Download Gravitational waves from extended theories of gravity

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