Terms crucial in the UV, but unimportant astrophysically, ie error scales as

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Constraints on the coupling constants: the Parametrized Post-Newtonian expansion

● At 1PN, theories = to GR except for preferred-frame parameters α1 and α

2

which are zero in GR but not in LV gravity

● Solar system & pulsar experiments require

● Imposing α1 = α

2 = 0 reduces couplings from 4 to 2 (AE theory) ….

… and from 3 to 2 (khronometric theory)

● c+, c_ and λ, β enter at PN order > 1 (they are “stronger-field”)

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Constraints on the coupling constants: stability

● AE theory has propagating spin-0, spin-1 and spin-2 gravitational modes

● Khronometric theory has spin-0, spin-2 modes● For stability, propagation speeds need to be real (no

tachyonic instability)● Propagation speed must be larger than speed of light to

avoid gravitational Cherenkov radiation

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Stability+Cherenkov constraints

AE theory Khronometrictheory

GR

GR

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How about cosmological constraints?

● Weak for AE theory

● For khronometric theory,

and BBN requires

● No constraints from CMB in khronometric theory yet

∣G N /GC−1∣< 1 /8

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Why are astrophysical effects expected?

● Matter couples minimally to metric, but metric couples non-minimally to aether effective matter-aether coupling in strong-field regimes

● For strongly gravitating body (e.g. neutron star), binding energy depends on velocity relative to the aether (i.e. structure depends on motion relative to preferred frame, as expected from Lorentz violation!)

● Gravitational mass depends on velocity relative to the aether

Violations of strong equivalence principle (aka Nordtvedt effect in Brans Dicke theory, scalar tensor theories, etc)

Smatter=Σi∫mi (γ)d τi

γ=Uμuμ

uaμ ∇μ(mau

ν)=−d mad γ

uμ ∇ νU μ

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Whenever strong equivalence principle (SEP) is violated, dipolar gravitational-wave emission may be produced

● In GR, dipolar emission not present because of SEP + conservation of linear momentum

● If SEP is violated,

● Dipolar mode might be observable directly by interferometers, or indirectly via its backreaction on a binary's evolution

Why are astrophysical effects expected?

h∼ 1Rddt

[m1(γ) x1+ m2(γ) x2]∼μRv( d logm1

d logγ−d logm2

d log γ )

not a wave!

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A PN analysis: the violation of the SEP

Define “active” gravitational mass

and “strong-field” gravitational constant

Modified Newton's law:

body's “sensitivities”

Foster 2007

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A PN analysis: the dissipative dynamics● GWs carry energy away from binaries

● As binary's binding energy decreases, period decreases

S=(s1m2+ s2m1)/MM=m1+ m2

μ=m1m2

M, sA=σ A/(1+ σ A)

are functions of the coupling constants or ;

Dipole

Quadrupole

in GR (Foster 2007, Yagi, Blas, Yunes, EB 2013)

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Why is this interesting?

Binary pulsars are the strongest test of GR to date

PSR B1913+16 (Weisberg & Taylor 2004)

To calculate rate of change of orbital period we need sensitivities

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A strong-field derivation of the sensitivities

● Consider stationary configuration descriving NS and aether moving with small velocity v against it, i.e. asymptotically

● All fields are time-independent, so

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A strong-field derivation of the sensitivities

● Taking difference between configurations with v and v+δv, bulk terms disappear (because we are on shell).

● Surface terms from metric vanish because of boundary conditions, and those from matter vanish because no matter at spatial infinity

● Surface integral can be evaluate asymptotically to get

To get sensitivity, we only need slowly-moving NS solution!

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The PN way to the sensitivities

Solving the field equations in the standard PPN gauge for a system of bodies with velocities v

A relative to the aether (which is asymptotically at rest):

Specialize to one body: we can extract sensitivity from the asymptotic metric of a slowly moving [i.e. O(v)] neutron star solution

e.g. in AE theory

AE theory: Foster 2007; khronometric theory: Yagi, Blas, Yunes & EB 2013

σ=2 c1(2 A−4−α1)(c1−c3)(8+ α1)

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Neutron stars at order O(v)0

● Static spherically symmetric, asymptotically flat solutions, same in AE and khronometric theory

● Aether and fluid are at rest

● Solutions found by imposing - regularity at center - junction condition at surface (i.e. no jumps in potentials whose derivatives enter field eqs)- asymptotic flatness

● Various EOS

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Neutron stars at order O(v)

● Stationary and axisymmetric around velocity direction (z)

● By symmetry under t → - t, z → - z, v → - v, most generic ansatz is

● x' = x + v δx sets A = B = 0 (comoving gauge)

● t' = t + v δt sets Q = 0 (AE) or Q = W = 0 (khrono)

Boundary conditions: - regularity at center- asymptotically,

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Neutron stars at order O(v)

● System of 3 (AE) or 2 (krono) coupled PDEs in r and θ

● Can be solved by expanding in Legendre polynomials

● Eqs for kn(r), s

n(r), w

n(r) decouple: system reduces to an infinite

number of ODEs

● Coefficient B+ and B_ needed to calculate sensitivity appears for n=1:

we can just solve 3 (AE) or 2 (krono) coupled ODEs and read off asymptotic behavior

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Results: the sensitivity of neutron stars

C* = M

* / R

*

Red = weak field prediction (Foster 2007)

We fit sensitivities as functions of couplings

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Constraints from binary pulsars

We choose pulsar-pulsar and pulsar-WD binaries with small eccentricities (PSR J1141-6545, PSR J0348+0432, PSR J0737-3039, PSR J1738+0333), and impose that difference from GR is < data uncertainties

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Constraints from binary pulsars

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Constraints on Lorentz violation in gravity

● Red = weak field prediction for α1= α

2=0 (by requiring exactly same fluxes as GR)

● Combined constraints from WD-pulsar and pulsar-pulsar systems (PSR J1141-6545, PSR J0348+0432, PSR J0737-3039, PSR J1738+0333)● Includes observational uncertainties (masses, spins, eccentricity, EOS)

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Are BHs possible in LV gravity?

● BHs in GR rely on causal structure of spacetime

eg in static spherical spacetime, horizon lies where light cones “tilt inwards” (cf Eddington Finkelstein coordinates).

● In GR, matter (photons) and gravitons have same speed c● In LV gravity, photon, spin-2, spin-1 and spin-0 gravitons

have different propagation speeds different propagation cones multiple horizons

● If higher-order terms included in the action, non linear dispersion relations for gravitons infinite speed in the UV limit do BHs exist at all?

ω2=k 2+α k 4+...

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Spherical BHs in infrared LV gravity

● Field equations are different in AE and Horava gravity, but spherically symmetric, static, asymptotically flat solutions coincide

● F1 plays role of BH mass, A

2 is aether charge!

● Aether has radial component (has to fall into horizon) unlike in a stellar solution Birkhoff theorem is not satisfied

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Spherical BHs in infrared LV gravity

● F1 = 1 by fixing units integrate field eqs inwards from spatial

infinity for various values of A2

● … but A2≠ A

2reg gives naked curvature singularity at spin-0 horizon

solutions probably unphysical because

- Nothing special happens at spin-0 horizon - Gravitational collapse picks regular solution A

2≠ A

2reg (Garfinkle,

Eling & Jacobson 2007)

● Impose regularity at spin-0 horizon by solving field eqs perturbatively, and pick asymptotically flat solution by shooting method (asymptotic flatness does not follow from field eqs, unlike in GR)

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BH exterior structure

● Because of Cherenkov bound, spin-0 horizon is inside matter horizon (“metric horizon”)

● Outside metric horizons, BHs similar to Schwarzschild

AE Horava

Δ(ΩISCO r g)ΩISCO r g

f (r)=1−r gr

+...

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BH exterior structure

Δ(b ph /r g)b ph /r g

=Δ(Ω ph rg)

Ω r g

AE Horava

Measurable with eLISA!

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BH interior structure

● Once picked up (by shooting) solution that is regular at spin-0 horizon and asymptotically flat, we can study region inside spin-0 horizon

● Metric qualitatively similar to Schwarzschild (curvature singularity at r=0), aether oscillates

is aether's Lorentz factor relativeto observer orthogonal to (spacelike) hypersurface r = const

γ r

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BH interior structure

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Implications for causal structure in BH interior

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aether orthogonal to (spacelike) hypersurface r = constθr=0



● Aether is hypersurface orthogonal (in Horava gravity as feature of theory, in AE due to spherical symmetry)

● Aether defines future though its orientation● ( ) correspond to aether orthogonal to (spacelike)

r=const=ru hypersurface and pointing inwards

● Any signal r<ru can only propagate inwards, whatever its

propagation speed, because future=inwards

r=ru is a Universal Horizon

(EB, Jacobson & Sotiriou 2011; Blas and Sibiryakov 2011)

defines “preferred time” foliation

θr=0 γ r=1

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Figure from Cropp, Liberati and Mohd,arXiv:1312.0405

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Is the Universal Horizon robust vs UV corrections?

● Universal Horizon is a significant concept only if higher-order UV terms included in action (because those are terms giving non-linear dispersion relations and infinite progation speeds in UV limit)

● BH solution derived with infrared theory, but universal horizon lies very close to metric horizon, i.e. in low curvature region (for astrophysical BHs)

UV corrections to infrared BH solutions are of relative order

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Further developments about Universal Horizons

● Universal horizon seems to exists also in slowly rotating BHs (at least in Horava gravity, Barausse and Sotiriou 2013)

● Universal horizons satisfy first law of BH thermodynamics (Berglund, Bhattacharyya, Mattingly 2012, 2013; Mohd 2013), and evidence that Hawking radiation is associated to it (Cropp, Liberati and Mohd 2013)

● Confirmation that universal horizons form in gravitational collapse (Saravani, Afshordi, Mann 2013)

● Universal horizon linearly stable, but clues of non-linear instability (Blas and Sibiryakov 2011)

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Conclusions

● Lorentz violations in gravity generically introduces violations of strong equivalence principle and thus dipole emission

● Placing precise constraints with binary pulsars requires exact values of sensitivities, which can be obtained exactly from slowly moving, strong-field neutron star solutions

● Resulting constraints are strong-field and ~ order of magnitude stronger than previous ones

● BH solutions very similar to GR in the “exterior”, but causal structure is very different in the “interior” (universal horizon acts as boudary for perturbations with infinite speed)

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in collaboration with diego blas, ted jacobson, thomas ...berti/testgrtalks/barausse_v2.pdf ·...

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