Note on galaxy catalogues in UHECR flux modelling
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Mon. Not. R. Astron. Soc. 403, 2131–2136 (2010) doi:10.1111/j.1365-2966.2010.16249.x
Note on galaxy catalogues in UHECR flux modelling
Hylke B. J. Koers1 and Peter Tinyakov1,2
1 Service de Physique Théorique, Université Libre de Bruxelles (U.L.B.), CP225, Bld. du Triomphe, B-1050 Bruxelles, Belgium
2 Institute for Nuclear Research, 60th October Anniversary Prospect 7a, 117312, Moscow, Russia
Accepted 2009 December 23. Received 2009 November 25; in original form 2009 September 16
ABSTRACT
We consider the dependence of ultra-high energy cosmic ray (UHECR) flux predictions on the
choice of galaxy catalogue. We demonstrate that model predictions by Koers and Tinyakov,
based on the so-called KKKST catalogue, are in good agreement with predictions based on the
XSCz catalogue, a recently compiled catalogue that contains spectroscopic redshifts for a large
fraction of galaxies. This agreement refutes the claim by Kashti that the KKKST catalogue
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is not suited for studies of UHECR anisotropy due to its dependence on photometric redshift
estimates. In order to quantify the effect of galaxy catalogues on flux predictions, we develop
a measure of anisotropies associated with model flux maps. This measure offers a general
criterion to study the effect of model parameters and assumptions on the predicted strength of
UHECR anisotropies.
Key words: catalogues – cosmic rays – large-scale structure of Universe.
The adequacy of the KKKST catalogue for UHECR flux mod-
1 I N T RO D U C T I O N
elling is questioned by Kashti (2009) because of its dependence on
Galaxy catalogues are an indispensable tool to model the structure photometric redshift estimates. The errors in these estimates give
of the local Universe, as required for studies of ultra-high energy rise to errors in flux estimates, which may distort model flux maps
cosmic ray (UHECR) anisotropy. Many such studies (e.g. Cuoco and may lead to inaccurate model predictions. The uncertainty in
et al. 2006; Kashti & Waxman 2008; Kashti 2009; Takami et al. photometric redshift estimates is indeed fairly large: Jarrett (2004a)
2009) have used the PSCz catalogue (Saunders et al. 2000) for quotes an error of 10–20 per cent for most normal galaxies in the
this purpose. This catalogue, however, has its drawbacks: it suffers 2MASS survey [Kashti (2009) claims an ∼30 per cent systematic
from incomplete sky coverage, may underestimate galaxy counts uncertainty]. In Paper I, we deemed these uncertainties acceptable
in high-density regions (Huchra et al. 2007) and has limited statis- for two reasons. First, large-scale anisotropies in flux maps arise
tics. The Two Micron All Sky Survey (2MASS) galaxy catalogue as a collective effect of many sources. Adding flux contributions of
(Jarrett et al. 2000, 2004b; Skrutskie et al. 2006), offering complete many individual galaxies reduces the relative strength of fluctua-
sky coverage (except for the galactic plane) and excellent statis- tions (for the KKKST catalogue with a smearing angle of θ s = 6◦ ,
tics, improves on these issues. However, this catalogue does not as adopted in Paper I, a model flux is composed of individual con-
contain spectroscopic redshifts so redshifts have to be estimated by tributions of ∼600 galaxies). Secondly, the remaining uncertainties
photometry, i.e. from the observed brightness under the assump- only affect the model flux beyond 30 Mpc, while the imprint of local
tion of a standard intrinsic brightness in a specific band. Efforts structure is strongest at close distances.
to obtain spectroscopic redshifts for 2MASS galaxies are ongoing. In this paper, we compare model fluxes based on the KKKST
The resulting catalogue, termed XSCz, is presently being compiled catalogue to model fluxes based on a preliminary version of the
(Jarrett et al. 2006). XSCz catalogue.1 This allows us to determine to which extent the
Motivated by the imperfection of existing galaxy catalogues, flux predictions in Paper I are contaminated by the inaccuracies
Kalashev et al. (2008) have compiled a hybrid catalogue that con- in the catalogue used. The XSCz catalogue (Jarrett et al. 2006)
tains galaxies with spectroscopic redshifts from the (HYPER)LEDA is a full-sky galaxy catalogue based on the 2MASS XSC, which
data base (Paturel et al. 2003) within 30 Mpc and galaxies from the contains resolved sources from the 2MASS galaxy survey (Jarrett
2MASS Extended Source catalogue (XSC; Jarrett et al. 2000) at et al. 2000, 2004b; Skrutskie et al. 2006). The XSCz catalogue con-
larger distances, for which photometric redshift estimates are used tains over a million galaxies with an observed Ks magnitude of 14
(the two contributions are of course normalized appropriately). This or brighter. Spectroscopic redshift measurements are available for
catalogue, which we have termed the KKKST catalogue, is used in
Koers & Tinyakov (2009b, hereafter Paper I).
1 We
thank Tom Jarrett for providing us with a preliminary version of the
E-mail: hkoers@ulb.ac.be (HBJK); Petr.Tiniakov@ulb.ac.be (PT) XSCz catalogue.
C 2010 The Authors. Journal compilation
C 2010 RAS2132 H. B. J. Koers and P. Tinyakov
27 per cent of these galaxies, but this fraction increases with in- 3 1
creasing brightness. In this work, we will only consider galaxies 0.8
with Ks magnitude below 12.5. Spectroscopic redshifts are avail- 2.5 0.6
Distribution (arbitrary units)
able for ∼70 per cent of these relatively bright galaxies. The XSCz 0.4
catalogue is very well suited to model the matter distribution in the 2 0.2
Universe because of its completeness, large statistics and spectro- 0
0 0.2 0.4 0.6 0.8 1
scopic redshift measurements. We therefore consider the resulting 1.5
|Δf|
model fluxes as a benchmark against which UHECR flux predic-
tions derived with the KKKST catalogue (as well as with the PSCz 1
catalogue) can be cross checked. As we will demonstrate, flux pre-
dictions based on the KKKST catalogue, as well as those based on 0.5
the PSCz catalogue, are in good agreement with the XSCz predic-
tion (see Section 2 for a quantitative statement). This provides an a 0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
posteriori verification of the accuracy of our models in Paper I. Δf
We also investigate, in a general set-up, the relationship be-
tween galaxy catalogues and the predicted strength of UHECR Figure 2. Distribution of f for individual galaxies in the XSCz catalogue.
anisotropies. For this purpose we develop a measure that quan- In producing this plot, we approximated fluxes as φ ∝ D −2 and we only
tifies the relation between model flux maps, which represent the considered galaxies with D > 5 Mpc.
model predictions for a given galaxy catalogue, and the test statistic
D (introduced in Paper I), which is a measure of the strength of large: we find that the average value of |δz/z|, the absolute relative
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UHECR anisotropies. This measure can be used to assess the pre- difference in redshift determinations, is 0.36.
dicted strength of anisotropies from the ‘contrast’ that is exhibited The UHECR luminosity at the source being unknown, model flux
in model flux maps and is independent of the event number. Because predictions are conveniently formulated in terms of the normalized
of its generality, this measure may also be useful to study the effect flux
of other model parameters and assumptions (e.g. threshold energy, φ
f = , (1)
deflection angle or injection spectrum) on the predicted strength of φ
UHECR anisotropies.
where φ denotes a single-source flux and φ is the average flux of
This paper is organized as follows. In Section 2, we compare
all galaxies. (Note that the overall flux normalization is determined
model predictions from the KKKST and PSCz catalogues to pre-
by observations.) To get a rough estimate of the inaccuracies in flux
dictions based on the XSCz catalogue. Section 3 concerns the re-
predictions at the level of individual sources, we approximate the
lationship between galaxy catalogues and the predicted strength of
flux associated with every galaxy as φ ∝ D −2 , D being the source
UHECR anisotropies. We summarize our findings in Section 4.
distance (we take D ∝ z here). Here, and throughout this paper, we
assume that all sources have the same intrinsic luminosity. We then
2 AC C U R AC Y O F M O D E L F L U X M A P S compute the quantity f = f photo − f spec , where f photo is the nor-
malized flux estimate based on the photometric redshift estimate and
We begin our analysis by considering the accuracy of photomet- f spec is the normalized flux derived from the spectroscopic redshift.
ric redshift determinations in the 2MASS catalogue. We compute The distribution of f is shown in Fig. 2. Here we only consider
the relative difference δz/z := (zphoto − zspec )/zspec , where zphoto galaxies beyond 5 Mpc because very close-by sources outshine the
denotes a photometric redshift estimate and zspec denotes a spec- bulk even though they do not provide a good statistical description
troscopic redshift, for all galaxies in the XSCz catalogue that have of the local source density. As a consequence, errors in photometric
spectroscopic redshifts. The distribution of δz/z is shown in Fig. 1. redshift estimates, which are already large at small distances, would
The inaccuracy in photometric redshift estimates is indeed fairly be strongly amplified and spoil the accuracy of model flux maps.2
We now consider model UHECR flux maps, which are generated
1 using the methods described in Paper I. Throughout this work, we
1.6
0.8 assume a proton composition and a power-law injection spectrum
1.4 0.6 with spectral index p = 2.2 extending to very high energies. We
Distribution (arbitrary units)
0.4 adopt a cold dark matter (CDM) concordance model with Hub-
1.2
0.2 ble constant 72 km s−1 Mpc−1 and cosmological density parameters
1 0
0 0.2 0.4 0.6 0.8 1 m = 0.27 and = 0.73. Following the set-up in Paper I, we
0.8 | δz / z | adopt a smearing angle θ s = 6◦ . Using the method described in
Koers & Tinyakov (2009a), we have verified that all catalogues
0.6
contain sufficient galaxies to provide a good statistical description
0.4 on this angular scale. We remove the region |b| < 15◦ from our
0.2 analysis because of incompleteness near the galactic plane. (We
choose the same region for every catalogue for comparison; for the
0
-1 -0.5 0 0.5 1 1.5 2 2.5 3 XSCz skymap, this choice is overly conservative.) Fig. 3 shows the
δz / z
Figure 1. Distribution of the relative difference δz/z between photometric 2 Note that we also excluded sources closer than 5 Mpc in the search for
and spectroscopic redshift estimates for individual galaxies in the XSCz correlations between UHECR arrival directions and the large-scale structure
catalogue. Inset: cumulative distribution of |δz/z|. In producing this figure, of the Universe reported in Paper I. Other methods are more suitable to test
we only used galaxies with D > 5 Mpc. for the existence of bright, close-by sources.
C 2010 The Authors. Journal compilation
C 2010 RAS, MNRAS 403, 2131–2136Galaxy catalogues in UHECR flux modelling 2133
1.8 1
KKKST
1.6 PSCZ 0.8
0.6
Distribution (arbitrary units)
1.4
0.4
1.2 0.2
0
1 0 0.2 0.4 0.6 0.8 1
0.8 |Δf|
0.6
0.4
0.2
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Δf
Figure 4. Distribution of f for the model fluxes shown in Fig. 3.
similar. The average value of |f | is 0.17 (0.19) for the KKKST
(PSCz) catalogue.
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Are differences in model flux predictions of roughly 20 per cent
of the average flux, as found between the model flux maps shown
in Fig. 3, acceptable? To answer this question we will investigate
the effect on test statistics, since these ultimately encode model
predictions in statistical tests such as those used in Paper I. We
consider here the D-test, to which we will refer as the ‘flux sampling
method’. The test defines a test statistic D as the Kolmogorov–
Smirnov (KS) distance DKS between the cumulative distribution of
flux values sampled by a set of UHECR events and a reference
Figure 3. UHECR model flux maps for E 0 = 60 EeV, θ s = 6◦ and uni- distribution that corresponds to the model that is tested (see Paper I
form exposure. The three panels show model predictions based on different for details and discussion). Here and in the following we consider
galaxy catalogues: the KKKST catalogue (top panel), the XSCz catalogue two models: the ‘isotropy’ model (denoted I), which states that
(middle panel) and the PSCz catalogue (bottom panel). The grey bands are UHECR events are distributed isotropically (we do not consider
chosen such that each contains one-fifth of the model flux, with darker grey experimental exposure here), and the ‘structure’ model (denoted S),
indicating a larger flux. which states that UHECR sources trace the distribution of matter
in the Universe as modelled from a particular galaxy catalogue.
As a case study, we show in Fig. 5 the distribution of the test
flux maps for a threshold energy E 0 = 60 EeV. We observe a good statistic D for 20 UHECR events with energies in excess of 60 EeV.
overall resemblance between the maps in the location of over- and Here the reference distribution corresponds to an isotropic flux so
underdense regions. that the curve labelled ‘isotropy’ follows the universal KS self-
The strength of anisotropies contained in a flux map can be correlation distribution (it is thus the same for all catalogues). The
expressed in terms of a flux distribution obtained by sampling the three ‘structure’ distributions represent model predictions under
map uniformly over the sky. This distribution encodes the relevant
intrinsic properties of the flux map, i.e. properties relating to the
10
strength and area of over- and underdense regions, but not to their Isotropy 1
position on the sky. Postponing a more thorough discussion of the Structure, KKKST 0.8
Structure, PSCz
flux distribution to the next section, we point out here that the 8 Structure, XSCz 0.6
Distribution (arbitrary units)
flux distribution corresponding to the KKKST flux map is broader 0.4
0.2
(displaying more contrast) than the flux distribution obtained with 6 0
the XSCz catalogue, which in turn is broader than the PSCz one. 0 0.2 0.4 0.6
This is reflected in the fact that the band of highest flux (darkest D
4
grey) in Fig. 3 occupies the smallest area on the sphere: 7.5 per cent
for the KKKST flux map, 8.2 per cent for the XSCz flux map and
10.4 per cent for the PSCz flux map. For an isotropic flux map, this 2
number would be 1/5 = 20 per cent. We thus find that the KKKST
map deviates most strongly from isotropy. 0
In Fig. 4 we show the distribution of flux differences f = 0 0.1 0.2 0.3 0.4 0.5 0.6
f alt − f XSCz , where f = φ/φ denotes the normalized flux and D
the subscript refers to the catalogue that was used in the modelling Figure 5. Distribution of test statistic D for an isotropic reference distri-
(‘alt’ standing for either KKKST or PSCz). Here φ represents the bution and test UHECR event sets following the distribution of matter as
integral UHECR flux in a given direction (as represented in Fig. 3) modelled from different catalogues. This figure applies to 20 events with a
and φ is the average flux on the sphere. We observe from Fig. 4 threshold energy of 60 EeV, and the distributions are obtained by sampling
that the distribution of f for the KKKST and PSCz catalogues is over 104 independent Monte Carlo realizations.
C 2010 The Authors. Journal compilation
C 2010 RAS, MNRAS 403, 2131–21362134 H. B. J. Koers and P. Tinyakov
S using one of the three galaxy catalogues under consideration. 1
We observe from the figure that the differences between the three
‘structure’ distributions are small in comparison to the differences 0.8
between the ‘structure’ distributions and the ‘isotropy’ one. Note
that the ‘structure’ distribution obtained from the KKKST catalogue
0.6
Cum. dist.
is furthest away from the ‘isotropy’ distribution, which reflects the
more pronounced anisotropies in the KKKST flux map.
The difference between the ‘isotropy’ and ‘structure’ distribu- 0.4
tions is quantified by the statistical power P, representing the prob-
ability of rejecting I when S is true. For the three cases shown in 0.2
Fig. 5, and assuming a significance α = 0.05, we find that P = 0.54 KKKST
XSCz
for the KKKST catalogue, P = 0.46 for the XSCz catalogue and PSCz
P = 0.37 for the PSCz catalogue. We thus find an ∼20 per cent 0
0 0.5 1 1.5 2 2.5 3
difference in power between the XSCz catalogue and the KKKST f
(or the PSCz). Inaccuracies of this order do not spoil the useful-
ness of the statistical power computation. The statistical power is Figure 6. The cumulative distribution Ciso (f ) for the KKKST, PSCz and
interpreted as a figure of merit of a test’s discriminatory power: XSCz catalogues. This figure applies to threshold energy E 0 = 60 EeV and
if the power is close to 1, the test is very sensitive (in this case, smearing angle θ s = 6◦ .
we may expect data to put strong constraints on models); if it is
close to 0, the test is completely insensitive; values in between in-
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dicate a sensitivity somewhere between very poor and very strong. The outcome of the flux sampling test depends on the number of
Hence, an uncertainty of ∼20 per cent does not dramatically change events as well as on the distribution of model fluxes over the sky,
the interpretation. Furthermore, the accuracy of statistical powers i.e. the flux map. The first parameter is obviously independent of
is limited by other factors inherent to UHECR flux modelling, the choice of galaxy catalogue. All intrinsic properties of a galaxy
such as poor statistics and uncertainties in energy calibration. The catalogue that may affect the strength of anisotropies are contained
∼20 per cent uncertainty in statistical power due to different galaxy in the flux map. Here and in the following, the expression ‘flux
catalogues is comparable to the uncertainty induced by a 10 per cent map’ is used to refer to the flux map derived under S (i.e. assuming
uncertainty in experimental energy calibration (near 60 EeV). This that UHECR sources trace the distribution of matter in the Uni-
is significantly smaller than the typical uncertainty in the energy verse). The flux map also depends on the threshold energy, UHECR
calibration of large-scale UHECR detectors, which is ∼20 per cent. injection spectrum and average deflection angle.
The above discussion pertains to the case of 20 events with a The connection between flux maps and the strength of UHECR
threshold energy of 60 EeV. To corroborate the results for the three anisotropies can be illustrated by two limiting cases. First, consider
scenarios considered in Paper I, we have recomputed statistical pow- the case that S would predict a uniform flux, so that the flux distri-
ers using the XSCz catalogue in the broad parameter range of E 0 = bution would be a delta-function (minimal contrast in the flux map).
40–100 EeV and N ev = 10–100, where N ev denotes the number of In that case, UHECR events sample the sky uniformly under both S
events. Within this energy range, we find that the KKKST catalogue and I and model predictions become identical. In the opposite case
leads to powers 20 per cent higher than the XSCz. At the high- of a very wide flux distribution (strong contrast in the flux map),
est energies, E 0 100 EeV, this is comparable to the uncertainty events will have a strong tendency to cluster in high-flux regions
induced by the poor energy resolution in UHECR experiments. At under S, exhibiting strong anisotropy. The connection between flux
lower energies, energy calibration is the dominant source of uncer- distributions and the strength of anisotropies can be made explicit
tainty. We have also recomputed the p-values for the Pierre Auger using the D-test. As shown in Appendix A, under S the test statis-
Observatory (PAO) data with E 0 = 57 EeV and for the Akeno Gi- tic D approaches the following limiting value as the number of
ant Air Shower Array (AGASA) data with E 0 = 40 EeV that were events goes to infinity (recall that it approaches 0 in the same limit
reported in Paper I. We found no significant difference between under I):
the KKKST and XSCz catalogues. We thus conclude that, for the 1
parameter values used in Paper I, the accuracy of model prediction D→ df Ciso (f ) , (2)
building on the KKKST catalogue is sufficient to obtain meaningful 0
results. where Ciso (f ) denotes the normalized cumulative flux distribution
under S and we recall that f = φ/φ, φ being the integral UHECR
flux and φ the average value on the sphere. Equation (2) identifies
the intrinsic property of a flux map that is important in determin-
3 G A L A X Y C ATA L O G U E S A N D P R E D I C T E D
ing the strength of UHECR anisotropies. The measure is related
S T R E N G T H O F A N I S OT RO P I E S
to the width of the flux distribution (the contrast in a flux map): a
In this section, we investigate how the choice of catalogue affects narrow flux distribution corresponds to a steeply rising cumulative
the predicted strength of large-scale UHECR anisotropies. In order distribution and, via equation (2), a value of D close to 0.
to quantify this strength, we use the flux sampling test to mea- The distribution Ciso is shown in Fig. 6 for the three flux maps
sure deviations from isotropy. Although other tests will measure presented in Fig. 3. Comparing the surfaces under the curves for
anisotropies in a different manner, a galaxy catalogue that yields a 0 < f < 1, it is clear that the PSCz catalogue yields the smallest
value for the D-test statistic close to the isotropic prediction will asymptotic value of D and the KKKST catalogue the largest. The
in general also yield outcomes close to the isotropic prediction in larger contrast in the KKKST catalogue implies stronger signatures
other tests (and similarly for predictions far away from the isotropic of anisotropy. This explains the ordering of statistical powers that
one). was found in the previous section.
C 2010 The Authors. Journal compilation
C 2010 RAS, MNRAS 403, 2131–2136Galaxy catalogues in UHECR flux modelling 2135
0.45 When N ev becomes very large, D approaches the asymptotic value
KKKST
0.4 XSCz computed with equation (2): 0.22 for the KKKST catalogue and
PSCz 0.17 for the PSCz catalogue.
0.35
0.3
0.25
4 S U M M A RY
D
0.2
0.15 In this paper we have investigated how the choice of galaxy cat-
alogue affects UHECR model fluxes in a scenario where UHECR
0.1
sources trace the distribution of matter in the Universe. The differ-
0.05 ences between the three catalogues considered here, the KKKST
0 catalogue (Kalashev et al. 2008), the XSCz catalogue (Jarrett et al.
10 20 30 40 50 60 70 80 90 100
E0 (EeV)
2006) and the PSCz catalogue (Saunders et al. 2000), are reasonably
small. This is reassuring because all catalogues are supposed to be
Figure 7. Asymptotic value of D for N ev → ∞ for the KKKST, XSCz and sampling the same underlying density field (barring biases induced
PSCz catalogues as a function of E0 . by selection effects).
The KKKST catalogue, which relies on photometric redshift esti-
mates for galaxies beyond 30 Mpc, was used by us earlier in Paper I.
In Section 2 we have shown that, for a threshold energy of 60 EeV,
the difference in model fluxes between the KKKST and XSCz cat-
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alogues is typically within 20 per cent of the average flux. As a
consequence, power estimates obtained with the KKKST catalogue
are within ∼20 per cent of the values obtained with the XSCz cat-
alogue. We have verified that this accuracy is typical for threshold
energies ranging from 40 to 100 EeV and for event numbers ranging
from 10 to 100. This level of accuracy is acceptable because dif-
ferences of this size hardly affect the interpretation of the statistical
power. Furthermore, it is subdominant to the uncertainty associated
with the energy calibration in UHECR experiments. We have also
verified that the p-values reported in Paper I do not change signif-
icantly when the XSCz catalogue is used instead of the KKKST.
Figure 8. Average value of D (thick lines) and 1σ contours (thin) as a We have thus quantified the errors associated with the photometric
function of the event number N ev for the KKKST and PSCz catalogues and redshift estimates in the KKKST catalogue and shown that they are
for E 0 = 60 EeV. The full distribution for N ev = 20 is shown in Fig. 5. small enough to provide meaningful results. This then refutes the
recent statement by Kashti (2009) that the KKKST catalogue is not
In Fig. 7, we show the asymptotic value of D from equation (2) suited for studies of UHECR anisotropy because of its dependence
for different catalogues and threshold energies. We observe that on photometric redshift estimates.
the KKKST catalogue systematically predicts somewhat stronger We have investigated the relation between the predicted strength
anisotropies compared to the PSCz catalogue, in keeping with the re- of large-scale UHECR anisotropies and model flux maps from a
sults presented above and with the discussion in Paper I.3 The XSCz general point of view in Section 3. The intrinsic properties of a
prediction lies in between. For threshold energies E 0 60 EeV, UHECR flux map, i.e. those properties relating to the strength and
the XSCz curve is virtually identical to the KKKST one, while it area of over- and underdense regions but not to their position on
approaches the PSCz curve at higher energies. The differences be- the sky, are contained in the flux distribution. Equation (2) demon-
tween the three curves are, however, small. In particular, they are strates how this distribution can be used to determine the value of
smaller than the uncertainty induced by systematic errors in energy the D-test statistic in the limit of infinite events. This asymptotic
determination. (One may check that the curves corresponding to value provides a measure of the expected anisotropy in UHECR
the KKKST and PSCz catalogues are within a 20 per cent shift in arrival directions for sources tracing the distribution of matter in
energy applied to the XSCz prediction.) the Universe. We have compared these values for the KKKST,
In reality, the number of UHECR events is of course finite and XSCz and PSCz catalogues as a function of energy (see Fig. 7).
often not very large. The discriminatory power then depends not The comparison shows that the KKKST catalogue typically yields
only on the flux map but also on the event number. In this case, stronger anisotropies than the PSCz catalogue, the XSCz catalogue
equation (2) provides a figure of merit that governs the asymptotic being in between. The difference is, however, small in the light of
value of D. This is illustrated in Fig. 8, where we show the average the uncertainties induced by systematical errors in UHECR energy
value of D and the 1σ bands as a function of the event number. determination.
3 Unfortunately, the power estimates for the PSCz catalogue as reported in AC K N OW L E D G M E N T S
Paper I are inaccurate due to an erroneous application of the PSCz selection
function in our numerical routines. In table 3 the power estimates for the We would like to thank Tom Jarrett for providing us with a prelim-
PSCz catalogue should read 0.37, 0.28 and 0.72 for scenarios I, II and inary version of the XSCz catalogue, and Tamar Kashti and Sergey
III, respectively. With these changes, the statistical powers for the PSCz Troitsky for stimulating discussions. We would also like to thank
catalogue remain smaller than those for the KKKST catalogue so that the the referee for her/his constructive comments, which have helped to
conclusions remain qualitatively unchanged. improve this paper. HBJK and PT are supported by Belgian Science
C 2010 The Authors. Journal compilation
C 2010 RAS, MNRAS 403, 2131–21362136 H. B. J. Koers and P. Tinyakov
Policy under IUAP VI/11 and by IISN. The work of PT is supported normalized, i.e.
in part by the FNRS, contract 1.5.335.08. ∞
df D A (f ) = 1 . (A1)
0
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Dubois P., Cambrésy L., 2003, A&A, 412, 45 D KS = max |D(f )|, where D(f ) = C A (f ) − C B (f ) denotes the
Saunders W. et al., 2000, MNRAS, 317, 55 difference between the two cumulative distribution functions. This
Skrutskie M. F. et al., 2006, AJ, 131, 1163 difference can be expressed as follows:
Takami H., Nishimichi T., Yahata K., Sato K., 2009, J. Cosmology Astropar- f
ticle Phys., 0906, 031 D(f ) = df D A (f )(1 − cf ) . (A5)
0
Since D A , c and f are strictly positive, the integral is maximal
A P P E N D I X A : KO L M O G O ROV – S M I R N OV
when f = 1/c = f¯ A . Hence
D I S TA N C E I N T H E L I M I T O F I N F I N I T E
f¯ A
S TAT I S T I C S 1
DKS = D(f¯ A ) = ¯ df CA (f ) , (A6)
In this appendix, we derive equation (2). We choose a general set- fA 0
up because the result is not limited to UHECR anisotropy tests, where the last equality has been obtained by integration by parts.
but is applicable to any experimental test that can be appropriately Normalizing the function f such that f¯ A = 1, the rhs of equa-
formulated. tion (A6) reduces to the integral of the cumulative distribution
Consider an experiment that records events which are character- function C A up to f = 1.
ized by a set of quantities x (these quantities may be observables Two remarks are in order. First, equation (A6) is a general result
or derived quantities). The space of all possible experimental re- that is applicable to any statistical test for a model that can be formu-
sults is denoted as V. Now consider two models, termed model A lated in terms of a probability function f (x), where x is distributed
and model B. Model A asserts that events sample V uniformly, i.e. uniformly in V under the alternative model. The result applies to
the probability of registering quantities in the range x, . . . , x + dx the UHECR anisotropy tests considered in this work by associating
is independent of x (as long as x ∈ V ). Within model B, on the x = (l, sin b), where l and b denote galactic coordinates, and f
other hand, the probability that an event has quantities in the range = φ/φ, where φ denotes the model flux when UHECR sources
x, . . . , x + dx is proportional to f (x)dx, where f is a non-trivial, trace the distribution of matter and φ is the average value on the
known function. This function defines a map x → f , which as- sphere. Note that, by definition, f¯ A = 1 in this case. Secondly,
sociates a real number f with every event. Note that we leave the equation (A6) determines DKS in the limit of an infinite number of
normalization of f arbitrary. events. For finite event numbers, it is still a useful figure of merit
Our aim is now to differentiate between models A and B. We will because it determines the asymptotic behaviour (cf. Fig. 8).
do this by considering the distribution of f over an observation, i.e. a
series of registered events. We consider the limit of infinite statistics.
The distribution of f under model A, denoted as D A (f ), represents
the distribution of f over V. We assume that this distribution is This paper has been typeset from a TEX/LATEX file prepared by the author.
C 2010 The Authors. Journal compilation
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