The traditional way of studying instabilities of non-linear dynamical systems is via linear stability analysis. In many cases, it leads to much simpler system of linearized equations, which can be solved analytically. However, in many other cases even the linearized system does not allow general solutions in terms of analytic functions for arbitrary equilibrium configuration. One way to overcome this problem is to restrict the analysis to special equilibrium configurations which allow us to simplify the linearized system of equations even further. Accordingly, most early studies of jet stability assumed simplified jet structure, including the magnetic field topology (e.g. Hardee 1979, 1982; Cohn 1983; Payne & Cohn 1985; Istomin & Pariev 1996; Begelman 1998; Lyubarskii 1999; Tomimatsu, Matsuoka & Takahashi 2001; Narayan et al. 2009). This allowed to obtain the solution in terms of Bessel and hypergeometric functions. Although very useful in many respects, this approach still cannot address the stability of jets with more complex and more realistic structure. In particular, all these equilibrium jets included surface currents, whereas Gourgouliatos et al. (2012) argued that such current sheets are likely to promote resistive jet instabilities. A jet that is free of current sheets would be free of resistive instabilities in computer simulations. As an alternative, they constructed equilibrium solutions which are current-sheet-free structures. There is an intuitive reason for expecting a magnetized jet without a current sheet to be more stable – such jets carry no volumetric net current. If the surface of a magnetized jet with current sheet is perturbed, two neighbouring perturbations could behave analogously to two parallel current carrying wires that are carrying current in the same direction. Since such wires attract, one might analogously expect the surface of such a jet to become more corrugated, i.e. the perturbations can grow. By avoiding current sheets on the surface, a current-sheet-free jet avoids this mode of destabilization. Another interesting feature of these solutions is that the jets carry zero net current and magnetic flux. In some part of the jet cross-section the poloidal electric current flows outwards and in the rest of the cross-section exactly the same amount of the current flows in the opposite direction. Thus, one does not have to worry of having a return current outside of the jet on large scales. The same applies to the poloidal magnetic field. Unfortunately, the magnetic structure of these solutions is too complex for the linearized equations to allow analytical solutions.


Type II supernovae (SNe) are defined by the prominent hydrogen lines in their spectra. They are believed to originate from the collapse of an iron core of massive stars (≳8 M⊙) that retain their hydrogen envelope. The most common sub-type, comprising ∼70 per cent of all Type II SNe, is characterized by a phase of a roughly constant magnitude in the optical bands, hence their name type II Plateau (II-P). This plateau phase typically starts 1–2 weeks after the explosion and lasts for ∼100 d. Pre-explosion images have revealed that the progenitors of this class are red supergiants, in the mass range of (7–16 M⊙) (Smartt 2015; for individual progenitor detections see e.g. Van Dyk, Li & Filippenko 2003a; Van Dyk, Li & Filippenko 2003b; Van Dyk et al. 2012). Type II Linear (II-L) SNe constitutes another subclass of type II SNe (e.g. Patat et al. 1994; Arcavi et al. 2012; Faran et al. 2014a,b). They are spectroscopically very similar to type II-P events (Faran et al. 2014b, see), but their light curves are declining in all bands. In both types (II-P and II-L), there is typically a sharp drop in the luminosity after ∼100 d and the luminosity starts to follow roughly the exponential decay expected from emission powered by the decay of 56Ni. The distinction between these two classes is not well defined, and studies have used different definitions for II-L SNe. However, several recent works have shown that there exists a continuum of decline rates between slow declining and fast declining SNe (Anderson et al. 2014; Faran et al. 2014a), which suggests that a separation into two different classes may be artificial. Other type II sub-classes will not be discussed here. In this paper, we focus on the light curves of type II-P and type II-L SNe, without making the distinction between the two types, and refer to them in short as Type II SNe. Our goal here is to perform a uniform analysis of the bolometric luminosity and temperature evolution of a large sample of Type II SNe and to compare our findings to theoretical models, focusing on the transition to the plateau, which takes place during the first 2 weeks.


We fit the radial dispersion profiles of each UCD to dynamical models using the Jeans Anisotropic Models (JAM) method with the corresponding code discussed in detail in Cappellari (2008). To briefly summarize, the dynamical models are made in a series of steps making two general assumptions: (1) the velocity ellipsoid is aligned with the cylindrical coordinate system (

R
,
z
,
ϕ

), (2) the anisotropy is constant. Here, the anisotropy is defined as

β

z

=
1
−

(

σ

z

/

σ

R

)

2

 where

σ

z

 is the velocity dispersion parallel to the rotation axis and

σ

r

 is the velocity dispersion in the radial direction in the plane of the galaxy. The first step in the dynamical modeling process is to construct a three-dimensional mass model by deprojecting the two-dimensional mass model MGEs discussed in the previous section. In the self-consistent case, the luminosity and mass profile are the same. However, in our case, we used the mass profile to construct the potential and we used the light profile to calculate the observable properties of the model, both described below. The choice to parameterize the light profile with MGEs is motivated by the ease of deprojecting Gaussians and the accuracy in reproducing the surface brightness profiles (Emsellem et al. 1994; Cappellari 2002). The second step in the dynamical modeling process is to construct a gravitational potential using our mass model. This potential also contains a Gaussian to represent a supermassive BH with the axis ratio, q = 1, and width,

σ
≲

r

min

/

3

, where rmin is the smallest distance from the BH that needs to be accurately modeled. Although a supermassive BH can be modeled by adding a Keplerian potential, it is much simpler to model the BH as this small Gaussian (Emsellem et al. 1994). Next, the MGE formalism is applied to the solution of the axisymmetric anisotropic Jeans equations (see Section 3.1.1 of Cappellari 2008). Finally, the intrinsic quantities are integrated along the LOS and convolved with the PSF from the kinematic data to generate observables that can be compared with the radially binned dispersion profiles. Supermassive BH masses are frequently measured with dynamical models that allow for fully general distribution functions (e.g., Schwarzschild), which is important to include because of the BH mass-anisotropy degeneracy in explaining central dispersion peaks in galaxies. Since plunging radial orbits have an average radius that is far from the center of the galaxy, these orbits can raise the central dispersion without significantly enhancing the central mass density. Similarly, a supermassive BH also raises the dispersion near the center of the galaxy. Other dynamical modeling techniques break this degeneracy by fitting the full orbital distribution without assumptions about the anisotropy. However, given the quality of our kinematic data, a more sophisticated dynamical modeling technique is not feasible; we further discuss the assumptions and limitations of our modeling at the beginning of Section 5.1.


Here we report the X-ray spectral and timing analysis of the joint XMM–Newton and NuSTAR observations of an IRAS 04416+1215, a nearby (z = 0.0889; Boller et al. 1992) hyper-Eddington AGN. The source is part of a XMM–Newton/NuSTAR campaign that aims to constrain the broad-band X-ray properties of eight super-Eddington AGN from the best sample of bona fide super-Eddington sources available, i.e. super-Eddington accreting massive black holes (SEAMBHs; Du et al. 2014, 2015; Wang et al. 2014) that contains exclusively objects with black hole masses estimated from reverberation mapping. In this campaign we are carrying out to study the broad-band X-ray properties of super-Eddington AGN, all the sources have new NuSTAR observations performed simultaneously with XMM–Newton or Swift-X-ray Telescope (XRT). IRAS 04416+1215 has bolometric luminosity $\log (L_{\rm bol}/\rm erg\, s^{-1})=47.55$, according to Castelló-Mor, Netzer & Kaspi (2016), and $\log (L_{\rm bol}/\rm erg\, s^{-1})=45.52$, according to Liu et al. (2021). The former estimate is computed using, for the SED fitting procedure, the Slone & Netzer (2012) code, including the comparison of the observed SED with various combinations of disc SEDs covering the range of mass, accretion rate, spins, and taking into account the correction for intrinsic reddening and host galaxy contribution. In the latter estimate, the SED fitting is done using the more semplicistic templates from Krawczyk et al. (2013). The dimensionless accretion rate (Du et al. 2014) and black hole mass of the source are $\log (\dot{\mathscr {M}})$ = $2.63^{+0.16}_{-0.67}$ and log (MBH/M⊙) = $6.78^{+0.31}_{-0.06}$ with the reverberation mapping technique (Du et al. 2015), respectively, where $\dot{\mathscr {M}}\equiv \dot{M}_{\bullet }c^2/L_{\rm Edd}$, $\dot{M}_{\bullet }$ is mass accretion rates, c is speed of light, and LEdd is the Eddington luminosity. The dimensionless accretion rate is estimated by $\dot{\mathscr {M}}=20.1\, \ell _{44}^{3/2}M_7^{-2}$ from the Shakura–Sunyaev disc model (Du et al. 2015), where ℓ44 is the 5100 Å luminosity in units of $10^{44}\, {\rm erg\, s^{-1}}$ and $M_7=M_{\bullet }/10^7\, \mathrm{M}_{\odot }$. This approximation is valid for $\dot{\mathscr {M}}\lesssim 10^3$. To compute the Eddington ratio we assumed the bolometric luminosity value from Castelló-Mor et al. (2016), which is a better and more trustable estimate of the bolometric luminosity of the source, obtaining λEdd ∼ 472. This value is in perfect agreement with the dimensionless accretion rate from Du et al. (2014). However even assuming the luminosity from Liu et al. (2021), with which the value of the accretion rate would be λEdd ∼ 4.40, the source would remain a super-Eddington accreting AGN. IRAS 04416+1215 turned out to be the most peculiar of our sample, it is classified as NLS1 galaxy, showing narrow Hβ line [full width at half-maximum (FWHM) = $1670 \, \rm km \, \rm s^{-1}$; Moran, Halpern & Helfand 1996] and very broad [O iii] (FWHM = $1150 \, \rm km \, \rm s^{-1}$; Véron-Cetty, Véron & Gonçalves 2001) lines, which is typically found in sources accreting at such high Eddington accretion rates (Greene & Ho 2005; Ho 2009). The source shows a photon index in the Roentgen Satellite (ROSAT) (0.1–2.4 keV) energy band, of Γ = 2.96 ± 0.50 (Boller et al. 1992) and of $\Gamma =2.46^{+0.27}_{-0.26}$ for the rest-frame >2 keV spectrum, according to Liu et al. (2021).


The strong radio flare observed on MJD 58523 is likely to be associated with the ejection of RK1, but the mechanism capable of launching and accelerating these relativistic plasma knots is still not understood. The current picture links the launch of the jets with the transition from the HS to the SS, i.e. during the IMS, where usually ejections take place prior to strong radio flares (e.g. Fender 2006; Miller-Jones et al. 2012) and are marked by changes in the X-ray timing and spectral properties (Belloni et al. 2005). RK1 fits very well in this picture, as the inferred ejection date tej, lin + Sedov = MJD 58518.9 ± 2.4 (see Section 3.3) is only 4 d before the strong radio flare observed with MeerKAT, and when MAXI J1348–630 was in the IMS. The presence of Type-B QPOs (tentatively linked to discrete ejections, e.g. Soleri et al. 2008; Fender et al. 2009; Miller-Jones et al. 2012; Homan et al. 2020) has been detected with NICER close to the first hard-to-soft state transition during the main outburst of MAXI J1348–630 (Belloni et al. 2020; Zhang et al. 2020a). In particular, Type-B QPOs started to be detected on MJD 58522.6 (Zhang et al. 2020a). This is ≳3 d after our inferred ejection date, similar to what was already observed for H1743–322 (Miller-Jones et al. 2012) and MAXI J1535–571 (Russell et al. 2019a). However, a detailed discussion of the NICER timing results in relation to the inferred ejection dates will be discussed in a forthcoming paper. The ∼5 arcsec MeerKAT resolution does not allow us to resolve the two components soon after the ejection, while the ATCA observations were short (providing a non-optimal uv-coverage). As the synchrotron-emitting component is launched and expands, it is predicted to have an optically thick rising phase due to an increasing surface area, followed by an optically thin decay produced by adiabatic expansion losses (van der Laan 1966). Flares produced by discrete ejecta are observed to rise on very different time-scales (from minutes or hours to days, e.g. Brocksopp et al. 2007; Tetarenko et al. 2017; Bright et al. 2020), depending on the size and energy of the ejected component. It is possible that the flare had a fast rise, and thus we missed the optically thick phase with our coverage. Another possibility is that the flare was entirely optically thin, as already observed for some sources (e.g. Fender et al. 1997). The ATCA observation taken on MJD 58524 and the decreasing flux observed on MJD 58523 (see Section 3.2) seem to confirm that we are observing the optically thin decay of the transient jet (see Fig. 10).


The basic idea, summarized in Dekel et al. (2009), is that during VDI, the high surface density of gas and ‘cold’ young stars, Σ, drives the Toomre Q parameter below unity, Q ∼ σΩ/(πGΣ) ≲ 1, where σ is the 1D velocity dispersion and Ω is the angular frequency, a proxy to the epicyclic frequency κ, which is related to the potential well (Toomre 1964). It has been established that under such conditions, the disc will fragment and produce large star-forming clumps. This has been shown using idealized simulations of isolated galaxies (Noguchi 1999; Gammie 2001; Immeli et al. 2004a,b; Bournaud, Elmegreen & Elmegreen 2007; Elmegreen, Bournaud & Elmegreen 2008; Bournaud & Elmegreen 2009; Hopkins et al. 2012b), as well as cosmological simulations (Agertz, Teyssier & Moore 2009; Ceverino et al. 2010; Ceverino et al. 2012; Genel et al. 2012; Mandelker et al. 2014; Oklopcic et al. 2016). The ratio of clump mass to the mass of the cold disc scales as Mc/Md ∝ δ2, where δ = Md/Mtot is the ratio of the cold disc mass to the total mass within the disc radius, which includes the bulge and dark matter halo (e.g. Dekel et al. 2009). This leads to much larger clumps at z ∼ 2 than the low-redshift giant molecular clouds (GMCs). Gravitational interactions in the perturbed disc drive turbulence causing the disc to self-regulate in a marginally stable state with Q ≲ 1 (Dekel et al. 2009; Ceverino et al. 2010; Krumholz & Burkert 2010; Cacciato, Dekel & Genel 2012; Forbes, Krumholz & Burkert 2012; Forbes et al. 2014) that can last for more than a Gyr so long as the accretion is not interrupted. Some recent works have called into question the validity of linear Toomre analysis in the context of these highly non-linear galaxies (Behrendt, Burkert & Schartmann 2015; Tamburello et al. 2015; Inoue et al. 2016) and others have suggested alternate fragmentation mechanisms related to turbulence (e.g. Hopkins 2013). However, since clump formation is largely determined by the balance between self-gravity, turbulent pressure and the centrifugal force, the largest clumps are always roughly at the Toomre scale. Larger clumps would be disrupted due to the shear and/or tidal forces within the disc, or would not collapse in the first place due to the centrifugal force. Therefore, regardless of the full validity of linear Toomre analysis, it is plausible that the Toomre Q parameter can serve as a crude criterion for instability, possibly with a critical value that is larger than unity.


The kinetic energy stored in the CO shell can be estimated as $E_{\rm kin} = 0.5\, M_{\rm shell}\, V^2_{\rm exp}$, where Vexp is the expansion velocity of the shell and Mshell is the total (molecular, atomic, and ionized) shell mass. Adopting an expansion velocity equal to half the velocity interval where the structure is observed, Vexp = 7.0 ± 1.3 km s− 1 , the molecular mass given in Table 1 and the atomic and ionized masses estimated by Cichowolski et al. (2001), 1450 and 3000 M⊙, respectively, we obtain Ekin = (2.5 ± 1.0) × 1049 erg, assuming a 40 per cent error for the masses.. Although Cichowolski et al. (2001) concluded that WR 130 could have alone created the observed structure, it is important to note that they did not take into account the molecular mass present in the shell, which considerably increases the kinetic shell energy. Thus, we can compare now the new value obtained for Ekin with the mechanical energy deposited in the ISM by the wind of the WR star, Ew = (0.7–2.2) × 1050 erg (Cichowolski et al. 2001). We obtain ϵ = Ekin/Ew = 0.007–0.5. The ratio ϵ measures the energy conversion efficiency in the shell, and according to evolutionary models ϵ ≤ 0.2 (Koo & McKee 1992). Thus, not all the possible values of ϵ are compatible with the scenario where the energy injected during the WR phase is enough to create the structure. In this case, the contribution of the energy injected during the O-star phase and/or other massive stars, should be considered. As mentioned in the Introduction, WR 130 is a WNH star, and according to Smith & Conti (2008) its age would be of about 2–3 Myr and its initial mass of at least 60 M⊙. A rough estimation of the energy injected by such a star during its main sequence yields Ew = (2.5–3.5) × 1050 erg (de Jager, Nieuwenhuijzen & van der Hucht 1988), which would be enough to create the observed structure. We have nevertheless looked for the presence of other massive stars in the region. We queried the available catalogues such as the Galactic O-Star Catalog (Maíz Apellániz et al. 2013), the Early-Type Emission-Line Stars Catalogue (Wackerling 1970), the Catalogue of Be stars (Jaschek & Egret 1982), the H-alpha Stars in the Northern Milky Way Catalogue (Kohoutek & Wehmeyer 1997), and the Catalog of Galactic OB Stars (Reed 2003), for early-type and emission stars. No stars were found in any catalogue. The only massive star located nearby is, as mentioned by Cichowolski et al. (2001), an OB star, which has an uncertain spectral type and no distance estimate (Stock, Nassau & Stephenson 1960). It is located in projection not in the centre of the structure but on to the shell (there is a second OB star mentioned by Cichowolski et al. 2001 but its location is actually outside the structure, see fig. 1 of Cichowolski et al. 2001). Although we cannot completely rule out the possibility that the OB star may be playing a role in creating the shell structure, we think that the action of WR 130 is sufficient and most likely dominant in the region.


We introduce our target object, MC27/L1521F (e.g., Mizuno et al. 1994; Onishi et al. 1996, 1998, 1999, 2002; Codella et al. 1997), which is one of the densest cores among nearby low-mass star-forming regions, and it contains a very low-luminosity (Lint  0.07 L⊙) protostar (Bourke et al. 2006; Terebey et al. 2009). Earlier studies based on single-dish observations showed both the gas temperature derived from molecular line observations (e.g., Codella et al. 1997; Tatematsu et al. 2004) and the dust temperature obtained from multiband dust continuum observations (e.g., Kirk et al. 2007; Sadavoy et al. 2018) are as cold as ∼10 K, indicating that most of the gas and dust contents at the line of site are in the cold environment. The high-deuterium fractionation estimated by the N2D+/N2H+ ratio also supported the cold and evolved nature of the core (Crapsi et al. 2004, 2005). On the other hand, submillimeter single-dish observations detected CO (J = 6–5, 7–6) lines, which trace the warm (∼30–70 K) and dense (∼105 cm−3) component at the center of the core (Shinnaga et al. 2009). Although the filling factor of this warm gas relative to the entire core is quite small, these facts demonstrated that multiple temperature mediums are coexisting within the core. Tokuda et al. (2014) (hereafter, Paper I), and Tokuda et al. (2016; hereafter, Paper II) observed this core with ALMA and demonstrated the dynamical nature in this system. We found a few starless high-density cores, one of which has a very high density of ∼107 cm−3 (MMS-2), within a region of several hundred astronomical units around a very low-luminosity protostar (MMS-1) detected by Spitzer (Bourke et al. 2006; Terebey et al. 2009). The molecular line observation showed several cores with arc-like structures, possibly due to the dynamical gas interaction. Similar arc-like structures have also been reproduced by hydrodynamical simulations both with and without a magnetic field (Matsumoto et al. 2015b, 2017), including the different mechanisms. These complex velocity/spatial structures indicate that in MC27/L1521F the turbulence may play an essential role in undergoing fragmentation in the central part of the cloud core, which is different from the classic scenarios of fragmentation in massive disks (Larson 1978; Boss 2002; Machida et al. 2008). More recently, Tokuda et al. (2017; hereafter, Paper III) found that the central very low-luminosity protostar has a dynamical mass of ∼0.2 M⊙ and the associated disk is extremely compact with the disk radius of ∼10 au with an indication of detachment nature from the surrounding dense environment. Paper III mainly focused on gas/dust distributions of the central protostar (MMS-1) seen in the 0.87 mm continuum and 12CO (J = 3–2) observations to determine its evolutionary stage. In this paper, we investigate the properties of the surrounding gas around the central protostar to better understand the dynamical nature of this system by using the high-resolution (∼02) 12CO (J = 3–2) observations as well as 13CO (J =2–1) and C18O (J = 2–1) observations with a moderate angular resolution, ∼1″.


The short-plateau SNe 2006Y, 2006ai, and 2016egz most likely come from partially stripped massive progenitors,36

36
The lack of nebular spectra for SNe 2006Y and 2006ai remains a caveat.
 but a remaining question is their exact formation channel. If it is single-star evolution as assumed in this work, the main theoretical uncertainties are RSG wind mass-loss rates and stellar rotation (e.g., Hirschi et al. 2004; Georgy 2012; Chieffi & Limongi 2013; Meynet et al. 2015; Renzo et al. 2017). We assume no rotation and tweak the wind efficiency by hand, but it is debatable whether such high mass-loss rates are physically plausible. Observationally, there is indeed a wide range of measured RSG wind mass-loss rates (e.g., de Jager et al. 1988; van Loon et al. 2005; Mauron & Josselin 2011; Goldman et al. 2017; Beasor et al. 2020). In addition, recent observational and theoretical studies on RSGs and SNe II indicate that RSG wind mass-loss rates may be independent from metallicity (Goldman et al. 2017; Chun et al. 2018; Gutiérrez et al. 2018). Thus, it could be possible that the short-plateau SNe 2006Y, 2006ai, and 2016egz originate from single-star evolution. However, it is unlikely the case if RSG mass-loss rates are metallicity dependent (as in the main-sequence O/B stars; e.g., Vink et al. 2000, 2001; Mokiem et al. 2007), given the estimated subsolar host metallicities (Table 2). In such a case, interacting binary evolution is more plausible, as Eldridge et al. (2017, 2018) indeed show some interacting binary products also result in short-plateau SNe. It is also important to note that any mass-loss models need to reproduce the observed populations of not only SNe II but also RSGs. For example, Neugent et al. (2020) recently show that the luminosity function of RSGs can be used to constrain their mass-loss rates. Future statistical studies with both RSG and SN II populations at various metallicities are required to distinguish the formation channels of short-plateau SNe.


In order to explain clustering of galaxies and its evolution, a linear bias factor, a parameter representing the difference between galaxy distribution and underlying matter distribution, is often introduced. The linear bias factor b of sgzKs at a scale of 8 h−1 Mpc is expressed as follows:
(15)$$\begin{eqnarray}
b = \frac{\sigma _{8, \rm sgzK}}{\sigma _{8}(z)}.
\end{eqnarray}$$The rms fluctuation of sgzK distribution $\sigma _{8, \rm sgzK}$ (Peebles 1980) and that of underlying matter distribution at a given redshift σ8(z) can be expressed as follows:
(16)$$\begin{eqnarray}
\sigma _{8, \rm sgzK} = \sqrt{J_{2}(\gamma) \biggl (\frac{r_{0}}{8\,h^{-1} ~\rm Mpc}\biggr)^{\gamma } }
\end{eqnarray}$$and
(17)$$\begin{eqnarray}
\sigma _{8}(z) = \sigma _{8}(0)D(z),
\end{eqnarray}$$where
(18)$$\begin{eqnarray}
D(z) = g(z)/[g(0)(1+z)],
\end{eqnarray}$$(19)$$\begin{eqnarray}
g(z) = \frac{5}{2}\Omega _{\mathrm{ m}}\biggl [\Omega _{\mathrm{ m}}^{4/7}-\Omega _{\Lambda }+\biggl (1+\frac{\Omega _{\mathrm{ m}}}{2}\biggr)\biggl (1+\frac{\Omega _{\Lambda }}{70}\biggr)\biggr ]^{-1},
\end{eqnarray}$$(20)$$\begin{eqnarray}
\Omega _{\mathrm{ m}} = \frac{\Omega _{\mathrm{ m},0}(1+z)^3}{E^{2}(z)},~~~ \Omega _{\Lambda } = \frac{\Omega _{\Lambda ,0}}{E^{2}(z)};
\end{eqnarray}$$the present rms mass fluctuation is σ8(0) = 0.83 (Planck Collaboration XVI 2014) and J2(γ) = 72/[2γ(3 − γ)(4 − γ)(6 − γ)]. Fig. 8 shows the bias factor of sgzKs along with those of other populations in the literature.2 The bias of sgzKs with Ks  21.5 is larger than those of SMGs with z  2, while it is similar to those of ULIRGs at comparable redshift, EROs, DOGs with z ∼ 1.2, and SMGs with z = 2–3. Galaxies and their host dark matter haloes can be connected through the correlation length or the bias factor under the assumption that each dark matter halo is occupied by a single galaxy. Therefore, in order to find present-day descendants of high-redshift galaxy populations, we can use a redshift evolution of the bias for dark matter haloes hosting high-redshift galaxies. For example, Hickox et al. (2012) and Wilkinson et al. (2017) investigated the evolution of either correlation length or bias for the haloes of SMGs to find an evolutionary link between high-redshift SMGs and local galaxies. Seo et al. (2019) also used the same method to find descendants of EROs. Thus, we investigated the evolution of bias for dark matter haloes hosting sgzKs with Ks  21.5. By the ellipsoidal collapse model, Mo & White (2002) estimated the bias for dark matter haloes with mass M at a given redshift as follows:
(21)$$\begin{eqnarray}
b (M, z) = 1 + \frac{1}{\delta _{\mathrm{ c}}}~\biggl [\nu ^{\prime 2}+b\nu ^{\prime 2(1-c)}-\frac{\nu ^{\prime 2c}/\sqrt{a}}{\nu ^{\prime 2c}+b(1-c)(1-c/2)}\biggr ], \\
\end{eqnarray}$$where
(22)$$\begin{eqnarray}
\nu ^{\prime } = \sqrt{a}\nu ~~~ \mathrm{with} ~~~ \nu = \frac{\delta _{\mathrm{ c}}}{\sigma (R)D(z)},
\end{eqnarray}$$a = 0.707, b = 0.5, c = 0.6, the critical overdensity is δc = 1.686 (Sheth, Mo & Tormen 2001), and σ(R) is the variance of density field fluctuation.3 To trace the bias of haloes over redshift, we calculated halo mass M at a given redshift using the formalism for mass growth rate of a halo. Fakhouri, Ma & Boylan-Kolchin (2010) suggested two kinds of mass growth rates, mean growth rate $\langle \dot{M}\rangle _{\mathrm{mean}}$ and median growth rate $\langle \dot{M}\rangle _{\mathrm{median}}$, as follows:
(23)$$\begin{eqnarray}
\langle \dot{M}\rangle _{\mathrm{mean}} = 46.1~\mathrm{M_{\odot }\,yr^{-1}}\biggl (\frac{M}{10^{12}~\mathrm{M_{\odot }}}\biggr)^{1.1} \,\,\,\,(1+1.11z)E(z)
\end{eqnarray}$$and
(24)$$\begin{eqnarray}
\langle \dot{M}\rangle _{\mathrm{median}} = 25.3~\mathrm{M_{\odot }\,yr^{-1}}\biggl (\frac{M}{10^{12}~\mathrm{M_{\odot }}}\biggr)^{1.1} \,\,\,\,(1+1.65z)E(z),
\end{eqnarray}$$where the former is larger than the latter. We estimated each mass growth rate separately, and took their average as the mass growth rate of the halo with M at a given z. In Fig. 8, dashed lines represent the evolution of bias for haloes, which are the results considering the mass growth of haloes. On the other hand, solid lines show the evolution of bias for haloes with fixed halo mass.


We repeated the plasma pressure calculation presented by Schwadron et al. (2011) and Fuselier et al. (2012) for the new ENA energy spectrum. The results for the downwind hemisphere and for the Voyager 1 region are summarized in Table 3. The measured intensity

j

ENA

 of neutralized hydrogen at a given energy translates into a pressure of the parent ion population in the heliosheath times the integration length along the line of sight, ΔP × l, in the following way:
3

Δ
P
×
l
=

4
π

3

n

H

m

H

v

j

ENA

(
E
)

σ
(
E
)

Δ
E

c

f

4

c

f

=

(
v
+

u

R

)

2

v

4

(

v

2

+
4

u

R

2

+
2

u

R

v
)
.

In Equation (3), ΔE denotes the width of the respective energy bin; for the typical radial velocity of solar wind in the flanks and the downwind hemisphere of the inner heliosheath, we assumed uR = 140 km s−1 as measured by Voyager 2, whereas uR = 40 km s−1 for the heliosheath in the Voyager 1 direction (Schwadron et al. 2011; Gloeckler & Fisk 2015). For the density of neutral hydrogen in the inner heliosheath a constant nH = 0.1 cm−3 is assumed (Schwadron et al. 2011; Gloeckler & Fisk 2015). The charge-exchange cross section between protons and neutral hydrogen decreases from (4 to 2) × 10−15 cm−2 for 0.015 to 1.821 keV (Lindsay & Stebbings 2005). The integration length l for ENA production in the plasma is approximately the thickness of the inner heliosheath. The part of Equation (3) without the velocity factor cf can be interpreted as stationary pressure. The total pressure or dynamic pressure is the stationary pressure times this factor. Integrating over all energy bins in Table 3, we obtain the total plasma pressure times heliosheath thickness as P × l = 304 pdyn cm−2 au for the downwind hemisphere and 66 pdyn cm−2 au for the Voyager 1 region (1 pdyn cm−2 au = 0.015 N m−1). If we want to put these numbers into the context of other studies, we face two problems. First, the uncertainty of the total pressure is large given the upper limits in the two lowest energy bins. Second, heliosheath plasma more energetic than 2 keV will produce ENAs that cannot be detected with IBEX-Lo. We therefore used the observed median j = 0 cm−2 sr−1 s−1 keV−1 for heliospheric ENAs in the two lowest energy bins of IBEX-Lo and relied on the study by Livadiotis et al. (2013). They compared the expected plasma pressure from a kappa distribution of protons with the plasma pressure derived from IBEX-Hi energy spectra: the energy range between 0.03 and 2 keV, roughly corresponding to the IBEX-Lo range, covered more than half of the total plasma pressure predicted from a kappa distribution. The authors found a total plasma pressure of P = 2.1 pdyn cm−2 for all sky directions except for the ENA Ribbon. Gloeckler & Fisk (2015) presented a multi-component plasma model for the heliosheath to explain Voyager and IBEX observations. At low energies they assumed the ENA energy spectra provided by Fuselier et al. (2012). They derived a total pressure of 2.5 pdyn cm−2 in all three plasma regions in the nose of the heliotail (Gloeckler & Fisk 2015). Pressure contributions from the slowed solar wind, magnetic pressure, and the pressure exerted from pickup ions and anomalous cosmic rays all had to be taken into account to obtain this total pressure.


Figure 9 demonstrates that the environment plays an important role in shaping the buildup of the passive population. The mild redshift dependence in the field ratios suggests that newly quenched galaxies are being added in both the bright end and the faint end of the population at similar fractions. Indeed, the number density of the bright and faint galaxies in the field have both grown by a factor of ∼4 from the highest redshift bin (z ∼ 1.15) to the lowest redshift bin (z ∼ 0.15). Note that the split magnitude between the bright and faint galaxies (

) roughly corresponds to

.24

24
This M/L conversion is derived using a range of SSP models, assuming zf = 0.5−4.0 and Z = Z⊙.
 The mild redshift dependence in the field ratios is therefore consistent with previous studies on the stellar mass function of passive galaxies in the field, which have seen significant growth in both the high-mass end and low-mass end from z ∼ 1 to z ∼ 0 (e.g., Muzzin et al. 2013b; Tomczak et al. 2014). The growth is commonly attributed to various mass-quenching processes internal to the galaxies, such as energetic feedback from supernovae and stellar winds for low-mass galaxies (e.g., Dekel & Silk 1986; Hopkins et al. 2014) and ejective feedback from AGNs for more massive galaxies (e.g., Bower et al. 2006; Terrazas et al. 2016). Given that the field ratios do not evolve strongly with redshift, it is evident that the strong redshift dependence of the cluster ratios is a result of the high-density environment and that preferentially low (or moderate) mass galaxies are quenched by the environment. Additionally, the difference between the cluster and field ratios suggests that the quenching effects induced by the environment are clearer at low redshifts. This is consistent with the mostly mass-independent environmental quenching scenario established at low redshift (e.g., Peng et al. 2010). Because the LF of star-forming galaxies is steeper at the faint end, a population of environmentally quenched galaxies would have a relatively high faint-to-luminous ratio. Recent studies of infalling galaxies in local groups and clusters have demonstrated that the ram pressure stripping of the cold gas in the galaxies when it passes through the ICM is the dominant mechanism (e.g., Boselli et al. 2016; Fillingham et al. 2016; Fossati et al. 2018), although strangulation may also play a significant role (for galaxies with

; see, e.g., Fillingham et al. 2015; Peng et al. 2015).


Specific examples of IR/maser flare correlations include the 22-GHz H2O and 6.7-GHz methanol masers associated with the ucH ii region G025.65+1.05 (Stecklum et al. 2017; Sobolev et al. 2019). The H2O maser variability in this object may be correlated with IR K-band variability in a source approximately 1500 au from the maser position (Sobolev et al. 2019). To be precise, the H2O emission appears to exhibit an anticorrelation, with bright 22-GHz episodes corresponding to times of low K band (most observations centred on 2.20 μm wavelength) flux density. This is to be expected for typical 22-GHz masers in star-forming regions, where computational models, for example Gray et al. (2016), show that strong IR emission from dust tends to destroy the 22-GHz inversion, probably by inhibiting the loss of population from the lower maser level (Strelnitskii 1981; Sobolev & Gray 2012). The 6.7-GHz methanol masers in this source have been described as showing a moderate flux rise at the time of a 22-GHz H2O maser flare (Sugiyama et al. 2017). A radiative mechanism is not the only explanation for H2O flaring in G025.65+1.05. VLBI observations identify the flaring object with a single compact spot of maser emission (Burns et al. 2020). An image from spectral channels in the blue-shifted wing of this spot shows a double structure, leading to an interpretation as a possible line-of-sight overlap of two masing objects. Preliminary modelling results for spatial overlap (Gray 2018) indicate that this mechanism can produce flares of significantly higher variability index than is typical of radiative pumping mechanisms, and the overlap interpretation therefore becomes more likely if the source is at the further of the two possible distances (2.7 and 12.5 kpc) mentioned in Burns et al. (2020). Another possible example of this overlap mechanism is a very high brightness H2O flare from Orion KL (Shimoikura et al. 2005). The 22-GHz maser emission of G025.65+1.05 has been monitored by a combination of the Simeiz, Torun, and Hartebeesthoek telescopes since 2000 (Vol’vach et al. 2019). Observations were approximately monthly for early data, but on an almost daily basis since a giant flare in 2017–2018. During the period 2000–2019, it has flared strongly (peak flux density >2000 Jy) three times: the peak flux density of each flare shows an increasing trend, while the flare duration and inter-flare interval have decreased. The shortest, brightest, and most recent flare (2017–2018) showed a double-peaked light curve, with rising sections well approximated by exponentials with characteristic times of approximately 10 d. The exponential shape, backed by evidence from polarization and spectral width, has led Vol’vach et al. (2019) to conclude that the flaring maser is unsaturated. The continuum flux at 870 μm has generally been rising towards G025.65+1.05 since 2008, but the data are very sparse compared to the maser observations, including an interval of 8 yr with no data (Vol’vach et al. 2019).


The most widely adopted parametrization of the observed Universe is based on the so-called Λ cold dark matter (ΛCDM) model (Peebles 1984), relying on the existence of cold dark matter and dark energy (Λ) associated with a cosmological constant (Carroll 2001) in a spatially flat geometry. Predictions from this model have been found to agree with most of the observational probes such as the cosmic microwave background (CMB; e.g. Planck Collaboration 2020), the baryon acoustic oscillations (BAO; e.g. Alam et al. 2021), and the present accelerated expansion of the Hubble flow, based on the distance modulus–redshift relation (the so-called Hubble–Lemaître, or simply Hubble diagram) of type Ia supernovae (SNe Ia; e.g. Riess et al. 1998; Perlmutter et al. 1999), where a dominant dynamical contribution, dubbed dark energy (DE) and related to the cosmological constant, should drive such an acceleration. However, the fundamental physical origin and the properties of DE are still unknown, as the interpretation of Λ is plagued by a severe fine-tuning issue to obtain the right amount of DE observed today. Moreover, the data sets listed above do not fully fit the evolution of DE ranging from early to late epochs (Benetti et al. 2019; Yang et al. 2021) and do not fully rule out a spatially non-flat Universe (Park & Ratra 2019; Di Valentino, Melchiorri & Silk 2020, 2021; Handley 2021; Yang et al. 2021). The latter possibility has raised a remarkable debate about the importance of properly combining CMB data to infer significant statistical interpretations from the analysis (Efstathiou & Gratton 2020; Planck Collaboration 2020) and, by extension, the importance of combining data sets that do not reveal manifest tension (Gonzalez et al. 2021; Vagnozzi, Loeb & Moresco 2021). Deviations from the spatially flat ΛCDM model would imply important theoretical and observational consequences and a change in our current understanding of cosmic evolution (e.g. Capozziello, Benetti & Spallicci 2020). Statistically significant deviations in this directions have already been found in cosmological analyses with high-redshift probes such as Gamma-Ray Bursts (see Dainotti, Cardone & Capozziello 2008; Dainotti et al. 2011b, 2013a,b, 2015, 2017, 2020a; Dainotti, Ostrowaki & Willingale 2011a; Dainotti et al. 2020b for the standardization of these sources as cosmological candles) and quasars (QSOs) combined with SNe Ia (Risaliti & Lusso 2019; Lusso et al. 2019, 2020; Bargiacchi et al. 2021). Such a joint analysis (SNe + QSO) makes use of the observed non-linear relation between the ultraviolet and the X-ray luminosity in QSOs (e.g. Steffen et al. 2006; Just et al. 2007; Lusso et al. 2010; Lusso & Risaliti 2016; Bisogni et al. 2021; Dainotti et al. 2022) to provide an independent measurement of their distance (see e.g. Risaliti & Lusso 2015, 2019; Lusso et al. 2020, for details). The methodology is complementary to the traditional resort to type Ia SNe to estimate the cosmological parameters, yet it extends the Hubble–Lemaître diagram to a redshift range currently inaccessible to SNe ($\mathit{ z}$ = 2.4–7.5). Within a model where an evolution of the DE equation of state (EoS) in form w($\mathit{ z}$) = w0 + wa × $\mathit{ z}$/(1 + $\mathit{ z}$) is assumed, the data suggest that the DE parameter is increasing with time (Risaliti & Lusso 2019; Lusso et al. 2020). Therefore, it is compelling to further study extensions of the ΛCDM model that could produce such behaviour of DE.


With the introduction of the first integral-field spectrographs (IFS, see e.g. Bacon et al. 1995), it became feasible to perform such observations in a spatially resolved manner and for larger samples of galaxies. The SAURON survey (Bacon et al. 2001; de Zeeuw et al. 2002) was one of the first projects to make extensive use of this technology. Based on their representative sample of 72 nearby early-type galaxies, the project investigated stellar and gaseous kinematics (Emsellem et al. 2004; Sarzi et al. 2006; Falcón-Barroso et al. 2006; Ganda et al. 2006) and stellar population properties (Peletier et al. 2007; Kuntschner et al. 2010), and distinguished between fast and slow rotating early-type galaxies (Emsellem et al. 2007). Subsequently, the ATLAS3D project (Cappellari et al. 2011) continued this endeavour by further investigating kinematic properties, for instance the global specific angular momentum (Emsellem et al. 2011), based on a volume complete sample of 260 early-type galaxies. The CALIFA survey (Sánchez et al. 2012) advanced these previous studies towards a morphologically unbiased sample of 667 galaxies. In addition, it provides a unique combination of a large spatial coverage of a few effective radii and high spatial sampling (see also García-Benito et al. 2015). Other surveys observe even larger samples: while the SAMI survey (Croom et al. 2012; Bryant et al. 2015) includes ∼3000 galaxies across different environments, MaNGA (Bundy et al. 2015) will contain approximately 10 000 nearby galaxies. Most recently, these projects are being complemented by IFS studies of local galaxies in unprecedented spatial resolution (e.g. MUSE, Bacon et al. 2010). For instance, the TIMER project (Gadotti et al. 2019) analyses the central structures of 24 barred local disc galaxies in order to study the formation histories of these structures and infer constraints on the related secular evolution processes. Fornax3D (Sarzi et al. 2018; Iodice et al. 2019) investigates mostly early-type galaxies in the Fornax cluster environment, spatially covering galaxies up to four effective radii. The PHANGS survey (Leroy et al., in prep.)1 aims to connect the physics of gas and star formation with the large-scale galactic structure by complementing IFS data from MUSE with interferometric data from ALMA.


A large

 survey in nearby spiral galaxies and (U)LIRGs performed by Gao & Solomon (2004a, 2004b) revealed a tight linear correlation between the infrared (IR) and the HCN luminosities for normal star-forming galaxies and starbursts. This linearity seemingly extends down to the scale of Galactic massive cores in the Milky Way and holds over a total range of luminosity of about eight orders of magnitude (Wu et al. 2005). These results imply that the dense molecular gas (i.e.,

) as traced by the

 line, rather than the total molecular gas, is the direct fuel for star formation. High-resolution simulation of dense clouds found that the HCN luminosity is related to mass of dense gas of ≳104 cm−3 (Onus et al. 2018). In addition, Spitzer studies of Galactic molecular clouds also show evidence that star formation is restricted to the dense cores of GMCs (e.g., Evans 2008; Lada et al. 2010). The critical density ncrit40

40
The critical density of rotational level j is defined as

, where

 is the Einstein coefficient for spontaneous emission,

 in units of s−1, and

 is the collision rate coefficient that depends on the gas temperature, in units of cm3 s−1. All critical densities in this work are calculated on the assumption of Tkin = 100 K and optically thin conditions. The critical densities will decrease if the lines are optically thick.
 of rotational transitions is proportional to μ2ν3 (for optically thin lines at frequency ν; μ is the dipole moment of the molecule); therefore, molecules with high dipole moment are expected to trace high-density molecular gas (e.g., μHCN ∼ 2.98 D,

 D, and μCS ∼ 1.96 D vs. μCO ∼ 0.11 D; see Schöier et al. 2005). Subsequently, a number of studies have explored the link between molecular lines of dense gas (e.g., HCN, HCO+, and CS) and IR luminosities in different populations of galaxies (e.g., Gao et al. 2007; Papadopoulos 2007; Baan et al. 2008; Liu & Gao 2010; Wu et al. 2010; Wang et al. 2011; García-Burillo et al. 2012; Zhang et al. 2014; Chen et al. 2015; Usero et al. 2015; Liu et al. 2016).


From the excitation maps as shown in Fig. 9, we determine the maximum distance of AGN-ionized regions (RENLR, max) with respect to the AGN location that we can cover with our depth and area. To suppress the impact of noise on misclassification close to the borders of the demarcation lines, we require that at least six surrounding unbinned spaxels share the same AGN-ionization classification or the classification is present in the spatially binned data with its intrinsically higher S/N. We need to exclude the LINER and the intermediate BPT regions, because those are not necessarily associated with AGN photoionization and can originate from post-AGN stars (e.g., Binette et al. 1994; Singh et al. 2013), shocked gas from stellar winds (e.g., Ho et al. 2016; López-Cobá et al. 2019), or the diffuse ionized gas inbetween star forming regions (e.g., Lacerda et al. 2018; Levy et al. 2019). We define the maximum ENLR size, RENLR, max, as the maximum projected distance of a robust AGN region to the AGN location detectable within the IFU FoV at the observational depth. In general, the maximum ENLR extent can be biased due to surface brightness dimming and observational depth. This is not a concern for the large FoV of MUSE and the narrow redshift distribution of the CARS sample. Nevertheless, we also determine two additional ENLR radii, RENLR, 15 and RENLR, 16, corresponding to the ENLR size out to a surface brightness limit of Σ15 = 10−15 erg s−1cm−2 arcsec−2(1 + z)−4 (e.g Liu et al. 2013b, 2014; Hainline et al. 2013, 2014), or Σ16 = 10−16 erg s−1cm−2 arcsec−2(1 + z)−4 (Chen et al. 2019b), respectively. All these measurements are listed in Table 5. The ENLR size can become unresolved for the CARS targets, in particular, at high intrinsic ENLR surface brightnesses. For such nondetections we set the upper limits for the ENLR size to be the angular size of the binned spaxels. The error on the sizes are generally driven by the spatial resolution of a given observations, so that we assume the error on all sizes to be the FWHM of seeing for each observation as reported in Table 2.


The general form of the velocity-dependent cross-section is given by $\bar{\sigma }=\sigma _0 (v/c)^n$, where the index n depends on different physical DM processes and c is the velocity of light in space (in natural unit $\bar{\sigma }=\sigma _0 v^n$). In the case of DM with magnetic and/or electric dipole moment, n = +2, −2 are considered. n = 2, 1, 0, −1 are applicable for scattering in presence of Yukawa potential (Buckley & Fox 2010), n = −4 is attributed for millicharged DM (Holdom 1986; Chun, Park & Scopel 2011). In Ref. Dvorkin, Lin & Schutz (2021), the nature of the DM–baryon cross-section is discussed for a wide mass range of DM. Similar investigations are also carried out in Ref. Nadler et al. (2019), Bhoonah et al. (2018), Kovetz et al. (2018), and Mack, Beacom & Bertone (2007). In this work, the DM–baryon interaction cross-section ($\bar{\sigma }$) is parametrized as $\bar{\sigma }=\sigma _0 v^{-4}$ (Muñoz, Kovetz & Ali-Haïmoud 2015; Barkana 2018; Mukhopadhyay et al. 2021). The term σ0 is the DM scalar scattering cross-section with baryons (of the type $\alpha _q \bar{\chi }\chi \bar{q}q$ for dark DM particle χ with coupling αq). It may be mentioned in some earlier works like Bhoonah et al. (2018) and Kovetz et al. (2018), where millicharged DM is considered. But here, we assume a particle DM candidate and adopt value of $\sigma _0 \sim 10^{-41}\,\rm {cm^2}$ consistent with the scalar cross-section bound obtained from ongoing direct DM search experiments [extrapolating the allowed region for 0.1 GeV ≤mχ ≤ 3 GeV from recent experiments like Aprile et al. (2018), Akerib et al. (2017), and Cui et al. (2017)] in the mass range discussed in this work. Several recent investigations on EDGES 21-cm signal also suggest the similar velocity dependence (n = −4) of the cross-section like Muñoz et al. (2015), Mahdawi & Farrar (2018), and Barkana (2018). Moreover, n = −4 is chosen in many DM-related cases, namely, hadronically interacting DM, millicharge DM, the baryon acoustic oscillations (BAO) signal, etc.


The required stellar population mass of ≳104 M⊙ represents a hard limit for the detection of direct star light from z ≳ 7 Pop III star clusters at magnification μ ∼ 1000 in the foreseeable future. As discussed in Section 1, large numbers of Pop III stars can form in atomic cooling haloes that manage to remain chemically pristine, a scenario that likely requires a background Lyman–Werner (LW) radiation field that suppresses molecular hydrogen (H2) formation and, consequently, Pop III star formation in minihaloes. This scenario has been explored predominantly in the context of supermassive star and direct collapse black hole (Bromm & Loeb 2003; Haiman 2013) formation in regions of the Universe with an LW radiation field high enough above the cosmic mean to entirely suppress H2 formation. It thought the onset of efficient H2 cooling is required for star cluster, rather than supermassive star, formation (Regan & Haehnelt 2009), requiring a UV background intensity below J21 ∼ 103 for H2 to form in sufficient abundance but above J21 ∼ 10 to suppress star formation in minihaloes (here, J21 denotes the radiation intensity at the Lyman-limit in units of 10−21 erg s−1 cm−1 Hz−1 sr−1). Explicitly focusing on Pop III star cluster formation, Safranek-Shrader et al. (2012) argued that total stellar masses of 104 M⊙ may be achievable given a low enough LW background intensity, though unaccounted for internal LW feedback may be significant in reducing the star formation efficiency. The inclusion of ionizing radiation further increases the propensity for Pop III galaxies with total stellar masses ≳104 M⊙ (Johnson et al. 2014). Related studies (Regan & Haehnelt 2009; Shang, Bryan & Haiman 2010; Latif et al. 2014; Regan, Johansson & Wise 2014; Visbal, Haiman & Bryan 2014) have routinely found central gravitationally unstable clumps with masses in excess of 104−5 M⊙ and central accretion rates of 0.01–1 M⊙ yr−1 to form in chemically pristine, high-redshift atomic cooling haloes. The results of these studies additionally suggest that fragmentation, if it does occur, happens on small scales (≲1–10 pc) favouring the formation of a compact Pop III stellar cluster. While the limit of ≳104 M⊙ adopted in this work seems marginally achievable for the total stellar mass of a Pop III galaxy, additional cosmological simulations are needed that focus on the possibility and characteristics of a metal-free stellar cluster forming in an atomic cooling halo.


Among the suggestedmechanisms responsible for the strong damping of the coronal loop oscillations (e.g., Ruderman & Roberts 2002; Ofman 2005, 2009; Morton & Erdélyi 2009) resonant absorption of the MHD waves, which was established first by Ionson (1978), is a strong candidate. Several works developed this theory (e.g., Davila 1987; Sakurai et al. 1991a, 1991b; Goossens et al. 1995; Goossens & Ruderman 1995; Erdélyi 1997; Cally & Andries 2010). The necessary condition for the resonant absorption is a continuum of Alfvén or slow frequency across the loop (Ionson 1978; Hollweg 1984, 1987; Davila 1987; Sakurai et al. 1991a). Resonant absorption occurs when the frequency of the global MHD mode matches at least with one of the frequencies of the background Alfvén or slow continuum at a location called he resonance point. As a result, the energy of the global MHD mode transfers to the local Alfvén modes in a layer around the resonance point, named the resonance layer (Lee & Roberts 1986; see also Goossens et al. 2013; Soler & Terradas 2015). In the absence of dissipation mechanisms, the amplitude of the oscillations diverges at the resonance point. Dissipation is important in the resonance layer, where the oscillations make large gradients. The background Alfvén or slow continuum can be due to the variation of the plasma density (e.g., Davila 1987; Ofman et al. 1994; Ruderman & Roberts 2002; Terradas et al. 2006; Soler & Terradas 2015), twisted magnetic field (Ebrahimi & Karami 2016), or both of them together (Karami & Bahari 2010; Giagkiozis et al. 2016). There are a variety of theoretical works related to the damping of the coronal loop oscillations based on the theory of resonant absorption of MHD waves (e.g., Goossens et al. 2002, 2009; Ruderman & Roberts 2002; Van Doorsselaere et al. 2004; Andries et al. 2005; Terradas et al. 2006; Karami et al. 2009; Karami & Bahari 2010; Soler et al. 2013; Soler & Terradas 2015; Ebrahimi & Karami 2016; Giagkiozis et al. 2016; Jung Yu & Van Doorsselaere 2016). For a good review about the theory of resonant absorption, see also Goossens et al. (2011).


In Fig. 20, we show two volumes containing filaments which are almost aligned to the sky plane, as an example to illustrate the implications of this effect on the rotation measurement of such objects. The top panels show the projected maps of density, RM and ε for the four Chronos simulations. In presence of a large degree of alignment between magnetic fields and filaments, we therefore expect the magnetic field to mostly lie in the sky plane as well, with a very small line-of-sight component, thereby reducing the observable |RM| towards the observer. Current instruments (e.g. VLA and LOFAR) are able to detect values of |RM| ≳ 5 rad/m2 (e.g. Bonafede et al. 2013; O’Sullivan et al. 2019; Locatelli, Vazza & Domínguez-Fernández 2018). The Figure suggests that, on one hand, clusters easily meet this requirement, while filaments would only be marginally detected, even for the runs in which the magnetic field is stronger (baseline and Z), due to the large degree of $\boldsymbol {B}$-filament alignment, which implies low values of ε, as can be seen in the third column. Therefore, the small line-of-sight component yields only little |RM|, typically below the detection threshold of present instruments, especially for runs in which the magnetization is weak already (DYN5 and CSFBH2). On the bottom panels we give, for each of the two selected areas, the median value of ε as a function of the rotation measure (in absolute value). The highest values of |RM|, mostly associated to clusters, correspond to higher values of ε, although never approaching ε ≈ 1; at the lower side of |RM|, corresponding to the areas populated by filaments, lower values of ε are found, as expected from our previous considerations. The simulations are in overall agreement, except for CSFBH2, where AGN and star formation feedback introduces additional effects: although the impact of a quasi-parallel $\boldsymbol {B}$-filament configuration is noticeable for a larger sample of objects (e.g. see Fig. 18), the bursty and random occurrence of star forming/AGN events may strongly affect the local magnetic field topology and cause the statistic over a small volume to deviate from the expected trend.


Using the constrained hyperbolic divergence cleaning scheme with variable cleaning speed from Tricco et al. (2016), we can keep the divergence error low in all cases. The mean normalized divergence error, ⟨ϵdivB⟩=⟨h|∇ ⋅ B|/|B|⟩, is typically of order 10−5 − 10−3. In Fig. 17, we show the normalized divergence error maps for several test problems. Again we see that the divergence cleaning works extremely well here, the maximum error is generally around 10−2. Comparing to Hopkins (2016, their Fig. 4), we find that the errors are smaller than their MFM simulations with the Dedner et al. (2002) cleaning in general, with an exception for the outskirt of the advection loop (where the magnetic field is essentially zero and thus not important for the result). The improvement is probably due to the more advanced constrained cleaning method (Tricco et al. 2016). The normalized divergence error for the μ = 10 cloud-collapse case at the jet launching time is shown in Fig. 18. Here, the divergence cleaning still performs very well in the disc, along the jets and for the majority of the regions where the outflow interacts with the ambient gas, especially when the divergence error is compared to the total gas pressure (right panel). The result is similar to the Dedner cleaning in Hopkins (2016), although our error is somewhat larger at the tip of the jets where the gas is shocked. However, we note that the comparison is not direct in this case as the jets may develop differently. Overall, the result from cleaning is still worse than the constrained transport or constrained gradient schemes (Hopkins 2016). For SPMHD, as shown in Tricco & Price (2012), divergence errors can be reduced to machine precision (or more practically to a certain tolerance value) using cleaning, with the help of a sub-cycling routine. However, local adjustments are required to determine the number of iterations for each particle to efficiently subcycle the cleaning in the simulation. This is because certain regions are more affected than others and because divergence is spread to nearby neighbours. Conceivably, if vector potentials (Stasyszyn & Elstner 2015) could work for a wider range of problems this could be an interesting avenue as well. However, the exploration of these methods in detail is beyond the scope of this work.


To check for any spectral variation in the optical/IR bands, we looked for variation in the V − J band colour against the V-band brightness. This colour variation was analysed for the epochs A, B, D, and E. During epochs A and B, the source showed a ‘redder when brighter’ (RWB) behaviour. During epoch E, a bluer when brighter behaviour was observed. During epoch D, we observed a complex behaviour. Upto a V-band brightness of around 15 mag, the source showed a ‘bluer when brighter’ behaviour, but for optical brightness fainter than 15.0 mag, a ‘redder when brighter’ behaviour was observed. The colour–magnitude diagrams for all the four epochs are shown in Fig. 10. The spectral variations shown by the source are thus complex. From studies on the optical–IR colour–magnitude diagram, it is known that FSRQs in general show an RWB trend, which is attributed to them having a luminous accretion disc (Gu et al. 2006; Bonning et al. 2012). The observed optical emission is a combination of thermal blue emission from the accretion disc and non-thermal red emission from the jet. As the source gets brighter, the non-thermal emission has a more dominant contribution to the total flux, giving rise to the RWB behaviour (Bonning et al. 2012). During epochs A and B, there is a trend of the object to become RWB, irrespective of its optical brightness. The optical flares dominated by synchrotron emission processes during A and B have corresponding γ-ray flares that are produced by EC processes. However, during epochs D and E, the colour variations were found to depend on the optical brightness. During the epochs when this complex spectral behaviour was noticed, the source showed an optical/IR flare with no or a weak corresponding flare in the γ-ray band. The source showed a much larger amplitude of variability in the optical/IR bands, while in the γ-ray band it was either faint or below the detection limit of Fermi. This definitely points to some complex physical changes and could be due to a combination of changes in the bulk Lorentz factor, electron energy density, and magnetic field as seen from our SED modelling of the multiband data.


Dense gas-fraction and the star-forming fraction. As in Paper I we define the dense gas-fraction, Mfrac, as the fraction of gas having density, n ≳ 103 cm−3 and being colder than ∼50 K. Although somewhat ad hoc, this density and temperature threshold is good enough to estimate the fraction of gas likely to end up in putative star-forming clumps (e.g. Ragan et al. 2016 and other references therein). Furthermore, we also use a higher density threshold, n ≳ 104 cm−3, to trace the fraction of strongly self-gravitating gas in a pristine cloud. In typical star-forming clouds such dense pockets of gas are traced due to emission from molecules such as HCN, HNC and HCO+ (e.g. Gao & Solomon 2004, Usero et al. 2015 and Biegel et al. 2016). The dense gas fraction, Mfrac$\equiv \frac{M_{\text{thresh}}(n \gtrsim 10^{3} \mathrm{cm}^{-3}, 10^{4} \mathrm{cm}^{-3}; 50 \mathrm{K})}{M_{\text{gas}}}$, for each choice of bimpact is shown on the upper panel of Fig. 6. Irrespective of the choice of the impact parameter and the density threshold, n, the fraction, Mfrac, increases steadily with time, but at best cycled a little under ∼ 10 per cent of the gas into the dense-phase at the time calculations were terminated. Indeed, this fraction is significantly smaller in the realization where the inflows merely grazed past each other. While the fraction of gas cycled into the dense phase could possibly increase as the post-collision slab continues to accrete gas from the colliding flows, it is unlikely to exhibit a dramatic increase. This is because the NTSI destabilises the shocked slab and causes fluid layers to mix, thereby rupturing the dense pockets of gas. In any case, it is evident that grazing inflows are most inefficient in cycling gas into the dense phase. The fraction of gas cycled to densities higher than ∼104 cm−3 is typically an order of magnitude smaller. Evidently, star formation in a pristine cloud must be quite inefficient. We also note, irrespective of the choice of the density threshold and the impact parameter, the characteristics of the dense gas fraction look mutually similar. This is because whatever the choice of the pre-collision impact parameter, the fraction of gas cycled into the dense phase is limited by shearing interaction between layers of gas in the shocked slab. The source of shear, however, is different in the extreme scenarios of near head-on collision and one in which the gas flows merely graze past each other. In the former, shear is induced by the growth of the NTSI while in the latter, it is the obliqueness of the collision that induces a strong shearing interaction between layers of gas.


The radius estimate of the X-ray emitting area is small compared to the size of a neutron star, which typically has a radius of 10–14 km. The emission is therefore likely to be coming from the polar caps of PSR J1909–3744, as is expected from such old objects where the surface of the neutron star has cooled so that it is no longer detectable in the X-ray band. The emitting radius is somewhat smaller than the classical radius of a polar cap,

R
pc

=

2
π
R

cP

R

$ R_{\mathrm{pc}}= \sqrt{\frac{2 \pi R}{c P}} R $

 (e.g. Dermer & Sturner 1994), where R is the neutron star radius, c the speed of light in a vacuum and P the rotation period of the neutron star. The latter is ∼2.9 km if we suppose a 10.6 km radius for the PSR J1909–3744 neutron star, the average of the best radius values for a neutron star of M = 1.5 M⊙ as determined by Özel et al. (2016b) who undertook a comprehensive study of 12 neutron stars. Considering the range of typical radii of neutron stars (10–14 km), the classical polar cap radius is between 2.7 and 4.4 km. Bogdanov et al. (2006) state that the radius determined through fitting the pulsar spectrum can be smaller than expected when the spectrum has been fitted with a single-temperature model, but that a higher signal-to-noise spectrum would require a two-temperature model, as for PSR J0030+0451, PSR J2124–3358 (e.g. Bogdanov et al. 2008), and PSR J0437–4715 (Bogdanov 2013). Fitting the spectrum with two black bodies does not improve the fit, again probably due to the low signal-to-noise ratio. Further, it is known that the radii of neutron stars are underestimated when using a black-body model as opposed to a realistic neutron star atmosphere model (e.g. Heinke et al. 2006). However, assuming the classical radius of the polar cap for this neutron star (2.9 km) in the nsatmos model provides a poor fit to the data with a C-statistic = 171.6, 33 degrees of freedom, suggesting that such a radius is too large. In fact, the nsatmos model produces a polar cap with size ∼100–400 m, slightly larger than the ∼50–300 m radius obtained with the black-body model (see Fig. 4).


Figure 3 (left) shows the moment zero map of the SiO (5–4, Eu = 31.26 K) emission integrated over the central ~3 km s−1 of the line covering the VLSR, between 21.4 and 24.0 km s−1. In contrast to Cesaroni et al. (2017), here we integrate the SiO emission over a narrower velocity range to highlight the elongated structure at near VLSR velocities. The elongated structure of the emission is clearly visible and perpendicular to the large-scale outflow (Fig. 2). Fitting the integrated SiO emission with a 2D Gaussian in the image plane indicates a PA of ~ 47 ± 7° and a major axis of ~0.28″ (~600 au). The deconvolved PA is 30 ± 10° consistent with the PA of the dust continuum emission. The SiO emission is broad in velocity, ranging from −3.0 to 45.7 km s−1 considering emission above 3 σ, this is broader than presented in Cesaroni et al. (2017), 4–39.1 km s−1, owing to our improved images after self calibration. In most massive YSOs, SiO is very spatially extended, typically tracing emission at shock fronts (e.g. Schilke et al. 1997) created by active jets and outflows driven by the energetic central sources (e.g. Cabrit et al. 2007; Sánchez-Monge et al. 2013b; Duarte-Cabral et al. 2014; Klaassen et al. 2015; Cunningham et al. 2016; Cesaroni et al. 2017; Beltrán et al. 2018; Moscadelli et al. 2018). In at least one massive YSO, Orion source I, the SiO emission is rather compact, associated with SiO masers and is shown to trace a rotating disc and disc wind structure (Goddi et al. 2009; Ginsburg et al. 2018). The SiO emission from G17.64 is also relatively compact, unlike a jet. There is also compact emission from other silicon and sulphur bearing species in our observations, emission from SiS (v = 0 12–11, Eu = 69.95 K) and 33SO (65 –54, Eu = 34.67 K) are centrally peaked at the location of the continuum and SiO emission. The strong detection of SiO points to a reservoir of silicon in the gas phase, which, in addition to the detected sulphur-bearing species, suggest an association with ionised, hot (>100 K), turbulent and shocked gas (e.g. Minh et al. 2010; Minh 2016) where these species are released from the grain material.


In recent decades, radio interferometers, such as the Karl G. Jansky Very Large Array (VLA), Australia Telescope Compact Array (ATCA), and Giant Meterwave Radio Telescope (GMRT), have surveyed fields of different sizes (ranging from tens of square arcminutes to thousands of square degrees), depths (microjansky to jansky), and multiwavelength coverage (e.g., Becker et al. 1995; Condon et al. 1998, 2003, 2012; Ciliegi et al. 1999; Georgakakis et al. 1999; Bock et al. 1999; Prandoni et al. 2001; Hopkins et al. 2003; Schinnerer et al. 2004; Bondi et al. 2003, 2007; Norris et al. 2005; Schinnerer et al. 2007, 2010; Afonso et al. 2005; Tasse et al. 2007; Smolčić et al. 2008, 2014; Owen & Morrison 2008; Miller et al. 2008, 2013; Owen et al. 2009; Hales et al. 2014). These past surveys have shown that deep observations at high angular resolution (≲1″) with exquisite panchromatic coverage are critical to comprehensively study the radio properties of the main galaxy populations, avoiding cosmic variance with large area coverage (e.g., Padovani et al. 2009; Padovani 2011; Smolčić et al. 2008, 2009b,a; Smolčić 2009; Smolčić & Riechers 2011; Seymour et al. 2008; Bonzini et al. 2012, 2013). In this context, large area surveys down to unprecedented depths are planned with new and upgraded facilities (e.g., VLA, Westerbork, Australian Square Kilometre Array Pathfinder, MeerKAT, and Square Kilometre Array; e.g., Jarvis 2012; Norris et al. 2011, 2013, 2015; Prandoni & Seymour 2015). Figure 1 shows the 1σ sensitivity of each survey as a function of the area covered for past, current, and future radio continuum surveys. The VLA-Cosmic Evolution Survey (COSMOS) 3 GHz Large Project bridges the gap between past and future radio continuum surveys by covering an area as large as two square degrees down to a sensitivity reached to date only for single pointing observations. This allows for individual detections of >10 000 radio sources, further building on the already extensive radio coverage of the COSMOS field at 1.4 GHz VLA (VLA-COSMOS Large, Deep and Joint projects; Schinnerer et al. 2004, 2007, 2010), 320 MHz VLA (Smolčić et al. 2014), 325 MHz and 616 MHz GMRT data (Tisanić et al., in prep.), 6 GHz VLA (Myers et al., in prep.), and the deep multiwavelength X-ray to mm photometry (Scoville et al. 2007; Koekemoer et al. 2007; Hasinger et al. 2007; Capak et al. 2007; Sanders et al. 2007; Bertoldi et al. 2007; Elvis et al. 2009; Ilbert et al. 2013; McCracken et al. 2012; Scott et al. 2008; Aretxaga et al. 2011; Smolčić et al. 2012; Miettinen et al. 2015; Civano et al. 2016; Laigle et al. 2016, Capak et al., in prep.) and more than 97 000 optical spectroscopic redshifts (Salvato et al., in prep.; zCOSMOS, Lilly et al. 2007, 2009; Trump et al. 2007; Prescott et al. 2006; Le Fèvre et al. 2015; Aihara et al. 2011; Nagao et al., priv. comm.). This further makes the survey part of one of the richest multiwavelength data sets currently available.


In this paper, our primary interest is in determination of plasma temperatures, in particular the characterization and measurement of high-temperature lines and continuum in the Chandra/HETGS spectrum of ζ Pup. If X-ray emission originates solely from embedded-wind shocks (Lucy & White 1980; Feldmeier et al. 1995), then the maximum temperature is determined by the highest relative velocities of the colliding structures, and the amount of emission is indicative of the emitting volume. Hot thermal plasmas emit strongly in bound–bound lines of highly ionized states, in thermal Bremsstrahlung radiation, and in bound-free continuum emission. Bremsstrahlung and bound-free emission drop exponentially for photon energies larger than about kT of the plasma (in which k is Boltzmann’s constant and T is the plasma temperature). For some plasma temperatures, bound-free emission can exceed that from thermal Bremsstrahlung at high energies (Landi 2007; Kaastra et al. 2008). However, the continuum shape is similar to that from Bremsstrahlung. The drop in the spectrum at high energies (short wavelengths) is sensitive to the hottest temperatures present in a multithermal plasma. Thermal emission lines are also very sensitive to temperature, having temperatures of peak emissivities that increase with atomic number; they typically emit most strongly over a range in temperature of about 0.3 dex (defined by the full width at half maximum of their emissivity curves). Hence, we concentrate here on the 2–9 Å region due to the relative sparseness of emission lines, the high-temperature sensitivity, and the reduced importance of wind absorption. The temperature sensitivity occurs because for the expected plasma temperatures from embedded-wind shocks (2–10  MK), the spectrum will drop accordingly somewhere in the 1–10  Å range. In this regime, the wind absorption is relatively unimportant since the continuum opacity (due mainly to K-shell absorption by metals) drops very steeply to short wavelengths, so we do not have to be concerned with the wind density structure to interpret the spectral shape.


NGC 3079 is a well-known nearby (z = 0.003723) Seyfert 2/LINER galaxy with a nuclear starburst, which is a member of an interacting galaxy pair. It is an edge-on spiral galaxy with several regions of star formation along the disk. Its eight-shaped radio structure with a ring-like feature (ring hereafter) inside its northeast lobe (Duric & Seaquist 1988; Irwin & Saikia 2003; Wiegert et al. 2015; Irwin et al. 2019), its slow-moving (v ∼ 0.1c) parsec-scale radio jet (Irwin & Seaquist 1988; Baan & Irwin 1995; Middelberg et al. 2005), and copious amount of hot X-ray gas and emission-line filaments (Cecil et al. 2001, 2002) have been well documented. NGC 3079 is a Compton-thick AGN (Iyomoto et al. 2001) having a disk-like pseudo-bulge. The origin of the double-lobed morphology observed in NGC 3079 is not yet completely understood. Duric & Seaquist (1988) argue that the lobe material is moving at supersonic velocities, though they dismiss the possibility that these are jets that may be precessing based on the difficulty in reproducing the closed-loop morphology using these models. They suggest that these are probably winds originating from a nuclear starburst or accretion activity. X-ray emission and emission-line imaging shows that NGC 3079 hosts a superwind (Veilleux et al. 1994; Cecil et al. 2001, 2002). Superwinds are believed to be generated when the kinetic energy from supernovae and massive stellar winds are transformed into thermal energy. The heated gas then expands into the lower pressure ambient medium. The direction of the steepest pressure gradient is along the minor axis. The hot gas is easily traced by the X-ray emission, which is a more direct tracer compared to the emission-line gas, which is generated at shock fronts as a result of the interaction with the ambient medium. The emission-line ratios seen in superwinds are typical of shock-ionized gas as also seen in the case of NGC 3079 (Veilleux et al. 1994). Although these authors favored a starburst-wind origin for the radio lobes, they were not able to rule out AGN-driven winds. Several authors favored the jet origin of the lobes (Irwin & Seaquist 1988; Irwin & Saikia 2003). Irwin et al. (2017) find that NGC 3079 is one of the few edge-on galaxies in the CHANG-ES sample (Irwin et al. 2012) that shows radio lobes that stand out in polarized intensity compared to the galactic disk emission that is not as highly polarized.


In summary, the different components show similar evolutionary trends for the fractional rms and characteristic frequency with the hardness ratio in the LE/ME/HE energy bands. The evolutionary trend can be explained under the truncated disk/corona model (Esin et al. 1997; Done et al. 2007). Actually, there is still debate on the truncation of the accretion disk in MAXI J1820+070. Buisson et al. (2019) discovered a steady inner accretion disk measured by relativistic reflection. Kara et al. (2019) found that the reverberation time lags between the continuum-emitting corona and the irradiated accretion disk are much shorter than previously seen in truncated accretion disks, and the timescale of the reverberation lags is shortened by an order of magnitude over a period of weeks, whereas the shape of the Fe Kα emission line remains remarkably constant. Similar results are also obtained from spectral analysis. Meanwhile, there are some other studies that support the truncated accretion disk argument by either spectral analysis or timing analysis (De Marco et al. 2021; Marino et al. 2021; Zdziarski et al. 2021a, 2021b). De Marco et al. (2021) found that the frequency of the thermal reverberation lags increases steadily, and, on the other hand, the temperature of the quasi-thermal component grows as the source softens, which can be explained in terms of a decrease in the disk inner radius. Moreover, De Marco et al. (2021) measured that the values of the lag amplitude are a factor of 3 longer than those reported in Kara et al. (2019). The longer lags might not be easily reconciled with the conclusion of a disk extending close to the innermost stable circular orbit. Zdziarski et al. (2021a, 2021b) confirmed the existence of an optically thick disk of at least >10R

g
 from joint spectral analysis. To sum up, so far, all arguments in favor of the nontruncated disk model can be reasonably explained under the truncated disk model. Except for the methods mentioned above, in the present paper, we confirm the truncated accretion disk model in MAXI J1820+070 from the evolutionary trend of broadband noise components with a new perspective by means of the correspondence relation between break frequency and radiation region radius. Quantitatively, we can calculate the radiation region at a different energy band of L
2, which represents the variable emission from the outermost region (see Figure 13). We take a parameter set for a standard α-disk (α = 0.1, M
BH = 10M
⊙, scale height H/r = 0.1) to calculate the viscous frequency at a certain radius (Kato et al. 2008). As Figure 13 shows, the characteristic frequency of the L
2 component shows an energy dependence: the emitting region spans from ∼34R

g
 to ∼27R

g
 corresponding to 1–150 keV photon energy. Like Dziełak et al. (2021) and Kawamura et al. (2022) found using frequency-resolved spectral analysis that the L
2 component is supposed to come from variable disk emission. However, making use of the ME and HE data from Insight-HXMT, we actually detect high-energy emission >100 keV from the L
2 component, which cannot be attributed to a simple standard accretion disk. Considering the change of radius plotted in Figure 13, even though we can attribute the high-energy emission to the propagation of fluctuation from the disk to the hot flow, we should also expect a constant characteristic frequency for L
2 at the high-energy band according to fluctuation propagation. The characteristic frequency of L
2 remains unchanged below 20–30 keV as Kawamura et al. (2022) found the constant peak frequency for P1. However, when the photon energy is greater than 20–30 keV, the radiation radius of L
2 starts to decrease to 27R

g
. This phenomenon cannot be easily interpreted by fluctuation from a disk propagating to the hot flow. Therefore, we speculate that L
2 may originate in a warm extended variable disk region. We therefore should consider a more complicated accretion flow geometry where a standard accretion disk transits to a hot advection-dominated accretion flow geometry.


The community searches for extensions to the ΛCDM model to resolve some of the mentioned problems. The approaches to face them are divided into two main branches: (i) assume a DE fluid with the capability to accelerate the Universe or (ii) modify General Relativity (GR) to obtain the cosmic acceleration without DE (Motta et al. 2021). This paper will focus on the second point under the formalism known as fractional calculus, which consists of a generalization of the classical integer order calculus, whose derivatives and integrals are of (real or complex) arbitrary order. These fractional operators are not local. In many cases can model real-world phenomena in a better fashion than those obtained by classical calculation. For example, it is coined fractional dynamics as a field of study in physics and mechanics investigating the behaviour of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by us and differentiation of non-integer orders, i.e. by methods in the fractional calculus (see the review by Tarasov 2013). Fractional calculus is a field with multiple applications and a great deal of research activity. Fractional quantum mechanics is employed as a tool within quantum field theory and gravity for fractional spacetime (Calcagni 2010a, b) and the fractional quantum field theory at positive temperature (Lim 2006; Lim & Eab 2019) and other applications of quantum cosmology (Moniz & Jalalzadeh 2020a, b; Jalalzadeh, da Silva & Moniz 2021; Rasouli, Jalalzadeh & Moniz 2021). Recently, the community explores the fractional calculus to tackle problems associated in cosmology (Shchigolev 2011, 2013a, b, 2016, 2021; Calcagni 2013, 2017a, 2021a, b; Calcagni & De Felice 2020; Jalalzadeh, Costa & Moniz 2022), stochastic GW background (Calcagni & Kuroyanagi 2021), luminosity distance (Calcagni et al. 2019), inflation and CMB spectrum (Calcagni, Kuroyanagi & Tsujikawa 2016; Calcagni 2017b), Fractional Action Cosmology (El-Nabulsi 2012, 2016a; Jamil, Momeni & Rashid 2012), fractional geodesic equation, complex GR, and discrete gravity El-Nabulsi (2013b), minimal couplings (El-Nabulsi 2013a), phantom (Rami 2015), Ornstein–Uhlenbeck-like fractional differential equation in cosmology (El-Nabulsi 2016b), a variable Order Parameter (El-Nabulsi 2017a), wormholes in fractional action cosmology (El-Nabulsi 2017c). New metrics were considered (El-Nabulsi 2017b), as well as some DE models in emergent, logamediate, and intermediate scenarios of the universe (Debnath, Jamil & Chattopadhyay 2012; Debnath, Chattopadhyay & Jamil 2013). For instance, Shchigolev (2016, 2021) found α = 0.926 (where α is the order of the Riemann–Liouville fractional integral). In Shchigolev (2011,2013a, b) were obtained several exact solutions for cosmological models, which differ significantly from the standard model due to the fractal nature of spacetime (Calcagni 2010a, b). Jalalzadeh et al. (2022) explore the interval 1  α  2 using Riesz’s fractional derivative (that is not related to the index of Riemann–Liouville fractional integral) to obtain the non-boundary and tunnelling wave functions for a closed de Sitter geometry. Another example, Barrientos, Mendoza & Padilla (2021), studies the Universe dynamics without DM and DE components by modifying the mathematical structure of Friedmann equations with fractional calculus. Another approach is calculating the value of the Λ (due to the well-known ultraviolet divergence in the standard quantum field theory), which needs to restructure the theory using the fractional calculus (Calcagni 2021c). Giusti (2020) and Torres et al. (2020) explore Modified Newtonian Dynamics Theories and quantum cosmology in this fractional approach (Barrientos et al. 2021). Finally, notice that there are several definitions of fractional derivatives and fractional integrals, such as those of Riemann–Liouville, Caputo, Riesz, Hadamard, Marchand, and Griinwald–Letnikov, among other more recent ones (see Kilbas, Srivastava & Trujillo 2006; Podlubny 1998, and references therein). Even though these operators are already well studied, some of the usual features related to function differentiation fails, such as Leibniz’s rule, the chain rule, and the semigroup property (Podlubny 1998; Kilbas et al. 2006).


There are four possible examples of where Lyα absorption is taking place. In Figures 3(b) and (c) we see that the high-velocity component of the double profile seen in both Hβ and [C ii] is significantly suppressed compared with the low-velocity component. For Figure 2(b) in particular, the peak in the Hβ profile falls close to a strong dip in the Lyα profile, which appears as two small wings on either side of the dip. For Figure 3(c), the feature seen in both [C ii] and Hβ is largely suppressed above 6500 km s−1. Also, in Figures 2(d) and 3(d), the Lyα profile is shifted blueward of the main [C ii] and Hβ peaks, suggestive of absorbing gas centered at V
helio = 6500–6700 km s−1. Asymmetric Lyα profiles like this are often associated with radial outflows in galaxies (e.g., Heckman et al. 2011). The COS pointing (HCG92-5) samples the gas in the so-called “AGN bridge” (Cluver et al. 2010), a linear H2 filament that is apparently separate from the main collisional shock in SQ. A strong shear in the velocity field of the [C ii] emission was noted by Appleton et al. (2013) in that region. The spectrum of the CO (1–0) emission from that direction (see Figures 2(d) and 3(d)) shows significant CO emission at the high-velocity side of the profile, which would be consistent with absorption. Finally, HCG92-7 shows, in Figure 3(e), the opposite effect. In this case, again considering the high-velocity component of the double-horned profile only, we see that the Lyα is significantly redshifted with respect to the Hβ emission, with a sharp drop in emission as one approaches the peak of the Hβ (around V
helio = 6600–6700 km s−1). This may be another example of asymmetric absorption, with resonant scattering to the red side of the wings of the kinematics. In summary, we see that two regions in the main emission-line filament are almost free of absorption, whereas other regions show strong absorption. Even in those cases, at least some of the Lyα emission is able to resonantly scatter and undergo many scatterings within the gas before eventually escaping into the wings of the velocity profile, where the optical depth is much lower. In Section 5.2, we estimate the escape fraction of the Lyα photons and the number of scatterings, and we discuss the multiphase structure of the intragroup medium gas. We also remind that the differences between the Lyα and Hβ spectral profiles may not only be due to Lyα scattering, because collisional excitation could also contribute.


Accretion disks around compact objects. Compact low-luminous massive objects are expected to lurk in the CGM of galaxies. From lone low-mass stars (e.g., Helmi 2020), to black holes (BHs) arising from stellar evolution (e.g., Fender et al. 2013), including remnants of Population III stars (e.g., Madau & Rees 2001; Filho & Sánchez Almeida 2018), and primordial BHs (e.g., Carr & Hawking 1974; Clesse & García-Bellido 2015). The number density of these objects is assumed to be high; for example, there are claims that primordial BHs account for all the dark matter in the universe (Clesse & García-Bellido 2015). Even if this is not the case (e.g., Carr & Kühnel 2020), the expectations clearly overwhelm the density required to account for all the emission signals that we observe, of the order of one clump per central galaxy (Section 4.1). When these compact objects are surrounded by accretion disks, they should emit in Hα, with the rotation of the disk giving rise to two-horn Hα profiles when observed with the appropriate viewing angle (Smak 1969). This kind of double-peak Hα emission is observed in X-ray binaries (Grundstrom et al. 2007; Zamanov et al. 2013; Casares & Torres 2018; Monageng et al. 2020) and cataclysmic variables (e.g., Zolotukhin & Chilingarian 2011), with peak separations of up to hundreds of km s−1 (Zamanov et al. 2013). At a completely different mass scale, double-peak Hα emission is sometimes observed in the broad-line region of AGN, where it is also supposed to trace a rotating disk (e.g., Eracleous & Halpern 1994; Ho 2008). The double-peak Hα emission observed around stellar-mass compact objects comes together with continuum emission (e.g., McSwain et al. 2010; Zamanov et al. 2019). There are hints of continuum emission in our stacks of observed spectra (Figure 8), but the question arises as to whether this very faint continuum is consistent or not with an accretion disk around a compact object. A number of arguments show that the observed lack of continuum emission does not discard the accretion disk scenario. First, large Hα equivalent widths (EWs) are sometimes observed in X-ray binaries. Hα EWs larger than 100 Å are not rare during quiescence (Fender et al. 2009; Casares & Torres 2018), with maxima reaching 2000 Å (Muñoz-Darias et al. 2016), which implies line to continuum flux ratios enough to account for our observations (Figure 8). Second, Hα photons are to be produced by recombination of H atoms photoionized by the accretion disk (e.g., Matthews et al. 2015). Under particular circumstances, the nebular emission produced by a photoionizing source can emit an Hα line with very little underlying continuum. For example, young starbursts produce Hα with an EW in excess of 103 Å (Leitherer et al. 1999), and galaxy-integrated spectra with EW of hundreds of angstrom are not uncommon (e.g., Morales-Luis et al. 2011). The main physical ingredient for the photoionization source to produce little continuum is presenting a hard spectrum with an ionization flux greatly exceeding the flux in the optical. Accretion disks can easily match this requirement since they can be extremely hot with spectra peaking in the far-UV and X-ray.


GCs are the most easily noticeable objects in the haloes of galaxies. However, their surface density in the inner parts always outnumbers that in the haloes (see e.g. Hargis & Rhode 2014; Kartha et al. 2014). In fact, the radial surface density distribution can be well described by the Sérsic function (Sersic 1968) in its classical form:
(2)$$\begin{eqnarray*}
N(R) = N_{\textrm {e}}\exp \left[-b_{n}\left(\frac{R}{R_{\textrm {e}}}\right)^{(1/n)}-1\right],
\end{eqnarray*}$$where, Re is the effective radius enclosing half the population, n is the Sérsic index that controls the shape of the profile, and bn = 1.992n − 0.3271. We show the radial distribution of surface number density in Fig. 13. The radial axis is plotted in units of R25. For two of our sample galaxies (M81: Perelmuter & Racine 1995, and NGC 4258: González-Lópezlira et al. 2017), GC searches have been carried out using the ground-based data that complement the HST data in the outer parts, which have also been shown (green empty circles). The observed and completeness-corrected values are shown in solid circles of black and magenta colours, respectively, with the fits to these data shown by solid lines of corresponding colours. The numbers inside and outside the Re are corrected by the incompleteness factors corresponding to high and low surface brightness fields, respectively. For M81, the inner most number is corrected by the completeness factor obtained from simulation on the frame that included the bulge and the nucleus. The Re values before and after the incompleteness corrections are shown by dashed vertical lines of black and magenta colours, respectively. The effect of correction is to marginally shift the Re to lower values, which is noticeable only for NGC 4258 and M51 in the figure. In Table 8, we show the values for Sérsic fit. $N_{{\rm GC},R_{\rm e}}$ and $N_{{\rm GC},R_{\rm e}}^\prime$ are the numbers of GCs inside Re before and after correction for contaminants. To estimate the total number of GCs (see Section 5.6), we use the number of GCs within Re, ($N_{{\rm GC},R_{\rm e}}^\prime$).


In order to connect with cosmic ray data, we extrapolate now from the previously studied single collision zone to a population of alike GRBs with a fixed duration of ΔT = 10 s, which implies that the emission from each GRB comes from ΔT∕tv such collisions. We perform a fit of UHECR observations from the Pierre Auger Observatory, combining the modeling of interactions in the source (computed as described in the previous sections) with the propagation of cosmic rays in the extragalactic space. For the propagation, the SimProp code (Aloisio et al. 2012) has been used with Gilmore EBL (Gilmore et al. 2012) and PSB cross-section model (as defined in Alves Batista et al. 2015). We also consider the extensive air shower in the Earth’s atmosphere and EPOS-LHC (Pierog et al. 2015) is assumed for UHECR-air interactions. We find that a good description of the data is obtained by distributing the GRBs as sources of cosmic rays following the star formation rate Yuksel et al. (2008), assuming a pure 28 Si at the injection. The primary spectrum is described in Eq. (10) and we use here k = 1.8 and P = 2. We fix the following parameters: source evolution, spectral index and cutoff shape at injection, and the nuclear species at the injection. In the present work, we keep this procedure as simple as possible in order to show the power of the method, while leaving a more detailed analysis for a future work. The fit is performed above 1018 eV (Mixed Composition Dip Model) and above 1019 eV (Mixed Composition Ankle Model) by using the combined spectrum (Valiño et al. 2015) and the shower depth (Xmax) distributions (Aab et al. 2014), which contain information about the mass of the nucleus interacting with the atmosphere, with a similar procedure as used in di Matteo et al. (2015) and Aab et al. (2017). A scan over (R, Lγ) is performed and for each pair the normalization to the experimental flux is found. For each point of the parameter space the number of expected prompt and cosmogenic neutrino events is calculated following Baerwald et al. (2015). For the prompt neutrino flux, the exposure for muon neutrinos is calculated by summing the exposure relative to the IceCube analyses of Aartsen et al. (2015,3 yr) and that of Aartsen et al. (2017, 3 yr) for a total of 1014 GRBs that occurred in the Northern Hemisphere, and that of Aartsen et al. (2017, 5 yr) for 664 GRBs that occurred in the Southern Hemisphere, and comparing the total number of bursts of the combined sample with the assumed 667 bursts per year as in Baerwald et al. (2015). For the cosmogenic neutrino flux, the exposure for muon neutrinos is taken from Aartsen et al. (2016). The exclusion regions (90% C.L.) of the prompt and cosmogenic neutrino analyses are calculated assuming both analyses as background free.


These three dimensions in momentum space and three dimensions in real space constitute the 6D phase space. The distribution of neutrinos in this phase space is given by f(r, θ, ϕ, ϵ, μ, Φ). The four-current of particles in the phase space cell dΓcom is given by
(22)$$\begin{eqnarray*}
j^\mu = \int f u^\mu \, \mathrm{d}\Gamma _\mathrm{com}.
\end{eqnarray*}$$The number ΔN of particles in dΓcom within some three volume is
(23)$$\begin{eqnarray*}
\Delta N &=& \int \sqrt{-g}\, \langle \boldsymbol{j}, \mathrm{d}x^\mu \rangle \wedge \mathrm{d}x^\nu \wedge \mathrm{d}x^\xi \wedge \mathrm{d}x^\sigma \nonumber \\
&=& \int \sqrt{-g} j^\mu \, \mathrm{d}x^\nu \, \mathrm{d}x^\xi \, \mathrm{d}x^\sigma ,
\end{eqnarray*}$$where $\langle \boldsymbol {j},dx^\mu \rangle$ denotes the contraction of the four-vector $\boldsymbol {j}$ with the volume form. This general form can be used to obtain both the number of particles within a space-like three-volume cell d3x as well as the number of particles flowing through cell interfaces per coordinate time dt (two space and one time dimensions). The number of particles within the phase space volume dΓcom in a cell $\mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3$ is
(24)$$\begin{eqnarray*}
N = \iiiint \sqrt{-g} f u^0 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3,
\end{eqnarray*}$$and the flux across a cell interface with x1 = const. is
(25)$$\begin{eqnarray*}
F = \iiiint \sqrt{-g} f u^1 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^2 \, \mathrm{d}x^3 \, \mathrm{d}t.
\end{eqnarray*}$$For a conformally flat metric (Isenberg 2008,used in coconut), with
(26)$$\begin{eqnarray*}
g_{\mu \nu }= \begin{pmatrix}-\alpha ^2+\beta _i \beta ^i & \quad \beta _x & \quad \beta _y & \quad \beta _z \\
\beta _x & \quad \phi ^4 & \quad 0 & \quad 0\\
\beta _y & \quad 0 & \quad \phi ^4 & \quad 0\\
\beta _z & \quad 0 & \quad 0 & \quad \phi ^4 \\
\end{pmatrix},
\end{eqnarray*}$$in terms of the lapse function α, the conformal factor ϕ, the shift vector βi, and βi = ϕ4βi, these can be simplified to
(27)$$\begin{eqnarray*}
N = \iiiint \phi ^6 f \hat{u}^0 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3,
\end{eqnarray*}$$(28)$$\begin{eqnarray*}
F = \iiiint \phi ^6 f \left(\frac{\alpha }{\phi ^2}\hat{u}^1+\beta ^r \hat{u}^0\right) \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^2 \, \mathrm{d}x^3 \, \mathrm{d}t.
\end{eqnarray*}$$in terms of the Eulerian velocity components $\hat{u}^\mu$. It is critical to recognize that because of the invariance of the distribution function we can directly use f in the comoving frame to compute the number of neutrinos within phase space cell dΓcomv that cross a zone boundary during the time interval dt. Only the neutrino four-velocity in equation (22) must be transformed into the coordinate frame. The problem, however, is that the current jμ only obeys a conservation law ∇μjμ = 0 if we define dΓcom away from a given reference point xμ as the phase space volume filled by particles from dΓcom at xμ (or in technical terms if we define dΓcom away from xμ by an exponential map). In general, the exponential map of dΓcom(xμ) from xμ to another point x′μ will differ from the momentum space volume dΓcom(x′μ) defined by the same comoving-frame momentum space coordinates, which necessitates the remapping step that we outlined in Section 2.1 and will specify in detail in the next section.