Spectroscopic source redshifts and parameter constraints from weak lensing and CMB

0.28 MB·English

by  Mustapha Ishak

Spectroscopic source redshifts and parameter constraints from weak lensing and CMB
Mustapha Ishak1 and Christopher M. Hirata2
1Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
2Department of Physics, Princeton University, Princeton, NJ 08544, USA
(Dated: February 2, 2008)
Weaklensingisapotentiallyrobustandmodel-independentcosmological probe,butitsaccuracy
isdependentonknowledgeoftheredshiftdistributionofthesourcegalaxies used. Themostrobust
waytodeterminetheredshiftdistributionisviaspectroscopyofasubsampleofthesourcegalaxies.
We forecast constraints from combining CMB anisotropies with cosmic shear using a spectroscopi-
callydeterminedredshiftdistribution,varyingthenumberofspectraNspec obtained from64to∞.
ThesourceredshiftdistributionisexpandedinaFourierseries,andtheamplitudesofeachmodeare
considered as parameters to be constrained via both the spectroscopic and weak lensing data. We
assumeindependentsourceredshifts,andconsiderinwhatcircumstancesthisisagood approxima-
5 tion (thesources are clustered and for narrow spectroscopic surveyswith many objects this results
0 intheredshiftsbeingcorrelated). Itisfound thatforthesurveysconsidered and forapriorof0.04
0 on the calibration parameters, the addition of redshift information make significant improvements
2 on the constraints on the cosmological parameters; however, beyond Nspec ∼few×103 the addition
of further spectra will make only a very small improvement to the cosmological parameters. We
n
a findthatabettercalibration makeslarge Nspec moreuseful. Usinganeigenvectoranalysis, wefind
J that the improvement continues with even higher Nspec, but not in directions that dominate the
uncertainties on the standard cosmological parameters.
3

PACSnumbers: 98.80.Es,98.65.Dx,98.62.Sb
2
v
2
I. INTRODUCTION Inthispaper,weforecastFisher-matrixconstraintsfrom
4
cosmic shear using a spectroscopically determined red-
0
5 Weak lensing (WL) is a promising tool for an era of shift distribution, varying the number of spectra Nspec
0 precision cosmology (for reviews, see [1, 2, 3, 4] and ref- obtained. The source redshift distribution is expanded
4 in a Fourier series, and the amplitudes of each mode are
erences therein). Many recentstudies showedthe poten-
0 tial of this established technique in constraining various considered as parameters to be constrained via both the
/
h cosmological parameters [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, spectroscopic and WL data. We address the effect of
p 15, 16]. Using current data, Refs. [17, 18, 19, 20, 21] galaxy clustering on our analysis and discuss the effect
- of a better calibration.
o showed that WL can provide constraints that are com-
r petitive with other cosmological probes. Many larger
t
s andmoreambitioussurveysareongoing,plannedorpro-
a posed. These include the Deep Lens Survey [22]; the II. MODEL PARAMETERS
:
v NOAODeepSurvey;theCFHTLegacySurvey[23];Pan-
i STARRS; SNAP [24, 25, 26]; and LSST [27]. A major The following basic parameter set for WL is consid-
X
challenge for the future of WL studies is to have a very ered: Ω h2, the physical matter density; Ω and w, re-
m Λ
r tight control of systematic errors. These include incom- spectively the fraction of the critical density in a dark
a
pleteknowledgeofthesourceredshiftdistribution[1,28]. energy component and its equation of state; n (k =
s 0
Recent cosmic shear studies have obtained their redshift 0.05h/Mpc) and α , the spectral index and running of
s
distributions from joint redshift-magnitude distributions the primordial scalar power spectrum at k ; σlin, the
0 8
measured in independent surveys [20]; from photometric amplitude oflinearfluctuations; q ...q ,the leading
{ 1 jmax}
redshifts (“photo-z’s”) [29]; or some combination. The coefficients of a Fourier expansion of the source galaxy
former approach suffers from the difficulty that the se- redshift distribution (see Eq. 4 in the see next section);
lection function (and relative weighting) of galaxies may in the case of two bin tomography we use q ,q
jA jB
{ }
be differentinthelensingandredshiftsurveys,whilethe with j = 1..j . We also include ζ and ζ as de-
max s r
latter approach suffers from possible photo-z errors that fined in [16] to parametrize the shear calibration bias
are difficult to constrain in the absence of spectroscopic [2, 30, 31, 32, 33, 34], in which the gravitationalshear is
confirmation, especially if only a small number of col- systematically over- or under-estimated by a multiplica-
orsaremeasured. Severalstudieshavemarginalizedover tive factor,i.e. Pˆ (ℓ)=(1+ζ )P (ℓ), whereP (ℓ)is the
κ s κ κ
the source redshift distribution. Ultimately the most ro- convergence power spectrum obtained in the absence of
bustandmodel-independentdeterminationofthe source calibrationerrors. ζ referstothecalibrationerrorofthe
s
redshift distribution would be via spectroscopy of a ran- power spectrum, which is twice the calibration error of
domly chosen sub-sample of the source catalog. How- the amplitude as the power spectrum is proportional to
ever, spectroscopy of faint galaxies can be very time- amplitude squared. When we consider tomography, we
consuming even with today’s large-aperture telescopes. mustalsoconsidertherelativecalibrationζ betweenthe
r
2
two redshift bins. This error affects the measured power The q arethusthecoefficientsintheFourierexpansion
j
spectrum P˜ (ℓ) in accordance with: of n({z)/}n (z) in the interval 0 P < 1. We have used
κ 0 0
≤
the cumulative fiducial probability P (z) instead of z as
0
P˜AA = (1+f ζ )PˆAA(ℓ), the independent variable because this will result in un-
κ B r κ
P˜κAB = (1+ fB−2 fAζr)PˆκAB(ℓ), jco=rre0latceodefficocniesnttravinantsisohnestbheecqajusferoomf tshpeecntorormscaolpizya;ttiohne
constraint n(z)dz = n (z)dz = 1. The selection of
P˜BB = (1 f ζ )PˆBB(ℓ), (1) 0
κ − A r κ the cosines is arbitrary and any complete set of orthog-
R R
onal functions would work just as well. The distribution
where f and f are the fraction of the source galaxies
A B
is completely specified by the q , although in this
in bin A and B respectively. { j}∞j=1
paperwecutoffthe seriesatsomej (weshowresults
In order to combine this with information from the max
for 5 and 100). For tomography, the normalized distri-
CMB we include Ω h2, the physical baryon density; τ,
b
butions and the respective Fourier expansions are given
the optical depth to reionization; and T/S, the tensor-
as above, except for bin A we replace n (z) with
to-scalar fluctuation ratio. We assume a spatially flat 0
UniversewithΩ +Ω =1,therebyfixingΩ andH as n (z)
m Λ m 0 nA(z)= 0 for z 2z ,
functions ofourbasic parameters,andwe donotinclude 0 1 5/e2 p ≤ 0
massive neutrinos, or primordial isocurvature perturba- −
nA(z)=0 for z >2z , (6)
tions. We use as fiducial model (e.g. Ref. [35], with w 0 p 0
and T/S added): Ω h2 = 0.0224, Ω h2 = 0.135, Ω = and for bin B, we replace n (z) with
b m Λ 0
0.73,w = 1.0,n =0.93,α =0.0,σ =0.84,τ =0.17,
T/S =0.2−, ζs =0s.0, ζr =0.0s, and qi,q8jA,qjB =0.0. nB0(z)=0 for zp ≤2z0,
n (z)
We will consider surveys with fsky = 0.01 and 0.1, nB(z)= 0 for z >2z . (7)
andhavenumberdensity-to-shapenoiseration¯/ γ2 = 0 5/e2 p 0
4.1 109 sr 1, corresponding to a number dehnsiintytiof
− The cumulative probabilities for the two bins are
n¯ =×30 galaxies/arcmin2 and shape+measurement noise PA(z) = znA(z )dz and PB(z) = znB(z )dz . We
γ2 = (0.3)2. (Note that this is the shape noise in 0 0 0 ′ ′ 0 0 0 ′ ′
thhientsihear γ, rather than the ellipticity which is roughly use the paRrameters {q1..q5,q1A..q5A,q1BR..q5B} to vary the
redshift distribution, see Fig. 1.
e 2γ if isophotal or adaptive ellipticities are used
≈
[33, 36].)
1
III. PARAMETDRIISZTARTIIBOUNTOIOFNTHE REDSHIFT 0.8 nnnnnnn11+02345-(((((((zzzzzzz)))))))
0.6
Thesourceaverageddistanceratioappearsasaweight-
ing function in the convergence power spectrum [37, 38, n(z)
39] and is given by 0.4
χH sin (χ χ)
K ′
g(χ)= n(χ′) sin (χ−) dχ′, (2) 0.2
Zχ K ′
where n(χ(z))is the normalizedsourceredshift distribu-
0
tion. Deviations from the fiducial distribution, [28], 0 0.5 1 1.5 2 2.5 3 3.5 4
z
n (z)= z2 e z/z0, (3) FIG. 1: Variations of the redshift distribution. The fiducial
0 2z03 − distribution n0(z) (which peaks at zp =2z0) and the Fourier
expansion terms are plotted. The distributions n1±(z) rep-
(which peaks at zp = 2z0 = 0.70 and has zmed 0.94 ) resent q1 varied by ±25%. We also plot n2+(z) to n5+(z).
are parameterized via a Fourier series, ≈ Whilewevariedtheqj′sby±0.05intheanalysis,weplotthe
25% variations just for theclarity of theplot.
1 ∞
n(z)=n (z) 1+ q cos(jπP (z)) , (4)
0 j 0
√2
h Xj=1 i
IV. FISHER-MATRIX ANALYSIS
where P is the cumulative redshift distribution in the
0
fiducial model,
The statistical error on a given parameter pα is given
z z2 z by
P (z)=1 1+ + e z/z0 = n (z )dz . (5)
0 −(cid:16) z0 2z02(cid:17) − Z0 0 ′ ′ σ2(pα)≈[(FCMB +FWL+Fspec+Π)−1]αα, (8)
3
where Π is the prior curvature matrix, andF , F V. SPECTROSCOPY
CMB WL
and F are the Fisher matrices from CMB, WL, and
spec
spectroscopy,respectively. WeuseforFWL theapproach In the discussion above, it has been assumed that the
described in Ref. [16], with a cutoff at ℓmax =3000 since spectroscopically obtained n(z) is on average the same
onsmallerscalesthetrispectrumcontributiontotheWL as the redshift distribution n(z) of the sources used for
covariance (neglected in Eq. 8) dominates [40, 41]. For lensing,andthattheredshiftsofthegalaxiestargetedfor
CMB, we use the 4 year WMAP parameter constraints spectroscopyareindependent. Theseassumptionswillbe
including TT, TE, and EE power spectra, assuming literally true in the idealized case that the spectroscopic
fsky = 0.768 (the Kp0 mask of Ref. [42]), tempera- galaxiesarechosenindependentlyfromthelensingsource
ture noise of 400, 480, and 580µK arcmin in Q, V, and catalog, and there are no failures to obtain spectro-z’s.
W bands respectively (the rms noise was multiplied by In practice, spectro-zs would probably be obtained by a
√2 for polarization), and the beam transfer functions of multi-object spectrograph attached to a large telescope.
Ref. [43]. The spectroscopy Fisher matrix is obtained as Thespectro-z failureratewillbeminimizedbyusingthe
follows. If a galaxy is chosen at random from the lens- longest practical integration time to maximize signal-to-
ing catalog, and is spectroscopically determined to have noise for each spectrum, and the demands on telescope
redshift z, then the log-likelihood for the redshift distri- time are thus reduced if multiple objects in the same
bution parameters qj is field of view can be targetedsimultaneously. Howeverin
this casethe galaxiesarenotbeing drawnindependently
1
ln (q )=lnn (z)+ln 1+ q cos(jπP (z)) . fromthe source catalog,in particular large-scalecluster-
j 0 j 0
L √2 ingcancausetheredshiftsofneighboringsourcegalaxies
h Xj i(9) to become correlated. The feasibility of obtaining many
The contribution to the Fisher matrix from this single independent spectro-z’s is directly tied to the clustering
galaxy is of the sourcesand the field of view of the telescope since
these determine the maximum number of galaxies that
∂ln ∂ln canbe simultaneouslytargetedfor spectroscopywithout
F(1) (q ,q ) = L L
spec j k ∂q ∂q the results being strongly correlated.
j k
D E The effect of source clustering can be understood
cos(jπP (z))cos(kπP (z)) δ
0 0 jk
= = (1.0) within the context of Fisher matrix theory as follows.
2 4
If the fiducial model is correct, and the analysis is done
D E
The last equality follows from the orthonormality of the assumingthatthespectro-zsareindependent, thediffer-
Fourier modes, combined with the fact that P (z) is ence δpα =pα(est) pα(fid) between the estimated and
0 −
uniformly distributed between 0 and 1. If N galax- fiducial model parameters is roughly
spec
ies have their spectra measured, and these galaxies are
∂ln
drawn independently from the source catalog, then the δpα [F−1]αβ L , (13)
above Fisher matrix is multiplied by Nspec: ≈ ∂pβ (cid:12)fid
(cid:12)
(cid:12)
1 where is the likelihood assuming in(cid:12)dependent spectro-
Fs(pNescpec)(qj,qk)= 4Nspecδjk. (11) zs (i.e.L the sum of Eq. 9 over the spectroscopic tar-
gets), F is the Fisher matrix (also assuming indepen-
In the caseof tomography,we have only N /2 spectra dentspectro-zs),andthegradientistakenatthefiducial
spec
in each of the two tomography bins A and B, and so: model. So long as each galaxy in the lensing source cat-
alog has an equal probability of being targeted for spec-
1
F(Nspec)(qA,qA)= N δ , (12) troscopy, it is easy to see that the spectroscopy contri-
spec j k 8 spec jk bution to ∂ln = 1 Nspec cos(jπP (z )) = 0 and
h ∂qjLi √2 a=1 h 0 a i
andsimilarlyforbinB. Inprincipleitispossibletotake hence δpα = 0, i.e. the inclusion of the spectroscopy
h i P
different numbers of spectra in the two bins; we have likelihood function introduces no bias in the parameters
notattempted anyoptimizationofthis. We considerthe even if the spectroscopic galaxies are not chosen inde-
cases of N = 0, 64, 512 and 4096 respectively. The pendently. However, the clustering of the sources does
spec
results can be compared to the case where the source increase the covariancematrix of the δpα; taking the co-
redshiftdistributionisknownexactlybytakingthelimit variance of Eq. (13) gives
N .
spec
→∞ ∂ln ∂ln
The Fisher matrix is an asymptotic expansion, and δpαδpβ [F−1]αγ[F−1]βδ L L
while it is the standard tool of parameter forecasting, it h i ≈ ∂pγ ∂pδ
(cid:28) (cid:29)fid
sometimesleadstoover-optimisticparameterconstraints
= [F 1]αγ[F 1]βδ [F +F +Π]
(see Ref. [44] for an extreme example). We have not − − CMB WL γδ
testedits validitywhenusedwiththis manyparameters, Nspec cnos(jπP (z ))cos(kπP (z ))
although this could be done by Monte Carlo methods as +δqjδqk h 0 a 0 b (i14).
γ δ 2
used by Ref. [45].
aX,b=1 o
4
If we included only the a = b terms in Eq. (14), the WLsurveys. ForN =512andafieldofviewof0.5de-
spec
last term in would simply be the spectroscopy Fisher greeradius,wefindfromEq.(17)therequirementJ 6
{} ≫
matrix for independent spectro-zs, F . The a = b (θ = 0.0002 deg) or J 1.3 (θ = 2.3 10 5 deg). In
spec,γδ 0 0 −
6 ≫ ×
termsareonlynon-zeroduetocorrelationsofthegalaxies eithercasethenumberoffieldsthatmustbe observedto
witheachother,andtheycontributetoquantityin by achieve N = 512 ranges from a few to a few dozen,
spec
{}
an amount ∆ . Since the probability of two galaxies and for N =4096 we find that the minimum number
qjqk spec
separated by angle θ being physically associated with offieldsJ isafewdozentoafewhundred. Thenumberof
ab
each other is ω(θ )/[1+ω(θ )], where ω is the angular spectratobeobtainedperfieldisM =N /J 100.
ab ab spec
≤∼
correlation function, and the cosine is bounded in the Theproblemofspectro-zfailuresismoredifficulttoas-
range 1 to +1, it follows that the a=b contribution is sessthanthesourceclustering. Inordertoaccuratelyre-
− 6
producethen(z)ofthelensingsourcecatalog,thetargets
1 ω(θ ) 1
ab in the associated spectroscopic survey must be selected
∆ ω(θ ), (15)
| qjqk|≤ 2 1+ω(θab) ≤ 2 ab randomly from the lensing catalog, or at least have the
a=b a=b
X6 X6 same selectioncriteria. Spectro-z failuresare notpartof
as compared with F = 1N δ . Therefore if the lensing catalog selection criteria, and hence can bias
ω(θ ) N s/pe2c,,qtjhqeknthe4cluspsetcerjikngcontribution the n(z) determination. The spectro-z failure rate can
∆a6=bwill babe s≪mallspceocmpared to the spectroscopy Fisher be reduced by using large-aperture telescopes and very
γδ
mPatrix F and hence will contribute negligibly to long integration times, which may prove feasible if r is
spec,γδ
the parameter uncertainties according to Eq. (14). large,θ0 is small, andhence the number offields J to be
Let us consider the implications of this result for an observedis only a few. It may also be possible to reduce
idealized spectroscopic survey strategy that consists of failuresbyimposingappropriatecolorcutsonthesource
observing J widely separated fields, and obtaining M = sample. Nevertheless, the failure rate will never be ex-
N /J spectra in each field. We suppose the telescope actly zero. The treatment of systematic errors in n(z)
spec
has a circular field of view of angular radius r, that the due to these failures is beyond the scope of this paper,
number density of galaxies targeted for selection in each but clearly deserves consideration in future work.
field is n = M/πr2, and that the angular correlation
function is ω(θ)=(θ /θ)0.7. We then have
0
VI. RESULTS
0.7
θ
ω(θ ) n2J 0 d~θ d~θ
Xa6=b ab ≈ Z Z |~θ1−~θ2|! 1 2 1
N2 θ 0.7
= 13.25n2Jθ0.7r3.3 =1.34 spec 0 (16.) 0.95
0 J r
(cid:18) (cid:19)
0.9
wheretheintegralsaretakenoveracirculardiskofradius
r. (The replacement of the galaxy-galaxy pair summa- 0.85
tion by an integral is appropriate because the integral
is convergent at small separation ~θ ~θ .) The fac- σ8 0.8
1 2
| − |
tor of J in the first line occurs because we have to re- 0.75
peat the sum for each of the J fields. In order to have
0.7 CMB only
nuam6=bbeωr(oθfabfi)e≪ldsNgsivpeecn/2b,yEq. (16) tells us that we need a CMB+WL+40C9M6 sBp+eWctrLa
0.65
P
θ 0.7 0.6
J 2.68N 0 . (17) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
≫ spec r ΩΛ
(cid:18) (cid:19)
Not surprisingly, the number of fields that need to be FIG. 2: The 68.3% confidence ellipses (assuming Fisher er-
rors)forCMBandweaklensing. Notetheimprovementinthe
observed depends on the ratio of the field-of-view ra-
constraints whenspectra are available, versusthecase where
dius r to the angular clustering scale θ . An angular
0 we marginalize overtheredshift distribution parameters.
clustering scale of θ 0.0002 degrees is observed in
0
∼
SDSS [46] for the magnitude range 21 < r < 22 (valid
at separations of 1–30 arcmin). Even smaller θ ap- Our results are summarized in Tables I and II, and
0
plies to fainter samples, e.g. the CFDF survey [47] finds Figs. 2 and 3. As expected, we find that combination of
ω(1) 0.01 for 18.5 < I < 25 galaxies, correspond- constraints from WL and from CMB leads to significant
′ AB
ing to≈θ0 = 2.3 10−5 degrees (for slope 0.7, which improvements in parameter estimation, notably for σ8,
is consistent with×the ω(θ) data from < 1 to−several ar- Ω h2, n , α , w and Ω .
m s s Λ
cminutes). The fainter CFDF sample is probably more As shown in Table I, the increase of the number of
representative of the galaxies that will be used in future expansiontermsfromj =5to100haslittle effecton
max
5
TABLE I: Parameter estimation errors for WL+CMB: For WL, no-tomography, fsky = 0.01, 0.1, and ℓmax = 3000. We use
Nspec =0, 64, 512, 4096 and the limit Nspec →∞, and a priors of 0.04 on ζs and ζr (unless indicated fixed). In the last part
of the table, we show the results with thenumberof terms in theseries increased from 5 to a 100.
fsky Ωmh2 Ωbh2 ΩΛ σ8 ns αs τ T/S w q1 q2 q3 q4 q5 σ(ζs) σ(ζr)
CMBonly 0.0118 0.0013 0.1981 0.086 0.059 0.036 0.018 0.187 0.872 - - - - - - -
→
Nspec=0 0.01 0.0096 0.0009 0.1320 0.053 0.033 0.017 0.018 0.142 0.514 4.342 19.912 34.086 66.902 58.275 0.040 0.040
0.10 0.0092 0.0009 0.0895 0.039 0.027 0.012 0.018 0.129 0.373 1.595 6.482 13.804 21.929 18.556 0.040 0.040
Nspec=64 0.01 0.0083 0.0008 0.0248 0.022 0.022 0.012 0.017 0.120 0.226 0.162 0.244 0.250 0.249 0.250 0.040 0.040
0.10 0.0075 0.0007 0.0141 0.018 0.019 0.010 0.017 0.114 0.186 0.121 0.218 0.249 0.248 0.250 0.040 0.039
Nspec=512 0.01 0.0076 0.0008 0.0219 0.021 0.021 0.011 0.016 0.115 0.220 0.081 0.088 0.088 0.088 0.088 0.040 0.039
0.10 0.0058 0.0007 0.0116 0.014 0.016 0.010 0.015 0.098 0.140 0.074 0.087 0.088 0.088 0.088 0.039 0.038
Nspec=4096 0.01 0.0075 0.0007 0.0211 0.021 0.021 0.011 0.016 0.114 0.219 0.031 0.031 0.031 0.031 0.031 0.040 0.039
0.10 0.0049 0.0006 0.0111 0.013 0.014 0.009 0.015 0.090 0.118 0.030 0.031 0.031 0.031 0.031 0.039 0.038
Nspec→∞ 0.01 0.0074 0.0007 0.0210 0.021 0.020 0.011 0.016 0.113 0.219 - - - - - 0.040 0.039
0.10 0.0047 0.0006 0.0110 0.012 0.013 0.009 0.015 0.088 0.113 - - - - - 0.039 0.038
Nspec=0 0.01 0.0096 0.0009 0.1320 0.053 0.033 0.017 0.018 0.142 0.514 4.341 19.912 34.086 66.902 58.275 - -
ζs,r fixed 0.10 0.0092 0.0009 0.0895 0.039 0.027 0.012 0.018 0.129 0.373 1.594 6.482 13.804 21.928 18.555 - -
Nspec=64 0.01 0.0083 0.0008 0.0243 0.022 0.022 0.012 0.017 0.120 0.226 0.159 0.244 0.250 0.249 0.250 - -
ζs,r fixed 0.10 0.0075 0.0007 0.0137 0.018 0.019 0.010 0.017 0.114 0.185 0.114 0.217 0.249 0.248 0.250 - -
Nspec=512 0.01 0.0076 0.0008 0.0211 0.021 0.021 0.011 0.016 0.115 0.220 0.081 0.088 0.088 0.088 0.088 - -
ζs,r fixed 0.10 0.0056 0.0007 0.0110 0.014 0.015 0.009 0.015 0.096 0.137 0.073 0.087 0.088 0.088 0.088 - -
Nspec=4096 0.01 0.0074 0.0007 0.0202 0.021 0.020 0.011 0.016 0.113 0.219 0.031 0.031 0.031 0.031 0.031 - -
ζs,r fixed 0.10 0.0045 0.0006 0.0105 0.012 0.013 0.009 0.014 0.085 0.110 0.030 0.031 0.031 0.031 0.031 - -
Nspec→∞ 0.01 0.0073 0.0007 0.0201 0.021 0.020 0.011 0.016 0.113 0.218 - - - - - - -
ζs,r fixed 0.10 0.0042 0.0006 0.0104 0.012 0.012 0.009 0.014 0.083 0.104 - - - - - - -
fsky Ωmh2 Ωbh2 ΩΛ σ8 ns αs τ T/S w q1 q2 to q99 q100 σ(ζs) σ(ζr)
Nspec=64 0.01 0.0083 0.0008 0.0249 0.022 0.022 0.012 0.017 0.120 0.227 0.162 0.244 ... 0.250 0.250 0.040 0.040
0.10 0.0075 0.0007 0.0144 0.018 0.019 0.010 0.017 0.114 0.186 0.122 0.218 ... 0.250 0.250 0.040 0.039
Nspec=512 0.01 0.0076 0.0008 0.0219 0.021 0.021 0.011 0.016 0.115 0.220 0.081 0.088 ... 0.088 0.088 0.040 0.039
0.10 0.0058 0.0007 0.0116 0.014 0.016 0.010 0.015 0.098 0.140 0.074 0.087 ... 0.088 0.088 0.039 0.038
Nspec=4096 0.01 0.0075 0.0007 0.0211 0.021 0.021 0.011 0.016 0.114 0.219 0.031 0.031 ... 0.031 0.031 0.040 0.039
0.10 0.0049 0.0006 0.0111 0.013 0.014 0.009 0.015 0.090 0.118 0.030 0.031 ... 0.031 0.031 0.039 0.038
Nspec→∞ 0.01 0.0074 0.0007 0.0210 0.021 0.020 0.011 0.016 0.113 0.219 - - ... - - 0.040 0.039
0.10 0.0047 0.0006 0.0110 0.012 0.013 0.009 0.015 0.088 0.113 - - ... - - 0.039 0.038
0.9
TABLE II: Parameter estimation errors for WL+CMB: WL
fsky is varied from 0.01 to 0.1, and ℓmax = 3000. We use
0.88 Nspec =0,64,512,4096andthelimitNspec →∞,andpriors
of0.04onζsandζr (unlessindicatedfixed). Tomgraphycase.
0.86
fsky ΩΛ σ8 w
CMBonly 0.1981 0.086 0.872
→
σ8 0.84 Nspec=0 00..0110 00..00738450 00..003254 00..323490
Nspec=64 0.01 0.0169 0.018 0.185
0.10 0.0102 0.013 0.118
0.82
Nspec=512 0.01 0.0160 0.017 0.173
0.10 0.0084 0.010 0.088
0.8 CMB+WL Nspec=4096 0.01 0.0154 0.016 0.162
CMB+WL+4096 spectra 0.10 0.0078 0.009 0.074
CMB+WL tomCoMgrBa+phWyL+ 4t0o9m6o sgpraepcthrya Nspec→∞ 00..0110 00..00104781 00..001058 00..105646
0.78
0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 Nspec=0 0.01 0.0684 0.032 0.311
ΩΛ ζs,rfixed 0.10 0.0288 0.023 0.230
Nspec=64 0.01 0.0147 0.017 0.168
FIG. 3: Same as Fig. 2, but including tomography. ζs,rfixed 0.10 0.0091 0.012 0.108
Nspec=512 0.01 0.0137 0.016 0.153
ζs,rfixed 0.10 0.0069 0.009 0.075
Nspec=4096 0.01 0.0130 0.014 0.140
this result. This suggests that the parameter estimates ζs,rfixed 0.10 0.0062 0.008 0.063
haveconvergedand,sinceEq.(4)canmodelanarbitrary Nspec→∞ 0.01 0.0118 0.012 0.122
ζs,rfixed 0.10 0.0043 0.005 0.043
function for sufficiently large j , it suggests that the
max
form(Eq.4)fortheredshiftdistributionisnotartificially
constraining the cosmologicalparameters.
the cosmological parameters such as Ω , σ , n , and w.
Λ 8 s
The optical depth τ is less improved by lensing because
VII. DISCUSSION most of the statistical power on τ is coming from the
CMB polarizationreionizationpeak, whichis notdegen-
In agreement with previous results, we find that the erate with any lensing-related quantities. The baryon
addition of redshift information is helpful for several of density is well-constrained by the CMB, however by im-
6
proving our constraints on several cosmological param- correlation coefficient ρ(x,q ) = 0.64. Consequently,
1
−
eters, lensing information breaks the relatively weak re- although x is degraded by imperfect knowledge of the
maining degeneracies in the CMB and reduces the error source redshift distribution, and is degenerate with the
from σ(Ω h2) = 0.0013 to 0.0009. Even lensing with redshift distribution parameters, it is not the direction
b
N =0 is sufficient to break this degeneracy,so inclu- that dominates the uncertainties on the individual cos-
spec
sion of redshift information adds little for Ω h2. mological parameters. The second-largest eigenvalue of
b
The addition of redshift information leads to further [C−1( ))C(Nspec)] is 1.2, indicating that the directions
∞
significant improvements in the parameter estimation as other than x are not significantly degraded. When we
expected. The N = 0 constraints are substantially fix the calibration parameters, the eigenvector analysis
spec
worse than N = 64, because in the N = 0 case shows an increase of almost an order of magnitude on
spec spec
evenwildlyoscillatingredshiftdistributionsn(z)arefor- the degradation with √λ 16.6, but again, not in
max
≈
mally allowed, as is evidenced by the large σ(q ) 1 thedirectionthatdominatestheuncertaintiesonthecon-
j
≫
in Table I. Although the gradual addition of spectro- ventional cosmological parameters. As seen from Table
scopic redshifts does make additional improvements on I and Table II, a better calibration will make a larger
the constraints on the cosmological parameters, we find numbers of spectra slightly more useful but a significant
that there is a certain number of N (several thou- improvementrequirearatheridealizedperfectknowledge
spec
sands for the surveys consideredhere) beyond which the of the calibration.
addition of further spectra will make only a very small The most significant improvement in the cosmological
improvement to the cosmologicalparameters. parameterswithlargenumbersofspectracomesfromthe
In order to try to explain this, let us recall that error caseswith tomography,since this helps breakthe degen-
bars are correlated, and hence there can still be com- eracies that dominate the parameter uncertainties and
binations of parameters that are degraded by incom- allows the directions well constrained by WL to become
plete knowledge of the source redshift distribution. In- more important. One can see from Table II that the
formation on this degradation is contained within the uncertainties on Ω , σ , and w are degraded by 50%
Λ 8
9 9 covariance matrix C of the cosmological parame- for N = 4096 versus N = with the large∼r area
te×rs{Ωmh2,Ωbh2,ΩΛ,σ8,ns,αs,τ,T/S,w}. We examine fsky =spe0c.1. In all the othespreccase∞s the degradation with
the degradation factor R of the combination of parame- N =4096 is much less.
spec
ters x=k pα given by
α Itispossiblethatdifferentresultswouldbeobtainedif
σ2(x;N ) [C(N )]αβk k WL were combined with additional cosmological probes
spec spec α β
R(x)= = . (18) such as supernovae,which could break remaining degen-
σ2(x,N = ) [C( )]αβk k
spec ∞ ∞ α β eracies in the data and therefore make the degradation
of x more important. A general way to investigate this
The maximum value of R can be found by setting
possibilityis to examine λ forthe WL-onlymatrices,
∂R/∂k =0, which leads to the eigenvalue equation max
α
since combining WL with cosmological probes that do
[C[C(N(spe)c]α)]βαkβkkαkβkγ =[C−1(∞)]γβ[C(Nspec)]βαkα. λnomtaxinscmluadleletrhteh{aqnj}thmatusftorreWsuLltainloaned.egWrahdeantiwonefaapcptolyr
α β
∞ theeigenvalueanalysistotheweaklensingonlymatrices,
(19)
we obtain the degradation factors of √λ 19.98 for
Thus the k that maximizes the degradation R(x) is an max
α ≈
eigenvector of [C−1( )C(Nspec)], and the degradation the fixed calibration case and Nspec = 512. The lat-
∞ ter number indicates that there are directions in cos-
factor R(x) is the eigenvalue. (The maximum degrada-
mological parameter space which are dramatically im-
tion factor corresponds to the maximum eigenvalue of
[C−1( ))C(Nspec)]; the other eigenvectors are minima proved by exact knowledge of the redshift distribution
∞ rather than only 512 spectra. We can reduce the degra-
or saddle points of R.) For N = 512, f = 0.1,
spec sky
dationfactorto λ =2by increasingN to 200000,
and no-tomography, we find a maximum eigenvalue of max spec
and it is thus only for N 200000 that every di-
λ = 2.2 with the corresponding combination of pa- spec
max ≥
rection in parameter space is limited by lensing statis-
rameters
ticsratherthanredshiftdistributionuncertainties. How-
x = 4.37Ω h2+8.89Ω 5.39σ 2.57n +4.32α ever,this improvementis rapidlylostdue to the approx-
m Λ 8 s s
−+0.53w+3.11Ω h2 −4.59τ 7−.18T/S. (20) imatecalibration-redshiftdegeneracyifthecalibrationis
b
− − uncertain: if the prior on the calibration parameters is
Havingonlyafinitenumberofspectraisdegradingthe1σ widened to σ(ζs) = σ(ζr) = 0.02 we achieve λmax = 2
WL+CMB constraint on x by a factor of √λmax 1.5, at Nspec = 4000, and with σ(ζs) = σ(ζr) = 0.04 we
but from Table I we can see that the constraints o≈n the achieve λmax = 2 at Nspec = 1100. Thus having either
standard set of cosmological parameters is degraded by good calibration or a well-determined redshift distribu-
< 10%. This is because x is a direction in parameter tionindividuallymaynotbeveryuseful,buthavingboth
space that is very well-constrained by WL+CMB: with combined can significantly improve some constraints.
CMB only, we have σ(x) = 1.02, whereas with WL and We conclude that, though significant improvement is
N =512addedtotheCMBwehaveσ(x)=0.098and obtainedfromtheadditionofredshiftsinformation,there
spec
7
isacertainnumberofN –oforder103forthesurveys distribution due to galaxy clustering so long as enough
spec
considered here – beyond which the addition of further spectroscopic fields are observed, as detailed in Sec. V,
spectra will make only a very small improvement to the while future work is required to address the spectro-z
cosmologicalparameters. Wedofindthattherearedirec- failures.
tionsinparameterspacethatcontinuetoimproveanddo
notsaturateuntilN isverylarge,especiallyifthecali-
spec
brationisverywelldetermined; howeverthesedirections
correspond to very large eigenvalues of the WL Fisher
Acknowledgments
matrixanddonotdominatethe parameteruncertainties
in the WL+CMB combinations we have discussed. The
results presented here indicate that if N few 103 We thank Uroˇs Seljak and David Spergel for useful
spec
∼ ×
spectra of representative sources can be obtained, cos- comments. M.I. acknowledges the support of the Natu-
mological constraints can potentially be improved rela- ralSciencesandEngineeringResearchCouncilofCanada
tive to the CMB-only case without model-dependent as- (NSERC).C.H.acknowledgesthesupportoftheNational
sumptions about the source redshift distribution. These Aeronautics and Space Administration (NASA) Gradu-
results are robust against fluctuations in the spectro-z ate Student Researchers Program(GSRP).
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