# On the size of temporal cliques in subcritical random temporal graphs
††thanks: Luc Devroye acknowledges support of NSERC grant A3450. Gábor Lugosi
acknowledges the support of Ayudas Fundación BBVA a Proyectos de Investigación
Científica 2021 and the Spanish Ministry of Economy and Competitiveness grant
PID2022-138268NB-I00, financed by MCIN/AEI/10.13039/501100011033,
FSE+MTM2015-67304-P, and FEDER, EU.).

Caelan Atamanchuk
Department of Mathematics and Statistics
McGill University
<EMAIL_ADDRESS>Luc Devroye
School of Computer Science, McGill University,
Montreal, Canada<EMAIL_ADDRESS>Gábor Lugosi
Department of Economics and Business,
Pompeu Fabra University, Barcelona, Spain
ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
Barcelona Graduate School of Economics
<EMAIL_ADDRESS>
###### Abstract

A _random temporal graph_ is an Erdős-Rényi random graph $G(n,p)$, together
with a random ordering of its edges. A path in the graph is called
_increasing_ if the edges on the path appear in increasing order. A set $S$ of
vertices forms a _temporal clique_ if for all $u,v\in S$, there is an
increasing path from $u$ to $v$. [Becker et al., 2023] proved that if $p=c\log
n/n$ for $c>1$, then, with high probability, there is a temporal clique of
size $n-o(n)$. On the other hand, for $c<1$, with high probability, the
largest temporal clique is of size $o(n)$. In this note we improve the latter
bound by showing that, for $c<1$, the largest temporal clique is of _constant_
size with high probability.

##  1 Introduction

A temporal graph $G=(V,E,\pi)$ is a graph $G=(V,E)$ together with an ordering
$\pi:E\to\\{1,\ldots,|E|\\}$ on the edge set, interpreted as the times where
the edges appear in the graph, often called the time stamps of $G$. We say
that an edge $e\in E$ precedes an edge $e^{\prime}\in E$ if
$\pi(e)<\pi(e^{\prime})$. A path from $u$ to $v$ is called increasing if each
edge used in the path precedes the edge that is used after it, and we say that
$v$ is reachable from $u$ (or that $u$ can reach $v$) if an increasing path
exists from $u$ to $v$. A set of vertices $S\subseteq V$ is called a _temporal
clique_ if for all dinstinct vetrices $u,v\in S$, there is an increasing path
from $u$ to $v$ (and vice versa).

In this note we discuss temporal graphs where $\pi$ is a uniform permutation
on the edges and $G$ is an Erdős-Rényi random graph. The resulting temporal
graph is called a random simple temporal graph; RSTG for short. Motivated by
modeling time-dependent propagation processes, this model was introduced by
Casteigts, Raskin, Renken, and Zamaraev [Casteigts et al., 2023].

One may generate RSTGs by a simple method: start with the complete graph
$K_{n}$, then assign each edge an independent $\operatorname{uniform}(0,1)$
random variable $(U_{e}:e\in E)$ and delete every edge with $U_{e}>(1-p)$. In
this construction, we say that $e$ precedes $e^{\prime}$ if
$U_{e}<U_{e^{\prime}}$. We also call the labels $U_{e}$ the time stamps.
Importantly, creating i.i.d. uniform time stamps like this allows us to extend
the notion of a temporal graph to infinite graphs which is needed for our
analysis.

[Casteigts et al., 2023] studied connectivity properties of RSTGs. They
identified the thresholds for different strengths of connectivity to be in the
region where $p=\frac{c\log(n)}{n}$ for some constant $c>0$. Furthering this
work, [Broutin et al., 2023] identified the asymptotic lengths of the longest
and shortest increasing paths in RSTGs with high probability for values of $p$
in this range. (We say that an event $E=E(n)$ happens with high probability if
$\mathbf{P}(E)\to 1$ as $n\to\infty$). [Becker et al., 2023] identified
$p=\frac{\log(n)}{n}$ as the threshold for the appearance of large temporal
cliques. In particular, they showed that for every $\epsilon>0$, when
$p\geq\frac{(1+\epsilon)\log(n)}{n}$, then there is a temporal clique of size
$n-o(n)$ with high probability, while if $p\leq\frac{(1-\epsilon)\log(n)}{n}$,
then every temporal clique is of size $o(n)$ with high probability.

RSTGs are a natural way to model time dependent processes on networks like
social interactions and infection spread. A closely related model is the
random gossip protocol model, in which a sequence of edges
$e_{1},\ldots,e_{k}$ are chosen uniformly from the edges of $K_{n}$ and
construct them into a graph $G_{n,k}$. Increasing paths are defined as for
temporal graphs. Papers studying this model include [Moon, 1972, Boyd and
Steele, 1979, Haigh, 1981].

For deterministic temporal graph models with random time stamps, see [Chvátal
and Komlós, 1971, Graham and Kleitman, 1973], [Lavrov and Loh, 2016, Angel et
al., 2020].

Our contribution is summarized in the following theorem. It shows that in the
subcritical regime $p=c\log n/n$ with $c<1$, the size of the largest temporal
clique is not only $o(n)$ but in fact, of size $O(1)$, improving the upper
bound of [Becker et al., 2023]. This reveals a quite spectacular phase
transition around $p=\log n/n$, since for $c>1$, there is a temporal clique of
size $n-o(n)$. The behavior of the size of the largest temporal clique near
the critical regime remains an intriguing research problem.

######  Theorem 1.

Let $p=\frac{c\log(n)}{n}$, and let $G$ be an RSTG with edge probability $p$.
If $c\in(0,1)$, then the largest temporal clique in $G$ is of size at most
$\lceil\frac{1}{1-c}+1\rceil$ with high probability.

Note that for $c\leq\frac{1}{2}$, Theorem 1 asserts that $G$ has no temporal
clique of size $4$. This bound is not improvable, since for
$p=\omega(\frac{1}{n})$ the static Erdős-Rényi graph contains a triangle with
high probability. Moreover, every triangle is trivially a temporal clique of
size $3$. We conjecture that the upper bound of Theorem 1 is sharp for all
$c\in(0,1)$.

The proof of Theorem 1 is based on relating the number of vertices that are
reachable by monotone paths from a typical vertex to the total progeny of a
certain “temporal” branching process. We utilize the temporal branching
process bounds to assert that the number of vertices that a collection of
$m\geq\lceil 1+\frac{1}{1-c}\rceil$ vertices can reach is small enough son
that the chance of them forming a component unlikely enough that expected
number of components of size $m$ tends to zero.

##  2 Temporal branching processes

We begin this section by introducing temporal branching processes that are the
key tool in the proof of Theorem 1. We only focus on branching processes with
binomial offspring distribution as this is the degree distribution of a
typical vertex in an Erdős-Rényi random graph. One way to generate these
processes as follows. Start with an infinite roted $n$-ary tree, add an
independent $\operatorname{uniform}(0,1)$ time stamp $U_{e}$ to every edge.
Delete any edge with $U_{e}>(1-p)$. This decomposes the tree into a forest and
we only focus on the component that contains the root vertex. We say that a
vertex $v$ is reachable from the root if the unique path from the root to $v$
in the infinite $n$-ary tree has edge labels $U_{e}\leq p$ for each edge on
the path and these labels are increasing on the path. The subtree consisting
of only vertices reachable from the root is a temporal branching process with
a $\operatorname{binomial}(n,p)$ offspring distribution. Throughout the rest
of the paper, $T$ is always an infinite $n$-ary tree with such a labelling on
the edges. The following sequence of results provides us the necessary upper
bounds for the size of the reachable set of $T$.

######  Lemma 2.

Let $P_{1},\ldots,P_{q}$ be a finite collection of distinct infinite paths in
$T$, and let $(X_{k})_{k\geq 0}$ be a random walk down the tree, that is,
$X_{0}$ is the root and $X_{k}$ is uniformly distributed over the children of
$X_{k-1}$ for all $k\geq 1$. Then,
$\mathbf{P}(\tau\geq\ell)\leq\frac{q}{n^{\ell}}$, where

$\tau=\max\\{k>0:X_{0},\ldots,X_{k}\text{ coincides with one of the $q$
paths}\\}.$

###### Proof.

If $\tau\geq\ell$, then $X_{0},\ldots,X_{\ell}$ coincides with one of the
$P_{1},\ldots,P_{q}$. Since $X_{k}$ is uniform over the children of $X_{k-1}$
and the tree is $n$-ary, the result follows from the union bound yields. ∎

######  Lemma 3.

Let $T^{*}$ be the set of all vertices that are reachable from the root of
$T$. If $v_{1},\ldots,v_{q}$ are uniform vertices chosen from the $\ell$-th
generation of $T$, then

$\mathbf{P}(v_{1},\ldots,v_{q}\in T^{*})\leq\frac{(q-1)!e^{npq}}{n^{\ell q}}.$

Furthermore, when $\ell\geq(np)^{2}$ and $np\to\infty$ as $n\to\infty$,

$\mathbf{P}(v_{1},\ldots,v_{q}\in T^{*})=o\left(\frac{1}{n^{\ell q}}\right),$

for any fixed $q\geq 1$ (not depending on $n$).

###### Proof.

Suppose that $v_{1},\ldots,v_{r}\in T^{*}$ and let $T^{\prime}$ be a subtree
of $T$ consisting of $r$ distinct infinite paths starting at the root through
$v_{1},\ldots,v_{r}$. Let $(X_{k})_{k\geq 0}$ be a random walk down the tree
(independent of $v_{1},\ldots,v_{r}$) and let $\tau$ be as in Lemma 2. If
$\tau=j$, then there are $\ell-j$ edges left that need to both exist and be
increasing to have $X_{\ell}\in T^{*}$. Hence, if
$V_{r}=\\{v_{1},\ldots,v_{r}\in T^{*}\\}$,

$\mathbf{P}(X_{\ell}\in
T^{*}|V_{r}\cap\\{\tau=j\\})\leq\frac{p^{\ell-j}}{(\ell-j)!}.$

Combining this with Lemma 2 yields

$\displaystyle\mathbf{P}(X_{\ell}\in
T^{*}|V_{r})\leq\sum_{j=0}^{\ell}\mathbf{P}(\tau=j)\mathbf{P}(X_{\ell}\in
T^{*}|V_{r}\cap\\{\tau=j\\})\leq\sum_{j=0}^{\ell}\left(\frac{r}{n^{j}}\right)\frac{p^{\ell-j}}{(\ell-j)!}=\frac{r}{n^{\ell}}\sum_{j=0}^{\ell}\frac{(np)^{\ell-j}}{(\ell-j)!}\leq\frac{re^{np}}{n^{\ell}}.$
(1)

Then, since $X_{\ell}$ is distributed uniformly across the $\ell$-th
generation, applying the above inequality repeatedly,

$\displaystyle\mathbf{P}(v_{1},\ldots,v_{q}\in T^{*})$
$\displaystyle=\mathbf{P}(v_{1}\in T^{*})\cdot\mathbf{P}(v_{2}\in
T^{*}|V_{1})\cdots\mathbf{P}(v_{q}\in T^{*}|V_{q-1})$
$\displaystyle\leq\frac{p^{\ell}}{\ell!}\prod_{r=1}^{q-1}\left(\frac{re^{np}}{n^{\ell}}\right)=\left(\frac{(np)^{\ell}}{\ell!}\right)\frac{(q-1)!e^{np(q-1)}}{n^{\ell
q}}\leq\frac{(q-1)!e^{npq}}{n^{\ell q}}.$

For $\ell\geq(np)^{2}$, $(np)^{\ell}/\ell!$ is decreasing, so by Stirling’s
approximation we have

$\frac{(np)^{\ell}}{\ell!}\leq\frac{(np)^{(np)^{2}}e^{(np)^{2}}}{(np)^{2(np)^{2}}}=\left(\frac{e}{np}\right)^{(np)^{2}}.$

Thus, if $\ell\geq(np)^{2}$ and $np\to\infty$ as $n\to\infty$, re-using the
bound from (1) yields

$\displaystyle\mathbf{P}(v_{1},\ldots,v_{q}\in T^{*})$
$\displaystyle=\mathbf{P}(v_{1}\in T^{*})\cdot\mathbf{P}(v_{2}\in
T^{*}|V_{1})\cdots\mathbf{P}(v_{q}\in T^{*}|V_{q-1})$
$\displaystyle\leq\frac{p^{\ell}}{\ell!}\frac{(q-1)!e^{np(q-1)}}{n^{\ell(q-1)}}\leq\left(\frac{e}{np}\right)^{(np)^{2}}\frac{(q-1)!e^{np(q-1)}}{n^{\ell
q}}=o\left(\frac{1}{n^{\ell q}}\right),$

where in the last equality we used the fact that
$e^{(np)^{2}+np(q-1)}=o((np)^{(np)^{2}})$ when $np\to\infty$ and $q>0$ is
fixed. ∎

We may use Lemma 3 to bound the moments of the number of vertices reachable in
a particular generation:

######  Corollary 4.

Let $(Z_{\ell})_{\ell\geq 0}$ be the number of vertices in $T$ that are
reachable from the root in the $\ell$-th generation. Then for all integers
$\ell\geq 0$ and $q\geq 1$, $\mathbf{E}[Z_{\ell}^{q}]\leq(q-1)!e^{npq}$.
Furthermore, when $\ell\geq(np)^{2}$ and $np\to\infty$, we further have that
$\mathbf{E}[Z_{\ell}^{q}]=o(1)$ for any fixed $q\geq 1$ (not depending on
$n$).

The above sequence of results are all in preparation for the next bound. It is
used to control the number of vertices that a typical vertex in an RSTG can
reach, which further give us control over how large temporal cliques can be.

######  Theorem 5.

Let $T^{*}$ be the set of vertices in $T$ that are reachable from the root and
suppose that $np\to\infty$. Then, for any integer $q\geq 1$, there is a
constant $C(q)$ such that

$\mathbf{E}[|T^{*}|^{q}]\leq C(q)(np)^{2q}e^{npq}.$

###### Proof.

Using Hölder’s inequality along with the identity $\mathbf{E}[X^{q}]=\int
qt^{q-1}\mathbf{P}(X>t)dt$ yields

$\displaystyle\mathbf{E}|T^{*}|^{q}=\mathbf{E}\left(\sum_{i=0}^{(np)^{2}}Z_{i}+\sum_{i>(np)^{2}}Z_{i}\right)^{q}\leq\underbrace{2^{q-1}(np)^{2(q-1)}\mathbf{E}\left[\sum_{i=0}^{(np)^{2}}Z_{i}^{q}\right]}_{:=I}+\underbrace{2^{q-1}q\sum_{t>0}t^{q-1}\mathbf{P}\left(\sum_{i>(np)^{2}}Z_{i}\geq
t\right)}_{:=II}.$

The first inequality of Corollary 4 may be used to bound $I$, as

$\mathbf{E}\left[\sum_{i=0}^{(np)^{2}}Z_{i}^{q}\right]\leq(np)^{2}\left(\sup_{i\geq
0}\mathbf{E}[Z_{i}^{q}]\right)\leq(np)^{2}(q-1)!e^{npq}.$

To bound $II$, we may write

$\displaystyle t^{q-1}\mathbf{P}\left(\sum_{i>(np)^{2}}Z_{i}\geq t\right)$
$\displaystyle\leq t^{q-1}\mathbf{P}\left(Z_{(np)^{2}+\log
t}>0\right)+t^{q-1}\mathbf{P}\left(\bigcup_{i=(np)^{2}+1}^{(np)^{2}+\log
t}\left\\{Z_{i}\geq\frac{t}{\log t}\right\\}\right)$
$\displaystyle\leq\underbrace{t^{q-1}\mathbf{P}\left(Z_{(np)^{2}+\log
t}>0\right)}_{:=III}+\underbrace{t^{q-1}(\log t)\max_{(np)^{2}\leq
i\leq(np)^{2}+\log t}\mathbf{P}\left(Z_{i}\geq\frac{t}{\log
t}\right)}_{:=IV},$

and we can bound the two terms separately. To bound $III$, note that at level
$(np)^{2}+\log t$, there are $n^{(np)^{2}+\log t}$ vertices, and they are each
reachable with probability $p^{(np)^{2}+\log t}/((np)^{2}+\log t)!$. Thus, by
Stirling’s approximation,

$\displaystyle III$ $\displaystyle\leq\frac{t^{q-1}(np)^{(np)^{2}+\log
t}}{((np)^{2}+\log t)!}$ $\displaystyle\leq\frac{t^{q-1}(enp)^{(np)^{2}+\log
t}}{(np)^{2(np)^{2}+2\log t}}$ $\displaystyle\leq
t^{q-1}\left(\frac{e}{(np)}\right)^{(np)^{2}+\log t}$ $\displaystyle\leq
t^{q-\log(np)}\left(\frac{e}{(np)}\right)^{(np)^{2}}$

for any $t\geq 0$. In particular, this implies that $III$ is summable and
converges to 0 when $np\to\infty$. Applying Markov’s inequality and the second
inequality in Corollary 4 gives

$IV\leq o\left(1\right)\left(\frac{\log^{k+1}(t)}{t^{k-q+1}}\right),$

for any $k,t\geq 0$ where $k$ does not depend on $n$. Choosing $k=q+1$ results
in $IV$ being summable and also converging to $0$ as $n\to\infty$ and
$np\to\infty$. Grouping up all that only depends on $q$ and upper bounding by
some dominating constant $C(q)$ we get

$\mathbf{E}|T^{*}|^{q}\leq I+III+IV\leq C(q)(np)^{2q}e^{npq}.$

∎

##  3 Proof of Theorem 1

We are now prepared to prove Theorem 1. In order for labelled vertices
$\\{1,\ldots,m\\}$ to form a temporal clique in an RSTG they need to all be
reachable from one another. This can happen if and only if for all distinct
$u,v\in\\{1,\ldots,m\\}$, there is a vertex $w$ that can reach $v$ with only
edges that have time stamps above $p/2$, and is reachable from $u$ with only
edges that have time stamps below $p/2$. With this in mind, for an interval
$[a,b]$ we define $G_{[a,b]}$ be the subgraph obtained from $G$ by only
keeping edges with time stamps in $[a,b]$. Set $A_{1},\ldots,A_{m}$ to be the
collection of all vertices that are reachable from $1,\ldots,m$ in
$G_{[0,p/2]}$ and $B_{1},\ldots,B_{m}$ to be the collection of all vertices in
$G_{[p/2,p]}$ that can reach $1,\ldots,m$. With this new notation, we can say
that $\\{1,\ldots,m\\}$ form a temporal clique if for all
$i,j\in\\{1,\ldots,m\\}$ distinct, the set $A_{i}\cap B_{j}$ is nonempty.

It is important to note that $G_{[p/2,p]}$ and $G_{[0,p/2]}$ are identically
distributed RSTGs, but are not independent. The next lemma deals with the
dependence between the two subgraphs.

######  Lemma 6.

Let $G$ be an RSTG with vertices labelled $\\{1,\ldots,n\\}$, and let $0\leq
a<b<c<d\leq 1$. Set $X_{i}=\mathbf{1}_{e_{i}\in G_{[a,b]}}$,
$Y_{i}=\mathbf{1}_{e_{i}\in G_{[c,d]}}$ for some enumeration of the edges of
$K_{n}$, $e_{1},\ldots,e_{\binom{n}{2}}$, and let
$X=(X_{1},\ldots,X_{\binom{n}{2}})$ and $Y=(Y_{1},\ldots,Y_{\binom{n}{2}})$.
Let $O_{[a,b]}$ and $O_{[c,d]}$ be the permutation that denotes the relative
orderings of the edges in the two graphs. If
$f(x_{1},\ldots,x_{\binom{n}{2}},y)$ and $g(x_{1},\ldots,x_{\binom{n}{2}},y)$
are two non-decreasing functions in $x_{1},\ldots,x_{\binom{n}{2}}$ for any
fixed $y$, then

$\mathbf{E}\left[f\left(X,O_{[a,b]}\right)g\left(Y,O_{[c,d]}\right)\right]\leq\mathbf{E}\left[f\left(X,O_{[a,b]}\right)\right]\mathbf{E}\left[g\left(Y,O_{[c,d]}\right)\right].$

In particular,

$\mathbf{E}\left[|A_{1}|^{q}\cdots|A_{m}|^{q}\cdot|B_{1}|^{q}\cdots|B_{m}|^{q}\right]\leq\mathbf{E}\left[|A_{1}|^{q}\cdots|A_{m}|^{q}\right]\mathbf{E}\left[|B_{1}|\cdots|B_{m}|^{q}\right]=\mathbf{E}\left[|A_{1}|^{q}\cdots|A_{m}|^{q}\right]^{2},$

for all $q\geq 0$ and $A_{1},B_{1},\ldots,A_{m},B_{m}$ defined as above.

###### Proof.

Conditioned on $Z=(X_{1},Y_{1},\ldots,X_{\binom{n}{2}},Y_{\binom{n}{2}})$, all
of the randomness of $f\left(X,O_{[a,b]}\right)$ and
$g\left(Y,O_{[c,d]}\right)$ comes from the random relative orderings. Since
$a<b<c<d$, the two random variables $O_{[a,b]}$ and $O_{[c,d]}$ are
independent, which implies that $f\left(X,O_{[a,b]}\right)$ and
$g\left(Y,O_{[c,d]}\right)$ must also be conditionally independent on $Z$.
Hence, by the tower property of conditional expectation,

$\displaystyle\mathbf{E}\left[f\left(X,O_{[a,b]}\right)g\left(Y,O_{[c,d]}\right)\right]$
$\displaystyle=\mathbf{E}\left[\mathbf{E}\left[f\left(X,O_{[a,b]}\right)g\left(Y,O_{[c,d]}\right)\Big{|}Z\right]\right]$
$\displaystyle=\mathbf{E}\left[\mathbf{E}\left[f\left(X,O_{[a,b]}\right)\Big{|}Z\right]\mathbf{E}\left[g\left(Y,O_{[c,d]}\right)\Big{|}Z\right]\right].$
$\displaystyle=\mathbf{E}\left[\mathbf{E}\left[f\left(X,O_{[a,b]}\right)\Big{|}X\right]\mathbf{E}\left[g\left(Y,O_{[c,d]}\right)\Big{|}Y\right]\right],$

where the final equality just follows from the fact that, once we condition on
$X$, knowing $Y$ tells us nothing about $f(X,O_{[a,b]})$ and vice versa. The
random variables

$\mathbf{E}\left[f\left(X,O_{[a,b]}\right)\Big{|}X=(z_{1},\ldots,z_{\binom{n}{2}})\right],\
\text{and}\
\mathbf{E}\left[g\left(Y,O_{[c,d]}\right)\Big{|}Y=(z_{1},\ldots,z_{\binom{n}{2}})\right]$

are non-decreasing functions in $z_{1},\ldots,z_{\binom{n}{2}}$ by the
definitions of $f$ and $g$. Furthermore, the collection of random variables
$\\{X_{i}:i\in\\{1,\ldots,\binom{n}{2}\\}\\}\cup\\{Y_{i}:i\in\\{1,\ldots,\binom{n}{2}\\}\\}$
are negatively associated (this can be seen by combining Proposition 7 and
Lemma 8 from [Dubhashi and Ranjan, 1998]). With this, applying the tower
property again gives

$\displaystyle\mathbf{E}\left[f\left(X,O_{[a,b]}\right)g\left(Y,O_{[c,d]}\right)\right]\leq\mathbf{E}\left[f\left(X,O_{[a,b]}\right)\right]\mathbf{E}\left[g\left(Y,O_{[c,d]}\right)\right].$

Observing that $|A_{1}|\cdots|A_{m}|$ and $|B_{1}|\cdots|B_{m}|$ satisfy the
conditions of the first statement and are identically distributed is enough to
complete the proof of the second inequality. ∎

The next lemma acts as a bridge between RSTGs and the temporal branching
processes explored in the previous section. The idea behind the proof hinges
on the fact that the sizes of $\operatorname{binomial}(n,p)$ branching
processes upper bounds the sizes of neighbourhoods around vertices in an Erdős
Rényi graph, though formalizing this idea takes some work. Equipped with this
and Theorem 5, the proof of Theorem 1 is reduced to a routine use of the first
moment method.

######  Lemma 7.

$|A_{1}|$ is stochastically dominated by $|T^{*}|$, where $T^{*}$ is the set
of vertices reachable from the root in a temporal branching process $T$ with
offspring distribution $\operatorname{binomial}(n,p/2)$. In particular,
$\mathbf{E}[|A_{1}|^{q}]\leq\mathbf{E}[|T^{*}|^{q}]$ for all $q\geq 0$.

###### Proof.

We can determine $A_{1}$ via the foremost tree algorithm from [Casteigts et
al., 2023]. The algorithm goes as follows: First initialize with $\tau_{0}=1$
and $G_{0}$ as the isolated vertex labelled $1$. Then, while $\tau_{k}\leq
p/2$ set $\tau_{k+1}$ to be the smallest time stamp of an edge between $G_{k}$
and $K_{n}\setminus G_{k}$ that is larger than $\tau_{k}$ and add the
corresponding edge $e_{k+1}$ and vertex $v_{k+1}$ to make $G_{k+1}$ if
$\tau_{k+1}\leq p/2$. Whenever we first hit a $k^{\prime}$ such that
$\tau_{k^{\prime}}>p/2$, then we output $G_{k^{\prime}}$. Moreover,
$|A_{1}|=|G_{k^{\prime}}|$, that is,

$|A_{1}|=\inf\left\\{k\geq 0:\tau_{k}>\frac{p}{2}\right\\}.$ (2)

This foremost tree algorithm can also be run on the tree $T$ as a way to
generate $T^{*}$ with the same procedure, and we denote the sequence of time
stamps in this graph as $\tau^{*}_{k}$. Additionally, we denote by $E_{k}$ and
$E^{*}_{k}$ the collection of all viable edges that could be added during step
$k$, i.e., it is all edges from $G_{k}$ to $K_{n}$ that, if added, keep the
graph $G_{k+1}$ as an increasing tree (all vertices reachable from the root).
Note that by the definition of the algorithm, every edge in $E_{k}$ must have
a time stamp that is at least $\tau_{k}$ and similarly for $E_{k}^{*}$ and
$\tau^{*}_{k}$. Moreover, the time stamps of edges in $E_{k}$ and $E^{*}_{k}$
are uniformly distributed on $[\tau_{k-1},1]$ and $[\tau_{k-1}^{*},1]$
respectively. By means of a direct inductive coupling we show that $\tau$
stochastically dominates $\tau^{*}$, $|E_{k}^{*}|$ stochastically dominates
$|E_{k}|$, and hence $|T^{*}|$ must stochastically dominate $|A_{1}|$ by the
characterization of (2).

The base case of the induction is easy to see. By definition $E_{1}=n-1$,
$E_{1}^{*}=n$, $\tau_{1}\sim\min_{1\leq i\leq n-1}U_{1,i}$, and
$\tau_{1}^{*}\sim\min_{1\leq n}U_{1,i}$. Thus, just using the same uniforms to
generate both $\tau_{1}$ and $\tau_{1}^{*}$ is enough. Now suppose that there
is some probability space $(\Omega_{k-1},\mathcal{F}_{k-1},\mathbf{P}_{k-1})$
and random variables distributed as
$|E_{k-1}|,|E_{k-1}^{*}|,\tau_{k-1},\tau^{*}_{k-1}$ (we just use the same
symbols to denote these random variables) such that
$|E_{k-1}|(\omega)\leq|E_{k-1}^{*}|(\omega)$ and
$\tau^{*}_{k-1}(\omega)\leq\tau_{k-1}(\omega)$ for all
$\omega\in\Omega_{k-1}$. In the $(k-1)$-th step of the algorithm we added a
new vertex to both graphs, resulting in $n-k$ possible new edges to $G$ and
$n$ edges to $T$ for the $k$-th step. Hence, since we cannot add edges that
are below $\tau_{k-1}$ and $\tau_{k-1}^{*}$ respectively

$|E_{k}|\sim|E_{k-1}|+\operatorname{binomial}(n-k,1-\tau_{k-1}),\quad|E_{k}^{*}|\sim|E_{k-1}^{*}|+\operatorname{binomial}(n,1-\tau_{k}^{*}),$

and, recalling the distribution of time stamps in $E_{k}$ and $E_{k}^{*}$,

$\tau_{k}\sim\tau_{k-1}+(1-\tau_{k-1})\min_{1\leq
i\leq|E_{k}|}U_{k,i},\quad\tau_{k}^{*}\sim\tau^{*}_{k-1}+(1-\tau^{*}_{k-1})\min_{1\leq
i\leq|E_{k}^{*}|}U_{k,i}.$

Let $(\Omega_{k},\mathcal{F}_{k},\mathbf{P}_{k})$ be the product of
$(\Omega_{k-1},\mathcal{F}_{k-1},\mathbf{P}_{k-1})$ with
$(\Omega^{\prime},\mathcal{F}^{\prime},\mathbf{P}^{\prime})$, the probability
space of $\binom{n}{2}+n$ independent uniform random variables,
$(U_{k,i}:1\leq i\leq\binom{n}{2}+n)$. Here we can couple the binomial random
variables by generating them as
$\sum_{i=1}^{k}\mathbf{1}_{\\{U_{k,i}\leq(1-\tau_{k-1})\\}}$ and
$\sum_{i=1}^{k}\mathbf{1}_{\\{U_{k,i}\leq(1-\tau^{*}_{k-1}\\}}$ respectively.
Then, if we generate $|E_{k}|$ and $|E_{k}^{*}|$ with these binomials, by the
inductive hypothesis, it must be the case that
$|E_{k}|(\omega)\leq|E_{k}^{*}|(\omega)$ for all $\omega\in\Omega_{k}$.
Similarly, using the uniforms $(U_{k,i}:n+1\leq i\leq|E_{k}^{*}|)$ to generate
both $\tau_{k}$ and $\tau_{k}^{*}$ according to their distributions results in
also having $\tau_{k}^{*}(\omega)\leq\tau_{k}(\omega)$ for all
$\omega\in\Omega_{k}$. ∎

With the lemmas out of the way we now prove our main result.

###### Proof of Theorem 1.

Let $m\geq 0$ and let $A_{1},\ldots,A_{m},B_{1},\ldots,B_{m}$ be as defined in
the beginning of this section. Let $N$ be the number of temporal cliques of
size $m$ in $G$. Then, if we take $(v_{ij})_{i,j=1}^{m}$ to be independently
and uniformly chosen random vertices from the labelled set $\\{1,\ldots,n\\}$,
we may apply Lemma 6 to get that

$\displaystyle\mathbf{E}[N]$ $\displaystyle\leq
n^{m}\mathbf{P}(\\{1,\ldots,m\\}\text{ form a temporal clique})$
$\displaystyle=n^{m+m(m-1)}\mathbf{P}(v_{ij}\in A_{i},v_{ij}\in B_{j}\ \forall
i\neq j)$
$\displaystyle=n^{m+m(m-1)}\mathbf{E}\left[\mathbf{P}\Big{(}v_{ij}\in
A_{i},v_{ij}\in B_{j}\ \forall i\neq
j\Big{|}|A_{1}|,\ldots,|A_{m}|,|B_{1}|,\ldots,|B_{m}|\Big{)}\right]$
$\displaystyle\leq
n^{m+m(m-1)}\mathbf{E}\left[\left(\frac{|A_{1}|}{n}\right)^{m-1}\cdots\left(\frac{|A_{m}|}{n}\right)^{m-1}\left(\frac{|B_{1}|}{n}\right)^{m-1}\cdots\left(\frac{|B_{m}|}{n}\right)^{m-1}\right]$
$\displaystyle\leq
n^{m+m(m-1)}\mathbf{E}\left[\left(\frac{|A_{1}|}{n}\right)^{m-1}\cdots\left(\frac{|A_{m}|}{n}\right)^{m-1}\right]^{2}$
$\displaystyle\leq
n^{m-m(m-1)}\mathbf{E}[|A_{1}|^{m-1}\cdots|A_{m}|^{m-1}]^{2}.$

Applying Hölder’s inequality along with Lemma 7 and Theorem 5 gives us the
upper bound

$\displaystyle\mathbf{E}[N]$ $\displaystyle\leq
n^{m-m(m-1)}\mathbf{E}[|A_{1}|^{m(m-1)}]^{2}\leq
C(m(m-1))^{2}(c\log(n))^{2m(m-1)}n^{m-m(m-1)+c(m(m-1))}.$

If $m-m(m-1)+cm(m-1)<0$, then $\mathbf{E}[N]\to 0$ as $n\to\infty$.
Rearranging, this inequality is equivalent to
$m\geq\lceil\frac{1}{1-c}+1\rceil$ as $m$ is an integer. ∎

## References

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  * [Dubhashi and Ranjan, 1998] Dubhashi, D. and Ranjan, D. (1998).  Balls and bins: a study in negative dependence.  Random Structures Algorithms, 13(2):99–124.
  * [Graham and Kleitman, 1973] Graham, R. L. and Kleitman, D. J. (1973).  Increasing paths in edge ordered graphs.  Period. Math. Hungar., 3:141–148.
  * [Haigh, 1981] Haigh, J. (1981).  Random exchanges of information.  Journal of Applied Probability, 18(3):743–746.
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# On Conformal Transformation with Multiple Scalar Fields and Geometric
Property of Field Space with Einstein-like Solutions

Yong Tanga,b,c,d and Yue-Liang Wub,c,e aSchool of Astronomy and Space
Sciences, University of Chinese Academy of Sciences (UCAS), Beijing, China
bSchool of Fundamental Physics and Mathematical Sciences,
Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
cInternational Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou,
China
dNational Astronomical Observatories, Chinese Academy of Sciences, Beijing,
China
eInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China

###### Abstract

Multiple scalar fields appear in vast modern particle physics and gravity
models. When they couple to gravity non-minimally, conformal transformation is
utilized to bring the theory into Einstein frame. However, the kinetic terms
of scalar fields are usually not canonical, which makes analytic treatment
difficult. Here we investigate under what conditions the theories can be
transformed to the quasi-canonical form, in which case the effective metric
tensor in field space is conformally flat. We solve the relevant nonlinear
partial differential equations for arbitrary number of scalar fields and
present several solutions that may be useful for future phenomenological model
building, including the $\sigma$-model with a particular non-minimal coupling.
We also find conformal flatness can always be achieved in some modified
gravity theories, for example, Starobinsky model.

##  I Introduction

Scalar fields are ubiquitous in modern particle physics and gravity models,
including inflation theories and dark energy scenarios Lyth:1998xn ;
Zee:1978wi ; Adler:1982ri ; Wu:2015wwa ; Wu:2017urh ; Copeland:2006wr ;
Nojiri:2017ncd ; Cai:2009zp . In many cases Wetterich:1987fm ; Kaiser:1994vs ;
Ferrara:2010in ; GarciaBellido:2011de ; Bars:2013yba ; Csaki:2014bua ;
Hamada:2014wna ; Kannike:2015apa ; Ferreira:2018qss ; Tang:2018mhn ;
Ghilencea:2018thl ; Ishiwata:2018dxg ; Barnaveli:2018dxo ; Karam:2018mft ;
Ema:2017rqn , the scalar fields are non-minimally coupled to gravity through a
term $f(\phi^{i})R$ where $f$ is some function of scalar fields $\phi^{i}$ and
$R$ is the Ricci scalar curvature. For example, the function $f(\phi)$ is
proportional to $\phi^{2}$ in Jordan-Brans-Dicke theory Jordan:1959eg ;
Brans:1961sx  and Higgs inflation model Bezrukov:2007ep . In modified gravity
theories where only functions of $R$ are introduced Sotiriou:2008rp , it is
equivalent to treat as introducing a scalar field, for instance,
$f(\phi)\sim\phi^{2}$ in the Starobinsky model Starobinsky:1980te .

The Lagrangian with $f(\phi^{i})R$ term is usually referred as the one in
Jordan frame, in which the kinetic terms of scalar fields are canonical. To
compare with experimental observations, it is standard to perform a conformal
transformation Maeda:1988ab ; Shapiro:1995kt ; Faraoni:1998qx  on the metric
tensor to Einstein frame such that $f(\phi^{i})R$ is transformed into
$\tilde{R}$. However, conformal transformation would induce non-canonical
kinetic term of scalars in Einstein frame, which makes analytic treatment
rather complicated and various approximate methods have to be utilized. If
there is only one scalar field in the theory, it is always possible to
redefine the field variable and make the kinetic term canonical. However, it
is not clear whether such redefinition always exists in theories with multiple
scalar fields.

The systematic investigation on conformal transformation with multiple scalar
fields was conducted in Ref. Kaiser:2010ps , in which the analysis was done
with the effective metric tensor $\mathcal{G}_{ij}$ in the field space defined
by the kinetic term $\mathcal{G}_{ij}d\phi^{i}d\phi^{j}$. Only one solution
was found for $f(\phi^{i})$ with two scalar fields such that the associated
$\mathcal{G}_{ij}$ is conformally flat. This finding seems to pose an obstacle
for analytic analysis in phenomenological model building involving multiple
scalar fields. For example, standard model Higgs doublet composes four real
scalar fields, and $\sigma$-model has $N$ fields with SO$(N)$ symmetry.

In this paper, we present several new solutions for $f(\phi^{i})$ with the
corresponding $\mathcal{G}_{ij}$ conformally flat. We solve the relevant
nonlinear partial differential equations for the requirements on $f(\phi^{i})$
for any number of scalar fields and tabulate the solutions in Table. 1, which
might be useful for future model building. Our results suggest that for
$\sigma$-model with $N$ scalar fields the field space can be conformally flat
if the coupling has a particular form, in which local scaling symmetry is
evident. We also find in some modified gravity theories that involve function
of $R$ and scalar fields, the associated field spaces are always conformally
flat.

This paper is organized as follows. In section II, we establish our
theoretical formalism along with the notations. Then in section III we analyze
the structure of field space and solve the differential equations for
conformally flat metric tensor. Later in section IV we discuss a particular
case in which a local scaling symmetry is present for $\sigma$-model that
couples to gravity non-minimally. After that, in section V we show in modified
gravity theories, such as Starobinsky model, the field space is always
conformally flat. Finally, we give our conclusion.

Throughout the paper, we use the 4-dimensional space-time metric $g_{\mu\nu}$
with a sign convention $(-1,1,1,1)$, and the natural unit, $M_{p}\equiv
1/\sqrt{8\pi G}=1$. Greek letters $(\mu,\nu,\rho,\cdots)$ denotes the space-
time indices while Latin letters $(i,j,I,J,\cdots)$ refer to field variables
in the field space. Riemann tensor is defined by
$R_{\;\;\sigma\mu\nu}^{\rho}=\partial_{\mu}\Gamma_{\sigma\nu}^{\rho}-\partial_{\nu}\Gamma_{\sigma\mu}^{\rho}+\Gamma_{\mu\tau}^{\rho}\Gamma_{\sigma\nu}^{\tau}-\Gamma_{\nu\tau}^{\rho}\Gamma_{\sigma\mu}^{\tau},$
where the connection is given by
$\Gamma_{\mu\nu}^{\rho}=\frac{1}{2}g^{\rho\tau}\left[\partial_{\mu}g_{\nu\tau}+\partial_{\nu}g_{\mu\tau}-\partial_{\tau}g_{\mu\nu}\right]$,
and Ricci scalar $R$ is defined through $R_{\sigma\nu}\equiv
R_{\;\;\sigma\rho\nu}^{\rho}$, $R\equiv g^{\sigma\nu}R_{\sigma\nu}$. We may
easily check $R_{\;\;\sigma\mu\nu}^{\rho}=R_{\nu\mu\sigma}{}^{\rho}$, where
the latter is also widely used in literature.

##  II Formalism and Notations

We shall first consider the following general Lagrangian $\mathcal{L}$ in
4-dimensional space-time for $N$ non-minimally coupled scalar fields,
$\phi^{i},i=1,...,N$,

$\frac{\mathcal{L}}{\sqrt{-g}}=f(\phi^{i})R-\frac{1}{2}g^{\mu\nu}\delta_{ij}\nabla_{\mu}\phi^{i}\nabla_{\nu}\phi^{j}-V(\phi^{i}),$
(1)

where $g$ are the determinant of $g_{\mu\nu}$ and the covariant derivative is
denoted by $\nabla$. $V$ is the scalar potential that can be neglected in our
main theoretical discussions but would be relevant for phenomenological
studies. This Lagrangian is referred to as the one in Jordan frame where non-
minimal couplings $f(\phi^{i})R$ is present.

We make the standard conformal transformation on the metric tensor,

$\tilde{g}_{\mu\nu}=\Omega^{2}(x)g_{\mu\nu},\;\Omega^{2}(x)=2f(\phi^{i}).$ (2)

After using the transformation relations in the Appendix and denoting
$\omega\equiv\ln\Omega$, we can get the Lagrangian in Einstein frame,

$\displaystyle\frac{\mathcal{L}}{\sqrt{-\tilde{g}}}$
$\displaystyle\supseteq\frac{1}{2}\tilde{R}-3\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\omega\tilde{\nabla}_{\nu}\omega-\frac{1}{2\Omega^{2}}\tilde{g}^{\mu\nu}\delta_{ij}\tilde{\nabla}_{\mu}\phi^{i}\tilde{\nabla}_{\nu}\phi^{j}$
$\displaystyle=\frac{1}{2}\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\mathcal{G}_{ij}\tilde{\nabla}_{\mu}\phi^{i}\tilde{\nabla}_{\nu}\phi^{j},$
(3)

where an effective metric tensor in field space $\mathcal{G}_{ij}$ appears and
is given by

$\mathcal{G}_{ij}=\frac{1}{2f}\left(\delta_{ij}+\frac{3}{f}f_{i}f_{j}\right).$
(4)

Here and after, we use the short notation for derivatives of $f$,
$f_{i}=\frac{\partial
f}{\partial\phi^{i}},f_{ij}=\frac{\partial^{2}f}{\partial\phi^{i}\partial\phi^{j}}$,
etc.

Due to the curved metric $\mathcal{G}_{ij}$ in field space, the scalar fields
generally have non-canonical kinetic terms (unless we begin with non-canonical
ones in Eq. 1, $\delta_{ij}\rightarrow 2f\delta_{ij}-\frac{3}{f}f_{i}f_{j}$).
If there is only one scalar field $\phi$ in the theory and if
$\mathcal{G}_{11}$ is positive definite, we can always redefine a new field
variable $\Phi$ by the differential equation,
$d\Phi(x)/d\phi(x)=\pm\sqrt{\mathcal{G}_{11}},$ therefore make $\Phi$’s
kinetic term canonical. For the cases with multiple scalars, it would be much
more complicated, as we shall present below.

##  III Structure of The Field Space

With the metric tensor $\mathcal{G}_{ij}(\phi^{k})$ in field space, we would
like to know whether there exists a set of field coordinates
$\varphi^{I}=\varphi^{I}(\phi^{k})$ such that the associated metric tensor is
flat, $\mathcal{G}_{JK}(\varphi^{I})=\delta_{JK}$. One necessary condition for
the existence is that the Ricci scalar curvature
$\mathcal{R}\left(\mathcal{G}_{ij}\right)=0$ because $\mathcal{R}$ is an
invariant quantity under coordinate transformation. We can easily check that
$\mathcal{R}\equiv 0$ for $N=1$ due to the anti-symmetric properties of
$\mathcal{R}_{ijkl}$, which confirms our analysis above that canonical kinetic
term can always be achieved after the redefinition of the field for $N=1$.
However, in general $\mathcal{R}\neq 0$ for $N>1$, therefore field coordinates
$\varphi^{i}$ may not exist such that their kinetic terms are canonical.

It is modest to ask whether the geometry of field space is conformally flat,
$\mathcal{G}_{JK}(\varphi^{I})\propto\delta_{JK}$. In such cases, the kinetic
terms are quasi-canonical. As we have learned from Riemannian geometry,
manifolds with $N\leq 3$ are always conformally flat. But for $N>3$ it is no
longer true and conformal flatness is determined by the Weyl tensor, which is
defined as

$\displaystyle\mathcal{C}_{ijkl}\equiv\mathcal{R}_{ijkl}-\frac{2}{N-2}\left(\mathcal{G}_{i[k}\mathcal{R}_{l]j}-\mathcal{G}_{j[k}\mathcal{R}_{l]i}\right)+\frac{2\mathcal{R}}{(N-1)(N-2)}\mathcal{G}_{i[k}\mathcal{G}_{l]j}.$
(5)

If $\mathcal{C}_{ijkl}=0$, the field space is conformally flat. To compute
$\mathcal{C}_{ijkl}$ in the cases of our interest, we conformally transform
$\mathcal{G}_{ij}$ into

$\hat{\mathcal{G}}_{ij}=2f\times\mathcal{G}_{ij}=\delta_{ij}+\frac{3}{f}f_{i}f_{j},$
(6)

and calculate the corresponding
$\hat{\mathcal{R}}_{ijkl},\hat{\mathcal{R}}_{ij},\hat{\mathcal{R}}$ and
$\hat{\mathcal{C}}_{ijkl}$. The relation
$\hat{\mathcal{C}}^{i}{}_{jkl}=\mathcal{C}^{i}{}_{jkl}$ enables us to reach
the condition that the field space with metric $\mathcal{G}_{ij}$ is
conformally flat if $\hat{\mathcal{C}}_{ijkl}=0$.

First, we compute the determinant of $\hat{\mathcal{G}}_{ij}$,
$\hat{\mathcal{G}}$, and the inverse metric $\hat{\mathcal{G}}^{ij}$,

$\displaystyle\hat{\mathcal{G}}=1+\frac{3}{f}\sum_{i=1}^{N}f_{i}^{2},\;\hat{\mathcal{G}}^{ij}=\delta_{ij}-\frac{3}{f\hat{\mathcal{G}}}f_{i}f_{j},$
(7)

which are surprisingly simple. The calculation details can be found in the
Appendix. Note that the indices of inverse metric are in the subscript, which
allows us to do tensor analysis just as the usual matrix manipulation. We can
easily check that
$\hat{\mathcal{G}}^{ij}\hat{\mathcal{G}}_{jk}=\delta_{ik}\equiv\delta^{i}_{k}$.

After tedious calculations, we obtain the following geometric quantities,

$\displaystyle\hat{\mathit{\Gamma}}_{jk}^{i}=$
$\displaystyle\frac{1}{2}\mathcal{G}^{il}\left(\partial_{j}\mathcal{G}_{kl}+\partial_{k}\mathcal{G}_{jl}-\partial_{l}\mathcal{G}_{jk}\right)=\frac{3}{f\hat{\mathcal{G}}}f_{i}\left(f_{jk}-\frac{1}{2f}f_{j}f_{k}\right),$
(8) $\displaystyle\hat{\mathcal{R}}_{ijkl}=$
$\displaystyle\frac{3}{f\hat{\mathcal{G}}}\left[\left(f_{ik}f_{jl}-f_{il}f_{jk}\right)+\frac{1}{2f}\left(f_{i}f_{l}f_{jk}+f_{j}f_{k}f_{il}-f_{i}f_{k}f_{jl}-f_{j}f_{l}f_{ik}\right)\right],$
(9) $\displaystyle=$
$\displaystyle\frac{6}{f\hat{\mathcal{G}}}\left(f_{i[k}f_{l]j}+\frac{1}{f}f_{[i}f_{j][k}f_{l]}\right),$
$\displaystyle\hat{\mathcal{R}}_{ij}=$
$\displaystyle\frac{3}{f\hat{\mathcal{G}}}\left[\left(f_{ij}f_{kk}-f_{ik}f_{jk}\right)+\frac{1}{2f}\left(f_{i}f_{k}f_{jk}+f_{j}f_{k}f_{ik}-f_{i}f_{j}f_{kk}-f_{ij}f_{k}^{2}\right)\right.$
$\displaystyle\left.+\frac{3}{f\hat{\mathcal{G}}}f_{k}f_{l}\left(f_{ik}f_{jl}-f_{ij}f_{lk}\right)-\frac{3}{2f^{2}\hat{\mathcal{G}}}f_{k}f_{l}\left(f_{j}f_{l}f_{ik}+f_{i}f_{k}f_{lj}-f_{i}f_{j}f_{lk}-f_{k}f_{l}f_{ij}\right)\right],$
(10) $\displaystyle\hat{\mathcal{R}}=$
$\displaystyle\frac{3}{f^{2}\hat{\mathcal{G}}^{2}}\left[f\hat{\mathcal{G}}\left(f_{ii}f_{jj}-f_{ij}^{2}\right)+\left(f_{i}f_{j}f_{ij}-f_{k}^{2}f_{jj}\right)+6\left(f_{ij}f_{jk}-f_{ik}f_{jj}\right)f_{i}f_{k}\right],$
(11)

where $f_{k}^{2}=\sum_{k}f_{k}f_{k},f_{ij}^{2}=\sum_{ij}f_{ij}f_{ij}$, and all
the repeated indices are summed. $\hat{\mathcal{C}}_{ijkl}$ can be obtained
straightforwardly with Eq. (5).

Now we are in a position to discuss the conditions for conformal flatness of
the field space. One solution with $\hat{\mathcal{R}}_{ijkl}=0$ was found in
the literature Kaiser:2010ps , with
$f(\phi_{1},\phi_{2})=\xi_{1}(\phi^{1})^{2}+\xi_{2}(\phi^{2})^{2}$ for $N=2$,
where $\xi_{i}$ are arbitrary constants. This can easily be checked by
calculating the $\hat{\mathcal{R}}_{1212}=0$, which is the only independent
component for $N=2$. One is tempted to extend the case to $N>2$ since
$\sigma$-model would fall in this category. Unfortunately, extension of such a
form for $N>2$ gives $\hat{\mathcal{R}}_{1212}\neq 0$ in general, except all
$\xi_{i}$ are equal to some particular value, as we shall show below.

Here we present new solutions with $\hat{\mathcal{R}}_{ijkl}=0$, or
$\hat{\mathcal{R}}_{ijkl}\neq 0$ but $\hat{\mathcal{C}}_{ijkl}=0$. Completely
solving $f$ from the nonlinear partial differential equation
$\hat{\mathcal{R}}_{ijkl}=0$ is notoriously difficult and unpractical.
Besides, there is no unique solution for such nonlinear equations. We have
known a similar case from solving the Einstein’s field equation, which is also
nonlinear and has multiple solutions. In this paper, we shall present several
solutions based on the symmetric properties of $\hat{\mathcal{R}}_{ijkl}$. We
enumerate several cases below.

  1. 1.

$\hat{\mathit{\Gamma}}_{jk}^{i}=0$ and $\hat{\mathcal{R}}_{ijkl}=0$.
This is the simplest case and can be easily verified from the definition of
$\hat{\mathcal{R}}_{ijkl}$. We may further divide this category into two
cases, after observing the feature in Eq. (8).

    1. (a)

$f_{i}=0.$
This is the trivial solution with $f=\mathrm{constant}$, in which case the
scalar fields are minimally coupled with gravity.

    2. (b)

$f_{jk}=\dfrac{1}{2f}f_{j}f_{k}.$
We can solve the equation by taking further derivative, using the above
relation recursively and getting an additional condition $f_{ijk}=0$, which
indicates $f$ is a quadratic function of $\phi_{i}$,

$f(\phi^{i})=a\left(c+b_{i}\phi^{i}\right)^{2},$ (12)

where $a,b_{i}$ and $c$ are arbitrary constants.

  2. 2.

$\hat{\mathit{\Gamma}}_{jk}^{i}\neq 0$ but $\hat{\mathcal{R}}_{ijkl}=0$.
Inspired by the second case above, we notice that taking a form
$f_{ij}=\dfrac{p}{f}f_{i}f_{j}$ would give vanishing Riemann tensor. Solving
the differential equation gives the solutions,

$f(\phi^{i})=a\left(c+b_{i}\phi^{i}\right)^{\frac{1}{1-p}},\;f_{i}=\frac{ab_{i}}{1-p}\left(c+b_{j}\phi^{j}\right)^{\frac{p}{1-p}},\;\textrm{
for }p\neq 1,$ (13)

and

$f(\phi^{i})=a\exp\left(b_{i}\phi^{i}\right),\;f_{i}=ab_{i}\exp\left(b_{j}\phi^{j}\right),\;\textrm{
for }p=1.$ (14)

The solution for $p=0$ is included above, in which case $f$ is a linear
function of $\phi^{i}$. The metrics in this category solve Einstein-like
equations in vacuum at any dimension $N$,

$\hat{\mathcal{G}}_{ij}=\begin{cases}\delta_{ij}+\frac{3ab_{i}b_{j}}{(1-p)^{2}}\left(c+b_{k}\phi^{k}\right)^{\frac{2p-1}{1-p}},\;&p\neq
1,\\\
\delta_{ij}+3ab_{i}b_{j}\exp\left(b_{k}\phi^{k}\right),\;&p=1.\end{cases}$
(15)

  3. 3.

$\hat{\mathcal{R}}_{ijkl}\neq 0$ but $\hat{\mathcal{C}}_{ijkl}=0$.
Even if the Riemann tensor does not vanish, but has the following structure,

$\hat{\mathcal{R}}_{ijkl}\propto\hat{\mathcal{G}}_{i[k}\hat{\mathcal{G}}_{l]j},$
(16)

we would obtain $\hat{\mathcal{C}}_{ijkl}=0$ as well. Contracting with
$\mathcal{G}^{ik}\mathcal{G}^{jl}$ gives the proportional factor
$\hat{\mathcal{R}}/[N(N-1)]$. Observing that

$\hat{\mathcal{G}}_{i[k}\hat{\mathcal{G}}_{l]j}=\delta_{i[k}\delta_{l]j}-\frac{6}{f}f_{[i}\delta_{j][k}f_{l]},$
(17)

and comparing with $\hat{\mathcal{R}}_{ijkl}$, we would have the following
relation,

$f_{ij}=-\frac{1}{6}\delta_{ij}.$ (18)

The general solution of the above equation would be

$f\left(\phi_{i}\right)=a+b_{i}\phi^{i}-\frac{1}{12}\delta_{ij}\phi^{i}\phi^{j},\;f_{i}=b_{i}-\frac{1}{6}\phi^{i}.$
(19)

We point out that by redefining $a,b_{i}$ and $c$, the forms of $f(\phi^{i})$
in all the above solutions do not change under the field shift,
$\phi^{i}\rightarrow\phi^{i}+d^{i}$, where $d^{i}$ are arbitrary constant. The
solutions are summarized in Tab. 1 where $f(\phi^{i})$ and its corresponding
$\hat{\mathcal{G}}_{ij},\hat{\mathit{\Gamma}}_{jk}^{i},\hat{\mathcal{R}}_{ijkl}$
and $\hat{\mathcal{C}}_{ijkl}$ are listed. These $f(\phi^{i})$ might be useful
for future model building due to their simple forms.

So far we have focused on the Riemannian metric. If one of the scalar fields
has opposite sign for the kinetic term, we would get the Lorentzian metric,
$\hat{\mathcal{G}}_{ij}=\eta_{ij}+3f_{i}f_{j}/f$, where
$\eta_{11}=-1,\eta_{ii}=1$ for $i\neq 1$, and $\eta_{ij}=0$ if $i\neq j$.
Similarly, we calculate

$\displaystyle\mathcal{G}$
$\displaystyle=\det\hat{\mathcal{G}}_{ij}=-\left(1+\frac{3}{f}\eta_{ij}f_{i}f_{j}\right)=-\left(1+\frac{3}{f}f_{i}f^{i}\right),$
(20) $\displaystyle\mathcal{G}^{ij}$
$\displaystyle=\eta_{ij}+\frac{3}{f\mathcal{G}}\left(-1\right)^{\delta_{1i}+\delta_{1j}}f_{i}f_{j}=\eta^{ij}+\frac{3}{f\mathcal{G}}f^{i}f^{j},$
(21) $\displaystyle\hat{\it\Gamma}_{jk}^{i}$
$\displaystyle=-\frac{1}{\mathcal{G}}\eta_{il}f_{l}\left(f_{jk}-\frac{1}{2f}f_{j}f_{k}\right)=-\frac{1}{\mathcal{G}}f^{i}\left(f_{jk}-\frac{1}{2f}f_{j}f_{k}\right).$
(22)

Here the supscripts of $f$ denote the usual derivatives, upgraded by
$\eta^{ij}$. Note that the similar tensor structures of
$\hat{\mathcal{G}}_{ij}$ and $\hat{\it\Gamma}_{jk}^{i}$ to previous cases lead
to the same tensor structure of $\hat{\mathcal{R}}_{ijkl}$. Therefore, the
solutions to $\hat{\mathcal{R}}_{ijkl}=0$ barely change, except the
replacement $\delta_{ij}\rightarrow\eta_{ij}$.

$\;f\left(\phi^{i}\right)=\;$ | $\;\mathrm{constant}\;$ | $\;\begin{cases}a\left(c+b_{i}\phi^{i}\right)^{\frac{1}{1-p}},\;&p\neq 1\\\ a\exp\left(b_{i}\phi^{i}\right),\;&p=1\end{cases}\;$ | $a+b_{i}\phi^{i}-\dfrac{1}{12}\delta_{ij}\phi^{i}\phi^{j}$
---|---|---|---
$\hat{\mathcal{G}}_{ij}=\;$ | $\delta_{ij}$ | $\;\begin{cases}\delta_{ij}+\frac{3ab_{i}b_{j}}{(1-p)^{2}}\left(c+b_{k}\phi^{k}\right)^{\frac{2p-1}{1-p}},\;&p\neq 1\\\ \delta_{ij}+3ab_{i}b_{j}\exp\left(b_{k}\phi^{k}\right),\;&p=1\end{cases}\;$ | $\delta_{ij}+\frac{3(b_{i}-\frac{\phi^{i}}{6})(b_{j}-\frac{\phi^{j}}{6})}{a+b_{k}\phi^{k}-\frac{1}{12}\delta_{kl}\phi^{k}\phi^{l}}$
$\hat{\mathit{\Gamma}}_{jk}^{i}=\;$ | $0$ | $\propto\left(p-\frac{1}{2}\right)b_{i}b_{j}b_{k}$ | $\propto\left(-\dfrac{1}{6}f_{i}\delta_{jk}-\dfrac{1}{2f}f_{i}f_{j}f_{k}\right)$
$\hat{\mathcal{R}}_{ijkl}=\;$ | $0$ | $0$ | $\;\propto\left(\delta_{i[k}\delta_{l]j}-\dfrac{6}{f}f_{[i}\delta_{j][k}f_{l]}\right)\;$
$\hat{\mathcal{C}}_{ijkl}=\;$ | $0$ | $0$ | $0$
Table 1: The analytic solutions $f\left(\phi^{i}\right)$ in the first row that
give conformally flat metrics, with the corresponding
$\hat{\mathcal{G}}_{ij},\hat{\mathit{\Gamma}}_{jk}^{i},\hat{\mathcal{R}}_{ijkl}$
and $\hat{\mathcal{C}}_{ijkl}$. The metrics in the third column are solutions
to Einstein-like equation in vacuum of field space.

##  IV Local Scaling Symmetry

In previous sections we have presented several $f(\phi^{i})$ with conformally
flat field space by showing the associated $\mathcal{C}_{ijkl}=0$. However, we
have not demonstrated explicitly the new field coordinate
$\varphi^{I}(\phi^{i})$ of the field space whose metric
$\mathcal{G}_{IJ}(\varphi^{K})$ is proportional to $\delta_{IJ}$. In most
cases, such coordinate transformations $\phi^{i}\rightarrow\varphi^{I}$ have
no compact analytic solutions. Here, we have found an interesting case,

$f(\phi^{i})=a-\frac{1}{12}\delta_{ij}\phi^{i}\phi^{j},$ (23)

where the system has a global SO$(N)$ symmetry for $\phi^{i}$ and is also
called as $\sigma$-model in field theory. We can introduce an auxiliary field
$\chi$ and rewrite the relevant Lagrangian as

$\frac{\mathcal{L}}{\sqrt{-g}}\supseteq
a\left[g^{\mu\nu}\nabla_{\mu}\chi\nabla_{\nu}\chi+\frac{1}{6}R\chi^{2}\right]-\frac{1}{2}\left[g^{\mu\nu}\delta_{ij}\nabla_{\mu}\phi^{i}\nabla_{\nu}\phi^{j}+\frac{1}{6}R\delta_{ij}\phi^{i}\phi^{j}\right].$
(24)

Because there is a local scaling symmetry in the above Lagrangian,

$\displaystyle g_{\mu\nu}\left(x\right)$
$\displaystyle\rightarrow\bar{g}_{\mu\nu}\left(x\right)=\lambda^{2}\left(x\right)g_{\mu\nu}\left(x\right),$
$\displaystyle\phi^{i}\left(x\right)$
$\displaystyle\rightarrow\bar{\phi}^{i}\left(x\right)=\lambda^{-1}\left(x\right)\phi\left(x\right),$
(25) $\displaystyle\chi\left(x\right)$
$\displaystyle\rightarrow\overline{\chi}\left(x\right)=\lambda^{-1}\left(x\right)\chi\left(x\right),$

where $\lambda(x)$ is an arbitrary positive function, the original theory can
be recovered by setting $\lambda(x)=\chi(x)/\sqrt{6}$. This symmetry is also
called Weyl/conformal symmetry in the literature and has wide applications in
model building, see Refs. Wu:2021ign ; Wu:2021ucc  for recent example.

When $a=1/2$, this theory has an extended global symmetry SO$(1,N)$,
$\chi^{2}-\delta_{ij}\phi^{i}\phi^{j}=6.$ The geometry of field space or
kinetic term is related with the distance element in the field space,
$ds^{2}=d\chi^{2}-\delta_{ij}d\phi^{i}d\phi^{j}.$ Introducing field
coordinates $(T,\varphi^{i})$, we can parameterize

$\displaystyle\chi$
$\displaystyle=\sqrt{6}\left(\frac{1}{2}\delta_{ij}\phi^{i}\phi^{j}\mathrm{e}^{T}+\cosh
T\right),$ $\displaystyle\varphi^{N}$
$\displaystyle=\sqrt{6}\left(\frac{1}{2}\delta_{ij}\phi^{i}\phi^{j}\mathrm{e}^{T}-\sinh
T\right),$ $\displaystyle\varphi^{i}$
$\displaystyle=\sqrt{6}\phi^{i}\mathrm{e}^{T},\;i,j=1,2,\cdots,N-1.$ (26)

We can easily show

$\displaystyle d\chi$
$\displaystyle=\sqrt{6}\left[\left(\frac{1}{2}\delta_{ij}\phi^{i}\phi^{j}\mathrm{e}^{T}+\sinh
T\right)dT+\delta_{ij}\phi^{i}\mathrm{e}^{T}d\phi^{j}\right],$ $\displaystyle
d\varphi^{N}$
$\displaystyle=\sqrt{6}\left[\left(\frac{1}{2}\delta_{ij}\phi^{i}\phi^{j}\mathrm{e}^{T}-\cosh
T\right)dT+\delta_{ij}\phi^{i}\mathrm{e}^{T}d\phi^{j}\right],$ $\displaystyle
d\varphi^{i}$
$\displaystyle=\sqrt{6}\left(\mathrm{e}^{T}d\phi^{i}+\phi^{i}\mathrm{e}^{T}dT\right).$
(27)

In the new field coordinates, $(T,\varphi^{i})$, the element is given by
$ds^{2}=-dT^{2}-\mathrm{e}^{2T}\delta_{ij}d\varphi^{i}d\varphi^{j}.$ Refine
$dT=-e^{T}d\tau$ or $\tau=e^{-T}$, we obtain
$ds^{2}=-\frac{1}{\tau^{2}}\left(d\tau^{2}+\delta_{ij}d\varphi^{i}d\varphi^{j}\right)$.
Therefore, the geometry of field space is shown explicitly to be conformally
flat in the coordinate of $(\tau,\varphi^{i})$.

Note that although the geometric structure is determined by the hypersurface
in field space, $\chi^{2}-\delta_{ij}\phi^{i}\phi^{j}=6$, in phenomenological
studies one usually adopts a particular parametrization in which only one
field variable is responsible for the physical effects of interests. For
example, if we are only interested in the radial part as the inflaton, we can
use the following parametrization of the field coordinates
$(\varphi,\theta^{i})$,

$\displaystyle\chi$ $\displaystyle=\sqrt{6}\cosh\varphi,$
$\displaystyle\phi^{1}$ $\displaystyle=\sqrt{6}\sinh\varphi\cos\theta^{1},$
$\displaystyle\phi^{2}$
$\displaystyle=\sqrt{6}\sinh\varphi\sin\theta^{1}\cos\theta^{2},$
$\displaystyle\vdots$ $\displaystyle\phi^{N-1}$
$\displaystyle=\sqrt{6}\sinh\varphi\sin\theta^{1}\cdots\sin\theta^{N-2}\cos\theta^{N-1},$
$\displaystyle\phi^{N}$
$\displaystyle=\sqrt{6}\sinh\varphi\sin\theta^{1}\cdots\sin\theta^{N-2}\sin\theta^{N-1}.$
(28)

In this case, radial field $\varphi$ would have canonical kinetic term from
the very beginning while the angular field variables $\theta^{i}$ would not.
Analysis of the dynamics of $\theta^{i}$ might involve approximations such as
expanding by the $\theta^{i}/M_{p}$ as small parameters, etc. We have actually
encountered the special case for $N=1$ in $\alpha$-attractor inflation model
Kallosh:2013hoa ; Kallosh:2013yoa  and its extensions with Weyl gauge field
Tang:2019olx ; Tang:2019uex ; Tang:2020ovf . The confirmation from $N=1$ case
also partially suggests the correctness of our calculation for general $N$.

##  V $F(R)$ Gravity

In this section, we extend our discussions by considering the following
Lagrangian of modified gravity in 4-dimensional space-time for $N$ non-
minimally coupled scalar fields, $\phi^{i},i=1,2,...,N$,

$\frac{\mathcal{L}}{\sqrt{-g}}\supseteq
F(R,\phi^{i})-\frac{1}{2}g^{\mu\nu}\delta_{ij}\nabla_{\mu}\phi^{i}\nabla_{\nu}\phi^{j},$
(29)

where $F(R,\phi^{i})$ is a function of $R$ and $\phi^{i}$ without derivatives.
Such terms are motivated from quantum corrections and cosmological models. For
example, in Starobinsky-like model, we have
$F(R,\phi^{i})=f\left(\phi^{i}\right)R+{\alpha}R^{2}/2$, which can give viable
inflation scenarios.

We can similarly use the auxiliary field $\chi$ and rewrite the Lagrangian

$\frac{\mathcal{L}}{\sqrt{-g}}=F(\chi^{2},\phi^{i})+F_{R}\left(\chi^{2},\phi^{i}\right)\left(R-\chi^{2}\right)-\frac{1}{2}g^{\mu\nu}\delta_{ij}\nabla_{\mu}\phi^{i}\nabla_{\nu}\phi^{j},$
(30)

where $F_{R}$ denotes the derivative of $F(R,\phi)$ over $R$ and
$F(\chi^{2},\phi)\equiv F(R\rightarrow\chi^{2},\phi)$. One may check that the
equation of motion for $\chi$ still gives $\chi^{2}=R$. Then denoting the
$0$-component $\phi^{0}\equiv\chi$ and new $f$ function,

$f(\phi^{I})=F_{R}\left(\chi^{2},\phi^{i}\right),I=0,1,2,\cdots,N,$ (31)

we have reduced the system into the case we discussed in previous sections.
However, there is a crucial difference that $\phi^{0}$ has no kinetic term. It
turns out that this difference leads to significantly different results.

Following the similar procedures and omitting the potential term, we can
obtain

$\displaystyle\frac{\mathcal{L}}{\sqrt{-\tilde{g}}}$
$\displaystyle\supseteq\frac{1}{2}\tilde{R}-3\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\omega\tilde{\nabla}_{\nu}\omega-\frac{1}{2\Omega^{2}}\tilde{g}^{\mu\nu}\delta_{ij}\tilde{\nabla}_{\mu}\phi^{i}\tilde{\nabla}_{\nu}\phi^{j}$
$\displaystyle=\frac{1}{2}\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\mathcal{G}_{IJ}\tilde{\nabla}_{\mu}\phi^{I}\tilde{\nabla}_{\nu}\phi^{J},$
(32)

where $\omega=\ln\Omega,\Omega^{2}=2f(\phi^{I})$ and
$\mathcal{G}_{IJ}=\dfrac{1}{2f}\overline{\mathcal{G}}_{IJ}$,

$\overline{\mathcal{G}}_{IJ}=\delta_{IJ}+\frac{3}{f}f_{I}f_{J}-\delta_{0I}\delta_{0J}.$
(33)

Here we have used $\overline{\mathcal{G}}_{IJ}$ to distinguish from
$\hat{\mathcal{G}}_{ij}$ in the $R$-gravity case. We calculate

$\displaystyle\overline{\mathcal{G}}$
$\displaystyle=\frac{3}{f}f^{2}_{0},\;\overline{\mathcal{G}}^{IJ}=\delta_{IJ}-\frac{3}{f\overline{\mathcal{G}}}f_{0}\left(f_{I}\delta_{0J}+f_{J}\delta_{0I}\right)+\frac{\hat{\mathcal{G}}}{\overline{\mathcal{G}}}\delta_{0I}\delta_{0J},$
(34) $\displaystyle\overline{\mathit{\Gamma}}^{I}_{JK}$
$\displaystyle=\frac{1}{f_{0}}\left(f_{JK}-\frac{1}{2f}f_{J}f_{K}\right)\delta_{0I},$
(35) $\displaystyle\overline{\mathcal{R}}_{IJKL}$
$\displaystyle=\overline{\mathcal{R}}_{IJ}=\overline{\mathcal{R}}=0,\;\overline{\mathcal{C}}_{IJKL}=0.$
(36)

All the curvature components vanish identically, which means the
$\overline{\mathcal{G}}_{IJ}$ in Eq. (33) with any nonzero $f(\phi^{I})$
solves the Einstein-like equations in vacuum. This conclusion does not depend
on the form of $F(R,\phi^{i})$ as long as $F(R,\phi^{i})$ depends on $R$
nonlinearly, a surprising result at first glance. Actually, there is a
transparent way to understand this result by observing the first line in Eq.
(V). Because $F(R,\phi^{i})$ gravity is associated with a additional scalar
degree of freedom, it is justified to introduce a new field variable $\varphi$
with ${d\varphi}/{d\omega}=\sqrt{6}\Omega\left(\phi^{I}\right).$ The kinetic
terms can be organized as

$\frac{1}{2\Omega^{2}}\tilde{g}^{\mu\nu}\delta_{IJ}\tilde{\nabla}_{\mu}\phi^{I}\tilde{\nabla}_{\nu}\phi^{J},\;I,J=1,2,\cdots
N+1,$ (37)

where $\phi^{N+1}=\varphi$. It is obvious that metric tensor in field space
$\delta_{IJ}/\Omega^{2}$ is conformally flat.

##  VI Conclusion

We have investigated the conformal transformation with multiple scalar fields
that non-minimally couple with gravity. These theories are ubiquitous in
modern particle physics and cosmological models. Conformal transformation is
employed to transform the Lagrangian from Jordan frame to Einstein frame,
which also makes the kinetic terms of scalar fields non-canonical. We have
found that if the number of scalar fields is larger than one, in general it is
not possible to redefine the field variables to make all the kinetic terms
canonical.

We have discussed under what conditions the kinetic terms could be brought
into quasi-canonical, namely different from canonical by a common factor. It
is equivalent to the problem of finding conformally flat metric tensor in the
field space of scalars. We have solved the nonlinear partial differential
equations in arbitrary dimensions and presented several solutions in Table. 1
that give conformally flat metric tensors. $\sigma$-model with a particular
non-minimal coupling is one of the solutions. These solutions may be useful
for future phenomenological model building for inflation and dark energy. We
have also shown that in some modified gravity theories, including Starobinsky
model, the metric tensor in field space is always conformally flat.

Acknowledgments YT is supported by National Natural Science Foundation of
China (NSFC) under Grants No. 11851302 and by the Fundamental Research Funds
for the Central Universities. YLW is supported in part by the National Key
Research and Development Program of China under Grant No.2020YFC2201501, and
NSFC under Grants No. 11851302, No. 11851303, No. 11690022, No. 11747601, and
the Strategic Priority Research Program of the Chinese Academy of Sciences
under Grant No. XDB23030100.

##  VII Appendix

###  VII.1 Conformal Transformation

The conformal transformation $\tilde{g}_{\mu\nu}=\Omega^{2}(x)g_{\mu\nu}$
leads to various relations between geometric quantities,

$\displaystyle\tilde{\Gamma}_{\mu\nu}^{\rho}$
$\displaystyle=\Gamma_{\mu\nu}^{\rho}+\left(\delta_{\mu}^{\rho}\nabla_{\nu}\omega+\delta_{\nu}^{\rho}\nabla_{\mu}\omega-
g_{\mu\nu}\nabla^{\rho}\omega\right),$ (38)
$\displaystyle\tilde{R}_{\;\;\sigma\mu\nu}^{\rho}$
$\displaystyle=R_{\;\;\sigma\mu\nu}^{\rho}+2\delta_{[\nu}^{\rho}\nabla_{\mu]}\nabla_{\sigma}\omega-2g^{\rho\alpha}g_{\sigma[\nu}\nabla_{\mu]}\nabla_{\alpha}\omega+2\nabla_{[\nu}\omega\delta_{\mu]}^{\rho}\nabla_{\sigma}\omega$
$\displaystyle\quad-2\nabla_{[\nu}\omega
g_{\mu]\sigma}g^{\rho\beta}\nabla_{\beta}\omega-2g_{\sigma[\nu}\delta_{\mu]}^{\rho}g^{\alpha\beta}\nabla_{\alpha}\omega\nabla_{\beta}\omega,$
(39) $\displaystyle\tilde{R}_{\mu\nu}$
$\displaystyle=R_{\mu\nu}-2\nabla_{\mu}\nabla_{\nu}\omega-
g_{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\omega+2\nabla_{\mu}\omega\nabla_{\nu}\omega-2g_{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}\omega\nabla_{\beta}\omega,$
(40) $\displaystyle\tilde{R}$
$\displaystyle=\tilde{g}^{\mu\nu}\tilde{R}_{\mu\nu}=\Omega^{-2}\left[R-6\square\omega-6g^{\mu\nu}\nabla_{\mu}\omega\nabla_{\nu}\omega\right],$
(41)

where $\omega\equiv\ln\Omega$, $\square\equiv
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ and [⋯] in the subscripts indicate anti-
symmetrization of the included indices. Note that conformal transformation
does not have effects on the coordinates, $x^{\mu}$, and the usual partial
derivative $\partial_{\mu}\equiv\frac{\partial}{\partial x^{\mu}}$. Then using
the relation between $\tilde{R}$ and $R$, we can rewrite the Lagrangian
$\mathcal{L}$ as

$\frac{\mathcal{L}}{\sqrt{-\tilde{g}}}=\frac{1}{2}\tilde{R}+\frac{3}{\Omega^{2}}\square\omega+3\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\omega\tilde{\nabla}_{\nu}\omega-\frac{1}{2\Omega^{2}}\tilde{g}^{\mu\nu}\delta_{ij}\tilde{\nabla}_{\mu}\phi^{i}\tilde{\nabla}_{\nu}\phi^{j}-\frac{V\left(\phi^{i}\right)}{\Omega^{4}}.$
(42)

The second term in the right-handed side of the equation can be written as

$\Omega^{-2}\square\omega=\tilde{\square}\omega-2\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\omega\tilde{\nabla}_{\nu}\omega.$
(43)

In the action which is the space-time integral of $\mathcal{L}$,
$\tilde{\square}\omega$’s contribution is a surface term due to the following
identity,

$\tilde{\square}\omega=\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\omega\right),$
(44)

therefore can be neglected in the cases we are considering in this paper.

###  VII.2 Determinant and Inverse Metric

Here we present the details of calculating the determinant $\mathcal{G}$ and
the inverse metric tensor $\hat{\mathcal{G}}^{ij}$. The computation of
$\mathcal{G}$ is done as follows. Writing
$\hat{\mathcal{G}}_{ij}=\delta_{ij}+Af_{i}f_{j},\;A\equiv{3}/{f}$, we have

$\displaystyle\hat{\mathcal{G}}\equiv\det\hat{\mathcal{G}}_{ij}$
$\displaystyle=\frac{1}{\Pi_{i}f_{i}}\det\left(\begin{array}[]{cccc}\frac{1}{f_{1}}+Af_{1}&Af_{2}&\cdots\\\
Af_{1}&\frac{1}{f_{2}}+Af_{2}&\cdots\\\
\vdots&\vdots&\ddots\end{array}\right)=\frac{1}{\Pi_{i}f_{i}}\det\left(\begin{array}[]{cccc}\frac{1}{f_{1}}+Af_{1}&Af_{2}&\cdots\\\
-1/f_{1}&\frac{1}{f_{2}}&\cdots\\\ \vdots&\vdots&\ddots\end{array}\right)$
(51)
$\displaystyle=\frac{1}{\Pi_{i}f_{i}}\det\left(\begin{array}[]{cccc}\frac{1}{f_{1}}+Af_{1}+\frac{A}{f_{1}}\left(f_{2}^{2}+\cdots+f_{N}^{2}\right)&0&\cdots\\\
-1/f_{1}&\frac{1}{f_{2}}&\cdots\\\
\vdots&\vdots&\ddots\end{array}\right)=1+A\sum_{i=1}^{N}f_{i}^{2}.$ (55)

For the inverse metric, we first compute the diagonal elements
$\hat{\mathcal{G}}^{ii}$. It is straightforward to obtain by using the
analogue to the determinant of $\hat{\mathcal{G}}_{ij}$,

$\hat{\mathcal{G}}^{ii}=\frac{1}{\hat{\mathcal{G}}}\left(1+A\sum_{j\neq
i}f_{j}^{2}\right).$ (56)

For the off-diagonal elements $\hat{\mathcal{G}}^{ij}\left(i\neq j\right)$,
without showing all the elements in the matrix, we have

$\hat{\mathcal{G}}^{ji}=\frac{\left(-1\right)^{i+j}}{\hat{\mathcal{G}}}\det\left(\begin{array}[]{cccccccc}\cdots&1+Af_{i-1}^{2}&Af_{i-1}f_{i}&Af_{i-1}f_{i+1}&\cdots\\\
\cdots&Af_{i+1}f_{i-1}&Af_{i+1}f_{i}&1+Af_{i+1}^{2}&\cdots&\iddots\\\
&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
&\vdots&\vdots&\vdots&\cdots&1+Af_{j-1}^{2}&Af_{j-1}f_{j+1}&\cdots\\\
\cdots&Af_{j}f_{i-1}&Af_{j}f_{i}&Af_{j}f_{i+1}&\cdots&Af_{j}f_{j-1}&Af_{j}f_{j+1}&\cdots\\\
&\vdots&\vdots&\vdots&\cdots&Af_{j+1}f_{j-1}&1+Af_{j+1}^{2}&\cdots\end{array}\right)$
$=\frac{\prod_{k\neq
i}f_{k}}{\left(-1\right)^{i+j}\hat{\mathcal{G}}}\det\left(\begin{array}[]{cccccccc}\cdots&\frac{1}{f_{i-1}}+Af_{i-1}&Af_{i}&Af_{i+1}&\cdots\\\
\cdots&Af_{i-1}&Af_{i}&\frac{1}{f_{i+1}}+Af_{i+1}&\cdots&\iddots\\\
&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
&\vdots&\vdots&\vdots&\cdots&\frac{1}{f_{j-1}}+Af_{j-1}&Af_{j+1}&\cdots\\\
\cdots&Af_{i-1}&Af_{i}&Af_{i+1}&\cdots&Af_{j-1}&Af_{j+1}&\cdots\\\
&\vdots&\vdots&\vdots&\cdots&Af_{j-1}&\frac{1}{f_{j+1}}+Af_{j+1}&\cdots\end{array}\right)$
$=\frac{\prod_{k\neq
i}f_{k}}{\left(-1\right)^{i+j}\hat{\mathcal{G}}}\det\left(\begin{array}[]{cccccccc}0&\frac{1}{f_{i-1}}&0&0&\cdots\\\
0&0&0&\frac{1}{f_{i+1}}&\cdots&\iddots\\\
&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
&\vdots&\vdots&\vdots&\cdots&\frac{1}{f_{j-1}}&0&\cdots\\\
\cdots&Af_{i-1}&Af_{i}&Af_{i+1}&\cdots&Af_{j-1}&Af_{j+1}&\cdots\\\
&\vdots&\vdots&\vdots&\cdots&0&\frac{1}{f_{j+1}}&\cdots\\\
\end{array}\right)=-\frac{Af_{i}f_{j}}{\hat{\mathcal{G}}}.$ (57)

The above two results can be written in a unified form,

$\hat{\mathcal{G}}^{ij}=\delta_{ij}-\frac{A}{\hat{\mathcal{G}}}f_{i}f_{j}.$
(58)

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