SIMULATIONS OF CLUSTERED
EXTRAGALACTIC POINT SOURCES
AT PLANCK FREQUENCIES
1,
L. Toffolatti
1
2
1,
J. Gonzalez-Nuevo
2

F. Argueso
Departamento de Fisica, Universidad de Oviedo, c.
Calvo Sotelo, s/n, 33007 Oviedo
Departamento de Matematicas,
Universidad de Oviedo, c.
Calvo Sotelo, s/n, 33007 Oviedo
[email protected]
Abstract
We
present
predictions
on
the
angular
power
spectrum
of
cosmic
microwave
background (CMB) uctuations due to extragalactic point sources (EPS) by using
a method for simulating realistic 2D distributions of clustered EPS. Both radio and
far–IR selected source populations are taken into account.
To analyze different
clustering scenarios, we exploit angular power spectra of EPS,
P (k)
, estimated
either by data coming from currently available surveys or by means of theoretical
predictions.
By adopting the source number counts predicted by the Toffolatti et al.
(1998)
evolution model – capable of accounting well for the available data at radio cm
wavelengths – we are able to reproduce current data on the two–point angular
correlation functions,
w(θ)
, of radio sources. We can conrm that the detection of
primordial CMB anisotropies is not hampered by undetected clustered sources at
frequencies
≤ 150 − 200
GHz.
On the other hand, our current ndings show that
at higher frequencies the clustering signal could severely reduce the detectability
of intrinsic CMB anisotropies, thus conrming previous theoretical predictions.
Keywords:
2D simulations; extragalactic point sources:
clustering; CMB anisotropies:
an-
gular power spectrum.
Introduction
Extragalactic
through
the
point
beam,
sources
being
their
–
i.e.,
typical
galaxies
seen
projected
isotropically distributed all over the sky.
as
angular
a
`point–like'
size
¿ θbeam
object
–
are
Thus, they give rise to a contaminat-
ing signal which presents the same average level all over the sky,
frequency (Tegmark and Efstathiou, 1996).
at a given
This signal can be only reduced by
the identication and detection of as many sources as possible.
On the other
hand, the relatively large beam sizes and high ux detection limits of current and
forthcoming earth, balloon and space–borne experiments (e.g., BOOMERanG,
2
VSA, CBI, DASI, ACBAR, Archeops, NASA WMAP and ESA Planck missions) imply that only relatively bright sources can be detected and removed
from current as well as future CMB sky maps (Vielva et al., 2003).
the
contribution
CMB
of
temperature
the
highly
undetected
uctuations
must
be
extragalactic
accurately
source
estimated
Therefore,
populations
to
avoid
an
to
un-
wanted incorrect reconstruction of the angular power spectrum of primordial
anisotropies.
This
multipoles, i.e.
Among
all
problem
is
particularly
important
at
intermediate
to
high
` ≥ 1000
, where the intrinsic CMB anisotropies are damped.
the
analyses
on
the
extragalactic
foreground
contributions
to
small–scale uctuations, a thorough one, over the full wavelength range from
∼1
cm to
∼ 300 µ
m, which improved on previous ones, has been presented by
Toffolatti et al.
(1998).
Assuming a Poisson distribution of point sources in the
sky, they found that the central frequency channels of the Planck mission will be
`clean' (i.e, only a few high latitude pixels will be contaminated by bright undetected sources).
As for radio selected extragalactic sources, which contaminate
CMB anisotropies at Planck Low Frequency Instrument (LFI) channels (Mandolesi et al., 1998), their clustering signal was found to give a generally small
contribution to temperature uctuations,
thanks to the broadness of the local
luminosity function (Dunlop and Peacock, 1990) and of the redshift distribution
of sources which dilute the clustering signal (Blake and Wall, 2002).
At higher
frequencies, the clustering of far–IR selected dusty galaxies was found to give a
more relevant – albeit not dominant – contribution to temperature anisotropies.
More recently,
the new results coming from the Sub–millimeter Common
Use Bolometric Array (SCUBA) surveys have been giving increasing evidence
that many of the sources detected in this frequency region of the e.m.
are
ultraluminous
star–forming
galaxies
at
z ≥ 2
(see,
e.g.,
spectrum
Dunlop,
2001).
These ndings are explained by recent models of Galaxy formation in which
SCUBA sources mainly correspond to the phases of intense star formation in
large spheroidal galaxies at substantial to high redshift.
As a consequence, the
relevant redshift range for these sources is highly limited and the dilution of
the clustering signal results relatively reduced; correspondingly, the amplitude
of
∆T /T
uctuations due to clustering is expected to be large, in agreement
with recent
galaxies
at
ndings.
Moreover,
intermediate–
(see, e.g., Webb et al.
to
all
recent
high–redshifts
studies
are
clearly indicate
strongly
correlated
that
in
dusty
the
sky
2003) and, thus, they are giving rise to a clustering signal
which probably dominates over the Poisson one at all multipoles and in almost
all Planck High Frequency Instrument (HFI) channels (Puget et al., 1998).
In view of the above and for having a useful tool for the analysis of current as
well as future all–sky maps, we decided to try a different approach for estimating
the power spectrum of EPS temperature uctuations.
This approach is based
on a fast algorithm capable of simulating all–sky maps as well as sky patches
on which EPS are distributed following a given two–point angular correlation
3
Simulations of clustered extragalactic point sources at Planck frequencies
function,
w(θ)
.
Moreover, with the purpose of producing all–sky maps which
could be used by the CMB community in general and by the Planck Consortia,
in particular, we adopted the standard HEALPIX pixelization scheme of Gorski
et al.
(2002) throughout the paper.
A at cosmological model with
adopted.
H0 = 70
km/s/Mpc,
ΩΛ = 0.7
has been
Anyway, the present results are almost unaffected by the underlying
cosmology.
1.
2D simulations of EPS: making the density eld
First of all, we remind here that the current approach is a purely phenomenological one, aimed at simulating 2D all–sky maps of clustered EPS in a fast way
with the guarantee that the input angular correlation function is well recovered
from the map.
As a preliminary step towards the realization of a at (a sky patch) or spherical
(all–sky) map of clustered sources, we distribute point sources by adopting a
n(x)
hni = N (> Smin )/Npixels
simple Poisson distribution for the number,
of
n(x)
is then
, of sources per pixel. The mean
, i.e.
the average number of sources
per pixel, which is determined by the total number counts
ν
frequency
.
As usual,
N (> Smin ) =
Slim
the differential source counts and
We
then
δ(x) =
dene
the
projected
R Slim
Smin N (S)dS
N (> Smin )
N (S)
, where
at a given
indicates
the ux limit for source detection.
density
contrast
at
a
given
point
n(x)−hni
, being hni the average number of sources per pixel.
hni
(pixel)
as
Therefore,
the covariance function of the projected density contrast is the usual two–point
angular correlation function (see, e.g., Peebles, 1993; Peacock 1997)
w(θ) = hδ(x)δ(x + θ)i
where
θ
(1)
is the angular separation in the sky between the two positions and
the brackets indicate an ensemble average.
As a following step, we calculate
the Fourier transform of the density contrast
1
L2
δ(k) =
where
It
is
Fourier
L
Z
δ(x)e−ikx dx
(2)
is the angular size of the map we are actually using.
very
easy
transform
to
of
show
the
that
the
angular
angular
power
correlation
angular power spectrum depends only on
spectrum
function
h|δ(k)|2 i
(Peacock,
is
1997).
the
The
k = |k|
, being the eld homogeneous
and isotropic.
Thus,
the basic idea of our method is to obtain a density eld determined
by a given power spectrum or, equivalently, by the suitable angular correlation
function.
trast,
We do this by calculating the Fourier transform of the density con-
δ(k)
, and obtaining its power spectrum, which is constant for all modes,
4
P (k)P oiss = const
, if sources are Poisson distributed in the sky.
Subsequently,
we introduce the chosen angular power spectrum of correlated sources,
§
suitable for the particular source population adopted (see
3.2.),
P (k)cl
,
by applying
the following formula
p
P (k)cl + P (k)P oiss
p
P (k)P oiss
δcorr (k) = δ(k)
Then we apply the inverse Fourier transform to
recovering of a new density eld,
P (k)cl
w(θ)
modied by
function,
.
, i.e.
δcorr (k)
(3)
which allows us the
δnew (x)
, in which the Poisson term has been
the Fourier transform of the chosen angular correlation
Finally, we calculate
nnew (x) = hni(1 + δnew (x))
which
map,
gives
the
according
P (k)cl = 0
2.
to
modied
the
number
previously
of
point
calculated
sources
new
(4)
at
each
density
position
eld,
in
the
δnew (x)
.
If
in equation (3) we obtain, again, a pure Poisson distribution.
The EPSS-2D algorithm:
an efcient way for
distributing uxes
We
have
shown
how
it
is
possible
to
their average number density per pixel,
number counts at each frequency.
T (x)
hni
, of sources per pixel.
EPS
in
the
sky,
by
using
, which is determined only by the
In this way we have obtained a pure density
eld, characterized by the particular
number,
distribute
hni
P (k)
used in each map and by the average
Now, for converting it to a temperature map,
N (> Smin )
S(Jy) → T (K)
, we have to distribute rst the uxes corresponding to the
sources
of
the
total
counts.
Then,
the
usual
conversion
is
applied.
To do this, we have to know the differential counts at a given frequency –
and, in principle, for each source population – which are giving us the number,
N (S)
, of extragalactic sources in each ux interval.
Then, it is necessary to nd
an efcient algorithm for distributing uxes in the map and which satises two
fundamental requirements:
to the ux limit,
Slim
a) the differential counts should be recovered down
, allowed by the adopted resolution element (the pixel size
or the FWHM of the beam); b) the “input” angular correlation function should
also be reconstructed, at least down to the ux limit of the sample by which that
specic
In
P (k)
has been determined.
principle,
among
pixels:
one
e.g.,
can
by
choose
simply
a pixel in which there are
n
of
uxes,
taken
at
n
many
different
distributing
ways
uxes
for
under
distributing
the
condition
uxes
that
in
sources we have to put a corresponding number
random
from
the
distribution
given
by
the
differential
counts; or, e.g., by imposing that the brightest uxes fall always in the highest
5
Simulations of clustered extragalactic point sources at Planck frequencies
density pixels, thus strengthening, like imposing a “bias” factor, the correlation
function of sources.
Anyway, no matter the method of distributing uxes one
chooses, the constraint has to be always the same:
a) and b),
fundamental requirements.
Thus,
to full the above quoted,
for testing the method and with
the purpose of simulating maps of EPS at CMB frequencies, we have exploited
the published data on the
w(θ)
s (or the
P (k)
s) of radio sources coming from
the analysis of the NRAO VLA 1.4 GHz Sky Survey (NVSS) (Blake and Wall,
2002) and of the Parkes-MIT-NRAO survey at 5 GHz (Loan, Wall and Lahav,
1997).
As a rst approach, we made the choice of distributing the uxes at random
×
among the pixels (512
512 pixels of
1.5
arcmin side in a at patch of the sky).
This is the simplest way of distributing uxes but it works well, at least in the
case of source populations not strongly clustered or for which the clustering
signal is washed out by the very broad redshift distribution, as in the case of
radio selected sources (see, e.g., Blake and Wall, 2002).
on the comparison of the above quoted observed
The detailed results
P (k)
s and the corresponding
ones recovered from our 2D simulated maps have been discussed extensively
by
Gonzalez-Nuevo
et
al.
summarized as follows:
(2004).
a) the input
The
main
outcomes
P (k)cl
Slim
, converted to
from the simulated maps for every chosen
of
C`
that
work
can
be
, is well recovered
, in agreement with the results
of the original articles; b) the Poisson term gives rise to a much greater signal
than the clustering term, in all the cases:
is
two/three
orders
dependence on
k
of
(or
`
magnitude
above
the total power,
the
clustering
P (k)cl + P (k)P oiss
term.
Moreover,
,
the
) of the Poisson and of the clustering term is different:
this well known result is a direct consequence of the increase of the clusteringto-Poisson ratio at low multipoles (De Zotti et al., 1996).
3.
Source counts and correlation functions at CMB
frequencies
All the most recent surveys of radio sources are conrming – with a small
offset – the predictions on number counts made by Toffolatti et al.
least up to
2004).
as
well
∼ 40
At these frequencies, the EPS relevant at uxes of interest for current
as
future
α ' 0.0 S(ν) ∝
(
(1998), at
GHz (see, e.g., Bennett et al., 2003; Gonzalez-Nuevo
et al.,
,
CMB
ν −α )
anisotropy
experiments
sources, i.e.
are
mainly
“at”—spectrum
QSOs, BL Lacs and local AGNs and the
contributions coming from other fainter source populations can be neglected.
Moreover, the number of “inverted” spectrum sources – which could have represented a threat for CMB
(De
Zotti
et
al.,
2000).
∆T /T
measurements – is found to be always small
Therefore,
we
can
adopt
the
Toffolatti
et
al.
(1998)
model counts and the angular correlation function determined by the ParkesMIT-NRAO survey at 5 GHz (see above) for simulating sky maps of clustered
6
EPS sources at Planck LFI frequencies and angular resolutions.
On the other
hand, when analyzing CMB sky maps coming from high resolution experiments
– which probe fainter uxes – the most suitable
w(θ)
, for simulating sky maps
of clustered radio EPS, is the one determined by Blake and Wall (2002) from
the NVSS survey.
As for dusty galaxies, which dominate the source counts at HFI frequencies,
the measurements of their clustering properties is made difcult by the poor
statistics.
However, Peacock et al.
(2000) found some evidence for clustering
of the background SCUBA source population in the Hubble Deep Field observed
at 850
et al.
µ
m.
The power in excess over Poisson uctuations detected by Peacock
(2000) is well accounted for by a two–point angular correlation function
of the form
w(θ) = (θ/θ0 )−0.8
by Perrotta et al.
with
θ0
in the range 1–2 arcsec.
(2003) the above value of
θ0
As discussed
is consistent with a number of
data on clustering of SCUBA sources and can be well accounted for by a
w(θ)
derived from physical assumptions on the evolution of clustering (Matarrese et
al.,
1997).
In the framework of this physical evolution model for clustering,
Perrotta et al.
power
(2003) worked out a complete set of predictions on the angular
spectrum
due
to
EPS
sources
at
HFI
frequencies.
Given
that
we
are
currently interested in testing our algorithm for distributing clustered sources
in CMB sky maps, in the following we adopt the
number counts of spheroids of Granato et al.
w(θ)
of Perrotta et al.
and the
(2001) for simulating all-sky maps
of far–IR selected EPS. We have to remind that for reproducing the correlated
distribution in the sky of these highly clustered source populations, like dusty
proto-spheroidal galaxies at high redshift, we have to force the brightest sources
to fall in the highest density bins (see Gonzalez-Nuevo
et al., 2004).
Finally, it is important to stress that the proposed method allows to add up as
many source populations as needed.
However, it shall be generally sufcient
to simulate only one source population.
The one which dominate the counts
in the ux interval of interest, except for some very specic frequency channel in which two, or more, source populations – showing different clustering
properties – are giving a comparable contribution to the counts.
This is a great
advantage, which reduces the total required CPU time, and is determined by the
fact that the total variance of intensity uctuations due to EPS,
2
σN
, is obtained
by adding up in quadrature all the contributions coming from different source
populations
(see,
e.g.,
Negrello
et
faint EPS populations to the total
4.
al.,
σN
2004).
Therefore,
the
contribution
of
is generally very small.
Angular power spectra of clustered EPS at CMB
frequencies
As an example of the goodness of the proposed method, Figure 1 displays our
current results on the temperature angular power spectrum,
C`
, due to clustered
7
Simulations of clustered extragalactic point sources at Planck frequencies
EPS at 30 and 353 GHz.
The
C`
have been recovered from the simulated maps
obtained by the EPSS–2D algorithm.
for
two
main
reasons:
a)
they
These two frequencies have been chosen
corresponds
to
the
central
frequencies
of
two
Planck channels; b) the source populations, radio sources and dusty galaxies,
which dominate the bright counts in each channel are showing quite different
clustering
properties.
Notice
that
each
plotted
C`
has
been
calculated
only one all-sky map of EPS, following the method explained in
the
assumptions
of
at
p
sky
δT` (ν) =
made
±1σ
also plot the
§
in
3.
However,
for
giving
a
condence
§
from
2 and with
interval,
we
levels around the mean, having performed 100 simulations
patches.
As
in
Toffolatti
`(` + 1)C` /2π
et
al.
(1998)
we
represent
the
quantity
(in units of K).
From the upper panel of Figure 1 it is clear that the contribution of clustered
radio EPS to temperature CMB uctuations at these frequencies is found to be
negligible at all angular scales, “if sources are not subtracted down to uxes
well below the detection limit of the survey, thus greatly decreasing the Poisson uctuations, while the contribution arising from clustering is only weakly
affected” (Toffolatti et al., 1998).
In fact, if only bright sources at
S>1
Jy are
subtracted out from the map – e.g., the detection limit of the WMAP survey –
the
C`
of clustered EPS still match very well the predictions of Toffolatti et al.
(1998).
These results are, again, in agreement with the well known observa-
tional result:
“the wide redshift range of radio sources washes out much of the
clustering signal” (Blake and Wall, 2002).
From the same panel it is also pos-
sible to appreciate a small excess at the largest angular scales in the estimated
C`
s of clustered EPS, in comparison with a pure Poisson source distribution in
the sky.
This is in agreement with the well known outcome that the ratio of
clustering–to–Poisson uctuations increases with increasing angular scale, i.e.
at decreasing multipole number (De Zotti et al., 1996).
In the bottom panel of Figure 1 we plot our current outcomes at 353 GHz
by using the EPS counts of spheroidal galaxies of Granato et al.
by applying the
w(θ)
of Perrotta et al.
(2003).
In this case,
(2001) and
due to the very
strong angular correlation function detected in SCUBA elds, for recovering
the input
pixels.
w(θ)
, we have to force the brightest uxes to fall in the highest density
From Figure 1 it is clear that a very good agreement is found between
the recovered
C`
all multipoles.
±1σ
s and the theoretical predictions of Perrotta et al.
The small scatter around the average value,
condence intervals, shows that we can be condent in the
from each simulated map.
at very bright uxes and,
SCUBA surveys, their
C`
(2003), at
displayed by the
C`
s recovered
At 353 GHz spheroids starts to dominate the counts
thus,
if they are strongly clustered as suggested by
s are no doubt the dominant ones.
As for other far–
IR selected source populations, they give a small to negligible contribution to
CMB temperature uctuations, since they are much less clustered (Toffolatti et
al., 2004).
8
−4
δ Tl(ν)[K]
10
−5
10
−6
30 GHz
10
1
2
10
3
10
10
Multipole(l)
−4
δ Tl(ν)[K]
10
−5
10
353 GHz
1
2
10
3
10
10
Multipole(l)
Figure 1.
Poisson
from
Top panel:
distributed
the
simulated
angular power spectrum (
EPS
at
30
GHz.
map
without
The
any
thick
source
δT` =
dashed
subtraction.
p
line
`(` + 1)C` /2π
δT`
shows
The
thin
the
dashed
) of clustered and
values
line
same quantity calculated after having subtracted from the map all sources with
recovered
represents
S≥1
Jy.
the
The
thin continuous line shows, for comparison, the prediction of TO98 obtained applying the same
detection limit for sources,
simulations
spectrum
of
on
2D
EPS
sky
at
Slim = 1
patches
353
GHz.
is
Jy, as before.
also
The
shown
thick
dashed
(2003) for the case
dotted lines show, again, the
line shows the
δT`
1σ
line
1σ
condence level obtained by 100
lines).
shows
Bottom
the
Slim
Mhalo /Msph = 100
the map by applying the same detection limit,
obtained by Perotta et al.
The
(dotted
,
δT`
angular
power
recovered
from
for EPS as in the theoretical predictions
(thin continuous line).
condence level obtained by 100 simulations.
The
The thin dashed
values obtained with Poisson distributed sources and by adopting the same
detection limit as in Perrotta et al (2003).
In each panel, the thick continuous line represents, for
comparison, the primordial CMB power spectrum calculated for a at
5.
panel:
values
Λ
CDM model.
Summary
We have presented predictions on the angular power spectrum of EPS by 2D
simulated maps in which extragalactic sources are distributed with correlated
positions in the sky.
The code, named EPSS-2D, is able to simulate both sky
patches and all–sky maps taking short CPU process times in a common Work-
9
Simulations of clustered extragalactic point sources at Planck frequencies
station.
We want to stress again that we adopted a purely phenomenological
approach given that our main purpose was the denition of a fast and exible
tool for simulating 2D maps of EPS, under the most general assumptions on
source counts and on the angular correlation functions of point sources.
present outcomes are, obviously, model dependent.
to
take
into
account
all
current
data
on
source clustering at CMB frequencies.
number
The
On the other hand, they try
counts
of
EPS
as
well
as
on
The main results can be summarized as
follows:
a) we create 2D maps by rst making a Poisson density eld, corresponding
to
the
differential
subsequently,
we
counts,
modify
N (S)
,
–
in
of
the
the
specic
Fourier
space
EPS
–
eld according to some angular power spectrum,
population
this
“white
of
interest;
noise”
density
P (k)
, the most reliable one
for that specic source population; as a nal step, we distribute uxes on the
density eld map, under the condition that
n
uxes – taken from the differential
counts – fall in a pixel whose number of sources is
in detail in
source
n
.
This method, discussed
§
2, proves “safe” given that we are always able to recover the input
counts
and
the
input
P (k)
.
Moreover,
the
proposed
method
allows
us to estimate also the contribution of clustered EPS to the CMB bispectrum

(Argueso
et al., 2003).
b) By using the Toffolatti et al. (1998) cosmological evolution model for EPS
and by applying the
w(θ)
of Loan, Wall and Lahav (1997) we have estimated the
temperature angular power spectrum of clustered EPS at WMAP and Planck LFI
frequencies.
Our current results conrm that the extra power due to clustering
of EPS is always small in comparison with the Poisson term in all cases for
which no subtraction of bright sources is applied.
As extensively discussed in
the body of the paper, the present outcome is in full agreement with all the main
studies on the subject.
c) On the other hand, we nd that at frequencies
contribution of the clustering term,
σC
ν ≥ 150 − 200
GHz, the
, to the total confusion noise can be, very
probably, the dominant one, thus conrming previous theoretical predictions.
This result is mainly determined by the very steep slope of EPS counts at sub–
mm
wavelengths
combined
with
the
strong
clustering
signal
-
not
diluted
in
redshift - inferred for high redshift spheroidal galaxies and SCUBA sources.
Late-type dusty galaxies are found to be weakly clustered (Madgwick et al.,
2003) and, thus, their contribution to confusion noise is much lower and can be
neglected.
Acknowledgments
The authors wish to thank the Spanish MCYT for nancial support under
project ESP2002-04141-C03-01.
10
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