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
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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
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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
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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
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Tuning with observations
Capozziello, Cardone,FrancavigliaGen. Rel. Grav. 38, 5 (2006)
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Correct theory from observations
Interferometric detection of gravitational waves
One more polarization is present with respect standard general relativity
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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
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The Virgo-Minigrail cross-correlationfor scalar relic GWs
One more polarization (scalar) in f(R) theories of gravity
massless case: the overlap reductionfunction
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Overlap reduction function very small, but a maximum is present
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The R-1 proposal
Einstein-Hilbert action
Modified action
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Field equations
Klein-Gordon equation
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Linearized theory in vacuum
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Production of mass from space-time curvature
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Observation: gravitational waves in the “Lorenz” gauge
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No transverse – traceless gauge
Third polarization
Line element
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Analysis in the frame of the local observer
Longitudinal component
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Two effects
Motion of test masses
Propagation in a curved space-time
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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
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Second contribution: the travel of photons in curved space-time
Computation in the Fourier domain using the translation and derivation Fourier theorems
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Longitudinal response function
Relation mass-velocity
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Correlation response function Ricci curvature scalar
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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
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