# Evolving laws and cosmological energy

João Magueijo<EMAIL_ADDRESS>Theoretical Physics Group, The
Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ,
United Kingdom

###### Abstract

We couple the issue of evolution in the laws of physics with that of
violations of energy conservation. Avoiding a time dependence in terms of
coordinate time, we define evolution as a function of time variables
canonically dual to “constants” (such as $\Lambda$, the Planck mass or the
gravitational coupling), mimicking a procedure associated with one formulation
of unimodular gravity. We then introduce variability via a dependence of other
fundamental “constants” on these clocks. Although this is not needed, sharper
results are obtained if this procedure violates local Lorentz invariance,
which we define in the spirit of Horava-Lifshitz theories (modifying a $3+1$
split action, so that a Lorentz invariant 4D reassembly is no longer
possible). We find that variability in the “laws of physics” generically leads
to violations of energy conservation if either a matter parameter varies as a
function of a gravitational clock, or a gravity parameter depends on a matter
clock, with the other combinations sterile. In general the matter components
associated with the varying parameter or the clock absorb or give off the
violated energy. We illustrate this with a variety of clocks (ticking
unimodular time, Ricci time, etc) and parameters (mainly the gravitational and
matter speed of light, but also $\Lambda$). We can accommodate in this
construction (and improve on) several early Varying Speed of Light models,
allowing for variability effects related to the spatial curvature and
$\Lambda$ to cause creation of radiation and a Hot Big Bang.

##  I Introduction

John Wheeler reputedly stated that “everything comes out of higgledy-
piggledy”, to convey the view that the laws of Nature are subject to (possibly
random) mutation, rather than being set in stone PaulDavis . In this paper we
investigate whether such a mutability could be the origin of the matter and
energy in our Universe. Symmetries and conservation laws are intimately
connected (as enshrined in Noether’s theorem); specifically the time-
translation symmetry of the laws of physics implies conservation of matter and
energy. This suggests a logical connection between Wheeler’s view and cosmic
creation out of nothing. A major question hanging over this possibility is how
to define the “time in terms of which” the laws evolve. The definition of time
is closely related to the laws one accepts (to the extent that many
propositions in any system are circular), so the construction of evolving laws
and their associated times may indeed be higgledy-piggledy.

In this paper we take a rather conservative view on the matter (for more
radical alternatives see, for example, Smolin ; Nielsen ). We propose
interweaving two strands of thought. Along one strand, it has for long been
known that one can frame evolution in the physical laws in terms of
variability in the fundamental constants they employ. This goes back to at
least Dirac’s seminal paper dirac  (see anth  and references therein).
Following another strand, one can obtain robust definitions of time by
demoting the constants of nature to mere constants of motion (as done for the
cosmological constant in unimodular gravity unimod1 ; unimod ; UnimodLee1 ;
alan ; daughton ; sorkin1 ; sorkin2 ; Bombelli ; UnimodLee2 , specifically
according to the procedure of unimod ). Employing the recipe in unimod , one
finds that the canonical conjugates of the demoted physical constants are
excellent candidates for physical, relational clocks, capable of surviving
quantum gravity, among other blessings (see gielen ; gielen1 , for example).

As an exploratory hypothesis, in this paper we propose a “parting of
constants”: perhaps the fate of some constants is to provide clocks via the
procedure of unimod1 ; unimod ; UnimodLee1 ; alan ; daughton ; sorkin1 ;
sorkin2 ; Bombelli ; UnimodLee2 , whereas the fate of others is to vary in
terms of such clocks. This orderly evolution might not have been to Wheeler’s
taste, but, as we will see, it will allow us a measure of pragmatic progress.

##  II A possible stage for evolution

We consider variability within a set of target parameters $\bm{\beta}$, which
could be any fundamental constant. The starting point is an action $S_{0}$
(which can be standard General Relativity or not) with all “constants” usually
set to 1 restored, placed inside the space-time integral, and their different
roles dissociated in the following sense. For example, $c$, colloquially the
“speed of light”, plays several roles that are usually conflated, but which
have no reason to be identified once we consider variability in “$c$”. This
was stressed in EllisVSL , and applies to other “constants” demoted from their
status: to give another example, Newton’s constant $G$ has the double role of
defining the Planck scale (appearing in the gravitational commutation
relations) and of gravitational coupling to matter, and the two can be
dissociated vikman ; JoaoLetter . We may thus consider a target $\bm{\beta}$
which includes $c_{P}$ and $G_{P}$ (appearing in the Planck scale, multiplying
the gravity action), $c_{g}$ (the $c$ in the gravity metric), $c_{m}$ (the $c$
in the matter metric) and the coupling between matter and gravity, $G_{M}$. We
may then fix some of these, identify others, or impose a constraint between
them. We could also consider further parameters, such as the electron charge
$e$, and further dissociations of roles.

###  II.1 The unimodular time prototype

As already stressed, the question then is: variability as a function of what?
Rather than allowing a coordinate time $t$ to provide the brutal answer
(thereby breaking time reparameterization invariance, as in AM ), we propose
the use of physical, “relational” time(s). These may be defined mimicking the
Henneaux and Teitelboim (HT) formulation unimod  of unimodular gravity unimod1
; unimod ; UnimodLee1 ; alan ; daughton ; sorkin1 ; sorkin2 , well known for
producing a physical measure of time dual to the cosmological constant
$\Lambda$: the so-called 4-volume or unimodular time unimod ; Bombelli ;
UnimodLee2 .

In the HT formulation of unimodular theory full diffeomorphism invariance is
preserved (i.e. they are not restricted to volume preserving ones), but one
adds to the base action $S_{0}$ an additional term:

$S_{0}\rightarrow S=S_{0}+S_{U}=S_{0}-\int
d^{4}x\,\rho_{\Lambda}(\partial_{\mu}{\cal T}^{\mu}_{\Lambda}).$ (1)

Here ${\cal T}^{\mu}_{\Lambda}$ is a density, so that the added term is
diffeomorphism invariant without the need of the metric or the connection.
Since the metric and connection do not appear in the new term, the Einstein
equations (and other field equations) are left unchanged. In the standard
theory $S_{0}$ does not depend on ${\cal T}^{\mu}_{\Lambda}$; hence, one
equation of motion states the on-shell constancy of $\rho_{\Lambda}$. In
addition, the zero-mode of the zero component of ${\cal T}^{\mu}_{\Lambda}$:

$T_{\Lambda}(\Sigma)\equiv\int_{\Sigma}d^{3}x\,{\cal T}^{0}_{\Lambda}$ (2)

(where $\Sigma$ is a spatial leaf within a 3+1 Hamiltonian splitting) provides
a definition of time, canonically dual to $\rho_{\Lambda}$. For standard
General Relativity we have:

$T_{\Lambda}(\Sigma)=-\int^{\Sigma}_{\Sigma_{0}}d^{4}x\sqrt{-g}$ (3)

that is, minus the 4-volume between $\Sigma$ and a conventional “zero-time”
$\Sigma_{0}$ leaf to its past, or unimodular time unimod . We will often omit
the $\Sigma$ in our expressions.

More generally JoaoLetter ; JoaoPaper , we may select a set of $D$ constants
$\bm{\alpha}$ and perform a similar exercise:

$S_{0}\rightarrow S=S_{0}+S_{U}=S_{0}-\int
d^{4}x\,{\bm{\alpha}}\cdot\partial_{\mu}\mathbf{\cal T}_{\bm{\alpha}}^{\mu}$
(4)

where the dot denotes the Euclidean inner product in $D$ dimensional space.
Just as it happens for unimodular theory, the zero-modes of the zero
components of the density $\mathbf{\cal T}_{\bm{\alpha}}^{\mu}$ provide
definitions of time $\bm{T}_{\bm{\alpha}}$, dual to the on-shell constants
$\bm{\alpha}$. Besides including $\rho_{\Lambda}$ in $\bm{\alpha}$ (as in
unimodular theory) we can consider a more general setting, including the
“sequester” pad ; pad1 , where $c_{P}^{2}/G_{P}$ (responsible for the Planck
mass) is an element of $\bm{\alpha}$, its canonical conjugate providing a
Ricci time clock.

###  II.2 Evolution potentials

We define “evolution in the laws of physics” by making the parameters
${\bm{\beta}}$ vary according to specified functions of the relational times
$\bm{T}_{\bm{\alpha}}$:

$\bm{\beta}=\bm{\beta}(\bm{T}_{\bm{\alpha}}).$ (5)

If $\bm{\beta}$ overlap with $\bm{\alpha}$ this amounts to imposing a second-
class constraint, but we do not consider this situation here. (The
implications of such constraints vary from a technical nuissance to downright
inconsistency, depending on the case; this is the reason for our “parting” the
constants into 2 classes.) Instead, we investigate a sub-class of theories
where some constants (the $\bm{\alpha}$) are deconstantized so that their
duals provide clocks, $\bm{T}_{\bm{\alpha}}$; and where a distinct set of
“constants” (the $\bm{\beta}$) are allowed to vary as functions of the
physical clocks provided by the former.

There are many combinations and choices for $\bm{\alpha}$ and $\bm{\beta}$ (to
be explored in this paper and in a sequel), but given our emphasis on Lorentz
invariance violation (LLIV) in this paper we will take as a starting point the
selection (to be changed later):

$\displaystyle{\bm{\beta}}$ $\displaystyle=$
$\displaystyle\left(c_{g}^{2},c_{m}^{2},...\right)$ (6)
$\displaystyle{\bm{\alpha}}$ $\displaystyle=$
$\displaystyle\left(\rho_{\Lambda},\frac{3c_{P}^{2}}{8\pi
G_{P}},\frac{G_{M}}{G_{P}},...\right),$ (7)

where the extra factor of 6 in $\alpha_{2}$ is included for later convenience.
For standard General Relativity:

$S_{0}=\int d^{4}x\,\sqrt{-g}\left[\frac{\alpha_{2}}{6}R+\alpha_{3}{\cal
L}_{M}-\rho_{\Lambda}\right],$ (8)

and the expressions for $T_{\bm{\alpha}}$ are well-known. Beside $T_{1}\equiv
T_{\Lambda}$, we have

$\displaystyle T_{2}\equiv T_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\int d^{4}x\sqrt{-g}R$ (9) $\displaystyle T_{3}\equiv
T_{N}$ $\displaystyle=$ $\displaystyle\int d^{4}x\sqrt{-g}{\cal L}_{M}$ (10)

i.e.: Ricci and Newton111Since $T_{3}$ is dual to Newton’s $G_{N}$ we propose
this nomenclature. times (see pad ; pad1 ; lomb ; vikman ; vikman1  in
addition to unimod1 ; unimod ; UnimodLee1 ; alan ; daughton ; sorkin1 ;
sorkin2 ; Bombelli ; UnimodLee2 ). $T_{\Lambda}$ and $T_{R}$ are nothing but
the “fluxes” used in the sequester model JoaoPaper ; JoaoLetter ; pad ; pad1 .
We will also toy with $c_{m}$ being included in the $\bm{\alpha}$ rather than
$\bm{\beta}$ and other combinations.

The evolution functions $\bm{\beta}(\bm{T}_{\bm{\alpha}})$ are “givens”, in
the same sense that classical potentials in field theory are givens. This is
supported by analogy, as we shall see in Sec. V.3. If LLIV is present, the
$\bm{\beta}$ functions can be seen as “LLIV-potentials”. In most cases we will
assume they are step functions, modelling abrupt phase transitions, just as in
MoffatVSL ; AM ; VSLreview .

###  II.3 Gauge-invariance

The unimodular-like extensions do not add new local degrees of freedom. This
was remarked in unimod  for $\Lambda$ (see also vikaxion ; unimodProca ) and
is a general feature of all extensions (4). It follows from the invariance of
$S_{U}$ under the gauge symmetry:

${\cal T}_{\Lambda}^{\mu}\rightarrow{\cal
T}_{\Lambda}^{\mu}+\epsilon^{\mu}\quad{\rm
with}\quad\partial_{\mu}\epsilon^{\mu}=0.$ (11)

It leads to first class constraints cancelling out the apparent new local
degrees of freedom. However, the zero-mode of the zeroth component of ${\cal
T}^{0}_{\Lambda}$ given by (2) is gauge-invariant, so globally (and in the
minisuperspace approximation) we do have an extra degree of freedom. For
example, in standard General Relativity with a pure $\Lambda$ we have no
degrees of freedom under homogeneity and isotropy (de Sitter with the pre-
given $\Lambda$ is the only solution), whereas in unimodular theory we can
specify the value of $\Lambda$.

Our $\bm{\beta}$, therefore, do not break gauge-invariance. Had we made the
$\bm{\beta}$ (now functions of space as well) depend on the local ${\cal
T}^{0}({\mathbf{x}})$ (and not just its zero mode) we would have broken the
gauge symmetry. This is interesting (and was investigated in a different
context in unimodProca ), but it will not be considered here. Finally, we note
that, although the gauge-invariant zero mode depends on the foliation
$\Sigma$, the theory does not need to break local Lorentz invariance. However,
it can do so, and the presence of a preferred $\Sigma$ may be a motivation for
doing so. We now explain what we mean by this, and how it relates to the
choice of $\bm{\beta}$.

##  III Breaking local Lorentz invariance

We define local Lorentz invariance violation (LLIV) in the spirit of Horava-
Lifshitz (HL) theory HL . The idea is to take a local Lorentz invariant
action, choose a foliation $\Sigma$, perform a 3+1 split as in the Hamiltonian
formulation, and then change an aspect of the split action so that a local
Lorentz invariant 4D reassembly is no longer possible. The kinetic term of HL
theory is an example of this. One considers a generic kinetic term of the form
HL :

$S_{K}=\frac{c^{3}}{16\pi G}\int
dt\,d^{3}x\,\sqrt{|g|}[K_{ij}K^{ij}-\lambda(K_{i}^{i})^{2}],$ (12)

so that when $\lambda=1$ it reduces to the kinetic term of General Relativity,
and so complies with local Lorentz invariance. One then allows $\lambda\neq
1$, so that the theory no longer derives from the Einstein-Hilbert action.
Although the action is still invariant under foliation-preserving (3D)
diffeomorphisms and time reparameterization, it lost full 4D diffeomorphism
invariance. This is merely an illustration of the process we have in mind, and
we ignore other aspects of HL theory, such as detailed balance.

###  III.1 A gravitational example

In the same spirit, we take the Einstein-Cartan action subject to the
operations described at the start Section II:

$S_{EC}=\int\frac{c_{P}^{2}}{32\pi
G_{P}}\epsilon_{ABCD}e^{A}e^{B}R^{CD}[\Gamma]$ (13)

(here $e^{A}$ is the tetrad and $\Gamma^{A}_{\;\;B}$ is the spin-connection).
We then split it as Thiemann :

$\displaystyle S_{EC}$ $\displaystyle=$ $\displaystyle\int
dt\,d^{3}x\,\frac{c_{P}^{2}}{16\pi
G_{P}}[2\dot{K}^{i}_{a}E^{a}_{i}-(NH+N^{a}H_{a}+$ (14)
$\displaystyle+N^{ij}G_{ij})],$

where $K^{i}_{a}$ is the extrinsic curvature connection222From which the
Ashtekar connection can be built by a canonical transformation Thiemann ,
should we wish to generalize the theory we are about to propose., $E^{a}_{i}$
is the densitized inverse triad, and the last three terms are the Hamiltonian,
Diffeomorphism and Gauss constraints:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{h}}(K^{j}_{a}K^{i}_{b}-K^{i}_{a}K^{j}_{b})E^{a}_{i}E^{b}_{j}-c_{g}^{2}\sqrt{h}R_{3}$
(15) $\displaystyle H_{a}$ $\displaystyle=$
$\displaystyle-2D_{b}(K^{i}_{a}E^{b}_{i}-\delta^{a}_{b}K^{i}_{c}E^{c}_{i})$
(16) $\displaystyle G_{ij}$ $\displaystyle=$ $\displaystyle K_{a[j}E^{a}_{k]}$
(17)

(where $R_{3}$ is the 3D curvature, $h$ is the determinant of the 3 metric and
$D_{a}$ the 3D covariant derivative) enforced by corresponding Lagrange
multipliers $N$, $N^{a}$ and $N^{ij}$. In writing (14) we have chosen units as
follows: the coordinates satisfy $[x^{0}]=T$ and $[x^{i}]=L$; the metric is
dimensionless and so $[e^{A}_{\mu}]=[E^{a}]=[N]=[N^{i}]=L^{0}$; the extrinsic
curvature has units of inverse time squared $[K^{a}_{b}]=1/T^{2}$ (derivatives
with respect to time), but the 3D curvature has units of inverse length
squared, $[R_{3}]=1/L^{2}$; the volume element is written in terms of time and
3D spatial volume. With these choices the pre-factor of the action is
proportional to $c_{P}^{2}/G_{P}$, as written in (14), and a factor of
$c_{g}^{2}$ appears in the last term of the Hamiltonian constraint (15) (with
the Hamiltonian constraint having units of $[H]=1/T^{2}$).

If $c_{g}$ is constant none of this matters. But if we allow its variation
(even if only as a function of a global time variable,
$c_{g}=c_{g}(\bm{T}_{\bm{\alpha}})$, as in Section II), then we have an
analogy with the kinetic term in HL for $\lambda\neq 1$. Although we started
from (13), once we allow $c_{g}=c_{g}(\bm{T}_{\bm{\alpha}})$ the split action
(14) can no longer arise from (13). Any attempt to do so would produce extra
terms in the derivatives of $c_{g}$. Since the connection $\Gamma^{A}_{\;\;B}$
has units of $1/T$, the extra terms in the action arise from:

$\int\epsilon_{ABCD}e^{A}e^{B}\Delta R^{CD}$ (18)

where:

$\Delta
R^{A}_{Bi}=\partial_{\mu}\frac{1}{c_{g}}\Gamma^{A}_{Bi}dx^{\mu}dx^{i}.$ (19)

These terms break local Lorentz invariance. Hence the action (14) obtained by
neglecting these terms when starting from the LI Eq. (13) must be LLIV.

The above choice of units is crucial to LLIV. If $K^{i}_{a}$ and $R_{3}$ are
defined with the same units, then an overall common factor of $c_{g}$ appears
in (15), so that the time variations in $c_{g}$ can be absorbed into a
redefinition of $N$. Also, if we define time as $x^{0}$, that is time with
units of length, no extra terms appear going from (13) to (14) (but see CovVSL
). With such choices, in effect the notation has absorbed the assumption of a
fixed speed of light, with time and space rendered equivalent and the speed of
light losing its meaning.

Upon quantization the gravitational commutation relations inferred from (14)
are:

$\displaystyle[K^{i}_{a}(\bm{x}),E_{j}^{b}(\bm{y})]$ $\displaystyle=$
$\displaystyle i\frac{8\pi
G_{P}\hbar}{3c_{P}^{2}}\delta^{b}_{a}\delta^{i}_{j}\delta(\bm{x}-\bm{y})$ (20)
$\displaystyle=$ $\displaystyle
il_{P}^{2}c_{P}\delta^{b}_{a}\delta^{i}_{j}\delta(\bm{x}-\bm{y}),$

where $l_{P}^{2}=8\pi G_{P}\hbar/c_{P}^{3}$ is the Planck length (note that
the delta function has units of $1/L^{3}$, so it all matches). This justifies
the statement that $G_{P}$ and $c_{P}$ are the relevant variables setting the
Planck scale. Indeed this “speed of light”, $c_{P}$, is nothing but a
conversion factor between different parameterizations of the Planck scale with
different dimensions, and it will be absorbed into a single parameter
($\alpha_{2}$, as in Eq.(7)), and ignored for the rest of this paper.

###  III.2 A matter example

Theories where $\alpha_{3}=G_{N}/G_{P}$ and its dual, $T_{N}$, come into play
offer another example of LLIV in the same spirit (see (7) and (8) for
definitions). Integrations by parts in ${\cal L}_{M}$, which are innocuous in
the standard theory, lead to different theories if $\alpha_{3}$ is included
and there is $T_{N}$ dependence. This induces LLIV if the integration by parts
is designed to do so.

Let us consider extension (4) applied to $\alpha_{3}$, and a foliation
$\Sigma_{t}$ used to define the associated gauge-invariant $T_{N}$ (as in
(2)). The 3+1 split of this term in $S_{U}$ is:

$S=S_{0}-\int dt\,\alpha_{3}(t)\dot{T}_{N}$ (21)

where we have not assumed homogeneity. If there is $T_{N}$ dependence in
$S_{0}$, then $\alpha_{3}$ will change in time but not in space in this
foliation, according to the corresponding Hamilton equation,
$\dot{\alpha}_{3}=\partial H/\partial T_{N}$.

Now take a massless scalar field in flat spacetime as an example. Upon a 3+1
split in the same foliation its action becomes:

$S_{\phi}=\int
d^{3}x\,dt\left(\dot{\phi}\Pi_{\phi}-\frac{N}{2}\left(\Pi_{\phi}^{2}+(\partial_{i}\phi)^{2}\right)\right)$
(22)

and we could perform an integration by parts in time in the first term to
bring this to:

$S_{\phi}=\int
d^{3}x\,dt\left(\dot{\Pi}_{\phi}\phi-\frac{N}{2}\left(\Pi_{\phi}^{2}+(\partial_{i}\phi)^{2}\right)\right)$
(23)

(with a $\Pi_{\phi}\rightarrow-\Pi_{\phi}$ redefinition). Both actions are (at
least classically) equivalent if $\alpha_{3}$, as defined in (8), is not
deconstantized. Both lead to a massless Klein-Gordon equation. Under (21),
however, it matters if in (8) we take (22) or (23) for ${\cal L}_{M}$. If
$S_{0}$ depends on $T_{N}$, so that $\alpha_{3}=\alpha_{3}(t)$ in this
foliation, they are different theories. The first is still Lorentz invariant
and has equation of motion $\partial_{\mu}(\alpha_{3}\partial^{\mu}\phi)=0$.
The second has LLIV, and satisfies:

$-\alpha_{3}(t)\ddot{\phi}+\nabla\phi=0$ (24)

in this foliation. Thus, $\alpha_{3}$ theories may be used to implement LLIV
in the matter sector, even though they do not have to.

Note that even when $\alpha_{3}$ is a constant, the on-shell value of the
action is different in the two theories. In the first, the on-shell Lagrangian
is the pressure (should there be a potential, the Lagrangian would be
$p=K-V$). In the second, it is minus the energy density (minus the
Hamiltonian, since the canonical pair terms vanish). This affects the on-shell
expression for $T_{N}$.

##  IV Reduction to homogeneity and isotropy

Reducing the action $S=S_{0}+S_{U}=S_{G}+S_{M}+S_{U}$ (where $S_{G}$ and
$S_{M}$ are the gravity and the matter base actions) to homogeneity and
isotropy we find:

$\displaystyle S_{G}$ $\displaystyle=$ $\displaystyle V_{c}\int
dt\,\alpha_{2}\bigg{(}\dot{b}a^{2}+Na(b^{2}+kc^{2}_{g})\bigg{)}$ (25)
$\displaystyle S_{M}$ $\displaystyle=$ $\displaystyle V_{c}\int
dt\,\alpha_{3}\bigg{(}\dot{m}_{i}\phi_{i}-Na^{3}(\rho_{\Lambda}+\rho_{M})\bigg{)}$
(26) $\displaystyle S_{U}$ $\displaystyle=$ $\displaystyle V_{c}\int
dt\,\dot{\bm{\alpha}}\cdot{\bm{T}}=V_{c}\int
dt\,\dot{\rho}_{\Lambda}T_{\Lambda}+...\,.$ (27)

Here $a$ is the expansion factor, $b$ is the minisuperspace connection
variable (an off-shell version of the Hubble parameter, since $b=\dot{a}$ on-
shell, if there is no torsion), $k$ is the spatial curvature, $N$ is the lapse
function and $V_{c}=\int d^{3}x$ is the comoving volume of the region under
study, assumed finite throughout this paper (in the quantum cosmology
classical literature one usually chooses $k=1$ and $V_{c}=2\pi^{2}$; see
Afshordi  for a discussion of the criteria for the choice of $V_{c}$). We
allow for a matter action $S_{M}$ with multi-fluids and possibly scalar fields
(see later applications), with $\rho_{M}=\sum_{i}\rho_{i}$ with

$\rho_{i}=\frac{m_{i}}{a^{3(1+w_{i})}}$ (28)

with $w_{i}$ the fluid’s equation of state (here assumed constant), $m_{i}$ a
constant of motion in the usual theory, dual to variables $\phi_{i}$ as
defined in the Lagrangian formulation of perfect fluids of brown , reduced to
MSS as in gielen ; gielen1 ; twot . We will generalize this in Section V.2.

For this action, we therefore have total Hamiltonian:

$H=NaV_{c}\bigg{[}-\alpha_{2}(b^{2}+kc_{g}^{2})+\alpha_{3}(\rho_{\Lambda}+\rho_{M})a^{2}\bigg{]}$
(29)

and Poisson brackets,

$\displaystyle\\{b,A^{2}\\}$ $\displaystyle=$ $\displaystyle\frac{1}{V_{c}}$
(30) $\displaystyle\\{\alpha_{i},T_{i}\\}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{c}}$ (31)

with the conjugate of $b$ given by

$A^{2}=a^{2}\alpha_{2}.$ (32)

Upon quantization the gravitational commutation relation in MSS can be written
as:

$\displaystyle\left[\hat{b},\hat{a^{2}}\right]$ $\displaystyle=$
$\displaystyle i\frac{8\pi
G_{P}\hbar}{3V_{c}c_{P}^{2}}=\frac{il_{P}^{2}c_{P}}{3V_{c}}\equiv
i\mathfrak{h}c_{p}$ (33)

which is a reduced version of (20). The Planck constant $\mathfrak{h}$ is the
relevant quantity for minisuperspace quantum cosmology.

We first assume that $\alpha_{2}$ is kept constant (this leads to a fork in
the formalism, as we will investigate in Section VI.1). Then $a$ and $A$ can
be used interchangeably. In finding the Poisson equations, we stress that, as
per the usual definition of the Poisson bracket, $\bm{\alpha}$ and $\bm{T}$
(and so also $\bm{\beta}$) are kept fixed when evaluating Poisson brackets
involving $b$ and $a$. As a result their equations of motion are obtained from
the usual ones simply by inserting varying constants where usually they appear
as fixed parameters. Hence, for all theories based on this set up (with the
exclusion of those with varying $\alpha_{2}$, as we will see), we have a
Hamilton constraint (obtained by varying with respect to $N$) and Poisson
equations for for $a$ and $b$:

$\displaystyle b^{2}+kc^{2}_{g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{3}}{\alpha_{2}}\rho a^{2}$ $\displaystyle\dot{a}$
$\displaystyle=$ $\displaystyle\\{a,H\\}=Nb$ $\displaystyle\dot{b}$
$\displaystyle=$
$\displaystyle\\{b,H\\}=-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na.$ (34)

with

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\rho_{\Lambda}+\rho_{M}=\rho_{\Lambda}+\sum_{i}\rho_{i}$ (35)
$\displaystyle p$ $\displaystyle=$ $\displaystyle
p_{\Lambda}+p_{M}=-\rho_{\Lambda}+\sum_{i}w_{i}\rho_{i}.$ (36)

This is equivalent to directly inserting varying constants into the unmodified
Friedman and Raychaudhuri equations, as in AM :

$\displaystyle\dot{a}^{2}+kc^{2}_{g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{3}}{\alpha_{2}}\rho a^{2}$ $\displaystyle\ddot{a}$
$\displaystyle=$ $\displaystyle-\frac{\alpha_{3}}{\alpha_{2}}(\rho+3p)]a.$

(in the $N=1$ gauge for simplicity). Assuming further a fixed $\alpha_{3}$
(not necessarily equal to 1), by dotting the first equation and comparing with
the second, a consistency relation can be obtained:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{k\alpha_{2}}{a^{2}\alpha_{3}}\frac{dc_{g}^{2}}{dt}$
(37)

(valid in any gauge). This the central Eq.(5) in AM :

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=\frac{3kc_{g}^{2}}{4\pi
G_{M}a^{2}}\frac{\dot{c}_{g}}{c_{g}}$ (38)

given that we fixed $c_{P}$ (together with $\alpha_{2}$) so the barred
densities in AM  there can be dropped.

It is interesting that the most abstruse assumptions in AM  are in fact
natural from the point of view of the construction in Section II. In AM  it
was postulated that varying constants did not produce stress-energy-momentum
$T_{\mu\nu}$, or changed the Einstein equations, something undoubtedly odd,
say, from the perspective of scalar-tensor theories. This is just what happens
here for two reasons. First, the unimodular-like procedure designed in Section
II.1 for producing clocks is such that the metric does not enter $S_{U}$, so
no $T_{\mu\nu}$ is produced. Second the theory has a Hamiltonian structure, so
that the Poisson equations for one set of variables are obtained fixing all
other variables. Hence the Einstein equations in their 3+1 split form are
obtained fixing all other phase space variables including the time variables
${\bm{T}}_{\bm{\alpha}}$ and by extension the varying constants $\bm{\beta}$.

We add that this is not a miracle and in fact it is not true in general. If we
do not fix $\alpha_{2}$, then new terms do appear in the Einstein equations,
unless we go beyond minimal assumptions regarding the sympletic/Hamiltonian
structure of the theory, as we will see in Section VI.1. In order to avoid
extra terms under a varying $\alpha_{2}$ one would need to fix the
$\alpha_{2}$ multiplying first term in (25), $\dot{b}a^{2}$, whilst allowing
deconstantization for the $\alpha_{2}$ multiplying the second term and the
gravitational Hamiltonian. Such a procedure goes beyond Section VI.1 and
clearly requires LLIV.

##  V Matter-based clocks

We thus justify the central equations of AM , but we can go beyond. A major
weakness of AM  was an absence of information on how the energy conservation
violating last term in (37) would be shared between the various matter
components (including $\Lambda$). This was not an issue in scenarios where
only radiation is present, as envisaged in AM . It turns out we can be more
predictive in this respect, opening up the doors to other scenarios. As a
first rule of thumb, this energy is dumped into (or extracted from) whatever
matter form is conjugate to the relational time variables upon which $c_{g}$
depends (should the clocks be “matter clocks”, as we now illustrate.
Throughout this Section we fix $\alpha_{2}$ and $\alpha_{3}$ (see next Section
for the associated issues).

###  V.1 Pure unimodular varying speed of light (VSL)

The simplest theory follows from $\bm{\alpha}=\rho_{\Lambda}$ (pure unimodular
$S_{U}$) and $\bm{\beta}=c_{g}$, so that $c_{g}=c_{g}(T_{\Lambda})$. Then,
beside Equations (34) (which always apply), we have two more Hamilton
equations:

$\displaystyle\dot{T}_{\Lambda}$ $\displaystyle=$
$\displaystyle\\{T_{\Lambda},H\\}=-N\alpha_{3}a^{3}$ (39)
$\displaystyle\dot{\rho}_{\Lambda}$ $\displaystyle=$
$\displaystyle\\{\rho_{\Lambda},H\\}=\frac{\partial H}{\partial
T_{\Lambda}}=-Na\alpha_{2}k\frac{dc_{g}^{2}}{dT_{\Lambda}}.$ (40)

The first equation just states that $T_{\Lambda}$ is the 4-volume time if
$\alpha_{3}=1$ and agrees with (3) modulo a factor of $V_{c}$. By introducing
$V_{c}$ in Poisson bracket (31) we are defining intensive times, i.e. time per
unit of spatial volume. In this case we are defining time as the 4-volume per
unit of 3-volume, so that:

$T_{\Lambda}=-\alpha_{3}\int dtN\,a^{3},$ (41)

with units of time. The time dependence of $c_{g}$ does not affect this
equation, since it only cares about the Hamiltonian dependence on
$\rho_{\Lambda}$.

The second equation implies that the energy source/sink term contained in (37)
goes fully into the vacuum energy, since it implies (when combined with (39)):

$\dot{\rho}_{\Lambda}+3\frac{\dot{a}}{a}(\rho_{\Lambda}+p_{\Lambda})=\dot{\rho}_{\Lambda}=\frac{k\alpha_{2}}{a^{2}\alpha_{3}}\frac{dc_{g}^{2}}{dt}.$
(42)

This is true even if there are other forms of matter $\rho_{i}$ and is the
simplest example of a pattern which we now uncover by generalization: the
violations of energy conservation are absorbed by the matter ingredient dual
to the clock.

###  V.2 The matter speed of light, $c_{m}$

Turning now to the matter speed of light, $c_{m}$, we note that it can vary
(or be fixed) independently from $c_{g}$ (e.g. EllisVSL ; Coleman ). For
definiteness, we model matter as a mixture of isentropic fluids according to
brown ; gielen ; gielen1 ; twot . Imposing homogeneity and isotropy the action
becomes:

$\displaystyle S_{M}$ $\displaystyle=$ $\displaystyle V_{c}\int
dt\sum_{i}\left(a^{3}n_{i}\dot{\Theta}_{i}-Na^{3}\rho_{i}(n_{i})\right).$ (43)

For each specie $i$, $\Theta_{i}$ is a Lagrange multiplier, so that $n_{i}$ is
the (spatial) volume density of a conserved particle number (i.e.:
$\Pi_{i}=n_{i}a^{3}$ is conserved). Thermodynamics shows that the chemical
potential and pressure are given by:

$\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\frac{p_{i}+\rho_{i}}{n_{i}}=\frac{\partial\rho_{i}}{\partial
n_{i}}$ (44) $\displaystyle p_{i}$ $\displaystyle=$ $\displaystyle
n_{i}\frac{\partial\rho_{i}}{\partial n_{i}}-\rho_{i}.$ (45)

Restoring $c_{m}$ we have in general:

$\rho_{i}=\rho_{i}(n_{i},c_{m}),$ (46)

so that:

$\mu_{ic}=\frac{\partial\rho_{i}}{\partial c_{m}^{2}}$ (47)

can be thought of as the speed of light “chemical” potentials, in analogy with
(44). For example, for dust $\rho=nm_{0}(c_{m})c_{m}^{2}$ (with $m_{0}$ a
constant in some cases); for black body radiation $\rho=n^{4/3}\hbar c_{m}$.
The $c_{m}$ chemical potentials are a property of each matter specie just like
$w_{i}$. Thus, detailed particle physics enters the cosmological
argument333Obviously we could complicate the model by attributing different
$c_{mi}$ to different species as in Carroll ..

If we include $c_{m}^{2}$ in $\bm{\alpha}$, then (43) becomes:

$S_{M}=V_{c}\int
dt\left[\dot{c}_{m}^{2}T_{c}+\sum_{i}\left(\Pi_{i}\dot{\Theta}_{i}-Na^{3}\rho_{i}\left(\frac{\Pi_{i}}{a^{3}},c_{m}^{2}\right)\right)\right].$
(48)

If $w_{i}\neq-1$ is constant, then $\rho_{i}=f_{i}n_{i}^{1+w_{i}}$, where
$f_{i}$ is a dimensionful proportionality constant brown . Hence:

$\rho_{i}=n_{i}^{1+w_{i}}f_{i}(c_{m}^{2})$ (49)

and we can perform a canonical transformation from $\\{\Pi_{i},\Theta_{i}\\}$
to $\\{m_{i},T_{mi}\\}$ so that:

$S_{M}=V_{c}\int
dt\left[\dot{c}_{m}^{2}T_{c}+\sum_{i}\left(\dot{m}_{i}T_{mi}-N\frac{m_{i}f_{i}(c_{m}^{2})}{a^{3w_{i}}}\right)\right]$
(50)

leading to the time formula:

$\displaystyle\dot{T}_{c}$ $\displaystyle=$
$\displaystyle-N\sum_{i}\frac{m_{i}f_{i}^{\prime}}{a^{3w_{i}}}.$ (51)

For a model where $c_{g}\in\beta$ only depends on $T_{c}$ we have:

$\displaystyle\dot{c}_{m}^{2}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial T_{c}}=-Na\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial T_{c}}$
(52)

resulting in:

$\displaystyle\dot{\rho}_{i}+3\frac{\dot{a}}{a}(\rho_{i}+p_{i})$
$\displaystyle=$
$\displaystyle\frac{m_{i}\dot{f}_{i}}{a^{3(1+w_{i})}}=\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt}$
(53) $\displaystyle=$
$\displaystyle\frac{k\alpha_{2}}{a^{2}}\frac{-\frac{Nf_{i}^{\prime}m_{i}}{a^{3w_{i}}}}{\dot{T}_{c}}\frac{dc_{g}^{2}}{dt}$

which summed over $i$ leads to Eq. (37), as it should. But we have learnt
more: in this theory the more a specie depends on $c_{m}$ the more it is
affected by energy conservation violations, an extension of the pattern in the
previous subsection.

We could instead have $c_{m}\in\bm{\beta}$, and in particularly set
$c=c_{g}=c_{m}$. We could also write
$\rho_{\Lambda}=m_{\Lambda}f_{\Lambda}(c_{m}^{2})$, and take $m_{\Lambda}$ for
$\bm{\alpha}$ leading to a slightly modified unimodular clock, with
$\dot{T}_{\Lambda}=-Na^{3}f_{\Lambda}^{\prime}$. Then:

$\displaystyle\dot{\rho}_{\Lambda}=$
$\displaystyle(m_{\Lambda}f_{\Lambda})^{.}=\dot{m}_{\Lambda}f_{\Lambda}+m_{\Lambda}f^{\prime}_{\Lambda}\frac{dc_{m}^{2}}{dt},$
(54)

whereas for the other matter components ($i\neq\Lambda$):

$\displaystyle\dot{\rho}_{i}+3\frac{\dot{a}}{a}(\rho_{i}+p_{i})$
$\displaystyle=$
$\displaystyle\frac{m_{i}\dot{f}_{i}}{a^{3(1+w_{i})}}=\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt}.$

The extra terms in $\dot{c}_{m}$ are
$\partial{\bm{\beta}}/\partial{\bm{T}}_{\bm{\alpha}}$ terms. The first term in
(54) can be expanded using:

$\displaystyle\dot{m}_{\Lambda}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial T_{\Lambda}}=\frac{\partial H_{g}}{\partial
T_{\Lambda}}+\frac{\partial H_{m}}{\partial T_{\Lambda}}$ (55)
$\displaystyle=$ $\displaystyle-Nak\alpha_{2}\frac{\partial
c_{g}^{2}}{\partial
T_{\Lambda}}+Na^{3}\sum_{i}\frac{\partial\rho_{i}}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{\Lambda}}$

so we find after some algebra:

$\dot{\rho}_{\Lambda}=\left(\frac{k\alpha_{2}}{a^{2}}-\sum_{i\neq\Lambda}\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\right)\frac{dc^{2}}{dt},$
(56)

(where we have set $c_{m}=c_{g}$). Again, summing over all the $i$ we get
(37). Even though the net production of matter is not affected by
$c_{m}(T_{\Lambda})$, the energy equations for each specie receive a
source/sink term dependent on their $c_{m}$ chemical potentials. If $c_{m}$ is
part of $\bm{\beta}$ it can be seen as an interaction potential between the
different components, allowing for interchange of matter.

###  V.3 An analogy

We could use the fluids as clocks ($m_{i}\in\bm{\alpha}$ with conjugate
$T_{mi}$ ticking “temperature time” for radiation, proper time for dust
particles, for example) and likewise with any scalar field (as was done in
gielen ; gielen1 ). We will not explore this option in this paper because it
is less fundamental than targeting proper constants; however here we draw
attention to an interesting analogy. Take a massless scalar field:

$S_{\phi}=V_{c}\int
dt\left(\dot{\phi}\Pi_{\phi}-N\frac{\Pi_{\phi}^{2}}{2a^{3}}\right),$ (57)

so the field $\phi$ and its conjugate $\Pi_{\phi}$ satisfy:

$\displaystyle\dot{T}_{\phi}\equiv\dot{\phi}$ $\displaystyle=$ $\displaystyle
N\frac{\Pi_{\phi}}{a^{3}}$ (58) $\displaystyle\dot{\Pi}_{\phi}$
$\displaystyle=$ $\displaystyle-\frac{\partial H}{\partial\phi}.$ (59)

It then looks as if we can use $\Pi_{\phi}$ as a constant $\bm{\alpha}$ and
$\phi$ as its conjugate time, but this is at best an effective description.
Nonetheless if for example we set $c_{g}=c_{g}(T_{\phi})$ we find that the
non-conservation equation for $\phi$ can be written as:

$\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}=\alpha_{2}k\frac{\partial
c_{g}^{2}}{\partial T_{\phi}}.$ (60)

since $\rho_{\phi}=p_{\phi}=\dot{\phi}^{2}/2$. It suggests interpreting:

$U_{LLIV}=-\frac{3kc_{P}^{2}c_{g}^{2}(\phi)}{8\pi G_{P}}$ (61)

as a LLIV potential, source of energy conservation violations. It makes sense
to look at $\bm{\beta}(\bm{T}_{\bm{\alpha}})$ as “potentials”.

##  VI Gravity clocks

The examples in the Section V show a preliminary pattern. Suppose a physical
clock is made of matter: it has a matter constant of motion, which is the
canonical dual variable to its variable “moving hands”. If the laws of nature
depend on the time ticked by this clock via a gravitational parameter ($c_{g}$
in our example), then there are energy conservation violations. This
surplus/deficit of energy goes into the clock’s constant of motion, and so
into the clock system. We saw this happen if the clock is unimodular time,
with $\Lambda$ collecting or giving off the energy. In the case of a $c_{m}$
clock, the dual constant of motion is shared by several fluids, with the total
energy violation apportioned between the matter components depending on how
much each contributes to the clock.

However, we already saw this pattern break down if a matter parameter,
instead, depends on a matter clock (as with $c_{m}(T_{\Lambda}$)). The pattern
also fails to apply to gravitational parameters depending on gravitational
clocks. To explore this, we need to develop formalism for $T_{R}$ and $T_{N}$,
bringing $\alpha_{2}$ and $\alpha_{3}$ into the calculation.

###  VI.1 LLIV theories with Ricci and Newton clocks

The calculation in Section IV is different if the Planck scale $\alpha_{2}$ is
one of the $\bm{\alpha}$ and the $\bm{\beta}$ depend on its associated time
(Ricci time, $T_{R}$, see (9)). Then, it is $A$ (as defined in (32)) and not
$a$ that should be used when computing Poisson brackets. It is useful to write
the Hamiltonian as a function of $A$ instead of $a$ (or simply to evaluate the
Euler-Lagrange equations, in this case). The equations of motion are:

$\displaystyle b^{2}+kc^{2}_{g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{3}}{\alpha_{2}}\rho a^{2}$ (62)
$\displaystyle\dot{a}+\frac{\dot{\alpha}_{2}}{2\alpha_{2}}a$ $\displaystyle=$
$\displaystyle Nb$ (63) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{3}}{2\alpha_{2}}(\rho+3p)Na$ (64)

and the central assumption in AM  is violated (a new term appears in the
second equation). We remark that for $\alpha_{3}=1$ these equations of motion
are the same as those for a non-dynamical scalar field in the Jordan frame
(i.e. Brans-Dicke theory in the first order formulation with $\omega_{BD}=0$)
as studied in flanagan ; vollick ; vollick2 444We can also rephrase the new
term in the equations as a theory with torsion, due to a non-fixed Lambda
potential, (as in Lee2 ; tom , in this case with the even-parity torsion given
by $T=\dot{\alpha}/(2N\alpha)$). The crucial difference here is the unimodular
addition, and the fact that the potentials are non-local functions of the
conjugate of the dilaton, rather than of the dilaton itself..

A non-conservation equation can be obtained as before, dotting the Hamilton
constraint (62), and using (63) and (64) to eliminate $b$ and $\dot{b}$. It
may then be convenient to define densities converted into geometrical
quantitities (with units of $1/T^{2}$):

$\tilde{\rho}_{i}=\frac{\alpha_{3}}{\alpha_{2}}\rho_{i}$ (65)

(and likewise for $p$), to find:

$\dot{\tilde{\rho}}+3\frac{\dot{a}}{a}({\tilde{\rho}}+\tilde{p})=-\frac{\dot{\alpha}_{2}}{\alpha_{2}}\frac{{\tilde{\rho}}+3\tilde{p}}{2}+\frac{k}{a^{2}}\frac{dc_{g}^{2}}{dt}.$
(66)

Reverting to the original variables, we have:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=-\frac{\dot{\alpha}_{3}}{\alpha_{3}}\rho+\frac{\dot{\alpha}_{2}}{\alpha_{2}}\frac{\rho-3p}{2}+\frac{k\alpha_{2}}{a^{2}\alpha_{3}}\frac{dc_{g}^{2}}{dt}.$
(67)

The first and last terms in the RHS are the same as in (37), for fixed
$G_{P}$. The second term is new, and is not predicted by AM . We also have the
extra Hamilton equations:

$\displaystyle\dot{T}_{R}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial\alpha_{2}}=\frac{\alpha_{3}}{\alpha_{2}}Na^{3}\frac{\rho-3p}{2}$
(68) $\displaystyle\dot{T}_{N}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial\alpha_{3}}=-Na^{3}\rho$ (69) $\displaystyle\dot{\alpha}_{2}$
$\displaystyle=$ $\displaystyle\frac{\partial H}{\partial
T_{R}}=-Na\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial T_{R}}$ (70)
$\displaystyle\dot{\alpha}_{3}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial T_{N}}=-Na\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial T_{N}},$
(71)

where, we stress, one should keep $A^{2}$ (and not $a^{2}$) fixed evaluating
the derivatives. This applies to the $a$ appearing in the matter Hamiltonian,
and it makes a crucial difference for the $\dot{T}_{R}$ formula. The first
formula agrees with (9) reduced to MSS. The second agrees with (10), modulo
the issues discussed in Section III.2 (if we do not perform a LLIV integration
by parts, $\dot{T}_{N}=Na^{3}p$, or pressure time). In the last two equations
we have assumed that only $c_{g}$ enters $\bm{\beta}$, but these can (and
will) be generalized.

Since we can replace:

$\frac{dc_{g}^{2}}{dt}=\sum_{i}\frac{\partial c_{g}^{2}}{\partial
T_{i}}\dot{T}_{i}$ (72)

in the last term of (67), we find that the terms due to the dependence of
$c_{g}$ on $T_{R}$ and $T_{N}$ cancel out the first two terms. That is, the
dependence of $c_{g}$ on Ricci or Newtonian time does not lead to violations
of energy conservation, with:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=0$ (73)

if $c_{g}$ only depends on $T_{R}$ and $T_{N}$ (if there are other
dependences, these still contribute the usual terms).

This result it makes sense. A gravitational parameter depending on a clock
leads to energy violations absorbed by that clock, but if the clock is
gravitational there is nothing “material” to absorb the violations of energy
conservation. So, there are none. In the language of natural selection, the
evolution is sterile.

###  VI.2 More general theories with varying Planck scale and gravitational
coupling

There is a final twist in the pattern: a matter parameter depending on a
gravitational clock. The cancellations in the previous subsection do not apply
if, for example, $c_{m}(T_{R},T_{N})$. Then, Eq. (67) still holds true (since
it only depends on the Einstein/gravity equations), but (70) and (71) receive
a new term:

$\displaystyle\dot{\alpha}_{2}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial T_{R}}=-Na\alpha_{2}k\frac{\partial c_{g}^{2}}{\partial
T_{R}}+Na^{3}\alpha_{3}\frac{\partial\rho}{\partial c_{m}^{2}}\frac{\partial
c_{m}^{2}}{\partial T_{R}}$ $\displaystyle\dot{\alpha}_{3}$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial T_{N}}=-Na\alpha_{2}k\frac{\partial
c_{g}^{2}}{\partial T_{N}}+Na^{3}\alpha_{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\frac{\partial c_{m}^{2}}{\partial T_{N}},$

resulting in:

$\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)=Na^{3}\frac{\partial\rho}{\partial
c_{m}^{2}}\left(\frac{\alpha_{3}}{\alpha_{2}}\frac{\partial
c_{m}^{2}}{\partial T_{R}}\frac{\rho-3p}{2}-\frac{\partial c_{m}^{2}}{\partial
T_{N}}\rho\right).$

Had we considered multiple components, we would find that the general pattern
is that if a matter $\bm{\beta}$ depends on a gravitational clock, then there
are violations of energy conservation, apportioned between the different
components by the dependence of their energy density on the relevant
$\bm{\beta}$.

##  VII General formula

We now collect the various example we have studied in a single general
formula. We should stress that the cancellations in Section VI.1 are not
mysterious. As we will show in a sequel to this paper sequel , they arise from
the sympletic Hamiltonian structure of the theory alone. The same applies to
the general formula we will derive in this Section. It results from that
structure plus the secondary constraint:

$\dot{H}\approx 0$ (74)

following from $H\approx 0$. This constraint usually amounts to local
matter/energy density conservation and the Bianchi identities for the
geometry, whereas here it will lead to a generalized non-conservation formula.

###  VII.1 General formula in Einstein-Cartan theory

We first derive the general formula with reference to the Einstein-Cartan
action subject to LLIV as defined in Section III (and the assumption of
homogeneity and isotropy). The idea is to repeat the calculations in Section
VI for a general ${\bm{\beta}}(T_{\bm{\alpha}})$. Defining:

$H=H_{G}+H_{M}=\alpha_{2}H_{g}+\alpha_{3}H_{m}=\alpha_{2}H_{g}+\alpha_{3}Na^{3}\rho$

so that $H_{g}$ does not depend on $\alpha_{2}$ (or $\alpha_{3})$, and
evaluating $\dot{H}=0$ we find555Note that we need to use Hamilton’s equations
to eliminate $b$, and this contains new terms in $\dot{\alpha}_{2}$; hence the
need to separate them. :

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)$ $\displaystyle=$
$\displaystyle-\frac{\dot{\alpha}_{3}}{\alpha_{3}}\rho+\frac{\dot{\alpha}_{2}}{\alpha_{2}}\frac{\rho-3p}{2}$
$\displaystyle-\frac{\alpha_{2}}{Na^{3}\alpha_{3}}\left(\sum_{I}\frac{\partial
H_{g}}{\partial\alpha_{I}}\dot{\alpha}_{I}+\sum_{K}\frac{\partial
H_{g}}{\partial\beta_{K}}\dot{\beta}_{K}\right).$

Using the Poisson equations, the term in brackets can be written as:

$\displaystyle\sum_{IK}\frac{\partial
H_{g}}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\beta_{K}}-\frac{\partial
H_{g}}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}},$

and for $I=2,3$ rearranged as:

$\displaystyle-\frac{1}{\alpha_{2}}\sum_{\begin{subarray}{c}I=2,3\\\
K\end{subarray}}\frac{\partial}{\partial\beta_{K}}(H-\alpha_{3}Na^{3}\rho)\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\alpha_{2}}\left(\dot{\alpha}_{2}\dot{T}_{2}+\dot{\alpha}_{3}\dot{T}_{3}-\alpha_{3}Na^{3}\sum_{\begin{subarray}{c}I=2,3\\\
K\end{subarray}}\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}\right)$

so that the first two terms generally cancel the new terms in the continuity
equation. For $I\neq 2,3$ only the matter component in $H$ leads to a non-zero
contribution. Thus:

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)$ $\displaystyle=$
$\displaystyle\alpha_{2}\sum_{\begin{subarray}{c}I\neq 2,3\\\
K\end{subarray}}\frac{\partial\rho}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial
H_{g}}{\partial\beta_{K}}-\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H_{g}}{\partial\alpha_{I}}$ (75)
$\displaystyle-\sum_{\begin{subarray}{c}I=2,3\\\
K\end{subarray}}\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}$

with the last term being nothing but a generalization of the terms discussed
in Section VI.2. In fact, since $\rho$ does not depend on $\alpha_{2}$ or
$\alpha_{3}$, and $\frac{\partial H}{\partial\alpha_{2}}=\frac{\partial
H_{G}}{\partial\alpha_{2}}=H_{g}$, this can be compressed into:

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(\rho+p)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}IK\end{subarray}}\frac{\partial\rho}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial
H_{G}}{\partial\beta_{K}}-\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H_{G}}{\partial\alpha_{I}}$ (76) $\displaystyle-
Na^{3}\rho\sum_{\begin{subarray}{c}K\end{subarray}}\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{N}},$

the last term being the only one that does not fit the simple pattern.

###  VII.2 Generalization

But this result is general and does not even require an action formulation:
the sympletic/Hamiltonian structure is enough sequel . An abridged proof is
provided here (with a more general proof presented elsewhere), by ignoring the
details of gravity and deriving the same result directly from the matter
degrees of freedom.

We assume a set of perfect fluids under homogeneity but that is not needed.
The crucial inputs are the dependence of each specie energy density $\rho_{i}$
on $\alpha_{I}$ and $\beta_{K}$:

$\rho_{i}=\rho_{i}\left(\alpha_{I},\beta_{K}\right)=\rho_{i}\left(\frac{\Pi_{i}}{a^{3}},\alpha_{j},\beta_{K}\right).$
(77)

The diagonal dependence on $\Pi_{i}$ is non-negotiable, so we separated it and
gave it a different index ($i$) from the other $\alpha_{j}$ (within a global
$I$ index for $\bm{\alpha}$). The other dependences are specific to each
theory (see Section V.2 for examples). For $\alpha_{3}$ we assume an
implementation where $\rho_{i}$ does not depend on it; instead the
$\Theta_{i}$ or $T_{mi}$ absorb it in their definition (cf. Eqs. (48) and
(50)).

Then:

$\displaystyle\dot{\rho}_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\rho_{i}}{\partial
n_{i}}\dot{n}_{i}+\sum_{j}\frac{\partial\rho_{i}}{\partial\alpha_{j}}\dot{\alpha}_{j}+\sum_{K}\frac{\partial\rho_{i}}{\partial\beta_{K}}\dot{\beta}_{K}$

with

$\displaystyle\frac{\partial\rho_{i}}{\partial n_{i}}\dot{n}_{i}$
$\displaystyle=$
$\displaystyle(p_{i}+\rho_{i})\left(\frac{\dot{\Pi}_{i}}{\Pi_{i}}-3\frac{\dot{a}}{a}\right)$
$\displaystyle\frac{\partial\rho_{i}}{\partial\alpha_{j}}\dot{\alpha}_{j}$
$\displaystyle=$
$\displaystyle\frac{\partial\rho_{i}}{\partial\alpha_{j}}\frac{\partial
H}{\partial
T_{j}}=\sum_{K}\frac{\partial\rho_{i}}{\partial\alpha_{j}}\frac{\partial
H}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial T_{j}}$
$\displaystyle\frac{\partial\rho_{i}}{\partial\beta_{K}}\dot{\beta}_{K}$
$\displaystyle=$
$\displaystyle-\sum_{I}\frac{\partial\rho_{i}}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}$ (78)

where we have used (45) and $\Pi_{i}=n_{i}a^{3}$ in the first equation, and
Hamilton’s equations for $\dot{\alpha}_{j}$ and $\dot{T}_{I}$ in the others.
But the Hamilton equation for $\Pi_{i}$ implies666Or an equivalent expression
for $m_{i}$, had we used it instead of $\Pi_{i}$; canonical transformations do
not affect this formula.:

$(p_{i}+\rho_{i})\frac{\dot{\Pi}_{i}}{\Pi_{i}}=\frac{\partial\rho_{i}}{\partial\Pi_{i}}\frac{\partial
H}{\partial T_{\pi i}}$ (79)

so this term fits into the same pattern as other $\bm{\alpha}$, once the
metric dependence (the term in $\dot{a}/a$) has been separated. Hence:

$\dot{\rho}_{i}+3\frac{\dot{a}}{a}(p_{i}+\rho_{i})=\sum_{IK}\frac{\partial\rho_{i}}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial
H}{\partial\beta_{K}}-\frac{\partial\rho_{i}}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}.$ (80)

This is the most informative equation we have. It tells us, specie by specie,
the violations of energy conservation for arbitrary $\bm{\beta}$ depending on
arbitrary times $\bm{T}_{\bm{\alpha}}$.

Summing over $i$ we finally get:

$\displaystyle\dot{\rho}+3\frac{\dot{a}}{a}(p+\rho)$ $\displaystyle=$
$\displaystyle\sum_{IK}\frac{\partial\rho}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial
H}{\partial\beta_{K}}-\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H}{\partial\alpha_{I}}$ (81) $\displaystyle=$
$\displaystyle\sum_{IK}\frac{\partial\rho}{\partial\alpha_{I}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial
H_{G}}{\partial\beta_{K}}-\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{I}}\frac{\partial H_{G}}{\partial\alpha_{I}}$ $\displaystyle-
Na^{3}\rho\sum_{\begin{subarray}{c}K\end{subarray}}\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{N}},$

that is, Eq. 76, derived without reference to the gravity action, and so
independent of the gravitational dynamics.

The above can be written covariantly with replacements
$a^{3}\rightarrow\sqrt{h}$ and $3\dot{a}/a\rightarrow\theta=\dot{h}/h$.
Defining $H_{m}=N\sqrt{h}\rho=N\sqrt{h}\sum_{i}\rho_{i}$ the general covariant
expression for non-conservation is:

$\boxed{n^{\nu}\nabla_{\mu}T^{\mu}_{\;\;\nu}=\frac{1}{N\sqrt{h}\alpha_{3}}\sum_{IK}\frac{\partial\beta_{K}}{\partial
T_{I}}\left(\frac{\partial H_{m}}{\partial\alpha_{I}}\frac{\partial
H_{G}}{\partial\beta_{K}}-\frac{\partial
H_{m}}{\partial\beta_{K}}\frac{\partial
H_{G}}{\partial\alpha_{I}}\right)-N\sqrt{h}\rho\sum_{\begin{subarray}{c}K\end{subarray}}\frac{\partial\rho}{\partial\beta_{K}}\frac{\partial\beta_{K}}{\partial
T_{N}}.}$ (82)

where $n^{\mu}$ is the normalized normal to $\Sigma$. Modulo $\alpha_{3}$
issues the right hand side is just the non-local component of the Poisson
bracket $\\{H_{m},H_{g}\\}$ (as explored further in sequel ).

###  VII.3 General pattern

In view of this general formula we can collate and confirm the various
patterns we found in the previous Sections.

As its first term indicates, to get net non-conservation of energy one
possibility is for a gravitational parameter (a $\bm{\beta}$ appearing in
$H_{G}$) to depend on one or more matter clocks (i.e. clocks dual to
$\bm{\alpha}$ appearing in at least one matter variable). This is the case in
Sections V, where $c_{g}$ depends on matter clocks ($T_{\Lambda}$ and
$T_{c}$). The more a specie $i$ contributes to the dual constant of these
clocks (according to a measure which is like a chemical potential for that
constant), the more of a share of violations it gets. The gravitational
parameter can also depend jointly on many clocks (e.g. via functions of the
form $c_{g}(T_{\Lambda},T_{c})$), in which case the partial derivatives with
respect to the clock time controls the apportionment.

Its second and third terms open up another possibility for a net energy
conservation violation: a matter parameter (a $\bm{\beta}$ in $H_{M}$)
dependent on the time ticked by a gravity clock (dual to a $\alpha$ appearing
in $H_{G}$, or to the prefactor $\alpha_{3}$ of $H_{m}$, describing the
strength of the gravitational coupling). This is the case in Section VI.2,
where $c_{m}=c_{m}(T_{R},T_{N})$. The violations are shared by the different
species according to how much their energy density depends on this matter
parameter.

No other combination generates net energy. A purely matter parameter dependent
on a purely matter clock is similar to an interaction term, energy exchanging
between different components, but with no net gain. This is the case in
Section V.2, with $c_{m}(T_{\Lambda})$ if $c_{g}$ is left fixed (or no other
gravity parameter depended on $T_{\Lambda}$). As we will see in Section
VIII.2, this is not useless: for example Lambda could exchange energy with
other forms of matter via this mechanism, leading to an interesting cosmogony
and opportunities to “solve” the $\Lambda$ problem.

Only a purely gravity parameter depending on a gravity clock would lead to no
energy conservation violations in any matter component. This is the case
explored in Section VI.1, with $c_{g}=c_{g}(T_{R},T_{N})$ (and $\alpha_{3}$
treated so that it does not appear in $H_{m}$). Such situation is entirely
sterile from the point of view of matter production.

##  VIII Illustrative LLIV cosmogonies

We finally provide some examples of scenarios for the origin of the matter in
the Universe, focusing on LLIV (deferring non-LLIV scenarios to sequel ). We
can think of the potentials $\bm{\beta}(\bm{\alpha})$ as random choices for
chaotic evolution, selected in/out by their ultimate effects, specifically by
whether they lead to “viable” offspring.

###  VIII.1 VSL scenarios

Foremost we find VSL scenarios relating matter production to departures from
spatial flatness. For example we could take $c^{2}_{m}\in\bm{\alpha}$,
$c^{2}_{g}\in\bm{\beta}$ (with $c_{g}(T_{c})$) as in Section V.2, and
investigate a phase transition with:

$c_{g}^{2}=c^{2}_{g-}H(T_{c\star}-T_{c})+c^{2}_{g+}H(T_{c}-T_{c\star})$ (83)

where $H$ is the Heaviside step function and $T_{\star}$ is a critical time
(possibly, but not necessarily of the order of the Planck scale). From the
Hamilton equations for $a$ and $b$ we know that for such a shock there can
only be delta functions in $\dddot{a}$, so $a$ and $\dot{a}$ must be
continuous. From the Hamiltonian constraint, $a_{-}=a_{+}=a_{\star}$ and
$\dot{a}_{-}=\dot{a}_{+}=\dot{a}_{\star}$ implies:

$\Delta\rho=\frac{k\alpha_{2}}{a_{\star}^{2}\alpha_{3}}\Delta c_{g}^{2},$ (84)

which could also be derived directly from (37). In such a scenario:

$\displaystyle a$ $\displaystyle=$ $\displaystyle
c_{g-}\sqrt{|k|}t\qquad\qquad\qquad(t<t_{\star})$ (85) $\displaystyle=$
$\displaystyle
c_{g-}\sqrt{|k|}\left(\frac{t}{t_{\star}}\right)^{1/2}\qquad(t>t_{\star})$

the only discontinuity happening in $\ddot{a}$.

This much just mimics the calculations in AM , but we can say more. The jump
in $c_{m}^{2}$ can be computed as:

$\frac{dc_{m}^{2}}{dt}=\frac{\partial H}{\partial
T_{c}}=-\frac{Na\alpha_{2}k}{\dot{T}_{c}}\frac{dc_{g}^{2}}{dt}=\frac{k\alpha_{2}}{a^{2}\alpha_{3}\mu_{c}}\frac{dc_{g}^{2}}{dt}.$
(86)

This is a non-linear equation (since $\mu_{c}=\mu_{c}(c_{m}^{2})$) with
solution:

$\int_{c_{m-}^{2}}^{c_{m+}^{2}}\mu_{c}\,dc_{m}^{2}=\frac{k\alpha_{2}}{a_{\star}^{2}\alpha_{3}}\Delta
c_{g}^{2}$

so that indeed:

$\Delta\rho=\rho(c_{m+}^{2})-\rho(c_{m-}^{2})=\frac{k\alpha_{2}}{a_{\star}^{2}\alpha_{3}}\Delta
c_{g}^{2}$ (87)

i.e. Eq. (84). But we also learn that the violations are shared according to:

$\Delta\rho_{i}=\rho_{i}(m_{i},c_{m+}^{2})-\rho_{i}(m_{i},c_{m-}^{2})=\int_{c_{m-}^{2}}^{c_{m+}^{2}}\mu_{ic}\,dc_{m}^{2}.$
(88)

Notice that at least one $m_{i}$ has to be non-zero, otherwise $\dot{T}_{c}=0$
(see Eq.(51)) and $c_{g}$ would never feel the step predicted in
$c_{g}^{2}=c_{g}^{2}(T_{c})$. More generally $\mu_{c}\propto\rho$, so the less
matter there is to begin with, the bigger the jump in $c_{m}$. In this theory
the net violations are (84), since this only depends on the gravitational
action/equations, but in addition $c_{m}$ must change so that this
$\Delta\rho$ is produced given the overall dependence $\rho(c_{m}^{2})$, with
the various species receiving a contribution in proportion to their individual
dependence $\rho_{i}(c_{m}^{2})$. We could have played this game with any
other fundamental matter parameter and its clock, say $e$, $m_{e}$, etc.

We could instead have used the reverse situations, with
$c^{2}_{g}\in\bm{\alpha}$ and $c^{2}_{m}\in\bm{\beta}$. The gravitational
clock associated with $c_{g}^{2}$ ticks as

$\dot{T}_{cg}=\frac{\partial H}{\partial c_{g}^{2}}=Na\alpha_{2}k,$ (89)

and we could consider a step function $c_{m}^{2}=c_{m}^{2}(T_{cg})$. As it
happens there is a formal symmetry between $c_{m}^{2}=c_{m}^{2}(T_{cg})$ and
$c_{g}^{2}=c_{g}^{2}(T_{c})$, because:

$\frac{dc_{m}^{2}}{dc_{g}^{2}}=\frac{\frac{dc_{m}^{2}}{dt}}{\frac{dc_{g}^{2}}{dt}}=-\frac{\frac{\partial
H}{\partial c_{g}^{2}}}{\frac{\partial H}{\partial
c_{m}^{2}}}=\frac{k\alpha_{2}}{a^{2}\alpha_{3}\mu}$ (90)

i.e. $d\beta/d\alpha$ drops out of the calculation leading to (86), whatever
choice we make for $\alpha$ and $\beta$. We then have the same end results
(87) and (88).

The physics is very different, however. Whereas for $c_{g}^{2}(T_{c})$ we need
some matter for the step function to be felt ($T_{c}\propto\rho$), here we
need $k\neq 0$, i.e. some spatial curvature. Otherwise $T_{cg}$ would stop and
$c_{m}$ never feel the jump. Hence, the $T_{cg}$ interpretations sheds light
on the flatness problem. Such a curvature clock stops if $k=0$ or if curvature
becomes negligible (just as a Ricci clock stops for radiation). It also flows
in opposite directions for $k=\pm 1$. Flatness therefore becomes an attractor,
with $\Delta c_{m}^{2}/\Delta c_{g}^{2}$ having the sign of $k/\mu$.

###  VIII.2 Lambda remittances

In the above model the $\Lambda$ problem is simply the statement that
$\mu_{\Lambda c}$ is much smaller than the other $\mu_{ic}$ if $\Lambda=0$
originally (with further fine tuning needed to address the quasi-Lambda
problem quasi-Barrow ). Whilst it is possible that this can be proved from
first principles, we should also explore models allowing the vacuum energy to
change as part of the evolution itself, dumping its energy into a form of
matter. Such vacuum remittances can be achieved either with $\Lambda$ the
generator of a clock ($\rho_{\Lambda}\in\bm{\alpha}$), with a matter
$\bm{\beta}$ depend on $T_{\Lambda}$ (e.g. $c_{m}=c_{m}(T_{\Lambda})$); or
with $\rho_{\Lambda}\in\bm{\beta}$ and dependent on a matter clock, such as
with $\rho_{\Lambda}(T_{c})$.

For example, with $c_{m}=c_{m}(T_{\Lambda})$, we can adapt the calculations in
Section V.2 to find:

$\displaystyle\dot{\rho}_{i}+3\frac{\dot{a}}{a}(\rho_{i}+p_{i})$
$\displaystyle=$
$\displaystyle\frac{m_{i}\dot{f}_{i}}{a^{3(1+w_{i})}}=\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt};\;(i\neq\Lambda)$
$\displaystyle\dot{\rho}_{\Lambda}$ $\displaystyle=$
$\displaystyle-\sum_{i\neq\Lambda}\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt},$
(91)

so that for a shock of the form (83):

$\displaystyle\Delta\rho_{\Lambda}$ $\displaystyle=$
$\displaystyle-\sum_{i\neq\Lambda}\Delta\rho_{i}$
$\displaystyle\Delta\rho_{i}$ $\displaystyle=$
$\displaystyle\rho_{i}(c_{m+}^{2})-\rho_{i}(c_{m-}^{2})=\int_{c_{m-}^{2}}^{c_{m+}^{2}}\mu_{ic}\,dc_{m}^{2};\;(i\neq\Lambda).$

As expected, if we include $\Lambda$ in the accounts, then no net energy is
produced; however there is energy exchange, with a $\Lambda$ remittance into
the rest of the matter, its energy distributed by species according to their
$\mu_{ic}$. This could be done with any other matter parameter elevated to
$\bm{\beta}$.

Alternatively, we could have
$\rho_{\Lambda}=m_{\Lambda}f_{\Lambda}(c_{m}^{2})$ and
$m_{\Lambda}\in\bm{\beta}$ and $c_{m}^{2}\in\bm{\alpha}$ (i.e. a
$m_{\Lambda}(T_{c})$ potential). Then:

$\displaystyle\frac{dc_{m}^{2}}{dt}=\frac{\partial H}{\partial T_{c}}$
$\displaystyle=$ $\displaystyle Na^{3}\frac{\partial\rho_{\Lambda}}{\partial
T_{c}}=Na^{3}f_{\Lambda}m^{\prime}_{\Lambda}$ (92)

so that instead of (53) we find for $i\neq\Lambda$:

$\displaystyle\dot{\rho}_{i}+3\frac{\dot{a}}{a}(\rho_{i}+p_{i})$
$\displaystyle=$
$\displaystyle\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt}=\frac{Nm_{i}f^{\prime}_{i}}{a^{3w_{i}}}f_{\Lambda}m^{\prime}_{\Lambda}$

which, summed over $i$, leads to
$-\dot{\rho}_{\Lambda}=-(\dot{m}_{\Lambda}f_{\Lambda}+m_{\Lambda}\dot{f}_{\Lambda})$
(recall that $\Lambda$ may contribute to (51)).

In both scenarios, whatever caused the hot Big Bang suppressed the vacuum
energy. The Big Bang is a sign of variability involving Lambda and a matter
parameter, with one of them playing the role of clock.

###  VIII.3 Latter-day Big Bangs?

Naturally within this framework one may ask why would the Big Bang be unique?
And could this be related to why Lambda is so small compared to the Planck
scale? The simplest scenario in this respect is a cascade of Big Bangs
siphoning matter out of Lambda when Lambda resurfaces. A Big Bang in the past,
originated by a phase transition due to vacuum domination, would be reflected
in a Big Bang in the future when similar circumstances arise (as seems to be
the case around now).

However, in such a cyclic scenario, the Big Bang temperature would be
$10^{-32}$ smaller each time, which is nonsense. We should therefore
renormalize the Planck mass at the start of each cycle. This could happen in
two ways: either via a dependence on Ricci time, $T_{R}$, or by making
$\alpha_{2}$ a $\bm{\beta}$. Imagine a $\rho_{\Lambda}(T_{c},T_{R})$. Then

$\dot{\alpha}_{2}=Na^{3}\frac{\partial\rho_{\Lambda}}{\partial T_{R}}$ (93)

and beside the remittance (91), we have the correction:

$\displaystyle\dot{\rho}_{\Lambda}$ $\displaystyle=$
$\displaystyle-\sum_{i\neq\Lambda}\frac{m_{i}f^{\prime}_{i}}{a^{3(1+w_{i})}}\frac{dc_{m}^{2}}{dt}+2\frac{Na^{3}}{\alpha_{2}}\rho_{\Lambda}\frac{\partial\rho_{\Lambda}}{\partial
T_{R}}.$ (94)

We can design $\rho_{\Lambda}(T_{c},T_{R})$ so that the Planck scale changes
at each event and any new Big Bang event is at the Planck scale.

##  IX Conclusions and outlook

Where does all the matter in the Universe come from? In this paper we proposed
that the Universe acquired its energy by virtue of “changing the rules of the
game”, viz. evolution in the laws of physics. Rather than implementing
evolution in terms of a coordinate time, we defined it in terms of time
variables, $\bm{T}_{\bm{\alpha}}$, dual to constants, $\bm{\alpha}$, using as
a blueprint one formulation of unimodular gravity unimod  (in which $\Lambda$
plays the role of $\bm{\alpha}$). Variability was then introduced by making
other constants, $\bm{\beta}$, functions of these times,
$\bm{\beta}(\bm{T}_{\bm{\alpha}})$. The main practical advantage over
coordinate time dependence, is that this approach specifies where the energy
is credited. In general there is energy production if either a matter
parameter varies in terms of a gravitational clock, or a gravity parameter
evolves in terms of a matter clock, with that matter component acquiring the
energy violation balance. Other combinations are sterile.

As in Wheeler’s view, we can think of the evolution potentials
$\bm{\beta}(\bm{T}_{\bm{\alpha}})$ as random choices for chaotic evolution,
selected in or out by their ultimate effects. These could be whether “viable”
offspring is produced: only evolution that implies creation of matter is
beneficial, with sterile evolution edited out. In this paper we
“geocentrically” qualified this statement as creation of semi-classical
matter, but this is not strictly necessary. Perhaps the quantum cosmology of
this process is far more interesting, with classical matter creation replaced
by an $S$-matrix process. In Appendix A we sketch the first steps in this
direction, to show that progress is possible, but defer full treatment to
future work. Another offshoot is a classical realization of the Hartle-Hawking
initial conditions, as sketched in Appendix B. This leads to signature change,
a possibly essential feature for more detailed model building, since it
affects the horizon structure of the Universe (see Appendix B).

We stress that the proposal in this paper is distinct from multidimensional
time. On-shell, the various ${\bm{T}}_{\bm{\alpha}}$ are all a function of
each other, so time is classically one dimensional. But the
${\bm{T}}_{\bm{\alpha}}$ start off as independent variables (something which
has dramatic effects in the quantum theory gielen ; gielen1 ; JoaoLetter ;
JoaoPaper ), and index evolution differently should there be a
$\bm{\beta}(\bm{T}_{\bm{\alpha}})$. If there are violations of energy
conservation due to evolution, it is crucial to specify in terms of which of
these clocks evolution takes place (even though on-shell they are all
related). This has physical effects, namely it determines into which matter
component the energy goes/is taken off.

We close with a few comments on future developments. In this paper we focused
on theories exhibiting local Lorentz invariance violations, defined in the
spirit of Horava-Lifshitz theory: a $4=3+1$ split that leads to a $3+1\neq 4$
after one fiddles with some aspect of the split theory. This author started
off this article with the hunch that Lorentz invariance violation would be
favoured by cosmic procreation of matter, but was disabuse by hard
calculations. The shattered remains of this dogma lead to the sequel sequel :
a repeat of this paper without recourse to local Lorentz invariance violation.
We should stress that in our theories, even under LLIV and with time-
dependence, we have a sympletic/Hamiltonian structure. This is very powerful,
and is responsible for most of the results in this paper.

Finally, there is the question of falsifiability. The various scenarios with
matter progeny considered in Section VIII are all equally possible, so one may
raise questions about testability. But producing copious amounts on matter on
near homogeneous patches is just the starting point: one must add
fluctuations. We gave special attention to scenarios with bimetric VSL
($c_{m}\neq c_{g}$) because these are known to produce very distinctive
predictions for the cosmic fluctuations nia ; nia1  (see also Moffatbim ;
cssound ; bim1 ; bim2 ). More directly, we speculate on whether violations of
energy conservation might be accessible on not-so-large scales, for example
during black hole formation. This is particularly relevant if there is
evolution in the laws of physics in our future, as predicted Section VIII.3.
In the presence of foliation-dependent evolution, the region near a black hole
horizon acts as a crystal ball for events in our future. Any future or latter-
day Big Bangs that might be implied, will indeed happen near the black hole
horizon, with possible observational consequences. This matter is currently
under investigation.

##  X Acknowledgments

We thank Bruno Alexandre, Michele Arzano, Giulia Gubitosi, John Moffat, Alex
Vikman and an anonymous referee for shaping this paper. This work was partly
supported by the STFC Consolidated Grants ST/T000791/1 and ST/X00575/1.

##  Appendix A Quantum cosmology

We briefly sketch how a quantum treatment might start. Standard theories with
relational times $\bm{T}_{\bm{\alpha}}$ are known to convert the Wheeler-
DeWitt equation into a Schrödinger equation (e.g. gielen ; gielen1 ;
JoaoLetter ; JoaoPaper ). An independent claim to the same effect was made for
VSL scenarios with hard break of diffeomorphism invariance MoffatQC . For the
VSL scenarios we have proposed in this paper we obtain a time-dependent
Schrödinger equation, similar to what one gets in the interaction picture.
Hence the classical matter creation we have reported can also be seen as
quantum matter creation, and an $S$-matrix approach could be developed, as we
now show.

To fix ideas, we take unimodular VSL (see Section V.1). One can put the
Hamiltonian constraint in the form JoaoPaper :

$0=\frac{1}{b^{2}+kc_{g}^{2}}a^{2}-\frac{3}{\Lambda}\equiv H_{0}-\phi$

with $\Lambda$ replaced by $\phi=3/\Lambda$ and the dynamical Hamiltonian
$H_{0}$ defined by this expression. Choosing $\phi$ (instead of $\Lambda$) for
$\alpha$ and allowing $c_{g}^{2}$ to become a function of $T_{\phi}$, upon
quantization with suitable ordering we obtain the time-dependent Schrödinger
equation:

$\left[\hat{H}_{0}(b,T_{\phi})-i\mathfrak{h}\frac{\partial}{\partial
T_{\phi}}\right]\psi(b,T_{\phi})=0,$

with

$\hat{H}_{0}(b,T_{\phi})=\frac{-i\mathfrak{h}}{b^{2}+kc_{g}^{2}(T_{\phi})}\frac{\partial}{\partial
b}$

following from (33) (where $\mathfrak{h}$ is defined). As is well known, if
$c_{g}$ does not change, one possible solution is:

$\psi(b,T_{\phi};\phi)={\cal
N}e^{-i\frac{\phi}{\mathfrak{h}}T_{\phi}}\psi_{s}(b,\phi)$

where

$\psi_{s}(b,\phi)=e^{i\frac{\phi}{\mathfrak{h}}X_{CS}(b)}=e^{i\frac{\phi}{\mathfrak{h}}\left(\frac{b^{3}}{3}+kc_{g}^{2}b\right)}$

is a minisuperspace version of the Chern-Simons-Kodama state kodama ; realkod
; stephkod . This is the monochromatic solution and superpositions with
different $\Lambda$ are possible (see JoaoPaper ; stephkod  for a discussion
of normalizability and inner product). If $c_{g}^{2}$ changes, the solution is
instead a time-dependent version of the Chern-Simons-Kodama state:

$\psi(b,T_{\phi})={\cal N}{\cal
T}\exp{\left[-\frac{i}{\mathfrak{h}}\int^{T_{\phi}}_{T_{\phi
0}}H_{0}(\tilde{T}_{\phi})d\tilde{T}_{\phi}\right]}\psi(b,T_{\phi 0})$

where $\cal T$ denotes time-ordering in $T_{\phi}$, thus opening up a parallel
with the $S$ matrix framework.

##  Appendix B Signature change and evolution

The fact that the natural variable in the dynamics is often $c^{2}$ suggests
that $c$ could appear in $\bm{\alpha}$ as a square. We could then entertain
scenarios in which the sign of $c^{2}$, and so the signature, changes and this
has a curious relation to the Hartle-Hawking Hartle and Hawking (1983) no-
boundary proposal, in a process reminiscent of VilUni . This could be done
with:

$c_{g}^{2}(T_{\Lambda})=c_{0}^{2}H(T_{\Lambda})-c_{0}^{2}H(-T_{\Lambda})$ (95)

for $k=1$ and $c_{0}^{2}>0$. Then, Eq. (42) implies:

$\Delta\rho_{\Lambda}=\frac{k\alpha_{2}}{a_{\star}^{2}\alpha_{3}}\Delta
c_{g}^{2}=2c_{0}^{2}\frac{k\alpha_{2}}{a_{\star}^{2}\alpha_{3}}$ (96)

which combined with the Hamiltonian constraint implies a swap in the sign of
$\rho_{\Lambda}=\pm\rho_{\Lambda 0}$, equivalent to a redefinition of $N$ and:

$\alpha_{2}(-b^{2}+kc_{0}^{2})+\alpha_{3}\rho_{\Lambda 0}a^{2}\approx 0.$

For $T_{\Lambda}<0$ we therefore obtain a solution to the (real) Euclidean
theory: the 4-sphere. In this scenario, rather than changing signature as a
function of $b$ (as in VilUni ), the change happens as a function of
$T_{\Lambda}$, that is, with proper time evolution. It is curious that Hartle
and Hawking’s “creation out of nothing”, usually seen as an instanton and so a
purely quantum process, could in fact be a classical phenomenon, within a
theory with varying laws and classical signature change. This would
dramatically change the usual discussion of the horizon problem, which would
thus become disconnected from the flatness problem. Options in which the
matter speed of light increases (rather than decreases, as in the usual
solution to the horizon problem) could then be considered, as long as the sign
of $c^{2}$ changed. We could also have early Universe situations where
$c_{m}^{2}>0$ but $c_{g}^{2}<0$, that is Lorentzian spacetime for matter (or
some of matter) and Euclidean gravity, or vice versa.

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11institutetext: Graduate School of Information Science, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8603, Japan
11email<EMAIL_ADDRESS>
22institutetext: School of Information Science, Aichi Institute of
Technology,33institutetext: Department of Gastroenterology and Hepatology,
Nagoya University Graduate School of Medicine,44institutetext: Department of
Endoscopy, Nagoya University Hospital,55institutetext: Technische Universität
München,66institutetext: Strategy Office, Information and Communications,
Nagoya University,
%\\\%Furo-cho,␣Chikusa-
ku,␣Nagoya,␣Aichi,␣464-8601,␣Japanhttp://www.mori.m.is.nagoya-u.ac.jp/~moda/index-e.html

Authors’ Instructions

# Tracking accuracy evaluation of electromagnetic sensor-based colonoscope
tracking method

Masahiro Oda 11 Hiroaki Kondo 11 Takayuki Kitasaka 22 Kazuhiro Furukawa 33
Ryoji Miyahara 33 Yoshiki Hirooka 44 Hidemi Goto 33 Nassir Navab 55 Kensaku
Mori 6611

###### Abstract

This paper reports a detailed evaluation results of a colonoscope tracking
method. A colonoscope tracking method utilizing electromagnetic sensors and a
CT volume has been proposed. Tracking accuracy of this method was evaluated by
using a colon phantom. In the previously proposed paper, tracking errors were
measured only at six points on the colon phantom for the accuracy evaluation.
The point number is not enough to evaluate relationships between the tracking
errors and positions in the colon. In this paper, we evaluated the colonoscope
tracking method based on more detailed measurement results of the tracking
errors. We measured tracking errors at 52 points on the colon phantom and
visualized magnitudes of the tracking errors. From our experiments, tracking
errors in the ascending and descending colons were enough small to perform
colonoscope navigations. However, tracking errors in the transverse and
descending colons were large due to colon deformations.

###### Keywords:

Colon, Colonoscope tracking, CT image, Evaluation

##  1 Introduction

Colonoscopy is conventionally performed as a colon diagnosis or inspection
method. However, colonoscopy may cause discomfort for patients while
diagnosis. Also, colonoscopy has a risk of complication including perforations
of the colon. Success of colon diagnosis under a colonoscope is heavily
depends on physician’s skill.

CT colonography (CTC) is a colon diagnosis method that reduces discomfort and
a risk of complication on patients. Diagnosis is performed by using CT images
of a patient in CTC. Some computer aided diagnosis (CAD) systems for CTC are
commercially available. These systems commonly display 2D or 3D or unfolded
views of the colon for observation. Physicians diagnose the colon to find
polyps or cancers.

Polyps or cancers of the early stages found by CTC CAD systems can be removed
in colonoscopic polypectomy. Colonoscopic polypectomy is a surgery to remove
polyps or cancers. In colonoscopic examinations including colonoscopic
polypectomy, a physician controls a colonoscope based on his/her experience.
Experienced physicians remove polyps or cancers minimizing patient discomfort.
However, colonoscopic examinations performed by inexperienced physicians may
painful for patients. Utilization of a navigation system for colonoscope is
one solution for such problem. Colonoscope navigation systems indicate
positions of the colonoscope tip and targets such as polyp positions while
performing colonoscopic examinations. Colonoscope navigation systems can be
used to reduce overlooking of polyps and to assist inexperienced physicians.
Conventionally, information obtained from CT volumes of patients is utilized
only for the diagnosis stage including diagnosis using CTC CAD systems.
Information obtained from CT volumes contains the polyp positions and the
colon shapes. Such information is useful for the treatment stage including
colonoscopic examinations. The colon shapes obtained from CT volumes can be
used as maps of the colon in colonoscope navigation systems. Also, the polyp
positions can be used as target point in navigations. A colonoscope will be
navigated to polyp positions while performing colonoscope examinations by
utilizing information obtained from CT volumes of patients.

To achieve colonoscope navigation systems, tracking method for colonoscope is
required. Tracking methods of endoscopes including colonoscope have been
proposed by many research groups, which estimate an endoscope tip position in
the organs. For bronchoscope tracking, image-based[1, 2, 3, 4] and sensor-
based[5, 6] tracking methods were reported. Colonoscope tracking is difficult
compared to the tracking for other hollow organs because the colon greatly
deforms during an insertion of the colonoscope. Liu et al.[7] tried to
estimate colonoscope tip movements from the optical flow of colonoscope
videos. This method can track a colonoscope tip without using additional
equipments for the tracking. However, tracking using colonoscope videos is
easily fails when unclear video frames appear. Unclear video frames frequently
appear in colonoscope video because fluid, feces, and bubbles exist in the
colon. The colonoscope tip touches the wall of the colon many times while
colonoscope examinations. It causes black video frames that make interruptions
of tracking. A colonoscope shape tracking system, the Olympus ScopeGuide
(UPD-3), is commercially available. The system detects colonoscope shape in
the colon using electromagnetic (EM) position sensors. Clinical reports about
utilization of the system in colonoscopy have been reported[8]. The system
just displays the shape of the colonoscope without combining CT volumes or CTC
information. Oda et al.[9, 10] and Kondo et al.[11] proposed colonoscope
tracking method using EM position sensors with combination of CT volumes. They
attach EM sensors to colonoscope to obtain the colonoscope shape. They obtain
the colon shape of a patient from a CT volume. Correspondences between the
colonoscope and colon shapes by applying two steps correspondence finding
processes. Based on the correspondences, they find a point in the CT volume
which corresponds to the colonoscope tip position. These methods can track the
colonoscope tip position even if the viewing fields of the colonoscope are not
clear. In their tracking error evaluations, they measured tracking errors only
at six points in a colon phantom. Behaviors of the colon deformation are
differ according to position. Therefore, tracking errors should evaluated at
many positions in the colon.

In this paper, we perform detailed evaluations of the tracking errors of the
colonoscope tracking method using EM sensors[11]. We measured tracking errors
at 52 points in a colon phantom by using the tracking method. Based on the
measurement results, we discuss relationships between the colon deformations
and the tracking errors.

In the section 2, we briefly introduce the colonoscope tracking method
proposed in the reference[11]. Experimental results including tracking error
measurement results are shown in the section 3. Discussion about the
experimental results are described in the section 4.

##  2 Method

###  2.1 Colon centerline and colonoscope line generation

A colon centerline is obtained from a CT volume. We use a region growing
method to extract a colon region from the CT volume. A thinning and a line
smoothing processes are applied to generate a colon centerline.

A colonoscope line that represents the colonoscope shape is obtained by using
EM position sensors. We insert an Aurora 5/6 DOF Shape Tool Type 1 (NDI) to
the colonoscope working channel. The shape tool gives positions and directions
of the colonoscope at seven points. The colonoscope line is calculated by
applying the Hermite spline interpolation to positions and directions measured
by the shape tool.

###  2.2 Coordinate system registration

We generate a modified colon centerline that simulates the shape of the colon
while an insertion of the colonoscope. This process is required because the
colon largely deformed while colonoscope insertions. To generate the modified
colon centerline, we detect sections on the colon centerline that corresponds
the transverse and sigmoid colons. The transverse and sigmoid colon sections
on the colon centerline are replaced with straight line sections. The
transverse and sigmoid colon sections are identified based on the positions
and the shape of the colon centerline.

We register the CT and sensor coordinate systems by using the ICP
algorithm[12]. The ICP algorithm finds a rigid transformation matrix that
minimizes the Euclidean distance between the modified colon centerline and
colonoscope line. The colonoscope line is transformed to the CT coordinate
system by using the rigid transformation matrix.

###  2.3 Colonoscope tip position finding

We find correspondences between each point on the colon centerline and
colonoscope line. This process consists of two steps including a landmark-
based coarse correspondence finding and a length-based fine correspondence
finding. The landmark-based coarse correspondence finding process finds
corresponding point pairs on the colon centerline and colonoscope line at five
anatomical landmarks. The five anatomical landmarks are detected based on
their positions and the shape of the colon. After performing the coarse
correspondence finding, the length-based fine correspondence finding is
applied. The length-based fine correspondence finding process finds
corresponding point pairs on the colon centerline and colonoscope line at all
points on them. This process finds correspondences by using lengths along the
lines. Finally, a point on the colon centerline that corresponds to the tip of
the colonoscope line is defined as the colonoscope tip position in the CT
coordinate system.

##  3 Experiments

The proposed method was evaluated in experiments using a colon phantom. A
colon phantom (KOKEN colonoscopy training model type I-B) (Fig. 1 (a)) and its
CT volume are utilized in our experiments. We used a colonoscope (Olympus
CF-Q260AI). An Aurora 5/6 DOF Shape Tool Type 1 (NDI) is inserted to the
working channel of the colonoscope. The Aurora 5/6 DOF Shape Tool Type 1 has
seven EM sensors.

In colonoscopic examinations, physicians observe the colon by using a
colonoscope while pulling back the colonoscope after insertion up to the
cecum. To simulate this situation, we inserted the colonoscope up to the cecum
of the colon phantom before performing the colonoscope tracking. After the
insertion, we started the colonoscope tracking and measured tracking errors
while pulling back the colonoscope. Definition of the tracking error is
described below.

We evaluated the performance of the proposed method by using a tracking error.
We defined evaluation points (EPs) on the surface of the colon phantom. Points
on the colon phantom surface which have characteristic shapes (such as parts
of the haustral folds or taeniae coli) were selected as EPs. The EPs are
visually identifiable from both of the colon phantom and its CT volume.
Positions of the EPs and indices of them are shown in the Fig. 1 (b). The
position of each EP was projected to the closest point on the colon
centerline. The tracking error is the length along the colon centerline
between an estimated position of the colonoscope tip (estimated by the
colonoscope tracking method) and a position of a projected EP when the real
colonoscope tip comes to the closest position to the marker.

|
---|---
(a) | (b)
Figure 1: (a) Colon phantom. (b) Positions of EPs placed on the surface of the
colon phantom. Purple points are EPs and numbers are indices of EPs. Black
line is colon centerline.

In the reference[11], tracking errors were measured at six EPs. We performed a
detailed evaluation of the colonoscope tracking method by using 52 EPs. We
measured tracking errors of colonoscope insertions in three trials. Figure 2
shows the average tracking errors at each EPs. The segments of the colon
(ascending, transverse, descending, and sigmoid colons) are also shown in this
figure. Measurements were failed at the EPs of indices from 46 to 50 because
we could not find these EPs due to large deformations of the colon phantom
while pulling the colonoscope. We showed the average tracking errors by using
colors on the colon phantom as Fig. 3. In this figure, blue and red colors
indicate small and large average tracking errors. Small tracking errors were
obtained in the ascending and descending colons. Large tracking errors were
obtained in the transverse and sigmoid colons.

Figure 2: Average tracking errors at EPs from three trials. Areas correspond
to the ascending, transverse, descending, and sigmoid colons are indicated by
yellow and blue colors. Average tracking errors of EPs having indices from 46
to 50 are not shown here because measurements were failed. Figure 3: Average
tracking errors at EPs indicated by colors on the colon phantom. Blue and red
colors indicate small and large average tracking errors. White color means
measurement was failed at the region due to deformations of the colon phantom
while pulling the colonoscope. This image is rendered from a CT volume of the
colon phantom.

##  4 Discussion

From the experimental results, relations between regions in the colon and
tracking errors were clearly shown. We measured tracking errors at 52 EPs. The
number of EPs was significantly larger than the reference[11]. Our result is
useful to investigate causes of the tracking errors.

The tracking errors were quite small in the ascending and descending colons.
Most of the tracking errors in the regions were smaller than 40 mm. Tracking
errors in the regions were enough small to perform colonoscope navigations. A
physician who specializes in gastroenterology said that tracking errors
smaller than 50 mm are acceptable for navigations of the colonoscope tip to
polyps. If a polyp comes to a position near the colonoscope tip (about 50 mm
or closer), it is observable from the colonoscope camera. The tracking method
is applicable for colon navigations to find polyps in the ascending and
descending colons. The tracking errors were small in the ascending and
descending colons because these regions not deform largely. The ascending and
descending colons are fixed to the other tissues. It makes small tracking
errors in these regions.

Unlike the ascending and descending colons, the transverse and sigmoid colons
largely moves in the abdominal cavity. The transverse and sigmoid colons
largely deform during colonoscope insertions. The shapes of the transverse and
sigmoid colons during the colonoscope insertions are nearly straight. It
caused the large tracking errors in the transverse and sigmoid colons.
Estimation method of colon deformations during colonoscope insertions is
required to reduce tracking errors.

##  5 Conclusions

This paper reported a detailed evaluation results of the tracking errors of
the colonoscope tracking method. The colonoscope tracking method estimates the
colonoscope tip position in the colon by using EM sensors and a CT volume. We
measured average tracking errors at 52 points in a colon phantom by using the
tracking method. Three trials of measurements were performed. The average
tracking errors in the ascending and descending colons were enough small to
perform colonoscope navigations. However, the average tracking errors in the
transverse and sigmoid colons were large due to deformations of the colon.

#### Acknowledgments.

Parts of this research were supported by the MEXT, the JSPS KAKENHI Grant
Numbers 24700494, 25242047, 26108006, 26560255, and the Kayamori Foundation of
Informational Science Advancement.

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