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Tests of Alternative Theories of Gravity Gilles Esposito-Farese (GReCO, IAP, France) General relativity passes all tests with flying colors ⇒ WHY considering alternative theories? ∃ theoretical motivations for alternatives to G.R.: Quantization of gravity & unification with other forces [strings] predict the existence of PARTNERS to graviton Useful to contrast their predictions with G.R.: – What theoretical information can we extract from experimental data? – What can be further tested? ∃ some puzzling exprimental issues: Dark energy (72%), dark matter (24%), Pioneer anomaly GENERAL RELATIVITY c3 S= 16πG action Z √ matter, g ] d x − gR + Sstandard [allfields µν 4 dynamics of gravity SPIN 2 FIELD model coupling of matter to gravity MINIMAL COUPLING TO gµν MATTER–GRAVITY COUPLING Smatter [ matter , gmn ] Metric coupling chosen to satisfy the (weak) equivalence principle Impossible to determine from a local experiment if there is acceleration or gravitation (Einstein 1907) acceleration € gravitation MATTER–GRAVITY COUPLING Smatter [ matter , gmn ] Metric coupling chosen to satisfy the (weak) equivalence principle freely falling elevator € (special relativity) Earth MATTER–GRAVITY COUPLING Metric coupling: Smatter [ matter , gµν ] ⇒ Freely falling elevator (= Fermi coordinate system) 1 gµν = 1 1 1 λ µν =0 1 Constancy of the constants 2 Local Lorentz invariance Space & time independence of coupling constants and mass scales of the Standard Model Local non-gravitational experiments are Lorentz invariant Oklo natural fission reactor . |α/α| < 7×10–17 yr–1 << 10–10 yr–1 (cosmo) Isotropy of space verified at the 10–27 level [Shlyakhter 76, Damour & Dyson 96] [Prestage et al. 85, Lamoreaux et al. 86, Chupp et al. 89] 3 Universality of free fall 4 Universality of gravitational redshift Non self-gravitating bodies fall with the same acceleration in an external gravitational field In a static Newtonian potential g00 = –1 + 2 U(x)/c2 + O(1/c4) the time measured by two clocks is τ1/τ2 = 1 + [U(x1)–U(x2)]/c2 + O(1/c4) Flying hydrogen maser clock: 2×10–4 level [Vessot et al. 79–80, Pharao/Aces will give 5×10–6] Laboratory: 4×10–13 level [Baessler et al. 99] : 2×10–13 level [Williams et al. 04] 4 Universality of gravitational redshift (time dilation) acceleration Doppler effect (cf. fire-truck siren) gravitation ⇒ Whatever their composition, lower clocks are slower (⇒ impossible to synchronize even static clocks) Theoretical motivations for non-metric coupling? ∃ dilaton ϕ, partner of graviton in 10 dimensions • SUPERstrings: • Dimensional reduction: ∃ moduli ϕ •… ⇒ ∃ dilatonic coupling of a scalar field to gauge fields ⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x ⇒ Masses m(ϕ) depend also on x ⇒ Violations of universality of free fall: Newton force = spin 2 mA + spin 0 Z SEM = gmn = ( ) gµν Aµ Aν ϕ eϕ 2 √ 4 − d x F µν g 2 4 k0 ϕ conformal invariant ⇒ e cannot be eliminated by redefining g~µν = f(ϕ) gµν δ a = –∇ ln m( x ) depends on composition of mA and mB mB ∂ϕmA ∂ϕmB Theoretical motivations for non-metric coupling? ∃ dilaton ϕ, partner of graviton in 10 dimensions • SUPERstrings: • Dimensional reduction: ∃ moduli ϕ •… ⇒ ∃ dilatonic coupling of a scalar field to gauge fields ⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x ⇒ Masses m(ϕ) depend also on x ⇒ Violations of universality of free fall: Newton force = Newton Einstein Strings spin 2 mA geometry coupling constants rigid soft soft rigid rigid soft + spin 0 Z SEM = gmn = ( ) gµν Aµ Aν ϕ eϕ 2 √ 4 − d x F µν g 2 4 k0 ϕ conformal invariant ⇒ e cannot be eliminated by redefining g~µν = f(ϕ) gµν δ a = –∇ ln m( x ) depends on composition of mA and mB mB ∂ϕmA ∂ϕmB Theoretical motivations for non-metric coupling? ∃ dilaton ϕ, partner of graviton in 10 dimensions • SUPERstrings: • Dimensional reduction: ∃ moduli ϕ •… Z ⇒ ∃ dilatonic coupling of a scalar field to gauge fields SEM = ⇒ Effective coupling constant k eff = k 0 e−ϕ/2 depends on x ⇒ Masses m(ϕ) depend also on x ⇒ Violations of universality of free fall: Newton force = Newton Einstein Strings spin 2 mA geometry coupling constants rigid soft soft rigid rigid soft + spin 0 gmn = ( ) gµν Aµ Aν ϕ eϕ 2 √ 4 − d x F µν g 2 4 k0 ϕ conformal invariant ⇒ e cannot be eliminated by redefining g~µν = f(ϕ) gµν δ a = –∇ ln m( x ) depends on composition of mA and mB mB ∂ϕmA ∂ϕmB Tree-level predictions of strings Experiment ∆a/a ~ 10–5 . α/α ~ H0 ~ 10–10 yr–1 >> >> γPPN–1 ~ O(1) βPPN–1 ~ O(40) >> 5×10–13 7×10–17 yr–1 10–5 How can strings be saved? • Add a mass to dilaton? BUT – no natural mechanism to generate masses for all scalar fields in the theory – difficult cosmological problems [e.g. Polonyi: too much energy stored in cosmological oscillations of ϕ(t)] • String loops! a(ϕ) Transform eϕ into ea(ϕ) Large slope Cosmological evolution Small : slope ϕmin ϕln m(ϕmin) ≈ 0 ϕ If ≈ same ϕmin for all elementary particles [cf. S-duality, i.e., symmetry under gstring = eϕ → 1/gstring = e−ϕ], then expected deviations from general relativity are [ ϕln m(ϕnow)]2 ∼ 10–10 → 10–19 [Damour & Polyakov 94, Damour & Vilenkin 95–96, Damour, Piazza, Veneziano 02] Experimental data in the context of this string model Composition independent tests Coupling strength to a dilaton ∼ [ ϕln m(ϕ)]2 10–3.5 Gravity Probe B (orbiting gyroscope) SORT (heliocentric clocks time delay) LATOR (light deflection by Sun) 10–5 10–5.5 10–6 10–7 10–8 10–9 possible 10–11 most probable 10–14 Composition dependent tests . Oklo reactor: |α/α| < 7×10–17 yr–1 ground clocks geocentric clocks: redshifts at 10–4 level equivalence principle tests: |∆a/a| < 2×10–13 heliocentric clocks (PHARAO, ASTROD, …) MICROSCOPE: 10–15 accuracy in ∆a/a [Damour, Piazza, Veneziano] EXPECTED DILATONIC EFFECTS [Damour & Polyakov] (satellite test of the equivalence principle): 10–18 accuracy in ∆a/a STEP ⇒ Within this string-inspired framework, free-fall experiments are the most precise DYNAMICS OF GRAVITY Now, assume metric coupling of matter to gravity: S = Sgravity + Smatter [ matter , gµν ] ? Phenomenological approach: PPN formalism • Do not assume anything about Sgravity • Write the most general form that gµν can take in presence of matter, at the first post-Newtonian order [Newton × 1/c2] Basic idea [Eddington 1923]: – g00 = 1 – 2 gij = δij " Gm rc2 PPN + 2β Gm rc2 PPN 1 + 2 γ Gm rc2 2 + … + … # Generalization [Will & Nordtvedt 1972]: 10 parameters including βPPN and γPPN Conclusion of experimental tests in the Parametrized Post-Newtonian formalism γPPN Lunar Laser Ranging 2 Mercury perihelion shift General Relativity 1.5 Mars radar ranging & Very Long Baseline Interferometry & Time delay for Cassini spacecraft 1 LLR 1.004 0.5 PPN 0 0.5 1 1.5 2 β 1.002 1 ξ α1,2,3 ζ1,2,3,4 γPPN Cassini VLBI GENERAL RELATIVITY is essentially the only theory consistent with weak-field experiments 0.998 general relativity 0.996 0.996 0.998 1 1.002 1.004 β PPN [Table from C.M. Will gr-qc/0504086] space curvature created by mass nonlinearity in superposition law preferred-location effects 10–4 preferred-frame effects combination of other parameters violation of conservation of total momentum 5¥10–4 Tests of the “strong equivalence principle” and of preferred-frame effects • The different accelerations (due to a third C.M. body or to their absolute velocity with respect to a preferred frame) induce a polarization of the periastron towards a precise direction A aA . wR e eF • $ several binary pulsars with e ª 0 . wR eR fixed direction |eFixed| |aA – aB| eR eF fi statistical argument to constrain PPN parameters [Damour, Schäfer, GEF, Bell, Camilo, Wex, …] B fixed direction e aB aA Tests of the “strong equivalence principle” and of preferred-frame effects B • The different accelerations (due to a third C.M. body or to their absolute velocity with respect to a preferred frame) induce a polarization of the periastron towards a precise direction eF aA eR fixed direction |eFixed| |aA – aB| • Earth-Moon-Sun system [Nordtvedt] G • $ several binary pulsars with e ª 0 . wR e A . wR e aB aA G = G (1 + d + d ) eR eF fi statistical argument to constrain PPN parameters [Damour, Schäfer, GEF, Bell, Camilo, Wex, …] = G (1 + d + d ) dA = grav. ^ dA + (4b–g–3) EA equivalence due to principle dilaton violation coupling –13 |Da/a| < 2¥10 experiment PPN contribution fi /mAc2 ~ 10–10 dilaton coupling < 10–8 PPN constraint |4b–g–3| < 10–3 DYNAMICS OF GRAVITY (continued) Brane models imply (long and) short-distance modifications of Newton’s law GM V = ( 1 + α e − r/λ ) r 5th dimension gravitation our 4-dimensional space-time (maybe other parallel spaces) [C.D. Hoyle et al., Phys. Rev. D70 (2004) 042004, hep-ph/0405262] Constraining the graviton mass? ! ∃ no clean theory of massive graviton (“vDVZ” discontinuity, ghosts, or predictions not yet worked out) ⇒ phenomenological point of view… • Solar system [C. Talmadge et al., Phys. Rev. Lett. 61 (1988) 1159] Gm –r/λg e r ⇒ mg < 4×10–22 eV/c2 ⇔ λg = h/(mgc) > 3×1012 km Yukawa-type Newtonian potential VN = • Binary pulsars [L.S. Finn and P.J. Sutton, Phys. Rev. D 65 (2002) 044022; Class. Quantum Grav. 19 (2002) 1355] ⇒ mg < 10–19 eV/c2 ⇔ λg = h/(mgc) > 1010 km • LISA correlated with optical observations photons E = γ mc2 ⇔ vg2/c2 = 1 – mg2c4/E2 (dispersion relation) ⇒ mg < 6×10–24 eV/c2 ⇔ λg = h/(mgc) > 2×1014 km gravitons [S.L. Larson, W.A. Hiscock, Phys. Rev. D 61 (2000) 104008; C. Cutler, W.A. Hiscock, S.L. Larson, Phys. Rev. D 67 (2003) 024015; A. Cooray, N. Seto, Phys. Rev. D 69 (2004) 103502] • GW interferometers alone vg≈ c [C.M. Will, Phys. Rev. D 57 (1998) 2061; E. Berti, A. Buonanno, C.M. Will, Phys. Rev. D 71 (2005) 084025] LIGO/VIRGO ⇒ mg < 2×10–22 eV/c2 ⇔ λg = h/(mgc) > 6×1012 km LISA (BH-BH) ⇒ mg < 2×10–26 eV/c2 ⇔ λg = h/(mgc) > 6×1016 km vg< c high frequency gravitational waves low frequency gravitational waves DYNAMICS OF GRAVITY (continued) S = Sgravity + Smatter [ matter , gµν ] ? Field-theoretical approach • Now, gµν is assumed to be a combination of fields which propagate in a consistent way: * + a1 BµBν + a2 gµν * BρB*ρ + a3 BµρBν* ρ + …] gµν = A2(ϕ)[gµν spin 0 spin 2 antisymmetric tensor field vector fields (spin 1) * ) lead in general to many flaws: • However, vector (Bµ) or tensor (Bµν) partners to graviton (gµν – discontinuities in the field degrees of freedom – negative-energy modes – causality violations – ill-posedness of the Cauchy problem – no theoretical motivations for being coupled to matter in the “metric” way Smatter [ matter , gµν ] –… ⇒ best-motivated and consistent alternatives to General Relativity: S= 1 16 π G ∫ Tensor−scalar theories −g* { R*− 2 spin 2 ( µϕ)2 } + S spin 0 [matter , g matter µν * A2(ϕ) gµν physical metric [N.B.: more than one scalar and massive scalars are also possible] ] Simplest example: Nordström’s purely scalar theory (1913) 1 S= 8πG Z d4 x η µν ∂µ ϕ∂ν ϕ + Smatter [matter, gµν ≡ ϕ2 ηµν ] spin 0 conformally flat physical metric – Correct Newtonian limit – Satisfies weak and strong equivalence principles (⇒ no Nordtvedt effect on Moon’s orbit) – But post-Newtonian predictions inconsistent with experiment: ds2 = gµν dxµ dxν = 0 ⇒ ηµν dxµ dxν = 0 Light rays follow geodesics of flat spacetime ⇒ no light deflection! Light deflection and the equivalence principle ⇒ Modification of the stars’ apparent position acceleration gravitation Sun Earth Light deflection and the equivalence principle ⇒ Modification of the stars’ apparent position acceleration gravitation Sun Earth Light deflection and the equivalence principle ⇒ Modification of the stars’ apparent position acceleration gravitation Sun Earth In 1914, Einstein predicts half the correct value [Eddington 1919] This is because ∃ also a deformation of space: Nordström’s theory 1913 Einstein’s general relativity 1915 Earth Sun Sun Earth For general tensor−scalar theories 1 16 π G ∫ −g* { R*− 2 ( µϕ)2 } + S spin 2 • Matter−scalar interaction: Effective Newton’s constant: 1 β 2 0 µν matter spin 0 ln A(ϕ) = α0 (ϕ–ϕ0) + matter [matter , g ] ln A(ϕ) 2 (ϕ–ϕ0) + … ϕ ϕ ϕ ϕ * A2(ϕ) gµν physical metric ... S= Geff = G ( 1 + α02 ) graviton scalar α0 α0 ϕ ϕ curvature β0 slope α0 ϕ0 ϕ For general tensor−scalar theories ∫ −g* { R*− 2 ( µϕ)2 } + S spin 2 ln A(ϕ) = α0 (ϕ–ϕ0) + Effective Newton’s constant: 1 β 2 0 ϕ ϕ ϕ SEM = ⇒ no photon−scalar vertex ! ⇒ light deflection ϕ ϕ curvature β0 Geff = G ( 1 + α02 ) scalar α0 conformal invariance ] ln A(ϕ) 2 (ϕ–ϕ0) + … ϕ * A2(ϕ) gµν physical metric graviton • Photon−scalar interaction: µν matter spin 0 • Matter−scalar interaction: matter [matter , g ... 1 16 π G Z √ µρ νσ − g g g Fµν Fρσ = ϕ , ϕ ϕ slope α0 α0 , GM Geff M = < G.R. result rc2 (1 + α02 )rc2 Z ϕ0 √ − g∗ g∗µρ g∗νσ Fµν Fρσ ϕ ... S= ϕ ϕ , … =0 ϕ Other post-Newtonian predictions ln A(ϕ) = α0 (ϕ–ϕ0) + 12 β0 (ϕ–ϕ0)2 + … ϕ ϕ ϕ ϕ ... matter ϕ ϕ matter ln A(ϕ) |α0| perihelion shift curvature β0 0.035 0.030 slope α0 0.025 LLR ϕ ϕ0 0.020 0.015 VLBI 0.010 Geff = G ( 1 + α02 ) graviton PPN γ –1 βPPN– 1 α02 α02 β0 ϕ scalar α0 LLR 0.005 Cassini α0 β0 α0 α0 −6 −4 −2 0 2 4 6 matter β0 ϕ ϕ General Relativity Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory α02 = Horizontal axis (α0 = 0) : perturbatively equivalent to G.R. 1 2 ωBD + 3 Higher-order deviations from G.R. • At any order in 1 , the deviations involve at least two α factors: 0 cn graviton α0 α0 α0 … α0 = small deviations! scalar α0 α0 • But nonperturbative strong-field effects may occur: " deviations = α20 × a0 + a1 < 10−5 Gm Rc2 + a2 LARGE for Gm Rc2 Gm Rc2 2 + … ≈ 0.2 ? # matter-scalar No deviation from coupling function General Relativity ln A(ϕ) in weak-field conditions α0 = 0 ϕ 0 β0 < 0 neutron star ϕc scalar charge |αA| ϕ 0.6 large slope ~ αA ⇒ large deviations from 0.4 General Relativity for neutron stars all sm Energy E≈ ∫[ 0.2 0 1 2 — (∇ϕ) 2 0.5 β0ϕ2/ 2 +ρe ] R( m/ n) Su /R al m tic cri R m/ r) ge sta lar tron u (ne 1 R — 2 “spontaneous scalarization” 2 ϕc2 + m eβ0ϕc / 2 parabola ϕ0 ϕ Gaussian if β0< 0 ϕc (at the center of the star) [T. Damour & G.E-F 1993] 1 1.5 critical mass 2 2.5 maximum mass in GR 3 maximum mass baryonic mass — mA/m Weak-field experiments – g00 = 1 – 2 gij = dij " Gm rc2 PPN + 2b 1 + 2 g Gm rc2 PPN Gm rc2 2 + … + … # Strong-field tests ? solar system –10 0 7¥10 –6 2¥10 neutron star black hole 0.2 0.5 ⇒ binary pulsars companion moving clock giving information about this stronggravity region pulsar radio waves observer self-gravity Gm " Rc2 deviation from flat space # Binary-pulsar tests pulses pulsar = (very stable) clock binary = pulsar t pulses moving clock t P • Time of flight across orbit – orbital period – eccentricity – periastron angular position –… • Redshift size of orbit c P e w G mB rAB c2 + second order Doppler effect – parameter “Keplerian” parameters vA 2 2 c2 (“Einstein time delay”) gTiming • Time evolution of Keplerian parameters – periastron advance (“Roemer time delay”) . 1 w (order c2 ) . 1 – gravitational radiation damping P (order c5 ) “post-Keplerian” observables [PSR B1913+16 • Hulse & Taylor] 3 observables – 2 unknown masses mA, mB = 1 test Plot the three curves [strips] theory observed gTiming(mA, mB) = gTiming . theory . observed w (mA, mB) = w . theory .observed P (mA, mB) = P . . “ g - w - P test ” PSR B1913+16 in general relativity companion mB/m . th . exp P (mA,mB) = P 2.5 2 intersection 1.5 γthT (mA,mB) = γexp T 1 . . ωth(mA,mB) = ωexp . ω = 4.22661°/yr GR γT = 4.294 ms . P = –2.421 × 10–12 ⇒ 0.5 s≤1 0 0.5 1 1.5 2 Discovered by R. Hulse and J. Taylor in 1974 2.5 mA/m pulsar mA = 1.4408 m mB = 1.3873 m PSR B1534+12 in general relativity companion mB/m . w 2.5 . P 2 1.5 1 intersection . w r, s gT s 5 observables - 2 masses = 3 tests . “Galactic” contribution to P r [Damour–Taylor 1991] g Doppler n.v 0.5 0 . P 0.5 1 1.5 Discovered by A. Wolszczan in 1991 2 2.5 mA/m pulsar fi d Doppler v2^ n.a + d PSR dt PSR J1141–6545 in general relativity companion mB/m 2.5 Asymmetrical system neutron star – white dwarf . P . ω Neutron star born after white dwarf ⇒ eccentricity e = 0.17 large and nonrecycled pulsar 2 intersection 1.5 s≤1 . P = –4 × 10–13 Mass function 1 3 (mB sin i ) γ 2 (mA+ mB) 0.5 0 0 0.5 1 1.5 2 2.5 Discovery Kaspi et al. 1999, Timing Bailes et al. 2003 mA/m pulsar = 2 3 2 π (x c) P G PSR J0737–3039 in general relativity 2nd pulsar mB/m pulsar A pulsar B . ω 2.5 . P xA/xB observer 2 s P = 2 h 27 min 14.5350 s 1.5 . ω = 16.90°/yr r 1 γ xB mA xA = mB = 1.07 intersection 0.5 6 observables − 2 masses = 4 tests 0 0 0.5 1 1.5 2 2.5 Timing Burgay et al. 2003, Double pulsar Lyne et al. 2004 mA/m 1st pulsar neutron star ϕ scalar charge |αA| 0.6 Strong-field effects 0.4 0.2 0 eff GAB = G ( 1 + αA αB ) A B graviton A B αA αB scalar depends on internal structure of bodies A & B Similarly for (γPPN– 1) and (βPPN– 1) A B αA αB B A βB αA ⇒ all post-Newtonian effects A αA Quadrupole +O c5 2 Dipole Monopole Quadrupole 1 + 0+ 2 + + +O c c c3 c5 Energy flux = (αA–αB)2 1 c7 spin 2 1 c7 spin 0 0.5 1 1.5 critical mass 2 2.5 maximum mass in GR 3 maximum mass baryonic mass — mA/m mB/m PSR B1913+16 in scalar-tensor theories . ω γ 1.5 1 mB/m 0 . P γ General relativity passes the test 0.5 0.5 1 1.5 mA/m 1 mA/m 1.5 mB/m 2.5 0.5 0 . P 0.5 . ω 1.5 1 A tensor–scalar theory which passes the test (β0 = –4.5, α0 small enough) γ γ 2 1.5 . ω A tensor–scalar theory which does not pass the test (β0 = –6, any α0) 1 0.5 0 . P 0.5 1 1.5 2 2.5 mA/m Solar-system & PSR B1913+16 constraints on scalar-tensor theories of gravity matter matter-scalar coupling function |α0| ln A(ϕ) ϕ PSR B1913+16 α0 β0 < 0 0.040 β0 > 0 0.035 α0 ϕ 0.025 0.020 0.015 −6 VLBI 0.010 Cassini 0.005 −4 −2 binary pulsars impose β0 > −4.5 i.e. βPPN– 1 < 1.1 PPN γ –1 0 2 4 6 β0 matter ϕ ϕ general relativity [T. Damour & G.E-F 1998] Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory Horizontal axis (α0 = 0) : perturbatively equivalent to G.R. α02 = 1 2 ωBD + 3 The four accurately timed binary pulsars in general relativity PSR B1913+16 mB . P . w 2.5 . w 2.5 2 s£1 0 0 0.5 1 2 1.5 2 2.5 mA 0 0 0.5 1 1.5 2 2.5 . w . P intersection s£1 mA xA/xB 2 1.5 2.5 PSR J0737-3039 mB . P . w g 0.5 PSR J1141-6545 mB 2.5 r 1 g s intersection 1.5 1.5 0.5 . P 2 intersection 1 PSR B1534+12 mB s 1.5 r 1 1 g g 0.5 0 intersection 0.5 0 0.5 1 1.5 2 2.5 mA 0 0 0.5 1 1.5 2 2.5 mA Solar-system & best binary-pulsar constraints on scalar-tensor theories of gravity matter-scalar coupling function matter ln A(ϕ) |α0| α0 β0 < 0 ϕ 0.050 β0 > 0 PSR B1913+16 0.045 α0 ϕ 0.040 0.035 0.025 0.020 0.015 VLBI Cassini −6 0.005 −4 −2 binary pulsars impose β0 > −4.5 i.e. PSR J1141–6545 0.010 βPPN– 1 γPPN– 1 < 1.1 0 2 4 6 β0 matter general relativity ϕ ϕ [T. Damour & G.E-F 2005] Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory Horizontal axis (α0 = 0) : perturbatively equivalent to G.R. α02 = 1 2 ωBD + 3 Solar-system and best binary-pulsar constraints on tensor–scalar theories (updated July 2005) matter matter Logarithmic scale for a0 j |a0| 100 j |a0| 0.2 SEP B1534+12 SEP 10-1 J0737–3039 0.175 B1913+16 B1534+12 0.15 All pulsars 10-2 0.125 J1141–6545 LLR Cassini 0.1 0.075 J0737–3039 0.05 B1913+16 J1141–6545 0.025 -6 -4 -2 0 10-3 2 4 All pulsars general relativity 6 matter matter b0 10-4 -6 -4 -2 0 2 j j general relativity (a0 = b0 = 0) 4 6 j j b0 |Φ0| 10−1 J1141–6545 J0737–3039 Experimental constraints on another natural class of scalar-tensor theories: S= 1 16 π G ∫ [matter , g ] + Smatter −g {( )−( R 1+ξΦ2 LLR B1534+12 10−2 )2 } All pulsars µΦ SEP Cassini µν 10−3 B1913+16 10−4 0 1 N.B.: ∃ other classes of scalar-tensor theories (e.g., some where the SEP tests are the most constraining, whereas PSR J1141–6545 does not tell us much!) general relativity 2 3 4 ξ Gravitational wave antennas LIGO/VIRGO/LISA signal Signal emitted by an inspiralling binary system: time ••• Effect on a detector: LIGO (depends on hundreds of post-Newtonian coefficients) VIRGO Gravitational wave antennas LIGO/VIRGO/LISA One needs accurate (3.5 PN) templates to extract the signal from the noise Gravitational waves in scalar-tensor gravity Quadrupole 1 + O c5 c7 Monopole . 1 2 Dipole Quadrupole 1 + σ+ 2 + + + O c c c3 c5 c7 Energy flux = Collapsing star Earth Factor α0 = Energy flux = (strong field)2 = Monopole/c >> usual Quadrupole/c5 1 1−γPPN ≈ < 0.003 2ωBD+3 2 Detection = (strong field) × (weak field) = too small for LIGO/VIRGO [J. Novak's thesis, PRD 57, 4789; 58, 064019 (1998)] and not in LISA's frequency band spin 2 spin 0 Gravitational waves in scalar-tensor gravity Quadrupole 1 + O c5 c7 Monopole . 1 2 Dipole Quadrupole 1 + s+ 2 + + + O c c c3 c5 c7 Energy flux = Collapsing star Earth Factor a0 = Energy flux = (strong field)2 = Monopole/c >> usual Quadrupole/c5 1 1-gPPN ª < 0.003 2wBD+3 2 Detection = (strong field) ¥ (weak field) = too small for LIGO/VIRGO [J. Novak's thesis, PRD 57, 4789; 58, 064019 (1998)] and not in LISA's frequency band Inspiralling binary Even if no helicity-0 wave is detected, the time-evolution of the (helicity-2) chirp depends on the Energy flux = (strong field)2 fi A priori possible to detect indirectly the presence of j: If binary inspiral detected with GR templates fi bound on matter–scalar coupling strength [matched-filter analysis: C.M. Will, Phys.Rev. D 50 (1994) 6058] spin 2 spin 0 Chirp evolution in general relativity signal time For a given binary system signal time Chirp evolution in a tensor–scalar theory Chirp evolution in general relativity signal time For an unknown mass of the system signal in phase out of phase time Chirp evolution in a tensor–scalar theory matter Solar-system and possible LIGO/VIRGO constraints on scalar-tensor gravity |α0| ϕ [Damour & GEF 1998] 0.050 LIGO/VIRGO 0.045 NS-BH 0.040 0.035 0.030 0.025 0.020 0.015 VLBI 0.010 Cassini −6 −4 −2 0.005 0 2 4 6 β0 matter ϕ ϕ general relativity Vertical axis (β0 = 0) : Jordan–Fierz–Brans–Dicke theory Horizontal axis (α0 = 0) : perturbatively equivalent to G.R. α02 = 1 2 ωBD + 3 Solar-system, possible LIGO/VIRGO, and binary-pulsar constraints on scalar–tensor theories of gravity matter ϕ |α0| LIGO/VIRGO 0.2 NS-BH 0.175 B1534+12 0.15 LIGO/VIRGO NS-NS 0.125 0.1 0.075 J0737–3039 0.05 B1913+16 J1141–6545 0.025 −6 −4 Bad news: LIGO/VIRGO will not probe scalar effects Good news! ⇒ GR templates can be used securely −2 0 2 4 All pulsars general relativity 6 matter β0 ϕ ϕ Possible LISA constraints on scalar-tensor theories of gravity matter |α0| PSR J1141-6545 LISA will probe |α0| ~ 1.5 × 10–3 if 1.4 m NS – 1000 m BH observed with S/N = 10 0.050 0.040 0.035 -15 10 0.030 -16 No scalar-field effect 10 compar able ma ) -1/2 1/2 PSR B1913+16 0.045 [Scharre & Will 2002; Will & Yunes 2004; Berti, Buonanno & Will 2005] ss BH-B -17 Sn (f) (Hz ϕ 10 0.025 H binar 0.020 ies Strong scalar-field effects NS-IM BH bin aries -18 10 -19 10 LISA with spin-orbit and spin-spin effects LISA with spin-orbit effects 0.015 VLBI 0.010 Cassini -20 10 0.005 LISA -21 10 -5 10 -4 10 -3 10 -2 10 f (Hz) -1 10 0 10 −6 −4 −2 ⇒ Tight constraints if detection of binary inspirals with GR templates But if no detection, what would we conclude? 0 2 4 general relativity 6 β0 matter ϕ ϕ matter |α0| Future binary-pulsar constraints on scalar-tensor theories of gravity ϕ 0.050 0.045 0.040 0.035 PSR J1141−6545. 0.030 LISA with spin-orbit and spin-spin effects 0.025 + 1% accurate P 0.020 LISA with spin-orbit effects 0.015 0.010 Cassini 0.005 LISA −6 Binary pulsars will probably probe such scalar-tensor theories before LISA is launched Good news: GR templates can be used securely −4 −2 0 2 4 general relativity 6 β0 matter ϕ ϕ Logarithmic scale for a0 |a0| 100 10-1 LISA NS-BH All pulsars spin-orbit & spin-spin spin-orbit 10-2 Cassini 10-3 J1141–6545 + 1% Pdot 10-4 -6 -4 -2 0 2 4 6 b0 Logarithmic scale for a0 |a0| Future binary-pulsar constraints on scalar-tensor theories of gravity 100 10-1 LISA NS-BH All pulsars spin-orbit & spin-spin spin-orbit 10-2 PSR-BH 1% Pdot Cassini Pulsar-white dwarf and Pulsar-black hole are the most constraining systems for alternative theories (large dipolar radiation) 10-3 J1141–6545 + 1% Pdot 10-4 -6 -4 -2 0 2 4 6 b0 Puzzling issues • Pioneer 10 & 11 anomaly in solar system (~70 AU): extra acceleration ~ 8.5 ¥10–10 m s–2 • Cosmological observations: 72% of “dark energy” 24% of “dark matter” 4% of baryonic matter “photograph” of Universe at 380 000 years (now 13.7 billion years) Dark energy – Why ΩΛ = 0.72 ~ Ωm = 0.28 today? (type Ia supernovae combined with CMB) ∃ hints from some models but no clean answer yet −122 – Why is Λ ≈ 3 × 10 c3 so small? (type Ia supernovae notably) /hG V(ϕ) Possible explanation via “quintessence” t today Λ ϕ ϕ0 • New qualitative difference between cosmological observations and solar-system/binary-pulsar ones ϕ ... matter ϕ ϕ matter ϕ matter ϕ ϕ Cosmological observations give access to the full shape of matter-scalar coupling A(ϕ) and/or scalar-field potential V(ϕ) • Usual cosmology: – Assume particular forms of V(ϕ) [and A(ϕ)] for theoretical reasons – Predict all observable quantities – Compare them to experimental data • Phenomenological approach: Reconstruct A(ϕ) & V(ϕ) from observational data. Result: If luminosity distance DL(z) and δρ density fluctuations δm(z) = ρ are both known as functions of the redshift z, then A(ϕ) & V(ϕ) can be reconstructed. [Boisseau, GEF, Polarski & Starobinsky 2000] N.B.: A priori obvious, since one “fits” two observed functions [DL(z) & δm(z)] with two unknown ones [A(ϕ) & V(ϕ)] ! • Semi-phenomenological approach: [δm(z) not yet well measured] – Theoretical hypotheses on V(ϕ) or A(ϕ) – Reconstruct the other one from DL(z) N.B.: A priori obvious too, since one fits one observed function [DL(z)] with one unknown function [A(ϕ) or V(ϕ)]. However, this naive reasoning works only locally (small interval). Result: ∃ tight constraints if DL(z) measured on a wide interval z ∈ [0, ∼2], even with large error bars! [GEF & Polarski 2001] Constraints come mainly from positivity of energy : Egraviton ≥ 0 ⇔ A2 > 0 ⇔ ΦBD > 0 Eϕ ≥ 0 ⇔ − ( µϕ)2 ⇔ ωBD > −3/2 Dark matter = pressureless and noninteracting component of matter • Imposed notably by rotation curves of galaxies and clusters: fi $ really some dark matter (many theoretical candidates notably from SUperSYmmetry), or modification of Newton’s law at large distances? • Milgrom’s phenomenological “MOND” proposal: = GM/r 2 √ √ a = a0 aN = GMa0 /r a= aN (OK for galaxies, not for clusters) if a > a0 ≈ 1.2 × 10- 10 m.s- 2 if a < a0 Relativistic formulations of MOND? $ various attempts but – some change the field-theory action depending on galaxy ! √ – some predict e.g. a = kM 2 /r instead of M/r , and assume then k = M - 3/2 ! – many contain tachyons or ghosts (fi unstable) ! Relativistic formulations of MOND? Best present candidate: Bekenstein-Sanders model – Many years of research fi seems very fine tuned but works (even for Pioneer anomaly if one wishes). • 1 usual (spin-2) graviton • 1 (spin-1) vector field • 2 (spin-0) scalar fields Idea: use one of the scalar fields as a “Lagrange parameter” to create a nonlinear F[( f)2] √ for the other one fi obtain the right M dependence Relativistic formulations of MOND? Best present candidate: Bekenstein-Sanders model – Many years of research fi seems very fine tuned but works (even for Pioneer anomaly if one wishes). • 1 usual (spin-2) graviton • 1 (spin-1) vector field • 2 (spin-0) scalar fields Idea: use one of the scalar fields as a “Lagrange parameter” to create a nonlinear F[( f)2] √ for the other one fi obtain the right M dependence – Stability not yet fully clear (tachyons & ghosts?). F local effects cosmology 1 -2 -4 -6 -8 2 3 4 Relativistic formulations of MOND? Best present candidate: Bekenstein-Sanders model – Many years of research fi seems very fine tuned but works (even for Pioneer anomaly if one wishes). • 1 usual (spin-2) graviton • 1 (spin-1) vector field • 2 (spin-0) scalar fields Idea: use one of the scalar fields as a “Lagrange parameter” to create a nonlinear F[( f)2] √ for the other one fi obtain the right M dependence – Stability not yet fully clear (tachyons & ghosts?). – Matter coupled to a physical metric involving the vector field fi $ preferred frame (ether). Idea: couple differently the scalar field f to “g00” and “gij” in order to get the right light deflection (cf. conformal coupling fi no extra deflection) F local effects cosmology 1 -2 -4 -6 -8 2 3 4 Relativistic formulations of MOND? Best present candidate: Bekenstein-Sanders model – Many years of research fi seems very fine tuned but works (even for Pioneer anomaly if one wishes). • 1 usual (spin-2) graviton • 1 (spin-1) vector field • 2 (spin-0) scalar fields Idea: use one of the scalar fields as a “Lagrange parameter” to create a nonlinear F[( f)2] √ for the other one fi obtain the right M dependence – Stability not yet fully clear (tachyons & ghosts?). F local effects cosmology 1 2 3 -2 -4 -6 -8 – Matter coupled to a physical metric involving the vector field fi $ preferred frame (ether). Idea: couple differently the scalar field f to “g00” and “gij” in order to get the right light deflection Bekenstein + large Wn (cf. conformal coupling fi no extra deflection) – CMB predictions [Skordis et al., astro-ph/0505519] need Wn = 0.17 (not far from WDM = 0.24) fi $ dark matter! Bekenstein without Wn Standard LCDM 4 Conclusions • ∃ theoretical motivations for considering alternatives to General Relativity. • Contrasting G.R. with alternatives ⇒ understand which features have been tested ⇒ compare probing power of different observations solar-system, –5 [weak field regime tested at the 10 level] binary-pulsar, [strong field regime] and cosmological observations. [time evolution] first derivative of ln A(ϕ) matter nonperturbative effects matter ϕ ϕ ϕ ⇒ second derivative of ln A(ϕ) a priori full shape of ln A(ϕ) but much more noisy • General Relativity passes all tests with flying colors. • ∃ still some puzzling experimental facts ⇒ understand them either theoretically or experimentally matter ϕ ... Qualitative difference between ϕ ϕ