#  5-D thermal field theory, Einstein field equations and spontaneous symmetry
breaking

S. Ganesh111Corresponding author:
Email<EMAIL_ADDRESS>
###### Abstract

It has been shown previously, that the spatial thermal variation of a thermal
medium can be recast as a variation in the Euclidean metric. It is now
extended to temporal variations in temperature, for a non-relativistic thermal
bath, which remains in local thermal equilibrium. This is achieved by
examining the thermal field theory in a five-dimensional space-time-
temperature. The bulk thermodynamic quantity, namely the energy density, is
calculated for a neutral scalar field with a time-dependent Hamiltonian.
Furthermore, the concept of recasting thermal variations as a variation in the
metric is extended to thermal systems in a gravitational field. The Einstein
field equations, in the 5-D space-time-temperature, is determined. It is shown
that the resulting Ricci scalar can then lead to spontaneous symmetry
breaking, leading to the Higgs mechanism. In essence, the asymmetry in the
distribution of temperature in space-time can translate to spontaneous
symmetry breaking of particle fields, in a very strong gravitational field.

Keywords  : Thermal gradient, 5-D Thermal field theory, Einstein field
equation, Gravity, Spontaneous symmetry breaking, Higgs
PACS numbers  : 11.10.Wx, 04.50.Kd, 11.15.Ex

##  I Introduction

The thermal field theory incorporates a Euclidean space-time, obtained by an
analytical continuation of time in Minkowski space-time to an imaginary time,
to model thermal systems matsubara ; martin ; gorkov ; adas . Numerous
research has been done using the framework of thermal field theory in some
form. Some examples include Refs. misc1 ; misc2 ; Kraemmer ; Kraemmer2 ; misc3
; misc7 ; misc4 ; misc5 ; misc6 ; tft ; epja . There are, however thermal
systems, where there are significant thermal variations in both spatial and
temporal dimensions, e.g., the Quark Gluon Plasma (QGP) nature ; naturephy .

The idea was first mooted in Ref. gans5 , that spatial thermal variations can
be modeled by recasting the variation in the temperature as a variation in the
metric. This idea was used to determine the quark anti-quark potential in a
thermal system with spatial variations, using AdS-CFT correspondence. The
thermal medium was assumed to be under local thermal equilibrium. In Ref.
gans6 , the concept that the spatial thermal variations can be recast as a
variation in the Euclidean metric, was placed on firmer grounds, by analyzing
the Polyakov loop, the partition function, the correlation function, and the
geodesic equation. It has been shown that the partition function for thermal
systems with spatial thermal variations, naturally leads to the notion of a
curved Euclidean space. While most of the analysis was carried out for a
canonical ensemble, the framework was also touched upon, for a grand canonical
ensemble. It is now shown, that the concept of the 5-dimensional space first
introduced in Ref. gans6 , enables the generalization of the concept to
temporal variations in temperature. The temporal variations must be
sufficiently slow to maintain local thermal equilibrium.

The 5-D space-time-temperature approach further enables the extension of the
concept to a thermal bath in a gravitational field. The Einstein field
equations are determined in the 5-D space. The resultant field equations are
then expressed in terms of the usual 4-D space-time covariant operators. This
process of dimension reduction leads to additional terms in the Ricci tensor,
and subsequently, the Ricci scalar.

There has been significant research on the source of spontaneous symmetry
breaking. Reference ssb1  has indicated that radiative corrections can be a
possible source. Reference ssb2  has explored the production of Higgs at the
Large Hadron Collider (LHC). A crucial outcome of the proposed framework is
that the modified Ricci scalar may be a source of spontaneous symmetry
breaking. A negative Ricci scalar can make the term, $(m^{2}+R)\phi^{2}$, in
the Lagrangian for a scalar field, become negative, i.e.,
$(m^{2}+R)\phi^{2}\rightarrow-\mu^{2}\phi^{2}$, for some real valued $\mu$.
Thus, a negative Ricci scalar curvature may be a viable candidate for the
negative potential necessary for spontaneous symmetry breaking for a scalar
field. The spontaneous symmetry breaking leads to the Higgs mechanism higgs1 ;
higgs2 ; goldstone .

For the sake of clarity, let us briefly revisit some of the relevant concepts
that were discussed in Ref. gans6 . An 8-D space, $\beta^{\mu}\times x^{\nu}$,
was considered to model thermal systems. Under conditions that the thermal
system is stationary, i.e., the four-velocity of the thermal medium,
$u^{\mu}=(1,0,0,0)$, it reduces to a 5-D space, $(i\beta,t,{\bf{x}})$.
Moreover, if the thermal system is time-invariant, it may suffice to consider
the 4-D sub-space, $(i\beta,{\bf{x}})$, leading to the imaginary time
formalism. If the temperature varies in space, with spatial variation
$s({\bf{x}})$, i.e., $\beta\equiv\beta({\bf{x}})=s({\bf{x}})\beta_{0}$, where
$\beta_{0}$ is a constant, then it leads to a curved Euclidean space, with the
spatial variation, $h_{00}=s({\bf{x}})^{2}$. A 5-D metric tensor along with
the $h_{00}$ term is shown in Eq. 1.

$G^{(5)}_{ab}=\left[\begin{array}[]{c c}h_{00}&0\\\ 0&g^{(4)}_{\mu\nu}\\\
\end{array}\right],$ (1)

where, $g^{(4)}_{\mu\nu}$, is the usual metric tensor in 4-D space-time,
resulting purely as a result of gravitational fields. The 5-D space
consideration was necessitated to allow separate metric components, $h_{00}$,
due to thermal variations, and $g_{00}$, due to gravity. Furthermore, the 5-D
space was again shown as a necessity, in order to model an external Dirac
spinor, with energy $E$, traversing a thermal medium. The 3-momentum
observable for the Dirac spinor, traversing a thermal bath, can be made
manifest only with a 5-D representation of the Dirac equation, instead of the
4-D imaginary time formalism. The conjugate momentum variables to time and
temperature, namely the energy, $E$, and the Matsubara frequency,
$\omega_{n}=\frac{C_{n}}{h_{00}}$, capture the intrinsic energy, $E$, of the
particle, and the interaction energy with the thermal medium,
$iE^{c}=\frac{C_{n}}{h_{00}}$, respectively. The curvature of Euclidean space
was considered in the complete absence of gravity.

In gans6 , the analysis was primarily for a time-invariant system. However,
since time and inverse temperature, $\beta$, are separate dimensions, it is
tempting, to extend the formalism to time-varying systems. We now develop the
formalism for time-varying systems. In addition, in the current work, the
curvature due to the effects of temperature variations and gravity is
encapsulated in a common framework. Furthermore, the criterion for the Ricci
scalar, $R$, to be negative, under the combined influence of the thermal
variations and gravity, is determined.

The rest of the paper is as follows. Section II, develops the formalism for a
5-D time-varying thermal system, that remains in local thermal equilibrium.
The expectation value of energy density, for a time-varying Hamiltonian, is
then determined in Sec. III. The 5-D Einstein field equations are developed in
Sec. IV. The criterion for spontaneous symmetry breaking is also developed in
this section. Finally, the results are summarized in Sec. V.

##  II The time variation in temperature

A time-varying thermal system would also have a four-velocity, $u^{\mu}$, of
the thermal medium, leading to an 8-D modeling as mentioned in gans6 .
However, in the non-relativistic limit, $u^{\mu}=(1,\bf{u})\approx(1,0,0,0)$,
the analysis can be approximated by an analysis in 5-D space itself. The 8-D
space can be divided into two 4-D Lorentz invariant sub-manifolds gans6 .
Thus, a 5-D sub-space may not lend itself to a full-fledged Lorentz invariant
treatment. A complete Lorentz invariant treatment would require the entire 8-D
space. It is planned to extend the current framework to a complete Lorentz
invariant treatment in 8-D space in the future.

In order to model partition function in a 5-D space-time-temperature, it would
be required to take care that the system is not acausal or non-local in
nature. This rules out an Hamiltonian of the form $H=\int\mathcal{H}dtd^{3}x$.
Instead, one may analyze the system in each time slice. A partition function
imbibes the statistical properties of the system. For a time-varying system,
it is possible to take time slices and define the statistical properties in
each time slice. The process of taking time slices, however, does not construe
that each time slice is modeled as a static system, with the effects of time
derivatives being ignored. The Hamiltonian would now be time-dependent, and in
general, be different from the Hamiltonian of a static system. The time-
dependent Hamiltonian should capture the non-trivial effects due to the time
variations. Consequently, we define the partition function at each time slice
for a time-varying system, which continues to be in local thermal equilibrium.
We now proceed on similar lines as Ref. gans6 , for each time slice, albeit
with a time-dependent Hamiltonian density.

Let us first develop the field theory for a non-interacting Lagrangian, in 5-D
space-time-temperature. The Lagrangian density in a 5-D space, for a neutral
scalar field would be,

$\mathcal{L}(\hat{\phi},\partial_{a}\hat{\phi})=\frac{1}{2}\left(\partial_{a}\hat{\phi}\partial^{a}\hat{\phi}-m^{2}\hat{\phi}^{2}\right),$
(2)

with the index $a=0,1,2,3,4$ corresponding to the dimensions $[\tau,t,x,y,z]$.
$\tau$ is the inverse temperature and varies from $0$ to $\beta$. The sign
convention used is (-,+,-,-,-). This gives rise to the equation of motion:

$\partial_{a}\partial^{a}\hat{\phi}+m^{2}\hat{\phi}=0$ (3)

The constraining 5 momentum delta function would be,
$\delta(E^{2}-\omega^{2}-{\bf{p}}^{2}-m^{2})$, and the 5-D integral measure
is:

$\frac{1}{\beta}\sum_{n}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{dE}{2\pi}2\pi\delta(E^{2}-\omega_{n}^{2}-{\bf{p}}^{2}-m^{2})\\\
=\frac{1}{\beta}\sum_{n}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2E},$ (4)

where, the Matsubara frequency $=\omega_{n}=\frac{2n\pi}{\beta}$, for a boson.
Let us look at the physical interpretation of the delta function,
$\delta(E^{2}-\omega_{n}^{2}-{\bf{p}}^{2}-m^{2})$. As mentioned in Ref. gans6
, $E$, may be considered the original intrinsic energy of a particle, and
$\omega_{n}=iE_{c}$, can be considered as the interaction energy of the
particle with the thermal medium. The variable, $\omega_{n}$, determines the
decay or enhancement of a particle wave-function with temperature (for
example, the Dirac spinor in Ref. gans6 ). The magnitude of the total energy
is then, $|E+iE_{c}|=\sqrt{E^{2}+E_{c}^{2}}=\sqrt{E^{2}-\omega_{n}^{2}}$. It
is intuitive, that a particle’s 3-momentum would be affected by both $E$ and
$E_{c}$, and not just $E$. Thus, one may consider
$E^{2}-\omega_{n}^{2}={\bf{p}}^{2}+m^{2}$. A portion of the particle’s
original energy, $E$, is lost due to interaction with the thermal medium. This
provides an intuition behind the delta function,
$\delta(E^{2}-\omega_{n}^{2}-{\bf{p}}^{2}-m^{2})$.

The operator for a neutral scalar field in 5-D space-time-temperature is,

$\hat{\phi}({\bf{x}},\tau,t)=\frac{1}{\beta}\sum_{n}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}\\\
\sum_{s}\left(a^{\dagger}_{{\bf{p}},\omega_{n}}e^{-ipx}e^{-i\omega_{n}\tau}+a_{{\bf{p}},\omega_{n}}e^{ipx}e^{i\omega_{n}\tau}\right)$
(5)

The operator, $a^{\dagger}_{{\bf{p}},\omega_{n}}$, creates a particle with
3-momentum ${\bf{p}}$, and Matsubara frequency $\omega_{n}$.

One may premise the below commutation relation:

$[a_{{\bf{p}}_{1},\omega_{n1}},a^{\dagger}_{{\bf{p}}_{2},\omega_{n2}}]=(2\pi)^{3}\delta^{3}({\bf{p}}_{1}-{\bf{p}}_{2})\zeta(\beta)\delta_{n1,n2},$
(6)

where, $\zeta(\beta)$ is a scalar normalization function, and needs to be
determined. Let us define,

$\displaystyle
a_{{\bf{p}}}(\tau)=\sum_{n}f(\omega_{n})a_{{\bf{p}},\omega_{n}},$
$\displaystyle
a^{\dagger}_{{\bf{p}}}(\tau)=\sum_{n}f^{*}(\omega_{n})a^{\dagger}_{{\bf{p}},\omega_{n}}.$
(7)

Equation II can be interpreted in the following way. When a momentum state
$|{\bf{p}}\rangle$ is created, then $|{\bf{p}}\rangle$ itself can be treated
as a superposition of the momentum-Matsubara eigenstates
$|{\bf{p}},\omega_{n}\rangle$, with probability amplitudes $f(\omega_{n})$.
Since $f(\omega_{n})$ is a probability amplitude,
$\sum_{n}f^{2}(\omega_{n})=1$. Then, the equal $\tau$ commutator,

$[a_{{\bf{p}}_{1}}(\tau),a^{\dagger}_{{\bf{p}}_{2}}(\tau)]\\\
=\sum_{n1}\sum_{n2}[a_{{\bf{p}}_{1},\omega_{n1}},a^{\dagger}_{{\bf{p}}_{2},\omega_{n2}}]f(\omega_{n1})f^{*}(\omega_{n2})\\\
=\sum_{n1}\sum_{n2}(2\pi)^{3}\delta^{3}({\bf{p}}_{1}-{\bf{p}}_{2})\zeta(\beta)\delta_{n1,n2}f(\omega_{n1})f^{*}(\omega_{n2})\\\
=\sum_{n1}(2\pi)^{3}\zeta(\beta)\delta^{3}({\bf{p}}_{1}-{\bf{p}}_{2})f^{2}(\omega_{n1}).$
(8)

Since $\sum_{n1}f^{2}(\omega_{n1})=1$, let us assign $\zeta(\beta)=1$, in Eq.
8, to obtain,

$[a_{{\bf{p}}_{1}}(\tau),a^{\dagger}_{{\bf{p}}_{2}}(\tau)]=(2\pi)^{3}\delta^{3}({\bf{p}}_{1}-{\bf{p}}_{2}).$
(9)

Thus, the usual commutation relation between the 3-momentum annihilation and
creation operator is recovered. For large $\beta$, i.e.,
$\beta\rightarrow\infty$, one may follow a similar procedure as above, but in
the continuous domain. The commutation relation in Eq. 6, can be seen to be,

$[a_{{\bf{p}}_{1},\omega_{n1}},a^{\dagger}_{{\bf{p}}_{2},\omega_{n2}}]=(2\pi)^{4}\delta^{3}({\bf{p}}_{1}-{\bf{p}}_{2})\delta(\omega_{n2}-\omega_{n1}),$
(10)

with $\zeta(\beta)=2\pi$. The relation between Eq. 6 and Eq. 10 can be
understood by noting that the 4-D space-temperature manifold is a
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ manifold. As
$\beta\rightarrow\infty$,
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}\rightarrow{\mathbb{R}}^{4}$.

We now proceed to determine the conjugate momenta and the Hamiltonian. There
can be a conjugate momenta w.r.t. either the time variable or the temperature
variable, i.e.,

$\hat{\pi}_{t}=\frac{\delta\mathcal{L}}{\delta\frac{\partial\hat{\phi}}{\partial
t}};~{}~{}~{}\hat{\pi}_{\beta}=i\frac{\delta\mathcal{L}}{\delta\frac{\partial\hat{\phi}}{\partial\tau}}.$
(11)

The corresponding Hamiltonian densities are:

$\mathcal{H}_{t}=\hat{\pi}_{t}\frac{\partial\hat{\phi}}{\partial
t}-\mathcal{L};~{}~{}\mathcal{H}_{\beta}=-i\hat{\pi}_{\beta}\frac{\partial\hat{\phi}}{\partial\tau}-\mathcal{L}.$
(12)

They would obey the evolution equations:

$i\frac{\partial\hat{\phi}}{\partial
t}=[\hat{\phi},H_{t}];~{}~{}~{}\frac{\partial\hat{\phi}}{\partial\tau}=[\hat{\phi},H_{\beta}].$
(13)

Since we are modeling a thermal system, and are interested in evolution in
$\tau$, the main object of interest would be $H_{\beta}$ and $\pi_{\beta}$.
For convenience, we now drop the subscript $\beta$. In the rest of the paper,
unless otherwise mentioned, $H$ and $\hat{\pi}$ refer to $H_{\beta}$ and
$\hat{\pi}_{\beta}$. We now follow a similar procedure as Ref gans6 ; kapusta
, albeit modified for a 5-D space with thermal variations.

Let $\phi({\bf{x}})$ and $|\phi({\bf{x}})\rangle$ be the eigenfunction and the
eigenket of the Schrodinger picture field operator $\hat{\phi}({\bf{x}},0,0)$,
while, $\pi({\bf{x}})$ and $|\pi({\bf{x}})\rangle$ be the eigenfunction and
the eigenket of the conjugate momentum field operator
$\hat{\pi}({\bf{x}},0,0)$. In other words,

$\displaystyle\hat{\phi}({\bf{x}},0,0)|\phi\rangle=\phi({\bf{x}})|\phi\rangle,$
$\displaystyle\hat{\pi}({\bf{x}},0,0)|\pi\rangle=\pi({\bf{x}})|\pi\rangle.$
(14)

The eigenkets, $|\phi\rangle$ and $|\pi\rangle$, obey the following relation:

$\langle\phi|\pi\rangle=\exp\left(i\int
d^{3}x\pi({\bf{x}})\phi({\bf{x}})\right).$ (15)

For a time-dependent system, the time-dependent Hamiltonian can be written as:

$H(t)=H_{0}+H_{I}(t),$ (16)

where $H_{0}$ is the time independent part, and $H_{I}(t)$ be the time
dependent part. As mentioned earlier, $H_{I}(t)$ should capture the non-
trivial effects of time variation in a time-varying system. $H_{0}$ is written
in terms of the Schrodinger picture operators $\hat{\pi}({\bf{x}},0,0)$ and
$\hat{\phi}({\bf{x}},0,0)$.

$H_{0}=\int
d^{3}x\mathcal{H}_{0}(\hat{\pi}({\bf{x}},0,0),\hat{\phi}({\bf{x}},0,0))\equiv\int
d^{3}x\mathcal{H}_{0}({\bf{x}}).$ (17)

For simplicity of notation, the form, $\mathcal{H}_{0}({\bf{x}})$, is used as
a representation of
$\mathcal{H}_{0}(\hat{\pi}({\bf{x}},0,0),\hat{\phi}({\bf{x}},0,0))$.

The free Hamiltonian density, corresponding to the Lagrangian in Eq. 2, would
then be,

$\mathcal{H}_{0}=\frac{1}{2}\left(\hat{\pi}^{2}+(\nabla\hat{\phi})^{2}-\left(\frac{\partial\hat{\phi}}{\partial
t}\right)^{2}+m^{2}\hat{\phi}^{2}\right).$ (18)

But, $\frac{\partial\hat{\phi}}{\partial t}=-iE\hat{\phi}$. In a gas composed
of scalar fields, which is equilibrated, $E\rightarrow 0$, as only the
ensemble interaction energy, captured by the Matsubara frequency,
$\omega_{n}$, is non zero gans6 . We are then left with the standard imaginary
time formalism. On similar lines, in a vacuum, as $\beta\rightarrow\infty$,
$\omega_{n}\rightarrow 0$. Then, the only energy left is the particle energy,
$E$. The Lagrangian in Eq. 2, then boils down to the normal 4-D space-time
Quantum Field Theory (QFT). When both $E$ and $\omega_{n}$ are non-zero, Eq. 2
can be used to model particles that are not yet fully equilibrated with the
thermal medium. A case in point is the external Dirac spinor traversing a
thermal medium, which was modeled in Ref. gans6 . The external Dirac Spinor
(not equilibrated with the thermal medium), will have its intrinsic energy,
$E$, as well as a non-zero $\omega_{n}$, due to interaction with the thermal
medium. With, $E\rightarrow 0$, the Hamiltonian density in Eq. 18 becomes,

$\mathcal{H}_{0}=\frac{1}{2}\left(\hat{\pi}^{2}+(\nabla\hat{\phi})^{2}+m^{2}\hat{\phi}^{2}\right).$
(19)

Equations 9 and 19, indicate that the generalizations to 5-D, characterized by
Eqs. 5, 6, 11, 12, are backward compatible with the existing 4-D imaginary
time formalism. A fairly generic time-dependent Hamiltonian density can be
written as:

$H_{I}(t)=\int\mathcal{H}_{I}({\bf{x}},t)d^{3}x,$ (20)

where,

$\mathcal{H}_{I}({\bf{x}},t)=\sum_{i}c_{i}({\bf{x}},t)f_{i}(\hat{\phi},\partial_{\mu}\hat{\phi},\hat{\pi},\partial_{\mu}\hat{\pi}),$
(21)

and $c_{i}({\bf{x}},t)$ are arbitrary scalar functions. However, variations in
$c_{i}({\bf{x}},t)$, should not be sharp enough to throw the system out of
local thermal equilibrium. In this paper, the thermodynamic properties are
evaluated for the below specific cases:

  1. 1.

$\mathcal{H}_{I}({\bf{x}},t)=V({\bf{x}},t)\hat{\phi}^{2}$,

  2. 2.

$\mathcal{H}_{I}({\bf{x}},t)=\lambda({\bf{x}},t)\hat{\phi}^{4}$.

In the case of a thermal bath composed of scalar particles,
$V({\bf{x}},t)\hat{\phi}^{2}$ can represent the coupling of an external field,
along with its derivatives, with $\hat{\phi}^{2}$.

The evolution operator $U(H_{\beta},\beta,t)$ provides the evolution w.r.t.
$\beta$, i.e.,

$U_{\beta}(H_{\beta},\beta,t)=\exp\left(-\int\int_{0}^{\beta({\bf{x}},t)}\mathcal{H}_{\beta}({\bf{x}},t)d\tau
d^{3}x\right).$ (22)

As mentioned earlier, we drop the subscript $\beta$ from $U$, $H$ and
$\mathcal{H}$, and obtain,

$U(H,\beta,t)=\exp\left(-\int\int_{0}^{\beta({\bf{x}},t)}\mathcal{H}({\bf{x}},t)d\tau
d^{3}x\right),\\\
=\exp\left(-\int\int_{0}^{\beta_{0}}s({\bf{x}},t)\mathcal{H}({\bf{x}},t)d\tau
d^{3}x\right),$ (23)

where,
$\mathcal{H}({\bf{x}},t)=\mathcal{H}_{0}({\bf{x}})+\mathcal{H}_{I}({\bf{x}},t)$.
The integrand in the exponent of Eq. 23, indicates a volume element $d\tau
d^{3}x$ of a 4-D slice, at time $t$, within the 5-D space, with metric,
$diag[-s({\bf{x}},t)^{2},1,-1,-1,-1]$, and determinant,
$\sqrt{g_{5}}=s({\bf{x}},t)$. Thus, it describes a curved 5-D space-time-
temperature.

We can then define the partition function $Z(t)$, at a time slice, $t$, in 5-D
space-time-temperature as:

$Z(\beta_{0},t)=tr\big{[}\langle\phi_{f}|U(\mathcal{H},\beta_{0},t)|\phi_{0}\rangle\big{]},$
(24)

with $|\phi_{0}\rangle$ and $|\phi_{f}\rangle$ being the eigenkets at $\tau=0$
and $\tau=\beta_{0}$ respectively.

The procedure to evaluate the partition function, $Z(\beta_{0},t)$, is now
straight forward and similar to Ref. gans6 . We first evaluate
$U(\mathcal{H},\beta,t)$. Let $\beta_{0}$ be sliced into $N$ slices, i.e.,
$\beta_{0}=N\Delta\beta$, with $N\rightarrow\infty$ and
$\Delta\beta\rightarrow 0$. This gives,

$U(\mathcal{H},\beta_{0},t)=\\\ \lim_{\Delta\beta\rightarrow
0}\exp\left(-\sum\limits_{n}\left[\int
s({\bf{x}},t)\mathcal{H}({\bf{x}},t)d^{3}x\right]\Delta\beta\right)$ (25)
$=\lim_{\Delta\beta\rightarrow 0}\prod\limits_{n}\exp\left(-\left[\int
s({\bf{x}},t)\mathcal{H}({\bf{x}},t)d^{3}x\right]\Delta\beta\right).$ (26)

Let $|\pi_{j}\rangle$ be the $\beta$ based conjugate momentum eigenstate.
Inserting a complete set of eigenstates:

$I=\int|\phi_{j}\rangle\langle\phi_{j}|d\phi_{j}\times\int|\pi_{j-1}\rangle\langle\pi_{j-1}|\frac{d\pi_{j-1}}{2\pi},$

between each product term in Eq. 26, and

$I=\int|\pi_{N-1}\rangle\langle\pi_{N-1}|\frac{d\pi_{N-1}}{2\pi};~{}~{}~{}I=\int|\phi_{1}\rangle\langle\phi_{1}|d\phi_{1},$

in the beginning and the end of the R.H.S. of Eq. 26, respectively, we get the
expression for
$K(\phi_{f},\phi_{0},\beta_{0},t)=\langle\phi_{f}|U(\mathcal{H},\beta_{0},t)|\phi_{0}\rangle$:

$K(\phi_{f},\phi_{0},\beta_{0},t)=\lim_{\Delta\beta\rightarrow
0}\int\prod\limits_{j=1}^{N-1}\langle\phi_{j+1}|\pi_{j}\rangle\\\
\times\langle\pi_{j}|\exp\left(-\int
s({\bf{x}},t)\mathcal{H}({\bf{x}},t)\Delta\beta
d^{3}x\right)|\phi_{j}\rangle\\\ \times\langle\phi_{1}|\phi_{0}\rangle
d\phi_{j}\frac{d\pi_{j}}{2\pi}.$ (27)

It is possible to evaluate the expression in Eq. 27, using the relations:

  * •

$\langle\phi_{j+1}|\pi_{j}\rangle=\exp\left(i\int
d^{3}x\pi_{j}({\bf{x}})\phi_{j+1}({\bf{x}})\right),$ (28)

  * •

$\langle\pi_{j}|\exp\left(-i\int
s({\bf{x}},t)\mathcal{H}d^{3}x\Delta\beta\right)|\phi_{j}\rangle\\\
=\langle\pi_{j}|\phi_{j}\rangle\exp\left(-i\int
s({\bf{x}},t)\mathcal{H}_{j}d^{3}x\Delta\beta\right),$ (29)

with,
$\mathcal{H}_{j}=\mathcal{H}(\pi_{j},\phi_{j},t)=\mathcal{H}_{0}(\pi_{j},\phi_{j})+\mathcal{H}_{I}(\pi_{j},\phi_{j},t)$,

  * •

and

$\langle\phi_{1}|\phi_{0}\rangle=\prod_{{\bf{x}}}\delta\left(\phi_{1}({\bf{x}})-\phi_{0}({\bf{x}})\right).$
(30)

It is to be noted that the form of $\mathcal{H}_{I}({\bf{x}},t)$, used in Eq.
21, would satisfy the relation specified in Eq. 29. This is because, the
operator arguments of $\mathcal{H}_{I}$, namely, $\hat{\phi}({\bf{x}})$ and
$\hat{\pi}({\bf{x}})$, continue to be Schrodinger picture operators, and the
temporal aspect is captured by the scalar functions, $c_{i}({\bf{x}},t)$. If a
different form of $\mathcal{H}_{I}({\bf{x}},t)$ is used, then Eq. 29 needs to
be checked for validity. Inserting Eqs. 28, 29 and 30 in Eq. 27, we obtain,

$K(\phi_{f},\phi_{0},\beta_{0},t)=\lim_{\Delta\beta\rightarrow
0}\int\prod\limits_{j}d\phi_{j}\frac{d\pi_{j}}{2\pi}\exp\Bigg{\\{}\int
d^{3}x\\\
\times\Big{[}-i\pi_{j}({\bf{x}})\Big{\\{}\phi_{j+1}({\bf{x}})-\phi_{j}({\bf{x}})\Big{\\}}-s({\bf{x}},t)\mathcal{H}_{j}\Delta\beta\Big{]}\Bigg{\\}}$
(31) $=\lim_{\Delta\beta\rightarrow
0}\int\left(\prod\limits_{j}d\phi_{j}\frac{d\pi_{j}}{2\pi}\right)\exp\Bigg{\\{}\sum_{k}\Big{(}\Delta\beta\int
d^{3}x\\\ \times
s({\bf{x}},t)\left[-i\pi_{k}({\bf{x}})\frac{\left\\{\phi_{k+1}({\bf{x}})-\phi_{k}({\bf{x}})\right\\}}{s({\bf{x}},t)\Delta\beta}-\mathcal{H}_{k}\right]\Big{)}\Bigg{\\}}.$
(32)

We recollect that, $s({\bf{x}},t)=\sqrt{h_{00}({\bf{x}},t)}$ (Eq. 1). Further,
for the metric, $G^{5}=diag[-s({\bf{x}},t)^{2},1,-1,-1,-1]$, we have,
$\sqrt{|G^{5}|}=s({\bf{x}},t)$. We denote the determinant of the Euclidean
metric, $|G^{5}|$, by $g_{5}$, and obtain in the continuum limit:

$K(\phi_{f},\phi_{0},\beta_{0},t)=\\\ \int
D\phi\int\frac{D\pi}{2\pi}\exp\Bigg{\\{}\int_{0}^{\beta_{0}}d\tau\int
d^{3}x\sqrt{g_{5}({\bf{x}},t)}\\\
\times\Big{[}-i\pi({\bf{x}},\tau)\frac{1}{\sqrt{h_{00}({\bf{x}},t)}}D_{\tau}\phi({\bf{x}},\tau)\\\
-\mathcal{H}(\pi({\bf{x}},\tau),\phi({\bf{x}},\tau),t)\Big{]}\Bigg{\\}},$ (33)

where, $D_{\tau}$ is the covariant derivative w.r.t. $\tau$. For a scalar
$\phi({\bf{x}},\tau)$,
$D_{\tau}\phi({\bf{x}},\tau)=\frac{\partial\phi({\bf{x}},\tau)}{\partial\tau}$.
Finally, the partition function becomes,

$Z(\beta_{0},t)=trK(\phi_{f},\phi_{0},\beta_{0},t)=\\\
\int_{periodic}D\phi\int\frac{D\pi}{2\pi}\exp\Bigg{\\{}\int_{0}^{\beta_{0}}d\tau\int
d^{3}x\sqrt{g_{5}({\bf{x}},t)}\\\
\times\Big{[}i\pi({\bf{x}},\tau)\frac{1}{\sqrt{h_{00}({\bf{x}},t)}}D_{\tau}\phi({\bf{x}},\tau)\\\
-\mathcal{H}(\pi({\bf{x}},\tau),\phi({\bf{x}},\tau),t)\Big{]}\Bigg{\\}}.$ (34)

##  III Evaluation of $\langle E(t)\rangle$

In this section, the expectation value of the energy density at different
instances of time, $\langle E(t)\rangle$, is evaluated for
$\mathcal{H}_{I}=V(t)\phi^{2}$, and $\mathcal{H}_{I}=\lambda(t)\phi^{4}$.

###  III.1 $\langle E(t)\rangle$ when $\mathcal{H}_{I}=V(t)\phi^{2}$

Since only $\phi^{2}$ is involved in the partition function, this is easily
evaluated as Gaussian integrals. Using the procedure outlined in gans6 , the
expression for the energy expectation value, $\langle E(t)\rangle$ is:

$\langle
E(t)\rangle=-\frac{\partial\ln(Z)}{\partial\beta_{0}}={\mathbb{V}}\int\frac{d^{3}p}{(2\pi)^{3}}\Bigg{[}\frac{\omega_{p}}{2}\coth\left(\frac{\beta_{0}\omega_{p}}{2}\right)\\\
+\left(\langle\eta({\bf{x}},t)\rangle+\frac{1}{2\omega_{p}^{2}}\langle
s({\bf{x}},t)V({\bf{x}},t)\rangle\right)\\\
\times\Big{\\{}\frac{\omega_{p}}{2}\coth\left(\frac{\beta_{0}\omega_{p}}{2}\right)+\frac{\beta_{0}\omega_{p}^{2}}{4}\operatorname{cosech}^{2}\left(\frac{\beta_{0}\omega_{p}}{2}\right)\Big{\\}}\Bigg{]},$
(35)

where, $\langle s({\bf{x}},t)V(t)\rangle=\frac{\int
d^{3}xs({\bf{x}},t)V(t)}{\int d^{3}x}$, $s({\bf{x}},t)=1+\eta({\bf{x}},t)$,
$\langle\eta({\bf{x}},t)\rangle=\frac{\int d^{3}x\eta({\bf{x}},t)}{\int
d^{3}x}$ and $\omega_{p}=\sqrt{{\bf{p}}^{2}+m^{2}}$.

###  III.2 $\langle E(t)\rangle$ when
$\mathcal{H}_{I}=\lambda({\bf{x}},t)\phi^{4}$

The Lagrangian density for a neutral scalar field would now read:

$\mathcal{L}=\frac{1}{2}\left(\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}-m^{2}\hat{\phi}^{2}\right)-\lambda({\bf{x}},t)\hat{\phi}^{4}.$
(36)

The corresponding interaction Hamiltonian is,
$H_{I}=\lambda({\bf{x}},t)\hat{\phi}^{4}$. For small $\lambda({\bf{x}},t)$,
the partition function can be written as:

$\displaystyle Z(\beta_{0},t)=\int_{periodic}D\phi\int\frac{D\pi}{2\pi}$
$\displaystyle\times\left(1+\sum_{r}\frac{1}{r!}\left[\int d\tau
d^{3}x\sqrt{g_{5}}\lambda({\bf{x}},t)\phi^{4}\right]^{r}\right)$
$\displaystyle\times\exp\Bigg{\\{}\int_{0}^{\beta_{0}}d\tau\int
d^{3}x\sqrt{g_{5}}\Big{[}-i\pi({\bf{x}},\tau)\frac{1}{\sqrt{h_{00}({\bf{x}})}}\frac{\partial\phi({\bf{x}},\tau)}{\partial\tau}$
$\displaystyle-\frac{1}{2}\left(\pi^{2}+(\nabla\phi({\bf{x}},\tau))^{2}+m^{2}\phi({\bf{x}},\tau)^{2}\right)\Big{]}\Bigg{\\}}.$

Then, one may represent the above equation as:

$\ln
Z(\beta_{0},t)=\ln(Z_{0}(\beta_{0},t))+\ln\left[1+\sum_{r}\frac{1}{r!}\frac{Z_{I}^{r}(\beta_{0},t)}{Z_{0}(\beta_{0},t)}\right].$
(38)

The calculation of $\ln(Z_{0}(\beta_{0},t))$, at time slice $t$, is identical
to the calculation of $\ln(Z_{0}(\beta_{0}))$, in Ref. gans6 .

Figure 1: Feynman diagram depicting the first order $\hat{\phi}^{4}$
interaction.

For $r=1$, the expression for $Z^{1}_{I}$, corresponding to the Feynman
diagram in Fig. 1 is:

$\displaystyle
Z_{I}^{1}(\beta_{0},t)=\int_{periodic}D\phi\int\frac{D\pi}{2\pi}\left(\int
d\tau d^{3}x\sqrt{g_{5}}\lambda({\bf{x}},t)\phi^{4}\right)$
$\displaystyle\times\exp\Bigg{\\{}\int_{0}^{\beta_{0}}d\tau\int
d^{3}x\sqrt{g_{5}}\Big{[}-i\pi({\bf{x}},\tau)\frac{1}{\sqrt{h_{00}({\bf{x}})}}\frac{\partial\phi({\bf{x}},\tau)}{\partial\tau}$
$\displaystyle-\frac{1}{2}\left(\pi^{2}+(\nabla\phi({\bf{x}},\tau))^{2}+m^{2}\phi({\bf{x}},\tau)^{2}\right)\Big{]}\Bigg{\\}}.$

The factor $\frac{Z_{I}^{1}}{Z_{0}}$, is now calculated. The $\int D\pi$
integrals can be done in exactly the same manner as the non-interacting case.
We consider a space of volume ${\mathbb{V}}$, discretize it, and divide it
into $M^{3}$ cubes of length $\Delta a$. Further, express $\phi({\bf{x}})$ in
terms of its momentum domain representation, $\phi_{n,{\bf{p}}}$ gans6 ,

$\phi({\bf{x}},\tau)=\sqrt{\frac{\beta_{0}}{{\mathbb{V}}}}\sum_{i}\sum_{n=-\infty}^{\infty}e^{iC_{n}\tau}e^{i{\bf{p}}_{i}.{\bf{x}}}\phi_{n,{\bf{p}}_{i}},$
(40)

where, $C_{n}$ is related to $\omega_{n}$. We follow a procedure similar to
the non-interacting case in Ref gans6 . Then, after replacing
$\left(\frac{\Delta p}{2\pi}\right)^{3}$ as ${\mathbb{V}}$ in the exponent,
one may obtain $Z_{I}^{1}$ as:

$Z_{I}^{1}=\Bigg{[}\mathcal{N}\left(\frac{1}{2\pi}\right)^{NM^{3}/2}\Bigg{]}\\\
\times\int D\phi_{n,{\bf{p}}}\left(\int d\tau
d^{3}xs({\bf{x}},t)\lambda({\bf{x}},t)\phi({\bf{x}})^{4}\right)\\\
\times\exp\Bigg{\\{}\frac{-1}{2}\sum_{i}\sum_{j}\sum_{n=-\infty}^{\infty}\phi_{n,{\bf{p}}_{j}}^{*}\Bigg{[}\beta_{0}^{2}\Big{\\{}(C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2})\delta_{ij}\\\
+\eta_{f}({\bf{p}}_{j}-{\bf{p}}_{i},t)\frac{1}{{\mathbb{V}}}(-C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2})\Big{\\}}\Bigg{]}\phi_{n,{\bf{p}}_{i}}\Bigg{\\}},$
(41)

where, we have used $\sqrt{g_{5}}=s({\bf{x}},t)$, and $\mathcal{N}$ is an
inconsequential multiplication factor and does not affect the dynamics of the
system kapusta ; Nbeta . The length, $\Delta a$, gets absorbed in
$\mathcal{N}$. The variable, $\eta_{f}({\bf{p}},t)$, is the 3-D Fourier
transform of $\eta({\bf{x}},t)$. Based on Eq. 40, $\phi({\bf{x}})^{4}$ can be
represented in terms of its Fourier components.

$\displaystyle\int d\tau
d^{3}xs({\bf{x}},t)\lambda({\bf{x}},t)\phi^{4}({\bf{x}})=\left(\sqrt{\frac{\beta_{0}}{{\mathbb{V}}}}\right)^{4}$
$\displaystyle\times\sum_{{\bf{p}}_{u},n_{u}}\sum_{{\bf{p}}_{v},n_{v}}\sum_{{\bf{p}}_{k},n_{k}}\sum_{{\bf{p}}_{l},n_{l}}\beta_{0}\delta\left(C_{n_{u}}+C_{n_{v}}+C_{n_{k}}+C_{n_{l}}\right)$
$\displaystyle\times s_{\lambda
f}(-{\bf{p}}_{u}-{\bf{p}}_{v}-{\bf{p}}_{k}-{\bf{p}}_{l},t)\phi_{n_{u},{\bf{p}}_{u}}\phi_{n_{v},{\bf{p}}_{v}}\phi_{n_{k},{\bf{p}}_{k}}\phi_{n_{l},{\bf{p}}_{l}},$

where, $s_{\lambda f}({\bf{p}},t)$ is the momentum domain representation of
the product term, $s({\bf{x}},t)\lambda({\bf{x}},t)$. The delta function
forces, $n_{u}=-n_{v}=n_{1}(say)$, and $n_{k}=-n_{l}=n_{2}(say)$. Secondly, we
make the transformation ${\bf{p}}_{v}\rightarrow-{\bf{p}}_{v}$ and
${\bf{p}}_{l}\rightarrow-{\bf{p}}_{l}$. With these, Eq. III.2, becomes

$\int d\tau
d^{3}xs({\bf{x}},t)\phi^{4}({\bf{x}})=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\sum_{n1}\sum_{n2}\sum_{u}\sum_{v}\sum_{k}\sum_{l}\\\
\left\\{s_{\lambda,f}\left({\bf{p}}_{v}-{\bf{p}}_{u}+{\bf{p}}_{l}-{\bf{p}}_{k},t\right)\phi^{*}_{n1,{\bf{p}}_{v}}\phi_{n1,{\bf{p}}_{u}}\phi^{*}_{n2,{\bf{p}}_{l}}\phi_{n2,{\bf{p}}_{k}}\right\\}.\\\
$ (43)

Let the exponent enclosed between $\phi_{n,{\bf{p}}_{j}}^{*}$ and
$\phi_{n,{\bf{p}}_{i}}$ in Eq. 41 be $M_{n}$, i.e.,

$\displaystyle
M_{n}=\Bigg{[}\beta_{0}^{2}\Big{\\{}(C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2})\delta_{ij}$
$\displaystyle+\eta_{f}({\bf{p}}_{j}-{\bf{p}}_{i},t)`\frac{1}{{\mathbb{V}}}(-C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2})\Big{\\}}\Bigg{]}.$
(44)

Let $\Lambda_{n}$ diagonalize $M_{n}$. Then
$M_{n}=\Lambda_{n}^{-1}D_{n}\Lambda_{n}$, where $D_{n}$ is a diagonal matrix.
Let $\psi_{n,a}=\sum_{i}\Lambda_{n,a,i}\phi_{n,{\bf{p}}_{i}}$. This means
$\phi_{n,{\bf{p}}_{i}}=\sum_{a}\Lambda_{n,i,a}^{-1}\psi_{n,a}$.

Inserting the above transformation of $\phi$ in Eq. 43, and making use of the
fact that
$(\Lambda^{-1}_{n,i,a}\psi_{n,a})^{*}=\psi_{n,a}^{*}\Lambda^{t}_{n,i,a}$,

$\int d\tau d^{3}x\,s({\bf{x}},t)\lambda({\bf{x}},t)\phi^{4}({\bf{x}})\\\
=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\sum_{n1}\sum_{n2}\sum_{b}\sum_{a}\sum_{d}\sum_{c}\psi^{*}_{n1,b}\psi^{*}_{n2,d}\\\
\times\Big{[}\sum_{v}\sum_{u}\sum_{l}\sum_{k}\Lambda_{n1,b,v}\Lambda_{n2,d,l}\\\
\times
s_{\lambda,f}({\bf{p}}_{v}-{\bf{p}}_{u}+{\bf{p}}_{l}-{\bf{p}}_{k},t)\Lambda^{-1}_{n1,u,a}\Lambda^{-1}_{n2,k,c}\Big{]}\psi_{n1,a}\psi_{n2,c}.\\\
$ (45)

Or,

$\int d\tau d^{3}x\,s({\bf{x}},t)\lambda({\bf{x}},t)\phi^{4}({\bf{x}})\\\
=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\sum_{n1}\sum_{n2}\sum_{b}\sum_{a}\sum_{d}\sum_{c}\\\
\times\Big{(}\psi^{*}_{n1,b}\psi^{*}_{n2,d}\Lambda^{s}_{bdacn_{1}n_{2}}\psi_{n1,a}\psi_{n2,c}\Big{)}.$
(46)

Equation 46 is same as Eq. 45, but with a more compact notation, by
introducing, $\Lambda^{s}_{bdacn_{1}n_{2}}$, for the term enclosed within the
$[\,]$ brackets.

Also, we have,

$\sum_{i}\sum_{j}\phi^{*}_{n,{\bf{p}}_{j}}M_{n,j,i}\phi_{n,{\bf{p}}_{i}}=\sum_{l}\psi^{*}_{n,l}D_{n,l,l}\psi_{n,l}.$
(47)

Since the transformation, $\phi\rightarrow\psi$, is unitary, we can replace
the measure $\int D\phi$ by $\int D\psi$. Substituting Eq, 46, and Eq. 47, in
Eq. 41, we obtain:

$\displaystyle
Z_{I}^{1}=\lambda(t)\Bigg{[}\mathcal{N}\left(\frac{1}{2\pi}\right)^{NM^{3}/2}\Bigg{]}$
$\displaystyle\times\sum_{n1}\sum_{n2}\sum_{b}\sum_{a}\sum_{d}\sum_{c}\left(3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\right)$
$\displaystyle\times\int
D\psi_{n,i}\left(\psi^{*}_{n1,b}\psi^{*}_{n2,d}\Lambda^{s}_{bdacn_{1}n_{2}}\psi_{n1,a}\psi_{n2,c}\right)$
$\displaystyle\times\exp\Bigg{\\{}-\frac{1}{2}\sum_{i}\sum_{n=-\infty}^{\infty}\psi_{n,i}^{*}D_{n,i,i}\psi_{n,i}\Bigg{\\}}.$
(48)

The functional integrals where, $b\neq a$ and $d\neq c$, are zero since the
integrands are odd. The path integral then simplifies to:

$Z_{I}^{1}=\Bigg{[}\mathcal{N}\left(\frac{1}{2\pi}\right)^{NM^{3}/2}\Bigg{]}\\\
\times\sum_{n1}\sum_{n2}\sum_{a}\sum_{c}\Lambda^{s}_{acacn_{1}n_{2}}\left(3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\right)\\\
\times\int
d\psi_{n1,a}\psi_{n1,a}^{2}\exp\left(-\frac{1}{2}\psi^{*}_{n1,a}D_{n,a,a}\psi_{n1,a}\right)\\\
\times\int
d\psi_{n2,c}\psi_{n2,c}^{2}\exp\left(-\frac{1}{2}\psi^{*}_{n2,c}D_{n,c,c}\psi_{n2,c}\right)\\\
\times\int_{remaining}D\psi_{n,i}\exp\Bigg{\\{}-\frac{1}{2}\sum_{\begin{subarray}{c}(n,i)\neq\\\
(n1,a),(n2,c)\end{subarray}}\psi_{n,i}^{*}D_{n,i,i}\psi_{n,i}\Bigg{\\}}.$ (49)

This finally gives,

$Z_{I}^{1}=\Bigg{[}\mathcal{N}\left(\frac{1}{2\pi}\right)^{NM^{3}/2}\Bigg{]}\\\
\times\left(3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\right)\sum_{n1}\sum_{n2}\sum_{a}\sum_{c}\Bigg{\\{}\Lambda^{s}_{acacn_{1}n_{2}}\\\
\times\left[\frac{\sqrt{2\pi}}{2D_{n1,a,a}^{3/2}}\right]\left[\frac{\sqrt{2\pi}}{2D_{n2,c,c}^{3/2}}\right]\prod_{\begin{subarray}{c}(n,i)\neq\\\
(n1,a),(n2,c)\end{subarray}}\left[\frac{\sqrt{2\pi}}{2D_{n,i,i}^{1/2}}\right]\Bigg{\\}},$
(50)

Since, $\Pi_{k}D_{n,k,k}=\det(D_{n})=\det(M_{n})$, it is possible to write:

$\displaystyle\frac{Z_{I}^{1}}{Z_{0}}=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}\sum_{n1}\sum_{n2}\sum_{a}\sum_{c}\Bigg{\\{}\Lambda^{s}_{acacn_{1}n_{2}}$
$\displaystyle\times\left[\frac{1}{2D_{n1,a,a}}\right]\left[\frac{1}{2D_{n2,c,c}}\right]\Bigg{\\}}.$
(51)

We now estimate, $\Lambda^{s}_{acacn_{1}n_{2}}$, to the first level of
approximation. Better approximations using numerical methods or superior
techniques could lead to higher accuracy. For matrices, $M_{n}$, $A_{n}$ and
$E_{n}$, where $M_{n}=A_{n}+E_{n}$, and if eigenvalues, $\lambda_{A}$, of
$A_{n}$ is known, it is possible to estimate the eigenvalues, $\lambda_{M}$,
of $M_{n}$ using the relation eigen :

$\lambda_{M}=\lambda_{A}+\frac{x^{t}E_{n}x}{x^{t}x}+\mathcal{O}(\|E\|^{2}),$
(52)

where, $x^{t}$ is eigenvector of $A_{n}$. Since,
$\mathcal{O}(\|E\|)\sim\mathcal{O}(\|n_{f}\|)$, this should estimate
eigenvalues of $M_{n}$ to order $\mathcal{O}(\|n_{f}\|^{2})$. The matrix,
$M_{n}$, is given in Eq. III.2. Let us take $A_{n}$ and $E_{n}$ to be the
matrices:

$\displaystyle
A_{n,i,j}=\beta_{0}^{2}(C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2})\delta_{ij},$
(53) $\displaystyle
E_{n,i,j}=\beta_{0}^{2}\eta_{f}({\bf{p}}_{j}-{\bf{p}}_{i},t)\frac{1}{{\mathbb{V}}}(-C_{n}^{2}+{\bf{p}}_{i}{\bf{p}}_{j}+m^{2}).$
(54)

Since $A_{n}$ is diagonal,
$\lambda_{A}=\beta_{0}^{2}(C_{n}^{2}+{\bf{p}}^{2}+m^{2})$, and the
eigenvectors, $x^{t}$, are unit vectors with only one non-zero entry. This
gives,

$\lambda_{M;n,i}\\\
\approx\beta_{0}^{2}\left\\{(C_{n}^{2}+{\bf{p}}_{i}^{2}+m^{2})+\frac{\eta_{f}(0,t)}{{\mathbb{V}}}(-C_{n}^{2}+{\bf{p}}_{i}^{2}+m^{2})\right\\}.$
(55)

Thus,

$D_{n,a,a}=\lambda_{M;n,a}\\\
\approx\beta_{0}^{2}\left\\{(C_{n}^{2}+{\bf{p}}_{a}^{2}+m^{2})+\frac{\eta_{f}(0,t)}{{\mathbb{V}}}(-C_{n}^{2}+{\bf{p}}_{a}^{2}+m^{2})\right\\}.$
(56)

This approximation has extracted the diagonal elements of $E$ as the
contribution from $E$ to $\lambda_{M}$. Consequently, the estimated
eigenvector of $M_{n}$ continues to be $x$, at this level of approximation.
Ergo, $\Lambda_{n,a}=\Lambda_{n,c}\approx I$, leading to,
$\Lambda^{s}_{bdacn_{1}n_{2}}\approx s_{\lambda f}(0,t)$. With all this, Eq.
III.2 becomes,

$\displaystyle\frac{Z_{I}^{1}}{Z_{0}}=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}s_{\lambda
f}(0,t)\left(\left[\sum_{a}\sum_{n1}\frac{1}{2D_{n1,a,a}}\right]\right)^{2}.$
(57)

The value of $D_{n,a,a}$ in Eq. 56, can be simplified, by recognizing that

  * •

$\frac{\eta_{f}(0,t)}{{\mathbb{V}}}=\langle\eta({\bf{x}},t)\rangle$,

  * •

$1+\langle\eta({\bf{x}},t)\rangle=\langle s({\bf{x}},t)\rangle$ and,

  * •

$1-\langle\eta({\bf{x}},t)\rangle\approx\frac{1}{\langle
s({\bf{x}},t)\rangle}$.

Substituting the value of $D_{n,a,a}$ thus obtained, in Eq. 57,

$\frac{Z_{I}^{1}}{Z_{0}}=3\frac{\beta_{0}^{3}}{{\mathbb{V}}^{2}}s_{\lambda
f}(0,t)\\\
\times\left[\sum_{a}\sum_{n}\frac{1}{\beta_{0}^{2}\left\\{(\frac{C_{n}^{2}}{\langle
s({\bf{x}},t)\rangle}+\langle
s({\bf{x}},t)\rangle\left({\bf{p}}_{a}^{2}+m^{2})\right)\right\\}}\right]^{2}.$
(58)

Finally,

$\frac{Z_{I}^{1}}{Z_{0}}=\frac{3}{4}(\beta_{0}{\mathbb{V}})\langle
s({\bf{x}},t)\lambda({\bf{x}},t)\rangle\\\
\times\Bigg{[}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\omega_{p}}\coth\left(\frac{\omega_{p}\beta_{0}\langle
s({\bf{x}},t)\rangle}{2}\right)\Bigg{]}^{2},$ (59)

where, $\omega_{p}=\sqrt{{\bf{p}}^{2}+m^{2}}$ and $\langle
s({\bf{x}},t)\lambda({\bf{x}},t)\rangle=s_{\lambda f}(0,t)$.

For, $r=1$, one may then determine, $\langle
E(t)\rangle=\frac{\partial\ln(Z(t))}{\partial\beta_{0}}$.

##  IV The Ricci tensor and negative potential

The Einstein field equations are now determined in the 5-D space-time-
temperature. We use the letters, $a$, $b$, as indices for the 5-D space-time,
i.e., $a$, $b$ = 0, 1, 2, 3, 4, with the index 0 referring to the temperature
dimension, while, the index 1 refers to the time dimension, We use the
letters, $\mu$, $\nu$, as indices for the 4-D Lorentzian space-time, i.e.,
$\mu$, $\nu$ = 1, 2, 3, 4. A superscript, $(N)$, within brackets, refers to
$N$ dimensional space. For example, $\nabla^{(4)}_{\mu}$, refers to the
covariant derivative in 4-D space-time. Let us consider the Lagrangian,

$\mathcal{L}(\phi,\partial_{a}\phi)=\frac{1}{2}\left(-\partial_{a}\phi\partial^{a}\phi-m^{2}\phi^{2}-\zeta
R\phi^{2}\right),$

with the sign convention (+,-,+,+,+), and the corresponding action,

$S=\int\sqrt{-g^{(5)}}\mathcal{L}(\phi,\partial_{a}\phi)d^{5}x.$ (60)

Let us consider the 5-D metric, $g^{(5)}_{ab}$, as:

$g^{(5)}_{ab}=\left[\begin{array}[]{c
c}s^{2}(g^{(4)}_{\mu\nu}({\bf{x}},t),{\bf{x}},t)&0\\\ 0&g^{(4)}_{\mu\nu}\\\
\end{array}\right],$ (61)

where, $g^{(4)}_{\mu\nu}$, is the usual metric tensor in 4-D space-time, and
is purely due to gravitational fields. It is also noted that $s$ would now
additionally be a function of $g^{(4)}_{\mu\nu}$ also. As an example, based on
the Ehrenfest Tolman effect ET1 ; ET2 , in the case of a uniform temperature
(system in global equilibrium), in a gravitational field,
$s(g^{(4)}_{\mu\nu}({\bf{x}},t),{\bf{x}},t)=\sqrt{g_{\mu\nu}\zeta^{\mu}\zeta^{\nu}}$,
where $\zeta^{\mu}$ is the Killing vector in 4-D space-time. We speculate
that, the function, $s()$, can be factored, i.e.,

$s(g_{\mu\nu}({\bf{x}},t),{\bf{x}},t)=s_{1}(g_{\mu\nu}({\bf{x}},t))\,s_{2}({\bf{x}},t),$
(62)

where, $s_{1}(g_{\mu\nu}({\bf{x}},t))$ is due to the Ehrenfest Tolman effect,
and $s_{2}({\bf{x}},t)$ is due to thermal gradients. It is the simplest
function that has the following limits:

  1. 1.

When the thermal system is in global equilibrium,
$s(g_{\mu\nu}({\bf{x}},t),{\bf{x}},t)\rightarrow
s_{1}(g_{\mu\nu}({\bf{x}},t))$, and,

  2. 2.

when $g_{\mu\nu}\rightarrow\eta_{\mu\nu}$,
$s(g_{\mu\nu}({\bf{x}},t),{\bf{x}},t)\rightarrow s_{2}({\bf{x}},t)$.

It is, however desirable to prove or improvise upon Eq. 62, based on first
principles.

For the metric, $g^{(5)}_{ab}$, the Einstein field equations in the 5-D space
become:

$R^{(5)}_{ab}-\frac{1}{2}g^{(5)}_{ab}R^{(5)}=\frac{8\pi
G}{c^{4}}T^{(5)}_{ab},$ (63)

where,

$T^{(5)}_{ab}=-\frac{2}{\sqrt{g^{(5)}}}\frac{\delta S}{\delta g^{(5)}_{ab}}.$
(64)

The 5-D Ricci tensor, $R^{(5)}_{ab}$, can be expressed in terms of the 4-D
covariant derivative operator, $\nabla^{(4)}_{\mu}$, and the 4-D Ricci tensor,
$R^{(4)}_{\mu\nu}$, as:

$R^{(5)}_{ab}=\left[\begin{array}[]{c c}R_{\beta\beta}&0\\\
0&R^{(4)}_{\mu\nu}-\frac{1}{s}\nabla^{(4)}_{\mu}\nabla^{(4)}_{\nu}s\\\
\end{array}\right].$ (65)

where, $R_{\beta\beta}=s\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s$. The treatment of
the geodesic equation in Ref. gans6 , gives an insight into the physical
interpretation of how thermal gradients, effects an apparent curvature in the
Lorentzian space-time. It can be easily seen that the expression for
$R^{(5)}_{ab}$ reduces to the 4-D Ricci tensor, for the uniform temperature
case, i.e., $\partial_{\mu}s\rightarrow 0$. Another thing to note is that in
the metric $s({\bf{x}},t)^{2}d\beta^{2}+g_{\mu\nu}dx^{\mu}x^{\nu}$, the
temperature dimension can be eliminated by taking the limit, $s\rightarrow 0$.
Thus, for a vacuum, one may take the limits $\partial_{\mu}s\rightarrow 0$,
followed by $s\rightarrow 0$, in which case, the Einstein field equations (Eq.
63) reduce to the usual 4-D space-time equations. Finally the 5-D Ricci
scalar, $R^{(5)}$ can be expressed as:

$R^{(5)}=R^{(5)a}_{~{}~{}a}=R^{(4)}-\frac{2}{s}\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s,$
(66)

where, $R^{(4)}$ is the conventional 4-D Ricci scalar, in the Lorentzian
space, due to gravitational fields, in the absence of any temperature
gradients. When the 5-D Ricci scalar is sufficiently negative, it can lead to
spontaneous symmetry breaking, i.e., when, $m^{2}+\zeta R^{(5)}<0$. This
translates to:

$\displaystyle R^{(5)}<-\frac{m^{2}}{\zeta},$
$\displaystyle\Rightarrow\frac{2}{s}\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s>R^{(4)}+\frac{m^{2}}{\zeta}.$
(67)

It is to be noted that the quantity,
$\frac{2}{s}\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s$, is scale-invariant, i.e., the
absolute temperature is inconsequential. Only the variations in temperature
matter. Since, $s({\bf{x}},t)$ is a scalar,
$\frac{2}{s}\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s$ simplifies to
$\frac{2}{s}\nabla^{(4)\mu}\partial_{\mu}s$.

In the case of a thermal medium formed in a collider, e.g., like the QGP at
the LHC, the gravitational effects can be neglected. This gives $R^{(4)}=0$.
Equation IV, then simplifies to:

$\displaystyle\frac{2}{s}\partial^{\mu}\partial_{\mu}s>\frac{m^{2}}{\zeta}.$
(68)

One may, however, consider the Eq. 68, with caution. For Eq. 68 to be
satisfied, $\frac{2}{s}\partial^{\mu}\partial_{\mu}s$ needs to be
significantly large. But if thermal gradients are very large, the thermal
medium may cease to be in local thermal equilibrium, and the current formalism
ceases to be valid.

The scenario in Eq. IV is however different. To see why, the value of
$\frac{2}{s}\nabla^{(4)\mu}\nabla^{(4)}_{\mu}s$ is affected by three factors:

  1. 1.

The thermal gradients, $\partial_{\mu}s_{2}(\bf{x},t)$ (refer Eq. 62 for the
factorization of $s=s_{1}s_{2}$),

  2. 2.

The Ehrenfest Tolman effect, represented by $s_{1}(\bf{x},t)$, and,

  3. 3.

The terms containing the Christoffel symbols, arising as part of the covariant
derivative.

Whereas the first factor may be small for a medium in local thermal
equilibrium, the second and third factors depend on the strength of the
gravitational field.

##  V Summary

In this work, it is seen that the concept of 5-D space-time-temperature,
permits the incorporation of time-dependent temperature effects, for systems
that continue to be in local thermal equilibrium. The bulk thermodynamic
property, namely, the energy density expectation value, has been calculated
perturbatively, for a thermal bath with both spatial and temporal variations,
in the absence of gravity. The Einstein field equations have been determined
in 5-D space-time-temperature, which incorporates both gravitational and
thermal effects. In the presence of a thermal variation, both temperature
gradients and gravity determine the net scalar curvature. The net curvature in
the 5-D space can be negative. If the d’Alembertian of the temperature
variation is sufficiently high, spontaneous symmetry breaking may occur in a
scalar field, particularly in the presence of a strong gravitational field.

As mentioned in Ref. gans6 , the curvature of the 5-D space due to temperature
variations can be validated experimentally, in terrestrial experiments, due to
the much lower energy scales involved. Thus, if the concept of the curved 5-D
space is confirmed, then it may have important consequences, such as the
spontaneous symmetry breaking as described in this work. This current
formalism may be useful in situations where both gravity and thermal
variations are present, like in the interior of a neutron star neutron1 ;
neutron2 . The rotation curves of galaxies have been a subject of intensive
research galaxy1 ; galaxy2 . It might even be possible to consider a galaxy as
a thermal medium, with the stars as point particles. However, a galaxy is not
in equilibrium, which might pose challenges for such an approach.

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# A Tale of Two Peas-In-A-Pod: The Kepler-323 and Kepler-104 Systems

C. Alexander Thomas Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, IN 46556, USA  Lauren M. Weiss Department of Physics and
Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA  Howard
Isaacson Department of Astronomy, University of California Berkeley, Berkeley
CA 94720, USA  Centre for Astrophysics, University of Southern Queensland,
Toowoomba, QLD, Australia  Hilke E. Schlichting Department of Earth,
Planetary, and Space Sciences, The University of California, Los Angeles, 595
Charles E. Young Drive East, Los Angeles, CA 90095, USA  Corey Beard NASA
FINESST Fellow  Department of Physics & Astronomy, The University of
California, Irvine, Irvine, CA 92697, USA  Casey L. Brinkman Institute for
Astronomy, University of Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822 USA
Ashley Chontos Henry Norris Russell Fellow  Department of Astrophysical
Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08540, USA
Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu,
HI 96822, USA  Paul Dalba Department of Earth & Planetary Sciences, University
of California Riverside, 900 University Ave, Riverside, CA 92521, USA  Fei Dai
NASA Sagan Fellow  Division of Geological and Planetary Sciences, 1200 E
California Blvd, Pasadena, CA, 91125, USA  Department of Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Steven Giacalone Department
of Astronomy, University of California Berkeley, Berkeley CA 94720, USA
Department of Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Jack Lubin Department of Physics & Astronomy, The University of
California Irvine, Irvine, CA 92697, USA  Judah Van Zandt Department of
Physics & Astronomy, University of California Los Angeles, Los Angeles, CA
90095, USA  Malena Rice Department of Astronomy, Yale University, New Haven,
CT 06511, USA

###### Abstract

In order to understand the relationship between planet multiplicity, mass, and
composition, we present newly measured masses of five planets in two planetary
systems: Kepler-323 and Kepler-104. We used the HIRES instrument at the W.M.
Keck Observatory to collect 79 new radial velocity measurements (RVs) for
Kepler-323, which we combined with 48 literature RVs from TNG/HARPS-N. We also
conducted a reanalysis of the Kepler-104 system, using 44 previously published
RV measurements. Kepler-323 b and c have masses of $2.0^{+1.2}_{-1.1}$ M⊕ and
6.5$\pm 1.6$ M⊕, respectively, whereas the three Kepler-104 planets are more
massive (10.0$\pm 2.8$ M⊕, $7.1^{+3.8}_{-3.5}$ M⊕, and $5.5^{+4.6}_{-3.5}$ M⊕
for planets b, c, and d, respectively). The Kepler-104 planets have densities
consistent with rocky cores overlaid with gaseous envelopes
($4.1^{+1.2}_{-1.1}$ g/cc, $2.9^{+1.7}_{-1.5}$ g/cc, and $1.6^{+1.5}_{-1.1}$
g/cc respectively), whereas the Kepler-323 planets are consistent with having
rocky compositions ($4.5^{+2.8}_{-2.4}$ g/cc and $9.9^{+2.7}_{-2.5}$ g/cc).
The Kepler-104 system has among the lowest values for gap complexity
($\mathcal{C}$ = 0.004) and mass partitioning ($\mathcal{Q}$ = 0.03); whereas,
the Kepler-323 planets have a mass partitioning similar to that of the Inner
Solar System ($\mathcal{Q}$ = 0.28 and $\mathcal{Q}$ = 0.24, respectively).
For both exoplanet systems, the uncertainty in the mass partitioning is
affected equally by (1) individual mass errors of the planets and (2) the
possible existence of undetected low-mass planets, meaning that both improved
mass characterization and improved sensitivity to low-mass planets in these
systems would better elucidate the mass distribution among the planets.

…

††software: RADVEL (Fulton et al. 2018)

##  1 Introduction

Of the 190,000 stars observed as part of the Kepler Mission, approximately 25%
host multiple transiting planets with orbital periods less than 100 days,
suggesting 30-80% of Kepler stars host multiple planets (Zhu et al., 2018;
Mulders et al., 2018). One intriguing finding is that the planets in Kepler’s
multi-planet systems tend to be similar in size and regularly spaced, like
“peas-in-a-pod” (Weiss et al., 2018). Measuring the masses of planets in peas-
in-a-pod like systems allows us to address several overarching questions about
the nature of multi-planet systems. (1) What are the masses and bulk
compositions of the individual planets? (2) Are the masses and radii of the
planets typical for their orbital periods? (3) Within a given system, what is
the mass diversity and spacing distribution of transiting planets? (4) How
could the presence of undetected, low-mass planets affect the statistics of
the intra-system size and mass uniformity?

A major advantage of characterizing transiting exoplanets is that the planet
radii and orbital periods are typically well-determined from transits, while
the host star properties can be characterized through spectroscopic, imaging,
and astrometric follow-up. In this paper, we investigate two systems of
multiple transiting planets that were discovered as part of the NASA Kepler
Mission: Kepler-323 and Kepler-104. In terms of their size and spacing, both
Kepler-323 and Kepler-104 follow the peas-in-a-pod pattern, making them ideal
test-beds for the questions posed above. However, a key physical property that
is not readily determined from photometry alone is the planet masses. We
selected Kepler-323 and Kepler-104 as part of a multi-semester, magnitude-
limited radial velocity (RV) survey to measure planet masses in multi-planet
systems. In this paper, we present new measurements of the Kepler-323 system
and conduct an analysis of the properties of the Kepler-323 and Kepler-104
systems.

###  1.1 Kepler-323

Kepler-323 is a sun-like star (0.95 $\pm$ 0.03 M⊙, 1.09 $\pm$0.29 R⊙). The
system consists of two Earth-sized planets, 1.35 $\pm$ 0.04 R⊕ and 1.53 $\pm$
0.05 R⊕, in short orbits of 1.67 days and 3.55 days respectively (Fulton &
Petigura, 2018). Bonomo et al. (2023) collected 48 radial velocities (RVs) of
Kepler-323 with the HARPS-N spectrograph on the 3.6m Telescopio Nazionale
Galileo at the Observatorio Roque de Los Muchachos in in La Plama, Spain. In
this work, we combine those RVs with 79 new RVs from the HIRES spectrograph on
the W. M. Keck Observatory 10m telescope Keck I on Maunakea, Hawaii (Vogt et
al., 1994) to better characterize the Kepler-323.

###  1.2 Kepler-104

Kepler-104 is another Sun-like star ($0.82\pm 0.01$ M⊙, 1.05 $\pm 0.03$ R⊙).
Its planetary system consists of three sub-Neptunes, Rb = 2.38 $\pm$ 0.06 R⊕,
Rc = 2.36 $\pm$ 0.08 R⊕, and R${}_{d}=$ 2.64 $\pm$ 0.14 R⊕ at orbital periods
of 11.43, 23.67, and 51.76 days (Fulton & Petigura, 2018). Kepler-104 has a
stellar companion reported in Lillo-Box et al. (2014) and Furlan & Howell
(2017), at an angular separation of approximately 1.8″. There is also a
companion at an angular separation of 17″ identified based on GAIA data.
(Mugrauer, 2019). For an angular separation of 1.8″, a distance of 400 pc
(Gaia Collaboration, 2020) corresponds to a semi-major axis of 720 AU and an
orbital period of 20,000 years, and so we do not expect this companion to
affect our analysis. We leverage 44 published RVs of Kepler-104 from Weiss et
al. (2024) to provide a detailed analysis of the mass diversity of the system.

##  2 New Observations

We collected 79 spectra of Kepler-323 using HIRES between May 20th, 2019 and
October 17th, 2021. To measure RVs, we used the standard California Planet
Search (CPS) data reduction pipeline as described in Howard et al. (2010).
This method involves using a warm iodine cell mounted in front of the slit to
imprint absorption features at reference wavelengths determined from a high-
resolution atlas of iodine lines (Butler et al., 1996). Since clouds and/or
moonlight can contaminate the spectra of stars with $V>10.5$ (Marcy et al.,
2014), we gathered observations in a manner that enabled sky subtraction.
Specifically, observations were taken with a 14″ slit, which is large compared
to the typical seeing-limited point-spread function (PSF) of the star ($\sim
1\arcsec$). Because the slit width is comparable to the seeing-limited PSF,
the HIRES PSF varies over time with changes in seeing, weather, and
aberrations of the wavefront. The CPS Doppler routine involves forward-
modeling the iodine-imprinted spectrum of a star as a combination of a high-
resolution iodine library spectrum and an iodine-free, velocity-shifted, PSF-
deconvolved template spectrum of the target star. The deconvolved template
spectrum is obtained by observing rapidly-rotating B stars with the iodine
cell in the light path immediately before and after an iodine-free observation
of the star, which effectively samples the PSF at the time of the template.
The individual spectra with iodine are then convolved with a PSF of HIRES that
is determined for each observation. These new RVs are presented in Table 1.

##  3 Orbit Analysis and Planet Properties

We used the open source python package RadVel (Fulton et al., 2018) to model
our RVs with a Keplerian orbit. We set the periods and time of conjunctions
for all five planets to the values reported in Morton et al. (2016), which are
presented in Table 2. Because the RV semi-amplitudes were small compared to
the per-measurement error, making it difficult to accurately determine the
eccentricities, we fixed the eccentricity of all planets at zero. Our choice
to fix the eccentricities at zero is further motivated by the observation that
most small planets in multi-planet systems have small eccentricities (median
$e=0.02$, Yee et al. 2021). Thus, our only free parameters were the RV semi-
amplitudes, RV zero-point ($\gamma$), and RV jitter ($\sigma_{i}$) for each
instrument. We adopted priors ensuring that the RV semi-amplitude and the
jitter were positive. We performed a maximum likelihood fit using RadVel. To
determine the uncertainties in our parameters, we used RadVel’s Markov Chain
Monte Carlo (MCMC) method with 500 walkers and 100,000 steps. The maximum
Gelman-Rubin value for the convergence of our MCMC is $<$ 1.001 for both
Kepler-323 and Kepler-104. The resulting confidence intervals for each
parameter are displayed in Table 2. The RVs and the best-fit models are shown
in Figures 1 and 2.

Our Keplerian orbital fit yields masses of $2.0^{+1.2}_{-1.1}$ M⊕ and 6.5$\pm
1.6$ M⊕ for Kepler-323 b and c and masses of 10.0$\pm 2.8$ M⊕,
$7.1^{+3.8}_{-3.5}$ M⊕, and $5.5^{+4.6}_{-3.5}$ M⊕ for Kepler-104 b, c, and d
(Figures 3 and 4). We compare these to a sample of known exoplanets, selected
by the process described in Appendix A, and objects from the Solar System. In
Figures 3 and 4, we focused solely on planets with calculated densities; we
removed any planets with a mass derived from a mass radius relation. Figures 3
and 4 include lines of constant composition and collisionally stripped planets
(Zeng et al., 2019; Marcus et al., 2010).

Kepler-323 c’s density of $9.9^{+2.7}_{-2.5}$ g/cc is consistent with a rocky
composition. The 1$\sigma$ credible interval spans an Earth-like composition
or a more iron-rich planet. Kepler-323 b, however, has a density of
$4.5^{+2.8}_{-2.4}$ g/cc, which is lower than what we would expect for a
planet of pure silicates. One possible interpretation of a low density is that
the planet could have a gaseous envelope of high mean molecular weight (such
as has been suggested for TOI-561 b in Brinkman et al. 2023). However, the
1$\sigma$ confidence interval spans densities of purely silicate and Earth-
like compositions.

The architecture of the Kepler-323 system provides an interesting example to
test atmospheric mass loss processes. Planets b and c have similar radii, but
their masses appear to differ by a factor of 3, with the outer planet being
more massive. The short orbital period of Kepler-323 c, combined with a
density consistent with a rocky composition, indicates that it would be
unlikely to retain a gaseous envelope in the presence of intense stellar X-ray
and ultraviolet radiation from the star(Owen & Schlichting, 2023). In
addition, we would expect Kepler-323 c to lose its atmosphere through core-
powered mass loss due to the short orbital period and high equilibrium
temperature (Ginzburg et al., 2018). If we assume that Kepler-323 c, which is
rocky, lost its envelope, then Kepler-323 b, which is closer to its star and
(apparently) lower mass, should also have lost its envelope. Follow-up
observations of Kepler-323 b are needed to better constrain its mass and
composition.

The three Kepler-104 planets, with densities of $4.1^{+1.2}_{-1.1}$,
$2.9^{+1.7}_{-1.5}$ g/cc, and $1.6^{+1.5}_{-1.1}$ g/cc respectively, are all
consistent with bulk compositions that include a gaseous envelope. Rocky
compositions are ruled out for all three planets with $>3\sigma$ confidence,
which is not surprising given that their radii are all $>2\,R_{\oplus}$
(Fulton et al., 2017; Rogers, 2015; Weiss & Marcy, 2014).

To explore the influence of orbital period on planet masses and radii, in
Figures 3 and 4 we split our sample of planets with $R<5R_{\earth}$ into
“short period” ($P<8$ days, purple circles) and “long period” ($P>8$ days,
blue triangles) groups 111This is an integer near the median orbital period of
our sample: 7.97 days. For planets that are “short period,” 59 of 134,
including Kepler-323, have a best-fit density that is consistent with a rocky
composition. For the “long-period” group, 119 planets of 130, including
Kepler-104, have best-fit densities and radii that indicate a gaseous
envelope.

There are at least two plausible explanations for this pattern: one
astrophysical, and one related to selection bias. The prevalence of gas
envelopes at longer orbital periods vs. bare rocky planets at short periods is
consistent with predictions from atmospheric mass-loss models (Owen &
Schlichting, 2023; Ginzburg et al., 2018). For photoevaporation, planets at
longer periods are less irradiated by stellar X-ray and UV radiation, meaning
they are less likely to lose their envelopes. For core-powered mass loss,
hotter equilibrium temperatures increases the odds of losing an atmosphere
from the cooling of the planet. A mass-radius-period relationship was
developed in Mills & Mazeh (2017). In addition to these astrophysical
phenomenon, the relationship between planet mass, radius, and orbital period
is sculpted, at least partially, by selection bias. For instance, the masses
of small planets ($R<1.5R_{\oplus}$) are typically only measured when the
planet is sufficiently close to the star to produce a detectable RV amplitude,
and so the masses of small planets at long orbital periods have rarely been
measured. To further clarify the relative impact of selection biases and
atmospheric mass-loss on sculpting this relationship, more observations of
small planets at long periods are required. Transit timing variations provide
a viable method for measuring the masses of some rocky planets at $P>8$ days
and will yield additional insights on the relationship between planet mass,
radius, and orbital period, especially in multi-planet systems e.g. (Agol et
al., 2021).

##  4 Planetary System Properties

###  4.1 Mass Partitioning

In addition to contextualizing the properties of the individual planets, we
can characterize these planetary systems with metrics of their architectures.
Mass partitioning and gap complexity describe the mass diversity and spacing
distribution of a system, respectively. These metrics have been used to
evaluate both observed and synthetic planetary system architectures (Gilbert &
Fabrycky, 2020).

The mass partitioning of a system is given by:

$\mathcal{Q}\equiv\biggl{(}\frac{N}{N-1}\biggl{)}\cdot\biggl{(}\sum_{i=1}^{N}\biggl{(}m_{i}^{*}-\frac{1}{N}\biggl{)}^{2}\biggl{)}$
(1)

where

$m^{*}_{i}=m_{i}/\sum^{N}_{i}m_{i}$ (2)

Here, $N$ is the number of planets in a system and $m_{i}$ is the mass of the
$i^{\mathrm{th}}$ planet (order does not matter). The normalizing coefficient,
$\biggl{(}\frac{N}{N-1}\biggl{)}$, restricts $\mathcal{Q}$ to be from 0 to 1,
where a value of 0 indicates equal masses throughout the system and a value
that approaches 1 represents a system with a dominant giant planet and $N-1$
tiny planets. Note that mass partitioning is agnostic to the ordering of the
planets.

Figure 5 shows the planet radius (R⊕) and orbital period (days) for
Kepler-323, Kepler-104, and other small exoplanets selected from the NASA
Exoplanet Archive (2019); Appendix A). Each planet is colored by the mass
partitioning for its system, based on the masses of all the known planets in
that system. In order for a system to appear in Figure 5 in color, it must
have all planets survive the selection process outlined in Appendix A and it
must have a mass measurement for every known planet in the system. Otherwise,
the system is colored gray; an example is Kepler-11 as Kepler-11 c’s mass
error is too large for our selection criteria and Kepler-11 g does not have a
measured mass.

We calculated the distribution of the possible mass partitioning values for
each system in our sample based on the mass measurement values and errors. For
each system we selected planet masses from a normal distribution defined by
the masses and errors reported on the NASA Exoplanet Archive. We used
rejection sampling to discard any negative mass draws. We iterated until each
planet had 1000 draws from our reconstructed mass posterior and calculated the
mass partitioning for each iteration. This resulted in 49,000 mass
partitioning values across 49 systems, the distribution of which is shown in
Figure 6. The median mass partitioning in this sample is $\mathcal{Q}=0.08$.

Kepler-104, with a mass partitioning of $\mathcal{Q}=0.03$, is consistent with
many of the systems in our sample (median $\mathcal{Q}=0.06$ for three-planet
systems). Kepler-323, on the other hand, has the twelfth highest mass
partitioning in our 52 system sample ($\mathcal{Q}=0.28$), which is slightly
larger than the Inner Solar System ($\mathcal{Q}=0.24$) and is higher than
typical (median $\mathcal{Q}=0.10$ for two-planet systems). 222We define the
Inner Solar System as Mercury, Venus, Earth, and Mars, the region consistent
with the $\lesssim 1$ a.u. orbits typical of transiting exoplanets. In order
to probe the relationship between the mass diversity and architecture of a
multi-planet system, we highlight LTT 1445 A (Winters et al. 2022; Pass et al.
2023) and TOI-125 (Nielsen et al. 2020) in Figure 5 as systems with similar
architectures to Kepler-323 and Kepler-104 respectively. TOI-125, which has
three planets of similar radii to Kepler-104 but with shorter orbital periods,
has a mass partitioning of $\mathcal{Q}=0.04$, similar to both Kepler-104 and
the median mass partitioning for three-planet systems in our sample. LTT 1445
A, a two-planet system with smaller radii and longer orbits than Kepler-323,
has a mass partitioning of $\mathcal{Q}=0.11$, which is typical for the two-
planet systems and lower than the mass partition of Kepler-323. While
Kepler-323 has a higher mass partitioning than typical, it is not a strong
example of a high mass partitioning system; Kepler-30 (Sanchis-Ojeda et al.,
2012), a system with two Neptune-mass planets and one super-Jupiter, has a
mass partitioning of $\mathcal{Q}=0.85$. A more detailed treatment of the
distribution of mass partitioning in various planetary architectures would be
enlightening, but is beyond the scope of this paper.

We investigated how the errors on the planet masses propagate to the mass
partitioning of the system to determine the accuracy of the calculated mass
partitioning and our confidence in the characterization of the mass diversity.
We selected 5000 possible mass combinations for Kepler-323 and Kepler-104 from
the masses defined by the mass and error for each planet333Formally, it would
be better to draw from the joint posterior distribution, but we see no
covariance between the planet masses (Figures 7,8) and used rejection sampling
to ensure all masses were positive. For each combination of planet masses, we
calculated the mass partitioning. Figure 9 and Figure 10 show histograms of
these calculated possible mass partitioning values for Kepler-323 and
Kepler-104 respectively. Each plot has a solid black line to indicate the
calculated median value of the credible interval.

The Kepler-323 mass partitioning posterior spans the full range of zero to one
as the error bars on the planet masses are large. There is a peak in the
posterior at a mass partitioning near zero due to more cases where Kepler-323
b and Kepler-323 c have very similar masses than very different masses, as
seen in Figure 11.

It is possible that there could be additional, undetected planets in a system.
This produces another source of uncertainty in our assessment of the mass
partitioning because a planet with significantly different mass from the
detected planets could strongly affect the mass partitioning. Figures 9 and 10
have lines that represent the mass partitioning for the system if one
(orange), two (gold), or three (brown) zero-mass planets are undetected in the
system. To explore the opposite extreme (an undetected planet of similar mass
to the known planets), we added an extra planet and varied its mass to
minimize the mass partitioning. The mass of this extra planet is 5.5 M$\earth$
for Kepler-323 and 8.0 M$\earth$ for Kepler-104. The mass partitioning for all
of these possibilities are included in Table 4. The presence of a Jupiter in
either Kepler-323 or Kepler-104 would be intriguing as it would dramatically
impact the mass partitioning of the system. Based on a decade of radial
velocity measurements of Kepler-323, we have no evidence of the presence of a
Jupiter out to 10 AU and we find a 3$\sigma$ upper limit of 1 $M_{J}$. The
presence of a Jupiter in Kepler-104 was ruled out in Weiss et al. (2024).

For Kepler-104, 92% of the mass partitioning posterior is below the mass
partitioning of the inner Solar System, 98% is below the mass partitioning of
the gaseous Solar System ($\mathcal{Q}=0.41$), and 99% is below the mass
partitioning of the full Solar System ($\mathcal{Q}=0.49$). Even if Kepler-104
has three undetected planets of zero mass, its mass partitioning is below that
of the inner Solar System. We find that the Kepler-104 mass partitioning,
compared to Kepler-323, is well-constrained and that there is low mass
diversity in this system.

###  4.2 Gap Complexity

Another metric we calculated to analyze the architectures of multi-planet
systems is the gap complexity (Gilbert & Fabrycky, 2020). Gap complexity
characterizes the irregularity of the orbital periods of planets in a single
system:

$\mathcal{C}=-K\biggl{(}\sum_{i=1}^{n}p_{i}^{*}\log
p_{i}^{*}\biggl{)}\cdot\biggl{(}\sum_{i=1}^{n}\biggl{(}p_{i}^{*}-\frac{1}{n}\biggl{)}^{2}\biggl{)}$
(3)

where $K$ is a normalization constant, $n$ is the number of gaps between
planets, $P^{{}^{\prime}}$ and $P$ are the outer and inner periods of pairs of
adjacent planets, and $p_{i}^{*}$ are defined as the “pseudo-probabilities”
and are normalized to one:

$p^{*}_{i}\equiv\frac{\log(P^{{}^{\prime}}/P)}{\log(P_{\max}/P_{\min})}$ (4)

where $P_{\max}$ and $P_{\min}$ are respectively defined as the maximum period
and the minimum period of any planet in the system. Like mass partitioning,
gap complexity ranges from zero (evenly spaced planets) to one (maximum
$p_{i}^{*}\approx\frac{2}{3}$).

The gap complexity of Kepler-104, along with a sample of known multi-planet
systems from the NASA Exoplanet Archive (see Appendix A), is shown in Figure
12. In order to calculate the gap complexity, there must be at least three
planets in the system. Any system with only two known planets, such as
Kepler-323 and LTT 1445 A, is colored gray. In addition, any system that has a
planet that was removed from the data set by the process described in Appendix
A is also colored gray. Unlike Figure 5, we do not exclude planets with a mass
derived from a mass-radius relation. Kepler-104 ($\mathcal{C}=0.004$) and
TOI-125 ($\mathcal{C}=0.01$) have low gap complexities, indicating that they
are almost perfectly evenly spaced. In contrast, Kepler-167 (Chachan et al.,
2022) has high gap complexity ($\mathcal{C}=0.80$). Kepler-104’s low gap
complexity, which is the 6th lowest among the systems examined here (66 total
systems), is consistent with the other exoplanet systems, whereas systems with
architectures like that of Kepler-167 are rarely detected. The inner Solar
System has a gap complexity of $\mathcal{C}=0.13$ and has been noted for its
regular spacing (Nieto, 1970).

##  5 Conclusion

We conducted an analysis of 79 newly gathered RVs, combined with 48 RVs from
Bonomo et al. (2023), of Kepler-323 and 44 RVs from Weiss et al. (2024) of
Kepler-104 in order to quantify the masses and densities of their planets.
Using these newly calculated masses, we computed the mass diversity and period
spacing of these systems and compared those results to other multi-planet
transiting systems. We also compared the individual planet masses, radii, and
period to planets from other multi-planet transiting systems.

Based on the orbital periods and planet radii of the Kepler-323 planets, we
find that these planets are, mostly, as we would expect when compared to the
general exoplanet population. In addition, when looking at the relationship
between orbital period and physical radius, 79% of planets in our sample with
radii less than 1.8 R⊕ have an orbital period less than 8 days. This includes
the Kepler-323 planets.

The calculated mass partitioning of Kepler-323 is $0.28^{+0.25}_{-0.18}$,
which is 0.04 larger than the Inner Solar System and is the twelfth largest in
our sample. Due to the errors on the individual planet masses, the mass
partitioning posterior for Kepler-323 contains values from 0 to 1, which
covers the entire range of possible values for mass partitioning. In addition,
the presence of undetected planets could also shift the calculated mass
partitioning value by a factor of two. Since we cannot calculate a gap
complexity for systems with only two planets, we cannot comment on the spacing
of the Kepler-323 system.

The Kepler-104 planets, based on their radii and orbital periods, are similar
to the other exoplanets with an orbital period of longer than 8 days have
radii larger than 1.8 R⊕ shown in Figure 3 and Figure 4.

Kepler-104 has a low mass partitioning of $\mathcal{Q}=0.03^{+0.15}_{-0.02}$.
If there were three undetected (zero mass) planets in the Kepler-104 system,
the mass partitioning for Kepler-104 would remain lower than that of the inner
Solar System. This indicates that Kepler-104 has very little mass diversity
between the planets. The gap complexity of Kepler-104 is 0.004, which is the
6th lowest of the 66 systems in our sample with three or more planets and can
have a calculated gap complexity. This further reinforces Kepler-104 as a
”peas-in-a-pod” system.

In this paper, we considered how the following questions applied to the peas-
in-a-pod systems of Kepler-323 and Kepler-104: 1) What are the masses and bulk
compositions of the individual planets? (2) Are the masses and radii of the
planets typical for their orbital periods? (3) Within a given system, what is
the mass diversity and spacing distribution of transiting planets? (4) How
could the presence of undetected, low-mass planets affect the statistics of
the intra-system size and mass uniformity? While Kepler-323 and Kepler-104
both display radius uniformity, they do not share the same level of mass
uniformity as Kepler-104 appears to have much more uniform masses. Applying
these questions to a larger sample of high multiplicity systems that exhibit
radius uniformity or diversity will provide valuable constraints for planet
formation and will serve as a basis for future work.

We acknowledge support in the form of observational resources at W. M. Keck
Observatory from NASA and the University of Hawai‘i. L.M.W. acknowledges
support from the NASA-Keck Key Strategic Mission Support program (grant no.
80NSSC19K1475) and the NASA Exoplanet Research Program (grant no.
80NSSC23K0269). This work was supported by a NASA Keck PI Data Award,
administered by the NASA Exoplanet Science Institute. Data presented herein
were obtained at the W. M. Keck Observatory from telescope time allocated to
the National Aeronautics and Space Administration through the agency’s
scientific partnership with the California Institute of Technology and the
University of California. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. This dataset made use of the
NASA Exoplanet Science Institute at IPAC, which is operated by the California
Institute of Technology under contract with the National Aeronautics and Space
Administration. We acknowledge contributions from B.J. Fulton for maintaining
the Jump database in which data for this project were curated. We acknowledge
L. A. Rogers for contributing to the proposals that led to the collection of
RVs presented here and for useful discussions and comments on the manuscript.
We would like to thank Greg Gilbert for investigating the presence of TTVs in
these systems. The authors wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Maunakea has always
had within the indigenous Hawaiian community. We are most fortunate to have
the opportunity to conduct observations from this mountain. Keck:I (HIRES)

##  Appendix A NASA Exoplanet Archive Sample Selection

In order to place the Kepler-323 and Kepler-104 planets into the context of
the general exoplanet population, we used a curated sample of planets from the
NASA Exoplanet Archive Confirmed Planets Table as comparison:

  1. 1.

We started with the largest possible sample by downloading all of the
confirmed planets from the NASA Exoplanet Archive, as of November 8th, 2022.
We included all the rows from the database, which sometimes resulted in
multiple entries for a single planet.

  2. 2.

We removed each row that did not include an entry for the orbital period,
planet physical radius, or planet physical radius error.

  3. 3.

We removed each row that had a non-transiting flag (indicating a non-
transiting planet).

  4. 4.

We removed each row that had a controversial flag.

  5. 5.

We removed each row that had a mass value but no mass error, as these rows
corresponded to mass upper limits.

  6. 6.

We removed rows where the planet masses were determined from Hadden & Lithwick
(2014) because the masses from that study were sometimes biased (Hadden &
Lithwick, 2017).

  7. 7.

Following these cuts, we sorted by the NASA Exoplanet Archive’s default
parameter. If a planet still had an entry in our sample but the NASA Exoplanet
Archive-determined “default” row was already dropped, we sorted by the date of
update. We then removed any duplicate row, keeping either the default
parameter set or the most recent data for each planet.

  8. 8.

For rows that did not have an entry for planet mass, we used the mass radius
relation from Weiss & Marcy (2014) to estimate a mass. This was done to
preserve the system’s architectures when calculating gap complexity.

  9. 9.

For rows that did not have an entry for planet density, we computed the planet
density.

  10. 10.

We calculated the percent error for the radius, mass, and density of each
planet. We then removed planets with a radius error $>$10%, a mass error
$>$85%, or a density error $>$104%. These figures were chosen based on the
calculated values and errors of Kepler-323 and Kepler-104. We wanted to ensure
that we selected planets with as precise measurements as possible, while being
sure that our planets would not be excluded from this population.

##  Appendix B RV Table

Table 1: HIRES RVs of Kepler-323

Time | RV | RV Error
---|---|---
(BJD-2450000 ) | (m/s) | (m/s)
8624.079912 | 2.69 | 2.80
8652.076300 | 8.57 | 2.79
8679.936895 | -4.56 | 2.09
8714.811564 | 2.57 | 2.25
8724.815753 | 6.04 | 2.28
8774.870351 | -1.78 | 2.92
8787.831759 | 0.75 | 3.43
8797.796716 | 5.63 | 3.35
8919.108473 | 3.66 | 2.98
9003.034178 | 3.07 | 2.46
9003.943462 | 0.94 | 2.73
9006.929700 | 2.32 | 2.82
9007.965294 | 5.97 | 2.68
9010.977440 | 9.39 | 2.50
9011.964465 | 3.06 | 2.52
9012.940900 | -9.46 | 2.80
9013.914590 | 0.47 | 2.54
9016.896354 | -3.01 | 2.74
9024.905287 | 3.46 | 2.51
9025.987166 | 5.80 | 2.76
9027.872397 | -2.96 | 2.28
9028.839619 | 4.50 | 2.67
9029.901252 | -1.64 | 2.32
9030.934017 | 1.31 | 2.70
9031.878713 | -0.15 | 2.62
9034.919478 | 0.42 | 2.42
9035.940978 | -5.07 | 2.73
9036.819639 | 5.98 | 2.58
9038.913723 | -2.15 | 2.48
9039.908818 | 0.23 | 2.95
9041.058788 | -10.17 | 3.19
9041.854713 | -2.08 | 2.69
9042.981926 | -2.69 | 5.08
9057.995943 | -4.91 | 3.75
9072.969537 | -5.76 | 2.80
9077.945189 | 3.14 | 2.54
9078.961976 | -0.20 | 3.03
9086.924455 | -7.42 | 3.86
9088.943940 | 4.70 | 3.37
9089.937189 | 0.53 | 2.52
9090.875231 | -1.63 | 2.43
9091.831883 | -6.28 | 2.82
9092.896238 | -0.61 | 2.41
9094.904686 | -18.34 | 2.99
9099.855638 | -4.06 | 2.97
9101.846391 | 3.78 | 2.41
9114.782555 | 2.55 | 2.80
9115.894902 | -6.32 | 2.83
9117.798061 | -0.99 | 2.49
9118.789695 | 7.57 | 2.82
9119.808424 | -1.43 | 3.14
9120.799605 | 9.92 | 2.61
9121.829744 | -1.28 | 2.72
9122.828842 | 7.92 | 2.64
9123.756542 | 3.83 | 2.42
9153.731898 | -2.53 | 2.78
9377.958508 | 4.03 | 2.50
9379.032190 | -2.76 | 2.51
9379.985243 | 2.96 | 2.44
9384.046876 | -0.91 | 2.60
9386.055153 | -3.60 | 2.55
9387.978216 | 5.34 | 3.45
9396.017499 | -11.21 | 2.76
9399.979627 | 0.88 | 2.59
9406.054052 | -3.32 | 2.70
9409.077191 | 4.38 | 2.95
9414.024875 | 0.23 | 3.56
9420.991303 | -20.42 | 3.72
9449.967276 | 7.55 | 2.95
9455.879048 | 1.56 | 2.73
9469.855090 | 3.90 | 2.75
9478.887201 | -5.90 | 3.55
9481.833186 | -8.02 | 2.67
9482.889332 | -2.07 | 2.90
9484.799051 | -3.52 | 2.43
9497.827107 | -4.62 | 3.82
9502.837147 | -0.81 | 2.72
9503.815084 | -5.29 | 3.20
9504.824258 | 5.03 | 3.68

Note. — These values come from the HIRES data alone and are presented before
any gamma offset or jitter have been applied. Times are in BJD-TDB.

Figure 1: Top: RVs for Kepler-323, from HIRES (yellow) over 80 nights between May 2019 and October 2021 combined with 48 RVs from HARPS-N (dark red) (Bonomo et al., 2023). The blue line represents the best fit Keplerian orbits. The residuals are shown in panel b). Bottom: Phase folded RVs and the corresponding best-fit Keplerian orbit for each individual planet. The red circles are the RVs binned in orbital phase. Figure 2: Same as Figure 1, but for Kepler-104 Figure 3: Planet densities vs physical radii for Kepler-104 (dark blue), Kepler-323 (dark purple), Solar System objects (red squares), and planets selected from the NASA Exoplanet Archive according to Appendix A (translucent). The Solar System objects include Mercury, Venus, Earth, Mars, Uranus, Neptune, Pluto and the Moon. Non-Solar System objects are colored by their orbital period (see legend). The composition curves consistent with 100% Fe, 100% MgSiO3, and 67.5% MgSiO3 \+ 32.5% Fe are taken from Zeng et al. (2019), while the maximal collisionally stripped composition curve is taken from Marcus et al. (2010). Figure 4: Same as Figure 3, but with the y-axis corresponding to mass instead of density. Figure 5: Individual planet radii against their orbital periods and colored by their mass partitioning (Gilbert & Fabrycky, 2020) for 649 planets (229 planetary systems). For planets colored gray (459 planets in 121 systems), we do not provide an estimate for the mass partitioning because the system has a planet missing from the steps outlined in Appendix A. In addition to our measurements for Kepler-323 and Kepler-104 (squares), LTT 1445 A, TOI-125, Kepler-30 and the Inner Solar System are highlighted for comparison. Figure 6: The posterior distribution of mass partitioning values, based on 1000 draws from the posteriors of the planet masses, for the systems described in Section 4.1. The solid black line represents the median mass partitioning of the sample. The dashed lines represent the best fit mass partitioning value for Kepler-104 (blue), Kepler-323 (purple), and the Inner Solar System (red). The shaded regions represent the 1$\sigma$ errors. Figure 7: Posterior distributions for all derived parameters from the Keplerian best-fit orbit of Kepler-323. There is no strong covariance between the masses of planets b and c Figure 8: Same as Figure 7, but for Kepler-104. Figure 9: The posterior distribution of mass partitioning values for the Kepler-323 system, based on 5000 draws from the posteriors of the planet masses. The solid black line represents the calculated mass partitioning from the maximum likelihood mass probabilities. The three dotted lines represent the mass partitioning value in hypothetical scenarios where one (orange), two (gold), or three (brown) planets of zero mass are also present in the system. The green dashed line represents another hypothetical scenario, where there is a third planet with a mass of $5.5\,M_{\earth}$, which minimizes the mass partitioning value. The peak in the posterior near zero is real. Figure 10: Same as Figure 9, but for Kepler-104 Figure 11: Potential masses of Kepler-323 b vs potential masses of Kepler-323 c for 5000 random draws from the posteriors of the individual planets, colored by the mass partitioning (Gilbert & Fabrycky, 2020). The yellow dot shows the calculated value from the maximum likelihood values and the black line represents a mass partitioning of exactly zero. Figure 12: Individual planet radii against their orbital periods and colored by their gap complexity (Gilbert & Fabrycky, 2020) for 649 planets (229 planetary systems). For planets colored gray (353 planets in 161 systems), we cannot provide an estimate for the gap complexity because either the system only has two planets, of which gap complexity is undefined, or the system has a planet missing due to the steps outlined in Appendix A. In addition to our measurements for Kepler-323 and Kepler-104 (squares), LTT 1445 A, TOI-125, Kepler-167 and the Inner Solar System are highlighted for comparison. Table 2:  Planet Orbital Properties |  $P$ (days) |  $T_{c}$ (BJD) | $K(m/s)$
---|---|---|---
Kepler-323 b | $1.67832545\pm 0.000002$ | $2454969.052935\pm 0.0009$ | $1.37^{+0.73}_{-0.69}$
Kepler-323 c | $3.553829671\pm 0.000005$ | $2454967.64145\pm 0.001$ | $2.78\pm 0.74$
Kepler-104 b | $11.427545631\pm 0.000009$ | $2454970.613679\pm 0.0006$ | $3.24^{+0.91}_{-0.89}$
Kepler-104 c | $23.66836427\pm 0.00003$ | $2454965.71366\pm 0.001$ | $1.79^{+0.98}_{-0.91}$
Kepler-104 d | $51.75529576\pm 0.00008$ | $2455104.08787\pm 0.001$ | $1.07^{+0.92}_{-0.7}$
Table 3: Planet Physical Properties  | Msini (M$\earth$) | Radius (R$\earth$) | Density (g/cc)
---|---|---|---
Kepler-323 b | $2.0^{+1.2}_{-1.1}$ | $1.348\pm 0.042$ | $4.5^{+2.8}_{-2.4}$
Kepler-323 c | $6.5\pm 1.6$ | $1.533\pm 0.050$ | $9.9^{+2.7}_{-2.5}$
Kepler-104 b | $10.0\pm 2.8$ | $2.380\pm 0.060$ | $4.1^{+1.2}_{-1.1}$
Kepler-104 c | $7.1^{+3.8}_{-3.5}$ | $2.360\pm 0.077$ | $2.9^{+1.7}_{-1.5}$
Kepler-104 d | $5.5^{+4.6}_{-3.5}$ | $2.635\pm 0.144$ | $1.6^{+1.5}_{-1.1}$
Table 4: Values of Mass Partitioning | Best-Fit | Minimized w/Extra Planet | 1 Massless Planet | 2 Massless Planets | 3 Massless Planets
---|---|---|---|---|---
Kepler-323 | $0.28^{+0.25}_{-0.18}$ | 0.09 | 0.46 | 0.52 | 0.55
Kepler-104 | $0.03^{+0.15}_{-0.02}$ | 0.02 | 0.14 | 0.19 | 0.22

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