Magnification-Temperature Correlation: the Dark Side of ISW Measurements

0.39 MB·English

by  Marilena LoVerde

Magnification–Temperature Correlation: the Dark Side of ISW Measurements
Marilena LoVerde1,2, Lam Hui1,2, Enrique Gaztan˜aga3,4
1Institute for Strings, Cosmology and Astro-particle Physics (ISCAP)
2Department of Physics, Columbia University, New York, NY 10027
3Institut de Ci`encies de l’Espai, CSIC/IEEC, Campus UAB,
F. de Ci`encies, Torre C5 par-2, Barcelona 08193, Spain
4INAOE, Astrof´ısica, Tonantzintla, Puebla 7200, Mexico
[email protected], [email protected], [email protected]
(Dated: February 5, 2008)
IntegratedSachs-Wolfe(ISW)measurements,whichinvolvecross-correlatingthemicrowaveback-
groundanisotropieswiththeforegroundlarge-scalestructure(e.g. tracedbygalaxies/quasars),have
proven to be an interesting probe of dark energy. We show that magnification bias, which is the
inevitable modulation of the foreground number counts by gravitational lensing, alters both the
7
scaledependenceandamplitudeoftheobservedISWsignal. Thisistrueespeciallyathighredshifts
0
because (1) the intrinsic galaxy-temperature signal diminishes greatly back in the matter domi-
0
natedera,(2)thelensingefficiencyincreaseswithredshiftand(3)thenumbercountslopegenerally
2
steepens with redshift in a magnitude limited sample. At z > 2, the magnification-temperature
∼
n correlation dominates over the intrinsic galaxy-temperature correlation and causes the observed
a ISW signal to increase with redshift, despite dark energy subdominance – a result of the fact that
J magnification probes structures all the way from the observer to the sources. Ignoring magnifica-

0 tion bias therefore can lead to (significantly) erroneous conclusions about dark energy. While the
3 lensing modulation opens up an interesting high z window for ISW measurements, high redshift
measurementsarenot expectedtoadd muchnewinformation to lowredshift onesif darkenergy is

3 indeed the cosmological constant. This is because lensing introduces significant covariance across
v redshifts. The most compelling reasons for pursuing high redshift ISW measurements are to look
9 for potential surprises such as early dark energy domination or signatures of modified gravity. We
3 concludewith a discussion of existing measurements, thehighest redshift of which is at themargin
5 of being sensitive to themagnification effect. We also develop a formalism which might beof more
1
generalinterest: topredictbiasesinestimatingparameterswhencertainphysicaleffectsareignored
1
in interpreting observations.
6
0
PACSnumbers: 98.80.-k,98.80.Es,98.70.Vc,95.36.+x,98.62.Sb
/
h
p
- I. INTRODUCTION mous: essentially all statements about one apply to the
o
other.) Cross-correlating the temperature anisotropy in
tr direction θˆ, δ (θˆ) = δT(θˆ)/T¯, with the galaxy fluctua-
as tatIinonaalupnoivteernsteiawlsitahssoaccicaetleedrawtiintghelaxrpgaen–ssicoanle, stthreucgtruarveis- tion at redshifTt z in direction θˆ′, δn(θˆ′,z)=δn(θˆ′,z)/n¯,
v: decay. A photon traveling through a decaying potential gives
i will experience a net change in energy. This leads to
X a secondary anisotropy in the cosmic microwave back- δT(θˆ)δn(θˆ′,z) =wnT(θ,z) (1)
r D E
ground (CMB) called the integrated Sachs–Wolfe (ISW)
a
effect [1, 2]. The ISW effect has been proposed as a tool where cosθ = θˆ θˆ′. This scale and redshift dependent
toexaminedarkenergy[3]. Recentstudieshavedetected signal provides i·nformation about the growth rates of
the ISW signature providing an additional confirmation large–scalestructureattheredshiftofthegalaxysample.
ofthepresenceofdarkenergy[4,5,6,7,8,9,10,11,12]. Gravitational lensing alters the observed galaxy fluc-
Since inhomogeneities in the matter distribution trace tuation in two ways. First, lensing alters the area of
gravitationalpotentials,oneexpectsasignificantcorrela- the patch of sky being observed thus modifying the ap-
tionbetweenthe CMBtemperatureanisotropiesandthe parent number density. Second, lensing can focus light
distribution of large–scale structure at redshifts where promotingintrinsicallyfaintobjectsabovethemagnitude
there is a non–negligible fraction of dark energy. A uni- threshold thus increasing the measured number density
versewithnon–negligiblecurvatureorcontributionsfrom [15, 16]. Including these effects changes the observed
anythingotherthanpressurelessmattercandothesame. galaxy fluctuation to
Bearinginmindthe evidenceforspatialflatness[13,14],
δ =δ +δ (2)
we will only consider dark energy. n g µ
The ISW anisotropy can be isolated from the primor- where δ is the intrinsic galaxy fluctuation and δ is the
g µ
dial anisotropies in the CMB by cross–correlating the magnification bias correction due to gravitational lens-
temperatureandgalaxy/quasarfluctuations. (Hereafter, ing. The effect of magnification bias on the galaxy–
galaxy and quasar can be considered roughly synony- galaxy and galaxy–quasar correlation functions is well
2
with. We define two illustrative samples in IV. The
§
cross–correlationsignalsfor these two samples, with and
without magnification, are discussed in V. In VI, we
§ §
investigatehowerroneousconclusionsregardingdarken-
ergy would be reached if one ignores magnification bias
wheninterpretingISWmeasurements. In VII,westudy
§
how projections for the signal–to–noise and parameter
estimation might be altered by the presence of magnifi-
cation bias. We conclude with a discussion of existing
measurements in VIII. Appendix A is devoted to a
§
technical discussion of how to estimate error and bias.
In particular, we develop a formalism for predicting the
estimation–bias for parameters of interest when certain
physical effects (such as magnification bias) are ignored
in interpreting data. We keep the discussion there quite
general with a view towards possible applications other
than ISW measurements.
Two comments are in order before we proceed. Some
of the existing ISW measurements come from cross-
correlating the microwave backgroundtemperature with
other diffuse backgrounds such as the X-ray background
FIG.1: Thederivative(d/dz)[D(z)(1+z)](solid line) where e.g. [7]. Magnificationbias,aneffectthataffectsnumber
Disthelineargrowthfactor,agalaxyselectionfunction(dot-
counts, does not affect these measurements. This is not
ted line) and the corresponding lensing weight function di-
to say that lensing has no effect on diffuse backgrounds,
vided by c/H0 (dashed line). The galaxy–temperature cor- gravitationallensingdoeshaveaneffectthroughstochas-
relation is proportional to the integral of the product of the
tic deflections. In particular, one might wonder how the
solid and dotted lines, while the magnification–temperature
correlation is proportional to the integral of the product of lensing of the microwavebackgrounditself might impact
thesolid and dashed lines. the observed galaxy–temperaturecross–correlation. The
effect appears to be small on the large scales where the
ISW measurements are generally considered interesting
studied [16, 17, 18, 19, 20, 21]. The most recent mea- (we consider ℓ = 2 100). The reader is referred to [4]
surementsofthis effect arediscussedin[22, 23,24]. Dis- and references there−in for more discussions.
cussionsofearliermeasurementscanbe found inthe ref-
erences therein.
With magnification bias the cross–correlation signal II. GALAXY, TEMPERATURE AND
becomes MAGNIFICATION BIAS ANISOTROPIES
w (θ,z)=w (θ,z)+w (θ,z). (3)
nT gT µT The temperature anisotropy due to the ISW effect is
expressed as an integral over conformal time from 0 to
The galaxy–temperature term, w (θ,z), is significant
gT today η [25]
onlywhendarkenergyis non–negligible(assumingaflat 0
universe, as is throughout this paper). Thus for high δISW(θˆ)=2 η0dηe−τ(η)∂φ (4)
redshift galaxysamples this termis very small. However T Z ∂η
0
the magnification–temperatureterm, w (θ,z), depends
µT
onthelower–redshiftdistributionoflensingobjectssoat where τ(η) is the optical depth between η0 and η. For
high redshifts it may dominate over the w (θ,z) term. perturbations sufficiently within the horizon, the gravi-
gT
Figure 1 illustrates this. tational potential φ is related to the mass (or matter)
Weexaminetheeffectofthemagnificationbiastermon fluctuation δ =δρ/ρ¯in Fourier space by [25]
measurements of the ISW effect from cross–correlation. 3H2 δ(k,z)
The magnification–temperature correlation has a dif- φ(k,z)= 0Ω (1+z) (5)
−2 c2 m k2
ferent scale and redshift dependence from the galaxy–
temperature correlation. We will demonstrate that the whereΩ istheratioofthematterdensitytothecritical
m
magnificationtermcanbelargethoughitsmagnitudede- density today, H is the Hubble constant today, c is the
0
pends onthe populationofgalaxiesunder consideration. speed of light, z is the redshift, and k is the comoving
This paper is organizedasfollows. We presentthe ba- wavenumber. Themassfluctuationδ onsufficientlylarge
sic expressions that govern the anisotropies and correla- scalesgrowsaccordingto linear theory: δ D(z), where
∝
tionfunctionsin IIand III. TheISWcross–correlation D(z) is commonly referred to as the growth factor.
§ §
signal,especiallyinthe presenceofmagnificationbias,is Weareinterestedincross–correlatingthe temperature
sensitive to the sample of objectsone is cross-correlating anisotropies, δ , with the observed galaxy overdensity,
T
3
δn. Withmagnification,themeasuredgalaxyfluctuation [δ ] (k,z )=3Ω H02(2.5s 1) (12)
is a sum of two terms (eq. [2]). The first is the intrinsic µ ℓ i m c2 i−
galaxy fluctuation c
dz g(z,z )(1+z)D(z)j (kχ(z))
i ℓ
×Z H(z)
δ (θˆ,z )= dzb(z)W(z,z )δ(χ(z)θˆ,z), (6)
g i i
Z
H2
δISW (k)=3 0Ω (13)
where b(z) is an assumed scale–independent bias factor T ℓ c2 m
(cid:2) (cid:3)
relating the galaxy overdensity to the mass overdensity d jℓ(kχ(z))
dz [D(z)(1+z)]
i.e. δg = bδ, W(z,zi) is a normalized selection function ×Z dz k2
about some mean redshift z , and χ(z) is the comoving
i The symbol j denotes the spherical Bessel function.
distance to redshift z. ℓ
Here we have elected to assume a constant galaxy–bias
The magnification bias term is also expressed as an
b and slope s for each redshift bin centered around z .
integral over redshift [16] i i i
In the temperature integralwe have neglected the factor
containingthe opticaldepth. This leadsto anerrorofat
δ (θˆ,z ) = 3Ω H02(2.5s(z ) 1) most about 3% at the highest redshift we consider.
µ i m c2 i −
Notice that the temperature function [δ ] (k) is in-
T ℓ
dz c g(z,z )(1+z)δ(χ(z)θˆ,z). (7) dependent of redshift, so the cross–correlation functions
i
× Z H(z) CgT and CµT are functions of only a single redshift z .
ℓ ℓ i
This is in contrast to the other correlation functions
Thelensingweightfunction,g(z,z ),canbethoughtofas
i which depend on the redshifts of the two samples being
being proportional to the probability of sources around correlated, e.g. Cgg(z ,z ), Cgµ(z ,z ), Cµg(z ,z ) and
zi to be lensed by intervening matter at z [26]. Cµµ(z ,z ). Note aℓlsoithajt eq.ℓ [10i] anjd eqℓ. [13i] ajre not
ℓ i j
suitableforcomputingCTT sincethatwouldaccountfor
∞ χ(z′) χ(z) ℓ
g(z,z )=χ(z) dz′ − W(z′,z ) (8) only the ISW contribution to CTT.
i Zz χ(z′) i Examining the correlation ℓfunctions, we can see
that the relative magnitude of the intrinsic galaxy–
For sources at a distance χ, the lensing weight function
temperature correlation, CgT, and the magnification–
peaksat χ/2. The prefactorofthe magnificationterm ℓ
∼ temperature correlation, CµT, is redshift and scale de-
depends on the slope of the number count of the source ℓ
pendent. The lensing efficiency increases with the red-
galaxies. This is defined as
shiftofthesourcegalaxiescausingthemagnificationbias
dlog N(<m) effect to increase with redshift as well. The scale depen-
s 10 (9)
dence arises because each term depends on the matter
≡ dm
distribution at different distances: the galaxy term de-
where m is the limiting magnitude of one’s sample, and pends on the matter distribution at the source redshifts
N(< m) represents the count of objects brighter than whilethemagnificationtermdependsonthedistribution
m. Magnification bias vanishes in the case that s =0.4. at the lenses.
The values of the slope s, as well as the galaxy bias b Additionally, the coefficients of CgT and CµT depend
ℓ ℓ
in eq. [6], depend on the population of galaxies under onpropertiesofthe galaxiesviab ands. The magnitude
consideration. ofthemagnificationtermcomparedwiththegalaxyterm
depends on the ratio (2.5s 1)/b. Thus the correction
−
may be negative for (s < 0.4) and in any case increases
III. CROSS–CORRELATION FUNCTIONS in magnitude as s 0.4. The bias and slope depend on
| − |
the choice of galaxy sample and are redshift dependent
themselves.
For the most part, we focus onthe sphericalharmonic
The Limber approximation, which is quite accurate
transform of the various angular correlation functions
whenℓisnottoosmall(ℓ>10),canbeobtainedfromeq.
w (θ), w (θ) and so on. They can all be neatly de-
gT µT
[10] by setting P(k) = P(∼k = (ℓ+1/2)/χ(z)) and using
scribed by the following expression:
the fact that (2/π) k2dkj (kχ)j (kχ′) = (1/χ2)δ(χ
ℓ ℓ
2 χ′). We find that tRhe substitution k = (ℓ+1/2)/χ(z−)
CAB(z ,z )= k2dkP(k)δA (k,z )δB (k,z ) (10)
ℓ i j π Z ℓ i ℓ j is a better approximation to the exact expressions than
k = ℓ/χ(z) (see also [9]). More explicitly, CgT and CµT
ℓ ℓ
where P(k) is the matter power spectrum today as a under the Limber approximationare given by:
function of the wavenumber k, and the functions δA
ℓ 3Ω H2 b H(z)
and δBℓ in the integrand can be one of the following: CℓgT(zi)= mc2 0 (ℓ+1i/2)2 Z dzW(z,zi) c D(z)
d
[δg]ℓ(k,zi)=biZ dzW(z,zi)D(z)jℓ(kχ(z)) (11) × dz[D(z)(1+z)]P(k⊥ =(ℓ+1/2)/χ) (14)
4
Sample-I
m<27 z-bin 1 z-bin 2 z-bin 3 z-bin 4 z-bin 5 z-bin 6
zi 0.49 1.14 1.93 2.74 3.54 4.35
bi 1.08 1.37 2.02 2.90 3.89 4.81
si 0.15 0.20 0.31 0.43 0.54 0.63
ni 34.6 29.0 10.1 3.89 1.68 0.81
Sample-II
m<25 z-bin 1 z-bin 2 z-bin 3 z-bin 4 z-bin 5 –
zi 0.48 1.07 1.85 2.67 3.46 –
bi 1.13 1.51 2.73 4.57 6.63 –
si 0.19 0.35 0.86 1.31 1.75 –
ni 16.1 8.6 0.87 0.11 0.02 –
TABLEI:Themeanredshiftzi,thegalaxy–biasbi,theslope
si and the number of galaxies per square arcminute ni are
givenforsamplesIandII,andforeachcorrespondingredshift
bin.
3Ω H2 2 2.5s 1
CµT(z )= m 0 i− dzg(z,z )(1+z)
ℓ i (cid:18) c2 (cid:19) (ℓ+1/2)2 Z i
d
D(z) [D(z)(1+z)]P(k⊥ =(ℓ+1/2)/χ) (15) FIG. 2: The number of galaxies per unit redshift over the
× dz
wholeskyasafunctionofredshift,fortwoillustrativegalaxy
Whendisplayingtheabovequantitiesinfigures,wefollow samples I and II (see text). The actual dN/dz for a given
surveyequals the sky coverage f times theabove.
the custom of multiplying them by 2.725 K. sky
Unless otherwisestated, we adoptthroughoutthis pa-
per the following values for the cosmological parame-
tribution of galaxies, the galaxy–bias, and the slope of
ters when making predictions for the various correla-
the number count (at the magnitude cut–off). For illus-
tions of interest: we assume a flat universe with a mat-
tration, we define two semi-realistic samples.
ter density of Ω = 0.27, a cosmological constant of
m
Sample I is defined by an observed I–band (centered
Ω =0.73, a Hubble constant of h = 0.7, a baryon den-
Λ around7994˚A)magnitude cut of 27which implies a red-
sity of Ω = 0.046, a scalar spectral index of n = 0.95
b s
shift distribution dN/dz shown as a solid line in Figure
and a fluctuation amplitude of σ = 0.8. The matter
8
2, adopting the observed redshift-dependent luminosity
power spectrum is computed using the transfer function
function given by [29]. This gives a net galaxy angular
of Eisenstein and Hu [27]. The microwave background
temperaturepowerspectrumCTT iscomputedusingthe number density of 80 per square arcminute. We assume
ℓ a sky coverageof f =0.5. These survey specifications
publiclyavailablecodeCAMB[28],withanopticaldepth sky
are similar to those of the Large Synoptic Survey Tele-
of τ = 0.09 and no tensor modes. Throughout this pa-
scope [30]. We divide these galaxies into 6 redshift bins,
per, we consider ISW measurements at ℓ = 2 100. In-
− following the procedure of [31]:
cluding higher ℓ modes does not significantly change our
conclusions. This is because the signal-to-noise of ISW
1dN(z) (i 1)∆ z
measurements is dominated by ℓ = 10 100 (see e.g. W(z,z ) erfc − − (16)
[31, 32], but also [33]). We adopt the L−imber approxi- i ∝ 2 dz h (cid:18) σ(z)√2 (cid:19)
mation for all discussions in VI and VII, which concern i∆ z
§ erfc −
issuesrelatedtosignal–to–noise,estimation–biasandpa- − (cid:18)σ(z)√2(cid:19)i
rameter forecast, since these issues are not significantly
affected by the very low ℓ modes. The discussions and where ∆ = 0.8, σ(z) = 0.02(1+z), and z denotes the
i
figuresin V,however,makeuseofthe exactexpressions mean redshift of the bin i. The complementary error
§
for the correlation functions (eq. [10]). function is defined as erfc(x) = (2/√π) ∞ exp( t2)dt.
x −
The normalization of W is fixed by dRemanding that
dzW(z,z )=1. Wehavecheckedthatnoneofourcon-
i
IV. SURVEY SAMPLES Rclusions are significantly altered by increasing the num-
ber of bins.
The predictions for the galaxy–temperature and Given the observed luminosity function from [29], the
magnification–temperature correlation depend on three slope s (eq. [9]) at the specified apparent magnitude
quantities that are sample dependent: the redshift dis- cut can be obtained in a straightforward manner. The
5
FIG.3: Thegalaxy–temperaturecorrelation(solidlines)com- FIG. 4: Theanalog of Figure 3 for sample II.
pared with the net galaxy–temperature plus magnification–
temperature correlation (dotted lines) for four different red-
shift bins. This is for sample I. 27. The corresponding dN/dz is shown as a dotted line
inFigure2. Thenetgalaxyangularnumberdensityis26
persquarearcminute. We assumethe sameskycoverage
galaxy–biasb, on the other hand, requires some theoret-
of f = 0.5. This is a shallower survey, and we divide
ical input. Here, we adopt the method of Kravtsov et sky
thesampleintoonly5redshiftbinsi.e. applyeq. [16]up
al. [34]. It works basically by number matching: one
to i=5.
matches the the number of galaxies brighter than some
Asummaryofthemeanredshift,thegalaxy–biasb,the
magnitude with the number ofhalos andsubhalos above
slope s and the galaxy number density for each redshift
somemassthreshold. Oncethismassthresholdisknown,
bin and each sample is given in Table I. The tabulated
the galaxy–bias can be computed. More specifically, the
dN/dz for each sample is available from [35].
number of (central plus satellite) galaxies within a halo
of a given mass M is
1 lnM lnM M V. THE CROSS–CORRELATION SIGNAL
0
erfc − (1+ ) (17)
2 (cid:18) 0.1√2 (cid:19)× 20M0
In Figures 3 and 4 we show the cross–correlations
CgT(z ) and CnT(z ) = CgT(z ) + CµT(z ) at several
Here, the complementary error function gives essentially ℓ i ℓ i ℓ i ℓ i
redshift bins for both sample I and sample II, calculated
astepfunctionwithamodestspread(of0.1),andM can
0
using equations (10) - (13).
be thoughtofasthe massthreshold. The quantity20M
0
There are several features in Figures 3 and 4 that
specifies the (parent) halo mass at which the expected
are worth commenting. As is expected, the galaxy–
number of satellites is unity. This factor is in princi-
pleredshiftdependent,butweignoresuchcomplications temperature correlationCgT decreases rapidly with red-
ℓ
here. We determine M by demanding that the integral shift: athighz,darkenergybecomessubdominantmak-
0
of the halo occupation number over the halo mass func- ingthetimederivativeofthegravitationalpotentialquite
tion equals the number of galaxies for the sample under small(eq. [4]). NotethatthisdecreaseofCgT occursde-
ℓ
consideration. The bias of this sample of galaxies can spitetheoverallincreaseofthegalaxy–biaswithredshift.
be obtained by integrating the product of the halo oc- The magnification–temperature correlation CµT, on
ℓ
cupation number and the halo–bias over the halo mass the other hand, generally increases with redshift be-
function,normalizedbythetotalnumberofgalaxies. We cause of the increase of the lensing efficiency. The
adoptthe massfunction andhalo-biasofSheth andTor- magnification–temperature correlation CµT can be high
men [36]. even as CgT becomes small, because the rather broad
SampleIIisdefinedinasimilarwaytosampleI,except lensingweightfunctionmakesCµT sensitivetostructures
thatthelimitingI–magnitudeistakentobe25insteadof at low redshifts where the gravitational potential has a
6
FIG. 5: The 2-point galaxy–temperature correlation FIG. 6: Theanalog of Figure 5 for sample II.
wgT(θ,zi) (solid lines) compared with the net galaxy–
temperature plus magnification–temperature correlation
wnT(θ,zi) (dotted lines) in several of our redshift bins. This signal, and the amplitude is preferentially increased at
is for sample I.
large angular scales.
Magnification bias can of course have either sign. At
significant time derivative. The general increase of CµT lowredshifts,oneisgenerallylookingatintrinsicallyfaint
ℓ galaxieswhichleadstoasmallsthatcanbelessthan0.4.
with redshift is aided also by the increase of the number
count slope s — at a higher z one is generally looking at This is why CℓnT <CℓgT atlow redshifts forour samples.
intrinsically brighter galaxies which reside in the steep The transition redshift from s < 0.4 to s > 0.4 is sam-
part of the luminosity function. The net result is that ple dependent. For a sample with an apparent limiting
the total cross-correlationCnT starts to climb with red- magnitude cut, the brighter the cut, the lower the tran-
ℓ
shift, for z > 2. In fact at a sufficiently high z, CnT sition redshift. It is conceivable that a sufficiently faint
ℓ
even become∼s comparable with CnT or CgT at the low- cut can be achieved such that s remains less than 0.4
ℓ ℓ out to redshifts where CµT > CgT. In such a case the
est redshift bin. Comparing sample I and sample II, one net cross-correlationCn|Tℓwo|uld bℓecome negative. This,
can see that a brighter magnitude cut leads to a more ℓ
however, requires an exceptionally deep survey: an I-
pronounced magnification bias effect. This is due to the
band magnitude limit of 28 or higher, according to the
steeper number count slope.
observed luminosity function [29]. Conversely, if a suffi-
AnotherfeaturevisibleinFigures3and4isthatmag-
ciently bright galaxy cut were used at low redshift it is
nification changes the shape of CnT with ℓ. The magni-
ℓ possible that magnification could become important at
fication term, being sensitive to structures at lower red-
lower redshifts than evidenced by our samples.
shift, peaks at a lower ℓ than the galaxy term. For posi-
In the literature the cross–correlation signal is of-
tive magnification contribution, the peak of CnT is at a
ℓ ten presentedin the normalizedform CgT(z )/ Cgg(z )
lower ℓ than the peak of CgT. ℓ i ℓ i
ℓ or w (θ,z )/ w (θ,z ) to remove the depenpdence on
In Figures 5 and 6 we plot the real space correlation gT i gg i
functionsw (θ,z )andw (θ,z )withoutthemonopole galaxy bias. Hpowever, when magnification bias is taken
gT i nT i
into account the auto–correlationis modified to be
and dipole contributions. These are calculated from
wX(θ,zi)=Xℓ5=002 2ℓ4+π1CℓX(zi)Pℓ(cosθ). (18) CℓnnC(zℓgig,(zzji),z=j)+Cℓgµ(zi,zj)+Cℓµg(zi,zj)+Cℓµµ(zi,zj)(.19)
whereX symbolizesgT ornT andP (cosθ) arethe Leg- Note,then,thatthegalaxy–biaswouldbecompletelyre-
ℓ
endre polynomials. We have extended the range of the movedbythedivisiononlyifmagnificationbiaswereab-
sum to ℓ=500 to insure the convergence of the sum for sent. Hadweplottedthenormalizedversionsofthecross-
small values of θ. The general discussion about CnT ap- correlation signals, the fractional difference between
ℓ
pliestow aswell;atz >2magnificationbiasgrowsto CgT/ Cgg and CnT/ Cnn would remain about the
nT ℓ ℓ ℓ ℓ
becomeanimportantcom∼ponentofthecross–correlation samepas that between tphe un-normalized versions. This
7
sureandenergydensityrespectively),thematterdensity
Ω (we assume a flat universe so that the dark energy
m
density is 1 Ω ), the amplitude of fluctuations σ , the
m 8
−
spectral index n , the Hubble constant h, the baryon
s
density Ω and the galaxy-bias for each redshift bin b .
b i
Constraints from other data (such as from the temper-
ature anisotropy itself CTT and from the galaxy power
ℓ
spectrum) are simulated by the inclusion of priors: the
fractional (1 σ) errors on Ω h2, Ω h2, h and σ are as-
m b 8
sumed to be 5%, the fractional error on b is 10%, and
i
n has an (absolute) error of 0.02. These are similar to
s
currentconstraints,dependingsomewhatonassumptions
[13]. A simple analytic expressioncan be derived for the
parameter estimation–bias, which is discussed in detail
in Appendix A (eq. [A14]).
The inferred w, together with its marginalized error,
for samples I and II are shown in Figure 7. The modes
used range from ℓ = 2 to ℓ = 100. The solid symbols
show the inferred w from each redshift bin separately.
The sign of the resulting estimation–bias is determined
by the slope s. If s<0.4, as is the case at low redshifts,
FIG. 7: The average inferred dark energy equation of state
the inferred w tends to be lower than the true value.
wfromISWmeasurementswhen(erroneously)ignoringmag-
This is understandable because the observed CnT is sup-
nification. The true w equals −1. The solid symbols show
pressed by magnification in that case, and the only way
the inferred w for each redshift bin separately. The dotted
tostickwithanerroneousmodelwithoutmagnificationis
symbols at z = 0 show the inferred w combining all redshift
bins. Theerror-barsare1σ —notethattheyaredependent to resort to a lower w to delay dark energy domination.
upon the inferred model (see text). The upper panel is for The reverse happens at high redshifts, where s > 0.4.
sample I, and the lower panel is for sample II. In fact, at the highest redshift bin, the bias can become
quite large. Ignoring magnification bias is simply unac-
ceptable.
Cisgbg(ezca,uzse) tfoorwtithheincaasbeosucto2n0s%id.ered here Cℓnn(zi,zi) ∼ depItenisdewnotrtih.e.emtphheaysizairnegctohmatputhteederursoirnbgartshaereinmfeorrdeedl
ℓ i i
model, which is different for each redshift bin. The gen-
eral trend is for the errorbar to shrink as one goes to
VI. ESTIMATION–BIAS DUE TO IGNORING higher redshifts, despite the drop in signal–to–noise (see
MAGNIFICATION —- A THOUGHT VII). This isdue tothe factthatdlnCgT/dw riseswith
§
EXPERIMENT both z and w. The generally positive bias in w at high
redshifts helps to diminish the corresponding errorbars.
Clearly, magnification bias should be taken into ac- The dotted symbols at z =0 show the inferred w and
count when interpreting ISW measurements, especially its associated errorbar when measurements from all the
at high redshifts where the magnification–temperature redshiftbinsarecombined. Thepreviousremarksonthe
correlation actually dominates over the usual galaxy– model dependent nature of the errorbar applies here as
temperature signal. Ignoring it would lead to erroneous well. In the case of sample II, one finds that the overall
conclusionsaboutdarkenergy. Wequantifythisbycom- inferred w is biased high by almost 3 σ.
putingtheestimation–biasinthedarkenergyequationof
state w if magnification were ignored when interpreting
measurements from samples I and II.
VII. SIGNAL–TO–NOISE AND PARAMETER
A thought experiment is set up as follows. Suppose FORECAST
the true dark energy equation of state is w = 1 i.e. a
−
cosmological constant (and other cosmological parame-
The above considerations suggest that magnification
ters take the values stated at the end of III). Suppose
§ couldcause the signal–to–noiseofISW measurementsto
further one were to infer w (and other parameters) from
remain favorable even at relatively high redshifts, deep
measurements of the total cross-correlation CnT by fit-
ℓ into the matter dominated regime. This is borne out by
ting them with a model that ignores magnification i.e.
Figure 8 [37], which shows (with f divided out)
CgT, and one were to assign errorbars based on the (er- sky
ℓ
roneous) model with no magnification. The fitted pa-
S 2 [CnT(z )]2f (2ℓ+1)
rameters include: the dark energy equation of state w (z ) = ℓ i sky (20)
(i.e. w = P/ρ where P and ρ are the dark energy pres- (cid:20)N i (cid:21) Xℓ (Cℓnn(zi,zi)+ n1i)CℓTT +[CℓnT(zi)]2
8
FIG. 8: Thesignal–to–noise (squared)of ISW measurements FIG.9: Thenetcumulativesignal–to–noise(squared)ofISW
from each redshift bin for samples I and II. The solid line measurements from all redshift bins up to zmax, for samples
ignores magnification (equivalent to setting s = 0.4) while I and II. The solid line ignores magnification (equivalent to
the dotted line includes magnification. Note (S/N)2 ∝ f setting s = 0.4) while the dotted line includes magnification
sky
i.e. the division by f takes out the dependence on sky (the two are very close to each other). The division by f
sky sky
coverage. removes thedependenceon sky coverage.
where n is the mean surface density of galaxies, and we
i It appears the cumulative signal–to–noise reaches a
includemodesfromℓ=2toℓ=100(asintherestofthe
plateau by z 2 whether or not magnification bias is
paper). The result for sample II is particularly striking: ∼
included. Asisexpected,magnificationbiasmakesmuch
the signal–to–noise remains more or less flat out to high
lessofadifferenceinthiscase: thesolidanddottedlines
redshifts(dottedline),insharpcontrastwiththe signal–
are very close together. This happens despite the signif-
to–noiseifmagnificationisignored(solidline; equivalent
icant difference in signal–to–noise on a redshift bin by
to putting s=0.4,or replacingthe superscriptn by g in
redshift bin basis, shown in Figure 8. The culprit is the
eq. [20]).
significant correlation between different redshift bins in-
Magnificationbias,therefore, opens up a highredshift
troduced by lensing — the high redshift measurements
window for ISW measurements which otherwise would
are in fact only probing large–scalestructure at low red-
not exist. That is the good news. The bad news, how-
shifts.
ever,isthatsincethemagnificationofredshiftz galaxies
i The reader might also wonder why the two samples
probes structures at redshifts z < z , the CnT measure-
i ℓ have such similar cumulative signal–to–noise. This is
ments at high redshifts are in fact quite correlated with
because at low redshifts (where most of the cumulative
those at low redshifts. Taking into account such correla-
signal–to–noisecomesfrom),neithersampleisshot-noise
tions,weshowinFigure9thenetaccumulatedsignal–to–
dominated i.e. for the most part, the noise is dominated
noise (squared) for measurements from all redshift bins
by the term associated with CnnCTT (eq. [22]).
up to a given z , defined as follows: ℓ ℓ
max
Ultimately, we are interested in cosmological con-
S 2 straints from ISW measurements on, for instance, the
(cid:20)N(zmax)(cid:21) = CℓnT(zi) Cov−ℓ1 ijCℓnT(zj) dark energy equation of state w. It is therefore useful to
zi,zXj≤zmaxXℓ (cid:2) (cid:3) showtheexpectederrorsforw, bothonaredshiftbinby
(21) redshift bin basis (as in Figure 8) and on a cumulative
where basis (as in Figure 9) (see also [32]). This is displayedin
Figures 10 and 11.
[Cov ] = (22)
ℓ ij The fiducial cosmologicalmodel here, as is in the case
(Cℓnn(zi,zj)+δij/ni)CℓTT +CℓnT(zi)CℓnT(zj). ofthecomputationofS/N,isthatdescribedattheendof
f (2ℓ+1) III. The cosmologicalparametersthat are marginalized
sky §
9
FIG. 10: The marginalized error on w, hδw2i, for ISW FIG. 11: The marginalized error on w, hδw2i, for ISW
measurements from each redshift bin. Thepsolid line ignores measurements from all redshift bins combpined up to zmax.
magnification(equivalenttosettings=0.4)whilethedotted Thesolidlineignoresmagnification(equivalenttosettings=
line includes magnification. 0.4) whilethedottedlineincludesmagnification (thetwoare
very close together).
VIII. DISCUSSION
over and the set of priors assumed are identical to those
described in VI (with the addition of s , the number A natural question: to what extent should we worry
i
countslopefo§reachredshiftbin,asanewparameter,for aboutmagnificationbiaswheninterpretingexistingISW
which we impose a prior of ∆s = 0.02), and the Fisher measurements? This is partially addressed in Figure 12,
i
matrix formalism for doing so is explained in Appendix whichshowsthecross-correlationsignalat6o separation.
A,inparticulareq. [A17]. Once again,weseethatwhile The points with errorbars are measurements taken
the presenceof magnificationbias improvesthe errorsat from a compilation in [42] with the addition of recent
high redshifts on a redshift bin by redshift bin basis, it results from [10] and [11]. All involve correlating the
does not in fact add much cumulative information if one microwave background with galaxies, except the highest
considers the errors from combining redshift bins. We redshiftone [11]whichinvolvescorrelationwith quasars.
have experimented with changing the priorsand varying The median redshifts of the measurements are: z¯
∼
sampledefinitions,anditappearsthisconclusionisquite 0.1,0.15,0.3,0.5,0.9,1.5. The ISW signal shown is nor-
robust. malized by the galaxy/quasar bias, with the bias values
takenfrom[10,11,42]. Theyare,intheorderofincreas-
ing redshift: 1.1,1.0,1.2,1.4,1.7,2.3.
The most compelling reason for doing high redshift
The solid lines show the theoretical prediction for
(z > 2) ISW measurements is therefore not that the
w /b (i.e. ignoring magnification) adopting the cosmo-
expe∼cted cumulative error on w would continue to im- gT
logical parameters stated in III. Because the selection
prove,butthattheremightbesurprisese.g. darkenergy §
function is somewhat uncertain, we choose to illustrate
might be quite different from the cosmological constant
the prediction with a ‘broad’ and a ‘narrow’ selection
and actually remain a significant component of the uni-
function, parametrized as follows:
verseevenathighredshifts,orgravitymaybemodifiedin
non-trivialwaysonlargescales[38,39,40,41,44]. Atthe β zα
very least, high redshift ISW measurements constitute a W(z)= e−(z/z0)β. (23)
consistency check that one should make. As explained Γ α+1 z0α+1
β
h i
in VI, suchhighredshiftmeasurementsshouldbe inter-
§
preted with care: magnification bias must be taken into whereα=2,β =1.5isbroad,andα=2(1+3z ),β =α
0
account. is narrow. For each selection function z is adjusted to
0
10
used,thegalaxy–temperaturecorrelationwilldeclineless
rapidly because low redshift sources will contribute. On
theotherhand,w depends onthe lensingweightfunc-
µT
tion (eq. [8]) which is broadly distributed regardless of
the width of the source distribution. In other words,
changing the width of the selection function from broad
to narrow causes the galaxy–temperature term to de-
crease,whileleavingthemagnification–temperatureterm
largely unchanged. Figure 12 illustrates this for a fixed
θ, however, the statement holds for other angles. Source
distributions usedinthe existing ISWmeasurementsare
probably closer to the broad selection function.
Toconclude,wefindthatmagnificationbiasaltersthe
observed galaxy/quasar–temperature cross–correlation
significantly at high redshifts (Figures 3 and 4). The
precise magnitude of the modification depends on the
sample of objects under consideration. Three facts more
or less guaranteemagnificationplays a non–negligible or
even dominant role at z > 2: the generally steepening
slope ofthe number count∼athighredshifts (because one
is looking at intrinsically brighter objects), the decay of
FIG. 12: The two panels show the 2-point correlation func-
the intrinsic galaxy–temperature correlation CgT as one
tion divided by the galaxy bias, w(θ,z¯)/b(z¯), at θ = 6o as a
entersthematter–dominatedera,andtheincreaseofthe
function of the median redshift of the sources, z¯. The points
lensing efficiency with redshift. Ignoring magnification
with errorbars are the existing measurements. The solid line
shows the theoretical prediction ignoring magnification for a bias when interpreting high redshift ISW measurements
flat cosmological constant model (see §III). The dotted lines would lead to erroneous conclusions about the nature
includemagnificationforthecasess=0.8(upperdottedline) of dark energy. For instance, the estimated equation of
ands=0.2(lowerdottedline). Theupperpanelusesabroad state w can differ from the true one by more than 3 σ in
galaxy selection function whilethelower panelusesanarrow some cases (Figure 7).
selection function (see text).
The boosting of the ISW signal by magnification (as-
suming s > 0.4 which is generally the case at high red-
shifts when one is looking at intrinsically brightobjects)
reproduce the median redshifts of the measurements. implies that, despite naive expectations, ISW measure-
The dotted lines [43]in Figure12show the theoretical ments remain viable even at z > 2. However, because
prediction for wnT/b (i.e. accounting for magnification) of the correlated nature of the le∼nsing signal (across dif-
for s=0.8 (upper) ands=0.2 (lower). These values for ferent redshifts), the cumulative information from low
the number countslopeseemto spanthe rangeobserved z’s to high z’s is not significantly enhanced by magni-
inquasarsamples[23]similartotheoneusedinthehigh- fication. The most compelling scientific justification for
est redshift ISW measurement, where magnificationbias pursuing high redshift ISW measurements is to look for
is most relevant. surprises: for instance, dark energy might remain signif-
Consideringallangularscalesandbothselectionfunc- icant out to high redshifts. At the very least, one would
tions, we find that for the four lowest redshift measure- liketo performa consistencycheckofthe verysuccessful
ments w /w <0.1, unless s>1.4 or s< 0.6. For cosmological constant model of dark energy. The large–
µT gT
| | −
the NVSS sample at z¯ 0.9, we estimate the slope of scale nature of the ISW signal also means it is a natural
∼
the NVSS radio sample to be s = 0.320 0.07 which place to look for spatial fluctuations in dark energy [31]
±
results in w /w < 0.1. This is small compared and possible signatures of modified gravity [44].
µT gT
| |
withthemeasuremente∼rror. Itappearsthatthequasar– When interpreting future high redshift ISW measure-
temperature correlation at z¯ 1.5 is the only measure- ments, accounting for the effect of magnification bias is
∼
ment for which magnification could be significant. To a must. The net cross–correlation signal depends not
determine precisely to what extent this measurement is only on the cosmological parameters (e.g. Ω , Ω , h,
m Λ
affectedby magnificationrequiresamoredetailedanaly- n , σ ) but also on the sample–dependent parameters b
s 8
sisofthequasarsample. Itisconceivablethatasuitably and s. The slope s can be estimated from the observed
chosen quasar subsample could exhibit a large magnifi- numbercounts. Thebiasb canbe inferredby comparing
cation effect. theamplitudeofthegalaxy/quasarauto–correlationwith
Figure12alsoshowsthatforz >1thereisasignificant the amplitude of the mass auto–correlationpredicted by
dependence on the width of the s∼election function. This the microwavebackground,modulo the factthat the ob-
is because the mass–temperature correlation decreases served galaxy/quasar auto–correlation function is itself
rapidly with redshift. If a broad selection function is affectedbymagnificationbias(thoughitappearstobeat