Directional limits on persistent gravitational waves using LIGO S5 science data
The gravitational-wave (GW) sky may include nearby pointlike sources as well as astrophysical and cosmological stochastic backgrounds. Since the relative strength and angular distribution of the many possible sources of GWs are not well constrained, searches for GW signals must be performed in a model-independent way. To that end we perform two directional searches for persistent GWs using data from the LIGO S5 science run: one optimized for pointlike sources and one for arbitrary extended sources. The latter result is the first of its kind. Finding no evidence to support the detection of GWs, we present 90% confidence level (CL) upper-limit maps of GW strain power with typical values between and for pointlike and extended sources respectively. The limits on pointlike sources constitute a factor of improvement over the previous best limits. We also set 90% CL limits on the narrow-band root-mean-square GW strain from interesting targets including Sco X-1, SN1987A and the Galactic Center as low as in the most sensitive frequency range near . These limits are the most constraining to date and constitute a factor of improvement over the previous best limits.
One of the most ambitious goals of gravitational-wave (GW) astronomy is to measure the stochastic cosmological gravitational-wave background (CGB), which can arise through a variety of mechanisms including amplification of vacuum fluctuations following inflation Kolb and Turner (1994), phase transitions in the early universe Starobinskii (1979); Bar-Kana (1994), cosmic strings Kibble (1976); Damour and Vilenkin (2005) and pre-Big Bang models Buonanno (1997); Mandic and Buonanno (2006). The CGB may be masked by an astrophysical gravitational-wave background (AGB), interesting in its own right, which can arise from the superposition of unresolved sources such as core-collapse supernovae Howell et al. (2004); Ferrari et al. (1999a), neutron-star excitations Ferrari et al. (1999b); Sigl (2006), binary mergers Regimbau and Chauvineaux (2007); Farmer and Phinney (2003), persistent emission from neutron stars Regimbau and de Freitas Pacheco (2001, 2006) and compact objects around supermassive black holes Barack and Cutler (2004); Sigl et al. (2007).
We present the results of two analyses using data from the LIGO S5 science run: a radiometer analysis optimized for pointlike sources and a spherical-harmonic decomposition analysis, which allows for arbitrary angular distributions. This work presents the first measurement of the GW sky in a framework consistent with an arbitrary extended source.
Ii LIGO Detectors and the S5 Science Run
We analyze data from LIGO’s and detectors (H1 and H2) in Hanford, WA and the detector (L1) in Livingston Parish, LA during the S5 science run, which took place between Nov. 5, 2005 and Sep. 30, 2007. During S5, both H1 and L1 reached a strain sensitivity of in the most sensitive region between Abbott et al. (2009a) and collected of coincident H1L1 and H2L1 data. S5 saw milestones in GW astronomy including limits on the emission of GWs from the Crab pulsar that surpass those inferred from the Crab’s spindown Abbott et al. (2008), as well as limits on the isotropic CGB that surpass indirect limits from Big Bang nucleosynthesis and the cosmic microwave background Abbott et al. (2009b). This work builds on Abbott et al. (2009b, 2007a).
Following Thrane et al. (2009); Abbott et al. (2007a) we present a framework for analyzing the angular distribution of GWs. We assume that the GW signal is stationary and unpolarized, but not necessarily isotropic. It follows that the GW energy density , can be expressed in terms of the GW power spectrum, :
Here is frequency, is sky location, is the critical density of the universe and is Hubble’s constant. We further assume that can be factored (in our analysis band) into an angular power spectrum, , and a spectral shape, , parameterized by the spectral index and reference frequency . We set to be in the sensitive range of the LIGO interferometers.
Our goal is to measure for two power-law signal models. In the cosmological model, (), which is predicted, e.g., for the amplification of vacuum fluctuations following inflation (see Maggiore (2000) and references therein). In the astrophysical model, (), which emphasizes the strain sensitivity of the LIGO detectors.
It is applicable to a GW sky dominated by a limited number of widely separated point sources. As the number of point sources is increased, however, the interferometer beam pattern will cause the signals to interfere and partly cancel. Thus, radiometer maps do not apply to extended sources. Since pointlike signals are expected to arise from astrophysical sources, we use for the radiometer analysis.
The spherical-harmonic decomposition (SHD) algorithm is used for both (cosmological) and (astrophysical) sources. It allows for the possibility of an extended source with an arbitrary angular distribution, characterized by spherical-harmonic coefficients such that
The series is cut off at , allowing for angular scale . The flexibility of the spherical-harmonic algorithm comes at the price of somewhat diminished sensitivity to point sources, and thus the two algorithms are complementary.
We choose so as to minimize the sky average of the product of and , where is the uncertainty associated with and is the typical angular area of a resolved patch of sky 111The data were first processed with ; the present method of choosing was then adopted a posteriori in order to more accurately model the angular resolution of the interferometer network.. Since (where is the number of independent parameters), this procedure amounts to choosing to maximize the sensitivity obtained by integrating over the typical search aperture (angular resolution). We obtain and for and , respectively. Since the search aperture becomes smaller at the higher frequencies emphasized by , is larger for than for .
Both algorithms can be framed in terms of a “dirty map”, , which represents the signal convolved the Fisher matrix, :
Here both the Greek indices and take on values of for the SHD algorithm and for the radiometer algorithm, for which we use the pixel basis. The two bases are related using spherical-harmonic basis functions:
, meanwhile, is the cross spectral density generated from the H1L1 or H2L1 pairs. and are the individual power spectral densities, and is the angular decomposition of the overlap reduction function , which characterizes the orientations and frequency response of the detectors Thrane et al. (2009):
Here characterizes the detector response of detector to a GW with polarization , is a basis function, is the speed of light and is the difference between the interferometer locations. A detailed discussion of these quantities can be found in Thrane et al. (2009).
In Thrane et al. (2009) it was shown that the maximum-likelihood estimators of GW power are given by . The inversion of is complicated by singular eigenvalues associated with modes to which the Hanford-Livingston (HL) detector network is insensitive. This singularity can be handled two ways. The radiometer algorithm assumes the signal is pointlike, implying that correlations between neighboring pixels can be ignored. Consequently, we can replace with to estimate the point source amplitude (see Eq. 2). (We note that pointlike sources create signatures in our sky maps that typically span several degrees or more; see Abbott et al. (2007a).)
The SHD algorithm, on the other hand, targets extended sources, so the full Fisher matrix must be taken into account. We regularize by removing a fraction, , of the modes associated with the smallest eigenvalues, to which the HL network is relatively insensitive. is known as the regularization cutoff. By removing some modes from the Fisher matrix, we obtain a regularized inverse Fisher matrix, , thereby introducing a bias, the implications of which are discussed below. For now, we note that the bias depends on the angular distribution of the signal.
We thereby obtain the estimators
We refer to as the “clean map” and as the “radiometer map.” We note that has units of whereas has units of .
In choosing one must balance the competing demands of reconstruction accuracy (sensitivity to the modes that are kept) with the bias associated with the modes that are removed. In practice, we do not know the bias associated with since it depends on the unknown signal distribution . Therefore, we choose a value of that tends to produce reliably reconstructed maps with minimal bias for simulated signals. Following Thrane et al. (2009), we use , which was shown to be a robust regularization cutoff for simulated signals including maps characterized by one or more point sources, dipoles, monopoles and an extended source clustered in the galactic plane (see Thrane et al. (2009)).
In the case of the SHD algorithm, we construct an additional statistic (see Thrane et al. (2009)),
which describes the angular scale of the clean map. The subtracted second term makes the estimator unbiased so that when no signal is present. The expected noise distribution of is highly non-Gaussian for small values of , and so the upper limits presented below are calculated numerically. The are analogous to similar quantities defined in the context of temperature fluctuations of the cosmic microwave background (see, e.g., Hinshaw et al. (2009)).
The analysis was performed using the S5 stochastic analysis pipeline. This pipeline has been tested with hardware and software injections, and the successful recovery of isotropic hardware injections is documented in Abbott et al. (2009b). The recovery of anisotropic software injections is demonstrated in Thrane et al. (2009). We parse time series into , Hann-windowed, 50%-overlapping segments, which are coarse-grained to achieve resolution. We apply a stationarity cut described in Abbott et al. (2007a), which rejects of the cross-correlated segments. We also mask frequency bins associated with instrumental lines (e.g., harmonics of the 60 Hz mains power, calibration lines and suspension-wire resonances) as well as injected, simulated pulsar signals. For we include frequency bins up to , so that is within of the minimum possible value. Thirty-three frequency bins are masked, corresponding to of the frequency bins between used in the broadband analyses. For additional details about the S5 stochastic pipeline, see Abbott et al. (2009b).
Iv Significance and upper limit calculations
In order to determine if there is a statistically significant GW signature, we are primarily interested in the significance of outliers—the highest signal-to-noise ratio (SNR) frequency bin or sky-map pixel. It is therefore necessary to calculate the expected noise probability distribution of the maximum SNR given many independent trials (when considering maximum SNR in a spectral band) and given many dependent trials (when considering maximum SNR for a sky map).
For independent frequency bins, the probability density function, , of maximum SNR, , is given by
Here we have assumed that the stochastic point estimate is Gaussian distributed. The Gaussianity of and , calculated by summing over many independent segments, is expected to arise due to the central limit theorem Allen and Romano (1999). Additionally, we find the Gaussian-noise hypothesis to be consistent with time-slide studies, wherein we perform the cross-correlation analysis with an unphysical time-shift in order to wash out astrophysical signals and thereby obtain different realizations of detector noise.
The distribution of maximum SNR for a sky map is more subtle due to the non-zero covariances that exist between different pixels (or patches) on the sky. For this case, we calculate numerically, by simulating many realizations of dirty maps that have expected covariances described by the Fisher matrix . Figure 1 shows the numerically determined for the clean map generated with Gaussian noise.
The likelihood function for at each point in the sky can be be described as a normal distribution with mean and a variance . In the case of the SHD algorithm, regularization introduces a signal-dependent bias. Without knowing the true distribution of , it is impossible to know the bias exactly, but it is possible to set a conservative upper limit by assuming that on average the modes removed through regularization contain no more GW power than the modes that are kept.
To implement this assumption, we calculate with a regularization scheme that sets eigenvalues of removed modes to zero, whereas is conservatively calculated using a regularization scheme that sets eigenvalues of removed modes to the average eigenvalue of the kept modes. This has the effect of widening the likelihood function at each sky location. The upper limits become on average larger than they would be if we had calculated using the same regularization scheme as .
Following the same procedure as in Abbott et al. (2009b), we marginalize over the H1, H2, and L1 calibration uncertainties, which were measured to be 10%, 10%, and 13%, respectively Abadie et al. (2010) 222We follow the marginalization scheme used in Abbott et al. (2009b), but note that this does not take into account covariance between baselines with a shared detector. Because the H2L1 baseline only contributes about 10% of the sensitivity, and the calibration uncertainty is on the order of 10% to start with, this effect is only on the order of 1%. Work is ongoing to take this effect into account, which we expect to be more important for baselines with comparable sensitivities. The posterior distribution is obtained by multiplying the marginalized likelihood function by a prior, which we take to be flat above 333A prior constructed from Abbott et al. (2007a) would be nearly flat anyway since the strain sensitivity has improved ten-fold since the S4 science run.. The Bayesian upper limits are then determined by integrating the posterior out to the value of which includes 90% of the total area under the distribution. The calculation of upper limits on is analogous except we need not take into account the effects of regularization.
Sky maps: Figure 2 presents sky map results for the different analyses: SHD algorithm with (left), SHD with (center), and radiometer with (right). The top row contains SNR maps. The maximum SNR values are (with significance ), (with ), and (with ) respectively. These -values take into account the number of search directions and covariances between different sky patches (see IV). Observing no evidence of GWs, we set upper limits on GW power as a function of direction. The 90% confidence level (CL) upper limit maps are given in the bottom row. For the SHD method with , the limits are between ; for SHD with , the limits are between ; and for the radiometer with , the limits are between .
The strain power limits can also be expressed in terms of the GW energy flux per unit frequency Abbott et al. (2007a):
(Radiometer energy flux is obtained by replacing with .) The corresponding values are and for the SHD method, and for the radiometer. The radiometer map constitutes a factor of improvement over the previous best strain power limits Abbott et al. (2007a).
When comparing the SHD analysis with and the radiometer upper limits obtained using the same spectrum, it is important to note that these maps have different units. The radiometer map has units of because the radiometer analysis effectively integrates the power from a GW point source over solid angle. The SHD maps, on the other hand, have units of . If we scale the SHD limit maps by the typical diffraction limited resolution (), then the limits are more comparable. The radiometer algorithm limits are lower (by a factor of ) because it requires a stronger assumption about the signal model (a single point source), whereas the SHD algorithm is model-independent.
Figure 3 show 90% CL upper limits on the . Since the have units of strain power (), the have the somewhat unusual units of .
Targeted searches: Sco X-1 is a nearby () low-mass X-ray binary likely to include a neutron star spun up through accretion. Its spin frequency is unknown. It has been suggested that this accretion torque is balanced by GW emission Chakrabarty et al. (2003). The Doppler line broadening due to the orbital motion is smaller than the chosen bin width for frequencies below Steeghs and Casares (2002). At higher frequencies, the signal is certain to span two bins. We determine the maximum value of SNR in the direction of Sco X-1 to be at , which has a significance of given the independent frequency bins. Thus in Fig. 4 (first panel) we present limits on root-mean-square (RMS) strain, , as a function of frequency in the direction of Sco X-1 . These limits improve on the previous best limits by a factor of Abbott et al. (2007a). RMS strain is related to narrow-band GW power via
and is better suited for comparison with searches for periodic GWs, which typically constrain the peak strain amplitude, , marginalized over neutron star parameters and (see, e.g., Abbott et al. (2007b)). Our limits on are for a circularly polarized signal from a pulsar whose spin axis is aligned with the line of sight. Marginalizing over and and converting from RMS to peak amplitude causes the limits to change by a sky-dependent factor of Messenger (2010). We note that these limits are on the RMS strain in each bin as opposed to the total RMS strain from Sco X-1, which might span as many as two bins. The frequency axis refers to the observed GW frequency as opposed to the intrinsic GW frequency.
We also look for statistically significant outliers associated with the Galactic Center and SN1987A . The maximum SNR values are at with and at with , respectively. Limits on RMS strain are given in the right panel of Fig. 4.
We performed two directional analyses for persistent GWs: the radiometer analysis, which is optimized for point sources, and the complementary spherical-harmonic decomposition (SHD) algorithm, which allows for arbitrary extended sources. Neither analysis finds evidence of GWs. Thus we present upper-limit maps of GW power and also limits on the RMS strain from Sco X-1, the Galactic Center and SN1987A. The radiometer map limits improve on the previous best limits Abbott et al. (2007a) by a factor of 30 in strain power, and limits on RMS strain from Sco X-1 constitute a factor of 5 improvement in strain over the previous best limits Abbott et al. (2007a). The SHD clean maps represent the first effort to look for anisotropic extended sources of GWs.
With the ongoing construction of second-generation GW interferometers, we are poised to enter a new era in GW astronomy. Advanced detectors G. Harry for the LIGO Science Collaboration (2010); J R Smith for the LIGO Science Collaboration (2009); F. Acernese for The Virgo Collaboration (2006); K Kuroda for the LCGT Collaboration (2010) are expected to achieve strain sensitivities approximately times lower than initial LIGO, and advances in seismic isolation are expected to extend the frequency band down from to G. Harry for the LIGO Science Collaboration (2010). By adding additional detectors to our network, we expect to reduce degeneracies in the Fisher matrix and improve angular resolution. These improvements will allow advanced detector networks to probe plausible models of astrophysical stochastic foregrounds and some cosmological models such as cosmic strings.
Acknowledgements.The authors gratefully acknowledge the support of the United States National Science Foundation for the construction and operation of the LIGO Laboratory, the Science and Technology Facilities Council of the United Kingdom, the Max-Planck-Society, and the State of Niedersachsen/Germany for support of the construction and operation of the GEO600 detector, and the Italian Istituto Nazionale di Fisica Nucleare and the French Centre National de la Recherche Scientifique for the construction and operation of the Virgo detector. The authors also gratefully acknowledge the support of the research by these agencies and by the Australian Research Council, the International Science Linkages program of the Commonwealth of Australia, the Council of Scientific and Industrial Research of India, the Istituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria d’Economia Hisenda i Innovació of the Govern de les Illes Balears, the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, the Polish Ministry of Science and Higher Education, the FOCUS Programme of Foundation for Polish Science, the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, The National Aeronautics and Space Administration, the Carnegie Trust, the Leverhulme Trust, the David and Lucile Packard Foundation, the Research Corporation, and the Alfred P. Sloan Foundation. This is LIGO document #P1000031.
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