Multigravity and the cosmological constant
E. T. Tomboulis111e-mail<EMAIL_ADDRESS>
Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547

Abstract

The possibility of flat metric solutions in the presence of arbitrary, in
particular, Planck mass sized cosmological constants is investigated within
multigravity theory, the ghost-free theory of $(N+1)$ rank-2 tensor fields
comprising one massless and $N$ massive spin-2 fields. It is found that,
indeed, for any $N$, no matter how large, no flat solutions exist without
fine-tuning parameters in the potential. For an infinite number of fields,
however, flat solutions exist within an infinite-dimensional parameter space.
Such a ground state may be viewed as a collective state that cannot be
approximated by any finite $N$.

The cosmological constant (c.c.) problem is one of the fundamental unsolved
problems of present-day physics (cf. [1] for a review). The observed vacuum
energy is $\rho_{V}\sim(10^{-12}{\rm Gev})^{4}$. If one defines the c.c.
$\Lambda$ in the standard manner through the term $(1/16\pi G)\Lambda\int
d^{4}x\sqrt{-g}$ in the gravity action, the observed $\rho_{V}$ correspond to
a c.c. $\Lambda\sim 10^{-84}{\rm Gev}^{2}$. Quantum field theory (QFT), on the
other hand, naturally yields c.c. of the order of the cutoff scale below which
the theory remains valid. Quantized matter fields in a classical gravitational
background thus result in c.c. that are huge compared to the observed value.
As General Relativity (GR) appears to be a good effective field theory up to
scales of the order of the Planck mass, and local QFT is believed to also
remain valid approximately up to such scales, one generally expects c.c. of
the order of $\Lambda\sim(10^{18}{\rm Gev})^{2}$. This is the infamous “120
orders of magnitude discrepancy problem”. It is a baffling problem as it
actually occurs at all scales, i.e., at whatever scale (cutoff) may be taken
as characteristic of the matter system. Furthermore, any phase transition
within a given system, such as the chiral or deconfinement transitions in QCD,
or the electroweak transition in the Higgs sector, results in contributions to
$\Lambda$ that are independent of any ultimate UV cutoff that may apply to the
system. These contributions are already huge compared to the observed value.

Since the observed $\Lambda$ is so tiny, it is natural to seek flat metric
solutions, which are subsequently subject to some small perturbations. The
problem is, of course, that flat solutions generically require extreme fine-
tuning of the parameters in the action - this in fact may be taken as the
field theoretic statement of the c.c. problem. Here we investigate the
possibility of flat metric solutions in multigravity theory (cf. [2], [3] for
reviews). The construction of the massive gravity potential [4], [5], [6],
[7], [8], [9] solved the problem of a consistent interacting massive spin-2
theory evading the famous BoulwareDeser ghost [10]. It was subsequently
extended first to bigravity [11] and then multigravity [12], the general
coordinate invariant theory of $N+1$ rank-2 fields comprising one massless and
$N$ massive spin-2 fields. We find that, indeed, for any finite number of
fields $N$, no matter how large, no flat solutions are possible without fine-
tuning of action parameters. For an infinite number of fields, however, there
are flat solutions. Such ground state solutions may be viewed as a kind of a
collective state that cannot be approximated by any finite $N$.

The multigravity action is defined by adding to the familiar Einstein -
Hilbert terms for each rank-2 tensor field $f_{i}$, $i=0,\ldots,N$, the
potential terms $U^{(i)}$ pairwise coupling adjacent fields $f_{i},f_{i+1}$,
$i=0,\ldots,N-1$. They are functions of the matrix

$\mathbb{X}_{i,i+1}\equiv\sqrt{f_{i}^{-1}f_{i+1}}\;.$ (1)

The square root is understood in the usual matrix sense:
$(f_{i}^{-1}f_{i+1})^{\mu}_{~{}\nu}=X^{\mu}_{~{}\alpha}X^{\alpha}_{~{}\nu}$.
The general form of the potential $U^{(i)}$ (in dimensionless units) is
explicitly given by

$U^{(i)}(\mathbb{X}_{i,i+1})=\sqrt{-f_{i}}\sum_{l=0}^{4}\beta^{(i)}_{l}{\rm
S}_{l}(\mathbb{X}_{i,i+1})\;,$ (2)

where the coefficients $\beta^{(i)}_{l}$ are arbitrary real numbers. The $\rm
S_{l}$ are the symmetric polynomials constructed out of the eigenvalues of the
matrix $\mathbb{X}$; they can be expressed in terms of the traces of powers of
$\mathbb{X}$. In $d=4$, with the common notation
$|\\![\mathbb{X}^{k}]\\!|\equiv{\rm tr}\mathbb{X}^{k}$, they are:

${\rm S}_{0}=1,\;{\rm S}_{1}=|\\![\mathbb{X}]\\!|,\;{\rm S}_{2}={1\over
2!}\big{(}|\\![\mathbb{X}]\\!|^{2}-|\\![\mathbb{X}^{2}]]\\!|\big{)},\;{\rm
S}_{3}={1\over
3!}\big{(}|\\![\mathbb{X}]\\!|^{3}-3|\\![\mathbb{X}]\\!||\\![\mathbb{X}^{2}]\\!|+2|\\![\mathbb{X}^{3}]\\!|\big{)},\;{\rm
S}_{4}=\det\mathbb{X}\,.$ (3)

In (2) the $\beta^{(i)}_{0}$ term is a cosmological constant term for $f_{i}$,
whereas, by (3), the $\beta^{(i)}_{4}$ term is a cosmological term for
$f_{i+1}$. When adding the potential terms $U^{(i)}(\mathbb{X}_{i,i+1})$,
$i=0,\ldots,N-1$, however, the $\beta^{(i)}_{4}$ term in
$U^{(i)}(\mathbb{X}_{i,i+1})$ can be absorbed in the $\beta^{(i+1)}_{0}$ term
of $U^{(i+1)}(\mathbb{X}_{i+1,i+2})$ except for the last $\beta^{(N-1)}_{4}$
term. Relabeling $\beta^{(N-1)}_{4}\equiv\beta^{(N)}_{0}$, the action is then
given by

$S=\sum_{i=0}^{N}\frac{1}{\kappa^{2}}\\!\int\\!d^{d}x\\!\sqrt{-f_{i}}R(f_{i})-\frac{1}{\kappa^{2}}\\!\int\\!d^{d}x\\!\Bigg{\\{}\sum_{i=0}^{N-1}M_{i}^{2}\\!\Big{(}\sqrt{-f_{i}}\sum_{l=0}^{3}\beta^{(i)}_{l}{\rm
S}_{l}(\mathbb{X}_{i,i+1})\Big{)}+\sqrt{-f_{N}}M_{N-1}^{2}\beta^{(N)}_{0}\Bigg{\\}}$
(4)

with $\kappa^{2}=16\pi G$.222We could allow different couplings
$(\kappa^{2}/\gamma_{i})$, $\gamma_{i}>0$, for each $f_{i}$ but such
generalizations make no qualitative difference in the following.  The masses
$M_{i}$ are taken to be of the order of Planck mass or higher. Having set the
scale of each potential term by $M_{i}$, all $\beta^{(i)}_{l}$ are taken to be
comparable in magnitude and of order unity. It would indeed be unnatural to
take $\beta^{(i)}_{l}$ for different $l$ varying greatly in magnitude.

It is clear from (4) that a flat metric solution $\bar{f}_{i}=\eta$, all $i$,
is not possible without fine-tuning the parameters $\beta_{l}^{(i)}$ by
imposing relations among them. Thus, in $d=4$, one must set
$\beta^{(0)}_{0}=(-3\beta^{(0)}_{1}-3\beta^{(0)}_{2}-\beta^{(0)}_{3})$ and
also fix the other $\beta^{(i)}_{0}$, $i=1,\ldots,N$, by similar linear
relations.

We now seek solutions of the form333Note that because of the ratio form of
$\mathbb{X}_{i,i+1}$, one may just set $C_{0}=1$.

$\bar{f}_{i}=\overline{C}_{i}^{2}\bar{g}\,,\quad\mbox{with}\quad\overline{C}^{2}_{i}=\prod_{j=0}^{i}C_{j}^{2}\,,\quad
C_{0}=1\,,\qquad i=0,\ldots,N\;,$ (5)

for some metric $\bar{g}$ and real constants $C_{i}$. Due to the unwieldy
square root form of (1) the equations of motion (e.o.m.) following from (4),
though straightforward to derive, are somewhat involved. For fields of the
assumed form (5), however, one simply has
$\mathbb{X}_{i,i+1}={\bf{1}}\bar{C}_{i}^{-1}\bar{C}_{i+1}={\bf 1}C_{i+1}$ for
any $\bar{g}$. Taking for the moment $\bar{g}$ to be flat, $\bar{g}=\eta$, the
e.o.m. in $d=4$ become the following algebraic equation system for the
$C_{i}$:

$\displaystyle\beta^{(0)}_{0}+3\beta^{(0)}_{1}C_{1}+3\beta^{(0)}_{2}C^{2}_{1}+\beta^{(0)}_{3}C^{3}_{1}=0\;.$
(6)
$\displaystyle\frac{M^{2}_{i-1}}{M^{2}_{i}}\Bigg{(}\beta^{(i-1)}_{1}C_{i}^{-3}+3\beta^{(i-1)}_{2}C_{i}^{-2}+3\beta^{(i-1)}_{3}C_{i}^{-1}\Bigg{)}$
$\displaystyle\qquad\qquad+\;\beta^{(i)}_{0}+3\beta^{(i)}_{1}C_{i+1}+3\beta^{(i)}_{2}C_{i+1}^{2}+\beta^{(i)}_{3}C_{i+1}^{3}=0\,,\qquad\quad
i=1,\dots,(N-1)\;.$ (7) and
$\displaystyle\beta^{(N-1)}_{1}C_{N}^{-3}+3\beta^{(N-1)}_{2}C_{N}^{-2}+3\beta^{(N-1)}_{3}C_{N}^{-1}+\beta^{(N)}_{0}=0\;.$
(8)

(6) - (8) is an algebraic system of cubic equations, which is already
decoupled. (6), from the e.o.m. for $f_{0}$, determines $C_{1}$; (7), from the
e.o.m. for $f_{i}$, $i=1,\ldots,N-1$, determine then successively
$C_{2},C_{3},\cdots,C_{N}$. In fact they can each be solved analytically by
the Cardano-Tartaglia (GT) formula for the solution of the cubic
equation.444One may, of course, consider less general models for illustrative
purposes, e.g., the ‘minimal model’ of [11], defined by setting
$\beta^{(i)}_{2}=\beta^{(i)}_{3}=0$. In this case (6)-(8) becomes a linear
system.  Note that with the coefficients $\beta^{(i)}_{l}$ of ‘order unity’,
the $C_{i}$ solving the system (6)-(7) are generally also of that order and,
by this iterative procedure, each $C_{i+i}$ depends on the set
$\\{\beta^{(k)}_{l}\\}$, $k\leq i$. Furthermore, for generic parameters
$\\{\beta^{(j}_{l}\\}$, the solutions $C_{i}\neq 0$, since $C_{i}=0$ requires
the vanishing of a discriminant, which requires relations among
$\beta^{(j)}_{l}$’s. It is important that, by the GT formula, there is always
at least one real solution.

This leaves (8) from the e.o.m. for $f_{N}$. Since all $C_{i}$ have been
determined by (6) - (7), it is clear that this equation cannot be satisfied
unless $\beta^{(N)}_{0}$ is fine-tuned. Alternatively, one may take $\bar{g}$
to be a metric of constant curvature $\lambda$. In the latter case, the system
(6) - (8), now augmented by the additional terms from the variation of the EH
terms in (4), determines the $(N+1)$ parameters $C_{i},\lambda$. With mass
parameters of Planck order, however, the residual c.c. is of the same
magnitude, so nothing is gained.

Returning to the $\bar{g}=\eta$ case, one may note that, actually, something
has been gained by using the Ansatz (5) versus simply setting
$\overline{f}_{i}=\eta$ for all $i$: instead of having to fine-tune all
$\beta^{(i)}_{0}$, only $\beta^{(N)}_{0}$ has to be fine-tuned. All other
$\beta^{(i)}_{l}$ remain arbitrary while all the e.o.m. are satisfied. This
one fine-tuning may be avoided by augmenting the last term in (4) to a full
potential term $U^{(N)}(\mathbb{X}_{N,N+1})$ by the addition of a field
$f_{N+1}$: the e..o.m. for $f_{N}$ now determines $C_{N+1}$. This of course
transfers the problem to fine-tuning the new $\beta^{(N+1)}_{0}$ parameter,
which can be evaded by adding the full $U^{(N+1)}(\mathbb{X}_{(N+1,N+2)})$
term, and so on ad-infinitum. A formal flat space solution is thus obtained by
replacing (4) by:

$S=\sum_{i=0}^{\infty}{1\over\kappa^{2}}\int
d^{d}x\sqrt{-f_{i}}\,R(f_{i})-\frac{1}{\kappa^{2}}\Bigg{\\{}\sum_{i=0}^{\infty}M_{i}^{2}\\!\int
d^{d}x\,\Big{(}\sqrt{-f_{i}}\sum_{l=0}^{3}\beta^{(i)}_{l}{\rm
S}_{l}(\mathbb{X}_{i,i+1})\Big{)}\Bigg{\\}}\;.$ (9)

It is crucial that an infinite tower of $f_{i}$ fields is present. Any
truncation to finite tower leaves a large residual c.c, i.e., a discrepancy
away from the flat solution.

There are other examples where such a situation arises. An elementary, but
fundamental, example is provided by the canonical commutation relation of non-
relativistic QM: ${\bf XP-PX}=i\hbar{\bf 1}$. This can hold only for infinite
matrices representing ${\bf X,P}$, since, as obvious at a glance, this
relation is mathematically nonsensical for any finite dimensional matrices.
Given matrices for ${\bf X,P}$ obtained from the e.o.m. for the system,
restricting to any $N\times N$ submatrix necessarily creates a discrepancy
between the two sides that grows with $N$.555This is well illustrated by the
exact harmonic oscillator solution in the energy representation (the original
matrix QM). Note that in the coordinate representation, where matrices have
continuously labeled indices, $<x^{\prime\prime}|{\bf
X}|x^{\prime}>=x^{\prime}\delta(x^{\prime\prime}-x^{\prime})$, the question of
a finite dimensional approximation does not arise.  To avoid the discrepancy
one has to enlarge to $(N+1)\times(N+1)$, which in turn necessitates
$(N+2)\times(N+2)$ matrices and so on ad infinitum. This is then closely
analogous to the case here, where the flat solution is obtained only by
extending to an infinite tower of $f_{i}$ fields.

The infinite tower of fields $f_{i}$ then collectively adjust their individual
ground states $\bar{f}_{i}$, which are correlated by the e.o.m., so that any
c.c. in (9) are absorbed leaving a flat metric. To better understand this
solution consider expanding about it:
$f_{i}=\overline{f}_{i}+\chi_{i}=\overline{C}_{i}^{2}\eta+\chi_{i}$,
$i=1,2,\ldots\ $. Substituting in (9), upon expanding the terms linear in
$\chi_{i}$ vanish since $\bar{f}_{i}$ satisfy the e.o.m. Noting that
$\sqrt{-f_{i}}=\overline{C}_{i}^{4}\det[\bf{1}+\overline{C}_{i}^{-2}\eta^{-1}\chi_{i}]$,
the part of the EH action quadratic in $\chi_{i}$ is of the form:

$S_{\scriptscriptstyle(2)}^{\scriptscriptstyle\rm
EH}=\sum_{i=0}^{\infty}\frac{1}{\kappa^{2}}\overline{C}^{-2}_{i}\int
d^{4}x\;\partial_{\alpha}\chi_{i\kappa\lambda}K^{\alpha\kappa\lambda\beta\mu\nu}(\eta^{-1})\partial_{\beta}\chi_{i\mu\nu}\;.$
(10)

In (10) $K^{\alpha\beta\kappa\lambda\mu\nu}(\eta)$ is the usual EH kinetic
tensor on flat metric. To obtain canonically normalized quadratic kinetic
terms then, each $\chi_{i}$ has to be rescaled:
$\chi_{i}\to\kappa\overline{C}_{i}$, so that the expansion assumes the form:

$f_{i}=\overline{C}_{i}^{2}\eta+\kappa\overline{C}_{i}\chi_{i}=\overline{C}_{i}^{2}\eta\,[1+\kappa\overline{C}_{i}^{-1}\eta^{-1}\chi_{i}]\;,\qquad
i=1,2,\ldots\;.$ (11)

The interactions are obtained by expansion of the “$\Gamma\Gamma$-form” of the
EH action, which gives the most efficient form for obtaining the interaction
vertices (for a recent discussion cf. [13]). It is obtained by simply
multiplying by $\ \det[\bf{1}+\overline{C}_{i}^{-1}\eta^{-1}\chi_{i}]$ and
replacing $K(\eta^{-1})$ by
$K([1+\kappa\overline{C}_{i}^{-1}\eta^{-1}\chi_{i}]^{-1})$ in the rescaled
(10). The EH interactions are then:

$S_{\scriptscriptstyle\rm I}^{\scriptscriptstyle\rm
EH}=\sum_{i=0}^{\infty}\,\sum_{n\geq 3}^{\infty}\kappa^{n-2}\int
d^{4}x\,V^{\alpha\beta\mu_{1}\nu_{1}\mu_{2}\nu_{2}\cdots\mu_{n}\nu_{n}}_{i,n}(\eta^{-1})\partial_{\alpha}\chi_{i\mu_{1}\nu_{1}}\partial_{\beta}\chi_{i\mu_{2}\nu_{2}}\overline{C}^{-1}_{i}\chi_{i\mu_{3}\nu_{3}}\ldots\overline{C}^{-1}_{i}\chi_{i\mu_{n}\nu_{n}}\;,$
(12)

where the $V_{i,n}$ are the usual EH $n$-point interaction vertices for the
field $\chi_{i}$.

The expansion is then an expansion in $\kappa\overline{C}_{i}^{-1}$. The
products $\overline{C_{i}}$, (5), are fully determined as functions of
$\\{\beta^{(i)}\\}$ by the e.o.m. as detailed above. There are three possible
behaviors for $\overline{C}_{i}$ as $i\to\infty$: (i)
$\overline{C_{i}}\to\infty$; (ii) $\overline{C_{i}}\to 0$; or (iii)
$\overline{C_{i}}\to{\rm constant}$. These correspond to the solution $C_{i}$
to (7) being: (i) $C_{i}>1$; (ii) $C_{i}<1$ and (iii) $C_{i}=1$, respectively,
for each $i\geq N_{0}$ for some finite $N_{0}$, which may be arbitrarily
large. (iii) implies relations between the potential parameters
$\beta^{(i)}_{k}$ for each $i\geq N_{0}$ and thus fine-tuning. As seen from
the CT formula, (i) and (ii) can be satisfied by an infinite subset of
$\\{\beta^{(i)}_{l}\\}$ of the same cardinality (order of infinity) as the set
$\\{\beta^{(i)}_{l}\\}$. (ii) has to be rejected since it would lead to
arbitrarily low masses and hence not viable range of validity as an effective
theory. Thus, (i) is the only possible option. It implies an infinite tower of
spin-2 fields with rising mass spectrum (see below).

Noting that $\mathbb{X}_{i,i+1}=C_{i+1}\Big{[}[{\bf
1}+\kappa\overline{C}_{i}^{-1}\eta^{-1}\chi_{i}]^{-1}[{\bf
1}+\kappa\overline{C}_{i+1}^{-1}\eta^{-1}\chi_{i+1}]\Big{]}^{1/2}$, the
expansion of the potential terms gives the quadratic mass term

$S^{\scriptscriptstyle\rm P}_{(2)}=M_{0}^{2}\int
d^{4}x\,\mathcal{M}^{\kappa\lambda\mu\nu}_{\scriptscriptstyle\rm
FP}(\eta^{-1})\,\sum_{i,j=0}^{\infty}\chi_{j\kappa\lambda}\mathbb{M}_{ji}\chi_{i\mu\nu}\;,$
(13)

where $\mathcal{M}^{\kappa\lambda\mu\nu}_{\scriptscriptstyle
FP}(\eta^{-1})=\Big{(}\eta^{\kappa\mu}\eta^{\lambda\nu}-\eta^{\kappa\lambda}\eta^{\mu\nu}\Big{)}$
is the Fierz - Pauli tensor structure. The (dimesionless) mass matrix
$\mathbb{M}_{ji}$ is a real symmetric tri-diagonal infinite matrix. Its
elements have the form
$\mathbb{M}_{ji}=\overline{C}_{j}b_{ji}\overline{C}_{i}$ with $b_{ji}=b_{ij}$
and $b_{ji}=0$ whenever $|i-j|\geq 2$. The $b_{ij}$ are functions of the
parameters $a_{k}\equiv(M^{2}_{k}/M^{2}_{0})$ and the set of $\beta^{(k)}_{l}$
coefficients with $k\leq i$.666The quantities $b_{ji}$, as obtained via the
expansion (11), are expressed in terms of $\\{\beta^{(k)}\\}$ and
$\\{C_{k+1}\\}$, but, of course, the latter are included as $\beta^{(k)}$
dependence since all $C_{k+1}$ have been fixed by the e.o.m (6) - (7) as
functions of the $\\{\beta^{(k)}\\}$, $k\leq i$.

Expansion of the potential beyond quadratic order gives the potential
interactions terms of the general form:

$S^{\scriptscriptstyle\rm P}_{\scriptscriptstyle\rm
I}=M_{0}^{2}\sum_{i=0}^{\infty}\sum_{n\geq 3}\kappa^{n-2}\int
d^{4}x\;\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\cdots\mu_{n}\nu_{n}}_{n}\Big{(}\eta^{-1};\\{a_{k},\beta^{(k)}\\}_{k\leq
i}\Big{)}\overline{C}_{i_{1}}\chi_{i_{1}\mu_{1}\nu_{1}}\overline{C}_{i_{2}}\chi_{i_{2}\mu_{2}\nu_{2}}\\\
\qquad\quad\cdot\overline{C}^{-1}_{i_{3}}\chi_{i_{3}\mu_{1}\nu_{3}}\cdots\overline{C}^{-1}_{i_{n}}\chi_{i_{n}\mu_{1}\nu_{n}}\,,\;\qquad
i_{j}\in\\{i,i+1\\}\;.$ (14)

For a finite system of $N+1$ fields $\chi_{i}$ the tridiagonal mass matrix
$\mathbb{M}_{ji}$ is a non-negative definite matrix possessing one zero and
$N$ non-zero eigenvalues. This follows from the general Hamiltonian analysis
in [9], [11], [12] and we assume that it holds in the limit $N\to\infty$. The
mass matrix $\mathbb{M}_{ji}$ is real symmetric and hence, trivially,
satisfies $\mathbb{M}_{i,i+1}\mathbb{M}_{i+1,i}>0$, which, for a tridiagonal
matrix, implies that the eigenvalues are distinct [14]. This property then
implies a rising eigenvalue spectrum. Note that this means that one has a
tower of rising mass eigenvalues even if one sets $a_{i}=1$, i.e.,
$M_{i}^{2}=M_{0}^{2}$, all $i$. This in fact makes little difference as the
rise is dominated by the $\overline{C}_{i}$ factors in the $\mathbb{M}$ matrix
elements, which grow asymptotically very rapidly. The size of the eigenvalues
can be estimated by taking finite ‘sections’ (submatrices) [15], [16] in
conjunction with the theory of Geršgorin disks [14]. Listing the eigenvalues
$\lambda^{(k)}$ in increasing order, one has the estimate
$\lambda^{(k)}\lesssim|\overline{C}_{k}\overline{C}_{k+1}|$.

The mass matrix $\mathbb{M}$ is diagonalizable by a real orthogonal matrix
$U$, whose $k$-th column is the column eigenvector $e^{(k)}$ belonging to
eigenvalue $\lambda^{(k)}$. Then, with $\chi$ and $\phi$ denoting the column
vectors of the $\chi_{i}$ fields and the mass eigenstate fields $\phi_{i}$,
$i=0,1,2,\ldots$, respectively, one has
$\chi_{i}=\sum_{j=0}^{\infty}e^{(j)}_{i}\phi_{j}$ and
$\phi_{i}=\sum_{j=0}^{\infty}e^{(i)}_{j}\chi_{j}$. Inserting unity in the form
$(U^{T}U)_{ij}=\delta_{ij}$ in the canonically normalized EH kinetic term,
i.e., in (10) after the rescaling $\chi_{i}\to\kappa\overline{C}_{i}\chi_{i}$,
one obtains the quadratic part of the action in terms of the mass eigenstate
fields:

$S^{\scriptscriptstyle\rm EH}_{(2)}+S^{\scriptscriptstyle\rm
P}_{(2)}=\sum_{i=0}^{\infty}\int
d^{4}x\;\left[\partial_{\alpha}\phi_{i\kappa\lambda}K^{\alpha\kappa\lambda\beta\mu\nu}(\eta^{-1})\partial_{\beta}\phi_{i\mu\nu}+\mathcal{M}^{\kappa\lambda\mu\nu}_{\scriptscriptstyle\rm
FP}(\eta^{-1})\,\mu^{2}_{i}\phi_{i\kappa\lambda}\phi_{i\mu\nu}\right]\;$ (15)

with $\mu^{2}_{i}=M_{0}^{2}\lambda^{(i)}$. The interaction parts can then be
expressed in terms of the mass eigenstate fields by substituting
$\chi_{i}=\sum_{j=0}^{\infty}e^{(j)}_{i}\phi_{j}$ in (12), (14).

In summary, we have obtained a flat solution of the form (5) in multi-gravity
with an infinite number of fields. As we saw it is absolute crucial that the
number of fields be infinite in order to obtain a flat solution, i.e.,
$\bar{g}=\eta$ in (5). The solution holds for arbitrary, in particular,
Planck-mass-sized cosmological constants for each field $f_{i}$, the scale
being set by the parameters $M^{2}_{i}$ in the action (9), and in the
infinite-dimensional space of the parameters $\\{\beta^{(i)}_{l}\\}$. Such
cosmological constants are expected to result from integrating out matter
fields in the multigravity background and are indeed necessary in order that
multi-gravity holds as an effective field theory up to Planck order scales
(just as GR does).

The flat solution (5): $\bar{f}_{i}=\overline{C}_{i}^{2}\eta$,
$i=0,1,2,\ldots$, (5), is the ground state of the infinite collection of
fields $f_{i}$. The constants $\\{C_{i}\\}$ are determined by the e.o.m (6),
(7) in terms of the parameters in the action. This ground state may be viewed
as a coherent state that, as we saw, cannot be approximated by any finite
number of fields, no matter how large. It collectively adjusts to accommodate
any changes in the action parameters maintaining a flat solution. Expanding
about this ground state gives a well-ordered perturbative expansion. Expressed
in terms of mass eigenstates upon diagonalization of the mass matrix, it gives
an interaction theory of a massless spin-2 and a tower of massive spin-2
excitations with asymptotically rapidly rising mass spectrum. Since all masses
are around Planck mass or higher, this implies that, for all practical
purposes, almost all these excitations above this ground state decouple and
one is left with a massless spin-2 and one or two massive spin-2 around Planck
scale.

Were one to be given this end result without knowing where it came from, one
would naturally view it as the usual manifestation of the cosmological
constant problem: a number of excitations propagating on a flat background
obtained from a potential dependent on seemingly fine-tuned parameters $C_{i}$
and $\hat{\beta}^{(i)}_{l}$. But these parameters are not fine-tuned:
$\hat{\beta}^{(i)}_{l}=\beta^{(i)}_{l}C_{i+1}^{l}$ and $\\{C_{i}\\}$ are
determined by the e.o.m (6), (7) for arbitrary parameters $\beta^{(i)}_{l}$ in
the starting action (9) as detailed above.

An infinite number of fields would indicate extended structure/nonlocality in
whatever fundamental theory underlies (9) viewed as an effective theory. There
have been some attempts to extract massive gravity or bigravity from string
theory, e.g [17], but so far they remain inconclusive.

The observed c.c. is of course not zero but extremely tiny. It is a striking
fact that it corresponds to cosmological mass (length) scales of the order of
the Hubble constant $\sim(10^{-42}{\rm Gev})$. This suggests that it might
arise due to some long distance mechanism operating on top of the flat space
solution, the latter absorbing all c.c. contributions due to quantum
fluctuations at short (molecular and shorter $\gtrsim 10^{-3}{\rm ev})$
scales. What such a mechanism might be, though, cannot be discussed here.

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# Soft X-ray Energy Spectra in the Wide-Field Galactic Disk Area Revealed with
HaloSat

Kazuki Ampuku  Graduate School of Science, Division of Particle and
Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi,
464-8602, Japan  Ikuyuki Mitsuishi  Graduate School of Science, Division of
Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, Aichi, 464-8602, Japan  Koki Sakuta  Graduate School of Science,
Division of Particle and Astrophysical Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya, Aichi, 464-8602, Japan  Philip Kaaret  NASA/Marshall Space
Flight Center, Huntsville, AL 35812  University of Iowa Department of Physics
and Astronomy, Van Allen Hall, 30 N. Dubuque Street, Iowa City, IA 52242, USA
Daniel M. LaRocca  University of Iowa Department of Physics and Astronomy, Van
Allen Hall, 30 N. Dubuque Street, Iowa City, IA 52242, USA  Lorella Angelini
NASA/Goddard Space Flight Center, Greenbelt, MD 20771

###### Abstract

We analyzed data from HaloSat observations for five fields in the Galactic
disk located far away from the Galactic center (135∘ $<$ $l$ $<$ 254∘) to
understand the nature of soft X-ray energy emission in the Galactic disk. The
fields have 14∘ diameter and were selected to contain no significant high-flux
X-ray sources. All five HaloSat soft X-ray energy spectra (0.4–7 keV with
energy resolution of $<$ 100 eV below 1 keV) show a possibility of the
presence of unresolved high-temperature plasma in the Galactic disk (UHTPGD)
with a temperature of 0.8–1.0 keV and an emission measure of (8–11)$\times
10^{-4}~{}\rm cm^{-6}~{}pc$ in addition to the soft X-ray diffuse background
components mainly studied at higher galactic latitudes (solar wind charge
exchange emission, local hot bubble, Milky Way halo emission, and the cosmic
X-ray background). This suggests that the UHTPGD is present across the whole
Galactic disk. We also observed UHTPGD emission in a region with no bright
sources in an XMM-Newton field contained within one of the HaloSat fields. The
temperature and emission measure are consistent with those measured with
HaloSat. Moreover, the stacked spectra of the X-ray point-like sources and
NIR-identified point sources such as stars in the XMM-Newton field also show a
spectral feature similar to the UHTPGD emission. This suggests that the UHTPGD
may partly originate from point-like sources such as stars.

X-rays: diffuse background — Galaxy: disk — X-rays: ISM — Galaxy: halo —
X-rays: stars

##  1 Introduction

Diffuse background emission across many wavebands provides important
astronomical information. Significant time and effort have been invested to
understand the nature of the diffuse backgrounds in the microwave, infrared,
and X-ray bands (e.g., Penzias & Wilson, 1965; Hauser & Dwek, 2001; Shanks et
al., 1991). In the X-ray band, the diffuse background above 2 keV is dominated
by the cosmic X-ray background (CXB) that originates mainly from discrete
extragalactic sources (e.g., Hickox & Markevitch (2006)), predominantly AGNs
at a wide range of redshifts. The CXB is well described empirically by a
powerlaw with a photon index of $\sim$1.4 in the 2-10 keV band (e.g., Vecchi
et al. (1999); Kushino et al. (2002); De Luca & Molendi (2004)). However,
McCammon et al. (2002) reveals that the CXB is responsible for only $\sim$40 %
of the total emission in the soft X-ray diffuse background (SXDB) in the
$\sim$0.4–1 keV band. This shows the existence of other components.
Contributors to the SXDB include: emission due to geocoronal and heliospheric
solar wind charge exchange (SWCX) processes (e.g., Snowden et al., 2004; Ezoe
et al., 2010; Yoshitake et al., 2013; Kuntz, 2019), hot gas in the solar
neighborhood referred to as the Local Hot Bubble (LHB) (e.g., Liu et al.,
2017; Farhang et al., 2019), and hot plasma in the Milky Way Halo (MWH) (e.g.,
Yoshino et al., 2009; Sakai et al., 2014; Nakashima et al., 2018).

However, the situation changes in the Galactic disk because dense neutral
material with $N_{{\rm H}}\sim 10^{21}\rm~{}cm^{-2}$ or more significantly
blocks soft X-ray photons below 1 keV arising from the MWH and CXB. Although
it was expected that the SXDB below 1 keV should decrease by at least by
$\sim$40%, the SXDB intensity in the $ROSAT$ $R45$ band ($\sim$0.44–1.0 keV)
is only $\sim$20% lower in the Galactic disk compared to higher Galactic
latitudes (McCammon et al., 2002). This suggests the existence of widely-
distributed excess emission that partly compensates for the absorption of the
MWH and CXB.

This mystery has been known as the “M band problem” (McCammon & Sanders, 1990;
Cox, 2005). The term M band is from an energy band used in the proportional
counters in Wisconsin and the Nagoya-Leiden rocket programs; the energy range
is almost equal to the $ROSAT$ $R45$ band. The origin of the excess emission
has been under debate for a long time. Possible candidates include: hot gas
with a temperature of $\sim$3$\times$106 K, dM stars, young expanding
superbubbles or supernova remnants in low density fields (Nousek et al., 1982;
Sanders et al., 1983; Rosner et al., 1981; Cox, 2005).

Masui et al. (2009) conducted a deep pointed observation of a 18′$\times$18′
blank sky field located in the Galactic disk ($l$, $b$) = (235∘, 0∘) to
investigate spectral characteristics of the excess emission. They used Suzaku
which had low and stable non-X-ray background especially in the soft energy
band and was thus suitable for study of low-surface brightness soft X-ray
sources such as the SXDB (Mitsuda et al., 2007). They removed point-like X-ray
sources to extract a spectrum of the diffuse emission and assumed that the MWH
component is completely blocked by dense neutral material associated with the
Galactic disk. They revealed an excess component that was well modeled by an
unabsorbed, optically-thin collisionally-ionized plasma in thermal equilibrium
with a temperature of $\sim$0.8 keV. In this paper, we defined it as
unresolved high-temperature plasma in the Galactic disk (UHTPGD). The lack of
absorption suggests that the excess X-ray photons come from relatively nearby
sources. Masui et al. (2009) considered a large fraction of the UHTPGD
originates from faint dM stars and their model spectrum for spatially
unresolved dM stars reproduces the observed features although it cannot
compensate entirely the decrease at high Galactic latitudes due to the small
scale height. However, the results are only for a single field. It is
important to test for the presence of UHTPGD across a wide span of the
Galactic disk.

HaloSat was a small satellite (CubeSat) designed to map the entire soft X-ray
sky in order to measure the MWH. It was deployed from the International Space
Station in 2018 (Kaaret et al., 2019) and re-entered Earth’s atmosphere in
early 2021. HaloSat had a large field of view (FoV) of near 100 square degrees
and a moderate energy resolution of $<$ 100 eV below 1 keV (Kaaret et al.,
2019; Zajczyk et al., 2020; LaRocca et al., 2020). It was non-imaging and had
no X-ray focusing optics on board. The large FoV and good soft X-ray
sensitivity of HaloSat enable us to study diffuse soft X-ray sources with
large angular extent such as the SXDB.

In this paper, we analyze HaloSat data to investigate whether UHTPGD emission
is found across the extent of the Galactic disk. We selected five HaloSat
fields located in the outer Galactic disk that avoid bright X-ray sources
enabling analysis of the diffuse emission. The rest of the paper is organized
as follows. Section 2 presents our field selection procedure and data
reduction. Section 3, describes our spectral analysis methods and results for
the HaloSat data. In Section 4, we discuss the origin of the detected high
temperature plasma and its relation to the spectra of point-like sources
measured with XMM-Newton. Finally, we summarize our results in Section 5.

Table 1: HaloSat and XMM-Newton observations

Obs. ID | Field name | Aim point [deg.] | Exposure
---|---|---|---
|  |  (R.A.J2000, Dec.J2000) |  ($l$, $b$) | [ksec]
HaloSat
HS0017 | HaloSat J0530+3400 | (05 30 33.12, +34 00 18.0) | (174.00, 0.00) |  32∗
HS0019 | HaloSat J0320+5715 | (03 20 05.52, +57 15 43.2) | (142.00, 0.00) |  23∗
HS0021 | HaloSat J0759-2954 | (07 59 59.28, -29 54 14.4) | (247.00, 0.00) |  28∗
HS0022 | HaloSat J0729-1746 | (07 29 16.08, -17 46 51.6) | (233.00, 0.00) |  43∗
HS0023 | HaloSat J0700-0430 | (07 00 42.00, -04 30 32.4) | (218.00, 0.00) |  27∗
XMM-Newton
0500240101 | V838 Mon | (07 04 04.85, -03 50 51.1) | (217.80, 1.05) |  49†

∗ The effective observation exposure after data cleaning of the S14 detector.

† Net exposure time of the PN detector after the standard data screening.

Figure 1: Example of the HaloSat spectral fitting results for the simple model
(model0) consisting of the LHB (blue) and CXB (orange). The instrumental
background (gray) was modelled with as a powerlaw with a diagonal response
matrix. Black, red, and green crosses show the observed data (upper) and
residuals subtracting the model from the data divided by error bars (lower)
for each detector. For simplicity, the best fit models are shown only for one
detector.

##  2 Field selection and Data reduction

To investigate the X-ray spectral properties of wide-field areas in the
Galactic disk, we searched for HaloSat fields in outer Galactic disk
satisfying the following conditions.

  * •

aim point with $|b|<2^{\circ}$ and $|l|>50^{\circ}$,

  * •

exposure times $>20\rm\,ksec$ to have good photon statistics,

  * •

containing no X-ray bright objects in the $ROSAT$ all sky survey (Boller et
al., 2016) or the MAXI/SSC all sky catalog (Tomida et al., 2016)

  * •

containing no extended soft X-ray sources such as the Cygnus superbubble or
Vela supernova remnant (e.g., Bluem et al., 2020; Silich et al., 2020).

The final sample contains a total of 5 fields that are presented in Table 1.
Data for the five fields were downloaded from the HaloSat archive at the
HEASARC. Spectra were obtained for each of the three detectors following
standard screening on the hard (3–7 keV) and very large event ($>$ 7 keV)
count rates as described in the HaloSat analysis document (see details in
https://heasarc.gsfc.nasa.gov/docs/halosat/analysis/).

We used HEAsoft v6.28 to extract the spectra and XSPEC version 12.11 for the
spectral fitting. The response matrix and ancillary response to apply in the
spectral data and as well the response for the particle-induced instrumental
background were obtianed from the HaloSat calibration at the HEASARC
(https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/halosat/halosat_caldb.html).
Throughout the paper, if not otherwise specified, errors are quoted at the 90%
confidence level and the solar abundance table of Wilms et al. (2000) was
utilized.

(a) Model1 (Obs. ID: HS0017) (b) Model2 (Obs. ID: HS0017)
---
(a) Model1 (Obs. ID: HS0019)  (b) Model2 (Obs. ID: HS0019)
(a) Model1 (Obs. ID: HS0021) (b) Model2 (Obs. ID: HS0021)
Figure 2: Observed HaloSat spectra in the wide fields of the Galactic disk
fitted with Model1 (left) and Model2 (right) for the selected five fields.
Black, red, and green crosses show the observed data (upper) and residuals
subtracting the model from the data divided by error bars (lower) for each
detector. Blue and magenta, and orange solid lines correspond to the
unabsorbed and absorbed optically thin thermal CIE plasmas corresponding to
the LHB and MWH, and the absorbed non-thermal model known as the CXB,
respectively. Red and gray lines indicate the excess high temperature plasma
and instrumental background models, respectively. For simplicity, the best fit
models are shown only for one detector. (a) Model1 (Obs. ID: HS0022) (b)
Model2 (Obs. ID: HS0022)
---
(a) Model1 (Obs. ID: HS0023) (b) Model2 (Obs. ID: HS0023)
Figure 2: (Continued)

##  3 Analysis and results

We analyzed the spectra of the five HaloSat fields. Our spectral fitting
procedure is based on previous works modeling the SXDB (e.g., Mitsuishi et
al., 2013). Each of the five HaloSat fields were fitted independently. We fit
the spectra for all three detectors simultaneously using common parameters for
the astrophysical emission.

The instrumental background was modeled as a powerlaw and allowed to vary
between detectors (Silich et al., 2020; Kaaret et al., 2020). The instrumental
background photon index and 1-sigma error were calculated for each detector
for each observation based on the empirical relation described in
https://heasarc.gsfc.nasa.gov/docs/halosat/analysis/back20210209.pdf obtained
using actual data. In our fitting, the instrumental background normalization
was a free parameter for each detector while the photon index was allowed to
vary within the 1-sigma uncertainty from the empirical relation. In the models
below, the instrumental background component is denoted as
$powerlaw_{\mathrm{IB}}$.

As a first step, we adopted a simple model consisting of only the SWCX, LHB
and CXB components, i.e. assuming that the UHTP emission is negligible and the
MWH is completely blocked by the cold dense neutral material of the Galactic
disk. We used one unabsorbed optically thin thermal collisionally-ionized
equilibrium (CIE) plasma model for the sum of the SWCX and LHB emission and an
absorbed powerlaw model for the CXB. We refer to this as Model0 which is
defined in XSPEC as $apec_{\mathrm{SWCX+LHB}}+tbabs_{\mathrm{Galactic}}\times
powerlaw_{\mathrm{CXB}}+powerlaw_{\mathrm{IB}}$, where
$tbabs_{\mathrm{Galactic}}$ gives the absorption along the line-of-sight
direction which is mainly associated with the Galactic disk. In this model,
the emission measure of the thermal plasma and the Galactic absorption column
density were free parameters. We fixed the apec redshift to 0, the apec
abundance to 1, and the powerlaw photon index to 1.4 (Kushino et al., 2002).
The plasma temperature was limited to the range $0.097\pm 0.014$ keV and the
surface brightness of the powerlaw was limited to $(6.38\pm 0.64)\times
10^{-8}\rm\,erg\,cm^{-2}\,s^{-1}\,sr^{-1}$ based on previous studies (Kushino
et al., 2002; Liu et al., 2017).

Fits to this model show significant residuals (see Fig. 1) for all fields
except HS0019, especially around 1 keV corresponding to Fe L-shell complex.
This suggests that additional thermal components are needed as found in Masui
et al. (2009). Additionally, residuals in the 0.5–0.7 keV band, corresponding
to He- and H- like oxygen emission lines, were found in all of the fields
except HS0019, which supports the above scenario. There seems to be no
significant residuals for HS0019 because the spectrum below $\sim$1 keV is
relatively flat. However, the fitted column density for HS0019 and all the
other fields is significantly smaller ($\ll 10^{21}\rm\,cm^{-2}$) than the
expected values based on multiwavelength observations (Kalberla et al., 2005;
Zhu et al., 2017), which indicates that the spectrum needs an additional
component to make up for soft emission that should be absorbed. Thus, an
additional soft component is also needed for HS0019. Note that we considered a
heliospheric SWCX scenario for the observed residuals around both 0.5-0.7 and
0.8-1.1 keV by adding multiple gaussian models corresponding to oxygen, neon,
and iron emission lines as shown in Ringuette et al. (2021) and Huang et al.
(2023). We found that addition of SWCX emission can reduce the residuals.
However the absorption column density remains much lower than the predicted
values, suggesting the need for contributions from an additional spectral
component. We also considered non-solar abundances for the LHB component as
discussed in McCammon et al. (2002) and reached the same conclusions.The
normalization of the instrumental background show typical values and no
significant difference is seen in the three detectors.

Therefore, as a next step, we added an unabsorbed thermal plasma component for
the UHTPGD emission, that we denote as $apec_{\mathrm{UHTPGD}}$. We denote
this as Model1 which is defined as:
$apec_{\mathrm{SWCX+LHB}}+tbabs_{\mathrm{Galactic}}\times
powerlaw_{\mathrm{CXB}}+apec_{\mathrm{UHTPGD}}+powerlaw_{\mathrm{IB}}$. This
model assumes that all of the hot gas associated with the MWH is blocked
completely by the dense neutral material as is the case with Masui et al.
(2009). Model1 provides a significantly better fit compared to Model0
including only the LHB and CXB components because the residuals around 1 keV
were well fitted by the additional thermal plasma component. The UHTPGD
components have temperatures of 0.7–0.8 keV and emission measure in the range
(8–15)$\times$10-4 cm-6 pc. The fitted column density became much larger, in
the range (0.1–0.7)$\times 10^{22}$ [cm-2], which is almost consistent with
the expected values for each field. The UHTPGD temperature is consistent with
that in Masui et al. (2009) and interestingly, the surface brightness values
are consistent with each other within the statistical errors even though the
surface brightness in Masui et al. (2009) is calculated from only one pointed
observation of Suzaku with a FoV of 18′$\times$18′. However, residuals around
0.7 keV persist in several fields (HS0021, HS0022, and HS0023). The residuals
could alternatively be explained by an LHB component with higher temperature
near 0.2 keV if the temperature constraint was not applied. However, the
degree of the deviation in the temperature of the LHB in these large fields
seems unreasonable considering previous work (Kushino et al., 2002; Liu et
al., 2017). We confirmed that the UHTPGD parameters do not change
significantly if a high-temperature LHB is allowed. As another possibility, we
tried another fitting model by setting an oxygen abundance free. We confirmed
that the results of the temperature and emission measure of the UHTPGD are
consistent with each other within the statistical errors even though the
resultant parameters are physically unacceptable because the oxygen abundance
values are too high over 3. These results suggest that one more softer thermal
component emitting H-like oxygen emissions such as the hot gas associated with
the MHW is needed.

Table 2: The results of the model fitting of the five selected HaloSat fields.

Obs. ID | Model | $N_{\rm H}$ | $kT_{\rm{UHTPGD}}$ | EM${}^{\ast}_{\rm{UHTPGD}}$ |  $\chi^{2}/$d.o.f
---|---|---|---|---|---
[$\times 10^{22}~{}{\rm cm^{-2}}$] | [keV] | [$\times{\rm 10^{-4}~{}cm^{-6}~{}pc}$] |
HS0017 | Model 1 | 0.31${}^{+0.11}_{-0.08}$ | 0.77${}^{+0.05}_{-0.05}$ | 11${}^{+3}_{-2}$ | 452/399
Model 2 | 0.42${}^{+0.43}_{-0.12}$ | 0.84${}^{+0.61}_{-0.08}$ | 9.6${}^{+3.2}_{-5.9}$ | 445/397
HS0019 | Model 1 | 0.74${}^{+0.20}_{-0.18}$ | 0.84${}^{+0.11}_{-0.11}$ | 8.2${}^{+2.6}_{-2.6}$ | 336/316
Model 2 | 0.90${}^{+0.23}_{-0.18}$ | 1.3${}^{+0.6}_{-0.3}$ | 6.7${}^{+7.6}_{-4.3}$ | 321/314
HS0021 | Model 1 | 0.098${}^{+0.046}_{-0.038}$ | 0.67${}^{+0.06}_{-0.07}$ | 15${}^{+2}_{-2}$ | 624/493
Model 2 | 0.32${}^{+0.15}_{-0.09}$ | 0.95${}^{+0.31}_{-0.13}$ | 8.4${}^{+2.2}_{-2.3}$ | 494/491
HS0022 | Model 1 | 0.22${}^{+0.07}_{-0.06}$ | 0.77${}^{+0.04}_{-0.04}$ | 13${}^{+2}_{-2}$ | 790/702
Model 2 | 0.37${}^{+0.10}_{-0.08}$ | 0.88${}^{+0.08}_{-0.06}$ | 11${}^{+2}_{-2}$ | 747/700
HS0023 | Model 1 | 0.27${}^{+0.13}_{-0.10}$ | 0.75${}^{+0.06}_{-0.06}$ | 13${}^{+3}_{-3}$ | 599/509
Model 2 | 0.46${}^{+0.17}_{-0.12}$ | 0.88${}^{+0.27}_{-0.09}$ | 10${}^{+2}_{-5}$ | 569/507

$\ast$ Emission Measure integrated over the line of sight, i.e. $\rm{EM}=\int
n_{e}n_{H}dl$.

Thus, we added another thermal plasma component, but with absorption equal to
that for the CXB, corresponding to the MWH hot gas. The temperature was
limited to $0.225\pm 0.023$ keV, the abundance was fixed to 0.3 solar, and
redshift was fixed to 0, following the results obtained in high-latitude
regions (Kaaret et al., 2020). This model, hereafter Model2, is described in
XSPEC as:
$apec_{\mathrm{SWCX+LHB}}+tbabs_{\mathrm{Galactic}}\times(powerlaw_{\mathrm{CXB}}+apec_{\mathrm{MWH}})+apec_{\mathrm{UHTPGD}}+powerlaw_{\mathrm{IB}}$.

Fitting with Model2 decreases the residuals around 0.7 keV due to the addition
of the MHW hot gas. The goodness of fit improves significantly ($>$99.9$\%$ in
the F-test) for the fields, HS0021, HS0022, and HS0023, and somewhat
($<$99.9$\%$ in the F-test) for the fields, HS0017 and HS0019. The spectral
analysis results for Model1 and Model2 are summarized in Table 2 and the
spectral plots and residuals in Figure 2. As for the other models, the
normalization of the instrumental background has typical values and no
significant difference is seen in the three detectors.

The best fitted column density, temperature, and emission measure of the
UHTPGD components are (0.3–0.9)$\times 10^{22}$ [cm-2], 0.8–1.3 keV, and
(7–11)$\times$10-4 cm-6 pc, respectively. The best fit values of the column
density and temperature are systematically higher than those of Model1 and the
emission measure decreases as expected with the addition of the MWH component.
The emission measure of the LHB is consistent with those of Model1 within the
statistical errors except HS0021 and the MWH emission measure is consistent
amongst the fields within the statistical errors. For HS0017, all of the
results are consistent within the statistical errors between Model1 and
Model2. For HS0019, the nominal value of the column density increases
significantly but is consistent with the expected value.

In summary, all five HaloSat soft X-ray energy spectra show a possibility of
the presence of UHTPGD with a temperature of 0.8–1.0 keV in the best fit
models and the parameters are consistent with those found in a smaller field
with Suzaku (Masui et al., 2009). These results suggest that UHTPGD is
distributed across much of the Galactic disk.

##  4 A Stellar Origin?

To investigate the relation of the observed excess high temperature plasma to
point-like X-ray sources including stars, we analyzed XMM-Newton archival data
because XMM-Newton has relatively large FoV and effective area and good
angular resolution which allow us to study properties of both diffuse and
point-like sources. We searched the XMM-Newton archive for observations within
our HaloSat fields with long exposure times. We excluded fields with bright
diffuse sources such as clusters of galaxies and supernova remnants. We
selected the observation (ObsID: 0500240101) that lies within the HaloSat
field HS0023. We use only the PN data since the PN has the largest effective
area. We applied the standard data screening on, e.g., time filtering to
remove flaring high-background periods (see (e.g., Mitsuishi et al., 2013)),
resulting in a net exposure time after filtering of 49 ks, long enough to
achieve our scientific goals. Information on the observation is shown in Table
1.

First, we extracted a spectrum of the entire area of the observation with a
radius of 12.5 arcmin from the aim point for comparison with the HaloSat
spectra. Fitting with Model0, we found the same type of residuals, e.g.
significant residuals around Fe-L complex and H-like oxygen emission lines, as
for the HaloSat spectra. The absorption column density was fixed to be
$\sim$0.53$\times$1022 cm-2 based on the Galactic HI survey (Kalberla et al.,
2005).

Thus, we fitted the XMM-Newton spectrum with Model1 and Model2 and found that
the fit improves significantly from Model0 to Model1 and then from Model1 and
to Model2 similarly to the HaloSat results. The spectrum fitted with Model2 is
shown in Fig. 4. We confirm the presence of high temperature plasma with a
temperature of 1.2 keV and emission measure of 12$\times$10-4 cm-6 pc in
Model2. The observed temperature in Model2 is consistent with that for HS0023.
The emission measure in Model2 is also consistent, even though the FoV of XMM-
Newton is approximately two orders of magnitude smaller than that of HaloSat.

To help understand the contribution of resolved point-like sources to the
observed high temperature plasma, we investigated a stacked spectrum made of
197 point-like source spectra detected in the XMM-Newton observation as
reported in the catalog (e.g., Webb et al., 2020). Source and background
spectra were found each point-like source in the catalog. The source regions
are circles with radius of 20 arcseconds while the background regions are
annuli with radii of 45 and 60 arcseconds, both are centered on the source
coordinates tabulated in the catalog. All of the source spectra were summed,
the same done for the background, and finally the stacked background spectrum
was subtracted from the stacked point-like sources spectrum.

As our first step in analysis of the stacked spectrum, we confirmed that the
spectrum was not represented well only by the non-thermal model corresponding
to the CXB because of large residuals below $\sim$1 keV including highly-
ionized oxygen emission lines and the Fe-L complex were found, suggesting that
an additional thermal plasma(s) is needed. We then added unabsorbed optically
thin thermal CIE plasma models until no significant (99.9%) improvement in the
goodness of fit was observed. The preferred model includes two plasma
components and the CXB.

The spectrum with the best fit model and the fitting results are indicated in
Figure 4 left and Table 3. The plasma temperatures are low ($\sim$0.3 keV) and
high ($\sim$1 keV). The latter temperature is consistent with that found for
the full observation area while the emission measure is $\sim$50 %. This
suggests that the UHTPGD partly originates from point-like sources. The
spectral shape of the low-temperature plasma is similar to that of the MWH.
Thus, the contribution of the plasma originated from the point-like sources is
included in the emission measure of the MWH of the entire FoV spectral
analysis. We confirmed the intensity of the CXB decreases as expected
considering the contribution of unresolved point sources.

To further study the contribution from stars to UHTPGD as discussed in Masui
et al. (2009), we used the Two Micron All Sky Survey (2MASS) Point Source
Catalog (PSC) (Skrutskie et al., 2006) because most of the 2MASS point sources
can be considered to be stars (Cambrésy et al., 2001). Approximately 60% of
the X-ray point-like sources were cross-matched with sources tabulated in the
2MASS PSC with a matching radius of $5\arcsec$. The cross-matched sources have
relatively high X-ray fluxes compared to the whole set of point-like X-ray
sources in the field. For example, of the 20 brightest X-ray point-like
sources tabulated in the catalog, 18 are cross matched in the 2MASS PSC. This
may suggest that the observed UHTPGD is mainly due to the contribution of
stars.

We extracted the stacked spectrum of X-ray point-like sources cross-matched
with the 2MASS PSC and conducted spectral analysis with the same model as
described above for the full set of X-ray point-like sources. There are no
significant differences between the temperature and emission measure of the
two plasmas in the stacked spectra of all versus the NIR-identified point-like
sources (see Figure 3 bottom and Table 3), suggesting that the contribution
from stars to the UHTPGD in the entire FoV of the pointed observation is
significant.

Figure 3: 0.4–5.0 keV XMM-Newton PN image with X-ray point-like sources (green
circles). The cross-matched X-ray point-like sources with 2MASS (filled orange
circle) is also shown. Spectral analysis was conducted within a circle with a
radius of 12.5 arcmin (magenta).
---
Figure 4: Spectrum of an XMM-Newton observation obtained from the entire area
(top) and stacked spectra of all of the X-ray point-like sources (bottom left)
and cross-matched X-ray point sources with 2MASS point sources (bottom right).
Top: blue, magenta, red, and orange lines correspond to the SWCX+LHB, MWH,
UHTPGD and the CXB, respectively. Bottom: blue, red, and orange lines show the
low- and high-temperature plasmas and the CXB, respectively. Table 3: The
results of the model fitting for a pointed XMM-Newton observation.

| $kT_{\rm{low}}$ |  EM${}^{\ast}_{\mathrm{low}}$ | $kT_{\rm{high}}$ |  EM${}^{\ast}_{\rm{high}}$ |  $\chi^{2}/$d.o.f
---|---|---|---|---|---
[keV] |  [$\times{\rm 10^{-4}~{}cm^{-6}~{}pc}$] | [keV] |  [$\times{\rm 10^{-4}~{}cm^{-6}~{}pc}$] |
FoV |  |  |  1.2${}^{+0.1}_{-0.1}$ |  12${}^{+4}_{-3}$ | 366/317
All point source |  0.28${}^{+0.03}_{-0.03}$ |  5.8${}^{+0.8}_{-0.9}$ |  0.99${}^{+0.08}_{-0.07}$ |  5.8${}^{+0.8}_{-0.8}$ | 152/145
2MASS point source |  0.31${}^{+0.03}_{-0.03}$ |  4.9${}^{+0.6}_{-0.6}$ |  1.0${}^{+0.1}_{-0.1}$ |  5.8${}^{+0.6}_{-0.6}$ | 130/87

$\ast$ Emission Measure integrated over the line of sight, i.e. $\rm{EM}=\int
n_{e}n_{H}dl$.

##  5 Summary and Conclusions

The composition of the SXDB in the Galactic disk is not well known even though
it is essential to our understanding of Galactic structure. The $ROSAT$ all
sky survey revealed excess emission in the R45 band ($\sim$0.44–1.0 keV)
widely distributed in the Galactic disk. Masui et al. (2009) showed that this
excess has an energy spectrum consistent with UHTPGD with a temperature of 0.8
keV in one field in the Galactic disk. Using HaloSat, we investigated the soft
X-ray energy spectra of five fields in the outer Galactic disk (135∘ $<$ $l$
$<$ 254∘) selected to exclude bright X-ray sources. Our analysis shows that
all of the spectra require a thermal plasma component with temperature in the
range of 0.8–1.0 keV. Thus, for the first time, we present evidence that
UHTPGD is pervasive in the outer Galactic disk. The observed temperatures are
not significantly different from the UHTPGD found previously in one field
(Masui et al., 2009), while the emission measure of the UHTPGD observed with
HaloSat varies from (8–11)$\times$10-4 cm-6 pc and the value is different by a
factor of $\sim$1.3. The spectrum of the total area of an XMM-Newton
observation contained within one HaloSat field shows UHTPGD temperature and
emission measure consistent with found with HaloSat.

To determine whether the observed UHTPGD may originate from stars, we examined
the same XMM-Newton observation and constructed stacked spectra of all X-ray
point-like sources detected in the FoV and of the X-ray sources with infrared
counterparts in the 2MASS catalog which are predominantly stars. The stacked
spectra are well fitted with a model including two thermal plasma components
and the CXB. There are no significant differences between the fit results in
the two thermal plasma components for the different stacked spectra. The
temperature of the highest temperature plasma component is consistent with
that observed in spectral analysis for the entire area while the emission
measure is $\sim$50 %. This suggests that the UHTPGD may partly originate from
point-like sources such as stars.

This study was financially supported by Grants-in-Aid for Scientific Research
(KAKENHI) of the Japanese Society for the Promotion of Science (JSPS, grant
Nos. 19H05609) and by NASA grant, 80NSSC22K0624, ”Final Archive of the HaloSat
Data”. This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France. This publication makes use of data products from the Two
Micron All Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis Center/California
Institute of Technology, funded by the National Aeronautics and Space
Administration and the National Science Foundation.

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