arXiv subjects

Bruce Allen

Publications and source records attributed to Bruce Allen.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the best lattice quantizers

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.

cs.IT↗

arXiv2023

The International Pulsar Timing Array checklist for the detection of nanohertz gravitational waves

Pulsar timing arrays (PTAs) provide a way to detect gravitational waves at nanohertz frequencies. In this band, the most likely signals are stochastic, with a power spectrum that rises steeply at lower frequencies. Indeed, the observation of a common red noise process in pulsar-timing data suggests that the first credible detection of nanohertz-frequency gravitational waves could take place within the next few years. The detection process is complicated by the nature of the signals and the noise: the first observational claims will be statistical inferences drawn at the threshold of detectability. To demonstrate that gravitational waves are creating some of the noise in the pulsar-timing data sets, observations must exhibit the Hellings and Downs curve -- the angular correlation function associated with gravitational waves -- as well as demonstrating that there are no other reasonable explanations. To ensure that detection claims are credible, the International Pulsar Timing Array (IPTA) has a formal process to vet results prior to publication. This includes internal sharing of data and processing pipelines between different PTAs, enabling independent cross-checks and validation of results. To oversee and validate any detection claim, the IPTA has also created an eight-member Detection Committee (DC) which includes four independent external members. IPTA members will only publish their results after a formal review process has concluded. This document is the initial DC checklist, describing some of the conditions that should be fulfilled by a credible detection. At the present time none of the PTAs have a detection claim; therefore this document serves as a road map for the future.

astro-ph.IM↗

arXiv2023

Variance of the Hellings-Downs Correlation

Gravitational waves (GWs) create correlations in the arrival times of pulses from different pulsars. The expected correlation $μ(γ)$ as a function of the angle $γ$ between the directions to two pulsars was calculated by Hellings and Downs for an isotropic and unpolarized GW background, and several pulsar timing array (PTA) collaborations are working to observe these. We ask: given a set of noise-free observations, are they consistent with that expectation? To answer this, we calculate the expected variance $σ^2(γ)$ in the correlation for a single GW point source, as pulsar pairs with fixed separation angle $γ$ are swept around the sky. We then use this to derive simple analytic expressions for the variance produced by a set of discrete point sources uniformly scattered in space for two cases of interest: (1) point sources radiating GWs at the same frequency, generating confusion noise, and (2) point sources radiating GWs at distinct non-overlapping frequencies. By averaging over all pulsar sky positions at fixed separation angle $γ$, we show how this variance may be cleanly split into cosmic variance and pulsar variance, also demonstrating that measurements of the variance can provide information about the nature of GW sources. In a series of technical appendices, we calculate the mean and variance of the Hellings-Downs correlation for an arbitrary (polarized) point source, quantify the impact of neglecting pulsar terms, and calculate the pulsar and cosmic variance for a Gaussian ensemble. The mean and variance of the Gaussian ensemble may also be obtained from the previous discrete-source confusion-noise model in the limit of a high density of weak sources.

gr-qc↗

arXiv2023

The Voronoi Region of the Barnes-Wall Lattice $Λ_{16}$

We give a detailed description of the Voronoi region of the Barnes-Wall lattice $Λ_{16}$, including its vertices, relevant vectors, and symmetry group. The exact value of its quantizer constant is calculated, which was previously only known approximately. To verify the result, we estimate the same constant numerically and propose a new very simple method to quantify the variance of such estimates, which is far more accurate than the commonly used jackknife estimator.

cs.IT↗

arXiv2022

Performance of Random Template Banks

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, called a template bank. These may be constructed systematically as a lattice, or alternatively, by placing templates at randomly selected points in parameter space. Here, we calculate the fraction of signals lost by an $n$-dimensional random template bank (compared to a very finely spaced bank). This fraction is compared to the corresponding loss fraction for the best possible lattice-based template banks containing the same number of grid points. For dimensions $n<4$ the lattice-based template banks significantly outperform the random ones. However, remarkably, for dimensions $n>8$, the difference is negligible. In high dimensions, random template banks outperform the best known lattices.

gr-qc↗

arXiv2021

New searches for continuous gravitational waves from seven fast pulsars

We conduct searches for continuous gravitational waves from seven pulsars, that have not been targeted in continuous wave searches of Advanced LIGO data before. We target emission at exactly twice the rotation frequency of the pulsars and in a small band around such frequency. The former search assumes that the gravitational wave quadrupole is changing phase-locked with the rotation of the pulsar. The search over a range of frequencies allows for differential rotation between the component emitting the radio signal and the component emitting the gravitational waves, for example the crust or magnetosphere versus the core. Timing solutions derived from the Arecibo 327-MHz Drift-Scan Pulsar Survey (AO327) observations are used. No evidence of a signal is found and upper limits are set on the gravitational wave amplitude. For one of the pulsars we probe gravitational wave intrinsic amplitudes just a factor of 3.8 higher than the spin-down limit, assuming a canonical moment of inertia of $10^{38}$ kg m$^2$. Our tightest ellipticity constraint is $1.5 \times 10^{-8}$, which is a value well within the range of what a neutron star crust could support.

astro-ph.HE↗

arXiv2021

The optimal lattice quantizer in nine dimensions

The optimal lattice quantizer is the lattice which minimizes the (dimensionless) second moment $G$. In dimensions $1$ to $8$, it has been proven that the optimal lattice quantizer is one of the classical lattices, or there is good evidence for this. In contrast, more than two decades ago, convincing numerical studies showed that in dimension $9$, a non-classical lattice is optimal. The structure and properties of this lattice depend upon a real parameter $a>0$, whose value was only known approximately. Here, we give a full description of this one-parameter family of lattices and their Voronoi cells, and calculate their (scalar and tensor) second moments analytically as a function of $a$. The value of $a$ which minimizes $G$ is an algebraic number, defined by the root of a $9$th order polynomial, with $a \approx 0.573223794$. For this value of $a$, the covariance matrix (second moment tensor) is proportional to the identity, consistent with a theorem of Zamir and Feder for optimal quantizers. The structure of the Voronoi cell depends upon $a$, and undergoes phase transitions at $a^2 = 1/2$, $1$ and $2$, where its geometry changes abruptly. At each transition, the analytic formula for the second moment changes in a very simple way. Our methods can be used for arbitrary one-parameter families of layered lattices, and may thus provide a useful tool to identify optimal quantizers in other dimensions as well.

math-ph↗

arXiv2021

Optimal Template Banks

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Voronoï) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

astro-ph.IM↗

arXiv2021

Results from an Einstein@Home search for continuous gravitational waves from G347.3 at low frequencies in LIGO O2 data

We present results of a search for periodic gravitational wave signals with frequency between 20 and 400 Hz, from the neutron star in the supernova remnant G347.3-0.5, using LIGO O2 public data. The search is deployed on the volunteer computing project Einstein@Home, with thousands of participants donating compute cycles to make this endevour possible. We find no significant signal candidate and set the most constraining upper limits to date on the amplitude of gravitational wave signals from the target, corresponding to deformations below $10^{-6}$ in a large part of the band. At the frequency of best strain sensitivity, near $166$ Hz, we set 90\%\ confidence upper limits on the gravitational wave intrinsic amplitude of $h_0^{90\%}\approx 7.0\times10^{-26}$. Over most of the frequency range our upper limits are a factor of 20 smaller than the indirect age-based upper limit.

gr-qc↗

arXiv2021

Template banks based on $\mathbb{Z}^n$ and $A_n^*$ lattices

Matched filtering is a traditional method used to search a data stream for signals. If the source (and hence its $n$ parameters) are unknown, many filters must be employed. These form a grid in the $n$-dimensional parameter space, known as a template bank. It is often convenient to construct these grids as a lattice. Here, we examine some of the properties of these template banks for $\mathbb{Z}^n$ and $A_n^*$ lattices. In particular, we focus on the distribution of the mismatch function, both in the traditional quadratic approximation and in the recently-proposed spherical approximation. The fraction of signals which are lost is determined by the even moments of this distribution, which we calculate. Many of these quantities we examine have a simple and well-defined $n\to\infty$ limit, which often gives an accurate estimate even for small $n$. Our main conclusions are the following: (i) a fairly effective template-based search can be constructed at mismatch values that are shockingly high in the quadratic approximation; (ii) the minor advantage offered by an $A_n^*$ template bank (compared to $\mathbb{Z}^n$) at small template separation becomes even less significant at large mismatch. So there is little motivation for using template banks based on the $A_n^*$ lattice.

astro-ph.IM↗

arXiv2020

Search for Continuous Gravitational Waves from the Central Compact Objects in Supernova Remnants Cassiopeia A, Vela Jr. and G347.3-0.5

We perform a sub-threshold follow-up search for continuous nearly-monochromatic gravitational waves from the central compact objects associated with the supernova remnants Vela Jr., Cassiopeia A, and SNR G347.3$-$0.5. Across the three targets, we investigate the most promising ~ 10,000 combinations of gravitational wave frequency and frequency derivative values, based on the results from an Einstein@Home search of the LIGO O1 observing run data, dedicated to these objects. The selection threshold is set so that a signal could be confirmed using the newly released O2 run LIGO data. In order to achieve best sensitivity we perform two separate follow-up searches, on two distinct stretches of the O2 data. Only one candidate survives the first O2 follow-up investigation, associated with the central compact object in SNR G347.3-0.5, but it is not conclusively confirmed. In order to assess a possible astrophysical origin we use archival X-ray observations and search for amplitude modulations of a pulsed signal at the putative rotation frequency of the neutron star and its harmonics. This is the first extensive electromagnetic follow-up of a continuous gravitational wave candidate performed to date. No significant associated signal is identified. New X-ray observations contemporaneous with the LIGO O3 run will enable a more sensitive search for an electromagnetic counterpart. A focused gravitational wave search in O3 data based on the parameters provided here should be easily able to shed light on the nature of this outlier. Noise investigations on the LIGO instruments could also reveal the presence of a coherent contamination.

astro-ph.HE↗

arXiv2020

Gravitational wave stochastic background from cosmological particle decay

We assume that the cosmological dark matter is composed of massive neutral scalar particles that decay into two massless particles. The decay produces a stochastic background of gravitational waves (GWs) via a 'memory effect' mechanism. We calculate the spectral amplitude and slope of the resulting background, which is frequency independent (flat). We discuss its potential observability and show that the resulting background might dominate the cosmological GW background at frequencies above about 10 GHz. Penrose has proposed a cosmological model in which dark matter particles have the Planck mass and decay into two gravitons. For these, the spectrum has an additional 'direct' contribution from the decay products, which we also calculate. At low frequencies, this direct contribution also has a flat spectrum but with a much smaller amplitude than the memory part.

gr-qc↗

arXiv2019

Spherical ansatz for parameter-space metrics

A fundamental quantity in signal analysis is the metric $g_{ab}$ on parameter space, which quantifies the fractional "mismatch" $m$ between two (time- or frequency-domain) waveforms. When searching for weak gravitational-wave or electromagnetic signals from sources with unknown parameters $λ$ (masses, sky locations, frequencies, etc.) the metric can be used to create and/or characterize "template banks". These are grids of points in parameter space; the metric is used to ensure that the points are correctly separated from one another. For small coordinate separations $dλ^a$ between two points in parameter space, the traditional ansatz for the mismatch is a quadratic form $m=g_{ab} dλ^a dλ^b$. This is a good approximation for small separations but at large separations it diverges, whereas the actual mismatch is bounded. Here we introduce and discuss a simple "spherical" ansatz for the mismatch $m=\sin^2(\sqrt{g_{ab} dλ^a dλ^b})$. This agrees with the metric ansatz for small separations, but we show that in simple cases it provides a better (and bounded) approximation for large separations, and argue that this is also true in the generic case. This ansatz should provide a more accurate approximation of the mismatch for semi-coherent searches, and may also be of use when creating grids for hierarchical searches that (in some stages) operate at relatively large mismatch.

gr-qc↗

arXiv2019

Results from an Einstein@Home search for continuous gravitational waves from Cassiopeia A, Vela Jr. and G347.3

We report results of the most sensitive search to date for periodic gravitational waves from Cassiopeia A, Vela Jr. and G347.3 with frequency between 20 and 1500 Hz. The search was made possible by the computing power provided by the volunteers of the Einstein@Home project and improves on previous results by a factor of 2 across the entire frequency range for all targets. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population, corresponding to sensitivity depths between 54 $[1/ {\sqrt{\textrm{Hz}}}]$ and 83 $[1/ {\sqrt{\textrm{Hz}}}]$, depending on the target and the frequency range. At the frequency of best strain sensitivity, near $172$ Hz, we set 90% confidence upper limits on the gravitational wave intrinsic amplitude of $h_0^{90\%}\approx 10^{-25}$, probing ellipticity values for Vela Jr. as low as $3\times 10^{-8}$, assuming a distance of 200 pc.

gr-qc↗

arXiv2019

Can a "pure vector" gravitational wave mimic a "pure tensor" one?

In the general theory of relativity, gravitational waves have two possible polarizations, which are transverse and traceless with helicity $\pm 2$. Some alternatives theories contain additional helicity $0$ and helicity $\pm 1$ polarization modes. Here, we consider a hypothetical "pure vector" theory in which gravitational waves have only two possible polarizations, with helicity $\pm 1$. We show that if these polarizations are allowed to rotate as the wave propagates, then for certain source locations on the sky, the strain outputs of three ideal interferometric gravitational wave detectors can exactly reproduce the strain outputs predicted by general relativity.

gr-qc↗

arXiv2017

Results of an all-sky high-frequency Einstein@Home search for continuous gravitational waves in LIGO's fifth science run

We present results of a high-frequency all-sky search for continuous gravitational waves from isolated compact objects in LIGO's 5th Science Run (S5) data, using the computing power of the Einstein@Home volunteer computing project. This is the only dedicated continuous gravitational wave search that probes this high frequency range on S5 data. We find no significant candidate signal, so we set 90%-confidence level upper-limits on continuous gravitational wave strain amplitudes. At the lower end of the search frequency range, around 1250 Hz, the most constraining upper-limit is $5.0\times 10^{-24}$, while at the higher end, around 1500 Hz, it is $6.2\times 10^{-24}$. Based on these upper-limits, and assuming a fiducial value of the principal moment of inertia of $10^{38}$kg$\,$m$^2$, we can exclude objects with ellipticities higher than roughly $2.8\times10^{-7}$ within 100 pc of Earth with rotation periods between 1.3 and 1.6 milliseconds.

gr-qc↗

arXiv2017

FINDCHIRP: an algorithm for detection of gravitational waves from inspiraling compact binaries

Matched-filter searches for gravitational waves from coalescing compact binaries by the LIGO Scientific Collaboration use the FINDCHIRP algorithm: an implementation of the optimal filter with innovations to account for unknown signal parameters and to improve performance on detector data that has nonstationary and non-Gaussian artifacts. We provide details on the FINDCHIRP algorithm as used in the search for subsolar mass binaries, binary neutron stars, neutron star-black hole binaries, and binary black holes.

gr-qc↗

arXiv2016

An Einstein@home search for continuous gravitational waves from Cassiopeia A

We report the results of a directed search for continuous gravitational-wave emission in a broad frequency range (between 50 and 1000 Hz) from the central compact object of the supernova remnant Cassiopeia A (Cas A). The data comes from the sixth science run of LIGO and the search is performed on the volunteer distributed computing network Einstein@Home. We find no significant signal candidate, and set the most constraining upper limits to date on the gravitational-wave emission from Cas A, which beat the indirect age-based upper limit across the entire search range. At around 170 Hz (the most sensitive frequency range), we set 90% confidence upper limits on the gravitational wave amplitude $h_0$ of $\sim\!\!~2.9\times 10^{-25}$, roughly twice as constraining as the upper limits from previous searches on Cas A. The upper limits can also be expressed as constraints on the ellipticity of Cas A; with a few reasonable assumptions, we show that at gravitational-wave frequencies greater than 300~Hz, we can exclude an ellipticity of $\gtrsim\!\!~10^{-5}$.

gr-qc↗