Correction to: The cosmic dipole in the Quaia sample of quasars: a Bayesian analysis

Abstract

In the original paper (Mittal, Oayda & Lewis 2024), an error was made in the calculation of the distribution of spectral indices α, as appearing in fig. 2 of that paper. The error arises from equation (3), which in the original paper reads While this equation is mathematically correct, the issue relates to the correct interpretation of the units of the passband flux density Sν when using Gaia magnitudes. These magnitudes, like the G band magnitude as appearing in the phot_g_mean_mag column of the Gaia DR3 release table, are defined such that Sν in equation (1) is in units of photoelectrons per second (e−s−1; Hambly et al. 2022). We noted this in our original paper at the beginning of section 4.2.2, at which stage we mentioned that to convert from e−s−1 into Jy, one must first multiply the passband flux density by a correction factor cν to give units of W m−2 Hz−1. However, since this factor was not applied in equation (4) in the original work, equation (5) is incorrect as one cannot use the relation Sν ∝ ν−α if Sν is in units of e−s−1. To correct this, let |$S^{\prime }_\nu$| denote the passband flux density in W m−2 Hz−1 such that |$S^{\prime }_\nu = c_\nu S_\nu$|⁠. Substituting this into equation (1) yields Thus, equations (4) and (5) in the original work should be replaced with these corrected versions: where ϵ is 2.5 log10(cG/cBP) and k is ZPG − ZPBP as defined originally. The value of these correction factors for each passband are given in section 5.4.1 of the Gaia DR3 documentation (Busso et al. 2022). With the correction applied, fig. 2 in the original work is to be replaced with Fig. 1. Since fig. 3 of the original work included the correction factor when converting Gaia magnitudes to flux densities, this figure does not need to be altered. Distribution of spectral indices α in the Quaia low and Quaia high samples computed from mG − BP. Blue and orange are used to denote Quaia low and Quaia high, respectively. The mean spectral indices |$\bar{\alpha }$| for each catalogue are indicated by the vertical lines. As the Ellis & Baldwin (1984) relation is dependent on α, the correction to the spectral index changes the expected dipole amplitude |$\mathcal {D}$|⁠. We repeated the methodology in section 4.2.2 with the amended values of α, finding a distribution of dipole amplitudes as shown in Fig. 2. This is to replace fig. 4 of the original work. We thus take the mean amplitude of |$\mathcal {\bar{D}} = 0.0048$| for Quaia low and |$\mathcal {\bar{D}} = 0.0043$| for Quaia high, using these as the dipole expectations. Probability distribution for the dipole amplitude assuming vCMB. This follows from the analysis in section 4.2.2 and was performed with both the Quaia low and Quaia high samples. Correcting for this error in the way described above impacts the relative Bayesian evidences of our different hypotheses. This is because models which incorporate the expected dipole amplitude will necessarily arrive at different marginal likelihoods. These are models M5 (kinematic velocity) and M6 (kinematic dipole) in our original paper. Accordingly, the last two rows of tables A1– A4 in the original paper are to be replaced with those given here in Tables 1, 2, 3 and 4. Table of Bayes Factors for different hypotheses and Galactic masks using the Quaia low catalogue with the point-by-point analysis. Here, 30* represents the combination of a 30° mask and a |$4\, \text{sr}$| circular mask centered at the (l°, b°) = (0, 0). The highlighted cell represents the model with the highest Bayes factor, indicating it has the strongest level of support. As for Table 1 but with the Poisson statistics. As for Table 1 but with Quaia high. As for Table 1 but with Quaia high and the Poisson statistics. Based on these tables, our conclusions are altered slightly. The relative ordering of each model in terms of evidential power is not changed. Namely, for the Quaia low sample with a 40° galactic plane mask, the kinematic dipole (model M6) is still the prevailing model. However, the difference in support between M6 and the next-favoured model, M4 (the marginal likelihood of which is given in the original work), becomes ln B64 = 1.5. This means that the kinematic dipole is somewhat less preeminent than as in the original work (ln B65 = 2.6). Additionally, the conclusion in section 6.3 in the original work needs to be replaced. We now find that our conclusions are, to a certain extent, sensitive to the choice of prior. As mentioned there, after adjusting the prior function for |$\mathcal {D}$| from |$\mathcal {U}[0,1.0]$| to |$\mathcal {U}[0,0.1]$|⁠, the marginal likelihood of model M4 increases to 8.0 (Quaia low; b° < |40| masked). Based on the corrected Bayes factors in Table 1, this means that model M4 (kinematic direction) is the prevailing model with ln B46 = 0.7, in which case we find |$\mathcal {D} \approx 11\substack{+5 \\-5} \times 10^{-3}$|⁠. Although, this Bayes factor is indicative of only a slight preference for M4 over M6. Meanwhile, the conclusions for Quaia high using the adjusted prior on |$\mathcal {D}$| are unchanged. Though these aspects have been altered, it still remains that the Quaia sample strongly favours a dipole aligning with the direction of the CMB dipole. The question of its amplitude is not as decisive; while we cannot rule out the possibility of a value larger than expected from the CMB dipole, we have also pointed towards support for it being equal to the CMB value. This leaves significant scope for future inquiry to attempt to resolve this matter. We extend our sincere gratitude to Nathan Secrest, who pointed out the steepness of our original spectral indices and generously shared some of their own results. This led us to uncover the error mentioned in this correction. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Topics & Concepts