1. Signals and interferometric response functions in the framework of gravitational waves arising from extended theories of gravity Speaker: Christian Corda Centro Scienze Naturali di Prato

2. Contents Motivations on the extension of general relativity Importance of gravitational waves for a potential discrimination between various theories

3. 3 The R -1 proposal The Scalar –Tensor Theory The magnetic component of gravitational waves Corda C. - Int. Journ Mod Phys. D 16, 9, 1497-1517 (2007); Corda C. - Int. Journ Mod Phys. A 22, 13, 2361 - 2381 (2007); Corda C. Topical Review on gr-qc 08062702 in press for Nova Science Publishers

4. Some misconceptions on gravitational waves clarified Difference in the response function between the TT gauge and the gauge of the local observer As both of the interferometer arm and the laser light are stretched by the gw, a signal is not present Corda C. gr-qc/07062412

5. 5 Connection between relic GWs and f(R) gravity Dark Matter and Dark Energy Problems Only 5% of the mass in the Universe is known

7. We have a snapshot of the Universe from electromagnetic waves Different snapshot from gravitational waves? The sound of the Universe Snapshot of Universe from GW

8. Gravitation: is it a mystery? Astrophysicists often perform computations with Newtonian theory! Is our understanding of Gravitation definitive? No one can say that GR is wrong! But, is it definitive?

9. SUN MOON EARTH STELLA REAL POSITION APPARENT POSITION In presence of a gravitational field lo space-time is curved Deflection of the light (Eddington 1919) Is Einsteins picture definitive? Einstein attempted a modification: Generalized Theory of Gravitation

10. 10 Is there an intrinsic curvature? Ricci Curvature R General Relativity Generic function of Ricci Curvature f(R) General Relativity + intrinsic curvature Extended theories of Gravitation: f(R) theories and scalar tensor theories which are coupled by conformal transformations

11. 11 Tuning with observations Capozziello, Cardone, Francaviglia Gen. Rel. Grav. 38, 5 (2006)

12. 12 Correct theory from observations Interferometric detection of gravitational waves One more polarization is present with respect standard general relativity

13. 13 The relic GWs – f(R) connection Amplification of vacuum fluctuations re-analyzed in the context of f(R) gravity theories using a conformal treatment Two important results 1) the purely tensorial part of GWs is conformally invariant 2) the amplitude of the background is tuned by the correct theory of gravity (i.e. the correct theory of gravity is printed in relic GWs)

14. Most important observative bound: the WMAP one old COBE bound (Allen, Turner '94) WMAP bound

15. Production mechanism and characteristic amplitude of the primordial GW stochastic background Amplification of vacuum fluctuations (Grishchuk 75; Starobinski 78; Allen '88..... Capozziello, Corda and De Laurentis in f(R) Gravity, 2007 )

16. Detection of the primordial background is very difficult Cross-correlation between the two LIGO WMAP bound We hope in advanced projects and in LISA old COBE bound

17. 17 The Virgo-Minigrail cross-correlation for scalar relic GWs One more polarization (scalar) in f(R) theories of gravity massless case: the overlap reduction function

18. 18 Overlap reduction function very small, but a maximum is present

19. 19 The R -1 proposal Einstein-Hilbert action Modified action

20. 20 Field equations Klein-Gordon equation

21. 21 Linearized theory in vacuum

22. 22 Production of mass from space-time curvature

23. 23 Observation: gravitational waves in the Lorenz gauge

24. 24 No transverse – traceless gauge Third polarization Line element

25. 25 Analysis in the frame of the local observer Longitudinal component

26. 26 Two effects Motion of test masses Propagation in a curved space-time

27. 27 Longitudinal response function Method of bouncing photon : the variation of space-time due to the massive polarization is computed in all the travel of the photon First contribution : the motion of test masses

28. 28 Second contribution: the travel of photons in curved space-time Computation in the Fourier domain using the translation and derivation Fourier theorems

29. 29 Longitudinal response function Relation mass-velocity

30. 30

31. 31

32. 32

33. 33 Correlation response function Ricci curvature scalar

34. 34 Conclusions 1)Is Dark Universe achieved by a modification of general relativity? 2)Importance of relic GWs 3) R -1 proposal: connection between the interferometer response function and the Ricci curvature scalar 4) Is a generalization possible? Is the correct theory of gravity imprinted in the interferometer response function?

35. The Scalar-Tensor Gravity 1)Mechanism of production of SGW from Scalar- Tensor Gravity 2) Massless case: invariance of the signal in three different gauges 3) Massless case: the frequency-dependent angular pattern 4) The small massive case Generalized previous results analyzed in the low- frequencies approximation

36. Mechanism of production of SGW from Scalar-Tensor Gravity Most general action for STG in literature

37. Considering the transformation previous action reads BD-like theory

38. Field equations Klein-Gordon

39. Linearized theory in vacuum Minkowski background + minimum for W We assume

40. obtaining with

41. Effective BD The massless case Most simple case: Gauge transforms (Lorenz condition)

42. Solutions are plan waves Purely scalar wave: line element TT gauge extended to scalar waves

43. The response of an interferometer Literature: low-frequencies approximation Method of bouncing photon : the variation of space-time due to the scalar field is computed in all the travel of the photon

44. Computation of the variation of proper time in presence of the SGW In the Fourier domain

45. The Shibata, Nakao and Nakamura gauge for SGW Purely scalar wave: line element Reanalyzed

46. Same results of the TT gauge In the Fourier domain Used a time transform

47. The local Lorentz gauge for SGW: three different effects The motion of test masses The travel of photons in curved spacetime

48. The shifting of time Gauge invariance recovered In the Fourier domain

49. Angular pattern for SGW

50. Line element in the u direction variation of proper time in presence of the SGW in the u direction

51. Response function in the u direction Same analysis: response function in the v direction

52. Total frequency-dependent response function Agrees with Low frequencies

56. The small massive case Totally equivalent to the R -1 Theory

57. Conclusions Realistic possibility to detect SGW in different gauges The investigation of scalar components of GW could be a tool to discriminate among several theories of gravity

58. The magnetic components of gravitational waves 1) Equations rewritten in different notations and spatial dependence 2) Used the bouncing photon method 3) Generalized previous results analyzed in the low-frequencies approximation: answer the question about an extension of the frequency range using the full theory of GWs Importance of magnetic components:

59. Coordinate transformation: analysis in the gauge of the local observer Line element in the TT gauge: Coordinate transformation

60. Equations of motion for test masses Not gauge artefact: equation directly obtained from geodesic deviation in the work of Baskaran and Grishchuk

61. Equations of motion for the pure magnetic components First polarizationSecond polarization

62. Coordinate transformation Distance Variation in distance

63. Variation in distance considering casuality Second effect: motion of the photon in a curved space-time Tidal acceleration of the test mass Equivalent to the presence of a Newtonian potential

64. Connection between GR and Newtonian theory Total variation of proper time from second effect

65. Total variation of proper time in the u arm In the Fourier domain

66. Response function in the u direction Same analysis: response function in the v direction

67. Total frequency-dependent response function

68. Low frequency approximation

69. Total frequency-dependent response function for the polarization

70. Low frequency approximation

71. High frequencies

73. Extension of the frequency range of interferometers?

75. The full theory of gravitational waves in the TT gauge: Corda C. Int. Journ. Mod. Phys D 16, 9, 1497-1517 (2007) Line element in the u direction for the + polarization variation of proper time in presence of the GW in the u direction

76. Response function in the u direction where

77. Same analysis: response function in the v direction where

78. Low frequencies Total response function for the + polarization

79. Low frequencies Similar analysis: total response function for the polarization

80. Drawn two response function in the frequency domain

82. The total response functions which take into account both of the electric and magnetic components decreases with frequency: no extension of the frequency range of interferometers. This is because the expansion used in the coordinate transformation breaks down at high frequencies and the distinction between electric and magnetic components becomes ambiguous at high frequencies. Thus the full theory has to be used, but if one uses the low frequencies approximation, magnetic contributions have to be taken into account Conclusions

83. Problems The distinction between high and low frequencies is not totally clear in the context of the magnetic components of GWs: where exactly the distinction between electric and magnetic components breaks down? Where exactly the response functions of Baskaran and Grishchuk have to be replaced with the ones today introduced? Gravito-magnetism in the GWs physics is a topic which is not totally understood, further and accurate studies are needed

84. Two misconceptions on gravitational waves clarified Difference in the response function between the TT gauge and the gauge of the local observer As both of the interferometer arm and the laser light are stretched by the gw, a signal is not present Corda C. gr-qc/07062412

85. Total response function for the + polarization in the TT gauge Difficulties to find the same response function in the frame of the local observer which is the frame of a laboratory environment on Earth, i.e. the local Lorentz gauge where we perform the data analysis

86. Gauge invariance only in the low frequency approximation and/or in the simplest interferometer - GW geometry Corda C. gr-qc/07062412 two effects considered in the u direction Motion of test masses Presence of curved spacetime

87. Adding the two effects Same analysis in the v direction The total response function in the frame of the local observer is the same calculated in the TT gauge

88. The total response functions which take into account both of the test masses motion and the redshift contributions is the same in the TT and in the local Lorentz gauges. As this response function is in general different to zero, the misconception which tells thatbecause both of the interferometer arm and the laser light are stretched by the GW a signal is not present is totally clarified Conclusions