Signal - Background .1 Polarizations.1Polarizations

DETECTING BEYOND-EINSTEIN POLARIZATIONS OF CONTINUOUS GRAVITATIONAL WAVES

7.2 Background .1 Polarizations.1Polarizations

7.2.2 Signal

Because of their persistence, continuous gravitational waves (CWs) provide the means to study GW polarizations without the need for multiple detectors. For the same reason, continuous signals can be integrated over long periods of time, thus

Table 7.1: LIGO detectors [147][148]

LHO LLO VIR

Latitude (λ) 46.45^◦N 30.56^◦N 43.63^◦N Longitude (φ) 119.41^◦W 90.77^◦W 10.5^◦E Orientation (γ) 125.99^◦ 198.0^◦ 71.5^◦

improving the likelihood of detection. Furthermore, these GWs are quasi-sinusoidal and present well-defined frequencies. This allows us to focus on the amplitude modulation, where the polarization information is contained.

CWs are produced by localized sources with periodic motion, such as binary systems or spinning neutron stars [146]. Throughout this paper, we target known pulsars (e.g., the Crab pulsar) and assume an asymmetry in their moment of inertia (rather than precession of the spin axis or other possible, but less likely, mechanisms) causes them to emit gravitational radiation. A source of this type can generate GWs only at multiples of its rotational frequencyν. In fact, it is expected that most power be radiated at twice this value [54]. For that reason, we take the GW frequency,ν_gw, to be 2ν. Moreover, the frequency evolution of these pulsars is well-known thanks to electromagnetic observations, mostly at radio wavelengths but also in gamma-rays.

Simulation of a CW from a triaxial neutron star is straightforward. The general form of a such a signal is:

h(t)= Õ

Ap(t;ψ|α, δ, λ, φ, γ, ξ)hp(t;ι,h₀, φ₀, ν,ν,Û νÜ), (7.1) where, for each polarization p, Ap is the detector response (antenna pattern) and hpa sinusoidal waveform of frequency ν_gw = 2ν. The detector parameters are: λ, longitude; φ, latitude; γ, angle of the detectorx-arm measured from East; and ξ, the angle between arms. Values for the LIGO Hanford Observatory (LHO), LIGO Livingston Observatory (LLO) and Virgo (VIR) detectors are presented in Table 7.1. The source parameters are: ψ, the signal polarization angle; ι, the inclination of the pulsar spin axis relative to the observer’s line-of-sight;h₀, an overall amplitude factor; φ₀, a phase offset; and ν, the rotational frequency, with νÛ, νÜ its first and second derivatives. Also,αis the right ascension andδ the declination of the pulsar in celestial coordinates.

Note that the inclination angleιis defined as is standard in astronomy, withι= 0 and ι= π respectively meaning that the angular momentum vector of the source points towards and opposite to the observer. The signal polarization angleψis related to

Table 7.2: Axis polarization (ψ) and inclination (ι) angles for known pulsars [149].

ψ (deg) ι(deg)

Crab 124.0 61.3 Vela 130.6 63.6

J1930+1852 91 147

J2229+6114 103 46

B1706−44 163.6 53.3

J2021+3651 45 79

ψ (deg) ι(deg)

J0205+6449 90.3 91.6 J0537−6910 131 92.8 B0540−69 144.1 92.9 J1124−5916 16 105

B1800−21 44 90

J1833−1034 45 85.4 the position angle of the source, which is in turn defined to be the East angle of the projection of the source’s spin axis onto the plane of the sky.

Although there are hundreds of pulsars in the LIGO band, in the majority of cases we lack accurate measurements of their inclination and polarization angles. The few exceptions, presented in Table 7.2, were obtained through the study of the pulsar spin nebula [149]. This process cannot determine the spin direction, only the orientation of the spin axis. Consequently, even for the best studied pulsars,ψ andιare only known modulo a reflection: we are unable to distinguish between ψ and −ψ or betweenιandπ−ι. As will be discussed in Sec. 7.3, our ignorance ofψandιmust be taken into account when searching for CWs.

Frequency evolution

In Eq. (7.1),hp(t)is a sinusoid carrying the frequency modulation of the signal:

hp(t)= apcos

φ(t)+φp+φ^gw

(7.2) φ(t)=4π

νt_b+ 1 2

νtÛ ²

b + 1 6

νtÜ ³

+φ^em

0 , (7.3)

wheret_bis the Solar System barycentric arrival time, which is the local arrival timet modulated by the standard Rømer∆_R, Einstein∆_E and Shapiro∆_S delays [56]:

t_b =t+∆_R+∆_E +∆_S. (7.4)

The leading factor of four in the r.h.s. of Eq. (7.3) comes from the substitution ν_gw = 2ν. For known pulsars,φ^em

0 is the phase of the radio pulse, whileφ^gw

0 is the phase difference between electromagnetic and gravitational waves. Both factors contribute to an overall phase offset of the signal (φ^em

0 +φ^gw

0 ). This is of astrophysical significance since it may provide insights about the relation between EM & GW radiation and provide information about the physical structure of the source.

The ap andφpcoefficients in Eq. (7.2) respectively encode the relative amplitude and phase of each polarization. These values are determined by the physical model.

For instance, GR predicts:

a₊ = h₀(1+cos²ι)/2, φ₊ = 0, (7.5) a× = h₀cosι , φ× = −π/2, (7.6) whilea_x= a_y= a_b=0. On the other hand, according to G4v [106]:

a_x= h₀sinι , φ_x= −π/2, (7.7) a_y= h₀sinιcosι , φ_x= 0. (7.8) whilea₊ =a× = a_b= 0. In both cases, the overall amplitudeh₀can be characterized by [54, 57, 106]:

h₀= 4π²G c⁴

Izzν²

r , (7.9)

where r is the distance to the source, Izz the pulsar’s moment of inertia along the principal axis, = (Ix x− Iy y)/Izz its equatorial ellipticity and, as before, ν is the rotational frequency. Choosing some canonical values,

h₀≈ 4.2×10⁻²⁶ Izz

10²⁸kg m² h ν

100 Hz

i₂1 kpc r

10⁻⁶

, (7.10)

it is easy to see that GWs from triaxial neutron stars are expected to be relatively weak [150]. However, the sensitivity to these waves grows with the observation time because the signal can be integrated over long periods of time [57].

As indicated in the introduction to this section, we have assumed CWs are caused by an asymmetry in the moment of inertia of the pulsar. Other mechanisms, such as precession of the spin axis, are expected to produce waves of different strengths and with dominant components at frequencies other than 2ν. Furthermore, these effects vary between theories: for instance, in G4v, if the asymmetry is not perpendicular to the rotation axis, there can be a significantνcomponent as well as the 2νcomponent.

In those cases, Eqs. (7.2, 7.9) do not hold (e.g., see [54] for precession models).

Amplitude modulation

At any given time, GW detectors are not equally sensitive to all polarizations. The response of a detector to a particular polarization pis encoded in a function Ap(t)

depending on the relative locations and orientations of the source and detector. As seen from Eq. (7.1), these functions provide the amplitude modulation of the signal.

A GW is best described in an orthogonal coordinate frame defined by wave vectors (wx, w_y, wz), withwz = wx ×w_ybeing the direction of propagation. Furthermore, the orientation of this wave-frame is fixed by requiring that the East angle between w_yand the celestial North beψ. In this gauge, the different polarizations act through six orthogonal basis strain tensors [107, 108]:2

e⁺_{j k} = ©

1 0 0

0 −1 0

0 0 0

, e^×_{j k} =©

0 1 0 1 0 0 0 0 0 ª

, (2,3)

e^x_{j k} = ©

0 0 1 0 0 0 1 0 0 ª

, e^y_{j k} = ©

0 0 0 0 0 1 0 1 0 ª

, (4,5)

e^b_{j k} =©

1 0 0 0 1 0 0 0 0 ª

, e^l_{j k} =√

0 0 0 0 0 0 0 0 1 ª

, (6,7)

with j,k indexing x, y and z components. These tensors can be written in an equivalent, frame-independent form

e⁺ = wx ⊗wx−wy ⊗wy, (7.17) e^× = wx ⊗wy+wy⊗wx, (7.18) e^x= wx⊗wz+wz⊗wx, (7.19) e^y =wy⊗wz+wz⊗wy, (7.20) e^b= wx ⊗wx+w_y ⊗w_y, (7.21)

e^l= √

2(wz⊗wz). (7.22)

If a detector is characterized by its unit arm-direction vectors (dx anddy, withdz the detector zenith), its differential-arm response Apto a wave of polarizationpis:

Ap= 1

2 dx ⊗dx −dy⊗dy

:e^p, (7.23)

2The normalization fore^lused here is unique to this chapter: in the rest of this thesis we define e^l=diag(0,0,0,1), i.e. without the

√

2 prefactor.

where the colon indicates double contraction. As a result, Eqs. (2-13) imply:

A₊ = 1 2

(wx ·dx)²− (wx ·d_y)²− (w_y ·dx)²+(w_y·d_y)², (7.24) A× =(wx·dx)(w_y·dx) − (wx·d_y)(w_y ·d_y), (7.25) A_x =(wx·dx)(wz·dx) − (wx·d_y)(wz ·d_y), (7.26) A_y=(wy·dx)(wz·dx) − (wy·dy)(wz·dy), (7.27) A_b= 1

(wx·dx)²− (wx ·dy)²+(wy ·dx)²− (wy ·dy)²

, (7.28) A_l= 1

√ 2

(wz·dx)²− (wz·d_y)². (7.29) Accounting for the time dependence of the arm vectors due to the rotation of the Earth, Eqs. (7.24-7.29) can be used to compute Ap(t)for any value oft. In Fig. 7.1 we plot these responses for the LIGO Hanford Observatory (LHO) observing the Crab pulsar, over a sidereal day (the pattern repeats itself every day). Note that the b and l patterns are degenerate (A_b =−√

2A_l3), which means they are indistinguishable up to an overall constant.

Although the antenna patterns areψ-dependent, a change in this angle amounts to a rotation of A₊ into A× or of Ax into Ay, and vice-versa. If the orientation of the source is changed such that the new polarization isψ⁰ = ψ+∆ψ, whereψ is the original polarization angle and∆ψ ∈ [0,2π], it is easy to check that the new antenna patterns can be written [108]:

A⁰₊ = A₊cos 2∆ψ+ A×sin 2∆ψ, (7.30) A⁰_× = A×cos 2∆ψ− A₊sin 2∆ψ, (7.31) A⁰_x = Axcos∆ψ+Aysin∆ψ, (7.32) A⁰_y = A_ycos∆ψ− Axsin∆ψ, (7.33)

A⁰_b = Ab, (7.34)

A⁰_l = Al, (7.35)

and the tensor, vector and scalar nature of each polarization becomes evident from theψ dependence.