EXTRAGALACTIC POINT SOURCES
AND THE COSMIC MICROWAVE BACKGROUND
BISPECTRUM.

F. Argueso
1
2
1,
2,
J. Gonzalez-Nuevo
2,
L. Toffolatti
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
Most analysis of cosmic microwave background (CMB) temperature maps up to
date show that CMB anisotropies follow a Gaussian distribution.
On the other
hand, astrophysical foregrounds, which hamper the detection of the CMB angular
power spectrum, are not Gaussian-distributed on the sky.
give
a
sizeable
contribution
to
the
CMB
bispectrum.
Therefore, they should
We
have
calculated
the
contribution of uncorrelated extragalactic point sources to the CMB bispectrum
for
all
the
WMAP
and
Planck
channels.
Our
calculations,
based
on
current
cosmological evolution models for sources, are in good agreement with the point
source
bispectrum
Finally,
the
our
detected
results
bispectrum
due
show
to
at
that
41
at
clustered
GHz
from
Planck
sources
LFI
is
the
analysis
frequencies,
similar
to
of
i.e.
that
of
the
WMAP
ν ≤ 100
the
data.
GHz,
uncorrelated
case, whereas at higher frequencies the clustering term can greatly enhance the
normalization of the bispectrum.
Keywords:
cosmic microwave background, point sources, bispectrum
Introduction
The analysis of the rst year Wilkinson Microwave Anisotropy probe (WMAP)
data clearly show that temperature uctuations of the Cosmic Microwave Background (CMB) follow a Gaussian distribution [1] in agreement with the standard
ination paradigm.
gular bispectrum,
function,
Therefore, the third moment of this distribution and its an-
Cl1 l2 l3
, the harmonic transform of the three–point correlation
should be zero and the angular power spectrum,
statistical properties of CMB anisotropy.
Cl
,
specify all the
Anyway, signicant non-Gaussianity
can be introduced, for instance, by second order relativistic effects or by features in the scalar eld potential [2].
All these effects can be summarized by
the following expression for the curvature perturbations,
2
Φ(x) = ΦL (x) + fnl (Φ2L (x) − hΦ2L (x)i).
ΦL (x)
where
is
the
Gaussian
hi
coupling parameter and
of
fnl
part
of
the
perturbations,
fnl
(1)
is
the
means the statistical ensemble average.
non-linear
High values
could give rise to a bispectrum signal potentially detectable in current as
well as future all sky CMB maps, anyway
level according to [1].
−58 < fnl < 134
at
95%
condence
These limits were established by considering the 1-year
WMAP data bispectrum.
A great challenge to precise measurements of non–Gaussianity in the CMB
is set by astrophysical foregrounds which constitute an unavoidable limitation.
In
this
paper,
we
focus
on
extragalactic
point
sources
whose
contribution
to
CMB anisotropies has been analyzed in detail [3], whereas their contribution
to
the
by
the
of
CMB
the
bispectrum
analysis
bispectrum
the
above,
it
of
the
due
is
to
has
rst
been
year
poorly
extragalactic
clearly
of
studied
WMAP
great
up
to
observations,
sources
interest
has
to
been
perform
now.
the
claimed
a
WMAP
the
the
and
data
point
Planck
available
source
bispectrum
missions,
so
far.
using
for
the
model
Moreover,
we
whole
number
take
also
[1].
In
of
view
analysis
of
Therefore, we will
frequency
counts
into
recently,
detection
thorough
the contribution of point sources to the CMB bispectrum.
calculate
Very
rst
able
account
range
to
of
the
reproduce
the
effect
of
correlated positions of point sources in the sky for studying how much source
clustering can enhance the value of the angular bispectrum.
1.
The bispectrum due to unclustered point sources
General formalism
CMB temperature uctuations
Z
harmonics
alm =
where
Ω
∆T (n)/T
Ω
d2 n
are usually expanded into spherical
∆T (n) ∗
Ylm (n)
T
(2)
denotes the whole sky. The angular third moment of CMB temperature
uctuations is dened as
1 m2 m3
Blm
≡ hal1 m1 al2 m2 al3 m3 i
1 l2 l3
where the averaging is over the ensemble of realizations.
(3)
If the uctuations are
Gaussian distributed, the third moment is zero.
Due to the rotational invariance of the universe
µ
1 m2 m3
Blm
1 l2 l3
where
Cl1 l2 l3
=
l1
l2
l3
m1 m2 m3
¶
Cl1 l2 l3
(4)
is the angular bispectrum and the matrix is the Wigner-3j symbol.
3
Extragalactic point sources and the cosmic microwave background bispectrum.
Another
related
quantity
is
the
reduced
bispectrum
bl1 l2 l3
,
which
can
be
expressed easily in terms of the angular bispectrum
s
(2l1 + 1)(2l2 + 1)(2l3 + 1)
×
4π
Cl1 l2 l3 =
and that contains all the physical information in
µ
l1 l2 l3
0 0 0
1 m2 m3
Blm
1 l2 l3
¶
bl1 l2 l3
(5)
.
Having summarized the general formalism, we remind here the calculation
of the reduced bispectrum expected in the case of unclustered point sources,
i.e.
extragalactic point–like sources which follow a Poisson distribution in the
sky.
Under this assumption, the point source bispectrum
bps ≡ bsources
=
l1 l2 l3
i.e.,
bps
equal
to
the
skewness
of
the
sky
bps
can be expressed
h(T − hT i)3 i
hT i3
temperature
(6)
distribution.
Therefore,
=const at all scales l.
The reduced bispectrum,
bps
, can be then calculated as follows
bps = g 3 (x)
where
dn
dS
Z Slim
0
dS S 3
dn
dS
(7)
is the differential source count per unit solid angle,
S
the ux and
g
the conversion factor from uxes to temperatures
g(x) ≡ 2
where
x ≡ hν/kB T
.
(hc)2 (sinh(x/2))2
(kB T )3
x4
(8)
The integral in eq.(7) has to be computed up to the ux
detection limit foreseen for the experiment,
are contributing to the estimated
bps
Slim
, since only undetected sources
.
Extragalactic point source contribution
To perform the integral in eq.(7) we have used the differential counts corresponding to the cosmological evolution models for radio and far–IR selected
sources discussed in [3].
The prediction of this model are very close to cur-
rent data on source number counts coming from different experiments at low
frequencies, such as WMAP [4], CBI [5] or VSA [6].
On the other hand, a lot of new data on far–IR/sub–mm source counts have
piled up in the last few years, in particular by means of the SCUBA and MAMBO
surveys.
models
These
of
current
far–IR
data
selected
are
better
sources,
as
explained
the
ones
by
new
presented
physical
in
[7]
.
evolution
Therefore,
4
we have also taken into account the counts in [7] in order to calculate our rst
estimate of
bps
at high frequencies.
Anyway, the integral in eq.(7) results not
much affected by the adopted evolution model.
In
bps
Figure
1
we
present
our
current
estimates
of
the
reduced
bispectrum,
, due to Poisson distributed extragalactic sources at various frequencies in
the
range
30–850
detection.
GHz
and
applying
different
ux
Slim
limits,
,
for
source
The frequencies and ux detection limits has been chosen to directly
compare with those of the Planck satellite mission.
All our estimates have been
calculated by taking into account both the radio and far–IR source populations at
each frequency channel.
model
in
our
We have considered a standard
calculations.
The
values
of
bps
are
at
a
ΛCDM
cosmological
minimum
close
to
the
CMB intensity peak where bright sources reach a minimum number.
bps
It is also interesting to compare
pectrum
obtained
in
non-Gaussian
potential given by eq.(1).
[8].
estimates with the reduced CMB bis-
models
with
uctuations
They have shown that the equilateral bispectrum,
bl1 l2 l3
l ' 200
angular bispectrum
at multipole
.
of
the
ination
This quantity has also been carefully calculated in
when
l1 = l2 = l3
blll
– i.e.,
– multiplied by
l4
the reduced
exhibits a peak
The height of this peak is proportional to the factor
fnl
,
the coupling parameter.
bps
l ' 200
In Figure 1 we compare our estimates of
lateral bispectrum,
blll
with the primordial CMB equi-
, at its peak value (
) by adopting
blll
Looking at Figure 1, the peak value of the primordial
bps
in
a
frequency
window
around
the
CMB
window sets larger by increasing the value of
detection limit,
fnl < 10
(
Slim
.
intensity
fnl
peak:
bps
(9.5±4.4)×10−5 µK 3
are in good agreement with this detection:
GHz differential counts up to
, by adopting
Slim = 0.75
Jy, we nd
' 0.7
Slim ' 1.0
we have to apply a correction factor of
the
more
realistic
bps = 22 × 10−5 µK 3
.
value
Our current ndings
bps = 14 × 10−5 µK 3
1σ
2.
.
level, whereas
for obtaining their best-t value.
Jy
for
source
detection
we
On the other hand, by integrating the counts at 61
Slim = 1.0
' 0.8
GHz (WMAP V band) up to
showing a reduced scatter
of
Slim ' 0.75
in fact, by directly integrating the 41
These results are compatible with the detected value at the
use
the
at microwave frequencies has been
Jy for source detection in the 41 GHz WMAP channel.
we
obviously,
and by reducing the source
In principle, very low values of the coupling parameter
Very recently [1], the rst detection of
If
.
) could be detectable by the Planck experiment.
claimed: the reported value is
nd
fnl = 100, 10, 1
appears greater than
Jy we obtain
bps = 1.4 × 10−5 µK 3
with the best-t value at this frequency.
The bispectrum due to clustered point sources
The clustering contribution to CMB uctuations due to extragalactic sources
is generally small in comparison with the Poisson one [3].
On the other hand,
5
Extragalactic point sources and the cosmic microwave background bispectrum.
if clustering of sources at high redshift is very strong it can give rise to a power
spectrum stronger than the Poisson one.
Anyway, in view of extracting the most
precise information from current and future CMB data, i.e.
pursuing precision
cosmology, it is also important to assess to which extent the clustering signal
can reduce the detectability of the primordial CMB bispectrum.
the clustering contribution to
bps
We estimate
as previously done for the Poisson case.
We
focus on WMAP and Planck LFI channels, at which extragalactic radio sources,
displaying a “at” energy spectrum, dominate the bright counts.
bps
a rst estimate of
We
briey
remind
here
the
adopted
procedure.
simulations in 2D–at patches of the sky of
2
deg
and
with
a
We also give
at 545 GHz (a Planck HFI channel).
pixel
size
of
∼
1.5
arcmin
We
have
12◦ .8 × 12◦ .8
for
Planck
carried
out
100
25◦ .6 × 25◦ .6
∼
nside = 512
and
and
of
WMAP, since maps are created in the HEALPix format with
6
arcmin
for
in
this latter case.
As discussed in more detail elsewhere [9], we rst distribute point sources at
random in the sky with their total number xed by the integral counts,
of the model [3] at each given frequency.
N (> S)
,
We then calculate the Fourier trans-
form of the density contrast map, obtaining a “white noise” power spectrum,
thus
normalized
next
step
is
to
to
the
modify
total
this
number
the angular correlation function,
population.
We
adopt
of
normalized
here
the
sources
foreseen
spectrum
w(θ)
w(θ)
S > 50
by
the
by
the
Fourier
model.
The
transform
of
, corresponding to the appropriate source
measured
source population of bright sources (
at
4.85
GHz
[10],
since
the
mJy) which has been sampled at
this frequency is representative also of bright sources seen at higher frequencies.
In this way the source density is modied by the adopted correlation function
whereas the normalization to the total effective number of sources predicted by
the
model
remains
xed.
By
the
2D
Fourier
antitransform
we
get
again
the
map in the real space (the sky patch); we then distribute the uxes at random
according to the differential source counts in [3] and obtain the simulated map
in
terms
of
∆T /T
.
Finally,
we
calculate
the
bispectrum
in
the
2D
at-sky
approximation.
We have calculated
bps
for all the WMAP channels and the Planck channels
from 30 to 100 GHz with different source detection limits.
we can conclude:
sources, e.g.
From the results
If we perform simulations with a correlated distribution of
by the
w(θ)
as in the Poissonian case.
of [10], the value of
bps
remains practically the same
This also happens if we choose another angular corre-
lation function among realistic ones which could be suitable for the underlying
source population.
As previously discussed, this is due to the very broad red-
shift distribution of sources contributing to the bright counts in this frequency
range.
On the other hand, to give a preliminary estimate of
bps
at Planck HFI fre-
quencies, we have also calculated the reduced equilateral bispectrum of a non
6
Poisson distribution of sources at 545 GHz. At this frequency the source populations dominating the number counts are dust–enshrouded elliptical galaxies and
spheroids at substantial redshift and, consequently, the clustering term should
not be negligible, given that the dilution of the signal is less effective than that
of sources showing a broad redshift distribution.
The results are obtained, as
before, by distributing sources in the sky using a Poisson distribution which is
then modied by the angular correlation functions appropiate for each source
populations.
As for sources whose emission is dominated by cold dust, the lack of direct
data at 545 GHz forced us to rely on SCUBA as well as on optical and near–
IR surveys.
By distinguishing the relevant populations in starburts/spirals at
low redshift and ellipticals and spheroids at intermediate to high redshift, we
then applied to them the
w(θ)
of [11] and [12] ,
respectively.
This is a very
preliminary estimate and, naturally, it will be possible to improve it in the future.
It is clear that at frequencies
value of
bps
≥ 300
GHz source clustering greatly enhances the
and, particularly, at multipoles
l≤
100.
More details can be found
in [13].
3.
Conclusions
The primordial bispectrum is a telling quantity of the non-Gaussianity level
of
the
CMB
and
its
the early Universe.
detection
should
be
of
great
importance
for
theories
of
Undetected extragalactic sources, which are not Gaussian
distributed in the sky are surely giving rise to a bispectrum detectable by current
as well as future CMB experiments.
Assuming that the sources are Poisson distributed in the sky their reduced
bispectrum
is
easy
to
calculate
by
integrating
relevant source populations at a given frequency.
the
differential
counts
of
the
In this paper, we have carried
out a detailed analysis of the bispectrum due to extragalactic point sources in
all Planck and WMAP frequency channels.
We used the evolution model in [3]
for radio and far–IR selected sources as a template to estimate the
in the case of a Poisson distribution of point sources.
bps
foreseen
Our main results can be
summarized as follows:
a) the bispectrum detected by [1] in the WMAP 1 year sky maps ( Q and
V bands) is compatible at the
by
the
band
number
can
be
counts
explained
in
1σ
[3].
by
Slim = 0.75Jy
(' 0.8
bps
mismatch
measured
as in [1].
best-t
undetected
model predictions multiplied by
to
level with the predictions on
The
value
of
extragalactic
' 0.70
bps
bps
calculated
measured
sources
in
according
the
to
Q
the
if we integrate the number counts up
On the other hand, in the V band we nd a smaller
with current WMAP measurements.
Both uncertainties in the
and errors in the predictions could explain the detected offsets.
7
Extragalactic point sources and the cosmic microwave background bispectrum.
b) The bispectrum due to extragalactic sources depends on the source detection threshold,
Slim
, and, thus, on the efciency of the detection algorithm.
The primordial CMB bispectrum,
poles close to
l ∼ 200
blll
,
appears higher than
where it reaches its peak value.
inside which the peak value of
blll
bps
only at multi-
The frequency window
is detectable sets larger by increasing
by reducing the source detection limit,
Slim
ciency of the sources detection algorithm.
, i.e.
fnl
and
it strongly depends on the ef-
In principle, very low values of the
fnl < 10
coupling parameter (
) could be detectable by the Planck experiment.
c) For estimating to which extent the clustering of sources can affect the value
of
of
bps
1.5
does
we have carried out at 2D simulations in sky patches with pixel sizes
(Planck) and
not
modify
6
the
hand, at frequencies
(WMAP) arcmin.
bispectrum
at
.
We conclude that source clustering
frequencies
≤ 100
GHz
.
On
the
other
ν≥
300 GHz, the number of sources in the sky at a given
ux limit is greatly enhanced by the rising energy spectra due to the emission
of the cold dust components.
Correspondingly, if sources do cluster down to
very low ux limits, the clustering term can dominate the Poisson one and
values
result
greatly
enhanced.
Therefore,
it
is
surely
of
great
bps
astrophysical
interest to study the clustering properties as well as the cosmological evolution
of extragalactic sources in this frequency range for obtaining a better assessment
of their contribution to the CMB angular power spectrum and bispectrum.
References
[1]
Komatsu, E., et al. 2003, ApJS,148,119
[2]
Bartolo N., Komatsu E., Matarrese S. and Riotto A., Physics Reports, 2004 in press
[3]

Toffolatti, L., Argueso
Gomez, F., de Zotti, G., Mazzei, P., Franceschini, A., Danese, L. and
Burigana, C., 1998, MNRAS, 297, 117
[4]
Bennett, C. L. et al. 2003b, ApJS, 148, 97
[5]
Mason, B. S. et al. 2003, Apj, 591, 540
[6]
Dickinson, C. et al. 2004, MNRAS, 353, 732
[7]
Granato, G. L., Silva, L., Monaco, P., Panuzzo, P., Salucci, P., De Zotti, G., and Danese, L.,
2001, MNRAS, 324, 757
[8]
Komatsu, E., and Spergel, D. N., 2001, Phys. Rev. D, 63, 063002
[9]

Gonzalez-Nuevo,
J., Toffolatti, L.,and Argueso,
F., 2004, Apj, accepted (astro-ph/0405553)
[10]
Loan, A. J., Wall, J. V. and Lahav, O., 1997, MNRAS, 286, 994
[11]
Tegmark M. et al., 2002, ApJ, 571, 191
[12]
Magliocchetti, M., Moscardini, L., Panuzzo, P., Granato, G.L., De Zotti, G., and Danese,
L., 2001, MNRAS, 325, 1553
[13]

Argueso,
F., Gonzalez-Nuevo,
J., and Toffolatti, L., 2003, 598, 86
8
−16
Slim=0.01 Jy
Slim=0.1 Jy
Slim=1 Jy
Slim=2 Jy
fnl=100
fnl=10
fnl=1
−18
log10(bps)
−20
−22
−24
−26
−28
−30
0
100
200
300
400
500
600
700
800
900
frequency (GHz)
Figure 1.
Reduced equilateral bispectrum,
bps
, due to Poisson distributed point sources for all
the Planck frequencies and for several ux detection limits,
the primordial CMB bispectrum (at
l ∼ 200
Slim
, for sources.
The peak value of
) generated by a quadratic potential is also plotted
for different values of the coupling parameter,
fnl
.