Table Of ContentCombining Supernovae and LSS Information with the
CMB
9 A.N. Lasenby, S.L. Bridle & M.P. Hobson
9
9
AstrophysicsGroup,Cavendish Laboratory,MadingleyRoad,Cambridge,CB3 0HE,
1
UK.
n
a
J ABSTRACTObservationsoftheCosmicMicrowaveBackground(CMB),largescalestructure
1 (LSS)andstandardcandlessuchasType1aSupernovae (SN)eachplacedifferentconstraints on
2 the values ofcosmological parameters. Weassumeaninflationary ColdDarkMatter model with
a cosmological constant, in which the initial density perturbations in the universe are adiabatic.
1
WediscusstheparameterdegeneraciesinherentininterpretingCMBorSNdata,andderivetheir
v
3 orthogonalnature. WethenpresentourpreliminaryresultsofcombiningCMBandSNlikelihood
0 functions. TheresultsofcombiningtheCMBandIRAS1.2Jysurveyinformationaregiven,with
3 marginalisedconfidenceregionsintheH0,Ωm,bIRASandQrms−psdirectionsassumingn=1,
1 ΩΛ+Ωm = 1 and Ωbh2 = 0.024. Finallywe combine all three likelihoodfunctions and find
0
that the three data sets are consistent and suitably orthogonal, leading to tight constraints on
9
9 H0,Ωm,bIRAS andQrms−ps,givenourassumptions.
/
h
p KEYWORDS:CMB,largescalestructure,supernovae, cosmology
-
o
r 1. INTRODUCTION
t
s By comparing the observed CMB power spectrum with predictions from cos-
a
: mological models one can estimate cosmological parameters. This has become an
v
area of great current interest, with many groups carrying out the analyses for a
i
X range of assumed models (e.g. Hancock et al. 1998, Lineweaver et al. 1997, Bond
r & Jaffe1998). Generally speaking, the results of using CMB data alone to do this
a
are broadly consistent with the expected range of cosmologicalparameters,though
perhaps with a tendency for H to come out rather low (assuming spatially flat
0
models). Inanindependentmanner,similarpredictionscanbe madebycomparing
Large Scale Structure (LSS) surveys with cosmologicalmodels (Willick et al. 1997,
Fisher&Nusser1996,Heavens&Taylor1995). Also,whendistantType1Asuper-
novae are used as a standard candle, one can assess the probability as a function
of Ω and Ω . We demonstrate that the results from the different data sets are
m Λ
compatible. However using each data set alone it is only possible to constrain
combinations of the cosmological parameters. We show how these combinations of
parametersare differentforeachdataset, thereforewhen wecombine the datasets
we obtain tight constraints.
1
FIGURE1PlotsshowingcontoursofconstantR0S(χ)atredshiftsof0.1,1,10and100.
2. THE COMPLEMENTARY NATURE OF SUPERNOVAE AND CMB DATA
There has recently been great interest in combining Type Ia supernovae (SN)
data with results from the CMB (e.g. Lineweaver 1998, Tegmark 1998). It is
instructiveto seehowthe complementaritybetweenthesupernovaeandCMB data
arises. ThekeyquantityforthisdiscussionisR S(χ),whichoccursinthedefinitions
0
of Luminosity Distance:
d =R S(χ)(1+z),
L 0
and Angular Diameter Distance:
d =R S(χ)/(1+z).
θ 0
Here R is the current scale factor of the Universe, χ is a comoving coordinate,
0
and S(χ) is sinh(χ), χ or sin(χ) depending on whether the universe is open, flat or
closed respectively. For a general Friedmann-Lemaitre model, one finds that
1 z dz′
R S(χ) sin(h) Ω 1/2
0 ∝ Ωk 1/2 (cid:26)| k| Z0 H(z′)(cid:27)
| |
where
Ω = 1 (Ω +Ω ),
k m Λ
−
H2(z) = H2 (1+Ω z)(1+z)2 Ω z(2+z) .
0 m − Λ
(cid:0) (cid:1)
For small z, it is easy to show that
1
d z+ (1 2q )z2,
L 0
∝ 2 −
where q = 1(Ω 2Ω ) is the usual deceleration parameter.
0 2 m− Λ
2
Contours of equal likelihooCd MinB the Ωm, ΩΛ plane (50 by 50 grid)
0.8
0.6
Λ
Ω
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ω
m
Supernovae
0.8
0.6
Λ
Ω
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ω
m
Supernovae + CMB
0.8
0.6
Λ
Ω
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ω
m
1 sigma, 2 sigma, ..., 5 sigma contours
Marginalised in 20 steps over 0.8 < n < 1.3 and 12 < Q< 25
Ωh2=0.145, Ωh2=0.0125
c b
FIGURE2LikelihoodsintheΩm,ΩΛ planefromCMBandSNdata
Therefore, for small z, SN results are degenerate along a line of constant q .
0
However, the contours of equal R0S(χ) shift around as z increases and for z∼>100
the contours are approximately orthogonal to those corresponding to q constant.
0
ThisisthereasonwhyCMBandSNresultsareideallycomplementary. Thecurrent
microwavebackgrounddataismainlysignificantindelimitingtheleft/rightposition
ofthefirstDopplerpeakinthepowerspectrum,andthisdependsonthecosmology
via the angular diameter distance formula, evaluated at z 1000. Thus the CMB
∼
resultswilltendtobe degeneratealonglines roughlyperpendicularto thoseforthe
supernovae in the (Ω ,Ω ) plane.
m Λ
We may calculate the likelihood of a given set of cosmological parameters for
CMBdataaloneusingthebandpowerapproachdescribedine.g.Hancocketal.1998.
We use Seljak and Zaldariagga’s CMBFAST code to calculate scalar mode CMB
3
power spectra for an adiabatic inflationary Cold Dark Matter universe and use the
CMB data points described in Webster et al. 1998. In this preliminary analysis we
fix Ω h2 = 0.145 and Ω h2 = 0.0125 (where h = H /100 and Ω = Ω Ω ). We
c b 0 c m b
−
marginalise over the initial power spectrum index, n, and CMB power spectrum
normalisation, C . The result is shown in the top panel of Figure 2. Contours are
2
at changes in log(Likelihood) of 0.5, 2, 4.5, 8 and 12.5 from the minimum value.
−
As expected, there is a degeneracy in a similar direction to the last plot in Figure
1. The SN likelihoods we use are based on data in Perlmutter et al. 1998 and are
plotted in the secondpanelof Figure 2. Again,the directionof degeneracyis along
the lines of the first two plots of Figure 1.
The lines of degeneracy are orthogonal and overlap, leaving a small patch of
parameter space that fits both data sets. However we may be more quantitative
than this. The probability of the Universe having a particular Ω , Ω given both
m Λ
theCMBandSNdatasetsissimplyfoundbymultiplyingtogethertheprobabilities
ofthoseΩ ,Ω foreachdataset. WetherebyobtainthefinalplotinFigure2. The
m Λ
preferred universe is close to flat with a low Ω . However, we have applied very
m
limiting assumptions. Detailed likelihoodcalculations coveringa muchwider range
of assumptions using both supernovae and CMB data are currently being carried
out by Efstathiou et al., and should be submitted shortly.
3. COMBINING CMB AND LSS
Recently, Webster et al. have combined the CMB likelihoods with IRAS likeli-
hoods obtained from the 1.2Jy galaxy redshift survey following the spherical har-
monic approach of Fisher, Scharf & Lahav. They assume a Harrison–Zel’dovich
primordial scalar power spectrum (n = 1) and the nucleosynthesis constraint
s
Ω h2 = 0.024 (Tytler, Fan & Burles 1996) in a flat universe with a cosmologi-
b
cal constant, although clearly it would be interesting to relax these constraints.
ThisapproachiscomplementarytothatofGawiser&Silk,whousedacompilation
oflargescalestructureandCMBdatatoassessthegoodnessoffitofawidevariety
of cosmologicalmodels.
Because the CMB and LSS predictions are degenerate with respect to different
parameters(roughly: Ω vsΩ forCMB;H andΩ vsb forLSS),thecombined
m Λ 0 m iras
data likelihood analysis allows the authors to break these degeneracies,giving new
parameter constraints. The different degeneracy directions in the Ω , h plane are
m
showninthetoptwopanelsofFigure4. Notethatthetwodatasetsagreewellinthe
region where the lines of degeneracy cross. Figure 3 shows the final 1-dimensional
probability distributions for the main cosmological parameters after marginalizing
over each of the others. The verticaldashed lines denote the 68% confidence limits
and the horizontal plot limits are at the 99% confidence limits.
The best fit results from the joint analysis of the two data sets on all the free
parametersareshowninTable1,forwhichwecanderiveΩ =0.085,σ =0.67and
b 8
shape parameter (Sugiyama 1995, Efstathiou, Bond & White 1992) Γ = 0.15. A
detaileddiscussionoftheseestimatesandcomparisonwithotherresultsiscontained
4
0.35
4
0.3
0.25 3
d d
o o
o 0.2 o
h h
eli eli2
Lik0.15 Lik
0.1
1
0.05
0 0
16 18 20 0.4 0.5 0.6 0.7
Qrms−ps h
3.5 1.4
3 1.2
2.5 1
d d
oo 2 oo0.8
h h
Likeli1.5 Likeli0.6
1 0.4
0.5 0.2
0 0
0.2 0.4 0.6 0.8 0.8 1 1.2 1.4 1.6 1.8
Ωm b
FIGURE3Theprobabilitydistributions,usingCMBandIRASdata,aftermarginalisingover
theotherfreeparameters.
inWebster et al. 1998,but inbroadtermsitis clearthatfairlysensible valueshave
resulted, which is encouraging for future prospects within this area.
Table 1. — Parameter values at the joint optimum. The 68% confidence limits are shown,
calculatedforeachparameterbymarginalisingthelikelihoodovertheother variables.
Ω 0.39 0.29< Ω < 0.53
m m
h 0.53 0.39< h < 0.58
Q (µK) 16.95 15.34< Q <17.60
b 1.21 0.98<b < 1.56
IRAS IRAS
For a spatially flat model, the age of the universe is given by:
2 tanh−1√Ω
Λ
t=
3H0 √ΩΛ
which evaluates to 16.5 Gyr in the current case, again compatible with previous
estimates.
4. COMBINING CMB, SUPERNOVAE AND LSS DATA
Finally, we combine all three data sets under the assumption applied in the
above CMB+IRAS analysis and find that the data sets agree well and tighten the
5
Contours of equal −log(likelihood)
CMB IRAS
0.8 0.8
0.7 0.7
h 0.6 h 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Ω Ω
m m
SN CMB, IRAS and SN
0.8 0.8
0.7 0.7
h 0.6 h 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Ω Ω
m m
Marginalised over 5 < Q < 30 and 0.7 < b < 2.0 for Ω h2=0.024, Ω =1−Ω , n=1
rms−ps b Λ m
1 sigma, 2 sigma, ..., 5 sigma contours shown
FIGURE4Likelihoodsintheh,Ωm planeaftermarginalisingoverQrms−ps andbIRAS for
CMB,IRAS,SNandallthreedatasetscombined.
constraintsonthefourparametersinvestigated. WefindthatQrms−ps =17.5 1µK,
H =59 8kms−1Mpc−1, Ω =0.33 0.07andb =1.05 0.2,whichm±ay be
0 m IRAS
± ± ±
compared to the results in Table 1. For a flat Universe, as considered here, the SN
results shift h up and Ω down slightly compared to the results found using CMB
m
and IRAS data alone.
5. CONCLUSIONS
6
We have combined CMB, LSS and SN data sets and found good agreement
between them, despite the restrictive nature of our assumptions. Due to the com-
plementary nature of the data sets the error bars on the cosmological parameters
are dramatically reduced by making such combinations.
ACKNOWLEDGMENTS We would like to acknowledge collaboration with Ofer Lahav (Racah
Institute ofPhysics, TheHebrew University,Jerusalem, IoA Cambridge), George Efstathiouand
Matthew Webster (IoA Cambridge) and Graca Rocha (Kansas). SLB acknowledges a PPARC
studentship.
REFERENCES
Bond,J.R.andJaffe, A.H.1998,preprint(astro-ph/9809043).
Efstathiou,G.,Bond,J.R.,andWhite,S.D.M.1992, MNRAS,258,1.
Fisher,K.andNusser,A.1996,MNRAS,279,L1.
Fisher,K.B.,Scharf,C.A.,andLahav,O.1994,MNRAS,266,219
Gawiser,E.,&Silk,J.1998, Science,280,1405
Hancock, S.,Rocha,G.,Lasenby,A.N.,andGutierrez,C.M.1998,MNRAS,294,L1
Heavens,A.andTaylor,A.1995,MNRAS,275,483
Lineweaver,C.1998, ApJ,inpress(astro-ph/9805326)
Lineweaver,C.,Barbosa,D.,Blanchard,A.,andBartlett,J.1997,Astron.Astrophys.,322,365
Perlmutter, S. et al. 1998 Poster displayed at the American Astronomical Society meeting in
Washington, D.C.,January9,1998,
http://panisse.lbl.gov/public/papers/aasposter198dir/aaasposter.html.
Seljak,U.andZaldarriaga,M.1996, ApJ,469,437
Sugiyama,N.1995ApJS,100,281
Tegmark,M.1998, ApJL,submitted(astro-ph/9809201)
Tegmark,M.,Eisenstein,D.,Hu,W.,andKron,R.1998,ApJ,submitted(astro-ph/9805117)
Tytler,D.,Fan,X.M.,andBurles,S.1996,Nature,381,207
Webster, A. M., Bridle, S. L., Hobson, M. P., Lasenby, A. N., Lahav, O., and Rocha, G. 1998,
ApJL,submitted(astro-ph/9802109)
Willick,J.,Strauss,M.,Dekel,A.,andKolatt,T.1997,ApJ,486,629
7
