Experimental Tests and Alternative Theories of Gravity

Transcription

1 Experimental Tests and Alternative Theories of Gravity Gonzalo J. Olmo Alba University of Valencia (Spain) & UW-Milwaukee Experimental Tests and Alternative Theories of Gravity p. 1/2

2 Motivation High-precision cosmological tests have improved our view of the Universe. The Universe is homogeneous, isotropic, spatially flat and is undergoing a period of accelerated expansion. The description given only ten years ago by General Relativity is not compatible with the observed accelerated expansion. Experimental Tests and Alternative Theories of Gravity p. 2/2

3 Motivation High-precision cosmological tests have improved our view of the Universe. The Universe is homogeneous, isotropic, spatially flat and is undergoing a period of accelerated expansion. The description given only ten years ago by General Relativity is not compatible with the observed accelerated expansion. Something has to be done to justify the acceleration. Experimental Tests and Alternative Theories of Gravity p. 2/2

4 New Ingredients in the cosmic pie? To justify the current acceleration we could... Introduce a new source of energy in T µν Experimental Tests and Alternative Theories of Gravity p. 3/2

5 New Ingredients in the cosmic pie? To justify the current acceleration we could... Introduce a new source of energy in T µν Cosmological constant, long-range fields,... It has been done before and tends to work (dark matter in galaxies, neutrino in β decay,... ). Experimental Tests and Alternative Theories of Gravity p. 3/2

6 New Ingredients or New Physics? To justify the current acceleration we could... Introduce a new source of energy in T µν Modify Einstein s equations. Experimental Tests and Alternative Theories of Gravity p. 3/2

7 New Ingredients or New Physics? To justify the current acceleration we could... Introduce a new source of energy in T µν Modify Einstein s equations. Because of quantum effects in curved space, string theory, higher dimensional theories,... GR could be the leading order of some effective gravity theory: R f(r) R+corrections. Experimental Tests and Alternative Theories of Gravity p. 3/2

8 New Ingredients or New Physics? To justify the current acceleration we could... Introduce a new source of energy in T µν Modify Einstein s equations. Today s choice... New Physics f(r) gravities Experimental Tests and Alternative Theories of Gravity p. 3/2

9 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S GR = 1 2κ 2 d 4 x gr + S m [g µν, ψ m ] Experimental Tests and Alternative Theories of Gravity p. 4/2

10 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S GR = 1 2κ 2 d 4 x g }{{} R + S m[g µν, ψ m ] 4D-Volume element Gravity Lagrangian Matter action Matter fields Experimental Tests and Alternative Theories of Gravity p. 4/2

11 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S GR = 1 2κ 2 d 4 x gr + S m [g µν, ψ m ] S f = 1 2κ 2 d 4 x gf(r) + S m [g µν, ψ m ] Experimental Tests and Alternative Theories of Gravity p. 4/2

12 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S f = 1 2κ 2 d 4 x gf(r) + S m [g µν, ψ m ] Two examples : f(r) = R µ4 R, f(r) = R + R2 M 2 Experimental Tests and Alternative Theories of Gravity p. 4/2

13 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S f = 1 2κ 2 d 4 x gf(r) + S m [g µν, ψ m ] Two examples : f(r) = R µ4 R, f(r) = R + R2 M 2 f(r) can be classified as Metric Theories of Gravity. Experimental Tests and Alternative Theories of Gravity p. 4/2

14 What are f(r) gravities? f(r) gravities can be seen as a generalization of GR S f = 1 2κ 2 d 4 x gf(r) + S m [g µν, ψ m ] Two examples : f(r) = R µ4 R, f(r) = R + R2 M 2 f(r) can be classified as Metric Theories of Gravity. MTG are the only theories of gravity that can embody the Einstein Equivalence Principle. Experimental Tests and Alternative Theories of Gravity p. 4/2

15 So far... We have seen that The accelerated expansion of the Universe is currently an unsolved problem. New theories have been proposed to justify the acceleration. Experimental Tests and Alternative Theories of Gravity p. 5/2

16 So far... We have seen that The accelerated expansion of the Universe is currently an unsolved problem. New theories have been proposed to justify the acceleration. How do these theories work? How do they change the gravitational physics? Experimental Tests and Alternative Theories of Gravity p. 5/2

17 So far... We have seen that The accelerated expansion of the Universe is currently an unsolved problem. New theories have been proposed to justify the acceleration. How do these theories work? How do they change the gravitational physics? Do they modify elementary-particle physics? Experimental Tests and Alternative Theories of Gravity p. 5/2

18 The Einstein Equivalence Principle The EEP states that Inertial and gravitational masses coincide, i.e., all bodies fall with the same acceleration WEP. The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed Local Lorentz Invariance. The outcome of any local non-gravitational experiment is independent of where and when it is performed Local Position Invariance. Experimental Tests and Alternative Theories of Gravity p. 6/2

19 Gravity as a curved-space effect If EEP is valid The non-gravitational laws of physics can be formulated by writing the laws of special relativity using the language of differential geometry: η µν g µν (x) Experimental Tests and Alternative Theories of Gravity p. 7/2

20 Gravity as a curved-space effect If EEP is valid The non-gravitational laws of physics can be formulated by writing the laws of special relativity using the language of differential geometry: η µν g µν (x) Gravitation would be described by an MTG S MTG = S G [g µν, φ, A µ, B µν, Γ α µν,...] + S m [g µν, ψ m ] Experimental Tests and Alternative Theories of Gravity p. 7/2

21 Gravity as a curved-space effect If EEP is valid The non-gravitational laws of physics can be formulated by writing the laws of special relativity using the language of differential geometry: η µν g µν (x) Gravitation would be described by an MTG S MTG = S G [g µν, φ, A µ, B µν, Γ α µν,...] + S m [g µν, ψ m ] However... Is the EEP really valid? Experimental Tests and Alternative Theories of Gravity p. 7/2

22 Weak Equivalence Principle Can be tested comparing the acceleration of two bodies in an external field: m I a = m p g m I and m p are made up of rest energy, e.m. energy, weak-interaction energy,... If m I and m p have different contributions m p = m I + i η iei c 2 E i Internal Energy generated by the i-th interaction η i strength of the violation Experimental Tests and Alternative Theories of Gravity p. 8/2

23 Weak Equivalence Principle Can be tested comparing the acceleration of two bodies in an external field: m I a = m p g m I and m p are made up of rest energy, e.m. energy, weak-interaction energy,... If m I and m p have different contributions η 2 a 1 a 2 a 1 + a 2 = i η i [ E i 1 m 1 c 2 Ei 2 m 2 c 2 ] E i Internal Energy generated by the i-th interaction η i strength of the violation Experimental Tests and Alternative Theories of Gravity p. 8/2

24 Weak Equivalence Principle Can be tested comparing the acceleration of two bodies in an external field: m I a = m p g m I and m p are made up of rest energy, e.m. energy, weak-interaction energy,... If m I and m p have different contributions η 2 a 1 a 2 a 1 + a 2 = i η i [ E i 1 m 1 c 2 Ei 2 m 2 c 2 ] E i Internal Energy generated by the i-th interaction η i strength of the violation The current bound is η = Experimental Tests and Alternative Theories of Gravity p. 8/2

25 Local Lorentz Invariance Are elementary-particle experiments tests of LLI? Experimental Tests and Alternative Theories of Gravity p. 9/2

26 Local Lorentz Invariance Are elementary-particle experiments tests of LLI? They are consistent, though not clean tests Experimental Tests and Alternative Theories of Gravity p. 9/2

27 Local Lorentz Invariance Are elementary-particle experiments tests of LLI? They are consistent, though not clean tests LLI would be violated if c would vary from one inertial reference frame to another. This violation would lead to shifts in the energy levels of atoms and nuclei depending on the direction of the quantization axis. Experimental Tests and Alternative Theories of Gravity p. 9/2

28 Local Lorentz Invariance Are elementary-particle experiments tests of LLI? They are consistent, though not clean tests LLI would be violated if c would vary from one inertial reference frame to another. This violation would lead to shifts in the energy levels of atoms and nuclei depending on the direction of the quantization axis. The current bound is δ = c 2 1 < Experimental Tests and Alternative Theories of Gravity p. 9/2

29 Local Position Invariance Can be tested by measuring the gravitational redshift of light. The comparison of the frequencies of two clocks at different locations boils down to the comparison of the velocities of two local Lorentz frames at rest at those positions. ν ν = (1 + α) U c 2 Experimental Tests and Alternative Theories of Gravity p. 10/2

30 Local Position Invariance Can be tested by measuring the gravitational redshift of light. The comparison of the frequencies of two clocks at different locations boils down to the comparison of the velocities of two local Lorentz frames at rest at those positions. ν ν = (1 + α) U c 2 The current bound is α < Experimental Tests and Alternative Theories of Gravity p. 10/2

31 Local Position Invariance Can be tested by measuring the gravitational redshift of light. The comparison of the frequencies of two clocks at different locations boils down to the comparison of the velocities of two local Lorentz frames at rest at those positions. ν ν = (1 + α) U c 2 The current bound is α < It can also be tested by measuring the constancy of the non-gravitational constants. Experimental Tests and Alternative Theories of Gravity p. 10/2

32 So far... We have seen that f(r) gravities, like GR and any other MTG, respect locally the physics of special relativity They do not modify elementary-particle physics Experimental Tests and Alternative Theories of Gravity p. 11/2

33 So far... We have seen that f(r) gravities, like GR and any other MTG, respect locally the physics of special relativity They do not modify elementary-particle physics The new interactions and phenomena predicted by string theory, higher dimensional theories,... are likely to violate the EEP Testing the EEP we could place bounds on the strength of those interactions Experimental Tests and Alternative Theories of Gravity p. 11/2

34 So far... We have seen that f(r) gravities, like GR and any other MTG, respect locally the physics of special relativity They do not modify elementary-particle physics The new interactions and phenomena predicted by string theory, higher dimensional theories,... are likely to violate the EEP Testing the EEP we could place bounds on the strength of those interactions We will study now the gravitational tests of MTG Experimental Tests and Alternative Theories of Gravity p. 11/2

35 Gravitational Tests Post-Newtonian gravity Stellar systems Cosmology Gravitational waves Experimental Tests and Alternative Theories of Gravity p. 12/2

36 Gravitational Tests Post-Newtonian gravity Deflection of light Time delay of light Perihelion Shift of Mercury Tests of SEP Conservation laws Geodesic precession (gyroscope) Gravitomagnetism Experimental Tests and Alternative Theories of Gravity p. 12/2

37 Gravitational Tests Post-Newtonian gravity Stellar systems Gravitational wave damping of orbital period Internal structure dependence Strong gravity effects Experimental Tests and Alternative Theories of Gravity p. 12/2

38 Gravitational Tests Post-Newtonian gravity Stellar systems Cosmology Distribution of anisotropies Hubble diagram of SNIa Age of the Universe Experimental Tests and Alternative Theories of Gravity p. 12/2

39 Gravitational Tests Post-Newtonian gravity Stellar systems Cosmology Gravitational waves Polarization Speed of grav.waves Experimental Tests and Alternative Theories of Gravity p. 12/2

40 Gravitational Tests Post-Newtonian gravity Stellar systems Cosmology Gravitational waves Experimental Tests and Alternative Theories of Gravity p. 12/2

41 Parametrized P-N Formalism In the weak-field, slow-motion limit, the metric of nearly every MTG has the same structure Experimental Tests and Alternative Theories of Gravity p. 13/2

42 Parametrized P-N Formalism In the weak-field, slow-motion limit, the metric of nearly every MTG has the same structure g 00 = 1 + 2U 2βU 2 2ξΦ W + (2γ α 3 + ζ 1 2ξ)Φ 1 +2(3γ 2β ζ 2 + ξ)φ 2 + 2(1 + ζ 3 )Φ 3 + 2(3γ + 3ζ 4 2ξ)Φ 4 (ζ 1 2ξ)A (α 1 α 2 α 3 )w 2 U α 2 w i w j U ij + (2α 3 α 1 )w i V i +O(ǫ 3 ) g 0i = 1 2 (4γ α 1 α 2 + ζ 1 2ξ)V i 1 2 (1 + α 2 ζ 1 + 2ξ)W i 1 2 (α 1 2α 2 )w i U α 2 w j U ij + O(ǫ 5/2 ) g ij = (1 + 2γU + O(ǫ 2 ))δ ij PPN parameters γ, β, ξ, α 1, α 2, α 3, ζ 1, ζ 2, ζ 3, ζ 4. Experimental Tests and Alternative Theories of Gravity p. 13/2

43 Parametrized P-N Formalism In the weak-field, slow-motion limit, the metric of nearly every MTG has the same structure With the potentials given by U = Φ W = A = Φ 2 = V i = ρ ρ x x d3 x (x x ) i (x x ) j, U ij = x x 3 d 3 x ρ ρ (x x ( ) x x x x 3 x x x ) x x x d 3 x d 3 x ρ [v (x x )] 2 x x 3 d 3 x ρ v 2, Φ 1 = x x d3 x ρ U x x d3 x ρ Π, Φ 3 = x x d3 x, Φ 4 = ρ v i x x d3 x, W i = p x x d3 x ρ [v (x x )](x x ) i x x 3 d 3 x Experimental Tests and Alternative Theories of Gravity p. 13/2

44 Parametrized P-N Formalism In the weak-fiel, slow-motion limit, the metric of nearly every MTG has the same structure It is characterized by a set of parameters: γ, β, ξ, α 1, α 2, α 3, ζ 1, ζ 2, ζ 3, ζ 4. The predictions of a particular theory depend on these parameters Experimental Tests and Alternative Theories of Gravity p. 13/2

45 Parametrized P-N Formalism In the weak-fiel, slow-motion limit, the metric of nearly every MTG has the same structure It is characterized by a set of parameters: γ, β, ξ, α 1, α 2, α 3, ζ 1, ζ 2, ζ 3, ζ 4. The predictions of a particular theory depend on these parameters We will need to obtain the PPN parameters of f(r) gravities. Experimental Tests and Alternative Theories of Gravity p. 13/2

46 P-N limit of f(r) gravities The Newtonian limit of these theories was recently discussed in R.Dick, Gen.Rel.Grav.36 (2004) The correct limit could be obtained if f(r 0 )f (R 0 ) 1. Experimental Tests and Alternative Theories of Gravity p. 14/2

47 P-N limit of f(r) gravities The Newtonian limit of these theories was recently discussed in R.Dick, Gen.Rel.Grav.36 (2004) The correct limit could be obtained if f(r 0 )f (R 0 ) 1. However, these theories are much more involved due to the cosmological evolution of the boundary conditions Experimental Tests and Alternative Theories of Gravity p. 14/2

48 P-N limit of f(r) gravities The Newtonian limit of these theories was recently discussed in R.Dick, Gen.Rel.Grav.36 (2004) The correct limit could be obtained if f(r 0 )f (R 0 ) 1. However, these theories are much more involved due to the cosmological evolution of the boundary conditions I have the Newtonian limit and the parameter γ Experimental Tests and Alternative Theories of Gravity p. 14/2

49 P-N limit of f(r) gravities The Newtonian limit of these theories was recently discussed in R.Dick, Gen.Rel.Grav.36 (2004) The correct limit could be obtained if f(r 0 )f (R 0 ) 1. However, these theories are much more involved due to the cosmological evolution of the boundary conditions I have the Newtonian limit and the parameter γ... I hope to have the remaining parameters next week or so Experimental Tests and Alternative Theories of Gravity p. 14/2

50 Details of f(r) gravities Defining a scalar field φ = df dr And a potential V (φ) = Rf (R) f(r) The first corrections of the P-N limit are ) g κ2 M (1 + e m φr 4πφ 0 r 3 ) g ij [1 + κ2 M (1 e m φr 4πφ 0 r 3 + V 0 6φ 0 r 2 V 0 6φ 0 r 2 ] δ ij Where Compare m 2 φ = φ 0V (φ 0 ) V (φ 0 ) 3 Experimental Tests and Alternative Theories of Gravity p. 15/2

51 Details of f(r) gravities Defining a scalar field φ = df dr And a potential V (φ) = Rf (R) f(r) The relevant parameters are G N = 1 φ 0 ( 1 + e m φr Where Compare γ = 3 e m φr 3 + e m φr 3 m 2 φ = φ 0V (φ 0 ) V (φ 0 ) 3 ) Experimental Tests and Alternative Theories of Gravity p. 15/2

52 Examples The Carroll et al. model f(r) = R µ4 R Experimental Tests and Alternative Theories of Gravity p. 16/2

53 Examples The Carroll et al. model f(r) = R µ4 R is characterized by m 2 φ = R 0 6µ 4(R µ 4 ) Experimental Tests and Alternative Theories of Gravity p. 16/2

54 Examples The Carroll et al. model f(r) = R µ4 R is characterized by m 2 φ = R 0 6µ 4(R2 0 3µ 4 ), R 0 1 t 2 Not valid in its original form If µ 4 µ 4, it is valid at early times only. Experimental Tests and Alternative Theories of Gravity p. 16/2

55 Examples The Carroll et al. model f(r) = R µ4 R is characterized by m 2 φ = R 0 6µ 4(R2 0 3µ 4 ), R 0 1 t 2 In general, f(r) = R + µ2n+2 R n are not viable models Experimental Tests and Alternative Theories of Gravity p. 16/2

56 Examples The Carroll et al. model f(r) = R µ4 R is characterized by m 2 φ = R 0 6µ 4(R2 0 3µ 4 ), R 0 1 t 2 In general, f(r) = R + µ2n+2 R n are not viable models The Starobinsky model f(r) = R + R2 M 2 It is characterized by m 2 φ = M2 6 Experimental Tests and Alternative Theories of Gravity p. 16/2

57 Examples The Carroll et al. model f(r) = R µ4 R is characterized by m 2 φ = R 0 6µ 4(R2 0 3µ 4 ), R 0 1 t 2 In general, f(r) = R + µ2n+2 R n are not viable models The Starobinsky model f(r) = R + R2 M 2 In general, f(r) = R + Rn M 2n+2 are always viable models m 2 φ = M2 6 Experimental Tests and Alternative Theories of Gravity p. 16/2

58 Summary and Conclusions f(r) gravities have been proposed to justify the cosmic speed-up Experimental Tests and Alternative Theories of Gravity p. 17/2

59 Summary and Conclusions f(r) gravities have been proposed to justify the cosmic speed-up They are MTG and, therefore, do not modify the physics of special relativity Experimental Tests and Alternative Theories of Gravity p. 17/2

60 Summary and Conclusions f(r) gravities have been proposed to justify the cosmic speed-up They are MTG and, therefore, do not modify the physics of special relativity Every experimental test of the EEP is potentially a deadly test for gravity as a curved spacetime phenomenon Experimental Tests and Alternative Theories of Gravity p. 17/2

61 Summary and Conclusions f(r) gravities have been proposed to justify the cosmic speed-up They are MTG and, therefore, do not modify the physics of special relativity Every experimental test of the EEP is potentially a deadly test for gravity as a curved spacetime phenomenon The post-newtonian regime is a good arena to test gravity theories Experimental Tests and Alternative Theories of Gravity p. 17/2

62 Summary and Conclusions f(r) gravities have been proposed to justify the cosmic speed-up They are MTG and, therefore, do not modify the physics of special relativity Every experimental test of the EEP is potentially a deadly test for gravity as a curved spacetime phenomenon The post-newtonian regime is a good arena to test gravity theories The post-newtonian limit of f(r) gravities is constrained by the cosmic evolution Experimental Tests and Alternative Theories of Gravity p. 17/2

63 ? There is nothing in this slide that can be of any interest to you Experimental Tests and Alternative Theories of Gravity p. 18/2

64 Parametrized P-N Formalism In the weak-fiel, slow-motion limit, the metric of nearly every MTG has the same structure g 00 = 1 + 2U 2βU 2 2ξΦ W + (2γ α 3 + ζ 1 2ξ)Φ 1 +2(3γ 2β ζ 2 + ξ)φ 2 + 2(1 + ζ 3 )Φ 3 + 2(3γ + 3ζ 4 2ξ)Φ 4 (ζ 1 2ξ)A (α 1 α 2 α 3 )w 2 U α 2 w i w j U ij + (2α 3 α 1 )w i V i +O(ǫ 3 ) g 0i = 1 2 (4γ α 1 α 2 + ζ 1 2ξ)V i 1 2 (1 + α 2 ζ 1 + 2ξ)W i 1 2 (α 1 2α 2 )w i U α 2 w j U ij + O(ǫ 5/2 ) g ij = (1 + 2γU + O(ǫ 2 ))δ ij PPN parameters γ, β, ξ, α 1, α 2, α 3, ζ 1, ζ 2, ζ 3, ζ 4. Back Experimental Tests and Alternative Theories of Gravity p. 19/2

65 Significance of the PPN parameters What it measures Value Parameter relative to GR in GR γ How much space-curvature 1 produced by unit rest mass? β How much nonlinearity 1 in the superposition law for gravity? ξ Preferred-location effects? 0 α i Preferred-frame effects? 0 α 3 Violation of conservation 0 ζ j of total momentum? 0 Back Experimental Tests and Alternative Theories of Gravity p. 20/2

66 Tests of the parameter γ The deflection of light δθ = 1 2 (1 + γ) [ 4m d cos χ + 4m d r ( 1 + cos Φr 2 )], where d and d r are the distances of closest approach of the source and reference rays respectively, Φ r is the angular separation between the Sun and the reference source, and χ is the angle between the Sun-source and the Sun-reference directions, projected on the plane of the sky. Back Experimental Tests and Alternative Theories of Gravity p. 21/2

67 Tests of the parameter γ The Time Delay of Light A radar signal sent across the solar system past the Sun to a planet or satellite and returned to the Earth suffers an additional non-newtonian delay in its round-trip travel time, given by δt = 2(1 + γ)m ln[(r + x n)(r e x e n)/d 2 ] Back Experimental Tests and Alternative Theories of Gravity p. 21/2

68 Tests of the parameter γ The perihelion shift of Mercury [ ] 1 ω (2 + 2γ β) (J 2 /10 7 ) Back Experimental Tests and Alternative Theories of Gravity p. 21/2

69 Gravitational waves polarization Six polarization modes permited in any MTG Gravitational Wave Polarization y x x (a) (b) y y x z (c) (d) x y z z y Shown displacement of each mode induced on a ring of test particles Propagation in +z direction No displacement out of the plane indicated In GR only (a) and (b) In scalar-tensor gravity, (c) is also possible (e) (f) Back Experimental Tests and Alternative Theories of Gravity p. 22/2

70 The binary pulsar PSR Magnetized NS rotating with T = 59 ms Orbital period 7 h 45 min Back Experimental Tests and Alternative Theories of Gravity p. 23/2

71 The binary pulsar PSR Magnetized NS rotating with T = 59 ms Orbital period 7 h 45 min Perihelion advance in 1 day the same as Mercury in 1 century Back Experimental Tests and Alternative Theories of Gravity p. 24/2

72 The binary pulsar PSR Magnetized NS rotating with T = 59 ms Orbital period 7 h 45 min Perihelion advance in 1 day the same as Mercury in 1 century The orbit is shrinking due to grav.wave radiation. Back Experimental Tests and Alternative Theories of Gravity p. 24/2

73 Ten years ago... Ten years ago, the standard cosmology was described by R µν 1 2 g µνr }{{} = κ 2 T µν Einstein equations Visible Matter Dark Matter Radiation Back Experimental Tests and Alternative Theories of Gravity p. 25/2

74 Cosmic Microwave Brackground Experimental Tests and Alternative Theories Back of Gravity p. 26/2