Draft version October 1, 2020

Typeset using LATEX twocolumn style in AASTeX63

A Test of the Cosmological Principle with Quasars

Nathan J. Secrest, Sebastian von Hausegger,2, 3, 4 Mohamed Rameez,5 Roya Mohayaee,3 Subir Sarkar,4 and
1

Jacques Colin3
1 U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA
arXiv:2009.14826v1 [astro-ph.CO] 30 Sep 2020

2 INRIA, 615 Rue du Jardin-Botanique, 54600 Nancy Grand-Est, France

3 Sorbonne Université, UPMC, CNRS, Institut d’Astrophysique de Paris, 98bis Bld Arago, Paris 75014, France
4 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom
5 Dept. of High Energy Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

ABSTRACT
We study the large-scale anisotropy of the Universe by measuring the dipole in the angular distribu-
tion of a flux-limited, all-sky sample of 1.3 million quasars observed by the Wide-field Infrared Survey
Explorer (WISE). This sample is derived from the new CatWISE2020 catalog, which contains deep
photometric measurements at 3.4 and 4.6 µm from the cryogenic, post-cryogenic, and reactivation
phases of the WISE mission. While the direction of the dipole in the quasar sky is similar to that of
the cosmic microwave background (CMB), its amplitude is over twice as large, rejecting the canonical,
exclusively kinematic interpretation of the CMB dipole with a p-value of 10−4 (3.9σ), the highest sig-
nificance achieved to date in such studies. Our results are in conflict with the cosmological principle,
a foundational assumption of the concordance ΛCDM model.

Keywords: cosmology: large-scale structure of universe — cosmology: cosmic background radiation —

cosmology: observations — quasars: general — galaxies: active

1. INTRODUCTION of matter on local scales, originally dubbed the “Great

The standard Friedmann-Lemaı̂tre-Robertson-Walker Attractor” (see, e.g., Dressler 1991).
(FLRW) cosmology is based on the cosmological princi- A consistency check would be to measure the concomi-
ple, which posits that the universe is homogeneous and tant effects on higher multipoles of the CMB anisotropy
isotropic on large scales. This assumption is supported (Challinor & van Leeuwen 2002); however, even the pre-
by the smoothness of the CMB, which has temperature cise measurements of these by Planck allow up to 40%
fluctuations of only ∼ 1 part in 100,000 on small angu- of the observed dipole to be due to effects other than the
lar scales. These higher multipoles of the CMB angular Solar System’s motion (see discussion in Schwarz et al.
power spectrum are attributed to Gaussian density fluc- 2016). According to galaxy counts in large-scale sur-
tuations created in the early universe with a nearly scale- veys the universe is sensibly homogeneous when aver-
invariant spectrum, which have grown through gravita- aged over scales larger than & 100 Mpc, as is indeed ex-
tional instability to create the large-scale structure in pected from considerations of structure formation in the
the present universe. The dipole anisotropy of the CMB concordance ΛCDM model. Hence the reference frame
is however much larger, being ∼ 1 part in 1000 as ob- of matter at still greater distances should converge to
served in the heliocentric rest frame. This is interpreted that of the CMB; i.e. the dipole in the distribution of
as due to our motion with respect to the rest frame in cosmologically distant sources, induced by our motion
which the CMB is isotropic, and is thus called the kine- via special relativistic aberration and Doppler shifting
matic dipole. This motion is usually attributed to the effects, should align both in direction and in amplitude
gravitational effect of the inhomogeneous distribution with the CMB dipole. Independent measurements of the
distant matter dipole are therefore an important test of
the cosmological principle, and equivalently of the stan-
Corresponding author: Nathan J. Secrest dard model of cosmology.
nathan.j.secrest.civ@mail.mil Ellis & Baldwin (1984) proposed that such a test be
done using counts of radio sources. These are typi-
2 Secrest et al.

cally active galactic nuclei (AGNs) at moderate redshift 2. QUASAR SAMPLE

(z ∼ 1), so locally clustered sources (z < 0.1), which Because of the unique power of mid-infrared photom-
can introduce an additional dipole in the distribution etry to pick out AGNs, WISE may be used to create re-
of matter (e.g., Tiwari & Nusser 2016), are not a sig- liable AGN/quasar catalogs based on mid-infrared color
nificant contaminant. Consider a population of sources alone (e.g., Secrest et al. 2015). We require an AGN
with power-law spectra Sν ∝ ν −α , and integral source sample optimized for cosmological studies, so the objects
counts per unit solid angle dN/dΩ (> Sν ) ∝ Sν−x , above should preferably be quasars: AGN-dominated and at
some limiting flux density Sν . If we are moving with ve- moderate or high-redshift (z & 0.1; cf., Tiwari & Nusser
locity v (≪ c) with respect to the frame in which these 2016). The sample should cover as much of the celestial
sources are isotropically distributed, then being “tilted sphere as is possible to minimize the impact of missing
observers” we should see a dipole anisotropy of ampli- (or masked) regions, and be as deep as possible to con-
tude (Ellis & Baldwin 1984): tain the largest number of objects and thus have the
greatest statistical power.
D = [2 + x(1 + α)]v/c. (1) We created a custom quasar sample from the new
The advent of the 1.4 GHz NRAO VLA Sky Sur- CatWISE2020 data release (Eisenhardt et al. 2020),
vey (NVSS; Condon et al. 1998), which contains ∼ which contains sources from the combined 4-band cryo,
1.8 million sources, enabled the first estimates of the 3-band cryo, post-cryo NEOWISE, and reactivation
matter dipole anisotropy (Blake & Wall 2002; Singal NEOWISE-R data. The CatWISE2020 catalog is
2011; Gibelyou & Huterer 2012; Tiwari et al. 2015; 0.71 mag and 0.45 mag deeper in W1 and W2 than the
Tiwari & Jain 2015; Tiwari & Nusser 2016). To im- previous AllWISE catalog. We select all sources in the
prove sky coverage, data was added from other radio CatWISE2020 catalog with valid measurements in W1
surveys, e.g. the 325 MHz Westerbork Northern Sky Sur- and W2, which are the most sensitive to AGN emis-
vey (WENSS; Rengelink et al. 1997; Rubart & Schwarz sion (e.g., Stern et al. 2012). We cut out any sources
2013), the 843 MHz Sydney University Molonglo Sky with possible saturation, as well as sources flagged as
Survey (SUMMS; Mauch et al. 2003; Colin et al. 2017; suffering from possible contaminants. To select AGNs,
Tiwari & Aluri 2019) and the 150 MHz TIFR GMRT we impose the color cut W1 − W2 ≥ 0.8 (Stern et al.
Sky Survey (TGSS; Bengaly et al. 2018; Singal 2019). 2012), which ensures that the spectral energy distribu-
However as was first noted by Singal (2011), while the tion is AGN-dominated, following a power-law distribu-
direction of the matter dipole is consistent with that of tion (Sν ∝ ν −α ) that is insensitive to heavy dust red-
the CMB, its amplitude is several times larger. dening at shorter wavelengths. This yields a raw sample
In this Letter, we report the first independent mea- of 174,701,084 objects.
surement of the dipole in the angular distribution of dis- We then remove low-redshift AGNs by excluding
tant quasars using mid-infrared data from the Wide-field sources in the 2MASS extended source catalog (XSC;
Infrared Survey Explorer (WISE; Wright et al. 2010), Jarrett et al. 2000), which contains nearly all galaxies
which surveyed the sky at 3.4 µm, 4.6 µm, 12µm, and not directly behind the Galactic plane out to z ∼ 0.1
22 µm (W1, W2, W3, and W4). This provides a mea- (Jarrett 2004). We made the sample uniform (equal
surement of the dipole that is independent of the radio depth) across the sky by addressing several known
survey-based results, as WISE is a space mission with causes of non-uniformity in the WISE data. The first
its own unique scanning pattern, not constrained by is a decrement of high quality measurements along the
the same observational systematics that affect ground- Galactic plane where source confusion is prevalent. This
based surveys, such as declination limits or atmo- can be mitigated by masking the sky below some Galac-
spheric effects. While WISE, along with 2MASS, has tic latitude; we find |b| > 30◦ to be effective in com-
been used before to set useful constraints on the mat- pletely removing non-uniformity because of the Galaxy.
ter dipole (Gibelyou & Huterer 2012; Yoon et al. 2014; The second is poor-quality photometry near clumpy and
Alonso et al. 2015; Bengaly et al. 2017; Rameez et al. resolved nebulae both in our Galaxy (e.g., planetary
2018), these studies were of relatively nearby galaxies nebulae) and in nearby galaxies such as the Magellanic
(z ∼ 0.05 − 0.1) where contamination from local sources Clouds and Andromeda. We remove these by masking
can be significant and has to be carefully accounted for. out 6 times the 20 mag arcsec−2 isophotal radii from the
In Section 2, we detail the quasar sample that we use, 2MASS Large Galaxy Atlas (LGA; Jarrett et al. 2003).
and we introduce our methodology in Section 3. Our The third is a decrement of sources, and the presence of
results are presented in Section 4, and we discuss their image artifacts, near bright stars, caused by density sup-
significance for cosmology in Section 5.
 The Quasar Dipole 3

pression in their vicinity.1 We find that circular masks

with 2MASS K band-dependent radii log10 (r/deg) =
−0.134K − 0.471 effectively remove these. In all, we
masked 265 sky regions, plus the Galactic plane. To
avoid any directional source count bias we mirror the
masks by 180◦ on the celestial sphere.
We calculate spectral indices α of our sources in the
W1 band by obtaining power-law fits of the form Sν =
kν −α , where k is the normalization. We produced a
lookup table to determine α based on W1 − W2, by cal-
culating synthetic AB magnitudes following Equation 2
of Bessell & Murphy (2012). The WISE magnitudes are
on the Vega magnitude system, so we convert from the
AB system using the offsets mAB −mVega = 2.673, 3.313
for W1 and W2, respectively. These WISE offsets corre- Figure 1. Distribution of flux densities Sν (∝ ν −α ) and
spond to the constant of −48.60 associated with the def- spectral indices α (W1 band) in the CatWISE AGN sample.
inition of the synthetic AB magnitude. The normalisa-
tion k is calculated by inverting the equation for the syn-
thetic magnitude and using the observed W1 AB mag-
nitude. Finally, we calculate the isophotal frequency, at
which the flux density Sν equals its mean value within
the passband, using Equation A19 in Bessell & Murphy
(2012). As our sample was constructed with the cut
W1 − W2 ≥ 0.8, the distribution peaks at α ∼ 1 and
extends to steeper slopes, with a mean value of 1.26.
Distributions of spectral indices and fluxes for our final
sample of sources are shown in Figure 1. The corre-
sponding mean isophotal frequency is 8.922 × 1013 Hz,
with a dispersion of 0.19%. We select a magnitude cut
of 9 > W1 > 16.4 (Vega), equivalent to a flux density
cut of 77.77 > Sν > 0.09 mJy, to fix the over-density
of fainter sources along overlaps in the WISE scanning Figure 2. Sky map of the CatWISE AGN sample, in Galac-
tic coordinates.
pattern, most prevalent at the ecliptic poles where they
converge. After removing low-z AGNs, applying the sky
2016), yielding even deeper spectral coverage. There are
masks, and making the flux density cut, our final sample
14,387 CatWISE AGNs in this region. For photometric
has 1,314,428 AGNs, which we show in Figure 2.
information, we cross-match these with the Dark En-
To estimate the distribution of AGN redshifts, we se-
ergy Survey (DES), Data Release 1 (Abbott et al. 2018),
lect those within SDSS Stripe 82, a 275 deg2 region of
which achieved an i-band depth of 23.44 mag (AB). Us-
the sky scanned repeatedly by the SDSS, thus achieving
ing a 10′′ match for completeness, we find counterparts
an increase of depth of ∼ 2 mag (Annis et al. 2014). In
for 14,343 (99.7%) of the CatWISE AGNs. Matching
the specObj table for SDSS DR16,2 Stripe 82 contains
the DES counterpart coordinates onto specObj table to
∼ 4.4 times more objects with spectroscopic r-band
within 1′′ for fiber coverage, we find 8612 matches (60%).
magnitudes fainter than 20 (AB) than a comparable sky
The unmatched objects are 0.3 mag fainter in W2 than
region in the SDSS main footprint. We use a sub-region
the matched objects on average, suggesting that they are
of Stripe 82 between −42◦ < R.A. < 45◦ , which lies
slightly less luminous or slightly more distant (or both).
outside the |b| < 30◦ Galactic plane mask we employ,
However, their mean r − W2 value, a measure of AGN
and which was observed by the Extended Baryon Os-
obscuration level (e.g., Yan et al. 2013), is ∼ 1.8 mag
cillation Spectroscopic Survey (eBOSS; Dawson et al.
redder than the mean of the matched sample, implying
that the unmatched objects are simply too faint at visual
1 http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec6 wavelengths
2.html#brt stars for SDSS. Indeed, while about one-third
2 https://www.sdss.org/dr16/spectro/spectro access of the full DES-matched sample has r − W2 > 6 mag
(Vega), in line with expectations from the literature for
 4 Secrest et al.

0.8 h  i2
~ q · r̂p
X np − n̄ 1 + D
0.7   (3)
0.6
~ q · r̂p
n̄ 1 + D
p

0.5 where np denotes the number of sources in sky pixel p

PDF

0.4 (r̂p being the unit vector to the pixel) and the sum is
0.3
to be taken over all unmasked pixels (in which n is the
average number of sources). Due to significantly higher
0.2
computational expense for the quadratic estimator, we
0.1 run simulations only with the linear estimator.
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3.2. Mock data and statistical significance
redshift
We generate mock samples of Ninit vectors drawn from
Figure 3. Redshift distribution (normalized as a probability a statistically isotropic distribution, whose directions are
density function) of the CatWISE AGN sample.
subsequently modified by special relativistic aberration
according to an observer boosted with velocity β. ~ Each
the prevalence of type-2 AGNs (Yan et al. 2013), 77% of sample is then masked with the same mask that was ap-
the unmatched sample have r − W2 > 6. This indicates plied to the data (Figure 2). In order to respect the ex-
that the objects in our sample without SDSS spectra are act distribution of flux values and spectral indices in the
predominantly type-2 systems, an effect of the orienta- data, we assign to each simulated source a flux density
tion of the AGN with respect to the line of sight, and so Sν and a spectral index α drawn at random from their
the matched objects may be used to estimate the distri- empirical distributions before applying the flux density
bution of redshifts for the full sample. We find a mean cut (Figure 1). The sampled fluxes are now modulated
redshift of 1.2, with 99% having z > 0.1, i.e. the Cat- depending on source position, velocity β, ~ and α. Lastly,
WISE AGN sample is not contaminated by low-redshift only sources with Sν > Sν,cut are retained, and the num-
AGNs. The redshift distribution of our sample is shown ber of remaining sources is finally reduced to that of the
in Figure 3. true sample, N , through random selection.
Under the null hypothesis that the measured dipole D ~l
3. METHOD
is a consequence of our motion with respect to a frame
3.1. Dipole Estimator shared by both quasars and the CMB, we generate a
We determine the dipole of our sample with the 3- set of mock skies according to the above recipe. For
dimensional linear estimator: each random choice we record D ~ sim , and correct for its
l
N directional bias using Equations A3 and A4. The frac-
~l = 3
X
D r̂i , (2) tion of mock skies with amplitude |D ~ l | larger than our
N i=1
empirical sample, and with angular distance from the
where r̂i is the unit vector pointing to source i, and CMB dipole closer than our sample, gives the p-value
N is the sample size. This estimator simply calcu- with which the null hypothesis is rejected. Note that
lates the mean resultant length and direction of the the effect on our results of the distributions of flux and
N unit vectors and is agnostic with regard to the spectral index (Figure 1) is automatically included via
true underlying signal (e.g., Fisher et al. 1987), as op- the bootstrap approach employed for our simulations.
posed to other estimators (e.g., Blake & Wall 2002;
Bengaly et al. 2019) which explicitly seek a dipolar pat- 4. RESULTS
tern. However, if the signal has a dipolar form then Our sample of 1,314,428 quasars exhibits a dipole
Equation 2 generally has a bias in both amplitude and with amplitude: Dl = 0.0173. Correcting for the di-
direction (Rubart & Schwarz 2013) induced by Poisson rectional bias induced by the mask employed, we find
noise and masking. We account for amplitude bias in that it points in the direction: (l, b) = (234.◦ 1, 29.◦ 2).
our results as well as in the estimates of their signif- This is 29.◦ 8 from the direction of the CMB dipole
icance using Monte Carlo methods, correcting for di- (l, b = 264.◦ 021, 48.◦253; Planck Collaboration et al.
rectional bias as discussed in Appendix A. We further 2018). However, when the expected dipole is simu-
confirm our results by employing the quadratic estima- lated assuming the kinematic interpretation of the CMB
~ q which does not suffer from bias and is evaluated
tor D dipole, only 4 out of 40,000 such simulations give D ~ sim
l
by minimising the quantity (e.g., Bengaly et al. 2019): with an amplitude larger than the observed value (left
 The Quasar Dipole 5

Figure 4. Left panel: Observed dipole amplitude Dl (solid vertical line) in the CatWISE AGN sample, versus the expectation
assuming the kinematic interpretation of the CMB dipole; the distribution of Dlsim from the simulations (Section 3.2) is shown
along with its median value (dashed vertical line). Right panel: Dipole direction of the CatWISE AGN sample in Galactic
~ l (circle) and the unbiased quadratic estimator D
coordinates using the bias-corrected linear estimator D ~ q (triangle); the shaded
area indicates the model-dependent 95% confidence level simulated using the velocity from the quadratic estimator.

panel, Figure 4) and within 29.◦ 8 of the CMB dipole di- son to suspect that the dipole we measure in the Cat-
rection as for our sample. We can therefore reject the WISE AGN catalog is an artifact of the survey.
null hypothesis with a p-value of 10−4 corresponding to After Ellis & Baldwin (1984) proposed this important
a significance of 3.9σ. observational test of the cosmological principle, agree-
If we assume that the anomalous quasar dipole is still ment was initially claimed between the dipole anisotropy
of kinematic origin, albeit with a velocity different from of the CMB and that of radio sources (Blake & Wall
that inferred from the CMB, we can estimate its di- 2002). If the rest frame of distant AGNs is indeed that
rectional uncertainty. To avoid bias, we first compute of the CMB, it would support the consensus that there
the dipole with the quadratic estimator Dq , which gives exists a cosmological standard of rest, related to quanti-
Dq = 0.01629 towards (l, b) = (234.◦ 0, 27.◦ 4). The corre- ties measured in our heliocentric frame via a local special
sponding velocity from Equation 1, with (median) α = relativistic boost. This underpins modern cosmology:
1.17 and index x = 1.7 at the flux density cut, is for example, the observed redshifts of Type Ia super-
v = 861 km s−1 . A set of 15,000 simulations with this novae are routinely transformed to the “CMB frame”.
input velocity is then performed to find the directional From this it is deduced that the Hubble expansion rate
uncertainty. The right panel of Figure 4 shows this as a is accelerating (isotropically), indicating dominance of a
patch around the (consistent) dipole direction obtained cosmological constant, and this has led to today’s con-
with both estimators. cordance ΛCDM model. If the purely kinematic inter-
pretation of the CMB dipole that underpins the above
procedure is in fact suspect, then so are the important
5. DISCUSSION
conclusions that follow from adopting it. In fact, as
The CatWISE AGN sample exhibits an anomalous observed in our heliocentric frame, the inferred acceler-
dipole, oriented similarly to the CMB dipole but over ation is essentially a dipole aligned approximately with
twice as large. Whereas a “clustering dipole” is ex- the local bulk flow of galaxies and towards the CMB
pected from correlations in the spatial distribution of dipole (Colin et al. 2019), so cannot be due to a cosmo-
the sources, this can be estimated knowing their auto- logical constant.
correlation function (or power spectrum) and distribu- If it is established that the distribution of distant mat-
tion in redshift (see Appendix B). It is smaller by a fac- ter in the large-scale universe does not share the same
tor of ∼ 60 than the dipole we observe in these higher reference frame as the CMB, then it will become im-
redshift quasars. perative to ask whether the differential expansion of
The unique statistical power of our study has allowed space produced by nearby nonlinear structures of voids
us to confirm the anomalously large matter dipole sug- and walls and filaments can indeed be reduced to just
gested in previous work, which used objects selected at a local boost (Wiltshire et al. 2013). Alternatively the
a different wavelength (radio), using surveys completely CMB dipole may need to be interpreted in terms of new
independent of WISE, viz. NVSS, WENNS, SUMMS, physics, e.g. as a remnant of the pre-inflationary uni-
and TGSS. The ecliptic scanning pattern of WISE has verse (Turner 1991). Gunn (1988) had noted that this
no relationship with the CMB dipole, so there is no rea-
6 Secrest et al.

issue is closely related to the bulk flow observed in the ACKNOWLEDGMENTS

local universe, which in fact extends out much further
than is expected in the concordance ΛCDM model (e.g., We thank Jean Souchay for helpful discussions.
Colin et al. 2011; Feindt et al. 2013). Further work is N.J.S., M.R. and S.S. gratefully acknowledge the hos-
needed to clarify these important issues. pitality of the Institut d’Astrophysique de Paris. S.v.H.
As Ellis & Baldwin (1984) emphasized, a serious dis- is supported by the EXPLORAGRAM Inria AeX grant
agreement between the standards of rest defined by dis- and by the Carlsberg Foundation with grant CF19 0456.
tant quasars and the CMB may require abandoning the
standard FLRW cosmology itself. The importance of
Facilities: WISE, Blanco, Sloan
the test we have carried out can thus not be overstated.

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8 Secrest et al.

APPENDIX

A. DIRECTIONAL BIAS OF THE LINEAR ESTIMATOR

Given a dipolar signal, Rubart & Schwarz (2013) demonstrated that the amplitude of the linear estimator (Equa-
tion 2) is biased, but not its direction. This can be seen by evaluating

~ l i = 3N
Z  
hD dΩ 1 + r̂ · d~ · r̂ ∝ d,
~ (A1)
4π
where the angular brackets denote the expectation value of the estimator given a dipolar probability distribution,
and d~ is the direction of the dipole. The amplitude bias stems from Poisson noise, always present in a sample of finite
size N . However, removing sources by masking alters the integral’s bounds and generally induces directional bias as
well. While the directional offset then caused by the first term in Equation A1 (the monopole) is alleviated by choosing
a mask that is symmetrical with respect to the observer, the contribution by the second term (the dipole) is not. This
effect was later worked out analytically for simple mask shapes by Rubart (2015), whose results we reproduce here for
reference.
The most prominent mask that we apply to our sample is the removal of the Galactic plane along lines of constant
latitude, b. Considering only this, the estimated direction is
"Z #
Z 2π π Z π/2+b
~ igal.mask = N 
~
 
~

hR dφ dθ sin(θ) 1 + r̂ · d · r̂ − dθ sin(θ) 1 + r̂ · d · r̂ (A2)
4π 0 0 π/2−b

~ The latitude, however, is affected

This shows that the estimated longitude equals the true longitude of the dipole d.
by a bias, B, that depends only on the latitude cut, b:3

tan(best. ) = B(b) · tan(btrue ), (A3)

1 − sin3 (b)
B(b) = 1 (A4)
1− 8 (9 sin(b) + sin(3b))
Note that the directional bias depends neither on the sample size N or dipole amplitude d, nor on the true dipole
~ as the
direction. It may also be of interest that the bias (Equation A4) is solely due to the dipolar contribution ∝ r̂ · d,
mask is chosen to be symmetric with respect to the observer. The true, unbiased dipole direction is therefore found
closer to the Galactic plane than is indicated by the uncorrected estimator, Equation 2.
The masks applied in this work carry small features in addition to the cut on Galactic latitude. It is not straight-
forward to analytically compute the bias arising from arbitrary mask shapes. However, by analysing simulations we
find the directional bias caused by these additional features to be negligible (< 1◦ ). For the results shown in Figure 4
we therefore show the dipole direction as corrected by Eqs. A3 and A4.

B. CLUSTERING DIPOLE WITHIN THE CONCORDANCE MODEL

The clustering dipole Dcls in a sample of objects as seen by a typical observer in the concordance ΛCDM cosmology
can be computed given the power spectrum P (k) of (dark) matter density perturbations (Gibelyou & Huterer 2012):
r
9
Dcls = C1 , (B5)
4π
where
Z ∞
22
Cl = b fl (k)2 P (k)k 2 dk. (B6)
π 0

3 The results are equivalent to those in Rubart (2015), but are

expressed here in terms of latitude rather than polar angle.
 The Quasar Dipole 9

Here b is the linear bias of the observed objects with respect to the dark matter and the filter function fl (k) is
Z ∞
fl (k) = jl (kr)f (r)dr, (B7)
0

where jl is the spherical Bessel function of order l and f (r) is the probability distribution for the comoving distance r
to a random object in the survey, given by

H(z) dN
f (r) = , (B8)
H0 r0 dz
R∞
normalised such that 0 f (r)dr = 1 and dN/dz is the redshift distribution of the observed objects. Employing
r0 = c/H0 = 3000h−1 Mpc, Planck 2015 cosmological parameters from Astropy, P (k) at z = 0 using camb (Lewis et al.
2000), and a cubic-spline fit to the redshift distributions shown in Figure 3 to determine dN/dz, we estimate Dcls to
be 0.00027 (taking b = 1) for the CatWISE AGN selection. Removing the 2MASS XSC sources reduces the clustering
dipole further to 0.00021, i.e. it is quite negligible compared to the observed quasar dipole.