Direction Dependent Non-gaussianity in High-z Supernova Data

0.13 MB·English

by  Shashikant Gupta

Direction Dependent Non-gaussianity in High-z Supernova Data
Shashikant Gupta1 and Tarun Deep Saini2
Indian Institute of Science, Bangalore, Karnataka, India, 560 012
and
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0 Tanmoy Laskar3
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St. Stephen’s College, University of Delhi, Delhi, India 110 007
2

n
a
J

5 ABSTRACT
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The most detailed constraints on the accelerating expansion of the universe and details of

2 nature of dark energy are derived from the high redshift supernova data, assuming that the
v
errorsinthemeasurementsareGaussianinnature. Thereisapossibilitythattherearedirection
3
dependentsystematicsinthedata,eitherduetouncorrected,knownphysicalprocessesorbecause
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6 therearetinydeparturesfromthecosmologicalprinciple,makingtheuniverseslightlyanisotropic.
1 Toinvestigatethispossibilityweintroduceastatisticbasedontheextremevaluetheoryandapply
0 it to the gold data set from Riess et al. (2004). Our analysis indicates a systematic, direction
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dependent non-gaussianity at about one sigma level.
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/ Subject headings: cosmology: cosmological parameters — cosmology: large-scale structure of universe
h
—supernovae: general
p
-
o 1. Introduction ence of a cosmological constant term, called the
r
Λ term. When combined with the usual mat-
t
s During the last decade the possibility that the ter term, the resultant model is referred to as
a
expansionofourUniverseisacceleratinghasbeen
: the ΛCDM model. In this model the Hubble pa-
v put on a firm footing. The combined analysis
rameter asymptotically approaches a constant at
i
X of high redshift supernova data (Perlmutter et al. late times, thus causing the universe to acceler-
1999; Riess et al. 1998, 2002, 2004), along with
r ate. There are compelling theoretical reasons to
a observations of cosmic microwave background
believe that the dark energy density may not be
(Benoit et al.2002;Page et al.2006)orlargescale
a strict constant. There are severalphysicalmod-
structure (Percival et al. 2002; Tegmark et al.
els where dark energy is generated dynamically
2004) indicates a spatially flat universe with low
from an evolving scalar field (called quintessence)
matter density (around one-third of the critical
or even from alternate theories of gravity (see
density),the restofthe closuredensityisbelieved
Sahni & Starobinsky 2006, for a recent review of
to be in an unknown form generically termed as
models and methods of reconstruction of cosmic
the dark energy. It is this component that drives
history). Due toits simplicity,ithasbecomepop-
the late time acceleration of the expansion of the
ular to phenomenologically model dark energy as
universe.
an ideal fluid with an equation of state given by
The simplest possibility, which fits the data p=wρ, wherew isallowedtobe negative. Inthis
well,isthattheaccelerationiscausedbythepres- model w = −1 gives the usual cosmological con-
stant. Models where w is a constant or a simple
1E-mail:[email protected] function of redshift have also been considered.
2E-mail:[email protected]
Cosmological data is rapidly approaching a
3E-mail:[email protected]
quality where we can start discriminating com-
1
peting models of dark energy. The effect of tiny of the above mentioned possibilities.
departures from a strict cosmological constant on Theoutlineofthispaperisasfollows. In§2we
observations is sufficiently small to render such describe our methodology in detail. We present
analysisunreliableifthenatureofstatisticalnoise our results in §3 and our conclusions in §4.
indataisnotwellunderstood. Since the mostde-
tailedconstraintsondarkenergyarederivedfrom 2. Methodology: Extreme Value Statistic
the luminosity distance to distant supernovae,
we would like to be certain that their statistics Throughout our analysis we have assumed a
is well understood. The central limit theorem flat FRW universe. Since ΛCDM model fits
(Kendall & Stuart 1977) ensures that, to a very the data quite well, we first obtain the best fit
good approximation, statistical noise due to a model to the full gold data set (Riess et al. 2004)
large number of random influences can be treated and calculate the dispersion normalized residu-
as Gaussian. There are severalpossible sources of als χi =[µi−µΛCDM(zi;Ωm)]/σµ(zi), where the
non-gaussianity in the supernova data; according distance modulus µ = 5 log(dL/Mpc) + 25, the
to Kolatt & Lahav (2001) these could be 1) sta- observed values being µi for a supernova at red-
tistical scatter due to location in the host galaxy shiftzi,andσµ(zi)istheobservedstandarderror.
and galaxytype 2) scatter due to dust absorption We shall consider subsets of the full data set to
inthehostgalaxy,intergalacticmediumorinour construct our statistic. We define the reduced χ2
own galaxy or 3) due to lensing along the line of in terms of χi as follows
sight. Some of these processes are corrected for
1
in the data reduction. It is also possible that the χ2 = χ2 , (1)
N i
observedanisotropyis aresultofcollationof data subset Xi
of disparate quality, perhaps due to differences in
where it should be noted that by ‘reduced’ we do
seeing condition or in the data reduction process.
not mean ‘per degree of freedom’, since we do not
Modern cosmology is based on the Cosmolog-
fit the model separately to the subsets of the full
ical Principle (CP) (Peebles 1993), which states
data. Here χ2 is an indicator of the statistical
thatonthelargescalestheuniverseisstatistically
scatterofthesubsetawayfromthebestfitΛCDM
homogenous and isotropic. Even if supernovae
model.
were perfect indicator of distance, and if statis-
If CP holds then the apparent magnitude of a
ticalnoise in the supernova distances is Gaussian,
supernova should not depend upon the direction
ananisotropicuniversecouldcontributeasystem-
in which it is observed but only on the cosmol-
atic, direction dependent modulation in the data.
ogy. We divide the data into two hemispheres
Another possibility is that our galaxy could con-
labeled by the direction vector nˆ, and take the
tain anisotropic, gray dust patches, and since the i
difference of the reduced χ2 computed for the
dust correctiondepends onreddening inthe spec-
two hemispheres separately to obtain ∆χ2 =
trum, this sort of dust could remain uncorrected. nˆi
χ2 −χ2 , where we have defined ’north’ as
Above arguments suggest that there is a strong north south
that hemisphere towards which the direction vec-
case for investigating direction dependence in the
tor nˆ points.
supernova data. i
We take the absolute value of ∆χ2 , since that
Kolatt & Lahav (2001) have investigated the nˆi
is the quantity which determines the plane across
possibility of detecting cosmic anisotropy with
which data is anisotropic, and then vary the di-
79 high-z supernovae obtained from Riess et al.
rection nˆ across the sky to obtain the maximum
(1998)andPerlmutter et al.(1999). Inthis paper
absolute difference
weuseadifferentstatisticthanusedbythem. We
use the gold data set containing 157 supernovae
∆=max{|∆χ2 |} . (2)
(Riess et al. 2004) for our analysis. Although our nˆi
methodology is derived from testing for isotropy,
Tointerpretourresultsweneedtoknowwhatval-
weshallusethetermnon-gaussianityinterchange-
uesof∆onemightexpect. Thedistributionofsu-
ably. Our results can be interpreted as a system-
pernovae is not uniform on the sky, therefore, the
aticdirectionaldependenceinthedataduetoany
numberofsupernovaeinthetwohemispheresfora
2
givendirectionvarieswiththedirectionnˆ. There- Oneproblemwiththe bootstrapdistributionis
fore one might expect the probability distribu- that we expect it to be shifted slightly to the left
tion function P(∆) to be extremely complicated, of the theoretical distribution. This is due to the
however, extreme value theory (Kendall & Stuart factthattheoreticaldistributionisobtainedbyas-
1977)showsthatthedistributionis,infact,asim- suming χ to be Gaussian random variates with a
i
ple, two parameter Gumbel distribution, charac- zero mean and unit variance. Therefore theoret-
teristic of extreme value distribution type I: ical χ s are unbounded. However, the bootstrap
i
distributionisobtainedbyshufflingthroughaspe-
1 ∆−m ∆−m cific realization of χ , and they have a maximum
P(∆)= exp − exp −exp − , i
s (cid:20) s (cid:21) (cid:20) (cid:18) s (cid:19)(cid:21) value such that |χ |<χ . Since the bootstrap re-
i 0
(3) alizations have bounded χ , they should produce,
i
where the position parameterm and the scale pa- on the average, slightly smaller values of ∆ as
rameter s completely determine the distribution. compared to what one expects from a Gaussian
To quantify departures from isotropy we need distributed χ . For a large number of supernovae
i
to know the theoretical distribution Ptheory(∆). thisbiasisexpectedtovanish. Sincethegolddata
Since it is difficult to obtain it analytically, we setcontainsonly157supernovae,anotherconcern
have calculated it numerically by simulating sev- is that our statistics may not be reliable enough
eralsetsofGaussiandistributedχi onthegoldset due to a lack of uniform sky coverage. We have
supernovapositionsandobtaining∆fromeachre- made a few checks to investigate these concerns:
alization. We plotthis distributioninFig1asthe We simulatethe samenumber ofsupernovaeas in
brokencurve. Wefindthatthedistributionclosely the gold data set with randomly chosen positions
resembles a Gumbel distribution. onthesky. Wethenprocessthe simulateddatain
If the data do have directional anisotropy then exactlythesamewayastheactualdata. Atypical
an independent possible test for non-gaussianity resultis showninFig2. Thereareafew thingsto
is obtained by constructing the bootstrap distri- be noted. 1) The simulated data givesa ∆ that is
bution P (∆) in the following manner. The ob- consistent with the theoretical distribution, indi-
BS
served χ are assumed to be drawn from some cating that a uniform sky coverage is not a strict
i
unknown, direction dependent probability distri- requirement in this statistic. 2) The theoretical
bution. We shuffle the data values z , m(z ) andbootstrapdistributionsforthesimulateddata
i i
and σm(zi) over the supernovae positions, thus lookidenticalinshapeand3)thepeakoftheboot-
destroying any directional alignment they might strapdistributionisshiftedleftwards,asdiscussed
have had due to anisotropy. We show in the next above. Thisshiftisontheorderof10percent. In
sectionthatthisgivesusyetanotherwayofquan- Fig 1 we showthat the bootstrapdistribution has
tifying non-gaussianity. a mean that is about 40 per cent shifted from the
theoreticaldistribution, andhas a different shape,
3. Results independentlyindicatingnon-gaussianity. Theex-
cessshiftcannotbereconciledwiththetheoretical
Our main result is plotted in the Fig 1. We
distribution by a simple scaling of the error bars.
find that the theoretical distribution P (∆)
theory To produce a rightward shift we would need to
(broken line) assuming Gaussian distributed χ
i increase ∆, which can be done by decreasing the
indicates that our universe has about one sigma
error bars on supernovae by a constant scale fac-
smaller anisotropy than the mean of the distri- tor. However,this would also produce a larger χ2
ν
bution. However, as mentioned in the last sec-
for the best fit ΛCDM model. The data actually
tion, if the residuals µi are non-Gaussian then a gives χ2 =1.14,and scaling wouldmake it larger,
ν
moreappropriateestimate ofdeparturefromnon-
thus making the primary fit worse.
gaussianity would be the bootstrap distribution
We find that the gold set is maximally asym-
PBS(∆). We have plotted PBS(∆) obtained by metric in the direction (l = 100◦,b = 45◦). We
randomlyshufflingχ onthegivensupernovaepo-
i designatethetwohemispheresas‘hot’or‘cold’ac-
sitions in the same figure. We find that the ob-
cording to largeness or smallness of their reduced
served value of ∆ is more than one sigma away
χ2 (as given in Eq. 1) with respect to the best fit
from the mean of this distribution as well.
3
ΛCDMmodel. InFig3we haveplottedthe resid- to fully understand the statistic that we have in-
uals ∆µ = µ −µ for the hot and cold troduced in our analysis and will be discussed in
data ΛCDM
subsetsofsupernovae. Thezeroline isthe bestfit greater detail in a future publication.
ΛCDM model. As expected, the cold supernovae
show slightly smaller dispersion compared to the
ShashikantthanksCSIRforprovidingfinancial
hot ones. Figure 3 also shows that the ΛCDM
assistance for this work. We thank Shiv Sethi for
model seems to fit the hot and cold supernovae
useful comments.
equally well. This is borne out by a parameter
estimation, which is tabulated in Table 1, where REFERENCES
we find that the best fit ΛCDM model for hot su-
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m
Ω = 0.31, so the difference is only a few per
m Perlmutter, S. et al. 1999,Astroph. J. , 517, 565
cent. However, the situation is not the same for
model where we have assumed a constant equa- Riess, A. G. et al. 1998, Astron. J. , 116, 1009
tion of state p = wρ. We find that the model
Riess, A. G. et al. 2002, Astroph. J. 560, 49.
parameters for the hot supernovae in this model
are Ω = 0.51 with w = −4.53 and for the cold
m Benoit, A., et al. 2002, astro-ph/0210306.
supernovae Ω =0.32 and w =−1.03. The value
m
of the Hubble constant is relatively quite robust, Page, L. et al. 2006, astro-ph/0603450
showing that most of the effect is coming from
Percival, W.J., et al. 2002, astro-ph/0206256.
high-z supernovae. Thelargedifferenceintheval-
ues for the constant w model shows that the level
Tegmark,M.etal.2004,Phys.Rev.D.,69,103501
of non-gaussianity indicates that constraints on a
more complicated dark energy model are not as Sahni V. & Starobinsky A. A., 2006,
robust. Perhapsthisexplainstheintriguingresult astro-ph/0610026.
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ΛCDM model as well as a model with a strongly
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4. Conclusions
Peebles,P.J.E.1993,PrincetonSeriesinPhysics,
We have used the extreme value statistics on
Princeton, NJ: Princeton University Press,
the gold data set and shown the presence of non-
c1993,
Gaussian features in the data. We find that there
is a direction of maximal asymmetry in the data Alam,U.,SahniV.,Saini,T.D.&StarobinskyA.
across which data seems to imply different cos- A., 2006,MNRAS, 344, 1057
mologicalmodels for a constantequationof state,
although, the constraints on the ΛCDM model
are found to be robust. We have discussed how
thiscouldeitherimplynon-gaussianityinthedata
due to various possible physical processes or due
to anisotropy in the universe. Our results can-
not be trivially understood by scaling the error
bars. Since this analysis has been done within
the framework of an FRW model, it is obviously
very difficult to quantify the precise meaning of
this anisotropy. We have discussed that we need
to be very careful in interpreting dark energy be-
yond the cosmological constant model since it is
possiblethatsystematicnoisemaymasqueradeas
This2-columnpreprintwaspreparedwiththeAASLATEX
evolving dark energy. Further work is required macrosv5.2.
4
Fig. 1.— Thesolidlineshowsthebootstrapprob-
ability distribution P (∆) obtained by shuffling
BS
supernovae data on the sky while keeping the su-
pernovae positions fixed. The broken line shows
thetheoreticaldistributionP (∆)obtainedby
theory
simulating Gaussian distributed χ , as described
i
in the text. The vertical lines show one and two
sigma regions. The observed value for ∆ for the
golddata setis seento be aroundone sigmaaway
from either distribution.
Fig. 2.— We plot the result of a typical simu-
lation where 157 supernovae positions were gen-
erated randomly and populated with Gaussian
noise. Similar to Fig 1 this plot shows the Gaus-
sianvs. thebootstrapdistribution. Thesimulated
universe is seen to be consistent with the theoret-
icaldistribution, thus indicating thatour statistic
Table 1: The model parameters for the hot and
does not require a uniform sky coverage
cold set are tabulated here. WCDM refers to a
modelwherethedarkenergyhasaconstantequa-
tion of state.
Model Subset Ω w H χ2
m 0 ν
ΛCDM hot 0.30 -1 64.80 1.55
ΛCDM cold 0.31 -1 63.88 0.70
WCDM hot 0.51 -4.53 68.46 1.49
WCDM cold 0.32 -1.03 63.90 0.71
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Fig. 3.— The toppanelshowsthe distributionof
coolsupernovaeandthebottomoneshowsthehot
supernovae. Horizontal line is the ΛCDM model
in both panels. The total number of supernovae
in the top panel is 74 and the bottom panel 83.
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