from the bottom of Fig. 18): $R_{NS}=10.3^{+1.2}_{-1.1}$ km gui16  or adding
also thermonuclear bursters: $10.1<R_{NS}<11.1$ km (1.5 M⊙) ozel16  and
$9.9<R_{NS}<11.2$ km (1.5 M⊙) bog16  on lower radius side, to
$12^{+1.9}_{-1.7}$ km (1.4 M⊙) shaw18  and $10.4<R_{NS}<13.7$ km (1.4 M⊙)
ste18  on the other side. In particular, Ref. kim21  shows a recent reanalysis
of the sources used in ozel16 , but takes in addition into account the
uncertainties in chemical composition of the photosphere and in the touchdown
radius. More specifically, the authors allow in their model that the
photosphere not necessarily falls back on the NS surface after expansion due
to a thermonuclear burst. The radius of the photosphere after fall back (at
touchdown) is allowed to be larger or equal to the radius of the NS. In
addition the authors vary the hydrogen mass fraction in the H-He plasma of the
photosphere in their analysis. The presented point
($R_{NS}=10.4^{+2.1}_{-2.2}$ km, M$=1.75^{+0.27}_{-0.30}$ M⊙) represents a
mean value of the results from Bayesian analysis for the six analysed sources.
The inferred radius is close to the result of ozel16  but has a much larger
uncertainty resulting from the additional degrees of freedom taken into
account in the analysis. The large error bar also almost covers the range of
radii spanned by the pioneering results of gui13  and Steiner:2012xt .

$8$$9$$10$$11$$12$$13$$14$$15$$16$Guillot (2013)gui13 Steiner
(2013)Steiner:2012xt Guillot (2016)gui16 Özel (2016)ozel16 Bogdanov
(2016)bog16 Nättilä (2017)nat17 Shaw (2018)shaw18 Steiner (2018)ste18 Baillot
d’Etivaux (2019)bai19 Kim (2021)kim21 Most (2018)mos18 Lim (2018)lim18 Abbott
(2018)LIGOScientific:2018cki Annala (2018)ann18 Tews (2018)tews18 De
(2018)de18 Radice (2019)rad19 Capano (2020)cap20 Essick GW (2020)ess20
González-Caniulef (2019)gon19 Riley (2019)Riley:2019yda Miller
(2019)Miller:2019cac Riley (2021)ril21 Miller (2021)mil21 Jiang (2020)jiang20
Landry (2020)lan20 Dietrich (2020)die20 Essick GW+NICER (2020)ess20
Raaijmakers (2021)raa21 Tang (2021)tan21 Biswas (2021)bis21 Al-Mamun
(2021)alm21 Raithel (2021)rai21 Huth Astro+HIC (2022)Huth:2021bsp Huth HIC
(2022)Huth:2021bsp Trautmann (2019)Trautmann:2019gmh LMXBGW170817MSPmultiple
sourcesASYEOS1.5901.4951.5901.5681.5681.9681.4901.4681.45681.75681.4951.4951.4901.4901.4901.4901.4901.4901.4901.44681.34681.44682.07682.08681.4901.4901.4901.4901.4951.4901.4681.4951.4951.4681.4681.495R${}_{\mbox{NS}}$
[km] Figure 18: Neutron star radii ($R_{NS}$) from different analyses of LMXB,
from GW170817 event, from millisecond pulsars, from mixed sources of
information and from the interpretation of the results of the heavy ion ASY-
EOS experiment, from bottom to top, giving the year of the publication and the
reference. The numbers to the right of the symbols denote the confidence
levels (in %) for the measured/inferred radii and the masses of the NS in
solar mass units. The masses emphasized in boldface are the inferred/measured
ones. The remaining masses are the assumed or averaged ones.

The point nat17  ($R_{NS}$ = 12.4 $\pm$ 0.4 km, M = 1.9 $\pm$ 0.3 M⊙) obtained
for an individual X-ray bursting NS in binary system (4U 1702-429)
demonstrates the achievable precision up to now for this kind of sources. It
shows quite a good agreement with the NICER result of ril21  (see also Fig.
19) .

Another very precise result for quiescent LMXB was obtained in bai19 . The
analysis included also the sources taken into account in gui13 ;
Steiner:2012xt ; gui14 ; shaw18 ; ste18 . The averaged value for two sets of
distances yields the $R_{NS}$ = 12.39 $\pm$ 0.35 km for M = 1.45 M⊙ and thus
joins the larger radius class of results. The analysis performed in bai19
allowed also to extract the most likely value of the $L$ parameter, which
amounts to 37.2${}_{-8.9}^{+9.2}$ MeV and points to a relatively soft symmetry
energy.

The results obtained for quiescent and bursting LMXB systems indicate the
importance of the model dependence in the data analysis and a strong
contribution of systematic uncertainties. Generally, looking at the bottom
part of Fig. 18, one might conclude that LMXBs provide radii in the range from
about 8 to 14 km with a tendency to converge towards a narrower range between
11 and 14 km (2$\sigma$). The results of nat17  and bai19  demonstrate that a
3% precision (1$\sigma$) is achievable, however overall, the LMXB results
suffer from a large systematic scatter.

The results for the NS radius inferred from the gravitational wave signal
GW170817 LIGOScientific:2017vwq  and its electromagnetic counterpart gwem17
are presented in the next to bottom part of Fig. 18 and refer to the canonical
NS mass of 1.4 M⊙. The main constraint from the GW170817 event comes from the
tidal deformability, $\Lambda$ found to be less than 800 in the original paper
LIGOScientific:2017vwq . This constraint together with the requirement that
the EoS should support the 2 M⊙ neutron stars yielded the radius range of
$9.9<R_{NS}<13.6$ km in ann18 .

Using a numerous family of the EoSs and the Bayesian inference with the
information on the lower and upper bounds on $\Lambda$ as well as on the
maximum mass of the NS, a most probable value of
$R_{NS}=12.39^{+1.06}_{-0.39}$ km has been obtained in mos18 . Similar
analysis in lim18  gave $R_{NS}=12.10^{+0.77}_{-1.74}$ km. Refs.
LIGOScientific:2018cki  and de18  attempted at deriving more tight constraints
on $\Lambda$ than in LIGOScientific:2017vwq  by applying an additional
condition on the maximum mass of the NS Dem10  and on the distribution of the
measured masses. They arrived at the values of $R_{NS}=11.9^{+1.4}_{-1.4}$ km
and $R_{NS}=10.8^{+2.4}_{-1.9}$ km, respectively.

Ref. tews18  uses the chiral EFT interactions and QMC methods to derive the
dense matter EoS, reduce the $\Lambda$ range and predict the corresponding NS
radius to fall between 9.0 km $<R_{NS}<$ 13.6 km.

By combining the gravitational wave and electromagnetic data and performing a
Bayesian parameter estimation of the GW170817 the authors of rad19  arrived at
the value of 12.2${}_{-0.8}^{+1.0}\pm 0.2$ km for the $R_{NS}$.

Finally, two more recent analyses yield the values of 11.0${}^{+0.9}_{-0.6}$
km cap20  and 11.40${}_{-1.04}^{+1.38}$ km ess20  for the $R_{NS}$, within
frameworks similar to that of tews18 .

Summarizing this non-exhaustive set of results based on GW170817 and its post
merger emissions, one can conclude that they span the range of radii from
about 9 to 13.5 km (1.6-2$\sigma$), consistent with the results from LMXBs.
The result from cap20  indicates that the NS merger events can provide the NS
radius estimates with about 6-7% precision (1$\sigma$). It is also worth to
notice that in Krastev:2018nwr ; Zhang:2018vbw  the authors stated that while
the tidal polarizability $\Lambda$ depends strongly on the details of the
symmetry energy, different trends of $E_{sym}(\rho)$ may lead to very similar
values of $\Lambda$. Thus, measuring $\Lambda$ alone may not uniquely
determine the density dependence of the symmetry energy; both nuclear
laboratory experiments and astrophysical observations are therefore necessary
to break this degeneracy. At similar conclusion was reached in Forbes-
PhysRevD.100.083010  where it was shown how observations of gravitational
waves from binary neutron star mergers can be combined with insights from
nuclear physics to obtain useful constraints on EoS of dense matter between
one and two times the nuclear saturation density.

The middle part of Fig. 18 shows the recent results obtained for three
millisecond pulsars (PSR J0437-4715 gon19 , PSR J0030+0451 Riley:2019yda ;
Miller:2019cac  and PSR J0740+6620 ril21 ; mil21 ) using the thermal emission
pulse profile modeling. The result of gon19 , $R_{NS}=13.6^{+0.9}_{-0.8}$ km,
(M=1.44 $\pm$ 0.07 M⊙) has been obtained from the HST and the ROSAT data,
while the radii and masses of the other two pulsars were obtained using the
NICER data (Riley:2019yda : $R_{NS}=12.71^{+1.14}_{-1.19}$ km,
M=1.34${}^{+0.15}_{-0.16}$ M⊙, Miller:2019cac : $R_{NS}=13.02_{-1.06}^{+1.24}$
km, M=1.44${}_{-0.14}^{+0.15}$ M⊙, ril21 : $R_{NS}=12.39^{+1.30}_{-0.98}$ km,
M=2.072${}_{-0.066}^{+0.067}$ M⊙ and mil21 : $R_{NS}=13.7_{-1.5}^{+2.6}$ km,
M=2.08 $\pm$ 0.07 M⊙). Including additional constraint in the latter result
the authors of mil21  arrive at the value of $R_{NS}=12.35\pm 0.75$ km. It is
more compatible with the result of ril21 , nevertheless the discrepancy
between the raw values indicates the importance of the systematic
uncertainties in MSP analyses, too. The results are compatible with the larger
radius class of LMXBs and with the upper side of the GW170817 results.
Overall, the pulse profile modeling for MSPs can provide the radii with the
precision of about 6-15% (1$\sigma$).

Multiple sources of information related to the symmetry energy from the low
energy nuclear experiments and data, from the high energy experiments as well
as the information from LMXBs, gravitational wave events or MSPs allow to
perform Bayesian type of analyses that use different combinations of these and
other constraints to infer the properties of the EoS and the NS radii. Some
recent results of such analyses are presented in the next to top part of Fig.
18 .

Generally, refs. jiang20 ; lan20 ; die20 ; ess20 ; raa21 ; tan21 ; bis21 ;
alm21 ; rai21 ; Huth:2021bsp  give consistent results, however as might be
expected, the results get easily biased depending on the constraint used. This
can be noticed for the two points from ess20 , where adding the NICER
constraint in addition to the GW one shifts the radius towards higher values.
Similar effect is visible from the results of rai21 , where the lower radius
value is constrained by the LMXB data and the higher value point is
constrained by the NICER data. We also emphasize the role of the heavy-ion
collision (HIC) constraint by presenting the two results of Huth:2021bsp , the
upper point constrained by HIC alone and the lower one constrained in addition
by astrophysical data. These results will be discussed in more detail later
on. The most constrained result, using the astrophysical and HIC information
presents the sharpest radius value so far: $R_{NS}=12.01_{-0.38}^{+0.37}$ km,
with about 3% precision.

Overall, the survey of astrophysical constraints on the $R_{NS}$ shows a
progress in the precision of the data, the quality of the models and in the
data analysis techniques. The uncertainty of inference of the $R_{NS}$ has
been reduced from about 25% down to 6-3% in the latest results. This denotes
the reduction of the uncertainty of $L$ parameter, when derived from R, from
about 100% to 24-12%. The factor of 4 in conversion of the relative
uncertainties comes from the approximate conversion formula (see
Lattimer:2015nhk  eq. (3), tews17  eq. (52) or rai21  eq. (6)), where
$L\propto R^{4}$ and thus $\Delta L/L=4\Delta R/R$. Thus, the accuracy of
derivation of $L$ from the astrophysical measurements of $R_{NS}$ becomes
comparable to the accuracy of $L$ derived from the heavy ion collision data,
as shown in the next paragraph.

The point at the top of Fig. 18 has been obtained in Trautmann:2019gmh  by
converting the symmetry energy slope parameter range from the ASY-EOS heavy-
ion experiment Russotto:2016ucm  to the NS radius range, using the known tight
correlation between the symmetry pressure $p_{0}$ and the radius $R_{1.4}$ of
a canonical neutron star of 1.4 $M_{\odot}$ Lattimer:2015nhk . The
corresponding $R_{NS}$ value amounts to $12.65\pm 0.75$ km . Its 6% precision
has been obtained from the $L$ value measured with about 20% accuracy. While
it may be difficult to improve the current 3% precision in the $R_{NS}$
extraction from the astrophysical data, improving the terrestrial results by
more than a factor of two in the $L$ extraction is within a reach with new
detectors, experimental techniques and models. It is, however, important to
stress that also this result relies on the two assumptions made in the first
UrQMD analysis of ASY-EOS data and discussed in sect. 4.2.2.

As a complement to Fig. 18, Fig. 19 shows the masses and radii for a few
selected individual NS from LMXB and MSP, including the latest NICER results.
In addition, the ASY-EOS radius range (see description of Fig. 18) obtained on
the basis of the heavy ion experiment is presented. The points are put on top
of some “typical” M-R relations for selected EoSs (see dege18 ; sot22 ; sot22d
for a detailed description of the lines). The M-R relations taken from sot22 ;
sot22d  are of particular interest since they provide a direct link to the
symmetry energy slope parameter $L$ values (see the numbers below the labels
in MeV).

Figure 19: Masses and radii for selected individual NS. The data for LMXB were
taken from kim21  (open symbols) and from nat17  (filled blue circle). The
data for millisecond pulsars measured by NICER (red symbols) were taken from
Riley:2019yda  and ril21  for J0030 and J0740 (filled crosses) and from
Miller:2019cac  and mil21  (filled triangles) for the same pulsars,
respectively). The data point for J0437 MSP measured by HST+ROSAT (green
filled square) was taken from gon19 . The radius range labeled as ASY-EOS was
obtained by translating the symmetry energy slope parameter range obtained
from the ASY-EOS heavy-ion experiment Russotto:2016ucm  to the NS radius range
Trautmann:2019gmh . It corresponds to a NS mass of 1.4 M⊙ but has been
displaced arbitrarily in vertical direction for clarity. The labeled gray
lines represent the M-R relations for selected EoSs. The lines for SQM2 and
WFF1 were taken from dege18 . The line for FPS was taken from sot22  and the
remaining lines were taken from sot22d  (see these references for description
of the lines). The numbers below the acronyms of the EoSs (where available)
denote the corresponding values of the symmetry energy $L$ parameter in MeV.

As stated above, the precision of the astrophysical results improves, however,
definitely more precise data points are needed to better constrain the nuclear
EoS. Despite the fact that the elaborate Bayesian techniques allow to combine
different data and models in optimal ways, and provide weights for the EoSs
that permit to construct the most likely M-R relations and to effectively
shrink the measured uncertainties, the presented results call for more
statistics and more coherence in the data.

Restricting to the most recent and most precise results presented in Fig. 19,
one can state that the range of the symmetry energy slope parameters
constrained by the data is still broad, from about 40 to about 110 MeV. Thus,
the uncertainty is still large, of the order of 40-50%.

What follows below is a more detailed discussion of some specific results.

Going back to the information obtained from by GW detection in binary NS
merger events, one of the main result is reported here in fig. 20, adapted
from fig. 2 of LIGOScientific:2018cki , showing the results for the pressure p
as a function of rest mass density of the NS interior as obtained in that
work. That resulted in an estimation of pressure of $21.8^{+16.9}_{-10.6}$
MeV/fm3 at 2 $\rho_{0}$. A second event of binary neutron star merger GW190425
has been more recently reported in LIGOScientific:2020aai ; there a pressure
of 19-80 MeV/fm3 was estimated for neutron star matter at $2\rho_{0}$. From
that figure we can see that the 90$\%$ confidence interval for the pressure of
neutron star matter at $\rho_{0}$ is about between 1.2 and 3.7 MeV/fm3. It
follows that the pressure estimation of 3.4$\pm$0.6 MeV/fm3, obtained in
Russotto:2016ucm  from the UrQMD analysis of the ASY-EOS experimental results,
is consistent with the GW170817, falling in the higher part of 90$\%$
confidence interval; a similar result was obtained by comparing to the results
from neutron-star observations of Ref. Steiner:2012xt , where the ASY-EOS
result was located within the upper half of the 95$\%$ confidence interval
obtained in that study.
In addition, another comparison of results obtained from GW170817 with the one
given by HICs is presented in fig 20. In fact, green and orange points of that
figure present constraints obtained from HIC collisions, fully discussed in
the previous sections of this paper, as deduced assuming a 3$\%$ proton
fraction. All points take into account FOPI constraints for the symmetric
nuclear matter contribution deduced from IQMD and UrQMD analysis of the
experimental data, as it has been validated between 0.7 up to $\approx
3\rho_{0}$. The symmetry energy contribution comes from two different
experimental HIC results, according to densities where they give the highest
precision. The green point at 0.7$\rho_{0}$ and the orange ones between 1 and
2 $\rho_{0}$ use respectively the ALADiN (as recently obtained by A. Le Fèvre
et al. in LeFevre22 ) and ASY-EOS constraints for $E_{sym}$. Apart from the 1
$\rho_{0}$ point, which lies at the high pressure boundary but within the
90$\%$ confidence interval, the other 3 points show a remarkable agreement.
And also the errors appear to be very competitive, providing a proof of the
effectiveness of HIC data.
As a further examples of comparison of the ASY-EOS results with recent results
triggered by multi-messenger astronomy, in Zhang:2018bwq  Zhang and Li
produced a restricted EoS parameter space using observational constraints on
the radius, maximum mass, tidal polarizability and causality condition of
neutron stars, resulting in an estimation of the $E_{sym}$ at $2\rho_{0}$ of
$46.9\pm 10.1$ MeV. It is interesting to note that in a subsequent paper
Zhang:2019fog , the authors show that the observation of a 2.17 $M_{\odot}$
neutron star reduces the error to $\pm 9$ MeV and point out that it is
unlikely that even heavier neutron stars will be observed because the value
2.17 $M_{\odot}$ is already close to the theoretical maximum according to
several studies. In Ref. Xie:2019sqb  a Bayesian analysis of GW170817 and
quiescent low-mass X-ray binaries radii suggested the value of $E_{sym}$ at
$2\rho_{0}$ to lie in the interval $31-51$ MeV. The ASY-EOS constraint of
Russotto:2016ucm  extracted using the UrQMD transport model suggests 51-60 MeV
interval for the $E_{sym}$ at $2\rho_{0}$, and this is in reasonable agreement
with above cited values of the Refs. Zhang:2018bwq ; Zhang:2019fog ;
Xie:2019sqb .

As written above, several studies, combining the constraints on the symmetry
energy from GW, astrophysical observations and the ones obtained in
terrestrial laboratory, have been recently published, showing the importance
and effectiveness of this ”multiple sources” approach. As a relevant example,
we will discuss here the results obtained in a very recent work of S. Huth et
al. Huth:2021bsp .

Figure 20: (adapted from Ref. LIGOScientific:2018cki ) Marginalized posterior
(green bands) and prior (purple dashed lines) for the pressure p as a function
of the rest-mass density $\rho$ of the NS interior as obtained in
LIGOScientific:2018cki , with superimposed (orange and green points)
constraints obtained from HICs discussed in this paper as deduced assuming a
3$\%$ proton fraction. All the points take into account FOPI constraints for
symmetric nuclear matter from both IQMD and UrQMD analyses, while the green
one at 0.7$\rho_{0}$ takes into account also the ALADiN constraint for
$E_{sym}$ at sub-saturation densities and the orange ones in the region
between 1 and 2 $\rho_{0}$ also the ASY-EOS constraint for $E_{sym}$. The dark
(light) shaded green regions corresponds to the 50$\%$ (90$\%$) posterior
credible level and the purple dashed lines show the 90$\%$ prior credible
interval. Vertical lines correspond to once, twice, and six times the nuclear
saturation density. Plotted in addition in gray are some representative EoS
models of Wiringa et al. Wiringa:1988tp , Akmal et al. Akmal:1998cf  and
Lackey et al Lackey:2005tk ; see fig. 2 of Ref. LIGOScientific:2018cki  for
more details.  Figure 21: (reprinted from Ref. Huth:2021bsp  under permission)
Evolution of the pressure as a function of baryon number density for the EoS
prior (A, gray), when including only data from multi-messenger neutron-star
observations (B, green), when including only HIC data (C, orange), and when
combining both (D, blue). The shading corresponds to the 95% and 68% credible
intervals (lightest to darkest). The impact of the HIC experimental constraint
(HIC Data, purple lines at 95% and 68%) on the EoS is shown in panel C. In
panels (B) through (D), the 95% prior bound is also shown for comparison (gray
dashed lines). Panels (E) and (F) show the distributions for the pressure at,
respectively, 1.5 and 2.5 $\rho_{0}$ at the different stages (lines colour
corresponds to the one used in the other four panels) of the analysis, with
the combined multi-messenger+HIC region shaded in light-blue. See Ref.
Huth:2021bsp  for more details.

In detail, Ref. Huth:2021bsp  uses Bayesian inference to combine data from
astrophysical multi-messenger observations of neutron stars and HICs, as
measured in FOPI and ASY-EOS experiments, with microscopic nuclear theory
calculations, to improve the constraints of dense matter. The authors start
from a ”prior” EoS, as determined by using local $\chi EFT$ interactions, as
shown in panel A of Fig. 21. The large uncertainties for the obtained EoS at
higher densities are then partially reduced by adding constraints from
astrophysics observations and HIC data. To that end, mass measurements of
massive neutron stars PSRJ0348+4042 and PSR J1614-2230, information from X-ray
pulse-profile modelling of PSR J0030+0451 and PSR J0740+6620 as obtained by
NICER and X-ray Multi-Mirror Mission (XMM-Newton) and information from the two
neutron-star mergers GW170817 and GW190425, observed by Advanced Ligo and
Advanced Virgo, are used to get multi-messenger astrophysics constraint on the
high-density neutron star EoS and reduce its uncertainties, as shown in panel
B of Fig. 21. In addition, HIC data from FOPI and ASY-EOS experiments are used
to impose a HIC constraint on the prior EoS, as shown in panel C of Fig. 21.
Finally, both astrophysical and HIC constraints are combined to get a final
EoS constraint, as shown in panel D of Fig. 21. The final result for the
radius of a 1.4 solar mass neutron stars is $12.01_{-0.38}^{+0.37}$ km at 68%
uncertainty. In panel (E) it is clearly seen how the inclusion of HIC data
leads to an increase in the pressure in the 1.5 $\rho_{0}$ region, where the
HIC experimental sensitivity is higher, shifting the neutron-star radii
towards larger values, consistent with recent NICER observations mil21 ; raa21
. Instead, pressure in the 2.5 $\rho_{0}$ region, shown in pane (F), is
essentially determined by the astrophysical observations, since the HIC
sensitivity at those densities is very small. However, this finding shows that
constraints from HIC experiments show a remarkable consistency with multi-
messenger observations and provide complementary information on nuclear matter
at intermediate densities. The work shows how joint analyses can shed light on
the properties of neutron-rich supra-nuclear matter over the density range
probed in neutron stars. The work also suggests that a model independent
analysis of HIC data and a reduction of experimental uncertainty have a great
potential to provide EoS information complementary to astrophysics; the larger
uncertainties in the EoS above 1.5 $\rho_{0}$, as clearly seen in Fig. 21,
indicates the importance of precise measurements of the symmetry energy in
that high-dense regime. This region of densities is also the one of most
importance for the modelling of neutron-star and the one where model
uncertainties are larger. In addition, constraints from HICs, as shown in this
paper, are scarce in this density region. This highlights the importance of
measurements of the symmetry energy around 1.5-2.5 $\rho_{0}$; plans in that
direction will be discussed in the next section.

Figure 22: (reprinted from Ref. Ghosh:2021bvw  under permission) Symmetry
energy, minus its value at saturation densities, as a function of reduced
density (nb is the baryon density and n0 indicates the nuclear saturation
density), of the prior EoSs (dashed red curves), used by Ghosh et al. in a
Bayesian scheme taking into account constraints from $\chi EFT$, observations
of mass and radius of neutron-stars and GW observations of binary neutron star
merger, lying inside the band allowed by the ASY-EOS results in the range of
1.1-2.0 $\rho_{0}$. See Ref. Ghosh:2021bvw  for more details.

In another recent paper by Ghosh et al. Ghosh:2021bvw , ASY-EOS data have been
again used to investigate the use of multi-physics constraints for studying
the neutron-matter EoS. Similarly to the work of Ref. Huth:2021bsp , Ghosh et
al. used a Bayesian scheme to investigate the EoS, with a Relativistic Mean
Field approach to determine the prior. The constraints taken into account in
the main analysis, in order to determine the posterior, were chiral effective
field theory results for low densities and multi-messenger astrophysical
observations for high densities. The HIC results, from KaoS, FOPI and ASY-EOS
experiments, were taken into accounts as constraints for intermediate
densities, but they were treated differently from $\chi EFT$ and astrophysical
ones, because of the problem of their model-dependence. As an example, Fig. 22
shows the set of selected EoSs (dashed red curves) when requiring agreement to
the ASY-EOS results in the range of 1.1-2.0 $\rho_{0}$ (blue band), amounting
to about $40\%$ of the prior EoSs. Posteriors with and without HIC results
were extracted and compared, finding few differences between the two cases,
but considering implications of HIC results on final EoSs with caution.
However, also this work emphasizes that precise and model-independent HIC data
can play a relevant role in determining the neutron-star EoS, especially in
the region of 1-2 $\rho_{0}$. Comparing the works of Huth et al. and Ghosh et
al., the former uses a softer EoS from $\chi EFT$ up to 1.5 $\rho_{0}$,
getting stiffer when including HIC constraints, with the final results being
mainly ruled by $E_{sym}$; instead the RMF model of Ghosh et al. favors
isospin-symmetric matter at high densities, and the final results is less
ruled by the $E_{sym}$ contribution.

The results on multi-physics analyses discussed here are surely not exhaustive
for what concerns recent advances in neutron-stars EoS determination. As we
have seen this is a very hot topic and new results are appearing at short
intervals. However, the works discussed emphasize that HIC data, such as those
made available by the FOPI and ASY-EOS experiment, can not only give
laboratory constraints on EoS asymmetric matter, to be compared with the
astrophysical ones, but can be used together with the latter to provide better
constraints on EoS. The development of model independent analyses of the HIC
data and new and more precise experiments able to investigate regions of high-
densities, around 2 $\rho_{0}$, thus appears to be mandatory.

###  4.4 Perspectives

In this section we will outline some possible future developments that
represent natural follow-ups of studies presented in this paper, with the
caveat of not representing an exhaustive report on future possibilities.

####  4.4.1 Pion studies

The up-to-date studies of the EoS of nuclear matter using pion production
close to threshold have relied on transport models that describe particle
emission exclusively via nucleon resonance excitation. This mechanism induces
a strong dependence of multiplicities and single ratios on the poorly known
in-medium resonance potentials, reducing the sensitivity to the symmetry
energy. The impact of the assumption of resonance production grows weaker for
energetic pions and, consequently, the study of the symmetry energy was
possible using the high transverse momentum region of pion spectra measured by
the S$\pi$RIT Collaboration SRIT:2021gcy . It has however been shown, using
microscopical models, that close to threshold pions are predominantly produced
in non-resonant nucleon-nucleon collisions Engel:1996ic ; Shyam:1996id .
Including such a production mechanism in transport models, while challenging
from a technical perspective, would reduce the sensitivity to resonance
potentials considerably. Consequently, the lower transverse momentum parts of
pion spectra could be used for more accurate constraints for the symmetry
energy above saturation.

Experimental data for pion production in Au+Au at an impact energy of 1.23
GeV/nucleon have been published recently by the HADES Collaboration
HADES:2020ver . Comparisons with five transport models for rapidity and
transverse mass spectra were also provided, revealing important discrepancies
among theoretical models but also deviations from experimental values. The
reasons for these difficulties may originate in different momentum dependent
interactions and resonance potentials considered in each model. Furthermore,
the inclusion of all relevant decay channels of heavier resonances, in
particular the two-pion ones, may be of relevance for a proper description of
rapidity spectra.

A recent study, employing the dcQMD transport model, has shown that the high
transverse momentum part of the pion single ratio is sensitive to the symmetry
energy even at impact energies far above threshold Cozma:2021tfu . The high
accuracy of the HADES pion transverse mass spectra opens the possibility of
studying the symmetry energy at densities above twice saturation. Preliminary
estimations using the dcQMD model dcQMD22  suggest a sensitivity of the pion
single ratio to the asy-EoS stiffness of the order of 30$\%$, which is
significantly larger than the experimental accuracy of single ratios of about
$\pm 2\%$. It is expected that data for Ag+Ag collisions at 1.58 AGeV will be
soon made available by the HADES collaboration. A comparison of the
predictions of four transport models (UrQMD, PHSD, PHQMD and SMASH) concerning
integrated multiplicities, rapidity distributions and transverse mass spectra
for proton, pions and other strange particles was presented in Ref.
Reichert:2021ljd . Despite different treatments of the HIC dynamics among the
models, consistent results were obtained for the bulk of investigated hadrons,
except for the $\Xi^{-}$ case. However, at such impact energies the pion
single ratio, such as other observables related to particle productions, is
sensitive to resonance potentials over the entire transverse mass range,
making such a study challenging though still feasible. For a successful
attempt, a transport model that can realistically describe pion production
from threshold up into the 1-2 GeV/nucleon region is required.

On the experimental side, the S$\pi$RIT Collaboration aims at enlarging the
experimental database for pion production close to threshold by measuring
136Xe+124Sn at 334 MeV/nucleon (RIKEN) and 200 MeV/nucleon (FRIB). A low
energy beam scan (200-800 MeV/nucleon) including Au+Au collisions currently
planned by the HADES Collaboration, would improve the situation considerably
if an accuracy comparable to one achieved for the same system at 1.23
GeV/nucleon can be reached. It can thus be foreseen that in the mid-term
future a rather accurate determination of the symmetry energy at densities
close to 1.5$\rho_{0}$ can be accomplished from studying pion production close
to threshold. A proper understanding of pion production above 1.0 GeV/nucleon
would add constraints in the 2.0-2.5$\rho_{0}$ density range.

####  4.4.2 Elliptic flow studies

The ASY-EOS experimental results proved the effectiveness of the
$v_{2}^{n}/v_{2}^{ch}$ ratio in constraining the high-density behaviour of the
$E_{sym}$, and validated the experimental approach used there. But, even if
the experiment was surely successful in reaching its main aim, two drawbacks
have to be noticed:

  * •

the problems of LAND electronics, briefly discussed here, but extensively
reported in Russotto:2016ucm , did not allow to get a better statistics and,
above all, did not allow neither discrimination of charged particles, nor to
implement analysis to separate hydrogen isotopes and extract the proton flow;

  * •

it was conceived as a first/exploratory experiment for a future extensive
campaign. In fact, although the ASY-EOS determination of $E_{sym}$ is
estimated to have reached the supra-saturation density region, which was
already a unique achievement, there remains a strong need for constraining it
further, with an emphasis on the density region around $\sim$2 $\rho_{0}$.
SIS@GSI provides a unique tool to probe such densities with HICs.
Unfortunately, because of the upgrade phase ongoing at GSI/FAIR in the last
years, strongly reducing beam-time opportunities for users and of the recent
difficulties related to the pandemic a new experimental campaign, expected to
be the natural follow-up of the ASY-EOS experiment, has still not been firmly
planned.

The next desirable step is to measure the elliptic flow of neutron, proton and
light clusters in Au+Au collisions over a wide range of beam energies from 250
to 1000-1500 MeV/nucleon, in order to provide tighter constraints on the slope
parameter $L$ and new limits on Ksym, the currently poorly constrained
symmetry energy curvature parameter. For that a new campaign Russotto:2021mpu
has been proposed representing a unique set of measurements, presently
possible only at the GSI/FAIR facility, in view of the available range of beam
energies and the existing instrumentation.
A relevant novelty of the proposed experiment is the direct measurement of n/p
elliptic flow ratio, made possible by the capabilities of the NeuLAND
detector. Some details about this new detector are presented below. We remind
that, as presented in Fig. 19 of Russotto:2016ucm , according to the TuQMD
model for the Au+Au case at 400 MeV/nucleon, the v2-ratio of neutrons with
respect to charged-particles or hydrogen is mainly sensitive to a region
centred slightly above saturation density, while the n/p ratio sensitivity is
centred around 1.4$\rho_{0}$. Thus, the direct measurement of n/p ratio,
instead of n/charged particles ratio, will enable probing of higher density
region than was done before, at the same incident energy of 400 MeV/nucleon.
This by itself will be a significant achievement. In addition, the n/p
elliptic-flow ratio is less affected by the procedure used to build the
clusters, the so called clusterization algorithms, from the final stage of the
HIC as simulated by the theoretical models, allowing thus the reduction of
model dependence related to that stage of the analysis.
The second important feature of the proposed campaign is the measurement of
excitation function in a relevant beam energy range, from 250 to 1000
MeV/nucleon. Microscopic transport calculations predict that for a short time
($\sim$20 fm/c) densities of up to $\sim$3 times the saturation density can be
reached in the central zone of a HIC even at moderate incident energies of
$\sim$1 AGeV Li:2002yda . This is shown in Fig. 23, presenting the time
evolution of the central baryon density in 132Sn+124Sn collisions at beam
energies from 200 to 2000 MeV/nucleon for b = 1 fm, as predicted by the
hadronic transport model of Ref. Li:2002yda  using a stiff (solid curves) and
a super-soft (dashed curves) $E_{sym}$. One can notice that the supersoft
$E_{sym}$ leads to slightly higher densities of about 10–15$\%$, due to the
softening of the nuclear EoS with the supersoft $E_{sym}$ relative to the
stiff one. Additional arguments related to the maximum probed density are
already given in the fig. 7. It shows that the maximum densities in the
innermost center of the collision may reach up to 3.5$\rho_{0}$ at 1.5 AGeV,
while the densities probed by the mid-rapidity protons during the heavy ion
collision may extend up to twice the saturation density, irrespective of the
transverse momentum cut LeFevre:2015paj . These IQMD calculations were used to
estimate the range of densities to which elliptic flow observable is most
sensitive to, that is the ”force-weighted” average density shown as error bars
in fig. 7. It shows that for Au+Au collisions at 1 AGeV, the typical densities
directly influencing the flow, through the impact of the mean-field on the
time evolution of the reaction, span between $\rho_{0}$ and 2.2$\rho_{0}$.
Thus, according to the models, raising the beam energy allows to probe
$E_{sym}$ at higher densities. Nevertheless, as demonstrated in
LeFevre:2016vpp , and already mentioned in Sect. 2.2, higher densities reached
in the fireball play also an indirect role in the strength of the elliptic
flow, because they determine how fast the later expansion of the fireball will
occur and modify the elliptic flow by the interplay between expanding fireball
and flowing away spectators. Therefore, at 1 AGeV, some influence of the EoS
at 3$\rho_{0}$ on $v_{2}$ cannot be ruled out. In the most compressed phase of
the collision, it is expected that due to the symmetry energy, the neutron
part in the fireball will expand faster than that of the protons. Thus they
will interact differently with the spectators, which will result in a
different elliptic flow. However, as the incident energy is increased the
fraction of nucleons excited into baryonic resonances, mainly $\Delta(1232)$
at energies of 1 GeV/nucleon and below, in the highly compressed phase of the
collision reaches values in the neighborhood of $20\%$ PhysRevC.47.R2467 .
This has a potential impact on the time evolution of the reaction that may
leave a comparable imprint on the $E_{sym}$ around 2 $\rho_{0}$, depending on
the chosen observable. Thus, one should be aware that the highest density
reached during a HIC is not necessarily equivalent to the density that can
effectively be probed, i.e. without the knowledge of additional systematic
theoretical uncertainties, for the purpose of constraining the high density
dependence of the symmetry energy. The proposed measurement could effectively
provide powerful data to explore also these delicate aspects.
In addition, with an unambiguous proton identification one should be able not
only to constrain the slope of $E_{sym}$, $L$, but also its curvature,
$K_{sym}$. The latter is the parameter least constrained experimentally and
theoretically so far. In Fig. 13 of MDCozEPJA18  it was shown that for Au+Au
semi-central collisions neutron-proton elliptic flow ratio sensitivities to
$L$ and $K_{sym}$ reach maxima at 600 and 250 MeV/nucleon, respectively, thus
at quite different energies. Note that the potential terms that are
proportional to $L$ and $K_{sym}$ have different dependencies on density and,
consequently, the forces generated by these two terms attain their maximum
effectiveness at different regions of density; in a simplified way, the $L$
term is proportional to the isospin asymmetry $\delta$, while the $K_{sym}$
term to $\delta\rho$. This explains why the maximum sensitivity for $L$ and
$K_{sym}$ do not occur for the same incident energies. This is another strong
reason for measuring excitation functions of neutron-proton elliptic flow
observable.

It was also proposed to measure, at the same time, yields and isotopic
compositions of clusters. This is of importance for advanced tuning of
clusterization algorithms used in the transport models in order to get more
realistic predictions. Ideally, cluster degrees of freedom should be included
explicitly in transport models as they have direct impact on predictions of
neutron-to-proton yields and flows, with clusters acting as absorbers of
otherwise free nucleons, as well as pionic observables Ikeno:2016xpr .
Nevertheless, such approaches are very challenging from a technical point of
view. As a consequence, only the effects of light clusters, up to at most 4He,
have been included explicitly in the collision integral of only a few existing
transport models Danielewicz:1991dh ; Ono:2013aaa . Alternative methods, such
as coalescence invariant proton or neutron spectra, which are defined as
weighted sums of proton or neutron content of light clusters with $Z\leq$2,
have been proposed Famiano:2006rb  and used successfully Coupland-
PhysRevC.94.011601 ; morf19  in studying the EoS. It is worth to notice that
transport models also predict a sensitivity of the ratio of yields and flows
of light isobar nuclei to the high density behavior of the symmetry energy
Famiano:2006rb ; Giordano:2010pv ; Zhang:2014sva ; Coupland-PhysRevC.94.011601
, that could be measured and used as an additional probe of the EoS. It
follows that precise data on the inter-related phenomena of clustering and
neutron and proton emissions as well as correlations between them could
represent strong constraints to nuclear transport theories.

Figure 23: (adapted from Ref. Li:2002yda  under permission) Evolution of the
central baryon density in 132Sn+124Sn collisions at beam energies from 200 to
2000 MeV/nucleon for b = 1 fm, as predicted by the hadronic transport model of
Li:2002yda  using a stiff (solid curves) and a super-soft (dashed curves)
$E_{sym}$.

Figure 24: Left panel: Excitation functions of neutron-to-proton elliptic flow
ratios, $v2_{n}/v2_{p}$, at mid-rapidity for semi-central Au+Au, 132Sn+124Sn,
124Sn+24Sn and Pb+Pb collisions, as predicted by the UrQMD model for stiff and
soft $E_{sym}(\rho)$. Right panel: differences between the stiff and soft
results, providing a quantitative evaluation of the $v2_{n}/v2_{p}$
sensitivity to the symmetry energy.

Simulations of semi-central Au+Au collisions at 250, 400, 600, 800 and 1000
MeV/nucleon and, for comparison, neutron rich 132Sn+124Sn, 124Sn+124Sn and
Pb+Pb systems at 400, 600 and 800 MeV/nucleon have been carried out by using
the same version of the UrQMD transport model used in Russotto:2016ucm . The
neutron-to-proton elliptic flow ratio, $v_{2}^{n}/v_{2}^{p}$, at mid-rapidity
($0.4<y_{lab}/y_{proj}<0.6$), with a stiff ($\gamma$=1.5) and a soft
($\gamma$=0.5) parametrizations of the potential part of the $E_{sym}$ for
semi-central $(b_{0}<0.54)$ collisions is shown, as a function of the incident
beam energy, in the left panel of Fig. 24. The difference of predictions, made
by using soft and stiff choices for the symmetry energy, for this quantity
provides a measure of the sensitivity of the proposed observable to the
symmetry energy, and is shown in the right panel of the same figure. The
obtained sensitivity is at the level of $\sim$30% at 250 and 400 MeV/nucleon
decreasing at higher beam energies. This reflects the fact that the mean-field
contribution decreases at higher energies and the two-body collisions start to
dominate. There, multiple rescattering of the nucleons in the more dense
interaction region and the stronger contribution of $\Delta$ resonance
excitation, with related opening of the pionic degree of freedom, start to
play the major role in determining the asymptotic pattern of the neutron ad
proton distributions, reducing the strength of the symmetry energy effect.
Nevertheless, up to 1 GeV/nucleon the sensitivity of the proposed observable
is $\sim$15%, while a measurement can easily reach a $\sim$5% accuracy,
allowing a discrimination between stiff and soft choices. A similar conclusion
can be drawn from a recent paper WANG2020135249  where UrQMD simulations with
11 selected Skyrme forces were performed. In Fig. 3 of that paper, the ratio
between the elliptic flow parameter of free neutrons and protons was plotted
as a function of the slope parameter $L$ for Au+Au collision from 400 to 1000
MeV/nucleon. The highest sensitivity was obtained at 400 MeV/nucleon, while
the sensitivity at 1000 MeV/nucleon was reduced by a factor 2. According to
the results shown in Fig. 24, the sensitivity of the Au+Au systems is very
similar to the one of the other neutron rich systems, as in the case of 132Sn
radioactive ion beams. Heavier systems, such as Pb and Au, allow for smaller
statistical uncertainties given the highest neutron and proton multiplicities
per event and an equal number of simulated events for each system. This
suggest that Au+Au choice still represents one of the best choice for future
experiments, given also the detailed studies available in literature, as the
ones performed by using the FOPI detector, to be used as reference data for
check and comparison. It is also important to stress the differences in trends
(slopes) observed in left panel of Fig. 24. For the soft $E_{sym}$, the ratios
increase with the energy while, for the stiff EoS the trend is opposite. This
proves the needs for measuring the excitation functions of these observable.
Similar sensitivity studies have been performed using two additional transport
models, IQMD Hartnack:1997ez  and TuQMD Cozma:2011nr , providing similarly
strong arguments for extending measurement of flow observable toward higher
energies. In a preliminary study, shown in sect. 4.2.2, the constraints
extracted for the slope parameter $L$ by comparing predictions of each of the
three transport models to ASY-EOS data, were observed to agree with one
another at a $\sim$5-15$\%$ level. Larger discrepancies between model
predictions have been observed at higher impact energies, requiring further
investigations. It is clear that the divergence of results among the different
transport codes constitutes a relevant issue to be investigated and solved,
before unambiguous and accurate information regarding the EoS at high-
densities can be extracted from experimental data. A collaborative effort
devoted to understanding differences between model predictions for
neutron/proton elliptic flow like observables, in a similar fashion to the one
initiated for pionic observables Xu:2016lue ; Zhang:2017esm ; Ono:2019ndq ;
Colonna:2021xuh , would be extremely beneficial/relevant for a positive
outcome of the ASY-EOS experimental program.
A strong argument in favor of a new experimental measurement is provided by
the possibility of relevant improvements in the experimental setup. In fact,
as a relevant novelty, the R3B collaboration at GSI has finalized in these
years the building of a new next-generation neutron detector, NeuLAND
R:2021lxa , being an improved device with respect to the old LAND detector. In
fact, the NeuLAND detector features a higher detection efficiency, a higher
resolution, and a larger multi-neutron-hit resolving power. This is achieved
by a highly granular design of plastic scintillators, avoiding insensitive
converter material. The final implementation of the detector will consist of
3000 individual sub-modules with a size of 5$\times$5$\times$250 cm3, arranged
in 30 double planes with 100 sub-modules, with alternate horizontal and
vertical positioning, providing an active face size of 250$\times$250 cm2 and
a total depth of 3 m. Given the reached performance, the NeuLAND detector will
give a unique opportunity to measure the neutron and LCPs in the same angular
regions. The outstanding calorimetric properties of NeuLAND will allow protons
and other hydrogen isotopes to be relatively well separated, and will give
access to the neutron vs proton observables. The NeuLAND demonstrator was a
part of the S$\pi$IRIT experiment SpRIT:2020blg  carried out at RIKEN in 2016
and the capability of resolving both protons and neutrons was clearly
demonstrated there. The identification plot of hydrogen isotopes in the
demonstrator (4 double planes, 40 cm total thickness) is presented in the left
panel of Fig. 25. The p, d, t lines are clearly resolved up to the punch-
through energy (about 260 MeV for protons) above which the characteristic
back-bendings occur. The 13 double planes of NeuLAND now available, resulting
in a total depth of 130 cm, will assure stopping of protons up to about 500
MeV. A simulated identification plot for the Au+Au collisions at 400
MeV/nucleon is presented in the right panel of Fig. 25. Indeed, no punch-
through segments are observed at this energy and the p, d, t lines clearly
stick out of the secondary reaction and multi-hit background. The simulations
include tracking, the secondary reaction losses, multiple Coulomb scattering,
light propagation in plastic scintillators and quenching effects. The
estimated efficiency for proton identification amounts to about 64% at 200 MeV
and 36% at 400 MeV. Taking into account the thickness of the NeuLAND
calorimeter and the secondary reaction and scattering probability, the
estimated efficiencies are still impressive. The one neutron interaction
probability is about 70% at 400MeV. Taking into account also the
reconstruction efficiency a five-neutron event is recognized with correct
neutron multiplicity with a probability of about 20 to 30% (200 to 1000 MeV).

Figure 25: Left panel: Time-of-flight (in ns) vs total light deposit in the
NeuLAND demonstrator as obtained from Sn+Sn @ 270 MeV/nucleon measurement at
RIKEN; Right panel: Time-of-flight vs total light collected in 12 double-
planes of NeuLAND from the simulation of Au+Au reaction at 400 MeV/nucleon. In
both panels, separated ridges corresponding to proton, deuteron and triton can
be clearly seen. In the simulation, also the low populated He region is
recognizable (courtesy of I. Gašparić).

Another advantage of a new experiment will be obtained by using a new device,
the KRAkow Barrel (KRAB), explicitly developed in view of future flow
measurements, Fig. 26. The KRAB, covering polar angles from $30^{\circ}$ to
$165^{\circ}$ with $\sim$87% geometrical efficiency and with $\sim$5% multi-
hit probability, will be used similarly to the MicroBall detector of the ASY-
EOS experiment, but will be also able to provide a fast trigger signal based
on the multiplicity threshold as well as, thanks to its high segmentation, a
very precise azimuthal distributions for charged particles in the c.m.
backward hemisphere, indispensable for high resolution estimates of the
reaction plane. The main features of the KRAB detector are: 5 rings of
4$\times$4 mm2 fast scintillating fibers read out by the SiPMs. This new
device will be sufficiently large for radioactive beams and sufficiently small
and lightweight in order not to disturb neutrons, having the minimum and
maximum internal radii of 6.9 and 11.5 cm and a length of $\sim$50 cm. It will
consist of 4$\times$160 segments in forward rings and 96 segments in the
backward ring with a total of 736 channels. It will then be able to produce a
fast trigger based on total multiplicity in an angular region of
$\theta>30^{\circ}$, where a strong correlation between the multiplicity and
the magnitude of the impact parameter is expected from model predictions. It
is expected that KRAB, as NeuLAND, should greatly improve the quality of the
data with respect to the previous ASY-EOS experiment.

Figure 26 presents the design and the current status of construction of the
KRAB detector.

Figure 26: Left panel: The design of the KRAB detector including the CITIROC
boards; Right panel: The actual view of the device as of the end of March
2022.

In addition to these new devices, the set-up will be completed by the use of
CHIMERA CsI(Tl) rings and of a wall of plastic scintillators, like the R3B New
Time-of-Flight Wall (TOFD) detector of the R3B collaboration, at small angles,
providing information needed to characterize the reaction centrality and
reaction plane orientation, and of KraTTA hodoscope at mid-rapidity, placed
symmetrically to the NeuLAND detector, but on the opposite side, to measure
yields and flows of proton and light clusters. Using the above described
devices, a set-up similar to the ones employed by the ASY-EOS experiment can
be configured, but with overall improved performances, being, as said before,
a strong reason motivating future experimental efforts.

##  5 Conclusions

The study of the nuclear matter EoS is an important topic in modern nuclear
physics. It has triggered a considerable amount of works in the last decades
on, both, theoretical and experimental sides. Among the latter, studies
performed by means of HICs play a relevant role, being able to provide very
accurate information.

In this review, we have focused on that subset of results concerning the high-
density part of the EoS, carried out at the beam energies around the 1
GeV/nucleon as typical of the SIS18 accelerator of the GSI laboratory. This
field of research has also seen in the last years the arrival of new and very
interesting data coming from astrophysical observations of neutron stars by
means of gravitational waves and satellite-based X-rays observations.

The isospin-symmetric contribution of the EoS has been extensively studied by
means of HICs. Studies on kaon production showed high-sensitivity and
stability against model ingredients, and allowed to get clear indications in
favour of a soft EoS. In contrast, pion studies were not effective in
constraining the isospin-symmetric contribution of the EoS, due to the
difficulties of the transport models to exhaustively reproduce different
aspects of the experimental data. The direced flow measurements carried out at
LBL, AGS and BNL and at energies between 1 and 10 GeV/nucleon gave further
indications for a soft EoS at densities around 3-4 $\rho_{0}$. The softness of
the EoS at densities around 2$\rho_{0}$ was precisely confirmed by the FOPI
experiment, by means of extensive studies on the elliptic flows of protons and
light clusters for Au+Au collisions at several beam energies between 400 and
1500 MeV/nucleon. Values of $K_{0}$ compressibility modulus of in the range
190-220 MeV with uncertainties of the order of $\sim 20\%$ were obtained and
can be considered as a well established result of HIC studies.

In contrast, a full knowledge of the isospin dependent component of the EoS
with a comparable accuracy requires additional effort, both on the
experimental and theoretical sides. Most of the studies devoted to investigate
the symmetry energy at densities below the saturation one show a significant
agreement for a $E_{sym}$ of 23-26 MeV around 0.7 $\rho_{0}$ and values of the
$E_{sym}$ slope around $\rho_{0}$, $L$, of $\sim 70\pm 20$ MeV. The recent
work of Lynch and Tsang lyn21 , evidencing the importance of using each
$E_{sym}$ constraints at the specific sensitive density, has given a further
strong indication in favour of symmetry energy being well constrained from
nuclear structure and reactions studies below saturation density. Latest PREX-
II results suggests a stiffer slope of the symmetry energy at $\rho_{0}$,
thus, in tension with the current understanding of the EoS. And this is
currently a matter of debate. However, further works will be advantageous to
get an, even better, coherent and exhaustive description of the symmetry
energy at sub-saturation densities. Additional work on models is required in
order to get a complete and coherent description of the several reaction
observables and to reach a better agreement among constraints obtained by
using different observables and/or models. Instead, only few works explicitly
constraining symmetry energy above the saturation densities by using HIC data
have been performed so far.

First studies based on integrated pion multiplicity measured by the FOPI
experiment close to the threshold reached contradictory results when using
different models, which could be often traced back to different treatments of
in-medium pion production. Later, works of the S$\pi$RIT collaboration and the
code-comparison project have allowed to get a better understanding of the
requirements for realistic pion production modelling and to select
observables, related to the high-kinetic energy part of the pion spectra, able
to provide reliable constraint of the symmetry energy at high densities.
Recent results, interpreting the high transverse momentum tail of the pion
single ratio spectra by using the dcQMD trasport model, indicated $L=79.9\pm
37.6$ MeV and a sensitivity around 1.5 $\rho_{0}$. A confirmation of this
result using additional transport models is of great importance and subject of
current efforts of the code comparison collaboration.

Elliptic flow studies, based on the ratio of neutron-to-proton and neutron-to-
charged particles, proved to be very effective for investigating the high-
density behaviour of symmetry energy. Transport model simulation with QMD
models indicated the robustness of the elliptic flow ratio observable with
respect to many ingredients, such as compressibility of symmetric matter EoS,
width of wave-packet, implementation of the optical potential, of the
transport models. Extensive Studies using BUU or SMF type of models compared
to the existing set of data have not yet been published, but are necessary for
a thorough study of possible model dependence of obtained results. First
studies based on Au+Au data at 400 MeV/nucleon of FOPI-LAND experiment,
employing both neutron-to-proton and neutron-to-hydrogen flow ratios,
suggested $L=83\pm 26$ MeV when using the UrQMD transport model, and $L=122\pm
57$ MeV and $K_{sym}=229\pm 363$ MeV when using the TuQMD model. The two
results, despite the different approaches used in the two models, have a
substantial overlap. A new data set for the same system and energy, with a
better statistics, was made available with the ASY-EOS experiment. A first
study based on the neutron-to-charged particles elliptic flow ratio and the
UrQMD model indicated $L=72\pm 13$ MeV. The effectiveness in testing the
symmetry energy at densities above the saturation one was also demonstrated.
The analysis based on the TuQMD model suggested $L$=85$~{}\pm$22(exp) $\pm$
20(th) $\pm$ 12(sys) MeV and $K_{sym}$=96 $\pm$315(exp) $\pm$ 170(th) $\pm$
166(sys) MeV. In this paper we have also presented a comparison of results
obtained by using IQMD, TuQMD and a new UrQMD version, using a common analysis
scheme, investigating the dependence of the slope of the symmetry energy on
the $E_{sym,0}$ value. Obtained values for $L$ from the 3 models increase as a
function of $E_{sym,0}$ and show a remarkable agreement. For $E_{sym,0}=32$
MeV the resulting $L$ range is about $70_{-25}^{+35}$ MeV.

ASY-EOS and FOPI data have allowed a HIC based estimation of pressure-density
relation for neutron star matter and radius of a canonical neutron star. An
overview of recent progress in neutron star radius measurement has been given
where the HIC results have been compared to results obtained from low-mass
X-ray binary system, X-ray modelling for millisecond pulsars, gravitational
waves and multiple source analyses. The noticeable progress made in the last
years has been highlighted with the uncertainty being pushed down to the level
of $\sim 5\%$. HIC based results present a remarkable agreement with
astrophysical ones and underline the potential of new and even more accurate
future studies. In addition, a recent study of the EoS of neutron matter,
based on a joint Bayesian analysis of microscopic chiral effective field
calculations, astrophysical constraints and FOPI and ASY-EOS constraints has
been discussed in some detail. This study has given the most accurate radius
value among the reviewed results, $R_{NS}=12.01_{-0.38}^{+0.37}$ km, with
about 3% precision. This study has shown also that HIC studies play a relevant
role in determining the neutron matter EoS in the multi-messenger astronomy
era. The need of new heavy-ion studies has been argued, probing the region
around 1-2 $\rho_{0}$, where theoretical model uncertainties are larger, where
laboratory constraints are still scarce, and where HICs have a large impact on
the neutron matter EoS.

In view of that, the potential of new experimental measurements has been
discussed. For the case of pion production, more accurate experimental
measurements close to production threshold are planned at several laboratories
worldwide. Regarding flow, needs of new measurements at energies different
from the 400 MeV/nucleon explored until now, would allow to extend the
symmetry energy studies to higher densities, constraining both slope and
curvature with improved precision.

It is expected that the huge amount of work in this field of research, of
which a small but relevant part has been reviewed in this paper, will allow to
get even more precise constraints on the EoS in the forthcoming years.

###### Acknowledgements.

Work supported by the Polish National Science Centre, contract No.
UMO-2017/25/B/ST2/02550, by the French-German Collaboration Agreements 03-45
and 13-70 between IN2P3 - DSM/CEA and GSI.
The authors wish to thank I. Gašparić (RBI, Zagreb, Croatia) for providing
NeuLAND plots in sect. 4.4.2, QingFeng Li (Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou, China and School of Science, Huzhou
University, Huzhou, China) and Yongjia Wang (School of Science, Huzhou
University, Huzhou, China) for providing through the years UrQMD calculations,
and Wolfgang Trautmann (GSI, Darmstadt, Germany) for his fundamental role in
the whole ASY-EOS project. We wish also to thank the Referee for the critical
reading of the manuscript and valuable suggestions.

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