# Minimum Variability Time Scales of Long and Short GRBs

G. A. MacLachlan1, A. Shenoy1, E. Sonbas2,3, K. S. Dhuga1, B. E. Cobb1, T. N.
Ukwatta1,3,4, D. C. Morris1,3,5, A. Eskandarian1, L. C. Maximon1, and W. C.
Parke1
1Department of Physics, The George Washington University, Washington, D.C.
20052, USA.
2University of Adiyaman, Department of Physics, 02040, Adiyaman, Turkey.
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
4Department of Physics and Astronomy, Michigan State University, East Lansing,
MI 48824, USA.
5Department of Physics, University of Virgin Islands, Virgin Islands, USA.
E-mail<EMAIL_ADDRESS>(GAM)

###### Abstract

We have investigated the time variations in the light curves from a sample of
long and short Fermi/GBM Gamma ray bursts (GRBs) using an impartial wavelet
analysis. The results indicate that in the source frame, that the variability
time scales for long bursts differ from that for short bursts, that
variabilities on the order of a few milliseconds are not uncommon, and that an
intriguing relationship exists between the minimum variability time and the
burst duration.

###### keywords:

Gamma-ray bursts

##  1 Introduction

The prompt emission from Gamma-ray Bursts (GRBs) shows complex time profiles
that have eluded a generally accepted explanation. Fenimore & Ramirez-Ruiz
(2000) reported a correlation between variability of GRBs and the peak
isotropic luminosity. The existence of the variability-luminosity correlation
suggests that the prompt emission light curves have embedded temporal
information related to the microphysics of GRBs. Several models have been
proposed to explain the observed temporal variability of GRB light curves.
Leading models such as the internal shock model (Kobayashi, Piran, and Sari,
1997) and the photospheric model (Ryde, 2004) link the rapid variability
directly to the activity of the central engine. Others invoke relativistic
outflow mechanisms to suggest that local turbulence amplified through Lorentz
boosting leads to causally disconnected regions which in turn act as
independent centers for the observed prompt emission. Within more recent
models, both Morsony et al. (2010) and Zhang & Yan (2011) argue that the
temporal variability may show two different scales depending on the physical
mechanisms generating the prompt emission.

In order to further our understanding of the prompt emission phase of GRBs and
to explicitly test some of the key ingredients in the various models, it is
clearly important to extract the variability for both short and long gamma-ray
bursts in a robust and unbiased manner. It is also clear that the chosen
methodology should not only be mathematically rigorous but also be
sufficiently flexible to apply to transient phenomena with multiple time
scales and a wide dynamic range. A wide dynamic range is naturally provided by
the bimodal separation of GRB duration occurring at $T_{90}=2$ sec as observed
by Kouveliotou et al. (1993) to distinguish between long and short duration
GRBs.

In this paper, we extract variability time scales for GRBs using a method
based on wavelets. The technique for such a temporal analysis is universal,
and has the advantage over Fourier analysis that transients and frequency
correlations can be more easily picked out in the data. Results presented
herein were compared with Bhat et al. (2012) who gave pulse parameters for
approximately 400 pulses obtained from 34 GRBs. It was shown by MacLachlan et
al. (2012) that the minimum variablility time scale tracks the rise times of
pulses very well for over three orders of magnitude in time scale. The
relation of minimum variability time scales to pulse parameters has been
extended to four orders of magnitude by Sonbas et al. (2012) who applied the
present technique to analyze X-ray flares.

The time scales being investigated here have power densities very near to that
of the noise in the data which makes extracting these time scales nontrivial.
A somewhat older but still interesting discussion of extracting signal in a
noisy environment can be found in Scargle (1982) and a more recent discussion
found in Kostelich & Schreiber (1993). The technique we offer is not
necessarily new but is different from previous published wavelet analyses
(Fritz & Bruch, 1998; Walker et al., 2000; Tamburini et al., 2009; Anzolin et
al., 2010)111Note that Fritz & Bruch (1998) analyzed optical data and
Tamburini et al. (2009); Anzolin et al. (2010) X-ray data from Cataclysmic
Variables. in that we apply a number of modifications suggested by various
authors (Addison, 2002; Coifman, 1995; Percival, 2000; Strang and Nguyen,
1997) as explained further in Sec 4.

The layout of the paper is as follows: the source of the data is described in
section 2; the main aspects of the wavelet methodology are outlined in section
3; in section 4 we provide the details of the data analysis; in section 5 we
present and discuss our main findings; finally, in section 6, we summarize our
conclusions.

##  2 Data

The Gamma-Ray Burst Monitor (GBM) on board Fermi observes GRBs in the energy
range 8 keV to 40 MeV. The GBM is composed of 12 thallium-activated sodium
iodide (NaI) scintillation detectors (12.7 cm in diameter by 1.27 cm thick)
that are sensitive to energies in the range of 8 keV to 1 MeV, and two bismuth
germanate (BGO) scintillation detectors (12.7 cm diameter by 12.7 cm thick)
with energy coverage between 200 keV and 40 MeV. The GBM detectors are
arranged in such a way that they provide a significant view of the sky (Meegan
et al., 2009).

In this work, we have extracted light curves for the GBM NaI detectors over
the entire energy range (8 keV - 1 MeV, also including the overflow beyond 1
MeV). Typically, the brightest three NaI detectors were chosen for the
extraction. Lightcurves for both long and short GRBs were extracted at a time
binning of 200 microseconds. The long GRBs were extracted over a duration
starting from 20 seconds before the trigger and up to about 50 seconds after
the $T_{90}$ for the burst without any background subtraction. For short GRBs,
durations were chosen to be 20 seconds before the trigger and 10 seconds after
the $T_{90}$. The $T_{90}$ durations were obtained from the Fermi GBM-Burst
Catalog (Paciesas et al., 2012).

Figure 1: GBM GRB080925775. Preburst portion of the light curve, used for
background removal, is shown in gray. The burst portion, from which a time
scale is extracted, is shown in black.

##  3 Methodology

We report on our model-independent statistical investigation of the
variability of Fermi/GBM long and short GRBs. We extract this information by
using a fast wavelet transformation to encode GRB light curves into a wavelet
representation and then compute a statistical measure of the variance of
wavelet coefficients over multiple time-scales.

###  3.1 Minimum Variability Time Scales

It is often the case when multiple processes are present that one process will
dominate the others at certain time scales but those same processes may
exchange dominance at other time scales. A wavelet technique is useful in
these situations because the variances of wavelet coefficients are sensitive
to whichever processes dominate the light curve at a given time scale.
Moreover, the technique can be used to classify those dominant processes as
well as provide a means to determine the characteristic time scale,
$\tau_{\beta}$, for which the processes exchange dominance. Determination of
$\tau_{\beta}$ helps in the development of theoretical models and the
understanding of observational data. Indeed, if there is a transition from a
time-scaling region to that of white noise then there is a smallest
variability time for the physical processes present.

###  3.2 Wavelet Transforms

Wavelet transformations have been shown to be a natural tool for multi-
resolution analysis of non-stationary time-series (Flandrin, 1989, 1992;
Mallat, 1989). Wavelet analysis is similar to Fourier analysis in many
respects but differs in that wavelet basis functions are well-localized,
_i.e._ have compact support, while Fourier basis functions are global. Compact
support means that outside some finite range the amplitude of wavelet basis
functions goes to zero or is otherwise negligibly small (Percival, 2000). In
principle, a wavelet expansion forms a faithful representation of the original
data, in that the basis set is orthonormal and complete.

####  3.2.1 Discrete Dyadic Wavelet Transforms

Given the discrete nature of the data, we employ a discrete wavelet analysis.
The rescaled-translated nature of the wavelet basis functions make the wavelet
transform well-localized in both frequency and time, which results in an
insensitivity to background photon counts expressed by polynomial fits. The
level of insensitivity, formally known as the vanishing moment condition, can
be adjusted by the choice of wavelet basis function. By construction, the
discrete wavelet transform is a multi-resolution operation (Mallat, 1989).
Such wavelets, denoted $\psi_{j,k}(t)$, form a dyadic basis set, i.e. wavelets
in the set have variable widths and variable central time positions.

The wavelet analysis employed in this study, as with the fast Fourier
transform, begins with a light curve with $N$ elements,

$X_{i}=\\{X_{0}\mathellipsis X_{N-1}\\},$ (1)

where $N$ is an integer power of two. The light curve is convolved with a
scaling function, $\phi_{j,k}(t_{i})$, and wavelet function,
$\psi_{j,k}(t_{i})$ which are rescaled and translated versions of the original
scaling and wavelet functions $\phi(t_{i})=\phi_{0,0}$, and
$\psi(t_{i})=\psi_{0,0}$. Translation is indexed by $k$ and rescaling is
indexed by $j$. The rescaling and translation relation is given by

$\psi_{j,k}(t)=2^{-j/2}\psi(2^{-j}t-k).$ (2)

The precise forms of the scaling and wavelet functions are not unique. The
choices are made according to the features one wishes to exploit (Percival,
2000; Addison, 2002). The scaling function acts as a smoothing filter for the
input time-series and the wavelet function probes the time-series for detail
information at some time scale, $\Delta t$, which is twice that of the finest
binning of the data, $T_{\rm bin}$. In the analysis, the time scale is doubled

$\Delta t\rightarrow 2\Delta t$

and the transform is repeated until

$\Delta t=NT_{\rm bin}.$

In this analysis we choose the Haar (Addison, 2002) scaling/wavelet basis
because it has the smallest possible support, has one vanishing moment, and is
equivalent to the Allan variance (Howe & Percival, 1995), allowing for a
straightforward interpretation.

####  3.2.2 The Haar Wavelet Basis

Convolving the light curve, $X$, with the scaling functions yields
approximation coefficients,

$a_{j,k}=\langle\phi_{j,k},X\rangle.$ (3)

Interrogating $X$ with the wavelet basis functions yields scale and position
dependent detail coefficients,

$d_{j,k}=\langle\psi_{j,k},X\rangle,$ (4)

It is interesting to point out that for the trivial $N=2$ case the Haar
wavelet transform and the Fourier transform are identical.

###  3.3 Logscale Diagrams and Scaling

Logscale diagrams are useful for identifying scaling and noise regions.
Construction of a logscale diagram for each GRB proceeds from the variance of
detail coefficients (Flandrin, 1992),

$\beta_{j}=\frac{1}{n_{j}}\sum_{k=0}^{n_{j}-1}|d_{j,k}|^{2},$ (5)

where the $n_{j}$ are the number of detail coefficients at a particular scale,
$j$. A plot of $\log_{2}$ variances versus scale, $j$, takes the general form

$\log_{2}\ \beta_{j}=\alpha j+{\rm constant},$ (6)

and is known as a logscale diagram. A linear regression is made of each
logscale diagram and the slope parameter, $\alpha$, (depicting a measure of
scaling) is estimated. White-noise processes appear in logscale diagrams as
flat regions while non-stationary processes appear as sloped regions with the
following condition on the slope parameter, $\alpha>1$ (Abry et al., 2003;
Percival, 2000; Flandrin, 1989).

##  4 Data Analysis

###  4.1 Background Subtraction

We now present a method for removing photometric background due to noise not
intrinsic to the GRB so that physical variability arising from the GRB remains
for further analysis. Background subtraction for a statistical analysis of
variability via wavelet transforms should proceed in the space variances as
opposed to a traditional flat or linear subtraction of counts. This owes to
the fact that Haar detail coefficients are insensitive to polynomial trends in
the signal up to first order. Subtraction of a flat or linear background from
a light curve is an operation under which the wavelet transform is invariant
(as are Fourier transforms) apart from the mean signal coefficient.

The GRB light curves show power at various variablity time scales. Most often,
there is a region of the logscale diagram (log-power verses log-varibility
time) with a single slope, indicating scaling in the power over those
variability-times, and a flat region at the shortest variability times,
indicating the presence of white-noise. Some of this white-noise may be
intrinsic to the GRB. Some may be attributed to instrumental noise and to
background emissions from sources not including the GRB in question. We
therefore express the variability of the burst, $\beta_{j}^{\rm burst}$, at
time scales $j$ as comprising of individual variances: a scaling component,
$\beta^{\rm scaling}$; an intrinsic noise component, $\beta^{\rm noise}$; and
a background component, $\beta^{\rm background}$. The variability of the burst
can then be described as a linear combination of the component variances so
long as the components have vanishing covariances. In this event we write,

$\beta_{j}^{\rm burst}=\beta_{j}^{\rm scaling}+\beta_{j}^{\rm
noise}+\beta_{j}^{\rm background}.$ (7)

The minimum variability time scale, $\tau_{\beta}$, is identified from a
logscale diagram by the octave, $j_{\rm{intersection}}$, of the intersection
of the flat intrinsic noise domain, $\beta_{j}^{\rm noise}$, with the sloped
scaling domain, $\beta_{j}^{\rm scaling}$,

$\tau_{\beta}\equiv T_{\rm bin}\times 2^{j_{\rm intersection}}.$ (8)

In practice, the octave at which the intersection occurs is determined by
equating the polynomial fits to the flat intrinsic noise domain and the sloped
scaling domain and solving for $j_{\rm intersection}$. The uncertainty in
$\tau_{\beta}$ is determined by propagating the uncertainty in the parameters
from the fits to the $\beta_{j}$ which in turn follow from a bootstrap
procedure described in Sec. 4.1.2 and Sec. 4.1.3. It is at this time scale,
$\tau_{\beta}$, that a structured physical process appears to give way to one
that is stochastic and unstructured. Clearly one seeks to remove
$\beta_{j}^{\rm background}$ from Eq. 7 to arrive at the cleanest possible
signal,

$\beta_{j}^{\rm burst}\rightarrow\beta_{j}^{\rm clean}\equiv\beta_{j}^{\rm
burst}-\beta_{j}^{\rm background}=\beta_{j}^{\rm scaling}+\beta_{j}^{\rm
noise}.$ (9)

In order to estimate the variance of the background during the burst, we will
assume that the variance obtained from a preburst portion of the light curve
can serve as an acceptable surrogate for the background variance. That is,

$\beta_{j}^{\rm preburst}\equiv\beta_{j}^{\rm background},$ (10)

and then the background is removed from the signal according to the relation,

$\log_{2}(\beta_{j}^{\rm clean})=\log_{2}(\beta_{j}^{\rm burst}-\beta_{j}^{\rm
preburst}).$ (11)

A simple algebraic manipulation of Eq. 11 gives a form,

$\log_{2}(\beta_{j}^{\rm clean})=\log_{2}(\beta_{j}^{\rm
burst})+\log_{2}\left(1-\frac{\beta_{j}^{\rm preburst}}{\beta_{j}^{\rm
burst}}\right).$ (12)

For long GRBs, the preburst is defined relative to a 0 s trigger time as T-20
s to T-5 s and for short GRBs the preburst is defined to be from T-15 s to T-1
s. Here T is the trigger time of the burst.

####  4.1.1 Statistical Uncertainties

We have considered the statistical uncertainties in the light curve by a
typical bootstrap approach in which the square root of the number of counts
per bin is used to generate an additive poisson noise. A new poisson noise is
considered for each iteration through the bootstrap process. More significant
contributions to the uncertaintaies are discussed in Sec. 4.1.2 and Sec.
4.1.3.

####  4.1.2 Circular Permutation

Spurious artifacts due to incidental symmetries resulting from accidental
misalignment (Percival, 2000; Coifman, 1995) of light curves with wavelet
basis functions are minimized by circularly shifting the light curve against
the basis functions. Circular shifting is a form of translation invariant de-
noising (Coifman, 1995). It is possible a shift will introduce additional
artifacts by moving a different symmetry into a susceptible location. Thus,
our approach is to circulate the signal through all possible values, or at
least a representative sampling, and then take an average over the cases which
do not show spurious correlations.

####  4.1.3 Reverse-Tail Concatenation

Both discrete Fourier and discrete wavelet transformations imply an overall
periodicity equal to the full time-range of the input data. This can be
interpreted to mean that for a series of $N$ elements,
$\\{X_{0},X_{1}\mathellipsis X_{N-1}\\}$ then $X_{0}$ is made a surrogate for
$X_{N}$ and $X_{1}$ is made a surrogate for $X_{N+1}$, and so forth. This
assumption may lead to trouble if $X_{0}$ is much different from $X_{N-1}$. In
this case, artificially large variances may be computed. Reverse-tail
concatenation minimizes this problem by making a copy of the series which is
then reversed and concatenated onto the end of the original series resulting
in a new series with a length twice that of the original. Instead of matching
boundary conditions like,

$X_{0},X_{1},\ldots,X_{N-1},X_{0},$ (13)

we match boundaries as,

$X_{0},X_{1},\ldots X_{N-1},X_{N-1},\ldots,X_{1},X_{0}.$ (14)

Note that the series length has thus artificially been increased to $2N$ by
reversing and doubling of the original series. Consequently, the wavelet
variances at the largest scale in a logscale diagram reflect this redundancy.
This is the reason the wavelet variances at the largest scale are excluded
from least-squares fits of the scaling region.

Another difficulty in wavelet expansions is that the initialization procedure
of the multi-resolution algorithm may pollute the detail coefficients at the
finest scale (see Strang and Nguyen, 1997; Abry et al., 2003). For this reason
we follow the advice of Abry et al. (2003) and discard the detail coefficients
at the finest scale.

###  4.2 Simulation

Figure 2: In the left hand panel are simulated light curves and noise
processes: an _ideal_ fBm process (green) and a _background_ Poisson noise
(black). The sum of the fBm and Poisson processes is shown in red and is
labeled _observed_. The _observed_ light curve (red) is the sum of the fBm and
Poisson noise. The right hand panel shows the results of the background
subtraction procedure. Red and black points show the logscale diagrams
corresponding to the _observed_ light curve and _background_ , respectively.
The green data shows the logscale diagram for the _ideal_ light curve and the
blue data are the logscale with background removed. The agreement between
green and blue data demonstrates the merit of the background removal
procedure.  Figure 3: Results of signal to noise sensitivity test. We
generated 1000 simulated light curves with expected $j_{\rm{intersection}}$
equal to 6 and 7 for various brightness. We show in gray the region along the
$\xi$-axis where we find the GRBs analyzed in this paper, $0.3<\xi<0.74$. We
note that in the region covered by the GRBs presented herein, brightness does
not affect $j_{\rm{intersection}}$ greatly.

The efficacy of this background subtraction method and the sensitivity to
signal to noise was tested using simulated data in the form of fractional
Brownian motion (fBm) time series that were first discussed by Mandelbrot
(1968). One advantage of using fBms for simulation of time series data is that
short duration, statistically significant fluctuations which trigger the
identification of a minimum variability time scale arise naturally as part of
the random process which produces them. Another is that fBms have a scaling
parameter, $\alpha$, which is easily varied.

The outline of the simulation procedure begins with using the numerical
computing environment MATLAB to produce 1000 realizations of fBms with scaling
parameter $\alpha$ randomly chosen from the range $1.0\leq\alpha\leq 2.0$ by
using a uniform random number generator. The fBms were then acted upon by a
Poisson operator which transformed each time series into a Poisson-distributed
series but left other properties of the fBm intact, e.g., $\alpha$.

The fBms were then combined with a Poisson noise with variance, $\lambda_{B}$.
These Poisson noises were regarded as intrinsic to the GRB. Another set of
Poisson noises with variances, $\lambda_{I}$, were generated and these noise
signals were interpreted as external _background_ meant to be removed by the
subtraction procedure.

The idealized light curves were then combined with external backgrounds
resulting in pseudo-_observed_ light curves shown in red on the left panel of
Fig. 2. The pseudo-observed light curves and the external background noises
were transformed into wavelet coefficients and wavelet variances were computed
according to Eq. 5. The variances of the pseudo-observed light curve (labeled
_actual_) and the background are plotted in the right panel of Fig. 2 in red
and black, respectively. The background was subtracted from the pseudo-
observed light curve as detailed in Eq. 12 and the resulting corrected
variances are plotted in blue in the right panel of Fig. 2. The corrected
variances are to be compared to the variances of the ideal light curve which
are plotted in green.

##### Signal to Noise Sensitivity

We define a brightness parameter, $\xi$, such that

$\xi\equiv 2^{-\delta/2},$ (15)

where

$\delta\equiv\log_{2}\lambda_{B}-\log_{2}\lambda_{I}.$ (16)

Ideally the octave, $j_{\rm{intersection}}$, which is related to the minimum
variability time scale, $\tau_{\beta}$, as defined in Eq. 8 is completely
determined by $\xi$, $\lambda_{I}$, $\alpha$, and the standard deviation of
the increments of the fBm, $\sigma$. We fixed the values of $\lambda$,
$\sigma$, and varied $\alpha$ and $\xi$ so that the expected values of
$j_{\rm{intersection}}$ are $\langle j_{\rm{intersection}}\rangle=\\{6,7\\}.$
The time series thus produced were then analyzed as described in Sec. 4.1. The
results of the simulation are given in Fig. 3. The horizontal red and blue
lines show the expected $j_{\rm{intersection}}$ for octaves 6 and 7,
respectively, and are given as a guide. Brightness, $\xi$, increases to the
right. We show in gray the region along the $\xi$-axis where we find the GRBs
analyzed in this paper based on the background noise level and the estimated
noise level instrinsic to the GRB. We have noted by our own experience that
for $\xi<0.1$ the technique discussed in this paper performs poorly. However,
in the range of signal to noise sampled by the GRBs used here, $0.3<\xi<0.74$,
the background subtraction technique does not suffer from a large systematic
response to variations in brightness, as can be seen in Fig. 3.

##### Flux Sensitivity

We also investigated the reliability of the analysis as a function of flux by
removing randomly selected counts from the original simulated signal component
while leaving the background noise level undisturbed.The analysis is then
repeated for the newly _dimmed_ simulated light curves and comparision is made
to the original un-dimmed version. This brightness comparison is similar to
the one described in Norris et al. (1995) but with a different normalization.
In Norris et al. (1995) light curves were normalized by peak intensities. In
this study the simulated light curves were normalized by signal power at the
time scales specified by the dyadic partitioning of the wavelet transform.

Dimming of the simulated light curves was done by removing 0-10% and
reanalyzing then repeating by removing 10-20% and so forth up to 70-80%. We
also considered the effect of larger variations in count removal, i.e.,
removing 0-25% 25-50% and 50-75%. We find that a decrease in flux has
essentially the same effect on $\tau_{\beta}$ as increasing the noise level.
However, the largest effect was by the wider bite of counts. For example, one
can expect more accurate results from this analysis by removing 30-40% of
counts on a bin to bin basis than by removing 25-50%. We conclude that flux
related effects are more serious when it varies widely throughout the duration
of the light curve. In all cases we have studied removing 80% or more of the
counts in the signal was a reliable way to make the method fail. Fortunately
when the method fails in this way it does so in an obvious way, i.e., the
white noise signal power coefficients in the logscale diagram become highly
irregular. As discussed in Sec. 4.3, we test for this effect in the GRB data
by performing a chi-squared test on the white noise signal power coefficients
and rejecting any GRB that fails.

In summary, 1000 simulated light curves were generated and background noise
was added. The light curves with background noise were then denoised using the
same algorithm applied to actual GRB data in which preburst data were used as
a surrogate for background. The simulated background subtracted variances were
then compared to the variances of the ideal light curves, i.e., light curves
without external background noise. Signal to noise effects on the reliability
of the method were also considered and found either to be small compared to
our quoted errors or large enough that $\tau_{\beta}$ could not be determined.
In the case of the latter the GRB was removed from the analysis. The results
indicate that the background subtraction method is robust and gives confidence
that external background noise can be subtracted from the GRB light curves
with the assumption that preburst data can serve as a surrogate for background
noise.

Figure 4: A histogram of minimum variability time scales, in the observer
frame, for long and short GRBs. It is clear that the distribution of long GRBs
is displaced from the distribution of short GRBs.

###  4.3 Selection Criteria

We analyzed 122 GRBs (61 long and 61 short) listed in the Fermi GBM-Burst
Catalog (Paciesas et al., 2012) for the first two years of the GBM mission. As
discussed in Sec. 4.2, the signal-to-background ratio is a factor to be
considered in recovering the intrinsic light curve (see Eq. 12). We required
the following condition on the ratio of variances,

$\frac{\beta_{j}^{\rm preburst}}{\beta_{j}^{\rm burst}}<0.75,$ (17)

for one or more octaves, $j$. In addition, we also required that the first
order polynomial fits to the noise region and to the scaling region each had a
$\chi^{2}$/d.f. that was less than 2. This reduced the sample to 14 short GRBs
(Tab. 1) and 46 long GRBs (Tab. 2) for a total of 60 and it is these GRBs
which are used to create Figs. 4, 5, and 8. For boosting into the source frame
(Figs. 6 and 7) a known $z$ is obviously required and this cut further reduced
the data set to 2 short GRBs and 16 long GRBs for a total of 18 GRBs
considered in the source frame (see Tab. 3).

Figure 5: Minimum variability time scale versus $T_{90}$ in the Observer
frame. Figure 6: Minimum variability time scale versus $T_{90}$ in Source
frame. The correction for time dilation shortens $T_{90}$ and decreases the
minimum variability time scale of each burst.  Figure 7: Minimum variability
time scale versus $T_{90}$ with symbol size determined by luminosity (larger
symbols for higher luminosity). No obvious relation between minimum
variability time scale and luminosity is apparent. See Fig. 6 for error bars.
Figure 8: The ratio of duration-to-minimum variability time scale
($T_{90}/\tau_{\beta}$) versus $T_{90}$.

##  5 RESULTS and DISCUSSION

For a large sample of short and long GBM bursts, we have used a technique
based on wavelets to determine the minimum time scale ($\tau_{\beta}$) at
which scaling processes dominate over random noise processes. The
$\tau_{\beta}$ is the intersection of the scaling region (red-noise) of the
spectrum in the logscale diagram with that of the ‘flat’ portion representing
the (white-noise) random noise component. This transition time scale is the
shortest resolvable variability time for physical processes intrinsic to the
GRB. Histograms of the extracted $\tau_{\beta}$ values for long and short GRBs
are shown in Fig. 4. We make two observations regarding these histograms: (1)
There is a clear temporal offset in the extracted mean $\tau_{\beta}$ values
for long and short GRBs. We believe this is the first clear demonstration of
this temporal difference. Walker et al. (2000), who studied the temporal
variability of long and short bursts using the BATSE data set did not report a
systematic difference between the two types of bursts. (2) The two histograms
are quite broad and very similar in dispersion. While the difference in the
mean $\tau_{\beta}$ is understandable (a point we discuss further elsewhere)
the similarity of the dispersion is somewhat surprising since the progenitors
and the environment for the two types of bursts are presumably very different.
The comparison is qualitative at best however because the $\tau_{\beta}$ scale
has not been corrected for redshift ($z$), an effect that impacts the long
bursts more than the short bursts. In passing we note that the dispersion of
the $\tau_{\beta}$ histogram (for long bursts) is in agreement with the
results of Ukwatta et al. (2011) who performed a power density spectral
analysis of a large sample of Swift long GRBs. In that work the authors
extracted threshold frequencies and related them to a variability scale.

In Fig. 5 we show a log-log plot of $\tau_{\beta}$ versus $T_{90}$ (the
duration of the bursts); long GRBs are indicated by circles, the short ones by
squares and both time scales are with respect to the observer frame. As in the
histograms above, the fact that short GRBs, in general, tend to have smaller
$\tau_{\beta}$ values compared to long GRBs, is evident in this figure. Also
shown in the figure (as a dash line) is the trajectory of $\tau_{\beta}$ equal
to $T_{90}$. As we expect, no long GRBs exhibit a $\tau_{\beta}$ longer than
$T_{90}$ although interestingly a few short GRBs of extremely short duration
appear to be approaching the limit of equality. In addition to establishing a
characteristic time scale for short and long bursts, this figure also hints at
a positive correlation between this time and the duration of bursts. We note
that the $\tau_{\beta}$ scale spans approximately two decades for both sets of
GRBs and that the two groups are fairly well clustered in the
$\tau_{\beta}$-$T_{90}$ plane. A closer examination of the two groups,
however, indicates that a correlation between $\tau_{\beta}$ and $T_{90}$, if
present, is marginal at best. This is certainly true for the short-GRB group,
especially given the large uncertainties in the $T_{90}$s for these bursts.
The situation for the long-burst group on the other hand is not immediately
clear. In order to explore this further we cast the $\tau_{\beta}$ and the
$T_{90}$ time scales into the source frame by applying the appropriate $(1+z)$
factor to the GRBs for which the z is known. Unfortunately the z is not
available for the majority of the short GRBs but we note that the correction
is the same for both axes and is, to first order, small for the short GRBs
since the mean z for this group is $<0.8$. The corrected results for long-GRBs
are shown as a log-log plot in Fig. 6. We see from this figure (and Fig. 5)
the appearance of a very intriguing feature: A plateau region in which the
$\tau_{\beta}$ is essentially independent of $T_{90}$ and a scaling region in
which the $\tau_{\beta}$ appears to increase with $T_{90}$, with the
transition occurring around $T_{90}$ on the order of a few seconds.

If one assumes a positive correlation between luminosity and variability as
suggested by a number of authors, then one might expect smaller $\tau_{\beta}$
values for higher luminosity bursts compared to those of lower luminosity. To
investigate this, the data (in Fig. 6) are re-plotted in Fig. 7 in which the
size of each datum symbol has been modulated by the gamma-ray luminosity of
the burst, i.e., a large symbol implies a high luminosity and a small symbol a
low luminosity. We see from Fig. 7 that no obvious connection between
$\tau_{\beta}$ and luminosity is evident.

Under the assumption that the $\tau_{\beta}$ is a measure proportional to the
smallest causally-connected structure associated with a GRB light curve, it is
then possible to interpret the scaling trend in terms of the internal shock
model in which the basic units of emission are assumed to be pulses that are
produced via the collision of relativistic shells emitted by the central
engine. Indeed, we note that Quilligan et al. (2002) in their study of the
brightest BATSE bursts with $T_{90}$ $>2$ sec explicitly identified and fitted
distinct pulses and demonstrated a strong positive correlation between the
number of pulses and the duration of the burst. More recent studies (Bhat &
Guiriec, 2011; Bhat et al., 2012; Hakkila & Cumbee, 2008; Hakkila & Preece,
2011) provide further evidence for the pulse paradigm view of the prompt
emission in GRBs. In our work we have not relied on identifying distinct
pulses but instead have used the multi-resolution capacity of the wavelet
technique to resolve the smallest temporal scale present in the prompt
emission. If the smallest temporal scale is made from pulse emissions from the
smallest structures, then we can get a measure of the number of pulses in a
given burst through the ratio $T_{90}$/$\tau_{\beta}$ . In the simple model in
which a pulse is produced every time two shells collide then the ratio,
$T_{90}$/$\tau_{\beta},$ should show a correlation with the duration of the
burst. A plot of this ratio versus $T_{90}$ is shown for a sample of short and
long bursts in Fig. 8. The correlation is apparent.

It is now widely accepted that the progenitors for the two classes of GRBs are
quite distinct i.e., the merger of compact objects in the case of short GRBs
and the collapse of rapidly rotating massive stars in the case of long GRBs.
Formation of an accretion disk in the two cases is posed in a number of models
but important factors such as the size of the disk, the mass of the disk, the
strength of the magnetic field, in addition to the magnitude of the accretion
rate during the prompt phase, remain largely uncertain. With contributions
from intrinsic variability of the central engine or nearby shock-wave
interactions within a jet, we should not be surprised to observe a systematic
difference in the extracted variability time scales for long and short bursts,
since the progenitors have different spatial scales. Knowing the variability
timescales, we can estimate the size of an assumed emission region. From Fig.
5, we note that the smallest temporal-variability scale for the short bursts
is approximately 3 ms and that for the long bursts is approximately 30 ms:
These times translate to emission scales of approximately $10^{8}$ and
$10^{9}$ cm respectively. Our variability times and size scales are generally
consistent with the findings of Walker et al. (2000) although these authors
also reported observing time scales as small as few microseconds. We find no
evidence for variability times as low as a few microseconds.

Morsony et al. (2010) modeled the behavior of a jet propagating through the
progenitor and the surrounding circumstellar material and showed that the
resulting light curves exhibited both short-term and long–term variability.
They attribute the long-term variability, at the scale of few seconds, to the
interaction of the jet with the progenitor. The short-term scale, at the level
of milliseconds, they attribute to the variation in the activity of the
central engine itself. Alternatively, Zhang & Yan (2011) consider a model in
which the prompt emission is the result of a magnetically powered outflow
which is self-interacting and triggers rapid turbulent reconnections to power
the observed GRBs. This model also predicts two variability components but
interestingly and in contrast to the findings of Morsony et al. (2010) , it is
the slow component that is associated with the activity of the central engine,
and the fast component is linked to relativistic magnetic turbulence. While we
are not in a position to distinguish between these two models it is intriguing
nonetheless to note (see Fig. 5) that indeed there do appear to be two
distinct time domains for the $\tau_{\beta}$: a plateau region dominated
primarily by short bursts although it includes some long bursts too, and a
scaling region (i.e., a hint of a correlation between $\tau_{\beta}$ and
$T_{90}$) that is comprised solely of long bursts. In addition, we observe
that the time scale in the plateau region is the order of milliseconds whereas
that for the scaling region is approaching seconds.

There is considerable dispersion in the extracted $\tau_{\beta}$. The
variation is evident for both short and long-duration GRBs. The main cause of
this dispersion is not fully understood but one factor that may play a
significant role is angular momentum. As Lindner et al. (2010) note, the basic
features of the prompt emission can be understood in terms of accretion that
results via a simple ballistic infall of material from a rapidly rotating
progenitor. Material with low angular momentum will radially accrete across
the event horizon whereas the material with sufficient angular momentum will
tend to circularize outside the innermost stable circular orbit and form an
accretion disk. Simulations that go beyond the simple radial infall model
(Lindner et al., 2010, 2011) suggest that the formation of the disk leads to
an accretion shock that traverses outwards through the infalling material. If
one assumes that the initiation of such an accretion shock and the subsequent
emission of the prompt gamma-rays are associated with a particular time scale,
the variability of this scale then (as given by the dispersion in
$\tau_{\beta}$ for example) may reflect the different dynamics (initial
angular momentum and the mass of the black hole) of each GRB in our sample. In
the case of long GRBs, the mass of the central black hole can vary by an order
of magnitude thus potentially explaining a large part of the dispersion seen
in the $\tau_{\beta}$. However a similar dispersion for short bursts is
difficult to reconcile using the same arguments since the mass range for the
central black hole in standard merger models (at least for NS-NS mergers) is
expected to be significantly smaller.

##  6 CONCLUSIONS

We have studied the temporal properties of a sample of prompt-emission light
curves for short and long-duration GRBs detected by the Fermi/GBM mission. By
using a technique based on wavelets we have extracted the variability
timescales for these bursts. Our main results are summarized as follows:

a) Both short and long-duration bursts indicate a temporal variability at the
level of a few milliseconds. Variability of this order appears to be a common
feature of GRBs. This finding is consistent with the work of Walker et al.
(2000). However, unlike these authors we do not find evidence of variability
at a time scale of few microseconds.

b) In general the short-duration bursts have a variability time scale that is
significantly shorter than long-duration bursts. In addition, the
$\tau_{\beta}$ values seem not to depend in any obvious way on the luminosity
of the bursts. The dispersion over different GRBs in the extracted time scale
for short-duration bursts is an order of magnitude within the smallest
variability time, that time being approximately 3 milliseconds. The dispersion
for the long-duration bursts is somewhat larger. The origin of the dispersion
in either case is not known, although we should expect that the size of the
initial angular momentum and the mass of the system play significant roles. We
note in passing that the 3 millisecond time scale may not be a physical lower
limit and may be a result of signal to noise and the set of GRBs used in this
analysis. We remind the reader that our light curve resolution was 200 $\mu$s
and if a strong enough signal within a range of time scales between 0.5 - 3
milliseconds were present we would expect our technique to be sensitive to it.

c) The ratio of $T_{90}/\tau_{\beta}$ appears to be positively correlated with
the minimum variability time scale. This suggests further support for the
pulse paradigm view of the prompt emission as being the result of shell
collisions. In this respect, the minimum variability time scale is likely
related to key pulse parameters such as risetimes and/or widths.

d) For short-duration bursts, the variability parameter $\tau_{\beta}$ shows
negligible dependence on the duration of the bursts (characterized by
$T_{90}$). In contrast, the long-duration bursts indicate evidence for two
variability time scales: a plateau region (at the shortest time scale) which
is essentially independent of burst duration and a scaling region (at the
higher time scale) that shows a positive correlation with burst duration. The
transition between the two regions occurs around $T_{90}$ on the order of a
few seconds in the source frame.

## ACKNOWLEDGEMENTS

The NASA grant NNX11AE36G provided partial support for this work and is
gratefully acknowledged. The authors, in particular GAM and KSD, acknowledge
very useful discussions with Jon Hakkila and Narayan Bhat early in the
manuscript development. GAM and KSD also acknowledge helpful correspondences
with Jeffery Scargle.

Table 1: Short GRBs (Observer Frame). GRB |  $T_{90}$[sec] |  $\delta T_{90}$[sec] |  $\tau_{\beta}$ [sec] |  $\delta\tau^{-}_{\beta}$ [sec] |  $\delta\tau^{+}_{\beta}$ [sec]
---|---|---|---|---|---
080723913 | 0.192 | 0.345 | 0.0307 | 0.0192 | 0.0510
081012045 | 1.216 | 1.748 | 0.0052 | 0.0024 | 0.0044
081102365 | 1.728 | 0.231 | 0.0258 | 0.0100 | 0.0165
081105614 | 1.280 | 1.368 | 0.0306 | 0.0147 | 0.0282
081107321 | 1.664 | 0.234 | 0.0504 | 0.0129 | 0.0173
081216531 | 0.768 | 0.429 | 0.0138 | 0.0037 | 0.0050
090108020 | 0.704 | 0.143 | 0.0241 | 0.0064 | 0.0088
090206620 | 0.320 | 0.143 | 0.0143 | 0.0063 | 0.0112
090227772 | 1.280 | 1.026 | 0.0053 | 0.0009 | 0.0011
090228204 | 0.448 | 0.143 | 0.0028 | 0.0005 | 0.0005
090308734 | 1.664 | 0.286 | 0.0120 | 0.0040 | 0.0059
090429753 | 0.640 | 0.466 | 0.0285 | 0.0115 | 0.0193
090510016 | 0.960 | 0.138 | 0.0049 | 0.0009 | 0.0011
100117879 | 0.256 | 0.834 | 0.0331 | 0.0122 | 0.0192
Table 2: Long GRBs (Observer Frame). GRB |  $T_{90}$[sec] |  $\delta T_{90}$[sec] |  $\tau_{\beta}$ [sec] |  $\delta\tau^{-}_{\beta}$ [sec] |  $\delta\tau^{+}_{\beta}$ [sec]
---|---|---|---|---|---
080723557 | 58.369 | 1.985 | 0.0440 | 0.0113 | 0.0151
080723985 | 42.817 | 0.659 | 0.1894 | 0.0557 | 0.0789
080724401 | 379.397 | 2.202 | 0.0741 | 0.0208 | 0.0290
080804972 | 24.704 | 1.460 | 0.4306 | 0.1336 | 0.1937
080806896 | 75.777 | 4.185 | 0.4189 | 0.1471 | 0.2268
080807993 | 19.072 | 0.181 | 0.0232 | 0.0096 | 0.0164
080810549 | 107.457 | 15.413 | 0.1353 | 0.0648 | 0.1243
080816503 | 64.769 | 1.810 | 0.1067 | 0.0428 | 0.0715
080817161 | 60.289 | 0.466 | 0.1919 | 0.0402 | 0.0509
080825593 | 20.992 | 0.231 | 0.0775 | 0.0138 | 0.0168
080906212 | 2.875 | 0.767 | 0.1011 | 0.0182 | 0.0222
080916009 | 62.977 | 0.810 | 0.2266 | 0.0630 | 0.0872
080925775 | 31.744 | 3.167 | 0.1748 | 0.0425 | 0.0562
081009140 | 41.345 | 0.264 | 0.1095 | 0.0170 | 0.0201
081101532 | 8.256 | 0.889 | 0.0948 | 0.0302 | 0.0444
081125496 | 9.280 | 0.607 | 0.2182 | 0.0504 | 0.0656
081129161 | 62.657 | 7.318 | 0.0912 | 0.0292 | 0.0429
081215784 | 5.568 | 0.143 | 0.0319 | 0.0043 | 0.0050
081221681 | 29.697 | 0.410 | 0.2701 | 0.0641 | 0.0841
081222204 | 18.880 | 2.318 | 0.1956 | 0.0533 | 0.0732
081224887 | 16.448 | 1.159 | 0.2055 | 0.0356 | 0.0431
090102122 | 26.624 | 0.810 | 0.0347 | 0.0111 | 0.0164
090131090 | 35.073 | 1.056 | 0.0733 | 0.0169 | 0.0220
090202347 | 12.608 | 0.345 | 0.1444 | 0.0575 | 0.0954
090323002 | 135.170 | 1.448 | 0.1598 | 0.0436 | 0.0599
090328401 | 61.697 | 1.810 | 0.0682 | 0.0139 | 0.0175
090411991 | 14.336 | 1.086 | 0.0673 | 0.0391 | 0.0935
090424592 | 14.144 | 0.264 | 0.0249 | 0.0031 | 0.0036
090425377 | 75.393 | 2.450 | 0.1346 | 0.0369 | 0.0508
090516137 | 118.018 | 4.028 | 0.4938 | 0.2063 | 0.3544
090516353 | 123.074 | 2.896 | 0.7992 | 0.5686 | 1.9711
090528516 | 79.041 | 1.088 | 0.1314 | 0.0320 | 0.0423
090618353 | 112.386 | 1.086 | 0.2631 | 0.0536 | 0.0673
090620400 | 13.568 | 0.724 | 0.1667 | 0.0422 | 0.0564
090626189 | 48.897 | 2.828 | 0.0498 | 0.0078 | 0.0093
090718762 | 23.744 | 0.802 | 0.1621 | 0.0482 | 0.0686
090809978 | 11.008 | 0.320 | 0.2436 | 0.0515 | 0.0652
090810659 | 123.458 | 1.747 | 0.7319 | 0.3027 | 0.5161
090829672 | 67.585 | 2.896 | 0.0678 | 0.0141 | 0.0177
090831317 | 39.424 | 0.572 | 0.0266 | 0.0103 | 0.0169
090902462 | 19.328 | 0.286 | 0.0223 | 0.0026 | 0.0029
090926181 | 13.760 | 0.286 | 0.0435 | 0.0061 | 0.0070
091003191 | 20.224 | 0.362 | 0.0300 | 0.0051 | 0.0062
091127976 | 8.701 | 0.571 | 0.0395 | 0.0059 | 0.0069
091208410 | 12.480 | 5.018 | 0.0621 | 0.0180 | 0.0254
100414097 | 26.497 | 2.073 | 0.0418 | 0.0074 | 0.0090
Table 3: Long and Short GRBs ($T_{90}$ and $\tau_{\beta}$ in Observer Frame). Luminosities are taken from references given in footnotes.  GRB | $z$ |  $T_{90}$[sec] |  $\delta T_{90}$[sec] |  $\tau_{\beta}$ [sec] |  $\delta\tau^{-}_{\beta}$ [sec] |  $\delta\tau^{+}_{\beta}$ [sec] |  $L_{iso}$ [ergs/s] |  $\delta L_{iso}^{-}$ [ergs/s] |  $\log$ $\delta L_{iso}^{+}$ [ergs/s]
---|---|---|---|---|---|---|---|---|---
080804972 | 2.204 | 24.704 | 1.460 | 0.4306 | 0.1336 | 0.1937 |  111Nava et al. 2011 (A&A 530, A21 (2011))3.58$\cdot 10^{52}$ | 5.82$\cdot 10^{51}$ | 7.85$\cdot 10^{51}$
080810549 | 3.350 | 107.457 | 15.413 | 0.1353 | 0.0648 | 0.1243 |  222GCN #81009.59$\cdot 10^{52}$ | 1.28$\cdot 10^{52}$ | 1.28$\cdot 10^{52}$
080916009 | 4.350 | 62.977 | 0.810 | 0.2266 | 0.0630 | 0.0872 |  333Ghirlanda et al. 2011 (arXiv:1107.4096)1.04$\cdot 10^{54}$ | 8.79$\cdot 10^{52}$ | 8.79$\cdot 10^{52}$
081222204 | 2.770 | 18.880 | 2.318 | 0.1956 | 0.0533 | 0.0732 |  111Nava et al. 2011 (A&A 530, A21 (2011))1.26$\cdot 10^{53}$ | 7$\cdot 10^{51}$ | 6$\cdot 10^{51}$
090102122 | 1.547 | 26.624 | 0.810 | 0.0347 | 0.0111 | 0.0164 |  333Ghirlanda et al. 2011 (arXiv:1107.4096)8.71$\cdot 10^{52}$ | 5.6$\cdot 10^{51}$ | 5.6$\cdot 10^{51}$
090323002 | 3.570 | 135.170 | 1.448 | 0.1598 | 0.0436 | 0.0599 |  111Nava et al. 2011 (A&A 530, A21 (2011))6.87$\cdot 10^{53}$ | 6.55$\cdot 10^{53}$ | 4.45$\cdot 10^{52}$
090328401 | 0.736 | 61.697 | 1.810 | 0.0682 | 0.0139 | 0.0175 |  444GCN #90571.79$\cdot 10^{52}$ | 1.42$\cdot 10^{51}$ | 1.11$\cdot 10^{51}$
090424592 | 0.544 | 14.144 | 0.264 | 0.0249 | 0.0031 | 0.0036 |  555Ukwatta, T. N., et al. 2010, _ApJ._ , 711, 10731.62$\cdot 10^{52}$ | 4$\cdot 10^{50}$ | 5$\cdot 10^{50}$
090510016 | 0.903 | 0.960 | 0.138 | 0.0049 | 0.0009 | 0.0011 |  333Ghirlanda et al. 2011 (arXiv:1107.4096)1.78$\cdot 10^{53}$ | 1.2$\cdot 10^{51}$ | 1.2$\cdot 10^{51}$
090516353 | 4.100 | 123.074 | 2.896 | 0.7992 | 0.5686 | 1.9711 |  666GCN #94158.17$\cdot 10^{52}$ | 2.85$\cdot 10^{52}$ | 6.1$\cdot 10^{51}$
090618353 | 0.540 | 112.386 | 1.086 | 0.2631 | 0.0536 | 0.0673 |  555Ukwatta, T. N., et al. 2010, _ApJ._ , 711, 10738.47$\cdot 10^{51}$ | 1.17$\cdot 10^{51}$ | 3.4$\cdot 10^{50}$
090902462 | 1.822 | 19.328 | 0.286 | 0.0223 | 0.0026 | 0.0029 |  333Ghirlanda et al. 2011 (arXiv:1107.4096)5.89$\cdot 10^{53}$ | 9.71$\cdot 10^{51}$ | 9.71$\cdot 10^{51}$
090926181 | 2.106 | 13.760 | 0.286 | 0.0435 | 0.0061 | 0.0070 |  333Ghirlanda et al. 2011 (arXiv:1107.4096)7.40$\cdot 10^{53}$ | 1.45$\cdot 10^{52}$ | 1.45$\cdot 10^{52}$
091003191 | 0.897 | 20.224 | 0.362 | 0.0300 | 0.0051 | 0.0062 |  111Nava et al. 2011 (A&A 530, A21 (2011))4.53$\cdot 10^{52}$ | 3.71$\cdot 10^{51}$ | 6.55$\cdot 10^{51}$
091127976 | 0.490 | 8.701 | 0.571 | 0.0395 | 0.0059 | 0.0069 |  777GCN #102043.70$\cdot 10^{51}$ | 1.38$\cdot 10^{50}$ | 1.06$\cdot 10^{50}$
091208410 | 1.063 | 12.480 | 5.018 | 0.0621 | 0.0180 | 0.0254 |  111Nava et al. 2011 (A&A 530, A21 (2011))1.45$\cdot 10^{52}$ | 1.48$\cdot 10^{51}$ | 3.45$\cdot 10^{51}$
100117879 | 0.920 | 0.256 | 0.834 | 0.0331 | 0.0122 | 0.0192 |  111Nava et al. 2011 (A&A 530, A21 (2011))2.63$\cdot 10^{52}$ | 5.01$\cdot 10^{51}$ | 1.08$\cdot 10^{52}$
100414097 | 1.368 | 26.497 | 2.073 | 0.0418 | 0.0074 | 0.0090 |  888GCN #105951.00$\cdot 10^{53}$ | 1.58$\cdot 10^{52}$ | 7.6$\cdot 10^{51}$

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