Implications from GW170817 for $\Delta$-isobar admixed hypernuclear compact stars

TL;DR

This paper investigates how $\Delta$ isobars influence the dense matter equation of state and the structure of compact stars, demonstrating compatibility with GW170817 tidal deformability data and suggesting hypernuclear stars with $\Delta$ isobars are plausible.

Contribution

It introduces a covariant density functional approach to include $\Delta$ isobars in hypernuclear star models and compares predictions with gravitational wave observations.

Findings

$\Delta$ isobars soften the equation of state, aligning models with small star radii.

GW170817 tidal deformability data supports the existence of hypernuclear stars with $\Delta$ isobars.

The event is consistent with mergers involving stars containing both hyperons and $\Delta$ isobars.

Abstract

The effects of $\Delta$ isobars on the equation of state of dense matter and structure of compact stars (CSs) are explored within the covariant density functional theory and confronted with the data on tidal deformability (TD) extracted from the GW170817 event. We show that the presence of $\Delta$ isobars substantially softens the tension between the predictions of the hypernuclear density functionals and the inference from the observations of relatively small radius and small TD of canonical mass CSs. The TDs deduced from GW170817 are compatible with the existence of hypernuclear CSs containing an admixture of $\Delta$ isobars. We thus argue that the GW170817 event is consistent with a merger of a binary CS system having both strangeness (hyperons) and $\Delta$ isobars in the stellar core.

Figures

Tables

Table 1. Table 1 Properties of CSs assuming purely hyperonic ( Y 𝑌 Y ) and hyperon- Δ Δ \Delta admixed ( Δ Δ \Delta ) composition for selected EoS models. The first three columns identify the EoS by the isoscalar skewness Q sat subscript 𝑄 sat Q_{\text{sat}} (MeV) and isovector slope L sym subscript 𝐿 sym L_{\text{sym}} (MeV). The remaining columns display: maximum mass M max subscript 𝑀 max M_{\text{max}} ( M ⊙ subscript 𝑀 direct-product M_{\odot} ), radius R 1.4 subscript 𝑅 1.4 R_{1.4} (km), TDs λ 1.4 subscript 𝜆 1.4 \lambda_{1.4} ( 10 36 superscript 10 36 10^{36} gr cm 2 s 2 ) and Λ 1.4 subscript Λ 1.4 \Lambda_{1.4} of canonical-mass CS. The results for hyperon- Δ Δ \Delta admixed matter are obtained by tuning the Δ Δ \Delta -potential V Δ / V N subscript 𝑉 Δ subscript 𝑉 𝑁 V_{\Delta}/V_{N} from 1 to 5/3.

DF	$Q_{\text{sat}}$	$L_{\text{sym}}$		$M_{\text{max}}^{\left(Y\right)}$	$R_{1.4}^{\left(Y\right)}$	${\Lambda }_{1.4}^{\left(Y\right)}$		$M_{\text{max}}^{\left(\Delta \right)}$	$R_{1.4}^{\left(\Delta \right)}$	${\lambda }_{1.4}^{\left(\Delta \right)}$	${\Lambda }_{1.4}^{\left(\Delta \right)}$
1	480	40		2.02	13.06	636		$2.03-2.06$	$12.91-12.08$	$3.25-1.97$	$575-348$
2	480	60		2.00	13.35	754		$2.01-2.04$	$13.24-12.38$	$3.97-2.42$	$701-426$
3	300	50		1.97	13.10	655		$1.98-2.01$	$12.94-12.07$	$3.30-1.94$	$584-342$
4	800	50		2.06	13.31	772		$2.07-2.10$	$13.22-12.39$	$4.08-2.56$	$722-454$

Equations

$$\Lambda =\lambda /M^5=\frac{2}{3}\frac{k_2}{C^5}.$$

$$\tilde{\Lambda}=\frac{16}{13}\Bigg{[}\frac{(M_{1}+12M_{2})M^{4}_{1}\Lambda_{1}}{(M_{1}+M_{2})^{5}}+1\leftrightarrow 2\Bigg{]},$$

$$E(\rho ,\delta )$$

Full text

1. Introduction

The first multi-messenger observations of gravitational waves (GW) from the binary neutron star (NS) merger event, GW170817, marks the start of a new era in astronomy and astrophysics (Abbott et al., 2017a, b, c). The significance of this and future similar events lies, in part, in the opportunity of gaining insight into the equation of state (EoS) and composition of dense matter (Abbott et al., 2017b, 2018, 2019; Margalit & Metzger, 2017; Bauswein et al., 2017; Ruiz et al., 2018; Rezzolla et al., 2018; Soares-Santos et al., 2017; Villar et al., 2017). In particular, the TD (or polarizability) of NSs deduced from GW170817 puts already additional constraints on stellar radii and ultimately on the details of the EoS (Fattoyev et al., 2018; Annala et al., 2018; De et al., 2018; Most et al., 2018; Paschalidis et al., 2018; Tews et al., 2018; Zhang et al., 2018).

The potential implications of the GW170817 event cover a wide range of fundamental questions associated with NSs, ranging from their interior composition to their role in nucleosynthesis. In fact, a number of works have explored the possibility of using the observations of GW170817 to probe the occurrence of a hadron-quark phase transition (Paschalidis et al., 2018; Most et al., 2018; Burgio et al., 2018; Blaschke & Chamel, 2018; Alvarez-Castillo et al., 2019).

Even at the level of the hadronic matter the composition of stellar matter could be rather complex. There have been intense studies of a number of possibilities of new degrees of freedom, such as hyperons (Glendenning, 1985; Prakash et al., 1992; Weissenborn et al., 2012; van Dalen et al., 2014; Oertel et al., 2015; Tolos et al., 2016; Fortin et al., 2017; Li et al., 2018a), $\Delta$ isobars (Prakash et al., 1992; Schürhoff et al., 2010; Drago et al., 2014; Cai et al., 2015; Kolomeitsev et al., 2017; Li et al., 2018b) or other exotic hadronic states such as $d^{*}(2380)$ resonance (Vidaña et al., 2018). The existence of hyperons inside NSs has been questioned for a long time because a class of models based on non-relativistic microscopic treatments of dense matter were not able to produce massive enough ($M\sim 2M_{\odot}$) stars. However, the hyperonization cannot be simply ruled out by the existence of $2M_{\odot}$ stars. In particular, covariant density functional (CDF) based models are versatile enough to resolve the problem by tuning the interactions in the hyperonic sector to hypernuclear data and CS masses (van Dalen et al., 2014; Oertel et al., 2015; Tolos et al., 2016; Weissenborn et al., 2012; Tolos et al., 2016; Fortin et al., 2017; Li et al., 2018a). Of course, one may also conjecture a layer of hypernuclear matter in-between the nucleonic and quark matter phases (Bonanno & Sedrakian, 2012; Masuda et al., 2013; Zdunik & Haensel, 2013; Dexheimer et al., 2015). These models based on purely hyperonic EoS predict CS sequences with $M_{\text{max}}\gtrsim 2.0M_{\odot}$ and radii of the canonical-mass $M\sim 1.4M_{\odot}$ star $R_{1.4}\gtrsim 13$ km (Katayama & Saito, 2015; Fortin et al., 2016; Tolos et al., 2016; Li et al., 2018a; Li & Sedrakian, 2019).

The possibility of hypernuclear stars with small radii ($R_{1.4}\lesssim 13$ km) is as exciting as it is challenging for nuclear theory. This requires a sufficiently soft EoS below 2-3 $\rho_{\text{sat}}$, where $\rho_{\text{sat}}$ is the nuclear saturation density, while the observed large masses require that the same EoS must evolve into a stiff one at high densities. In this work, we construct such a model which is based on purely hadronic forms of stellar matter. We use the CDF theory (Meng et al., 2006) to explore the effects of the $\Delta$ isobars on the TDs of CSs. We compare our results with the recent limits placed by the GW170817 event and discuss their implications for the interpretation and detection of the current and future GW signals from NS merger.

2. Tidal deformability

Consider a static, spherically symmetric star, placed in a static external quadrupolar tidal field of the companion. As two stars approach each other during the early stages of an inspiral due to their mutual gravitational attraction they experience tidal deformation effects that can be quantified in terms of the TD $\lambda$. It can be expressed in terms of the dimensionless tidal Love number $k_{2}$ and the star’s radius $R$ as $\lambda=({2}/{3})k_{2}R^{5}.$ The tidal Love number $k_{2}$ is calculated along with the solution of the Tolman-Oppenheimer-Volkov equations (Hinderer, 2008; Flanagan & Hinderer, 2008) and measures how easily the bulk of the matter in a CS is deformed (Hinderer, 2008; Flanagan & Hinderer, 2008; Binnington & Poisson, 2009). It is more convenient to work with the dimensionless TD $\Lambda$, which is related to the Love number $k_{2}$ and the compactness parameter $C=M/R$ through

[TABLE]

The total tidal effect of two CSs in an inspiraling binary system is given by the mass-weighted TD

[TABLE]

where $\Lambda_{1}(M_{1})$ and $\Lambda_{2}(M_{2})$ are the TDs of the individual binary components. The quantity $\tilde{\Lambda}$ is usually evaluated as a function of the chirp mass $\mathcal{M}=(M_{1}M_{2})^{3/5}/M^{1/5}_{T}$ for various values of the mass ratio $q=M_{2}/M_{1}$, where $M_{T}=M_{1}+M_{2}$ is the total mass of the binary.

3. Hadronic matter equation of state

Below we shall concentrate on the EoS of hypernuclear matter obtained from the CDF theory in its version which uses density-dependent meson-baryon couplings (Typel & Wolter, 1999). The extension to the hyperonic sector is described elsewhere (Li et al., 2018a, b), see also (Weissenborn et al., 2012; Fortin et al., 2017). A useful parameterization of the nucleonic EoS is given by the formula

[TABLE]

where $n=(\rho-\rho_{\text{sat}})/3\rho_{\text{sat}}$ and $\delta=(\rho_{\text{n}}-\rho_{\text{p}})/\rho$. The coefficients entering the parameterization (3) are the nuclear characteristic parameters $E_{\text{sat}}$ (binding energy), $K_{\text{sat}}$ (compressibility), $E_{\text{sym}}$ (symmetry energy), $Q_{\text{sat}}$ (isoscalar skewness coefficient) and $L_{\text{sym}}$ (isovector slope coefficient), all defined at $\rho_{\text{sat}}$. The low-order characteristics $E_{\text{sat}}$, $K_{\text{sat}}$ and $E_{\text{sym}}$ are either strongly constrained or have no noticeable impact on the gross properties of CSs (Li & Sedrakian, 2019; Margueron et al., 2018). Therefore, the remaining higher-order characteristic parameters are those that can be used to calibrate the parameters of the density functional to the desired properties of the system at hand (Li & Sedrakian, 2019). Their values are weakly constrained by the conventional fitting protocol used in constructing the CDF. Specifically, their ranges can be constrained by $\chi$EFT computations of neutron matter (Drischler et al., 2016) and the requirement that the EoS reproduces the observed maximum mass of a CS $M_{\text{max}}\gtrsim 2M_{\odot}$ (Antoniadis et al., 2013). The low-order characteristic parameters for our models are the same as the values predicted by the DD-ME2 parametrization (Li et al., 2018b). The observations (Antoniadis et al., 2013) and recent inferences of $M_{\text{max}}$ (Margalit & Metzger, 2017; Ruiz et al., 2018; Rezzolla et al., 2018), enable one to limit the range of $Q_{\text{sat}}$. However, it should be stressed that such range will essentially depend on the composition of matter (Li & Sedrakian, 2019). For instance, we have checked that the constraint $1.97\lesssim M_{\text{max}}/M_{\odot}\lesssim 2.17$ limits $Q_{\text{sat}}$ to the range $\sim[-650,-400]$ for purely nucleonic matter, and $\sim[300,800]$ for hyperonic matter.

In Table 1 we present integral parameters of hyperonic CSs (maximum mass $M_{\text{max}}$, the radius $R_{1.4}$ and dimensionless TD $\Lambda_{1.4}$ of canonical-mass star) for pairs of values of $L_{\text{sym}}$ and $Q_{\text{sat}}$. To obtain massive enough hypernuclear CSs, $Q_{\text{sat}}$ values must be large (Li & Sedrakian, 2019); as a result the radii for canonical stars obtained with these values of $Q_{\text{sat}}$ are $R_{1.4}\gtrsim 13$ km (Katayama & Saito, 2015; Fortin et al., 2016; Tolos et al., 2016; Li et al., 2018a; Li & Sedrakian, 2019). The corresponding TDs are $\Lambda_{1.4}>600$. To obtain smaller values of these parameters, which are favored by the observational data, we next explore the effect of $\Delta$ isobars by varying the $\Delta$-potential. The underlying EoS is parameterized in terms of the potential $V_{\Delta}$ of $\Delta$-isobar in symmetric nuclear matter at $\rho_{\text{sat}}$.

To set the stage for the discussion of TDs of CSs in the next section we show in Fig. 1 the mass-radius (MR) relations for purely nucleonic, hyperonic and hyperon-$\Delta$ admixed stellar matter for a number of parameter values, as indicated in the figure. It is seen that the allowance of $\Delta$-isobars in matter shifts the radii of configurations to smaller values without affecting seriously the value of the maximum mass ($M_{\text{max}}\gtrsim 2.0M_{\odot}$), an observation already made using alternative parametrizations of CDF in (Drago et al., 2014; Li et al., 2018b). We now turn to the discussion of our results for TDs.

4. Results and Discussions

The analysis of the GW170817 and complementary electromagnetic data could provide an important bound on the TDs. In particular, the LIGO/Virgo data analysis from GW170817 (hereafter LV constraint) placed an upper bound for the case of low-spin priors (Abbott et al., 2017b, 2019). We note that the LV detection of GW170817 derived an upper bound $\tilde{\Lambda}\lesssim 900$ (Abbott et al., 2017b) from the phase-shift analysis of the observed signal, and was recently reanalyzed to be $\tilde{\Lambda}\lesssim 720$ (Abbott et al., 2019). However, since this boundary is somewhat dependent on the waveform models, we use here all the data reported in (Abbott et al., 2017b, 2019). A lower bound of $\tilde{\Lambda}\gtrsim 400$ is imposed by combining optical/infrared and GW data with new numerical relativity results (Radice et al., 2018). More recently, an alternative approach involving radiative transfer simulations for the electromagnetic transient AT2017gfo predicts the lower bound to be $\tilde{\Lambda}\gtrsim 197$ (Coughlin et al., 2018). In the equal-mass scenario, these limits translate to constraints on the TD of a single CS itself.

In Fig. 2 we show the tidal Love number $k_{2}$, the polarizability $\lambda$, and the dimensionless TD $\Lambda$ of a CS as a function of its mass $M$. It is seen that the value of $k_{2}$ peaks for stellar configurations with mass near $0.8M_{\odot}$, while it decreases rapidly for both higher and lower mass configurations. The behavior of $k_{2}$ as a function of $M$ as seen in Fig. 2 (a, b) can be understood by noting that the more centrally condensed stellar models have smaller $k_{2}$ values (Hinderer et al., 2010).

The TD $\lambda$ for the same set of EoS is shown in Fig. 2 (c, d). This parameter has a direct astrophysical significance because it is proportional to a quantity directly measurable by GW observations of inspiraling star binaries. As can be seen from Fig. 2 (c, d), for each EoS $\lambda$ follows a trend that is very similar to that of $k_{2}$. However, $\lambda$ is, in addition, proportional to $R^{5}$ and thus it shows a more pronounced variation compared to $k_{2}$. The error bars that are shown in Fig. 2 (c, d) denote the range of probable values of the TD $\lambda$ for a star with $M=1.362M_{\odot}$. The relatively small $\lambda$ suggested by GW170817 implies that both the isoscalar and the isovector sectors of the EoS up to 2-3$\rho_{\text{sat}}$ should be soft. This, in turn, provides additional constraints for the nuclear characteristic parameters, in particular, the combinations of $Q_{\text{sat}}$ and $L_{\text{sym}}$. The dimensionless TD $\Lambda$ for selected EoS models is presented in Fig. 2 (e, f).

We now explore how the variations of $Q_{\text{sat}}$ and $L_{\text{sym}}$ affect the TDs of CSs. We concentrate only on the hyperonic models, but the conclusions also apply for $\Delta$-admixed models. It is seen from Fig. 2 (c, d) that the variation of $L_{\text{sym}}$ from 40 to 60 MeV (with $Q_{\text{sat}}$ fixed) as allowed by the $\chi$EFT calculations has appreciable effect on $\lambda$ for less massive stars ($M\lesssim 1.4M_{\odot}$), whereas the variation of $Q_{\text{sat}}$ from 300 to 800 MeV (with $L_{\text{sym}}$ fixed) as allowed by both the $\chi$EFT and the maximum mass constraints has a more significant effect on $\lambda$ of heavier stars ($M\gtrsim 1.4M_{\odot}$). The former observations on $L_{\text{sym}}$ is consistent with previous studies (Fattoyev et al., 2013). Here we report, for the first time, a broad analysis of the effects of variations of $Q_{\text{sat}}$. The stars of interest ($M\approx 1.1-1.6M_{\odot}$) are just at the intersection where the effects of $Q_{\text{sat}}$ and $L_{\text{sym}}$ on the TD are comparable. Therefore, $Q_{\text{sat}}$ or $L_{\text{sym}}$ values alone are insufficient to characterize the low-density (up to $\sim 2\rho_{\text{sat}}$) behavior of EoS (Li & Sedrakian, 2019). Further observations for binary merger events with smaller chirp mass will allow one to narrow down the uncertainty in $L_{\text{sym}}$ (important for low-mass stars); larger chirp mass observations will constrain more tightly $Q_{\text{sat}}$ (important for high-mass stars).

Below we discuss results for a single set of EoS models with fixed $Q_{\text{sat}}$ and $L_{\text{sym}}$, but the conclusions also apply for other sets of models specified by these parameters. As already seen from the MR relation, accounting for $\Delta$ isobars reduces the radii of models from their values obtained for purely hyperonic (or nucleonic) EoSs. As expected, the deeper the potential $V_{\Delta}$, the larger is the observed shift in the radius is and, accordingly, the larger the reduction of the TD. For example, for $V_{\Delta}/V_{N}=5/3$ the radius of a canonical $1.4M_{\odot}$ CS is by about 1 km smaller than the radius of its purely hyperonic (or nucleonic) counterpart. Correspondingly, the dimensionless TD of a canonical $1.4M_{\odot}$CS is by about 300 smaller than the value of its purely hyperonic counterpart; see Fig. 2 (e, f).

Table 1 shows how the properties of CS (including the maximum mass, radius, and TD for a $1.4M_{\odot}$ star) change with the value of the $\Delta$-potential $V_{\Delta}$. The differences in the values of $\Lambda_{1.4}$ can be mapped on the differences in the compactness using the scaling $\Lambda\varpropto C^{-6}$ valid for moderate-mass stars (Hinderer et al., 2010; Postnikov et al., 2010). This scaling follows from the proportionality $k_{2}\varpropto C^{-1}$, which is observed for a wide variety of EoS in the mass range $1.1\lesssim M/M_{\odot}\lesssim 1.6$ relevant for GW170817. This mass range corresponds roughly to $0.11\lesssim C\lesssim 0.20$ (Hinderer et al., 2010; Postnikov et al., 2010). We note that for the same mass stars (e.g. $M=1.4M_{\odot}$) the larger is the $\Delta$-potential the smaller the radii are and, therefore, the larger is the compactness and the smaller is the value of $\Lambda$.

We also observe in Fig. 2 (c, d) that the mass corresponding to the maximum value of $\lambda$ becomes smaller for EoS models that have a deeper potential $V_{\Delta}$. As the increase in the maximum masses of CSs due to $\Delta$ isobars is marginal, it, more importantly, affects medium-mass stars. For large enough $\Delta$-potentials the isobars may appear already at about $2\rho_{\text{sat}}$, which implies that even low-mass CSs can be affected by the populations of $\Delta$ isobars, see Fig. 2 (c, d). It is worth noticing that the reduction of TD caused by the inclusion of $\Delta$ isobars in the composition of matter is not very sensitive to the exact values of $Q_{\text{sat}}$ and $L_{\text{sym}}$. However, the onset density of $\Delta$ isobars and their fraction in the matter are sensitive to the values of $Q_{\text{sat}}$ and $L_{\text{sym}}$.

We now directly compare our modeling with the observational analysis from the GW170817 event, assuming a chirp mass $\mathcal{M}=1.186M_{\odot}$. We only investigate the more realistic low-spin case, because large spins are not expected from the observed galactic binary NS population (Abbott et al., 2018).

In Fig. 3 we display predictions from all EoS models for the individual tidal polarizabilities $\Lambda_{1}$-$\Lambda_{2}$ associated with the $M_{1}$-$M_{2}$ components of the binary. The diagonal line corresponds to the case of an equal-mass binary, i.e., $M_{1}=M_{2}=1.362M_{\odot}$. The shaded areas correspond to the 90% and 50% confidence limits, which are obtained from the improved analysis of the GW170817 event (Abbott et al., 2019). The previously reported data (Abbott et al., 2017b) are also shown as they have been widely used to constrain EoS models.

It is clearly seen from Fig. 3 that all the hyperonic (or nucleonic) EoS models satisfy the original 90% confidence (Abbott et al., 2017b), whereas the two sets of hyperonic EoS models, $(Q_{\text{sat}},L_{\text{sym}})=(800,50)$ and $(480,60)$ [MeV], are ruled out by the updated data (Abbott et al., 2019). The remaining two sets of hyperonic EoS models with $(Q_{\text{sat}},L_{\text{sym}})=(300,50)$ and $(480,40)$ [MeV], closely follow the updated 90% confidence (Abbott et al., 2019). As soon as the $\Delta$ isobars appear in matter with reasonably attractive $\Delta$-potential in nuclear matter ($V_{\Delta}<V_{N}$) all EoS models are completely inside the region of compatibility with the data from GW170817. Thus, if the binary in the GW170817 event contained hadronic stars, the updated analysis would strongly favor the presence of $\Delta$ isobars in addition to hypernuclear matter, as our purely hyperonic EoS models are rather representative of this class of models. An alternative is the strong phase transition to a quark matter phase, as it has been shown that such a transition also helps in producing more compact objects than those composed of nucleons only (Kojo et al., 2016; Alford & Sedrakian, 2017; Blaschke & Chamel, 2018; Alvarez-Castillo et al., 2019; Montana et al., 2018; Xia et al., 2019).

Interestingly, the $\Lambda_{1}$-$\Lambda_{2}$ curves predicted by EoS models in the parameter spaces $(Q_{\text{sat}},L_{\text{sym}},V_{\Delta})=(480,40,V_{N})$ and $(480,60,4/3V_{N})$ MeV are almost identical, but their MR relations are noticeably different. This is because the two MR curves cross around $M=1.4M_{\odot}$, and converge to each other at $M\simeq 2.0M_{\odot}$, see Fig. 1 (a). It is thus possible that different $M_{1}-M_{2}$ components of the binary have almost the same polarizabilities $\Lambda_{1}-\Lambda_{2}$.

For the sake of completeness, in Fig. 4 we show the mass weighted average TD $\tilde{\Lambda}$ with the ratio of masses of merger components for a fixed chirp mass $\mathcal{M}=1.186M_{\odot}$. It is seen that $\tilde{\Lambda}$ depends weakly on the mass ratio $q$. The boundaries on $\tilde{\Lambda}$, which were set by the GW and electromagnetic spectrum observations, respectively, are also shown. We should mention that the estimates of both boundaries are strongly model dependent (Abbott et al., 2017c, 2019; Radice et al., 2018; Coughlin et al., 2018; Most et al., 2018), especially the lower limit. The lower limit calculated from the $\Delta$-isobar featuring EoS models is about 350. Further reduction of the radius by up to 2 km can be obtained for larger values of $V_{\Delta}$, see (Li et al., 2018b) for a detailed discussion. In turn, $\tilde{\Lambda}$ could be decreased in this case.

5. Summary

In this work, we have explored the TDs of CSs featuring hypernuclear matter with an admixture of $\Delta$ isobars. As a consequence of including $\Delta$ isobar degrees of freedom, the dimensionless TD $\Lambda_{1.4}$ for canonical mass stars is reduced by about 300 for reasonably attractive $\Delta$-potential. Thus, the presence of $\Delta$ isobars lifts the tension between the predictions of the hypernuclear density functionals (which predict large TD) and the observations, which imply small TD.

In addition, we found that the nuclear characteristics $Q_{\text{sat}}$ and $L_{\text{sym}}$ control the TDs of large- and small-mass stars, respectively, whereas they are equally important for the TDs of canonical mass $M\simeq 1.4M_{\odot}$ stars. Thus, one may conclude that the GW170817 event is highly useful in constraining these universal parameters.

J.L. is supported by the Alexander von Humboldt Foundation. A.S. acknowledges the support by the DFG (grant No. SE 1836/4-1), the European COST Action “PHAROS” (CA16214) and the State of Hesse LOEWE-Program in HIC for FAIR.