patrick r brady

• Home
• Calendar
• CurriculumVitae
• Photos
• Research
• Talks
• Teaching
• Travel

Preprint list

This is a list of recent preprints which have been submitted for publication.

---

Upper limits on burst sources of gravitational waves

Patrick R Brady, Jolien D E Creighton, L Samuel Finn and Alan G Wiseman

We present statistical methods for determining upper limit event rates on burst sources using gravitational wave detectors. By assuming that burst events are Poisson distributed in time, we devise two methods to estimate the event rate. The first is a generalization of the loudest event statistic used by Allen \textit{et al}.~\mbox{[}B. Allen \textit{et al}., Phys.\@ Rev.\@ Lett.\@ \textbf{83}, 1498 (1999)\protect\mbox{]} to determine an upper limit on the number of binary neutron star inspirals in the Galaxy using data taken in 1994 using the LIGO 40-meter prototype interferometer at Caltech. If the probability distribution of background events can be determined, this method could be used to improve the event rate limit. However, in the absence of accurate knowledge of the distribution in signal to noise ratio of the false alarms, we show that the loudest event statistic determines a robust upper bound. The second method assumes that one sets a detection threshold before performing the data analysis, and that the number of events one observes with signal to noise exceeding this threshold is $n$. If the threshold has been chosen so that the probability of a false event is low, then this method determines a lower limit than the loudest event statistic. Nevertheless, we argue that the loudest event statistic provides a robust estimator, and, as such, is to be preferred when the distribution of signal to noise due to background events is not well understood.

Preprint  number XXX,  to be submitted.

This work is supported in part by NSF grant PHY 9970821

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Black hole binaries in the corotating frame: techniques in a toy problem

Patrick R Brady, Jolien D E Creighton and Alan G Wiseman

Abstract

Preprint  number XXX,  in preparation.

This work is supported in part by NSF grant PHY 9970821

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

An excess power statistic for detection of burst sources of gravitational radiation

Warren G Anderson, Patrick R Brady, Jolien D E Creighton and Eanna Flanagan

We examine the properties of the excess power method proposed by Flanagan and Hughes \mbox{[}{\'E}.~{\'E}. Flanagan and S.~A. Hughes, Phys.\@ Rev.\@ D \textbf{57}, 4535 (1998)\mbox{]} to detect gravitational waves in interferometric detector data. This method is designed to detect short-duration ($\lesssim$ 0.1 s) burst signals of unkown waveform, such as those from supernovae or black hole mergers. If only the burst's duration and frequency band are known, the method is an optimal detection strategy in both Bayseian and frequentist senses. It consists of summing the data power over the known time interval and frequency band of the burst. If the detector noise is stationary and Gaussian, this sum is distributed as a $\chi^2$ (non-central $\chi^2$) deviate in the absence (presence) of a signal. These distributions provide a frequentist detection threshold for the measured power. Bayesian thresholds are also computed. More generically, only upper and/or lower bounds on the burst's duration and frequency band are known, and one must search for excess power in all concordant durations and bands. Two search schemes are presented and their computational efficiency compared. We find that given reasonable constraints on the effective duration and bandwidth of the signal, the search for excess power can be performed on a single workstation. Furthermore, the excess power method's effectiveness is comparable to a bank of matched filters. It can detect sources to more than half the distance seen by a bank of matched filters in all practical cases, and to approximately the same distance in a subtantial subset of them. Finally, we extend the method to a network of interferometers, first in the simple aligned interferometer case and then to more complicated unaligned networks.

Preprint  number XXX,  to be submitted.

This work is supported in part by NSF grant PHY 9970821

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Black-hole threshold solutions in stiff fluid collapse

Patrick R Brady, Carsten Gundlach and David Nielsen

Abstract

Preprint  number XXX,  in preparation.

This work is supported in part by NSF grant PHY 9970821

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Observational limit on gravitational waves from binary neutron stars in the Galaxy

B. Allen, K. Blackburn, P. Brady, J. Creighton, T. Creighton, S. Droz, A. Gillespie, S. Hughes, T. Lyons, J. Mason, B. Owen, F. Raab, B. Sathyaprakash, S. Whitcomb and A. Wiseman

We analyze 25 hours of data from the LIGO 40-meter prototype laser interferometric gravitational wave detector taken during the third week of November 1994. The instrument was sensitive enough to detect the gravitational-wave chirps that would be emitted by coalescing compact binary systems within our Galaxy. The data stream was searched using optimal matched filtering --- the first implementation/test of this technique on real interferometric data. These methods will be an important part of upcoming LIGO data analysis. An upper limit on the rate $R$ of neutron star binary inspirals in our Galaxy is obtained: with $90\%$ confidence, $R< 0.5/\mathrm{hour}$. Similar experiments with LIGO interferometers will provide constraints on the population of tight binary neutron star systems in the Universe.

Preprint  number NSF-ITP-99-39,  Physical  Review  D

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Radiative falloff in Schwarzschild-de Sitter spacetime

Patrick R. Brady, Chris M. Chambers, William G. Laarakkers and Eric Poisson
 

We consider the evolution of a scalar field propagating in Schwarzschild-de Sitter spacetime. The field is non-minimally coupled to curvature through a coupling constant $\xi$. The spacetime has two distinct timescales, $t_e = r_e/c$ and $t_c = r_c/c$, where $r_e$ is the radius of the black-hole horizon, $r_c$ the radius of the cosmological horizon, and $c$ the speed of light. When $r_c \gg r_e$, the field's time evolution can be separated into three epochs. At times $t \ll t_c$, the field behaves as if it is in pure Schwarzschild spacetime; the nature of spacetime far from the black hole has no influence on the evolution. In this early epoch, the field's initial outburst is followed by quasi-normal oscillations, and then by an inverse power-law decay. At times $t \lesssim t_c$, the power-law behavior gives way to a faster, exponential decay. In this intermediate epoch, the conditions at radii $r \gtrsim r_e$ and $r \lesssim r_c$ both play an important role. Finally, at times $t \gg t_c$, the field behaves as if it is in pure de Sitter spacetime; the nature of spacetime near the black hole no longer influences the evolution in a significant way. In this late epoch, the field's behavior depends on the value of the curvature-coupling constant $\xi$. If $\xi$ is less than a critical value $\xi_c = 3/16$, the field decays exponentially, with a decay constant that increases with increasing $\xi$. If $\xi > \xi_c$, the field oscillates with a frequency that increases with increasing $\xi$; the amplitude of the field still decays exponentially, but the decay constant is independent of $\xi$. We establish these properties using a combination of numerical and analytical methods.

Preprint  number NSF-ITP-99-07,  Physical  Review  D

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Searching for periodic sources with LIGO. II: Hierarchical searches

Patrick R. Brady and Teviet Creighton

The detection of quasi-periodic sources of gravitational waves requires the accumulation of signal-to-noise over long observation times. This represents the most difficult data analysis problem facing experimenters with detectors like those at LIGO. If not removed, Earth-motion induced Doppler modulations and intrinsic variations of the gravitational-wave frequency make the signals impossible to detect. These effects can be corrected (removed) using a parameterized model for the frequency evolution. In a previous paper, we introduced such a model and computed the number of independent parameter space points for which corrections must be applied to the data stream in a coherent search. Since this number increases with the observation time, the sensitivity of a search for continuous gravitational-wave signals is computationally bound when data analysis proceeds at a similar rate to data acquisition. In this paper, we extend the formalism developed by Brady \textit{et al.} \mbox{[}Phys. Rev. D {\bf 57}, 2101 (1998)\mbox{]}, and we compute the number of independent corrections $N_p(\Delta T, N)$ required for incoherent search strategies. These strategies rely on the method of \emph{stacked} power spectra---a demodulated time series is divided into $N$ segments of length $\Delta T$, each segment is Fourier transformed, a power spectrum is computed, and the $N$ spectra are summed up. This method is incoherent; phase information is lost from segment to segment. Nevertheless, power from a signal with fixed frequency (in the corrected time series) is accumulated in a single frequency bin, and amplitude signal-to-noise accumulates as $\sim N^{1/4}$ (assuming the segment length $\Delta T$ is held fixed). For fixed available computing power, there are optimal values for $N$ and $\Delta T$ which maximize the sensitivity of a search in which data analysis takes a total time $N \Delta T$. We estimate that the optimal sensitivity of an all-sky search that uses incoherent stacks is a factor of $2$--$4$ better than achieved using coherent Fourier transforms, assuming the same available computing power; incoherent methods are computationally efficient at exploring large parameter spaces. We also consider a two-stage hierarchical search in which candidate events from a search using short data segments are followed up in a search using longer data segments. This hierarchical strategy yields a further $20$--$60\%$ improvement in sensitivity in all-sky (or directed) searches for old ($\geq 1000$yr) slow ($\leq 200$Hz) pulsars, and for young ($\geq 40$yr) fast ($\leq 1000$Hz) pulsars. Assuming enhanced LIGO detectors (LIGO-II) and $10^{12}$~flops of effective computing power, we examine the sensitivity to sources in three specialized classes. A limited area search for pulsars in the Galactic core would detect objects with gravitational ellipticities of $\epsilon\agt5\times 10^{-6}$ at 200~Hz; such limits provide information about the strength of the crust in neutron stars. Gravitational waves emitted by unstable $r$-modes of newborn neutron stars would be detected out to distances of $\sim 8$~Mpc, if the $r$-modes saturate at a dimensionless amplitude of order unity and an optical supernova provides the position of the source on the sky. In searches targeting low-mass x-ray binary systems (in which accretion-driven spin up is balanced by gravitational-wave spin down), it is important to use information from electromagnetic observations to determine the orbital parameters as accurately as possible. An estimate of the difficulty of these searches suggests that objects with x-ray fluxes exceeding $2\times10^{-8}\mathrm{erg\,cm}^{-2}\mathrm{s}^{-1}$ would be detected using the enhanced interferometers in their broadband configuration. This puts Sco~X-1 on the verge of detectability in a broad-band search; the amplitude signal-to-noise would be increased by a factor of order $\sim 5$--$10$ by operating the interferometer in a signal-recycled, narrow-band configuration. Further work is needed to determine the optimal search strategy when limited information is available about the frequency evolution of a source in a targeted search.

Preprint  number GRP-508, NSF-ITP-98-120  To appear in Physical  Review  D

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Computing the merger of black-hole binaries: The IBBH problem

Patrick R. Brady, Jolien D. E. Creighton, and Kip S. Thorne
 

Gravitational radiation arising from the inspiral and merger of binary black holes (BBH's) is a promising candidate for detection by kilometer-scale interferometric gravitational wave observatories. This paper discusses a serious obstacle to searches for such radiation and to the interpretation of any observed waves: the inability of current computational techniques to evolve a BBH through its last $\sim 10$ orbits of inspiral ($\sim 100$ radians of gravitational-wave phase).  A new set of numerical-relativity techniques is proposed for solving this ``Intermediate Binary Black Hole'' (IBBH) problem: (i)~numerical evolutions performed in coordinates co-rotating with the BBH, in which the metric coefficients evolve on the long timescale of inspiral, and (ii)~techniques for mathematically freezing out gravitational degrees of freedom that are not excited by the waves.

Preprint  number GRP-498,  Physical  Review  D 58, 061501 (1998)

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Quantum corrections to critical phenomena in gravitational collapse

Patrick R Brady and Adrian C Ottewill

We investigate conformally coupled quantum matter fields on spherically symmetric, continuously self-similar  backgrounds.   By exploiting the symmetry associated with the self-similarity the general structure of the renormalized quantum stress-energy tensor can be derived.   As an immediate application we consider a combination of classical, and quantum perturbations about exactly critical collapse.   Generalizing the standard argument which explains the scaling law for black hole mass, $M \propto |\eta-\eta^*|^\beta$, we demonstrate the existence of a quantum mass gap when the classical critical exponent satisfies $\beta \geq 0.5$.  When  $\beta < 0.5$ our argument is inconclusive; the semi-classical approximation breaks down in the spacetime region of interest.

Preprint number GRP-492,  Physical Review  D 58, 024006 (1998)

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Cosmic censorship: as strong as ever

Patrick R Brady, Ian G Moss and Robert C Myers

Spacetimes which have been considered counter-examples to strong cosmic censorship are revisited. We demonstrate the classical instability of the Cauchy horizon inside charged black holes embedded in de Sitter spacetime for all values of the physical parameters. The relevant modes which maintain the instability, in the regime which was previously considered stable, originate as outgoing modes near to the black hole event horizon. This same mechanism is also relevant for the instability of Cauchy horizons in other proposed counter-examples of strong cosmic censorship.

Preprint number GRP-490 and NSF-ITP-98-001,  Physical Review Letters 80, 3432 (1998)

Download a postscript version of this manuscript, or the PDF version direct from PRL-o (subscription may be required).

---

The generic black hole singularity

Patrick R Brady, Serge Droz and Sharon M Morsink

It was long believed that the singularity inside a real black hole must be spacelike. However, studies of the internal geometry of black holes indicate a more complicated structure is typical. While an observer falling into a black hole with the collapsing star encounters a crushing spacelike singularity, an observer falling in at late times generally reaches a null singularity which is vastly different in character to the standard Belinsky, Khalatnikov and Lifschitz (BKL) spacelike singularity~\mbox{[}V.~A. Belinsky, I.~M. Khalatnikov, and E.~M. Lifshitz, Sov. Phys. JETP \mbox{\bf 32}, 169 (1970)\mbox{]}. In the spirit of the classic work of BKL we present an asymptotic analysis of the null singularity inside a realistic black hole. Motivated by current understanding of spherical models, we argue that the Einstein equations reduce to a simple form in the neighborhood of the null singularity. The main results arising from this approach are demonstrated using an almost plane symmetric model. The analysis shows that the null singularity results from the blueshift and non-linear interaction of the late time gravitational wave tail; the amplitude of these gravitational waves is taken to decay as an inverse power of advanced time as suggested by perturbation theory. In the most general case, the divergence of the Weyl curvature at the null singularity is dominated by the propagating modes of the gravitational field, that is C_{\alpha\beta\gamma\delta} C^{\alpha\beta\gamma\delta} \sim \Psi_0\Psi_4 \sim v^{-(2l+3)} e^{2 \kappa v} as $v\rightarrow \infty$ at the Cauchy horizon. Here, $\Psi_0$ and $\Psi_4$ are the Newman-Penrose Weyl scalars, and $l\geq 2$ is the multipole order of the perturbations crossing the event horizon. The null singularity is weak in the sense that tidal distortion remains bounded along timelike geodesics crossing the Cauchy horizon. We briefly discuss the outstanding problems which must be resolved before the picture of the generic black hole interior is complete.

Preprint number GRP-491,  Physical Review D 58, 084034 (1998)

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Phases of massive scalar field collapse

P. R. Brady, C. M. Chambers, and S. M. C. V. Goncalves

We study critical behavior in the collapse of massive spherically symmetric scalar fields. We observe two distinct types of phase transition at the threshold of black hole formation. Type II phase transitions occur when the radial extent $(\lambda)$ of the initial pulse is less than the Compton wavelength ($\mu^{-1}$) of the scalar field. The critical solution is that found by Choptuik in the collapse of massless scalar fields. Type I phase transitions, where the black hole formation turns on at finite mass, occur when $\lambda \mu \gg 1$. The critical solutions are unstable soliton stars with masses $\alt 0.6 \mu^{-1}$. Our results in combination with those obtained for the collapse of a Yang-Mills field~{[M.~W. Choptuik, T. Chmaj, and P. Bizon, Phys. Rev. Lett. {\bf 77}, 424 (1996)]} suggest that unstable, confined solutions to the Einstein-matter equations may be relevant to the critical point of other matter models.

Preprint number GRP-475, Physical Review D56, R6057 (1996)

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

The central density of a neutron star is unaffected by a binary companion at linear order in $\mu/R$

Patrick R. Brady and Scott A. Hughes

Recent numerical work by Wilson, Mathews, and Marronetti [J.R.\ Wilson, G.\ J.\ Mathews and P.\ Marronetti, Phys.\ Rev.\ D {\bf 54}, 1317 (1996)] on the coalescence of massive binary neutron stars shows a striking instability as the stars come close together: Each star's central density increases by an amount proportional to 1/(orbital radius). This overwhelms any stabilizing effects of tidal coupling [which are proportional to 1/(orbital radius$)^6$] and causes the stars to collapse before they merge. Since the claimed increase of density scales with the stars' mass, it should also show up in a perturbation limit where a point particle of mass $\mu$ orbits a neutron star. We prove analytically that this does {\it not} happen; the neutron star's central density is {\it unaffected} by the companion's presence to linear order in $\mu/R$. We show, further, that the density increase observed by Wilson et.\ al.\ could arise as a consequence of not faithfully maintaining boundary conditions.

Preprint number GRP-466, Physical Review Letters 79, 1186 (1997)

Download a postscript version of this manuscript, or the PDF version direct from PRL-online (subscription may be required).

---

Searching for periodic sources with LIGO

Patrick R Brady, Teviet Creighton, Curt Cutler and Bernard F Schutz

We investigate the computational requirements for all-sky, all-frequency searches for gravitational waves from spinning neutron stars, using archived data from interferometric gravitational wave detectors such as LIGO. These sources are expected to be weak, so the optimal strategy involves coherent accumulaton of signal-to-noise using Fourier transforms of long stretches of data (months to years). Earth-motion-induced Doppler shifts, and intrinsic pulsar spindown, will reduce the narrow-band signal-to-noise by spreading power across many frequency bins; therefore, it is necessary to correct for these effects before performing the Fourier transform. The corrections can be implemented by a parametrized model, in which one does a search over a discrete set of parameter values ({\em points} in the parameter space of corrections). We define a metric on this parameter space, which can be used to determine the optimal spacing between points in a search; the metric is used to compute the number of independent parameter-space points $N_p$ that must be searched, as a function of observation time $T$. This method accounts automatically for correlations between the spindown and Doppler corrections. The number $N_p(T)$ depends on the maximum gravitational wave frequency and the minimum spindown age $\tau=f/\dot{f}$ that the search can detect. The signal-to-noise ratio required, in order to have $99\%$ confidence of a detection, also depends on $N_p(T)$. We find that for an all-sky, all-frequency search lasting $T=10^7$ s, this detection threshhold is $h_c \approx (4--5) h_{\rm \scriptsize 3/yr}$, where $h_{\rm \scriptsize 3/yr}$ is the corresponding $99\%$ confidence threshhold if one knows in advance the pulsar position and spin period.

We define a coherent search, over some data stream of length $T$, to be one where we apply a correction, followed by an FFT of the data, for every independent point in the parameter space. Given realistic limits on computing power, and assuming that data analysis proceeds at the same rate as data acquisition (e.g. $10$ days of data gets analysed in $\sim 10$ days), we can place limitations on how much data can be searched coherently. In an all-sky search for pulsars having gravity-wave frequencies $f\le 200\mbox{\rm Hz}$ and spindown ages $\tau\ge 1000\mbox{\rm Yrs}$, one can coherently search $\sim 18$ days of data on a teraflops computer. In contrast, a teraflops computer can only perform a $\sim 0.8$-day coherent search for pulsars with frequencies $f \le 1\mbox{\rm kHz}$ and spindown ages as low as $40\mbox{\rm Yrs}$.

In addition to all-sky searches we consider coherent directed searches, where one knows in advance the source position but not the period. (Nearby supernova remnants and the galactic center are obvious places to look.) We show that for such a search, one gains a factor of $\sim 10$ in observation time over the case of an all-sky search, given a $1\mbox{\rm Tflops}$ computer.

The enormous computational burden involved in coherent searches indicates a need for alternative data analysis strategies. As an example we briefly discuss the implementation of a simple hierarchical search in the last section of the paper. Further work is required to determine the optimal approach.

Preprint number GRP-460, to appear in Physical Review D 57, 2102 (1998)

Download a postscript version of this manuscript, or the PDF version direct from PRD-o (subscription may be required).

---

Telling Tails in the Presence of a Cosmological Constant

P. R. Brady, C. M. Chambers, William Krivan and Pablo Laguna

We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a non-zero cosmological constant. Considering test fields on both Schwarzschild-de~Sitter and Reissner-Nordstrom-de Sitter backgrounds, we demonstrate the existence of  exponentially decaying tails at late times. Interestingly the l=0 mode asymptotes to a non-zero value, contrasting the asymptotically flat situation. We also compare these results, for l=0, with a numerical integration of the Einstein-scalar field equations, finding good agreement between the two. Finally, the significance of these results to the study of the Cauchy horizon stability in black hole-de Sitter spacetimes is discussed.

Preprint number GRP-462, Physical Review D 55, 7538 (1997)

Download a postscript version of this manuscript, or the PDF version from PRD-o (subscription may be required).

---

Covariant double-null dynamics: (2+2)-splitting of the Einstein equations

P. R. Brady, S. Droz , W. Israel , S. M. Morsink

The paper develops a $(2+2)$-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein-Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity.

Preprint number EFI-94-36, Class. Quantum Grav. 13, 2211 (1996)

Download a postscript version of this manuscript.

---

Patrick Brady

Last modified: Tue Nov 10 16:31:34 PST 1998