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increases, assisting further heating and expansion. When a SN injects energy into the ISM, the resulting pressure in the hot bubble greatly exceeds the surrounding pressure. Provided that the energy is not quickly radiated away, the bubble expands only if it is not confined by self-gravity. This second condition was fo... |
Ceverino & Klypin ( 2007 in terms of the pressure difference between the bubble and the surrounding gas [EQUATION] where [MATH] is the radius of the bubble, and [MATH] is the ambient gas density. Using the ideal gas equation of state and our SF density threshold, we can rewrite eq. to obtain the minimum resolution nece... |
[EQUATION] Since our models start to resolve individual SNe, the mass of a resolution element should be small enough in order for it to get heated by the typical [MATH] explosion energy. A single SN explosion in a dense cloud may have a hydrodynamical impact only if it can heat its host cell to high ( [MATH] ) temperat... |
[EQUATION] which gives us an estimate of the minimum resolution necessary to heat up the host cell [EQUATION] At first glance, this constraint requires much higher resolution than the self-gravity condition (eq. ). Fortunately, type II SN explosions have a [MATH] delay after the initial starburst, and many of them expl... |
Once conversion of feedback energy into hydrodynamical expansion becomes efficient, lack of spatial resolution can present an additional problem. If the density contrast in the ISM is not resolved in the simulation, heating and hydrodynamical stirring of the star-forming clouds might erase high-density peaks bringing t... |
3.3. Neutral gas kinematics in quasar absorption lines Fig. shows the [MATH] diagram of the entire simulation volume in all three runs at [MATH] . In cells hosting stellar particles, stellar winds and SNe return both mass and energy. Assuming that winds remove all ambient gas, the minimum density in such cells can be e... |
[EQUATION] where [MATH] is the fiducial outflow speed, and [MATH] is the cell size. These feedback regions can be easily seen in the starburst model in Fig. at [MATH] , and to a much lesser degree in the quiescent model. |
Since we model isolated systems without external accretion, winds from the disk do not experience any ram pressure of the infalling material and can be stopped only by gravity and collision with the gas previously blown off the disk. The wind speeds often exceed several hundred [MATH] , consequently a large fraction of... |
To answer this question, we constructed a set of low-ionization metal line spectra. At each time output, we projected 200 random lines of sight within [MATH] of the center of each disk and calculated absorption line profiles of an unsaturated low-ion transition along sight-lines with HI column density above [MATH] . Fo... |
of the total optical depth of all components in the line. Note that this diagnostic measures the neutral gas velocity dispersion, not the typical outflow velocities, and is dominated by clouds with large optical depths – its more detailed discussion and the comparison to the equivalent width can be found in |
Prochaska et al. ( 2008 . Fig. shows the distribution of [MATH] widths for each model as a function of time, along with the median value. Although these velocities cannot be compared to DLAs statistics directly, since we do not have a cosmological sample and do not account for gas accretion which would regulate SF epis... |
Razoumov et al. ( 2007 and close to [MATH] in (Pontzen et al., 2008 . Note that in cosmological models the simulated widths are also sensitive to the DLA cross-sections, as, for instance, the velocity dispersion in “puffy” galaxies with extended radial profiles will have an additional weight. |
We argue that if a substantial fraction of clumps along the quasar line of sight in a host halo experience an active starburst, it could result in a much larger velocity dispersion possibly explaining the observed incidence rate of high-velocity DLAs. The fraction of such active star-forming galaxies is poorly constrai... |
Here instead we focus on individual systems. In the starburst model [MATH] approximately matches the observed value for [MATH] whereas the quiescent model has [MATH] for most of the disk evolution. The low-resolution model features a delayed and much weaker SF resulting in a brief episode during which [MATH] reaches th... |
We can see the formation of hot galactic chimneys driving the outflows in the vertical slice in Fig. . The visual structure of the outflows is very different from the single-source models of winds from high-redshift dwarf galaxies |
(e.g., Fujita et al., 2004 , more resembling the high-resolution multiphase models of Ceverino & Klypin ( 2007 ); Wada ( 2008 . Our current spatial resolution is not yet sufficient to model instabilities in the shells created by hot bubbles (Ferrara & Ricotti, 2006 or even follow these shells farther away from the disk... |
Fig. shows the range of velocities and densities in the wind in the starburst model. Ionized, low-density wind moves with velocities up to several thousand [MATH] , whereas neutral gas exhibits velocities of few hundred [MATH] . Cold gas absorption comes from [MATH] ; there is a clear asymmetry above and below the disk... |
4. Conclusions We use high-resolution hydrodynamic simulations of isolated [MATH] protogalactic clumps to show for the first time that the high-end tail of the DLA neutral gas velocity width distribution can be naturally produced if a substantial fraction of clumps in [MATH] halos experience a starburst event at the ra... |
[MATH] lasting roughly for the entire duration of the starburst. Similar to other multiphase disk models, the cold component of the ISM is distributed in a complex network of filamentary structures confined by hot bubbles and voids. Inside these filaments dense clouds form as gravitational instabilities the growth of w... |
In this paper we chose the typical clump masses and initial conditions representative of protogalactic environments at [MATH] . We expect similar winds to arise in grid-based cosmological simulations at |
[MATH] spatial resolution. Although none of the current cosmological models have such resolution, we argue that using AMR techniques to zoom in only on those protogalactic clumps that have a high gas infall rate will allow to obtain galactic winds from thermal feedback in the cosmological context, without suppressing c... |
Acknowledgments I thank Jesper Sommer-Larsen and Eduard Vorobyov for many useful discussions. I am grateful to Marc Schartmann for providing me with the cooling data. Computational facilities for this work were provided by ACEnet, the regional high performance computing consortium for universities in Atlantic Canada. A... |
# Source: arxiv 0808.2685 # Title: Statistical Tools for Analyzing the Cosmic Ray Energy Spectrum # Sections: all # Downloaded: 2026-03-02T07:58:20.981988+00:00 |
Statistical Tools for Analyzing the Cosmic Ray Energy Spectrum Abstract In this paper un-binned statistical tools for analyzing the cosmic ray energy spectrum are developed and illustrated with a simulated data set. The methods are designed to extract accurate and precise model parameter estimators in the presence of s... |
keywords: cosmic ray spectrum , power-law , CRPropa , TP-statistic , flux suppression Introduction The observation of suppression in the flux of the highest energy cosmic rays (CRs) has been of central interest to astro-particle physics since the prediction of the GZK-effect |
in 1966. Most recently both the Auger and the HiRes detectors have released results favoring the observation of flux suppression at a [MATH] and [MATH] level of confidence, respectively. |
With this in mind, we describe a set of statistical tools designed to extract the most accurate and precise information concerning the flux of the highest energy cosmic rays. By binning the data we can only lose information |
(see [MATH] and therefore our statistical tools use an un-binned maximum likelihood approach to answer two related statistical questions: |
Is there flux suppression at the highest energies? and, if yes, What are the characteristic cut-off energy and shape parameters? |
In detail we first generate a toy data set using the CRPropa package , as in [MATH] 2.2 We then fit this simulated data to the three models described in [MATH] 2.3 The un-binned maximum likelihood fit is outlined in [MATH] 3.1 and methods for incorporating systematic and statistical energy errors are described in [MATH... |
and a likelihood ratio test Though we cast our discussion in terms of cosmic ray energies, it is worth noting that these tools can be applied to any astrophysical data set where deviations from the power-law hypothesis are relevant, e.g. the galaxy correlation function |
or gamma ray astronomy CRPropa Data Set and Models 2.1 Input from the HiRes and Auger Observatories Both the HiRes and Auger observatories have reported spectra and fit parameters for various power-law models. The collaborations use binned fitting methods. They fit the spectrum over many orders of magnitude in energy b... |
relevant only to the highest energies. The best fit double power-law parameters reported by HiRes are [MATH] (stat) [MATH] (sys), |
[MATH] (stat) and [MATH] (stat). For the same model Auger reports [MATH] (stat) [MATH] (sys), [MATH] (fixed) and [MATH] (stat). Fitting to the Fermi power-law Auger |
finds [MATH] (stat), [MATH] (stat) and [MATH] (stat). 2.2 A Toy CR Data Set To illustrate the methods in this note we use un-binned proton primary cosmic ray, CR, arrival energies (in EeV [MATH] eV) as simulated by the package CRPropa |
with input spectral index [MATH] [MATH] EeV and [MATH] EeV. We draw [MATH] events to act as a toy data set from a modern CR detector. |
The CRPropa toy data set is similar size and shape to the flux reported by these observatories but the results of this study do not, otherwise, reflect any information about any physical data set. The probability distribution function (p.d.f.) of the best fit double power-laws reported by HiRes |
and Auger are shown in Figure along with the CRPropa toy data. The CRPropa propagation simulation is implemented by first generating proton CR primaries with initial energies according to a power-law “at the source,” propagating them through a simulated Universe and then observing the final energy. The spacial extent o... |
2.3 Power-Law Models The fundamental probability distribution function governing the pure power-law assumption, denoted [MATH] , is shown in Table |
[MATH] The parameter [MATH] is referred to as the spectral index Here the sub-scripted-P stands for Pure-power-law. For the highest energy CRs, the interesting observation would be to confirm or deny deviation from the power-law form at the highest magnitudes, i.e. the GZK-cutoff. We therefore study two toy models that... |
[MATH] (“b” for bend or break) and [MATH] above. The point at which this p.d.f. reaches half the value it would have if the pure power-law continued above [MATH] is given by |
[MATH] , see for a discussion of this quantity. Both HiRes and Auger have analyzed their data using this model. We also study a toy p.d.f. where the cut-off is a “Fermi-like” Power-law (FP) |
The advantage of fitting with this toy model is that the location parameter [MATH] is a parameter in the fit. All three p.d.f.’s are normalized on the interval [MATH] , i.e. |
[MATH] for each of the models [MATH] The first element of the parameter vector [MATH] is fixed for the fit (see [MATH] ) and then varied to estimate the stability (see [MATH] 4.1 ). Thus the power-law has one free parameter and the other models have three; low energy spectral index, location of cut-off and “steepness” ... |
Fitting the Data We take an un-binned maximum log-likelihood approach to estimating the best-fit parameters of each model. The method constructed here is designed to extract the maximum possible statistical information about these parameters. For the ideal detector we assume that the observed energies are known with in... |
3.1 Ideal Detector We find estimates of the parameters in each model by maximizing, [EQUATION] where the sum is carried out over the event energies and |
[MATH] is fixed. The global maximum of this function [MATH] determines the best parameter estimates, [MATH] The the function is maximized using Minuit |
with the MIGrad option. To determine the one degree of freedom error estimate for a parameter we vary the parameter (with the others fixed at [MATH] until [MATH] The two degrees of freedom error estimates |
are determined by varying two parameters with the other fixed and choosing the contour such that [MATH] For the toy data set, we plot these contours and the asymmetric one degree of freedom error estimates in [MATH] : Figure 11 and 12 |
3.2 Systematic Energy Error The errors on the observed energy [MATH] of an event from a real CR detector are considerable and must be included in any realistic analysis of a spectrum. For our purposes, these errors take the two canonical forms; |
statistical and systematic i.e. [MATH] The systematic errors energy errors of a CR detector reflect the uncertainties in the absolute calibration of the detector. At the highest energies the systematics are the dominant contribution to the overall uncertainty of an event’s energy. For example, the two fluorescence dete... |
and Hires report uncertainties of 22% and 17% respectively. The shift in energy due to the systematic error can be asymmetric, i.e. [MATH] and energy dependent, see Eq( ), but it effects every event at a given energy the same way; a shift up or down. For the Monte-Carlo (MC) data sets we model the systematic detect... |
[EQUATION] Here we choose symmetric systematically-shifted energies such that the energy of the [MATH] event is [MATH] For the systematic errors we choose [MATH] and [MATH] |
To account for this in the parameter estimation procedure, we shift each energy up or down and carry out the methods in [MATH] 3.1 The difference between the parameter estimates of a shifted set and those of the centered set gives “systematic” errors of the parameter estimates. |
3.3 Statistical Energy Error To model the statistical energy errors of the detector we assume that the true energy of the cosmic ray has a [MATH] chance of being within the interval [MATH] The observed energy has been “smeared” from the true value; |
[MATH] where [MATH] is drawn from a normal distribution with mean [MATH] and variance [MATH] Note that while the true energies can only be found on [MATH] , there is a nonzero probability for the (after smearing) observed energy to be less than |
[MATH] [MATH] lives on the interval [MATH] This edge effect near [MATH] can be accounted for by assuming that the true distribution of energies follows a power-law well below [MATH] and then re-normalizing the convolution technique used in Howell |
See [MATH] for further discussion. For the integrand, three factors are necessary: 1. The model to be fitted, [MATH] (see [MATH] 2.3 ). By letting [MATH] we are assuming that the power-law extends below the observed [MATH] |
2. A normal distribution [MATH] with mean [MATH] and variance [MATH] to reflect the statistical energy errors. 3. The acceptance of the CR detector as a function of the true energies [MATH] Since we are using MC data we choose [MATH] for simplicity. |
The convolution is calculated by integrating over all possible true energies ( [MATH] ): [EQUATION] Re-normalizing so that the observed energies define a p.d.f., we numerically calculate the p.d.f. to be: |
[EQUATION] and we must modify the likelihood found in Eq( ) accordingly: [EQUATION] By finding the parameters [MATH] which maximize Eq( ) we can be confident that we are accounting for the statistical uncertainty inherent in data collected by a realistic detector. To model statistical errors in our toy data set, we par... |
as in Eq( ) with [MATH] and [MATH] Evaluating the Fit In this section we outline ways to evaluate the fit of a candidate model to the data set. The Kolmogorov-Smirnov statistic can be used to extract a best fit minimum energy [MATH] |
and, with its corresponding [MATH] -value, evaluate the “absolute goodness of fit” of a candidate model (see [MATH] 4.1 ). The relevant question for CR physics is not whether a particular model is a good fit to the data but rather whether the flux exhibits suppression (relative to the single power-law form) at the high... |
4.1 Kolmogorov Statistic While the minimum value of the likelihood function will indeed give the best value of the fit parameters, this fit may nonetheless be poor. The typical |
method for evaluating goodness of fit is the Kolmogorov-Smirnov test The relevant statistic for this test is the KS distance: [EQUATION] |
where, [MATH] and [MATH] are the cumulative distribution functions (c.d.f.) of the best fit model and the data respectively. The maximum distance between the c.d.f.’s is taken over all energies in the fitted data set, |
[MATH] By stepping over [MATH] and re-minimizing Eq( ) at each step to determine the best fit parameters, we can calculate [MATH] as a function of [MATH] The value of [MATH] that minimizes [MATH] can be taken as the best estimate of the minimum energy above which the model holds |
To test how well a particular model fits the data we must simulate many MC data sets drawn from the best fit model p.d.f. with the same number of events as the original data. The fraction of sets [MATH] with [MATH] greater than that of the data gives the suitable [MATH] -value; if [MATH] then it is unlikely that the da... |
4.2 Tail Power Statistic The Tail-Power (TP) statistic is similar to the KS statistic discussed above, however it has, at least, three advantages over [MATH] when testing the power-law assumption; |
1. The TP statistic and it’s corresponding [MATH] -value [MATH] are nearly independent of the value of the spectral index [MATH] |
2. The asymptotic behavior of the TP statistic is known, and therefore no simulations are required to calculate the corresponding [MATH] -value [MATH] |
3. If [MATH] the deviation suggests flux suppression in the tail and if [MATH] the deviation suggests flux enhancement in the tail |
and 4. [MATH] offers an unambiguous [MATH] -value in standard deviations. This “measure of power-law-ness” has been developed and studied elsewhere (see |
and here we expand its use to the un-binned case. The sample TP statistic is defined as [EQUATION] where: [EQUATION] and the sum is carried out over all [MATH] events with energy greater than a given minimum. If the data are drawn from a pure power-law then [MATH] will tend to zero as [MATH] , regardless of the value o... |
We may approximate the asymptotic joint distribution of [MATH] and [MATH] as a bivariate Gaussian [MATH] The asymptotic mean and variance of [MATH] are [MATH] |
and [MATH] , and of [MATH] are [MATH] and [MATH] . The random variables [MATH] and [MATH] are highly correlated; the correlation coefficient is [MATH] independent of [MATH] Thus, for a given [MATH] and [MATH] , we calculate the p.d.f. of [MATH] to be, |
[EQUATION] The analytic “location” [MATH] and “shape” [MATH] parameters of this distribution are consistent with simulation generated values. We measure the [MATH] -value [MATH] for the TP statistic in units of standardized deviation, |
[EQUATION] A spectrum with flux suppression in the tail (like that in the Fermi-like model) will result in a positive significance |
The application of Eq( 10 ) to the toy CR data set (see [MATH] 2.2 is plotted in Figure The top panel shows the (pure power-law) spectral index as a function of |
[MATH] A spectral index which increases as [MATH] increases is indicative of flux suppression. The red, left leaning hatching shows the variation of [MATH] due to a |
[MATH] systematic shift in the energies (see [MATH] 3.2 ) while the opposite, blue hatching shows the statistical error of the estimator [MATH] , see |
[MATH] 3.1 The bottom panel shows the resulting TP statistic significance [MATH] in standard deviations. Notice that while the systematic errors can be significant for the measured spectral index, they do not effect the TP statistic. Since we must estimate the spectral index to compute [MATH] , we also propagate the st... |
To test the effectiveness of this statistic, we apply it to a series of simulated data sets drawn from both the Fermi and double power-law models. For all the models we set |
[MATH] EeV, [MATH] and either [MATH] or [MATH] We vary each characteristic cut-off energy, either [MATH] or [MATH] in three steps [MATH] The total number of events in the data set is varied in four steps |
[MATH] For each of these twelve sets of parameter choices we make [MATH] Monte-Carlo realizations and plot the mean and RMS of [MATH] in Figure |
Based on Figure we can see that the best way to evaluate a data set with a potential for tail suppression is to collect as much data with [MATH] as close to the expected cut-off as possible. The experimenter may use Figure , or one like it, to help tune observation parameters, i.e. collecting time on a gamma ray source... |
prior to analyzing a data set to avoid a penalty for scanning in this parameter. 4.3 Model Discrimination Here we introduce a likelihood ratio test designed to discriminate candidate suppressed models (DP and FP) from the pure power-law. We define two log-likelihood ratios; for each model M: |
[EQUATION] where [MATH] with M either DP (double power-law) or FP (Fermi-like), and [MATH] for the pure power-law likelihood per event (see Table and Eq( )). Note that each suppressed model is fit independently of the pure power-law best fit. The asymptotic variance of [MATH] can be estimated by the sample value: |
[EQUATION] The hypothesis of the pure power-law is nested within the hypothesis of a suppressed power-law. As a consequence, [MATH] as [MATH] |
and the distribution of [MATH] is not Gaussian The correct [MATH] -value is calculated as the integral of a [MATH] function [EQUATION] |
where [MATH] We interpret this [MATH] -value in the following way: if [MATH] is “small” then the best fit model M may be preferred over the best fit pure power-law. By small we mean that, a priori and rather arbitrarily, we may choose to reject the single power-law in favor of the model if [MATH] . This quantity tells ... |
For each of the twelve sets of parameter choices used in Figure we plot the mean and RMS of [MATH] in Figure As before, we see that the best way to reject the power-law in favor of the suppressed model is to collect as much data with [MATH] as close to the expected cut-off as possible. Note that for [MATH] the distribu... |
Summary and Conclusion In this paper we describe a set of statistical tools designed to extract the most accurate and precise information about the flux of the highest energy cosmic rays. We show how to use the un-binned likelihood method described in [MATH] 3.1 to fit a data set to the three model distributions descri... |
[MATH] 3.2 and [MATH] 3.3 respectively. In [MATH] we describe [MATH] -values useful for extracting information about flux suppression. We show in [MATH] 4.2 and [MATH] 4.3 how an experimenter might use an |
a priori estimate of the cut-off energy to maximize an observational setup for detecting flux suppression. The collection of these statistical tools are the primary result of this paper. To answer the questions posed in the introduction for a given data set we suggest the following steps: |
1. Estimate the best fit parameters [MATH] of the model; (a) The estimates [MATH] [MATH] or [MATH] and [MATH] or [MATH] are determined via the likelihood Eq( ), |
(b) The estimate of the minimum energy [MATH] is that which minimizes the Kolmogorov distance [MATH] (see [MATH] 4.1 ). 2. Shift the energies up and down according to the systematic uncertainty described in [MATH] 3.2 and repeat step ( ). The resulting shift in parameter estimates gives the systematic uncertainty of th... |
3. Obtain the model parameter estimates using the methods in [MATH] 3.3 to incorporate the statistical error of each event energy. |
4. Test the model hypothesis; (a) The absolute goodness of fit for any of the models can be evaluated using [MATH] in [MATH] 4.1 |
(b) The Tail-Power statistic [MATH] can be used to reject the single power-law hypothesis (nearly independently of the spectral index estimate, see [MATH] 4.2 |
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