11institutetext: School of Physics and Astronomy, University Of Nottingham,
Nottingham NG7 2RD, UK22institutetext: Nottingham Centre Of Gravity,
Nottingham NG7 2RD, UK

# Clockwork Cosmology

Kieran Wood 1,2 Paul M. Saffin 1,2 and Anastasios Avgoustidis
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The higher order generalisation of the clockwork mechanism to gravitational
interactions provides a means to generate an exponentially suppressed coupling
to matter from a fundamental theory of multiple interacting gravitons, without
introducing large hierarchies in the underlying potential and without the need
for a dilaton, suggesting a possible application to the hierarchy problem. We
work in the framework of ghost free multi-gravity with “nearest-neighbour”
interactions, and present a formalism by which one is able to construct
potentials such that the theory will always exhibit this clockwork effect. We
also consider cosmological solutions to the general theory, where all metrics
are of FRW form, with site-dependent scale factors/lapses. We demonstrate the
existence of multiple deSitter vacua where all metrics share the same Hubble
parameter, and we solve the modified Einstein equations numerically for an
example clockwork model constructed using our formalism, finding that the
evolution of the metric that matter couples to is essentially equivalent to
that of general relativity at the modified Planck scale. It is important to
stress that while we focus on the application to clockwork theories, our work
is entirely general and facilitates finding cosmological solutions to any
ghost free multi-gravity theory with “nearest-neighbour” interactions.
Moreover, we clarify previous work on the continuum limit of the theory, which
is generically a scalar-tensor braneworld, using the Randall-Sundrum model as
a special case and showing how the discrete-clockwork cosmological results map
to the continuum results in the appropriate limit.

##  1 Introduction

The existence of an exponentially large hierarchy between the interaction
scales of electroweak and gravitational physics remains somewhat shrouded in
mystery. Generically, the mass of the Higgs boson is quadratically unstable to
radiative corrections arising from any new physics with a mass scale in this
large UV window; this is the _hierarchy problem_. Over the years, myriad
potential solutions to this problem have been proposed, though crucially no
experimental evidence that favours any one particular model has been
forthcoming Beyond_SM_lectures ; Criteria_for_natural_hierarchies , and so it
remains important to consider new ideas. Solutions are often based on
supersymmetry Supergravity , where loop cancellations protect the Higgs mass
from such corrections. However, there is another school of thought who purport
that the apparent exponential hierarchy we observe is deceptive, and that the
mass scale of gravity is actually much closer to the electroweak scale.
Historically, this deception has been attributed to (e.g.) the presence of a
warped RS1  or large Hierarchy_at_mm  extra dimension, but more recently a new
set of _clockwork_ models ClockworkGrav ; Deconstructing  have emerged, which
show signs of promise.

Underpinning the clockwork ethos is the realisation that the usual
identification one would make between new physics effects (i.e. UV completion)
and their corresponding interaction scales is not not necessarily a correct
one – a hidden assumption is present. Interaction scales characterise the
strength of some effective interaction, whereas UV completion refers to the
mass scale at which new degrees of freedom must enter, and these two
quantities are _incommensurable_. If one were to take some arbitrary
Lagrangian in natural units and reinsert factors of $\hbar$ and $c$, it would
be immediately clear that masses and interaction scales have different
dimensions, and in fact that the commensurable quantities are masses with
_products_ of scales and couplings (see Giudice  and related discussions in
hbar_expansion  for an explicit demonstration). Indeed, this is why we have
been careful thus far when referring to mass scales and interaction scales, so
as to emphasise their distinction. In natural units, most couplings are
typically $\mathcal{O}(1)$, so the identification usually works in practice,
but it could be the case that UV completion occurs at a much lower energy than
the associated interaction scale, if we were to have particularly small
couplings. It is in this sense that one is able to solve the hierarchy
problem, by using a small enough coupling so that quantum gravity effects can
enter at a mass scale small enough to not bother the Higgs, _while still
maintaining Planck scale interactions_. The question then is how we might
obtain such a small coupling in a natural manner; this is the purpose of the
clockwork.

The clockwork mechanism was initially proposed to construct an axion setup
where the effective axion decay constant becomes super-Planckian Rattazzi , as
is required by cosmological relaxation models Relaxation , but the general
idea has since been generalised to a much wider class of fields Giudice . The
premise is to use a chain of pairwise-interacting fields to generate a
hierarchy between the parameters of the fundamental theory and the effective
coupling to an external matter source, in a manner akin to the mechanism of
gears in a clock (hence the name).

The framework is as follows Giudice : suppose we have a system which contains
$N$ fields, $\phi_{i}$, referred to as ‘gears’ (following the analogy),
arranged in a 1D-lattice in theory space. The gears describe $N$ particles,
which remain massless due to $N$ copies of some symmetry $\mathcal{S}$.
Neither the explicit nature of the fields nor that of the symmetries are
important, but we know that the full symmetry group of the theory contains at
least the product $\mathcal{S}^{N}$ . Now suppose we introduce an interaction
potential between the fields linking nearest-neighbours, characterised by some
parameter $q>1$ which treats the sites _asymmetrically_. Since we are working
on a 1D-lattice with boundaries, we have only $N-1$ interactions, and since
each interaction breaks only the symmetry corresponding to each individual
site, one diagonal copy of $\mathcal{S}$ survives the breaking. As a result,
the system possesses a massless zero-mode i.e. some linear combination of the
original $\phi_{i}$ fields which has mass eigenvalue 0, as well as a tower of
massive modes on top. This zero-mode is, however, not uniformly distributed
throughout the lattice. Because the interaction treats the sites
asymmetrically, the distribution of the zero-mode throughout the lattice is
also asymmetric, and becomes exponentially suppressed at one end. Thus, by
coupling some matter fields to the gear at the suppressed end of the lattice,
one can engineer an exponentially suppressed coupling to the zero-mode. This
idea has seen various interesting applications in recent years Clockwork_WIMP
; Flavour_puzzle ; Clockwork_inflation ; Clockwork_goldstones , and has since
been generalised to allow also for non-nearest-neighbour interactions Non-
nearest_neighbour . A schematic diagram of the classic setup is shown in Fig.
1.

Figure 1: Schematic diagram depicting the classic clockwork setup. The gears
are arranged in a 1D lattice, and we have nearest-neighbour interactions of
strength $q$, which break all but one of the individual symmetries
$\mathcal{S}$. The result is a zero-mode, $a_{0}$, associated with the
unbroken subgroup $\mathcal{S}_{0}$, which is asymmetrically distributed
through the lattice, scaling as
$a_{0}\sim\phi_{0}+\phi_{1}/q+\ldots+\phi_{N-1}/q^{N-1}$. We are then able to
engineer a hierarchy of scales by coupling matter to $\phi_{N-1}$, as the
overlap with $a_{0}$ is exponentially suppressed.

If we apply the clockwork philosophy to gravitational physics, taking our
clockwork gears to be gravitons and our symmetries to be $N$ copies of
diffeomorphism invariance, then we naively have a solution to the
aforementioned hierarchy problem, with the clockwork interactions breaking the
overall symmetry down to one asymmetrically distributed diagonal subgroup of
diffeomorphisms associated with the surviving massless graviton.

As ever with gravity, things are not quite so simple. In
Disassembling_the_clockwork , it was demonstrated that one cannot apply the
clockwork mechanism in the traditional sense Giudice  to non-Abelian theories,
by using a series of elegant group-theoretic arguments to show that such an
asymmetric structure in the unbroken diagonal subgroup is forbidden. The
argument rested upon the assumption of a lack of site dependence in the
couplings between gears, as was the case in the original proposal. The authors
of Giudice  retorted in Giudice_rebuttal , saying that such site independence
could be little more than a statement about the full UV model, if one views
the clockwork as a low-energy EFT. Regardless, if we allow for site dependence
in the couplings then we can once again obtain interesting phenomenology,
though there is some disagreement about whether this defeats the object
Disassembling_the_clockwork , as including site dependence necessarily means
that we must have a degree of hierarchy in the underlying fundamental theory.
Such hierarchies in the fundamental theory have in the past been accounted for
by introducing a dilaton Dilaton ; here we instead take the view that a small
hierarchy in the fundamental theory can be acceptable so long as the couplings
$q$ remain roughly of order unity.

With this in mind, more opportunities avail themselves. In ClockworkGrav , it
was shown that the desired asymmetric distribution of the graviton zero-mode
could be obtained in an entirely new manner through a higher order
generalisation of the standard clockwork mechanism, from a fundamental theory
with only hierarchies up to $q^{2}$, and so no large parameters, provided that
$q\sim\mathcal{O}(1)$. The unbroken diagonal subgroup of diffeomorphisms is
symmetric, as it must be Disassembling_the_clockwork , but the underlying
_background_ vacua follow an asymmetric distribution – the action of the
symmetric subgroup on their fluctuations then results in the desired zero-mode
distribution. This situation was not considered in Disassembling_the_clockwork
, whose background was Minkowski at all sites, and it offers us a very
enticing new prospect to solve the hierarchy problem. Later, it was shown in
Deconstructing , following the dimensional deconstruction philosophy of
Discrete_gravitational_dimensions , that the general ghost free multi-gravity
action used in ClockworkGrav ; Interacting_spin_2  can be viewed as the
deconstruction of a general 5-dimensional continuum theory, where the discrete
set of gears are considered simply as the induced metrics on 4D hypersurfaces
at discrete locations along the extra dimension. In particular, it was shown
that within the class of continuum gravitational clockwork theories reside
scalar-tensor theories where the scalar should _not_ be identified with a
dilaton, distinguishing them from previous approaches. The continuum picture
naturally has the interpretation of a braneworld model, where coupling matter
to one of the gears becomes analogous to placing matter on a brane at the
corresponding location in the 5 dimension. It is no surprise, then, that
amongst the special cases of the general theory put forward in Deconstructing
resides the quintessential braneworld model of Randall and Sundrum, RS1 RS1
(though without any $\mathbb{Z}_{2}$ symmetry – we will discuss this in
Sections 4 and 5.1).

Clearly, much progress has been made surrounding the application of the
clockwork mechanism to gravity in recent years, but still there is much to be
discovered. Indeed, we do not actually know much about which explicit forms of
interaction potential are able to produce the asymmetric background vacuum
structure needed to clockwork the graviton gears, besides the existence of the
particular case considered in ClockworkGrav . Even less is known about the
cosmology of such models, though one might expect some clockworks to possess
interesting phenomenology worth studying, given the presence of RS1 as a
special case in the continuum version of the theory. The aim of the present
work is to shed light on both of these issues, as we develop a general
formalism utilising symmetric polynomials to determine which interactions give
a valid clockwork vacuum, and begin to look at the cosmology by determining
the background evolution for two example models. One of these is essentially a
deconstruction of RS1, which serves as a good consistency check for our
formalism as the phenomenological consequences are well-studied in the
continuum limit Carsten_review ; Langlois_review . The formalism we present is
entirely general and provides a means to determine the background cosmological
solutions to _any_ modified theory of gravity utilising multiple pairwise
interacting metric fields, although in the present work we specialise to
clockwork gravity models due to their potential application to the hierarchy
problem.

The structure of the paper, then, is as follows: in Section 2 we outline the
specifics of clockwork theory and develop our symmetric polynomial formalism;
in Section 3 we specialise to clockwork gravity; in Section 4 we show how the
continuum theory arises and explain how the construction of the 5 dimension
must differ from the usual RS1 orbifold; in Section 5 we look at the
background cosmology of two example models, and finally we conclude in Section
6.

Throughout, we work in natural units $\hbar=c=1$ and use a mostly-plus metric
signature.

##  2 The Clockwork Mechanism

In order to implement the clockwork mechanism as outlined above, we need an
action for a chain of $N$ fields originally exhibiting copies of some symmetry
$\mathcal{S}$, which will be broken by introducing interactions. Of course,
the ultimate goal here is to choose our fields to be gravitons and our
symmetries to be diffeomorphisms, but in this Section we shall work only with
scalar fields, whose associated symmetries are Goldstone shifts inherited from
$N$ copies of $U(1)$. We do this for two reasons: first, for clarity – the
formalism we introduce is much more intuitive in the language of scalars, and
second, as we will see later on, the clockwork gravity potential is identical
to the scalar potential when we look for the background vacuum solution, with
the respective conformal factors in the metrics (which we said fully determine
the structure of the graviton zero-mode ClockworkGrav ) playing the role of
the scalar fields.

The Lagrangian we choose is one for $N$ real scalar fields
$\boldsymbol{\phi}=(\phi_{0},\phi_{1},...,\phi_{N-1})$, which interact via
some homogeneous polynomial potential of degree “$\deg$”, in $D$ dimensions:

$\displaystyle S$ $\displaystyle=\int
d^{D}x\,\mathcal{L}(\boldsymbol{\phi},\partial\boldsymbol{\phi})$ (2.1)
$\displaystyle\mathcal{L}(\boldsymbol{\phi},\partial\boldsymbol{\phi})$
$\displaystyle=-\frac{1}{2}\sum_{i=0}^{N-1}\partial_{\mu}\phi_{i}\partial^{\mu}\phi_{i}-V(\boldsymbol{\phi})\,,$
(2.2)

where the potential is given by:

$V(\boldsymbol{\phi})=\sum_{i_{1},i_{2},...,i_{\deg}=0}^{N-1}T_{i_{1}i_{2}...i_{\deg}}\phi_{i_{1}}\phi_{i_{2}}...\phi_{i_{\deg}}\,,$
(2.3)

and the coefficients $T_{i_{1}i_{2}...i_{\deg}}=T_{(i_{1}i_{2}...i_{\deg})}$
are totally symmetric. The action has a scaling symmetry when we take
$\deg=2D/(D-2)$. Although the clockwork has been generalised to allow for non-
nearest-neighbour interactions Non-nearest_neighbour , we will restrict
ourselves to nearest-neighbour anyway in the knowledge that, when working with
gravity, non-nearest neighbour (loop-type) interactions generically lead to
undesirable Boulware-Deser ghosts BD_Ghosts ; GhostFreedom ; Cycles . This
means that our coefficients $T_{i_{1}i_{2}...i_{\deg}}$are restricted to only
terms of the form $T_{iii...}$, $T_{i+1,ii...}$, $T_{i+1,i+1,i...}$ and so on,
i.e. the potential contains only terms that look like $\phi_{0}^{4}$,
$\phi_{0}^{3}\phi_{1}$, $\phi_{0}^{2}\phi_{1}^{2}$ etc.

The equations of motion that result from the Lagrangian Eq. (2.2) are:

$\partial_{\mu}\partial^{\mu}\phi_{a}-\deg\sum_{i_{2},i_{3},...,i_{\deg}=0}^{N-1}T_{ai_{2}i_{3}...i_{\deg}}\phi_{i_{2}}\phi_{i_{3}}...\phi_{i_{\deg}}=0\,,$
(2.4)

where the factor $\deg$ arises as a result of the symmetry in the
coefficients. If we want there to exist a vacuum solution at
$\boldsymbol{\phi}=\textbf{c}=(c_{0},c_{1},...,c_{\deg})$, we require:

$\sum_{i_{2},i_{3},...,i_{\deg}=0}^{N-1}T_{ai_{2}i_{3}...i_{\deg}}c_{i_{2}}c_{i_{3}}...c_{i_{\deg}}=0\,.$
(2.5)

Perturbing the vacuum solution,
$\boldsymbol{\phi}=\textbf{c}+\delta\boldsymbol{\phi}$, the fluctuations have
dynamics determined via the following second-order action:

$S^{(2)}=\int
d^{D}x\,\left[-\frac{1}{2}\sum_{i=0}^{N-1}\partial_{\mu}\delta\phi_{i}\partial_{\mu}\delta\phi_{i}-\frac{1}{2}\deg(\deg-1)\sum_{i_{3},i_{4},...,i_{\deg}=0}^{N-1}T_{abi_{3}...i_{\deg}}c_{i_{3}}c_{i_{4}}...c_{i_{\deg}}\delta\phi_{a}\delta\phi_{b}\right]\,,$
(2.6)

giving the mass matrix:

$M^{2}_{ab}=\deg(\deg-1)\sum_{i_{3},i_{4},...,i_{\deg}=0}^{N-1}T_{abi_{3}...i_{\deg}}c_{i_{3}}c_{i_{4}}...c_{i_{\deg}}\,.$
(2.7)

Immediately we can see the presence of the zero-mode from equations (2.5) and
(2.7),

$\sum_{b=0}^{N-1}M^{2}_{ab}c_{b}=0\,,$ (2.8)

and hence there is a flat direction along transformations for which
$\delta\phi_{a}\propto c_{a}$.

We would like to impose a hierarchy on the vacuum structure such that one end
of the chain of fields is exponentially suppressed compared to the other; this
is the defining feature of a clockwork model. That is, we want to take our
vacuum solution $\boldsymbol{\phi}=\mathbf{c}$ to look something like:

$c_{a}=\frac{c}{q^{a}}\,,$ (2.9)

for some constant $q\gtrapprox 1$ and some universal scale $c$. This choice of
vacuum is not unique; in principle any vacuum possessing a hierarchy with
$c_{i+1}/c_{i}<1$ would suffice. Eq. (2.9) is simply a natural choice and
serves well to develop our formalism. The question we then want to answer is
whether we can determine a set of symmetric coefficients
$T_{i_{1}i_{2}...i_{\deg}}$ in the potential Eq. (2.3) such that the vacuum
has this hierarchy but the coefficients themselves do not. To begin to answer
this question, we turn to symmetric polynomials to reformulate the problem.

###  2.1 Symmetric Polynomial Formalism

We start this section by defining objects called the _elementary symmetric
polynomials_ :

$\begin{split}e_{0}(x_{1},x_{2},...,x_{\deg})&=1\\\
e_{1}(x_{1},x_{2},...,x_{\deg})&=\sum_{1\leq i\leq\deg}x_{i}\\\
e_{2}(x_{1},x_{2},...,x_{\deg})&=\sum_{1\leq i<j\leq\deg}x_{i}x_{j}\\\
\vdots\\\ e_{k}(x_{1},x_{2},...,x_{\deg})&=\sum_{1\leq
j_{1}<j_{2}<...<j_{k}\leq\deg}x_{j_{1}}...x_{j_{k}}\\\ \vdots\\\
e_{\deg}(x_{1},x_{2},...,x_{\deg})&=x_{1}x_{2}...x_{\deg}\,.\end{split}$
(2.10)

Each $e_{k}$ contains $\binom{\deg}{k}=\frac{\deg!}{k!(\deg-k)!}$ terms in
total. The elementary symmetric polynomials are special because _any_
symmetric polynomial of degree $\deg$ can be expressed in terms of sums and
products of the elementary symmetric polynomials up to $e_{\deg}$; in essence
they act as a basis for general symmetric polynomials (hence their name).

The $e_{k}$ are useful because we know that our coefficients
$T_{i_{1}i_{2}...i_{\deg}}$in the potential are totally symmetric, so we can
introduce the potential polynomial,

$T(x_{1},...,x_{\deg})=\sum_{i_{1},i_{2},...,i_{\deg}=0}^{N-1}T_{i_{1}i_{2}...i_{\deg}}(x_{1})^{i_{1}}(x_{2})^{i_{2}}...(x_{\deg})^{i_{\deg}}\,,$
(2.11)

which is then manifestly symmetric in all of its arguments, and hence can be
expressed in terms of the elementary symmetric polynomials Eqs. (2.10). Then,
when combined with Eqs. (2.5) and (2.9), we get the requirement on the
potential polynomial, and hence on $T_{i_{1}i_{2}...i_{\deg}}$, that will give
us the desired vacuum structure:

$T(x_{1},q^{-1},q^{-1},...,q^{-1})=0\,.$ (2.12)

In addition to this condition, we want $T$ to contain only low powers of $q$
so that there is no fundamental hierarchy in the coefficients
$T_{i_{1}i_{2}...i_{\deg}}$.

We can do a similar thing for the mass matrix, and introduce the mass
polynomial,

$\begin{split}M(x_{1},x_{2})&=\sum_{i_{1},i_{2}=0}^{N-1}M_{i_{1}i_{2}}^{2}(x_{1})^{i_{1}}(x_{2})^{i_{2}}\\\
&=c^{\deg-2}\deg(\deg-1)\sum_{i_{1},i_{2},...,i_{\deg}=0}^{N-1}T_{i_{1}i_{2}...i_{\deg}}(x_{1})^{i_{1}}(x_{2})^{i_{2}}q^{-i_{3}}q^{-i_{4}}...q^{-i_{\deg}}\\\
&=c^{\deg-2}\deg(\deg-1)T(x_{1},x_{2},q^{-1},...,q^{-1})\,,\end{split}$ (2.13)

from which we should readily be able to determine the components of the mass
matrix, once the coefficients $T_{i_{1}i_{2}...i_{\deg}}$have been determined.

Currently, there is a lot of freedom in choosing a potential polynomial that
may do the job for us. However, our restriction to nearest-neighbour
interactions constrains the form of $T$ to _only_ a linear combination of
terms of the form

$T\supset\kappa(e_{\deg})^{p}e_{i}\;,$ (2.14)

for constant $\kappa$ and some power $0\leq p\leq N-1$. Given this
restriction, we can construct the most general possible potential polynomial
as:

$T(x_{1},...,x_{\deg})=\sum_{n=0}^{N-2}\alpha_{n}\left(e_{\deg}(x_{1},...,x_{\deg})\right)^{n}\sum_{m=0}^{\deg}\beta_{m}e_{m}(x_{1},...,x_{\deg})\,,$
(2.15)

where $\alpha_{n}$ and $\beta_{m}$ are constants. This is nothing more than a
general linear combination of terms of the form Eq. (2.14). The coefficients
can be read off as:

$\begin{split}T_{pppp...p}&=\alpha_{p}\beta_{0}+\alpha_{p-1}\beta_{\deg}\\\
T_{\\{p+1\\}^{q}\\{p\\}^{\deg-q}}&=\alpha_{p}\beta_{q}\,,\end{split}$ (2.16)

for $p=0,...,N-1$ and $q=1,...,\deg-1$, with $\alpha_{N-1}=\alpha_{-1}=0$, and
all other $T_{i_{1}i_{2}...i_{\deg}}=0$. Then, all we need to construct a
theory with the desired vacuum hierarchy is to find a set of $\alpha_{n}$ and
$\beta_{m}$ such that Eq. (2.12) is satisfied, with sufficiently small powers
of $q$ to avoid that same hierarchy in $T_{i_{1}i_{2}...i_{\deg}}$.

We can go further here, because we can evaluate the elementary symmetric
polynomials at $(x_{1},q^{-1},...,q^{-1})$ – they are:

$e_{k}(x_{1},q^{-1},...,q^{-1})=\binom{\deg-1}{k-1}x_{1}q^{1-k}+\binom{\deg-1}{k}q^{-k}\;,$
(2.17)

so, the vacuum condition Eq. (2.12) reads:

$\sum_{n=0}^{N-2}\alpha_{n}x_{1}^{n}q^{n(1-\deg)}\sum_{m=0}^{\deg}\beta_{m}\left[\binom{\deg-1}{m-1}x_{1}q^{1-m}+\binom{\deg-1}{m}q^{-m}\right]=0\,.$
(2.18)

Then, comparing coefficients either side, we see that the requirement is that
the coefficients of all powers of $x_{1}$ must vanish separately, which leads
to the following:

$\alpha_{i}\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-1}{m}q^{-m}+q^{\deg}\alpha_{i-1}\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-1}{m-1}q^{-m}=0\;\;\;\forall
i$ (2.19)

If we can find a set of $\alpha_{n}$ and $\beta_{m}$ that satisfies Eq.
(2.19), then we can build a clockwork potential that provides the desired
asymmetrically distributed vacuum solution111Note that here, because of the
two equations at the end points i.e. $i=0$ and $i=N-1$ where one of the
$\alpha$’s vanishes, both of the two sums involving the $\beta_{m}$’s must
vanish separately. However, later in Section 3 we will see that including
matter on the boundaries adds an extra term to Eq. (2.19) that stops this from
being true, so we leave the condition in full here.. One is free to make this
even simpler, by setting all of the non-zero $\alpha$’s to be equal, since
this is just a choice of potential, which turns the above into a condition
only on the $\beta$’s. We will indeed do this going forward to make our lives
easier, but for now the $\alpha_{n}$ remain for completeness.

We can do a similar thing for the mass polynomial Eq. (2.13), where we need to
evaluate $e_{k}(x_{1},x_{2},q^{-1},...,q^{-1})$. These follow a similar
pattern:

$e_{k}(x_{1},x_{2},q^{-1},...,q^{-1})=\binom{\deg-2}{k-2}x_{1}x_{2}q^{2-k}+\binom{\deg-2}{k-1}(x_{1}+x_{2})q^{1-k}+\binom{\deg-2}{k}q^{-k}\,.$
(2.20)

Substituting into our mass polynomial Eq. (2.13) yields the following:

$M(x_{1},x_{2})=K\sum_{n=0}^{N-2}\alpha_{n}\left(e_{\deg}(x_{1},x_{2},q^{-1},...,q^{-1})\right)^{n}\sum_{m=0}^{\deg}\beta_{m}e_{m}(x_{1},x_{2},q^{-1},...,q^{-1})\,,$
(2.21)

defining the constant $K=c^{\deg-2}\deg(\deg-1)$. Expanded out in full, the
mass polynomial reads:

$\begin{split}M(x_{1},x_{2})=&K\sum_{n=0}^{N-2}\alpha_{n}(x_{1}x_{2})^{n}q^{n(2-\deg)}\\\
&\times\sum_{m=0}^{\deg}\beta_{m}\left[\binom{\deg-2}{m-2}x_{1}x_{2}q^{2-m}+\binom{\deg-2}{m-1}(x_{1}+x_{2})q^{1-m}+\binom{\deg-2}{m}q^{-m}\right]\,.\end{split}$
(2.22)

From this expression, we can read off the components of the mass matrix,

$\begin{split}M_{pp}^{2}&=K\left[\alpha_{p}q^{p(2-\deg)}\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m}q^{-m}+\alpha_{p-1}q^{(p-1)(2-\deg)}\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m-2}q^{2-m}\right]\\\
M_{p+1,p}^{2}&=M_{p,p+1}^{2}=K\alpha_{p}q^{p(2-\deg)}\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m-1}q^{1-m}\;,\end{split}$
(2.23)

with the indices on $M^{2}$ running from 0 to $N-1$, again with
$\alpha_{N-1}=\alpha_{-1}=0$, and all other $M^{2}_{i_{1},i_{2}}=0$. We can
write this in a nicer and more symmetric manner by factoring out some of the
$q$’s:

$\begin{split}M_{ij}^{2}&=K(q^{\frac{2-\deg}{2}})^{i+j-1}\\\
&\times\bigg{\\{}\delta_{ij}\left[\alpha_{i}q^{\frac{2-\deg}{2}}A+\alpha_{i-1}q^{-\frac{2-\deg}{2}}B\right]+\alpha_{i+j-1}(\delta_{i,j-1}+\delta_{i-1,j})C\bigg{\\}}\,,\end{split}$
(2.24)

where we have defined the constants:

  * •

$A=\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m}q^{-m}$

  * •

$B=\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m-2}q^{2-m}$

  * •

$C=\sum_{m=0}^{\deg}\beta_{m}\binom{\deg-2}{m-1}q^{1-m}$

for brevity.

Written out in full matrix form (and taking $\alpha_{n}=1\;\forall\;n\neq
0,N-1$ for simplicity), this looks like:

$M^{2}=K\begin{bmatrix}aQ^{-1}&C&0&0&\dots&0&0\\\
C&(a+b)Q&CQ^{2}&0&\dots&0&0\\\ 0&CQ^{2}&(a+b)Q^{3}&CQ^{4}&\dots&0&0\\\
0&0&CQ^{4}&(a+b)Q^{5}&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\dots&(a+b)Q^{2N-5}&CQ^{2N-4}\\\
0&0&0&0&\dots&CQ^{2N-4}&bQ^{2N-3}\end{bmatrix}$ (2.25)

where:

  * •

$Q=q^{\frac{2-\deg}{2}}$

  * •

$a=QA$

  * •

$b=Q^{-1}B$

The symmetric polynomial formalism is powerful; just by determining a set of
numbers that satisfy some relatively simple condition, Eq. (2.19), we are able
to determine a potential that will give the required asymmetric vacuum
structure that characterises the clockwork, as well as the matrix encoding the
masses of the gears. One could hope to find an analytic form for the
eigenvalues of Eq. (2.25), in order to determine the mass gap between the
zero-mode and first massive mode, as this would be the first to show up in
collider experiments. However, thus far this looks impossible except for when
$Q=1$ i.e. when the potential is quadratic Rattazzi ; Giudice ; TridiagEigs .
Nevertheless, it is easy enough to calculate eigenvalues numerically on a
case-by-case basis.

###  2.2 Examples

To demonstrate the usefulness of the formalism, we now use it to reproduce
some results from the clockwork literature, to write down a new clockwork, and
to make a statement about shift symmetric potentials with $\deg>2$.

####  2.2.1 Original (quadratic) clockwork scalar

The original clockwork proposal Rattazzi ; Giudice , as stated in Section 1,
was an axion setup that used a $\deg=2$ shift-symmetric potential for $N$
Goldstone bosons, $\pi_{i}$. Here, we will start from their potential and try
to obtain their mass matrix, using the techniques developed thus far (this
method to find the mass matrix is perhaps a bit overkill, given that we
already know the simple form of the potential, but it serves well as an
illustration of the procedure).

The potential used in Rattazzi ; Giudice  is the following:

$V(\pi)=\frac{m^{2}}{2}\sum_{j=0}^{N-1}(\pi_{j}-q\pi_{j+1})^{2}\;.$ (2.26)

Expanding out the sum, we can extract from this potential the coefficients
$T_{i_{1}i_{2}...i_{\deg}}$:

$T_{00}=\frac{m^{2}}{2}\,,\;\;\;T_{(N-1)(N-1)}=\frac{m^{2}}{2}q^{2}\,,\;\;\;T_{pp}=\frac{m^{2}}{2}(1+q^{2})\,,\;\;\;T_{p(p+1)}=-\frac{m^{2}}{2}q\,,$
(2.27)

for $p=1,...,N-2$. We can then use Eq. (2.16) to determine the values of
$\alpha_{n}$ and $\beta_{m}$ that should comprise our potential polynomial. A
quick check shows that a good choice is $\alpha_{n}=\alpha=m^{2}/2$, with

$\beta_{0}=1\,,\;\;\;\beta_{1}=-q\,,\;\;\;\beta_{2}=q^{2}\,.$ (2.28)

In our formalism, choosing these numbers would be the starting point, and we
would check that they satisfy the vacuum condition and subsequently construct
the potential $V(\pi)$. With these $\alpha$’s and $\beta$’s, the potential
polynomial $T(x_{1},x_{2})$ reads:

$T(x_{1},x_{2})=\frac{m^{2}}{2}\sum_{n=0}^{N-2}\left(e_{2}(x_{1},x_{2})\right)^{n}\left[1-qe_{1}(x_{1},x_{2})+q^{2}e_{2}(x_{1},x_{2})\right]\,.$
(2.29)

Substitution into Eq. (2.19) returns 0, with both sums vanishing separately,
and so the vacuum condition is indeed satisfied as we expect.

Finally, all that remains to determine the form of the mass matrix is to
calculate the constants $A,B$, $C$ and $Q$ that appear in Eq. (2.25). This is
simple in this case, since the potential is quadratic so we have $Q=1$. The
remaining constants are

$A=\beta_{0}\binom{0}{0}q^{0}=1\;,\;\;\;\;B=\beta_{2}\binom{0}{2-2}q^{2-2}=q^{2}\;,\;\;\;\;C=\beta_{1}\binom{0}{1-1}q^{1-1}=-q\,.$
(2.30)

Substituting into Eq. (2.25) gives:

$M=2\,\frac{m^{2}}{2}\begin{bmatrix}1&-q&0&\dots&0\\\ -q&1+q^{2}&-q&\dots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\dots&1+q^{2}&-q\\\
0&0&\dots&-q&q^{2}\end{bmatrix}\,,$ (2.31)

which is exactly the mass matrix in Giudice , and all is well. The authors of
the original paper go on to find closed form solutions for the eigenvalues and
eigenvectors of this matrix, finding in particular the presence of a zero-mode
which follows exactly the structure we have set out ($c_{a}=c/q^{a}$).

####  2.2.2 Quartic clockwork scalar

The second example we consider is a new clockwork theory, whose potential we
build from the ground up using our basic assumptions about the coefficients
$T_{i_{1}i_{2}...i_{\deg}}$. In particular, we stated that we would like to
produce the desired vacuum hierarchy from a general $\deg$ potential without
introducing a similar hierarchy in $T_{i_{1}i_{2}...i_{\deg}}$. Ideally, we
would like this hierarchy to be only up to $q^{2}$, if $q\sim\mathcal{O}(1)$.
In terms of the $\alpha$’s and $\beta$’s, this means that we would like to
ideally have all $\alpha$’s equal and $\beta_{m}\propto q^{\pm 1}$ at most.

Can we still satisfy the vacuum condition Eq. (2.19) for some $\beta$’s with
this property? The vacuum condition tells us that, if we assert that
$\beta_{m}\propto q^{\pm 1}$, for _any_ $\deg$ only terms up to $m=2$ can be
non-vanishing in order for the sums to cancel. In this case, it turns out that
we can indeed satisfy Eq. (2.19), with the choice
$\alpha_{n}=\alpha=1\;\forall\;n$, and,

$\begin{split}\beta_{0}&=\frac{1}{2}\deg(\deg-1)q^{-1}\\\ \beta_{1}&=1-\deg\\\
\beta_{2}&=q\,,\end{split}$ (2.32)

with all other $\beta_{m}=0$222We could also flip everything by a minus sign,
and the choice would still satisfy Eq. (2.19). The original proposal is just
the $\deg=2$ version of this particular choice (with everything multiplied up
by $q$); this is the generalisation to arbitrary $\deg$.

We shall consider the $\deg=4$ case as a simple albeit as yet unstudied
example, and we shall come back to study the gravitational version of this
theory later in Section 5. For $\deg=4$, we have
$\beta_{0}=6q^{-1},\beta_{1}=-3$, and $\beta_{2}=q$. This means that the
potential coefficients are:

  * •

$T_{iiii}=6q^{-1}$

  * •

$T_{i+1,iii}=-3$

  * •

$T_{i+1,i+1,ii}=q$

Recalling that the coefficients are symmetric on exchange of indices, we
reconstruct the potential which produces the desired vacuum solution:

$\begin{split}V(\boldsymbol{\phi})&=T_{0000}\phi_{0}^{4}+4T_{1000}\phi_{0}^{3}\phi_{1}+6T_{1100}\phi_{0}^{2}\phi_{1}^{2}+\ldots\\\
&=6q^{-1}\phi_{0}^{4}-12\phi_{0}^{3}\phi_{1}+6q\phi_{0}^{2}\phi_{1}^{2}+\ldots\\\
&=\frac{6}{q}\sum_{i=0}^{N-2}\phi_{i}^{2}(\phi_{i}-q\phi_{i+1})^{2}\;.\end{split}$
(2.33)

The next step is to determine the mass matrix. For $\deg=4$ we have
$Q=q^{-1}$, and for our choice of $\beta$’s the other constants are:

$A=q^{-1}\;,\;\;\;\;B=q\;,\;\;\;\;C=-1\,,$ (2.34)

which means that the mass matrix has components:

$\begin{split}M_{ij}^{2}&=K(q^{-1})^{i+j-1}\\\
&\times\bigg{\\{}\delta_{ij}\left[(1-\delta_{i,N-1})q^{-2}+(1-\delta_{i,0})q^{2}\right]-(\delta_{i,j-1}+\delta_{i-1,j})\bigg{\\}}\,.\end{split}$
(2.35)

We can diagonalise this mass matrix via an orthogonal field space
transformation, $\phi_{i}=\sum_{j}O_{ij}a_{j}$, where the orthogonal matrix
$O_{ij}$ has its columns given by the mass eigenvectors. In particular,
numerical investigations show the presence of the zero-mode with the correct
structure, $O_{i0}\propto q^{-i}$, and so we have used the formalism to build
a valid clockwork from a $\deg=4$ potential.

We note that this mass matrix is of precisely the same form as the mass matrix
derived for the orignal ‘higher order clockwork gravity’ theory ClockworkGrav
, after a bit of massaging to get their mass matrix into the form used here
(although we work with scalars, we will see in Section 3 that the symmetric
polynomial results carry over to the static vacuum of the gravitational
theory). The model of ClockworkGrav  used a rather convoluted choice for the
coefficients $T_{i_{1}i_{2}...i_{\deg}}$, which consequently lead to there
being a function $F(q)$ out in front of $M^{2}_{ij}$. This just corresponds to
a more complicated choice of $\alpha_{n}$ for us.

####  2.2.3 Shift symmetric potential

Suppose we now have a potential of arbitrary degree which carries a shift
symmetry between adjacent gears. That is, consider potentials of the form:

$V(\boldsymbol{\phi})=\sum_{i=0}^{N-1}(\phi_{i}-q\phi_{i+1})^{\deg}\;,$ (2.36)

which have a valley along $\phi_{i}=q\phi_{i+1}$. The original proposal is the
$\deg=2$ case of this type of potential, but here we work with general $\deg$.

This type of theory cannot work as a clockwork in nature: if we define the
theory in terms of new fields $\chi_{i}=\phi_{i}-q\phi_{i+1}$, whose potential
consists of only self-interactions of the form $\chi_{i}^{\deg}$, then all the
gears are massless unless $\deg=2$ (when the self-interaction is a mass term).
We would like to see how our formalism comes to the same conclusion.

It is not difficult to check that the set of $\beta_{m}$ that produces this
potential is:

$\beta_{m}=(-q)^{m}\beta_{0}\;,$ (2.37)

for arbitrary $\beta_{0}$. Naively, this is a nice choice because it
automatically satisfies the vacuum condition, due to the binomial coefficient
identity,

$\sum_{m=0}^{n}(-1)^{m}\binom{n}{m}=0\;\;\;\;\;(n>0)\;,$ (2.38)

which forces both sums in Eq. (2.19) to vanish.

However, when it comes to the mass matrix, if $\deg>2$, then the same identity
forces the constants $A$, $B$ and $C$ to vanish also, and so the mass matrix
is populated entirely by zeroes, as we expect333When $\deg>2$, the sums for
$A,B$ and $C$ have multiple terms which always mutually cancel, but when
$\deg=2$, which corresponds to the $n=0$ case of the identity (2.38), the sums
have only a single term, which clearly survives – we saw this explicitly for
the original scalar case..

##  3 Discrete Clockwork Gravity

Now we have all we need to begin to turn our attention to gravity. We stated
at the start of Section 2 that the action for clockwork gravity is equivalent
to the scalar case in vacuum, and we will see this explicitly in a short
while, though some groundwork is required before then. The starting point is
the standard multi-metric gravity action in the tetrad formalism (see
Deconstructing ; Interacting_spin_2 ; Cosmo_spin_2 ), with $N$ Einstein-
Hilbert kinetic terms and a $\deg=4$ interaction coupling the various basis
1-forms:

$\displaystyle S$ $\displaystyle=S_{K}+S_{V}+S_{M}$ (3.1) $\displaystyle
S_{K}$ $\displaystyle=\sum_{i=0}^{N-1}\frac{M_{(4)i}^{2}}{4}\int
e^{(i)a}\wedge e^{(i)b}\wedge{\star}{R^{(i)}_{ab}}$ (3.2) $\displaystyle
S_{V}$
$\displaystyle=\sum_{i,j,k,l=0}^{N-1}\int\,T_{ijkl}\varepsilon_{abcd}\,e^{(i)a}\wedge
e^{(j)b}\wedge e^{(k)c}\wedge e^{(l)d}\;,$ (3.3)

where the $T_{ijkl}=T_{(ijkl)}$ are our symmetric coefficients from Section 2,
and the tetrad basis 1-forms are $e^{(i)a}=e_{\mu}^{(i)a}\text{d}{x}^{\mu}$,
with the vierbeins defined through
$g^{(i)}_{\mu\nu}=e_{\mu}^{(i)a}e_{\nu}^{(i)b}\eta_{ab}$. $S_{M}$ is the
action for the collective matter fields coupled to the theory. Indices are
raised/lowered site-wise, Latin indices with $\eta^{(i)}_{ab}$ and Greek
indices with $g^{(i)}_{\mu\nu}$, while we can swap between Latin and Greek
indices using the vierbeins (via change of basis).

$R^{(i)}_{ab}$ is the curvature 2-form associated with the $i$-th tetrad, with
one index lowered by $\eta_{ab}$, and ${\star}{R^{(i)}_{ab}}$ is its Hodge
dual (also a 2-form in $D=4$ dimensions). We have:

$\displaystyle R^{(i)}_{ab}$
$\displaystyle=\frac{1}{2}R^{(i)}_{ab\mu\nu}\text{d}{x}^{\mu}\wedge\text{d}{x}^{\nu}$
(3.4) $\displaystyle{\star}{R^{(i)}_{ab}}$
$\displaystyle=\frac{1}{2}\sqrt{-g}R^{(i)\alpha\beta}_{\;\;\;\;\;\;\;\;ab}\varepsilon_{\alpha\beta\gamma\delta}\text{d}{x}^{\gamma}\wedge\text{d}{x}^{\delta}\;,$
(3.5)

where the $R$’s with 4 indices are components of the $i$-th Riemann tensor.
This kinetic term is nothing more than the usual Einstein-Hilbert action, just
written in a nicer way using differential forms, so as to make computing the
equations of motion simpler – we can see this as follows (suppressing the
$(i)$ indices):

$\begin{split}S_{K}&=\frac{M_{(4)}^{2}}{8}\int
e_{\mu}^{\;a}e_{\nu}^{\;b}\sqrt{-g}R^{\alpha\beta}_{\;\;\;\;ab}\varepsilon_{\alpha\beta\gamma\delta}\text{d}{x}^{\mu}\wedge\text{d}{x}^{\nu}\wedge\text{d}{x}^{\gamma}\wedge\text{d}{x}^{\delta}\\\
&=\frac{M_{(4)}^{2}}{8}\int
e_{\mu}^{\;a}e_{\nu}^{\;b}\sqrt{-g}R^{\alpha\beta}_{\;\;\;\;ab}\varepsilon_{\alpha\beta\gamma\delta}\frac{\varepsilon^{\mu\nu\gamma\delta}}{\sqrt{-g}}({\star}{1})\\\
&=\frac{M_{(4)}^{2}}{8}\int
R^{\alpha\beta}_{\;\;\;\;\mu\nu}\left(4\delta^{\mu}_{[\alpha}\delta^{\nu}_{\beta]}\right)({\star}{1})\\\
&=\frac{M_{(4)}^{2}}{2}\int
R^{\alpha\beta}_{\;\;\;\;\alpha\beta}({\star}{1})\\\
&=\frac{M_{(4)}^{2}}{2}\int\text{d}^{4}{x}\sqrt{-g}R\;,\end{split}$

where in the 4 line we used the fact that the Riemann tensor is antisymmetric
on its last two (2-form) indices.

The action Eq. (3.1) is the same as the standard dRGT action for ghost-free
multi-metric gravity dRGT_1 ; dRGT_2 ; Cycles ; GhostFreedom  in the metric
formulation provided that the Deser-van Niewenhuizen symmetric vierbein
condition,

$e_{\;\;\;\;\;a}^{(i)\mu}e_{\mu}^{(j)b}=e^{(i)\mu b}e^{(j)}_{\mu a}\;,$ (3.6)

is satisfied. Taking nearest neighbour interactions ensures that this is true,
and so avoids the presence of Boulware-Deser ghosts in the clockwork theory
GhostFreedom ; Cycles .

It is important to stress that the multi-gravity theory we are considering is
_entirely_ specified by a choice for both the number of sites and the
potential coefficients $T_{ijkl}$. Therefore, all the results we present in
this Section will hold for a general multi-gravity theory with nearest
neighbour interactions (and so any multi-gravity devoid of the Boulware-Deser
ghost), although we shall later specialise to some choice for the coefficients
corresponding to a clockwork model.

###  3.1 Modified Einstein Equations

We get the equations of motion by varying the action with respect to the
$i$-th tetrad Cosmo_spin_2 , $e^{(i)a}$, resulting in:

$\frac{M_{(4)i}^{2}}{2}e^{(i)b}\wedge{\star}{R^{(i)}_{ab}}+\varepsilon_{abcd}\sum_{jkl}\mathcal{P}(i)T_{ijkl}e^{(j)b}\wedge
e^{(k)c}\wedge e^{(l)c}={\star}T^{(i)}_{a}\;.$ (3.7)

$\mathcal{P}(i)$ counts the number of times $(i)$ appears in the coefficient
$T_{ijkl}$ i.e. terms of the form

  * •

$T_{ijkl}\implies\mathcal{P}(i)=1$

  * •

$T_{iijk}\implies\mathcal{P}(i)=2$

  * •

$T_{iiij}\implies\mathcal{P}(i)=3$

  * •

$T_{iiii}\implies\mathcal{P}(i)=4$ ,

and ${\star}T^{(i)}_{a}$ is the dual of the energy-momentum 1-form
$T^{(i)}_{a}$ associated with matter coupled to the $i$-th site, defined as:

${\star}T^{(i)}_{a}=\frac{\delta S_{M}}{\delta
e^{(i)a}}=\absolutevalue{e^{(i)}}T^{(i)\mu}_{\;\;\;\;\;\;a}\varepsilon_{\mu\nu\alpha\beta}\text{d}{x}^{\nu}\wedge\text{d}{x}^{\alpha}\wedge\text{d}{x}^{\beta}\;,$
(3.8)

with $T^{(i)\mu}_{\;\;\;\;\;\;\nu}=T^{(i)\mu}_{\;\;\;\;\;\;a}e^{(i)a}_{\nu}$
and $\absolutevalue{e^{(i)}}$ the vierbein determinant. If the matter sector
has a metric formulation, this is just the standard energy-momentum tensor of
GR.

In components, after applying the Hodge star, Eq. (3.7) reads:

$M^{2}_{(4)i}G^{(i)\mu}_{\;\;\;\;\;\nu}+24e_{\nu}^{({i}){a}}e_{\;\;\;\;\;{[a}}^{({i}){\mu}}e_{\;\;\;\;\;{b}}^{({i}){\lambda_{1}}}e_{\;\;\;\;\;{c}}^{({i}){\lambda_{2}}}e_{\;\;\;\;\;{d]}}^{({i}){\lambda_{3}}}\sum_{jkl}\mathcal{P}(i)T_{ijkl}e_{\lambda_{1}}^{({j}){b}}e_{\lambda_{2}}^{({k}){c}}e_{\lambda_{3}}^{({l}){d}}=T^{(i)\mu}_{\;\;\;\;\;\;\nu}\;,$
(3.9)

and for brevity we can package up the interaction term into a single tensor
$W^{(i)\mu}_{\;\;\;\;\;\;\nu}$, so that the Einstein equations are:

$M^{2}_{(4)i}G^{(i)\mu}_{\;\;\;\;\;\nu}+W^{(i)\mu}_{\;\;\;\;\;\;\nu}=T^{(i)\mu}_{\;\;\;\;\;\;\nu}\;,$
(3.10)

with, explicitly:

$W^{(i)\mu}_{\;\;\;\;\;\;\nu}=24e_{\nu}^{({i}){a}}e_{\;\;\;\;\;{[a}}^{({i}){\mu}}e_{\;\;\;\;\;{b}}^{({i}){\lambda_{1}}}e_{\;\;\;\;\;{c}}^{({i}){\lambda_{2}}}e_{\;\;\;\;\;{d]}}^{({i}){\lambda_{3}}}\sum_{jkl}\mathcal{P}(i)T_{ijkl}e_{\lambda_{1}}^{({j}){b}}e_{\lambda_{2}}^{({k}){c}}e_{\lambda_{3}}^{({l}){d}}\;.$
(3.11)

With our nearest neighbour restriction for the interactions, the $W$-tensor
expands out as:

$\begin{split}W^{(i)\mu}_{\;\;\;\;\;\;\nu}&=24\times
4e_{\nu}^{({i}){a}}e_{\;\;\;\;\;{[a}}^{({i}){\mu}}e_{\;\;\;\;\;{b}}^{({i}){\lambda_{1}}}e_{\;\;\;\;\;{c}}^{({i}){\lambda_{2}}}e_{\;\;\;\;\;{d]}}^{({i}){\lambda_{3}}}\\\
\times&\Bigl{[}T_{iiii}e_{\lambda_{1}}^{({i}){b}}e_{\lambda_{2}}^{({i}){c}}e_{\lambda_{3}}^{({i}){d}}\\\
&+3T_{iii,i+1}e_{\lambda_{1}}^{({i+1}){b}}e_{\lambda_{2}}^{({i}){c}}e_{\lambda_{3}}^{({i}){d}}\\\
&+3T_{ii,i+1,i+1}e_{\lambda_{1}}^{({i+1}){b}}e_{\lambda_{2}}^{({i+1}){c}}e_{\lambda_{3}}^{({i}){d}}\\\
&+T_{i,i+1,i+1,i+1}e_{\lambda_{1}}^{({i+1}){b}}e_{\lambda_{2}}^{({i+1}){c}}e_{\lambda_{3}}^{({i+1}){d}}\\\
&+T_{i,i-1,i-1,i-1}e_{\lambda_{1}}^{({i-1}){b}}e_{\lambda_{2}}^{({i-1}){c}}e_{\lambda_{3}}^{({i-1}){d}}\\\
&+3T_{ii,i-1,i-1}e_{\lambda_{1}}^{({i-1}){b}}e_{\lambda_{2}}^{({i-1}){c}}e_{\lambda_{3}}^{({i}){d}}\\\
&+3T_{iii,i-1}e_{\lambda_{1}}^{({i-1}){b}}e_{\lambda_{2}}^{({i}){c}}e_{\lambda_{3}}^{({i}){d}}\Bigr{]}\;.\end{split}$
(3.12)

###  3.2 Constraints and Energy Conservation

Because of the individual diffeomorphisms associated with each site, each
Einstein tensor is covariantly conserved with respect to its own (Levi-Civita)
connection:

$\nabla^{(i)\mu}G^{(i)}_{\mu\nu}=0\;,$ (3.13)

but because of our new interaction terms, this means that, individually, the
energy-momentum tensors are not covariantly conserved, but satisfy instead:

$\nabla^{(i)\mu}T^{(i)}_{\mu\nu}=\nabla^{(i)\mu}W^{(i)}_{\mu\nu}\;.$ (3.14)

However, because we also have our overall diagonal diffeomorphism invariance
(which gives rise to the clockwork zero-mode), the matter sector as a whole,
over all sites, is conserved:

$\sum_{i=0}^{N-1}\absolutevalue{e^{(i)}}\nabla^{(i)\mu}T^{(i)}_{\mu\nu}=0\;,$
(3.15)

which therefore implies the following constraint on the interactions due to
Eq. (3.14), which we shall henceforth refer to as the _Bianchi constraint_ :

$\sum_{i=0}^{N-1}\absolutevalue{e^{(i)}}\nabla^{(i)\mu}W^{(i)}_{\mu\nu}=0\;.$
(3.16)

The Bianchi constraint tells us something about what is going on physically.
For example, if we have matter coupled to only one site, then by virtue of Eq.
(3.15) the energy-momentum tensor on that site _is_ conserved individually and
so a free test particle would follow the geodesics of that site’s metric. By
Eq. (3.14), the corresponding $W$-tensor, and as a consequence every other
$W$-tensor, is also conserved individually in this case. The force between two
sources, however, will comprise a contribution from the zero-mode, as well as
suppressed contributions from the additional massive modes. If matter couples
to more than one site, it is not obvious what the ‘physical’ metric should be
and there is more work to be done to try and understand this.

Regardless, we can now begin to use all this technology to try to solve our
modified Einstein equations for cosmological FRW-like solutions.

###  3.3 Vacuum solutions

The first situation we consider is the vacuum case, where there is no matter
on any of the sites i.e. $S_{M}=0$. We shall take the ansatz for the gear
metrics to be:

$\text{d}{s}^{2}_{(i)}=-c_{i}^{2}(t)\text{d}{t}^{2}+a_{i}^{2}(t)\eta_{jk}\text{d}{x}^{j}\text{d}{x}^{k}\;,$
(3.17)

and eventually look for deSitter vacuum solutions of the form
$a_{i}(t)=a_{i,0}e^{H_{i}t}$ for some set of constant Hubble parameters
$H_{i}$, and $a_{i,0}=a_{i}(0)$. Since all of the metrics live on the same
space, we can only rescale the coordinates to fix the lapse and scale factor
of _one_ of the metrics, while the rest must remain free (normally we will
choose to fix them to 1 on an appropriate site e.g. the first site, in vacuum,
or the site where matter couples, if there is a matter coupling).

With this choice for our metric, the Einstein tensor has the following non-
vanishing components:

$\displaystyle G^{(i)0}_{\;\;\;\;\;0}$
$\displaystyle=-3\left(\frac{\dot{a}_{i}}{a_{i}c_{i}}\right)^{2}$ (3.18)
$\displaystyle G^{(i)j}_{\;\;\;\;\;k}$
$\displaystyle=\frac{1}{c_{i}^{2}}\left(-\frac{\dot{a}^{2}_{i}}{a^{2}_{i}}-2\frac{\ddot{a}_{i}}{a_{i}}+2\frac{\dot{a}_{i}}{a_{i}}\frac{\dot{c}_{i}}{c_{i}}\right)\delta^{j}_{k}\;.$
(3.19)

Also, the vierbeins are:

$\displaystyle e_{0}^{({i}){0}}$ $\displaystyle=c_{i}$ (3.20) $\displaystyle
e_{j}^{({i}){k}}$ $\displaystyle=a_{i}\delta^{k}_{j}\;,$ (3.21)

which lead to the following non-vanishing $W$-tensor components, in terms of
the $\alpha$’s and $\beta$’s defined in Section 2 for the potential
coefficients:

$W^{(i)0}_{\;\;\;\;\;\;0}=\alpha_{i}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m}a_{i+1}^{m}a_{i}^{-m}+\alpha_{i-1}\left(\frac{a_{i-1}}{a_{i}}\right)^{4}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m-1}a_{i-1}^{-m}a_{i}^{m}$
(3.22)
$\begin{split}W^{(i)j}_{\;\;\;\;\;\;k}=24\delta^{j}_{k}\Bigl{\\{}\alpha_{i}&\bigl{[}\beta_{0}+\beta_{1}(c_{i+1}c_{i}^{-1}+2a_{i+1}a_{i}^{-1})+\beta_{2}(2c_{i+1}c_{i}^{-1}a_{i+1}a_{i}^{-1}+a_{i+1}^{2}a_{i}^{-2})\\\
&+\beta_{3}(c_{i+1}c_{i}^{-1}a_{i+1}^{2}a_{i}^{-2})\bigr{]}\\\
&+\alpha_{i-1}\bigl{[}\beta_{1}(c_{i-1}c_{i}^{-1}a_{i-1}^{2}a_{i}^{-2})+\beta_{2}(2c_{i-1}c_{i}^{-1}a_{i-1}a_{i}^{-1}+a_{i-1}^{2}a_{i}^{-2})\\\
&+\beta_{3}(c_{i-1}c_{i}^{-1}+2a_{i-1}a_{i}^{-1})+\beta_{4}\bigr{]}\Big{\\}}\;.\end{split}$
(3.23)

So, we have 2 sets of Einstein equations: a modified Friedmann equation coming
from all of the 00-type terms, and a modified Raychaudhuri equation coming
from the $jk$-type terms.

Considering first the modified Friedmann equation, and substituting in our
deSitter ansatz for the vacuum solution, we get:

$\begin{split}3M_{(4)i}^{2}\left(\frac{H_{i}}{c_{i}}\right)^{2}=&\alpha_{i}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m}a_{i+1,0}^{m}a_{i,0}^{-m}e^{m(H_{i+1}-H_{i})t}\\\
&+\alpha_{i-1}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m-1}a_{i-1,0}^{4-m}a_{i,0}^{m-4}e^{(4-m)(H_{i-1}-H_{i})t}\;.\end{split}$
(3.24)

Note that we still we have the lapse on the left hand side that we must deal
with. Thankfully, we are able to make some progress here due to the Bianchi
constraint, Eq. (3.16). In Appendix A, we show that if there is no matter
coupling, or matter couples to one site only, then the only way to satisfy the
constraint is to take the lapses to be given in terms of the scale factors as,

$c_{i}=\frac{\dot{a}_{i}}{\dot{a}_{I}}\;,$ (3.25)

where $I$ is the site whose lapse we fix to 1 via coordinate rescaling. In
terms of Eq. (3.16) this is the case where each term in the sum vanishes
separately i.e. where every $W$-tensor is covariantly conserved with respect
to its own connection, which must necessarily be the case in vacuum. This
directly generalises the solution for 2 sites given in Cosmo_spin_2 ; Bimetric
and for 3 sites in Cosmo_3_spin2  (who work in conformal time so that the
above becomes $c_{i}=(\dot{a}_{i}/\dot{a}_{I})a_{I}$, but it describes the
same situation). Physically, it means that there is no flow of energy-momentum
across the sites. The solution also automatically satisfies the Bianchi
constraint _even if_ matter couples to more than one site; it is just that
more complicated solutions could also exist, in that scenario.

With this expression for the lapses, something nice happens to the Friedmann
equation. Substituting into Eq. (3.24) yields:

$\begin{split}3M_{(4)i}^{2}H_{0}^{2}=a_{i,0}^{2}e^{2(H_{i}-H_{0})t}\bigg{[}&\alpha_{i}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m}a_{i+1,0}^{m}a_{i,0}^{-m}e^{m(H_{i+1}-H_{i})t}\\\
&+\alpha_{i-1}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m-1}a_{i-1,0}^{4-m}a_{i,0}^{m-4}e^{(4-m)(H_{i-1}-H_{i})t}\bigg{]}\;,\end{split}$
(3.26)

and we see that the LHS is now a constant, while the RHS is time-dependent.
The only way that these equations can be satisfied is if all sites have the
same Hubble parameter, thus killing the time dependence by forcing the
exponentials on the RHS to 1! We find, therefore, that the theory possesses
deSitter vacua,

$a_{i}(t)=a_{i,0}e^{Ht}\;,$ (3.27)

for some constant $H$ (so the lapses are, explicitly, $c_{i}=a_{i,0}/a_{I,0}$
– in particular this means that the Friedmann and Raychaudhuri equations
become equivalent), where the $a_{i,0}$’s and $H$ satisfy the _algebraic_
equations:

$3M_{(4)i}^{2}H^{2}=a_{i,0}^{2}\bigg{[}\alpha_{i}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m}a_{i+1,0}^{m}a_{i,0}^{-m}\\\
+\alpha_{i-1}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m-1}a_{i-1,0}^{4-m}a_{i,0}^{m-4}\bigg{]}\;.$
(3.28)

Since we are free to set $a_{I,0}=1$, this is a system of $N$ equations for
$N$ variables ($H$ and the $N-1$ remaining $a_{i,0}$’s) and is hence in
principle solvable, with a number of solutions, corresponding to deSitter
vacua with different values of $H$. The number of physical deSitter vacua
depends only on the number of solutions to these equations that have real
scale factors, which in general is dependent on both the number of sites and
the potential coefficients $T_{ijkl}$.

Naturally, the vacuum condition Eq. (2.19) we derived in Section 2 is simply a
special case of Eq. (3.28)444It is important to stress, however, that without
the symmetric polynomial formalism we would not have possessed the intuition
about splitting the $T_{ijkl}$ into $\alpha$’s and $\beta$’s that was crucial
in all of this.. Namely, it is the static (i.e. $H=0$) solution where we
impose the desired clockwork vacuum structure $a_{i,0}=a_{0,0}/q^{i}$ (if we
set $a_{I,0}=1$ then this also fixes the overall normalisation to be
$a_{0,0}=q^{I}$), and then choose our $\alpha$’s and $\beta$’s to ensure that
this vacuum is indeed a solution to the Einstein equations555A choice of
$\alpha$’s and $\beta$’s completely specifies the theory. We choose them such
that the clockwork vacuum is a static solution to our theory, but in principle
one could choose them however they wish – the results of this section are
entirely general.. In fact, if we work in conformal time (so that the lapses
become $c_{i}=a_{i,0}$, and all metrics are conformally flat) one can see
directly that the clockwork gravity potential Eq. (3.3) is identical to the
$\deg=4$ scalar potential Eq. (2.3) we used throughout Section 2, with the
conformal factors $a_{i,0}$ playing the role of the scalars $\phi_{i}$.

The upshot is that we are free to use all of the techniques we have already
developed in order to pick a good set of coefficients $T_{ijkl}$ for the
gravitational theory. This carries over to the graviton mass matrix too –
which we can determine by expanding around the static vacuum solution,

$g^{(i)}_{\mu\nu}=a_{i,0}^{2}\eta_{\mu\nu}+\frac{a_{i,0}}{M_{(4)i}}h^{(i)}_{\mu\nu}\;,$
(3.29)

where the normalisation by $M_{(4)i}$ is to ensure the Fierz-Pauli kinetic
term for $h^{(i)}_{\mu\nu}$ is canonical FierzPauli . The second-order
variation of the potential is then ClockworkGrav :

$S^{(2)}_{V}=\frac{1}{2}\int\text{d}^{4}{x}\sum^{N-1}_{i,j=0}\frac{1}{2M_{(4)i}M_{(4)j}}M^{2}_{ij}\left[h^{(i)}h^{(j)}-h^{(i)\mu}_{\;\;\;\;\;\nu}h^{(i)\nu}_{\;\;\;\;\;\mu}\right]\;,$
(3.30)

where $h^{(i)}=h^{(i)\mu}_{\;\;\;\;\;\mu}$, and the mass matrix $M^{2}_{ij}$
is as in Section 2, so we are able to calculate its components given a set of
$T_{ijkl}$, and hence determine the spectrum of graviton masses.

####  3.3.1 The physical Planck scale

We have thus far not said anything about how we should interpret the quantity
$M_{(4)i}$, which looks like a Planck scale for each site, so might naively be
assumed to be just that. The physical Planck scale $M_{\text{eff}}$, however,
is the one associated with the clockwork zero-mode, which is related to
$M_{(4)i}$ but is crucially _not the same thing_.

To figure out what this scale should be, we must work in terms of the graviton
mass eigenstates, which we obtain as in Section 2.2.2 via an orthogonal
rotation of the field basis,

$h^{(i)}_{\mu\nu}=\sum_{j=0}^{N-1}O^{ij}\tilde{h}^{(j)}_{\mu\nu}\;,$ (3.31)

where the columns of the orthogonal matrix $O$ are given by the mass
eigenvectors. In particular, we have the zero-mode,
$\tilde{h}_{\mu\nu}^{(0)}=h^{(i)}_{\mu\nu}O^{i0}=\frac{\mathcal{N}}{q^{i}}h^{(i)}_{\mu\nu}$,
where the normalisation is:

$\mathcal{N}=\frac{1}{\sqrt{\sum_{i=0}^{N-1}q^{-2i}}}=\sqrt{\frac{1-q^{-2}}{1-q^{-2N}}}\;.$
(3.32)

Note that for $q>1$ we see that the contribution to the zero-mode from the
$i$-th metric diminishes as $i$ increases towards $N-1$. It also follows that
the gear metrics may be written as:

$g^{(i)}_{\mu\nu}=a_{i,0}^{2}\eta_{\mu\nu}+\frac{a_{i,0}}{M_{(4)i}}\frac{\mathcal{N}}{q^{i}}\tilde{h}^{(0)}_{\mu\nu}+\frac{a_{i,0}}{M_{(4)i}}\sum_{j=1}^{N-1}O^{ij}\tilde{h}^{(j)}_{\mu\nu}\;.$
(3.33)

So, if we include a minimal coupling to matter on the $I$-th site, and assume
(as in ClockworkGrav ) that the fundamental clockwork scale $M_{(4)i}=M_{(4)}$
is the same on all sites, then the variation in the action becomes (fixing
$a_{I,0}=1$ so that $a_{0,0}=q^{I}$):

$\begin{split}\delta S_{M}&=\frac{1}{2}\int\text{d}^{4}{x}\;\delta
g^{(I)}_{\mu\nu}T^{(I)\mu\nu}\\\
&=\int\text{d}^{4}{x}\;\frac{1}{2M_{\text{eff}}}\tilde{h}^{(0)}_{\mu\nu}T^{(I)\mu\nu}+\frac{1}{2M_{(4)}}\sum_{j=1}^{N-1}O^{ij}\tilde{h}^{(j)}_{\mu\nu}\;,\end{split}$
(3.34)

and we can identify the physical Planck scale as:

$M^{2}_{\text{eff}}=\frac{1-q^{-2N}}{1-q^{-2}}q^{2I}M_{(4)}^{2}\;,$ (3.35)

which can be much larger than $M_{(4)}$ if the number of fields in the chain
is big enough (indeed, this is the purpose of using the clockwork).

####  3.3.2 Adding a cosmological constant

The full treatment of the Einstein equations given above is more general than
the derivation of our vacuum condition Eq. (2.19) from Section 2, and allows
for modification by a cosmological constant. We see this by taking on some
sites $j$ a non-zero energy-momentum tensor of the form:

$T^{(j)}_{\mu\nu}=-\sigma_{j}g^{(j)}_{\mu\nu}\;,$ (3.36)

for some constants $\sigma_{j}$, which will subsequently appear on the RHS of
our Einstein equations. In particular, the static $H=0$ clockwork vacuum
solution which we use as a means of choosing a good set of potential
coefficients $T_{ijkl}$ is modified to:

$\alpha_{i}\sum_{m=0}^{\deg}\deg!\beta_{m}\binom{\deg-1}{m}q^{-m}+q^{\deg}\alpha_{i-1}\sum_{m=0}^{\deg}\deg!\beta_{m}\binom{\deg-1}{m-1}q^{-m}+\sum_{j}\sigma_{j}\delta^{j}_{i}=0\;\;\;\forall
i\;,$ (3.37)

where we write the result for general $\deg$, but of course we have $\deg=4$.

A particularly interesting case is the one where we have $j=0,N-1$ i.e. we
place a cosmological constant on only the first and last sites of the
clockwork lattice. In this case, since we have (by definition) that
$\alpha_{-1}=\alpha_{N-1}=0$, the two $\sigma$’s play the role of the missing
sum on each of the end sites, that is:

$\displaystyle\sigma_{0}$
$\displaystyle=-\alpha_{0}\sum_{m=0}^{\deg}\deg!\beta_{m}\binom{\deg-1}{m}q^{-m}\;,$
(3.38) $\displaystyle\sigma_{N-1}$
$\displaystyle=-q^{\deg}\alpha_{N-2}\sum_{m=0}^{\deg}\deg!\beta_{m}\binom{\deg-1}{m-1}q^{-m}\;.$
(3.39)

If all of the nonzero $\alpha$’s are equal, the vanishing of the bulk
equations implies that the $\sigma$’s must be equal and opposite:

$\alpha_{n}=\alpha\;\forall n\implies\sigma_{0}=-\sigma_{N-1}\;.$ (3.40)

In Section 4, we will see that in the continuum limit these $\sigma$’s are
identified with the respective tensions of branes placed either end of the
clockwork lattice, so this result is not surprising.

###  3.4 Matter solutions

Now we wish to add a minimal matter coupling to some of the sites. Since we
are interested in cosmology, we assume that each energy-momentum tensor is of
perfect fluid form, but keep in the possibility of an additional cosmological
constant $\sigma$, i.e.

$T^{(i)}_{\mu\nu}=(\rho_{i}+\sigma_{i})u_{\mu}u_{\nu}+(p_{i}-\sigma_{i})\gamma^{(i)}_{\mu\nu}\;,$
(3.41)

for 4-velocity $u_{\mu}$ and
$\gamma^{(i)}_{\mu\nu}=u_{\mu}u_{\nu}+g^{(i)}_{\mu\nu}$.

When $T^{(i)}_{\mu\nu}$ refers to any form of matter other than a cosmological
constant (e.g. radiation, pressureless dust), it is necessarily time-dependent
due to the conservation equation (3.15). This means that our exponential
ansatz for the vacuum solutions no longer works, because the RHS of our
Einstein equations will _always_ be time-dependent, which is inconsistent with
the constancy of the LHS. Therefore, we must work with the equations in
general, and figure out a way to solve them.

Written out in full, the modified Friedmann equation reads:

$\begin{split}3M_{(4)i}^{2}\left(\frac{\dot{a}_{i}}{a_{i}c_{i}}\right)^{2}=&\alpha_{i}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m}a_{i+1}^{m}a_{i}^{-m}\\\
&+\alpha_{i-1}\left(\frac{a_{i-1}}{a_{i}}\right)^{4}\sum_{m=0}^{4}24\beta_{m}\binom{3}{m-1}a_{i-1}^{-m}a_{i}^{m}+(\rho_{i}+\sigma_{i})\;,\end{split}$
(3.42)

and the modified Raychaudhuri equation is:

$\begin{split}\frac{M_{(4)i}^{2}}{c_{i}^{2}}\left(\frac{\dot{a}^{2}_{i}}{a^{2}_{i}}+2\frac{\ddot{a}_{i}}{a_{i}}-2\frac{\dot{a}_{i}}{a_{i}}\frac{\dot{c}_{i}}{c_{i}}\right)\\\
=24\Bigl{\\{}\alpha_{i}&\bigl{[}\beta_{0}+\beta_{1}(c_{i+1}c_{i}^{-1}+2a_{i+1}a_{i}^{-1})+\beta_{2}(2c_{i+1}c_{i}^{-1}a_{i+1}a_{i}^{-1}+a_{i+1}^{2}a_{i}^{-2})\\\
&+\beta_{3}(c_{i+1}c_{i}^{-1}a_{i+1}^{2}a_{i}^{-2})\bigr{]}\\\
&+\alpha_{i-1}\bigl{[}\beta_{1}(c_{i-1}c_{i}^{-1}a_{i-1}^{2}a_{i}^{-2})+\beta_{2}(2c_{i-1}c_{i}^{-1}a_{i-1}a_{i}^{-1}+a_{i-1}^{2}a_{i}^{-2})\\\
&+\beta_{3}(c_{i-1}c_{i}^{-1}+2a_{i-1}a_{i}^{-1})+\beta_{4}\bigr{]}\Big{\\}}-(p_{i}-\sigma_{i})\;.\end{split}$
(3.43)

At first glance, these equations appear very ugly: they are $N$ highly
nonlinear coupled differential equations, which are difficult to solve (and
potentially impossible to do so analytically in general). To make things more
tractable, we can use the same result for the lapses that we used in deriving
the vacuum solutions, since this came directly from the Bianchi constraint,
which still holds here. Therefore, we fix the lapse/scale factor to 1 on some
site $I$ where there is a minimal matter coupling and so take
$c_{i}=\dot{a}_{i}/\dot{a}_{I}$.

Substituting these lapses into the Friedmann equations, Eqs. (3.42), as
before, yields a series of equations for $\dot{a}_{I}$, which we package
together as:

$3M_{(4)i}^{2}\left(\frac{\dot{a}_{I}}{a_{i}}\right)^{2}=f_{i}(\mathbf{a})\;,$
(3.44)

where each $f_{i}$ is understood as the RHS of the $i$-th Friedmann equation.
That is, $f_{i}$ essentially represents the 00-component of the $i$-th
$W$-tensor encoding the clockwork interactions, plus any additional matter
minimally coupled to the $i$-th site. Each $f_{i}$ is a function of $a_{i}$
and its nearest-neighbours only i.e.
$f_{i}(\mathbf{a})=f_{i}(a_{i},a_{i-1},a_{i+1})$. While we only have an
evolution equation for $a_{I}$, taking the ratio of $f_{i}$ and $f_{j}$ gives
a set of algebraic conditions that the $a$’s must satisfy throughout the
evolution, namely, that:

$a_{j}^{2}f_{j}(\mathbf{a})=a_{i}^{2}f_{i}(\mathbf{a})\;.$ (3.45)

So, we should (at least numerically) be able to evolve $a_{I}$ via Eq. (3.44),
and at each time step ensure that the rest of the scale factors obey Eq.
(3.45). This will then implicitly track the evolution of all $a_{i}$, and so
solve for the background evolution. We will do this to solve the evolution
equations for two example models in Section 5.

As eluded to in Section 1, one of the models we use as a consistency check for
our work is a deconstructed version of the RS1 braneworld. It is not
immediately obvious that we can do this, so it is useful to first review how
we can relate our discrete clockwork to a continuum theory in 5D, of which RS1
exists as a special case, with suitable modifications to the system
boundaries. This is explored in the next Section, clarifying the work of
Deconstructing .

##  4 Continuum Clockwork Gravity

Following Deconstructing , we can relate our 4D clockwork quantities to an
underlying 5D geometry into which they are embedded. We take the continuum 5D
line element to be:

$\text{d}{s}^{2}=g_{\mu\nu}(x,y)+\text{d}{y}^{2}\;,$ (4.1)

which corresponds to a 5D geometry given by
$\mathcal{M}=\mathbf{M}_{4}\times[0,L]$, where $\mathbf{M}_{4}$ is our usual
4D spacetime, parametrised by coordinates $x^{\mu}$, and the new coordinate
$y\in[0,L]$ parametrises the compact extra dimension, which lives on an
interval from 0 to $L$. Our bulk spacetime $\mathcal{M}$ possesses a boundary
$\partial\mathcal{M}$ that has two components: one at $y=0$ and one at $y=L$.
The component at $y=0$ is negatively oriented, whereas the component at $y=L$
is postively oriented, in the sense that integrating over the boundary
component in question comes equipped with the appropriate sign. Usually, when
one talks of these kind of 5D deconstructions, the extra dimension is
orbifolded on $S_{1}/\mathbb{Z}_{2}$ RS1 ; RS2 ; Carsten_review ;
Langlois_review , with fixed points of $\mathbb{Z}_{2}$ at $y=0$ and $y=L$
rather than true boundaries, with the $\mathbb{Z}_{2}$-symmetry being used to
ease calculations at these special points. With the clockwork, we are not
afforded this luxury. The reason for this will become clear very shortly, but
for now we work with the geometry as described. From now on we will use
$M,N=0,1,2,3,5$ to refer to 5D indices and $\mu,\nu=0,1,2,3$ to refer to 4D
indices.

With our choice of line element, 4D hypersurfaces of constant $y$ have a very
simple unit normal, $n^{M}=(\mathbf{0},1)$, and so the induced metric
$h_{MN}=g_{MN}-n_{M}n_{N}$ has only $h_{\mu\nu}=g_{\mu\nu}$ as its non-zero
components. We can define the constant-$y$ hypersurfaces’ extrinsic curvature
for arbitrary vector fields $X$ and $Y$ by:

$K(X,Y)=g(\nabla_{X}n,Y)\;.$ (4.2)

The components are given by the Lie derivative of $h_{MN}$ along the normal
vector, $K_{MN}=\frac{1}{2}\mathcal{L}_{n}h_{MN}$, so only the following
components are non-vanishing:

$K_{\mu\nu}=\frac{1}{2}\partial_{y}g_{\mu\nu}\;,$ (4.3)

which in terms of the vierbeins reads:

$K^{\mu}_{\;\;\nu}=\frac{1}{2}e^{\mu b}e^{\sigma}_{\;\;b}e_{\nu
a}\partial_{y}e_{\sigma}^{\;\;a}+\frac{1}{2}e^{\mu a}\partial_{y}e_{\nu a}\;.$
(4.4)

We connect the discrete theory to the continuum by interpreting the $(i)$
indices as corresponding to discrete locations in the 5 dimension, separated
by some distance $\delta y$. That is, we have $y_{i}=i\delta y$, and the
$i$-th discrete clockwork metric is the induced metric on the hypersurface at
$y_{i}$:

$g^{(i)}_{\mu\nu}(x)=g_{\mu\nu}(x,y_{i})\;.$ (4.5)

This picture makes it clear why we cannot orbifold our 5 dimension on
$S_{1}/\mathbb{Z}_{2}$: the first and last sites of the clockwork lattice in
the discrete theory are just that, endpoints of the system, so in the
continuum limit they must become true boundaries of the extra dimension,
without $\mathbb{Z}_{2}$-symmetry. When we come to talk about RS1, we shall
see the effect of this lack of $\mathbb{Z}_{2}$-symmetry on calculations
explicitly.

We are lead to introduce finite difference expressions for derivatives in the
$y$ direction,

$\partial_{y}e_{\mu a}\rightarrow\frac{1}{\delta y}\left[e^{(i+1)}_{\mu
a}-e^{(i)}_{\mu a}\right]\;,$ (4.6)

which we can then use, along with the symmetric vierbein condition (and a lot
of algebra), to rewrite the discrete action Eq. (3.1) in terms of these
extrinsic curvatures. The continuum limit is achieved by sending $\delta
y\rightarrow 0$ and the number of sites $N\rightarrow\infty$, while keeping
the product $(N-1)\delta y=L$ fixed. In this limit, the clockwork action Eq.
(3.1) becomes:

$S_{\text{bulk}}=\int_{\mathcal{M}}\left[\frac{M^{3}_{(5)}}{2}R_{(5)}-2\Lambda_{5}(y)+\alpha_{1}(y)M^{4}_{(5)}K+\alpha_{2}(y)M^{3}_{(5)}K_{2}+\alpha_{3}(y)M^{2}_{(5)}K_{3}\right]\;,$
(4.7)

where

$\displaystyle K$ $\displaystyle=K^{\mu}_{\;\;\mu}$ (4.8) $\displaystyle
K_{2}$
$\displaystyle=\delta^{[\mu}_{\alpha}\delta^{\nu]}_{\beta}K^{\alpha}_{\;\;\mu}K^{\beta}_{\;\;\nu}$
(4.9) $\displaystyle K_{3}$
$\displaystyle=\delta^{[\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\rho]}_{\gamma}K^{\alpha}_{\;\;\mu}K^{\beta}_{\;\;\nu}K^{\gamma}_{\;\;\rho}$
(4.10) $\displaystyle 2\Lambda_{5}(y)$ $\displaystyle=\frac{24}{\delta
y}\left(T_{iiii}+4T_{iii,i+1}+6T_{ii,i+1,i+1}+4T_{i,i+1,i+1,i+1}\right)$
(4.11) $\displaystyle\alpha_{1}(y)M^{4}_{(5)}$
$\displaystyle=-24\left(T_{iii,i+1}+3T_{ii,i+1,i+1}-3T_{i,i+1,i+1,i+1}\right)$
(4.12) $\displaystyle\alpha_{2}(y)M^{3}_{(5)}+M^{3}_{(5)}$
$\displaystyle=-24\delta y\left(T_{ii,i+1,i+1}+2T_{i,i+1,i+1,i+1}\right)$
(4.13) $\displaystyle\alpha_{3}(y)M^{2}_{(5)}$ $\displaystyle=-24\delta
y^{2}T_{i,i+1,i+1,i+1}$ (4.14) $\displaystyle M^{3}_{(5)}$
$\displaystyle=\frac{M^{2}_{(4)}}{\delta y}\;,$ (4.15)

and $R_{(5)}$ is the Ricci scalar constructed from the 5D metric with
components $g_{MN}$. If one wishes, the extrinsic curvature terms can instead
be realised as an extra scalar degree of freedom, which is not a dilaton
Deconstructing . We can also invert these to give us the coefficients
$T_{ijkl}$ in terms of the 5D quantities666We corrected a minus sign error in
Deconstructing  here.:

$\displaystyle 24T_{iiii}$ $\displaystyle=2\Lambda_{5}\delta
y+28\alpha_{3}\frac{M^{2}_{(5)}}{\delta
y^{2}}-6\frac{\alpha_{2}M^{3}_{(5)}+M^{3}_{(5)}}{\delta
y}+4\alpha_{1}M^{4}_{(5)}$ (4.16) $\displaystyle 24T_{iii,i+1}$
$\displaystyle=-9\alpha_{3}\frac{M^{2}_{(5)}}{\delta
y^{2}}+3\frac{\alpha_{2}M^{3}_{(5)}+M^{3}_{(5)}}{\delta
y}-\alpha_{1}M^{4}_{(5)}$ (4.17) $\displaystyle 24T_{ii,i+1,i+1}$
$\displaystyle=2\alpha_{3}\frac{M^{2}_{(5)}}{\delta
y^{2}}-\frac{\alpha_{2}M^{3}_{(5)}+M^{3}_{(5)}}{\delta y}$ (4.18)
$\displaystyle 24T_{i,i+1,i+1,i+1}$
$\displaystyle=-\alpha_{3}\frac{M^{2}_{(5)}}{\delta y^{2}}\;.$ (4.19)

Eq. (4.7) is the action that describes the 5D bulk in the continuum theory. To
account for the boundary $\partial\mathcal{M}$, we must include a Gibbons-
Hawking term York ; GH  to ensure that the variational problem is well-posed:

$S_{\text{boundary}}=\int_{\partial\mathcal{M}}M_{(5)}^{3}K\;,$ (4.20)

where $K$ is the trace of the extrinsic curvature on the boundary
$\partial\mathcal{M}$.

In principle, we may also place a brane at each of the respective boundary
components, each with some tension $\sigma_{i}$ and matter Lagrangian
$\mathcal{L}_{m,i}$:

$S_{\text{branes}}=\sum_{i=L,R}\int\text{d}^{4}x\,\sqrt{-h^{(i)}}\,\left(-\sigma_{i}+\mathcal{L}_{m,i}\right)\;,$
(4.21)

where $h^{(i)}$ are the induced metrics on each of the left and right branes –
which we identify with the first and last site of clockwork i.e.
$h^{(L)}_{\mu\nu}(x)=g_{\mu\nu}(x,0)=g_{\mu\nu}^{(0)}$, and
$h^{(R)}_{\mu\nu}(x)=g_{\mu\nu}(x,L)=g_{\mu\nu}^{(N-1)}$. The full action for
the continuum clockwork is then just the sum of the three pieces outlined
above,

$S=S_{\text{bulk}}+S_{\text{boundary}}+S_{\text{branes}}\;.$ (4.22)

###  4.1 Randall-Sundrum (Continuum)

The Randall-Sundrum model is the simplest possible special case of the
continuum theory (4.22), which has $\alpha_{1}=\alpha_{2}=\alpha_{3}=0$ and
$\Lambda_{5}=\text{const}$, so that the bulk is just pure 5D gravity i.e.

$S_{\text{bulk}}=\int_{\mathcal{M}}\frac{M_{(5)}^{3}}{2}R_{(5)}-2\Lambda_{5}\;.$
(4.23)

To determine the equations of motion, we vary the action with respect to the
metrics in question. In standard GR, variations of the boundary metric $h$
vanish, but here we must allow them to be arbitrary, as we are interested in
what they do (they correspond to the first and last sites of the clockwork).
The result is that we get two sets of equations – one set for the bulk due to
the Ricci scalar variation:

$M_{(5)}^{3}G_{MN}+2\Lambda_{5}g_{MN}=0\;,$ (4.24)

which is just standard GR in 5D, and one set for the boundary due to the
Gibbons-Hawking variation, which reads:

$\begin{split}K_{MN}-Kh_{MN}&=-\kappa^{2}S_{MN}\;\;\;\;\;(y=0)\\\
K_{MN}-Kh_{MN}&=+\kappa^{2}S_{MN}\;\;\;\;\;(y=L)\;,\end{split}$ (4.25)

where $S_{MN}$ is the brane energy-momentum tensor associated with
$\mathcal{L}_{m,i}$ and $\sigma_{i}$, $\kappa^{2}=1/M_{(5)}^{3}$, and the sign
change on the RHS is due to the change in orientation between the two boundary
components.

We can understand these boundary equations as being _one side_ of the usual
Israel junction conditions across a singular hypersurface embedded in an
underlying spacetime manifold Israel ; Shtanov . Indeed, we can reconstruct
the usual $S_{1}/\mathbb{Z}_{2}$ orbifold by thinking of the full spacetime in
that case to be constructed from two separate intervals, $[-L,0]$ and $[0,L]$,
which share a common boundary at $y=0$, and also at $y=L$ due to the
$\mathbb{Z}_{2}$ symmetry. Variation of the Gibbons-Hawking term with this
construction leads to precisely the Israel conditions across the branes at
$y=0$ and $y=L$, and the $\mathbb{Z}_{2}$ symmetry manifests as an additional
factor 1/2 on the RHS of Eqs. (4.25). For us, we _only_ have the interval
$[0,L]$, so there is no notion of jumping across a brane, and hence our
boundary equations have no factor 1/2.

Since the bulk is just 5D GR, the Bianchi identity says that the Einstein
tensor is covariantly conserved. By Eq. (4.24), so too is the energy-momentum
tensor:

$\nabla^{M}G_{MN}=\nabla^{M}T_{MN}=0\;.$ (4.26)

As a result of the Codazzi equation, which relates the 5D quantities to their
projection onto a 4D hypersurface Wald ; Langlois_review , the brane energy-
momentum tensor is also covariantly conserved with respect to its associated
covariant derivative i.e.

$\nabla_{(4)}^{\mu}S_{\mu\nu}=0\;.$ (4.27)

The solutions to this system are well-studied in the literature (see e.g.
Langlois_review ; Carsten_review ; Full_EFEs_brane  and refs therein), so we
shall only give a very brief overview.

####  4.1.1 Static vacuum solution

This is the solution where we have no matter, only tension, on the brane i.e.
$S_{MN}=-\sigma h_{MN}$, which is the continuum version of the situation in
the discete theory where we have only a cosmological constant on each of the
end sites. In full analogy with the discrete case, we take our metric to
depend only on $y$, and make it such that the hypersurfaces are conformally
flat. That is, the bulk is $\text{AdS}_{5}$, and we have:

$\text{d}{s}^{2}=e^{-2A(y)}\eta_{\mu\nu}\text{d}{x}^{\mu}\text{d}{x}^{\nu}+\text{d}{y}^{2}\;,$
(4.28)

where for convenience we work with the _warp factor_ $A(y)$ rather than the
usual scale factor $a(y)$, but of course we can go between them with the
identification $a=e^{-A}$. With this metric, the bulk equations become:

$\displaystyle G_{\mu\nu}$ $\displaystyle=(6A^{\prime
2}-3A^{\prime\prime})g_{\mu\nu}$ (4.29) $\displaystyle G_{\mu 5}$
$\displaystyle=G_{5\mu}=0$ (4.30) $\displaystyle G_{55}$
$\displaystyle=6A^{\prime 2}\;,$ (4.31)

where primes denote derivatives with respect to $y$. From the 55-equation we
get777We could, of course, also have had $A=-ky$. All this amounts to is
changing perspective on which brane you are looking from.:

$A=ky\;,\;\;\;k^{2}=-\frac{\Lambda_{5}}{3M_{(5)}^{3}}\;,$ (4.32)

and the $\mu\nu$-equation is internally consistent with this result.

Regarding the boundary, the only surviving components of the extrinsic
curvature are $K_{\mu\nu}=-A^{\prime}h_{\mu\nu}$, so we get:

$\begin{split}3A^{\prime}h_{\mu\nu}&=+\kappa^{2}\sigma_{0}h_{\mu\nu}\;\;\;\;\;(y=0)\\\
3A^{\prime}h_{\mu\nu}&=-\kappa^{2}\sigma_{L}h_{\mu\nu}\;\;\;\;\;(y=L)\;,\end{split}$
(4.33)

which implies that our brane tensions must satisfy:

$\sigma_{0}=-\sigma_{L}=3kM_{(5)}^{3}\;.$ (4.34)

This is the standard RS1 solution, adapted to our lack of
$\mathbb{Z}_{2}$-symmetry (including $\mathbb{Z}_{2}$-symmetry would give
$\sigma_{0}=-\sigma_{L}=6kM_{(5)}^{3}$ instead, which is the usual result RS1
).

####  4.1.2 Matter solution

As with the discrete theory, when we add a dynamical matter fluid to the
system, we no longer have the freedom to solve the system for a conformally
flat hypersurface metric, and need to keep things general. The metric ansatz
that does the job for us is:

$\text{d}{s}^{2}=-c^{2}(t,y)\text{d}{t}^{2}+a^{2}(t,y)\eta_{ij}\text{d}{x}^{i}\text{d}{x}^{j}+b^{2}(t,y)\text{d}{y}^{2}\;,$
(4.35)

where now it proves more convient to work with the conventional scale factors
rather than warp factors. With this choice of metric, we get our 5D Einstein
equations, of which there are four:

$\displaystyle G_{00}$
$\displaystyle=3\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\dot{a}}{a}\frac{\dot{b}}{b}\right)+3\frac{c^{2}}{b^{2}}\left(\frac{a^{\prime}}{a}\frac{b^{\prime}}{b}-\frac{a^{\prime\prime}}{a}-\frac{a^{\prime
2}}{a^{2}}\right)=\kappa^{2}T_{00}$ (4.36)
$\displaystyle\begin{split}G_{ij}&=\frac{a^{2}}{c^{2}}\eta_{ij}\left(-\frac{\dot{a}^{2}}{a^{2}}+2\frac{\dot{a}}{a}\frac{\dot{c}}{c}-2\frac{\dot{a}}{a}\frac{\dot{b}}{b}+\frac{\dot{b}}{b}\frac{\dot{c}}{c}-2\frac{\ddot{a}}{a}-\frac{\ddot{b}}{b}\right)\\\
&+\frac{a^{2}}{b^{2}}\eta_{ij}\left(\frac{a^{\prime
2}}{a^{2}}-2\frac{a^{\prime}}{a}\frac{b^{\prime}}{b}+2\frac{a^{\prime}}{a}\frac{c^{\prime}}{c}-\frac{b^{\prime}}{b}\frac{c^{\prime}}{c}+2\frac{a^{\prime\prime}}{a}+\frac{c^{\prime\prime}}{c}\right)=\kappa^{2}T_{ij}\end{split}$
(4.37) $\displaystyle G_{05}$
$\displaystyle=3\left(\frac{\dot{a}}{a}\frac{c^{\prime}}{c}+\frac{a^{\prime}}{a}\frac{\dot{b}}{b}-\frac{\dot{a}^{\prime}}{a}\right)=\kappa^{2}T_{05}$
(4.38) $\displaystyle G_{55}$ $\displaystyle=3\left(\frac{a^{\prime
2}}{a^{2}}+\frac{a^{\prime}}{a}\frac{c^{\prime}}{c}\right)+3\frac{b^{2}}{c^{2}}\left(\frac{\dot{a}}{a}\frac{\dot{c}}{c}-\frac{\dot{a}^{2}}{a^{2}}-\frac{\ddot{a}}{a}\right)=\kappa^{2}T_{55}\;.$
(4.39)

With bulk energy momentum tensor $T_{MN}=-2\Lambda_{5}g_{MN}$, we immediately
see that $G_{05}=0$. Remarkably, using this equation for $G_{05}$, one can
show that any set of functions $a,b$ and $c$ which satisfy both $G_{05}=0$ and

$\left(\frac{\dot{a}}{ac}\right)^{2}=\left(\frac{a^{\prime}}{ab}\right)^{2}+\frac{\kappa^{2}\Lambda_{5}}{3}+\frac{C}{a^{4}}\;.$
(4.40)

will solve _all_ of Einstein’s equations, locally in the bulk Full_EFEs_brane
. The term scaling as $a^{-4}$ is the so-called ‘dark radiation’ term, which
arises as a result of bulk Weyl curvature when the bulk is not AdS but AdS-
Schwarzchild GeometryofBraneWorld , so we are safe to set the constant $C$,
which is proportional to the mass of the bulk black hole, to be 0.

If the matter on the brane (say, at $y=0$) is of perfect fluid form i.e
$S_{\mu\nu}$ is given by Eq. (3.41), then the boundary equation Eq. (4.25)
allows us to substitute in for $\left.(a^{\prime}/ab)^{2}\right|_{0}$ to
obtain a modified Friedmann equation for the dynamics on the brane888We could
have also obtained this equation by using the Gauss-Codazzi equations Wald  to
directly project the 5D Einstein tensor onto the brane at $y=0$.:

$\left(\frac{\dot{a}_{0}}{a_{0}}\right)^{2}=\frac{8\pi
G}{3}\rho_{0}\left(1+\frac{\rho_{0}}{2\sigma_{0}}\right)+\frac{\Lambda_{4}}{3}+\frac{C}{a_{0}^{4}}\;,$
(4.41)

where Newton’s constant and the 4D effective cosmological constant are,
respectively, in terms of the 5D parameters, $8\pi G=2\kappa^{4}\sigma/3$ and
$\Lambda_{4}=\kappa^{2}(\Lambda_{5}+\kappa^{2}\sigma^{2}/3)$. Again these are
adapted from the standard RS identifications to our lack of
$\mathbb{Z}_{2}$-symmetry – though the dynamics remain the same (since
$\rho_{0}$ also is affected by the $\mathbb{Z}_{2}$ factor). In particular, at
early times we get a modification to the usual Friedmannn equation by a
$\rho^{2}$ term, but since $\rho$ decays, at late times the evolution is as in
standard FRW cosmology.

Since we have that the energy-momentum tensor on the brane is conserved,
$\rho_{0}$ just follows the usual fluid equation:

$\dot{\rho_{0}}+3\frac{\dot{a}_{0}}{a_{0}}(\rho_{0}+p_{0})=0\;,$ (4.42)

so we have all the information we need to solve for the dynamics.

We can also extend the solution on the brane to the whole bulk, provided we
take $b=1$ for all time (with some suitable radion stabilisation mechanism to
allow for this, see e.g. Radion_stab ). The procedure is outlined in
Full_EFEs_brane , but here we will simply state the solution, which reads
(when $C=0$):

$\displaystyle a(t,y)$
$\displaystyle=a_{0}(t)(\cosh{ky}-\eta\sinh{k\absolutevalue{y}})$ (4.43)
$\displaystyle c(t,y)$
$\displaystyle=\frac{\dot{a}(t,y)}{\dot{a}_{0}(t)}=\cosh{ky}-\tilde{\eta}\sinh{k\absolutevalue{y}}\;,$
(4.44)

where $k$ is as in the static solution, and

$\eta=1+\frac{\rho_{0}}{\sigma}\;,\;\;\;\;\;\tilde{\eta}=\eta+\frac{\dot{\rho}_{0}a_{0}}{\dot{a}_{0}}\;.$
(4.45)

The equation (4.44) for the lapse in terms of the scale factor is a direct
result of the Einstein equation $G_{05}=0$, which tells us that there is no
flow of energy along the 5 dimension. In the discrete theory, the lapses have
the same form – see Eq. (3.25) – but there it is due to the Bianchi constraint
on the $W$-tensor interaction terms, which tells us that there can be no
energy flow across the sites of the clockwork. Clearly there is some
correspondence between these two ideas – more on this very shortly, but for
now we continue on.

##  5 Example Models

The upshot to all of the work in the previous Section is that in RS1 we have a
fully solvable system in the continuum limit which we can use as a consistency
check for our discrete clockwork gravity formalism. We shall show now
explicitly how the discrete results map exactly to the RS results after taking
the appropriate limit, and then go on to examine the solutions to the
clockwork equations for the gravitational version of the toy quartic model
introduced in Section 2.2.2.

###  5.1 Randall-Sundrum (Discrete)

Here we shall use the formalism developed in Section 3 to reproduce the
results of Section 4 in the continuum limit. Recall that RS1 is the simplest
possible continuum theory, with the continuum $\alpha$ parameters, which
determine the size of the extrinsic curvature contributions to the bulk
action, all set to 0. Using the dictionary of Eqs. (4.16)-(4.19) for going
between the discrete potential coefficients $T_{ijkl}$ and the continuum
parameters, we see that the RS1 case corresponds to following choice of
clockwork couplings (setting all discrete $\alpha_{n}=1$):

$24\beta_{0}=-\frac{6M^{3}_{(5)}}{\delta
y}\;,\;\;\;24\beta_{1}=\frac{3M^{3}_{(5)}}{\delta
y}\;,\;\;\;24\beta_{2}=-\frac{M^{3}_{(5)}}{\delta
y}\;,\;\;\;\beta_{3}=0\;,\;\;\;24\beta_{4}=2\Lambda_{5}\delta y\;.$ (5.1)

This choice for the $\beta$’s completely specifies the discrete theory.

We shall consider the static and dynamical solutions in turn.

####  5.1.1 Static vacuum solution

Comparison of our $H=0$ clockwork vacuum solution from Section 3 with the RS1
warped metric Eq. (4.28) tells us that we should identify $q=e^{k\delta y}$,
so that we have the dictionary for going between clockwork and RS1 parameters:

$q=e^{k\delta y}\;,\;\;\;\;\;M_{(4)}^{2}=M_{(5)}^{3}\delta
y\;,\;\;\;\;\;L=(N-1)\delta y\;.$ (5.2)

The continuum limit is achieved, as we said, by taking the limit
$N\rightarrow\infty$ and $\delta y\rightarrow 0$ while keeping $L=(N-1)\delta
y$ fixed. In terms of the clockwork parameters, this is equivalent to taking
$N\rightarrow\infty$, $q\rightarrow 1$ and $M_{(4)}^{2}\rightarrow 0$ while
keeping $(N-1)M_{(4)}^{2}$ and $(N-1)\ln{q}$ fixed.

Substituting the RS $\beta$’s into the cosmological-constant-modified vacuum
condition Eq. (3.37), and then taking the appropriate limit described above,
to first order in $\delta y$ we find that in order for the vacuum condition to
be satisfied (in the bulk) we require:

$-\Lambda_{5}=3k^{2}M_{(5)}^{3}+6k^{3}M_{(5)}^{3}\delta y\;,$ (5.3)

and so we recover $k^{2}=-\Lambda_{5}/3M_{(5)}^{3}$, as we wanted, once we
send $\delta y\rightarrow 0$.

The brane tensions, also to first order in $\delta y$, and utilising the above
Eq. (5.3), are:

$\displaystyle\sigma_{1}$
$\displaystyle=3kM^{3}_{(5)}+\frac{3}{2}k^{2}M^{3}_{(5)}\delta y$ (5.4)
$\displaystyle\sigma_{2}$
$\displaystyle=-3kM^{3}_{(5)}-\frac{3}{2}k^{2}M^{3}_{(5)}\delta y\;,$ (5.5)

also in agreement with our continuum solution.

The zero-mode coupling Eq. (3.35), in this limit, becomes:

$M^{2}_{\text{Pl}}=\frac{1}{2k}\left(e^{2kL}-1\right)M_{(5)}^{3}\;,$ (5.6)

which again is the usual association one would make between the 4D and 5D
Planck masses in static RS1.

####  5.1.2 Matter solution

We now wish to add matter to the brane at $y=0$, corresponding to the first
site of the clockwork. While in the context of a clockwork model we would
normally couple to the end site, to engineer the smallest coupling to the
zero-mode, brane cosmology literature typically puts matter on the $y=0$
brane, so we do this too here for comparative purposes. In Appendix B, we show
explicitly that, in the continuum limit, our discrete Einstein equations from
Section 3 become the continuum Einstein equations from Section 4. To be more
precise, for sites in the bulk, the discrete equations (3.42) and (3.43) for
$G^{0}_{\;0}$ and $G^{i}_{\;j}$ become the continuum $G^{0}_{\;0}$ and
$G^{i}_{\;j}$ equations (4.36) and (4.37), respectively, while for the two end
sites the discrete equations for $G^{0}_{\;0}$ and $G^{i}_{\;j}$ respectively
become the continuum Gibbons-Hawking equations (4.25) for $K^{0}_{\;0}$ and
$K^{i}_{\;j}$, corresponding to the component of $\partial\mathcal{M}$ in
question. The Bianchi constraint (3.16) also maps directly to the continuum
$G^{0}_{\;5}$ equation, solidifying the correspondence eluded to at the end of
Section 4.

However, since the discrete theory is 4 dimensional, we have no notion of the
$G^{5}_{\;5}$ equation, which was important when solving the continuum
equations. Therefore, we need to check that the solutions to the discrete
Einstein equations coincide with the solutions to the continuum Einstein
equations, in the appropriate limit.

To do this, we used Python to solve the discrete Friedmann equations (3.44)
using the framework set out at the end of Section 3, i.e. we set the lapse on
the first site to 1 and evolve the one dynamical equation for $\dot{a}_{0}$
via 4 order Runge-Kutta, ensuring that the algebraic conditions (3.45) are
satisfied at every time step of the evolution. To simplify things numerically,
we choose units such that the parameters $M_{(5)}^{3}=1$, $\Lambda_{5}=-1$,
and the size of the 5 dimension is also $L=1$. We use these parameters also in
the continuum Friedmann equation on the brane, Eq. (4.41), and solve it using
a standard numerical ODE integrator, the hope being that the solutions to the
continuum and discrete equations match in the limit where $N\rightarrow\infty$
and $\delta y=L/(N-1)\rightarrow 0$.

In Figure 2 we show in black the solutions to the discrete equations when we
add pressureless dust (i.e. with $\rho_{0}=1/a_{0}^{3}$ and $p_{0}=0$) to the
first site of the clockwork, for $N=50,N=500$ and $N=5000$ sites, with the RS
continuum solution overlaid in red with presureless dust on the brane at
$y=0$.

(a) $N=50$ sites

(b) $N=500$ sites

(c) $N=5000$ sites

Figure 2: In black: solutions to the discrete Friedmann equations with
potential couplings given by the RS $\beta$’s (5.1); in red: solution to the
continuum RS equations. Since the discrete system involves very many scale
factors, whose solutions sit on top of one another sequentially, we only show
explicitly the evolution of the first (solid line) and last (dashed line)
sites and shade the region in between where the other scale factors lie. We
see that as the number of sites is increased, the solution to the discrete
equations indeed approaches that of the continuum – this convergence is better
seen in Fig 3.

Indeed, we see that the solutions begin to overlap as we get closer to the
continuum limit, so we can be confident that our formalism works as intended,
and that while we have no analogue for $G^{5}_{\;5}$ in the discrete system,
the solutions are nevertheless equivalent. In Figure 3 we show the convergence
as the number of sites increases explicitly.

Figure 3: Difference between the continuum and discrete solutions displayed
for the increasing number of sites as plotted in Fig 2, showing more clearly
the convergence as we approach the continuum limit. Solid/dashed lines and the
shaded regions have the same meaning as in Fig 2.

We note that while the solution for small $N$ deviates markedly from the
continuum solution, it is nevertheless a real, physical solution to the multi-
gravity theory specified by the RS $\beta$’s. It is simply that this theory
does not constitute a clockwork when one is away from the continuum limit. We
can see this as follows: recall that the defining feature of clockwork gravity
is that there exists a static vacuum solution where all metrics are
conformally flat, and the corresponding conformal factors are asymmetrically
distributed throughout the clockwork lattice, the canonical choice being
$a_{i,0}=a_{0,0}/q^{i}$. For this solution to exist, the potential
coefficients must satisfy the (in this case cosmological-constant-modified)
vacuum condition (3.37), and for the RS $\beta$’s this is only true in the
limit $\delta y\rightarrow 0$. We stress this point to make clear the fact
that our formalism is entirely general, and can in principle be used to
determine cosmological solutions to _any_ theory of gravity involving multiple
pairwise interacting metrics – in this case (i.e. for the theory with the RS
$\beta$’s but away from the continuum limit) the solution is an accelerating
one at late times. The novelty of clockwork gravity models, as a subclass of
these general theories, is of course their potential to solve the hierarchy
problem, so we pay them special attention.

As an example unstudied clockwork scenario, we now return to the quartic model
introduced in Section 2.2.2 for scalars, and construct its gravitational
equivalent.

###  5.2 Quartic clockwork gravity

We introduced this model in Section 2.2.2 as a simple $\deg=4$ clockwork
theory with the nice feature of having the smallest possible hierarchy in the
potential coefficients, with $\beta_{m}\propto q^{\pm 1}$ only, where $q>1$ is
$\mathcal{O}(1)$. The potential coefficients specifying the theory are:

$24\beta_{0}=6q^{-1}\;,\;\;\;24\beta_{1}=-3\;,\;\;\;24\beta_{2}=q\;,\;\;\;\beta_{3}=0\;,\;\;\;\beta_{4}=0\;,$
(5.7)

with all of the nonzero $\alpha_{n}=1\;\forall n\neq 0,N-1$. We rescale the
$\beta$’s here by a factor 24 compared with those stated in Section 2.2.2, in
order to force them in line with the Einstein equations. We are allowed to do
this since we already showed that the $\beta$’s from Section 2.2.2 indeed
satisfy the vacuum condition (now without the need to include any $\sigma$’s
on the end sites), so multiplying through by a common factor will not affect
this. This time, in accordance with our clockwork philosophy, we will
eventually place our matter on the end ($i=N-1$) site – as this produces the
greatest suppression of scales – so we choose to fix the initial lapse and
scale factor on that site to 1, which means we must have
$c_{i}=\dot{a}_{i}/\dot{a}_{N-1}$ (see Eq. (3.25) and Appendix A).

With this choice of potential coefficients and lapse, the modified Friedmann
equations take the form:

$3M_{(4)}^{2}\left(\frac{\dot{a}_{N-1}}{a_{i}}\right)^{2}=\alpha_{i}\frac{6}{q}\left(1-\frac{q}{2}\frac{a_{i+1}}{a_{i}}\right)\left(1-q\frac{a_{i+1}}{a_{i}}\right)-3\alpha_{i-1}\left(\frac{a_{i-1}}{a_{i}}\right)^{3}\left(1-q\frac{a_{i}}{a_{i-1}}\right)+\rho_{i}\;.$
(5.8)

Written in this form, the presence of the $H=0$ clockwork vacuum is manifest –
if $\rho_{i}=0$ on all sites then the RHS vanishes when $a_{i+1,0}=a_{i,0}/q$.
In general we cannot find analytic forms for the dS vacua for more than 2
sites, as increasing the number of sites increases the order of the polynomial
equation, Eq. (3.28), which one must solve to determine the conformal factors.
Indeed, we find numerically that the number of distinct (i.e. with different
values for $H$) physical dS vacua increases with the number of sites, although
the high degree of nonlinearity in the system prevents us from determining the
exact quantity.

To solve for the dynamics when we add pressureless dust to the $(N-1)$-th
site, we follow our usual procedure and evolve the one dynamical equation for
$\dot{a}_{N-1}$ while ensuring that the algebraic conditions (3.45) are
satisfied at all time steps. Since the theory possesses multiple vacua, this
process is numerically sensitive to initial data – indeed, if we set up the
initial set of scale factors we feed into Python to be close to one of the dS
vacua, then the addition of matter essentially acts as a perturbation to the
system which quickly dilutes away (scaling as $a^{-3}$) as the system returns
to the corresponding vacuum. However, if we choose initial scale factors close
to the clockwork $H=0$ vacuum, then we do get interesting cosmological
evolution.

In Figure 4, the background evolution of the scale factors, with presureless
dust on the end site, is displayed for $N=10$ sites, starting off at the
clockwork vacuum, and for simplicity taking the parameters as
$M_{(4)}^{2}=1/3$, $q=1.2$, and $\rho_{N-1}=1/a_{N-1}^{3}$. We are free to
make these choices for $M_{(4)}^{2}$ and $\rho_{N-1}$ as these are essentially
unit choices which correspond to a rescaling of $a_{N-1}$, so long as one also
rescales the other scale factors in a consistent manner (taken care of by the
algebraic equations). For comparison, we also plot in dashed black the
effective evolution of a Universe whose gravity is described by GR with only a
single FRW metric, with effective Planck mass $M_{\text{eff}}$ given by the
zero-mode coupling Eq. (3.35). That is, the solution to the standard Friedmann
equation:

$3M_{\text{eff}}^{2}\left(\frac{\dot{a}}{a}\right)^{2}=\rho\;,$ (5.9)

with $\rho=1/a^{3}$ as before.

Figure 4: Solutions to the discrete Friedmann equations for $N=10$ sites with
potential couplings given by the set of $\beta$’s (5.7). The dashed black line
is the effective evolution of the clockwork zero-mode (i.e. the solution to
Eq. (5.9)) which we see matches almost exactly the evolution of $a_{N-1}$.

We see that the system is solved by a bunch of successive scale factors
undergoing decelerated expansions, but more importantly that the dynamics of
the metric to which matter couples minimally are completely dominated by the
contribution of the zero-mode. For the parameter choices given above –
numerically it appears that the following statement depends on the value of
$q$, but it is certainly true here – the effect of the clockwork interactions
is such that, as far as the matter on the end site is concerned, the evolution
is equivalent to that of a single copy of GR, whose strength is characterised
by the Planck mass of the zero-mode (with some small corrections from the
massive modes). This Planck mass, as we have seen, can be made much larger
than the fundamental scale $M_{(4)}$ depending on one’s choice for $q$ and $N$
(keeping in mind that we would like to avoid large hierarchies in the
potential coefficients), and so in principle we see the potential route to
solving the hierarchy problem – in ClockworkGrav , the authors showed that
with $M_{(4)}\sim\mathcal{O}(\text{TeV})$ one can generate a Planck scale
coupling to the zero-mode with $q=4$ and $N=26$ sites. This result should not
be surprising, of course, since we are looking for cosmological solutions, and
it is clear that the massless mode should dominate over large distance scales.
The mass gap to the heavier modes is in general dependent on the choice of
potential, but typically the mass gap to the lightest massive mode is roughly
of order $qM_{(4)}$999Remember from Section 1, masses are products of scales
($M_{(4)}$) with couplings ($q$)., with the heavier modes distributed
exponentially above this. The strengths of their couplings to matter,
determined by the components of the orthogonal matrix in Eq. (3.34), are also
generally model-dependent; for this model we find that the heaviest modes are
even more weakly coupled than the zero-mode, which is favourable
experimentally as their effects will not show up until above the Planck scale.

Of course, as mentioned at the end of the original clockwork gravity paper
ClockworkGrav , we still cannot yet present this as a robust solution to the
hierarchy problem without considering radiative corrections to the potential.
Graviton loops, for example, could generate non-nearest neighbour
interactions, which would resurrect the undesirable Boulware-Deser ghost
BD_Ghosts ; Cycles . Matter loops would not have this problem, since we chose
to couple to only one site. We would naively expect that loops involving the
graviton zero-mode would be safe due to the surviving diffeomorphism
invariance, though loops involving the heavier modes could prove more
dangerous, and a fuller analysis is required. Indeed, this discussion speaks
to a more general question regarding the radiative stability of ghost-free
multigravity theories (see drgt_corrections ; bigravity_corrections  for work
regarding this in the context of dRGT massive gravity and bigravity, to 1-loop
level), which is outside the scope of the present work but important for
future investigations.

##  6 Conclusion

To summarise, the clockwork mechanism provides an efficient means to generate
exponentially suppressed couplings from a fundamental theory containing only
$\mathcal{O}(1)$ parameters, and as a result obtain exponentially large
interaction scales without new physics appearing at these large energies.
Applied to gravitational physics, one can naively solve the electroweak
hierarchy problem through a higher order generalisation of the standard
clockwork mechanism, involving nearest-neighbour interactions in the ghost
free multi-gravity theory, where although the $N$ copies of diffeomorphism
invariance are broken to the diagonal subgroup, an exponential suppression of
the coupling to the graviton zero-mode is nevertheless achieved thanks to an
asymmetric distribution of conformal factors in the background vacua.

In this work, we have done two main things. Firstly: we have developed a
formalism by which one is able to construct potentials for which the general
multi-gravity theory necessarily possesses the desired clockwork vacuum as a
solution. To achieve this, we started with a general theory with nearest-
neighbour interactions and used symmetric polynomials to constrain the choices
of coefficients which are allowed in the potential. As an added bonus, we also
get the matrix encoding the masses of the higher mass modes for free, and we
used the formalism to reproduce some standard results in the clockwork
literature, as well as to introduce a new model which has the nice feature of
possessing the smallest possible hierarchy between the parameters of the
fundamental theory.

Secondly: we derived the Einstein equations of the general ghost free multi-
gravity theory, using the results from the symmetric polynomial formalism
regarding the potential coefficients to aid in making them tractable. With the
help of energy conservation considerations, we are able to solve the equations
for a bunch of pairwise interacting FRW metrics whose lapse and scale factor
are site-dependent. In vacuum, we can do this analytically, and find that
there are a number of deSitter vacua where all scale factors evolve
exponentially with time and every site shares the same Hubble parameter, the
number of such solutions being in general dependent on the number of sites and
a choice for the potential coefficients. In particular, if one imposes that
there should be an asymmetrically distributed static vacuum solution with
$H=0$, as required by the clockwork, then one recovers the condition on the
allowed potential coefficients from the symmetric polynomials. When we add a
minimally coupled perfect fluid to one of the sites, we must solve the
equations numerically, and we provide a means to do this in an entirely
general manner which works for _any_ multi-gravity theory devoid of the
Boulware-Deser ghost, although we focus ourselves on clockwork theories i.e.
those with the desired vacuum structure. We solved the equations explicitly
for the new model constructed from the symmetric polynomial machinery and
found that there is a cosmological solution where the evolution of the scale
factor, as seen by a minimally coupled observer on the end site, looks like a
single copy of GR whose strength is characterised by the exponentially
enhanced Planck mass of the zero-mode.

We also revisited the continuum limit of the ghost free multi-gravity theory,
which has a natural interpretation as a braneworld model, and made some
clarifications about the nature of the extra dimension. Namely, the extra
dimension cannot live on $S_{1}/\mathbb{Z}_{2}$, as is usually the case in
braneworld literature, but rather, it must live on an interval, and so one
must invoke a Gibbons-Hawking term to deal properly with the system
boundaries. We paid special attention to the original Randall-Sundrum model
(RS1) as a special case of the continuum theory, using it as a consistency
check for our work on the discrete theory, and found that as one approaches
the continuum limit, the solutions to the corresponding discrete theory (which
is only a clockwork in this limit, otherwise it is just some generic multi-
gravity theory) do indeed match up with the continuum results.

However, we cannot present this as a full resolution to the hierarchy problem
just yet, as we still need to compute the radiative corrections, including
those arising from graviton loops. We also need to investigate more of the
cosmological phenomenology associated to the theory, in regards to
perturbations and structure growth, black holes, gravitational waves etc.
Since we only studied the explicit FRW solutions to a single model, these
phenomenological considerations could be highly model-dependent. We have,
however, provided a general framework within which one can work if one wishes
to study such interesting questions; we certainly intend to do so in the near
future.

## Acknowledgements

We would like to thank A. Padilla, S.S. Mishra and S. Sevillano Muñoz for
useful discussions. K.W. is supported by a UK Science and Technology
Facilities Council studentship. P.M.S. and A.A. are supported by a STFC
Consolidated Grant [Grant No. ST/T000732/1]. For the purpose of open access,
the authors have applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising.

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##  Appendix A Satisfying the Bianchi constraint

We would like to see what the Bianchi constraint, Eq. (3.16), looks like
explicitly with our FRW+lapse ansatz (3.17) for the gear metrics.

Recall that the Bianchi constraint is

$\sum_{i=0}^{N-1}\absolutevalue{e^{(i)}}\nabla^{(i)}_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;\nu}=0\;.$
(A.1)

The index structure is written slightly differently to Eq. (3.16), but we are
free to write it in this way since we use the Levi-Civita connection on each
site, which is metric-compatible.

The covariant divergence of the $i$-th $W$-tensor is

$\nabla^{(i)}_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;\nu}=\partial_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;\nu}+\Gamma^{(i)\mu}_{\;\;\;\;\mu\lambda}W^{(i)\lambda}_{\;\;\;\;\;\;\nu}-\Gamma^{(i)\lambda}_{\;\;\;\;\mu\nu}W^{(i)\mu}_{\;\;\;\;\;\;\lambda}\;.$
(A.2)

The only non-vanishing Christoffel symbols for our metrics are:

$\displaystyle\Gamma^{(i)0}_{\;\;\;\;00}$
$\displaystyle=\frac{\dot{c}_{i}}{c_{i}}$ (A.3)
$\displaystyle\Gamma^{(i)0}_{\;\;\;\;jk}$
$\displaystyle=\frac{a_{i}\dot{a}_{i}}{c_{i}^{2}}\eta_{jk}$ (A.4)
$\displaystyle\Gamma^{(i)j}_{\;\;\;\;k0}$
$\displaystyle=\frac{\dot{a}_{i}}{a_{i}}\delta^{j}_{k}\;,$ (A.5)

and the only non-vanishing components of the $W$-tensors are
$W^{(i)0}_{\;\;\;\;\;\;0}$ and $W^{(i)j}_{\;\;\;\;\;\;k}$, given respectively
by Eqs. (3.22) and (3.23). Substituting into Eq. (A.2), we get

$\displaystyle\nabla^{(i)}_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;0}$
$\displaystyle=\partial_{0}W^{(i)0}_{\;\;\;\;\;\;0}+3\frac{\dot{a}_{i}}{a_{i}}W^{(i)0}_{\;\;\;\;\;\;0}-\frac{\dot{a}_{i}}{a_{i}}W^{(i)k}_{\;\;\;\;\;\;k}$
(A.6) $\displaystyle\nabla^{(i)}_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;j}$
$\displaystyle=0\;,$ (A.7)

so we only need to consider the $\nu=0$ component.

Substituting in our explicit expressions for the $W$-tensor components yields
(after some enjoyable algebra) the following final expression for the
covariant divergence:

$\begin{split}\nabla^{(i)}_{\mu}W^{(i)\mu}_{\;\;\;\;\;\;0}=3\times
24&\bigg{[}\alpha_{i}(\dot{a}_{i+1}a_{i}^{-1}-c_{i+1}\dot{a}_{i}a_{i}^{-1}c_{i}^{-1})(\beta_{1}+2\beta_{2}a_{i+1}a_{i}^{-1}+\beta_{3}a_{i+1}^{2}a_{i}^{-2})\\\
&+\alpha_{i-1}(\dot{a}_{i-1}a_{i}^{-1}-c_{i-1}\dot{a}_{i}a_{i}^{-1}c_{i}^{-1})(\beta_{1}a_{i-1}^{2}a_{i}^{-2}+2\beta_{2}a_{i-1}a_{i}^{-1}+\beta_{3})\bigg{]}\;.\end{split}$
(A.8)

Clearly, the Bianchi constraint will be automatically satisfied when _all_ of
these covariant divergences vanish individually; in fact, we showed in Section
3 that this _has_ to be the situation when matter only couples to one site (or
when there is no matter at all). Since we want (A.8) to vanish for every site,
it must in particular vanish on the boundary sites ($i=0,N-1$), where one of
the $\alpha$’s is 0. This means that both terms inside the square brackets
must vanish individually, which tells us that we must have (both the $(i+1)$
and $(i-1)$ term give the same result),

$\frac{c_{i+1}}{c_{i}}=\frac{\dot{a}_{i+1}}{\dot{a}_{i}}\implies
c_{i}=\frac{\dot{a}_{i}}{f(t)}\;.$ (A.9)

To determine what the function of time $f(t)$ must be, we need only use the
freedom to rescale our coordinates to fix the lapse on one of the sites. For
example, if we want to set $c_{0}=1$, then we have
$c_{1}/c_{0}=\dot{a}_{1}/\dot{a}_{0}$, and so $f(t)=\dot{a}_{0}$. If we wish
to work in conformal time and set $c_{0}=a_{0}$, then we get
$f(t)=\dot{a}_{0}/a_{0}$. If we wish to fix the lapse on the other end, i.e.
take $c_{N-1}=1$, then we get $f(t)=\dot{a}_{N-1}$.

If matter couples to more than one site, there are of course in principle
other non-trivial solutions to Eq. (A.1), involving more complicated
cancellations across the sums depending on the specific choice of $\beta$
coefficients. However, these will not help us to do anything about the unknown
lapses in our Einstein equations, and we only place matter on one of the sites
anyway, so we use the solution (A.9) going forward. In fact, in Appendix B we
show that the situation in Eq. (A.9) is the discrete analogue of the continuum
$G^{0}_{\;5}$ Einstein equation (see Section 4), so it is probably a good
choice to make.

##  Appendix B RS continuum equations from the discrete equations

We wish to show that, for the RS $\beta$’s, when we take the continuum limit,
the discrete Einstein equations for the bulk scale factors become the
corresponding 5D Einstein equations, the discrete equations for the boundary
scale factors become the Gibbons-Hawking equations, and the Bianchi constraint
becomes the $G^{0}_{\;5}$ equation.

Starting with $G^{0}_{\;0}$ for one of the bulk equations, and substituting in
the RS $\beta$’s, we have explicitly:

$\begin{split}-3M_{(4)}^{2}\left(\frac{\dot{a}_{i}}{a_{i}c_{i}}\right)^{2}+\Bigg{[}&\left(-\frac{6M_{(5)}^{3}}{\delta
y}+\frac{9M_{(5)}^{3}}{\delta y}a_{i+1}a_{i}^{-1}-\frac{3M_{(5)}^{3}}{\delta
y}a_{i+1}^{2}a_{i}^{-2}\right)\\\ &+\left(\frac{3M_{(5)}^{3}}{\delta
y}a_{i-1}^{3}a_{i}^{-3}-\frac{3M_{(5)}^{3}}{\delta
y}a_{i-1}^{2}a_{i}^{-2}+2\Lambda_{5}\delta y\right)\Bigg{]}=0\;,\end{split}$
(B.1)

and we can Taylor expand the nearest neighbour scale factors to second order
in $\delta y$:

$\displaystyle a_{i-1}$ $\displaystyle=a_{i}-\delta
ya_{i}^{\prime}+\frac{1}{2}\delta y^{2}a_{i}^{\prime\prime}$ (B.2)
$\displaystyle a_{i+1}$ $\displaystyle=a_{i}+\delta
ya_{i}^{\prime}+\frac{1}{2}\delta y^{2}a_{i}^{\prime\prime}\;.$ (B.3)

From now on we will suppress the $(i)$ indices for brevity. Substituting in
the expansions, the first term in brackets becomes, to first order in $\delta
y$,

$\left(\text{Bracket 1}\right)\simeq
3M_{(5)}^{3}\frac{a^{\prime}}{a}-3M_{(5)}^{3}\delta y\frac{a^{\prime
2}}{a^{2}}+\frac{3}{2}M_{(5)}^{3}\delta y\frac{a^{\prime\prime}}{a}\;,$ (B.4)

and the second term becomes:

$\left(\text{Bracket
2}\right)\simeq-3M_{(5)}^{3}\frac{a^{\prime}}{a}+6M_{(5)}^{3}\delta
y\frac{a^{\prime 2}}{a^{2}}+\frac{3}{2}M_{(5)}^{3}\delta
y\frac{a^{\prime\prime}}{a}+2\Lambda_{5}\delta y\;.$ (B.5)

Adding the two together, and taking $\delta y\rightarrow 0$, we get the
continuum equation:

$3\left(\frac{\dot{a}}{ac}\right)^{2}-3\left(\frac{a^{\prime
2}}{a^{2}}+\frac{a^{\prime\prime}}{a}\right)=\frac{2\Lambda_{5}}{M_{(5)}^{3}}\;,$
(B.6)

which is exactly $G^{0}_{\;0}=\kappa^{2}T^{0}_{\;0}$.

On the boundaries, only one of the brackets is present, and the other is
compensated for by a $(\sigma+\rho)$-type term. At $y=0$, we get

$-3M_{(5)}^{3}\delta
y\left(\frac{\dot{a}_{0}}{a_{0}c_{0}}\right)^{2}+3M_{(5)}^{3}\frac{a^{\prime}}{a}-3M_{(5)}^{3}\delta
y\frac{a^{\prime 2}}{a^{2}}+\frac{3}{2}M_{(5)}^{3}\delta
y\frac{a^{\prime\prime}}{a}+(\sigma_{0}+\rho_{0})=0\;,$ (B.7)

which in the limit yields:

$\left.\frac{a^{\prime}}{a}\right|_{0}=-\frac{\sigma_{0}+\rho_{0}}{3M_{(5)}^{3}}\;,$
(B.8)

which is, as expected, $K^{0}_{\;0}-Kh^{0}_{\;0}=-\kappa^{2}S^{0}_{\;0}$.

On the opposite boundary, we get

$-3M_{(5)}^{3}\delta
y\left(\frac{\dot{a}_{N-1}}{a_{N-1}c_{N-1}}\right)^{2}-3M_{(5)}^{3}\frac{a^{\prime}}{a}+6M_{(5)}^{3}\delta
y\frac{a^{\prime 2}}{a^{2}}+\frac{3}{2}M_{(5)}^{3}\delta
y\frac{a^{\prime\prime}}{a}+2\Lambda_{5}\delta
y+(\sigma_{N-1}+\rho_{N-1})=0\;,$ (B.9)

which in the limit yields:

$\left.\frac{a^{\prime}}{a}\right|_{N-1}=+\frac{\sigma_{N-1}+\rho_{N-1}}{3M_{(5)}^{3}}\;,$
(B.10)

which is, as expected, $K^{0}_{\;0}-Kh^{0}_{\;0}=+\kappa^{2}S^{0}_{\;0}$. All
is well thus far.

Next, we move to the considerably more complicated $G^{i}_{\;j}$ equation. For
this, we also need to Taylor expand the lapses, as $c_{i+1}$ and $c_{i-1}$
terms now appear in the $W$-tensor as well. Explicitly, the equations are:

$\begin{split}\frac{M_{(4)i}^{2}}{c_{i}^{2}}\left(\frac{\dot{a}^{2}_{i}}{a^{2}_{i}}+2\frac{\ddot{a}_{i}}{a_{i}}-2\frac{\dot{a}_{i}}{a_{i}}\frac{\dot{c}_{i}}{c_{i}}\right)\\\
=24\Biggl{\\{}&\biggl{[}-\frac{6M_{(5)}^{3}}{\delta
y}+\frac{3M_{(5)}^{3}}{\delta y}(c_{i+1}c_{i}^{-1}+2a_{i+1}a_{i}^{-1})\\\
&-\frac{M_{(5)}^{3}}{\delta
y}(2c_{i+1}c_{i}^{-1}a_{i+1}a_{i}^{-1}+a_{i+1}^{2}a_{i}^{-2})\biggr{]}\\\
&+\biggl{[}\frac{3M_{(5)}^{3}}{\delta
y}(c_{i-1}c_{i}^{-1}a_{i-1}^{2}a_{i}^{-2})\\\ &-\frac{M_{(5)}^{3}}{\delta
y}(2c_{i-1}c_{i}^{-1}a_{i-1}a_{i}^{-1}+a_{i-1}^{2}a_{i}^{-2})+2\Lambda_{5}\delta
y\biggr{]}\Bigg{\\}}\;.\end{split}$ (B.11)

Performing our Taylor expansion on both $a$ and $c$, the first term in the
square brackets becomes:

$\begin{split}(\text{Sq. bracket 1})&\simeq
M_{(5)}^{3}\frac{c^{\prime}}{c}+2M_{(5)}^{3}\frac{a^{\prime}}{a}+M_{(5)}^{3}\delta
y\frac{a^{\prime\prime}}{a}+\frac{1}{2}M_{(5)}^{3}\delta
y\frac{c^{\prime\prime}}{c}\\\ &-2M_{(5)}^{3}\delta
y\frac{a^{\prime}}{a}\frac{c^{\prime}}{c}-M_{(5)}^{3}\delta y\frac{a^{\prime
2}}{a^{2}}\;,\end{split}$ (B.12)

and the second one is:

$\begin{split}(\text{Sq. bracket 2})&\simeq-
M_{(5)}^{3}\frac{c^{\prime}}{c}-2M_{(5)}^{3}\frac{a^{\prime}}{a}+M_{(5)}^{3}\delta
y\frac{a^{\prime\prime}}{a}+\frac{1}{2}M_{(5)}^{3}\delta
y\frac{c^{\prime\prime}}{c}\\\ &+4M_{(5)}^{3}\delta
y\frac{a^{\prime}}{a}\frac{c^{\prime}}{c}+2M_{(5)}^{3}\delta y\frac{a^{\prime
2}}{a^{2}}+2\Lambda_{5}\delta y\;.\end{split}$ (B.13)

Summing and taking $\delta y\rightarrow 0$, we get our continuum equation:

$\frac{1}{c^{2}}\left(2\frac{\dot{a}}{a}\frac{\dot{c}}{c}-\frac{\dot{a}^{2}}{a^{2}}-2\frac{\ddot{a}}{a}\right)+\left(2\frac{a^{\prime\prime}}{a}+\frac{a^{\prime
2}}{a^{2}}+2\frac{a^{\prime}}{a}\frac{c^{\prime}}{c}+\frac{c^{\prime\prime}}{c}\right)=-\frac{2\Lambda_{5}}{M_{(5)}^{3}}\;,$
(B.14)

which is, of course, $G^{i}_{\;j}=\kappa^{2}T^{i}_{\;j}$.

On the boundaries, we get the following at $y=0$,

$\left.\left(-2\frac{a^{\prime}}{a}-\frac{c^{\prime}}{c}\right)\right|_{0}=-\frac{p_{0}-\sigma_{0}}{M_{(5)}^{3}}\;,$
(B.15)

which is precisely $K^{i}_{\;j}-Kh^{i}_{\;j}=-\kappa^{2}S^{i}_{\;j}$. At the
$y=L$ boundary, we get the respective minus sign accounting for the change in
orientation. So, once again, all is well.

Finally, we come to the Bianchi constraint. For the RS $\beta$’s, Eq. (A.8)
for the vanishing of the individual covariant divergences of the $W$-tensors
becomes:

$\begin{split}3&\Bigg{[}\alpha_{i}\left(\dot{a}_{i+1}a_{i}^{-1}-c_{i+1}\dot{a}_{i}a_{i}^{-1}c_{i}^{-1}\right)\left(\frac{3M_{(5)}^{3}}{\delta
y}-2\frac{M_{(5)}^{3}}{\delta y}a_{i+1}a_{i}^{-1}\right)\\\
&+\alpha_{i-1}\left(\dot{a}_{i-1}a_{i}^{-1}-c_{i-1}\dot{a}_{i}a_{i}^{-1}c_{i}^{-1}\right)\left(\frac{3M_{(5)}^{3}}{\delta
y}a_{i-1}^{2}a_{i}^{-2}-2\frac{M_{(5)}^{3}}{\delta
y}a_{i-1}a_{i}^{-1}\right)\Bigg{]}=0\;.\end{split}$ (B.16)

Taylor expanding to first order in $\delta y$,

$3\Bigg{[}\alpha_{i}\left(\frac{\dot{a}^{\prime}}{a}-\frac{\dot{a}}{a}\frac{c^{\prime}}{c}\right)\delta
y\left(\frac{M_{(5)}^{3}}{\delta
y}+\mathcal{O}(1)\right)+\alpha_{i-1}\left(-\frac{\dot{a}^{\prime}}{a}+\frac{\dot{a}}{a}\frac{c^{\prime}}{c}\right)\delta
y\left(\frac{M_{(5)}^{3}}{\delta y}+\mathcal{O}(1)\right)\Bigg{]}=0\;.$ (B.17)

Taking $\delta y\rightarrow 0$, we get the continuum equation:

$3M_{(5)}^{3}\left(\frac{\dot{a}^{\prime}}{a}-\frac{\dot{a}}{a}\frac{c^{\prime}}{c}\right)(\alpha_{i}-\alpha_{i-1})=0\;.$
(B.18)

Obviously this is true in the bulk where all $\alpha$’s are equal, but it must
also hold on the boundaries, where one of the $\alpha$’s is 0. Hence, we get
(in complete analogy with the process in Appendix A)

$3M_{(5)}^{3}\left(\frac{\dot{a}^{\prime}}{a}-\frac{\dot{a}}{a}\frac{c^{\prime}}{c}\right)=0\;,$
(B.19)

which is precisely $G^{0}_{\;5}=0$ – in other words, covariant conservation of
the interactions between the gears in the discrete theory is the analogue of
there being no flow of energy-momentum along the 5 dimension.