be expected at a distance of 5000 au or more. Sana et al. (2014) show that at a separation of 8, the cumulative multiplicity frequency reaches 91%. It would correspond to a distance of 16 000 au at a typical distance of 2 kpc. For deprojected distances above 10 000 au the neighbour star is probably not bound to the main star ( Sana et al. 2014 , actually indicate that at large separation only the brightest companions are bound to the main star). The maximum projected separation in arcsec corresponding to the linear separation of 10 000 au in the image is also provided (see Table 5 ). The results are indicated in Table 5 , Cols. 3 to 5. Stars P4; HD 168625b; HD 168607b; MWC314b,c; MWC930c; WR102e; HD 152234b; and HD 164794b,c,d are separated from the main component by less than 10 000 au. This test eliminates 50% of the potential companions visible in the images as bound in a binary system. According to the study by Sana et al. (2014) among massive stars, it is not rare to find binaries with a magnitude difference in K or Lp of 88.5. The limiting magnitude would ensure that almost all potential companions are caught. In addition, they also used NACO to find binaries in O stars, as this threshold would be adequate for LBV stars and corresponds to a late B-type star (using the models from Schaller et al. 1992 ). The results are provided in Table 5 , Col. 6, while we give the angular projected separation corresponding to 10 000 au as an indication in Col. 5. This test allows us to discard 30% of the potential companions visible in the images as bound in a binary system. The results are discussed in Sect. 3.4 . 3.4. Binarity assessment and discussion To summarize, a companion is bound to the main star if it complies with the following criteria: The deprojected/probable separation must be smaller than10 000 au. This value is given inTable 5 , Col. 4. The projectedseparation (from Col. 2) is corrected with the mostprobable angle of projection to provide a more realistic angularseparation (Col. 3) converted to au inCol. 4. The difference in magnitudes must be mag 8.5 (Table 5 , Col. 6). The chance projection probability (field) must be smaller than 10%, (field) smaller than the cut-off probability, or (field) smaller than 5%. In Col. 7 we indicate the chance projection probability for the potential companion considering the stars counted in the NACO images (except for WR102ka for which SINFONI image is used). In Col. 8, we provide the chance projection probability for the potential companion considering the stars counted in the results of simulations with the trilegal code ( Girardi et al. 2005 ). They differ by the magnitude ranges used: all trilegal stars, stars with a maximum difference of 2 and 1. In Col. 9, the chance projection probability following the formula of Correia et al. (2006) is provided and the scaled cut-off probability is given in Col. 10. In Col. 11, the chance projection probabilities (field), (field), and (field) respectively

O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject B-CelestialObject B-CelestialObject B-CelestialObject I-CelestialObject O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Wavelength O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Instrument O O O B-CelestialObject O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O


its multiple imaging cross section at the redshift of the LAE in our sample will be comparable to or larger than the diameter of a BOSS fiber. As a result, the multiply imaged source search region associated with each CMASS galaxy is essentially the size of the fiber. This is very different from general studies of lens statistics (e.g.,Kochanek 1993a , 1993b , 1996 ; Maoz Rix 1993 ; Jaunsen et al. 1995 ; King Browne 1996 ; Möller et al. 2007 ) which must account for variations in the multiple imaging cross section with lens redshift, lens mass, and source redshift. Given this fact, our estimate of the magnification bias can be checked observationally by comparing the incidence of LAE detections in BOSS galaxy spectroscopic fibers to their incidence within the blank-sky fibers of the survey. To explore this, we rerun the full candidate selection procedure on BOSS sky fibers and obtain 9 candidates with - σ detection significance. This number can be compared to the 187 candidates found to this same significance threshold in the galaxy-fiber sample. Since there are approximately six times more galaxy fibers than sky fibers in the BOSS survey, we find that our high- z LAE detections occur with a frequency times higher in the galaxy fibers than in the sky fibers. This is consistent with the boost factor expected based on our simple magnification-bias calculation above: the higher relative incidence of LAE detections in the galaxy fibers thus suggests that the galaxy-fiber detections are highly magnified by strong lensing. We stress, however, that the close quantitative agreement between the predicted and observed detection-rate boost of galaxy fibers relative to sky fibers must be regarded as coincidental considering the rough approximations involved. 5. DISCUSSION We first emphasize that our spectroscopic selection technique is powerful and generic, and can be easily applied to a variety of scientific considerations with appropriate modifications. For instance, by changing the single-line detection strategy to detections for at least two of the lines [O ii ], [O iii ] 4959, 5007, H α , and H β at a common redshift, one can pick out lens candidates with star-forming galaxies as the background sources. That is exactly the approach adopted in the successful SLACS, BELLS, and S4TM surveys (Bolton et al. 2006a ; Brownstein et al. 2012 ; Shu et al. 2015 ). One can also specifically search for lensed quasars by tuning the selection criteria to the presence of more than one prominent quasar emission lines such as Ly α , C iv , C iii ], and Mg ii . Interested readers are encouraged to adopt the spectroscopic selection technique according to their own research interests. Although the size of the sample of galaxy–LAE lenses from our initial HST program seems to be relatively modest when compared to other galaxy-scale strong-lens samples, there are another 166 galaxy–LAE lens candidates remaining in the parent sample. We applied very stringent

O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation O B-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength B-Wavelength B-Wavelength I-Wavelength O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey B-Survey O B-Survey O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O B-Telescope O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O


which starts forming BHs in galaxies with . While modelling BH formation in low-mass galaxies is crucial to understand BH formation in the high-redshift Universe, as well as to understand the current populations of BHs in local dwarf galaxies, the regime of low-mass galaxies is barely resolved in such large-scale simulations of on a side (but see Habouzit et al. 2017 , and references therein for BH formation in low-mass galaxies from high redshift to low redshift). All the other simulations employ lower seeding masses than the TNG simulations. In Horizon-AGN fewer BHs of are present at z = 0 than at higher redshift. This is because BH formation stops at z = 1.5 by design in this simulation. In simulations, the initial mass of BHs was shown to affect the low-mass end of the M BH − M ⋆ diagram, and the overall normalization of the scaling relation (e.g. Bower et al. 2017 ). For higher stellar masses, the relations for different seeding masses converge to the same relation (whose normalization depends on the BH accretion efficiency parameter) as a result of self-regulation. The seed mass is important for the simulations using the Bondi accretion model (e.g. Illustris, TNG, Horizon-AGN, EAGLE), for which there is a degeneracy between the seeding mass and the boost factor to produce the same normalization of the M BH − M ⋆ mean relation. In simulations using a torque accretion model (SIMBA), the accretion rate on to the BHs does not strongly depend on their masses (, contrary to the Bondi model ). Therefore, the low-mass BHs can grow more efficiently than in the Bondi model, and they converge into the M BH − M ⋆ mean relation regardless of their seed mass (Anglés-Alcázar, Özel Davé 2013 ; Anglés-Alcázar et al. 2015 ). We point here again that the seeding in large-scale cosmological simulations is often not physically motivated (e.g. fixed BH mass, threshold in galaxy, or halo mass to form a BH), but instead parameters of the seeding models are chosen to reproduce the local scaling relation. 3.3.2 On the growth of BHs Getting the BHs on the main scaling relation depends both on the seeding BH mass and BH accretion rate. The parametrizations of these models are degenerate. Therefore, different choices allow the BH population to get on to the same empirical scaling relation. For example, the TNG model seeds BHs with a higher initial BH mass than the Illustris model (by one order of magnitude) but also does not include any boost factor in the accretion model (while Illustris does include a boost factor of 100). The colour code of Fig. 3 provides information on the accretion rates on to the BHs. At z ∼ 2, it is clear that the accretion properties are very different among the different simulations. For example, BHs of embedded in galaxies of have high bolometric luminosities of (limit commonly employed to define an AGN) in TNG, Horizon-AGN, and SIMBA, while a large fraction of these BHs in Illustris and EAGLE have much lower luminosities.

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In Figure 13 we also compare the number density of our red nondusty massive galaxies with the number density of passive galaxies selected in the literature for the same (or a very similar) stellar mass cut (Mancini et al. 2009 ; Muzzin et al. 2013 ; Straatman et al. 2014 ; Tomczak et al. 2014 ; Davidzon et al. 2017 ; Merlin et al. 2018 ). These passive galaxy selections are based on rest-frame colors, and their number densities are directly quoted by the authors, or we have obtained them by integrating their corresponding GSMF. 10 10 In the latter case we integrate the Schechter fits to the GSMF. The uncertainties on the number densities reflect the uncertainties on the Schechter parameters. This method was implemented by Muzzin et al. ( 2013 ), Tomczak et al. ( 2014 ), and Davidzon et al. ( 2017 ). Our comparison shows that the number density of our red nondusty galaxies is significantly higher than the number density of passive galaxies derived from Muzzin et al. ( 2013 ) and Davidzon et al. ( 2017 ) GSMFs. These differences are mainly produced by the shallower depths of the data sets used by these authors with respect to our own. In the case of the comparison with Davidzon et al. ( 2017 ), there is the effect of (a) their choice of dust extinction law and (b) their specific criterion to select passive galaxies, which is color based, but different from that applied by other authors. Davidzon et al. ( 2017 ) also correct for Eddington bias (Eddington 1913 ), which causes a drop in number densities at the massive end of the GSMF. Our red nondusty galaxy number densities are broadly consistent with the value derived by Merlin et al. ( 2018 ) and the upper limit from Mancini et al. ( 2009 ). They are also in excellent agreement with the number densities derived by Straatman et al. ( 2015 ) at z ∼ 2–3, although this might be somewhat fortuitous, given their very different methodology to select quiescent galaxies. At higher redshifts, instead, our results indicate a much faster decline in the number density of passive galaxies than that obtained by these authors. This difference does not seem to be the product of the different methodologies (note that we would select less rather than more passive galaxies using the quiescent wedge of the UVJ diagram, according to Figure 12 ). The observed differences could in part be the result of cosmic variance, as Straatman et al. ( 2014 ) analyzed images over an area ∼6.5 times smaller than that considered here. Figure 14 shows the fractions of our different populations (dusty, nondusty/blue, and nondusty/red, i.e., quiescent) among all massive () galaxies, versus redshift. We compare these fractions to those obtained in the literature. In these other works, quiescent galaxies have been selected using the UVJ color–color diagram wedge. In addition, Martis et al. ( 2016 ) have also determined regions to segregate dusty and nondusty star-forming galaxies on this color–color plane. Figure 14. Comparison of our

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O B-Model I-Model O O B-Model O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O


that gravitational instabilities drive the formation of molecular clouds within the Galactic Centre (hereafter, GC) gas stream. 2 DATA This Letter makes use of data from the Mopra CMZ survey, first published by Jones et al. ( 2012 ). We focus on the J = 1 → 0 transition of N 2 H + (the rest frequency of the main J, F 1 , F = 1, 2, 3 → 0, 1, 2 hyperfine component is ∼93.174 GHz), the emission from which is both bright and contiguous over the region of interest (Henshaw et al. 2016 ). The presented data cubes have an effective spatial resolution of ∼60 arcsec (corresponding to ∼2.4 pc), a pixel size of 30 arcsec × 30 arcsec (1.2 pc × 1.2 pc), a spectral resolution of 2 km s −1 , and the typical rms noise level is ∼0.01–0.02 K. Fig. 1 is a map of the inner CMZ. The colour-scale represents the Herschel -derived column density (Battersby et al., in preparation). The present investigation focuses on the region −07 l 00, −01 b 01, which is indicated by the black dashed rectangle. The black contours highlight integrated emission of N 2 H + . The data enclosed within the black dashed rectangle were analysed using the Semi-automated multi-COmponent Universal Spectral-line fitting Engine ( scouse 1 ; Henshaw et al. 2016 ). 3 RESULTS Panel A of Fig. 2 provides a close-up view of the Herschel -derived column density map (Fig. 1 ). It focuses on the location of the upstream velocity field (depicted by the blue dots). To identify the main structures we use dendrograms (Rosolowsky et al. 2008 ). Each contour in panel A represents a leaf, the highest level in the dendrogram hierarchy. 2 Red contours depict structures that are spatially coincident with the velocity pattern. The column density map of Battersby et al. (in preparation) is used to derive physical properties of each leaf, including the peak column density ( N ), total mass ( M ), equivalent number density and corresponding equivalent free-fall time ( n eq and t ff,eq ). 3 These properties are presented in Table 1 . Panel B of Fig. 2 showcases the velocity pattern first identified by Henshaw et al. ( 2016 ), which appears as a corrugation in { l , v LSR } space. Each data point refers to the location ( l ) and centroid velocity ( v 0 ) of a Gaussian component extracted using scouse . We corroborated these values using the hyperfine structure (HFS) fitting functionality available in class . 4 This method accounts for the effects of opacity on the line-profiles (whereas Gaussian fitting does not), leading to a more accurate measure of the velocity dispersion and line centroid where the opacity is high. We find that the velocity dispersions derived using scouse are systematically larger than those obtained using class (). However, the derived centroid velocities are very similar, km s −1 . Both the scouse and class derived kinematic properties are included in Table 1 . Figure 2. Galactic longitude versus: (A) Galactic latitude. A column density map of the CMZ focusing on the region of interest

O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O B-CelestialObjectRegion O O O O O O O O O O O O B-Survey I-Survey I-Survey O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O O O O B-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength I-Wavelength O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software I-Software I-Software I-Software I-Software O B-Software I-Software O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O B-Software O O O O O O O B-Model I-Model B-Model B-Model O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O B-CelestialObject O O O O O O O O O O O O O O


be measured, then this would provide constraints on the field strength and distribution. We also explored the possibility of non-WKBJ effects relevant to mixed modes of low degree. Our results demonstrate that the form of the quantization condition remains very similar to that in the case where WKBJ is applicable throughout the star. However, in our more general expression, an additional offset g appears, given in equation( 90 ). This is unrelated to magnetic effects and has its origins in the non-WKBJ nature of strongly coupled cavities. Measurements of this may provide information about the structure of the evanescent zone and base of the convective envelope. The effects of rotation have not yet been incorporated and would be the subject of future work. APPENDIX A: TREATMENT OF THE UPPER LAYERS In this appendix, we discuss the convective envelope and evanescent zone (regions P and E), for the purposes of connecting the solutions in these regions to those in the g-mode cavity (treated in Section 4 ), and thereby obtaining the mixed-mode quantization condition. A1 Region P Here we still shall assume N 2 0, making the region isentropic. Equations( 15 )( 16 ) then combine to give where . Making a change of variable , we obtain the DE where The first term in the expression for is expected to dominate, since this corresponds to the squared acoustic wavenumber, which greatly exceeds 1/ r for typical of stochastically driven oscillations in the outer parts of the convection zone. Retaining only the first term in when considering amplitude factors in the WKBJ approximation, the WKBJ solution to ( A2 ) is corresponding to where C 3 and are arbitrary constants. Here r t is the lower turning point where vanishes. We shall assume for simplicity that this exceeds r 0 , where N 2 drops to zero from a positive value further in. Otherwise, the lower limit of the integral should be replaced by r 0 , as ( A1 ) does not apply where N 2 0. The constants C 3 and are in the first instance determined by the requirement that the solution( A4 ) is a continuation of the one that is specified just interior to r e . Asserting the continuity of the solutions multiplying C 1 and C 2 in equation( 34 ), we write The amplitude factors A 1 and A 2 and the phases 1 and 2 in general have to be determined from numerical integrations if the WKBJ approach cannot be used throughout region E, as will be the case for small . A1.1 Incorporating a surface boundary condition The solution representing a mode of oscillation must also satisfy a boundary condition at the surface. Following the usual procedure when constructing WKBJ solutions, we assume that this condition can be met with a solution of the form ( A5 ) by specifying the phase . The value of 0 is determined by the surface boundary condition. Given the above continuations, it is readily found that the condition that the linear combination occurring in equation( 34 ) leads to the correct phase condition specified above in this

O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


plot the branching ratio as a function of the NSC density (, β = 2.3); in the bottom panel, we show as a function of the the slope of the BH mass function ( and ). The general trend with a hard discussed previously is in part smeared out by the fact that we sample the binary semimajor axis in the range ( a GW , a hard ), rather than having it fixed to a hard . Moreover, BHs do not have a fixed mass, as in Mod1 and Mod2 in equal-mass binaries, but their masses are sampled from a distribution. Additionally, we find that SBH–SBH binaries produce the largest branching ratio, while IMBH–IMBH lead to the lowest one. This can again be ascribed to the fact that hard binaries of two SBHs have typically smaller semimajor axes than binaries comprised of two IMBHs. As a consequence, stars that undergo a resonant encounter with an SBH–SBH binary are more likely to pass within the tidal radius of one of the two BHs and get tidally disrupted. Branching ratios of binary IMBH–SBHs are in between the SBH–SBH and IMBH–SBH cases. Figure 2. Top panel: branching ratio as a function of the NSC mass (, β = 2.3) for SBH–SBH (orange), IMBH–SBH (red), and IMBH–IMBH (green). Central panel: branching ratio as a function of the NSC density (, β = 2.3) for SBH–SBH (orange), IMBH–SBH (red), and IMBH–IMBH (green). Bottom panel: branching ratio as a function of the the slope of the BH mass function (, ) for SBH–SBH (orange), IMBH–SBH (red), and IMBH–IMBH (green). 3.3 Orbital properties of binaries that exhibit tidal disruption events The semimajor axis of the binaries that undergo a TDE depends on the host NSC mass and density, and on the slope of the BH mass function β. We illustrate in Fig. 3 the distribution of semimajor axes of the SBH–SBH binaries that undergo a TDE. In the top panel, we show the case for and different values of the NSC mass (Mod3), while in the bottom panel, we plot the case for and different values of the NSC density (Mod4). The slope of the SBH mass function is fixed at β = 2.3. We find that the NSC mass has only a modest effect in shaping the semimajor axis of the binaries that undergo a TDE, unlike the host NSC density. The reason for this trend can be explained considering that binaries have orbital semimajor axis in the range ( a GW , a hard ), where Larger densities correspond to smaller values of a GW , thus a wider range of possible binary semimajor axes, while the NSC mass does not have a significant effect. Figure 3. Distribution of semimajor axis ( a ) of the SBH–SBH binaries that undergo a TDE. Top panel: and different values of the NSC mass; and bottom panel: and different values of the NSC density. The slope of the SBH mass function is fixed to β = 2.3. Fig. 4 shows the distribution of semimajor axes ( a ) of the SBH–SBH binaries (top) and masses ( m SBH ) of the SBHs (bottom) that undergo a TDE, for different values of the slope of the SBH mass function. The mass and density of the NSC are fixed to and , respectively. We find that the

O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


best fit and subsequently adjusting the result. The best-fit is obtained for T eff = 10, 140 K, and log g = 8.10. Statistical uncertainties are negligible due to the high SNR of the optical spectrum. Subsequently, we performed a photometric fit of the available APASS and Gaia DR2 photometry (Table 2 ), in which we used the Gaia DR2 parallax and the white dwarf mass–radius relation (Fontaine, Brassard Bergeron 2001 ) as additional priors. We have not used VHS photometry to be sure that the IR excess has no effect in the fit. The photometric fit delivers a 0.1 dex-lower surface gravity. Hence, we iterated between spectroscopic (log g fixed) and photometric ( fixed) fits, until both methods converged to a final solution. The difference between the initial spectroscopic fit and the final result is adopted as the error estimate, resulting in T eff = 10, 090 ± 50 K and log g = 8.00 ± 0.13. The interstellar reddening obtained as a result of our fitting procedure is negligible, and of the order of 0.01 mag. We measured the radial velocities of the strongest metal lines via fitting Gaussian models and estimated the uncertainty as the scatter between the measurements of the different lines. For the 2011 spectrum, we obtained an average radial velocity of = 44 ± 5 km s −1 and for the 2019 spectrum, = 47 ± 8 km s −1 . The individual radial velocities for the different lines are presented in table 6. The atmospheric parameters of WDJ1814−7354 correspond to a white dwarf mass of M ⊙ and a cooling age of Gyr. Using the Cummings et al. ( 2018 ) initial-to-final-mass relation (IFMR), we infer a progenitor mass of M ⊙ . To estimate the progenitor age, we used the evolutionary tracks of Choi et al. ( 2016 ). For a progenitor mass of 1.3 M ⊙ (late F-type star), the progenitor age is τ MS = 5.2 Gyr, and the total age of the white dwarf would be τ MS + τ cool = 5.8 Gyr. The large systematic uncertainty implied by the empirical IFMR translates into a large uncertainty on the progenitor age. For the upper limit of the progenitor mass of 1.8 M ⊙ (a late A-type star), the progenitor age would be τ MS = 3.4 Gyr, so the lower limit on the total age of the white dwarf is 4 Gyr. The upper limit on the total age is unconstrained, because the lower limit of the progenitor mass is 0.9 M ⊙ (an early K dwarf star), whose lifetime could be comparable or longer than the age of the Universe. 4.2 Composition of the accreted material and diffusion time-scales The abundances of the three detected elements (Mg, Ca, and Fe) are given in Table 7 . The listed abundances and uncertainties are the averages and the standard deviations inferred from the modelling of the strongest lines for each detected element: lines at 3933.66, 3968.47, 8498.02, 8542.09, 8662.14 (Ca ii ), and 4226.73 Å (Ca i ) for Calcium; lines at 4481.13 (Mg ii ), 3838.29, 5167.32, 5172.68, and 5183.69 Å (Mg i ) for Magnesium; and lines at 3609, 3619, 3632, 3720, 3738, 3750, 3759, 3816, 3821, and 3826 Å for iron (Fe i ).

O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Wavelength O O O O O O O O O O B-Survey O B-Dataset I-Dataset O O O O O O O O O B-Dataset I-Dataset O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Survey O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Model I-Model B-Model O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength O O O O B-Wavelength I-Wavelength O O O O O O O B-Wavelength O O O B-Wavelength B-Wavelength B-Wavelength O B-Wavelength I-Wavelength O O O O O O O O B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength B-Wavelength O B-Wavelength I-Wavelength O O O O O


with the MW (Mayer et al. 2006 ; see also Section 4 ). Crucial to the identification of t sat through features of Δ F ( t ) are the uncertainties in ψ( t ). Ideally, the permitted smooth ψ( t ) should be obtained from the raw observational data by taking into account correlated errors in each time bin. Such an analysis is beyond our scope here. Upon advice from T. J. L. de Boer, we give instead an approximate treatment as follows. We first make a Monte Carlo realization of the data by sampling for each time bin according to a Gaussian distribution with the mean and standard deviation specified by the corresponding mean and 1σ uncertainty shown in Fig. 1 . The realization is accepted only if it gives (3.12 ± 0.05) × 10 7 M ⊙ for the total mass of stars formed (personal communications from T. J. L. de Boer). Out of a total of 10 4 realizations, only 3806 satisfy the constraint. For each accepted realization, we obtain a quadratic-spline fit to ψ( t ) and use it to calculate M g ( t ) and Δ F ( t ) from equations ( 15 ) and ( 16 ), respectively. 3.1 Baseline case We first discuss our baseline case of α = 1.5 and λ * ( t sat ) = 10 − 14 M ⊙ yr −1 . The M g ( t ) calculated from the smooth ψ( t ) shown in Fig. 1 assuming that λ * ( t ) = λ * ( t sat ) is shown as the top curve in Fig. 2 . The corresponding Δ F ( t ) is shown as the curve in Fig. 3 (a). The shaded region in this figure indicates the 68 per cent confidence interval for Δ F ( t ) estimated from the Monte Carlo simulations constrained by the total mass of stars formed. Figure 2. Total mass M g for cold gas in Fornax's star-forming disc as a function of time t derived from equation ( 15 ) and the smooth ψ( t ) shown in Fig. 1 for the baseline case of α = 1.5 and λ * ( t sat ) = 10 − 14 M ⊙ yr −1 . The top curve assumes a disc of fixed size with λ * ( t ) = λ * ( t sat ). The middle (bottom) curve is for a disc growing until t sat ≈ 4.8 (7.8) Gyr. The middle and top (bottom) curves have the same M g ( t ) for t > 4.8 (7.8) Gyr. Note that all curves reach M g = 0 at t ≈ 13.6 Gyr, the end of star formation in Fornax. Figure 3. Rate Δ F of net gas flow to or from Fornax's star-forming disc as a function of time t estimated from equations ( 15 ) and ( 16 ) and the data shown in Fig. 1 for the baseline case of α = 1.5 and λ * ( t sat ) = 10 − 14 M ⊙ yr −1 . A net inflow or outflow corresponds to Δ F > 0 and 0, respectively. In each panel, the curve is calculated from the smooth ψ( t ) shown in Fig. 1 and the shaded region indicates the 68 per cent confidence interval. Panel (a) assumes a disc of fixed size with λ * ( t ) = λ * ( t sat ). Panels (b) and (c) are for a disc growing until t sat ≈ 4.8 and 7.8 Gyr, respectively. We attribute the onset of suppression of the net gas inflow at t ≈ 4.8 or 7.8 Gyr to effects from Fornax's becoming an MW satellite. See text for details. We take t sat to coincide with the onset of suppression of the net gas inflow. Based on the Δ F ( t ) shown in Fig. 3

O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Person I-Person I-Person I-Person I-Person O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Person I-Person I-Person I-Person I-Person O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O B-CelestialObject O O O O O O O O O O O O B-CelestialObject B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-CelestialObject O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


of the blue curves with the (ortho+para) HO abundance profile deduced by Keto et al. (2014) (black curve in Fig. 3 ) shows that the total NH is about 30 times more abundant than HO within the central 1000 AU, while it is two orders of magnitude less abundant than HO from about 8000 AU to 20 000 AU. Fig. 4 MOLLIE radiative transfer results using the o-NH fractional abundance profiles from Fig. 3 , overlapped with the observed spectrum: ( top ) blue solid and dashed curves refer to the Crapsi et al. (2007) profiles, assuming an ortho-to-para NH ratio of 0.7 and 1.0, respectively. ( Bottom ) The red thick curve refers to the Sipilä et al. (2015a , b) profile; the red thin curve is the same as the thick red curve, but divided by 2. The o-NH profile has a very different shape when the chemical model of Sipilä et al. (2015a , b) , which includes spin-state chemistry, is considered (red curve in Fig. 3 ). The chemical abundance gradients in L1544 were simulated using the same procedure as described in Sipilä et al. (2016) : the physical model for L1544 ( Keto Caselli 2010 ; Keto et al. 2014 ) was separated into concentric shells and the chemical evolution was calculated separately in each shell. The gas was initially atomic with the exception of H (all in H) and D (in HD). The o-NH abundance profile (Fig. 3 ) was extracted from the model at the time when the CO column density was comparable to the observed value (; see Caselli et al. 1999 ), taking into account telescope beam convolution. Other details of the chemical model are extensively discussed in Sipilä et al. (2015a , b) . The predicted o-NH profile (red curve in Fig. 3 ) presents a broad peak between 5000 AU and 15 000 AU, it drops at larger radii because of photodissociation and at lower radii because of freeze-out. The o-NH() spectra simulated with MOLLIE are shown in Fig. 4 for the three o-NH abundance profiles in Fig. 3 , together with the observed spectrum. This figure shows that the empirical abundance profile of Crapsi et al. (2007) reproduces the observed spectrum better than the abundance profile predicted by Sipilä et al. (2015a , b) , although the simulated spectrum underestimates the brightness temperature at the lowest velocities of all groups of hyperfine components, as also found in the case of o-HO ( Caselli et al. 2012 ), which suggests that the contraction velocity within the core centre is slightly higher than the velocity used in our model. The red spectrum (based on the chemical model of Sipilä et al. 2015a , b ) is significantly brighter than the detected line and the ratios between the low- and high-velocity peaks of the various hyperfine components are not well reproduced (as better shown by the red thin curve in Fig. 4 , which is the red thick profile scaled down by a factor of two). It is interesting to note that the red curve is shifted to lower velocities than the blue curves. This is due to the large abundance of NH predicted at radii in the range of 400012

O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-CelestialObject O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


been performed) remain candidate stellar cloaks in the recoil scenario. A2261 has two Chandra exposures of 10 and 25 ks (ObsIDs 550 and 5007, respectively), which do not conclusively reveal any accretion onto an SMBH. The X-ray flux in those data is dominated by 0.5–2 keV emission from hot cluster gas, showing a cooled core (Bauer et al. 2005 ). Above 2 keV, there is some emission, possibly a point source consistent with knot 4, although it is too faint to reliably distinguish it from cluster gas emission. 4.2. Assessment of the Radio Relic in the A2261-BCG Core The age of the relic radio source can potentially be understood as a time marker for an event related to the disturbance of the central SMBH, and thus we are interested in understanding the age and nature of this relic. We limited the timescale of last electron energization in the radio source to . The existence of a relic in this BCG is interesting in itself and has a number of potential origins, as discussed in Section 3.3 . To investigate how atypical A2261-BCG’s radio source properties are among BCGs, we compare it with the sample of Hogan et al. ( 2015 ), who systematically studied the radio properties of a large sample of BCGs (including A2261-BCG). In comparison with that sample, we find that the 1.4 GHz radio luminosity is typical of ∼50% of BCGs. Strikingly, we appear to not detect a distinct core in this object down to a 3 σ limit of at 10 GHz. Although the non-detection of a core component in the Hogan ( 2014 ) sample was relatively common, their typical upper limit on core emission was on average a factor of >1000 less stringent than our limit. Nevertheless, they found that ∼30% of their BCGs had clear spectrally identified radio cores, while 30% consisted of only non-core components (i.e., had upper limits on core emission, as in our case), and 30% were complete radio non-detections; see their Table C.1. The most unusual thing about the radio relic in A2261-BCG is its central position combined with its small size and steep spectral index. We show the spectral index distribution in Figure 9 of 30% of the non-core BCGs in the Hogan sample. There are only a handful of targets in the sample with spectra comparable to or steeper than the central relic in A2261-BCG, and it is likely that given the large selection in Hogan ( 2014 ), there is some contamination from BCG-unrelated cluster relics included in this distribution. The distribution of core-dominated BCGs tend to center on indices closer to . Therefore, it seems that the radio spectrum of A2261-BCG lies at the extremity of implied activity and jet ages for BCGs. Figure 9. Distribution of the spectral index for a subset of the Hogan ( 2014 ) BCG sample. Here we show only the ∼30% of BCGs in that sample that are similar to A2261-BCG, in that only a non-radio-core component has been spectroscopically identified in the BCG. Note that Hogan et al.'s measurement of the A2261-BCG is included in the histogram. Our measurement is

O O O O O O O O O O B-CelestialObject O O B-Telescope O O O O O O B-Identifier I-Identifier O B-Identifier O O O O O O O O O O O O B-Wavelength O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Wavelength I-Wavelength O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O B-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-Wavelength I-Wavelength B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation B-Dataset I-Dataset O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O B-CelestialObject O O O O O O O O


absorption or emission-line features can be present alongside smoothly varying blackbody or other continuum radiation (see e.g. Tennyson 2019 ). In each of these general settings, we observe a signal plus noise and would like to denoise the signal as accurately as possible. Indeed the set of statistical tools available for addressing this general problem is quite vast. Commonly used non-parametric regression methods include kernel smoothers (e.g. Hall et al. 2002 ; Croft et al. 2002 ), local polynomial regression (LOESS; e.g. Maron Howes 2003 ; Persson et al. 2004 ), splines (e.g. Peiris Verde 2010 ; Contreras et al. 2010 ; Dhawan et al. 2015 ), Gaussian process regression (e.g. Gibson et al. 2012 ; Aigrain, Parviainen Pope 2016 ; Gómez-Valent Amendola 2018 ), and wavelet decompositions (e.g. Fligge Solanki 1997 ; Theuns Zaroubi 2000 ; Golkhou Butler 2014 ). A rich and elegant statistical literature exists on the theoretical and practical achievements of these methods (see e.g. Györfi et al. 2002 ; Wasserman 2006 ; Hastie, Tibshirani Friedman 2009 for general references). However, when the underlying signal is spatially heterogeneous , i.e. exhibits varying degrees of smoothness, the power of classical statistical literature is quite limited. Kernels, LOESS, smoothing splines, and Gaussian process regression belong to a broad family of non-parametric methods called linear smoothers , which has been shown to be uniformly suboptimal for estimating spatially heterogeneous signals (Nemirovskii, Polyak Tsybakov 1985 ; Nemirovskii 1985 ; Donoho Johnstone 1998 ). The common limitation of these methods is that they are not locally adaptive; i.e. by construction, they do not adapt to local degrees of smoothness in a signal. In particular, continuing with the example of a smoothly varying signal with occasional sharp features, a linear smoother will tend to oversmooth the sharp features and/or overfit the smooth regions in its effort to optimally balance statistical bias and variance. Considerable effort has been made to address this problem by locally varying the hyperparameter(s) of a linear smoother. For example, locally varying the kernel bandwidth (e.g. Muller Stadtmuller 1987 ; Fan Gijbels 1992 , 1995 ; Lepski, Mammen Spokoiny 1997 ; Gijbels Mammen 1998 ) irregularly varying spline knot locations (e.g. De Boor 1974 ; Jupp 1978 ; Dimatteo, Genovese Kass 2001 ), and constructing non-stationary covariance functions for Gaussian process regression (e.g. Schmidt O’Hagan 2003 ; Paciorek Schervish 2004 , 2006 ). However, since hyperparameters typically need to be estimated from the data, such exponential increases in the hyperparameter complexity severely limit the practicality of choosing the hyperparameters in a fully data-driven, generalizable, and computationally efficient fashion. Wavelet decompositions offer an elegant solution to the problem of estimating spatially heterogeneous signals, providing both statistical optimality (e.g. Donoho

O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Model I-Model I-Model B-Model O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Model O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Model I-Model I-Model O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model B-Model B-Model I-Model O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Model I-Model I-Model O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation


; Grimes et al. 2009 ; Tripp et al. 2011 ; Tumlinson et al. 2011 ; Muzahid 2014 ). This is not the case for either of our O vi absorbers. The sub-solar metallicity, which we obtain for the low- and high-ionization gas phases, is also not usually found in metal-rich outflows (Muzahid 2014 ). Thus, there is no definitive evidence for the absorbers discussed in this paper being directly associated with a galactic-scale wind. These warm absorbers may be tracers of ancient outflows (wind age >1 Gyr), which, as the simulations of Ford et al. ( 2014 ) suggest, can end up diffusely distributed at galactocentric distances of d ≳ 500 kpc where absorption from the ambient intergalactic medium also dominates. With the limited SDSS sampling of galaxies discussed in this paper, it is difficult to distinguish an outflow-generated O vi from the several other possible scenarios that can give rise to warm O vi in galaxy haloes (see Heckman et al. 2002 ; Sembach et al. 2003 ; Maller Bullock 2004 ; Tripp et al. 2008 ). Based on models of structure formation, it has been hypothesized that galaxy groups can have intragroup gas at T ∼ 10 5 –10 6 K, temperatures too low to discriminate their emission from the soft X-ray background (Mulchaey et al. 1996 ). Presently, such a medium can be probed only through high-ionization absorption lines such as O vi , Ne viii and thermally broadened Ly α against the light from background QSOs. Exploring this gas reservoir is important, as it potentially harbours as much as ∼4 per cent of the total baryons in the low- z Universe, on a par with the fraction of baryons trapped inside galaxies (Savage et al. 2014 ). A recent example is given by Stocke et al. ( 2014 ), whose sample of BLA–O vi warm absorbers is more likely to be from intragroup gas than individual galaxy haloes. The warm O vi is hypothesized to arise in conductive interface layers between cold T ≲ 3 × 10 4 K clouds and a hitherto undetected diffuse T ∼ 10 6.5 K intragroup medium. Hydrodynamic simulations also predict that a substantial fraction of baryons are to be found along large-scale filaments connecting galaxy overdensity regions. In Narayanan et al. ( 2011 ), a BLA–O vi absorber is identified as tracing T ∼ 1.4 × 10 5 K intergalactic gas along a nearby galaxy filament ( v HELIO = 3081 km s −1 ), with several loose groups of galaxies beyond ρ ∼ 1.5 Mpc of projected separation. Another example of baryonic reservoirs along galaxy filaments is given by Wakker et al. ( 2015 ). About 85 per cent of the Ly α systems from their sample of 15 absorbers, which includes BLAs, appear to be tracing intergalactic filament gas that lies far outside the virial radii of the nearest galaxies that trace the filament. Along similar lines, Tejos et al. ( 2016 ) found a ∼4–7 times higher covering fraction for BLAs near galaxy overdensities, compared to random fields. In that study also, the warm gas was identified as being away from individual galaxy haloes, but possibly along cluster

O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Wavelength I-Wavelength O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObjectRegion I-CelestialObjectRegion I-CelestialObjectRegion O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


core-helium burning in the blue part of the HRD or crosses the HRD and starts core-helium burning as a RSG. This depends on a variety of parameters, which can be interdependent, such as the following: rotation (and its implementation), mass-loss rates, chemical gradients in the radiative zones, activity of the intermediate H-burning shell ( Langer Maeder 1995 ; Maeder Meynet 2001 ). Models at low metallicity tend to remain in the blue region of the HRD longer (e.g. Georgy et al. 2013 ), and this is also what we observe in the computations used in this work. A similarity between our models at both metallicities is that the luminosity of the hydrogen burning-shell at the onset of shell hydrogen-burning is systematically higher for models computed with the Schwarzschild criterion compared to models computed with the Ledoux one (see Fig. 4 ). This is due to chemical gradients in the region which was previously occupied by the hydrogen burning-core during the MS (the region below the dot-dashed line in Fig. 3 ), which prevent a convective zone from appearing at the same location where hydrogen is burnt. Since hydrogen-shell burning starts in a radiative condition with the Ledoux criterion, there is no refueling in fresh hydrogen by convection and then the energy production remains lower. In the Ledoux models, the convective zone appears higher inside the star, where no burning occurs. It then grows at a deeper level by eroding the chemical gradient below the convective zone. The difference between solar and LMC metallicity is that at solar metallicity, the first crossing of the HRD can be fast: All the Ledoux models and the Schwarzschild 25 M ⊙ models ignite central helium burning on the RSG branch after a quick crossing of the HRD. Only the Schwarzschild 20 M ⊙ model starts burning its helium as a BSG before slowly crossing the HRD for the first time: The luminosity of the intermediate hydrogen-burning shell is sufficient to maintain the model on the blue side of the HRD, and it remains almost constant; whereas in other cases, it decreases over time. Figure 4 shows the time-evolution of the shell luminosity for 3 × 10 5 years after the end of the MS. Three different kinds of behaviours can be seen. The first type are models with an abrupt decrease in the shell luminosity shortly after the shell ignites (Ledoux models at solar metallicity). These are the models which cross the Hertzsprung gap very quickly after the end of the MS. The second type are models with a slow decrease in the shell luminosity over time (models at Z = 0.006 and Schwarzschild solar metallicity 20 M ⊙ model). These models ignite core-He burning in the blue part of the HRD and cross the gap slowly. The third type is the model with an intermediate behaviour (Schwarzschild solar metallicity 25 M ⊙ model). This model crosses the HRD quite quickly, but not as fast as the first type of models. For the models that are quickly crossing the HRD, the intermediate convective zone of the

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-Model I-Model O O B-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Model I-Model I-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


we refer to the subset of haloes that share the same ΛCDM companion across all the simulations as the common sample. For geometrical reasons, we set the limit value for to be 0.5: this represents the case in which two haloes with the same major axis a have centres separated exactly by the same amount a . The configurations that are selected by point ( i ) are the ones for which the distance between the halo centres is less or equal to the smallest major axis between the two. A higher value for would include genuine small haloes that have been more subject to dynamical QP drifting but would also result in a spurious match of bigger haloes. For these reasons, we adopt , checking that the selected sample gains or loses of components if values 0.45 and 0.55 are used, without modifying the overall statistical properties of the sample itself. With respect to at point ( ii ), instead, we applied the matching algorithm using several values, each denoting a specific threshold of the minimal value allowed for the mass ratio of halo couples in order to be considered as a match. As reported in Table 3 , more than of all the matching haloes across LCDM and FDM simulations without mass selection case have a mass ratio in the ratio range and almost in the range. In order not to spoil our matching catalogue, especially with very close but highly different in mass halo couples, we choose the limiting value of . Table 3. Number of common matches across LCDM and FDM simulations, using different values of the parameter representing the minimum allowed ratio between the minimum and maximum masses of each candidate couple. 4 RESULTS In this section we present the results obtained from our simulations in decreasing order of scale involved, starting from the matter power spectrum, to the simulated Lyman α forest observations, to the statistical characterization of halo properties, and their density profiles. 4.1 Matter power spectrum The relative difference of the matter power spectrum of the various FDM models with respect to ΛCDM at four different redshifts is displayed in Fig. 2 . Figure 2. Matter power spectrum of FDM models contrasted with LCDM at different redshifts. As already found in the literature (see e.g. Marsh 2016a ; Nori Baldi 2018 ), the evolution of the matter power spectrum shows that the initial suppression encoded in the transfer functions used to build up the initial conditions is restored at intermediate scales to the unsuppressed level, eventually, by the non-linear gravitational evolution. At the redshifts and scales that are relevant for Lyman α forest observations, however, the relative suppression with respect to ΛCDM is still important and ranges from for the lowest FDM mass considered. The relative difference of the matter power spectrum, displayed in Fig. 3 , shows an additional suppression with respect to ΛCDM when the QP is included in the dynamical evolution in the comparison between the FDM and FDMnoQP simulations, e.g. comparing Figs

O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O B-Model O O O O O O O O O O O O O O O O B-Model I-Model O O B-Model O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength O O O O O O O O B-Model O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O B-Model O B-Model I-Model O O O


2 (Lanzafame et al. 2018 ). The extended Kepler mission, K2 (Howell et al. 2014 ), has now observed more stars than the original Kepler sample, potentially doubling the number of high precision stellar rotation periods available. Similarly, by the end of the Gaia mission, Lanzafame et al. ( 2018 ) estimate that a sample of 3–20 million rotation periods will be recovered. The era of statistical studies in stellar rotation has therefore begun. Stellar rotation has long been noted as a means to possibly age-date stars due to their constant angular momentum loss via winds (Skumanich 1972 ). While studies of open clusters give hope that this “gyrochronology” model broadly works for solar-type and lower-mass stars, many uncertainties exist about the details of this spindown and its utility as a clock. These include questions about the initial rotation period distribution for stars (e.g., Barnes 2010 ; Matt et al. 2015 ), the specific analytic prescription for modeling the spindown (Angus et al. 2015 ), and exploring the efficiency of this angular momentum loss mechanism at older ages (van Saders et al. 2016 ). One of the most compelling results from the rotating star sample in Kepler is the discovery of a bimodal period distribution. McQuillan et al. ( 2013 ) first found a bimodality in the distribution of field M dwarfs with periods between ∼10 and ∼50 days. This feature was also found in the Kepler field K dwarfs in McQuillan et al. ( 2014 ), but was not seen in the bluer stars. Using Gaia DR1, Davenport ( 2017 ) was able to remove contaminating subgiants from the rotation sample, and found the bimodality extended to the G dwarfs as well. This bimodal surface rotation period distribution is either a new short-lived transition or instability phase of rapid angular momentum loss, or a signature of star formation history imprinted in the present-day rotation period distribution. However, as Davenport ( 2017 ) notes, this feature has only been observed in the Kepler rotation period catalog, and most critically only for stars within ∼300 pc. The faint M dwarfs in the Kepler sample studied in McQuillan et al. ( 2013 ) have an average distance of 250 pc. Gaia DR1 was also only able to provide precise distances for the nearest stars, and Davenport ( 2017 ) report a median distance of 285 pc for the blue stars in their sample. In this paper, we extend the work of Davenport ( 2017 ) in studying the Kepler rotation period distribution as a function of distance from the Sun using new astrometric data from the Gaia mission. By matching the McQuillan et al. ( 2014 ) rotation period catalog to the newest data from Gaia Data Release 2 (Gaia Collaboration et al. 2018b ), we can use precise distances for essentially every star in the McQuillan catalog to select the most-likely main-sequence dwarfs out to distances >2 kpc. Importantly, this filters out both subgiants, the main contaminant noted by Davenport ( 2017 ), and unresolved binary stars. Here we demonstrate

O B-Citation I-Citation I-Citation I-Citation O O B-Mission I-Mission I-Mission I-Mission B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Telescope O O O O O O O O O O O O O O O O O O B-Telescope O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Telescope O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset O O O O O O O O O O O O O O O B-Telescope O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Telescope O O O O O O O O O O B-CelestialObject O O O O O O B-Telescope O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Dataset I-Dataset I-Dataset I-Dataset B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O


( 2015 ). 2.1. Data Data for these studies were collected by reviewing published papers and searching both exoplanet.eu (Schneider et al. 2011 ) and exoplanets.org (Han et al. 2014 ). We start with the six planets in Table 2 of Schwartz Cowan ( 2015 ) that have thermal eclipses and phase amplitudes (at wavelengths longward of 0.8 μ m): HD 149026b, HD 189733b, HD 209458b, WASP-12b, WASP-18b, and WASP-43b. Then we add WASP-14b (Wong et al. 2015 ), HAT-P-7b, and WASP-19b (Wong et al. 2016 ) to our sample. We also incorporate new data from Zhou et al. ( 2015 ), Evans et al. ( 2015 ), and Line et al. ( 2016 ). We collect first-order phase offsets (Sections 2.2 and 2.2.1 ) from Knutson et al. ( 2009a , 2009b ) and Wong et al. ( 2015 , 2016 ). Knutson et al. ( 2009a ) concluded the offset they found was not statistically significant, so we use their largest uncertainty (72° ± 61°). Phase offsets through second order come from Cowan et al. ( 2012 ), Knutson et al. ( 2012 ), Maxted et al. ( 2013 ), Zellem et al. ( 2014 ), and Stevenson et al. ( 2014c , 2017 ). However, the second-order components in Zellem et al. ( 2014 ) were found to be unnecessary, and those at 4.5 micron in Cowan et al. ( 2012 ) are disputed, so we use neither in our fits. Also, Cowan et al. ( 2012 ) reported two sets of fit parameters for WASP-12b based on different models for detector systematics 7 7 Stevenson et al. ( 2014a ) fit eclipse depths for WASP-12b but not phase parameters, so we do not use their values in our analysis. —we use parameters from their preferred polynomial model but test the other scenario in Section 3.1 . 8 8 WASP-12 has binary companion stars that affect photometry of the system (Bechter et al. 2014 ). We use dilution factors from Stevenson et al. ( 2014b ) to correct observations of WASP-12b when appropriate. All nine planets in our study have non-zero offsets in at least one waveband. 2.2. Model We take an energy balance approach to interpreting thermal phase variations (e.g., Cowan Agol 2011b ): we compare the radiation going into and coming out from a planet to infer bulk energetics of that planet’s atmosphere. For our study, we use the energy balance model described in Section 3.1 of Schwartz Cowan ( 2015 ), which accounts for uncertainties in system parameters, as well as reflected light contamination, meridional heat transport, and other sources of uncertainty. Those authors treated stars as blackbodies; we estimate better stellar brightness temperatures for each observation by using BT-NextGen spectra (Allard et al. 2012 ). Then we use our compiled data to calculate the relative intensity of planets and their host stars at each observed wavelength. In Cowan Agol ( 2011b ) and Schwartz Cowan ( 2015 ), this nightside intensity ratio, ψ n ( λ ), is defined aswhere δ ecl is the eclipse depth, δ var is the peak-to-trough phase amplitude of the full light curve, and δ tr is the transit depth. This is exact only when an observation has no phase offset

B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-URL B-Citation I-Citation I-Citation I-Citation O O B-URL B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Wavelength I-Wavelength I-Wavelength B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject I-CelestialObject B-CelestialObject B-CelestialObject O B-CelestialObject O O O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O B-CelestialObject O B-CelestialObject B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Wavelength I-Wavelength O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O B-CelestialObject O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


ALMA data (Decarli et al. 2017 ; Champagne et al. 2018 ). In Figure 9 we show the postage stamps of our follow-up observations. Figure 9. Source adjacent to the quasar J2211, detected solely in the dust-continuum emission, i.e., with no secure redshift measure. Top: postage stamps (20′′ × 20′′) of our follow-up observations; labels are as in Figure 3 . Bottom: spectral energy distribution of J2211c. We assume that the source is located at the same redshift of the quasar. We report our photometric measurements/limits and, for comparison, various templates of local galaxies and the best SED fit from MAGPHYS -high- z . The labels and templates are as in Figure 6 . J2211c SED results to be intermediate between the low- z ULIRG Arp220 and the starbursting galaxy M82 (see Section 3.1 ). On the basis of our follow-up observations, and considering the predicted density of millimeter-sources, we are not able to exclude that this source is a fore/background (see the text for details, and Champagne et al. 2018 ). In analogy with the companions discussed in the main body of the paper, we compare the SED of J2211c with those of local galaxies, and we fit our photometric data with MAGPHYS -high- z (see Figure 9 ). From the latter, we find that the SED of J2211c is better reproduced by a galaxy model in between Arp 220 and M82 (i.e., a powerful local ULIRG and a starburst), with M * ∼ 3 × 10 10 M ⊙ and SFR ∼ 130 M ⊙ yr −1 . We further measure the obscured/unobscured SFR ratio of J2211c, following the procedure used for PJ167c (see Section 3.2 ). The star formation rate is dominated by the obscured contribution (SFR UV ∼ 2 M ⊙ yr −1 and f obscured ∼ 0.99). We report all these estimates in Table 6 . The lack of a secure redshift confirmation prevents us from drawing further conclusions on the nature of this source, or from placing it in the context of previous observations. Appendix B Quasar Photometry In the framework of our study of companion galaxies, we also perform forced photometry at the position of the quasars in the Spitzer /IRAC and HST /WFC3 images (see Section 2 for methodology). In Table 7 we report the derived quasars’ photometry. In Figure 10 , we show the quasars SEDs. The fluxes measured in our follow-up data are consistent with those expected from the observed optical/NIR spectra, when available, and/or from a lower- z quasar template (Selsing et al. 2016 ) rescaled to match the observed J band magnitude. Figure 10. Spectral energy distribution of the quasars in our sample. The observed photometric measurements (filled points) are obtained from our new follow-up data and from the literature (see Table 7 ; the filter responses are reported in the lower right panel). We also show the available optical/NIR spectra (light gray; see also Mazzucchelli et al. 2017b ), and a lower-redshift composite template shifted at the redshift of the quasar (black solid line; Selsing et al. 2016 ). The location of the Ly α line is highlighted with a light blue

B-Observatory O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Software I-Software I-Software O O O O O O O O O O O B-CelestialObject O O O O O O O O O O B-CelestialObject O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O B-Software I-Software I-Software O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O B-CelestialObject I-CelestialObject O B-CelestialObject O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-CelestialObject O O O O O B-CelestialObject O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope B-Instrument O B-Telescope B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Wavelength I-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Wavelength I-Wavelength O O O O O O O


for % of the total instantaneous flux, and % of the total maximum flux, in good agreement with the results of Lamb et al. (2013) . Here and in the next paragraph again, we assume that, the term “total instantaneous (maximum) flux” corresponds to the sum of instantaneous (maximum) fluxes in those events considered in the context. 8. Cancellation in Lamb et al. (2013) accounts for % of the total flux. If we leave out the fluxes contained in surviving features in semi and partial cancellation (as done in Lamb et al. 2013 ), we obtain % total instantaneous flux and % total maximum flux that corresponds to cancellation event, which is comparable to the results obtained by Lamb et al. (2013) . 4.5. Lifetimes of the features In Fig. 11 we plot histograms of lifetimes of the features, where the lifetime is determined as described in Sect. 3.3 . A correction factor was proposed by Danilovic et al. (2010) to avoid underestimating the lifetimes for the same observations, as the duration of the time-series is itself comparable to the lifetimes of longer-lived features. The suggested weight to be multiplied to each bin of the lifetime distribution is , where is the number of frames a feature lives, and is the total number of frames. We show lifetime histograms before (yellow-shaded) and after (gray-shaded) applying the correction factor. The plotted histograms are for the two extreme ratios of the areas of the children or parents, 10:1 and 2:1 respectively. Nearly % and % of the features for the 10:1 and 2:1 cases respectively, in the lifetime histograms live for just one time step. The more restrictive the area-ratio criterion restricting the death of features, the larger the number of long lived features, since a higher number of splitting and merging events are not counted as contributing to the birth or death of the larger feature (although for the smaller features they are involved in these events). Fig. 12 Lifetimes of features for an area ratio criterion 5:1 vs. maximum fluxes (panel a) ) and maximum, average magnetic field strength values from COG technique (panel b) ). Here maximum refers to the maximum value of the magnetic flux or the field strength, gained during the entire lifetime of the feature. In panel a) the range of plotted maximum flux of feature is limited to Mx to enhance the visibility of flux-lifetime relationship. The over-plotted red cross symbols represent the average value of the maximum flux of features at each lifetime (see main text for details). The maximum averaged over each lifetime bin are displayed in the figure and are represented by over-plotted red cross symbols. The mean and median lifetimes are and min ( and s) respectively for the 10:1 case and are and min ( and s) respectively for the 2:1 case. Similar lifetimes were also reported by Zhou et al. (2010) . These small mean and median lifetimes are clearly dominated by the large number of features that live for just one time step. Thus, the mean lifetimes we

O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-ObservationalTechniques I-ObservationalTechniques O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O


in Section 3 and the recipe given in González Delgado et al. ( 2016 ). In a nutshell, for any given galaxy we compute the spaxel-to-spaxel mass of stars younger than 32 Myr. Then, by co-adding these masses over a given aperture we can estimate the average rate of star formation during this period. An extensive discussion about the optimal limiting age of the stellar population used to define the recent SFR is given in González Delgado et al. ( 2016 ). We tested the robustness of this assumption by repeating Fig. 1 using a limit of both 100 and 200 Myr. We find that the results are qualitatively in agreement. Our stellar population approach to compute the SFR is different from the usual H α luminosity proxy used in the literature (Kennicutt 1998 ; Catalán-Torrecilla et al. 2015 ). Nevertheless, it is most powerful for ETGs since it does not depend on the detection of emission lines. Fig. 1 also shows the SFR versus M ⋆ relation for our comparison sample of ellipticals (see Section 2 ) and for our ETGs without considering the bulge and disc components separately. It is worth noticing that, due to the optical coverage of the spectroscopic data, our SFR might be considered as upper limits for very low values of the SFR (see discussion in López Fernández et al. 2018 ; Bitsakis et al. 2019 ). This issue is more acute for our bulge SFR estimations, while it is almost negligible for the star-forming discs. However, this does not affect the main conclusions of this paper. Figure 1. Star formation rate (SFR) versus stellar mass ( M ⋆ ) distribution for our sample of ETGs. The measurements for bulges and discs are shown with red circles and blue stars, respectively. Black squares represent measurements of the same ETGs, but for the galaxy as a whole. Yellow triangles display the position of the sample of elliptical galaxies described in the text. The best fit to the SFMS from Elbaz et al. (Grey; 2007 ), Renzini Peng (Orange; 2015 ), González Delgado et al. (Navy; 2016 ), and Sánchez et al. (Green; 2019 ) are also shown for comparison. Downward arrows mark where the measured SFR is an upper limit (see text for details). It is clear from Fig. 1 that, whenever there is star formation in our ETGs, this is happening in the disc component and not in the bulge. We find that on average the specific SFR (sSFR) is smaller in bulges than in discs. The mean difference in logarithm accounts to at least 1.3 dex when integrated across the whole FoV of CALIFA. In addition, a qualitative comparison with literature values of the best fit to the SFMS at z ∼ 0 (Elbaz et al. 2007 ; Renzini Peng 2015 ; González Delgado et al. 2016 ; Sánchez et al. 2019 ) shows that several discs of our ETG sample are compatible (or over) with the SFMS, whereas bulges occupy the region of the diagram defined by retired galaxies (Crocker et al. 2011 ; Cano-Díaz et al. 2016 ; Sánchez et al. 2018 ), or even lower SFR values for a given stellar mass. In particular, we find that 30 discs and only

O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength I-Wavelength O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Survey I-Survey I-Survey I-Survey O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O