# Cosmological Number Counts under Disformal Transformations

Basundhara Ghosh1 Jérémie Francfort2 and  Rajeev Kumar Jain1

###### Abstract

We investigate whether true physical observables associated with the
measurements of large scale structure in the universe are frame-independent.
In particular, we study if cosmological observables such as the galaxy number
counts and weak lensing observables are invariant under the disformal
transformations. In a previous work, it was shown that this frame-invariance
holds true for the case of conformal transformations. In this work, we find
that although the cosmological number counts remain invariant under the
disformal transformations, convergence and cosmic shear associated with the
weak lensing potential are generally not invariant. Since the lightcone
structure does not remain causal under disformal transformations, photon
geodesics do not remain null anymore, and as a result, weak lensing
observables are indeed affected. We also briefly comment on the disformal
invariance of other cosmological observables.

##  1 Introduction

Scalar-Tensor (ST) theories are considered viable alternatives to the general
theory of relativity (GR) and are usually at play while describing the
inflationary epoch in the early universe or the present accelerated expansion
in the late universe. While GR still remains the minimal theory of gravity
which is remarkably successful in explaining diverse cosmological
observations, there has been a long quest to explore various modifications of
GR as well as to formulate viable alternatives and test them with multiple
observations. One of the well-known examples of a ST theory is the Brans-Dicke
theory, in which the gravitational interaction is mediated by a scalar field
in addition to the tensor field of GR [1] (See, for instance, Refs. [2, 3, 4]
for reviews on modified gravity). Moreover, there usually exists a non-trivial
coupling between the gravitational sector and the matter sector in all such
theories, and they also often have more than two propagating degrees of
freedom [5]. In comparison, GR strictly has two such physical degrees of
freedom, i.e. two polarization modes of the graviton. In order to be
consistent with the observational tests on smaller scales such as the scales
of our solar system, a realistic modification of gravity should necessarily
contain a physical mechanism – a screening mechanism to suppress the extra
propagating scalar degrees of freedom on such scales [6, 7, 8, 9, 10].

In order to construct viable modified theories, one must ensure that the
additional degrees of freedom arising in such modifications do not lead to a
ghost behaviour or other unwanted instabilities, both at the background and
the perturbative order. These requirements led to the construction of the
Horndeski theory – the most general ST theory with a single scalar field
leading to second-order equations of motion, thereby avoiding ghost degrees of
freedom [11]. These theories have found numerous applications in cosmology,
particularly in constructing viable models of inflation and dark energy [12].
Lately, healthy extensions beyond Horndeski theories have also been studied
which are usually obtained by performing the so-called disformal
transformation of the field space metric tensor which is essentially a
generalization of the well-known conformal transformation (also sometimes
called the Weyl transformation). A theory invariant under the conformal
transformation is called conformally invariant which only involves rescaling
of the metric tensor by a conformal factor. It is important to note that such
a conformal transformation does not modify the causal structure of the
spacetime. In general, the disformal transformation contains a scalar field
$\phi$ as well as its first order derivatives111Note that the disformal
transformation is not just a simple field redefinition because it involves
derivatives of the scalar field. and can be written as

$g_{\mu\nu}=\mathcal{A}(\phi,X)\tilde{g}_{\mu\nu}+\mathcal{B}(\phi,X){\nabla}_{\mu}\phi{\nabla}_{\nu}\phi\,,$
(1.1)

where $\mathcal{A}$ and $\mathcal{B}$ are some general functions, referred to
as the conformal and disformal factors, respectively and
$X=-\frac{1}{2}{\nabla}_{\mu}\phi{\nabla}^{\mu}\phi$. Although
${\nabla}_{\mu}$ indicates the covariant derivative, for a scalar field it
becomes ${\nabla}_{\mu}\phi={\partial}_{\mu}\phi$. For $\mathcal{B}=0$, Eq.
(1.1) reduces to the ordinary conformal transformation while $\mathcal{A}=1$
represents the pure disformal transformation. In [13], the concept and
formulation of the disformal transformation was first introduced by Bekenstein
in order to relate geometries of the same gravitational theory which suggests
a richer set of possibilities for the transformed metric. Since some of these
possibilities may be unphysical, the functions $\mathcal{A}$ and $\mathcal{B}$
are generally subject to some constraints.

Over the years, disformal transformations have been applied in different
contexts in GR and cosmology. For the case of massless Klein-Gordon equation
[14] and the vacuum Maxwell’s equations [15], the disformal transformations
form a set of new symmetry transformations. Further, it has been shown that
the gauge-invariant primordial cosmological perturbations are invariant under
disformal transformation within the context of the Horndeski theory (in this
context, see Refs. [16, 17, 18, 19]). Moreover, they have been used to study
black holes in the context of ST theory [20], to investigate allowed regions
of the solution space and their symmetry [21], in the context of dark energy
[22, 23, 24], dark matter [25, 26] and more importantly, in generating a wider
class of healthy ST theories (without additional number of physical degrees of
freedom) beyond Horndeski theories, so called the DHOST theories [27, 28, 29,
30, 31, 32, 33, 34].

One may naively expect that true physical observables must be frame invariant.
However, it may still be a tedious task to explicitly show this invariance in
different physical frames. It is well known that conformal transformations
leave physical observables invariant. In an earlier paper [35], we had shown
that some specific observables associated with the distribution of large scale
structure remain invariant under a purely conformal transformation. It was
discussed that while the (unobservable) matter power spectrum is frame
dependent, the observable number counts do not depend on the choice of the
frame. In this paper, we study whether these observables such as the
cosmological number counts, lensing potential, convergence and shear are
invariant under the disformal transformations. Thus, the present work can be
considered as a disformal extension of the previous work.

Since time and spatial coordinates are rescaled by the same factor under
conformal transformations, they are usually referred to as the causality
preserving transformations as the lightcone structure remains preserved.
However, disformal transformations usually don’t preserve causality as the
time and spatial coordinates are rescaled by different factors. In order to
ensure causal behaviour for all particles, one therefore requires the
condition $\mathcal{B}<0$ everywhere. In fact, demanding that the disformally
transformed metric is healthy and well defined everywhere, there arises a set
of conditions that must be imposed on it: (i) It must be causal (ii) It must
preserve Lorentzian signature (iii) The inverse must exist and be non-singular
and (iv) The volume element must be non-singular. All these conditions will
ensure that the disformal frame is also a well-defined physical frame and the
physical observables, if correctly calculated, must therefore be independent
of the choice of any frame. However, one must keep in mind that disformal
transformations do not preserve the causal structure of lightcones. As a
result, the photons geodesics are not null anymore and therefore, the lensing
observables are affected. In other words, disformal transformations introduce
a new coupling of the scalar field to the stress energy tensor thereby leading
to modifications of gravitational lensing [36, 37] .

Conventions and notations: We work with the $(-,+,+,+)$ signature. Various
quantities in the Jordan frame are indicated with a tilde. As stated in Eq.
(1.1), the disformal transformation is given by

$g_{\mu\nu}=\mathcal{A}(\phi,X)\tilde{g}_{\mu\nu}+\mathcal{B}(\phi,X){\nabla}_{\mu}\phi{\nabla}_{\nu}\phi\,,\quad\mathrm{with}\quad
X=-\frac{1}{2}{\nabla}_{\mu}\phi{\nabla}^{\mu}\phi\,.$

Note that, as $\phi$ is a scalar, the covariant derivative is nothing but a
partial derivative, hence we do not put any tilde to avoid heavy notations. In
a cosmological context, the FRLW metrics are

$\displaystyle\boldsymbol{\tilde{g}}$
$\displaystyle=-\boldsymbol{\mathrm{d}}t^{2}+\tilde{a}^{2}\,\boldsymbol{\mathrm{d}}\boldsymbol{x}^{2}\,,$
(1.2) $\displaystyle\boldsymbol{{g}}$
$\displaystyle=-\alpha_{\mathrm{T}}^{2}\,\boldsymbol{\mathrm{d}}t^{2}+\alpha_{\mathrm{L}}^{2}\tilde{a}^{2}\,\boldsymbol{\mathrm{d}}\boldsymbol{x}^{2}\,,$
(1.3)

with

$\alpha_{\mathrm{T}}^{2}=\mathcal{A}-\mathcal{B}\dot{\phi}^{2}\,,\quad\alpha_{\mathrm{L}}^{2}=\mathcal{A}\,.$
(1.4)

This implies that the measurements of length and time in both frames are
related as

$L=\alpha_{\mathrm{L}}\tilde{L}\,,\quad T=\alpha_{\mathrm{T}}\tilde{T}\,.$
(1.5)

Note that the change in notation we have employed here compared to [35] is
slightly different in the sense that we have used upper-case $L$ and $T$
instead of the lower-case $\ell$.

This paper is organized as follows. In Section 2, we present a brief
dictionary of conformal and disformal transformations and point out the
scaling of various quantities, including the background variables and first
order perturbations. In Section 3, we obtain the transformations of the
density and volume perturbations, and prove that the cosmological number
counts are indeed equal in the Jordan and Einstein frames. Finally, in Section
4 we summarise our results and provide an outlook for future directions.

##  2 Einstein and Jordan frames - A dictionary for disformal transformations

###  2.1 A new formalism

In this section, we present a formalism to compute quantities in different
frames, if both the unit of length and the unit of time are different. For
simplicity, we will only consider physical quantities which are product of
length, time and masses (with a potential exponent). This is sufficient to
describe most of the quantities relevant in cosmology. We assume that in each
frame, there are three _basis units_ : length, time and action. This means
that, for example, a mass is of the form

$m\propto\frac{AT}{L^{2}}\,,$

where $L$, $T$ and $A$ are respectively a length, a time and an action. As
mentioned earlier, in the Einstein/Jordan frame there is a unit ruler
$L$/$\tilde{L}$ and a unit time $T$/$\tilde{T}$. Any physical quantity can be
expressed as a product of powers of this quantity and a pure number, and in
the Einstein frame, one finds

$X_{\mathrm{p}}=X\times\mathcal{L}^{n_{\mathrm{L}}}\mathcal{T}^{n_{\mathrm{T}}}A^{n_{\mathrm{A}}}\,.$
(2.1)

We assume that the action remains the same in both the frames, hence we only
write $A$. At this stage, it is important to make our notations clear.

  * •

$X_{\mathrm{p}}$ is a physical dimensionful quantity, and whose intrinsic
value is independent of the unit system

  * •

$X$ is a dimensionless number whose value depends on the unit system

  * •

$\mathcal{L}$, $\mathcal{T}$ and $A$ are given length, time and action which
form the unit system

  * •

${n_{\mathrm{L}}}$, ${n_{\mathrm{T}}}$ and ${n_{\mathrm{A}}}$ are rational
numbers222In principle we could also allow real exponents, but is it not
common to encounter, for example, a length to the power $\pi$, hence we will
only consider rational exponents.

In order to go from one frame to another, we define $\alpha_{\mathrm{L}}$ and
$\alpha_{\mathrm{T}}$ such that

$\mathcal{L}=\frac{1}{\alpha_{\mathrm{L}}}\tilde{\mathcal{L}}\,,\quad\text{and}\quad\mathcal{T}=\frac{1}{\alpha_{\mathrm{T}}}\tilde{\mathcal{T}}\,.$
(2.2)

We can get the transformation rules for the dimensionless numbers by imposing
the condition that any physical quantity should be invariant under a frame
transformation. For the sake of the argument, let’s consider a length
$L_{\mathrm{p}}$ given by

$L_{\mathrm{p}}=L\times\mathcal{L}=L\frac{1}{\alpha_{\mathrm{L}}}\tilde{\mathcal{L}}\equiv\tilde{L}\times\tilde{\mathcal{L}}\,,$
(2.3)

from which we deduce

$L=\alpha_{\mathrm{L}}\tilde{L}\,,$ (2.4)

and a similar argument yields

$T=\alpha_{\mathrm{T}}\tilde{T}\,,$ (2.5)

which is the same as Eq. (1.5).

From these two relations, it is straightforward to deduce (using that the unit
of action $\mathcal{A}$ is the same in both frames) that

$\displaystyle X_{\mathrm{p}}$
$\displaystyle=X\times\mathcal{L}^{n_{\mathrm{L}}}\mathcal{T}^{n_{\mathrm{T}}}A^{n_{\mathrm{A}}}\,,$
(2.6) $\displaystyle X_{\mathrm{p}}$
$\displaystyle=\tilde{X}\times\tilde{\mathcal{L}}^{n_{\mathrm{L}}}\tilde{\mathcal{T}}^{n_{\mathrm{T}}}A^{n_{\mathrm{A}}},$
(2.7) $\displaystyle X$
$\displaystyle=\tilde{X}\alpha_{\mathrm{L}}^{n_{\mathrm{L}}}\alpha_{\mathrm{T}}^{n_{\mathrm{T}}}\,.$
(2.8)

These relations provide us a recipe to write the _dictionary_ for any physical
quantity i.e. write it in terms of length, time and action and read off the
exponent of the length and time. In Table 1, we have constructed examples of
some physical quantities of interest.

Quantity Dictionary Length $L=\alpha_{\mathrm{L}}\tilde{L}$ Time
$T=\alpha_{\mathrm{T}}\tilde{T}$ Mass
$m=\frac{\alpha_{\mathrm{T}}}{\alpha_{\mathrm{L}}^{2}}\tilde{m}$ Velocity
$v=\frac{\alpha_{\mathrm{L}}}{\alpha_{\mathrm{T}}}\tilde{v}$ Volume
$V=\alpha_{\mathrm{L}}^{3}\tilde{V}$ Energy
$E=\frac{1}{\alpha_{\mathrm{T}}}\tilde{E}$ Mass density
$\rho=\frac{\alpha_{\mathrm{T}}}{\alpha_{\mathrm{L}}^{5}}\tilde{\rho}$

Table 1: Dictionary for various quantities between the two frames

###  2.2 Background variables

In order to understand the correspondence between the two frames, we shall
follow the same approach as adopted in [35]. We first proceed by obtaining the
relations between the background variables in the Jordan and Einstein frames,
followed by that of the perturbations as well in these two frames. In case of
the disformal transformation, we can get the relation between the comoving
Hubble parameters in the two frames as follows:

$\displaystyle\tilde{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\mathcal{A}_{0}^{1/2}}\mathcal{H}-\frac{1}{2}\frac{\mathcal{A}^{\prime}_{0}}{\mathcal{A}_{0}}\dot{\phi}_{0}$
(2.9) $\displaystyle=$
$\displaystyle\frac{\alpha_{0T}}{\alpha_{0L}}\mathcal{H}-\frac{\alpha_{0L}^{\prime}}{\alpha_{0L}}\dot{\phi}_{0}$

Here, the quantities with subscript $0$ indicate the background quantities and
an overprime denotes derivative with respect to $\phi_{0}$. From the energy-
momentum tensors in the two frames, given by

$\displaystyle T^{\mu}{}_{\nu}$ $\displaystyle=$ $\displaystyle\rho
u^{\mu}u_{\nu}+P\left(u^{\mu}u_{\nu}+\delta^{\mu}{}_{\nu}\right),$ (2.10)
$\displaystyle\tilde{T}^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle\tilde{\rho}\tilde{u}^{\mu}\tilde{u}_{\nu}+\tilde{P}\left(\tilde{u}^{\mu}\tilde{u}_{\nu}+\delta^{\mu}{}_{\nu}\right),$
(2.11)

we can obtain a generalised relation between the energy-momentum tensor
components, i.e., the energy density and the pressure in the two frames in
terms of the kinetic term $X$ as [38]

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\xi_{1}\tilde{\rho}+\xi_{2}\tilde{P}$ (2.12) $\displaystyle P$
$\displaystyle=$
$\displaystyle\frac{\tilde{P}}{\alpha\mathcal{A}^{3/2}}=\frac{\tilde{P}}{\alpha_{T}\alpha_{L}^{3}}$
(2.13)

where $\xi_{1}$ and $\xi_{2}$ are given by

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle\frac{(1-2\mathcal{B}X/\mathcal{A})^{1/2}}{\mathcal{A}\left(\mathcal{A}-\mathcal{A}_{,X}X+2\mathcal{B}_{,X}X^{2}\right)}$
(2.14) $\displaystyle\xi_{2}$ $\displaystyle=$
$\displaystyle-\frac{3(1-2\mathcal{B}X/\mathcal{A})^{1/2}\mathcal{A}_{,X}X}{\mathcal{A}^{2}\left(\mathcal{A}-\mathcal{A}_{,X}X+2\mathcal{B}_{,X}X^{2}\right)}$
(2.15)

where $\mathcal{A}_{,X}=\partial\mathcal{A}/\partial X$. However, in order to
preserve the Lorentzian signature of the metric, the dependence on $X$ is
often done away with [39, 27]. Also for our purpose, the computation of number
counts requires only the density and volume perturbations, so we can neglect
the pressure term above. Then we find

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle\alpha\mathcal{A}^{-5/2}=\frac{\alpha_{T}}{\alpha_{L}^{5}},\quad\text{and}$
(2.16) $\displaystyle\xi_{2}$ $\displaystyle=$ $\displaystyle 0$ (2.17)

These equations simply follow from the fact that
$\alpha_{T}=\sqrt{\mathcal{A}-\mathcal{B}\dot{\phi}^{2}}=\sqrt{\mathcal{A}-2\mathcal{B}X}$,
since $2X=-{\nabla}_{\mu}\phi{\nabla}^{\mu}\phi$.

In both the conformal and disformal cases, the conservation of the energy-
momentum tensor $\nabla^{\mu}\tilde{T}_{\mu\nu}=0$ gives (Eq. 2.18 of [35])

$\dot{\tilde{\rho}}=-(\tilde{\rho}+\tilde{P})3\tilde{\mathcal{H}}$ (2.18)

So, from Eq. (2.12) and the expression of $\xi_{1}$ above, we have,

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\dot{\xi}_{1}\tilde{\rho}+\xi_{1}\dot{\tilde{\rho}}$ (2.19)
$\displaystyle=$
$\displaystyle-3\mathcal{H}\rho\frac{\alpha_{T}}{\alpha_{L}}-\frac{2\alpha_{L}^{\prime}\dot{\phi}}{\alpha_{L}}\rho+\frac{\alpha_{T}^{\prime}\dot{\phi}}{\alpha_{T}}\rho$

In the conformal case, $\xi_{1}=\mathcal{A}^{-2}$, $\xi_{2}=0$ and
$\alpha=\sqrt{\mathcal{A}}$.

###  2.3 First order perturbations

We now turn to the first order perturbed quantities for scalar perturbations
and neglect the vector and tensor perturbations. The expression for a
perturbed $\xi_{1}$ turns out to be

$\displaystyle\xi_{1}$ $\displaystyle=$
$\displaystyle\xi_{10}\left(1-\frac{5}{2}\frac{\mathcal{A}_{0}^{\prime}\dot{\phi}}{\mathcal{A}_{0}}+\frac{\alpha^{\prime}_{0}\dot{\phi}}{\alpha_{0}}\right)$
(2.20) $\displaystyle=$
$\displaystyle\xi_{10}\left(1-\frac{5\alpha_{0L}^{\prime}\dot{\phi}}{\alpha_{0L}}+\frac{\alpha_{0T}^{\prime}\dot{\phi}}{\alpha_{0T}}\right)$

where $\xi_{10}$ is the background quantity. Hence, the perturbations in both
the frames can be related as,

$\displaystyle\delta(x)$ $\displaystyle=$
$\displaystyle\frac{\rho(x)-\rho_{0}(t)}{\rho_{0}(t)}=\frac{\xi_{1}\tilde{\rho}(x)-\xi_{10}\tilde{\rho}_{0}(t)}{\xi_{10}\tilde{\rho}_{0}(t)}$
(2.21) $\displaystyle=$
$\displaystyle\tilde{\delta}(x)-\left(\frac{5\alpha_{0L}^{\prime}\dot{\phi}}{\alpha_{0L}}-\frac{\alpha_{0T}^{\prime}\dot{\phi}}{\alpha_{0T}}\right)$
$\displaystyle=$ $\displaystyle\tilde{\delta}(x)-\left(\frac{5\delta
L}{L}-\frac{\delta T}{T}\right)$

As previously done in the conformal case, we can also check if the lensing
potential, given below, is disformal frame-invariant [40].

$\psi(\mathbf{n},z)=\int_{0}^{r(z)}\mathrm{d}r\frac{r(z)-r}{r(z)r}\left[\Phi\left(r\mathbf{n},t_{now}-r\right)+\Psi\left(r\mathbf{n},t_{now}-r\right)\right]$
(2.22)

However, very fundamentally speaking, weak gravitational lensing and its
observables are not expected to be disformally invariant because the causal
lightcone structure is not preserved and hence null geodesics do not exist in
disformal transformations [13, 41, 42]. This statement can be further
supported mathematically by verifying that the sum of the two Bardeen
potentials $\Phi+\Psi$, that the lensing potential is sensitive to, is not
invariant under disformal transformation. According to [16], the gauge-
invariant metric perturbations in the longitudinal gauge in terms of
$\alpha_{T}$ and $\alpha_{L}$ are related by:

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\tilde{\Phi}-\frac{1}{\alpha_{T}^{2}}\left((\alpha_{L}^{2}-\alpha_{T}^{2})\frac{\dot{a}}{a}+\frac{2\alpha_{L}\dot{\alpha}_{L}}{2}\right)\frac{\delta\phi}{\dot{\phi}}$
(2.23) $\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha_{T}^{2}}\left(\alpha_{L}^{2}\tilde{\Psi}+\frac{2\alpha_{L}\dot{\alpha}_{L}(2\alpha_{T}^{2}-\alpha_{L}^{2})+\alpha_{L}^{2}(2\alpha_{L}\dot{\alpha}_{L}-2\alpha_{T}\dot{\alpha}_{T})}{2\alpha_{T}^{2}}\frac{\delta\phi}{\dot{\phi}}\right)$
(2.24)

Also in the conformal case, $\alpha_{T}=\alpha_{L}$, and we recover the
conformal case relations [c.f. Eq. (2.27) of [35]]. However in general, we do
not find the sum of two Bardeen potentials $\Phi+\Psi$ to be invariant in the
disformal case.

This further means that other lensing observables such as the convergence
$\kappa$ and shear $\gamma$ that are dependent on the lensing potential cannot
be disformal frame-invariant.

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{1}\partial_{1}+\partial_{2}\partial_{2}\right)\psi=\frac{1}{2}\nabla^{2}\psi$
$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{1}\partial_{1}-\partial_{2}\partial_{2}\right)\psi;\quad\gamma_{2}=\partial_{1}\partial_{2}\psi;\quad\gamma=\gamma_{1}+i\gamma_{2}$
(2.25)

##  3 Cosmological number counts

The cosmological number counts are observables that are computed in terms of
the number $N$ of galaxies observed in a patch of the sky in a particular
direction $\mathbf{n}$ per unit solid angle and per unit redshift bin and are
quantified as the perturbation in the number density of galaxies as

$\Delta(\mathbf{n},z)=\frac{N(\mathbf{n},z)-\langle N\rangle(z)}{\langle
N\rangle(z)}$ (3.1)

We follow the notation of [43], and define the number counts in the Jordan
frame with a tilde

$\tilde{\Delta}(\mathbf{n},z)=\tilde{\delta}_{z}(\mathbf{n},z)+\frac{\delta\tilde{V}(\mathbf{n},z)}{\tilde{V}(z)}$
(3.2)

Here the first term $\tilde{\delta}_{z}$ is the redshift-space density
perturbation, and the second term is the volume perturbation divided by the
physical survey volume density per redshift bin, per solid angle. Our aim is
to show that $\tilde{\Delta}(\mathbf{n},z)$ is invariant under a disformal
transformation and is the same as the quantity in the Einstein frame, which we
will denote without a tilde, $\Delta(\mathbf{n},z)$.

From the fundamental definition of the density perturbation in the redshift
space, we have

$\displaystyle\delta_{z}(\mathbf{n},z)$ $\displaystyle=$
$\displaystyle\frac{\rho(\mathbf{n},z)-\rho_{0}(z)}{\rho_{0}(z)}$ (3.3)
$\displaystyle=$
$\displaystyle\frac{\frac{N(\mathbf{n},z)(\alpha_{T}/\alpha_{L}^{2})^{-1}}{V(\mathbf{n},z)}-\frac{N_{0}(z)(\alpha_{0T}/\alpha_{0L}^{2})^{-1}}{V_{0}(z)}}{\frac{N_{0}(z)(\alpha_{0T}/\alpha_{0L}^{2})^{-1}}{V_{0}(z)}}$
$\displaystyle=$ $\displaystyle\Delta(\mathbf{n},z)-\frac{\delta
V}{V_{0}}+2\left(\delta
z\frac{1}{L_{0}}\frac{\mathrm{d}L_{0}}{\mathrm{d}z}-\frac{\delta
L_{0}}{L_{0}}\right)-\left(\delta
z\frac{1}{T_{0}}\frac{\mathrm{d}T_{0}}{\mathrm{d}z}-\frac{\delta
T_{0}}{T_{0}}\right)$

where the subscript $0$ corresponds to background quantities. The second
equality follows from the fact that $\rho=mN/V$ and the mass $m$ scales as
$\alpha_{T}/\alpha_{L}^{2}$. The last equality is obtained in a manner similar
to that explained in Appendix C of [35]. The quantity $\delta z$ corresponds
to the splitting of the observable redshift $z$ into a background and a
perturbative part, that is, $z=z_{0}+\delta z$ (see Figure 1 of [35]).

For a general function $f$, we can assume that it has time and length
dimensions $n_{T}$ and $n_{L}$ respectively, and the generalization of the
formula becomes

$f=\tilde{f}L^{n_{L}}T^{n_{T}}.$ (3.4)

Again, as in [35], we define $L_{0}(z)$, $T_{0}(z)$, $\delta L$ and $\delta
T$. Following the same steps we gets

$\frac{\delta
f(\boldsymbol{n},z)}{f_{0}(z)}=\frac{\delta\tilde{f}(\boldsymbol{n},z)}{\tilde{f}_{0}(z)}+\left(n_{L}\frac{\delta
z}{L_{0}}\frac{\mathrm{d}L_{0}}{\mathrm{~{}d}z_{0}}-n_{L}\frac{\delta
L}{L}\right)+\left(n_{T}\frac{\delta
z}{T_{0}}\frac{\mathrm{d}T_{0}}{\mathrm{~{}d}z_{0}}-n_{T}\frac{\delta
T}{T}\right)\,,$ (3.5)

where $\delta f$ is the first order perturbation in the function $f$, as a
function of direction $n$ and redshift $z$ and calculated in the Einstein
frame and $\delta\tilde{f}$ is the corresponding quantity in the Jordan frame.

###  3.1 Density perturbations

In this section, we aim to obtain a relation for the density perturbations
between the two frames. Following the definition of the redshift-space density
perturbation in [43], we have

$\delta_{z}(\mathbf{n},z)=\frac{\delta\rho(\mathbf{n},z)}{\bar{\rho}\left(z_{0}\right)}-\frac{\mathrm{d}\rho_{0}}{\mathrm{~{}d}z_{0}}\frac{\delta
z(\mathbf{n},z)}{\rho_{0}(z_{0})}$ (3.6)

where the first term corresponds to the relation established by Eq. (2.21).
For the second term we first need to compute the expressions for
$\mathrm{d}z_{0}/dt$ and $\mathrm{d}\rho_{0}/dt$. The latter has already been
done in Eq. (2.19). In the Einstein frame we have,

$1+z_{0}=\frac{\alpha_{0T}}{a}$ (3.7)

which implies that

$\frac{dz_{0}}{dt}=-\frac{\dot{a}}{a^{2}}\alpha_{0T}+\frac{\dot{\alpha_{0T}}}{a}=-(1+z_{0})\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}_{0}}{\alpha_{0T}}\right)$
(3.8)

We can now use this expression to obtain

$\displaystyle\frac{\mathrm{d}\rho_{0}}{\mathrm{d}z_{0}}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}\rho_{0}}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}z_{0}}$
(3.9) $\displaystyle=$
$\displaystyle\frac{-3\mathcal{H}\rho_{0}\frac{\alpha_{0T}}{\alpha_{0L}}-\frac{2\alpha_{0L}^{\prime}\dot{\phi}_{0}}{\alpha_{0L}}\rho_{0}+\frac{\alpha_{0T}^{\prime}\dot{\phi_{0}}}{\alpha_{0T}}\rho_{0}}{-(1+z_{0})\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}_{0}}{\alpha_{0T}}\right)}$

Now, from Eq. (3.6), and realising that the first term on the right hand side
is essentially the quantity $\delta(x)$ as defined in Eq. (2.21), we get an
expression for the density contrast $\tilde{\delta}_{z}$ in the Jordan frame
as

$\tilde{\delta}_{z}(\mathbf{n},z)=\tilde{\delta}(x)-\left(\frac{\mathrm{d}\rho_{0}}{\mathrm{d}z_{0}}\right)\frac{\delta
z(\mathbf{n},z)}{\rho_{0}(z_{0})}$ (3.10)

which, using the previous equation, yields

$\tilde{\delta}(x)=\tilde{\delta}_{z}-\frac{3\mathcal{H}\frac{\alpha_{0T}}{\alpha_{0L}}+\frac{2\alpha_{0L}^{\prime}\dot{\phi}_{0}}{\alpha_{0L}}-\frac{\alpha_{0T}^{\prime}\dot{\phi_{0}}}{\alpha_{0T}}}{(1+z_{0})\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}_{0}}{\alpha_{0T}}\right)}\delta
z$ (3.11)

Upon using Eq. (2.21), we find

$\delta(x)=\tilde{\delta}_{z}-\frac{3\mathcal{H}\frac{\alpha_{0T}}{\alpha_{0L}}+\frac{2\alpha_{0L}^{\prime}\dot{\phi}_{0}}{\alpha_{0L}}-\frac{\alpha_{0T}^{\prime}\dot{\phi}_{0}}{\alpha_{0T}}}{(1+z_{0})\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}_{0}}{\alpha_{0T}}\right)}\delta
z-\left(\frac{5\delta L}{L}-\frac{\delta T}{T}\right)$ (3.12)

which leads to

$\displaystyle\delta_{z}$ $\displaystyle=$
$\displaystyle\tilde{\delta}_{z}-\left(\frac{5\delta L}{L}-\frac{\delta
T}{T}\right)$ (3.13) $\displaystyle=$
$\displaystyle\tilde{\delta}_{z}-5\frac{\delta
L}{L_{0}}+5\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z+\frac{\delta
T}{T_{0}}-\frac{\mathrm{d}T_{0}}{T_{0}\mathrm{d}z_{0}}\delta z$

This equation is a generalised version of Eq. (3.9) of [35] since

$\delta(x)-\tilde{\delta}(x)=\delta_{z}-\tilde{\delta}_{z}$

The relation in Eq. (3.13) gives us the “unobservable” density contrast
$\delta_{z}$ in the Einstein frame in terms of the same quantity in the Jordan
frame, and as is evident, they are unequal. This renders the quantity
$\delta_{z}$ frame-dependent. This also automatically implies that the
“unobservable” matter power spectrum $P_{\delta_{z}}(k,z)$ which is nothing
but a Fourier transform of the two-point correlation function of the density
contrast, is also frame-dependent, as can be seen from the following
definition of the matter power spectrum

$\langle\delta_{z}(\mathbf{k})\delta_{z}^{*}(\mathbf{k^{\prime}})\rangle=(2\pi)^{3}P_{\delta_{z}}(k,z)\delta^{3}(\mathbf{k-k^{\prime}})\neq(2\pi)^{3}P_{\tilde{\delta}_{z}}(k,z)\delta^{3}(\mathbf{k-k^{\prime}})$
(3.14)

where $\delta^{3}$ is the three-dimensional Kronecker delta function. Contrary
to this result, in the next section, we will show that the “observable” matter
power spectrum that includes both the density and volume perturbations is
indeed frame-invariant.

###  3.2 Volume perturbations

In order to proceed with the transformation of the volume perturbations, we
need to first understand what corrections are to be taken into consideration
in the Einstein frame. An obvious correction comes into play in the prefactor
for Eq. (14) of [43] that gives the expression for the volume element, which
in the conformal case is (assuming the Jordan frame, and hence the tilde over
$\mathcal{H}$),

$\frac{a^{4}}{\tilde{\mathcal{H}}}=\frac{a}{\tilde{\mathcal{H}}}a^{3}=\frac{a^{3}\alpha_{0L}}{1+z_{0}}\frac{1}{\mathcal{H}-\frac{\dot{\phi}_{0}\mathcal{A}_{0}^{\prime}}{2\mathcal{A}_{0}}}=\frac{\alpha_{0L}^{3}}{(1+z_{0})^{4}}\frac{1}{\mathcal{H}-\frac{\dot{\phi}_{0}\mathcal{A}_{0}^{\prime}}{2\mathcal{A}_{0}}}$
(3.15)

where the second and third equality come from the fact that
$a=L_{0}/(1+z_{0})=\alpha_{0L}/(1+z_{0})$ (assuming that the length scale in
the Jordan frame $\tilde{L}_{0}$ is unity) and the relation between the Jordan
and Einstein frame Hubble parameters (see Eq. (2.13) of [35]). Analogously,
for the disformal case, we should have the prefactor as

$\displaystyle\frac{a^{4}}{\tilde{\mathcal{H}}}$ $\displaystyle=$
$\displaystyle\frac{a^{3}}{1+z_{0}}\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}}{\alpha_{0T}}\right)^{-1}$
(3.16) $\displaystyle=$
$\displaystyle\frac{\alpha_{0T}^{3}}{(1+z_{0})^{4}}\left(\mathcal{H}-\frac{\alpha_{0T}^{\prime}\dot{\phi}}{\alpha_{0T}}\right)^{-1}$

The volume transformations work the same way as the conformal case since the
volume in the Einstein frame scales as $\alpha_{L}^{3}$. Following the steps
underlined in [43] in order to obtain the volume perturbation, we obtain

$\displaystyle\frac{\delta V}{V_{0}}$ $\displaystyle=$
$\displaystyle\frac{\delta\tilde{V}}{\tilde{V}_{0}}+3\frac{\alpha_{L}^{\prime}\dot{\phi}}{\alpha_{L}}=\frac{\delta\tilde{V}}{\tilde{V}_{0}}+3\frac{\delta
L}{L}$ (3.17) $\displaystyle=$
$\displaystyle\frac{\delta\tilde{V}}{\tilde{V}_{0}}+3\frac{\delta
L}{L_{0}}-3\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z$

From Eqs. (3.3), (3.13) and (3.17) we finally have

$\displaystyle\Delta_{E}=\Delta(\mathbf{n},z)$ $\displaystyle=$
$\displaystyle\delta_{z}+\frac{\delta V}{V_{0}}-2\left(\delta
z\frac{1}{L_{0}}\frac{\mathrm{d}L_{0}}{\mathrm{d}z_{0}}-\frac{\delta
L}{L_{0}}\right)+\left(\delta
z\frac{1}{T_{0}}\frac{\mathrm{d}T_{0}}{\mathrm{d}z_{0}}-\frac{\delta
T}{T_{0}}\right)$ (3.18) $\displaystyle=$
$\displaystyle\left(\tilde{\delta}_{z}-5\frac{\delta
L}{L_{0}}+5\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z+\frac{\delta
T}{T_{0}}-\frac{\mathrm{d}T_{0}}{T_{0}\mathrm{d}z_{0}}\delta
z\right)+\left(\frac{\delta\tilde{V}}{\tilde{V}_{0}}+3\frac{\delta
L}{L_{0}}-3\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z\right)$
$\displaystyle-2\left(\delta
z\frac{1}{L_{0}}\frac{\mathrm{d}L_{0}}{\mathrm{d}z_{0}}-\frac{\delta
L}{L_{0}}\right)+\left(\delta
z\frac{1}{T_{0}}\frac{\mathrm{d}T_{0}}{\mathrm{d}z_{0}}-\frac{\delta
T}{T_{0}}\right)$ $\displaystyle=$
$\displaystyle\tilde{\Delta}(\mathbf{n},z)=\Delta_{J}$

where the subscripts $E$ and $J$ stand for Einstein and Jordan frames,
respectively.

Therefore, we can show that in this case the matter power spectrum in both
frames would indeed be invariant

$\langle\Delta(\mathbf{k})\Delta^{*}(\mathbf{k^{\prime}})\rangle=\langle\tilde{\Delta}(\mathbf{k})\tilde{\Delta}^{*}(\mathbf{k^{\prime}})\rangle$

Or, equivalently,

$(2\pi)^{3}P_{\Delta}(k,z)\delta^{3}(\mathbf{k-k^{\prime}})=(2\pi)^{3}P_{\tilde{\Delta}}(k,z)\delta^{3}(\mathbf{k-k^{\prime}})$
(3.19)

This establishes an interesting and important fact that while the
“unobservable” power spectrum of the density contrast is frame-dependent, the
“observable” matter power spectrum is indeed frame-invariant.

##  4 Conclusions

Disformal transformations are considered a generalisation of the conformal
transformations which also involve derivative dependent terms of a
scalar/vector field. While the conformal transformations usually involve a
rescaling of the metric which also preserve causality, the disformal
transformations are more general which affect the particle geodesics and also
lead to a non-trivial coupling of the matter Lagrangian. These salient
features provide very rich phenomenology which has been explored in different
cosmological contexts, in black hole physics and more interestingly, in
constructing an even more general class of ST theories beyond Horndeski
theories.

In this paper, we have investigated how physical observables associated with
the galaxy surveys in the Jordan and Einstein frames are related by a
disformal transformations constructed by scalar fields. In particular, we have
successfully been able to show the frame-invariance of the cosmological number
counts for the case of disformal transformations which is an interesting
result. This establishes the property of number counts as a physical
observable even further. The notable difference with respect to the conformal
frame-invariance of cosmological observables is the fact that the lensing
potential as a standalone observable does not turn out to be disformally
invariant, because of the absence of null geodesics. As a result, various
quanties such as the convergence and shear which are derived from the lensing
potential are also not invariant under the disformal transformations. A
summary of our results that we have obtained in this work, has been outlined
in Table 2.

Quantity |  Gauge dependent | Frame dependent
---|---|---
Background |  |  |
Density $\rho_{0}$ |  Yes |  Yes |  $\rho_{0}=\frac{\alpha_{0T}}{\alpha_{0L}^{5}}\tilde{\rho}_{0}$
Pressure $P_{0}$ |  Yes |  Yes |  $P_{0}=\frac{\alpha_{0T}}{\alpha_{0L}^{3}}\tilde{P}_{0}$
Redshift $z_{0}$ |  Yes |  No |
Observed redshift $z$ |  No |  No |
Perturbations |  |  |
Density $\delta$ |  Yes |  Yes |  $\delta=\tilde{\delta}-\left(\frac{5\delta L}{L}-\frac{\delta T}{T}\right)$
Velocity ${\mathbf{v}}$ |  Yes |  No |
Bardeen potential $\Phi$ |  No |  Yes |  Eq. (2.23)
Bardeen potential $\Psi$ |  No |  Yes |  Eq. (2.24)
Lensing potential $\varphi$ |  No |  Yes |
Redshift density $\delta_{z}$ |  No |  Yes |  $\delta_{z}=\tilde{\delta}_{z}-5\frac{\delta L}{L_{0}}+5\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z+\frac{\delta T}{T_{0}}-\frac{\mathrm{d}T_{0}}{T_{0}\mathrm{d}z_{0}}\delta z$
Volume perturbation $\frac{\delta V}{V}$ |  No |  Yes |  $\delta V/V=\frac{\delta\tilde{V}}{\tilde{V}_{0}}+3\frac{\delta L}{L_{0}}-3\frac{\mathrm{d}L_{0}}{L_{0}\mathrm{d}z_{0}}\delta z$
Number counts $\Delta({\mathbf{n}},z)$ |  No |  No |
Table 2: Gauge and frame dependence of various quantities.

Although we have only considered the disformal factor $\cal B$ as a function
of $\phi$ only, and it might also seem non-trivial to carry out the same kind
of proof for a disformal transformation including the $X$ dependence, we
believe that our calculations provide a first step towards a necessary
generalisation, as we also observe that our results reduce to those of [35]
under the pure conformal limit. However, an apparent appearance of
Ostrogradsky ghosts for a $X$ dependent disformal transformation might happen,
no real ghost in the theory should actually be present in the transformed
frame due to the existence of hidden constraints [30]. Our work can, in fact,
be further extended to more complex transformations, for example to those in
[18, 19, 17, 42], employing much more rigorous and non-trivial calculations,
which is beyond the scope of the current work.

Another direction in which one can explore the frame-invariance of
cosmological number counts is by further generalisation of the Brans-Dicke ST
theory itself, via the Horndeski Lagrangian [11]. An attempt to establish the
invariance of the Horndeski Lagrangian under disformal transformations has
been made in [27], where it was found that the $X$ dependence is detrimental
to the frame invariance. However, it would be interesting to explore how the
frame-invariance of number counts holds up beyond the ST theories. We leave
these interesting directions for future work.

## Acknowledgments

We would like to thank Ruth Durrer for useful discussions and comments on the
draft. BG acknowledges partial financial support from the CV Raman
Postdoctoral Fellowship and the DST-INSPIRE Faculty Fellowship
DST/INSPIRE/04/2020/001534. JF acknowledges financial support from the Swiss
National Science Foundation. RKJ wishes to acknowledge financial support from
the new faculty seed start-up grant of the Indian Institute of Science,
Bengaluru, India, Science and Engineering Research Board, Department of
Science and Technology, Government of India, through the Core Research Grant
CRG/2018/002200 and the Infosys Foundation, Bengaluru, India through the
Infosys Young Investigator award.

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# Origin of the Very High Energy $\gamma$-rays in the Low-luminosity Active
Galactic Nucleus NGC 4278

Ji-Shun Lian School of Physics, Beijing Institute of Technology, Beijing
100081, People’s Republic of China<EMAIL_ADDRESS>Jia-Xuan Li  School of
Physics, Beijing Institute of Technology, Beijing 100081, People’s Republic of
China<EMAIL_ADDRESS>Xin-Ke Hu  Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, People’s Republic of China  Ying-Ying Gan School of Physics,
Beijing Institute of Technology, Beijing 100081, People’s Republic of China;
<EMAIL_ADDRESS>Tan-Zheng Wu  School of Physics, Beijing Institute of
Technology, Beijing 100081, People’s Republic of China<EMAIL_ADDRESS>Jin Zhang† School of Physics, Beijing Institute of Technology, Beijing 100081,
People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

NGC 4278, a Low-luminosity active galactic nucleus (AGN), is generally
classified as a low-ionization nuclear emission line region (LINER) type AGN.
Recently, it is reported to be associated with a very high energy (VHE)
$\gamma$-ray source 1LHAASO J1219+2915 in the first Large High Altitude Air
Shower Observatory (LHAASO) source catalog. However, no associated counterpart
has been detected by Fermi-LAT. By analyzing its X-ray observation data from
Swift-XRT, we find it is in a high-flux state on MJD 59546, with the X-ray
flux more than one order of magnitude higher than that observed $\sim$ 11.7
year earlier by Chandra. We propose that the detection of VHE $\gamma$-rays
from NGC 4278 may be attributed to the presence of an active nucleus
displaying behavior similar to that of a BL Lac. To reproduce its spectral
energy distribution (SED), we employ a one-zone leptonic model, typically used
for fitting broadband SEDs of BL Lacs, and find that smaller values for both
Doppler factor ($\delta$) and magnetic field strength ($B$) are required than
that of typical TeV BL Lacs. Furthermore, NGC 4278 exhibits significantly
lower luminosity in both radio and TeV bands when compared with typical TeV BL
Lacs. In the radio-luminosity vs. Eddington-ratio plane, NGC 4278 shows
greater similarity to Seyfert galaxies and LINERs rather than BL Lacs;
however, it still roughly follows the extension towards lower luminosity seen
in BL Lacs.

astroparticle physics – gamma rays: galaxies – galaxies: active – galaxies:
individual (NGC 4278)

##  1 Introduction

Radio-loud active galactic nuclei (RL-AGNs) account for about $15\%-20\%$ of
the AGN population (Urry & Padovani 1995), characterized as the powerful
relativistic jet radiations. Blazars, a special type of RL-AGNs, have their
jets pointing nearly along the line of sight and are divided into BL Lacertae
objects (BL Lacs) and flat spectrum radio quasars (FSRQs) based on their
emission line features in the optical band (Scarpa & Falomo 1997). Blazars are
the main extragalactic gigaelectronvolt-teraelectronvolt (GeV–TeV)
$\gamma$-ray emitting objects, especially in the TeV band. In recent years,
many other types of RL-AGNs have been confirmed as $\gamma$-ray emitters. For
example, radio galaxies (Abdollahi et al. 2020), Narrow-line Seyfert 1
galaxies (NLS1s, Abdo et al. 2009; Sun et al. 2015), compact steep-spectrum
sources (CSSs, Abdollahi et al. 2020; Zhang et al. 2020), compact symmetric
objects (CSOs, Migliori et al. 2016; Principe et al. 2020; Lister et al. 2020;
Gan et al. 2021, 2022, 2024), and low-luminosity FR0 type radio galaxies
(Grandi et al. 2016; Fu et al. 2022). The large-scale extended regions of jets
in AGNs also have been confirmed to emit detectable $\gamma$-rays (Abdo et al.
2010; Ackermann et al. 2016; Yu et al. 2024). It is worth noting that in the
TeV band, apart from several FSRQs and radio galaxies, BL Lacs are the main
type of detected extragalactic $\gamma$-ray emitting
AGNs111http://tevcat2.uchicago.edu/.

Recently, the Large High Altitude Air Shower Observatory (LHAASO)
collaboration reported their first catalog (Cao et al. 2024). One new
extragalactic TeV $\gamma$-ray emitting source was presented. The point-like
source 1LHAASO J1219+2915 located at high Galactic Latitude was detected with
a test statistic of TS=49.2 and suggested to be associated with the AGN NGC
4278. The distance between NGC 4278 and 1LHAASO J1219+2915 is 0.05$\arcdeg$.
NGC 4278 is previously classified as a low-ionization nuclear emission line
region (LINER, Heckman 1980). No AGNs of this type have been detected in the
$\gamma$-ray band.

NGC 4278, located at $z=0.00216$ (González-Martín et al., 2009), belongs to an
RL-AGN. However, the total radio luminosity of NGC 4278 is at least 2 orders
of magnitude less than the typical RL-AGNs (Condon et al. 1998). The Very Long
Baseline Array (VLBA) has resolved its two-sided symmetric steep-spectrum jets
on a sub-parsec scale emerging from a flat-spectrum core region (Giroletti et
al. 2005). Its radio flux density at 6 cm between 1972 and 2003 exhibits
significant variability ($\geq$100%) on timescale of years, and the outburst
around 1985 may be related to the presence of an active nucleus (Giroletti et
al. 2005). In the UV band, its nucleus is barely resolved and exhibits evident
variability. Its UV flux increases by a factor of 1.6 during a period of 6
months, and the observed UV flare features, amplitude and scale time, are
prominently similar to that observed in other low-luminosity AGNs (LLAGNs,
Cardullo et al. 2009). Using the Space Telescope Imaging Spectrograph,
Cardullo et al. (2009) obtained spectra of the nucleus, measured the broad
components of the emission lines, and estimated the mass of the central
supermassive black hole (SMBH) to be in the range of $7\times 10^{7}-2\times
10^{9}~{}M_{\bigodot}$. In X-rays, no evident short time-scale (hours and
days) variability is observed in the nucleus of NGC 4278 during the Chandra
observations. However, an increase in flux by a factor of $10\%$ is observed
over a period of 1 hour during the XMM-Newton observation. Younes et al.
(2010) also reported that the nucleus flux in NGC 4278 increases by a factor
of 3 over a few months and by a factor of 5 between the faintest and brightest
observations separated by $\sim$3 years. Additionally, NGC 4278 hosts numerous
resolved low-mass X-ray binaries (LMXBs), more than one hundred of which
exhibit long-term variability (Brassington et al. 2009). In $\gamma$-rays,
apart from a transient source, namely 1FLT J1219+2907, reported by the Fermi-
LAT transient sources catalog with its position consistent with NGC 4278 from
2009 March 5 to 2009 April 5 (Baldini et al. 2021), no other high-energy
$\gamma$-rays have been detected. Therefore, the origin of the detected
$\gamma$-ray emission from 1LHAASO J1219+2915 associated with NGC 4278 is
confusing.

In this paper, we aim to study the radiation physics and the origin of VHE
$\gamma$-rays from NGC 4278 by analysing its observational data in
multiwavelength. Data analysis of Fermi-LAT, Chandra, and Swift-XRT is
presented in Section 2. In Section 3, we describe the X-ray emission property
of the nucleus in NGC 4278. The construction and modeling of the broadband
spectral energy distribution (SED) for the nucleus of NGC 4278 are presented
in Section 4. Discussion on the origin of VHE $\gamma$-rays and the AGN type
of NGC 4278 is given in Section 5, while a summary is provided in Section 6.
Throughout, the cosmological parameters $H_{0}=70$ km s-1 Mpc-1, $\Omega_{\rm
m}=0.3$, and $\Omega_{\Lambda}=0.7$ are adopted.

##  2 Observations and Data Reduction

###  2.1 Fermi-LAT

We investigate the latest Fermi-LAT 14 yr Source Catalog (4FGL-DR4, Abdollahi
et al. 2022; Ballet et al. 2023) and find only two $\gamma$-ray sources
located within a 1$\arcdeg$ radius circle centered on the radio position of
NGC 4278, namely 4FGL J1221.3+3010 and 4FGL J1217.9+3007. They are
0.929$\arcdeg$ and 0.964$\arcdeg$ distance from NGC 4278, and are associated
with BL Lacs 1ES 1218+304 and B2 1215+30, respectively. Both of these two BL
Lacs have been detected in the TeV band and have been confirmed as TeV
$\gamma$-ray emitters (Albert et al. 2006; Acciari et al. 2009; Aleksić et al.
2012; Aliu et al. 2013).

To further investigate if there are $\gamma$-ray emission from Fermi-LAT
associated with NGC 4278, the Pass 8 data, covering $\sim$16 yr from 2008
August 4 to 2024 April 24 (MJD 54682–60424), are extracted from Fermi Science
Support Center222https://fermi.gsfc.nasa.gov/ssc/data/access/ for our
analysis. We select the region of interest (ROI) centered at the radio
position of NGC 4278 (R.A.=185.028$\arcdeg$, Decl.=29.281$\arcdeg$) with a
radius of 15$\arcdeg$. The publicly available software Fermitools (version
2.2.0) are used for our analysis. We use event class “SOURCE” (evclass=128)
and event type “FRONT+BACK” (evtype=3) for data analysis based on LAT data
selection recommendations. The $\gamma$-ray events in the energy range of
0.1–300 GeV are selected with a standard data quality selection criteria of
“(DATA_QUAL>0)&&(LAT_CONFIG==1)”. The instrument response function of
“P8R3_SOURCE_V3”(Bruel et al. 2018) is used. A maximum zenith angle of
$90\arcdeg$ is set to reduce the $\gamma$-ray contamination from the Earth
limb. The background model in the ROI includes the isotropic background model
iso_P8R3_V3_v1.txt and the diffuse Galactic interstellar emission with the
parameterized model gll_iem_v07.fits, as well as all the $\gamma$-ray sources
listed in the 4FGL-DR4. The normalization of the isotropic background emission
and the diffuse Galactic interstellar emission, together with the
normalization and spectral parameters of the $\gamma$-ray point sources within
a 6.5$\arcdeg$ radius centered on NGC 4278, are left free, whereas the
parameters of those point sources lying beyond $6.5^{\circ}$ are fixed to
their 4FGL-DR4 values.

The maximum likelihood test statistic (TS) is used to quantify the
significance of the detection of $\gamma$-ray sources, which is defined as
$\rm{TS=2log(\frac{\mathcal{L}_{src}}{\mathcal{L}_{null}})}$ (Mattox et al.
1996), where $\rm{\mathcal{L}_{src}}$ and $\rm{\mathcal{L}_{null}}$ are the
maximum likelihood values for the background with and without a source,
respectively. If TS$\geq$25, it indicates that there is a new source. We
generate a $5^{\circ}\times 5^{\circ}$ residual TS map centered on the radio
position of NGC 4278, and find that the maximum TS value in the residual TS
map is $\sim$13\. Therefore, no new $\gamma$-ray source is observed,
indicating that no GeV $\gamma$-ray emission from NGC 4278 is detected.

###  2.2 Chandra

NGC 4278 has been observed by the Chandra observatory using the Advanced CCD
Imaging Spectrometer S-array (ACIS-S) in nine epochs. The shortest and longest
exposure times are 1.4 ks and 110.7 ks, as listed in Table 1. The first seven
observations (Obs-ID: 398, 4741, 7077, 7078, 7079, 7080, 7081) have been
analyzed by Younes et al. (2010), and they found that all of these
observations suffered the pile-up effect, with a pile-up fraction of $>5\%$.
The first Chandra snapshot observation (Obs-ID: 398) with the short exposure
time cannot produce a useful spectrum after considering the pile-up effect
(Younes et al. 2010), and thus it is excluded.

In this paper, we only analyze the last two observation data from Chandra. The
data are reduced with CIAO (v.4.15) and CALDB (v.4.10.4), and then we generate
the level-2 event file in a standard procedure. We firstly check the pile-up
effect of the two observations by creating pile-up maps with a CIAO tool of
pileup_map333https://cxc.harvard.edu/ciao/ahelp/pileup_map.html. No pixel has
the count rate per frame greater than 0.1 (corresponding to a pile-up fraction
of $>5\%$), indicating no pile-up effect in the last two observations. We then
extract the source photons from a circle centered on the radio position of NGC
4278 with a radius of $7^{{}^{\prime\prime}}$. The background is taken from an
annulus region with inner and outer radii of $10^{{}^{\prime\prime}}$ and
$20^{{}^{\prime\prime}}$, respectively. The spectrum is grouped to have at
least 15 counts per bin and the $\chi^{2}$ minimization technique is used for
the spectral analysis. The spectral fitting is performed using XSPEC
(v.12.13.0c). We use the same model adopted in Younes et al. (2010) to fit the
spectra in the energy range of 0.5–8.0 keV, i.e., a single power-law absorbed
by Galactic and host galaxy plus a thermal component (mekal in XSPEC). The
neutral hydrogen column density of Galactic ($N^{\rm H}_{\rm gal}$) is fixed
as $2.07\times 10^{20}$ cm-2 (Kalberla et al. 2005) while the value of host
galaxy ($N^{\rm H}_{\rm int}$) is set to be free. The fitting parameters of
the two spectra are given in Table 1 and the results of the first seven
observations reported in Younes et al. (2010) are also presented in the table
for comparison. The derived temperatures of the thermal component in the two
spectra are almost identical to that reported in Younes et al. (2010).
However, we can only obtain an upper limit of $N^{\rm H}_{\rm int}$ for the
two observations.

###  2.3 Swift-XRT

Swift X-Ray Telescope (Swift-XRT) has performed 4 observations to NGC 4278 in
the Photon Counting readout mode between 2017 December and 2021 November.
However, the radio position of NGC 4278 is outside of the XRT’s view field
during two observations. Therefore, only the other two sets of observational
data are used in our analysis; one observation was on 2021 Feb 26 (MJD 59271)
with an exposure time of 173 s and another observation was on 2021 Nov 28 (MJD
59546) with an exposure time of 923 s. It is worth noting that the
observational data from MJD 59271 have a minimal impact on the spectral fit
due to their short exposure time. The result of spectral fit is primarily
influenced by the observational data from MJD 59546. The data are processed
using the XRTDAS software package (v.3.7.0). The softwave package was
developed by the ASI Space Science Data Center and released by the NASA High
Energy Astrophysics Science Archive Research Center (HEASARC) in the HEASoft
package (v.6.30.1). The calibration files from XRT CALDB (version 20220803)
are used within xrtpipeline to calibrate and clean the events. Considering the
short exposure time of the two observations, we merge the two XRT event files
together using the xselect package and produce a combined average spectrum.
The source photons are extracted from a circle centered on the radio position
of NCG 4278 with a radius of 20 pixels ($\sim 47^{\prime\prime}$). The
background is taken from an annulus with an inner and outer radii of 30 pixels
($\sim 71^{\prime\prime}$) and 45 pixels ($\sim 106^{\prime\prime}$),
respectively. The combined average spectrum is grouped to have at least 4
counts per bin. We use the public software XSPEC (v.12.13.0c) to fit the
spectrum and the C-statistic minimization is adopted for evaluating the
goodness of the fits. The energy channels below 0.5 keV and above 8 keV are
excluded during fitting. Same as the data analysis of Chandra, the spectrum is
fitted by a single power-law absorbed by two absorption components; the
neutral hydrogen column density of Galactic is fixed at $N^{\rm H}_{\rm
gal}=2.07\times 10^{20}\rm cm^{-2}$ (Kalberla et al. 2005) while the neutral
hydrogen column density of host galaxy ($N^{\rm H}_{\rm int}$) is set to be
free. We obtain a C-statistic value/d.o.f of $3.51/6$ and calculate the
corrected flux in the energy range of $0.5-8.0$ keV, as listed in Table 1.

##  3 X-ray Emission of the nucleus in NGC 4278

The X-ray light curve of NGC 4278 is compiled in Figure 1, incorporating the
archived X-ray data from Younes et al. (2010) and the analysis results from
the Chandra and Swift-XRT observations conducted in this study. Notably, NGC
4278 exhibits significant flux variations in the X-ray band. During the
observation on MJD 59546, Swift-XRT detected a remarkably high-flux state for
NGC 4278 in the X-ray band, surpassing even the historically highest flux
obtained from XMM-Newton observation on 2004 May 23. The ratio between the
highest flux observed by Swift-XRT in 2021 and the lowest flux observed by
Chandra in 2010 exceeds one order of magnitude. Unfortunately, no additional
data between 2010 to 2021 could be acquired to estimate variability timescales
accurately. While previous studies have reported a $\sim$10% increase in flux
over a period of $\sim$1 hr during XMM-Newton observations (Younes et al.
2010), due to limited exposure time during the Swift-XRT observations, further
investigation into short timescale variability is not feasible in this study.

The compact and symmetric two-sided radio morphology of NGC 4278 fulfills the
criteria for classification as a CSO, and it is even included in a ‘bona fide’
CSO sample (Kiehlmann et al., 2024). CSOs may be transient or episodic sources
(O’Dea & Saikia 2021 for a review), with some $\gamma$-ray emitting CSOs
reported to display episodic nuclear activity (Lister et al. 2020; Gan et al.
2021, 2022). It should be noted that radio observations at 6 cm from 1972 to
2003 indicate variability in the flux density of NGC 4278, including an
observed outburst around 1985, which may be associated with the presence of an
active nucleus (Giroletti et al. 2005). Therefore, the high-flux state
observed in X-rays for NGC 4278 could potentially be attributed to the
reactivation of its nucleus once again around 2021. This may be the reason
that NGC 4278 has recently been detected at the VHE $\gamma$-ray band by
LHAASO.

The photon spectral index ($\Gamma_{\rm X}$) against the X-ray flux is
depicted in Figure 2. Despite the presence of large errors in the data points,
significant spectral variations can still be observed. NGC 4278 shows the
_harder when brighter_ behavior, which is commonly observed among BL Lacs in
the X-ray band (e.g., Tramacere et al. 2009; Tramacere et al. 2011; Zhang et
al. 2013; Di Gesu et al. 2022; Hu et al. 2024).

##  4 Constructing and Modeling the SED of Nucleus

The reactivation of the nucleus, as mentioned above, likely result in the
detection of VHE $\gamma$-rays and a high-flux state in X-rays from NGC 4278.
Consequently, it is plausible that both the emission in X-rays and
$\gamma$-rays originate from the same region. In order to further investigate
the $\gamma$-ray emission properties of NGC 4278, we collect its
multiwavelength observation data from literature and construct a broadband
SED. However, only a few non-simultaneous observations from Younes et al.
(2010) are available in the radio-optical-UV bands, as shown in Figure 3. The
sensitivity curves of LHAASO (Wang et al. 2022), VERITAS (Bigongiari & CTA
Consortium 2016), and Fermi-LAT (Funk et al. 2013) are also presented in
Figure 3. The hard X-ray spectrum and soft TeV $\gamma$-ray spectrum should
correspond to two distinct radiation components, resembling those typical TeV-
emitting BL Lacs that have a high synchrotron radiation peak (e.g., Zhang et
al. 2012). We employ a simple one-zone leptonic model, widely used in modeling
the broadband SEDs of TeV BL Lacs (e.g., Zhang et al. 2012; Yan et al. 2014),
to reproduce the SED of NGC 4278, only considering synchrotron and
synchrotron-self-Compton (SSC) processes of the relativistic electrons.

The electron distribution is taken as a broken power law,

$N(\gamma)=N_{0}\left\\{\begin{array}[]{ll}\gamma^{-p_{1}}&\mbox{ $\gamma_{\rm
min}\leq\gamma\leq\gamma_{\rm b}$},\\\ \gamma_{\rm
b}^{p_{2}-p_{1}}\gamma^{-p_{2}}&\mbox{ $\gamma_{\rm b}<\gamma<\gamma_{\rm
max}$.}\end{array}\right.$ (1)

The emission region is assumed as a sphere with radius $R$, magnetic field
$B$, and a Doppler boosting factor $\delta$, where $R$ is taken as $R=c\Delta
t\delta/(1+z)$. BL Lacs generally show variability on daily or hour timescales
(e.g., Xie et al. 2001; Ravasio et al. 2002; Błażejowski et al. 2005; Fossati
et al. 2008; Acciari et al. 2011; Nalewajko 2013), we thus take $\Delta t=1$
day for NGC 4278. The VLBA observations indicate that NGC 4278 has mildly
relativistic jets ($\beta\sim 0.75$) with a viewing angle of $2\arcdeg\sim
4\arcdeg$, which result in a Doppler booting factor of 2.7 (Giroletti et al.
2005). We thus fix $\delta$ to be 2.7. The values of $p_{1}$ and $p_{2}$ are
constrained by the spectral indices at X-ray and TeV bands. Since
$\gamma_{\min}$ cannot be constrained and is fixed as $\gamma_{\min}=1$, while
$\gamma_{\max}$ is poorly constrained by the last observation point at TeV
band. We adjust the parameters $B$, $N_{0}$, and $\gamma_{\rm b}$ to represent
the observed SED of NGC 4278. No detection of VHE $\gamma$-rays from NGC 4278
has been reported by other TeV detectors; therefore, we consider the
sensitivity curve of VERITAS as an upper limit during SED modeling. The
Klein–Nishina effect (Ghisellini et al. 1998) and the absorption of
extragalactic background light (EBL; Finke et al. 2022) are also taken into
account in the SED modeling. The SED fitting parameters are given in table 2.

Reproducing the SED of NGC 4278, we obtain $B=0.015$ G, $\gamma_{\rm
b}=4\times 10^{6}$, and $N_{0}=3.5\times 10^{4}$ cm-3. It should be noted that
the fitting parameters $B$ and $\delta$ are degenerate (Zhang et al. 2012). If
a high magnetic field strength is required to represent the broadband SED of
NGC 4278, then a smaller $\delta$. For example, $\delta=1$ would be necessary
with $B=0.2$ G. Based on these SED fitting parameters, we also calculate the
jet power ($P^{e^{\pm}}_{\rm jet}$) in case of the e± pair jet for NGC 4278 by
assuming that the jet consist of electrons ($P_{\rm e}$), magnetic fields
($P_{B}$), and radiation ($P_{\rm r}$). The corresponding values are
$6.97\times 10^{42}$ erg s-1, $5.99\times 10^{38}$ erg s-1, and $5.56\times
10^{41}$ erg s-1, respectively (see Table 2).

##  5 Discussion

The angular resolution of LHAASO can achieve $<1\arcdeg$ around 10 TeV (Ma et
al. 2022). As stated in Section 2.1, there are two other GeV-TeV BL Lacs
located within a distance of less than $1\arcdeg$ from NGC 4278. Moreover, NGC
4278 harbors a substantial number of LMXBs (Brassington et al. 2009; D’Abrusco
et al. 2014), and $\gamma$-rays have been observed from several LMXBs in the
Galaxy. We are wondering whether the VHE $\gamma$-rays detected by LHAASO from
1LHAASO J1219+2915 could potentially originate from either the two
$\gamma$-ray emitting BL Lacs or these LMXBs in NGC 4278.

###  5.1 Two TeV BL Lacs Located near NGC 4278

The GeV-TeV emitting BL Lacs 1ES 1218+304 and B2 1215+30 are situated at a
distance of 0.929$\arcdeg$ and 0.964$\arcdeg$, respectively, from the radio
position of NGC 4278. Firstly, we conduct a literature search to identify the
highest observed flux at the TeV band for the two BL Lacs. We then extrapolate
their intrinsic $\gamma$-ray spectrum up to 100 TeV assuming a power-law
spectral form. Subsequently, we calculate the observable spectral shape by
taking into account the absorption of EBL and compare it with that of 1LHAASO
J1219+2915. Two EBL models are considered in this analysis, corresponding to
the strongest absorption (Franceschini & Rodighiero 2017) and weakest
absorption (Kneiske & Raue 2010), respectively.

For 1ES 1218+304, located at a redshift of $z=0.182$ (Véron-Cetty & Véron
2003), the intrinsic $\gamma$-ray spectrum in the TeV band is taken from
Acciari et al. (2009). We extrapolate this intrinsic $\gamma$-ray spectrum to
higher energy bands by assuming a power-law spectral shape, and subsequently
derive the observable spectrum in the TeV band by taking into account the
absorption of EBL, as illustrated in Figure 4(a). For B2 1215+30, we obtain
the observed spectrum in the TeV band from Abeysekara et al. (2017).
Considering that there is uncertainty regarding its redshift, i.e., either
$z=0.130$ (Akiyama et al. 2003) or $z=0.237$ (Lanzetta et al. 1993), we
consider both cases. Firstly, we correct for EBL absorption on the observation
data using both strongest and weakest EBL models to obtain an intrinsic
$\gamma$-ray spectrum and extrapolate this obtained intrinsic $\gamma$-ray
spectrum up to 100 TeV (the blue and magenta solid lines in Figure 4) by
assuming a power-law spectral form. Subsequently, considering again the effect
of EBL absorption, we calculate the observable flux in the TeV band,
represented by the blue and magenta dashed lines shown in Figures 4(b) and
4(c).

The LHAASO is capable of detecting both 1ES 1218+304 and B2 1215+30 when they
are in a high-flux state, as depicted in Figures 4. However, the estimated
observable spectra of the two BL Lacs fail to represent the observed highest
energy photon of 1LHAASO J1219+2915 when considering the strongest EBL
absorption model. In other words, due to their high redshift, it appears
impossible for the LHAASO to detect $\gamma$-ray photons with energies
exceeding 10 TeV from the two BL Lacs, even when considering the weakest EBL
absorption model. Therefore, the VHE $\gamma$-ray emission from 1LHAASO
J1219+2915 should not be contributed by the two BL Lacs.

###  5.2 LMXBs in NGC 4278

Several Galactic LMXBs have been detected at the $\gamma$-ray band (e.g., Xing
& Wang 2015; Harvey et al. 2022a; Harvey et al. 2022b; Kantzas et al. 2022).
In NGC 4278, there exist multiple LMXBs with an X-ray luminosity exceeding
$10^{38}$ erg s-1 (Brassington et al. 2009; Fabbiano et al. 2010), which are
likely to be sources of $\gamma$-ray emission as well. Therefore, we estimate
the potential $\gamma$-ray flux from these LMXBs in NGC 4278 by assuming that
they exhibit a similar X-ray to $\gamma$-ray flux ratio as the Galactic
$\gamma$-ray emitting LMXBs, i.e.,

$\frac{F_{\gamma,\rm LMBX}}{F_{\rm X,LMXB}}=\frac{F_{\gamma,\rm J1023}}{F_{\rm
X,J1023}},$ (2)

where $F_{\gamma,\rm LMBX}$ and $F_{\rm X,LMXB}$ are the fluxes at
$\gamma$-ray and X-ray bands of the LMXBs in NGC 4278, $F_{\gamma,\rm J1023}$
and $F_{\rm X,J1023}$ are the $\gamma$-ray and X-ray fluxes of the LMXB PSR
J1023+0038. PSR J1023+0038 is a well observed Galactic $\gamma$-ray emitting
LMXB. The observational data for PSR J1023+0038 are obtained from Li et al.
(2014) and Xing et al. (2018), as shown in Figure 5 (black symbols). Taking
$F_{\rm X,J1023}=4.01\times 10^{-13}\rm\ erg\ cm^{-2}\ s^{-1}$ in the $0.5-10$
keV band (Campana et al. 2016) and $F_{\rm X,LMBX}=7.79\times 10^{-14}\rm\
erg\ cm^{-2}\ s^{-1}$ in the $0.3-8$ keV band (Fabbiano et al. 2010), where
the maximum value of $F_{\rm X,LMXB}$ among these LMXBs in NGC 4278 is
considered, the $\gamma$-ray fluxes of the brightest LMXB in NGC 4278 can be
estimated as $F_{\gamma,\rm LMBX}=0.19F_{\gamma,\rm J1023}$.

We extrapolate the GeV spectrum of PSR J1023+0038 up to the TeV band by
assuming they follow the same power-law distribution (the black solid line in
Figure 5). Multiplying the black solid line by 0.19, we obtain the possible
$\gamma$-ray flux (the blue solid line) of the brightest LMXB in NGC 4278. The
blue dashed line in Figure 5 presents the result after considering EBL
absorption. It is evident that if there is only one $\gamma$-ray emitting LMXB
in NGC 4278, its emission would not be detectable by Fermi-LAT and VERITAS but
could potentially be detected by LHAASO. NGC 4278 hosts several LMXBs with
X-ray luminosities exceeding $10^{38}$ erg s-1, some of which exhibit
significant variability (Fabbiano et al. 2010). Therefore, it is plausible
that the observed $\gamma$-rays from 1LHAASO J1219+2915 can be attributed to
some LMXBs in NGC 4278. However, no detection has been made for any LMXB at
TeV energy band.

In addition, the spectrum at the TeV band of LMXB may not follow the same
power-law spectral shape as that at the GeV band; a flatter spectrum is
observed in high-mass X-ray binaries (HMXBs) at the TeV band, for example,
HESS J1018–589 A/1FGL J1018.6–5856 (H. E. S. S. Collaboration et al. 2015).
Considering the presence of multiple bright LMXBs in NGC 4278, we cannot
dismiss the possibility that the detection of 1LHAASO J1219+2915 could be
attributed to these LMXBs within NGC 4278.

###  5.3 NGC 4278: a BL Lac or a LINER

The compact radio morphology and flat radio spectrum of NGC 4278 suggest its
similarity to BL Lacs (Giroletti et al. 2005). The SED of NGC 4278 can be
represented by a one-zone synchrotron+SSC model, similar to the typical TeV BL
Lacs. We compare the derived jet physical parameters of NGC 4278 with that of
a TeV-BL Lac sample from Zhang et al. (2012). It seems that NGC 4278 does not
fit in with the characteristics of those typical TeV BL Lacs. As shown in
Figure 6, both the $B$ and $\delta$ values for NGC 4278 are considerably
smaller, while $\gamma_{\rm b}$ is slightly larger compared to typical TeV BL
Lacs. In the $P_{\rm jet}^{e^{\pm}}-P_{\rm r}$ plane (Figure 7), NGC 4278 well
follows the sequence of typical TeV BL Lacs but resides at the low-power end.
Nevertheless, when examining the $P_{\rm jet}^{e^{\pm}}-P_{B}$ plane, it
becomes evident that NGC 4278 stands apart from these TeV BL Lacs due to an
unusually smaller ratio of $P_{B}$ to $P_{\rm jet}^{e^{\pm}}$.

To further investigate this issue, we compile the radio luminosity at 8 GHz
($L_{\rm 8~{}GHz}$) of NGC 4278 and these typical TeV BL Lacs from the Radio
Fundamental Catalog (RFC)444https://astrogeo.smce.nasa.gov/rfc/, and plot the
$\gamma$-ray luminosity at 1 TeV ($L_{\rm 1~{}TeV}$) against $L_{\rm 8~{}GHz}$
in Figure 8(a). No significant correlation between $L_{\rm 8~{}GHz}$ and
$L_{\rm 1~{}TeV}$ for these typical TeV BL Lacs is observed. Moreover, NGC
4278 stands out as it is clearly separated from these typical BL Lacs and
positioned at the lower end of the luminosity scale. The luminosity of NGC
4278 is more than two orders of magnitude lower than that of typical TeV BL
Lacs in both radio and $\gamma$-ray bands.

As described above, NGC 4278 does exhibit distinctive characteristics that
differentiate it from typical TeV BL Lacs, and it is generally classified as a
LINER (Heckman 1980) or a ‘bona fide’ CSO (Kiehlmann et al. 2024). In Figure
8(b), we plot the radio luminosity ($L_{\rm R}$) against the Eddington ratio
($R_{\rm Edd}$) for a sample of Seyfert galaxies and LINERs from Sikora et al.
(2007), as well as for a sample of BL Lacs from Zhang et al. (2020). Note that
the BL Lac sample in Figure 8(b) differs partially from that in Figure 8(a).
Not all BL Lacs shown in Figure 8(b) are TeV sources, and not all BL Lacs
depicted in Figure 8(a) have available $R_{\rm Edd}$ data. However, there is
an overlap between the two samples. For Seyfert galaxies and LINERs, $L_{\rm
R}$ represents the luminosity at 5 GHz, while for NGC 4278 and these BL Lacs
it corresponds to $L_{\rm 8~{}GHz}$, where the $L_{\rm 8~{}GHz}$ values are
also derived from the observation data obtained in RFC. Both samples (neither
considering NGC 4278) show a clear correlation between $L_{\rm R}$ and $R_{\rm
Edd}$. The Pearson correlation analysis yields a coefficient $r=0.629$ and a
chance probability $p=2.34\times 10^{-5}$ for all Seyfert galaxies and LINERs,
while $r=0.769$ and $p=1.17\times 10^{-4}$ are obtained for these BL Lacs. In
the $R_{\rm Edd}-L_{\rm R}$ plane, NGC 4278 is situated within the region
occupied by Seyfert galaxies and LINERs, slightly deviating from the low-
luminosity extension of the 95% confidence band of the best linear fit for BL
Lacs. The Eddington ratio of NGC 4278 is more than one order of magnitude
lower than that of BL Lacs, which may be the reason for its low luminosity.
Considering all these aspects, NGC 4278 does not resemble a BL Lac; instead,
it exhibits more similarities to Seyfert galaxies and LINERs.

##  6 summary

NGC 4278 was classified as a LINER-type AGN and no associated $\gamma$-ray
source counterpart reported in the 4FGL-DR4 (Abdollahi et al. 2022; Ballet et
al. 2023). By reanalyzing all the Fermi-LAT observation data from 2008 Aug 4
to 2024 Apr 24 around the radio position of NGC 4278, no significant
$\gamma$-ray emission associated with NGC 4278 is found. Interestingly, NGC
4278 has been detected by LHAASO and is reported to be associated with 1LHAASO
J1219+2915 (Cao et al. 2024). To investigate the origin of VHE emission from
this source, we examine its X-ray observations obtained by Swift-XRT, Chandra,
and XMM-Newton, and find only one valid pointing (on MJD 59546) from Swift-XRT
during the LHAASO observations. Surprisingly, the derived X-ray flux from
Swift-XRT is more than one order of magnitude higher than the archive X-ray
flux obtained by Chandra observation $\sim$11.7 years ago, indicating that the
X-ray emission of NGC 4278 is currently in a historically high-flux state.
Together with the mildly relativistic jets detected in the radio band using
the VLBA observations, we propose that the TeV $\gamma$-rays of 1LHAASO
J1219+2915 originate from an atypical BL Lac-type AGN in the center of NGC
4278 that has recently become more active; however, other possibilities cannot
be completely ruled out yet. The derived parameters for its jet using a one-
zone synchrotron+SSC model are also significantly different from that of these
typical TeV BL Lacs, with smaller $\delta$ and $B$, but higher energies of
electrons. In both radio and TeV bands, NGC 4278 shows luminosity lower by
over two orders of magnitude when compared to these typical TeV BL Lacs. In
the $R_{\rm Edd}-L_{\rm R}$ plane, NGC 4287 slightly deviates from the low-
luminosity and low-Eddington-ratio extension of the 95% confidence band
representing best linear fit for BL Lacs; instead it exhibits similarities
with Seyfert galaxies and LINERs.

We acknowledge the use in our research the Radio Fundamental Catalogue
available at https://astrogeo.smce.nasa.gov/rfc/. This work is supported by
the National Key R&D Program of China (grant 2023YFE0117200) and the National
Natural Science Foundation of China (grants 12022305 and 11973050).

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Table 1: Observations of Chandra and Swift-XRT and Data Analysis Results for NGC 4278 Obs-date | Obs-ID | Exposure | $N^{\rm H}_{\rm int}$ | $\Gamma_{\rm X}$ | PL norm | kT | Corr.flux | Refs**‘a’ denotes the Chandra observation results obtained from Younes et al. (2010), while ‘b’ and ‘c’ respectively represent the analysis results of Chandra and Swift-XRT observations conducted in this work.
---|---|---|---|---|---|---|---|---
|  | $[\rm ks]$ | $[10^{20}\rm cm^{-2}]$ |  | $[\rm 10^{-4}\ Ph\ keV^{-1}\ cm^{-2}\ s^{-1}]$ | $[\rm keV]$ | $[\rm 10^{-13}\ erg\ cm^{-2}\ s^{-1}]$ |
2000-04-20 | 398 | 1.4 |  |  |  |  |  | a
2005-02-03 | 4741 | 37.5 | $<6.78$ | $2.13^{+0.15}_{-0.13}$ | $4.28^{+0.43}_{-0.36}$ | $0.62\pm 0.04$ | $18.5^{+1.6}_{-0.3}$ | a
2006-03-16 | 7077 | 110.3 |  | $2.26^{+0.13}_{-0.10}$ | $1.82^{+0.17}_{-0.14}$ |  | $8.21^{+0.79}_{-0.21}$ | a
2006-07-25 | 7078 | 51.4 |  | $2.34^{+0.13}_{-0.12}$ | $4.21^{+0.40}_{-0.31}$ |  | $16.5^{+2.0}_{-0.4}$ | a
2006-10-24 | 7079 | 105.1 |  | $2.38^{+0.12}_{-0.10}$ | $3.76^{+0.31}_{-0.22}$ |  | $15.05^{+1.34}_{-0.35}$ | a
2007-04-20 | 7080 | 55.8 |  | $2.02^{+0.18}_{-0.18}$ | $1.10^{+0.16}_{-0.14}$ |  | $6.0^{+0.8}_{-0.3}$ | a
2007-02-20 | 7081 | 110.7 |  | $2.12^{+0.13}_{-0.12}$ | $1.25^{+0.04}_{-0.11}$ |  | $6.4^{+0.7}_{-0.2}$ | a
2010-03-15 | 11269 | 81.9 | $<2.43$ | $2.14^{+0.13}_{-0.11}$ | $0.35^{+0.04}_{-0.03}$ | $0.62^{+0.04}_{-0.05}$ | $1.86\pm 0.06$ | b
2010-03-20 | 12124 | 25.8 | $<11.62$ | $2.29^{+0.41}_{-0.22}$ | $0.33^{+0.12}_{-0.08}$ | $0.57^{+0.10}_{-0.16}$ | $1.66\pm 0.11$ | b
2021 |  |  | $<52.7$ | $1.42^{+0.57}_{-0.55}$ | $5.88^{+3.24}_{-2.30}$ |  | $43.60^{+14.17}_{-11.69}$ | c
Table 2: SED Fitting Parameters and Derived Jet Powers of NGC 4278

$N_{0}$ | $p_{1}^{\star}$ | $p_{2}^{\star}$ | $\gamma_{\rm min}^{\star}$ | $\gamma_{\rm b}$ | $\gamma_{\rm max}^{\star}$ | $B$ | $\delta^{\star}$ | $R^{\star}$ | $P_{\rm e}$ | $P_{\rm B}$ | $P_{\rm r}$ | $P_{\rm jet}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$[\rm 1/cm^{3}]$ |  |  |  |  |  | $[\rm G]$ |  | $[\rm cm]$ | $[\rm erg\ s^{-1}]$ | $[\rm erg\ s^{-1}]$ | $[\rm erg\ s^{-1}]$ | $[\rm erg\ s^{-1}]$
$3.5\times 10^{4}$ | 2.27 | 3.8 | 1 | $4\times 10^{6}$ | $3\times 10^{7}$ | 0.015 | 2.7 | $6.98\times 10^{15}$ | $6.97\times 10^{42}$ | $5.99\times 10^{38}$ | $5.56\times 10^{41}$ | $7.52\times 10^{42}$

Note: The parameters marked with an asterisk in the superscript remain fixed
during SED modeling.

Figure 1: The X-ray light curve of NGC 4278 in the 0.5–8.0 keV band, obtained
with the observations from XMM-Newton, Chandra, and Swift-XRT. The data points
(black open symbols) before MJD 55000 are directly taken from Younes et al.
(2010), while two red circles and one green square represent the analysis
results conducted in this work. Figure 2: $\Gamma_{\rm X}$ vs. $F_{\rm
0.5-8~{}keV}$, where the symbols are same as in Figure 1. Figure 3: Observed
SEDs with model fitting for NGC 4278. The data in radio-optical-UV bands (gray
open circles) are taken from Younes et al. (2010). The X-ray spectrum (green
bowtie) is derived from the Swift-XRT observations in this work, while the TeV
spectrum (red bowtie) is obtained with the LHAASO observations taken from Cao
et al. (2024). The black solid line is the sum of each emission component,
synchrotron radiation (red line) and SSC process (blue line). The sensitivity
curves of LHAASO, VERITAS, and Fermi-LAT are also presented.

Figure 4: The TeV spectrum of NGC 4278 (red bowtie) is obtained with the
LHAASO observations taken from Cao et al. (2024). The blue and magenta dashed
lines represent the observable flux level, taking into account the absorption
from both the strongest and weakest models of EBL, respectively. The
sensitivity curves of LHAASO, VERITAS, and Fermi-LAT are also presented. Panel
(a): The intrinsic fluxes (black circles) of 1ES 1218+304 in the TeV band are
taken from Acciari et al. (2009). They are fitted with a power-law function
and extrapolated up to the 100 TeV band (black solid line). Panel (b): The
observed fluxes of B2 1215+30 in the TeV band are taken from Abeysekara et al.
(2017). Considering its redshift of $z=0.130$, correcting into the intrinsic
fluxes with the strongest and weakest models of EBL, fitting them with a
power-law function, and extrapolating the fitting lines up to the 100 TeV
band, we obtain the blue and magenta solid lines. Panel (c): Same as in Panel
(b), but with the redshift of $z=0.237$. Figure 5: The observed data (black
squares, circles, and arrows) of PSR J1023+0038 are taken from Li et al.
(2014) and Xing et al. (2018). The black solid line represents the power-law
fitting line for the last four detection data points, and it is extrapolated
to the 100 TeV band. The blue solid line indicates the possible flux level of
the brightest LMXB in MGC 4278, which is obtained by multiplying the black
solid line by a factor of 0.19. The green solid line is simply the solid blue
line multiplied by 10, assuming that these are several $\gamma$-ray emitting
LMXBs in NGC 4278. The blue and green dashed lines present the results after
considering the EBL absorption.

Figure 6: $\gamma_{\rm b}$ and $B$ as a function of $\delta$. The data of
these typical TeV BL Lacs are from Zhang et al. (2012).

Figure 7: $P_{\rm r}$ and $P_{B}$ as a function of $P^{e^{\pm}}_{\rm jet}$.
The data of these typical TeV BL Lacs are from Zhang et al. (2012).

Figure 8: Panel (a): $L_{\rm 1~{}TeV}$ vs. $L_{\rm 8~{}GHz}$. The data of
$L_{\rm 8~{}GHz}$ for both NGC 4278 and typical TeV BL Lacs are taken from the
RFC, while the $L_{\rm 1~{}TeV}$ values of the typical TeV BL Lacs are from
Zhang et al. (2012). Panel (b): $L_{\rm R}$ vs. $R_{\rm Edd}$, where $L_{\rm
R}$ is the luminosity at 8 GHz for NGC 4278 and BL Lacs taken from the RFC,
while it is the luminosity at 5 GHz for seyfert galaxies and LINERs taken from
Sikora et al. (2007). $R_{\rm Edd}$ for seyfert galaxies and LINERs is also
taken from Sikora et al. (2007), while it is taken from Zhang et al. (2020)
for BL Lacs. Note that the BL Lac sample in Panel (b) differs partially from
that in Panel (a); for more details, refer to Section 5.3.