Signals and interferometric response functions in the framework of gravitational waves arising from extended theories of gravity Speaker: Christian Corda.

Signals and interferometric response Signals and interferometric response functions in the framework of functions in the framework of

gravitational waves arising from gravitational waves arising from extended theories of gravityextended theories of gravity

Speaker: Christian CordaSpeaker: Christian Corda

Centro Scienze Naturali di Centro Scienze Naturali di PratoPrato

ContentsContents

Motivations on the extension of general Motivations on the extension of general relativityrelativity

Importance of gravitational waves for a Importance of gravitational waves for a potential discrimination between various potential discrimination between various theoriestheories

3

The R-1 proposal

The Scalar –Tensor Theory

The “magnetic” component of gravitational wavesCorda 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

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 thelaser light are stretched by the gw, a signalis not present

Corda C. gr-qc/07062412

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

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

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?

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 Einstein’s picture definitive?Einstein attempted a modification: Generalized Theory of Gravitation

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 arecoupled by conformal transformations

11

Tuning with observations

Capozziello, Cardone,FrancavigliaGen. 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) connectionAmplification of vacuum fluctuationsre-analyzed in the context of f(R) gravity theories using a conformal treatment

Two important results

1) the purely tensorial part of GWsis conformally invariant 2) the amplitude of the background istuned by the correct theory of gravity(i.e. the correct theory of gravity is printedin relic GWs)

Most important observative bound: the WMAP one

old COBE bound (Allen, Turner '94)

WMAP bound

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

We hope in advanced projects and in LISA

old COBE bound

17

The Virgo-Minigrail cross-correlationfor scalar relic GWs

One more polarization (scalar) in f(R) theories of gravity

massless case: the overlap reductionfunction

18

Overlap reduction function very small, but a maximum is present

19

The R-1 proposal

Einstein-Hilbert action

Modified action

20

Field equations

Klein-Gordon equation

21

Linearized theory in vacuum

22

Production of mass from space-time curvature

23

Observation: gravitational waves in the “Lorenz” gauge

24

No transverse – traceless gauge

Third polarization

Line element

25

Analysis in the frame of the local observer

Longitudinal component

26

Two effects

Motion of test masses

Propagation in a curved space-time

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

Second contribution: the travel of photons in curved space-time

Computation in the Fourier domain using the translation and derivation Fourier theorems

29

Longitudinal response function

Relation mass-velocity

30

31

32

33

Correlation response function Ricci curvature scalar

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?

The Scalar-Tensor Gravity1) 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

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 + minimum for W

We assume

obtaining

with

Effective BD

The massless case

Most simple case:

Gauge transforms (Lorenz condition)

Solutions are plan waves

Purely scalar wave: line element

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

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

Same results of the TT gauge

In the Fourier domain

Used a time transform

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

Gauge invariance recovered

In the Fourier domain

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

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”:

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

Low frequencies

Total response function for the + polarization

Low frequencies

Similar analysis: total response function for the polarization

Drawn two response function in the frequency domain

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

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 thelaser light are stretched by the gw, a signalis 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 frequencyapproximation and/or in the simplestinterferometer - 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 ofthe local observer is the same calculated inthe TT gauge

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

Conclusions