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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 Contents Motivations on the extension of general relativity Importance of gravitational waves for a potential discrimination between various theories 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 3 08062702 in press for Nova Science Publishers 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 Connection between relic GWs and f(R) gravity Dark Matter and Dark Energy Problems Only 5% of the mass in the Universe is known 5 Snapshot of Universe from GW We have a snapshot of the Universe from electromagnetic waves Different snapshot from gravitational waves? The sound of the Universe 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? REAL In presence of a gravitational field lo space-time is curved POSITION Deflection of the light (Eddington 1919) STELLA SUN APPARENT POSITION MOON Is Einstein’s picture definitive? Einstein attempted a modification: Generalized Theory of Gravitation EARTH 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 10 Tuning with observations Capozziello, Cardone, Francaviglia Gen. Rel. Grav. 38, 5 (2006) 11 Correct theory from observations Interferometric detection of gravitational waves One more polarization is present with respect standard general relativity 12 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) 13 Most important observative bound: the WMAP one WMAP bound old COBE bound (Allen, Turner '94) 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 ) 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 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 17 Overlap reduction function very small, but a maximum is present 18 The R-1 proposal Einstein-Hilbert action Modified action 19 Field equations Klein-Gordon equation 20 Linearized theory in vacuum 21 Production of mass from space-time curvature 22 Observation: gravitational waves in the “Lorenz” gauge 23 No transverse – traceless gauge Third polarization Line element 24 Analysis in the frame of the local observer Longitudinal component 25 Two effects Motion of test masses Propagation in a curved space-time 26 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 27 Second contribution: the travel of photons in curved space-time Computation in the Fourier domain using the translation and derivation Fourier theorems 28 Longitudinal response function Relation mass-velocity 29 30 31 32 Correlation response function Ricci curvature scalar 33 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 34 function? The Scalar-Tensor Gravity 1) Mechanism of production of SGW from ScalarTensor 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 lowfrequencies approximation Mechanism of production of SGW from Scalar-Tensor Gravity Most general action for STG in literature Considering the transformation previous action reads BD-like theory Field equations Klein-Gordon Linearized theory in vacuum Minkowski background + We assume minimum for W obtaining with The massless case Effective BD Most simple case: Gauge transforms (Lorenz condition) Solutions are plan waves TT gauge extended to scalar waves Purely scalar wave: line element 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 Computation of the variation of proper time in presence of the SGW In the Fourier domain The “Shibata, Nakao and Nakamura” gauge for SGW Purely scalar wave: line element Reanalyzed Used a time transform Same results of the TT gauge In the Fourier domain The local Lorentz gauge for SGW: three different effects The motion of test masses The travel of photons in curved spacetime The shifting of time In the Fourier domain Gauge invariance recovered Angular pattern for SGW Line element in the u direction variation of proper time in presence of the SGW in the u direction Response function in the u direction Same analysis: response function in the v direction Total frequency-dependent response function Agrees with Low frequencies The small massive case Totally equivalent to the R-1 Theory 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 The “magnetic components” of gravitational waves Importance of “magnetic components”: 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 Coordinate transformation: analysis in the gauge of the local observer Line element in the TT gauge: Coordinate transformation Equations of motion for test masses Not gauge artefact: equation directly obtained from geodesic deviation in the work of Baskaran and Grishchuk Equations of motion for the pure “magnetic” components First polarization Second polarization Coordinate transformation Distance Variation in distance 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 Connection between GR and Newtonian theory Total variation of proper time from second effect Total variation of proper time in the u arm In the Fourier domain Response function in the u direction Same analysis: response function in the v direction Total frequency-dependent response function Low frequency approximation Total frequency-dependent response function for the polarization Low frequency approximation High frequencies Extension of the frequency range of interferometers? 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 Response function in the u direction where Same analysis: response function in the v direction where Total response function for the + polarization Low frequencies Similar analysis: total response function for the polarization Low frequencies Drawn two response function in the frequency domain Conclusions 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 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 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 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 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 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 Conclusions 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 that “because both of the interferometer arm and the laser light are stretched by the GW a signal is not present” is totally clarified