# Hyperonic neutron stars: reconciliation between nuclear properties and NICER
and LIGO/VIRGO results

Luiz L. Lopes<EMAIL_ADDRESS>Centro Federal de Educação Tecnológica de
Minas Gerais Campus VIII; CEP 37.022-560, Varginha - MG - Brasil

###### Abstract

Using a extended version of the Quantum Hadrodynamics (QHD), I propose a new
microscopic equation of state (EoS) able to correctly reproduce the main
properties of symmetric nuclear matter at the saturation density, as well
produce massive neutron stars and satisfactory results for the radius and the
tidal parameter $\Lambda$. I show that even when hyperons are present, this
EoS is able to reproduce at least 2.00 solar masses neutron star. The
constraints about the radius of a $2.00M_{\odot}$ and the minimum mass that
enable direct Urca effect are also checked.

††preprint: APS/123-QED

##  I Introduction

Our knowledge of nuclear physics and nuclear astrophysics took a great leap in
the last decade. From nuclear masses analyze Wang , passing through nuclear
resonances Colo ; Reinhard , and heavy ion collisions (HIC) Pagano ; we are
able to constraint six parameters of the symmetric nuclear matter at the
saturation point: the saturation density itself ($n_{0}$), the effective
nucleon mass ($M_{N}^{*}/M_{N}$), the incompressibility ($K$), the symmetry
energy ($S_{0}$) and its slope ($L$), as well the binding energy per baryon
($B/A$) Glen .

On the other hand, quiescent analyzes Heinke  and Shapiro delay measurements
Antoniadis  constraint the radius of the canonical star, as well the minimum
mass that a microscopic EoS needs to reproduce, in order to be considered
valid. The constraints improved in the last years with the results coming for
the NICER x-ray telescope NICER1 , and also, from the analyzes of the so
called GW170817 event, detected by the LIGO/VIRGO gravitational wave
observatories GW17 . Moreover, the newly opened gravitational waves window,
give us an additional constraint: the dimensionless tidal deformation
parameter $\Lambda$. Another recent constraint is related to how squeezable
are the neutron stars. In the past, it was believed that very massive neutron
stars should be smaller than less massive ones. However, a very recent study
NICER2  shows that a two solar solar masses and a canonical mass of 1.4
$M_{\odot}$ have very similar radii. This fact puts additional constraint on
the microscopic EoS.

Nowadays one of the open issue in nuclear astrophysics is the content of the
inner core of massive neutron stars. Because of the Pauli principle, as the
number density increases, the Fermi energy of the nucleons exceeds the mass of
heavier baryons, and the conversion of some nucleons into hyperons become
energetically favorable. Among others, two extensive studies about the hyperon
threshold hyp2010 ; lopesnpa  show that hyperons are are ultimately
inevitable. Otherwise, it’s been well known for a long time that the hyperon
onset softens the EoS. In some cases, this softening of the EoS puts the
maximum mass below the observational limits of massive neutron stars. This
possible conflict between theory and observation is called hyperon puzzle.

The hyperon puzzle can be avoided assuming a very stiff EoS, as GM1 Glen2  or
NL3 Lala , which produce very massive neutron stars, despite the hyperon
threshold. However, due to the the refinement of the constraint in both,
nuclear properties, as well in astrophysical observations, these models cannot
be faced as realistic anymore.

In this work I present a new microscopic EoS via an extended version of the
QHD Lagrangian, where besides the traditional non-linear $\sigma\omega\rho$
mesons, also employs the strangeness hidden $\phi$ vector meson, which couples
only to the hyperons and a non-linear $\omega-\rho$ coupling, present in
models like the IUFSU IUFSU . Also, to fix the hyperon-meson coupling
constant, I use complete symmetry arguments as presented in ref. Lopes2013 .

The formalism, the parametrization and the constraints of symmetric nuclear
matter at the saturation point, and at supra-nuclear densities are presented
in section II; the astrophysical results and the astrophysical constraints are
presented in section III; and finally, the conclusions are drawn in section
IV.

##  II Formalism

The non-linear $\sigma\omega\rho$ QHD has the following Lagrangian Glen :

$\displaystyle\mathcal{L}_{QHD}=\sum_{B}\bar{\psi}_{B}[\gamma^{\mu}(i\partial_{\mu}-g_{B\omega}\omega_{\mu}-g_{B\rho}\frac{1}{2}\vec{\tau}\cdot\vec{\rho}_{\mu})+$
$\displaystyle-(M_{B}-g_{B\sigma}\sigma)]\psi_{B}-U(\sigma)+$
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{s}^{2}\sigma^{2})-\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\frac{1}{2}m_{v}^{2}\omega_{\mu}\omega^{\mu}+$
$\displaystyle+\frac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\;\mu}-\frac{1}{4}\bf{P}^{\mu\nu}\cdot\bf{P}_{\mu\nu},$
(1)

in natural units. The sum in $B$ can run only over the nucleons, or the entire
baryon octet; $\psi_{B}$ is the baryonic Dirac field. The $\sigma$,
$\omega_{\mu}$ and $\vec{\rho}_{\mu}$ are the mesonic fields. The
$g^{\prime}s$ are the Yukawa coupling constants that simulate the strong
interaction, $M_{B}$ is the baryon mass, $m_{s}$, $m_{v}$, and $m_{\rho}$ are
the masses of the $\sigma$, $\omega$, and $\rho$ mesons respectively. The
antisymmetric mesonic field strength tensors are given by their usual
expressions as presented in Glen . The $U(\sigma)$ is the self-interaction
term introduced in ref. Boguta  to fix the incompressibility, given by:

$U(\sigma)=\frac{\kappa
M_{N}(g_{\sigma}\sigma)^{3}}{3}+\frac{\lambda(g_{\sigma}\sigma)^{4}}{4},$ (2)

and , $\vec{\tau}$ are the Pauli matrices. Now, besides the traditional non-
linear $\sigma\omega\rho$ QHD, it is needed that we introduce two additional
terms. The first one is the strangeness hidden $\phi$ vector meson, which
couples only with the hyperons, not affecting the properties of symmetric
matter:

$\mathcal{L}_{\phi}=g_{Y,\phi}\bar{\psi}_{Y}(\gamma^{\mu}\phi_{\mu})\psi_{Y}+\frac{1}{2}m_{\phi}^{2}\phi_{\mu}\phi^{\mu}-\frac{1}{4}\Phi^{\mu\nu}\Phi_{\mu\nu},,$
(3)

as pointed in ref. lopesnpa ; Rafa2005 ; Lopes2020a ; Weiss1 , this vector
channel is crucial to obtain massive hyperonic neutron stars. The second one
is a non-linear $\omega$-$\rho$ coupling meson as present in the IUFSU model
IUFSU :

$\mathcal{L}_{\omega\rho}=\Lambda_{\omega\rho}(g_{\rho}^{2}\vec{\rho^{\mu}}\cdot\vec{\rho_{\mu}})(g_{\omega}^{2}\omega^{\mu}\omega_{\mu}),$
(4)

which is necessary to correct the slope of the symmetry energy ($L$) and has
strongly influence on the radii and tidal deformation of the neutron stars
Rafa2011 ; dex19jpg .

In order to produce beta stable matter, with zero net charge, I also add
leptons as a free Fermi gas. The detailed calculation of the EoS for symmetric
nuclear matter, as well for beta stable matter in the QHD formalism is out of
the scope of this work, but is well documented and can easily found in the
literature Glen ; Serot ; Menezes2021s . In the same sense, the calculation of
the six nuclear parameters at the saturation density
($n_{0},~{}M_{N}^{*}/M_{N},~{}K,~{}S_{0},~{}L,~{}B/A$) can be found in ref.
Glen ; Rafa2011  and the references therein.

###  II.1 Parametrization and nuclear constraints

The parametrization utilized in this work, as well the predictions of this
model for the symmetric nuclear matter are presented in Tab. 1. The nuclear
constraints at the saturation density are also in Tab. 1 and are taken from
two extensive review articles, ref. Dutra2014 ; Micaela2017 . Besides, the
masses of the particles are the physical ones. The meson masses are
$m_{\omega}$ = 783 MeV, $m_{\rho}$ = 770 MeV, $m_{\phi}$ = 1020 MeV,
$m_{\sigma}$ = 512 MeV, the baryon octed masses are $M_{N}$ = 939 MeV,
$M_{\Lambda}$ = 1116 MeV, $M_{\Sigma}$ = 1193 MeV, $M_{\Xi}$ = 1318 MeV, and
the lepton masses are $m_{e}$ = 0.51 MeV, $m_{\mu}$ = 105.6 MeV.

| Parameters |  | Constraints | This model
---|---|---|---|---
$(g_{N\sigma}/m_{s})^{2}$ | 12.108 $fm^{2}$ |  $n_{0}$ ($fm^{-3}$) | 0.148 - 0.170 | 0.156
$(g_{N\omega}/m_{v})^{2}$ | 7.132 $fm^{2}$ | $M^{*}/M$ | 0.6 - 0.8 | 0.69
$(g_{N\rho}/m_{\rho})^{2}$ | 4.801 $fm^{2}$ |  $K$ (MeV) | 220 - 260 | 256
$\kappa$ | 0.004138 |  $S_{0}$ (MeV) | 28.6 - 34.4 | 31.2
$\lambda$ | -0.00390 |  $L$ (MeV) | 36 - 86.8 | 74
$\Lambda_{\omega\rho}$ | 0.0185 |  $B/A$ (MeV) | 15.8 - 16.5 | 16.2
- | - |  $S(2n_{0})$ (MeV) | 38 - 64 | 52.4
- | - |  $p(2n_{0})$ (MeV/fm3) | 11.2 - 38.7 | 16.4
Table 1:  Parameters of the model utilized in this work and their prediction
for the symmetric nuclear matter at the saturation density; the
phenomenological constraints are taken from ref. Dutra2014 ; Micaela2017 ; s3
; tidal1 , as well the recent PREX2 results PREX2 .

As displayed, this new parametrization is able to fulfill all the constraints
at the saturation density. Notice also that the new PREX2 results points that
the slope is constrained between 106 $\pm$ 37 MeV PREX2 . Combined with ref.
Micaela2017 , the slope now lies between 69 and 86.8 MeV. The value $L=74$ MeV
fulfill it. As PREX2 still needs confirmation, I only point it as a curiosity.
Besides the saturation point, there are also a couple of tries in order to
constraint the pressure of symmetric nuclear matter at supra-nuclear
densities. In ref. Daniel , the pressure from 2 to 4.6 times the saturation
density was constrained by HIC analyzes. On the other hand, from transiently
accreting neutron stars in quiescence, ref. Steiner2013  constraint the
pressure for densities up to 1.0 $fm^{-3}$. The main problem is that for high
densities, the region constrained in ref. Steiner2013  is broader than the
region constrained in ref. Daniel . To overcome this issue, ref. Dutra2014
assume the region from ref. Daniel  plus an increase of 20$\%$. Here, I follow
this prescription and plot in Fig. 1 the density dependent pressure of the
presented model altogether with some well known relativistic parametrizations.
The hatched area in red is the constraint originally presented in ref. Daniel
. The increase by 20$\%$ is the hatched area in blue.

Figure 1: (Color online) The pressure as a function of the density; the
original constraint from ref. Daniel  in red and its value extended by 20$\%$
in blue, to reconcile with the constraint presented in ref. Steiner2013 .

As can be seen, the model is not flawless. If we assume the original
constraint in red, the EoS discussed in this work have a over-pressure, and
only fulfill this constraint in the range between 2.52 to 3.43 times the
saturation density, which correspond to 35$\%$ of the interval (2 - 4.6)
$n_{0}$. But, if we assume that the upper limit can be increased by 20$\%$, as
done in ref. Dutra2014 , therefore this EoS fulfill the constraint from 2 to
4.20 times the saturation density, which turns out to be 85$\%$ of the range.
When compared with others parametrization of the QHD, we see that the
presented model presents higher degree of agreement (85$\%$) in comparison
with very stiff EoS, as NL3 Lala  (0$\%$), FSU2H Tolos  (50$\%$), and GM1
Glen2  (68$\%$), but lower degree of agreement when compared with soft EoS as
GM3 Glen2  and IUFSU IUFSU  (both 100$\%$).

Another two recent constraints at supra-nuclear densities are the symmetry
energy at two times the saturation density, $S(2n_{0})$, and the pressure of
the symmetric matter at the same point $p(2n_{0})$. Ref. s3  bounds
$S(2n_{0})$ in the range between 51 $\pm$ 13 MeV at 68$\%$ confidence level,
while ref. tidal1  fixed $p(2n_{0})$ between 11.2 MeV/fm3 and 38.7 Mev/fm3 at
90$\%$ level. These values are also presented in Tab. 1, and the new
parametrization also agrees with both.

##  III Astrophysical results and constraints

Now I turn my attention to construct a beta stable, charge neutral EoS, with
and without hyperons. When hyperons are present, it is crucial to define the
strength of the hyperon-mesons coupling constants. The only well known
parameter is the $\Lambda^{0}$ hyperon potential depth, $U_{\Lambda}$ = -28
MeV. We can find in the literature several approaches to fix the hyperon-
mesons coupling constants, from universal couplings Glen2 , trough to fixed
potential depths lopesnpa ; Weiss2 , non-fixed potential depths Weiss1 ;
Lopes2013 ; Stone2021 ; Tsu , with and without symmetry group arguments
Rafa2011 ; Biswal  and so on. Here, I follow ref. Lopes2013  and assume that
both, vector and scalar mesons are constrained by SU(3) symmetry group Tsu ;
Swart . In this case, the relative strength of the hyperon-mesons coupling
constants are:

$\frac{g_{\Lambda,\omega}}{g_{N,\omega}}=\frac{4+2\alpha_{v}}{5+4\alpha_{v}};\quad\frac{g_{\Sigma,\omega}}{g_{N,\omega}}=\frac{8-2\alpha_{v}}{5+4\alpha_{v}};\quad\frac{g_{\Xi,\omega}}{g_{N,\omega}}=\frac{5-2\alpha_{v}}{5+4\alpha_{v}};$
(5)

for the $\omega$ meson,

$\displaystyle\frac{g_{\Lambda,\phi}}{g_{N,\omega}}=\sqrt{2}\bigg{(}\frac{2\alpha_{v}-5}{5+4\alpha_{v}}\bigg{)};\quad\frac{g_{\Sigma,\phi}}{g_{N,\omega}}=\sqrt{2}\bigg{(}\frac{-2\alpha_{v}-1}{5+4\alpha_{v}}\bigg{)};\quad$
$\displaystyle\frac{g_{\Xi,\phi}}{g_{N,\omega}}=\sqrt{2}\bigg{(}\frac{-2\alpha_{v}-4}{5+4\alpha_{v}}\bigg{)};$
(6)

for the $\phi$ meson,

$\frac{g_{\Lambda,\rho}}{g_{N,\rho}}=0;\quad\frac{g_{\Sigma,\rho}}{g_{N,\rho}}=2\alpha_{v};\quad\frac{g_{\Xi,\rho}}{g_{N,\rho}}=-(1-2\alpha_{v});$
(7)

for the $\rho$ meson, and finally:

$\displaystyle\frac{g_{\Lambda,\sigma}}{g_{N,\sigma}}=\frac{10+6\alpha_{s}}{13+12\alpha_{s}};\quad\frac{g_{\Sigma,\sigma}}{g_{N,\sigma}}=\frac{22-6\alpha_{s}}{13+12\alpha_{s}};\quad$
$\displaystyle\frac{g_{\Xi,\sigma}}{g_{N,\sigma}}=\frac{13-6\alpha_{s}}{13+12\alpha_{s}};$
(8)

for the $\sigma$ meson. When we set $\alpha_{v}$ = 1, we recover the SU(6)
parametrization for the vector mesons Pais . Now, to constraint $\alpha_{v}$
to $\alpha_{s}$ we use the well known $\Lambda^{0}$ potential depth.
Therefore, for a given value of $\alpha_{v}$, we determine $\alpha_{s}$ to
obtain $U_{\Lambda}$ = -28 MeV. This approach allow us to fix every hyperon-
meson coupling constant using only one free parameter: $\alpha_{v}$. Also, it
is worth to point that the results from eq. 5 to eq. 8 are fully model
independent. The results obtained presented in Tab. 2

$\alpha_{v}$ | 1.00 | 0.75 | 0.50
---|---|---|---
$\alpha_{s}$ | 1.582 | 1.240 | 0.911
Table 2: Relation of $\alpha_{v}$ with $\alpha_{s}$ in order to produce $U_{\Lambda}$ = -28 MeV. When $\alpha_{v}$ = 1.00 we recover the SU(6) symmetry group. |
---|---
Figure 2: (Color online) EoS (left) and the strangeness fraction $f_{s}$
(right) for different values of $\alpha_{v}$. Higher the value of
$\alpha_{v}$, higher is the value of $f_{s}$ and softer is the EoS.

In fig. 2, I plot the EoS for beta stable, charge neutral matter with and
without hyperons for different values of $\alpha_{v}$. I also plot the
strangeness fraction $f_{s}$ for each value of $\alpha_{v}$, once previous
works showed that its value is strongly related to the maximum mass Weiss2 ;
lopesnpa . The strangeness fraction is defined as:
($f_{s}=\sum(n_{i}|s_{i}|)/{3n}$) Rafa2011 . It is also worth to point that
the sequence of hyperon threshold for each value of $\alpha_{v}$ is always the
same: first one is the $\Lambda^{0}$, followed by the $\Sigma^{-}$ and the
$\Xi^{-}$, except when $\alpha_{v}$=0.50, where the $\Sigma^{-}$ is absent.

As expected, there is an inversely proportional relationship between the
stiffness of the EoS with the strangeness fraction at high densities. In turn,
the strangeness content of the EoS is directly linked to the value of
$\alpha_{v}$. Higher the value of $\alpha_{v}$, higher is the value of
$f_{s}$. Also, despite the value of $\alpha_{v}$, hyperons are always present,
corroborating the results present in ref. hyp2010 ; lopesnpa .

In relation to the neutron star properties and the astrophysical constraint,
one of the most robust constraint is the minimum mass that a EoS must
reproduce due to observational measures of massive stars via Shapiro delay.
Not long ago, the neutron star king was the PSR J0348+0432, with a mass of
2.01 $\pm 0.04$ $M_{\odot}$ Antoniadis . The old king was deposed, and now the
PSR J0740+6620 with a mass of approximately of 2.07 $\pm$ 0.07 $M_{\odot}$
NICER3  rules over the sky. Notwithstanding, both pulsars send the same
message: the EoS must be stiff enough to produce a two solar masses neutron
star. On the other hand, one of the most controversial contraint is the radius
of the canonical 1.4$M_{\odot}$. Just in the last couple of years, different
studies point to different values for its maximum allowed value. For instance,
a maximum value of 13.8 km, 14.2 km, 13.5 km, 12.6 km, 13.5 km, 12.8 km, 13.0
km, and 13.2 km was found respectively in ref. NICER1 ; NICER2 ; rad1 ; rad2 ;
rad3 ; rad4 ; rad5 ; rad6 . The average value of these studies is 13.32 km,
and I use it as a constraint.

Also, as pointed in ref. urcac , any acceptable EoS shall not allow the direct
Urca process to occur in NS with masses below 1.5$M_{\odot}$. The trigger to
nucleonic direct Urca channel is directly related to the leptonic fraction
$x_{DU}$. Rafa2011 ; urcac . On the other hand, when hyperons are present the
situation is more complex. Hyperons disfavours DU process due to the
deleptonization of the star, and the nucleonic DU process maybe forbidden.
However, channels for neutrino emission involving the hyperons are also
possible but with lower efficiency. As in this work the $\Lambda^{0}$ is
always the first and the most populous hyperon, I only consider its process.
The nucleonic, $x_{DU}=Y_{p}$ and hyperonic, $x_{\Lambda^{0}}=Y_{\Lambda}$
fraction to enable DU process are urcae :

$x_{DU}=\frac{1}{1+(1+x_{e}^{1/3})^{3}},\quad\mbox{and}\quad
x_{\Lambda^{0}}=0.032,$ (9)

where $x_{e}$ = $n_{e}/(n_{e}+n_{\mu})$. Notwithstanding, the efficiency of
the hyperonic DU process is only $4\%$ of the nucleonic one urcae .

Another important constraint comes from the GW170817 event, detected by the
LIGO/VIRGO gravitational wave telescopes: the dimensionless tidal deformation
parameter $\Lambda$. The tidal deformability of a compact object is a single
parameter that quantifies how easily the object is deformed when subjected to
an external tidal field. A larger tidal deformability indicates that the
object is easily deformable. On the opposite side, a compact object with a
smaller tidal deformability parameter is smaller, more compact, and it is more
difficult to deform it. Its value is defined as: $\Lambda={2k_{2}}/{3C^{5}}$,
where C = ($GM/R$) is the compactness. The parameter $k_{2}$ is called Love
number and is related to the metric perturbation. A complete discussion about
the Love number and its calculation is both, very extensive and well
documented in the literature. Therefore, it is out of the scope of this work.
I refer the interested reader to see tidal1 ; tidal2 ; tidal3 ; tidal4  and
the references therein. Ref. tidal1  constraint the dimensionless tidal
parameter for the canonical mass, $\Lambda_{1.4}$ in the range of 70-580.
Here, I use 580 as a superior limit.

|
---|---
Figure 3: (Color online) Mass radius relation (left) and the dimensionless
tidal parameter $\Lambda$ (right). The hatched area on the left (right) is
related to the uncertain about the mass of PSR J0740+6620 (uncertain about the
value of $\Lambda_{1.4}$ in the GW170817 event tidal1 ).

Finally, a very recent constraint comes from the measure of the radius of the
massive pulsar PSR J0740+6620 made by the NICER x-ray telescope. This is
related to how squeezable are the neutron stars and allows us to put a
inferior limit to the radius of a two solar mass object. A inferior bound of
11.4 km, 12.2 km, 11.6 km and 12.2 km was found respectively in ref. NICER3 ;
s1 ; s2 ; s3 . An average value of 11.85 km arises and I also use this value
as a constraint to the EoS.

In fig. 3, I display both, the TOV equation solution TOV , as well the
calculated value for the dimensionless tidal parameter $\Lambda$. The BPS EoS
is used to modeling the neutron star crust BPS . The hatched areas are the
uncertain about the mass value of PSR J0740+6620 NICER3 , as well the
uncertain about the value of $\Lambda_{1.4}$ tidal1 . The main neutron star
properties, and the astrophysical constraints are also displayed at Tab. 3.

$\alpha_{v}$ |  1.00 |  0.75 |  0.50 | No hyp. | Constraint
---|---|---|---|---|---
$M_{max}/M_{\odot}$ | 2.00 | 2.07 | 2.15 | 2.30 |  $>$ 2.00
$f_{s}$ ($M_{max}$) | 0.173 | 0.151 | 0.110 | 0.000 | -
$n_{max}$ (fm-3) | 1.01 | 1.02 | 1.00 | 0.94 | -
$R_{1.4}$ (km) | 12.96 | 12.96 | 12.96 | 12.96 |  $<$ 13.32
$\Lambda_{1.4}$ | 527 | 527 | 527 | 527 |  $<$ 580
$R_{2.0}$ (km) | 11.27 | 12.07 | 12.28 | 12.58 |  $>$ 11.85
$f_{s}$ (2.0$M_{\odot}$) | 0.173 | 0.103 | 0.064 | 0.000 | -
$n_{DU}$ (fm-3) | 0.42* | 0.43* | 0.43* | 0.47 | -
$M_{DU}/M_{\odot}$ | 1.50* | 1.52* | 1.55* | 1.73 |  $>$ 1.50
Table 3: Some neutron star properties and constraints for different values of
$\alpha_{v}$. Results marked with * indicates hyperonic DU process, with only
4$\%$ of the efficiency of the nucleonic DU.

As can be seen, even in the presence of hyperons, we are to produce a two
solar mass neutron star. Also, as hyperons are not present at masses around
1.4$M_{\odot}$, the $R_{1.4}$ and $\Lambda_{1.4}$ always give the value of
12.96 km and 527 respectively. This then means that besides correctly
reproduce the six parameter of symmetric nuclear matter, this EoS also fulfill
the main astrophysical constraints: $M_{max}>2.00M_{\odot}$, $R_{1.4}<13.32$,
$M_{DU}>1.50M_{\odot}$, and $\Lambda_{1.4}<580$. The only astrophysical
constraint that is not satisfied for all models is the recent measure of the
radius of the PSR J0740+6620, as $\alpha_{v}$ = 1.00 produces too low radius
for a 2.00$M_{\odot}$. Indeed, as can be seen in Tab. 3, there is a relation
between the maximum possible mass and the radius of the 2.00$M_{\odot}$.
Higher is the maximum mass, higher is the radius of 2.00$M_{\odot}$. This
relation can be explained due to the different strangeness content in the core
of the 2.00$M_{\odot}$ star. Higher the value of $f_{s}$, softer is the EoS,
lower is the radius of the 2.00$M_{\odot}$ as well the maximum mass. Also,
when hyperons are present, the nucleonic DU process is always forbidden and
only the hyperonic DU process is present. Without hyperons, nucleonic DU
happens for masses above 1.73$M_{\odot}$, which agrees with ref. urcac .

##  IV Final Remarks

In this work I build a new EoS which is able to satisfy all six main
constraint at the nuclear saturation point. This EoS also is able to fulfil
the experimental constraint about the pressure at supra-nuclear densities for
85$\%$ of the interval presented in the combined ref. Daniel ; Steiner2013
and produce good values for $S(2n_{0})$ and $p(2n_{0})$, in agreement with
ref. s3 ; tidal1 .

In the realm of astrophysical constraint, this EoS is able to reproduce masses
above 2.00$M_{\odot}$ even when hyperons are present. It also produces
$R_{1.4}~{}<~{}13.32$, km $M_{DU}>1.50M_{\odot}$, $\Lambda_{1.4}~{}<~{}580$
and $R_{2.0}~{}>11.85$, km; thus satisfying all the main astrophysical bounds.

Before I finish, it is worth to point that PREX2 experiments has pointed for
much lager values for both, the slope $L$, as well the tidal parameter
$\Lambda_{1.4}$ and the radius of the canonical star PREX2 . These results can
be far beyond the most acceptable values coming from several different studies
and still needs confirmation. Nevertheless, if the PREX2 results turn out to
be true, that does not imply in a strong modification in the presented EoS.
Only the $g_{\rho,N}$ coupling constant and the $\Lambda_{\omega\rho}$
parameter will needed to be modified. The same is true if the slope needs to
be lowered.

Acknowledgments: The author is eternally grateful to Debora P. Menezes for the
many (MANY) English correction, and for almost seven years of tutelage.

## References

  * (1) M. Wang, et al., Chin. Phys. C 36, 1603 (2012)
  * (2) S. Shlomo, V. Kolomietz, G. Colo, Eur. Phys. J. A 30, 23 (2006)
  * (3) P. Reinhard and W. Nazarewicz, Phys. Rev. C 87, 014324 (2013)
  * (4) E. De Filippo and A. Pagano, Eur. Phys. J. A 50, 32 (2014)
  * (5) N. K. Glendenning, _Compact Stars_ , Springer, New York - Second Edition (2000)
  * (6) C. Heinke et al., Mon. Not. Roy. Astron. Soc.444, 443 (2014)
  * (7) J. Antoniadis et al., Science 340, 1233232 (2013).
  * (8) T. Riley et al., Astroph. J. Lett. 887, L21 (2019)
  * (9) B. Abbot et al., Phys. Rev. Lett. 119, 161101 (2017)
  * (10) T. Riley et al., arXiv:2105.06980
  * (11) H. Dapo, B. Schaefer, J. Wambach, Phys. Rev. C 81, 035803 (2010)
  * (12) L. Lopes, D. Menezes, Nucl. Phys. A 1009, 122171 (2021)
  * (13) N. K. Glendenning, S. A. Moszkowski, Phys. Rev. Lett. $\mathbf{67}$, 2414 (1991)
  * (14) G. A. Lalazissis, J. Konig, P. Ring, Phys. Rev. C $\mathbf{55}$, 540 (1997)
  * (15) F. Fattoyev et al., Phys. Rev. C 82, 055803 (2010)
  * (16) L. L. Lopes, D. P. Menezes, Phys. Rev. C 89, 025805 (2013)
  * (17) J. Boguta, A. R. Bodmer, Nucl. Phys. A 292, 413 (1977)
  * (18) R. Cavagnoli, D.P. Menezes, Braz. J. Phys. 35, 869 (2005)
  * (19) L. Lopes, D. Menezes, Eur.Phys.J. A 56, 122 (2020)
  * (20) S. Weissenborn, D. Chatterjee, J. Schaffner-Bielich, Nucl. Phys. A 881, 62 (2012)
  * (21) R. Cavagnoli, D. P. Menezes, C. Providencia, Phys. Rev. C 84, 065810 (2011)
  * (22) V. Dexheimer et. al, J. Phys. G 46, 034002 (2019)
  * (23) B. D. Serot, Rep. Prog. Phys. $\mathbf{55}$, 1855 (1992)
  * (24) D. Menezes, arXiv:2106.09515
  * (25) M. Dutra et al., Phys. Rev. C 90, 055203 (2014)
  * (26) M. Oertel et al., Rev. Mod. Phys. 89, 015007 (2017)
  * (27) B. Li et al., Universe 7, 182 (2021)
  * (28) B. Abbott et al., Phys. Rev. Lett. $\mathbf{121}$, 161101 (2018)
  * (29) B. Reed, et al., Phys. Rev. Lett. $\mathbf{126}$, 172503 (2021)
  * (30) P. Danielewicz et al., Science 298, 1592 (2002)
  * (31) A. Steiner, J. Lattimer, E. Brown, Astropys. J. Lett. 765, L5 (2013)
  * (32) R. Negreiros et al., Astrophys. J. 863, 104 (2018)
  * (33) S. Weissenborn, D. Chatterjee, J. Schaffner-Bielich, Phys. Rev. C 85, 065802 (2012)
  * (34) J. Stone, et al., Mon. Not. Roy. Astron. Soc.502, 3476 (2021)
  * (35) T. Miyatsu and M. Cheoun† Phys. Rev. C 88, 015802 (2013)
  * (36) S. Biswal, S.Patra, S. Zhou, Astrophys. J. 885, 25 (2019)
  * (37) J. J. Swart, Rev. Mod. Phys. 35, 916 (1963)
  * (38) A. Pais, Rev. Mod. Phys. 38, 215 (1966)
  * (39) T. Riley et al, arXiv:2105.06980
  * (40) M. Cougghllin, et al., Mon. Not. Roy. Astron. Soc. Lett. 489, L91 (2019)
  * (41) T. Malik, et al., Phys. Rev. C 99, 052801(R) (2019)
  * (42) A. Mondal, G. Dewangan, B. Raychaudhuri, Mon. Not. Roy. Astron. Soc. 487, 5441 (2019)
  * (43) B. Kumar and P. Landry, Phys. Rev. D 99, 123026 (2019)
  * (44) M. Fasano et al., Phys. Rev. Lett. $\mathbf{123}$, 141101 (2019)
  * (45) Y. Li et al., Eur.Phys.J. A 57, 31 (2021)
  * (46) T. Klahn et al., Phys. Rev. C 74, 035802 (2006)
  * (47) M. Prakash et al., Astrophys. J. Lett. 390, L77 (1992)
  * (48) C. Flores et al., Eur. Phys. J. C 80, 1142 (2020)
  * (49) K. Chatziioannou, Gen. Rel. Grav. 52, 109 (2020)
  * (50) T. Hinderer, Astrophys. J. 677, 1216 (2008)
  * (51) N. Zhang and B. Li, arXiv:2105.11031
  * (52) M. Miller et al., arXiv:2105.06979
  * (53) J. R. Oppenheimer, G. M. Volkoff, Phys. Rev. 55, 374 (1939)
  * (54) G. Baym, C. Pethick, P. Sutherland, Astrophys. J. 170, 299 (1971)


# A Concept of Assessment of LIV Tests with THESEUS Using the Gamma-Ray Bursts
Detected by Fermi/GBM

Anastasia Tsvetkova1,2,*1,2,*affiliationmark:  , Luciano Burderi11affiliation:
Dipartimento di Fisica, Universitá degli Studi di Cagliari, SP Monserrato-
Sestu, km 0.7, I-09042 Monserrato, Italy 33affiliation: INFN, Sezione di
Cagliari, Cittadella Universitaria, 09042 Monserrato, CA, Italy 44affiliation:
INAF—Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy , Alessandro Riggio11affiliation: Dipartimento di
Fisica, Universitá degli Studi di Cagliari, SP Monserrato-Sestu, km 0.7,
I-09042 Monserrato, Italy 33affiliation: INFN, Sezione di Cagliari, Cittadella
Universitaria, 09042 Monserrato, CA, Italy 55affiliation: INAF—Istituto di
Astrofisica Spaziale e Fisica Cosmica di Palermo, Via U. La Malfa 153, 90146
Palermo, Italy , Andrea Sanna11affiliation: Dipartimento di Fisica, Universitá
degli Studi di Cagliari, SP Monserrato-Sestu, km 0.7, I-09042 Monserrato,
Italy 33affiliation: INFN, Sezione di Cagliari, Cittadella Universitaria,
09042 Monserrato, CA, Italy 44affiliation: INAF—Osservatorio Astronomico di
Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy , Tiziana Di
Salvo66affiliation: Dipartimento di Fisica e Chimica, Universitá degli Studi
di Palermo, Via Archirafi 36, 90123 Palermo, Italy

###### Abstract

According to Einstein’s special relativity theory, the speed of light in a
vacuum is constant for all observers. However, quantum gravity effects could
introduce its dispersion depending on the energy of photons. The investigation
of the spectral lags between the gamma-ray burst (GRB) light curves recorded
in distinct energy ranges could shed light on this phenomenon: the lags could
reflect the variation of the speed of light if it is linearly dependent on the
photon energy and a function of the GRB redshift. We propose a methodology to
start investigating the dispersion law of light propagation in a vacuum using
GRB light curves. This technique is intended to be fully exploited using the
GRB data collected with THESEUS.

gamma-ray bursts: general – methods: data analysis – gamma-ray bursts as
cosmological probes and test-bench for fundamental physics – gamma-ray bursts:
past, present and future experiments and missions

††slugcomment: Accepted by MDPI Universe22affiliationtext: Ioffe Institute,
Politekhnicheskaya 26, St. Petersburg 194021, Russia**affiliationtext:
<EMAIL_ADDRESS>
##  1 Introduction

According to Einstein’s special relativity theory, a proper length is Lorentz-
contracted by a factor of $\gamma^{-1}=[1-(v/c)^{2}]^{1/2}$ as observed from
the reference frame moving at speed $v$ relative to the rest frame. However,
various spacetime theories, e.g., some string or loop quantum gravity theories
(see, e.g., Rovelli & Smolin (1988, 1990); Rovelli (1998)), imply the
existence of a minimum spatial length of the order of the Plank length
$l_{\textrm{Pl}}=\sqrt{G\hbar/c^{3}}=1.6\times 10^{-33}$ cm Hossenfelder
(2013). Lorentz invariance is a fundamental property of both the standard
model of particle physics and general relativity. In general relativity, a
locally inertial reference frame where the Lorentz symmetry is fulfilled can
always be chosen. Some quantum gravity (QG) theories predict the Lorentz
invariance violation (LIV) at the Planck energy scale
($E_{\textrm{Pl}}=\sqrt{\hbar c^{5}/G}\simeq 1.22\times 10^{19}$ GeV) as there
exists a minimum spatial length $l_{\textrm{min}}=\alpha l_{\textrm{Pl}}$
(where $\alpha\sim 1$ is a dimensionless constant inherent to a particular
spacetime theory), which, e.g., in the string theories, corresponds to the
string length. Therefore, the Lorentz contraction is limited by this spatial
scale (see Hossenfelder (2013) for a review).

There are several frameworks implying LIV, e.g., string theory Kostelecký &
Samuel (1989a, b), noncommutative spacetime Carroll et al. (2001); Ferrari et
al. (2007), Brane worlds Santos & Almeida (2013), Hořava–Lifshitz gravity
Hořava (2009). LIV was considered in the gravitational context for the first
time in Kostelecký (2004), where the so-called standard model extension was
developed. The Bumblebee111The name of the model was inspired by the insect,
whose ability to fly has been questioned theoretically. models, which are the
simplest cases of theories including the spontaneous breaking of Lorentz
symmetry, are effective field QG theories describing a vector field with a
non-zero vacuum expectation value and involving the vacuum condensate
Kostelecký & Samuel (1989a, b). The spontaneous symmetry breaking preserves
both the geometric constraints and conservation laws or quantities required by
the general relativity theory or Riemannian geometry. Regarding gravity, LIV
can happen if a vector field ruled by a potential exhibiting a minimum rolls
to its vacuum expectation value, similar to the Higgs mechanism Kostelecký
(2004). This “bumblebee” vector thus takes an explicit (four-dimensional)
orientation, and preferred-frame effects may emerge Bertolami & Páramos
(2005). The generalized uncertainty principle (GUP) states that, in quantum
theory, if the quantities are incompatible, they are mutually dependent, and
measuring one observable may yield some information about its incompatible
partners. GUP is based on a momentum-dependent modification of the standard
dispersion relation, which is supposed to produce LIV Tawfik et al. (2016);
Lambiase & Scardigli (2018). See, e.g.,Kanzi & Sakallı (2019); Övgün et al.
(2019); Kanzi & Sakallı (2021); Delhom et al. (2021); Gogoi & Dev Goswami
(2022, 2023); Neves (2023) for recent advances in the Bumblebee and GUP
models.

One of the LIV effects, relevant to astrophysics, is the existence of a
dispersion law for the photon speed $c$ (see, e.g., Amelino-Camelia (2000)).
However, LIV is not a mandatory property of all QG theories: in some of them,
e.g., in the spacetime uncertainty principle Burderi et al. (2016) or in the
quantum spacetime Sanchez (2019), LIV is not expected. Despite photon velocity
dispersion, the Lorentz invariance is not violated and the dispersion law is a
second-order effect relative to the ratio of the photon energy to the QG
energy scale. To obtain an idea of the nature of the speed of light dispersion
due to LIV, e.g., within the Liouville string approach, one can consider a
vacuum as a non-trivial medium containing “foamy” quantum gravity fluctuations
whose origin can be imagined as processes that involve the pair creation of
virtual black holes. In this concept, one can verify that the massless
particles of different energies can excite vacuum fluctuations differently as
they propagate through the quantum gravity medium, producing a non-trivial
dispersion relation of Lorentz “non-covariant” form, similarly to the thermal
medium Amelino-Camelia et al. (1998a). For more details regarding this
concept, see, e.g., Amelino-Camelia et al. (1997).

Since gamma-ray bursts (GRBs) are characterized by high-energy emission, large
cosmological distances, and temporal variability at short ($\lesssim$10 ms)
timescales, they have been applied as powerful tools in LIV searches (see,
e.g., Amelino-Camelia et al. (1998a); Ellis et al. (2003, 2006); Abdo et al.
(2009a); Vasileiou et al. (2015); Zhang & Ma (2015); Pan et al. (2015); Xu &
Ma (2016a, b); Chang et al. (2016); Wei et al. (2017); Ganguly & Desai (2017);
Liu & Ma (2018); Zou et al. (2018); Ellis et al. (2019); Wei (2019); Pan et
al. (2020); Acciari et al. (2020); Du et al. (2021); Agrawal et al. (2021);
Wei & Wu (2021); Bartlett et al. (2021); Xiao et al. (2022); Desai et al.
(2023)) for more than two decades. In Amelino-Camelia et al. (1998a, b), it
was first suggested to test LIV using a comparison of the arrival times of GRB
photons detected in distinct energy ranges. In Abdo et al. (2009a); Vasileiou
et al. (2013, 2015), the authors exploited the spectral lag between high-
energy ($\sim$31 GeV) and low-energy photons of GRB 090510 to derive lower
limits on the linear and quadratic QG energy: $E_{\textrm{QG,1}}>(1-10)\times
E_{\textrm{Pl}}=1.22\times(10^{19}-10^{20})$ GeV and
$E_{\textrm{QG,2}}>(1-10)\times E_{\textrm{Pl}}=1.3\times 10^{11}$ GeV,
respectively. In Abdo et al. (2009b), $E_{\textrm{QG}}>10^{18}$ GeV was
obtained based on the observation of GRB 080916C. For GRB 190114C MAGIC
Collaboration et al. (2019), the linear and quadratic LIV were constrained
using the time delays of TeV photons: $E_{\textrm{QG,1}}>0.58\times 10^{19}$
GeV ($E_{\textrm{QG,1}}>0.55\times 10^{19}$ GeV) and
$E_{\textrm{QG,2}}>0.63\times 10^{11}$ GeV ($E_{\textrm{QG,1}}>0.56\times
10^{11}$ GeV) for the subluminal (superluminal) case Acciari et al. (2020). In
Liu et al. (2022), the authors constrained the linear and quadratic LIV for
the set of 32 Fermi/GBM GRBs with known redshifts and characterized by the
positive-to-negative transition of the spectral lag:
$E_{\textrm{QG,1}}=1.5\times 10^{14}$ GeV for the linear case and
$E_{\textrm{QG,2}}\leavevmode\nobreak\ =\leavevmode\nobreak\
8\leavevmode\nobreak\ \times\leavevmode\nobreak\ 10^{5}$ GeV for the quadratic
case.

The phenomenon of GRBs remains puzzling, although much progress has been made.
Both the light curves and the spectra among GRBs vary significantly. It is
generally believed that collisions between relativistic shells ejected from an
active central engine produce pulses in GRB light curves Rees & Meszaros
(1994). The collision of the slower-moving shell with the second, faster shell
ejected later produces a shock that dissipates internal energy and accelerates
the particles that emit the GRB radiation. As of now, two “physical” classes
of GRBs are distinguished (see, e.g., Zhang et al. (2009)): the merger-origin
Type I GRBs Blinnikov et al. (1984); Paczynski (1986); Eichler et al. (1989);
Paczynski (1991), which are usually short, with a duration of less than 2 s
Mazets et al. (1981); Kouveliotou et al. (1993), and spectrally hard, and the
collapser-origin Type II GRBs Woosley (1993); Paczyński (1998); MacFadyen &
Woosley (1999); Woosley & Bloom (2006), characterized by longer durations.

The spectral lag, which is known as the difference in arrival time between
high-energy and low-energy photons, is a common phenomenon occurring during
the GRB prompt emission phase Norris et al. (1986, 2000); Band (1997); Chen et
al. (2005) and in high-energy astrophysics in general (e.g., Norris et al.
(2000); Zhang et al. (2002)). The authors of Cheng et al. (1995) were the
first to analyze the spectral lags of GRBs. It was found that a soft lag,
i.e., the hard photons arriving first, dominates in long GRBs Norris et al.
(2000); Wu & Fenimore (2000); Chen et al. (2005); Norris et al. (2005), and
some GRBs have significantly different spectral lags in early and late epochs
Hakkila & Giblin (2004). It was shown that the lags are correlated with the
GRB luminosity Norris et al. (2000) and the jet break times in afterglow light
curves Salmonson & Galama (2002); the spectral lags of long GRBs are
correlated with the pulse duration, while the spectral lags of short GRBs are
not Yi et al. (2006). The GRB spectral lags can be explained within several
physical models, such as the curvature effect of a relativistic jet and
rapidly expanding spherical shell Ioka & Nakamura (2001); Shen et al. (2005);
Lu et al. (2006); Shenoy et al. (2013); Uhm & Zhang (2016). Regardless of its
physical origin, a spectral lag is an important GRB parameter as it may help
to distinguish between long and short GRBs: long bursts have large lags, while
short bursts have relatively smaller or negligible lags Norris (1995); Ryde
(2005); Norris & Bonnell (2006).

To show the feasibility of the search for the dispersion of the speed of the
light in the data collected by THESEUS Amati et al. (2018), one can start
performing similar research using the data of already commissioned missions.
In this paper, we describe a methodology for the testing of the dispersion of
the light speed, and we intend to apply it to the Fermi/GBM data and the set
of GRBs with known redshifts. The paper is organized as follows. We start with
a brief description of the instrumentation and data in Section 2. The
methodology that involves constraining LIV using GRBs is described in Section
3. In Section 4, we discuss the THESEUS mission in the context of the LIV
tests using GRBs. Section 5 concludes the paper.

##  2 Instrumentation and Data

Among several space-based detectors able to collect GRB data, the most
prolific are Swift/BAT (BAT; Gehrels et al. (2004)), Fermi/GBM (GBM; Meegan et
al. (2009)), and Konus-Wind (KW; Aptekar et al. (1995)). More details
regarding the BAT, GBM, KW, and other GRB detectors’ design and performance
can be found in Tsvetkova et al. (2022). However, since Swift/BAT collects GRB
data in a narrow energy band of 15–350 keV, and Konus-Wind records GRB light
curves in three fixed energy windows, the Fermi/GBM GRB time histories seem to
be the most suitable for the testing of the speed of light variance.

Launched in June 2008, the Fermi Gamma-Ray Space Telescope Thompson & Wilson-
Hodge (2022) harbors two scientific instruments: the Gamma-Ray Burst Monitor
(GBM) and the Large Area Telescope (LAT; Atwood et al. (2009)). The LAT covers
the 30 MeV–300 GeV band, while the GBM, intended to detect and study GRBs, is
sensitive within the 8 keV–30 MeV energy range, extending the spectral band
over which bursts are observed downwards to the hard X-ray range. GBM
comprises twelve NaI(Tl) detectors covering an energy range of 8 keV-1 MeV and
two bismuth-germanate (BGO) scintillation detectors sensitive within the 150
keV to 30 MeV band that observe the whole sky not occluded by the Earth (¿8
sr).

The primary scientific data produced by GBMs can be summarized as a time
history and spectra, which are provided as temporally pre-binned (CTIME and
CSPEC) or temporally unbinned time tagged events (TTE). These data types are
produced as “snippets” for every trigger and are also provided continuously.
The CTIME data are collected in 8 energy channels with a 256 ms time
resolution, while the CSPEC data are recorded in 128 channels with an 8.192 s
time resolution. TTEs for each detector are recorded with time precision down
to 2 $\mu$s, in 128 energy channels, matching the CSPEC ones, which gives an
excellent opportunity to bin the data in time and energy in a suitable way.
From 2008 through November 2012, TTE were available only during a 330 s
interval: from 30 s before the burst trigger to 300 s after the burst trigger.
Since November 2012, GBM flight software has produced a new data type,
continuous TTE (CTTE), available at all times that the instrument is
operating.

To date, the GBM has triggered almost 3500 GRBs, among which almost 3000 have
$T_{90}>2$ s.

##  3 Methodology

We start the research by computing the redshifts and the rest-frame spectral
lags for all GBM-triggered GRBs. Then, thanks to the central limit theorem, we
can consider the distribution of the lags as normal and compute the mean and
variance. The large number of “measurements” is expected to significantly
increase the accuracy of the spectral lags and constraints on
$E_{\textrm{QG}}$. Another approach would be to constrain the QG energy (see
Section 3.1) first and then estimate its mean values and variance.

###  3.1 Brief Introduction to the Basics of the Speed of Light Variance

The velocity dispersion law for photons can be expressed as a function of its
observer-frame energy $E_{\textrm{obs}}$ in units of the QG energy scale
$E_{\textrm{QG}}$, at which the quantum nature of gravity becomes important:

$v_{\textrm{phot}}/c-1=\xi\left(\frac{E_{\textrm{obs}}}{E_{\textrm{QG}}}\right)^{n},$
(1)

where $v_{\textrm{phot}}$ is the group velocity of a photon wave-packet,
$E_{\textrm{QG}}=\zeta m_{\textrm{Pl}}c^{2}=\zeta E_{\textrm{Pl}}$,
$m_{\textrm{Pl}}=2.176\times 10^{-5}$ g is the Planck mass,
$\zeta\sim\alpha^{-1}\sim 1$ expresses the significance of the QG effects,
$\xi\sim\pm 1$ is a dimensionless constant inherent to a particular QG theory,
and the index $n$ denotes the order of the first relevant term of the small
parameter $\left(\frac{E_{\textrm{obs}}}{E_{\textrm{QG}}}\right)$. This
expression takes into account that high-energy photons can travel faster
(superluminal, $\xi=+1$) or slower (subluminal, $\xi=-1$) than low-energy ones
Amelino-Camelia & Smolin (2009).

The difference in the arrival times of photons emitted at the same time in the
same place is

$\Delta
t_{\textrm{QG}}=\xi\left(\frac{D_{\textrm{trav}}}{c}\right)\left(\frac{\Delta
E_{\textrm{obs}}}{E_{\textrm{QG}}}\right)^{n},$ (2)

where $D_{\textrm{trav}}$ is the comoving distance traversed by a massless
particle, emitted at redshift $z$ and traveling down to redshift 0.

###  3.2 Observed Spectral Lag as a Function of Redshift

For a given observer-frame energy $E_{\rm obs}$, the total spectral lag
$\tau_{\rm total,\,obs}(E_{\rm obs},z)$ can be split into two terms:

$\tau_{\rm total,\,obs}(E_{\rm obs},z)=\tau_{\rm int,\,obs}(E_{\rm
obs},z)+\tau_{\rm QG,\,obs}(E_{\rm obs},z),$ (3)

where $\tau_{\rm int,\,obs}$ is the observed intrinsic spectral lag, which
corresponds to the intrinsic rest-frame lag

$\tau_{\rm int,\,rf}(E_{\textrm{obs}})=\tau_{\rm int,\,obs}(E_{\rm rf})/(1+z)$
(4)

induced by the GRB central engine emission mechanism and assumed to be
independent of the photon source redshift $z$. $\tau_{\rm QG,\,obs}$ is the
lag induced by the QG effects discussed above, and and $E_{\textrm{rf}}$ is
the photon energy in the rest frame of its source. Following Jacob & Piran
(2008), $\tau_{\rm QG,\,obs}$ can be expressed as a function of the GRB rest-
frame energy:

$\begin{split}\tau_{\rm QG}(E_{\rm
rf},z)=\xi\left(\frac{1}{H_{0}}\right)\left(\frac{E_{\rm rf}}{\zeta E_{\rm
pl}}\right)^{n}\left(\frac{1+n}{2}\right)\left(\frac{1}{1+z}\right)^{n}\times\\\
\times\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}{,}\end{split}$
(5)

where $H_{0}$ is the Hubble constant, $\Omega_{m}$ is the matter density
parameter, and $\Omega_{\Lambda}$ is the dark energy density parameter, i.e.,
the parameters of the standard $\Lambda$CDM model.

Experimentally, the total observed spectral lag $\tau_{\rm total,\,obs}(E_{\rm
rf},z)$ can be computed by cross-correlating the GRB light curves recorded in
the redshift-dependent energy windows corresponding to the fixed rest-frame
energy windows as $E_{\rm obs}=E_{\rm rf}/(1+z)$, with the GRB light curve
collected at the lowest possible energy channel, where the lag induced by QG
is negligible, e.g., it is $\sim$$\mu$s in the 5–20 keV energy range, while,
in the higher-energy bands, it is $\sim$ms.

The total observed spectral lag $\tau_{\rm total,\,obs}(E_{\rm rf},z)$ should
follow the relation obtained by inserting Equations (4) and (5) into Equation
(3):

$\begin{split}\tau_{\rm total,\,obs}(E_{\rm rf},z)=\tau_{\rm int,\,obs}(E_{\rm
rf})+\\\ \xi\left(\frac{1}{H_{0}}\right)\left(\frac{E_{\rm rf}}{\zeta E_{\rm
pl}}\right)^{n}\left(\frac{1+n}{2}\right)\left(\frac{1}{1+z}\right)^{n}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.\end{split}$
(6)

The transformation of spectral lags from the observer frame back to the rest
frame222This value reflects only the mathematical transformation from the
observer frame to the rest frame as, at the moment of emission of two photons,
the QG lag between them equals zero. can be performed by dividing by the
redshift factor $(1+z)$:

$\tau_{\rm total,\,rf}(E_{\rm rf},z)=\tau_{\rm total,\,obs}(E_{\rm
rf},z)/(1+z).$ (7)

Thus, the GRB rest-frame spectral lag obeys the following relation:

$\begin{split}\tau_{\rm total,\,rf}(E_{\rm rf},z)=\frac{\tau_{\rm
int,\,obs}(E_{\rm rf})}{(1+z)}+\\\
\xi\left(\frac{1}{H_{0}}\right)\left(\frac{E_{\rm rf}}{\zeta E_{\rm
pl}}\right)^{n}\left(\frac{1+n}{2}\right)\left(\frac{1}{1+z}\right)^{n+1}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.\end{split}$
(8)

Let us define a function

$\begin{split}u(z)=\left(\frac{1+n}{2}\right)\left(\frac{1}{1+z}\right)^{n+1}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.\end{split}$
(9)

Then, the experimentally determined lag $\tau_{\rm total,\,obs}(E_{\rm rf},z)$
will follow the relation

$\begin{split}\tau_{\rm total,rf}(E_{\rm rf},z)=\tau_{\textrm{int,rf}}(E_{\rm
rf})+\xi\left(\frac{1}{H_{0}}\right)\left(\frac{E_{\rm rf}}{\zeta E_{\rm
pl}}\right)^{n}u(z).\end{split}$ (10)

The dependence of $\tau_{\rm total,\,rf}(E_{\rm rf},z)$ on $u(z)$ is expected
to be linear, with the intercept corresponding to the intrinsic lag and the
slope proportional to the ratio of the rest-frame photon energy to the QG
energy $\zeta E_{\rm pl}$ raised to the $n$-th power, which is the first
significant term in the series expansion of the quantum gravity dispersion
relation. An argument in support of the independence of $\tau_{\rm int,\,rf}$
from $z$ is the absence of the prominent cosmological evolution of GRB
energetics Tsvetkova et al. (2017, 2022), which indicates that the GRB central
engine does not evolve significantly with $z$. However, if $\tau_{\rm
int,\,rf}$ is dependent on $z$, one can fit both $\tau_{\rm int,\,rf}(z)$ and
$\tau_{\textrm{QG}}(z)$ simultaneously to the data, as was done in Liu et al.
(2022).

In Liu et al. (2022), it was found that the behavior of the lags follows some
key statistical properties described in terms of log-normal and Gaussian
distributions around average values. These empirical functions describing the
spectral lags (as a function of energy) for each of the 32 Fermi/GBM GRBs in
the sample of bursts with known redshifts that exhibit the lag transition
phenomenon are shown in Figure 1 (as derived from Figure 1 of Liu et al.
(2022)). In the method proposed here, we make the reasonable assumption that
the intrinsic lags (that dominate in magnitude over the small delays induced
by QG effects; see the upper limits for the first- and second-order QG delays
shown as blue and orange curves in Figure 1) do not correlate with the
redshift as the distance of a given GRB from us does not affect the emission
properties in, e.g., the fireball model. This means that, averaging over a
large sample of GRBs at different redshifts, the intrinsic delays will cluster
around the common value that defines the intrinsic average rest-frame lag.

Figure 1:  The dependence of the spectral lag on the energy window for the GRB
sample studied in Liu et al. (2022). The fits with a smoothly broken power law
(SBPL) are shown by black solid lines. Blue and orange dotted lines denote the
maximally allowed LIV-induced lags in linear and quadratic cases, thereby
defining the lower limits on the QG energy. This figure is adapted from Figure
1 from Liu et al. (2022) (see Section 3 of this work for details). ©AAS.
Reproduced with permission.

###  3.3 The Technique of Averaging over the Sample

Relation (10) shows that, for a given GRB rest-frame energy $E_{\rm rf}$, the
lag $\tau_{\rm total,\,rf}(E_{\rm rf})$ depends only on the GRB redshift $z$
through the function $u(z)$ defined in Equation (9). Therefore, fixing $E_{\rm
rf}$ for an ensemble of $N$ GRBs with known redshift, one can compute the
function $\tau_{\rm total,\,rf}(E_{\rm rf})$ from the observed data for all
GRBs of the ensemble, i.e., it is possible to obtain a set of $N$
experimentally computed values of $\tau_{\rm total,\,rf}(E_{\rm rf})$. This
ensemble of $N$ values can be fitted as a function of $u(z)$ through Equation
(10) to obtain the best fit values of the intrinsic lag in the GRB rest frame
$\tau_{\rm int,\,rf}$, and the coefficient of the QG-induced delay at the GRB
rest-frame energy $E_{\rm rf}$

$\begin{split}\left\\{\begin{array}[]{lclcl}\left[\tau_{int,\,rf}(E_{\rm
rf})\right]_{\rm BEST}&&\vspace{6pt}\\\ \vspace{6pt}{\rm and}&&&&\\\
\vspace{6pt}\left[\xi\left(\frac{1}{H_{0}}\right)\left(\frac{E_{\rm rf}}{\zeta
E_{\rm pl}}\right)^{n}\right]_{\rm BEST}&=&&&\\\
\vspace{6pt}\left[\xi\left(\frac{1}{H_{0}}\right)\left(\alpha\frac{E_{\rm
rf}}{E_{\rm pl}}\right)^{n}\right]_{\rm BEST}&=&\left[\phi(\alpha E_{\rm
rf})\right]_{\rm BEST}&&\\\ \end{array}\right..\end{split}$ (11)

Once the values of $\left[\phi(\alpha E_{\rm rf})\right]_{\rm BEST}$ are
obtained for all $E_{\rm rf}$, these values can be plotted as a function of
$s(E_{\rm rf})=(E_{\rm rf}/E_{\rm pl})^{n}$ and subsequently linearly fitted
through the equation

$\begin{split}\phi(\alpha E_{\rm
rf})=\left(\frac{\alpha^{n}}{H_{0}}\right)s(E_{\rm rf})=\Delta_{\rm
QG}\,s(E_{\rm rf})\end{split},$ (12)

to obtain the best fit value of the strength of the QG effect $\Delta_{\rm
QG}=\alpha^{n}/H_{0}$. We note that this technique allows us to combine the
whole ensemble of $N$ GRBs to obtain a unique measure of the strength of the
QG effect, whose uncertainty $\sigma_{\Delta_{\rm QG}}$, in the absence of
other systematic errors, only depends on the (Poissonian) statistics of the
whole ensemble of $N$ GRBs and therefore improves as the inverse square root
of $N$:

$\begin{split}\sigma_{\Delta_{\rm QG}}\propto\frac{1}{\sqrt{N}}.\end{split}$
(13)

Consequently, the precision of the measurement of the QG effect’s strength can
be improved by increasing the size of the analyzed sample.

###  3.4 GRB Intrinsic Spectral Lags vs. Quantum Gravity Effects

The GRB spectral lag can be caused by a mixture of two effects: the QG one and
the one inherent to the fireball model. The latter is due to the curvature
effect, i.e., the kinematic effect caused by the fact that the observer looks
at an increasingly off-axis annulus area relative to the line-of-sight
Fenimore et al. (1996); Salmonson (2000); Kumar & Panaitescu (2000); Ioka &
Nakamura (2001); Qin (2002); Qin et al. (2004); Dermer (2004); Shen et al.
(2005); Lu et al. (2006). Softer low-energy radiation comes from the off-axis
annulus area with smaller Doppler factors and is delayed for the observer with
respect to on-axis emission due to the geometric curvature of the shell.

A competing hypothesis is that the traditional view based on the high-latitude
emission “curvature effect” of a relativistic jet cannot explain spectral
lags. Instead, spectral peaks should be swept across the observing energy
range in a specific manner to account for the observed spectral lags. A simple
physical model that implies synchrotron radiation from a rapidly expanding
outflow can explain GRB spectral lags Uhm & Zhang (2016). This model requires
the following conditions to be fulfilled: (1) the emission radius has to be
large (over several $10^{14}$ cm from the central engine), in the optically
thin region, well above the photosphere; (2) the $\gamma$-ray photon spectrum
is curved (as observed); (3) the magnetic field strength in the emitting
region decreases with the radius as the region expands in space, which is
consistent with an expanding jet; and (4) the emission region itself undergoes
rapid bulk acceleration as the prompt emission is produced. These requirements
are consistent with a Poynting-flux-dominated jet abruptly dissipating
magnetic energy at a large distance from the engine. The aforementioned
theories successfully explain the positive spectral lags. Nevertheless, the
rarely observed negative lags remain a more intriguing phenomenon that can be
used to infer the different radiation mechanisms Li (2010); Zhang et al.
(2011) or emission regions Toma et al. (2009) of low- and high-energy photons.

For a given GRB, the intrinsic delay inherent to the GRB emission could mimic
the genuine quantum gravity eﬀect, making these two effects difficult to
disentangle. However, currently, there is no evidence for a correlation
between the GRB intrinsic delays and the distances to its sources. For
example, in Tsvetkova et al. (2017, 2021), where the largest sample of GRBs
with known redshifts detected by a single instrument in a wide energy range is
studied, the significance of the cosmological evolution of GRB energetics is
$\lesssim$2$\sigma$. Meanwhile, the delays induced by a photon dispersion law
are proportional both to the light travel distance (a function of redshift)
and to the diﬀerences in the energy of the photons. This dual dependence on
energy and redshift could be the unique feature of a genuine QG eﬀect. As
suggested in Burderi et al. (2020), given an adequate collection area, GRBs,
once their redshifts are known, are potentially excellent tools to search for
the ﬁrst-order dispersion law for photons.

###  3.5 Computation of Spectral Lags

To avoid distortions due to the fact that the shape of the light curve changes
with the energy, we suggest fixing the energy channels in which the light
curves are recorded to certain values in the rest frame. In this case, the
corresponding observer-frame values of the channel boundaries will be
$E_{\textrm{obs}}=E_{\textrm{rest}}/(1+z)$, i.e., redshift-dependent. The
first step to test the LIV effects with the suggested technique would be apply
it to the GBM data. Thus, we propose to use the following energy bands to
record the light curves. Given the 9 keV lower boundary of the GBM spectral
window, one has to select the rest-frame channels starting from, at least, 60
keV, to allow for bursts with redshifts up to $z=5$. This boundary can be
shifted towards a higher value to allow high-redshift GRBs to contribute to
the study. However, we should mention that the majority of GRBs have redshifts
$z<5$ (see Figure 2). Some examples of the pseudo-logarithmic channels that
could be used are 60–100 keV, 100–160 keV, 160–250 keV, 250–400 keV, 400–600
keV, 600–900 keV, or 60–80 keV, 80–100 keV, 100–130 keV, 130–160 keV, 160–200
keV, 200–250 keV, 250–320 keV, 320–400 keV, 400–500 keV, 500–650 keV, 650–900
keV. We found these numbers of channels to be reasonable in terms of SNR based
on the research of Liu et al. (2022), carried out for 32 Fermi/GBM GRBs.

Since the GRB energetics are usually considered on the logarithmic scale, we
suggest adopting the geometric mean of the lower and upper boundaries of an
energy band, $E_{\textrm{phot}}=\sqrt{E_{\textrm{min}}\times
E_{\textrm{max}}}$, as a proxy for the average energy of photons in the given
range.

Figure 2: The cosmological GRB formation rate (GRBFR) derived in Tsvetkova et
al. (2021), superposed onto the star formation rate (SFR) data from the
literature. The gray points show the SFR data from Hopkins (2004); Bouwens et
al. (2011); Hanish et al. (2006); Thompson et al. (2006). The marked line
denotes the SFR approximation from Li (2008). The GRBFR normalization is equal
for all four data sets and the GRBFR points have been shifted arbitrarily to
match the SFR at $(1-z)\sim 3.5$. Figure 5b from Tsvetkova et al. (2021).
©AAS. Reproduced with permission.

Since the spectral lag distributions of the short and long GRBs significantly
differ (Yi et al., 2006), and these two types of bursts belong to distinct
classes of progenitors, they have different intrinsic spectral lags. Thus, we
suggest studying them separately.

###  3.6 Obtaining GRB Redshifts

Since the suggested technique of testing LIV using GRBs strongly relies on
prior knowledge of the burst redshift, it is necessary either to measure it
directly from the observations in optics or estimate it using the prompt
emission parameters. GRB redshift measurements based on the detection of
emission lines or absorption features of GRB host galaxies imposed on the
afterglow continuum, or performed photometrically, are widespread. However,
there are other methods to obtain the redshift estimates, e.g., the “pseudo-
redshift” (pseudo-z) technique based on the spectral properties of GRB prompt
high-energy emission Atteia (2003), using well-known correlations such as, for
example, the Norris correlation (spectral lag vs. isotropic peak luminosity;
Norris et al. (2000)), the Amati correlation (rest-frame peak energy vs.
isotropic energy release; Amati et al. (2002)), the isotropic peak luminosity
vs. temporal variability correlation Reichart et al. (2001); Fenimore &
Ramirez-Ruiz (2000), the Yonetoku (the rest-frame peak energy vs. the
isotropic peak luminosity; Yonetoku et al. (2004)) correlation, etc., or the
method of searching for a minimum on the intrinsic hydrogen column density
versus the redshift plane (see, e.g., Ghisellini et al. (1999)).

Nowadays, the machine learning (ML) approach to redshift estimation is
becoming popular in astrophysics (see, e.g., D´Isanto, A. & Polsterer, K. L.
(2018); Dainotti et al. (2019); Lee & Shin (2021); Momtaz et al. (2022)).
Supervised ML is a data mining method based on prior knowledge of a “training”
data set, on which we can build models predicting the parameter under
consideration, a “validation” set, which provides an unbiased evaluation of a
model’s fit while tuning the model’s hyperparameters, and a “test” data set
necessary for an unbiased evaluation of the final model fit.

Considering only spectroscopic and photometric redshifts, there were
$\gtrsim$420 GRBs with reliably measured redshifts by the middle of 2022 (for
a list of GRBs with measured redshifts, see Gruber et al. (2011); Atteia et
al. (2017); Tsvetkova et al. (2017); Minaev & Pozanenko (2020, 2021);
Tsvetkova et al. (2021), the Gamma-Ray Burst Online
Index333https://sites.astro.caltech.edu/grbox/grbox.php, Jochen Greiner’s GRB
table444https://www.mpe.mpg.de/~jcg/grbgen.html, and the references therein).
Using one of the aforementioned techniques, one can estimate the redshift of
any burst based on its temporal or spectral parameters and energetics. For
example, Lloyd-Ronning et al. (2002); Yonetoku et al. (2004); Kocevski & Liang
(2006) used various correlations to obtain unknown GRB redshifts from GRB
observables, while Ukwatta et al. (2016) used the random forest algorithm to
estimate GRB redshifts.

##  4 Discussion

THESEUS is a mission aimed at increasing the discovery rate of the high-energy
transient phenomena over the entirety of cosmic history and fully exploiting
GRBs to explore the early Universe Amati et al. (2018, 2021). THESEUS is
likely to become a cornerstone of multi-messenger and time-domain astrophysics
thanks to its exceptional payload, providing wide and deep sky monitoring in a
broad energy range (0.3 keV–20 MeV); focusing capabilities in the soft X-ray
band, providing large grasp and a high angular resolution; and onboard near-IR
capabilities for immediate transient identification and redshift
determination.

The THESEUS payload is planned to include the following instrumentation: (1)
the X-Gamma-Ray Imaging Spectrometer (XGIS, 2 keV–20 MeV): a set of two coded-
mask cameras using monolithic X-gamma-ray detectors based on bars of silicon
diodes coupled with a crystal scintillator, granting a $\sim$2 sr field of
view (FoV) and source location accuracy of $\sim$10′ in the 2–150 keV band, as
well as a ¿4 sr FoV at energies $>150$ keV, with a few $\mu$s timing
resolution; (2) a Soft X-Ray Imager (SXI, 0.3–5 keV): a set of two lobster-eye
telescope units, covering a total FOV of $\sim$0.5 sr with source location
accuracy $\lesssim 2^{\prime}$; (3) an infrared telescope (IRT, 0.7–1.8
$\mu$m): a 0.7 m class IR telescope with a $15^{\prime}\times 15^{\prime}$
FOV, for a fast response, with both imaging (I, Z, Y, J, and H) and
spectroscopic (resolving power, R$\sim$400, through $2^{\prime}\times
2^{\prime}$ grism) capabilities.

Thanks to the unique combination of a wide 0.3 keV–10 MeV energy range,
remarkable sensitivity, and exceptionally high counting statistics, THESEUS
heralds a new era in the multi-wavelength studies of GRBs, providing the
community with a sample of GRBs with known redshifts of unprecedented size,
which, in turn, will not only allow the use of GRBs as cosmological tools but
also shed light on one of the most challenging aspects of QG theory, the
systematic study of which is still beyond the current instrumental
capabilities. The capability of THESEUS to detect and localize GRBs, as well
as measure their redshifts, will essentially surpass those of the current
missions.

The left panel of Figure 3 shows the expected detection rate of long GRBs by
THESEUS compared with observed GRBs. The orange histogram depicts the
cumulative distribution of the GRBs detected by the SXI and/or XGIS (the
bursts with measured $z$ are marked in purple), while the blue histogram
presents the distribution of GRBs with known redshifts detected from 2005 to
the end of 2020. The distribution of the GRBs detected by THESEUS was acquired
based on the anticipated IRT capabilities and on the assumption of a ground
follow-up rate of 50% for the GRBs at $z<5$. It is noticeable that THESEUS is
expected to detect an order of magnitude more bursts than Swift does,
especially in the high-redshift domain ($z>6$) Ghirlanda et al. (2021). It is
expected that the redshifts for the majority of GRBs detected by THESEUS will
be measured (onboard or on the ground). The cumulative distribution plotted in
the right panel of Figure 3 represents the annual detection rate of short GRBs
by XGIS, not corrected for mission observation efficiency. THESEUS is supposed
to acquire a statistically significant sample of short GRBs, including high-
redshift ($z\lesssim$ 4–5) events. Considering that the distribution of the
spectral lags of short GRBs and long GRBs differs significantly, advances in
research on short GRBs are very important for such sophisticated studies of
the QG effects as we discuss in this paper. Thus, the sample of GRBs with
measured redshifts obtained by THESEUS is the most promising for the
application of the described technique to study LIV.

Figure 3:  Left plot: Observed GRBs with known redshifts measured in 2005–2020
(blue line and filled cyan area representing $1\sigma$ uncertainty)
superimposed on the anticipated frequency of detection of long GRBs by THESEUS
(orange histogram). The purple hatched histogram shows the GRBs that are
expected to have a redshift measured by either THESEUS or ground-based
facilities’ telescopes. The model that fits the observed distribution used to
make predictions for THESEUS is represented by the green curve. THESEUS is
expected to detect one to two orders of magnitude more GRBs than Swift at any
redshift, and most importantly in the high-redshift range ($z>6$). Right plot:
Cumulative redshift distribution of short GRBs detectable with THESEUS/XGIS
per year of mission. Theoretically, short GRBs can be detected at high
redshifts $z>4$ with a rate of $\sim$1 event per year. This figure is adopted
from Ghirlanda et al. (2021).

##  5 Conclusions

Various QG theories predict LIV, which can manifest itself as the dispersion
of the speed of light. The method that we propose to disentangle and constrain
the QG delays from the intrinsic spectral lags in GRB light curves is based on
the assumption of the constancy of the rest-frame intrinsic spectral lags and
on the linear dependence of the GRB spectral lag on both the photon energy and
function of the GRB redshift. The ability to collect a large sample of GRBs
with known redshifts is crucial for this type of study, as the precision of
the QG effect measurement can be improved by expanding the data set.
Currently, redshifts are measured spectroscopically or photometrically for
$\lesssim 500$ GRBs. Thus, indirect estimates of the redshifts from the prompt
emission observables are necessary to obtain a large GRB sample for LIV
studies using the commissioned instruments. The sample of GRBs collected by
Fermi/GBM could provide a promising opportunity to apply the aforementioned
technique, thanks to its extensive trigger statistics ($\gtrsim$3500 GRBs up
to date) and sophisticated data acquired with a high temporal and spectral
resolution, which could allow precise measurements of the rest-frame spectral
lags.

The THESEUS mission is likely to initiate a breakthrough in this field of
fundamental physics as, thanks to the combination of its unique
characteristics, the observatory will collect one order of magnitude more
samples of GRBs with known redshifts than are currently available, which will
not only allow the use of GRBs as cosmological tools but will also enable us
to constrain the QG theories. Moreover, thanks to its capability to detect the
GRB emission in the relatively soft energy band of 0.3–5 keV, THESEUS could
provide a unique opportunity not only to constrain the empirical and physical
GRB models but also to expand the data range, providing more accurate
constraints on the QG energy from the lag-energy plane. Due to its high
sensitivity, THESEUS will also allow advances in the study of short GRBs; in
particular, XGIS will be able to detect short GRBs up to z$\sim$4–5, which is
important as the spectral lags of short GRBs essentially differ from the ones
of long bursts. Thus, the THESEUS mission could make a significant
contribution to the study of the QG effects using GRBs.

We thank the anonymous referees for helpful comments on the manuscript. Some
of the authors wish to thank ASI and INAF (agreements ASI-UNI-Ca 2016- 13-U.O
and ASI- INAF 2018-10-hh.0), the Italian Ministry of Education, University and
Research (MIUR), Italy (HERMES-TP project) and the EU (HERMES-SP Horizon 2020
Research and Innovation Project under grant agreement 821896) for the
financial support within the HERMES project. L. B., A. S. and A. T.
acknowledge funding from the Italian Ministry of University and Research
(MUR), PRIN 2017 (prot. 20179ZF5KS), “The new frontier of multi-messenger
astrophysics: follow-up of electromagnetic transient counterparts of
gravitational wave sources” (PI: E. Capellaro). T.d.S., L.B. and A.S. also
acknowledge the financial support of PRIN-INAF 2019 within the project
“Probing the geometry of accretion: from theory to observations” (PI:
Belloni).

## References

  * Abdo et al. (2009a) Abdo, A. A., Ackermann, M., Ajello, M., et al. 2009a, Nature, 462, 331, doi: 10.1038/nature08574
  * Abdo et al. (2009b) Abdo, A. A., Ackermann, M., Arimoto, M., et al. 2009b, Science, 323, 1688, doi: 10.1126/science.1169101
  * Acciari et al. (2020) Acciari, V. A., Ansoldi, S., Antonelli, L. A., et al. 2020, Phys. Rev. Lett., 125, 021301, doi: 10.1103/PhysRevLett.125.021301
  * Agrawal et al. (2021) Agrawal, R., Singirikonda, H., & Desai, S. 2021, J. Cosmology Astropart. Phys, 2021, 029, doi: 10.1088/1475-7516/2021/05/029
  * Amati et al. (2002) Amati, L., Frontera, F., Tavani, M., et al. 2002, A&A, 390, 81, doi: 10.1051/0004-6361:20020722
  * Amati et al. (2018) Amati, L., O’Brien, P., Götz, D., et al. 2018, Advances in Space Research, 62, 191, doi: 10.1016/j.asr.2018.03.010
  * Amati et al. (2021) Amati, L., O’Brien, P. T., Götz, D., et al. 2021, Experimental Astronomy, 52, 183, doi: 10.1007/s10686-021-09807-8
  * Amelino-Camelia (2000) Amelino-Camelia, G. 2000, in Toward Quantum Gravity, ed. J. Kowalski-Glikman, Vol. 541, 1
  * Amelino-Camelia et al. (1997) Amelino-Camelia, G., Ellis, J., Mavromatos, N. E., & Nanopoulos, D. V. 1997, International Journal of Modern Physics A, 12, 607, doi: 10.1142/S0217751X97000566
  * Amelino-Camelia et al. (1998a) Amelino-Camelia, G., Ellis, J., Mavromatos, N. E., Nanopoulos, D. V., & Sarkar, S. 1998a, Nature, 393, 763, doi: 10.1038/31647
  * Amelino-Camelia et al. (1998b) —. 1998b, Nature, 395, 525
  * Amelino-Camelia & Smolin (2009) Amelino-Camelia, G., & Smolin, L. 2009, Phys. Rev. D, 80, 084017, doi: 10.1103/PhysRevD.80.084017
  * Aptekar et al. (1995) Aptekar, R. L., Frederiks, D. D., Golenetskii, S. V., et al. 1995, Space Sci. Rev., 71, 265, doi: 10.1007/BF00751332
  * Atteia (2003) Atteia, J. L. 2003, A&A, 407, L1, doi: 10.1051/0004-6361:20030958
  * Atteia et al. (2017) Atteia, J. L., Heussaff, V., Dezalay, J. P., et al. 2017, ApJ, 837, 119, doi: 10.3847/1538-4357/aa5ffa
  * Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
  * Band (1997) Band, D. L. 1997, ApJ, 486, 928, doi: 10.1086/304566
  * Bartlett et al. (2021) Bartlett, D. J., Desmond, H., Ferreira, P. G., & Jasche, J. 2021, Phys. Rev. D, 104, 103516, doi: 10.1103/PhysRevD.104.103516
  * Bertolami & Páramos (2005) Bertolami, O., & Páramos, J. 2005, Phys. Rev. D, 72, 044001, doi: 10.1103/PhysRevD.72.044001
  * Blinnikov et al. (1984) Blinnikov, S. I., Novikov, I. D., Perevodchikova, T. V., & Polnarev, A. G. 1984, Soviet Astronomy Letters, 10, 177.  https://arxiv.org/abs/1808.05287
  * Bouwens et al. (2011) Bouwens, R. J., Illingworth, G. D., Labbe, I., et al. 2011, Nature, 469, 504, doi: 10.1038/nature09717
  * Burderi et al. (2016) Burderi, L., Di Salvo, T., & Iaria, R. 2016, Phys. Rev. D, 93, 064017, doi: 10.1103/PhysRevD.93.064017
  * Burderi et al. (2020) Burderi, L., Di Salvo, T., Riggio, A., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11444, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 114444Y, doi: 10.1117/12.2561779
  * Carroll et al. (2001) Carroll, S. M., Harvey, J. A., Kostelecký, V. A., Lane, C. D., & Okamoto, T. 2001, Phys. Rev. Lett., 87, 141601, doi: 10.1103/PhysRevLett.87.141601
  * Chang et al. (2016) Chang, Z., Li, X., Lin, H.-N., et al. 2016, Chinese Physics C, 40, 045102, doi: 10.1088/1674-1137/40/4/045102
  * Chen et al. (2005) Chen, L., Lou, Y.-Q., Wu, M., et al. 2005, ApJ, 619, 983, doi: 10.1086/426774
  * Cheng et al. (1995) Cheng, L. X., Ma, Y. Q., Cheng, K. S., Lu, T., & Zhou, Y. Y. 1995, A&A, 300, 746
  * Dainotti et al. (2019) Dainotti, M., Petrosian, V., Bogdan, M., et al. 2019, arXiv e-prints, arXiv:1907.05074.  https://arxiv.org/abs/1907.05074
  * Delhom et al. (2021) Delhom, A., Nascimento, J. R., Olmo, G. J., Petrov, A. Y., & Porfírio, P. J. 2021, European Physical Journal C, 81, 287, doi: 10.1140/epjc/s10052-021-09083-y
  * Dermer (2004) Dermer, C. D. 2004, ApJ, 614, 284, doi: 10.1086/426532
  * Desai et al. (2023) Desai, S., Agrawal, R., & Singirikonda, H. 2023, European Physical Journal C, 83, 63, doi: 10.1140/epjc/s10052-023-11229-z
  * D´Isanto, A. & Polsterer, K. L. (2018) D´Isanto, A., & Polsterer, K. L. 2018, A&A, 609, A111, doi: 10.1051/0004-6361/201731326
  * Du et al. (2021) Du, S.-S., Lan, L., Wei, J.-J., et al. 2021, ApJ, 906, 8, doi: 10.3847/1538-4357/abc624
  * Eichler et al. (1989) Eichler, D., Livio, M., Piran, T., & Schramm, D. N. 1989, Nature, 340, 126, doi: 10.1038/340126a0
  * Ellis et al. (2019) Ellis, J., Konoplich, R., Mavromatos, N. E., et al. 2019, Phys. Rev. D, 99, 083009, doi: 10.1103/PhysRevD.99.083009
  * Ellis et al. (2003) Ellis, J., Mavromatos, N. E., Nanopoulos, D. V., & Sakharov, A. S. 2003, A&A, 402, 409, doi: 10.1051/0004-6361:20030263
  * Ellis et al. (2006) Ellis, J., Mavromatos, N. E., Nanopoulos, D. V., Sakharov, A. S., & Sarkisyan, E. K. G. 2006, Astroparticle Physics, 25, 402, doi: 10.1016/j.astropartphys.2006.04.001
  * Fenimore et al. (1996) Fenimore, E. E., Madras, C. D., & Nayakshin, S. 1996, ApJ, 473, 998, doi: 10.1086/178210
  * Fenimore & Ramirez-Ruiz (2000) Fenimore, E. E., & Ramirez-Ruiz, E. 2000, arXiv e-prints, astro.  https://arxiv.org/abs/astro-ph/0004176
  * Ferrari et al. (2007) Ferrari, A. F., Gomes, M., Nascimento, J. R., et al. 2007, Physics Letters B, 652, 174, doi: 10.1016/j.physletb.2007.07.013
  * Ganguly & Desai (2017) Ganguly, S., & Desai, S. 2017, Astroparticle Physics, 94, 17, doi: 10.1016/j.astropartphys.2017.07.003
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
  * Ghirlanda et al. (2021) Ghirlanda, G., Salvaterra, R., Toffano, M., et al. 2021, Experimental Astronomy, 52, 277, doi: 10.1007/s10686-021-09763-3
  * Ghisellini et al. (1999) Ghisellini, G., Haardt, F., Campana, S., Lazzati, D., & Covino, S. 1999, ApJ, 517, 168, doi: 10.1086/307182
  * Gogoi & Dev Goswami (2022) Gogoi, D. J., & Dev Goswami, U. 2022, J. Cosmology Astropart. Phys, 2022, 029, doi: 10.1088/1475-7516/2022/06/029
  * Gogoi & Dev Goswami (2023) —. 2023, J. Cosmology Astropart. Phys, 2023, 027, doi: 10.1088/1475-7516/2023/02/027
  * Gruber et al. (2011) Gruber, D., Greiner, J., von Kienlin, A., et al. 2011, A&A, 531, A20, doi: 10.1051/0004-6361/201116953
  * Hakkila & Giblin (2004) Hakkila, J., & Giblin, T. W. 2004, ApJ, 610, 361, doi: 10.1086/421060
  * Hanish et al. (2006) Hanish, D. J., Meurer, G. R., Ferguson, H. C., et al. 2006, ApJ, 649, 150, doi: 10.1086/504681
  * Hopkins (2004) Hopkins, A. M. 2004, ApJ, 615, 209, doi: 10.1086/424032
  * Hossenfelder (2013) Hossenfelder, S. 2013, Living Reviews in Relativity, 16, 2, doi: 10.12942/lrr-2013-2
  * Hořava (2009) Hořava, P. 2009, Phys. Rev. D, 79, 084008, doi: 10.1103/PhysRevD.79.084008
  * Ioka & Nakamura (2001) Ioka, K., & Nakamura, T. 2001, ApJ, 554, L163, doi: 10.1086/321717
  * Jacob & Piran (2008) Jacob, U., & Piran, T. 2008, J. Cosmology Astropart. Phys, 2008, 031, doi: 10.1088/1475-7516/2008/01/031
  * Kanzi & Sakallı (2019) Kanzi, S., & Sakallı, İ. 2019, Nuclear Physics B, 946, 114703, doi: 10.1016/j.nuclphysb.2019.114703
  * Kanzi & Sakallı (2021) Kanzi, S., & Sakallı, I. 2021, European Physical Journal C, 81, 501, doi: 10.1140/epjc/s10052-021-09299-y
  * Kocevski & Liang (2006) Kocevski, D., & Liang, E. 2006, ApJ, 642, 371, doi: 10.1086/500816
  * Kostelecký (2004) Kostelecký, V. A. 2004, Phys. Rev. D, 69, 105009, doi: 10.1103/PhysRevD.69.105009
  * Kostelecký & Samuel (1989a) Kostelecký, V. A., & Samuel, S. 1989a, Phys. Rev. D, 39, 683, doi: 10.1103/PhysRevD.39.683
  * Kostelecký & Samuel (1989b) —. 1989b, Phys. Rev. D, 40, 1886, doi: 10.1103/PhysRevD.40.1886
  * Kouveliotou et al. (1993) Kouveliotou, C., Meegan, C. A., Fishman, G. J., et al. 1993, ApJ, 413, L101, doi: 10.1086/186969
  * Kumar & Panaitescu (2000) Kumar, P., & Panaitescu, A. 2000, ApJ, 541, L51, doi: 10.1086/312905
  * Lambiase & Scardigli (2018) Lambiase, G., & Scardigli, F. 2018, Phys. Rev. D, 97, 075003, doi: 10.1103/PhysRevD.97.075003
  * Lee & Shin (2021) Lee, J., & Shin, M.-S. 2021, AJ, 162, 297, doi: 10.3847/1538-3881/ac2e96
  * Li (2008) Li, L.-X. 2008, MNRAS, 388, 1487, doi: 10.1111/j.1365-2966.2008.13488.x
  * Li (2010) Li, Z. 2010, ApJ, 709, 525, doi: 10.1088/0004-637X/709/1/525
  * Liu & Ma (2018) Liu, Y., & Ma, B.-Q. 2018, European Physical Journal C, 78, 825, doi: 10.1140/epjc/s10052-018-6294-y
  * Liu et al. (2022) Liu, Z.-K., Zhang, B.-B., & Meng, Y.-Z. 2022, ApJ, 935, 79, doi: 10.3847/1538-4357/ac81b9
  * Lloyd-Ronning et al. (2002) Lloyd-Ronning, N. M., Fryer, C. L., & Ramirez-Ruiz, E. 2002, ApJ, 574, 554, doi: 10.1086/341059
  * Lu et al. (2006) Lu, R. J., Qin, Y. P., Zhang, Z. B., & Yi, T. F. 2006, MNRAS, 367, 275, doi: 10.1111/j.1365-2966.2005.09951.x
  * MacFadyen & Woosley (1999) MacFadyen, A. I., & Woosley, S. E. 1999, ApJ, 524, 262, doi: 10.1086/307790
  * MAGIC Collaboration et al. (2019) MAGIC Collaboration, Acciari, V. A., Ansoldi, S., et al. 2019, Nature, 575, 455, doi: 10.1038/s41586-019-1750-x
  * Mazets et al. (1981) Mazets, E. P., Golenetskii, S. V., Ilinskii, V. N., et al. 1981, Ap&SS, 80, 3, doi: 10.1007/BF00649140
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ, 702, 791, doi: 10.1088/0004-637X/702/1/791
  * Minaev & Pozanenko (2020) Minaev, P. Y., & Pozanenko, A. S. 2020, MNRAS, 492, 1919, doi: 10.1093/mnras/stz3611
  * Minaev & Pozanenko (2021) —. 2021, MNRAS, 504, 926, doi: 10.1093/mnras/stab1031
  * Momtaz et al. (2022) Momtaz, A., Salimi, M. H., & Shakeri, S. 2022, arXiv e-prints, arXiv:2201.04391.  https://arxiv.org/abs/2201.04391
  * Neves (2023) Neves, J. C. S. 2023, Annals of Physics, 454, 169338, doi: 10.1016/j.aop.2023.169338
  * Norris (1995) Norris, J. P. 1995, Ap&SS, 231, 95, doi: 10.1007/BF00658595
  * Norris & Bonnell (2006) Norris, J. P., & Bonnell, J. T. 2006, ApJ, 643, 266, doi: 10.1086/502796
  * Norris et al. (2005) Norris, J. P., Bonnell, J. T., Kazanas, D., et al. 2005, ApJ, 627, 324, doi: 10.1086/430294
  * Norris et al. (2000) Norris, J. P., Marani, G. F., & Bonnell, J. T. 2000, ApJ, 534, 248, doi: 10.1086/308725
  * Norris et al. (1986) Norris, J. P., Share, G. H., Messina, D. C., et al. 1986, ApJ, 301, 213, doi: 10.1086/163889
  * Övgün et al. (2019) Övgün, A., Jusufi, K., & Sakallı, I. 2019, Phys. Rev. D, 99, 024042, doi: 10.1103/PhysRevD.99.024042
  * Paczynski (1986) Paczynski, B. 1986, ApJ, 308, L43, doi: 10.1086/184740
  * Paczynski (1991) —. 1991, Acta Astron., 41, 257
  * Paczyński (1998) Paczyński, B. 1998, ApJ, 494, L45, doi: 10.1086/311148
  * Pan et al. (2015) Pan, Y., Gong, Y., Cao, S., Gao, H., & Zhu, Z.-H. 2015, ApJ, 808, 78, doi: 10.1088/0004-637X/808/1/78
  * Pan et al. (2020) Pan, Y., Qi, J., Cao, S., et al. 2020, ApJ, 890, 169, doi: 10.3847/1538-4357/ab6ef5
  * Qin (2002) Qin, Y.-P. 2002, A&A, 396, 705, doi: 10.1051/0004-6361:20021439
  * Qin et al. (2004) Qin, Y.-P., Zhang, Z.-B., Zhang, F.-W., & Cui, X.-H. 2004, ApJ, 617, 439, doi: 10.1086/425335
  * Rees & Meszaros (1994) Rees, M. J., & Meszaros, P. 1994, ApJ, 430, L93, doi: 10.1086/187446
  * Reichart et al. (2001) Reichart, D. E., Lamb, D. Q., Fenimore, E. E., et al. 2001, ApJ, 552, 57, doi: 10.1086/320434
  * Rovelli (1998) Rovelli, C. 1998, Living Reviews in Relativity, 1, 1, doi: 10.12942/lrr-1998-1
  * Rovelli & Smolin (1988) Rovelli, C., & Smolin, L. 1988, Phys. Rev. Lett., 61, 1155, doi: 10.1103/PhysRevLett.61.1155
  * Rovelli & Smolin (1990) —. 1990, Nuclear Physics B, 331, 80, doi: 10.1016/0550-3213(90)90019-A
  * Ryde (2005) Ryde, F. 2005, A&A, 429, 869, doi: 10.1051/0004-6361:20041364
  * Salmonson (2000) Salmonson, J. D. 2000, ApJ, 544, L115, doi: 10.1086/317305
  * Salmonson & Galama (2002) Salmonson, J. D., & Galama, T. J. 2002, ApJ, 569, 682, doi: 10.1086/339391
  * Sanchez (2019) Sanchez, N. G. 2019, Gravitation and Cosmology, 25, 91, doi: 10.1134/S0202289319020142
  * Santos & Almeida (2013) Santos, V., & Almeida, C. A. S. 2013, Physics Letters B, 718, 1114, doi: 10.1016/j.physletb.2012.11.045
  * Shen et al. (2005) Shen, R.-F., Song, L.-M., & Li, Z. 2005, MNRAS, 362, 59, doi: 10.1111/j.1365-2966.2005.09163.x
  * Shenoy et al. (2013) Shenoy, A., Sonbas, E., Dermer, C., et al. 2013, ApJ, 778, 3, doi: 10.1088/0004-637X/778/1/3
  * Tawfik et al. (2016) Tawfik, A. N., Magdy, H., & Ali, A. F. 2016, Physics of Particles and Nuclei Letters, 13, 59, doi: 10.1134/S1547477116010179
  * Thompson & Wilson-Hodge (2022) Thompson, D. J., & Wilson-Hodge, C. A. 2022, arXiv e-prints, arXiv:2210.12875.  https://arxiv.org/abs/2210.12875
  * Thompson et al. (2006) Thompson, R. I., Eisenstein, D., Fan, X., et al. 2006, ApJ, 647, 787, doi: 10.1086/505568
  * Toma et al. (2009) Toma, K., Wu, X.-F., & Mészáros, P. 2009, ApJ, 707, 1404, doi: 10.1088/0004-637X/707/2/1404
  * Tsvetkova et al. (2022) Tsvetkova, A., Svinkin, D., Karpov, S., & Frederiks, D. 2022, Universe, 8, 373, doi: 10.3390/universe8070373
  * Tsvetkova et al. (2017) Tsvetkova, A., Frederiks, D., Golenetskii, S., et al. 2017, ApJ, 850, 161, doi: 10.3847/1538-4357/aa96af
  * Tsvetkova et al. (2021) Tsvetkova, A., Frederiks, D., Svinkin, D., et al. 2021, ApJ, 908, 83, doi: 10.3847/1538-4357/abd569
  * Uhm & Zhang (2016) Uhm, Z. L., & Zhang, B. 2016, ApJ, 825, 97, doi: 10.3847/0004-637X/825/2/97
  * Ukwatta et al. (2016) Ukwatta, T. N., Woźniak, P. R., & Gehrels, N. 2016, MNRAS, 458, 3821, doi: 10.1093/mnras/stw559
  * Vasileiou et al. (2015) Vasileiou, V., Granot, J., Piran, T., & Amelino-Camelia, G. 2015, Nature Physics, 11, 344, doi: 10.1038/nphys3270
  * Vasileiou et al. (2013) Vasileiou, V., Jacholkowska, A., Piron, F., et al. 2013, Phys. Rev. D, 87, 122001, doi: 10.1103/PhysRevD.87.122001
  * Wei (2019) Wei, J.-J. 2019, MNRAS, 485, 2401, doi: 10.1093/mnras/stz594
  * Wei & Wu (2021) Wei, J.-J., & Wu, X.-F. 2021, Frontiers of Physics, 16, 44300, doi: 10.1007/s11467-021-1049-x
  * Wei et al. (2017) Wei, J.-J., Zhang, B.-B., Shao, L., Wu, X.-F., & Mészáros, P. 2017, ApJ, 834, L13, doi: 10.3847/2041-8213/834/2/L13
  * Woosley (1993) Woosley, S. E. 1993, ApJ, 405, 273, doi: 10.1086/172359
  * Woosley & Bloom (2006) Woosley, S. E., & Bloom, J. S. 2006, ARA&A, 44, 507, doi: 10.1146/annurev.astro.43.072103.150558
  * Wu & Fenimore (2000) Wu, B., & Fenimore, E. 2000, ApJ, 535, L29, doi: 10.1086/312700
  * Xiao et al. (2022) Xiao, S., Xiong, S.-L., Wang, Y., et al. 2022, ApJ, 924, L29, doi: 10.3847/2041-8213/ac478a
  * Xu & Ma (2016a) Xu, H., & Ma, B.-Q. 2016a, Physics Letters B, 760, 602, doi: 10.1016/j.physletb.2016.07.044
  * Xu & Ma (2016b) —. 2016b, Astroparticle Physics, 82, 72, doi: 10.1016/j.astropartphys.2016.05.008
  * Yi et al. (2006) Yi, T., Liang, E., Qin, Y., & Lu, R. 2006, MNRAS, 367, 1751, doi: 10.1111/j.1365-2966.2006.10083.x
  * Yonetoku et al. (2004) Yonetoku, D., Murakami, T., Nakamura, T., et al. 2004, ApJ, 609, 935, doi: 10.1086/421285
  * Zhang et al. (2009) Zhang, B., Zhang, B.-B., Virgili, F. J., et al. 2009, ApJ, 703, 1696, doi: 10.1088/0004-637X/703/2/1696
  * Zhang et al. (2011) Zhang, B.-B., Zhang, B., Liang, E.-W., et al. 2011, ApJ, 730, 141, doi: 10.1088/0004-637X/730/2/141
  * Zhang & Ma (2015) Zhang, S., & Ma, B.-Q. 2015, Astroparticle Physics, 61, 108, doi: 10.1016/j.astropartphys.2014.04.008
  * Zhang et al. (2002) Zhang, Y. H., Treves, A., Celotti, A., et al. 2002, ApJ, 572, 762, doi: 10.1086/340349
  * Zou et al. (2018) Zou, X.-B., Deng, H.-K., Yin, Z.-Y., & Wei, H. 2018, Physics Letters B, 776, 284, doi: 10.1016/j.physletb.2017.11.053


The Primordial Black Hole Formation
from Single-Field Inflation is Still Not Ruled Out

Antonio<EMAIL_ADDRESS>
1 Department of Theoretical Physics and Gravitational Wave Science Center,
24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland

Abstract

In response to a recent criticism, appeared in Ref. yokoyamanew , we argue
that the standard scenario to form primordial black holes in the early
universe based on a phase of ultra-slow-roll in single-field inflation is not
ruled out.

##  I I. Introduction and Conclusion

A standard mechanism to create Primordial Black Holes (PBHs) in the early
universe is by enhancing the short scale curvature perturbation $\zeta_{S}$ s1
; s2 ; s3  during an Ultra Slow Roll (USR) period when the inflaton potential
$V(\phi)$ is quite flat. If loop corrections from the short modes are larger
than the tree-level contribution to the large-scale power spectrum
$P_{\zeta}(k_{L})$, the USR mechanism to generate PBHs will be inevitably
ruled out, as advocated in Ref. yokoyama , and more recently in Ref.
yokoyamanew . In a short note, we have argued that the PBH Formation from
single-field inflation is in fact not ruled out riotto . The argument was as
follows. Short modes propagate in a universe perturbed by the presence of a
long, basically static, CMB mode which can be absorbed by a short momentum
rescaling. The loop correction is therefore dictated by the correlation
between the short mode expectation value and the long wavelength mode or, in
other words, by the corresponding squeezed bispectrum

$\langle\zeta_{L}\zeta_{L}\rangle\sim\langle\langle\zeta_{S}\zeta_{S}\rangle_{\zeta_{L}}\zeta_{L}\rangle\sim\frac{{\rm
d}\ln{\cal P}_{\zeta}(k_{S})}{{\rm d}\ln
k_{S}}P_{\zeta}(k_{S})P_{\zeta}(k_{L}),$ (I.1)

Loop corrections are therefore suppressed by the breaking of scale invariance
in the short mode power spectrum or, in other words, by field derivatives of
the inflaton potential. In Ref. yokoyamanew  this point has been questioned
based on the fact that a non-suppressed (by slow-roll parameters) squeezed
bispectrum (corresponding to a non-linear non-Gaussianity parameter $f_{\rm
NL}=5/2$) is present once one recall that the cubic interaction

$S_{\rm int}[\zeta]\sim\int{\rm d}^{3}x\,{\rm
d}t\,a^{3}\epsilon\,\frac{1}{2}\dot{\eta}\dot{\zeta}\zeta^{2}$ (I.2)

is obtained after the field redefinition
$\zeta\rightarrow\zeta+\eta\,\zeta^{2}/4+\zeta\zeta^{\prime}$ necessary to
remove a boundary term m .

However, in Ref. riotto  (see end of Section III) it was explicitly stated
that this is not necessarily true when taking into account the details of the
transition into the slow-roll phase following the USR period. Let us elaborate
about this point further here. If the transition is not sudden, the non-
Gaussianity evolves to become suppressed by slow-roll parameters and the
argument of Ref. riotto  applies. To show this explicitly, we follow Ref. c
and write the interaction term (I.2) in the form

$S_{\rm int}[\zeta]\sim-\int{\rm d}^{3}x\,{\rm d}t\,\frac{{\rm d}}{{\rm
d}t}\left(\frac{a^{3}\epsilon\dot{\eta}}{6}\right)\,\zeta^{3}.$ (I.3)

Suppose that the transition is not sudden, in the sense that the slow-roll
parameter $\epsilon_{V}$ during the slow-roll phase might be smaller compared
to the inflaton velocity at the end of the USR phase. This can be achieved for
instance in the Starobinsky spiky model sta  where there is an abrupt change
in the slop of the linear potential of the inflaton field $\phi$. Even if one
is subjects to drastic variations of the $\eta$ parameter during the USR
phase, inserting the analytical solutions for $\epsilon$ and $\eta$ one finds
c

$-\frac{{\rm d}}{{\rm
d}t}\left(\frac{a^{3}\epsilon\dot{\eta}}{6}\right)=\frac{a^{3}\epsilon}{3}\sqrt{2\epsilon}V^{\prime\prime\prime}.$
(I.4)

Therefore, the contribution from this term is always tiny, no matter how large
is $\eta$ and $\dot{\eta}$ are during the transition, and in fact suppressed
by slow-roll parameters. Indeed, the corresponding squeezed bispectrum
$\langle\langle\zeta_{S}\zeta_{S}\rangle_{\zeta_{L}}\zeta_{L}\rangle$ is
suppressed by ${\cal O}(\epsilon_{V}^{1/2}\eta_{V})$ c , with $\epsilon_{V}$
small enough to generate PBHs. In other words, if the curvature perturbations
keep evolving during the transition from the USR phase into the slow-roll
phase (this happens in realistic cases of smooth transitions or some sharp
transition scenarios), the large local non-Gaussianity generated in the USR
phase is erased by the subsequent evolution and the argument of Ref. riotto
applies.

As a closing note, let us also add an important point that in the discussion
has never been touched upon. Much ambiguity in the literature has been present
regarding which perturbations one should use to compute the PBH abundance.
However, it is rather clear by now that the most correct correct quantity is
the smoothed density contrast Y

$\delta_{m}=\delta_{l}-\frac{3}{8}\delta_{l}^{2},\,\,\,\,\delta_{l}=-\frac{4}{3}r_{m}\zeta^{\prime}(r_{m}),$
(I.5)

where $r_{m}$ is the point where the compaction function has its maximum,
rather than the metric perturbation $\zeta$. Indeed, on superhorizon scales
one can always shift the comoving curvature perturbation by an arbitrary
constant by a coordinate transformation. This makes the calculation of the PBH
unphysical. This problem is avoided by using $\delta_{m}$ which depends on
spatial derivatives of the curvature perturbation.

The point is now that the PBH abundance calculated through the correct
variable $\delta_{m}$ requires much smaller values of the short scale power
spectrum amplitude ${\cal P}_{\zeta}(k_{S})$ than if the same PBH abundance is
calculated using the comoving curvature perturbation. For instance, for the
value adopted in Ref. yokoyamanew  ${\cal P}_{\zeta}(k_{S})={\cal
O}(10^{-1})$, the same abundance of PBHs, when calculated with the correct
variable $\delta_{m}$, requires ${\cal P}_{\zeta}(k_{S})={\cal O}(10^{-3})$ DR
, which reduces considerably the loop correction to the large scale CMB
spectrum even in the case of sudden transition.

We conclude that the one may not claim that the PBH formation scenario from
single-field inflation is ruled out in full generality as the fine details of
the transition from the USR phase into the standard slow-roll one matter, as
well as the adoption of the correct variable to calculate the PBH abundance.

Acknowledgements

A.R. is supported by the Boninchi Foundation for the project “PBHs in the Era
of GW Astronomy”.

## References

  * (1) P. Ivanov, P. Naselsky and I. Novikov, Phys. Rev. D 50, 7173 (1994).
  * (2) J. García-Bellido, A.D. Linde and D. Wands, Phys. Rev. D 54 (1996) 6040 [astro-ph/9605094].
  * (3) P. Ivanov, Phys. Rev. D 57, 7145 (1998) [astro-ph/9708224].
  * (4) J. Kristiano and J. Yokoyama, [hep-th/2211.03395].
  * (5) J. Kristiano and J. Yokoyama, [hep-th/2303.00341].
  * (6) A. Riotto, [astro-ph.CO/2301.00599].
  * (7) J. M. Maldacena, JHEP 05 (2003), 013 [astro-ph/0210603].
  * (8) Y. F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D. G. Wang and Z. Wang, JCAP 05 (2018), 012 [astro-ph.CO/1712.09998].
  * (9) A. A. Starobinsky, JETP Lett. 55, 489 (1992).
  * (10) S. Young, I. Musco and C. T. Byrnes, JCAP 11, 012 (2019) [astro-ph.CO/1904.00984].
  * (11) V. De Luca and A. Riotto, Phys. Lett. B 828, 137035 (2022) [astro-ph.CO/2201.09008].