#  Tsallis cosmology and its applications in dark matter physics with focus on
IceCube high-energy neutrino data

P. Jizba<EMAIL_ADDRESS>FNSPE, Czech Technical University in Prague,
Břehová 7, 115 19 Praha 1, Czech Republic
ITP, Freie Universitat Berlin, Arnimallee 14, D-14195 Berlin, Germany.  G.
Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica ”E.R. Caianiello”,
Universita’ di Salerno, I-84084 Fisciano (Sa), Italy &
INFN - Gruppo Collegato di Salerno, Italy

###### Abstract

In this paper we employ a recent proposal of C. Tsallis and formulate the
first law of thermodynamics for gravitating systems in terms of the extensive
but non-additive entropy. We pay a particular attention to an integrating
factor for the heat one-form and show that in contrast to conventional
thermodynamics it factorizes into thermal and entropic part. Ensuing first law
of thermodynamics implies Tsallis cosmology, which is then subsequently used
to address the observed discrepancy between current bound on the Dark Matter
relic abundance and present IceCube data on high-energy neutrinos. To resolve
this contradiction we keep the conventional minimal Yukawa-type interaction
between standard model and Dark Matter particles but replace the usual
Friedmann field equations with Tsallis-cosmology-based modified Friedmann
equations. We show that when the Tsallis scaling exponent $\delta\sim 1.57$
(or equivalently, the holographic scaling exponent $\alpha\sim 3.13$) the
aforementioned discrepancy disappears.

##  I Introduction

In his seminal paper [1], Jacobson showed that there is a deep connection
between gravity and thermodynamics, with the possibility to derive the
Einstein field equations from the first law of thermodynamics (see also Refs.
[2, 3, 4] for alternative approaches). The important upshot of this approach
is that one can infer the cosmological equations (the Friedmann equations)
from the first law of thermodynamics on the apparent horizon [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16]. In recent years, there has been an upsurge of
interest in extending this line of thoughts to cases where more general
entropies than just the conventional Boltzmann–Gibbs entropy are employed [17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. These models account for various
modifications of Bekenstein–Hawking’s entropy area law. For instance, in the
context of loop quantum gravity [28, 29, 30] or entropic cosmology [31] the
area law gets logarithmic corrections due to entanglement of quantum fields
inside and outside the horizon [32, 33, 34, 35, 36]. Similarly, the
generalized non-additive entropies [37, 38, 39] often lead to more generic
power-law instead of area law behavior. A simple but prominent example of the
latter is the so-called $\delta$-entropy

$\displaystyle S_{\delta}\ =\ \gamma_{\delta}A^{\delta}\,,$ (I.1)

where $A$ is the horizon area, $\delta$ is the scaling exponent and
$\gamma_{\delta}$ is a $\delta$-dependent constant, which for $\delta=1$
reduces to Hawking’s conventional form $\gamma=1/(4L_{p}^{2})$. Entropy
$S_{\delta}$ is a particular example of the entropy that was introduced by
Tsallis in Ref. [40] in order to define a correct thermodynamical entropy in 3
spatial dimensions for systems with the sub-extensive scaling, such as, e.g.
black holes. This issue was then further elaborated, e.g., in Refs. [37, 38].
Entropy $S_{\delta}$ is a special member of a two-parameter class of entropic
functionals known as $S_{q,\delta}$ (also introduced by Tsallis in [37]),
where $S_{\delta}\equiv S_{1,\delta}$. It should be emphasized that
$S_{\delta}$ has nothing to do with the popular Tsallis entropy [41, 42, 43]
that is widely used in statistical physics and theory of complex dynamical
systems. Strictly speaking, Tsallis’ entropy with the non-extensivity
parameter $q$ is the $S_{q,1}$ member in the previously mentioned two-
parameter class of entropies.

The so-called Tsallis cosmology is a particular approach that accommodates the
$S_{\delta}$ entropy directly into the first law of thermodynamics in order to
arrive at the modified cosmological Friedmann equations. The standard
cosmological model is then recovered in the limit $\delta=1$. It should be
noted that usage of $S_{\delta}$ in formulating the first law of thermodynamic
seem a bit ad hoc in the current literature. For this reason, we stick in this
paper to the original Tsallis proposal [37] for $S_{\delta}$ and formulate the
first law so that the entropy will be extensive but not additive. Due to a non
additive nature of the entropy a particular attention must be paid to the
integration factor of the heat one-form, which in this case is not a simple
inverse of thermodynamic temperature, but instead it factorises into entropic
and thermal part.

With the first law of thermodynamics at hand one can consider potential
implications of the ensuing Tsallis cosmology. In particular, here we show
that the Tsallis cosmology is capable to alleviate the discrepancy between the
current bound on the PeV Dark Matter (DM) relic abundance and recent IceCube
data [44, 45] about neutrino events with high energies ($\sim 1$ PeV) [45].
Though astrophysical sources are favorite candidates for the origin of these
high energy events [46, 47, 48, 49], another viable possibility is that these
neutrinos are the product of the decay of PeV mass Dark Matter [50, 51, 52,
53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68] (see also
Refs. [69, 70, 71, 72, 73, 69, 74, 75]). In this latter case, the minimal
extension of Standard Model (SM) describing such a decay is given by the
renormalizable, dimension four (in mass units) Yukawa-type interaction

${\cal L}_{4{\rm{-dim}}}\ =\ y_{\sigma\chi}\,\bar{L}_{\sigma}\cdot H\chi\,.$
(I.2)

Here $\sigma=e,\mu,\tau$ indicates the mass eigenstates of the three active
neutrinos, $H$ the Higgs doublet, $L_{\sigma}$ the left-handed lepton doublet,
$\chi$ the DM particle, and $y_{\sigma\chi}$ the (dimensionless) Yukawa
coupling constants. We calculate the freeze-in abundance of DM in Tsallis’
cosmology and fix the parameter $\delta$ so as to be consistent with the
observed relic abundance and the IceCube requirements. More precisely, the
IceCube high energy events and the DM relic abundance are not compatible with
the DM production if the latter is ascribed to the 4-dimensional operator
(I.2). This apparent tension can be resolved if one assumes that the Universe
evolves according to Tsallis-cosmology implied Friedmann equations, provided
one properly constrains the scaling exponent $\delta$.

The layout of the paper is as follows. In the next section we discuss the role
of $S_{\delta}$ entropy and place it in a proper thermodynamic framework. A
particular attention is paid to an integrating factor for the heat one-form.
It is show that the latter cannot be simply identified with inverse
thermodynamic temperature, but instead it factorises into entropic an thermal
part. With the first law of thermodynamics obtained we discuss in Section III
the ensuing modified Friedmann equations, which are obtain when one applies
the first law, at apparent horizon of a FRW (Friedmann–Robertson–Walker)
Universe. In Section IV we show that the DM relic abundance and the IceCube
data can be explained in a consistent way by using the minimal 4-dimensional
Yukawa-type interaction operator (I.2) and by assuming that the cosmological
background is described via Tsallis cosmology. Finally, Section V summarizes
our results and discusses possible extensions. For the reader’s convenience
the paper is supplemented with one appendix which clarifies more technical
aspects associated to discrepancy between PeV neutrinos and IceCube data in
the conventional FRW cosmology.

##  II Thermodynamic framework for the $S_{\delta}$ entropy

Application of laboratory thermodynamics in (self-)gravitating systems is
fraught with peril: from a formal point of view, it cannot even be defined,
because there is no thermodynamic limit. The technical reason why one cannot
scale the system to an infinitely large size stems from the fact that
gravitation is a long-range force, which implies that the gravitational
potential energy grows faster than a linear function of the mass of the
system.

In many cosmological systems, such as black holes, this is typically rectified
by employing entropies that have in 3 spatial dimensions a sub-extensive
scaling (such as area law scaling of the Hawking–Bekenstein (HB) entropy).
Recently Tsallis proposed [37, 38] an alternative viewpoint, namely that such
systems might still allow for a conventional thermodynamic description
provided the entropy involved is extensive but not additive. In this section
we will consider this proposal more seriously and compute the ensuing
integration factor for the heat one-form. We show that this not only allows to
define the temperature but it will also provide a heuristic justification for
the entropic form (I.1).

We start by observing that the key property in thermodynamic framework is the
Legendre transform which, for instance, for Gibbs free energy takes the form

$\displaystyle G(T,p,N,\ldots)\ =\ U(S,V,N,\ldots)\ +\ pV\ -\ TS\,,$ (II.1)

where $G$ and $U$ stand for Gibbs free energy and internal energy,
respectively. Both $G$ and $U$ are expressed in terms of their natural
variables and dots stand for prospective additional state variables.

By following [37, 38], we now define the length-scale independent
thermodynamic potentials $g=\lim_{L\rightarrow\infty}G/L^{\varepsilon}$ and
$u=\lim_{L\rightarrow\infty}U/L^{\varepsilon}$, where $L$ is the
characteristic linear scale of the system and $\varepsilon$ is a scaling
exponent (not necessarily identical with the spatial dimension $d$). Note that
$g$ and $u$ must satisfy (for large $L$)

$\displaystyle G(T,p,N,\ldots)\ =\
L^{\varepsilon}g(T/L^{\theta},p/L^{\theta},N/L^{d},\ldots)\,,$ $\displaystyle
U(S,V,N,\ldots)\ =\ L^{\varepsilon}u(S/L^{d},1,N/L^{d},\ldots)\,.$ (II.2)

Here we do not assume that the scaling exponent $\theta$ has a typical
laboratory value $\theta=0$. So, because (for large $L$)
$G(T,p,N,\ldots)\propto L^{\varepsilon}$, $U(S,V,N,\ldots)\propto
L^{\varepsilon}$, $p\propto L^{\theta}$ and $T\propto L^{\theta}$, then (II.1)
inevitably implies that $S\propto L^{d}$ (this was implicitly used in (II.2))
and $\varepsilon=\theta+d$. In this way one can (for large $L$) rewrite (II.1)
in the form

$\displaystyle g(T/L^{\theta},p/L^{\theta},N/L^{d},\ldots)\ =\
u(S/L^{d},1,N/L^{d},\ldots)\ +\ \frac{p}{L^{\theta}}\cdot 1\ -\
\frac{T}{L^{\theta}}\frac{S}{L^{d}}\,.$ (II.3)

hence, the structure of Legendre transform is satisfied also for length-scale
independent thermodynamic potentials. A lesson that can be drawn from this
analysis is that entropy should be extensive quantity (provided that $T$ and
$p$ scale in the same way) irrespective of the actual scaling of thermodynamic
potentials (which should be inevitably the same for all of them). It is clear
that one could repeat the same argument for other thermodynamic potentials.

Holographic principle posits that entropy of a black hole and more generally
the entropy of the Universe is a Shannon entropy with a peculiar area-law
scaling, namely

$\displaystyle S_{{\rm{HB}}}\ \propto\ -\sum_{i}p_{i}\log p_{i}\ =\ L^{2}\,,$
(II.4)

where $L$ is a characteristic length-scale in the problem. By the asymptotic
equipartition property [76] is

$\displaystyle S_{{\rm{HB}}}\ \propto\ \log W\,,$ (II.5)

where $W$ is a number of states (more precisely a volume of a typical set).
So, $W$ should scale exponentially so that

$\displaystyle W\ =\
\phi(L)\eta^{L^{2}}\,,\;\;\;\mbox{with}\;\;\;\eta>1\;\;\;\;\;\;\mbox{and}\;\;\;\lim_{L\rightarrow\infty}\phi(L)/L\
=\ 0\,.$ (II.6)

As argued before, the scaling of $S_{{\rm{BH}}}$ prevents to consider it as a
full-fledged thermodynamic entropy. In this connection, it was argued in [17]
that the entropy

$\displaystyle S_{\delta}\ \propto\
\sum_{i}p_{i}\left(\log\frac{1}{p_{i}}\right)^{\\!\\!\delta}\,,\;\;\;\;\delta\
>\ 0\,,$ (II.7)

which for equiprobable distribution behaves as 111Note that the positivity of
$\delta$ implies that $S_{\delta}$ grows with an increasing number of
available microstates. This might be viewed as a consistency condition on the
validity of the “generalized” second law of thermodynamics.

$\displaystyle S_{\delta}\ \propto\ \left(\log W\right)^{\delta}\,,$ (II.8)

might potentially represent a correct thermodynamic entropy in 3 spatial
dimensions for systems with the sub-extensive scaling (II.6). Though many
other non-additive and potentially useful entropies are available in the
literature, cf. e.g. [77, 78, 79, 80, 81, 82] our focus here will be on
$S_{\delta}$. Among other, this will allow us to make contact with other
results associated with Tsallis cosmology.

It is well known that the area-law formula for black hole entropy (II.4) holds
only in Einstein theory, i.e., when the ensuing action functional includes
only a linear term of scalar curvature $R$. On the other hand, the area-law
formula of black hole entropy no longer holds in generic higher-derivative
gravity theories [10], for instance in $f(R)$ gravity the entropy of a static
black hole acquires the form $S\propto L^{2}f^{\prime}(R)$, cf. e.g. [83]. It
is thus intriguing to consider entropy (II.4) with more general scaling law,
namely

$\displaystyle S_{{\rm{Gen.HB}}}\ \propto\ L^{\alpha}\,,$ (II.9)

in other words, we allow for a deformation in the holographic scaling. So,
$\delta$ from (I.1) equals to $\alpha/2$. In the spirit of Tsallis suggestion
we now assume that $S_{3/\alpha}$ is a thermodynamic entropy. There are two
immediate impediments associated with this assumption. First, $S_{3/\alpha}$
is not additive (not even in the $L\rightarrow\infty$ limit) but it satisfies
the pseudo-additivity rule

$\displaystyle S_{3/\alpha}(A+B)\ =\ \left[S_{3/\alpha}^{\alpha/3}(A)\ +\
S_{3/\alpha}^{\alpha/3}(B)\right]^{3/\alpha}\,,$ (II.10)

for any two independent subsystems $A$ and $B$. This is an inevitable
consequence of working with systems with the sub-extensive scaling — such as
gravity. Second, it is not clear what is a thermodynamic conjugate to such an
entropy. Caratheodory theorem [84, 85] ensures that heat one-form has an
integration factor but since the entropy is not additive one cannot use the
conventional Carnot cycle argument [86] in the proof of Clausius equality, to
simply identify the integration factor with inverse temperature.

Let us dwell a bit more on this last point. Since the exact differential
associated with the heat one-form is entropy, we can write

$\displaystyle dS_{3/\alpha}({\bf{a}},\theta)\ =\ \mu({\bf{a}},\theta)\delta
Q({\bf{a}},\theta)\,,$ (II.11)

where ${\bf{a}}$ represent a collection of relevant state variables and
$\theta$ is some empirical temperature whose existence is guaranteed by the
zeroth law of thermodynamics. We now divide the system in question into two
subsystems $A$ and $B$, that are respectively described by state variables
$\\{{\bf{a}}_{1},\theta\\}$ and $\\{{\bf{a}}_{2},\theta\\}$, respectively.
Then

$\displaystyle\delta Q_{A}({\bf{a}}_{1},\theta)\ =\
\frac{1}{\mu_{A}({\bf{a}}_{1},\theta)}\
\\!dS_{A,3/\alpha}({\bf{a}}_{1},\theta)\;\;\;\mbox{and}\;\;\;\delta
Q_{B}({\bf{a}}_{2},\theta)\ =\ \frac{1}{\mu_{B}({\bf{a}}_{2},\theta)}\
\\!dS_{B,3/\alpha}({\bf{a}}_{2},\theta)\,.$ (II.12)

So, for the whole system

$\displaystyle\delta Q_{A+B}\ =\ \delta Q_{A}\ +\ \delta
Q_{B}\;\;\;\mbox{with}\;\;\;\delta Q_{A+B}({\bf{a}}_{1},{\bf{a}}_{2},\theta)\
=\ \frac{1}{\mu_{A+B}({\bf{a}}_{1},{\bf{a}}_{2},\theta)}\
\\!dS_{(A+B),3/\alpha}({\bf{a}}_{1},{\bf{a}}_{2},\theta)\,,$ (II.13)

we can write

$\displaystyle dS_{(A+B),3/\alpha}({\bf{a}}_{1},{\bf{a}}_{2},\theta)\ =\
\frac{\mu_{A+B}({\bf{a}}_{1},{\bf{a}}_{2},\theta)}{\mu_{A}({\bf{a}}_{1},\theta)}\
\\!dS_{A,3/\alpha}({\bf{a}}_{1},\theta)\ +\
\frac{\mu_{A+B}({\bf{a}}_{1},{\bf{a}}_{2},\theta)}{\mu_{B}({\bf{a}}_{2},\theta)}\
\\!dS_{B,3/\alpha}({\bf{a}}_{2},\theta)\,.$ (II.14)

Let us now assume that there is only one state variable (apart from $\theta$),
so that ${\bf{a}}=a$. If that there would be more state variables, our
following argument would go through as well but we would need to employ more
than two subsystem. Under this assumption we can invert
$S_{A,3/\alpha}(a_{1},\theta)$ and $S_{B,3/\alpha}(a_{b},\theta)$ and write

$\displaystyle a_{1}\ =\
a_{1}(S_{A,3/\alpha},\theta)\;\;\;\mbox{and}\;\;\;a_{2}\ =\
a_{2}(S_{B,3/\alpha},\theta)\,.$ (II.15)

With this (II.14) can be rewritten as

$\displaystyle dS_{(A+B),3/\alpha}(S_{A,3/\alpha},S_{B,3/\alpha},\theta)\ $
$\displaystyle=$ $\displaystyle\
\frac{\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta)}{\mu_{A}(S_{A,3/\alpha},\theta)}\
\\!dS_{A,3/\alpha}$ (II.16) $\displaystyle+$ $\displaystyle\
\frac{\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta)}{\mu_{B}(S_{B,3/\alpha},\theta)}\
\\!dS_{B,3/\alpha}\ +\ 0\ \\!d\theta\,.$

Since $dS_{3/\alpha}$ (for all considered systems) must be a total
differential (so that $S_{3/\alpha}$ is a proper state function),
integrability conditions give

$\displaystyle\frac{\partial\log(\mu_{A}(S_{A,3/\alpha},\theta))}{\partial\theta}\
=\ \frac{\partial\log(\mu_{B}(S_{B,3/\alpha},\theta))}{\partial\theta}\ =\
\frac{\partial\log(\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta))}{\partial\theta}\,,$
(II.17)
$\displaystyle\frac{1}{\mu_{A}(S_{A,3/\alpha},\theta)}\frac{\partial\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta)}{\partial
S_{B,3/\alpha}}\ =\
\frac{1}{\mu_{B}(S_{B,3/\alpha},\theta)}\frac{\partial\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta)}{\partial
S_{A,3/\alpha}}\,.$ (II.18)

Note that in (II.17) the derivatives cannot depend on entropy but only on
$\theta$. By denoting the RHS of (II.17) as $-\omega(\theta)$ we might resolve
(II.17) in the form

$\displaystyle\mu_{A}(S_{A,3/\alpha},\theta)\ =\ \psi_{A}(S_{A,3/\alpha})\
\\!\exp\left(-\int\omega(\theta)d\theta\right)\,,$
$\displaystyle\mu_{B}(S_{B,3/\alpha},\theta)\ =\ \psi_{B}(S_{B,3/\alpha})\
\\!\exp\left(-\int\omega(\theta)d\theta\right)\,,$
$\displaystyle\mu_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha},\theta))\ =\
\psi_{A+B}(S_{A,3/\alpha},S_{B,3/\alpha})\
\\!\exp\left(-\int\omega(\theta)d\theta\right)\,,$ (II.19)

where $\psi$ are some arbitrary functions of the entropy. Note that the
temperature part of $\mu$ is $\alpha$ independent.

Let us now observe from (II.10) that

$\displaystyle dS_{(A+B),3/\alpha}\ =\
\frac{S_{A,3/\alpha}^{\alpha/3-1}}{S_{(A+B),3/\alpha}^{\alpha/3-1}}\
\\!dS_{A,3/\alpha}\ +\
\frac{S_{B,3/\alpha}^{\alpha/3-1}}{S_{(A+B),3/\alpha}^{\alpha/3-1}}\
\\!dS_{B,3/\alpha}\,.$ (II.20)

By comparing this with (II.16) and (II.19) with (II.20) we can make
identification $\psi(S_{\bullet,3/\alpha})=\varsigma
S^{1-\alpha/3}_{\bullet,3/\alpha}$, where $\varsigma$ is a constant and
$\bullet$ stands for $A$, $B$ and $A+B$, respectively. With this
identification one can easily check that also the integrability condition
(II.18) is satisfied. In conventional thermodynamics $\psi$ would be only a
constant and so the integration factor could be identified with a genuine
absolute temperature. In the context of non-additive entropy $S_{3/\alpha}$ we
see that this is not so. Fortunately $\mu$ has a simple factorized form.

Let us now call the temperature part in $\mu$ in (II.19) as $1/T$. So, the
corresponding thermal contribution that enters the first thermodynamic law
will have the form

$\displaystyle\frac{1}{\mu}\ \\!dS_{3/\alpha}\ =\
T\frac{S^{\alpha/3-1}_{3/\alpha}}{\varsigma}\ \\!dS_{3/\alpha}\ =\
\frac{T}{\varsigma\alpha/3}\ \\!dS_{3/\alpha}^{\alpha/3}\,.$ (II.21)

In the following we will denote $3S_{3/\alpha}^{\alpha/3}/\varsigma\alpha$ as
${\mathcal{S}}_{(\alpha)}$. Note that ${\mathcal{S}}_{(\alpha)}\propto
L^{\alpha}$ and so it behaves in the same way as $S_{\alpha/2}$ in Eq. (I.1).
By analogy with (I.1) we set the proportionality factor to be
$(4\pi)^{\alpha/2}\gamma_{\alpha/2}$, where $\gamma_{\alpha/2}$ still needs to
be determined. Finally, we can write the first law of thermodynamics in the
form

$\displaystyle dU\ =\ Td{\mathcal{S}}_{(\alpha)}\ -\ pdV\,,$ (II.22)

In passing we note that in the cosmology framework (similarly as, e.g. in
fluid dynamics) the role of pressure is taken over by the work density $W$.

##  III Tsallis cosmology from the First law of thermodynamics

We assume a homogeneous and isotropic Universe (i.e., FRW Universe), thus the
line element is the given by
$ds^{2}={h}_{\alpha\beta}dx^{\alpha}dx^{\beta}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$, where $\tilde{r}=a(t)r$, $x^{\alpha}=(t,r)$, $\alpha=0,1$,
$h_{\alpha\beta}=(-1,a^{2}/(1-kr^{2}))$ is a two-dimensional metric, and $k$
is a constant (spatial) curvature parameter with values $-1,0,+1$ representing
open, flat and closed geometry, respectively. The physical boundary of the
Universe is assumed to be given by the apparent horizon with radius [87]

$\displaystyle\tilde{r}_{A}\ =\ \frac{1}{\sqrt{H^{2}+k/a^{2}}}\,,$ (III.23)

($H=\dot{a}/a$ is the Hubble parameter) with the associated surface gravity
and horizon temperature [8, 87]

$\displaystyle{\kappa}\ =\ \left.\frac{1}{2\sqrt{-h}}\
\\!\partial_{\alpha}\left(\sqrt{-h}h^{\alpha\beta}\partial_{\beta}\tilde{r}\right)\right|_{\tilde{r}={\tilde{r}}_{A}}\
=\
-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\;\;\;\;\;\;\mbox{and
}\;\;\;\;\;T\ =\ \frac{|\kappa|}{2\pi}\,.$ (III.24)

Here $\dot{\tilde{r}}_{A}\equiv d{\tilde{r}}_{A}/dt$, $\kappa$ is the surface
gravity and $H$ is the Hubble parameter. Usually one assumes that the apparent
horizon radius is (almost) fixed [10, 87, 89], which allows to set
$\dot{\tilde{r}}_{A}\ll 2H\tilde{r}_{A}$. This implies that the volume
(almost) does not change and one may thus simply take
$T=1/(2\pi\tilde{r}_{A})$. In order to satisfy the field equations, the
symmetries of the Einstein tensor, imply that the energy-momentum tensor in
the FRW Universe must have the perfect fluid form:
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$, where $\rho$ and $p$ are the
energy density and pressure, respectively, which are bound together via the
continuity equation

$\displaystyle\dot{\rho}\ +\ 3H(\rho\ +\ p)\ =\ 0\,.$ (III.25)

The work density (which is due to the change in the apparent horizon radius),
for a FRW Universe assumes the form
$W=-\frac{1}{2}{\mbox{Tr}}(T^{\mu\nu})=\frac{1}{2}(\rho-p)$ where “Tr” denotes
the two-dimensional normal trace, i.e.
${\mbox{Tr}}(T^{\mu\nu})=T^{\alpha\beta}h_{\alpha\beta}$. The first law of
thermodynamics (II.22) then acquires the form

$\displaystyle dU\ =\ Td{\mathcal{S}}_{(\alpha)}\ -\ WdV\,.$ (III.26)

We now use the fact that the increase in internal energy $dU$ due to the
change of the apparent horizon volume (i.e. a $3$-sphere of radius
$\tilde{r}_{A}$) corresponds to the decrease in the total energy content $E$
of the Universe inside of the volume, so that $dU=-dE$, and rewrite (III.26)
in the form

$\displaystyle dE\ =\ -Td{\mathcal{S}}_{(\alpha)}\ +\ WdV\,,$ (III.27)

which formally coincides with the so-called the unified first law [88]. By
employing the relation $E=\rho V$ (with $V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$
being the apparent horizon volume) and by setting $L$ in the definition of
${\mathcal{S}}_{(\alpha)}$ to be $\tilde{r}_{A}$, we obtain from (III.27) that

$\displaystyle V\dot{\rho}dt\ +\ \rho dV\ =\
-Td\left[(4\pi)^{\alpha/2}\gamma_{\alpha/2}\ \\!\tilde{r}_{A}^{\alpha}\right]\
+\ \frac{1}{2}(\rho-p)dV\,.$ (III.28)

If we now employ (III.25), the fact that $\dot{\tilde{r}}_{A}/\tilde{r}_{A}\ll
2H$ and identity $dV=4\pi\tilde{r}_{A}^{2}d\tilde{r}_{A}$, we arrive at

$\displaystyle\frac{\alpha}{2\pi\tilde{r}_{A}^{3}}\ \\!\gamma_{\alpha/2}\
\\!(4\pi\tilde{r}_{A}^{2})^{\alpha/2-1}\ \\!d\tilde{r}_{A}\ =\ H(\rho+p)\
\\!dt\ =\ -\frac{1}{3}\ \\!d\rho\,.$ (III.29)

The last equality is due to (III.25). Eq. (III.29) is nothing but a
differential version of Friedmann equation. Ensuing differential equation can
be solved yielding

$\displaystyle\rho(\tilde{r}_{A})\ =\
\left(\tilde{r}_{A}^{2}\right)^{\alpha/2-2}\left[\frac{2(4\pi)^{\alpha/2-2}\
\\!\alpha\ \\!\gamma_{\alpha/2}}{4-\alpha}\right]\ +\ c\,.$ (III.30)

By requiring that the matter density is positive and that for
$\tilde{r}_{A}\rightarrow\infty$ the density $\rho\rightarrow 0$, we see that
$\alpha<4$, the integration constant $c$ should be set to zero, and in
addition $\gamma_{\alpha/2}>0$. If we now employ (III.23) we obtain the first
modified Friedmann equation

$\frac{8\pi M_{\rm{Pl}}^{-\alpha}}{3}\rho\ =\
\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\alpha/2}\,,$ (III.31)

where $\gamma_{\alpha/2}$ appearing in (I.1) assumes the form

$\displaystyle\gamma_{\alpha/2}\ =\
\frac{3(4-\alpha)(4\pi)^{1-\alpha/2}M_{\rm{Pl}}^{\alpha}}{4\alpha}\,,$
(III.32)

where $M_{\rm{Pl}}$ is the Planck mass.

The second modified Friedmann equation governing the evolution of the Universe
in Tsallis cosmology can be obtained by taking time derivative of (III.31) and
using (III.25). With this we get, cf. also [87]

$\displaystyle\frac{\ddot{a}}{a}\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\alpha/2}\
=\ \frac{8\pi
M_{{\rm{Pl}}}^{-\alpha}}{3(4-\alpha)}\left[(1-\alpha)\rho-3p\right]\,.$
(III.33)

Cosmological observations such as Type Ia SNe [90], CMB [91] and the large
scale structure [92, 93], indicate that the Universe is currently in an
accelerated phase, hence $\ddot{a}>0$. By using the equation of state
$p=\omega\rho$ we get from (III.33) that

$\displaystyle(1-\alpha)\rho\ -\ 3\omega\rho\ >\
0\;\;\;\Rightarrow\;\;\;\omega<(1-\alpha)/3\,.$ (III.34)

From this we see that for $\alpha\geq 1$ it always follows that $\omega<0$,
while for $\alpha<1$ one can have also $\omega\geq 0$, thus in Tsallis
cosmology the accelerated phase of the late time Universe is possible even
with the ordinary matter. In particular, for $\omega=0$ (ordinary dust matter)
the accelerated expansion can be obtained with the scaling exponent
$\alpha<1$. Here, however, we will not explore this interesting issue any
further.

##  IV PeV neutrinos and DM relic abundance in Tsallis cosmology

The conventional Yukawa-type interaction (I.2) fails to explain the IceCube
data on high-energy neutrinos events ($\sim 1$ PeV) and PeV DM relic abundance
if the cosmological background evolves according to Einstein field equations
(see Appendix, cf. also e.g., [74, 75]). We will now demonstrate that such a
discrepancy can be avoided provided that the cosmological background evolves
according to Tsallis cosmology. To this end, we first cast the modified
Friedmann equation (III.31) in the form (we set $k=0$)

$H(T)\ =\ Q(T)\,H_{{\rm{St.Cosm.}}}(T)\,,$ (IV.1)

where
$H_{{\rm{St.Cosm.}}}=\displaystyle{\sqrt{\frac{8\pi}{3M^{2}_{{\rm{Pl}}}}\,\rho(T)}}$
is the Hubble parameter in the standard cosmology, and $Q(T)$ represents the
amplification factor, which in the present case reads

$\displaystyle Q(T)\ $ $\displaystyle=$ $\displaystyle\
\left[\sqrt{\frac{8\pi}{3}}\,\frac{\rho^{1/2}}{M_{{\rm{Pl}}}^{2}}\right]^{\frac{\alpha-2}{4-\alpha}}\
=\ \eta\left(\frac{T}{T_{*}}\right)^{\\!\nu}\,,$ (IV.2) $\displaystyle\eta\ $
$\displaystyle=$ $\displaystyle\ \left[\frac{2\pi}{3}\sqrt{\frac{\pi\
\\!g_{*}(T)}{5}}\right]^{\frac{\alpha-2}{4-\alpha}}\,,$ (IV.3)
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{2(\alpha-2)}{4-\alpha}\,,\quad T_{*}\ =\ M_{{\rm{Pl}}}\,.$
(IV.4)

Here we have used the relation $\rho=\frac{\pi^{2}g_{*}(T)}{30}\,T^{4}$, with
$g_{*}(T)\sim 106$ the effective number of degrees of freedom. The equation of
state for the radiation, $p=\rho/3$, implies that the continuity equation
reads $\dot{\rho}(t)+4H\rho(t)=0$ from which one simply gets that
$\rho(t)={\rho_{0}}/{a^{4}(t)}$, where $\rho_{0}$ is a constant. Inserting
this expression for $\rho$ into (III.31), one obtains
$a(t)=a_{0}\left(\frac{t}{4-\alpha}\right)^{1-\alpha/4}$. The latter implies
that there is a simple relation between the cosmic time and the temperature,
namely ${t}\propto T^{\frac{4}{\alpha-4}}$, which in turn gives that the
product $Ta(t)={\rm{constant}}$.

The modified expansion rate (IV.1) allows to write the inverse decay processes
(see Eq. (A.14)) in the form

$\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{inv.dec.}}}\ =\
-\frac{m_{\chi}^{2}\Gamma_{\chi}}{\pi^{2}sH}\,K_{1}\left(\frac{m_{\chi}}{T}\right)\,,$
(IV.5)

where

$sH\ =\
\frac{3.32\pi^{2}}{45}g_{*}^{3/2}\eta\left(\frac{m_{\chi}}{T_{*}}\right)^{\nu}\frac{1}{z^{5-\nu}}\,,\quad\quad
z\ \equiv\ \frac{T}{T_{*}}\,.$ (IV.6)

The DM relic abundance follows by using222In this calculations we use
$\displaystyle{\int_{0}^{\infty}dxx^{3+\nu}K_{1}(x)=2^{2+\nu}\Gamma\left(\frac{5+\nu}{2}\right)\Gamma\left(\frac{3+\nu}{2}\right)}$,
where $\Gamma(z)$ are the Gamma functions. Notice also that
$\Gamma\left(\frac{5}{2}\right)\Gamma\left(\frac{3}{2}\right)=\frac{3\pi}{2}$.
(IV.5) and (IV.6) (see Eq. (A.11))

$\displaystyle\Omega_{{\rm{DM}}}h^{2}\ $ $\displaystyle=$ $\displaystyle\
\frac{2m_{\chi}^{2}s_{0}h^{2}}{\rho_{{\rm{cr}}}}\int_{0}^{\infty}\frac{dx}{x^{2}}\left(-\frac{dY_{\chi}}{dT}\Big{|}_{T=\frac{m_{\chi}}{x}}\right)\,,$
$\displaystyle=$
$\displaystyle\frac{45h^{2}}{1.66\pi^{2}g^{3/2}}\frac{s_{0}M_{Pl}}{\rho_{{\rm{cr}}}}\frac{\Gamma_{\chi}}{m_{\chi}}\frac{2^{2+\nu}}{\eta}\left(\frac{T_{*}}{m_{\chi}}\right)^{\nu}\Gamma\left(\frac{5+\nu}{2}\right)\Gamma\left(\frac{3+\nu}{2}\right)$
$\displaystyle\simeq$ $\displaystyle\
0.1188\left(\frac{106,7}{g_{*}}\right)^{3/2}\frac{\sum_{\sigma}|y_{\sigma\chi}|^{2}}{10^{-58}}\,\Pi\,,$

where $\rho_{{\rm{cr}}}$ is the critical density, $s_{0}$ is the present value
of the entropy density and $h$ is “little $h$”, i.e. the dimensionless Hubble
constant, and

$\displaystyle\Pi\ =\ 10^{-33}\,\frac{2^{\nu}}{7.5\
\\!\eta}\left(\frac{T_{*}}{m_{\chi}}\right)^{\nu}\frac{\Gamma\left(\frac{5+\nu}{2}\right)\Gamma\left(\frac{3+\nu}{2}\right)}{\Gamma\left(\frac{5}{2}\right)\Gamma\left(\frac{3}{2}\right)}\,.$
(IV.8)

The DM relic abundance ($\Omega_{{\rm{DM}}}h^{2}\sim 0.1188$) and the IceCube
data ($\sum_{\sigma}|y_{\sigma\chi}|^{2}\sim 10^{-58}$) can be consistently
explained provided

$\Pi\ \simeq\ 1\,.$ (IV.9)

Using for the DM mass $m_{\chi}\sim\,\,\,$PeV $\sim 10^{6}$GeV and for
$T_{*}=M_{{\rm{Pl}}}=10^{19}$GeV, it then follows that $\alpha\simeq 3.131$. A
numerical solution is shown in Fig. 1. Remarkably, such a value is $<4$, as
required by a positive matter-field density, and $>0$ as needed for the
validity of the generalized second law of thermodynamics.

Figure 1: $\Pi$ vs $\alpha$

##  V Conclusions

To reconcile both the current bound on DM relic abundance and IceCube high-
energy events of neutrinos one must either change the minimal Yukawa-type
interaction between standard model particles and DM particle or change the
standard cosmological model or possibly modify both. Our modus operandi in
this paper was to keep the Yukawa-type interaction
$\overline{L}_{\sigma}H\chi$ unchanged but instead replace the FRW cosmology
with the Tsallis cosmology and the Friedmann field equations with ensuing
modified Friedmann equations. This proved to be a useful strategy.

In order to provide a sound thermodynamic basis for $S_{\delta}$ entropy and
ensuing first thermodynamic law a particular attention has been paid to an
integrating factor for the heat one-form. We have shown that the latter cannot
be simply identified with inverse thermodynamic temperature, but instead it
factorises into entropic an thermal part.

With the first law of thermodynamics at hand we have shown that the Tsallis
cosmology is capable to resolve the aforementioned discrepancy with only one
additional adjustable parameter — Tsallis scaling exponent $\alpha$. The main
idea of our proof relies on the observation that in non-additive cosmology of
Tsallis the expansion rate of the Universe can be cast in the form
$H(T)=Q(T)H_{{\rm{St.Cosm.}}}(T)$, where $Q(T)$ encodes the parameters
characterizing the model of gravity. Consequently, also the thermal history of
Universe and DM in it are altered. With the modified expansion rate we have
solved the Boltzmann equation and obtained the abundance of DM particles.
Results attained are consistent with observed DM relic abundance provided
$\alpha\sim 3.131$ (or equivalently $\delta\sim 1.565$), which is well in the
range of validity of $\alpha\in(0,4)$.

##  VI Acknowledgement

GL thanks MUR and INFN for support. PJ was in part supported by the FNSPE CTU
grant RVO14000.

##  Appendix A PeV neutrinos and IceCube data in GR

In this Appendix, we recall the main features related to DM relic abundance
and IceCube data [74, 75] by using conventional FRW cosmology. The simplest
4-dimensional operator able to explain the IceCube high energy signal, is
given by the Yukawa-type interaction (I.2).

We consider the freeze-in production, that is the DM particles are never in
thermal equilibrium owing to the fact that their interaction is very weak, and
are produced from the hot thermal bath [94, 74] . As usual, the evolution of
the DM particles is described by the Boltzmann equation. Defining
$Y_{\chi}=n_{\chi}/s$ the DM abundance (here $n_{\chi}$ is the number density
of the DM particles and $s=\frac{2\pi^{2}}{45}g_{*}(T)T^{3}$ the entropy
density, with $g_{*}\simeq 106.75$ the degrees of freedom), the Boltzmann
equation gives [74]

$\frac{dY_{\chi}}{dT}\ =\
-\frac{1}{H_{{\rm{St.Cosm.}}}Ts}\left[\frac{g_{\chi}}{(2\pi)^{3}}\int
C\frac{d^{3}p_{\chi}}{E_{\chi}}\right]\,,$ (A.10)

where $H_{{\rm{St.Cosm.}}}$ is the expansion rate of the Universe described by
conventional Friedmann equation (i.e., the Hubble parameter in the standard
cosmology), $g_{\chi}$ is the number of degrees of freedom for DM (in our case
it corresponds to two helicity projections, i.e. $g_{\chi}=2$) and $C$ is the
general collision term. In the case in which $dg_{*}/dT=0$, the DM relic
abundance acquires the form [74]

$\Omega_{{\rm{DM}}}h^{2}\ =\
\frac{2m_{\chi}^{2}s_{0}h^{2}}{\rho_{{\rm{cr}}}}\int_{0}^{\infty}\frac{dx}{x^{2}}\left(-\frac{dY_{\chi}}{dT}\Big{|}_{T=\frac{m_{\chi}}{x}}\right)\,,$
(A.11)

where $x=m_{\chi}/T$, $s_{0}$ the present value of the entropy density
($s_{0}=\frac{2\pi^{2}}{45}g_{*}T_{0}^{3}\simeq 2891.2/$cm3), and
$\rho_{{\rm{cr}}}$ the critical density ($\rho_{{\rm{cr}}}=1.054\times
10^{-5}h^{2}$GeV/cm3). Eq. (A.11) gives the observed DM abundance, that is
[95]

$\Omega_{{\rm{DM}}}h^{2}\Big{|}_{{\rm{obs}}}\ =\ 0.1188\pm 0.0010\,,$ (A.12)

and explain the IceCube neutrino PeV signals. The dominant contributions to DM
production, induced by (I.2), are the inverse decay processes
$\nu_{\sigma}+H^{0}\to\chi$ and $l_{\sigma}+H^{+}\to\chi$ (they are
proportional to factor $|y_{\sigma\chi}|^{2}$ and kinematically allowed for
$m_{\chi}>m_{H}+m_{\nu,l}$), and the Yukawa production processes, such as
$t+{\bar{t}}\to{\bar{\nu}}_{\sigma}+\chi$ (they are proportional to
$|y_{\sigma\chi}y_{t}|^{2}$), where $t$ is the top quark and $y_{t}$
represents the Yukawa coupling constant between top quark and Higgs boson. The
explicit expressions of these processes are [74]

$\frac{dY_{\chi}}{dT}\ =\
\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{inv.dec.}}}+\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{Yuk.prod.}}}\,,$
(A.13)

where

$\displaystyle\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{inv.dec.}}}\ $
$\displaystyle=$ $\displaystyle\
-\frac{m_{\chi}^{2}\Gamma_{\chi}}{\pi^{2}sH_{{\rm{St.Cosm.}}}}\,K_{1}\left(\frac{m_{\chi}}{T}\right)\,,$
(A.14) $\displaystyle\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{Yuk.prod.}}}\ $
$\displaystyle=$ $\displaystyle\ -\frac{1}{512\pi^{6}sH_{{\rm{St.Cosm.}}}}\int
d{\tilde{s}}\ \\!d\Omega\
\\!\sum_{\sigma}\frac{W_{t{\bar{t}}\to{\bar{\nu}}_{\sigma}\chi}+2W_{t\nu_{\sigma}\to
t\chi}}{\sqrt{\tilde{s}}}K_{1}\left(\frac{\sqrt{\tilde{s}}}{T}\right)\,.$
(A.15)

Here ${\tilde{s}}$ is the centre-of-mass energy, $W_{ij\to kl}$ are related to
scattering probabilities of corresponding processes, $\Gamma_{\chi}$ is the
interaction rate
$\Gamma_{\chi}=\displaystyle{\sum_{\sigma}\frac{|y_{\sigma\chi}|^{2}}{8\pi}}\,m_{\chi}$,
$\sigma=e,\mu,\tau$, and $K_{1}(x)$ is the modified Bessel function of the
second kind. Since $\frac{dY_{\chi}}{dT}\Big{|}_{{\rm{inv.dec.}}}$ is the
dominant process (see [74] for details), one gets

$\Omega_{{\rm{DM}}}h^{2}|_{{\rm{inv.dec.}}}\ =\
0.1188\frac{\sum_{\sigma}|y_{\sigma\chi}|^{2}}{7.5\times 10^{-25}}\,.$ (A.16)

From (A.16) it follows that the correct DM relic abundance (A.12) occurs
provided

$\sum_{\sigma=e,\mu,\tau}|y_{\sigma\chi}|^{2}\ =\ 7.5\times 10^{-25}\,.$
(A.17)

Such a result disagrees with the value of
$\sum_{\sigma=e,\mu,\tau}|y_{\alpha\chi}|^{2}$ needed to explain the IceCube
signals. In fact, the DM lifetime $\tau_{\chi}=\Gamma_{\chi}^{-1}$ has to be
larger that the age of the Universe, that is $\tau_{\chi}>t_{U}\simeq
4.35\times 10^{17}$sec. However, lower bound on DM lifetime provided by
IceCube spectrum is $\tau_{\chi}^{b}\simeq 10^{28}$sec, i.e.
$\tau_{\chi}\gtrsim\tau_{\chi}^{b}$ [74]. Inserting (A.17) into
$\Gamma_{\chi}$ one gets $\Gamma_{\chi}\ \simeq\ 4.5\times
10^{4}\displaystyle{\frac{m_{\chi}}{\text{1PeV}}}\text{sec}^{-1}$, to which
corresponds $\tau_{\chi}\ \simeq\ 2.2\times
10^{-5}\displaystyle{\frac{\text{1PeV}}{m_{\chi}}}\text{sec}\ \ll\ t_{U}$. The
IceCube observations require the dark matter decay lifetime of the order
$\tau_{\chi}\simeq 10^{28}$ sec, which implies
$\sum_{\sigma=e,\mu,\tau}|y_{\sigma\chi}|^{2}\ \simeq\ 10^{-58}$. Such a value
is $\sim 33$ order of magnitudes smaller than the value of
$\sum_{\sigma=e,\mu,\tau}|y_{\sigma\chi}|^{2}\sim 10^{-25}$ that is required
for explaining the DM relic abundance, cf. Eq. (A.17). Therefore, the IceCube
events and the DM relic abundance turn out to be incompatible with the DM
production if the latter is described by the renormalizable Yukawa-type
operator $\overline{L}_{\alpha}H\chi$ and one works in the context of the
standard cosmological model. As discussed in the main body of the paper, this
discrepancy can be alleviated when Tsallis cosmology is employed without
changing Yukawa-type interaction for DM.

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# The CO-to-H2 Conversion Factor of Galactic Giant Molecular Clouds using CO
isotopologues: High-resolution $X_{\rm CO}$ maps

Mikito Kohno1,2, and Yoshiaki Sofue3
1Astronomy Section, Nagoya City Science Museum, 2-17-1 Sakae, Naka-ku, Nagoya,
Aichi 460-0008, Japan
2Department of Physics, Graduate School of Science, Nagoya University, Furo-
cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan
3Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015,
Japan E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted 2023 November 22. Received 2023 November 1; in original form 2023
May 9)

###### Abstract

We investigated the correlation between intensities of the 12CO and 13CO
($J=1$-0) lines toward the Galactic giant molecular clouds (GMCs) W51A, W33,
N35-N36 complex, W49A, M17SW, G12.02-00.03, W43, and M16 using the FUGIN
(FOREST Unbiased Galactic plane Imaging survey with the Nobeyama 45-m
telescope) CO line data. All the GMCs show intensity saturation in the 12CO
line when the brightness temperature of 13CO is higher than a threshold
temperature of about $\sim 5$ K. We obtained high-resolution ($\sim
20\arcsec$) distribution maps of the $X_{\rm CO}$ factor ($X_{\rm CO,{iso}}$)
in individual GMCs using correlation diagrams of the CO isotopologues. It is
shown that $X_{\rm CO,{iso}}$ is variable in each GMC within the range of
$X_{\rm CO,{iso}}\sim(0.9{\rm-}5)\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$.
Despite the variability in the GMCs, the average value among the GMCs is found
to be nearly constant at $X_{\rm CO,{iso}}={(2.17\pm 0.27)\times 10^{20}}$
cm-2 (K km s${}^{-1})^{-1}$, which is consistent with that from previous
studies in the Milky Way.

###### keywords:

ISM: clouds – ISM: general – ISM: molecules – radio lines: ISM

††pubyear: 2024

##  1 Introduction

Giant molecular clouds (GMC) mainly consist of hydrogen molecules (H2) and are
the sites of star formation in galaxies (Blitz et al., 2007; Fukui & Kawamura,
2010; Dobbs et al., 2014; Chevance et al., 2023). The hydrogen molecule is
difficult to observe directly because it has no an electric dipole moment for
its homonuclear diatomic nature. Therefore, we often observe carbon monoxide
(CO) with rotational transitions at millimeter wavelengths (Heyer & Dame,
2015) and convert the intensity to the H2 column density using the CO-to-H2
conversion factor (hereafter $X_{\rm CO}$), which is given by

$X_{\rm CO}=N_{\rm H_{2}}/W_{\rm{}^{12}CO}\ {\rm[cm^{-2}\ (K\ km\
s^{-1})^{-1}]},$ (1)

where $W_{\rm{}^{12}CO}$ and $N_{\rm H_{2}}$ are the 12CO $J=$1-0 integrated
intensity and column density of hydrogen molecules, respectively (Bolatto et
al., 2013).

The conversion factor has been obtained in various methods by comparing CO
luminosity with virial mass (e.g.,Solomon et al., 1987; Scoville et al.,
1987), visual or infrared extinction (e.g.,Frerking et al., 1982; Lombardi et
al., 2006; Pineda et al., 2008; Lee et al., 2018), X-ray absorption
(e.g.,Sofue & Kataoka, 2016), gamma-ray brightness(e.g.,Strong et al., 1988;
Abdo et al., 2010; Planck Collaboration et al., 2015; Hayashi et al., 2019b),
and dust emission (e.g.,Planck Collaboration et al., 2011; Leroy et al., 2011;
Sandstrom et al., 2013; Fukui et al., 2014; Okamoto et al., 2017; Hayashi et
al., 2019a; Yasuda et al., 2023). The values have been discussed in the Milky
Way and Local Group galaxies, and their dependence on metallicity has been
suggested (e.g.,Wilson, 1995; Arimoto et al., 1996; Sakamoto, 1996; Dame et
al., 2001; Mizuno et al., 2001; Rosolowsky et al., 2003; Fukui & Kawamura,
2010; Liszt et al., 2010; Leroy et al., 2011; Lee et al., 2014; Lin et al.,
2016; Muraoka et al., 2017; Pitts & Barnes, 2021; Ohno et al., 2023). The
often used value currently in the Milky Way is $X_{\rm CO}=2.0\times 10^{20}\
{\rm cm^{-2}\ (K\ km\ s^{-1})^{-1}}$, having an uncertainty of $\pm 30\%$
(Bolatto et al., 2013).

Recently, Sofue & Kohno (2020)(hereafter paper I) proposed a new method to
estimate the molecular cloud mass considering the variability of the $X_{\rm
CO}$ factor from the correlation between 12CO and 13CO. Paper I analyzed only
two Galactic GMCs of M16 (Sofue, 2020b; Nishimura et al., 2021) and W43(Sofue
et al., 2019; Sofue, 2021; Kohno et al., 2021), while the spatial
distributions of $X_{\rm CO}$ inside the GMCs have not yet been discussed.

In this paper, we extend the analysis of Paper I, aiming to reveal the
variability of $X_{\rm CO}$ within the molecular cloud of scales ($\sim 10$-50
pc) at sub-pc resolutions ($\sim 20^{\prime\prime}$). The target GMCs are W51A
(Fujita et al., 2021), W33 (Kohno et al., 2018; Murase et al., 2022), W49A
(Miyawaki et al., 2022), N35-N36 complex (Torii et al., 2018; Sofue, 2019a,
b), M17SW (Nishimura et al., 2018; Sofue, 2020a, 2022), G012.02-00.03(Sanna et
al., 2014), W43(Sofue et al., 2019; Kohno et al., 2021), and M16(Sofue, 2020b;
Nishimura et al., 2021). These GMCs are also known as massive star-forming
regions in the Milky Way. The basic parameters of these GMCs are summarized in
Table 1.

This paper is structured as follows: section 2 introduces the FUGIN (FOREST
Unbiased Galactic plane Imaging survey with the Nobeyama 45-m telescope) data;
section 3 presents the methods of analysis; in section 4, we demonstrate the
results; in section 5, we discuss the variability of the $X_{\rm CO}$ factor,
and in section 6, we summarize the results.

Table 1: Properties of the Galactic massive star forming regions

Name | $l$ | $b$ | $V_{\rm LSR}$ | $D$ | $R_{\rm G}$ |  $T_{\rm rms}$ (13CO$J=$1-0) |  $T_{\rm rms}$ (12CO$J=$3-2) | Reference
---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [km s-1] | [kpc] | [kpc] | [K] | [K] |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
W51A | $49.5$ | $-0.4$ | 57 | 5.4 | 6.1 | $\sim 0.6$ | $\sim 0.9$ | [1]
W33 | $12.8$ | $-0.2$ | 35 | 2.4 | 5.7 | $\sim 0.6$ | $\sim 0.4$ | [2]
N35-N36 complex | $24.5$ | $0.1$ | 110 | 7.3 | 3.3 | $\sim 0.6$ | $\sim 0.5$ | [3,4]
W49A | $43.16$ | $0.0$ | 11 | 11 | 7.5 | $\sim 0.5$ | $\sim 0.4$ | [5]
M17SW | $15.0$ | $-0.7$ | 20 | 2.0 | 6.1 | $\sim 0.7$ | — | [6,7,8]
G012.02-00.03 | $12.02$ | $-0.03$ | 112 | 9.4 | 2.3 | $\sim 0.7$ | $\sim 0.3$ | [9]
W43 | $30.8$ | $0.0$ | 95 | 5.5 | 4.3 | $\sim 0.9$ | $\sim 0.4$ | [10,11]
M16 | $17.1$ | $0.7$ | 23 | 2.0 | 6.1 | $\sim 0.7$ | — | [12,13]

Columns: (1) Name. (2) Galactic longitude (3) Galactic latitude (4) Radial
velocity (5) Distance from the solar system (6) Distance from the Galactic
center. The distance to the Galactic centre from the solar system is assumed
to be $R_{0}=8.0$ kpc obtained by the mean value of the very long baseline
interferometry (VLBI) astrometry results (VERA Collaboration et al., 2020;
Reid et al., 2019). (7) r.m.s noise levels of the 13CO $J=$1-0 data (8) r.m.s
noise levels of the 12CO $J=$3-2 data (9) References [1] Fujita et al. (2021)
[2] Kohno et al. (2018) [3] Torii et al. (2018) [4] Sofue (2019a) [5] Miyawaki
et al. (2022) [6] Nishimura et al. (2018) [7] Sofue (2020a) [8] Yamagishi et
al. (2016) [9] Sanna et al. (2014) [10] Sofue et al. (2019) [11]Kohno et al.
(2021) [12] Sofue (2020b)[13] Nishimura et al. (2021)

##  2 Data

We utilized the 12CO and 13CO $J=$1-0 line data obtained with the Nobeyama 45
m telescope (FUGIN: Umemoto et al., 2017; Torii et al., 2019; Fujita et al.,
2023). The rest frequencies of 12CO and 13CO $J=$1-0 are 115.271 GHz and
110.201 GHz, respectively. The front end was the FOur beam REceiver System on
the 45-m Telescope (FOREST: Minamidani et al., 2016; Nakajima et al., 2019),
which is the four-beam, side-band separating (2SB), and dual-polarization
superconductor-insulator-superconductor (SIS) receiver. The observations were
performed in the on-the-fly mapping mode (Sawada et al., 2008). The back-end
system was used an FX-type spectrometer named SAM 45 (Kuno et al., 2011;
Kamazaki et al., 2012). The half-power beam width (HPBW) of the 45 m telescope
is 14″and 15″at 115 GHz and 110 GHz, respectively. The data are gridded to
8.5″and 0.65 km s-1for the spatial and velocity space. The final 3D cube used
in this study has a voxel size of $(l,b,v)=(8.5$″$,8.5$″, $0.65$ km s-1). The
effective resolution convolved with a Bessel $\times$ Gaussian function is
20″and 21″for 12CO and 13CO $J=$1-0, respectively. We subtracted the baselines
using a first-order polynomial function from each spectrum. The baseline
ranges are adopted from $-200$ km s-1to $-50$ km s-1and $200$ km s-1to $350$
km s-1for the first Galactic quadrant with the FUGIN CO survey (Umemoto et
al., 2017). The data are calibrated from the antenna temperature ($T_{A}^{*}$)
to the main-beam temperature ($T_{\rm MB}$) by measuring the main-beam
efficiency of 0.43 for 12CO, 0.45 for 13CO, and C18O using standard
calibration sources. Detailed information on the FUGIN project is summarized
in the project overview paper by Umemoto et al. (2017). Cube data calibrated
to the main-beam temperature are available at the Japanese Virtual Observatory
(JVO). We also used the 12CO ($J=$3-2) High-Resolution Survey (COHRS) archival
data of the Galactic Plane obtained by the James Clark Maxwell Telescope
(Dempsey et al., 2013; Park et al., 2023). The cube data are converted
$T_{A}^{*}$ to the $T_{\rm MB}$ scale using a main-beam efficiency of 0.61
(Park et al., 2023). The spatial and velocity resolutions are 16.6″and 0.635
km s-1, respectively. Comparing the FUGIN CO survey data, we smoothed them to
20″with the same spatial resolution. The 12CO $J=$3-2 data are re-gridded to
the same voxel size as the FUGIN data using CASA (CASA Team et al., 2022).
Table 1 presents the root-mean-square noise level of each GMC at the $T_{\rm
MB}$ scale.

Figure 1: The 12CO peak intensity maps of (a) W51A at the radial velocity of
57.0 km s-1, (b) W33 at 34.9 km s-1, (c) N35-N36 complex at 110.3 km s-1, (d)
W49A at 10.8 km s-1, (e) M17SW at 19.9 km s-1, (f) G012.02-00.03 at 111.6 km
s-1,(g) W43 at 95.3 km s-1, and (h) M16 at 23.2 km s-1.

##  3 Methods

We derived the H2 column density per velocity channel using the local thermal
equilibrium (LTE) method and the $X_{\rm CO}$ factor, as described in Paper I
and Pineda et al. (2008). The brightness temperature ($T_{\rm B}$) of the CO
line intensity with the excitation temperature ($T_{\rm ex}$) and optical
depth ($\tau$) is given by

$T_{\rm B}=T_{0}\left(\frac{1}{e^{T_{0}/T_{\rm
ex}}-1}-\frac{1}{e^{T_{0}/T_{\rm bg}}-1}\right)\left(1-e^{-\tau}\right)\ [{\rm
K}],$ (2)

where $T_{\rm bg}=2.725$ K is the temperature of the cosmic microwave
background radiation. $T_{0}=h\nu/k$ is the Planck temperature with $h$,
$\nu$, and $k$ being the Planck constant, rest frequency, and Boltzmann
constant, respectively. If we assume that the 12CO line is optically thick,
the excitation temperature is given by

$T_{\rm ex}=T_{0}^{115}\bigg{/}{\rm ln}\left(1+\frac{T_{0}^{115}}{T_{\rm
B}({\rm{}^{12}CO})_{\rm max}+0.83632}\right)\ [{\rm K}],$ (3)

where $T_{\rm B}({\rm{}^{12}CO})_{\rm max}$ and $T_{0}^{115}=5.53194$
correspond to the 12CO peak intensity and the Planck temperature at the rest
frequency of 12CO $J=$1-0, respectively. We assume that $T_{\rm ex}$ is equal
in the 12CO and 13CO line emissions, and express the optical depth as follows:

$\tau(^{13}{\rm CO})=-{\rm ln}\left(1-\frac{T_{\rm B}({\rm{}^{13}CO})_{\rm
max}/T_{0}^{110}}{(e^{T_{0}^{110}/T_{\rm ex}}-1)^{-1}-0.167667}\right),$ (4)

where $T_{\rm B}({\rm{}^{13}CO})_{\rm max}$ and $T_{0}^{110}=5.28864$
represent the 13CO peak intensity and the Planck temperature at the rest
frequency of 13CO $J=$1-0, respectively (Pineda et al., 2008). According to
Wilson et al. (2009), the 13CO column density is given by

$N_{\rm{}^{13}CO}=3.0\times
10^{14}~{}\frac{\tau}{1-e^{-\tau}}\frac{1}{1-e^{-T_{0}^{110}/T_{\rm
ex}}}~{}I_{\rm{}^{13}CO}\ [{\rm cm^{-2}}],$ (5)

where $I_{\rm{}^{13}CO}$ is the 13CO integrated intensity. Then, we converted
$N_{\rm{}^{13}CO}$ to the H2 column density using the abundance ratio of H2 to
13CO molecules given by

$N_{\rm H_{2}}({\rm{}^{13}CO})=Y_{\rm{}^{13}CO}N_{\rm{}^{13}CO}\ [{\rm
cm^{-2}}].$ (6)

Here, $Y_{\rm{}^{13}CO}$ is adopted as $(5.0\pm 2.5)\times 10^{5}$ (Dickman,
1978) following Paper I.

Figure 2: Scatter plots between SCD13L and SCD12X of (a) W51A, (b) W33, (c)
the N35-N36 complex, (d) W49A, (e) M17SW, (f) G012.02-00.03, (g) W43, and (h)
M16. Red points show the averaged values of each bin, and the error bars are
standard deviations of SCD12X. Blue curves indicate the fitting results of
scatter plots. The black dashed lines indicate the linear relation of
SCD${}_{\rm 12X}=$SCD13L. The color bars show the number of data points in a
bin. ($N_{\rm data}/{\rm bin}$) Figure 3:  Spatial distributions of $X_{\rm
CO,{iso}}$ in (a) W51A, (b) W33, (c) the N35-N36 complex, (d) W49A, (e) M17SW,
(f) G012,02-00.03, (g) W43, and (h) M16. The points are plotted of $T_{\rm B}$
(13CO) $>3\sigma$. The $1\sigma$ noise level of each GMC is presented in Table
1.

We then calculated the H2 column density per velocity channel, which is
defined as the spectral column density, using the $X_{\rm CO}$ and LTE method
from the intensity of 12CO and 13CO, as described in Paper I. The spectral
column densities (SCD) of 12CO and 13CO are expressed as

${\rm SCD}_{\rm 12X}=\frac{dN_{\rm H_{2}}({\rm{}^{12}CO})}{dv}=X_{\rm
CO}T_{\rm B}({\rm{}^{12}CO})\ {\rm[cm^{-2}(km\ s^{-1})^{-1}]},$ (7)

and

$\begin{split}{\rm SCD}_{\rm 13L}&=\frac{dN_{\rm
H_{2}}({\rm{}^{13}CO})}{dv}\\\ &=3.0\times
10^{14}~{}\frac{\tau}{1-e^{-\tau}}\frac{Y_{\rm{}^{13}CO}T_{\rm
B}({\rm{}^{13}CO})}{1-e^{-T_{0}^{110}/T_{\rm ex}}}\ {\rm[cm^{-2}(km\
s^{-1})^{-1}]}.\end{split}$ (8)

Here, ${\rm SCD}_{\rm 12X}$ and ${\rm SCD}_{\rm 13L}$ are the H2 column
densities at the peak velocity channels, assuming a standard (normalization)
value of $X_{\rm CO}=2.0\times 10^{20}\ {\rm cm^{-2}\ (K\ km\
s^{-1})^{-1}}$(Bolatto et al., 2013) and $Y_{\rm{}^{13}CO}=5.0\times 10^{5}$
(Dickman, 1978), respectively.

##  4 Results

###  4.1 Correlation between SCD12X and SCD13L

Figure 1(a-h) shows the 12CO $J=$1-0 peak intensity maps of W51A, W33, N35-N36
complex, W49A, M17SW, G012.02-00.03, W43, and M16 GMC. W49A and G012.02-00.03
GMC show compact CO distributions of $\sim 20$ pc, while W33, N35-N36 complex,
W43, and M16 have more diffuse distributions of molecular gas with about the
half-intensity level of the peak brightness temperature.

Figure 2(a-h) presents scatter plots between SCD12X and SCD13L of GMCs
presented in Figure1(a-h). Black dashed lines show the linear relation of
SCD12X = SCD13L. SCD12X shows saturation at high SCD13L, and the nonlinear
relation of each GMC has different saturation levels. Indeed, previous studies
have reported that the 12CO intensity shows apparent saturation at the high-
intensity level of 13CO from the observation of Galactic molecular clouds
(Langer et al., 1989; Pineda et al., 2008; Yoda et al., 2010). Curve fitting
is useful for quantitatively evaluating the nonlinear relationship between the
12CO and 13CO intensities. Here, we performed curve fitting to the scatter
plots between SCD12X and SCD13L using the free parameters of $\alpha$ and
SCDc(spectral column density coefficient), as in Paper I. SCD12X is given by

$\rm SCD_{\rm 12X}={\rm SCD}_{\rm c}\left(\frac{\rm SCD_{\rm 13L}}{{\rm
SCD}_{\rm c}}\right)^{\alpha}.$ (9)

Blue curves show the fitting results for individual GMCs. To confirm whether
the scatter plots shown in Figure 2 depend on the physical resolutions due to
different distances from the GMCs, we smoothed the data of GMCs to the same
physical resolution and grid size of $\sim 1$ pc and $\sim 0.45$ pc. This
physical resolution corresponds to a beam size of $\sim 20$″at the distance of
W49A, the GMC furthest from us in this study. In Figures 7 and 8(a-h) of the
Appendix, we show the obtained peak intensity maps and scatter plots of GMCs
for equal physical resolution. The results are almost the same as those in
Figure 2. The fitting parameters have the range of $\alpha=0.30$-$0.54$ and
SCDc=$(1.6$-$4.4)\times 10^{21}{\rm cm^{-2}\ (km\ s^{-1})^{-1}}$. Thus, we
find that the opacity and internal clump sizes in a GMC do not affect the
relation of the 12CO and 13CO correlation. Our analysis also shows that the
saturation level of correlations does not depend on physical spatial
resolution and pixel size.

Figure 4: Scatter plots between $X_{\rm CO,{iso}}$ and $T_{\rm B}$ (12CO) in
(a) W51A, (b) W33, (c) the N35-N36 complex, (d) W49A, (e) M17SW, (f)
G012,02-00.03, (g) W43, and (h) M16. Black points show the averaged values of
each bin, and the error bars are standard deviations of $X_{\rm CO,{iso}}$.
The color bars show the number of data points in a bin. ($N_{\rm data}/{\rm
bin}$) Figure 5: Scatter plots between $X_{\rm CO,{iso}}$ and
$R^{12}_{3-2/1-0}$ of (a) W51A, (b) W33, (c) the N35-N36 complex, (d) W49A,
(e) G012,02-00.03, and (f) W43. The points are plotted of $T_{\rm B}$ (13CO)
$>3\sigma$. The color bars show the number of data points in a bin. ($N_{\rm
data}/{\rm bin}$)

###  4.2 Spatial distributions of $X_{\rm CO,{iso}}$ in the Galactic GMCs

We obtained the $X_{\rm CO}$ factor (hereafter $X_{\rm CO,{iso}}$) of each GMC
from the correlation between SCD12X and SCD13L. $X_{\rm CO,{iso}}$ at each
pixel of the molecular clouds is expressed as follows:

$X_{\rm CO,{iso}}={\rm SCD}_{\rm c}\left(\frac{\rm SCD_{\rm 13L}}{{\rm
SCD}_{\rm c}}\right)^{\alpha}\bigg{/}T_{\rm B}({\rm{}^{12}CO})\
{\rm[cm^{-2}(K\ km\ s^{-1})^{-1}]},$ (10)

where $\alpha$ and SCDc are the fitting parameters obtained in Figures 2 and
8.

Figure 3(a-h) shows the $X_{\rm CO,{iso}}$ maps of each GMC. $X_{\rm
CO,{iso}}$ has a variability with the range of $(0.9$-$5.0)\times 10^{20}$
[cm-2(K km s${}^{-1})^{-1}$] in a GMC. Comparing Figure 1 and Figure 3, we
find an anti-correlation between the 12CO peak intensity and $X_{\rm
CO,{iso}}$. Figure 4 shows the scatter plot between $X_{\rm CO,{iso}}$ and
$T_{B}$ (12CO) in each GMC. $X_{\rm CO,{iso}}$ monotonically decreases with
increasing brightness temperature, except for G012.02-00.03. It also shows a
lower limit around $X_{\rm CO}\sim 2\times 10^{20}$cm-2 (K km s${}^{-1})^{-1}$
above $T(^{12}{\rm CO})>20$ K. These results may correspond to the saturation
of the 12CO intensity in the dense cores of the GMC. This is consistent with
previous studies of Perseus molecular clouds obtained by the correlation
between the 12CO integrated intensity and HI column number density derived
from the optical depth of 353 GHz dust emission ($\tau_{\rm 353}$ ) by Planck
satellite observations (see Figure 12 and 15 in Okamoto et al., 2017).

Table 2: Fitting results and <$X_{\rm CO,{iso}}$> taken from the correlation
of SCD12X and SCD13L.

Name | $\alpha$ | SCDC | $<X_{\rm CO,{iso}}>$ | C.C
---|---|---|---|---
|  | [$\times 10^{21}\ {\rm cm^{-2}\ (km\ s^{-1})^{-1}}$] | [$\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$] |
(1) | (2) | (3) | (4) | (5)
W51A | 0.39 | 3.3 | 2.23 | 0.76
W33 | 0.46 | 1.9 | 2.09 | 0.82
N35-N36 | 0.34 | 1.8 | 2.20 | 0.79
W49A | 0.45 | 4.4 | 2.68 | 0.78
M17 SW | 0.36 | 4.4 | 2.10 | 0.77
G012.02-00.03 | 0.54 | 1.6 | 1.69 | 0.69
W43 | 0.36 | 3.0 | 2.19 | 0.82
M16 | 0.30 | 3.3 | 2.15 | 0.89
Average | $0.40\pm 0.08$ | $2.96\pm 1.11$ | $2.17\pm 0.27$ | $0.79\pm 0.06$

Columns: (1) GMC names (2) The fitting parameter of the non-linear relation.
(3) The coefficient of the spectrum column density (SCDC). (4) The mean value
of $X_{\rm CO,{iso}}$ in a GMC. (5) Correlation coefficient. The errors of
averaged value are adopted as the standard variation in all GMCs.

Table 2 lists the results of parameter fitting for $\alpha$, SCDc, $X_{\rm
CO,{iso}}$, and the correlation coefficient obtained from the resolution of
Figure 8. The mean value of $X_{\rm CO,{iso}}$ of all GMCs in this study is
$(2.17\pm 0.27)\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$. $X_{\rm CO,{iso}}$
shows the local variability in the Galactic GMCs (Figure3), while the mean
value is consistent within the margin of error in previous works of $2\times
10^{20}$ cm-2 (K km s${}^{-1})^{-1}$ (Bolatto et al., 2013), $(1.8\pm
0.3)\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$ (Dame et al., 2001), and
$(2.54\pm 0.13)\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$ (Planck
Collaboration et al., 2011).

##  5 Discussion

###  5.1 The variability of $X_{\rm CO,{iso}}$

Our fitting results show that the correlation of column densities between 12CO
and 13CO is not universal in Galactic GMCs. Each GMC has its own correlation
parameters (Figures 2 and 8). Weiß et al. (2001) reported that the $X_{\rm
CO}$ factor for a virialized GMC depends on the kinetic temperature ($T_{\rm
kin}$) and molecular hydrogen number density ($n({\rm H_{2}})$) of the
molecular gas. (see also Chapter 2.1 in Bolatto et al., 2013). To reveal the
origin of the variations in $X_{\rm CO,{iso}}$, we investigated the 12CO
$J=$3-2/1-0 intensity ratio (hereafter $R^{12}_{3-2/1-0}$) using the JCMT 12CO
$J=$3-2 archival data (Dempsey et al., 2013; Park et al., 2023).
$R^{12}_{3-2/1-0}$ depends on $T_{\rm kin}$ and $n({\rm H_{2}})$ in molecular
clouds assuming the large velocity gradient model (Goldreich & Kwan, 1974). We
analyzed the W51, W33, N35, W49A, G12.02-00.03, and W43 GMC because the JCMT
data are only covered with $|b|<0.5^{\circ}$. Figure 5 shows the scatter plot
between $X_{\rm CO,{iso}}$ and $R^{12}_{3-2/1-0}$. The plots of each GMC do
not show a clear correlation. Thus, we suggest that the origin of own
correlation parameters and variations of $X_{\rm CO,{iso}}$ might not be
simply density and/or temperature variations in a molecular cloud. In our
analysis, we assumed that $Y_{\rm{}^{13}CO}$ is constant in all GMCs. We point
out that the change of the isotope abundance ratio with density might explain
the variability of $X_{\rm CO,{iso}}$ in a GMC.

###  5.2 Radial gradient of the $X_{\rm CO}$ factor

Figure 6: The $X_{\rm CO,{iso}}$ gradient from the distance of the Galactic
Center. The blue line shows the fitting result adopted by the exponential
function. The green dashed line indicates the relation from Arimoto et al.
(1996).

In addition, we investigated the radial gradient of the $X_{\rm CO}$ factor in
the Galactic disc. The averaged $X_{\rm CO}$ factor has a high value in W49A
at Galactocentric distance $R=$7.5 kpc, while G012.02-00.03 has a low value
$R=$2.3 kpc. According to Arimoto et al. (1996), the $X_{\rm CO}$ factor
increases with the Galactocentric radius by an exponential function given by

$\displaystyle{\rm log_{10}}\left(\frac{X_{\rm CO}}{X_{0}}\right)$
$\displaystyle=$ $\displaystyle 0.41\frac{(R-R_{0})}{r_{e}},$ (11)

where $r_{e}=6.2$ kpc is the scale radius of the Galactic disc and $X_{0}$ is
$X_{\rm CO}$ at $R=R_{0}=8.0$ kpc (VERA Collaboration et al., 2020; Reid et
al., 2019). This equation can be deformed as follows:

$\displaystyle X_{\rm CO}={0.59\times 10^{20}\exp\left(\frac{R}{6.567\
{\rm[kpc]}}\right)}\ {\rm[cm^{-2}(K\ km\ s^{-1})^{-1}]}$ (12)

Figure 6 shows a plot of $X_{\rm CO,{iso}}$ obtained in this paper as a
function of the Galactocentric distance. It is found that $X_{\rm CO,{iso}}$
increases with the distance from the Galactic Centre, and the plot may be
fitted by an exponential function by

${X_{\rm CO,{iso}}=1.61\times 10^{20}\exp\left({R\over 17.85[{\rm
kpc}]}\right){\rm[cm^{-2}(K\ km\ s^{-1})^{-1}]}}.$ (13)

Previous studies reported that a value of the $X_{\rm CO}$ factor depends on
the metal abundance in galaxies (e.g., Arimoto et al., 1996; Israel, 1997;
Leroy et al., 2011; Genzel et al., 2012). We suggest that the galactocentric
dependency of $X_{\rm CO,{iso}}$ is caused by the radial metallicity gradient
from the Galactic center to the outer Galaxy in the Milky Way. Our fitting
result yields a significantly larger $X_{\rm CO}$ at the Galactic Centre by a
factor of 3 compared to that by Arimoto et al. (1996) indicated by the green
dashed line. This discrepancy may be caused by no data point within $R<2$ kpc
in our study as well as by the normalization of the local value to $2\times
10^{20}$ [H2 cm-2(K km s-1)-1 at 8 kpc in the previous study. Sodroski et al.
(1995) reported that the $X_{\rm CO}$ factor within 400 pc in the Galactic
Centre region is by a factor 3-10 lower than the Galactic disk. Therefore, the
value of $X_{\rm CO}$ in the Galactic Centre is still controversial. The
present method applied to the 12CO and 13CO $J=$1-0 line data in the Galactic
Centre region would provide a more precise determination of $X_{\rm CO}$ in
the innermost Milky Way. A detailed analysis of this point will be presented
in a separate paper using the CO survey data of the Galactic Centre (e.g., Oka
et al., 1998; Torii et al., 2010; Enokiya et al., 2014; Tokuyama et al.,
2019).

###  5.3 Comparison with the virial methods

Table 3: Physical parameters and $X_{\rm CO,{vir}}$ using virial method

Name | $R$ | $\sigma_{v}$ | $M_{\rm vir}$ | $L_{\rm CO}$ | $X_{\rm CO,{vir}}$ | $X_{\rm CO,iso}/X_{\rm CO,vir}$
---|---|---|---|---|---|---
| [pc] | [km s-1] | [$M_{\odot}$] | [K km s-1 pc2] | [$\times 10^{20}$ cm-2 (K km s${}^{-1})^{-1}$] |
(1) | (2) | (3) | (4) | (5) | (6) | (7)
W51A | 6.3 | 4.2 | $1.2\times 10^{5}$ | $3.5\times 10^{4}$ | 1.51 | 1.5
W33 | 6.5 | 2.5 | $4.3\times 10^{4}$ | $1.7\times 10^{4}$ | 1.15 | 1.8
N35-N36 | 32 | 4.4 | $6.2\times 10^{5}$ | $3.8\times 10^{5}$ | 0.73 | 3.0
W49A | 4.8 | 4.8 | $1.2\times 10^{5}$ | $2.3\times 10^{4}$ | 2.21 | 1.2
M17 SW | 2.6 | 3.0 | $2.4\times 10^{4}$ | $6.1\times 10^{3}$ | 1.74 | 1.2
G012.02-00.03 | 7.1 | 2.4 | $4.2\times 10^{4}$ | $7.7\times 10^{3}$ | 2.47 | 0.68
W43 | 20 | 4.0 | $3.2\times 10^{5}$ | $2.5\times 10^{5}$ | 0.58 | 3.8
M16 | 6.0 | 2.1 | $2.9\times 10^{4}$ | $1.3\times 10^{4}$ | 1.01 | 2.1
Average | $11$ | $3.4$ | $1.7\times 10^{5}$ | $9.2\times 10^{4}$ | $1.43\pm 0.68$ | $1.80\pm 0.52^{*}$

Columns: (1) GMC names (2) Cloud radius defined by the %30 levels of the peak
integrated intensity (3) Mean velocity dispersion in a molecular cloud (4)
Virial mass (5) 12CO luminosity (6) The $X_{\rm CO}$ factor derived by the
virial mass. (7) Ratio of $X_{\rm CO,iso}$ to $X_{\rm CO,vir}$. *The averaged
value is calculated from the ratio of averaged $X_{\rm CO,iso}$ and $X_{\rm
CO,vir}$.

Finally, we calculated the $X_{\rm CO}$ factor from the virial mass of each
GMC compared with our estimation. According to Solomon et al. (1987), the
virial mass of GMCs is given by

$M_{\rm vir}=1040\ R\ \sigma_{v}^{2}\ [M_{\odot}],$ (14)

where $R$ is the effective radius of a GMC in parsec and $\sigma_{v}$ is the
1D velocity dispersion in km s-1, assuming a density profile of
$\rho(r)\propto r^{-1}$. We estimated the radius of a GMC with
$R=\sqrt{S/\pi}$, where S is the cloud area within 30% levels of the 12CO peak
integrated intensity. The velocity dispersion is adopted as the averaged value
of each pixel in a GMC. Then, the $X_{\rm CO}$ factor using virial methods is
expressed as

$X_{\rm CO,vir}={M_{\rm vir}\over\mu_{\rm H_{2}}m_{\rm H}L_{\rm CO}}\
{\rm[cm^{-2}(K\ km\ s^{-1})^{-1}]},$ (15)

where $\mu_{\rm H_{2}}\sim 2.8$ is the mean molecular weight per hydrogen
molecule (Appendix A.1 in Kauffmann et al., 2008), $m_{\rm H}=1.67\times
10^{-24}$g is proton mass and $L_{\rm CO}$ is the 12CO total luminosity. Table
3 shows the results of the $X_{\rm CO}$ factor in each GMC obtained by the
virial methods. The ratio of averaged $X_{\rm CO}$ taken from the virial mass
and CO isotopologues is $1.80\pm 0.52$.

##  6 Summary

The conclusions of this paper are summarized as follows:

  1. 1.

We studied the correlation between 12CO and 13CO intensities toward the
Galactic GMCs W51A, W33, N35-N36 complex, W49A, M17SW, G12.02-00.03, W43, and
M16 using the FUGIN CO survey data taken with the Nobeyama 45 m telescope.

  2. 2.

All the GMCs show intensity saturation of the 12CO line in regions with high
brightness of 13CO.

  3. 3.

We also present high-resolution $X_{\rm CO,{iso}}$ maps made from the
correlations of the CO isotopologues, which revealed local variability of
$X_{\rm CO,{iso}}$ in each GMC. We also show that the $X_{\rm CO,{iso}}$
monotonically decreases with increasing 12CO brightness temperature.

  4. 4.

The averaged value of all GMCs is calculated to be $X_{\rm CO,{iso}}={(2.17\pm
0.27)\times 10^{20}}$ cm-2 (K km s${}^{-1})^{-1}$, which is consistent within
the margin of error with the reported values in the previous review.

  5. 5.

We show that $X_{\rm CO,{iso}}$ increases with the Galactocentric distance in
accordance with the previous works while suggesting a larger value for the
Galactic Centre by a factor of 3 compared to that by Arimoto et al. (1996).

  6. 6.

The averaged value of the $X_{\rm CO}$ factor taken from the virial mass and
CO isotopologues of GMCs is consistent within the error by a factor of 2.

## Acknowledgements

The authors are grateful to the anonymous referee for carefully reading our
manuscript and giving us thoughtful suggestions, which greatly improved this
paper. We are grateful to Dr. Shinji Fujita of the University of Tokyo and
Prof. Toshihiro Handa of Kagoshima University for useful comments on data
analysis and discussion. The authors would like to thank the FUGIN project
members for the CO data observed by the Nobeyama 45-m telescope. The Nobeyama
45-m radio telescope is operated by the Nobeyama Radio Observatory. We
utilized the Python software package for astronomy (Astropy Collaboration et
al., 2013), NumPy (van der Walt et al., 2011), Matplotlib (Hunter, 2007),
IPython (Perez, & Granger, 2007), CASA (CASA Team et al., 2022), and APLpy
(Robitaille & Bressert, 2012).

Data availability: The data underlying this article are available in
http://nro-fugin.github.io. The FUGIN CO data were retrieved from the JVO
portal http://jvo.nao.ac.jp/portal operated by ADC/NAOJ. The JCMT COHRS cube
data is downloaded from the CANFAR data archive(Gaudet et al., 2009).

Conflict of interest: The authors declare that there is no conflict of
interest.

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##  Appendix A Effect of the physical resolution

To confirm that the scatter plots shown in Figure 2 do not depend on the
physical resolutions due to different distances to the GMCs, we smoothed the
data and made the same physical resolution maps of $\sim 1$ pc (Figure 7). The
result is shown in Figure 8, which is almost identical to the original plots.
We thus confirm that the results are not affected by the different distances
of the objects.

Figure 7: Same as Figure 1, but the data was smoothed to the same physical
resolution of $\sim 1$ pc.  Figure 8: Same as Figure 2, but the data was
smoothed to the same physical resolution of $\sim 1$ pc.