however, need to find more stars with similar travel times as S5-HVS1. It is remarkable that the age of S5-HVS1 ejection is close to both the age of the disc of young stars around the GC (Lu etal. 2013 ) and the age of the Fermi bubbles (Bland-Hawthorn Cohen 2003 ; Su, Slatyer Finkbeiner 2010 ) which have been potentially associated with the recent accretion event in the GC (Zubovas, King Nayakshin 2011 ; Guo Mathews 2012 ), thus potentially linking these different astrophysical objects. An alternative scenario that would naturally produce a time-variable HVS spectrum is that involving an Intermediate Mass Black Hole (IMBH) orbiting the GC (Yu Tremaine 2003 ; Levin 2006 ). In this mechanism, during the inspiral of the IMBH, the HVS production rate peaks and then subsides due to dynamical friction around the SMBH (Baumgardt, Gualandris Portegies Zwart 2006 ; Darbha etal. 2019 ), with the fastest HVS being ejected in the final phase of the in-spiral. This mechanism produces a strongly anisotropic distribution with the fastest stars in the orbital plane of the IMBH (Rasskazov etal. 2019 ). There is also some indication that the HVSs produced by this mechanism tend to have higher velocities and a flatter velocity spectrum than the classical Hills mechanism (Sesana, Haardt Madau 2007 ). While there is currently not much evidence for the presence of an IMBH in the GC (Gualandris Merritt 2009 ) other than a shallow stellar density slope that can be produced by an IMBH scattering (Baumgardt etal. 2006 ), if an IMBH inspiral happened a few Myr ago, then it would produce an excess of nearby and fast HVSs with a narrow range of ejection times. To test for this possibility we need to search for other nearby HVS and see if there is an excess of stars that were ejected at roughly the same time as S5-HVS1 (5Myr ago), are strongly anisotropic and that have a velocity spectrum inconsistent with the Hills mechanism. An interesting consequence of the fact that S5-HVS1 has lower mass than most other HVS in the halo is that its expected lifetime given the stellar mass of 2.3M is quite long around a 1Gyr. By the end of its life the star would have travelled a distance of 2Mpc, traversing a large fraction of the Local Group. This suggests that searching for such ejected stars at large distances from the MW or Andromeda (Sherwin, Loeb OLeary 2008 ) is quite promising. On top of being well separated in colourmagnitude space from other contaminants, an S5-HVS1-like star would eventually evolve on to the red giant branch and thus be detectable much more easily. Searches for S5-HVS1-like stars within the whole Local Group will be possible with upcoming deep imaging surveys like LSST (LSST Science Collaboration etal. 2009 ). In this paper, we tried to use the positionvelocity information on S5-HVS1 to constrain the distance from the Sun to the GC and the Galactic Solar velocity. We have not been able to constrain those simultaneously, mainly due to the precision of


of this procedure at z = 6 (red) and z = 7 (blue) in Figure 4 as obtained by changing only the outflows (left) or changing only the column density (right). While both scenarios produce stacks that have consistent FWHM values with observations at z = 6, the changing outflows scenario yields also a consistent FWHM with z = 7 measurements. This also confirms that changing the column density predicts very high FWHM >400 km s −1 at z = 7. As quoted in the legend for changing the outflows scenario (left), z = 6 lines are broader than those at z = 7, and consistent with Pentericci et al. ( 2018 ). The lines are highly asymmetric with extended red wings. The blue peaks disappear due to the poor resolution. Figure 4. Example for stacks of randomly selected 52 mock spectra at z = 6 (red) and 19 mock spectra at z = 7 (blue) with f esc ( z = 7) ≈ 0.5 f esc ( z = 6) in agreement with the measured drop of X Ly α . We assembled the stacks following the method outlined in Pentericci et al. ( 2018 ) with spectral resolution R = 1390 (corresponding to Δ = 216 km s −1 ). All these spectra are either obtained by changing only outflows while keeping the column density N H I fixed (left) or changing only the column density N H I while keeping outflows fixed. In all cases, dust is kept fixed. In either scenario, the FWHM of stacks at z = 6 is consistent with values reported in Pentericci et al. ( 2018 , FWHM( z ∼ 6) = 300 ± 30). The FWHM of stacks obtained by changing outflows at z = 7 is also consistent with Pentericci et al. ( 2018 ) measurements (FWHM( z ∼ 7) = 220 ± 25) while those produced by changing the column density N H I are not, and hence indicating that changing the outflow can naturally lead to a higher Ly α escape as well as broader lines at lower redshifts as observations indicate. To quantify the width using the above recipe, we generate randomly 10,000 combination of stacks at z = 6 and z = 7 from the total number of model combinations (i.e., 143 and 201) in each scenario and compute their widths. We show the resulting width distribution at these redshifts in Figure 5 . Results from changing outflows and changing column density scenarios are presented by dashed and solid lines. Shaded red and blue areas show Pentericci et al. ( 2018 ) measurements at z ∼ 6 and z ∼ 7, respectively. From this exercise alone, we constrain the FWHM, using changing column density scenario, to 261.7 ± 14.3 km s −1 at z = 6 and 434.9 ± 47.8 km s −1 at z = 7, and using changing outflows scenario, to 291.5 ± 7.9 km s −1 z = 6 and 224.6 ± 22.8 km s −1 at z = 7. It is evident that, over all possible combinations and the prior range assumed, the changing outflows scenario produces consistent width distribution within the 1 σ level of Pentericci et al. ( 2018 ) measurements. This confirms that changing the column density predicts inconsistent width distributions. It might be noteworthy that while we use the Pentericci et al. ( 2018 ) results in this section, other studies of


( 1 ) returned best-fitting parameters of α = 2.29 ± 0.08 and log ( N ) = −7.3 ± 0.1. The total number of degrees of freedom was 10 970. These values along with the measured Whittle statistic are provided in Table 4 . Allowing both N and C to vary between data sets did improve the fit by Δ S = 90, but there was no change to the measured slope. Additionally, allowing the fit to be driven by the low-frequency ‘red-noise slope’ by truncating the PSDs at 10 −4 Hz returns best-fitting parameters that are consistent with those listed in Table 4 . We also fitted equation ( 1 ) to PSDs generated in ‘soft’ and ‘hard’ energy bands (0.3–1 and 1–10 keV, respectively). One may expect to observe differences in the PSD at different energies based on the energy dependence of the rms variability at low and high frequencies, as shown in Fig. 7 . For example, it is clear that larger variations are observed between the slow and fast variations in the soft band compared to the hard. Subsequently, we may expect the soft band to have a steeper PSD. However, we found that the PSD does not exhibit any significant energy-dependence with best-fitting values typically remaining consistent within the uncertainties. As such, these variations are most likely contained within the uncertainties of the PSD fitting. These values are reported in Table 4 . Table 4. The best-fitting parameters of the two models (power-law and bending-power-law) fitted to the PSDs of NGC 3227 in three energy bands: 0.3–10, 0.3–1, and 1–10 keV. The fits are applied to all six XMM–Newton EPIC-pn observations from 2016. See Section 3.5 for details. We then searched for evidence of a break (or bend) in the PSD by fitting a bending power law (plus a constant): where N is the normalization, ν b is the break frequency, and and are the spectral indices below and above the bend, respectively (see González-Martín Vaughan 2012 for further details, who find that a significant bend is detected in the PSDs of 15 out of their sample of 104 AGN observed with XMM–Newton ). Fitting equation ( 3 ) to the 0.3–10 keV PSD improves the fit by Δ S = 10 for two additional free parameters, suggesting the marginal detection of a bend. The low-frequency slope, , is consistent with a value of 1, consistent with the X-ray PSD slopes observed in other Seyfert galaxies. Meanwhile, the high-frequency slope above the bend is steeper, with , where the best-fitting bend frequency is Hz, largely consistent with previous results (Uttley McHardy 2005 ; González-Martín Vaughan 2012 ; Arévalo Markowitz 2014 ). Extending this fit to the soft and hard bands, we find consistent results (Δ S = 9 and Δ S = 10, respectively) with break frequencies of and Hz and high-frequency slopes of and , respectively. Again, the bend detection is marginal and we find no evidence of any energy-dependence of the PSD. The best-fitting values are summarized in Table 4 and the bending-power-law fit is shown in Fig. 9 . In Fourier-based PSD analysis, it is


panels. In the labels we show the average cross-power above 20′′. Figure 2. Stacked CIB–CXB cross-power spectra: between 3.6 μ m and [0.5–1], [1–2], [2–4], and [4–7] keV (top panels); between 4.5 μ m and CXB (bottom panels). Open circles with dashed error bars denote the absolute values of negative results. Note that the shown spectra are subtracted by the cross-power between IR and the stowed X-ray map, that is, the potential contamination from the X-ray particle background is removed from our final results. The final mean powers (used for further SED analysis) are summarized in Table 1 . As demonstrated in Cappelluti et al. ( 2013 , 2017 ), Mitchell-Wynne et al. ( 2016 ), and Li et al. ( 2018 ) the CIB versus CXB cross-power is in excess over shot noise at angular scales larger than 20′′. Since at those scales known foregrounds are significantly weaker than our signal (Helgason et al. 2014 ; Cappelluti et al. 2017 ; Kashlinsky et al. 2018 ), we therefore calculate the weighted mean cross-power above 20′′ asbetween each IR and X-ray band, for each field and for the stacked cross-power spectrum. As shown in Table 2 , the [1–2] keV band is significantly correlated with all IR wavelengths at ∼5 σ significance level, while the [0.5–1] and [2–4] keV bands are correlated with the IR wavelengths at ∼1–3 σ levels. There is no significant cross-power signal between the [4–7] keV band and any IR wavelengths so we cannot exclude an intrinsic correlation. By including the COSMOS field with a larger area, our new analysis extends the CIB–CXB cross-power analysis to ∼3000′′, doubling the largest scale reached in Cappelluti et al. ( 2017 ). The reason the highest significance is in the [1–2] keV band is due the higher S/N due to the peak in effective area of the Chandra mirror in that spectral region. Stacked results in the hard X-ray band (i.e., [2–4] keV + [4+7] keV) are not as significant as in Li et al. ( 2018 ) since deep fields add data affected by strong cosmic variance on very large scales. In the hard band most of the events are particles, so the low astrophysical hard X-ray photon statistics is another source of noise. As a cross-check we compared the significance of our stacks with those shown in Figure 17 of Li et al. ( 2018 ) for the [2–7] keV band versus 3.6 and 4.5 μ m and found consistent results. Table 2 Cross-power Spectrum Amplitude (>20′′), Calculated as a Note. a In units of 10 −11 photon s −1 cm −2 nW m −2 sr −1 . 4.2. Cross-power Hardness Ratios Similar to the X-ray color–color diagram, we look into three hardness ratios for the mean cross -power, which are the equivalent of photometric color indices, following the typical definition (Brunner et al. 2008 ) The statistical 1 σ hardness ratio errors are calculated from the mean power (Table 2 ) by error propagation. The mean 1 σ errors for the hardness ratios are ∼0.3 for HR1 and HR2 and ∼0.8 for HR3. Due to the larger errors in HR3, resulting from the uncertainties from harder


region that is magnetically connected to the bow shock and is filled with ions or electrons backstreaming from the bow shock (Eastwood et al. 2005 ). The backstreaming particles can excite many instabilities, generating various waves (Johlander et al. 2018 ; Shan et al. 2018 ; Shi et al. 2020 ) and transient structures (Schwartz et al. 2018 ; Bai et al. 2020 ; Turner et al. 2020 ). In recent years, a new type of magnetic structure, electron-scale magnetic holes, has been revealed to exist upstream of the terrestrial bow shock (Yao et al. 2019a ; Wang et al. 2020b , 2020c ) as well as in the magnetosheath (Huang et al. 2017a ; Yao et al. 2017 ) and current sheet (Ge et al. 2011 ; Sun et al. 2012 ). The sizes of these holes are within 2 ρ i (ion gyroradii) (Sun et al. 2012 ; Wang et al. 2020d ), and their cross sections have complicated shapes (Liu et al. 2020b )—some are circular (Liu et al. 2019 ) or like a two-dimensional sheet (Wang et al. 2020d ). Bipolar electron velocities are often observed in the cross section, suggesting that there are electron vortices inside the structures (Huang et al. 2017a ; Wang et al. 2020c ). In contrast, electron-scale magnetic holes have little effect on the ion population (Balikhin et al. 2012 ; Sundberg et al. 2015 ). These structures can trap electrons, and have rich wave activity (Huang et al. 2017b , 2019 ; Yao et al. 2019b ). Several mechanisms have been provided to explain their generation, such as electron mirror instabilities (Yao et al. 2019a ), tearing-mode instabilities (Balikhin et al. 2012 ), solitary waves (Ji et al. 2014 ; Li et al. 2016 ), and electron vortex magnetic holes (Haynes et al. 2015 ). Since they can survive up to 100 electron gyroperiods (Haynes et al. 2015 ) and they can propagate with the ambient flow (Wang et al. 2020d ), it is difficult to identify whether an electron-scale magnetic hole in the terrestrial foreshock is locally generated or originates from the upstream undisturbed solar wind. Thus, the generation mechanism of electron-scale magnetic holes upstream of the bow shock is still not fully understood. Magnetic peaks are another kind of magnetic structure, characterized by an enhancement of the magnetic field strength (Balikhin et al. 2010 ). They widely exist in the planetary magnetosheath (Horbury et al. 2004 ; Balikhin et al. 2010 ) and in the heliosheath (Burlaga et al. 2006 ). In the heliosheath, their sizes are of the order of 20 ρ i (Burlaga et al. 2006 ). They mainly appear in the mirror-unstable region, and they are suggested to be generated by ion mirror instabilities (Soucek et al. 2008 ). Using observations of the four Cluster spacecraft, Horbury et al. ( 2004 ) studied the propagation of magnetic peaks based on timing analysis (Harvey 1998 ). The authors found that their propagation velocities are scattered around zero within the timing errors in the plasma frame, and suggested that they are stationary in the plasma frame. The authors also found that their


amplitude than a denser sample (gray region). Second, if the galaxy sample is sufficiently sparse, then the genus amplitude becomes sensitive to galaxy bias (or more precisely the combination b 2 A s )—the width of the filled regions indicates the sensitivity of to bias, for given number density and fixing all other cosmological parameters. For our fiducial number density a ∼10% variation in the galaxy bias generates a ∼0.5% uncertainty in the genus amplitude (see red shaded area). One can also observe a faint redshift evolution in the red shaded region—this is due to the amplitude of the matter power spectrum decreasing with redshift relative to the shot-noise term, which we have fixed to be constant at each redshift. However, if we increase the number density of the galaxy sample by an order of magnitude (gray shaded region), the sensitivity of the genus amplitude to the bias becomes negligible—as the shot noise decreases, the genus amplitude becomes insensitive to the amplitude of the power spectrum. The redshift evolution is also suppressed by selecting a denser sample. A more general word of caution is required: the shot-noise contribution is not perfectly represented by a white-noise power spectrum with . In fact, as we stray into scales at which shot noise affects our results, the expansion of the genus in Hermite polynomials will not provide a good representation of the Poissonian signal, as the Poisson distribution has a different moment-generating function. , where is the mean galaxy separation of the catalog, is an important condition on our analysis. Shot noise has been discussed in Appleby et al. ( 2017 ) and further in S. Appleby et al. (2020, in preparation). We note that both the redshift-space distortion effect and shot noise introduce a redshift dependence to the genus amplitude. For the smoothing scales and redshifts used in this work, the redshift-space distortion effect systematically decreases the genus amplitude with increasing redshift. Conversely, the shot-noise contribution increases the genus amplitude with redshift. Both effects are and will act to practically cancel one another. To completely suppress the shot noise and bias effects, the condition is required. For the smoothing scales and number densities considered in this work, the genus amplitude is only weakly sensitive to the combination of b , A s , and P SN . Nevertheless, we must include the P SN contribution to the power spectrum to avoid systematic bias in our cosmological parameter reconstruction. Appendix B Effect of Variation of Cosmological Parameters on Measured Genus Amplitudes At numerous points in our analysis when measuring the genus curves from the BOSS galaxy catalog, we have been forced to fix the distance–redshift relation. Specifically, we smoothed the field with constant comoving scale R G = 20 Mpc, which corresponds to angular smoothing on the unit sphere. We measured the genus per unit area, and so divided the genus by the total area


to evolve only a slice spanning ( r , θ), and with constant φ, thus assuming that the system will remain rotationally symmetric during the evolution. In order to justify to what extent and under which conditions this assumption is valid the comparison with 3D computation was needed. We will comment on that in Section 4.3 . However, within the 2.5-D approach, the φ component of the four-velocity is considered. We prescribe this component in Boyer–Lindquist coordinates according to relations given by equations ( 4 ) and ( 5 ). We chose several exemplary simulations and show their results in the form of snapshots. Every snapshot contains four panels with the slices of and its equatorial profile, ρ in arbitrary units, and λ in geometrized units, and it is labelled by the time t in geometrized units, where M is the mass of central black hole. The axes show the position in geometrized units. The red colour in the slice of radial Mach number corresponds to the supersonic motion, and blue regions indicate the subsonic accretion. The shock front is located at the place where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre. The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre. On top of the radial Mach number map the velocity streamlines are plotted in blue. These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel. 4.2.1 Bondi solution equipped with angular momentum At the beginning we check that for pure Bondi solution with λ = 0 we get the stationary solution of the flow with the properties consistent with the analytical solution. After that, we start to study the slowly rotating flows. For the first simulation, we pick the parameters such that λ eq belongs to the multicritical region (the region in the parameter space where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular , λ eq = 3.6M and γ = 4/3. In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data.). In other words, we expect that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way that it gets very close to 1, but does not touch it. In Fig. 2 , the initial conditions at t = 0 M are shown and the final state at t f = 10 4 M of the gas is depicted in Fig. 7 . At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch. It is interesting to note that for t = t f the Mach number actually crosses the line, however only on a very short radial range. Usually, the presence of the


power-law template used by Wik et al. ( 2014b ) to actually fit the harder high-mass binary dominated sources) gives essentially the same results. Figure 2. NuSTAR spectrum (black) within a circular region of 30′′ centered on the X-ray peak joint-fitted with the Chandra ICM (blue) and LMXB (red) spectra. The dashed lines are the individual components discussed in the text. The vertical dashed blue lines below 3 keV are the Gaussian models, to take into account the line emission. Beyond ∼5 keV, the X-ray emission is dominated by the AGN-related activities (core, knots, and jet) and is completely dominated by these activities beyond ∼15 keV. Error bars are at 1 σ . The 3–40 keV NuSTAR spectrum was joint-fitted with the Chandra ICM and LMXB spectra by including these two component models as well as a single power-law model to represent the combined emission from the core, knots, and the jet. The same Galactic absorption model was also used for all the components. We tied all the parameters for the ICM and LMXB models, except for the ICM component. The ratio of the two APEC normalizations is tied to that of the Chandra model but not the overall normalizations to allow for the cross-calibration uncertainty of NuSTAR and Chandra and the difference in response due to the much larger PSF of NuSTAR . The difference in best-fit normalizations turns out to be small, with less than a 1% difference when constraining the overall hard X-ray emission from the AGN activities (this section) and about 6% when decomposing the core emission from the knots/jet (Section 4 ). The best-fit spectra with all the model components are shown in Figure 2 . The best-fit power-law index of the overall AGN-related emission is and the associated 20–40 keV flux, which completely dominates the spectrum, is erg cm −2 s −1 . The spectrum is well characterized by a single power law with no evidence of cut-off in energy below ∼100 keV (3 σ ). 11 11 The lower limit of the cut-off energy is determined by the fitting of a cut-off power-law model ( cutoffpl). The 20–40 keV flux measured is an order of magnitude lower than that determined by de Jong et al. ( 2015 ) with the Suzaku observation taken in 2006. It is believed that the X-ray emission above 20 keV detected near the end of 2006 was due to the HST -1 flare (de Jong et al. 2015 ; Prieto et al. 2016 ). The Chandra observations in 2017 show that the soft emission (≲10 keV) from HST -1 has become fainter by a factor of ∼40 since 2006 November. By simple scaling, the contribution from HST -1 in 20–40 keV should be a factor of a few lower than that from the core. This is indeed the case where HST -1 is a factor of six fainter than the core in 2017 measured in this band (Section 4 ). Therefore, for the first time, we have detected hard X-ray emission above ∼10 keV dominated by the core, which also gives the upper limit of its quiescent emission in hard X-rays. 4. Decomposing the Hard X-Ray Emission of the Core,


We perform this analysis for the outer and the inner (if present) spirals. On the one hand, we observe that all the outer spirals have similar pitch angles (|φ| ∼ 16°), but the simulation with T s = 20 kyr (|φ| ∼ 20°). On the other hand, we find that the inner spirals for the moving shadows (|φ| ∼ 23°) are bit larger than the planetary ones (|φ| ∼ 14°). Again, we see that the pitch angle increases with T s (see Table 1 ). In Fig. 3 , we show a synthetic image of the emission of the disc at a wavelength equal to 1 μm, where the intensity has been normalized and scaled by r 2 . This calculation is done by means of the radmc-3d code (Dullemond et al. 2012 ), modifying it to mimic the shadows by blocking the photons sent from the star in the shadowed regions. It is based on the density field for the simulation with prograde shadows with T s = 10 kyr, assuming a dust-to-gas ratio of 0.01. Figure 3. Radiative transfer calculation at 1 μm for the simulation of prograde shadows with T s = 10 kyr, shown in the third column of Fig. 2 . We assume a dust-to-gas ratio equal to 0.01. 4 DISCUSSION Moving shadows dramatically changes the disc structure and evolution. For retrograde rotations, we do not observe any spirals, despite the fact that there are two shadows cast in the disc. Because the shadows are always confronting the gas motion, there is not enough time for it to cool down in order to create a gradient pressure to act as a symmetry breaking. This could be different if a more efficient cooling mechanism is taken into account (e.g. β cooling prescription, Gammie 2001 ). Realistically, it is expected that upper layers of the gas cool down faster, resulting in a colder shadowed region able to produce a gradient pressure, promoting sharper spiral features for this mode. For static shadows, we obtain similar results as in Montesinos et al. ( 2016 ). However, in the prograde rotation case, a gas portion under the shadow moving at the same shadow's speed will always be in the shade. This corotating position can be obtained by equating the azimuthal velocity with the shadow speed v s = (2π/ T s ) r . For a Keplerian disc, where , we obtain R co = ( GM ⋆ ( T s /2π) 2 ) 1/3 . In Fig. 4 , we plot R co as a function of the shadow's period and the azimuthal velocity along with the shadow speed for the different periods T s . The intersection between both curves gives the corotating location for each model (red dots). These points are in excellent agreement with the location of the local perturbations observed in the density fields shown in Fig. 2 (see also Table 1 ). We also found that the perturbation development is independent of the stellar luminosity or the gas density. Figure 4. Left-hand panel: Shadow rotation speed ( v s = 2π/ T s r ) for different periods T s (colored dotted lines) along with the azimuthal velocity (solid curve). The intersections (red dots) give the corotating locations, in remarkable agreement with the location of the perturbations


consuming and may suffer from a low S/N (Grier et al. 2013 ) caused by the combination of weak absorption lines, surface brightness dimming, instrument sensitivity and atmospheric transmission. In the rest-frame optical regime, the QSO and host galaxy contribution can be separated in high S/N single-aperture spectra by directly modelling them with template spectra for each component (Shen et al. 2008a , 2015 ). While a large sample can be studied with this technique, the fixed aperture size prevents such studies from measuring σ * at a physical size matched to the spheroid component. Furthermore, the reliability of the measurements decreases with increasing AGN luminosity and contrast ratio, thus limiting this technique to AGN with L 5100 ≲ 10 45 erg s −1 . To probe more luminous AGN, deep off-axis spectroscopy a few arcsec away from the QSO has been obtained to reduce the contrast between QSO and host galaxy emission (Wolf Sheinis 2008 ; Hughes et al. 2000 ; Wold et al. 2010 ). While σ * can be reliably derived from these off-axis spectra (Wolf Sheinis 2008 ), an uncertain aperture correction needs to be applied to infer comparable measurements representative for the spheroidal component. A direct decomposition the stellar continuum of the host galaxy underneath very luminous QSOs with deep optical integral-field unit observations was previously attempted with VIMOS for HE 1029−1401 (Husemann et al. 2010 ). Although HE 1029−1401 is much closer with a redshift of z = 0.086 but has a similar luminosity as PG 1307+085, the achieved S/N in the host spectrum was significantly lower than what we obtained with MUSE for PG 1307+085. This can be clearly attributed to the superior sensitivity, spatial resolution/sampling, and image quality of MUSE compared to VIMOS. These capabilities allow direct measurements of σ * with MUSE even in a high contrast regime that has not been accessible to any other techniques so far. 4.2 The nature of the large extended emission line region 4.2.1 Ionized gas shells driven by AGN outflows? A number of recent studies report kpc-scale high-velocity outflows to be common around luminous AGN with line widths of about 1000 km s −1 (e.g. Cano-Díaz et al. 2012 ; Liu et al. 2013 ; Harrison et al. 2014 ; Liu, Zakamska Greene 2014 ). Those outflows have been interpreted as spherically symmetric radiation pressure-dominated winds (Liu et al. 2013 ) and are possibly common for AGN with L bol > 2 × 10 45 erg s −1 (Zakamska Greene 2014 ). However, not all QSOs above this luminosity threshold exhibit such powerful superbubble-like outflows (e.g. Husemann et al. 2010 , 2013 ). The ionized gas kinematics of the ENLR around PG 1307+085 exhibits only small line widths with a velocity dispersion of 50 km s −1 in all the bright extended knots without any asymmetric component. This confidently rules out the AGN-driven superbubble scenario at least for this particular QSO. Although PG 1307+085 is classified as radio-quiet QSO with R 0.1


RTE) applied to the spectra and polarized spectra emitted by the object. The RTE characterizes the interaction of the matter with the light that traverses through it. This interaction leaves its “fingerprints” in many different ways, for instance in the shape of the continuum of the spectra coming from the object, or in the presence or absence and the shape of spectral lines. Properly modeled, these features allow approximate characterization of the studied object. In recent decades, the advent of polarimetric observations has allowed the generalization of the RTE to polarized light ( Unno 1956 ; Rachkovsky 1967 ; Landi Degl’Innocenti Landi Degl’Innocenti 1972 ). This step was of great significance, as the polarization of the light encodes information on the magnetic field configuration as well as the geometry of the system, which allows a deeper insight into the system being studied. In solar physics, there are different approaches to solving the polarized RTE depending on the magnetic field regime adopted: atomic, Hanle, Zeeman, and/or Paschen (see e.g., Landi Degl’Innocenti Landolfi 2004 ). In this work, we focus on the solution of the polarimetric RTE under the Zeeman magnetic regime, which is usually the dominant one at this atmosphere layer. There are already several methods to address the solution of the RTE, each with its respective accuracy (see, e.g., del Toro Iniesta Ruiz Cobo 2016 ). There are several computational codes that allow one to infer the optical depth ( τ ) dependence of the physical properties in the solar atmosphere; for example, SIR ( Ruiz Cobo del Toro Iniesta 1992 ), SPINOR ( Frutiger et al. 2000 ), NICOLE ( Socas-Navarro et al. 2015 ), SNAPI ( Milić van Noort 2018 ), and STIC ( de la Cruz Rodríguez et al. 2019 ). These are of particular relevance here as they allow the furthest insight into the properties of the solar atmosphere in a three-dimensional volume. In these cases, the conversion between optical depth τ and geometrical height z is then performed by considering the relation d τ = − ρ κ c d z , where ρ is the density and κ c the continuum opacity, with the latter being a nonlinear function of the temperature T and gas pressure P g . Since the proper calculation of the gas pressure (and density) involves the whole three-dimensional volume, for which there is no account in the aforementioned codes, the gas pressure and density inferred by them are unrealistic, and hence also the conversion between τ and z . Deviations from hydrostatic equilibrium involve, among others, time-variation of the velocity, advection and viscous terms and, the most relevant at this atmospheric layer, the Lorentz force. The latter is responsible for the Wilson depression, a depression of the height at which photospheric spectral lines are formed with respect to the quiet-Sun because of the presence of magnetic field. Because the geometrical scale z is then inaccurate, so is the calculation of the spatial derivatives, such as those


et al. 2009 ; Rosario et al. 2013 , 2016 , 2018 , and references therein) or mid-infrared neon ([Ne ii ] 12.81 μ m and [Ne iii ] 15.56 μ m) emission lines (Zhuang, Ho Shangguan 2019 ). The infrared observations used in these works are obtained using large apertures, including the whole galaxy emission. As discussed in Rosario et al. ( 2016 ), despite finding a good correlation between far-infrared emission and SFR (which can be attributed to leaking of the SF regions to the whole galaxy or that the SF is constant over several hundreds of Myr) a substantial component of the cold dust luminosity that is associated with a diffuse interstellar radiation field can come from evolved stars. Even if the contribution from an old age stellar population is not dominant, the far-infrared may not be a good tracer of SF since dust can be heated by stars that are older than a few hundred Myr (Kennicutt et al. 2009 ; Hao et al. 2011 ; Rosario et al. 2016 ). Other recent effort on the bluer part of the spectrum has been made using a re-calibration [O ii ] λ3727 emission line (Zhuang et al. 2019 ; Zhuang, Ho Shangguan 2020 ), which is metallicity and electron density dependent. From the above, it is clear that it is thus necessary to find an independent way to obtain SFRs in AGN hosts. A powerful technique to disentangle the components summing up to a galaxy spectral energy distribution (SED) is stellar populations synthesis (e.g. Cid Fernandes et al. 2004 , 2005 ; Riffel et al. 2009 , 2008 ; Walcher et al. 2011 ; Baldwin et al. 2018; Cid Fernandes 2018 ; Salim, Boquien Lee 2018 ; Peterken et al. 2020 , and references therein). The synthesis is based on the simultaneous fit of different proportions of composite or simple stellar populations (SSPs) templates from a base of such spectra to the observed spectrum. In the case of the starlight fitting code (Cid Fernandes et al. 2004 , 2005 ; Cid Fernandes 2018 ), it returns the values of the gas mass rate that has been converted into stars throughout the galaxy life as well as this rate for each single SSP included in the base. These values can therefore be used to compute the SFR via stellar population synthesis, SFR ⋆ . In fact, this was already applied in Asari et al. (2007) using single fiber Sloan Digital Sky Survey (SDSS) data and older stellar population models generations (they used Bruzual Charlot 2003 models). From the discussion above, it is clear that it is of utmost importance to characterize the SFR of AGN hosts – in particular in the narrow-line region (NLR) and extended narrow-line region (ENLR), which are photoionized by the AGN. And in order to verify the actual role of the AGN on the SFR, it is necessary to compare the SFR values obtained for the NLR and ENLR with those obtained for a matched control sample of non-active galaxies at similar distances from the nucleus, as well as with predictions from simulations. In Rembold et al. ( 2017 ), we described the method we have used to select


polarized) radiation. Then, the radio spectral index S ν ∝ ν α can provide information as well, but it will often be ambiguous. Third, variability can be an indication, with rapid variability a sign of nonthermal emission (on timescales as short as a few minutes, e.g., Forbrich et al. 2017 ), but there is again some ambiguity on intermediate timescales. Fourth, radio brightness temperature is a rarely employed but powerful indicator, since this technique allows us to select sources with brightness temperatures that are above that of thermal emission. This can naturally be accomplished by using Very Long Baseline Interferometry (VLBI), with which it is thus possible to filter out any thermal emission, enabling a census of nonthermal radio emission that remains largely unaffected by thermal radio emission. Apparent brightness is inversely proportional to the synthesized beam size for an unresolved source, and for milli-arcsecond resolution VLBI observations in C -band (4–8 GHz), even faint detections have corresponding brightness temperatures of several 10 6 K, well above possible values of free–free emission ( ≈ 10 4 –10 5 K), even more so when considering that this is a lower limit. Even at this high resolution, the stellar radio emission remains unresolved, with a synthesized beam size corresponding to a linear scale of order 1 au in our case. Overall, the two most useful indicators thus are polarization and brightness temperature. In recent years, it has become possible to study populations of radio stars at unprecedented sensitivity, with a range of radio observatories. Our focus has been on observing YSOs in the Orion Nebula Cluster, the nearest young rich open cluster containing high-mass stars ( d ∼ 400 pc; Menten et al. 2007 ; Kounkel et al. 2017 ). Altogether, the ONC has 3500 stellar members that formed less that 2 Myr ago (Hillenbrand et al. 2013 ), many of which emit X-rays (e.g., Getman et al. 2005a ). Radio emission from YSOs in the ONC and the nearby Kleinmann–Low nebula was first observed when the NRAO Very Large Array became available (Churchwell et al. 1987 ; Garay et al. 1987 ), and prior to the VLA expansion, 77 radio sources had been observed in this cluster (Zapata et al. 2004 ). After the VLA expansion was completed, we obtained a deep pointing with what is now called the Karl G. Jansky Very Large Array, which allowed us to identify 556 radio sources in the ONC (Forbrich et al. 2016 ). Our analysis revealed extreme variability for many objects, strongly suggesting nonthermal emission (Forbrich et al. 2017 ). While this analysis included the derivation of in-band spectral indices for the brightest sources, it has been impossible so far to extract reliable wideband polarization information across the entire primary beam of this deep pointing. To obtain a more reliable census of the incidence of nonthermal emission, and to then use this information for a proper motion census of the inner ONC, we have utilized the


T icb = 6.36 × 10 4 K; the density ρ icb = 1.47 g cm −3 . The composition is uniform with 100% silicates out to a temperature T = 2.1 × 10 4 K and radius 2.72 × 10 9 cm, decreasing to 50% at T = 5.42 × 10 3 K and radius 3.39 × 10 9 cm, and to 1% at T = 2960 K and radius 4.84 × 10 9 cm. During the isolated phase, the parameter in Equation ( 7 ) is set to 0.04. During the first 10 8 yr, an additional 0.25 M ⊕ is lost, giving M XY = 0.42 M ⊕ , R p = 7.43 R ⊕ , L = 2.0 × 10 −10 L ⊙ , and a cooling time of 8 × 10 8 yr. At this time, M ⊕ yr −1 . As in Run 0.128 (Rev), a radiative zone extends inward to T = 1500 K. Later, the luminosity declines to ≈10 −11 L ⊙ , the cooling time increases by an order of magnitude and the rate of mass loss declines significantly, by a factor of 400, to 2 × 10 −12 M ⊕ yr −1 , by the final time. Between t = 10 8 yr and t = 7 × 10 9 yr, a further 0.13 M ⊕ is lost from the H/He envelope. The radius of the final model planet agrees quite well with that of the actual planet, as does the total mass. The total heavy-element mass (including the inner and outer cores) is M Z = 7.52 M ⊕ , and the H/He mass M XY = 0.29 M ⊕ . The inner core of 1.87 M ⊕ has R icore = 1.075 R ⊕ and mean density 8.35 g cm −3 . The region of almost 100% heavy elements has radius 1.84 R ⊕ , and the mean density of the inner plus outer cores is 5.86 g cm −3 . The mass of this final model is very close to that of the in situ model 0.128 (Rev). The temperature at the boundary between the inner and outer cores ( T icb ) is similar (1.54 × 10 4 K versus 1.75 × 10 4 K), and the corresponding density is slightly lower (7.59 versus 8.0 g cm −3 ). These differences are presumably caused primarily by the different masses of the inner cores. Both final models have outer radiative zones, extending inward to T = 1850 K in the in situ case and to 1500 K in the present case; they include less than 1% of the envelope mass. The structure of the final model for Run 1.00 (Rev) is plotted in Figure 12 . Figure 12. Structure of the model in Run 1.00 (Rev) at time 7.03 × 10 9 yr. Symbols and curves: as in Figure 11 . The mean density of the inner core is 8.35 g cm −3 . 3.4. Migration Model: Run 1.00(Old) Run 1.00(Old) starts at 1 au with σ init = 190 g cm −2 , slightly lower than that in Run 1.00(Rev). Outer densities and temperatures are the same in the two runs. The masses and radius as a function of time are given in Figure 13 . At first, M core increases very rapidly until, at t ≈ 7 × 10 3 yr, it reaches 6.78 M ⊕ , close to the isolation mass of 6.81 M ⊕ . At this point, M env = 0.084 M ⊕ . The temperature T cb is 9.91 × 10 3 K, much lower than the value of T icb = 6.1 × 10 4 K reached at a comparable evolutionary phase in Run 1.00 (Rev). The lower mean molecular weight in the H/He envelope accounts for much of this difference. The luminosity during the solid accretion phase averages about 10 −4 L ⊙ , corresponding to a mass accretion rate onto the core of 1 to 1.5 ×


8 . It can be seen from Fig. 8 that the age of red giants estimated by our model is usually younger than the age given by Pinsonneault et al. ( 2018 ). This is determined by the sample (Sanders Das 2018 ; Yu et al. 2018 ) we used to train the red giant age model. Pinsonneault et al. ( 2018 ) used the revised scaling relation when inferring stellar mass from seismic data, and they took the influence of the [ α /Fe] into account when estimating the age of red giants. Wu et al. ( 2018 ) also mentioned these two points when comparing the red giant ages of Sanders Das ( 2018 ). It should be mentioned that, in principle, using asteroseismology to estimate the age of red giants should be much more precise. In this work, we used the stellar age given by Sanders Das ( 2018 ) as a standard, established different stellar age models by using different stellar parameters, and compared the model results. By comparison, we infer which parameters are used to establish the stellar age model that is more accurate, that is, to filter out which stellar parameters are more relevant to the stellar age. Figure 8. The age of the red giant estimated using our model is compared with the age given by Pinsonneault et al. ( 2018 ). 4.2 EW of the magnetic activity lines and the age of red giants In Table 3 , by comparing the input parameter combinations ‘b’ and ‘j’, ‘d’ and ‘f’, ‘g’, ‘h’, ‘i’, we know that no matter adding all or part of the EW of the magnetic activity lines to the input parameters, the red giants age estimation cannot be improved. This suggest that in the red giants, the EW of the magnetic activity lines has no direct correlation with the age of the red giants. When studying the relationship between stellar chromospheric activity and age in the open clusters, Zhang et al. ( 2019 ) found that the chromospheric activity indices (log and log ) of low-mass stars with strong surface magnetic field are strongly correlated with the stellar age, while for those stars with relatively large mass and weak surface magnetic field, their stellar chromospheric activity indices are weakly correlated with the age. In other words, the relationships between stellar age and chromosphere activity indices are affected by the intensity of the stellar surface magnetic field. Red giants are formed when low- or intermediate-mass stars have finished burning the hydrogen in their cores. Stello et al. ( 2016 ) mentioned that in the red giants with asteroseismic signals, the outer layer dynamo has shut off at the end of core-hydrogen-burning, and most of these red giants have core-dynamo-generated fields. In Section 2 , by analysing the EW of the magnetic activity lines, we know that there is no strong magnetic activity in the chromosphere and photosphere of all red giants in this study sample. This is consistent with the conclusion of Stello et al. ( 2016 ). Moreover, the results of Chaplin et al. ( 2000 ) showed that the amplitude of the solar p mode decreases with the increase of


here but contains also a 1/ r dependence. Recently, Vigren Eriksson ( 2017 ) developed a 1D model to test the ability of H 2 O molecules to interrupt ion acceleration along weak electric fields in the radial direction. From the model results, they raised concerns about the correctness in assuming u I, flow = u for r r IN as given by Equation ( 1 ) or ( 2 ). As an example, for perihelion conditions, r IN from equation (1 ) is at ∼1000 km, while the estimated effective ion flow speed at 200 km was calculated as several km s −1 , assuming a rather weak electric field strength of the order of ∼0.03–0.1 mV m -1 , which may commence due to the electron pressure gradient force. Their finding would support the observations of H 3 O + /H 2 O + number density ratios observed at times to be lower than predicted by models running with the assumption of collisionally coupled ions (see fig. 7 of Fuselier et al. 2016 ). The observed ratios were observed to be variable and at times also at the level predicted by collisionally coupled models (see in particular, Fig. 6 of Fuselier et al. 2016 ), particularly so when the RPC-Ion Electron Sensor (RPC-IES, see Mandt et al. 2016 ) did not see clear signs of ion acceleration. Vigren Eriksson ( 2017 ) referred also to preliminary values of effective ion speeds (not restricted to ion flow speeds) of several km s −1 as calculated from combined measurements by the Mutual Impedance Probe (MIP, Trotignon et al. 2007 ) and the Langmuir Probe (LAP, Eriksson et al. 2007 ), both being subsystems of the Rosetta Plasma Consortium (RPC, Carr et al. 2007 ). In this work, we will first (Section 2 ) discuss the MIP and LAP measurements and how they can serve to estimate the mean (or rather the effective) ion speed. The selected time interval for the study, 2015 August 2–4, near perihelion, is discussed in Sec-tion 3 . Results are presented and discussed in Section 4 . The derived effective ion speeds are markedly higher than the neutral outflow velocity and also shown to at least periodically be in good agreement with effective ion drift speeds derived from an independent method; a simple flux conservation model relying on MIP and COPS measurements. The work is briefly summarized in Section 5 . 2 METHOD Our method utilized to derive effective ion speeds is based on combining electron number densities measured by the MIP with voltage–current characteristics from the ion side (negative voltages) of Langmuir probe bias voltage sweeps (we used the LAP1 sensor, see Eriksson et al. 2007 for details). The MIP measurements consist of determining the mutual impedance between two electric antennas – a transmitting and a receiving one. The transmitting electrode injects an oscillating current in the plasma, at a given frequency, while the receiving one measures the potential difference induced in the plasma at the same frequency. The frequency is increased step by step, in a frequency range that ideally contains the plasma frequency. The


magnification, which depends on the microlensing model and blending. Our method conserves the variability and noise in the original light curves, as well as information on the quality of individual measurements. Let be the flux of the source and the unmagnified flux from possible blended stars and/or the lens itself. The flux of the baseline object () is magnified during a microlensing event by a factorwhere A ( t ) is the model magnification and is the blending parameter. If there is no blending ( f s = 1), then A ′( t ) = A ( t ); if the blending is very strong (), the observed magnification . To inject a microlensing event into the database light curve, we needed to transform the observed flux and its uncertainty to . The naive transformation is incorrect because it preserves the original photon noise during the magnified part of the event. To illustrate this, let us consider a constant I = 19.5 star with a typical rms light-curve scatter of 0.1 mag (left panel of Figure 9 ). The transformation A ′( t ) = 100 would shift the mean magnitude to I ′ = 14.5, but it would preserve the original scatter, whereas the observed scatter of constant stars of that magnitude is much smaller (Figure 9 ). Figure 9. The rms light-curve scatter as a function of the mean magnitude of a star in the I band in the OGLE-IV field BLG521.12. Left panel: real data. Right panel: constant stars injected on top of real light curves (, where 0 ≤ Δ m ≤ 0.75 is drawn from a uniform distribution). The orange line is the best-fit model from Skowron et al. ( 2016 ). Assuming that every observed fluctuation in brightness of the initial light curve from its mean brightness ( F i − F 0 ) is due to the observational noise, the following transformation ensures that the photon noise is properly scaled:where F 0 and are the mean flux in the baseline and the magnified mean flux, respectively (the corresponding magnitudes are m 0 and ). The ratio can be calculated by assuming the photometric noise model of Skowron et al. ( 2016 ):In this simple model, the observed scatter is the sum of the Poisson noise contributions from the object and background (parameterized by m B ). To verify the proposed model, we injected “constant” stars on top of real light curves from the database (which correspond to the transformation ). The right panel of Figure 9 shows the rms scatter of simulated light curves, which is consistent with that of real data. Our simple model underpredicts the scatter of the brightest stars ( I ≲ 15), likely because the accuracy of their photometry is limited by the accuracy of modeling the PSF and thus the quality of image subtractions, not by the photon noise. This effect does not influence our detection efficiency simulations, because the vast majority of events are fainter than I = 15. Similarly, we used the noise model of Skowron et al. ( 2016 ) to transform the uncertainties:When the predicted uncertainties were equal to or below 0.003 mag, we assumed mag, as in the


: pseudo-ADI image of the inner regions obtained by differentiating the Q Φ image (see Ligi et al. 2018 , for more details). The white cross marks the position of the star, and the cyan circle shows the size of the coronagraphic mask. The other labels refer to the blobs we discuss in this paper that are also visible in these images. Thecolor scale of the differential image is five times less extended to show the faint structures better. In both panels, N is up and E to the left; a segment represents 1 and 0.5 arcsec in the left and right panel, respectively. Fig. 2 Signal-to-noise ratio (S/N) maps for the individual epochs for IFS. The individual S/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median over the wavelength (see Mesa et al. 2015 ). Upper row, left panel : JD = 57180.17, middle panel : JD = 57201.12, and right panel : JD = 57499.34. Lower row, left panel : JD = 57566.15, middle panel : JD = 57873.30, and right panel : JD = 58288.19. In all panels, the central 0.1 arcsec is masked, the solid white line at the bottom represents 1 arcsec, the white cross represents the position of the star. N is up and E to the left. The upper panels of Fig. 3 show images of HD 169142 obtained by combining the six individual images, assuming the distance given by Gaia DR2, a mass of 1.7 M ⊙ (see below), and circular orbits. 3.1 Coincidence images To improve our ability of discerning faint signals, we combined data from different epochs using a coincidence map (see the lower panel of Fig. 3 ). The principle of this coincidence map is to start with S/N maps for individual epochs. We used S/N maps after correcting for the small-number-statistics effect using the formula by Mawet et al. (2014) . The maps were then multiplied by each other. The S/N maps average to zero, with both positive and negative values for individual pixels. Of course, this may result in a false-positive signal for an even number of negative signals in the individual S/N maps. To avoid this problem, we arbitrarily set to negative the result for a given pixel when the signal for that pixel was negative for at least one epoch. Of course, this is not a realistic flux map: the aim is merely to identify consistent signals throughout all individual images. To consider the orbital motion around the star over the three years covered by our observations, we divided the field into 65 rings, each one 2 IFS pixels wide (15 mas, i.e., about 1.8 au at the distance of HD 169142). For each ring, we rotated the S/N maps obtained at different epochs with respect to the first reference image according to Kepler’s third law. When the distance to the star was fixed, the only free parameter remaining in this model is the (dynamical) stellar mass. If there is a companion orbiting the star, the signal is maximized for a value of the mass that, if the assumptions made (circular motion seen face on) are correct, is the dynamical mass of the star. If these hypotheses are not


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


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


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.


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


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


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


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


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 ).


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


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


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


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


; 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


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


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