Detecting Additional Polarization Modes with eLISA
L. Philippoz, P. Jetzer
Summary
Within the frame of Einstein’s General Relativity, gravitational waves are expected to possess two tensorial polarizations, namely the well-known
h+ and h× modes. Other metric theories of gravity however allow the existence of additional modes (two vector and/or two scalar modes), and
the (non-)observation of those additional polarizations could put constraints on the validity of all existing theories, which would consequently
provide a further test for General Relativity.
In its 2-arm-planned-configuration, eLISA only consists of one detector orbiting around the Sun, and we therefore investigate if there is a
possibility to still detect and separate additional modes of a given gravitational wave signal.
Polarization modes
Sensitivity to additional modes
Perturbed metric corresponding to a propagating gravitational wave:
• Time-delay interferometric combinations,
hij (ωt − k · x) =
X
hA(ωt − k ·
2-arm configuration (4 beams)
x)eA
ij
• Noise power spectrum : SX (f )
A
with A = ×, +, b, l, x, y the six possible polarization modes and the following tensors (tensor,
scalar and vector modes)
y



x
1
10

XRMS(f)/H
z
x
−1
10
−2
(d)
y
Vector
Scalar T
Scalar L
10
−24
10
−4
−3
10
−2
10
f(Hz)
−1
10
0
10
10
−4
10
−3
10
−2
10
f(Hz)
−1
10
0
10
more sensitive to (longitudinal) scalar modes)
z
z
• XRMS(f ) and the sensitivity are given for LISA (similar results expected with shorter arms;
to be confirmed)
(f)
Stochastic GW background : single detector [7]
• GW stochastic background in the low frequency limit
Z ∞
XZ
h(t, x) =
dΩ̂
df h̃A(f, Ω̂)e2πif (t−Ω̂x/c)FA(Ω̂)
S2
• If the output data of the single detector is written as h(t) + n(t), with n(t) the noise, the
autocorrelation of the signal reads
−∞
1
hh̃(f )h̃ (f )i = δ(f − f 0)Sh(|f |)
2
∗
with A = +, ×, x, y, b, l all possible polarizations, FA the antenna pattern function.
• Overlap reduction function (how much degree of correlation is preserved between detectors)
1
ij J
kj ˆ ˆ
ij kl ˆ ˆ ˆ ˆ
M
M
i
M
ρM
(α)D
D
+
ρ
(α)D
D
γIJ
(f ) = 2
d
d
+
ρ
(α)D
1
ij
2
I,k J i j
3
I
I DJ didj dk dl
sin χ
x
2πf |x|
ˆ
= f (j0(α), j2(α), j4(α)), di =
,α=
, M = T, V, S
|x|
c
• GW background energy density
−22
10
• Similar behaviour between LISA and eLISA in the high-frequency regime (where they are
Stochastic GW background : network of detectors [1]
ΩTgw
Tensor
−3
−4
(e)
with
f Hz
1
−23
10
0 0 1
0 0 0

 y 

exij =  0 0 0  eij  0 0 1 




1 0 0
0 1 0
ρM
i
0.1
10

A
0.01
−21
10
x

0.001
10
10

(c)

10-42
Scalar L
0
y
1 0 0
0 0 0




ebij =  0 1 0  elij  0 0 0 




0 0 0
0 0 1
Tensor
Vector
Scalar T
(b)
y

eLISA
x
(a)

LISA
10-40
B : 1 cycle/year, SNR=5
0 1 0
1 0 0


 ×
e+
=
e
 0 −1 0  ij  1 0 0 
ij




0 0 0
0 0 0

10-38
10-44
10-4
X Sensitivity

y
• Root-mean squared GW response : XRMS(f )
p
• Sensitivity : SNR · SX (f )B/XRMS(f )
Noise power spectrumHz ¹
10-36
=
Ω+gw
+
Ω×
gw
(similar for
ΩVgw
and
ΩSgw),
related to the
0
with Sh(|f |) the one-sided power spectral density; one can define Sn in a similar way for the noise.
• The maximum SNR given by this process is
Z ∞
2
T
S
(|f
|)
h
SN R2 =
df
2 −∞
[Sh(|f |) + Sn(|f |)]2
⇒ Also valid in the high-frequency limit, for a single detector, but for an unpolarized GWB
3 A
(f
)
∝
f
Sh (f )
one-sided power spectral density S(|f |) ∝ hh̃∗A(f, Ω̂)h̃A0 (f 0, Ω̂0)i by ΩM
gw
• The tensor, vector and scalar modes can then be separately detected via
#(1/2)
Z ∞ " M
2
(Ω
(f
))
det F(f )
gw
M
SN R ∝
df
6F (f )
f
M
0
Possible solutions for eLISA
• Cross-correlating the TDI combinations of a LISA-like single detector also correlates the noise
• The separation of all modes however requires a network of detectors
0
X
where the elements of the (3 × 3)-matrix F are given by FM M 0 =
det pairs
M
γIJ
(t, f )γiM (t, f )
dt
PI (f )PJ (f )
• Work in progress
– Letting eLISA evolve on its orbit and correlating the signals at different time
⇒ Valid for a network of independant detectors in space (3 LISA-like detectors would be suffi-
– Correlating eLISA signal with future earth-based detectors around 1Hz
cient), in the low frequency limit, and for a full polarized GWB
⇒ “Static” system (the relative position of the detectors in the network doesn’t change)
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