1. Aerospace Research and Technology Centre, Barcellona - SPAIN
Florence, January 28, SIGRAV School in Cosmology and INFN Formation School Signals 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 ' 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. Is Einstein’s picture definitive?
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-correlation for 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 response of an interferometer
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
Importance of “magnetic components”:
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

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:
Christian Corda ­ J. Cosmol. Astropart. Phys. JCAP04(2007)009 doi: /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: accepted by Int. Journ. Mod. Phys. A
Christian Corda – Gen.Rel. Grav.­0001­7701 (Print) 1572­9532 (Online) (2008) pre­print on arXiv:
Christian Corda – Astrophysics and Space Science 0004­640X (Print) 1572­946X (Online)‏
Christian Corda ­ Astropart. Phys., doi: /j.astrpartphys Avaible on line 10 September 2008
Salvatore Capozziello, Christian Corda and Maria Felicia De Laurentis ­ – Phys. Lett. B /j.physletb (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 ­2655 (2007)‏