Article
Reference
Bias and the power spectrum beyond the turn-over
DURRER, Ruth, et al.
Abstract
Threshold biasing of a Gaussian random field gives a linear amplification of the reduced two point correlation function at large distances. We show that for standard cosmological models this does not translate into a linear amplification of the power spectrum (PS) at small k. For standard CDM type models this means that the ``turn-over'' at small k of the original PS disappears in the PS of the biased field for the physically relevant range of the threshold parameter \nu. In real space this difference is manifest in the asymptotic behaviour of the normalised mass variance in spheres of radius R, which changes from the
``super-homogeneous'' behaviour \sigma^2(R) \sim R^{-4} to a Poisson-like behaviour
\sigma^2 {\nu}(R) \sim R^{-3}. This qualitative change results from the intrinsic stochasticity of the threshold sampling. While our quantitative results are specific to the simplest threshold biasing model, we argue that our qualitative conclusions should be valid generically for any biasing mechanism involving a scale-dependent amplification of the correlation function. One implication is that the real-space correlation [...]
DURRER, Ruth, et al . Bias and the power spectrum beyond the turn-over. Astrophysical Journal , 2003, vol. 585, p. L1-L4
arxiv : astro-ph/0211653
Available at:
http://archive-ouverte.unige.ch/unige:959
Disclaimer: layout of this document may differ from the published version.
1 / 1
Ruth Durrer , Andrea Gabrielli , Mihael Joye and Franeso SylosLabini
ABSTRACT
Threshold biasing of a Gaussian random eld gives a linear ampliation
of the redued two point orrelation funtion at large distanes. We show
that for standard osmologial models this does not translate into a linear
ampliation of the power spetrum (PS) neither at small k not at large k.
For standard CDM type models the \turn-over" at small k of the original
PS disappears in the PS of the biased eld for the physially relevant range
of threshold parameters . In real spae this dierene is manifest in the
asymptoti behaviour of the normalised mass variane in spheres of radius
R , whih hanges from the \super-homogeneous" behaviour 2
(R ) R 4
to
a Poisson-like behaviour 2
(R ) R 3
. This qualitative hange results from
the intrinsi stohastiity of the threshold sampling. While our quantitative
results are spei tothe simplest threshold biasing model, we argue that our
qualitative onlusions should be validgenerially for any biasing mehanism
involving a sale-dependent ampliation of the orrelation funtion. One
impliationisthatthe real-spaeorrelationfuntionwillbeabetterinstrument
to probe for the underlying Harrison Zeldovih spetrum inthe distribution of
visiblematter, asthe harateristiasymptoti negativepower-law(r) r 4
tailis undistortedby biasing.
1
ruth.durrerphysis.unige.h
2
DepartementdePhysiqueTheorique,UniversitedeGeneve,24quaiErnestAnsermet,CH-1211Geneve
4,Switzerland.
3
andreapil.phys.uniroma1.it
4
Dipartimentodi Fisia,Universita'diRoma"LaSapienza",P.leAldoMoro2,00185Rome,Italy.
5
CentroStudieRierhe"EnrioFermi"eMuseoStoriodellaFisia,ViaPanisperna89A, Compendio
delViminale,Palaz. F,00184RomeItaly.
6
joyelpnhep.in2p3.fr
7
LaboratoiredePhysiqueNuleaireetdeHautesEnergies,UniversitedeParisVI,4,PlaeJussieu,Tour
33-Rezdehausee,75252ParisCedex05,Frane.
8
franeso.sylos-labinith.u-psud.fr
9
Subjet headings: galaxies: general; galaxies: statistis, biasing; osmology:
large-sale struture of the universe
The onept of bias has been introdued by Kaiser (1984), primarily to explain the
observed dierene in amplitude between the orrelation funtion of galaxies and that of
galaxy lusters. In this ontext the underlying distribution of dark matter is treated as a
orrelated Gaussian density eld. The galaxies of dierent luminosities or galaxy lusters,
areinterpretedasthepeaksofthematterdistribution,whihhaveollapsedbygravitational
lustering. Dierent kind of objets are seleted as peaks above a given threshold, with
a hange in the threshold seleting dierent regions of the underlying Gaussian eld,
orresponding to utuations of diering amplitudes. The redued two-point orrelation
funtion of the seleted objets is then that of the peaks
(r), whih is enhaned with
respet to that of the underlying density eld (r). In a previous paper (Gabrielli,Sylos
Labini &Durrer, 2000- GSLD00)some of ushave disussed the problematiaspets of this
mehanism. In partiular the ampliation of the orrelation funtion is infat only linear
in the regime in whih
(r) 1 (see alsoPolitzer & Wise, 1984). In the region of most
observational relevane (where
(r) 1) the orrelation funtion is atually distorted at
least exponentially. Furthermore wehavedrawn attentionto the fatthat theampliation
of the orrelation funtion by biasing reets simplythat the distribution of peaks is more
lusteredbeausepeaksare exponentially sparser.
In this letter we disuss a dierent aspet of this model for bias. We are interested
in understanding the eet of biasing on the power spetrum (PS). In partiular we
address here a qualitative hange that is aused to the matter perturbations in in
standard osmologial models. In real spae this hange manifests itself ina hange from
sub-Poissonian behaviour of the mass variane at large sales in the underlying density
eld, to Poissonian behaviour of the same quantity for the biased eld. In k spae this
implies a distortion of the PS at smallk. Our analysis shows that this eet an be very
importantobservationally, asit an make the \turn-over" inthe dark matter PSdisappear
from the PS of visiblematter. Furthermore it shows the importane of measuring not just
not just the PS of visible objets, but also their real spae orrelation properties. Earlier
works about the eet of biasing inthe PS an be found e.g. Coles (1993)and Sherrer &
Weinberg (1998).
Before onsideringthreshold biasing,wereall therelevantpart ofthe analysis given in
Gabrielli,Joye & SylosLabini (2002)(hereafter GJSL02). In this paperwe havedisussed
osmologial models (with Harrison-Zeldovih like spetra P(k) k at smallk). While
this point is often noted in the osmologial literature (see e.g. Padmanabhan 1993), its
signiane and impliations are not orretly appreiated (see GJSL02 for disussion).
It implies the requirement that the integral over all spae of the orrelation funtion
vanishes, meaningthat in the system there is an exat balane between orrelations and
anti-orrelations at all sales. This is a highly non-trivial, non-loal, ondition on the
distribution. Itsspeiityan behighlightedby thefollowinglassiationofallstationary
stohasti proesses into three ategories: (i) For P(0) = 1 the utuations are like
those in a ritial long range orrelated system, (ii) for P(0)= onstant > 0 the system
is Poisson-like at large sales e.g. any short-range positively orrelated system suh as a
quasi-ideal gas at thermalequilibrium, and (iii)for P(0) =0 the system is what we have
termed \super-homogeneous". The reason for the use of this lastterm omes from the fat
thatthe threeategoriesare distinguishedmost strikinglyinrealspaeby thelarge distane
behaviour of the mass varianein spheres, as one an show that P(0)=lim
V!1
h(M(V)) 2
i
2
o V
where h(M(V)) 2
i isthe mass varianein a volume V (and
o
the mean mass density). In
the Poissontypedistributionthisvarianeisproportionaltothe volumeofthe sphere, while
inthe rstategory(ritialsystems)itgrowsmorerapidly(withalimitingbehaviourofthe
volume squared), while in the last (super-homogeneous distributions)the growth is slower
than the Poissonian one. In partiular the ase of the H-Z spetrum marks the transition
to the limiting slowest possible growth of this quantity for any stohasti distribution of
points (Bek 1987), whihis agrowth proportionalto the surfae of the sphere.
These super-homogeneous distributions are enountered in various ontexts in
statistialphysis. They are desribed in this ontext asglass-like: they are highlyordered
distributions like alattie, but with full statistial isotropy and homogeneity. In Gabrielli
etal. (2002)anexample of asystem with suh orrelationsatthermal equilibriumisgiven,
and a modiation of this same system whih should give preisely the orrelation of a
standard osmologial modelis desribed.
Wenowturntothethresholdbiasingmehanism. FollowingKaiser(1984)weonsidera
stationary, isotropiandorrelated ontinuousGaussianrandomeld, Æ(x), withzero mean
and variane 2
=hÆ(x) 2
i in a volume V as V ! 1. The marginalone-point probability
density funtion of Æ is P(Æ) = 1
p
2 e
Æ 2
2 2
. Using P(Æ), we an alulate the fration of
the volume V with Æ(x) , Q
1 ()=
R
1
P(Æ)dÆ: The orrelation funtion between the
values of Æ(x) in two points separated by a distane r is given by (r) =hÆ(x)Æ(x+rn)i.
By denition, (0)= 2
. In this ontext, stationarity means that the variane, 2
, and the
orrelation funtion, (r), do not depend on x. Statistial isotropy means that (r)does
not depend on the diretion n. The goalis to determine the orrelation funtion of loal
simplied(Kaiser 1984) by omputing the orrelation of regionsabove aertain threshold
insteadof the orrelations of maxima. However, these quantities are losely relatedfor
values of signiantly largerthan 1. We dene the threshold density,
(x) by
(x)(Æ(x) )= (
1 if Æ(x)
0 else.
(1)
Note the qualitative dierene between Æ whih is a weighted density eld, and
whih
just denes uniform domains,all havingequal weight,and h
(x)i =Q
1 .
Let us now onsider how the biasing hanges the distribution in relation to the
lassiation we have given interms of P(0). In what follows we show, for >
o
>0, (
o
given below)
P
(0) >P(0) (2)
where P
(k) and P(k) are the PS of the biased and underlying eld respetively, i.e.
the Fourier transform of
(r) and of the normalised underlying orrelation funtion
(r) (r)=
2
respetively. This result is independent of
(r). The orrelation funtion
(r)of the biasedeld is given (Kaiser1984) by the expression
Q
1 ()
2
(
(r)+1)=
1
2 q
1
2
(r)
Z
1
Z
1
dÆdÆ
0
exp (Æ
2
+Æ 02
) 2
(r)ÆÆ
0
2(1 2
(r))
!
(3)
where the integrand on the right hand side is the two point joint probability density for
the Gaussian eld Using the expression for Q
1
() given above, one an reast this after a
simple hange of variables intothe form
(r)=
R
1
dxe
x 2
=2 R
dye
y 2
=2
[ R
1
dxe
x 2
=2
℄ 2
: (4)
where=(
x)=
q
1
2
. Inthis formitis evident that
(r)=0,
(r)=0and that
sign[
(r)℄=sign[
(r)℄.
Taylorexpanding this expression about
=0, we nd 1
(r)=b
1 ()
(r)+b
2 ()
2
(r)+::: (5)
1
FortheexpansiontoallordersseeJensen&Szalay(1986).
with
b
1
()=e
2
=2 R
1
dxxe x
2
=2
[ R
1
dxe
x 2
=2
℄ 2
(6)
b
2 ()=
1
2 e
2
=2 R
1
dx(x
2
1)e x
2
=2
[ R
1
dxe
x 2
=2
℄ 2
: (7)
The rst term gives the linear relation obtained by Kaiser(1984), as b
1
() 2
for 1,
valid inthe regime j
j1 and j
j 1. It is easy to hek that b
2
() ispositive denite
for 0 (and b
2
() 4
=2for 1),so that tothis orderin
(r) one has the bound
(r)>b
1 ()
(r) for
(r)6=0and >0: (8)
Ifj
(r)j1 atallr this bound suÆes togivethe desiredresult (2)forallvaluesof
suhthat b
1
()1. As b
1
() isamonotonially inreasingfuntion of (withb
1
(0)=2=)
this isequivalent to the requirement
o
with
o
suh that b
1 (
o
) =1 (i.e.
o
'0:303).
To show that there is a value of above whih Eq. (2)is indeed satised for all permitted
values of
, itsuÆes to nd the threshold value
1
o
suh that for all
1
one has
sign
"
d
(r)
d
(r)
b
1 ()
#
=sign[
℄ 8r : (9)
In fat this is a suÆient ondition to have the exat urve
(
), given by Eq. (4), all
above the line
=
for all
. We have found numerially, using Eq. (4), that this
ondition is satisedfor
1
'0:38.
Note that, if the ondition (8) holds, this means simply that, relative to the
asymptoti (j
j 1 and j
j 1) linearly biased regime in whih
b
1 ()
, the
anti-orrelated regions are less amplied (j
j < b
1 ()j
j) than the positively orrelated
regions (j
j > b
1 ()j
j). Thus the integral over the biased orrelation funtion is always
positive, and the bound (2) thus holds. Further it is easyto see that P
(0)is nite if P(0)
is:
isbounded for any valueof , and,has the sameonvergene properties as
atlarge
distanes. This implies that, if the integral of
over allspae onverges, then alsothat of
does.
In terms of the lassiation of distributions by P(0) we thus draw the following
onlusion: Both the ritial type system (with P(0) = 1) and Poisson type system
(with P(0)=onstant>0) remain inthe same lass; the super-homogeneous distribution
(with P(0) = 0) however beomes Poissonian (P
(0) = onstant > 0). The essential
reason for these hanges is simple: as disussed above the behaviour of the PS is the
is stohasti in nature, and introdues a variane in the number of objets whih is
proportional to the volume. This new variane will dominate asymptotially over that
of the original distribution only if the latter is super-homogenous (i.e. its asymptoti
normalised variane issub-poissonian,deaying faster than Poisson). Consider for example
the ase of aperfet lattie, whih isa super-homogeneous distribution (P(0)=0)in whih
the normalised variane 2
(R ) = h(M(R )) 2
i=hM(R )i 2
in a sphere of radius R deays
asymptotially as 1=R 4
. The distribution obtained by keeping (or rejeting) eah point
with probabilityp (or 1 p) isdesribed by asimple binomialdistribution, with avariane
2
(R )/p(1 p)=N /1=R 3
(N beingthe mean number of pointsinside a sphere). Biasing
is not suh a purely randomsampling,but the eet of stohastiity as a soureof Poisson
variane atlarge sales is similar. Translated in terms of the PS itgives the result we have
derived.
Nowlet us turn to the impliations of this result for osmologial models. Sine suh
models have P(0)= 0 in the full matter spetrum, it is evident that we annot have the
behaviour P
(k) /b
1
()P(k) for smallk whihone might naively infer from the fat that
(r) b
1 ()
(r) for large separations. Inevitably a non-linear distortion of the biased
PS at smallk relative to the underlying one is indued. How importantan the eet be
qualitatively for a realisti osmologial model? To answer this question we onsider the
simple model PS P(k)=Ake k=k
. The dierenes with a old dark matter (CDM)model
- whih has the same linear Harrison-Zeldovihform at smallk but adierent (power-law)
funtional form for large k - are not fundamental here, and this PS allows us to alulate
the orrelation funtion
analytially (see GJSL02). This greatly simplies our numerial
alulation of the biased PS P
(k), whih we do by diret integration of
(r) alulated
using the approximation
(r)=
2
4 v
u
u
t 1+
(r)
1
(r)
exp 2
1+
!
1 3
5
(1+o(
1
)) (10)
whihisveryaurateovermostoftherangeoftheintegration 2
. InFigure1weshowP
(k)
for various values of the threshold =1;2;3. Wesee that the shape of the PSat smallk is
ompletelyhangedwithrespet totheunderlyingPS. Indeed themain featureofthe latter
in this range - the display of a lear maximum and \turn-over" - is ompletely modied.
2
Thisapproximationisobtainedbyexpandingthefullexpression for
(r)givenin Eq. (4)in 1=,and
furtherassumingonlythat p
(1
)=(1+
)1. ItisamuhbetterapproximationthanthatofPolitzer
andWisebothatsmallandlargervaluesof
. Inpartiularitgivesanasymptotibehaviour
(
2
+1)
for 2
1whihisamuh betterapproximationto theexatbehaviourattypiallyrelevantvaluesof
(b (1)2:4)
Qualitatively it is not diÆult tounderstand why this is so. The only harateristi sale
in the PS (and alsoin the orrelation funtion) is given by the turn-over (speied inour
ase by k = k
). On the other hand, the value of P
(0) is just the integral over all spae
of
whih is proportionalto the overall normalisationA and (sine it is stritly positive)
must be given on dimensional grounds by Ak
times some funtion whih depends on .
For '1 this funtion is oforder one, so that P
(0) max[P(k)℄.
ThislastpointisbetterillustratedbyonsideringtheintegralJ
3
(r;)=4 R
r
0 x
2
(x)dx
whih onverges to P
(0) = lim
r!1 J
3
(r;). In Figure 2 the value obtained for it by
numerial integration of the exat expression given by Eq. (4) for
(r) is shown for
= 1;2;3. We also show the same integral for
whih onverges to P(0) = 0. While
the latter dereases atlarge r, onverging very slowly to zero (as1=r sine
(r) / 1=r 4
at large sales), the former all onverge towards a onstant non-zero value We see that
the integral piks up its dominant ontribution from sales around (and above for =1)
r 10Mp(see aptionfor explanation of the normalisations,whih are irrelevantfor the
present onsiderations). From the inset in the gure, whih shows both
(r) and
(r),
we see that this is the sale below whih the orrelation funtion is non-linearlyamplied.
Moreover it is shown that the smaller sales at whih
(r) is most distorted relative to
(r) donot ontribute signiantlyto J
3
(beause of the r 2
fator). This fat alsoexplains
the auray of the PS obtained using the approximation Eq. (10) for
(r), whih an
be seen by omparing the asymptoti values of the integrals in Figure 2 with P
(0) in
Figure 1. Note that for = 1 the distortion away from linear is relatively weak in the
part of the orrelation funtion whih dominates the integral in J
3
(r;), and that there is
even a non-negligible ontributionfrom the larger sales at whih the orrelation funtion
ampliationis extremely lose tolinear.
Let us now draw our onlusions. We have alulated the eet of the distortion
at smallk of a biased PS relative to an underlying PS, whih we have shown to be an
inevitable eet of biasing on osmologialmodels. In partiular we have used a simplied
modelPSand the simplest(but referene) threshold biasingsheme of Kaiser. The latteris
not arealisti modelof biasingfor various reasons, most notably beause of the extremely
strong non-linear distortionof the two-point orrelation funtion at small sales whilst the
observed ratio of orrelation between galaxies and lusters is approximately linear. Our
results however should be qualitatively orretfor any biasing sheme and standard CDM
type model. The only thing whih we expet to depend on the biasing sheme is the
funtionalformof thePS tothe right ofthe turn-over(fork >k
),and orrespondingly the
shape of the orrelation funtion at small sales, whih willonly make a minor numerial
hange to our alulation. The essential feature of biasingwhih bringsabout the eet we
any biasing model. In fat the onverse of what we have arguedis that any biasing sheme
whihis stohasti (i.e. any proedurefor seleting sites for objets whih is probabilisti)
must give suh a non-linear distortion of the orrelation funtion for osmologial models:
if the orrelationfuntion were ampliedlinearly at allsales, the asymptotibehaviourof
the varianewillbe sub-Poissonianinstead of Poissonian.
Wenallydrawtwo onlusionswith respet to theomparison of osmologialmodels
with observational data. In order to make the link to observations of galaxies or lusters,
urrent theories make use normally of biasing inone form or another. Invariably however
the measured PS fromobservations ist toa modelspetrum whih is just aresaled dark
matter PS(for a reent example see Lahav et al. 2002). Our rst onlusion isthat suh a
behaviourannotbeobtained by biasingasthe PSis notlinearly ampliedneitheratsmall
nor atlarge wavenumbers. Our seond onlusion is that itwill be very useful to measure
not just the PS at small k, but also the real spae orrelation funtion at large distanes.
Invariably only the rst is onsidered in analysis of observations of galaxy distributions at
large sales, beause, it isargued, itis the natural probegiven that osmologial models of
struture formationare formulatedink spae. Wehave seenhoweverthatbiasingleadstoa
distortionof the underlyingdarkmatterPS, while (insertpanel of Figure2)the orrelation
funtionatsuÆientlylargedistane remainsundistorted. In partiularforstandard models
with a H-Z PS at smallk, the leanest way to detet this behaviour in biased objets is
by lookingat the orrelation funtion atlarge sales, whih shouldmaintain the behaviour
(r) r 4
assoiated with the small k behaviour of the underlying darkmatter PS. To
address the viability of measuring this behaviour in urrent and forthoming surveys of
large sale struture requires in partiular the detailed treatment of both the passage to
the disrete distribution (we have treated here always ontinuous density elds), and the
question of the variane of estimators of (r).
F.S.L. thanks the PostDotoralsupport of a Marie Curie Fellowship. R.D. and F.S.L.
aknowledge support of the Swiss National Siene Foundation. This work issupported by
the TMR network FMRXCT980183 onFratal Strutures and Self-Organization.
REFERENCES
Bek, J.,1987, AtaMathematia, 159, 1-878282.
Coles, P., 1993 Mon.Not.R.aad.So,264, 1065
Gabrielli,A., SylosLabini, F., and Durrer, R.,2000, Astrophys.J. Letters, 531, L1
Gabrielli, A., Janovoi, B., Joye, M., Lebowitz, J., Pietronero, L., and Sylos Labini, F.,
Phys.Rev. D(2003) inthe press (astro-ph/0210083)
Kaiser N., 1984,ApJ, 284, L9
Jensen L. &Szalay A.,1986, Astrophys.J., 305, L5.
Lahav O.et al.,2002, MNRAS, 333, 961
Padmanabhan T., 1993 Struture formation inthe universe, Cambridge University Press.
Politzer H.D., & WiseM.B., 1984, ApJ, 285, L1.
Sherrer R.J., Weinberg, D. H., 1998, ApJ,504, 607
Thispreprintwas preparedwiththeAASL A
TX marosv4.0.
10 −5 10 −4 10 −3 10 −2 10 −1 10 0
k
10 −4 10 −2 10 0 10 2 10 4 10 6
P(k)
ξ c (r) ν=1 ν=2 ν=3
Fig. 1.| The PS P
(k) derived from the biased orrelation funtion
(r) for values of
the threshold =1;2;3 is shown. The underlying orrelation funtion whih gives
(r) is
that derived from the P(k) = Ake k=k
and the approximation given in Eq. (10) for
is
used. The lear distortion of the PS at small k is seen, the \turn-over" in the underlying
PS essentially disappears already for =1. The onstants k
and A are xed by
(0)=1
and the requirement that
(r)=0 atr=38Mp. The latter istaken as inatypialCDM
model (see e.g. Padmanabhan 1993). We ould alternatively x the wavenumber at the
maximum of the PS.
0.1 1 10 100 1000
r
0.001 0.01 0.1 1 10 100 1000 10000 1e+05 1e+06
J 3 (r)
1 1000
0.001 1 1000 1e+06
ξ (r) ν=1 ν=2 ν=3
10 0 r 10 2
10 -6 10 -3 10 0 10 3
| ξ( r)|
Fig. 2.| The integral J
3
(r) for the same underlying orrelation funtion as in Figure 1
and for the same range of values of , alulated with the exat expression Eq. (4) for
(r). Also shown is the analagous integral of
. While the latter onverges slowly to its
asymptotivalueofP(0)=0,theother integralsonvergetoonstantnon-zero P
(0). They
are dominated by the range r 10 Mp where, as an be seen from the inset whih shows
both
and eahofthe
(r),the orrelationfuntions
(r)are ampliednon-linearly. The
ontributionfromtheextremelyampliedregionatsmallr issmall(beauseofther 2
fator
in the integral), whih also makes the approximation in alulating the PS with Eq. (10)
veryaurate, asan be heked by omparingthe asymptotivalueswith those ofP
(0)in
Figure1.
