A Test of the Cosmological Principle with Quasars
Nathan Secrest, Sebastian von Hausegger, Mohamed Rameez, Roya Mohayaee, Subir Sarkar, Jacques Colin
aa r X i v : . [ a s t r o - ph . C O ] S e p Draft version October 1, 2020
Typeset using L A TEX twocolumn style in AASTeX63
A Test of the Cosmological Principle with Quasars
Nathan J. Secrest, Sebastian von Hausegger,
2, 3,4
Mohamed Rameez, Roya Mohayaee, Subir Sarkar, andJacques Colin U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA INRIA, 615 Rue du Jardin-Botanique, 54600 Nancy Grand-Est, France Sorbonne Universit´e, UPMC, CNRS, Institut d’Astrophysique de Paris, 98bis Bld Arago, Paris 75014, France Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom Dept. of High Energy Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
ABSTRACTWe study the large-scale anisotropy of the Universe by measuring the dipole in the angular distribu-tion of a flux-limited, all-sky sample of 1.3 million quasars observed by the Wide-field Infrared SurveyExplorer (WISE). This sample is derived from the new CatWISE2020 catalog, which contains deepphotometric measurements at 3.4 and 4.6 µ m from the cryogenic, post-cryogenic, and reactivationphases of the WISE mission. While the direction of the dipole in the quasar sky is similar to that ofthe cosmic microwave background (CMB), its amplitude is over twice as large, rejecting the canonical,exclusively kinematic interpretation of the CMB dipole with a p-value of 10 − (3 . σ ), the highest sig-nificance achieved to date in such studies. Our results are in conflict with the cosmological principle,a foundational assumption of the concordance ΛCDM model. Keywords: cosmology: large-scale structure of universe — cosmology: cosmic background radiation —cosmology: observations — quasars: general — galaxies: active INTRODUCTIONThe standard Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) cosmology is based on the cosmological princi-ple, which posits that the universe is homogeneous andisotropic on large scales. This assumption is supportedby the smoothness of the CMB, which has temperaturefluctuations of only ∼ ∼ Corresponding author: Nathan J. [email protected] of matter on local scales, originally dubbed the “GreatAttractor” (see, e.g., Dressler 1991).A consistency check would be to measure the concomi-tant effects on higher multipoles of the CMB anisotropy(Challinor & van Leeuwen 2002); however, even the pre-cise measurements of these by Planck allow up to 40%of the observed dipole to be due to effects other than theSolar System’s motion (see discussion in Schwarz et al.2016). According to galaxy counts in large-scale sur-veys the universe is sensibly homogeneous when aver-aged over scales larger than &
100 Mpc, as is indeed ex-pected from considerations of structure formation in theconcordance ΛCDM model. Hence the reference frameof matter at still greater distances should converge tothat of the CMB; i.e. the dipole in the distribution ofcosmologically distant sources, induced by our motionvia special relativistic aberration and Doppler shiftingeffects, should align both in direction and in amplitudewith the CMB dipole. Independent measurements of thedistant matter dipole are therefore an important test ofthe cosmological principle, and equivalently of the stan-dard model of cosmology.Ellis & Baldwin (1984) proposed that such a test bedone using counts of radio sources. These are typi-
Secrest et al. cally active galactic nuclei (AGNs) at moderate redshift( z ∼ z < . S ν ∝ ν − α , and integral sourcecounts per unit solid angle d N/ dΩ ( > S ν ) ∝ S − xν , abovesome limiting flux density S ν . If we are moving with ve-locity v ( ≪ c ) with respect to the frame in which thesesources are isotropically distributed, then being “tiltedobservers” we should see a dipole anisotropy of ampli-tude (Ellis & Baldwin 1984): D = [2 + x (1 + α )] v/c. (1)The advent of the 1.4 GHz NRAO VLA Sky Sur-vey (NVSS; Condon et al. 1998), which contains ∼ . µ m, 4.6 µ m, 12 µ m, and22 µ m (W1, W2, W3, and W4). This provides a mea-surement of the dipole that is independent of the radiosurvey-based results, as WISE is a space mission withits own unique scanning pattern, not constrained bythe same observational systematics that affect ground-based surveys, such as declination limits or atmo-spheric effects. While WISE, along with 2MASS, hasbeen used before to set useful constraints on the mat-ter dipole (Gibelyou & Huterer 2012; Yoon et al. 2014;Alonso et al. 2015; Bengaly et al. 2017; Rameez et al.2018), these studies were of relatively nearby galaxies( z ∼ . − .
1) where contamination from local sourcescan be significant and has to be carefully accounted for.In Section 2, we detail the quasar sample that we use,and we introduce our methodology in Section 3. Ourresults are presented in Section 4, and we discuss theirsignificance for cosmology in Section 5. QUASAR SAMPLEBecause of the unique power of mid-infrared photom-etry to pick out AGNs, WISE may be used to create re-liable AGN/quasar catalogs based on mid-infrared coloralone (e.g., Secrest et al. 2015). We require an AGNsample optimized for cosmological studies, so the objectsshould preferably be quasars: AGN-dominated and atmoderate or high-redshift ( z & .
1; cf., Tiwari & Nusser2016). The sample should cover as much of the celestialsphere as is possible to minimize the impact of missing(or masked) regions, and be as deep as possible to con-tain the largest number of objects and thus have thegreatest statistical power.We created a custom quasar sample from the newCatWISE2020 data release (Eisenhardt et al. 2020),which contains sources from the combined 4-band cryo,3-band cryo, post-cryo NEOWISE, and reactivationNEOWISE-R data. The CatWISE2020 catalog is0.71 mag and 0.45 mag deeper in W1 and W2 than theprevious AllWISE catalog. We select all sources in theCatWISE2020 catalog with valid measurements in W1and W2, which are the most sensitive to AGN emis-sion (e.g., Stern et al. 2012). We cut out any sourceswith possible saturation, as well as sources flagged assuffering from possible contaminants. To select AGNs,we impose the color cut W1 − W2 ≥ . S ν ∝ ν − α ) that is insensitive to heavy dust red-dening at shorter wavelengths. This yields a raw sampleof 174,701,084 objects.We then remove low-redshift AGNs by excludingsources in the 2MASS extended source catalog (XSC;Jarrett et al. 2000), which contains nearly all galaxiesnot directly behind the Galactic plane out to z ∼ . | b | > ◦ to be effective in com-pletely removing non-uniformity because of the Galaxy.The second is poor-quality photometry near clumpy andresolved nebulae both in our Galaxy (e.g., planetarynebulae) and in nearby galaxies such as the MagellanicClouds and Andromeda. We remove these by maskingout 6 times the 20 mag arcsec − isophotal radii from the2MASS Large Galaxy Atlas (LGA; Jarrett et al. 2003).The third is a decrement of sources, and the presence ofimage artifacts, near bright stars, caused by density sup- he Quasar Dipole We find that circular maskswith 2MASS K band-dependent radii log ( r/ deg) = − . K − .
471 effectively remove these. In all, wemasked 265 sky regions, plus the Galactic plane. Toavoid any directional source count bias we mirror themasks by 180 ◦ on the celestial sphere.We calculate spectral indices α of our sources in theW1 band by obtaining power-law fits of the form S ν = kν − α , where k is the normalization. We produced alookup table to determine α based on W1 − W2, by cal-culating synthetic AB magnitudes following Equation 2of Bessell & Murphy (2012). The WISE magnitudes areon the Vega magnitude system, so we convert from theAB system using the offsets m AB − m Vega = 2 . − .
60 associated with the def-inition of the synthetic AB magnitude. The normalisa-tion k is calculated by inverting the equation for the syn-thetic magnitude and using the observed W1 AB mag-nitude. Finally, we calculate the isophotal frequency, atwhich the flux density S ν equals its mean value withinthe passband, using Equation A19 in Bessell & Murphy(2012). As our sample was constructed with the cutW1 − W2 ≥ .
8, the distribution peaks at α ∼ . × Hz,with a dispersion of 0.19%. We select a magnitude cutof 9 > W1 > . . > S ν > .
09 mJy, to fix the over-densityof fainter sources along overlaps in the WISE scanningpattern, most prevalent at the ecliptic poles where theyconverge. After removing low- z AGNs, applying the skymasks, and making the flux density cut, our final samplehas 1,314,428 AGNs, which we show in Figure 2.To estimate the distribution of AGN redshifts, we se-lect those within SDSS Stripe 82, a 275 deg region ofthe sky scanned repeatedly by the SDSS, thus achievingan increase of depth of ∼ specObj table for SDSS DR16, Stripe 82 contains ∼ . r -bandmagnitudes fainter than 20 (AB) than a comparable skyregion in the SDSS main footprint. We use a sub-regionof Stripe 82 between − ◦ < R . A . < ◦ , which liesoutside the | b | < ◦ Galactic plane mask we employ,and which was observed by the Extended Baryon Os-cillation Spectroscopic Survey (eBOSS; Dawson et al. http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec6 2.html Figure 1.
Distribution of flux densities S ν ( ∝ ν − α ) andspectral indices α (W1 band) in the CatWISE AGN sample. Figure 2.
Sky map of the CatWISE AGN sample, in Galac-tic coordinates. i -band depth of 23.44 mag (AB). Us-ing a 10 ′′ match for completeness, we find counterpartsfor 14,343 (99.7%) of the CatWISE AGNs. Matchingthe DES counterpart coordinates onto specObj table towithin 1 ′′ for fiber coverage, we find 8612 matches (60%).The unmatched objects are 0.3 mag fainter in W2 thanthe matched objects on average, suggesting that they areslightly less luminous or slightly more distant (or both).However, their mean r − W2 value, a measure of AGNobscuration level (e.g., Yan et al. 2013), is ∼ . r − W2 > Secrest et al. . . . . . . . . . . . . . . . . . P D F Figure 3.
Redshift distribution (normalized as a probabilitydensity function) of the CatWISE AGN sample. the prevalence of type-2 AGNs (Yan et al. 2013), 77% ofthe unmatched sample have r − W2 >
6. This indicatesthat the objects in our sample without SDSS spectra arepredominantly type-2 systems, an effect of the orienta-tion of the AGN with respect to the line of sight, and sothe matched objects may be used to estimate the distri-bution of redshifts for the full sample. We find a meanredshift of 1.2, with 99% having z > .
1, i.e. the Cat-WISE AGN sample is not contaminated by low-redshiftAGNs. The redshift distribution of our sample is shownin Figure 3. METHOD3.1.
Dipole Estimator
We determine the dipole of our sample with the 3-dimensional linear estimator: ~ D l = 3 N N X i =1 ˆ r i , (2)where ˆ r i is the unit vector pointing to source i , and N is the sample size. This estimator simply calcu-lates the mean resultant length and direction of the N unit vectors and is agnostic with regard to thetrue underlying signal (e.g., Fisher et al. 1987), as op-posed to other estimators (e.g., Blake & Wall 2002;Bengaly et al. 2019) which explicitly seek a dipolar pat-tern. However, if the signal has a dipolar form thenEquation 2 generally has a bias in both amplitude anddirection (Rubart & Schwarz 2013) induced by Poissonnoise and masking. We account for amplitude bias inour results as well as in the estimates of their signif-icance using Monte Carlo methods, correcting for di-rectional bias as discussed in Appendix A. We furtherconfirm our results by employing the quadratic estima-tor ~ D q which does not suffer from bias and is evaluatedby minimising the quantity (e.g., Bengaly et al. 2019): X p h n p − ¯ n (cid:16) ~ D q · ˆ r p (cid:17)i ¯ n (cid:16) ~ D q · ˆ r p (cid:17) (3)where n p denotes the number of sources in sky pixel p (ˆ r p being the unit vector to the pixel) and the sum isto be taken over all unmasked pixels (in which n is theaverage number of sources). Due to significantly highercomputational expense for the quadratic estimator, werun simulations only with the linear estimator.3.2. Mock data and statistical significance
We generate mock samples of N init vectors drawn froma statistically isotropic distribution, whose directions aresubsequently modified by special relativistic aberrationaccording to an observer boosted with velocity ~β . Eachsample is then masked with the same mask that was ap-plied to the data (Figure 2). In order to respect the ex-act distribution of flux values and spectral indices in thedata, we assign to each simulated source a flux density S ν and a spectral index α drawn at random from theirempirical distributions before applying the flux densitycut (Figure 1). The sampled fluxes are now modulateddepending on source position, velocity ~β , and α . Lastly,only sources with S ν > S ν, cut are retained, and the num-ber of remaining sources is finally reduced to that of thetrue sample, N , through random selection.Under the null hypothesis that the measured dipole ~ D l is a consequence of our motion with respect to a frameshared by both quasars and the CMB, we generate aset of mock skies according to the above recipe. Foreach random choice we record ~ D sim l , and correct for itsdirectional bias using Equations A3 and A4. The frac-tion of mock skies with amplitude | ~ D l | larger than ourempirical sample, and with angular distance from theCMB dipole closer than our sample, gives the p-valuewith which the null hypothesis is rejected. Note thatthe effect on our results of the distributions of flux andspectral index (Figure 1) is automatically included viathe bootstrap approach employed for our simulations. RESULTSOur sample of 1,314,428 quasars exhibits a dipolewith amplitude: D l = 0 . l, b ) = (234 . ◦ , . ◦ . ◦ l, b = 264 . ◦ , . ◦ ~ D sim l with an amplitude larger than the observed value (left he Quasar Dipole Figure 4.
Left panel:
Observed dipole amplitude D l (solid vertical line) in the CatWISE AGN sample, versus the expectationassuming the kinematic interpretation of the CMB dipole; the distribution of D sim l from the simulations (Section 3.2) is shownalong with its median value (dashed vertical line). Right panel:
Dipole direction of the CatWISE AGN sample in Galacticcoordinates using the bias-corrected linear estimator ~ D l (circle) and the unbiased quadratic estimator ~ D q (triangle); the shadedarea indicates the model-dependent 95% confidence level simulated using the velocity from the quadratic estimator. panel, Figure 4) and within 29 . ◦ − corresponding toa significance of 3 . σ .If we assume that the anomalous quasar dipole is stillof kinematic origin, albeit with a velocity different fromthat inferred from the CMB, we can estimate its di-rectional uncertainty. To avoid bias, we first computethe dipole with the quadratic estimator D q , which gives D q = 0 . l, b ) = (234 . ◦ , . ◦ α =1 .
17 and index x = 1 . v = 861 km s − . A set of 15,000 simulations with thisinput velocity is then performed to find the directionaluncertainty. The right panel of Figure 4 shows this as apatch around the (consistent) dipole direction obtainedwith both estimators. DISCUSSIONThe CatWISE AGN sample exhibits an anomalousdipole, oriented similarly to the CMB dipole but overtwice as large. Whereas a “clustering dipole” is ex-pected from correlations in the spatial distribution ofthe sources, this can be estimated knowing their auto-correlation function (or power spectrum) and distribu-tion in redshift (see Appendix B). It is smaller by a fac-tor of ∼
60 than the dipole we observe in these higherredshift quasars.The unique statistical power of our study has allowedus to confirm the anomalously large matter dipole sug-gested in previous work, which used objects selected ata different wavelength (radio), using surveys completelyindependent of WISE, viz. NVSS, WENNS, SUMMS,and TGSS. The ecliptic scanning pattern of WISE hasno relationship with the CMB dipole, so there is no rea- son to suspect that the dipole we measure in the Cat-WISE AGN catalog is an artifact of the survey.After Ellis & Baldwin (1984) proposed this importantobservational test of the cosmological principle, agree-ment was initially claimed between the dipole anisotropyof the CMB and that of radio sources (Blake & Wall2002). If the rest frame of distant AGNs is indeed thatof the CMB, it would support the consensus that thereexists a cosmological standard of rest, related to quanti-ties measured in our heliocentric frame via a local specialrelativistic boost. This underpins modern cosmology:for example, the observed redshifts of Type Ia super-novae are routinely transformed to the “CMB frame”.From this it is deduced that the Hubble expansion rateis accelerating (isotropically), indicating dominance of acosmological constant, and this has led to today’s con-cordance ΛCDM model. If the purely kinematic inter-pretation of the CMB dipole that underpins the aboveprocedure is in fact suspect, then so are the importantconclusions that follow from adopting it. In fact, asobserved in our heliocentric frame, the inferred acceler-ation is essentially a dipole aligned approximately withthe local bulk flow of galaxies and towards the CMBdipole (Colin et al. 2019), so cannot be due to a cosmo-logical constant.If it is established that the distribution of distant mat-ter in the large-scale universe does not share the samereference frame as the CMB, then it will become im-perative to ask whether the differential expansion ofspace produced by nearby nonlinear structures of voidsand walls and filaments can indeed be reduced to justa local boost (Wiltshire et al. 2013). Alternatively theCMB dipole may need to be interpreted in terms of newphysics, e.g. as a remnant of the pre-inflationary uni-verse (Turner 1991). Gunn (1988) had noted that this
Secrest et al. issue is closely related to the bulk flow observed in thelocal universe, which in fact extends out much furtherthan is expected in the concordance ΛCDM model (e.g.,Colin et al. 2011; Feindt et al. 2013). Further work isneeded to clarify these important issues.As Ellis & Baldwin (1984) emphasized, a serious dis-agreement between the standards of rest defined by dis-tant quasars and the CMB may require abandoning thestandard FLRW cosmology itself. The importance ofthe test we have carried out can thus not be overstated. ACKNOWLEDGMENTSWe thank Jean Souchay for helpful discussions.N.J.S., M.R. and S.S. gratefully acknowledge the hos-pitality of the Institut d’Astrophysique de Paris. S.v.H.is supported by the EXPLORAGRAM Inria AeX grantand by the Carlsberg Foundation with grant CF19 0456.
Facilities:
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Secrest et al.
APPENDIX A. DIRECTIONAL BIAS OF THE LINEAR ESTIMATORGiven a dipolar signal, Rubart & Schwarz (2013) demonstrated that the amplitude of the linear estimator (Equa-tion 2) is biased, but not its direction. This can be seen by evaluating h ~ D l i = 3 N π Z dΩ (cid:16) r · ~d (cid:17) · ˆ r ∝ ~d, (A1)where the angular brackets denote the expectation value of the estimator given a dipolar probability distribution,and ~d is the direction of the dipole. The amplitude bias stems from Poisson noise, always present in a sample of finitesize N . However, removing sources by masking alters the integral’s bounds and generally induces directional bias aswell. While the directional offset then caused by the first term in Equation A1 (the monopole) is alleviated by choosinga mask that is symmetrical with respect to the observer, the contribution by the second term (the dipole) is not. Thiseffect was later worked out analytically for simple mask shapes by Rubart (2015), whose results we reproduce here forreference.The most prominent mask that we apply to our sample is the removal of the Galactic plane along lines of constantlatitude, b . Considering only this, the estimated direction is h ~R i gal.mask = N π Z π d φ "Z π d θ sin( θ ) (cid:16) r · ~d (cid:17) · ˆ r − Z π/ bπ/ − b d θ sin( θ ) (cid:16) r · ~d (cid:17) · ˆ r (A2)This shows that the estimated longitude equals the true longitude of the dipole ~d . The latitude, however, is affectedby a bias, B , that depends only on the latitude cut, b : tan( b est . ) = B ( b ) · tan( b true ) , (A3) B ( b ) = 1 − sin ( b )1 − (9 sin( b ) + sin(3 b )) (A4)Note that the directional bias depends neither on the sample size N or dipole amplitude d , nor on the true dipoledirection. It may also be of interest that the bias (Equation A4) is solely due to the dipolar contribution ∝ ˆ r · ~d , as themask is chosen to be symmetric with respect to the observer. The true, unbiased dipole direction is therefore foundcloser to the Galactic plane than is indicated by the uncorrected estimator, Equation 2.The masks applied in this work carry small features in addition to the cut on Galactic latitude. It is not straight-forward to analytically compute the bias arising from arbitrary mask shapes. However, by analysing simulations wefind the directional bias caused by these additional features to be negligible ( < ◦ ). For the results shown in Figure 4we therefore show the dipole direction as corrected by Eqs. A3 and A4. B. CLUSTERING DIPOLE WITHIN THE CONCORDANCE MODELThe clustering dipole D cls in a sample of objects as seen by a typical observer in the concordance ΛCDM cosmologycan be computed given the power spectrum P ( k ) of (dark) matter density perturbations (Gibelyou & Huterer 2012): D cls = r π C , (B5)where C l = b π Z ∞ f l ( k ) P ( k ) k dk. (B6) The results are equivalent to those in Rubart (2015), but areexpressed here in terms of latitude rather than polar angle. he Quasar Dipole b is the linear bias of the observed objects with respect to the dark matter and the filter function f l ( k ) is f l ( k ) = Z ∞ j l ( kr ) f ( r ) dr, (B7)where j l is the spherical Bessel function of order l and f ( r ) is the probability distribution for the comoving distance r to a random object in the survey, given by f ( r ) = H ( z ) H r dNdz , (B8)normalised such that R ∞ f ( r ) dr = 1 and dN/dz is the redshift distribution of the observed objects. Employing r = c/H = 3000 h − Mpc, Planck 2015 cosmological parameters from Astropy, P ( k ) at z = 0 using camb (Lewis et al.2000), and a cubic-spline fit to the redshift distributions shown in Figure 3 to determine dN/dz , we estimate D cls tobe 0.00027 (taking bb