Christian Corda Aerospace Research and Technology Centre, Barcellona - SPAIN

1. Florence, January 28, 2009SIGRAV School in Cosmology and INFN Formation SchoolSignals and interferometric response functions in the framework of gravitational waves from extended theories of gravity Christian Corda Aerospace Research and Technology Centre, Barcellona - SPAIN

2. Contents General features of the stochastic background in standard GR and WMAP bound Extension to f(R) theories of gravity Gravitational waves from f(R) theories The Scalar –Tensor Theory The “magnetic” component of gravitational waves

3. Possible target of GW experiments Stochastic background of GW General assumptions: isotropic, stationary, gaussian Primordial: Parametric amplification of vacuum fluctuations during inflation, phase transitions non-equilibrium processes, topological defects Astrophysical: Large populations of binary systems of compact objects (wd,ns,bh), hot, young rapidly spinning NS …

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

5. 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)‏

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

7. Dark Matter and Dark Energy Problems Only 5% of the mass in the Universe is known

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

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

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

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

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. The Virgo-Minigrail cross-correlationfor scalar relic GWs One more polarization (scalar) in f(R) theories of gravity and Scalar-Tensor Gravity massless case: the overlap reduction function

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

16. f(R) theories Einstein-Hilbert action Modified action Observation of gravitational waves in the “Lorenz” gauge No transverse – traceless gauge

17. The massive mode gives a longitudinal effect which implies a longitudinal response function Relation mass-velocity

19. The Scalar-Tensor Gravity Massless case: invariance of the signal in three different gauges Massless case: the frequency-dependent angular pattern The small massive case is totally equivalent to f(R) theories Generalized previous results analyzed in the low-frequencies approximation

20. The massless case: TT gauge extended to scalar waves 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

21. Gauge invariance between the three gauges well known in the literature: the TT gauge, the gauge of the local observer and the SNN gauge

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

23. Scalar pattern of Virgo at high frequencies Scalar pattern of Virgo at low frequencies

24. 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 with more precise response functions using the full theory of GWs Importance of “magnetic components”:

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

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

27. Equations of motion for the pure “magnetic” components First polarization Second polarization

28. Coordinate transformation Distance Variation in distance

29. Magnetic pattern “plus” Virgo 8000 Hz

30. Magnetic pattern “cross” LIGO 8000 Hz

31. The full theory of gravitational waves in the TT gauge: total response function for the + polarization Low frequencies

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

33. Low-frequency angular pattern (“plus” polarization)‏ High-frequency angular pattern (“plus” polarization)‏

34. Final Remarks General features in standard GR and WMAP bound of relic GWs Connection between f(R) gravity and relic GWs Analised the Virgo-Minigrail cross-correlation for the third component in the massless case

35. Realistic possibility to detect gravitational waves from extended theories The investigation of the scalar component of GW could be a tool to discriminate among several theories of gravity Importance of the “magnetic” components and more accurate response functions at high frequencies

36. References Christian Corda – “Magnetic” components of gravitational waves and response functions of interferometers – in “Interferometers, Research, Technology and applications” ­ review commissioned by Nova Science Publishers, in press (2008), pre­print on arXiv:0806.2702 Christian Corda ­ J. Cosmol. Astropart. Phys. JCAP04(2007)009 doi:10.1088/1475­7516/2007/04/009 (2007)‏ Christian Corda ­ Astropart. Phys. 27, 539­549 (2007)‏ Christian Corda ­ Mod. Phys. Lett. A, vol. 22, No. 16 (2007) pp. 1167­1173 Christian Corda ­ Intern. Journ. Mod. Phys. D, 16, 8, 1497­1517 (2007) Christian Corda ­ Intern. Journ. Mod. Phys. A ,22, 13, 2361­2381 (2007)‏ Christian Corda ­ Astropart. Phys. 28, 2, 247­250 ( 2007)‏ Christian Corda – Intern. Journ. Mod. Phys. A, 22, 26, 4859­4881 (2007)‏ Christian Corda – Mod. Phys. Lett. A, vol. 22, No. 23 (2007) pp. 1727­1735 Christian Corda ­ arXiv:0711.4917 accepted by Int. Journ. Mod. Phys. A Christian Corda – Gen.Rel. Grav.­0001­7701 (Print) 1572­9532 (Online) (2008) pre­print on arXiv:0802.2523 Christian Corda – Astrophysics and Space Science 0004­640X (Print) 1572­946X (Online)‏ Christian Corda ­ Astropart. Phys., doi:10.1016/j.astrpartphys.2008.09.003 Avaible on line 10 September 2008 Salvatore Capozziello, Christian Corda and Maria Felicia De Laurentis ­ – Phys. Lett. B 10.1016/j.physletb.2008.10.001 (2008)‏ Salvatore Capozziello and Christian Corda ­ Intern. Journ. Mod. Phys. D, Vol. 15, No. 7 (2006) 1119­1150 Salvatore Capozziello, Christian Corda and Maria Felicia De Laurentis ­Modern Physics Letters A, Vol. 22, No. 15 (2007) 1097­1104 Salvatore Capozziello, Christian Corda and Maria Felicia De Laurentis ­ Modern Physics Letters A, Vol. 22, No. 35 2647­2655 (2007)‏