An improved method for estimating source densities using the temporal distribution of Cosmological Transients
aa r X i v : . [ a s t r o - ph ] J un An improved method for estimating source densities using thetemporal distribution of Cosmological Transients
E. Howell , D. Coward , R. Burman and D. Blair School of Physics, University of Western Australia, Crawley WA 6009, Australia [email protected]
ABSTRACT
It has been shown that the observed temporal distribution of transient eventsin the cosmos can be used to constrain their rate density. Here we show thatthe peak flux–observation time relation takes the form of a power law that isinvariant to the luminosity distribution of the sources, and that the methodcan be greatly improved by invoking time reversal invariance and the temporalcosmological principle. We demonstrate how the method can be used to constraindistributions of transient events, by applying it to
Swift gamma-ray burst dataand show that the peak flux–observation time relation is in good agreement withrecent estimates of source parameters. We additionally show that the intrinsictime dependence allows the method to be used as a predictive tool. Within thenext year of
Swift observation, we find a 50% chance of obtaining a peak flux School of Physics, University of Western Australia, Crawley WA 6009, Australia
Swift peak flux to date – andthe same probability of detecting a burst with peak flux >
100 photons s − cm − within 6 years. Subject headings: gamma-rays: bursts –gravitational waves – cosmology: miscel-laneous
1. INTRODUCTION
The brightness distribution of cosmological sources is conventionally used to constrainthe luminosity function of the sources, their evolution in density (Peebles 1993) and, fortransient sources, their rate density (Schmidt 2001; Sethi & Bhargavi 2001; Totani 1997).This method is applicable both to long-lived sources such as galaxies and to transient eventssuch as supernovae and gamma-ray bursts (GRBs). Estimates are obtained by fitting thenumber – brightness distribution to models that include luminosity, source density and evo-lution effects. In the case of transient events an additional parameter is available – the eventarrival times.The temporal distribution of transient astrophysical populations of events has beendescribed by the ‘probability event horizon’ (PEH) concept of Coward & Burman (2005).This method establishes a temporal dependence by noting the occurrences of successivelybrighter events in a time series. By utilizing the fact that the rarest events will preferentiallyoccur after the longest observational periods, it produces a data set with a unique statistical 3 –signature.Here we show that a well-defined observation-time dependence is an intrinsic feature ofthe source distribution of events. Using Swift GRB data we demonstrate how this propertycan be used to constrain source distributions. We start by presenting an analytical derivationof the peak flux–observation time relation, P ( T ), for sources which are uniformly distributedin Euclidean space and then describe its extension to cosmological models ( § P ( T ) that is invariant to the luminosity distribution of events.We then utilize the PEH technique to show how P ( T ) data can be extracted from adistribution of peak fluxes ( § P ( T ) relation to Swift long-GRB data. We demonstrate thatthe technique can be used as a probe of the event rate density and luminosity distribution ofthe sources ( § §
2. THE PEAK FLUX – OBSERVATION TIME RELATION
In this paper we will define an event to be an astrophysical transient occurrence witha duration much less than the period of observation. Examples are GRBs and gravitational 4 –wave burst sources such as coalescing compact binaries or core-collapse supernovae.Consider a distribution of events defined within a Euclidean space by an event ratedensity r and a luminosity function φ ( L ) ( L min ≤ L ≤ L max ). The observed peak flux, or‘brightness’, distribution of events over an observation time T is a convolution of the radialdistribution of the sources and their luminosity function. For peak fluxes (photons cm − s − )between P and P + d P : d N ( P ) = 4 πT Z L max L min φ ( L ) dL r r dr, (1)with r = ( L/ πP ) / . The total number of events observed in time T with a peak fluxgreater than P is given by : N ( > P ) = T ∆Ω Z L max L min φ ( L ) dL Z √ L/ πP r r dr , (2)where the average solid angle covered on the sky has been accounted for by ∆Ω / π . Theupper limit in the integration over r is the maximum distance for which an event withluminosity L produces a peak flux P .For r and φ ( L ) independent of position, integrating over the radial distance yields: We use here the luminosity function for GRB sources, φ ( L ), which includes a normalization constant toensure that it integrates to unity over the range of source luminosities. This means that φ ( L ) has units ofinverse luminosity–see for example, Porciani & Madau (2001). N ( > P ) = T r ∆Ω / π √ π P − / Z L max L min φ ( L ) L / dL . (3)This is the familiar log N –log P relation, N ( > P ) ∝ P − / , a power law independent of theform of the luminosity function (Horack, Emslie & Meegan 1994).To introduce the temporal distribution of events, we note that, as the events are in-dependent of each other, the individual events will follow a Poisson distribution in time.Therefore, the temporal separation between events will follow an exponential distribution,defined by a mean event rate R ( r ) = r (4 / πr for events out to r . The probability for atleast one event > P to occur in a volume bounded by r during an observation time T atconstant probability ǫ is given by: P ( n ≥ R ( r ) , T ) = 1 − e R ( r ) T = ǫ . (4)For this equation to remain satisfied with increasing observation time: N ( > P ) = R ( r ) T = | ln(1 − ǫ ) | . (5)Equations (3) and (5) for N ( > P ) combine to give the relation for the evolution ofbrightness as a function of observation time: P ( T ) = (cid:18) r ∆Ω / π √ π | ln(1 − ǫ ) | (cid:19) / (cid:20)Z L max L min φ ( L ) L / dL (cid:21) / T / . (6) 6 –This relation shows that for a simple Euclidean geometry, a log P –log T distribution willhave a slope of 2/3, independent of the form of the luminosity function. One can considerthat changes in r create a horizontal offset in the log P –log T distribution, while changesin the integrated luminosity create a vertical offset. However, the slope is fixed by the 3-DEuclidean geometry.We can use the log P –log T relation to produce curves defining the probability, ǫ , ofobtaining some value of peak flux, P , within an observation-time, T , for a given r and φ ( L ).For a cosmological distribution of sources, equation (6) must be modified to allow for cos-mic evolution. A standard Friedman cosmology can be used to define a differential event rate,d R ( z ), in the redshift shell z to z +d z . The luminosity and flux will be related through z by aluminosity distance d ( z ) (see for example Coward & Burman (2005) or Porciani & Madau(2001)). In this case, solving equation (5) numerically, with P = L/ πd ( z ), will yield thecosmological log P –log T relation.
3. AN ENHANCED PEH FILTER
To utilize the time domain, we use the probability event horizon (PEH) filter of Coward & Burman(2005) to produce P ( T ) time data. The PEH filter is a tool that exploits the temporal in-formation encoded in a time series of transient events and works by recording successivelybrighter events in a time series. Howell at al. (2007) demonstrated that the unique statis-tical signature of events filtered in this way could be exploited to obtain rate estimates of 7 –transient events. However, the significant probability of a bright event occurring early in anobservational period meant that only a small fraction of data was used by the method. Asa result, large uncertainties were obtained in the estimates. There are however, two ways inwhich the amount of usable data can be increased.Firstly, the temporal cosmological principle implies that the PEH signature of a transientpopulation of events is independent of when a detector is switched on. Secondly, time reversalinvariance allows the PEH filter to be applied to a data set in both temporal directions. Thus,a time series of events can be treated as a closed loop which can be interrogated in bothdirections. The observational period is now defined as the total length of the loop. The starttime for the PEH analysis is now arbitrary so, without loss of generality, we can choose anystart time. This allows the PEH filter to be applied in such a way that the brightest eventcan be set as the final event in a series. This ensures that the PEH filter is applied to thefull data stream and the process can be repeated in each direction, increasing the quantityof PEH data. We refer to these techniques as ‘from max’ plus ‘time reversal’ (FMTR). Weshow below how FMTR increases the PEH sample and significantly improves the statisticalresolution when applied to the Swift data.
4. APPLICATION TO
SWIFT
DATA
In this section, we will apply the log P –log T relation to a cosmological population oflong GRBs. To account for the event rate and luminosity function, we will use estimates 8 –based on recent studies. Using the FMTR method, we will extract a time-dependent sampleof GRB peak fluxes from the Swift data, and demonstrate how it can be constrained by alog P –log T fit.For our long-GRB peak flux sample we use data recorded by the Swift satellite between2004 December and 2007 April. We consider only bursts with confirmed peak fluxes detectedwithin the 15–150 keV band of the Burst Alert Telescope (BAT) and with T ≥ s (a T duration is the interval in which a signal contains 90% of its total observed counts). Thetotal sample consists of 190 peak fluxes.Figure 1A displays the Swift peak flux distribution of long GRBs as a time series. It isapparent that as observation time increases, there is an greater probability of a bright event.By extracting successively brighter events as a function of observation time, the PEH filtersamples events from the low probability tail of the distribution. The strong brightness–timedependence of these events creates a unique statistical signature which can be modeled bythe log P – log T relation.In Table 1 we show the PEH filtered data. It is apparent how the time intervals betweensuccessive events increase with observation time. This is a result of a progressive samplingof the rarer events of the distribution.To apply the log P –log T relation to the filtered data we must first set up a modelto account for the source rate evolution and luminosity distribution. We use a model fromthe recent study of Guetta & Piran (2007). They employ a ‘flat-Λ’ cosmological model and This data can be obtained from the
Swift website http://swift.gsfc.nasa.gov/docs/swift/archive/grb table.html H = 65 km s − Mpc − for the Hubble parameter at the present epoch. For the isotropicluminosity function of GRB peak luminosities, φ ( L ), they use a broken power law formbased on the work of Schmidt (2001). Assuming that the rate of GRBs traces the globalstar formation history of the Universe, they employ a number of different star formationrate models. For each, they determine best fitted values for the luminosity function andevent rate density. For this study, we use their model (i) parameters, which are based on theSF2 star formation rate model of Porciani & Madau (2001). These parameters include fittedvalues for the luminosity function and a local event rate density r = 0 . − yr − . Toaccount for the average solid angle covered on the sky by Swift , we use a value of ∆Ω = 1 . P – log T fit to the PEH filtered data (shown by squares), usingthe fitting parameters of Guetta & Piran (2007): i.e. there are no free parameters in thecomparison between theory and observation. We define a 90% confidence band – shown bythe shaded area – corresponding to the ǫ = 95% (top) and ǫ = 5% (bottom) probabilitiesof detecting at least one event within an observation time T . We see that the data is wellconstrained. The fit shows that by using only a small sample of the brightest events, it ispossible to extract the geometrical signature of the source population and to test estimatesof the luminosity function and rate density of events.The dashed lines of Figure 1B show the 90% confidence band corresponding to theEuclidean model using equation (6). We see that two of the first few events lie outside theEuclidean curves but are constrained by the cosmological model. These events, occurring 10 –at early observation times, most likely result from sources at large cosmological distances.The very bright events at late observation times are more probable – it is apparent that theEuclidean and cosmological curves begin to converge in this regime. To account for non-uniformly distributed sources, the method could be refined to take into account the spatialdistribution of potential host galaxies.To test the power-law dependence in equation (6), we have performed least-squaresfitting to the PEH filtered data using the Euclidean log P – log T curve as a linear regressionmodel. By setting the power as a single free parameter, we obtained a value of 0 . ± .
5. A PREDICTIVE APPLICATION OF THE LOG P – LOG T RELATION
Figure 2 illustrates how the log P –log T relation can be used as a predictive tool. Bymapping the temporal evolution of detection probability, the maximum brightness of futureevents can be constrained (Coward at al. 2005). As in Fig. 1B, the shaded areas show thelog P –log T detection confidence bands corresponding to different probability values (shownin the legend). The current Swift observation time (398 days) is shown by the vertical solidline. For predictive applications it is essential that the true temporal sequence of events isretained. Therefore, rather than using the FMTR technique, we apply the unmodified PEHfilter from the time of the first event. Comparing with Fig. 1B, we see that the FMTRmethod has increased the sample by 220%, of which 80% is gained by incorporating timereversal. Using 100 Monte-Carlo simulations, we find a mean fractional increase in data of 11 – ∼ Swift data, implies that the FMTR method can be further optimally tuned.The plot shows that after 3 days of operation, there was a <
5% probability of detectingan event with peak flux equal to the first in the PEH sample, GRB 041123. The Poissonprobability of detecting at least one event within a Gpc at this time is ∼ × − . Thisimplies that this event occurred at a considerable cosmological distance. The event GRB050525A is the brightest long GRB with a secure redshift, z = 0 .
606 – the Poisson probabilityof at least one event within this volume is 35%. The next brightest event, GRB 060017, isthe most intense burst, in terms of peak flux, detected by Swift.As a demonstration of the predictive nature of the the log P – log T relation, Figure 2shows that there is a 50% probability of obtaining an event with a peak flux greater thanthat of GRB 060017 within the next year and an 80% probability within 5 years.The curve predicts that there is a 50% (80%) probability of obtaining a burst with peakflux >
60 photons s − cm − within 2 (8) years. To determine the feasibility of this prediction,we consider again GRB 050525A, which had a peak flux of 42.3 photons s − cm − . Using thisburst’s redshift and converting to a peak luminosity, we find that an equivalent burst wouldhave to occur within z ≈ . ∼
90% (99%). If we considera burst with a peak flux of 100 photons s − cm − , we find that an event of this peak flux is50% probable within 6 years. Such an event would correspond to a GRB 050525A-equivalent 12 –burst occurring within z ≈ .
4, for which the probability is 99%.The log P – log T technique naturally uses the brightest events of a data set. As theprobability of obtaining a GRB afterglow increases with peak flux, the method can be usedto predict the expected occurrence of events at low z .
6. SUMMARY
We have provided a clear demonstration of the log P –log T relation by applying thePEH filter to the Swift
GRB peak flux distribution. A log P –log T model with no freeparameters was fitted to filtered data confirming the power law P ∝ T / in the Euclideanlimit; the power law is independent of the form of the luminosity distribution.The FMTR method significantly improves the PEH method, which was previously dis-advantaged by using only a small fraction of a data stream. We have shown that FMTRenables the PEH filter to use over 8% of the available data, making it a practical tool forcosmology.We have shown that the PEH technique can be used as a predictive tool. Comparingobservation with prediction provides an additional means to test rate estimates and evaluatesource parameters such as the limits of the luminosity distribution.In a future study, we intend to apply the FMTR method to both the Swift and
BATSE
GRB data. We will investigate the efficiency of the method in determining constraints on 13 –the rate density and limits of the luminosity function.
ACKNOWLEDGMENTS
E Howell and D Coward are supported by the Australian Research Council. 14 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
15 –GRB Peak Flux Redshift Observation Time(photons s − cm − ) (days)060202 0.5 8060203 0.6 9060204B 1.3 10060206 2.8 4 .
045 12060223B 2.9 29060306 6.1 42060418 6.7 1 .
49 84060510A 17.0 106061121 21.1 1 .
314 297050219B 25.4 506050525A 42.3 0 .
606 602060117 48.9 839060111A 1.72 4060110 1.9 5060105 7.5 10060603 27.6 2.821 227Table 1: Data extracted from the
Swift peak flux distribution of long GRBs using the PEHfilter. The observation time is determined by treating the time series as a closed loop andsetting the last event in the series to be the brightest burst. The lower set of data wasobtained by invoking time reversal. A log P – log T fit to this data is shown in Fig. 1 16 – log T (days) log P ( ph cm −2 s −1 ) (B) (A) CosmologicalEuclidean
Fig. 1.— Panel (A) shows the
Swift peak flux distribution as a time series. It is evident thatas observation time increases the probability of a bright event increases. Panel (B) uses aPEH filter to extract the geometrical signature of the GRB distribution (shown as squares).Assuming an event rate of 0 . − yr − (Guetta & Piran 2007), the shaded area shows thecosmology dependent log P –log T model corresponding to a (5 – 95)% confidence band. Theequivalent model for a Euclidean geometry is shown by the dashed curves – the two outliers,which have no associated redshifts, probably result from distant cosmological events. 17 – observation time (days) Peak Flux ( ph cm −2 s −1 ) −1 (2−98)%(5−95)%(50−80)% Fig. 2.— The log P –log T relation used as a predictive tool. The successive maximum peakfluxes detected by Swift since the start of operation are shown by squares. The shaded areasshow detection confidence bands corresponding to different probability values (shown in thelegend). The current Swift observation time (398 days) is shown by the solid line. The plotshows that after 3 days of operation, there was a <
5% probability of detecting an eventwith peak flux equal to the first in the PEH sample, GRB 041123. We see that within thenext year, there is a 50% chance of obtaining a peak flux greater than for GRB 060017 – themost intense burst (in peak flux) detected by Swift – and the same probability of obtaininga burst with peak flux >
100 photons s − cm −2