A frequency comb search for Sco X-1 L. Sammut1, A. Melatos1, C. Messenger2, B. Owen3 1University

of Melbourne, Australia, 2Albert Einstein Institute, Germany, 3Pennsylvania State University, USA [email protected]

Abstract We evaluate critically a semi-coherent gravitational wave search technique for continuous wave sources in binary systems, a technique called the frequency comb. The comb search is sensitive to the frequency and amplitude modulations produced by the orbital motion of the binary and the rotation of the earth, which generate sidebands in the detection statistic. It is implemented by summing incoherently the standard F-statistic, evaluated at each of these sidebands, with suitable weights. The sensitivity of the search is quantified with the aid of simulations on synthetic noisy data, and the results are compared with analytic estimates. The comb search is best suited to binaries with known sky position but unknown spin frequency, such as low mass x-ray binaries. Sco X-1, the brightest x-ray source in the sky, is a good candidate of this kind. Indirect limits on the signal from Sco X-1, e.g. from Bildsten's stalling hypothesis, are revised to include new effects related to the nuclear equation of state, surface magnetic geometry, and the accretion disk-magnetosphere interaction.

Comb Search

Application

The comb search [1,2] utilises the F-statistic as a first stage pass in a hierarchical search for known binary systems. It is sensitive to the amplitude and frequency modulation in the signal caused by the orbital motion of the binary, which generate sidebands in the detection statistic.

The search is well suited to binary systems with known sky positions but unknown spin frequencies, such as quasi periodic oscillation emitting low mass x-ray binaries (QPO LMXBs), in which a neutron star is orbited by a Roche lobe-filling companion. X-ray timing observations have revealed that the observed spin frequencies (νs) range from around 45Hz to a cut-off at around 700Hz, significantly less than the centrifugal break-up frequency for most equations of state [4]. A popular suggestion to explain this observation is that the angular momentum added by accretion is lost to gravitational radiation at the same rate [5]. This theory provides a reasonable explanation for the observed cut-off at 700Hz. The x-ray flux and spin frequency (which can only be guessed in QPO sources) are used to get a limit on the gravitational wave strain hc of GWs from LMXBs. Figure 1 shows the detectability of signals from several different types of GW sources. Sco X-1, marked by a star, is clearly seen above all the other sources.

The Detection Statistic The F-statistic is a detection statistic derived from the method of maximum likelihood, used in matched filtering for continuous gravitational wave (GW) searches [3]. It is defined by 2 F = Aˆ i ( x || hi ). (1) The data x(t) are represented as a combination of signal s(t)=ΣAihi(t) and noise n(t) where hi are the wave strain amplitudes and Ai are the time-independent signal amplitudes of the plus and cross polarizations in quadrature phases. Âi represents maximum likelihood estimators of these amplitudes. The inner T product is defined by 2 s ( x || hi ) = ∫ dtW (t ) x(t )hi (t ). (2) T0 0 where W(t) is the observational window function (0 or 1 when the detector is off or on), Ts is the observational time span for individual windows and T0 is the observational time. The F-statistic can also be described in terms of this window function by Z

2F =

T0 ~ Κ ∑ J n2 ( Z ) | W ( f − f n ) |2 2S h n =− Z

The Frequency Comb The frequency comb is shown in Figure 2. It shows the frequency modulated (FM) sidebands separated by 1/Porb Hz, where Porb is the orbital period. It also shows the amplitude modulation (AM) due to the rotation of the Earth with separation of 1/(siderealday)Hz. The position of the sidebands in frequency space are given by fn = f0 + n/Porb where n is an integer in the range [–Z to Z].

(3)

where Sh is the one-sided noise spectral density, Κ is a time and frequency independent constant containing the time averaged antenna pattern functions ~ and the amplitudes Ai, W is the Fourier transform of the window function and Z equals 2π f0 a, where f0 is the intrinsic GW frequency and a is the light crossing time of the orbital radius.

Input: GW timeseries data

Demodulate for sky position

FM

1/ day

Stage Stage 22

Stage Stage 33

Frequency comb

Coherent follow-up search on candidates indentified in second stage

Incoherent summation of F-statistic sidebands

Output: GW detection (or upper limits)

Frequency dependent detection statistic

Summary

AM

Search Pipeline Stage Stage 11

Figure 1: Best case detectability of GW sources for a coherent search over T0=2years. Source strengths assume the GW torque balances accretion. The three kinds of sources shown are accreting millisecond pulsars (boxes), type 1 x-ray bursters (green upright triangles) with confirmed (solid) and unconfirmed (open) frequencies, and QPO sources (red downward triangles). Sco X-1 is highlighted with a star. Detectability thresholds for Initial LIGO (I-LIGO, dotted), Enhanced LIGO (ELIGO), Advanced LIGO (A-LIGO) and the Einstein Telescope (ET) are shown along with those of Advanced LIGO in narrow band (A-LIGO NB) mode as well as a curve for the thermal noise floor alone (A-LIGO NB TH) [6].

2F

Hz Artificial data Wide bandwidth Fstatistic search Detection statistic

A successful detection of a source like Sco X-1 will help to determine the origin of the GW quadrupole, i.e. whether it is a mountain or r-mode. The polarisation and frequency content of the signal will help to determine: • a more accurate equation of state • the electrical resistivity of the neutron star crust • the surface magnetic field • the crustal breaking strain and • the thermal conductivity A detection will also shed light on the accretion disk-magnetospheric interaction, in particular • the size of the Alfven radius. A nondetection will yield upper or lower limits on many of these quantities. References

f − f0 (Hz)

1/P

orb

Hz

Figure 2: Detection statistic 2F of the comb search showing amplitude and frequency modulated sidebands where f0 = 0.01 Hz, a = 200 s, and T0 = 10 days.

1.

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3.

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4.

Chakrabarty D., et. al., 2003, Nautre, 424, 42

5.

Bildsten L., 1998, Astrophys. J., 501, L89

6.

Watts A., et. al., 2008, Mon. Not. R. Astron. Astrophys., 389(2), 839