A large disparity in cosmic reference frames determined from the sky distributions of radio sources and the microwave background radiation
aa r X i v : . [ phy s i c s . g e n - ph ] J u l A large disparity in cosmic reference frames determined from the sky distributions ofradio sources and the microwave background radiation
Ashok K. Singal ∗ Astronomy and Astrophysics Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380 009, India (Dated: July 3, 2019)The angular distribution of the Cosmic Microwave Background Radiation (CMBR) in sky shows adipole asymmetry, ascribed to the observer’s motion (peculiar velocity of the solar system!), relativeto the local comoving coordinates. The peculiar velocity thus determined turns out to be 370km s − in the direction RA= 168 ◦ , Dec= − ◦ . On the other hand, a dipole asymmetry in thesky distribution of radio sources in the NRAO VLA Sky Survey (NVSS) catalog, comprising 1.8million sources, yielded a value for the observer’s velocity to be ∼ I. INTRODUCTION
Due to the assumed isotropy of the Universe – `a lacosmological principle – an observer stationary with re-spect to the comoving coordinates of the cosmic fluid,should find the number counts of distant radio sources aswell as the sky brightness therefrom (i.e., an integratedemission from discrete sources per unit solid angle), tobe uniform over the sky. However, an observer movingwith a velocity v relative to the cosmic fluid will find,as a combined effect of aberration and Doppler boost-ing, the number counts and the sky brightness to varyas ∝ δ x (1+ α ) , where δ (= 1 + ( v/c ) cos θ , for a non-relativistic case) is the Doppler factor, c is the velocity oflight, α ( ≈ .
8) is the spectral index, defined by S ∝ ν − α ,and x is the index of the integral source counts of extra-galactic radio source population, which follows a powerlaw N ( > S ) ∝ S − x ( x ∼
1) [1–3]. The angular variationof the number counts as well as of the sky brightness canbe expressed as 1 + D cos θ , implying a dipole anisotropyover the sky with an amplitude [1–6] D = [2 + x (1 + α )] vc . (1)By observing such angular variation over the sky for asufficiently large dataset of distant radio sources, one cancompute the dipole D and thereby velocity v of the ob-server with respect to the comoving coordinates.Let ˆr i be the angular position of i th source of observedflux density S i with respect to the stationary observer, ∗ [email protected] who should find Σ S i ˆr i = . However, for a moving ob-server, due to the dipole anisotropy over the sky, it wouldyield a finite vector along the direction of the dipole. Let ˆd be a unit vector in the direction of the dipole, thenwriting ∆ F = Σ S i ˆd · ˆr i and F = Σ S i | ˆd · ˆr i | , a sum-mation over the whole sky determines magnitude of thedipole in the sky brightness as [2, 3, 6] D = D o k = 32 k ∆ FF = 32 k Σ S i cos θ i Σ S i | cos θ i | , (2)where θ i is the polar angle of the i th source with respectto the dipole direction. Here D o is the dipole determinedfrom observational data and might be affected by anygaps in the sky coverage and some other factors as dis-cussed later, and k is the correction factor, of the orderof unity, and as such would need to be determined nu-merically for individual samples.A study of the angular variation in the temperaturedistribution of the CMBR has given quite accurate mea-surements of a dipole anisotropy, supposedly arising fromthe observer’s motion (peculiar velocity of the solar sys-tem!) of 370 km s − , in the direction l = 264 ◦ , b = 48 ◦ or equivalently, RA= 168 ◦ , Dec= − ◦ [7–9]. Some ear-lier attempts to determine the dipole from the observedangular asymmetry in the sky distribution of distant ra-dio sources claimed the radio source dipole to matchthe CMBR dipole within statistical uncertainties [2, 4].However, Singal [3], from a study of the anisotropy inthe number counts as well as in the sky brightness fromdiscrete radio sources in the NVSS catalog [10], cover-ing whole sky north of declination − ◦ and containing ∼ . S > ∼ ∼ σ )level. At the same time, the direction of the velocityvector, though, was surprisingly found to be in agree-ment with the CMBR value. These unexpected findingsof the NVSS dipole being many times larger than theCMBR dipole, have since been confirmed in a numberof publications [11–15]. Such a difference between twodipoles would imply a relative motion between two cos-mic reference frames which will be against the cosmo-logical principle on which the whole modern cosmologyis based upon. Therefore it is imperative that an inves-tigation of the radio source dipole be made employingsome independent radio source samples. A recent esti-mate from the TGSS data [16], has yielded an even largeramplitude for the radio dipole [15], which is all the moredisturbing. Here we investigate this radio dipole in fur-ther details, by choosing different flux-density levels, toexamine the self-consistency of the dipole in the TGSSdata and relate the results to those from NVSS catalogby using the spectral index information between the twodatasets [17, 18]. II. TGSS DATASET
TIFR GMRT Sky Survey (TGSS) is a 150 MHz, con-tinuum survey, carried out between 2010 and 2012 us-ing the Giant Metrewave Radio Telescope (GMRT) [19],and the raw data are available at the GMRT archive.A First Alternative Data Release of the TGSS (TGSS-ADR1) [16], that includes direction-dependent calibra-tion and imaging, is available online in the public domain.The TGSS-ADR1, henceforth called TGSS, dataset cov-ers whole sky north of declination − ◦ , a total of 3 . π sr, amounting to 90% of the celestial sphere, with anrms noise below 5 mJy/beam and an approximate reso-lution of 25 ′′ × ′′ . Using a detection limit of 7-sigma,the TGSS catalog comprises 0.62 Million radio sourceswith an accuracy of about 2 ′′ or better in RA and Dec.From the spectral index data of common sources in theTGSS and NVSS catalogs [15, 17, 18], it has been demon-strated that the number counts in the two datasets couldbe compared above the flux-density limits of 100 mJy and20 mJy respectively, using a relation S NVSS ≈ . S TGSS ,and that the two catalogs are essentially complete abovethese respective flux-density limits.As the TGSS catalog [16] has a gap of sources for Dec < − ◦ , in that case our assumption of Σ S i ˆr i = fora stationary observer does not hold good. However ifwe drop all sources with Dec > ◦ as well, then withequal and opposite gaps on two opposite sides, Σ S i ˆr i = is valid for a stationary observer [3]. Exclusion of suchsky-strips, which affect the forward and backward mea-surements identically, to a first order do not have sys-tematic effects on the results [1]. We also exclude allsources from our sample which lie in the galactic plane( | b | < ◦ ), otherwise a large number of galactic sourcesin the galactic plane is likely to have an unwanted in-fluence on our determination of the radio source dipole. To ascertain effects of some systematics like local cluster-ing (mainly the Virgo super-cluster), we also examinedany alterations in our results by restricting our datasetto regions outside the super-galactic plane by rejectingsources with low super-galactic latitude ( | SGB | < ◦ ).We used Monte–Carlo simulations to create an artifi-cial radio sky with a similar number density of sourcesas in the TGSS catalog, distributed at random positionsin the sky. However, for the flux-density distributionwe took the observed TGSS sample, so that the sourcecounts remain unchanged. On this we superimposedDoppler boosting and aberration effects of an assumedmotion, choosing a different velocity vector for each sim-ulation. This artificial sky was then used to retrieve thevelocity vector under conditions similar as in our actualTGSS sample (e.g., with | Dec | > ◦ , | b | < ◦ gaps inthe sky), and compared with the input value in that par-ticular realization. This not only validated our procedureas well as the computer routine, but also helped us makean estimate of errors in the dipole co-ordinates from onehundred simulations we made, each time with randomlychosen radio source sky positions and a different velocityvector assumed for the solar peculiar motion. The errorin ∆ F / F is given by (Σ S i ) / / √ F [3], which allows usto compute error in the dipole magnitude D from Eq. (2). III. RESULTS AND DISCUSSIONA. Sky brightness
As a relatively small number of strong sources at highflux-density levels could introduce large statistical fluc-tuations in the sky brightness, we have restricted oursample here to below 5,000 mJy level. At the otherend we chose 100 mJy as the lowest cut-off limit sincethe TGSS catalog is essentially complete only above thatflux-density level [16, 17].Results for the dipole, determined from the anisotropyin sky brightness for the TGSS dataset, for sources invarious flux-density bins, are presented in Table I. Here N is the total number of sources in the correspondingflux-density bin, RA and Dec give the dipole direction insky, D o is the ’raw’ dipole value computed from 3∆ F / F that needs to be corrected for various effects as explainedbelow, k being the correction factor.The numerical factor k includes the effect of gaps insky coverage of the data as well as the effects, if any, ofthe variations of power index x and the spectral index α with flux density, and is therefore determined sepa-rately for each flux-density bin. Now, from Table I, ourestimate of the direction of the dipole (RA= 172 ◦ ± ◦ ,Dec= − ◦ ± ◦ ), is quite in agreement with that de-termined from the CMBR (RA= 168 ◦ , Dec= − ◦ , witherrors less than a degree) [7–9]. Taking this into accountwe can make our estimates of k better by using this in-formation in our simulations. Accordingly, these simu-lations differ from the ones above in the sense that the TABLE I. The dipole and velocity vector from the sky brightness for the TGSS datasetFlux-density Range N RA Dec D o k D v (mJy) ( ◦ ) ( ◦ ) (10 − ) (10 − ) (10 km s − )5000 > S ≥
250 098205 174 ± − ±
10 6 . ± .
73 1.14 5 . ± .
64 4 . ± . > S ≥
200 122549 174 ± − ±
10 6 . ± .
68 1.14 5 . ± .
60 4 . ± . > S ≥
150 160133 173 ± − ±
09 5 . ± .
63 1.11 5 . ± .
57 4 . ± . > S ≥
100 226242 172 ± − ±
09 5 . ± .
58 1.09 5 . ± .
53 4 . ± . | SGB | > ◦ Flux-density Range N RA Dec D o k D v (mJy) ( ◦ ) ( ◦ ) (10 − ) (10 − ) (10 km s − )5000 > S ≥
250 082336 170 ± − ±
11 6 . ± .
82 1.15 5 . ± .
71 4 . ± . > S ≥
200 102700 169 ± − ±
11 6 . ± .
76 1.14 5 . ± .
67 4 . ± . > S ≥
150 134106 167 ± − ±
10 6 . ± .
71 1.12 5 . ± .
63 4 . ± . > S ≥
100 189416 167 ± − ±
10 5 . ± .
65 1.10 5 . ± .
59 4 . ± . dipole used now is in the direction of the CMBR dipoledirection, superimposed on randomly assigned source po-sitions in the sky. It should be noted that this procedureallows us to make a more realistic assessment of the in-fluence of sky-gaps on our determination of the dipolemagnitude and does not introduce bias of any kind. Acomparison of the dipole derived from each simulationwith the input dipole magnitude yielded the correctionfor that simulation. A set of 100 different simulationswas used to determine an average value of the correctionfactor k , given in Table I. The corrected dipole D is ob-tained by dividing the ’raw’ dipole strength D o by k foreach flux-density bin.From Table I we see a trend that the dipole strength D falls systematically with a decrease in the lower flux-density cut-off. From 250 mJy to 100 mJy, D graduallyfalls as much as by ∼ v , the dipole D could get af-fected by two quantities, the power law index x and thespectral index α (Eq. (1)). To estimate effects of thepower index x on D , we have plotted in Fig. (1) theintegrated source counts, N ( > S ) for different S for theTGSS and NVSS samples. The index x in the powerlaw relation, N ( > S ) ∝ S − x , can be estimated from theslope of the log − log S plot in Fig. (1), where we findthat the index x steepens from low to high flux-densitylevels for both samples. From piece-wise straight linefits to the log N − log S data, we find that x steepensfrom − .
85 ( − .
95) at 100 (20) mJy to − . − . − .
05 ( − .
1) at 250(50) mJy in the TGSS (NVSS) data. It should be notedthat the value of x that enters into Eq. (1) is the one atthe lower cut-off flux density of the bin. On the otherhand variation in α is much smaller. From a comparison of TGSS and NVSS samples, the spectral index for theflux-density range 100 < S TGSS <
200 mJy was found tobe 0 . ± .
245 while for S TGSS >
200 mJy it turned outto be 0 . ± .
225 [18]. This slight steepening of α withflux density would affect the dipole value only by < ∼ α = 0 .
76 for the 100 (20) mJybin and α = 0 . x and α . The peculiarvelocity v , calculated accordingly from Eq. (1), is listedfor each flux-density bin in Table I, where no significanttrend with changing flux-density cut-off levels is seen.The underlying assumption throughout is that v repre-sents motion of the observer (solar system!), with respectto the corresponding reference frame of radio sources, inthe direction given by RA and Dec of the dipole, andits value should not vary from one flux-density bin toanother.As we mentioned above, the direction of the dipolefrom TGSS data is quite in agreement with that of theCMBR dipole. However, the strength of the dipole ( D ≃ . × − , the best estimate in Table I) appears an orderof magnitude larger than the CMBR dipole ( ≃ . × − ) [3], even though it is smaller by a factor of ∼ . × − ) [15]. When comparedwith the dipole determined from the sky brightness in theNVSS dataset [3], the TGSS dipole is a factor of ∼ . ≃ . × − ).To rule out the possibility that the excessive dipolestrength in Table I might be the result of some local clus-tering (e.g., the Virgo super-cluster), we determined thedipole from the sky brightness from radio sources out-side the super-galactic plane by dropping sources withlow super-galactic latitude, | SGB | < ◦ . We see thatthe computed dipole (Table II) is still an order of magni- TABLE III. The velocity vector from the number countsFlux-density cut-off N RA Dec D o k D v (mJy) ( ◦ ) ( ◦ ) (10 − ) (10 − ) (10 km s − ) ≥
250 099736 168 ± − ±
09 5 . ± .
53 1.05 5 . ± .
50 3 . ± . ≥
200 124080 166 ± − ±
09 4 . ± .
47 1.05 4 . ± .
45 3 . ± . ≥
150 161664 164 ± − ±
08 4 . ± .
42 1.00 4 . ± .
42 3 . ± . ≥
100 227773 162 ±
09 +03 ±
08 4 . ± .
35 0.96 4 . ± .
37 3 . ± . N ( > S ) against S , for the TGSS and NVSS samples, showing the power lawbehavior ( N ( > S ) ∝ S − x ) of the source counts. From piece-wise straight line fits to data in different flux-density ranges ineither sample, index x appears to steepen for stronger sources,as shown by continuous lines with the best-fit x values shownabove. tude larger than the CMBR dipole, but lies in the samedirection, and that this anomalous result is not due to alocal clustering. B. Number counts
We have determined the dipole and the solar peculiarvelocity from the number counts as well. First the direc-tion of the dipole was determined from Σ ˆr i . With ˆd asa unit vector in the direction of the dipole, we define thefractional difference as∆ NN = Σ ˆd · ˆr i Σ | ˆd · ˆr i | = Σ cos θ i Σ | cos θ i | , (3)where θ i is the polar angle of the i th source with respectto the dipole direction. The dipole magnitude is thencalculated from the fractional difference D = D o k = 32 k ∆ NN , (4)similar to that from ∆ F / F in the case of sky brightness(Eq. (2)). Since, unlike in the case of sky brightness, a small number of bright sources do not adversely affectthe number counts, in the latter case we have relaxedthe upper limit of 5000 mJy on the flux density.Like in the case of sky brightness, here too we createdan artificial radio sky with sources distributed at randompositions in the sky, but with a flux-density distributionas of the TGSS sample, so that the source counts remainunchanged. On this was superimposed a mock dipoleoriented in a random direction and of random magni-tude. One hundred such independent simulations weremade to estimate the expected errors in the dipole co-ordinates. The error in ∆ N / N is given by 2 / √ N , thenfrom Eq. (4), error in D o is p /N . For estimating k ,another set of 100 simulations were made by choosingdipoles oriented in the direction of the CMBR dipole di-rection, superimposed on randomly assigned source po-sitions in the sky. Such derived correction factor k wasused to divide D o to get D for each flux-density bin. Thepeculiar velocity v , was then calculated using appropriate x nd α values for each flux-density bin.Results from the number count are summarized in Ta-ble III. Comparing with Table I we notice that the di-rections of the dipoles determined both from the skybrightness as well as from the number counts, are con-sistent with that of the CMBR. However, the numbercounts yield a magnitude of the dipole (and the therebyinferred solar peculiar velocity) to be somewhat smaller( ∼ x , this may resultin a dipole estimate to be higher by a factor of ∼ . TABLE IV. The dipole magnitude and speed estimates for the TGSS and NVSS samples with respect to the CMBR dipoledirectionsample ν S N σ N N N δN δN/σ N D o k D v (MHz) (mJy) ( √ N ) ( N − N ) (10 − ) (10 − ) (10 km s − )TGSS 150 >
250 99736 316 51254 48482 2772 8.8 5 . ± .
63 1.14 4 . ± .
56 3.74 ± . >
200 124080 352 63648 60432 3216 9.1 5 . ± .
57 1.15 4 . ± .
49 3.59 ± . >
150 161664 402 82852 78812 4040 10.0 5 . ± .
50 1.09 4 . ± .
46 3.79 ± . >
100 227773 477 116532 111241 5291 11.1 4 . ± .
42 1.05 4 . ± .
40 3.77 ± . >
50 91652 303 46372 45280 1092 3.6 2 . ± .
66 1.29 1 . ± .
51 1.39 ± . >
40 115905 340 58547 57358 1189 3.5 2 . ± .
59 1.25 1 . ± .
47 1.26 ± . >
30 155110 394 78434 76676 1758 4.5 2 . ± .
51 1.21 1 . ± .
42 1.48 ± . >
20 229551 479 115932 113619 2313 4.8 2 . ± .
42 1.15 1 . ± .
36 1.43 ± . the NVSS dataset [3, 11–15], we find that the directionsof the dipole from both these datasets match well withthe CMBR measurements, implying that the cause ofthe dipoles is common and a peculiar motion of the solarsystem seems to be the only reasonable interpretation forthat. However such a statistically significant disparity intheir magnitudes, with the TGSS dipole being an orderof magnitude (a factor of ∼
10) larger than the CMBRdipole, while the NVSS dipole being ∼ C. Radio survey dipoles with respect to theCMBR dipole direction
The fact that the directions of the dipole from the radiosource data and the CMBR measurements are matchingwell, suggests that the direction of the CMBR dipole,known with high accuracy, could be taken to be the di-rection for the radio source dipoles too. However, weneed to first explicitly examine for both TGSS and NVSSdatasets if there exist indeed dipoles in the radio sourcesky distribution with respect to the CMBR dipole di-rection. For this we compute the dipole strength andthe inferred velocity for both radio source datasets, butnow with respect to the CMBR dipole direction, viz.RA= 168 ◦ , Dec= − ◦ . For this we employ an alternateprocedure which is more transparent, simpler in natureand more easily visualized.Using the great circle at 90 ◦ from the CMBR dipoledirection, we divide the sky in two equal hemispheres, Σ and Σ , with Σ containing the CMBR dipole, and Σ containing the direction opposite to the CMBR dipole.Then if there is indeed a motion of the observer along theCMBR dipole direction, due to a combined effect of theaberration and Doppler boosting, the number counts willhave a dipole anisotropy, 1 + D cos θ , over the sky withan amplitude D = [2 + x (1 + α )] v/c (Eq. (1)), θ being theangle measured from the CMBR dipole direction. Thenthe number of sources in the hemisphere Σ , should be larger than the number of sources in the hemisphere Σ .Let φ be a complementary angle to θ , i.e. φ = π/ − θ ,with φ measured towards the CMBR dipole direction,starting from the great circle that divides the sky intohemispheres Σ and Σ . Now, counting from the greatcircle, if we denote by N the number of sources between0 and φ in Σ , then we can write N = 2 πN Z φ (1 + D sin φ ) cos φ d φ, (5)where N is the number density per unit solid angle foran isotropic distribution, in the absence of any peculiarmotion. Similarly we can write the number of sources inthe opposite hemisphere Σ as N = 2 πN Z − φ (1 + D sin φ ) cos φ d φ, (6)Then the fractional excess in number of sources in skyregion between 0 and φ in Σ over the corresponding,symmetrically placed, opposite region in Σ will be δNN = N − N N + N = D sin φ D cos θ . (7)Thus the dipole D could then be determined from δN/N computed for the whole sky ( φ = π/ D = D o k = 2 k δNN . (8)where k is a constant, of the order of unity, to be deter-mined numerically for individual samples.For estimating k , Monte–Carlo simulations were madeby choosing mock dipoles, oriented in the direction of theCMBR dipole direction, superimposed on randomly as-signed source positions in the sky but with a flux-densitydistribution as of the TGSS sample, so that the sourcecounts remain unaffected. From ∼
100 such computersimulations, we estimated k for our TGSS sample for dif-ferent flux-density cut-offs.Our results are presented in Table IV, which is almostself-explanatory. Dipole D was estimated for samplescontaining all sources with flux-density levels > S , start-ing from S = 250 mJy and going down to S = 100 mJylevels. Of course the accuracy in our estimate improvesas we go to lower flux-density limits since the number ofsources increases as N ( > S ) ∝ S − x (with x ∼ ∼ ∼ × / (2 × ) = 2 × − radian is ∼ π × × − sror 10 − fraction of the sky, which for the N values in Ta-ble IV contains only one or two sources, with negligiblecontribution to δN (or even to σ N ) at any flux-densitylevel.Our estimate of the magnitude of the velocity vector( v ≃ ±
340 km s − ) from Table IV appears order ofmagnitude higher than the CMBR value (370 km s − ).The quoted errors for v in Table IV are from the expecteduncertainty σ N (= √ N ) in δN (= N − N ), the uncer-tainty here being that of a binomial distribution, similarto that of the random-walk problem (see, e.g. [20]). Thecorresponding 1 σ uncertainty in δN/N is 1 / √ N , thenfrom Eq. (8), error in D o is 2 / √ N .For a comparison, we also employed the same tech-nique to estimate the magnitude of the velocity vectorfrom the NVSS data, with respect to the CMBR dipoledirection. The results for NVSS dataset are also summa-rized in Table-IV, where v turns out to be ≃ ± − . In Table IV, the four rows of TGSS datasetat various flux-density levels could be compared to thecorresponding four rows of the NVSS dataset. We no-tice that while the total number N of sources at eachflux-density level do match reasonably well (within 1%to < ∼ δN between two hemispheresin respective flux-density bins, and the thereby derivedmagnitudes of dipole D and velocity v , differ as much asby a factor of ∼ .
5. Of course all estimates of dipole D and velocity v in either dataset are way above the valuesexpected from the CMBR.From the rms errors ( ∼
300 to 400 km s − , Table 1V),determined basically by the radio source density field, forthe peculiar velocity estimates for the two samples, detec-tion of a peculiar velocity like the CMBR value ( ∼ − ) would not have been possible as it would bewithin 1 σ level. However, because of the much largeramplitude of the peculiar velocity, by a factor of ∼ ∼
10 for the TGSS data, a posi-tive detection became possible at statistically significant, ∼ σ and ∼ σ levels, respectively. FIG. 2. A plot of the fractional cumulative excess δN /N against φ , as observed in the sky regions of Σ over those ofΣ , for the TGSS ( >
100 mJy) and NVSS ( >
20 mJy) sam-ples. The corresponding peculiar velocity of the solar systemis shown on the right hand scale. The dotted lines show theactual observed δN /N values, while the continuous lines showtheir expected ( ∝ cos θ ) behavior. Some representative datapoints are plotted, as circles (o) for the TGSS data and crosses(x) for the NVSS data, with error bars, calculated for a ran-dom (binomial) distribution. For a comparison, the dashedline shows δN/N , expected for the peculiar velocity equal tothe CMBR value, v = 370 km s − . Here we have explored the radio source dipole bystudying any excess of radio source density with respectto the CMBR direction, the latter itself incidentally didnot use any information from the radio survey datasets.Moreover, as we move to lower flux-density levels, δN seems to steadily increase, specifically, in none of the flux-density bin, for either dataset, we find N to be largerthan N . Now if an excess in the sources due to somelocal clustering in certain regions of the sky were indeedmasquerading as a radio source dipole, only in a verycontrived situation would one expect to get N > N atall flux-density levels and that too for both radio catalogs .From Eq. (7), we expect that the fractional excess, δN /N , should have a sin φ or cos θ dependence. We canverify this sin φ dependence of δN /N by making cumu-lative counts of N and N as a function of φ . We shouldexpect large departures from the expected sin φ behaviorfor small φ , not just because of larger statistical fluctu-ations in a Binomial distribution for small numbers, butalso because dipole strength builds up only at larger φ (number distribution having a dipole D cos θ = D sin φ ),i.e., when one approaches the dipole direction at smaller θ . Figure (2) shows the fractional excess, δN/N , for bothTGSS and NVSS data as a function of φ or θ . We haveused the number counts for the bins S TGSS >
100 and S NVSS >
20 as these have the largest number of sourcesin our samples. As expected, in Fig. (2), we see largefluctuations for smaller φ . While the fractional excess inthe NVSS data does stabilize at φ ≈ ◦ , in the TGSSdata it seems to stabilizes only around φ ≈ ◦ . However,from Fig. (2) it is clear that not only in both TGSS andNVSS cases is the fractional excess, δN /N , way abovethat expected from the CMBR value of peculiar veloc-ity, viz. 370 km s − , but also that TGSS dipole is muchstronger than the NVSS dipole.The evidence seems irrefutable that the peculiar veloc-ity of the solar system estimated from the distant radiosource distributions in sky is indeed much larger thanthat inferred from the CMBR sky distribution. Such astatistically significant difference in the estimates of themagnitude of the peculiar velocity is puzzling and onecannot escape the conclusion that there is a genuine dis-parity in the three reference frames defined by the radiosource populations selected at different frequencies andthe CMBR.From the clustering properties of radio sources in theTGSS angular spectrum on large angular scales, corre-sponding to multipoles 2 ≤ l ≤
30, the amplitude ofthe TGSS angular power spectrum is found to be sig-nificantly larger than that of the NVSS, and from thatquestions have been raised [21] that some unknown sys-tematic errors may be present in the TGSS dataset. Atthe same time, while the amplitude of the dipole ( l = 1)too is significantly larger than that of the NVSS, the self-consistency of the TGSS dipole in different flux-densitybins and the fact that the direction of the dipole coin-cides with that of the CMBR dipole, indicates that theTGSS dataset may not be affected to such a great extentby systematics.Now, unless one wants to disregard radio dipoles, de-rived from TGSS and NVSS datasets, altogether, at thisstage one may be left mainly with only two alternatives.One of them is to say that there may be something amissin the interpretation of the observed dipoles (includingthe CMBR) as reflecting observer’s motion (peculiar ve-locity of the solar system!) and that the strength of adipole may not be representing an observer’s peculiarspeed. In this line of thinking, one will then have toexplain the existence of a common direction of all thedipoles and that what is so peculiar about this direction and whether it represents some sort of an “axis” of theuniverse. The other alternative would be to still followthe conventional wisdom that these dipoles are arising asa result of observer’s motion and that these dipole mag-nitudes differing by as much as an order of magnitude,indicates that there may be a large relative motion of thevarious cosmic reference frames. Either alternative doesnot fit with the cosmological principle, which is the start-ing point for the standard modern cosmology. Perhaps itpoints out to the need for a fresh look at the role of thecosmological principle in the cosmological models. IV. CONCLUSIONS
From the dipole anisotropy computed for the TGSSdataset, it was found that the dipole strength and thethereby inferred peculiar motion of the solar system isan order of magnitude larger than that inferred from theCMBR dipole. The TGSS dipole is also larger than theNVSS dipole by a factor of ∼ .
5. But the direction of thedipole in all these cases turns out to be the same withinerrors. An obvious inference is that the reference framesdetermined from some of the most distant observables,viz. the CMBR, the NVSS 1400 MHz dataset of radiosources and the TGSS 150 MHz dataset of radio sources,somehow do not coincide with each other, which raisesuncomfortable questions about the cosmological princi-ple, the basis of the modern cosmology.
ACKNOWLEDGEMENTS
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