###### Abstract

The dipole anisotropy in the Cosmic Microwave Background Radiation (CMBR) has
given a peculiar velocity vector 370 km s-1 along
$l=264^{\circ},b=48^{\circ}$. However, some other dipoles, for instance, from
the number counts, sky brightness or redshift distributions in large samples
of distant Active Galactic Nuclei (AGNs), have yielded values of the peculiar
velocity many times larger than that from the CMBR, though surprisingly, in
all cases the directions agreed with the CMBR dipole. Here we determine our
peculiar motion from a sample of 0.28 million AGNs, selected from the Mid
Infra Red Active Galactic Nuclei (MIRAGN) sample comprising more than a
million sources. From this, we find a peculiar velocity, which is more than
four times the CMBR value, although the direction seems to be within $\sim
2\sigma$ of the CMBR dipole. A genuine value of the solar peculiar velocity
should be the same irrespective of the data or the technique employed to
estimate it. Therefore, such discordant dipole amplitudes, might mean that the
explanation for these dipoles, including that of the CMBR, might in fact be
something else. But, the observed fact that the direction in all cases, is the
same, though obtained from completely independent surveys using different
instruments and techniques, by different sets of people employing different
computing routines, might nonetheless indicate that these dipoles are not
merely due to some systematics, otherwise why would they all be pointing along
the same direction. It might instead suggest a preferred direction in the
Universe, implying a genuine anisotropy, which would violate the Cosmological
Principle, the core of the modern cosmology.

###### keywords:

active galactic nuclei surveys; cosmic background radiation; large-scale
structure of universe; solar system peculiar motion; cosmological principle

xx x x Received: date; Accepted: date; Published: date Our Peculiar Motion
Inferred from Number Counts of Mid Infra Red AGNs and the Discordance Seen
with the Cosmological Principle Ashok K. Singal Correspondence:
<EMAIL_ADDRESS>Tel.: +91-9427-631-833

##  1 Introduction

According to the Cosmological Principle, the Universe, when seen on a
sufficiently large scale (beyond a few hundred Mpc), should appear isotropic,
without any preferred directions, to a co-moving observer in the expanding
Universe. Such an observer is at rest with respect to the Universe at large
and the angular distribution of sources in sky should appear statistically to
be similar in all directions. However, if relative to the co-moving
coordinates the observer has a motion, called a peculiar motion, then because
of the Doppler boosting as well as aberration effects, the observer will find
the sky brightness as well as the number counts of distant extragalactic
objects to manifest a dipole anisotropy, proportional to the peculiar velocity
of the observer. The Cosmic Microwave Background Radiation (CMBR), shows such
a dipole anisotropy that, when ascribed to the peculiar motion of the Solar
system, yields for the peculiar velocity a value $370$ km s-1 along right
ascension (RA) $=168^{\circ}$, declination (Dec) $=-7^{\circ}$, or in galactic
coordinates, $l=264^{\circ},b=48^{\circ}$ 1 ; 2 ; 3 .

Figure 1: The sky distribution of 0.28 million AGNs of our MIRAGN sample (see
text), in the Hammer-Aitoff equal-area projection map, plotted in equatorial
coordinates with right ascension (RA) from $-180^{\circ}$ to $180^{\circ}$ and
declination (Dec) from $-90^{\circ}$ to $90^{\circ}$, shows the sources to be
spread quite uniformly across the sky, except for the gap seen in the $\pm
15^{\circ}$ wide strip about the galactic plane, a zone of exclusion. The sky
position of the CMBR pole is indicated by $\odot$.

On the other hand, our peculiar velocity has also been determined in recent
years from the anisotropy observed in the sky distribution of large samples of
discrete radio sources. The NRAO VLA Sky Survey (NVSS), comprising 1.8 million
radio sources 12 , showed a statistically significant dipole asymmetry
corresponding to a velocity $\sim 4$ times the CMBR value 4 , something that
was not only unexpected, but appeared initially almost preposterous, however,
confirmed subsequently by many independent groups 5 ; 6 ; 7 ; 8 . Further, in
the TIFR GMRT Sky Survey (TGSS) 14 , comprising 0.62 million sources 13 , a
very significant ($>10\sigma$) dipole anisotropy, amounting to a velocity
$\sim 10$ times the CMBR value, was detected 8 ; 9 . However, equally
surprising, the direction of motion in both cases has turned out to be along
the CMBR dipole. Recently, a homogeneously selected DR12Q sample of 103245
distant quasars has shown a redshift dipole along the CMBR dipole direction,
implying a velocity $\sim 6.5$ times though in a direction directly opposite
to, but nonetheless parallel to, the CMBR dipole 10 . A more recent
determination of the peculiar motion from a sample of quasars derived from the
Wide-field Infrared Survey Explorer (WISE), has shown an amplitude over twice
as large the CMBR value 11 . Now a genuine solar peculiar velocity cannot vary
from one set of measurements to another and such discordant dipoles could
imply that the explanation for the genesis of these dipoles, including that of
the CMBR, might lie elsewhere. At the same time a common direction for all
these dipoles, determined from completely independent surveys by different
groups, using independent computational routines, does indicate that the
differences in the dipoles are not merely random fluctuations or due to some
systematics in data or procedures, otherwise their directions too would be
different. Instead, it might suggest a preferred direction in the Universe
implying a genuine anisotropy, which would violate the Cosmological Principle,
the core of the modern cosmology. Because of the huge impact on the
cosmological models any genuine variations in the dipole magnitudes may
impart, further independent determinations of the dipole vectors are
warranted.

Here we determine our peculiar motion from a sample of 0.28 million AGNs
(Figure 1), selected from the Mid Infra Red Active Galactic Nuclei (MIRAGN)
sample comprising more than a million sources 17 . A preliminary account of
these results was presented in proceedings of the 1st Electronic Conference on
Universe si21 , where the genesis of a dipole anisotropy in the number
density, owing to the observer’s peculiar motion, in an otherwise isotropic
distribution of sources in the sky, and how one could estimate the peculiar
velocity vector from the observed dipole anisotropy and the sample selected by
us for this purpose, were discussed. Here we pursue and explore these findings
in a greater detail, discussing any possible pitfalls in the sample selection
and the techniques employed. Further, we exploit another alternate method to
extract the dipole, and do a comparison of our derived results vis-à-vis the
peculiar velocity information previously available in the literature, and
point out their cosmological implications.

Figure 2: The sky distribution of 0.27 million MIRAGNs in the range
$15.5<W1<16$, in the Hammer-Aitoff equal-area projection map, plotted in
equatorial coordinates with RA from $-180^{\circ}$ to $180^{\circ}$ and Dec
from $-90^{\circ}$ to $90^{\circ}$. The sky position of the CMBR pole is
indicated by $\odot$. The sky distribution of these sources, which are at
fainter levels than those in Figure 1, shows a non-uniformity that is quite
evident in the figure and seems to the result of a deeper coverage in certain
regions of sky.

##  2 Dipole Vector Due to the Motion of the Observer

Due to the assumed isotropy of the Universe – à la cosmological principle – an
observer stationary with respect to the comoving coordinates of the cosmic
fluid, should find the average number densities of distant AGNs as well as
their flux densities to be distributed uniformly over the sky. However, an
observer moving with a velocity $v$ relative to the cosmic fluid, will find a
source along an angle $\theta$ with respect to the direction of motion, to
appear brighter due to Doppler boosting by a factor
$\propto\delta^{(1+\alpha)}$ 15 , where $\delta=1+(v/c)\cos\theta$ is the
Doppler factor and $\alpha$ is the spectral index defined by
$S\propto\nu^{-\alpha}$, with $S$ being the flux density of the source at the
frequency $\nu$ of the observing instrument. Here we have used the non-
relativistic formula for the Doppler factor as all previous observations
indicate that $v\ll c$. As the integral source counts of the extragalactic
source population usually follow a power law, $N(>S)\propto S^{-x}$ 15 , the
number of sources observed by a telescope of a given sensitivity will be
higher by a factor $\propto\delta^{x(1+\alpha)}$, due to the Doppler boosting
15 . Additionally, due to the aberration of light, the apparent position of a
source will shift toward the direction of motion by a value,
$(v/c)\sin\theta$, thereby changing the number density by another factor
$\propto\delta^{2}$ 15 . Thus, as a combined effect of Doppler boosting and
aberration, the observed number counts will vary with direction as
$\propto\delta^{2+x(1+\alpha)}$, which, for $v\ll c$, can be expressed as a
dipole anisotropy, $1+{\cal D}\cos\theta$ 4 ; 15 ; 16 , with an amplitude

${\cal D}=\left[2+{x(1+\alpha)}\right]\frac{v}{c}\;.$ (1)

Let $\bf{\hat{r}}_{i}$ be the position vector of $i^{th}$ source, then a
stationary observer, due to the assumed isotropy of the Universe, should find
$\Sigma\bf{\hat{r}}_{i}=0$. However, for a moving observer,
$\Sigma\bf{\hat{r}}_{i}$ will yield a net vector along the direction of motion
16 . Then the peculiar speed $v$ of the observer could be obtained from 4 ; 9

${\cal
D}=\frac{3}{2}\frac{\Sigma\cos\theta_{i}}{\Sigma|\cos\theta_{i}|}=\frac{3\Sigma\cos\theta_{i}}{N}\>,$
(2)

where $\theta_{i}$ is the polar angle of the $i^{th}$ source with respect to
the dipole direction of motion 16  and $N$ is the total number of sources in
the sample.

Thus, exploiting the angular positions of extragalactic sources in a large
survey that covers the whole sky and is complete in the sense that it
comprises all sources above a certain flux-density limit, we can determine our
peculiar motion. It should be noted that exclusions of strips like the
galactic plane, $|b|<15^{\circ}$ (Figure 1), which affect the forward and
backward measurements identically, do not have systematic effects on the
determined direction of the peculiar motion 4 ; 15 .

Figure 3: The sky distribution of 0.32 million MIRAGNs in the range
$16<W1<17$, in the Hammer-Aitoff equal-area projection map, plotted in
equatorial coordinates with RA from $-180^{\circ}$ to $180^{\circ}$ and Dec
from $-90^{\circ}$ to $90^{\circ}$. The sky position of the CMBR pole is
indicated by $\odot$. The non-uniformity in the sky distribution of these
sources, which are at still fainter levels than those in Figure 2, is even
higher, and seems to arise from a deeper coverage in certain regions of sky.

##  3 Our Sample of MIRAGNs

The sample of AGNs used in this study is selected from a publicly made
available larger all-sky sample of 1.4 million active galactic nuclei (AGNs)
17 , in turn derived from the Wide-field Infrared Survey Explorer final
catalog release (AllWISE), that incorporates data from the WISE Full
Cryogenic, 3-Band Cryo, and NEOWISE Post-Cryo survey 18 ; 19 . The WISE survey
is an all-sky mid-IR survey at 3.4, 4.6, 12, and 22 $\mu$m (W1, W2, W3, and
W4) with angular resolutions 6.1, 6.4, 6.5 and 12 arcsec, respectively.
AllWISE comprises data for almost 748 million objects, out of these about 1.4
million objects met a two-color infrared photometric selection criteria for
AGNs, that formed the original MIRAGN sample 17 .

For our purpose we have restricted the MIRAGN sample to an upper limit of
magnitude, W1$<15.0$, mainly because of a differential increase in the number
density for weaker sources in various regions of the sky, due to deeper WISE
coverage. However, due to the completeness of the basic survey at strong
source levels, the number density distribution in the sky at low infrared
magnitudes remains unaffected as a deeper coverage adds only fainter sources,
which are at higher infrared magnitudes. From a detailed examination of the
original MIRAGN sample data in small-range magnitude slices at different W1
levels, we find that from W1$\approx 15.5$ onward, there is a non-uniform
distribution in sky that increases rapidly at higher magnitudes, i. e., for
weaker sources. Figures (2) and (3)) show the increasing non-uniformity for
magnitude levels W1$>15.5$. Accordingly, we have chosen W1=15.0 to be our
upper magnitude limit. On the lower side, we have restricted our sample to
W1$>12.0$. This is only to minimize the effects of the local bulk flows which
will affect sources at low redshifts, $z<0.05$, corresponding to W1<12\. In
any case, the number of sources for W1$<12$ is relatively very small and their
inclusion or exclusion hardly affects the results. Further, we have also
excluded all sources in the galactic plane with $|b|<15^{\circ}$ to avoid
contamination by galactic sources 17 . Our final sample then comprises 279139
AGNs. Figures (1) shows the sky distribution of sources in our sample. The
distribution seems to be quite uniform across the sky, except for the gap seen
in the $\pm 15^{\circ}$ wide strip about the galactic plane.

Figure 4: A plot of the integrated source counts N(<W1) against W1, for our
MIRAGN sample, showing the power law behaviour of the source counts.

##  4 Results

We determine the dipole using two alternate methods from the same data set of
angular sky positions of MIRAGNs in our sample. In the computation of the
dipole, the weight that a source gets in the two methods is different,
depending upon its angular position in the sky.

###  4.1 Dipole Vector Determined Directly from the Sum of Position Vectors of
AGNS

Figure 1 shows the sky distribution of all 0.28 million MIRAGN AGNs in our
sample, in the Hammer-Aitoff equal-area projection map, plotted in equatorial
coordinates. The source distribution looks quite uniform over the sky, except
for the gap in the galactic plane band, where we have removed a $\pm
15^{\circ}$ band about the galactic plane. As mentioned earlier, such
exclusions, which affect the forward and backward measurements identically, do
not have systematic effects on the results, as far as the direction of the
dipole is concerned. However, the estimated dipole amplitude $\cal D$ might be
affected by the $\pm 15^{\circ}$ gap in the sky coverage about the galactic
plane. Let $k_{1}$ be the factor, of the order of unity, to be determined
numerically, by which the dipole amplitude gets overestimated due to the
presence of gaps in the sky coverage, and with which $\cal D$ should be
divided while computing $v$.

Before proceeding with the actual source sample we made Monte-Carlo
simulations with source distributions similar to that in our sample. In each
simulation, for the magnitude (W1) distributions we took the actual values as
in our sample, however, the sky position was allotted randomly for each one of
the 0.28 million sources. On this were superimposed Doppler boosting and
aberration effects of an assumed peculiar motion of the observer, choosing a
different velocity vector for each simulation. The resultant artificial sky
was then used to recover back the velocity vector and compared with the value
actually used in that particular simulation. This not only verified our
procedure and the computation routine, but also allowed us to make an estimate
of errors in the dipole direction from the spread observed in 500 independent
simulations. The correction factor, determined from another set of 500 Monte-
Carlo simulations run without any gap about the galactic plane, turned out to
be only $k_{1}\approx 1.01$

The results for the dipole, determined from the anisotropy in number counts in
our sample of about 0.28 million MIRAGNs in the $12<$W1$<15$ range, are given
in Table 1, where the 1st column gives the magnitude range, 2nd column gives
the number of sources, 3rd and 4th columns give the direction of the dipole in
terms of Right Ascension and Declination, 5th column gives $\cal D$, the
dipole magnitude, and the last column gives the value of the speed estimated
from $\cal D$.

We find a peculiar speed $1.7\pm 0.2\times 10^{3}$ km s-1, which is more than
four times the peculiar speed estimated from the CMBR dipole. In order to
ensure that this excess in speed is not due to a skew distribution of sources
belonging to a particular magnitude, we divided our sample into three
magnitude bins, with approximately 0.1 million sources in each bin. The
results for the three bins are also presented in Table I. Here, N is the total
number of sources in the corresponding W1 bin, RA and Dec give the dipole
direction in the sky, $\cal D$ is the dipole value computed from ${\cal
D}={3\Sigma\cos\theta_{i}}/(2\Sigma|\cos\theta_{i})]$, with error $\Delta{\cal
D}=\sqrt{3/(2\Sigma|\cos\theta_{i}|)}$ 4 ; 9 ; 16 . Then the peculiar speed of
the observer, or rather of the Solar system, is computed from ${\cal
D}=[2+x(1+\alpha)](v/c)$. In order to determine $x$ we have made a plot of the
integrated source counts N(<W1) against W1, for our MIRAGN sample, in Figure
4, which shows a power law behaviour of the integrated source counts, with a
slope that varies between 0.71 and 0.65, with a mean value of 0.68. From this
we estimate the index of integral source counts in our sample to be
$x=2.5\times 0.68=1.7$, consistent with the value quoted in the literature 11
. For the spectral index, we have taken $\alpha\approx 1$, a value quoted in
the literature 11 , for the extragalactic population of AGNs. The peculiar
speed, accordingly, is given by, $v\approx cD/(5.4k_{1})\approx 5.5\times
10^{4}{\cal D}$ km/s.

Table 1: Peculiar velocity estimated from the dipole asymmetry in number counts, for $|{\rm b}|>15^{\circ}$. (1) | (2) |  | (3) |  | (4) |  | (5) |  | (6)
---|---|---|---|---|---|---|---|---|---
Magnitude Range | $N$ |  | RA |  | Dec |  | ${\cal D}$ |  | $v$
|  |  | (∘) |  | (∘) |  | ($10^{-2}$) |  | ($10^{3}$ km s-1)
$15.0>W1\geq 12.0$ | 279139 |  | $148\pm 19$ |  | $23\pm 17$ |  | $3.0\pm 0.4$ |  | $1.7\pm 0.2$
$15.0>W1\geq 14.7$ | 102822 |  | $157\pm 20$ |  | $23\pm 18$ |  | $4.1\pm 0.6$ |  | $2.3\pm 0.3$
$14.7>W1\geq 14.3$ | 086035 |  | $132\pm 21$ |  | $32\pm 19$ |  | $2.9\pm 0.6$ |  | $1.6\pm 0.3$
$14.3>W1\geq 12.0$ | 090282 |  | $143\pm 21$ |  | $11\pm 19$ |  | $2.1\pm 0.6$ |  | $1.2\pm 0.3$

In Table 1, we have also listed the direction of the dipole, along with the
estimated errors, as determined in each case. The direction of the velocity
vector (with our best estimate from Table-1), is given by the pole at
RA$=148^{\circ}\pm 19^{\circ}$, Dec= $23^{\circ}\pm 17^{\circ}$, within
$\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2\sigma$ of the CMBR pole at
RA$=168^{\circ}$, Dec$=-7^{\circ}$ (in galactic coordinates, the MIRAGN pole
lies at $l=209^{\circ},b=49^{\circ}$ while the CMBR pole is at
$l=264^{\circ},b=48^{\circ}$, with the CMBR pole positions being $~{}55\cos
49\approx 38^{\circ}$ away). However the estimates of $v(=1.7\pm 0.2\times
10^{3}$ km s-1) appear much higher than the CMBR value (370 km s-1) by a
factor $>4$ at a statistically significant
($\stackrel{{\scriptstyle>}}{{{}_{\sim}}}5\sigma$) level, which however, is in
agreement with the dipole derived from the NVSS radio source data 4 ; 5 ; 6 ;
7 ; 8 .

Although we have restricted our sample to W1$>12.0$ to minimize the effect of
local bulk flows, still, in order to estimate the influence on our results of
any local clustering, like the Virgo super-cluster, we determined dipole
vectors by excluding sources at low super-galactic latitudes, progressively in
steps of five degrees. Table 2 shows the results where the 1st column gives
the $|SGB|$ limit, 2nd column gives the number of sources, 3rd and 4th columns
give the direction of the dipole in terms of Right Ascension and Declination,
5th column gives $\cal D$, the dipole magnitude, and the last column gives the
value of the speed estimated from ${\cal D}$.

Table 2: Dipole estimates for various $|{\rm SGB}|$ limits (1) | (2) |  | (3) |  | (4) |  | (5) |  | (6)
---|---|---|---|---|---|---|---|---|---
$|{\rm SGB}|$ limit | $N$ |  | RA |  | Dec |  | ${\cal D}$ |  | $v$
|  |  | (∘) |  | (∘) |  | ($10^{-2}$) |  | ($10^{3}$ km s-1)
$|{\rm SGB}|\geq 0$ | 279139 |  | $148\pm 17$ |  | $23\pm 17$ |  | $3.0\pm 0.4$ |  | $1.7\pm 0.2$
$|{\rm SGB}|\geq 5$ | 251281 |  | $145\pm 18$ |  | $21\pm 18$ |  | $3.1\pm 0.4$ |  | $1.7\pm 0.2$
$|{\rm SGB}|\geq 10$ | 223230 |  | $142\pm 18$ |  | $20\pm 19$ |  | $3.4\pm 0.4$ |  | $1.9\pm 0.2$
$|{\rm SGB}|\geq 15$ | 195664 |  | $141\pm 19$ |  | $12\pm 20$ |  | $3.7\pm 0.4$ |  | $2.0\pm 0.2$
$|{\rm SGB}|\geq 20$ | 168826 |  | $136\pm 20$ |  | $09\pm 20$ |  | $3.7\pm 0.4$ |  | $2.0\pm 0.2$

From a comparison of the results in these cases
($|SGB|>0^{\circ},5^{\circ},10^{\circ},15^{\circ},20^{\circ}$; Table 2), no
unusually large variations, beyond the statistical uncertainties, were seen in
the computed dipole vectors.

###  4.2 Dipole determined from the hemisphere method

We can also determine the dipole employing an alternate method known as the
hemisphere method. Though this method appears to be simpler in nature, but
here, unlike the previous dipole vector method, one does not directly get the
direction of the dipole, which one has to assume or determine in some other
way. Suppose we know the pole direction in sky, then using the great circle at
$90^{\circ}$ from the pole direction, we divide the sky in two equal
hemispheres, ${\cal S}_{1}$ and ${\cal S}_{2}$; ${\cal S}_{1}$ containing the
assumed pole and ${\cal S}_{2}$, the opposite hemisphere, containing the
antipole.

As we saw earlier, a motion of the observer toward the pole will result in a
dipole anisotropy, $n_{0}(1+{\cal D}\cos\theta)$, in the number counts of
sources, where $n_{0}$ is the mean number density (per steradian) in sky and
$\theta$ is the angle measured from the pole. Then the number of sources in
the hemisphere ${\cal S}_{1}$ should be larger than the number of sources in
${\cal S}_{2}$.

If ${N}_{1}(\theta)$ be the cumulative number of sources in the sky zone
between the great circle at $\theta=90^{\circ}$ and a parallel circle at angle
$\theta$ in ${\cal S}_{1}$, while ${N}_{2}(\theta)$ is the cumulative number
of sources in the symmetrically placed, corresponding zone in the opposite
hemisphere ${\cal S}_{2}$, i.e., ${N}_{2}(\theta)$ is the cumulative number of
sources lying between $\theta=90^{\circ}$ and $\pi-\theta$. Then we obtain the
dipole component ${\cal D}_{\theta}$ along $\theta$, from 9

${\cal D}_{\theta}={\cal
D}\cos\theta=\frac{{N}_{1}(\theta)-{N}_{2}(\theta)}{[{N}_{1}(\theta)+{N}_{2}(\theta)]/2}.$
(3)

The $1\sigma$ uncertainty in ${\cal D}_{\theta}$ is
$2/\sqrt{{N}_{1}(\theta)+{N}_{2}(\theta)}$.

Amplitude of the dipole ${\cal D}$ could then be determined by counting ${N}1$
and ${N}2$ for the complete hemispheres, i.e., from $\theta=90^{\circ}$ to
$\theta=0^{\circ}$ in ${\cal S}_{1}$ and $\theta=90^{\circ}$ to
$\theta=180^{\circ}$ in ${\cal S}_{2}$, to get

${\cal D}={{\cal D}_{\rm
o}}=\frac{{N}_{1}-{N}_{2}}{{N}/2}=\frac{{N}_{1}-{N}_{2}}{2\pi n_{0}}\,,$ (4)

where ${N}={N}_{1}+{N}_{2}=4\pi n_{0}$ is the total number of sources
integrated over all directions in our sample, with the $1\sigma$ uncertainty
in ${\cal D}$ being $2/\sqrt{{N}}$.

Figure 5: A contour map of the dipole amplitudes, estimated for various
directions in the sky. The horizontal and vertical axes denote RA and Dec in
degrees. The true pole direction is expected to be closer to the higher
contour values, indicated by darker grey regions, surrounded by continuous
lines, while the the true antipole should lie closer to the lower contour
values, indicated by lighter grey regions, surrounded by dotted lines. The
symbol $\odot$ indicates the CMBR pole position, while $M$ indicates the pole
position for our MIRAGN sample, as determined by the Dipole Vector method
(Table 1). Symbol $A$ indicates the corresponding antipole position. The
dashed curve, representing the zero amplitude of the dipole, criss-crosses the
Great Circle, at $90^{\circ}$ from the poles at $M$ and $A$, dividing the sky
in two equal hemispheres, ${\cal S}_{1}$ and ${\cal S}_{2}$.

To determine the direction of the dipole, we have employed a ’brute force
method’, by dividing the sky into cells of $2^{\circ}\times 2^{\circ}$ with
minimal overlap, creating a grid of $10360$ pixels covering the sky. Then
assuming the centre of each of these 10360 pixels in turn to be the pole
direction, we computed the dipole magnitude from the sky positions of our 0.28
million MIRAGNs. This on the average yields only a projection of the peculiar
velocity in the direction of each pixel, peaking in the pixel that lies
closest to the true dipole direction.

Figure 6: A plot of dipole amplitudes (teal colour) estimated for various
directions in the sky. The horizontal axes denote Right Ascension (RA) and
Declination (Dec) in degrees. The positions (RA and Dec) of the peak values
could be read from the 2-d projections along each axis, shown in light grey.
Due to fluctuations in individual values it is not possible to determine a
unique peak, whose location would have unambiguously determined the direction
of our peculiar motion in the sky.

Figure 5 shows a contour map of the dipole amplitudes, estimated this way for
various pixels on the sky. We should expect the true pole direction to be
closer to the higher contour values, indicated by darker grey regions,
surrounded by continuous lines, while the the true antipole should lie closer
to the lower contour values, indicated by lighter grey regions, surrounded by
dotted lines. The pole position determined by the Dipole Vector method (Table
1) for our MIRAGN sample and denoted by $M$ here, lies almost in the middle of
the darkest region, while the corresponding antipole position, indicated by
the Symbol $A$, lies in the lighter-most region. The CMBR pole position,
indicated by the symbol $\odot$, is not far from the peak of the dipole
amplitude, indicated by the highest contour. The dashed curve, representing
the zero amplitude of the dipole, criss-crosses the Great Circle, at
$90^{\circ}$ from the poles at $M$ and $A$, dividing the sky in two equal
hemispheres, ${\cal S}_{1}$ and ${\cal S}_{2}$.

A 3-d representation of the dipole amplitude distribution across the sky is
shown in Figure 6. Although an overall trend, in agreement with the dipole
direction from Table 1, is seen in Figure 6, and where in principle, the
location of the peak value for the dipole amplitude should yield the true
direction of the dipole. However, due to fluctuations in individual values it
is not possible to determine the location of an unambiguous peak from Figure
6. At the same time, for an actual dipole, one would expect on the average a
$\cos\theta$ dependence of the determined dipole magnitudes with respect to
the true pole. To exploit this expectation, we have written a 3-d COSFIT
routine, where for each of the $n=10360$ pixel positions in turn, we applied a
3-d $\cos\theta$ fit to the remaining $n-1$ dipole values in the pixels around
it on the sky, and determined the resulting amplitude as well as a chi-square
value, for each of these $n$ fits.

Figure 7 shows the output of our 3-d COSFIT routine. The process converged
rapidly to show a clear unique peak at RA$=148^{\circ}$, Dec$=22^{\circ}$,
with a maximum value of ${\cal D}=3.2\times 10^{-2}$, accompanied by an ideal
minimum value of 1.0 for the reduced chi-square $\chi^{2}_{\nu}$, occurring
for the same pixel position as the peak. Figure 7(a) shows the 3-d plot of the
peak while a clear minimum in the reduced $\chi^{2}$ value is seen in Figure
7(b), both occurring at RA$=148^{\circ}$, Dec$=22^{\circ}$. In order to make
sure that nothing further is amiss in our procedure, we also tried a finer
grid with $1{{}^{\circ}}\times 1{{}^{\circ}}$, with more than $41000$ pixels,
or even a coarser one with a pixel size of $5{{}^{\circ}}\times
5{{}^{\circ}}$, but it made no perceptible difference in our final results.

To test our 3-d COSFIT procedure, we made Monte-Carlo simulations, with random
positions (RA and Dec) in sky allocated to AGNs in our sample and then a mock
dipole was superimposed to calculate Doppler boosting and aberration effects
for each source. Then on this mock catalogue of MIRAGNs, our 3-d COSFIT
procedure was applied to recover the dipole and compared with the input dipole
in that simulation. This not only validated our method but it also provided us
an estimate of errors from 500 independent simulations. The correction factor
was determined from another set of 500 Monte-Carlo simulations, run without a
gap about the galactic plane, to be $k_{2}\approx 1.15$ in this method.

Accordingly, for the direction of the peculiar velocity, along with estimated
errors, we arrive at RA$=148^{\circ}\pm 19^{\circ}$, Dec$=22^{\circ}\pm
18^{\circ}$ from the the Chi-square ($\chi^{2}_{\nu}$) fit which shows a
minimum (Figure 7b) at the same sky position as the peak in the dipole
magnitude (Figure 7a). The $\chi^{2}_{\nu}$ minimum value of 1.0, which is an
ideal value, indicates that the estimates of errors in the dipole amplitudes
are quite realistic. The value for the MIRAGN dipole direction agrees, within
the $2\sigma$ uncertainty, with the CMBR dipole direction, RA$=168^{\circ}$,
Dec$=-7^{\circ}$. However, the peculiar speed, determined from, $v\approx
cD/(5.4k_{2})\approx 4.8\times 10^{4}{\cal D}$ km s-1, gives a solar speed
value $1.6\pm 0.2\times 10^{3}$ km s-1,
$\stackrel{{\scriptstyle>}}{{{}_{\sim}}}4$ times the CMBR value.

Figure 7: A plot of 3-d COSFITs made to the dipole amplitudes estimated for
various trial dipole directions across the sky, showing (a) a unique peak
(blue colour) unambiguously indicating the optimum direction of the dipole (b)
reduced chi-square ($\chi^{2}_{\nu}$) values (magenta colour), having an ideal
minimum value of 1.0, at the same sky position as the blue colour peak in (a).
The horizontal axes denote Right Ascension (RA) and Declination (Dec) in
degrees. The positions (RA and Dec) of the extrema are determined more easily
from the 2-d projections, shown in light grey, thence we infer the direction
of the observer’s peculiar velocity as RA$=148^{\circ}$, and Dec$=22^{\circ}$.

Our results are presented in Table 3, where the 1st column gives the magnitude
range, 2nd column gives the number of sources, 3rd column gives the difference
(${N}1-{N}2$) between the numbers of sources in ${\cal S}_{1}$ and ${\cal
S}_{2}$, 4th and 5th columns give the direction of the dipole in terms of
Right Ascension and Declination, 6th column gives $\cal D$, the dipole
magnitude, and the last column gives the value of the speed estimated from
$\cal D$. Dipole ${D}$ was estimated for samples containing all sources in our
sample as well as for the three bins with different magnitude (W1) limits. The
direction (RA and Dec) of the dipole in each case was determined from a 3-d
COSFIT and the velocity $v$ then determined, taking the pole to be in that
direction.

From Tables 1 and 3 we notice that the uncertainty in the peculiar velocity
values ($0.2-0.4\times 10^{3}$ km s-1) determined from the MIRAGN data is of
the same order as the CMBR dipole amplitude ($0.37\times 10^{3}$ km s-1), from
that it is evident that a positive detection of the MIRAGN dipole has been
possible only because the strength of the signal has turned out to be much
larger ($>4$) than the CMBR dipole. It has been pointed out that due to
statistical fluctuations, there could be a bias in the dipole amplitudes
towards higher values 5 , and which, even in the absence of a genuine dipole
could yield a value approximately equal to the CMBR dipole value (of course in
a random direction), or for a dipole actually equal to the CMBR value, could
lead to an estimate of the dipole magnitude substantially higher (by a factor
$\sim 1.4$). However, at the level of much larger dipole amplitudes, like our
observed ${\cal D}\sim 3\times 10^{-2}$ (Tables 1 and 3), the effects of the
amplitude bias are at most $\stackrel{{\scriptstyle<}}{{{}_{\sim}}}3\%$, a
negligible fraction of the quoted uncertainties in the Tables.

##  5 Discussion

The view that the Solar system peculiar motion is 370 km s-1, as given by the
CMBR dipole, seems to be widely accepted. However, one needs to keep an open
mind about it, without sticking rigidly to the conventional view that only the
CMBR provides a reference frame which, somehow, is to be considered as more
fundamental than the other ones, e.g. AGNs, for establishing the peculiar
motion of the solar system. While the CMBR refers to the radiation era ($z\sim
10^{3}$), AGNs represent the much later matter era ($z\sim 1-3$). A common
direction for the two dipoles, determined from completely independent data,
does indicate that these dipole amplitudes differ not because of random
statistical fluctuations, or due to some systematics in the observations or in
the data analysis, otherwise even their directions would have turned out to be
different, and that the difference in their amplitudes might be a pointer to
some occurrence of cosmological significance.

In Figure 3, we saw that there are large scale inhomogeneities in the sky
distribution of the MIRAGN number densities at W1$>16$ levels. In fact, even
at somewhat brighter levels, $15.5<$W1$<16$, such large scale inhomogeneities
in the sky distribution are present (Figure 2), though at a reduced level. How
can we be sure that such large scale inhomogeneities in the sky distribution
are not present in our sample at W1$<15$ levels? Although we may not be able
to discern them from Figure 1 with our eyes, but how to make sure of the lack
of their underlying presence in our data, at least at levels which could have
influenced our intended determination of the dipole significantly. In other
words, could such large scale inhomogeneities or even some large scale
statistical fluctuations in certain directions be the cause of our observed
dipole being much larger than the CMBR dipole?

Table 3: The dipole magnitude estimated using differential number counts in the hemisphere method (1) |  | (2) | (3) | (4) | (5) | (6) | (7)
---|---|---|---|---|---|---|---
Magnitude Range |  | $N$ | $N_{1}-N_{2}$ | RA | Dec | $\cal D$ | $v$
|  |  |  | (∘) | (∘) | ($10^{-2}$) | ($10^{3}$ km s-1)
$15.0>W1\geq 12.0$ |  | 279139 | 4629 | $148\pm 19$ | $22\pm 18$ | $3.3\pm 0.4$ | 1.6 $\pm 0.2$
$15.0>W1\geq 14.7$ |  | 102822 | 2370 | $157\pm 20$ | $22\pm 19$ | $4.6\pm 0.6$ | 2.2 $\pm 0.3$
$14.7>W1\geq 14.3$ |  | 086035 | 1211 | $132\pm 20$ | $30\pm 19$ | $2.9\pm 0.7$ | 1.4 $\pm 0.4$
$14.3>W1\geq 12.0$ |  | 090282 | 1022 | $143\pm 20$ | $10\pm 19$ | $2.1\pm 0.7$ | 1.0 $\pm 0.4$
Figure 8: A plot of the dipole components ${\cal D}_{\theta}$ and the
equivalent peculiar velocity component $v\cos\theta$, computed for various
zones of the sky between the great circle and a parallel circle at $\theta$,
the angle with respect to the determined dipole direction, RA$=148^{\circ}$,
Dec$=22^{\circ}$. The latitude angle $\phi$ is measured from the great circle
at $\theta=90^{\circ}$. The observed values in various sky strips are plotted
as circles (o), with the error bars calculated for a random (binomial)
distribution. The continuous curve, corresponding to a peculiar velocity of
$1.7\pm 0.2\times 10^{3}$ km s-1, shows the expected ($\propto\cos\theta$)
behaviour for ${\cal D}_{\theta}$, which is a best fit to the data observed in
different sky strips, shown by the dotted curve. For a comparison, the
expectation for the CMBR value, $v=370$ km s-1, is shown by dashed curve,
which lies way below the values observed for the MIRAGNs. Figure 9: Plots of
the dipole components ${\cal D}_{\theta}$, observed for the TGSS (+) and NVSS
(x) data 9  for various zones of the sky between the great circle and a
parallel circle at $\theta$, the angle with respect to the CMBR dipole
direction. The uncertainties expected in random (binomial) distributions are
shown as error bars. The continuous curves, corresponding to the indicated
peculiar velocities, show the expected ($\cos\theta$) behaviour for ${\cal
D}_{\theta}$, which seem to fit well the observed values. The dashed curve
shows the plot, expected for the CMBR value, $v=370$ km s-1.

For one thing, such non-uniformities would have to be distributed over various
sky regions in such a way that these make the resulting excess in the dipole
amplitude to occur in the same direction as the CMBR dipole. Moreover, these
also have to be spread all across the magnitude band of our MIRAGN sample, as
our three bins, selected with no overlap in the magnitude band, yield very
similar dipole vectors. In fact, one can verify that it is a genuine dipole
distribution in the number densities and not merely a result of some
systematics in the data. From Eq. (3), we see that ${\cal D}_{\theta}$ yields
a component of the dipole along $\theta$. We can verify this $\cos\theta$
dependence of ${\cal D}_{\theta}$, by making cumulative counts of
$N_{1}(\theta)$ and $N_{2}(\theta)$ in two opposite hemispheres, ${\cal
S}_{1}$ and ${\cal S}_{2}$, for various $\theta$ values. Figure 8 shows a plot
of the ${\cal D}_{\theta}$, computed for our MIRAGN sample using Eq. (3), as a
function of $\theta$, the angle with respect to the determined dipole
direction, RA$=148^{\circ}$, Dec$=22^{\circ}$. The dotted curve shows ${\cal
D}_{\theta}$ calculated from cumulative counts which has the best fit by a
$v\cos\theta$ dipole curve, with $v=1.7\pm 0.2\times 10^{3}$ km s-1. We have
also plotted the error bars, calculated for a random (binomial) distribution,
at some representative points. The larger fluctuations seen for
${\theta}\rightarrow 90^{\circ}$, although still within statistical
uncertainties, are because of a smaller dipole component (${\cal
D}_{\theta}\propto\cos\theta$) vis-à-vis the relatively larger statistical
fluctuations in a binomial distribution due to a lesser number of sources in
the reduced sky zones. Also shown is the expected curve for the peculiar
velocity, $v=370$ km s-1 derived from the CMBR dipole, which lies way below
the dipole components estimated from the MIRAGNs data.

A vectorial summation of sky positions of all sources, as determined in Table
1, in itself does not guarantee that the resultant vector is necessarily a
dipole, after all we will always get some resulting RA, Dec and $\cal D$ from
such a summation, whatever might be the statistical uncertainties. However, a
systematic variation pattern of dipole amplitudes over sky, as seen in Figure
5, where the pole position $M$ from Table 1 is right in the middle of the
darkest region, while the corresponding antipole position $A$ lies in the
lighter most region, indicates the genuine nature of the dipole. Moreover, the
dashed curve, representing the zero amplitude of the dipole, overlaps the
great circle, at $90^{\circ}$ from the poles at $M$ and $A$, added to the fact
that we obtained the same particulars for the dipole from the hemisphere
method, where not only the direction of the dipole from the brute force method
(Table 3) turned out to be the same as was determined in Table 1, even the
amplitude of the dipole, where various sources, depending upon their sky
positions get different weights, gave the same value, confirms that it is a
genuine dipole. More so because of the $\cos\theta$ pattern seen in Figure 8,
as would be expected only for a dipole. Only in a rather contrived scenario
would one expect the differential number counts in two opposite hemispheres in
sky, to follow the $\cos\theta$ behaviour, unless it were the result of a
genuine dipole in the number density distribution, irrespective of the
ultimate cause of the dipole.

Though spectroscopic redshifts are not available for most of the sources in
the sample, however, from the sources where redshifts are available 17 , as
well as from the previous information in the literature, AGNs are known to lie
mostly at redshifts between 0.5 and 3 and are, in fact, the most distant
discrete objects seen in the universe. Therefore, the dipole anisotropy seen
among them is not a local effect, and has a genuine bearing on the
cosmological principle.

For a comparison with some earlier dipole determinations from the radio data
on AGNs 9 , in Figure 9 we show the plots of the ${\cal D}_{\theta}$ for both
TGSS and NVSS data as a function of $\theta$, measured with respect to the
CMBR dipole direction. It should be noted that the conversion factor from
$\cal D$ to peculiar velocity $v$ in Figure 9 is somewhat different in
comparison with that in Figure (7), because of the difference in indices $x$
and $\alpha$ between the radio and Infra red populations. It does seem that
the peculiar velocity of the Solar system estimated from the TGSS data is much
higher than the NVSS data 9 , the latter in agreement with the MIRAGN dipole
in Figure 8.

Figure 10: The sky position, in equatorial coordinates RA and Dec, of the pole
determined from our MIRAGN sample is indicated by $M$, along with the error
ellipse, while other pole positions for various dipoles shown on the map are:
$N$ (NVSS) 4 , $T$ (TGSS) 9 , $Z$ (DR12Q) 10 , $D$ (Dark Flow) Ka10 , $\alpha$
(Fine Structure constant) Ki12 ; Ma16  , and $E$ (Dark Energy) Ma12 . The CMBR
pole, indicated by $\odot$, has negligible errors 3 .

A recent determination of the peculiar motion of the Solar system from a
dipole anisotropy in the redshift distribution of distant quasars has also
yielded discrepant value of $-2350\pm 280$ km s-1 along the direction of the
CMBR dipole 10 , implying a solar peculiar motion in a direction opposite to
that derived from all other previous dipoles. Nevertheless, it is evident that
all AGN dipoles have much larger amplitudes than the CMBR dipole, even though
various AGN dipole directions in sky may be lying parallel to the CMBR dipole.

In Figure 10, the sky position of our MIRAGN pole is shown by M, along with
the $1\sigma$ error ellipse, while the CMBR pole position is indicated by
$\odot$, which has negligible errors 1 ; 2 ; 3 . Also plotted are the various
other determined pole positions, N for the NVSS dipole, T for the TGSS dipole
and D for the DR12Q dipole, along with their $1\sigma$ error ellipses 4 ; 9 ;
10 . From Figure 10, it does seem that various AGN dipoles are within $\sim
1\sigma$, except for the MIRAGN pole which is at
$\stackrel{{\scriptstyle<}}{{{}_{\sim}}}2\sigma$, of the CMBR pole, from which
we can surmise that the various AGN dipoles are pointing along the same
direction as the CMBR dipole.

Here it may be pointed out that a recent determination of the dipole 11  from
a flux-limited, all-sky sample of 1.36 million quasars observed by the Wide-
field Infrared Survey Explorer (WISE) gave the direction of the dipole to be
similar,
$l=238\\!\stackrel{{{}^{\circ}}}{{{}_{\cdot}}}\\!2,b=28\\!\stackrel{{{}^{\circ}}}{{{}_{\cdot}}}\\!8$
(RA=$140\\!\stackrel{{{}^{\circ}}}{{{}_{\cdot}}}\\!0$,
Dec=$-6\\!\stackrel{{{}^{\circ}}}{{{}_{\cdot}}}\\!5)$, about
$27\\!\stackrel{{{}^{\circ}}}{{{}_{\cdot}}}\\!8$ from the CMBR pole. However,
the amplitude of the peculiar velocity ($8.2\times 10^{2}$ km s-1) was over
twice as large the CMBR value ($3.7\times 10^{2}$ km s-1), while we find the
peculiar velocity ($1.7\times 10^{3}$ km s-1) to be over four times the CMBR
value. This mysterious factor of two difference in the peculiar velocity
values, estimated from these two Mid Infra Red AGN samples, needs to be
explored further, whether it is due to a difference in the basic samples
themselves or is it arising from the differences in the excluded zones, e.g.
the galactic latitudes, or caused by some other, as yet unknown, reason.
Nevertheless, one thing is clear, both the peculiar velocity estimates are
inconsistent with the CMBR peculiar velocity value, though the direction
estimate in each individual case does appear consistent with the CMBR dipole
direction, within statistical uncertainties.

Since the peculiar velocity of the Solar system should not dependent upon the
specific data or technique used to determine it, one obvious inference from
the discordant values of the inferred peculiar motion from observed dipoles
for the CMBR and the AGNs could be that there is a large relative motion
between various cosmic reference frames. Otherwise, one has to take the view
point that the observed discordant dipoles (including the CMBR), contrary to
the conventional wisdom, do not reflect a motion of the observer (or a
peculiar velocity of the Solar system), and that we should instead look for
some other possible cause for the genesis of these dipoles, including that of
the CMBR. However, one has to then explain the existence of a preferred
direction of all the dipoles and that what is so special about this particular
direction A not-too-far-fetched inference drawn could be that a common
direction for the dipoles, including of the CMBR, is a pointer toward the
presence of an inherently preferred cosmic direction (axis!), implying perhaps
an anisotropic Universe 4  in conflict with the Cosmological Principle.

There are also other dipoles seen at cosmological scales. For instance, the
“dark flow” dipole, which is a statistically significant dipole found at the
position of galaxy clusters in filtered maps of the CMBR temperature
anisotropies, implies the existence of a primordial CMBR dipole of non-
kinematic origin, presenting itself as an effective motion across the entire
cosmological horizon Ka08 ; Ka09 ; Ka10 ; At15 . The CMBR dipole measurements
are also consistent with some dynamical contributions to the CMBR dipole,
resulting possibly from an SU(2) gauge principle Sz08 ; Lu09 ; Ho13 . The
detection of a dipole in the spatial variation of the fine-structure constant,
$\alpha=e^{2}/\hbar c$, based on study of quasar absorption systems has also
been reported We11 ; Ki12 ; Be11 ; Be12 . These spatial variations of the fine
structure constant can be constrained using clusters of galaxies Ga13 ; Ma16 .
It seems that the fine structure constant cosmic dipole is aligned with the
corresponding dark energy dipole within 1$\sigma$ uncertainties Ma12 ; Ma13 .
In Figure 10, we have plotted positions of some of these poles to show their
correlation, if any, with the CMBR dipole as well as various AGN dipoles. Here
$D$ (Dark Flow) Ka10 , $\alpha$ (Fine Structure constant) Ki12 ; Ma16  , and
$E$ (Dark Energy) Ma12  are shown along with their quoted error ellipses.
While the direction of dark flow dipole could be correlating with the
direction of the CMBR dipole, the fine structure constant cosmic dipole and
the dark energy dipole seem to be pointing quite away from the direction of
the CMBR dipole. At the same time, because of large statistical uncertainties.
the direction of dark flow dipole being in the same direction as the dark
energy dipole cannot be ruled out. Further, there is a preference seen at
significant levels for odd multipoles in the large scale anisotropies of the
CMB temperature, a parity asymmetry, to be strongly aligned with the CMBR
dipole Na12 ; Zh14 ; Ch16 .

If the CMBR dipole is of kinematic origin due to the Solar system peculiar
motion, something resulting from the distribution of matter at local scales,
in contrast to dipoles originating at cosmic scales, these observed
correlations in the directions of various dipoles and asymmetries give rise to
doubts about whether the CMBR dipole is really caused by our peculiar motion.
And if the CMBR dipole is due to some intrinsic asymmetry at cosmic scales,
suggested by an order of magnitude difference in the dipole magnitudes as
inferred from various AGN datasets, even though all appear to be in the same
direction as the CMBR dipole, it gives rise to a larger question of the
ultimate reliability of the Cosmological Principle, a cornerstone of the
modern cosmology.

##  6 Conclusions

From the angular positions in sky of a sample of 0.28 million Mid Infra Red
AGNs, we found an anisotropy in their number densities in different
directions. Ascribing this anisotropy to the peculiar motion of the observer,
we determined the peculiar velocity of the Solar system that turned out to be,
like other earlier AGN dipoles, at least a factor of four larger that that
inferred from the CMBR dipole, but along the same direction. Since the
peculiar velocity of the Solar system should not depend upon the specific data
or the technique used to determine it, a question gets raised about the nature
of these dipoles seen in the sky and whether the genesis of some or all of
these dipoles indeed is due to the peculiar motion of the Solar system. A
common direction for all these dipoles, including the CMBR one, determined
from completely independent surveys by different groups, does indicate that
the differences in the dipole amplitudes are genuine and not because of random
statistical fluctuations, or due to some systematics in the observations or in
the data analysis, otherwise even the dipole directions obtained from
different data sets would have been different. An inference that could
possibly be drawn from a common direction for all the dipoles is that it might
be a pointer toward the presence of an inherently preferred cosmic direction
(axis!), implying perhaps an anisotropic Universe, in discordance with the
Cosmological Principle, a cornerstone of the modern cosmology.

No funds, grants, or other support of any kind was received from anywhere for
this research. The author has no conflicts of interest to declare that are
relevant to the content of this paper. References

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