LARGE PECULIAR MOTION OF THE SOLAR SYSTEM FROM THE DIPOLE ANISOTROPY IN SKY BRIGHTNESS DUE TO DISTANT RADIO SOURCES

The peculiar velocity of the solar system through the universe has been determined relative to the frame of reference provided by the cosmic microwave background radiation (CMBR) to be 369 km s⁻¹ in the direction l = 264°, b = 48° (Lineweaver et al. 1996; Hinshaw et al. 2009). A study using the number counts of radio sources (Blake & Wall 2002) had found the velocity to be consistent with that from the CMBR. Here we determine the peculiar motion from the anisotropy in sky brightness due to radio sources. This provides an independent check on the interpretation of CMBR dipole anisotropy being due to motion of the solar system. Also, CMBR provides information about the isotropy of the universe for redshift z ∼ 700, but the radio source population refers to a much later epoch z ∼ 1. Thus, it also provides an independent check on the cosmological principle where isotropy of the universe is assumed for all epochs.

We have used the NRAO VLA Sky Survey (NVSS) catalog (Condon et al. 1998) for our investigations. This survey covers whole sky north of declination −40°, a total of 82% of the celestial sphere, at 1.4 GHz. There are about 1.8 million sources in the catalog with a flux-density limit S > 3 mJy. Figure 1 shows a plot of the bright sources (>300 mJy) from the NVSS catalog in galactic coordinates. There is a strip of excess sources near the galactic equator. The large gap corresponds to the southern declination limit of the survey.

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An observer moving with a velocity v will find sources in the forward direction brighter by a factor δ^{1 + α} due to Doppler boosting, where δ = 1 + (v/c)cos θ is the Doppler factor, c is the velocity of light, and α (≈0.8) is the spectral index defined by S∝ν^−α. Here we have used the non-relativistic formula for the Doppler factor as CMBR observations indicate that v ≪ c. The increased flux density due to Doppler boosting in the forward direction will cause a telescope of a given sensitivity limit to detect a larger number of sources comparatively. The contribution to the observed sky brightness at a given flux-density level S comes from sources having the rest-frame flux-density S/δ^{1 + α}. With the integral source counts of extragalactic radio source population following a power law N(> S)∝S^−x (x ∼ 1; Ellis & Baldwin 1984), the number of the sources observed therefore will be higher by δ^{x(1 + α)}, a factor independent of S (as long as x and α can be deemed to be independent of S level). The observed sky brightness will thus change by an overall factor δ^{x(1 + α)} due to Doppler boosting.

Also, due to the aberration of light, the apparent position of a source at angle θ will be shifted in the forward direction by an angle vsin θ/c, and as a result there will be a higher number density per steradian in the forward direction as compared to that in the backward direction. This excess in number density ∝δ². Thus as a combined effect of Doppler boosting and the aberration, the observed sky brightness (an integrated emission from discrete sources per unit solid angle) will vary as ∝δ^{2 + x(1 + α)} which (for v ≪ c) can be written as $1+{\cal D}\cos \theta$, a dipole anisotropy over the sky with amplitude ${\cal D}=[2+x(1+\alpha)](v/c)$ (Ellis & Baldwin 1984; Crawford 2009).

To find the velocity of the solar system, we consider all sources to lie on the surface of a sphere of unit radius, and let $\textbf {r}_i$ be the position vector of the ith source with respect to the center of the sphere. An observer stationary at the center of the sphere will find the sky brightness to be uniformly distributed in all directions (due to the assumed isotropy of the universe) and therefore should get $\Sigma S_i \textbf {r}_i=0$. On the other hand, for a moving observer the forward shift in position due to aberration and the Doppler boosting of flux density in the forward direction implies that the vectorial sum $\Sigma S_i\textbf {r}_i$ will yield a net vector in the direction of motion, thereby fixing the direction of the dipole. If θ_i is the polar angle of the ith source with respect to the dipole direction, then the magnitude of the vectorial sum can be written as ${\Delta {\cal F}}=\Sigma S_i\: \cos \theta _i$. Writing ${\cal F}=\Sigma S_i\: |\cos \theta _i|$ and converting the summation into an integration over the sphere, we get

$$\begin{eqnarray} \frac{\Delta {\cal F}}{{\cal F}}&=&k\: \frac{\int ^{\pi }_{0}(1+{\cal D}\cos \theta)\cos \theta \: \sin \theta \:d\theta }{2\int ^{\pi /2}_{0}\cos \theta \:\sin \theta \:d\theta }\nonumber\\ &=&\frac{2k {\cal D}}{3}= \frac{2k}{3} \left[2+x(1+\alpha)\right]\frac{v}{c}. \end{eqnarray} \tag{ 1 }$$

The formula is equally valid for samples with finite upper and lower flux-density limits. Here k = 1 for a sky fully covered by the sample, and 1 < k ⩽ 3/2 when there are finite gaps in the sky coverage and may need to be determined numerically for individual cases.

As the NVSS catalog has a gap of sources for decl. < − 40°, in that case our assumption of $\Sigma S_i \textbf {r}_i=0$ for a stationary observer does not hold good. However, if we drop all sources with decl. >40° as well, then with equal and opposite gaps on opposite sides, $\Sigma S_i \textbf {r}_i=0$ is valid for a stationary observer. Further, we also excluded all sources from our sample which lie in the galactic plane (|b| < 10°) as the excess of galactic sources toward the galactic center (see Figure 1) is likely to contaminate the determination of velocity. Of course, exclusion of such strips, which affect the forward and backward measurements identically, to a first order does not have systematic effects on the results (Ellis & Baldwin 1984).

Before proceeding with the actual source sample we used the Monte Carlo technique to create an artificial radio sky with about two million sources (number density of sources similar to that in the NVSS catalog) distributed at random positions in the sky. While the sky positions for each simulation were allotted randomly for each source, for the flux-density distribution we took the observed NVSS sample, the latter justified because the source counts remain unchanged when integrated over the whole sky. The aberration merely shifts the apparent positions in the sky without adding or removing any sources and the effects of Doppler boosting on the differential source counts at any flux-density level are equal and opposite in the forward and backward hemispheres, and therefore the total source counts summed over the whole sky remain the same. On this we superimposed Doppler boosting and the aberration effects of our assumed motion, choosing a different velocity vector for each simulation (in a number of cases we also used the velocity vector from CMBR measurements (Lineweaver et al. 1996; Hinshaw et al. 2009) for Monte Carlo simulations). The resultant artificial sky was then used to recover back the velocity vector under conditions similar to our NVSS case (e.g., with |decl.| > 40°, |b| < 10° gaps in the sky), and thence obtained velocity vector was compared with the value actually used in that particular simulation. This not only verified our procedure but also allowed us to make an estimate of errors in the dipole coordinates as a large number of simulations (∼200) were run starting with different random sky positions and for a different velocity vector each time. From these simulations we also estimated k ∼ 1.1 (see Equation (1)), for the effect of the gaps in our samples.

Our results are presented in Table 1, which is almost self-explanatory. As a relatively small number of strong sources at high flux-density levels could introduce large statistical fluctuations in the sky brightness, we have restricted our sample to below 1000 mJy level. At the lower end we have restricted it to 15 mJy levels as the completeness of the sample at weaker flux-density levels could be doubtful (Blake & Wall 2002). The error in ${\cal D}$ due to statistical fluctuations comprises two components. The variance in the flux-density distribution about the average value S₀ among sources contributes an error $\sigma _1=(\Sigma (S_i - S_0)^2\cos ^2 \theta _i)^{1/2}=(\Sigma S_i^2 - N S_0^2)^{1/2}/{\sqrt{3}}$, while the statistical fluctuations in number density in the two hemispheres contribute $\sigma _2=\sqrt{N} S_0/{\sqrt{3}}$. As these two are statistically independent, the net error in ${\Delta {\cal F}}/{\cal F}$ then is $(\sigma _1^2+\sigma _2^2)^{1/2}/{\cal F}=(\Sigma S_i^2)^{1/2}/{\sqrt{3}}{\cal F}$. The effects of error in the dipole direction on ΔF can be estimated this way. A shift Δθ in the dipole direction causes an exchange of spherical wedges of solid angle 2Δθ near the equator (i.e., for θ ∼ π/2) between the forward and backward hemispheres. However, the dipole anisotropy being minimum there (∝cos θ), the effect on the speed estimate is minimal. It will result in a fractional change in ${\Delta {\cal F}}$ of about 4sin ²(Δθ/2)(2Δθ/2π)(3/2) ∼ 3(Δθ)³/2π. In all cases it will be a systematic effect, resulting in the speed being slightly underestimated (≲ 1% for a typical error of 12°–15° in the dipole direction).

Table 1. The Velocity Vector from the Dipole Asymmetry in Sky Brightness

Flux-density Range	N	${\cal D}$	R.A.	Decl.	v
(mJy)		(10−2)	(deg)	(deg)	(103 km s−1)
1000 > S ⩾ 50	090360	2.3 ± 0.7	163 ± 12	−11 ± 11	1.8 ± 0.5
1000 > S ⩾ 40	114600	2.2 ± 0.6	159 ± 12	−11 ± 11	1.7 ± 0.5
1000 > S ⩾ 35	131691	2.2 ± 0.6	159 ± 11	−10 ± 10	1.7 ± 0.5
1000 > S ⩾ 30	153759	2.2 ± 0.6	159 ± 11	−07 ± 10	1.8 ± 0.5
1000 > S ⩾ 25	184237	2.2 ± 0.6	159 ± 10	−07 ± 09	1.7 ± 0.4
1000 > S ⩾ 20	228128	2.1 ± 0.5	158 ± 10	−06 ± 09	1.7 ± 0.4
1000 > S ⩾ 15	296811	2.0 ± 0.5	157 ± 09	−03 ± 08	1.6 ± 0.4

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From Table 1, the direction of the velocity vector (with our best estimate R.A. =157°  ±  9°, decl. =−03°  ±  8° or in galactic coordinates l = 248°  ±  12°, b = 44°  ±  8°) is quite in agreement with those determined from the CMBR (R.A. =168°, decl. =−7° or l = 264°, b = 48° with errors less than a degree; Lineweaver et al. 1996; Hinshaw et al. 2009). However, the estimates of v (∼1.6 ± 0.4 × 10³ km s⁻¹) appear much higher than the CMBR value (369 ± 1 km s⁻¹) by a factor of ∼4 at a statistically significant (∼3σ) level.

To guard against the possibility that some systematic effects like local clustering (mainly the Virgo supercluster) might have affected the dipole magnitude, we restricted our region of the sky brightness to that outside the super-galactic plane by rejecting sources with low super-galactic latitude, |sgb|. We determined the dipole progressively excluding sources in the latitude steps of 5°, and from a comparison of all these cases (|sgb| ⩾ 0°, 5°, 10°, 15°, 20°) no systematic changes were seen in the computed dipole magnitude (Table 2). Thus, it does not seem that the observed v, a factor of ∼4 larger than the CMBR, resulted from a local clustering.

An earlier attempt using the number counts of radio sources (Blake & Wall 2002) had found a peculiar velocity seemingly consistent with that from the CMBR observations, but from the sky brightness anisotropy we got the velocity ∼4 times the CMBR value. To ascertain that the difference somehow is not between the dipoles arising from the sky brightness and the number counts, we have determined the velocity from the number counts as well, using a technique slightly different from that of Blake & Wall (2002). First, the direction of the dipole was determined from $\Sigma \textbf {r}_i$, the three vector components (x, y, z) being essentially the same as those of the dipole position determined from the three l = 1 spherical harmonic coefficients (Blake & Wall 2002). Then, the dipole magnitude was calculated from the fractional difference ${\Delta {\cal N}}/{{\cal N}}= \Sigma \cos \theta _i\;/\;\Sigma |\cos \theta _i|=(2k/3) [2+x(1+\alpha)](v/c)=2k {\cal D}/3$, similar to that for ${\Delta {\cal F}}/{\cal F}$ in the case of sky brightness. The results are summarized in Table 3. Comparing with Table 1, we note that the observed anisotropies in both the sky brightness and the number counts yield similar velocities with magnitudes ∼4 times the CMBR value in both cases. However, while in number counts the weaker sources, because of their much larger numbers (∝S^−x), dominate the dipole determination, in the case of sky brightness, the contribution of each source being proportional to its flux density, the dipole determination depends equally on the stronger sources, S^−x × S ∼ 1 (for x ∼ 1).

Table 3. The Velocity Vector from the Number Counts

S	N	$\cal D$	R.A.	Decl.	v
(mJy)		(10−2)	(deg)	(deg)	(103 km s−1)
⩾50	091957	2.1 ± 0.5	171 ± 13	−18 ± 14	1.7 ± 0.4
⩾40	115837	1.8 ± 0.4	158 ± 12	−19 ± 12	1.4 ± 0.4
⩾35	132930	1.9 ± 0.4	157 ± 11	−12 ± 11	1.5 ± 0.3
⩾30	154996	2.0 ± 0.4	156 ± 11	−02 ± 10	1.6 ± 0.3
⩾25	185474	1.8 ± 0.4	158 ± 11	−02 ± 10	1.4 ± 0.3
⩾20	229365	1.8 ± 0.3	153 ± 10	+02 ± 10	1.4 ± 0.3
⩾15	298048	1.6 ± 0.3	149 ± 09	+15 ± 09	1.3 ± 0.2

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A comparison of our results for source counts (Table 3) with those of Blake & Wall (2002) shows that the direction estimates of the dipole match exceedingly well, with an almost one-to-one correspondence for various bins. In fact, at a first look, even the magnitudes of dipoles, with average value ${\cal D}\sim 1.9\times 10^{-2}$ in our case as compared to ∼1.8 × 10⁻² in Blake & Wall (2002), seem to match very well. However, there is an essential difference. The dipole magnitude defined by Blake & Wall (2002) as 2(2 + x(1 + α))v/c, is a factor of two larger than the ${\cal D}$ defined in our case. Thus, while the dipole expected from the CMBR value in Blake & Wall (2002) is ∼0.9 × 10⁻², in our case it is only half of that (∼0.47 × 10⁻²). The tabulated values of Blake & Wall (2002) seem about a factor of two larger than the CMBR prediction in all bins (∼1.8 × 10⁻² versus ∼0.9 × 10⁻²) though only at the ∼1.5σ level. But, in our case, the observed dipole is ∼4 times the CMBR prediction (∼1.9 × 10⁻² versus ∼0.47 × 10⁻²) at ∼3σ level. Thus, effectively, we are finding dipole magnitude to be double of that by Blake & Wall (2002), which is quite surprising since the basic data used (NVSS) are the same even if the techniques differ. In a consistent case our tabulated ${\cal D}$ for all bins should have been ∼0.9 × 10⁻² (i.e., half of the tabulated values of Blake & Wall 2002), which definitely is not the case.

The major difference in the two techniques is in the masking out of certain areas in sky which may otherwise contribute excess local sources to cause an under- or overestimate of the dipole magnitude. However, it will be very surprising if this were making the difference of two for almost all flux bins. After all the local cluster sources should have different source counts than the truly far-off sources and should be affecting different flux bins quite differently; the difference should have been especially discernible if the overall source counts are going to get affected by a factor of ∼4 or so. It will be all the more intriguing if it had to happen without causing any shifts in the direction of the dipole which in both cases is found to be the same as that of the CMBR dipole.

To us it appears that more likely the actual dipole determination in both cases (Blake & Wall 2002 and ours) is essentially the same (with the different techniques and differential masking procedure making ≲ 10% difference) and that the difference of two creeps in while relating the dipole magnitude to the velocity as the two formulae differ by a factor of two. We can only state that because of the discrepancy we are finding with the already known results (Lineweaver et al. 1996; Blake & Wall 2002; Hinshaw et al. 2009) we had to be extra careful in the magnitude scaling as well as in checking our code thoroughly, through Monte Carlo simulations and otherwise.

One way to conclusively eliminate the possibility that our large dipole value could be a consequence of some unknown or ignored local clustering in certain regions of the sky is to determine the projection of the inferred velocity for different polar angles with respect to the dipole direction and to verify if the observed velocity components are following the vcos θ relation. This is because any local clustering would affect the magnitude of the components being determined from different regions of the sky very differently. We determined the velocity components for three different polar angles for the last flux-density bin (>15 mJy) which had the largest number of sources. Starting from the dipole direction we first divided the sky into six equal-area zones. Then, computing the fractional difference in source counts between symmetrically placed pairs of sky zones, we determined the peculiar velocity components for three different polar angles. Figure 2 shows a plot of the three components, which seem to fit very well with the expected cos θ variation of v = 1300 km s⁻¹, the estimated speed for that bin in Table 3. Also plotted is the projection of the CMBR value (369 km s⁻¹) for a comparison.

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Suppose a relatively local (but perhaps on a scale much larger than that of Virgo supercluster but at z ≪ 1) overdensity generating the gravitational field responsible for the large observed motion also yields an overdensity in the distribution of radio sources and it gives a significant contribution to the excess sky brightness and to the excess counts in the direction of motion. The observed dipole amplitude will then be the sum of the contribution due to the actual motion with the added contribution of the source overdensity masquerading as high solar motion in our analysis, in that case the derived velocity could be an upper limit. But Figure 2 would be still difficult to explain there. If some excess in the sources due to local clustering in certain regions of the sky were indeed instrumental in causing the large (v = 1300 km s⁻¹) tabulated value, then the three velocity component values could not have been influenced by the same factor almost identically by the local clustering as these were determined from different slices of the sky with absolutely no overlaps. This is a clinching evidence that some local clustering is not the cause of the inferred high velocity values. The NVSS survey could be affected by a number of systematics, caused by changes in observing conditions when sweeping large areas on the sky, that may produce excess power on large angular scales. For example, Blake & Wall (2002) have noted that dim sources in NVSS show some declination dependence which largely disappears above 15 mJy, but nevertheless one cannot be totally sure that this and other possible systematics (e.g., calibration changes in time during the observing months) do not introduce a spurious modulation of the source density on large scales that may partially mimic a dipole. Again, except in a very contrived situation we could not have obtained consistent results for the projected velocity values.

The expected CMBR values are way below the observed ones in Figure 2, but one must ensure that there are no scaling problems either. Actually, the plotted values are arrived at directly from the observed asymmetry in number counts in a straightforward calculation. For example, for the first plotted point (θ ∼ 32°) the observed number counts in the respective sky zones give a fractional number ${\Delta {\cal N}}/{{\cal N}}=0.0138$, which multiplied with c/(2 + x(1 + α)) gives 1089 (km s⁻¹) as the projected velocity value, which is plotted in Figure 2. The calculations for the other plotted points are similar, and it does not seem that there could be anything amiss in the scaling. The evidence seems irrefutable that the velocity inferred from the radio source distribution is indeed much larger than that from the CMBR.

The fact that the directions of the dipole from the radio source data and the CMBR measurements are matching well implies that the cause of the dipoles is common and the motion of the solar system seems to be the only reasonable explanation for that. But such a statistically significant difference in the estimates of the magnitude of the velocity vector is puzzling. Assuming that the CMBR dipole estimates do not suffer from any residual errors during subtraction of galactic and other contributions, one cannot escape the conclusion that there is a genuine discrepancy in the two dipoles and that the reference frame defined by the radio source population at z ∼ 1 does not coincide with that defined by the CMBR originating at z ∼ 700. Here we may add that there is some evidence that the motion of the local group of galaxies may be different when measured with respect to different reference frames (Lauer & Postman 1994; Giovanelli et al. 1998) at z ≲ 0.03 and at z ∼ 0.05. However, while an anisotropy at z ≲ 0.05 scale might still be called "local," an anisotropy at z ≳ 1 is nevertheless "global" as it encompasses a substantial fraction of the universe. On the other hand, if there are differential perturbations to the Hubble flow on the scale of the distribution of the radio source population vis-à-vis that of the CMBR, the implication will be serious as any such anomaly would imply anisotropy on a universal scale. This would violate the cosmological principle where the isotropy of the universe is assumed for all epochs, and on which the entire modern cosmology rests upon.