E Préau

A Pascal

† J Novak

M Oertel

What can be learned from a proto-neutron star's mass and radius?

Keywords:

We make extensive numerical studies of masses and radii of proto-neutron stars during the first second after their birth in core-collapse supernova events. We use a quasi-static approach for the computation of proto-neutron star structure, built on parameterized entropy and electron fraction profiles, that are then evolved with neutrino cooling processes. We vary the equation of state of nuclear matter, the proto-neutron star mass and the parameters of the initial profiles, to take into account our ignorance of the supernova progenitor properties. We show that if masses and radii of a proto-neutron star can be determined in the first second after the birth, e.g. from gravitational wave emission, no information could be obtained on the corresponding cold neutron star and therefore on the cold nuclear equation of state. Similarly, it seems unlikely that any property of the proto-neutron star equation of state (hot and not beta-equilibrated) could be determined either, mostly due to the lack of information on the entropy, or equivalently temperature, distribution in such objects.

INTRODUCTION

Neutron stars are fascinating objects as their understanding combines many aspects of physics under extreme conditions: relativistic gravitational field and matter at supra-nuclear densities, to list the most notable ones (for a more detailed description, see e.g. [START_REF] Haensel | Neutron Stars 1 : Equation of State and Structure[END_REF]. They can therefore be used as astrophysical laboratories to test our understanding of the laws of physics under conditions which cannot be reached by Earth-based experiments. Whereas the theory of General Relativity is tested to a high accuracy within the neutron star context (Wex & Kramer 2020;Kramer et al. 2006), the situation is not so favorable when probing the properties of nuclear matter. In this respect, one of the major objectives in the field is the determination of the equation of state (EoS) of cold nuclear matter in beta-equilibrium, relating e.g. pressure P and baryon number density n b . This EoS is relevant for the description of rather old (more than a few minutes) neutron stars at the end of their chemical evolution, i.e. in equilibrium with respect to weak interactions and transparent to neutrinos. Recent progress has been made with the first detection of gravitational waves (GW) from a binary neutron star coalescence, which has put some constraints on the neutron star tidal deformability (Abbott et al. 2018). This first detection shows the potential of future GW detections from binary neutron star mergers to constrain the EoS, see e.g. (De et al. 2018;Dexheimer et al. 2019;Capano et al. 2020;Malik et al. 2019;Güven et al. 2020). There exists in fact a one-to-one relation between a given EoS and the tidal deformability as function of the star's mass (Hinderer et al. 2010) similar to the mass-radius diagram (M-R diagram) of cold non-rotating neutron stars. The latter is the subject of the standard approach for inferring constraints on the nuclear matter EoS (Ozel et al. 2010;Steiner et al. 2010Steiner et al. , 2013a;;Ozel et al. 2016). Therefore, many efforts are underway to measure at least some points in this diagram, see for example recent results by NICER (Miller et al. 2019;Riley et al. 2019).

In a prospective way, Torres-Forné et al. (2019a), have numerically determined empirical relations between masses (MPNS), radii (RPNS) and oscillation frequencies of protoneutron stars (PNS). The PNS forms during core-collapse supernova (CCSN) events, which should be good sources of gravitational radiation, too. The detection of GW from such modes could thus directly be related to PNS properties. With a numerical perturbative approach in General Relativity Torres-Forné et al. (2018, 2019b) have identified oscillation modes of a newly born PNS during the first second of its life and related them to the PNS surface gravity (MPNS/R 2 PNS ) for so-called g-modes and shock properties for p-modes. With some additional assumptions, it is then possible to deduce MPNS and RPNS from observed mode frequencies in GW. However, the structure of newly born PNSs during the first seconds of their life is very different from that of cold neutron stars (Prakash et al. 1997;Pons et al. 1999). In particular, properties of nuclear matter cannot in general be cast into a simple function of one variable as P (n b ), but usually rely on a more complex 3-dimensional relation such as P (n b , T, Ye), with T , the temperature and Ye = (ne -nē) /n b the electron fraction, i.e. the ratio between electron number density (ne, minus the positron number density nē) and n b .

In this context, the aim of this work is to investigate two problems. (P1) how much would the determination of the mass and radius of a PNS born in core-collapse supernovae be able to constrain the M-R diagram of cold neutron stars, and therefore also their cold equation of state P (n b ) ?

The second one (P2) is related to the first one: is it possible, with such observations, to constrain at least the (3dimensional) EoS of hot newly born PNS?

We present the model and the methods that were used to tackle these two problems in Sec. 2. Our results are presented in Sec. 3, before some concluding remarks in Sec. 4.

QUASI-STATIC APPROACH TO PROTO-NEUTRON STAR EVOLUTION

In this work, we shall only consider models in spherical symmetry, i.e. non-rotating, non-magnetized PNS, and we develop in this section a quasi-static approach to derive their structure. Although a PNS is an evolving star through cooling processes, the quasi-static hypothesis is justified by the fact that the timescale necessary to establish hydrodynamic equilibrium is much shorter than the PNS evolution timescale, mostly determined by cooling through neutrino emission (Pons et al. 1999). Following this work, we consider models in hydrostatic equilibrium, meaning that we follow the evolution of the PNS driven by the neutrino cooling processes, but at each time-step, we solve for hydrostatic equilibrium.

It is in particular assumed here that the eventual supernova explosion is successful, so that matter outside the shock is excised from the numerical grid, once this shock has crossed the matter of the PNS (see the discussion e.g. by [START_REF] Roberts | Handbook of Supernovae[END_REF]. The starting point is the expression for the metric of a static, spherically symmetric spacetime, using the standard Schwarzschild-type coordinates:

ds 2 = -c 2 dt 2 e 2Φ + e 2λ dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) , (1)
where Φ(r) and λ(r) are the two gravitational potentials. It is often more convenient to introduce a new function m(r) such that:

e -λ = 1 -2Gm/(rc 2 ) . (2)
In the Newtonian limit, m(r) represents the gravitational mass inside a sphere with radial coordinate r. The total gravitational mass of the star Mg = m(R) is the quantity that appears in the Schwarzschild solution for the metric outside the star. Note that it is this mass that could be determined through GW observation of p-and g-modes (MPNS = Mg).

Determining the structure of a star amounts to deriving the stress-energy tensor field Tµν , related to the metric in Eq. (1) through Einstein Equations:

Rµν - 1 2 gµν R = 8πG c 4 Tµν , (3)
where Rµν is the Ricci tensor of the metric and R = g µν Rµν the Ricci scalar.

We then assume the neutron star matter to be made of a single perfect fluid and neglect any other contribution to the stress-energy tensor: Tµν = (P + E)uµuν + P gµν , where uµ is the 4-velocity of matter, E the total energy density and P the pressure. This approximation is justified for newly born neutron stars because shear stresses produced by strong magnetic fields are generally negligible compared with the pressure. On the other hand, the PNS is hot: crust is not formed yet and superfluid effects are not present, either.

Einstein equations (3) in this case yield the well known Tolman-Oppenheimer-Volkov (TOV) system for hydrostatic equilibrium for a static spherically symmetric star: where Eq. ( 6) corresponds to the projection along the r-axis of the conservation of the stress-energy tensor describing hydrostatic equilibrium. The final ingredient for the system constituted by Eqs. ( 4)-( 6) to be closed is an EoS, that is a relation of the form P = P (E) (equivalent to P (n b ) introduced above). Since in a PNS thermal effects are not negligible and weak (beta-) equilibrium is not reached, a generic EoS for our case depends additionally, as stated in the Introduction, on the temperature T and the composition of matter summarized by the electron fraction Ye (Oertel et al. 2017).

dm dr = 4π c 2 r 2 E , (4)
dΦ dr = Gm r 2 c 2 1 + 4πP r 3 mc 2 1 - 2Gm c 2 r -1 , (5)
dP dr = -(E + P ) dΦ dr , (6)
Note that in presence of muons, the EoS would additionally depend on a muon fraction. Due to their higher masses, they contribute, however, only marginally to the EoS and we will neglect them here. A generic EoS then takes the form P = P (E, T, Ye).

Consequently, in order to get an effective relation of the form P = P (E) and integrate the TOV system to derive the PNS structure, it is necessary to assume some profiles T (r) and Ye(r) or any equivalent ones inside the star. These profiles can be obtained from CCSN simulations and present a high variability, depending on the kind of progenitor that is considered for the supernova and on the time considered af-ter the formation of the PNS. Some examples are plotted in Fig. 1: temperature and electron fraction radial profiles coming from various CCSN simulations published in the literature are shown in order to give an idea of the existing uncertainties. To be more precise, the blue dashed-dotted curve corresponds to the profile at 0 s in Fig. 12 of Pons et al. (1999), the orange dashed curve to the model s40 of Liebendörfer et al. (2004) and Fischer et al. (2009) at 400 ms after the bounce, the green dashed curves to profiles obtained using the Co-CoNuT code (Dimmelmeier et al. 2005;Pascal et al. 2020) for a s15-type progenitor at 100 ms, 200 ms and 300 ms after the bounce, the magenta dotted curves to the profile for the model lsu40 of [START_REF] Pascal | Proto-neutron star cooling with improved neutrino-nucleon interactions[END_REF] and the red dashed curve to the model Λ-u40 of Peres et al. ( 2013), both just before the collapse of the star into a black hole.

Since such profiles can a priori not be directly measured in a supernova event, their variability should be taken into account when making estimates about the structure of PNS. In other words, it is necessary, for any given EoS, to test many realistic profiles for T and Ye in order to say something about the range of possible PNS structures that could arise for this EoS. In order to avoid performing a large number of computationally expensive CCSN simulations, we will use parameterized initial profiles at the bounce to investigate PNS structure.

These profiles are then evolved using a quasi-static evolution code [START_REF] Pascal | Proto-neutron star cooling with improved neutrino-nucleon interactions[END_REF]): between two timesteps the electron fraction Ye and the entropy per baryon s are evolved using equations ( 7)-( 8 where µe represents the electron chemical potential. Eq. ( 7) stems from lepton number conservation and Eq. ( 8) from energy conservation. For the latter, general thermodynamic relations have been used to express energy conservation in terms of entropy per baryon. The two source terms Sn and Se are respectively the source of electrons and the source of energy issued from neutrino-matter interactions with a 3-flavor neutrino transport. They are given by

Sn = - 1 c (Γν e -Γν e ) (9) Se = - 1 c (Qν e + Qν e + 4Qν x ) (10)
with Γν e -Γν e the net electron neutrino creation rate which, by lepton number conservation, is related to the net change in electron number and Qν the energy loss rate induced by neutrino species ν. Those rates are computed using a "Fast Multi-group Transport" (FMT) scheme (Müller & Janka 2015). This scheme is based on a stationary approximation of neutrino transport and uses a two-ray approximation in areas with a high optical depth and a two-moment closure in areas with a low optical depth. Considered reaction rates are the same as in the original paper Müller & Janka (2015).

For technical convenience, we choose here to parameterize the initial profiles that enter the PNS evolution code as the entropy per baryon as a function of the enclosed baryon mass s(m b ) and the electron fraction Ye(m b ). In spherical symmetry, it is equivalent to use the radial coordinate r or the enclosed baryon mass,

m b = 4π r 0 e λ n b mnr 2 dr . ( 11
)
The latter presents the advantage of being a Lagrangian coordinate. Fig. 2 represents typical s(m b ) and Ye(m b ) profiles obtained from CCSN simulations. Similar profiles can be found, e.g. in Fig. 1 of Roberts (2012). As can be seen, these profiles have the following characteristics: s(m b ) first increases in the inner core of the star, then slightly decreases in a region corresponding to the shocked material and increases again very fast towards the surface. Here, the "inner core" should not be confused with the NS inner core, but be understood as the inner regions of the progenitor star. At bounce the shock has formed at the border of the inner core. Note that a more massive inner core (a wider region of increasing entropy per baryon in the s(m b ) profile) implies that the kinetic energy of the shock is higher, such that it goes through more matter before exhaustion, which will result in a larger amount of shocked material (a wider region of decreasing entropy per baryon in the s(m b ) profile). This means that the relative mass contained in both regions should be almost independent of the initial conditions (Janka 2012), as in the profile of Fig. 2. Second, Ye(m b ) is nearly constant ∼ 0.2 -0.3 in the inner core, then decreases more sharply in the shocked region and goes up again near the surface. We find that the following parameterizations of s(m b ) and Ye(m b ) reproduce these main features. s(m b ) is modeled by a plateau followed by a Gaussian and a power-law divergence near the surface, with five parameters in total:

s(m b ) = sc × 1 + exp 2smax scσs (m b -ms + 2σs/3) -1 + smax exp - (m b -ms) 2 2 (σs/2) 2 + smax (M b -m b ) -0.3 1 + exp -2 σs (m b -(ms + 2σs/3)) , ( 12
)
where sc is the central entropy per baryon, smax the maximal entropy par baryon between inner core and shocked material, σs the width of the Gaussian, ms the mass at which smax is reached and M b the total baryon mass of the star. According to the remarks above, ms and M b -ms should be of the same order for consistency with respect to the shock physics.

Ye(m b ) is modeled by a plateau followed by an inverse asymmetric Gaussian, with four parameters in total:

Ye(m b ) = Y0 + Yc -Y0 1 + exp 5 2σ Y Y min -Y 0 Yc-Y 0 m b -M b + 2σ Y 3 + (Ymin -Y0) exp - (m b -M b ) 2 2 (σY /2) 2 × 1 2 1 + erf -15 m b -M b σY / √ 2 , (13)
where "erf" refers to the error function and Yc is the central electron fraction, Y0 = 0.5, Ymin the minimal Ye and σY the width of the Gaussian. Since the region where a depletion in Ye is observed, here described by the Gaussian, corresponds to the shocked region (see Fig. 2), σY should be of the same order as M b -ms.

NUMERICAL MODEL PARAMETERS AND RESULTS

We present in this section the results of a comparison between several simulations of PNS structure during the first second of their post-bounce evolution varying the initial entropy and electron fraction profiles as well as the employed EoS model. Let us start with detailing the procedure applied for that comparison.

The initial profiles for s(m b ) and Ye(m b ) at bounce are built upon the parameterizations discussed in the previous section, see Eqs. ( 12)-( 13). The parameters are chosen such that the resulting profiles remain in reasonable agreement with results from CCSN. In particular, in order to respect the core-shock picture discussed above and illustrated in Fig. 2, ms and σY should be of the same order as M b -ms. For simplicity, we choose σY = M b -ms. Note that M b is a parameter of both profiles, s(m b ) and Ye(m b ), so that different baryon masses correspond to different profiles. We say that two profiles obey the same parameterization if they have the same parameters, except for M b which can be chosen freely. ms is then given by the ratio ms/M b ≈ 1/2. The values of the remaining free parameters for the five parameterizations considered here are detailed in Tab. 1. For each parameterization, three different total baryon masses M b will be considered: 1.6 M , 1.8 M and 2.0 M . For illustration, the parameterized profiles are plotted in Fig. 3 Then, with five parameterizations and three baryon masses, we end up with a total of 15 different initial profiles. These profiles are evolved during the first second postbounce for a given EoS using the quasi-static PNS evolution code [START_REF] Pascal | Proto-neutron star cooling with improved neutrino-nucleon interactions[END_REF], see Section 2. Fig. 4 shows as an example the s(m b ), Ye(m b ) and T (m b ) profiles in the PNS every 0.1 s post-bounce, for the parameterization p3 of Tab. 1, a total baryon mass M b = 1.6 M and the HS(DD2) EoS (Hempel & Schaffner-Bielich 2010).

Finally, with evolved profiles taken every 0.1s post-bounce, we end up with 150 PNS profiles for each EoS. These profiles are used to solve the TOV system (4)-( 6) and to derive the PNS structure at fixed baryon mass at each time. The surface of the PNS is thereby defined as in Torres-Forné et al. (2019a) at a fixed baryon density of n b = 5.970 × 10 -5 fm -3 . We obtain in the end one point in the (RPNS, MPNS) plane for each of the 150 PNS profiles.

Based on the results of the procedure described above, we first address the problem (P1), concerning the possibility, from the observation of mass and radius of a hot PNS in the first second post-bounce, to constrain the corresponding quantities of a cold neutron star. To take into account the uncertainty on the entropy and electron fraction in the PNS, the PNS structure is computed for the five parameterizations of initial profiles given in Tab. 1. In Fig. 5 are displayed the time evolution of 60 different PNS: our 15 different initial profiles with four different choices 1. Different colors correspond to different EoS models: HS(DD2) (Hempel & Schaffner-Bielich 2010, magenta), RG(SLy4) (Raduta & Gulminelli 2019, blue), SFHo (Steiner et al. 2013b, green) and SRO(APR) (Schneider et al. 2019, cyan). Time runs from 0 s to 1 s post-bounce from right to left. The red dots indicate the points computed every 0.1 s and the black line the points for the PNS 1 s after the bounce. Bottom : same curve in the particular case of the HS(DD2) EoS, parameterization p3 of Tab. 1 and a total baryon mass M b = 1.8 M .

of EoS models. These are HS(DD2) (Hempel & Schaffner-Bielich 2010, magenta), RG(SLy4) (Raduta & Gulminelli 2019, blue), HS(SFHo) (Steiner et al. 2013b, green) and SRO(APR) (Schneider et al. 2019, cyan). The evolution is described in terms of the radius RPNS and the mass MPNS, each one divided by the corresponding value for the cold configuration (label T = 0), corresponding to a fully-evolved (cold, beta-equilibrated) NS.

It is apparent that for all parameterizations and EoSs, the PNS structure is still quite far from the cold configuration after 1 s of evolution. Quantitatively, we find that, after 1 s, the mass is larger by about 6 -10 % and the radius by about 40 -200 % compared with the cold configuration. Note that, as far as the practical observation is concerned, because the mass is almost constant1 during the first second of the PNS' life, its value is very little influenced by the time over which the observation is integrated, so that a good signal to noise ratio (SNR) is reachable in principle. On the contrary, integrating over a few tenths of a second would induce a large error on the radius measurement. All in all, our results imply that whereas measuring the mass of a PNS during the first second post-bounce would give an upper bound quite close to the actual value for the cold star, the obtained value of the radius is much less reliable. Indeed during the first few seconds of evolution of the PNS the shocked mantle cools down and shrinks very fast Prakash et al. (1997); Pons et al. (1999), such that the radius is evolving on a short timescale. As a consequence, the determination of MPNS/R 2 PNS proposed by Torres-Forné et al. (2019a) is not expected to give a reliable estimate of the same quantity in the cold configuration.

We now turn to the problem (P2), regarding the possibility, from the knowledge of RPNS and MPNS, to constrain its EoS, which is fully dependent on three parameters. Although the answer would be yes if we were able to know, for each measured point at a given post bounce time, both s and Ye profiles inside the PNS, the fact that they are not accessible implies that the actual answer depends on the influence of our limited knowledge of these profiles on MPNS and RPNS. If the dispersion in (RPNS, MPNS) due to the uncertainty in the profiles is smaller than the variation due to the difference induced by employing different EoS, then this measurement can in principle (provided a sufficient precision) constrain the EoS of the PNS. Fig. 6 shows the points obtained from the PNS evolution procedure for the four different EoS models considered before. In this figure, the condition for the (RPNS, MPNS) measurement being able to discriminate between different EoS is that the clouds of points (corresponding to the various parameterizations for the initial s and Ye profiles) corresponding to the different EoS models do not intersect. Actually, employing only four different EoS models, a positive answer to problem (P2) would actually be difficult to justify in this way. However, the clouds are largely mixed already for a small number of different EoS. We conclude that, measuring RPNS and MPNS alone can hardly constrain the EoS of the PNS. It is a priori possible to add some timing information to this picture, with the measure of the mass and radius at some given post-bounce times. To illustrate this purpose, Figs. 7 show snapshots of Fig. 6 at 0.5 s and 1 s after bounce. Here again, it is clear that the spreading of points (mainly along the RPNS direction) due to the use of different entropy or electron fraction profiles is much larger than that coming from the various EoS models. This means in particular that, even if we can measure mass and radius of the PNS with additional timing information, little could be deduced about the underlying EoS.

In this context, it it is interesting to ask which of the uncertainty on entropy or on electron fraction has the largest influence on the large spreading in the (MPNS, RPNS)-plane responsible for the negative answer to (P1) and (P2), or whether they both contribute equally. To answer that question, we have introduced four additional parameterizations based on p1: two with a modified entropy keeping the electron fraction profile the same, and two with modified electron fraction and the same entropy profile. For the former, we vary sc and smax by ± 50 20 % and ±50 %, respectively, compared with the p1 entropy profile. These variations remain still in reasonable agreement with results from simulations. For the latter, we use two rather extreme profiles, one with a Yc and a Ymin increased by roughly 15 % and 50 % respectively and another one reducing Yc and Ymin by 30 % and 70 %, respectively. Repeating the same exercise as before, i.e. following the evolution of the PNS over 1 s with these additional initial profiles and the same four different EoS models and analysing the obtained distribution in MPNS and RPNS, the conclusion is rather obvious. Varying the Ye-profile, even in an extreme way, we observe only a limited dispersion of points. As far as MPNS is concerned, it is less than a few percent and for RPNS, of the order of 10 %. On the contrary, the different entropy profiles induce a large uncertainty, in particular for the radii which spread between roughly 15 and 60 km. Taking snapshots at given post-bounce times does not alter the conclusion that mainly the uncertainty in the entropy profile of the PNS prevents us from obtaining information about the EoS and cold neutron star properties from the observation of the PNS' mass and radius. This finding can be understood from the fact that small variations in entropy per baryon induce large variations in pressure and energy density of matter in particular at lower densities, i.e. in the outer regions of the star, and that radii are very sensitive to these variations. That means, if we want to improve the situation and be able to extract information from the measurement of PNS mass and radii, we should better constrain the entropy profiles in the PNS.

It should be mentioned here, too, that the PNS is very likely unstable to convection, flattening the entropy per baryon profile in the shocked region (see e.g. the discussion in Roberts & Reddy 2017). This flattening is not reproduced by our profiles and the simulated PNS evolution does not treat convection. Including the latter would certainly change our results quantitatively, but the main result of our study remains true: without considerably reducing the uncertainty in entropy profiles, no useful information on EoS and cold NS mass and radius can be obtained from the corresponding quantities of a PNS.

SUMMARY AND DISCUSSION

As it seems to be possible to infer from future GW observations the mass and the radius of a newly born PNS, we have investigated here what information these data could bring about the nuclear EoS for NSs (one-parameter EoS) or for PNSs (3-parameter EoS). Thermal and composition effects cannot be neglected within a PNS. In addition, temperature and electron fraction in a PNS evolve with time and its mass and radius accordingly. We have thus considered neutrino cooling of the PNS using a quasi-static approach [START_REF] Pascal | Proto-neutron star cooling with improved neutrino-nucleon interactions[END_REF]) and determined entropy and electron fraction profiles in the PNS at several instants during the first second of postbounce evolution. Based on these profiles, we have built hydrostatic PNS models at each step giving mass and radius.

From CCSN simulations it appears that large uncertainties exist as far as entropy and electron fraction profiles at early post-bounce are concerned. These uncertainties result, among others, from different progenitor structure, uncertainties on weak reactions during infall, the employed EoS and convection in the newborn PNS. In order to take these uncertainties into account, we have set up analytic parameterized profiles for both quantities (s ans Ye), that can mimic most of the simulated profiles from the literature, enabling us to generate a large set of physically relevant evolutionary models of PNSs, in the first second after the bounce.

The study of this set of models showed that, after 1 s of evolution, the value of the radius is still far from the corresponding value of a cold NS. This indicates that the knowledge of PNS properties soon after the bounce cannot give much information on the properties of the resulting cold NS (question (P1)). Additionally, a large dispersion of evolutionary curves of PNS in the MPNS, RPNS plane and, above all, a mixing of points coming from various EoSs is observed. Conversely, this means that, even with some timing information (as illustrated by Figs. 6 and7), it is difficult to disentangle evolutionary tracks of models with different EoSs. This also leads to a negative answer to the question (P2), about the possibility of inferring the EoS of the PNS from its masses and radii.

Of course, there are several limitations to this study. First, the quasi-static approach may not be the most accurate, but we believe that, as discussed in previous works Pons et al. (1999); Villain et al. (2004), such an approximation is under control and is not the main source of uncertainties in our results. Another point which merits discussion is the choice of parameterized profiles. The choices done here, with formulas in Eqs. ( 12)-( 13), may be regarded as too broad, giving too much freedom in the dispersion of points in Figs. 5 and 6. However, our choices of parameters listed in Tab. 1 are such that they should reproduce previous results from CCSN simulations. As these simulations include not fully controlled ingredients, in particular initial data for the progenitor stars, it is important to consider a general set of profiles, unless future studies can bring techniques to determine temperature and electron fraction profiles in CCSN from astrophysical observable features. Last but not least, a limitation of our study is the fact that we have considered the evolution of the PNS only during the first second after its birth. In particular concerning the problem (P1) about information of cold NS, the situation would of course be quite different if longer evolution times were considered. The particular value of 1 s postbounce is justified by CCSN simulations showing that during this period, a quasi-stationary situation is achieved for the stalled accretion shock and instability-induced perturbations of the PNS. It is not clear to what extent strong-enough GW can be emitted later, and therefore whether it is still possible to determine MPNS and RPNS at later stages. This point would certainly be worth exploring more in details, as later information would certainly be more decisive in order to get insights on nuclear matter properties. A possibility in this direction is evoked in Gallo Rosso et al. ( 2018), estimating the PNS radius at about 6-10s after bounce from the neutrino signal. However, for the moment the uncertainties entering the neutrino signal, in particular those related to neutrino transport in the newborn hot PNS and to neutrino-matter interactions, render a reliable extraction of the PNS radius prohibitive.

Figure 1 .

 1 Figure 1. Temperature T (top) and electron fraction Ye (bottom) radial profiles obtained from various core-collapse supernova simulations (see text for details).

Figure 2 .

 2 Figure 2. Entropy per baryon s (top) and electron fraction Ye (bottom) as a function of the enclosed baryon mass m b for the configuration of profile A100ms from Fig. 1. The dashed line separates the inner core of the proto-neutron star from the shocked region.

  ) (see e.g.Pons et al.

Figure 3 .

 3 Figure 3. Entropy per baryon s (top) and electron fraction Ye (bottom) as a function of the enclosed baryon mass m b , for our chosen sets of parameterized profiles (12)-(13) defined at bounce, for a total baryon mass M b = 1.6 M . Ye profiles for the p2 and p5 parameterizations are identical.

Figure 4 .

 4 Figure 4. Entropy per baryon s(m b ) (top), electron fraction Ye(m b ) (middle) and temperature T (m b ) (bottom) profiles as a function of enclosed baryon mass m b , for several post-bounce times (darker blue corresponds to later times) employing the initial profile p3 of Tab. 1, total baryon mass M b = 1.6 M and the HS(DD2) EoS(Hempel & Schaffner-Bielich 2010). The time evolution is computed within a quasi-static approach, see Sec. 2.

  for a total baryon mass M b = 1.6 M , the upper panel showing s(m b ) and the lower one Ye(m b ). Profile p1 is thereby chosen to resemble the profiles of Fig. 2 and p2 those of Fig. 1 in Roberts (2012).

Figure 5 .

 5 Figure5. Top : PNS evolutions in the gravitational mass (M PNS )radius (R PNS ) plane, normalized to the zero temperature values (subscript T = 0), for the parameter sets p1 -p5 of Table1. Different colors correspond to different EoS models: HS(DD2)(Hempel & Schaffner-Bielich 2010, magenta), RG(SLy4)(Raduta & Gulminelli 2019, blue), SFHo(Steiner et al. 2013b, green) and SRO(APR)(Schneider et al. 2019, cyan). Time runs from 0 s to 1 s post-bounce from right to left. The red dots indicate the points computed every 0.1 s and the black line the points for the PNS 1 s after the bounce. Bottom : same curve in the particular case of the HS(DD2) EoS, parameterization p3 of Tab. 1 and a total baryon mass M b = 1.8 M .

Figure 6 .

 6 Figure 6. Points in the (R PNS , M PNS )-plane obtained from the evolution of the five initial parameterizations of Tab. (1), for three different total baryon masses, M b = 1.6 M (bottom cloud), M b = 1.8 M (middle cloud) and M b = 2 M (top cloud). 4 different EoSs are used: HS(DD2) (Hempel & Schaffner-Bielich 2010, red circles), RG(SLy4) (Raduta & Gulminelli 2019, blue crosses), SFHo (Steiner et al. 2013b, green triangles) and SRO(APR) (Schneider et al. 2019, cyan diamonds).

Figure 7 .

 7 Figure 7. Same as Fig. 6, but only snapshots at t = 0.5 s (top) and t = 1 s (bottom) are taken from the time evolution.

Table 1 .

 1 ). The time evolution is computed within a quasi-static approach, see Sec. 2. Parameters defining for a given M b the five parameterizations labeled p1 to p5 of the initial s(m b ) and Ye(m b ) profiles.

	Label	sc	smax	σs	ms/M b	Yc	Y min
	p1	1.3	4	0.7	0.5	0.26	0.1
	p2	1.5	5.5	0.4	0.6875	0.3	0.06
	p3	1.1	3	0.7	0.4375	0.32	0.12
	p4	1	5	0.6	0.5625	0.28	0.1
	p5	1.5	4.5	0.4	0.6875	0.3	0.06

MNRAS 000, 1-9 (2020)

Except in the first tenth of a second, when the mass should increase if accretion is properly taken into account MNRAS 000, 1-9 (2020)

This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-9 (2020)

ACKNOWLEDGEMENTS

We warmly thank Pablo Cerdá-Durán and Alejandro Torres-Forné for enlightening discussions, and Francesca Gulminelli for interesting suggestions. The research leading to these results has received funding from the PICS07889; it was also partially supported by the Observatoire de Paris through the action fédératrice "PhyFog".


Reliability and Low Latency: Impact of The Architecture.

Tania Alhajj

IMT Atlantique IRISA, UMR CNRS 6074 F-35700 Rennes, France tania.alhajj@imt-atlantique.fr

Xavier Lagrange

IMT Atlantique IRISA, UMR CNRS 6074 F-35700 Rennes, France xavier.lagrange@imt-atlantique.fr Abstract-Many use cases are meant to be supported by the fifth generation (5G) wireless technology. The one which is occupying the research area for its challenging requirements is the Ultra Reliable Low Latency Communications (URLLC). Hybrid Automatic Repeat reQuest (HARQ) protocol is used to ensure reliability but it induces delay. Furthermore, the transmission in the Radio Access Network (RAN) should be taken into account in the delay budget. In this paper, we jointly analyze the reliability and the delay with two RAN architectures: the legacy one where only one radio unit receives the packet from a terminal and a Centralized-RAN (C-RAN) architecture where several radio units can decode a packet. We propose to combine these approaches in a flexible architecture. The observed enhancement is a division by 850 of the packet erasure rate compared to the legacy architecture with a latency of 3 milliseconds.

Index Terms-5G, URLLC, RAN architecture, C-RAN, flexible architecture.

I. INTRODUCTION

The fifth generation (5G) communication technology is emerging after four previous generations to offer higher rates, lower latency, higher reliability and to serve a larger number of users simultaneously. One of the 5G usage scenarios is the Ultra reliable low latency communications (URLLC). Its importance lies in its use for critical applications. Some of these applications are telemedicine and driverless cars [START_REF]New services and applications with 5G ultra-reliable low latency communications[END_REF]. These applications require very high reliability because they concern human lives. The required reliability ranges from 10 -3 to 10 -7 packet error rate (PER) depending on the usage scenario [START_REF] Li | 5G URLLC : Design challenges and system concepts[END_REF]. Since they are critical, a very low latency (between 1 and 10 ms [START_REF] Li | 5G URLLC : Design challenges and system concepts[END_REF]) is required as well. Responding to these two strict requirements while maintaining acceptable network capacity, reasonable energy consumption, and satisfying overall network operation is challenging [START_REF] Soret | Fundamental tradeoffs among reliability, latency and throughput in cellular networks[END_REF].

Hybrid automatic repeat request (HARQ) is widely known and deployed in wireless networks. It is used to reduce errors and enhance the transmissions reliability [START_REF] Le Martret | Analytical performance derivation of hybrid ARQ schemes at IP layer[END_REF].

(Author's Accepted Version) (Final version of record is available at: https://doi.org/10.1109/ISCC50000.2020.9219636) ©2020 IEEE Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

However, packet repetition induces latency. Ensuring high reliability with fewer transmissions requires to reduce the interference. Consequently, this will reduce the network's capacity. Previous studies were made to reduce the latency caused by the HARQ re-transmissions. Cheng and Chen proposed to allocate resources for re-transmissions in advance [START_REF] Cheng | Dynamic pre-allocation HARQ (DP-HARQ) in IEEE 802.16j mobile multihop relay (MMR)[END_REF]. Other authors suggested predicting the decoding feedback (acknowledgement (ACK)/negative acknowledgement (NACK)) [START_REF] Berardinelli | Enabling early HARQ feedback in 5G networks[END_REF]. This prediction enables earlier re-transmissions. As a result, the delay is shortened.

In the literature, studies were made to respond to URLLC requirements. Some of them were based on varying the transmission time interval (TTI) size and the frame size to reduce the latency [START_REF] Li | 5G ultra-reliable and low-latency systems design[END_REF]. That way, more re-transmissions are allowed before exceeding the accepted delay. Other authors opted the grant free access to cut down the delay generated by the grant based access [START_REF] Mahmood | Uplink grant-free access solutions for URLLC services in 5G new radio[END_REF]. Diversity techniques were also studied to achieve high reliability. Time diversity is discarded because it produces latency. In [START_REF] Jacobsen | Multi-Cell reception for uplink grant-free ultrareliable low-latency communications[END_REF] and [START_REF] Mahmood | On the resource utilization of multi-connectivity transmission for URLLC services in 5G new radio[END_REF], the multiple reception concept was studied. It is shown that it increases reliability by providing spatial diversity. The authors were interested in studying resource utilization and the network's capacity using this technique.

All these papers consider only the radio interface. The latency perceived by users is also due to the delay in the radio access network. The radio access network (RAN) architecture should thus be included in the study of the latency. Centralized-RAN (C-RAN) is a new network architecture that emerged to be the evolution of the traditional RAN. This centralized architecture is a key element of the 5G technology. It consists of dividing the base station (BS) into two parts: a centralized unit (CU) and a radio unit (RU) [START_REF] Huang | Recent progress on C-RAN centralization and cloudification[END_REF] (Fig. 1). The connection between these two parts is usually a fiber optic link. High bandwidth, low losses, and interference insensitivity are the interesting advantages of this link. The C-RAN architecture presents many benefits. It helps the operators to reduce the sites' deployment costs. It ensures also the collaboration between different RUs connected to the same pool of CUs. It also provides the ability to choose which BS functions to centralize and which functions to keep implemented locally. This is called the functional split [START_REF] Larsen | A survey of the functional splits proposed for 5G mobile crosshaul networks[END_REF] (Fig. 1). Some researchers were interested in the flexibility of the functional split. This flexibility allows switching from an option to another upon the need: whether to reduce energy consumption, to diminish resource utilization, to respond to a latency requirement, or any other need. The adaptive split in [START_REF] Alfadhli | Realtime demonstration of adaptive functional split in 5g flexible mobile fronthaul networks[END_REF] is an example.

In this paper, we investigate the impact of the architecture on the latency and on the reliability. We compare the single receiving point to the multiple receiving case while adopting different functional splits. For each architecture, we derive the distributions of the number of transmissions and of the delay. Then we suggest a flexible switch between the two chosen functional split options. Our results are demonstrated analytically and by simulations.

The paper is structured as follows: Section II exposes the two different architectures studied, in addition to the propagation, transmission, and error models. Afterward, the delays for both architectures are detailed in section III. In section IV, we develop the transmissions distributions analytically. Then, in section V, we propose a flexible functional split between the two architectures. Finally, before concluding, we expose and discuss our results in section VI.

II. ARCHITECTURE AND MODEL

A. Architectures overview

We consider two RAN architectures with different functional splits: A and B.

In A, the medium access control (MAC) layer, responsible for the error correction (using HARQ), is implemented in the RU. This corresponds to functional split option 1 [START_REF] Larsen | A survey of the functional splits proposed for 5G mobile crosshaul networks[END_REF]. In this architecture, the data is close to the user [START_REF] Gpp | Study on new radio access technology; radio access architecture and interfaces[END_REF]. We consider here a user equipment (UE) transmitting and only one receiving RU (Fig. 2). During each transmission, the data is received at the RU. A decoding process occurs. In the case of a failed decoding, the MAC layer in the RU asks for a re-transmission. When the decoding succeeds, the data is transmitted to the CU.

In B, the error correction is triggered in the CU. The MAC layer is therefore implemented in the CU. This is referred to as split option 8 in [START_REF] Larsen | A survey of the functional splits proposed for 5G mobile crosshaul networks[END_REF]. Here, we consider that M RUs receive the data from the same UE (Fig. 3). We have a single transmission and multiple receptions. The decoding is based on the M receptions. The coordination between the M RUs is managed by the CU: different receiving RUs send the received signals to the CU where the signals can be combined. In this work, we consider that a reception is successful if the data from at least one of the M RUs is correctly decoded. In case of an error on all the receiving RUs, a re-transmission is requested from the CU. In A and B, we consider that a CU is connected via fiber optic links to multiple RUs located at the tower site. We consider these RUs to be equidistant from the CU. We define θ, the propagation delay between different RUs and the CU that they're connected to:

θ = ρ v ( 1
)
where ρ is the distance between the CU and the RUs connected to it and v is the velocity of light inside a fiber (v = 2 × 10 8 m/s). Then, for the same CU, θ is the same for all RUs.

B. System model

The channel model is considered as Cost-231 Hata [START_REF] Gpp | Evolved universal terrestrial radio access (e-utra); radio frequency (rf) system scenarios[END_REF]. The received power depends on the path-loss and the fading:

P r = P t r 0 r α χ (2)
where P r is the received power, P t the transmitted power, r 0 is a constant, r the distance between the transmitter and the receiver (the UE and the BS respectively in our case), α the path-loss exponent and χ is an exponential random variable representing the fading with mean = 1. The shadowing is not taken into consideration. The transmissions in both architectures are in the uplink (UL) direction. We consider successive transmissions of data packets. All transmitted packets have the same size. We consider that the transmission duration of a packet or an ACK/NACK is 1 TTI. Thereby, we have T A = T D with T A being the transmission duration of the ACK/NACK and T D the transmission duration of a packet of data. We consider a perfect downlink (DL), i.e. ACKs and NACKs are always successfully received.

We consider the background noise N as a constant. Parameter N includes the noise and the interference. The error in decoding is related to the signal to noise ratio (SNR) which is denoted by γ:

γ = P r N . (3)
With the previous considerations, we define γ, the average SNR as:

γ = P t N r 0 r α . ( 4
)
The PER is approximated and calculated as a function of the SNR similarly to [START_REF] Liu | Cross-layer combining of adaptive modulation and coding with truncated ARQ over wireless links[END_REF]:

h(γ) = 1 if 0 < γ < γ M ae -gγ if γ ≥ γ M (5)
where a and g are parameters that depend on the modulation and coding scheme (MCS) mode and γ M = ln a g .

III. DELAYS

The reliability is achieved by the HARQ chase combining (CC) mechanism in architecture A. For architecture B, both HARQ and spatial diversity are used to achieve reliability. Now, to compute the delay for each architecture, we expose its components. We assume that the processing duration is negligible. We consider that the propagation delay is absorbed by the guard time of the slot. We also suppose that we have a very high rate on the fiber link. Then, the transmission duration over the fiber is negligible. In the following, the number of transmissions is denoted by l: l -1 failed transmissions and then a successful one.

For architecture A, we define the cycles duration in both cases: good and bad decoding. In Fig. 4, d 1f denotes the delay of 1 cycle in which the decoding fails:

d 1f = T D + T A . (6)
Parameter d 1s represents the delay of 1 cycle where the decoding succeeds:

d 1s = T D + θ. (7)
Therefore, the total delay produced by l transmissions is:

d 1 = (l -1)(T D + T A ) + T D + θ. (8)
For architecture B, the cycles delays are shown in Fig. a UE and one BS (BS i ). Let d 2f and d 2s denote the delay of 1 cycle with an erroneous and a successful decoding respectively. Then, we have:

d 2f = T D + 2θ + T A (9)
and

d 2s = T D + θ. (10)
The total delay generated by l transmissions in B is:

d 2 = (l -1)(T D + 2θ + T A ) + T D + θ. (11)

IV. ANALYTIC FORMULATION

The delay detailed in III is a function of l, the number of transmissions. Consequently, the distribution of l is needed to know the delay and its distribution.

The UEs are considered to be uniformly distributed in an hexagonal cell of radius R c . The RUs are at the centers of the cells. In [START_REF] Lagrange | Throughput of HARQ protocols on a block fading channel[END_REF], the probability of having more than k transmissions as a function of γ was computed when the PER is given by (5):

P(l > k/γ) = Γ l (k, X) + e -X k-1 i=0 (X) i i! Γ( 1 gγ ) (gγ) k-i+1 Γ( 1 gγ + k -i + 1) (12)
where P(l > k/γ) is the probability of having more than k transmissions for a given γ in a fading channel, X = γ M γ , and Γ l (b, x) = 1 Γ(b) x 0 t b-1 e -t dt is the lower incomplete normalized gamma function. The probability distribution function (PDF) of the number of transmissions (l) is obtained by:

P(l = k/γ) = P(l > k -1/γ) -P(l > k/γ). ( 13
) RU 1 RU 2 3 ‾ √ R c R c RU 3
Fig. 6: Zone of study in the hexagonal cell.

For a fixed P t and N , γ varies with the distance between the transmitter and the receiver.

For symmetry reasons, our study is limited to the shadowed area of Fig. 6. We consider a UE at the position (x, y) relative to RU 1 . We have 0 < x < √ 3

2 R c and 0 < y < x √ 3 .

A. Architecture A

In A, the UE in position (x, y) has a corresponding

γ(x, y) = Pt N r0 √ x 2 +y 2 α
. Thus, for each UE, we have a distribution of l, the number of transmissions from ( 12) and ( 13). In order to get a distribution of l, the distribution depending on the position should be averaged over all the studied surface:

P(l > k) = 8 √ 3R 2 c √ 3 2 Rc 0 x √ 3 0 A(x, y)dydx ( 14
)
where A(x, y) = P (l > k/γ(x, y)).

B. Architecture B

In B, for M = 2, the UE in position (x, y) relative to RU 1 , is in position ( √ 3R c -x, y) relative to RU 2 . The average SNR relative to the second BS is γ( √ 3R c -x, y). A transmission is considered erroneous if its reception fails at the 2 RUs. So, the probability of having more than k transmissions is:

P(l > k) = 8 √ 3R 2 c √ 3 2 Rc 0 x √ 3 0 A(x, y)A( √ 3R c -x, y)dydx. ( 15
) For M = 3, the UE is in position ( √ 3Rc
2 -x, 3Rc 2 -y) relative to the third RU. Similarly to [START_REF] Gpp | Evolved universal terrestrial radio access (e-utra); radio frequency (rf) system scenarios[END_REF], we get:

P(l > k) = 8 √ 3R 2 c √ 3 2 Rc 0 x √ 3 0 A(x, y) A( √ 3R c -x, y)A √ 3R c 2 -x, 3R c 2 -y dydx. ( 16
)

V. FLEXIBLE SWITCH BETWEEN A AND B

After presenting both architectures, we propose to dynamically switch between them. In fact, when the UE is too close to the first RU, there is no need to let the second RU receive from it. In this case, architecture A and single reception are considered. On the other side, when the UE is near the cell edge, we switch to architecture B and we enable multiple receptions. We define R th , the radius of the limit zone outside which we allow two RUs to receive using architecture B (Fig. 7). If the UE is inside the mentioned zone, architecture A and single reception are enabled. This switch is done by the flexible functional split. The CU measures x and chooses the adopted architecture accordingly. For each transmission to/from the considered UE, each unit encodes/decodes the data until the corresponding layer (dashed lines in Fig. 8) and transmits to the other unit. The switching algorithm is the following:

Algorithm 1: Flexible switch between A and B if 0 < x < √ 3 2 R th then consider architecture A; else if x < √ 3
2 R c then consider architecture B; enable multiple receptions; end For the flexible split with M = 2, we have:

P(l > k) = 8 √ 3R 2 c √ 3 2 R th 0 x √ 3 0 A(x, y)dydx+ √ 3 2 Rc √ 3 2 R th x √ 3 0 A(x, y)A( √ 3R c -x, y)dydx . (17)

VI. RESULTS

In this section we present our analytic and simulations results. Our simulations follow the analytic steps. The propagation and error models from (2) and ( 5) are used. The parameters values for our numerical calculations and our simulations are summarized in table I.

A. Comparison of the two architectures

Fig. 9 shows the similarity between the mathematical computation and the simulation results for P(l = k). It also compares the single reception (with A) to the multiple receptions (with B). It is shown that for the case of 2 receiving BSs, we have higher chances to get a successful decoding from the first transmission. For further transmissions, the probability is lower for the case of 2 BSs. Thereby, in B, the number of transmissions needed to get a successful decoding is reduced. No significant improvement is observed for receiving from a third BS. This is because the third BS is too far. For this reason, we limit our study to M = 2.

The delay depends on the number of transmissions (l). We can get the delay's distribution based on the distribution of l. Fig. 10 illustrates the complementary cumulative distribution function (CCDF) of the delay. Having the number of transmissions with a certain probability, we get the latency's distribution. If we define the reliability as receiving the data successfully within a certain duration, then this CCDF represents the outage probability. For example, if we want to assure a delay less than 2 ms, we have higher reliability by a ACK/NACK transmission time 1 TTI=0.25 a a Numerology 2 of the 5G new radio (NR) [START_REF] Gpp | 5G NR physical channels and modulation[END_REF]. factor of 850 in the case of 2 BSs (architecture B) compared to 1 (architecture A). This improvement is larger if we allow higher delays. If we desire to get an outage probability of 10 -6 , we can see in Fig. 10 that we get lower latency with B (2.9 ms for B compared to 3.8 ms latency for A). We can notice that distancing the CU from the RUs can add more delay. This additional delay affects B more than A (which is seen in ( 8) and ( 11)). This leads the delay's CCDF of B to approach the CCDF of A. The difference between the 2 delays (for A and B) at 10 -6 PER is 0.83 ms. If we let ρ = 20 km, ( 8) and [START_REF] Huang | Recent progress on C-RAN centralization and cloudification[END_REF] give the same results for the mentioned PER. A distance higher than 20 km increases more the delay in B. That way, we observe higher delays in architecture A compared to architecture B for the same PER.

B. Flexible C-RAN architecture

Fig. 11 illustrates the distribution of l, the number of transmissions, for different R th . We note that R th = R c is equivalent to the case of adopting architecture A with one receiving BS and R th = 0 corresponds to architecture B with two receiving BSs. The threshold is chosen when we start getting the same performance as if we are receiving from 2 BSs. Accordingly, in our case, we choose R th = 0.6R c . This is also shown in Fig. 12, where we have the same performances for the case of R th = 0.6R c and R th = 0. The delays achieved by the switching algorithm are slightly lower than the case where B is only implemented (2.34 ms and 2.4 ms respectively for 10 -5 PER). The maximum observed difference is 0.13 ms. By flexibly splitting the BS's functions, we save approximately 40% of the use of the second RU.

VII. CONCLUSION

In this paper, we took two different C-RAN architectures. In the first one, we chose the re-transmissions to be triggered in the RU and we adopted single reception. In the second one, the re-transmission decision was made in the CU and we adopted multiple receptions. It was shown that lower delays can be achieved, for the same reliability, when receiving from more than one BS. These lower delays are reached even if we have higher round trip time (RTT) per transmission. The spatial diversity reduces the number of transmissions needed to get a successful decoding. As a result, lower delays can be reached. We proposed to dynamically switch between A and B. We proved that by enabling the reception from two BSs just when needed, we can reach the aimed target: lower latency and higher reliability.

In future work, we can consider a combining diversity technique that provides more reliability such as maximum ratio combining (MRC).

Fig. 1 :

 1 Fig. 1: C-RAN architecture.

Fig. 2 :

 2 Fig. 2: Architecture A-One receiving BS.

Fig. 3 :

 3 Fig. 3: Architecture B-Multiple receiving BSs (M = 2).

5 .Fig. 4 :Fig. 5 :

 545 Fig. 4: 1 cycle delay (a) failure and (b) success case (A).

Fig. 7 :Fig. 8 :

 78 Fig. 7: Zone delimitation to switch between A and B.

Fig. 9 :

 9 Fig. 9: Probability distribution of the number of transmissions for A and B.

Fig. 10 :

 10 Fig. 10: Delay CCDF for A (with 1 BS) and B (with 2 BSs).

Fig. 11 :Fig. 12 :

 1112 Fig. 11: Probability distribution of the number of transmissions for different R th .

TABLE I :

 I Parameters values.

	Symbol	Parameter	Values
	Pt (dBm)	UE's transmission power	23
	r 0 (m)	Reference distance	0.2
	α	Path-loss exponent	3.38
	N (dBm)	Noise power	-116.45
	a [16]	Parameter depending on the MCS	274.7
	g [16]	Parameter depending on the MCS	7.993
	Rc (km)	Cell radius	3.2
	M	Number of receiving BSs	1 for A, 2/3 for B
	ρ (Km)	CU-RUs distance	3.5
	T D (ms)	Data transmission duration	1 TTI=0.25 a
	T A (ms)