# Heintze-Kobayashi-Wolf theory for negatively curved homogeneous Finsler
manifolds

Ming Xu  School of Mathematical Sciences, Capital Normal University, Beijing
100048, P.R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we generalize the Heintze-Kobayashi-Wolf theory to homogeneous
Finsler geometry, by proving two main theorems. First, any connected
negatively curved homogeneous Finsler manifold is isometric to a Lie group
endowed with a left invariant metric, and that Lie group must be simply
connected and solvable. Second, the requirement in Heintze’s criterion is
necessary and sufficient for a real solvable Lie algebra to generate a Lie
group which admits negatively curved left invariant Finsler metrics.

Mathematics Subject Classification(2010): 53B40, 53C30, 53C60
Keywords: homogeneous flag curvature formula, homogeneous Finsler manifold,
Heintze-Kobayashi-Wolf theory, linear submersion, negative curvature, solvable
Lie algebra

##  1\. Introduction

Complete Riemannian manifold with strictly negative section curvature (we will
call it negatively curved for simplicity) is a hot topic, which has been
extensively studied [6, 13]. In homogeneous geometry, using Lie algebraic data
to classify negatively curved homogeneous manifolds is a natural thought.
However, unlike dealing with positively curved ones [7], there are too many
smooth coset spaces admitting negative curvature, so that explicitly
classifying them is impossible.

Fortunately, we have the following Heintze-Kobayashi-Wolf theory (HKW theory
in short) as a remedial measure:

  1. (1)

By a result of J.A. Wolf [22] and its refinement by E. Heintze [14], any
connected negatively curved homogeneous manifold is isometric to a Lie group,
which is endowed with a left invariant metric. So we only need to discuss
those negatively curved solvmanifolds.

  2. (2)

By a theorem of S. Kobayashi [19], any connected negatively curved homogeneous
Riemannian manifold must be simply connected. So the classification for
negatively curved solvmanifolds is a Lie algebraic problem.

  3. (3)

E. Heintze proved that a real solvable Lie algebra $\mathfrak{g}$ generates a
Lie group $G$ which admits negatively curved left invariant Riemannian metrics
if and only if
$\dim_{\mathbb{R}}\mathfrak{g}=\dim_{\mathbb{R}}[\mathfrak{g},\mathfrak{g}]+1$
and there exists $y_{0}\in\mathfrak{g}$ such that
$\mathrm{ad}(y_{0}):[\mathfrak{g},\mathfrak{g}]\rightarrow[\mathfrak{g},\mathfrak{g}]$
only has eigenvalues with positive real parts (see [15] or Theorem 2.7 below).

To summarize, for each negatively curved homogeneous Riemannian manifold, the
HKW theory provides a relatively simple representative for it. Later, R.
Azencott and E.N. Wilson proved similar results for homogeneous non-positive
curvature [3, 4].

In recent years, researchers studied negative curvature in Finsler geometry,
where the negatively curved property requires the flag curvature to be
strictly negative everywhere. For example, Akbar-Zadeh’s theorem tells us that
any compact or homogeneous Finsler manifold with negative constant curvature
must be Riemannian [1, 8]. Z. Shen proved that a compact Finsler manifold with
negative flag curvature and constant S-curvature must be Riemannian [21]. Deng
and his coworkers proved some rigidity results in the homogeneous context [11,
26].

It is natural to ask

######  Question 1.1.

Can the HKW theory be generalized to homogeneous Finsler geometry?

The progresses imply a positive answer to Question 1.1. For example, S. Deng
and Z. Hou proved that any connected negatively curved homogeneous Finsler
manifold is simply connected [10]. We proved that the criterion in [15] can be
generalized to some special Finsler solvmanifolds (see Theorem 1.3 in [26]).

In this paper, we completely answer Question 1.1 by two main theorems.

######  Theorem 1.2.

Any connected homogeneous Finsler manifold is isometric to a Lie group endowed
with a left invariant Finsler metric, and this Lie group must be simply
connected and solvable.

######  Theorem 1.3.

Let $G$ be a connected simply connected solvable Lie group with
$\dim_{\mathbb{R}}G\geq 2$, and we apply the notations
$\mathfrak{g}=\mathrm{Lie}(G)$, $\mathfrak{l}^{0}=[\mathfrak{g},\mathfrak{g}]$
and $\mathfrak{l}^{1}=[\mathfrak{l}^{0},\mathfrak{l}^{0}]$. Then the following
claims are equivalent:

  1. (1)

$G$ admits a negatively curved left invariant Finsler metric;

  2. (2)

$\dim_{\mathbb{R}}\mathfrak{g}=\dim_{\mathbb{R}}\mathfrak{l}^{0}+1$ and there
exists $y_{0}\in\mathfrak{g}$ such that the real linear endomorphism
$\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}({u}_{0})$ on
$\mathfrak{l}^{0}/\mathfrak{l}^{1}$ induced by
$\mathrm{ad}(y_{0})=[y_{0},\cdot]$ only has eigenvalues with positive real
parts;

  3. (3)

$\dim_{\mathbb{R}}\mathfrak{g}=\dim_{\mathbb{R}}\mathfrak{l}^{0}+1$ and there
exists $y_{0}\in\mathfrak{g}$ such that the real linear endomorphism
$\mathrm{ad}_{\mathfrak{l}^{0}}(y_{0})=[y_{0},\cdot]$ on $\mathfrak{l}^{0}$
only has eigenvalues with positive real parts.

To summarize, the HKW theory can still guide us study the negative curvature
problem in homogeneous Finsler geometry.

The proof of Theorem 1.2 is very similar to that for its analog in Riemannian
geometry, which is sketchy in the literatures. To make this paper more self
contained, we supply the details. The proof of Theorem 1.3 is very different
from that in [15], because most calculations there are no longer valid in the
Finsler context. Here we use a homogeneous flag curvature formula (see Theorem
2.3 or Theorem 4.1 in [27]), which qualitatively indicates where to find a
non-negative flag curvature, and we refine the argument which proves Theorem
1.3 in [26] with a linear submersion and careful algebraic discussion.

This paper is scheduled as follows. In Section 2, we summarize some necessary
knowledge in general and homogeneous Finsler geometry. In Section 3, we prove
Theorem 1.2. In Section 4, we prove Theorem 1.3.

##  2\. Preliminaries in general and homogeneous Finsler geometries

###  2.1. Minkowski norm and linear submersion

A Minkowski norm on a finite dimensional real vector space $\mathbf{V}$ is a
continuous function $F:\mathbf{V}\rightarrow[0,+\infty)$ satisfying [5]:

  1. (1)

Regularity: $F|_{\mathbf{V}\backslash\\{0\\}}$ is a positive smooth function;

  2. (2)

Positive 1-homogeneity: $F(\lambda y)=\lambda F(y)$, $\forall\lambda\geq
0,y\in\mathbf{V}$;

  3. (3)

Strong convexity: for each $y\in\mathbf{V}\backslash\\{0\\}$, the fundamental
tensor

$g_{y}(u,v)=\frac{1}{2}\frac{\partial^{2}}{\partial s\partial
t}|_{s=t=0}F^{2}(y+su+tv),\quad\forall u,v\in\mathfrak{m},$

is an inner product on $\mathbf{V}$.

Let $F$ and $\overline{F}$ be the Minkowski norms on $\mathbf{V}$ and
$\overline{\mathbf{V}}$ respectively. The surjective real linear map
$l:\mathbf{V}\rightarrow\overline{\mathbf{V}}$ is called a linear submersion
from $F$ to $\overline{F}$, if [2]

$\inf\\{F(v)|l(v)=\overline{v}\\}=\overline{F}(\overline{v}),\quad\forall\overline{v}\in\overline{\mathbf{V}}.$

For each $F$ and $l$ as mentioned above, there exists a unique $\overline{F}$
on $\mathbf{V}$ such that $l$ is a submersion between Minkowski norms. We call
this $\overline{F}$ the Minkowski norm induced by submersion from $F$ and $l$.
The following lemma is useful in later discussion.

Suppose $l:(\mathbf{V},F)\rightarrow(\overline{\mathbf{V}},\overline{F})$ is a
linear submersion between Minkowski norms. Then for each
$\overline{u}\in\overline{\mathbf{V}}\backslash\\{0\\}$, there exists a unique
$u\in l^{-1}(\overline{u})$ satisfying $F(u)=\overline{F}(\overline{u})$.
Denote by $g_{u}(\cdot,\cdot)$ and $\overline{g}_{\overline{u}}(\cdot,\cdot)$
the fundamental tensors for $F$ and $\overline{F}$ respectively, then this
$u\in l^{-1}(\overline{u})$, which is called the horizonal lifting of
$\overline{u}$, can be alternatively determined by $g_{u}(u,\ker l)=0$.
Further more, we have the following lemma, which is a reformulation of
Proposition 2.2 in [2].

######  Lemma 2.1.

The linear map $l$ induces a linear isometry from the inner product
$g_{u}(\cdot,\cdot)$ on the $g_{u}(\cdot,\cdot)$-orthogonal complement of
$\ker l$ to the inner product $\overline{g}_{\overline{u}}(\cdot,\cdot)$ on
$\overline{\mathbf{V}}$.

###  2.2. Finsler metric and flag curvature

A Finsler metric on a smooth manifold $M$ is a continuous function
$F:TM\rightarrow[0,+\infty)$, such that $F|_{TM\backslash 0}$ is a positive
smooth function and $F(x,\cdot)=F|_{T_{x}M}$ for each $x\in M$ is a Minkowski
norm on $T_{x}M$ [20].

At any point $x$ in a Finsler manifold $(M,F)$, the flag curvature for the
vector $y\in T_{x}M\backslash\\{0\\}$ (i.e., the flag pole) and the tangent
plane $\mathbf{P}=\mathrm{span}^{\mathbb{R}}\\{y,u\\}\subset T_{x}M$ (i.e.,
the flag) is defined as

$K(x,y,\mathbf{P})=\frac{g_{y}(R_{y}(u),u)}{g_{y}(y,y)g_{y}(u,u)-g_{y}(y,u)^{2}},$

in which $R_{y}:T_{x}M\rightarrow T_{x}M$ is the Riemann curvature operator.
When $F$ is Riemannian, the flag curvature coincides with the sectional
curvature, which is irrelevant to the choice of $y$. See [20] for the details.

###  2.3. Homogeneous Finsler manifold and a flag curvature formula

A Finsler manifold $(M,F)$ is called homogeneous if its isometry group
$I(M,F)$ acts transitively on $M$ [8]. Since $I(M,F)$ is a Lie transformation
group [9], we can present the homogeneous manifold $M$ as $M=G/H$. Here $G$ is
a Lie subgroup of $I(M,F)$ which acts transitively on $M$, and $H$ is the
isotropy subgroup at the origin $o=eH\in G/H=M$. When $M$ is connected, we may
require $G$ to be connected as well.

A decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ with
$\mathfrak{g}=\mathrm{Lie}(G)$ and $\mathfrak{h}=\mathrm{Lie}(H)$ is called a
reductive decomposition for $(G/H,F)$ if it is $\mathrm{Ad}(H)$-invariant (in
the Lie algebraic level, it implies
$[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$). The following lemma
provides a canonical reductive decomposition.

######  Lemma 2.2.

The orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ with
respect to the Killing form of $\mathfrak{g}$ is a reductive composition, such
that the maximal nilpotent ideal of $\mathfrak{g}$ is contained in
$\mathfrak{m}$.

The Riemannian analog of Lemma 2.2 can be found in [28]. Its proof can be
naturally transferred to the Finsler context.

The subspace $\mathfrak{m}$ in a reductive decomposition for $(G/H,F)$ can be
canonically identified with the tangent space $T_{o}(G/H)$, such that the
$\mathrm{Ad}(H)$-action on $\mathfrak{m}$ coincides with the isotropic
$H$-action on $T_{o}(G/H)$. Then the $G$-invariant Finsler metric $F$ on $G/H$
can be one-to-one corresponded to its restriction to $T_{o}(G/H)$, which is
any arbitrary $\mathrm{Ad}(H)$-invariant Minkowski norm on $\mathfrak{m}$. For
simplicity, we still use $F$ and $g_{y}(\cdot,\cdot)$ to denote this norm and
its fundamental tensor respectively. See [8] for more detailed discussion in
homogeneous Finsler geometry.

By homogeneity, we only need to discuss the curvatures of a homogeneous
Finsler manifold $(G/H,F)$ at the origin. The following homogeneous flag
curvature formula (see Theorem 4.1 in [27]) played an important role when we
classified positively curved homogeneous Finsler manifolds [12].

######  Theorem 2.3.

Let $(G/H,F)$ be a homogeneous Finsler manifold with the reductive
decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$. Suppose that $u$ and
$v$ are a commuting pair of linearly independent vectors in $\mathfrak{m}$ and
assume $g_{u}(u,[u,\mathfrak{m}]_{\mathfrak{m}})=0$. Then for
$\mathbf{P}=\mathrm{span}\\{u,v\\}$, we have

$K(o,u,\mathbf{P})=\frac{g_{y}(U(u,v),U(u,v))}{g_{y}(u,u)g_{y}(v,v)-g_{y}(u,v)^{2}},$

where $U(u,v)\in\mathfrak{m}$ is determined by

$2g_{u}(U(u,v),w)=g_{u}([u,w]_{\mathfrak{m}},v)+g_{u}(u,[v,w]_{\mathfrak{m}}),\quad\forall
w\in\mathfrak{m}.$

When a Lie group $G$ is viewed as the homogeneous manifold $G/H=G/\\{e\\}$,
the corresponding homogeneous Finsler metric is called left invariant. In this
situation, the reductive decomposition is unique, i.e.,
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}=0+\mathfrak{g}$, so we have the
following immediate corollary of Theorem 2.3.

######  Lemma 2.4.

Let $F$ be a left invariant Finsler metric on the Lie group $G$. Suppose that
there exist a commuting pair of linearly independent vectors $u$ and $v$ in
$\mathfrak{g}=\mathrm{Lie}(G)$, which satisfies $g_{u}(u,[\mathfrak{g},u])=0$,
then $(G,F)$ is not negatively curved.

See [16, 17, 23, 24, 25] for more homogeneous curvature formulae in
homogeneous Finsler geometry and homogeneous spray geometry.

###  2.4. Negatively curved homogeneous Finsler manifold

Let $(M,F)$ be a connected negatively curved homogeneous Finsler manifold.
Then it is complete. The main theorem in [10] tells us

######  Theorem 2.5.

Any connected negatively curved homogeneous Finsler manifold must be simply
connected.

By Cartan-Hadamard Theorem [5], $M$ is homeomorphic to an Euclidean space,
which implies

######  Lemma 2.6.

A connected negatively curved homogeneous Finsler manifold $(M,F)$ can be
presented as $M=G/H$, where $G$ is the connected isometry group $I_{0}(M,F)$,
and $H$ is a maximal compact subgroup of $G$.

The proof of Lemma 2.6 is contained in the argument proving Theorem 1.1 in
[26].

When $M$ is a connected solvable Lie group and $F$ is a left invariant
Riemannian metric, E. Heintze proved [15]

######  Theorem 2.7.

Let $G$ be a connected simply connected Lie group with a solvable Lie algebra
$\mathfrak{g}$, then it admits a negatively curved left invariant Riemannian
metric if and only if
$\dim_{\mathbb{R}}\mathfrak{g}=\dim_{\mathbb{R}}[\mathfrak{g},\mathfrak{g}]+1$
and there exists $y_{0}\in\mathfrak{g}$ such that
$\mathrm{ad}(y_{0})=[y_{0},\cdot]:[\mathfrak{g},\mathfrak{g}]\rightarrow[\mathfrak{g},\mathfrak{g}]$
only has eigenvalues with positive real parts.

##  3\. Proof of Theorem 1.2

Suppose that $(M,F)$ is a connected negatively curved homogeneous Finsler
manifold, then Lemma 2.6 provides $M=G/H$, where $H$ is a maximal compact
subgroup of $G=I_{0}(M,F)$.

######  Lemma 3.1.

$G$ has a trivial center.

Proof. Assume conversely that there exists $\rho\in C(G)$ which is not the
identity map. Then $\rho$ generates an infinite discrete subgroup
$\Gamma\subset C(G)$, which acts freely and isometrically on $(M,F)$. Indeed,
each $\rho^{i}$ is a Clifford-Wolf translation and each $\Gamma$-orbit is
contained in a geodesic on $(M,F)$. On the quotient manifold
$\overline{M}=M/\Gamma$, $F$ induces a metric $\overline{F}$, such that the
covering map $\pi:M\rightarrow\overline{M}$ is locally isometric everywhere.
So $(\overline{M},\overline{F})$ is also negatively curved. The $G$-action on
$\overline{M}$ is transitive and isometric, so $(\overline{M},\overline{F})$
is a homogeneous Finsler manifold. Because $\overline{M}$ is not simply
connected, we get a contradiction to Theorem 2.5.

Let $\mathfrak{g}=\mathfrak{s}+\mathfrak{r}$ be the Levi decomposition for
$\mathfrak{g}=\mathrm{Lie}(G)$, where $\mathfrak{r}$ is the maximal solvable
ideal and $\mathfrak{s}$ is a semi simple subalgebra of $\mathfrak{g}$
respectively. Let $\mathfrak{k}$ be a maximally compactly imbedded subalgebra
of $\mathfrak{s}$, i.e., $\mathfrak{k}$ generates a maximal compact connected
subgroup of $\mathrm{Int}(\mathfrak{s})$.

######  Lemma 3.2.

$\mathfrak{k}$ generates a compact connected subgroup $K$ in $G$.

Notice that a compactly imbedded subalgebra is compact, but generally
speaking, it may not generate a compact subgroup. Here the negative curvature
condition is crucial.

Proof. We have a Lie algebra direct sum decomposition
$\mathfrak{s}=\oplus_{i=1}^{m}\mathfrak{s}_{i}$, in which each
$\mathfrak{s}_{i}$ is a simple ideal. Correspondingly,
$\mathfrak{k}=\oplus_{i=1}^{m}\mathfrak{k}_{i}$, where each $\mathfrak{k}_{i}$
is a maximally compact subalgebra of $\mathfrak{s}_{i}$. Denote by $K_{i}$ the
connected Lie subgroup that $\mathfrak{k}_{i}$ generates. Then we have
$K=K_{1}\cdots K_{m}$. To prove Lemma 3.2, we only need to verify that each
$K_{i}$ is compact. There are two cases to consider.

Case 1: $\mathfrak{k}_{i}$ is semi simple. In this case, $\mathfrak{k}_{i}$ is
compact and semi simple, so $K_{i}$ is compact.

Case 2: $\mathfrak{k}_{i}$ is not semi-simple. In this case,
$\mathfrak{k}_{i}=\mathfrak{k}^{\prime}_{i}\oplus\mathfrak{c}(\mathfrak{k}_{i})$,
where $\mathfrak{k}^{\prime}_{i}=[\mathfrak{k}_{i},\mathfrak{k}_{i}]$ is semi
simple and $\dim_{\mathbb{R}}\mathfrak{c}(\mathfrak{k}_{i})=1$. Meanwhile,
$\mathfrak{s}_{i}$ is a simple ideal of non-compact type, its compact dual
$\mathfrak{s}^{\prime}_{i}\subset\mathfrak{g}\otimes\mathbb{C}$ is compact
simple and has the same rank as $\mathfrak{k}_{i}$. Indeed,
$(\mathfrak{s}^{\prime}_{i},\mathfrak{k}_{i})$ is an irreducible Hermitian
symmetric pair. Let $\mathfrak{t}_{i}$ be a Cartan subalgebra of
$\mathfrak{k}_{i}$, then it is also a Cartan subalgebra of
$\mathfrak{s}^{\prime}_{i}$, and it contains $\mathfrak{c}(\mathfrak{k}_{i})$.
The
$\mathrm{ad}_{\mathfrak{g}\otimes\mathbb{C}}(\mathfrak{s}^{\prime}_{i})$-action
on $\mathfrak{g}\otimes\mathbb{C}$ only has purely imaginary weights in
$\sqrt{-1}\mathfrak{t}_{i}^{*}$. In particular, for any
$u\in\mathfrak{c}(\mathfrak{k}_{i})$, the semi simple complex linear
endomorphism $\mathrm{ad}_{\mathfrak{g}\otimes\mathbb{C}}(u)$ on
$\mathfrak{g}\otimes\mathbb{C}$ only has purely imaginary eigenvalues. Since
the root system of $\mathfrak{k}_{i}$ is a subset in that of
$\mathfrak{s}^{\prime}_{i}$, and $\mathfrak{c}(\mathfrak{g})$ is the common
kernel for all roots of $\mathfrak{k}_{i}$, we can find a suitable
$u\in\mathfrak{c}(\mathfrak{g})\backslash\\{0\\}$, such that all eigenvalues
of $\mathrm{ad}(u)$ are contained in $2\mathbb{Z}\pi\sqrt{-1}$. Then
$\mathrm{Ad}(\exp u)=e^{\mathrm{ad}(u)}$ is the identity map on
$\mathfrak{g}$. Since $G$ is connected, $\exp u\in C(G)$, and it must be $e$
by Lemma 3.1.

To summarize, above argument indicates that $\mathfrak{c}(\mathfrak{k}_{i})$
generates a compact Lie subgroup $S^{1}$. Meanwhile, the compact semi simple
$\mathfrak{k}^{\prime}_{i}=[\mathfrak{k}_{i},\mathfrak{k}_{i}]$ generates a
compact connected $K^{\prime}_{i}$ in $G$. So $K_{i}=K^{\prime}_{i}S^{1}$ is
compact. The proof of Lemma 3.2 is finished.

######  Lemma 3.3.

There exists a connected solvable subgroup $G^{\prime}$ of $G$ acting
transitively on $M$.

Proof. By Lemma 2.6, $H$ is a maximal compact subgroup of $G$. Since any
compact subgroup of $G$ is contained by a maximal one, and all maximal compact
subgroups of $G$ are conjugate to each other (see Theorem 14.1.3 in [18]), we
may assume without loss of generality that $H$ contains the compact subgroup
$K$ in Lemma 3.2, i.e., in the Lie algebraic level,
$\mathfrak{k}\subset\mathfrak{h}$.

For each $\mathfrak{s}_{i}$, $1\leq i\leq m$, in the proof of Lemma 3.2, we
have an Iwasawa decomposition
$\mathfrak{s}_{i}=\mathfrak{k}_{i}+\mathfrak{n}_{i}+\mathfrak{a}_{i}$ (when
$\mathfrak{s}_{i}$ is compact, $\mathfrak{k}_{i}=\mathfrak{s}_{i}$ and
$\mathfrak{n}_{i}=\mathfrak{a}_{i}=0$). Then
$\mathfrak{g}^{\prime}=\oplus_{i=1}^{m}(\mathfrak{n}_{i}+\mathfrak{a}_{i})+\mathfrak{r}$
is a solvable subalgebra of $\mathfrak{g}$. Denote by $G^{\prime}$ the
solvable Lie subgroup $\mathfrak{g}^{\prime}$ generates in $G$. Since
$\mathfrak{k}\subset\mathfrak{h}$, i.e.,
$\mathfrak{g}^{\prime}+\mathfrak{h}=\mathfrak{g}$, so $G^{\prime}\cdot o$ is
an open submanifold in $G/H$. Since both $(G/H,F)$ and $(G^{\prime}\cdot
o,F|_{G^{\prime}\cdot o})$ are homogeneous, they are both complete. So we have
$G^{\prime}\cdot o=G/H=M$.

######  Lemma 3.4.

There exists a connected solvable subgroup $G^{\prime\prime}$ of $G$, which
acts freely and transitively on $M$.

Proof. Let $G^{\prime}$ be the connected solvable Lie subgroup of $G$ given by
Lemma 3.3. Then $M$ can be presented as $M=G^{\prime}/H^{\prime}$. Let
$\mathfrak{g}^{\prime}=\mathrm{Lie}(G^{\prime})$ and
$\mathfrak{h}^{\prime}=\mathrm{Lie}(H^{\prime})$. Then Lemma 2.2 tells that
$\mathfrak{h}^{\prime}\cap[\mathfrak{g}^{\prime},\mathfrak{g}^{\prime}]=0$.
This observation enables us to find a linear complement
$\mathfrak{g}^{\prime\prime}$ of $\mathfrak{h}^{\prime}$ in
$\mathfrak{g}^{\prime}$ which contains
$[\mathfrak{g}^{\prime},\mathfrak{g}^{\prime}]$. Indeed,
$\mathfrak{g}^{\prime\prime}$ is a Lie subalgebra of $\mathfrak{g}^{\prime}$,
and it generates a connected solvable Lie subgroup $G^{\prime\prime}$ in
$G^{\prime}$. By similar argument as in the proof of Lemma 3.3, we get
$G^{\prime\prime}\cdot o=M$. Because
$\dim_{\mathbb{R}}G^{\prime\prime}=\dim_{\mathbb{R}}M$, the map
$\pi:G^{\prime\prime}\rightarrow M$, $\pi(g)=g\cdot o$ is a smooth covering
map. Since $M$ is connected and simply connected, this covering map $\pi$ is a
diffeomorphism. So the $G^{\prime\prime}$-action on $(M,F)$ is free and
transitive, which proves Lemma 3.4.

Now we finish the proof of Theorem 1.2.

Proof of Theorem 1.2. Lemma 3.4 provides a connected solvable Lie subgroup
$G^{\prime\prime}\subset I_{0}(M,F)$. The map $\pi:G^{\prime\prime}\rightarrow
M$, $\pi(g)=g\cdot o$, is a diffeomorphism. Further more, it is equivariant
with respect to left translations on $G^{\prime\prime}$ and the natural action
of $G^{\prime\prime}\subset I(M,F)$ on $M$. So $\pi^{*}F$ is a left invariant
metric on $G^{\prime\prime}$, i.e., $\pi$ is an isometry between $(M,F)$ and
$(G^{\prime\prime},\pi^{*}F)$. The first statement in Theorem 1.2 is proved.

Suppose that the connected Lie group $G$ admits a negatively curved left
invariant Finsler mtric. By Lemma 2.5, $G$ must be connected. We prove the
solvability of $G$ by contradiction. Assume $G$ is not solvable, then in the
Levi decomposition $\mathfrak{g}=\mathfrak{s}+\mathfrak{r}$ for
$\mathfrak{g}=\mathrm{Lie}(G)$, the semi simple subalgebra $\mathfrak{s}$ has
a nonzero maximal compactly imbedded subalgebra $\mathfrak{k}$. By Lemma 3.2,
$\mathfrak{k}$ generates a compact subgroup $K$ in $G$, with $\dim K>0$. The
left translations of $K$ on $G$ is contained in a maximal compact subgroup of
$I_{0}(G,F)$, so Lemma 2.6 indicates that left translations of $K$ fixes some
element of $G$. This is an impossible because $K\neq\\{e\\}$. The second
statement in Theorem 1.2 is proved.

##  4\. Proof of Theorem 1.3

###  4.1. Some notations and preparation lemmas

Throughout this section, we apply the following notations and assumptions. Let
$G$ be a connected simply connected solvable Lie group, then
$\mathfrak{g}=\mathrm{Lie}(G)$ is solvable. We denote by

$\mathfrak{l}^{0}=[\mathfrak{g},\mathfrak{g}],\
\mathfrak{l}^{1}=[\mathfrak{l}^{0},\mathfrak{l}^{0}],\ \cdots,\
\mathfrak{l}^{i}=[\mathfrak{l}^{0},\mathfrak{l}^{i-1}],\cdots$

the descending sequence of $[\mathfrak{g},\mathfrak{g}]$. Because
$\mathfrak{g}$ is solvable, $\mathfrak{l}^{0}$ is nilpotent. We may assume it
is $k$-step nilpotent, i.e., $\mathfrak{l}^{k}=0$ and
$\mathfrak{c}(\mathfrak{l}^{0})\supset\mathfrak{l}^{k-1}\neq 0$. For any
$0\leq i<j$ and $y_{0}\in\mathfrak{g}$, $\mathrm{ad}(y_{0})$ induces a real
linear endomorphism $\mathrm{ad}_{\mathfrak{l}^{i}/\mathfrak{l}^{j}}(y_{0})$
on $\mathfrak{l}^{i}/\mathfrak{l}^{j}$. We denote by $\mathrm{pr}_{j}$ the
linear projection from $\mathfrak{l}^{0}$ to
$\mathfrak{l}^{0}/\mathfrak{l}^{j}$. Here are some obvious facts:

$\displaystyle\mathrm{ad}_{\mathfrak{l}^{i}/\mathfrak{l}^{j}}(y_{0})(\mathrm{pr}_{j}(v))=\mathrm{pr}_{j}([y_{0},v]),\
e^{\mathrm{ad}_{\mathfrak{l}^{i}/\mathfrak{l}^{j}}(y_{0})}(\mathrm{pr}_{j}(v))=\mathrm{pr}_{j}(e^{\mathrm{ad}(y_{0})}(v)),\
\forall v\in\mathfrak{l}^{i}.$ (4.1)

For simplicity, we use the same notations for real linear maps to denote their
complexifications (i.e., their complex linear extension maps). For example,
(4.1) is still valid when we choose $v$ from
$\mathfrak{l}^{i}\otimes\mathbb{C}$ and view the projection image
$\mathrm{pr}_{j}(v)$ as a vector in
$(\mathfrak{l}^{i}/\mathfrak{l}^{j})\otimes\mathbb{C}=\mathfrak{l}^{i}\otimes\mathbb{C}/\mathfrak{l}^{j}\otimes\mathbb{C}$.

######  Lemma 4.1.

Let $A$ be a real linear endomorphism on a finite dimensional real vector
space $\mathbf{V}$. Then the following statements are equivalent:

  1. (1)

$A$ only has eigenvalues with positive real parts;

  2. (2)

For each $v\in(\mathbf{V}\otimes\mathbb{C})\backslash\\{0\\}$,
$\lim_{t\rightarrow+\infty}e^{tA}(v)=\infty$;

  3. (3)

For each $v\in\mathbf{V}\backslash\\{0\\}$, there exits a sequence
$t_{n}\in\mathbb{R}$ satisfying $\lim_{n\rightarrow\infty}t_{n}=+\infty$ and
$\lim_{n\rightarrow\infty}e^{t_{n}A}(v)=\infty$.

Proof. First, we prove (1)$\Rightarrow$(2). Assume that $A$ only has
eigenvalues with positive real parts. Its complex linear extension on
$\mathbf{V}\otimes\mathbb{C}$ shares the same eigenvalues as $A$, which only
have positive real parts. Notice that $\mathbf{V}\otimes\mathbb{C}$ is the
linear direct sum of

$(\mathbf{V}\otimes\mathbb{C})_{\lambda}=\\{v\in\mathbf{V}\otimes\mathbb{C}|\exists
m>>0,\mbox{ s.t. }(\lambda I-A)^{m}(v)=0\\}$

for all eigenvalues $\lambda$ of $A$, and $e^{tA}$ preserves each
$(\mathbf{V}\otimes\mathbb{C})_{\lambda}$. So we only need to verify
$\lim_{t\rightarrow+\infty}e^{tA}(v)=\infty$ for each
$v\in(\mathbf{V}\times\mathbb{C})\backslash\\{0\\}$. Using the Jordan form of
$A$, we can find $m\in\mathbb{N}\cup\\{0\\}$ and
$v_{0},v_{1},\cdots,v_{m}\in(\mathbf{V}\otimes\mathbb{C})_{\lambda}$,
satisfying $v_{0}=v$, $v_{m}\neq 0$ and

$\displaystyle e^{tA}(v)=e^{\lambda
t}(v_{0}+tv_{1}+\cdots+t^{m}v_{m}),\quad\forall t\in\mathbb{R}.$ (4.2)

By the assumptions $v_{m}\neq 0$ and $\mathrm{Re}\lambda>0$, we get
$\lim_{t\rightarrow+\infty}e^{tA}(v)=\infty$ immediately. This ends the proof
for (1)$\Rightarrow$(2).

Next, (2)$\Rightarrow$(3) is a trivial fact.

Finally, we prove (3)$\Rightarrow$(1). Assume conversely that $A$ has an
eigenvalue $\lambda$ with $\mathrm{Re}\lambda\leq 0$. If
$\lambda\in\mathbb{R}$, $A$ has a nonzero eigenvector $v\in\mathbf{V}$
satisfying $A(v)=\lambda v$. Then we have
$\lim_{t\rightarrow+\infty}e^{tA}(v)=\lim_{t\rightarrow+\infty}e^{\lambda
t}v=0$ or $v$, which is a contradiction to (3). If $\lambda=a+b\sqrt{-1}$ with
$a\leq 0$ and $b\in\mathbb{R}\backslash\\{0\\}$, then we can find a linearly
independent pair $u,v\in\mathbf{V}$, such that $A(u)=au+bv$ and $A(v)=-bu+av$.
Then $e^{tA}(u)=e^{at}(\cos(bt)u+\sin(bt)v)$, which is periodic when $a=0$ and
converges to $0$ when $a<0$. In each situation, we can get a contradiction to
(3). This ends the proof of Lemma 4.1.

######  Lemma 4.2.

Let $\mathbf{V}$ be a finite dimensional real or complex vector space and $r$
a positive integer. Suppose that we have $k_{i}\in\mathbb{N}\cup\\{0\\}$,
$\xi_{i}=a_{i}+b_{i}\sqrt{-1}\in\mathbb{C}$ with $a_{i}>0$ and
$b_{i}\in\mathbb{R}$, and $w_{i}\in\mathbf{V}\backslash\\{0\\}$, $\forall
1\leq i\leq r$. Assume that the pairs in $\\{(k_{i},\xi_{i}),\forall 1\leq
i\leq r\\}$ are all distinct. Then there exists a sequence
$t_{n}\in\mathbb{R}$ satisfying $\lim_{n\rightarrow+\infty}t_{n}=+\infty$ and
$\lim_{n\rightarrow+\infty}f(t_{n})=\infty$, where $f(t)$ is the
$\mathbf{V}$-valued function $f(t)=\sum_{i=1}^{s}t^{k_{i}}e^{\xi_{i}t}w_{i}$.

Proof. Suppose that $\\{1,\cdots,s\\}$ is the set of all indices
$i\in\\{1,\cdots,r\\}$ which satisfies

$a_{i}=\max\\{a_{j},\forall 1\leq j\leq r\\}\quad and\quad
k_{i}=\max\\{k_{j}|a_{j}=\max\\{a_{k},\forall 1\leq k\leq r\\}\\}.$

Because $\\{(k_{i},\xi_{i}),\forall 1\leq i\leq r\\}$ are all distinct,
$\\{b_{1},\cdots,b_{s}\\}$ are all distinct. Direct calculation shows

$\displaystyle(t^{k_{1}}e^{\xi_{1}t})^{-1}f(t)=w_{1}+\sum_{i=2}^{s}e^{(b_{i}-b_{1})t\sqrt{-1}}w_{i}+o(1),$
(4.3)

where $o(1)$ is with respect to $t\rightarrow+\infty$.

Now we prove Lemma 4.2 by contradiction. Assume conversely that it is not
correct, then $f(t)$ is bounded for $t\in[0,+\infty)$, and the right side of
(4.3) converges to $0$ when $t$ goes to $+\infty$. Then we must have $s\geq 2$
and

$\lim_{t\rightarrow+\infty}\sum_{i=2}^{s}e^{(b_{i}-b_{1})t\sqrt{-1}}w_{i}=-w_{1}\neq
0.$

It implies

$\displaystyle\lim_{C\rightarrow+\infty}\int_{0}^{C}\sum_{i=2}^{s}e^{(b_{i}-b_{1})t\sqrt{-1}}w_{i}dt=\infty.$
(4.4)

However, because each $e^{(b_{i}-b_{1})t\sqrt{-1}}$ has zero integral in its
periods, we have the estimate

$\displaystyle||\int_{0}^{C}\sum_{i=2}^{s}e^{(b_{i}-b_{1})t\sqrt{-1}}w_{i}dt||\leq\sum_{i=2}^{s}\frac{2\pi||w_{i}||}{|b_{i}-b_{1}|}<+\infty,$
(4.5)

in which $||\cdot||$ is any arbitrary norm on $\mathbf{V}$. The contradiction
between (4.4) and (4.5) ends the proof of Lemma 4.2.

###  4.2. Proof of (1)$\Rightarrow$(2) in Theorem 1.3

Assume that there exists a negatively curved left invariant Finsler metric $F$
on $G$. The first statement in (2) of Theorem 1.3 has been proved in
Proposition 4.1 in [26]. To be self contained, we recall its proof here. Since
$\mathfrak{g}$ is solvable,
$\dim_{\mathbb{R}}\mathfrak{l}^{0}<\dim_{\mathbb{R}}\mathfrak{g}$. So we can
find $u\in\mathfrak{g}\backslash\mathfrak{l}^{0}$ satisfying
$g_{u}(u,\mathfrak{l}^{0})=0$. Obviously, we have
$\dim_{\mathbb{R}}[u,\mathfrak{g}]\leq\dim_{\mathbb{R}}\mathfrak{l}^{0}<\dim_{\mathbb{R}}\mathfrak{g}-1$.
To prove the first statement in (2) of Theorem 1.3, we only need to verify
$\dim_{\mathbb{R}}[u,\mathfrak{g}]=\dim_{\mathbb{R}}\mathfrak{g}-1$. Assume
conversely it is not true, then the kernel of
$\mathrm{ad}(u):\mathfrak{g}\rightarrow[u,\mathfrak{g}]$ contains a vector
$v\in\mathfrak{g}\backslash\mathbb{R}u$, i.e., $u$ and $v$ is a linear
independent commuting pair. By Lemma 2.4, $F$ can not be negatively curved.
This is a contradiction.

Next, we prove the second statement in (2) of Theorem 1.3. If
$\dim_{\mathbb{R}}\mathfrak{g}=2$, it can not be Abelian, otherwise it has
constant zero curvature. Then we can find a basis $\\{e_{1},e_{2}\\}$ for
$\mathfrak{g}$, such that $[e_{1},e_{2}]=e_{2}$. Choosing $y_{0}=e_{1}$, then
the second statement in (2) is proved. In the discussion below, we assume
$\dim_{\mathbb{R}}\mathfrak{g}\geq 3$.

By linear submersion, the projection map
$\mathrm{pr}_{1}:\mathfrak{l}^{0}\rightarrow\mathfrak{l}^{0}/\mathfrak{l}^{1}$
and the Minkowski norm $F|_{\mathfrak{l}^{0}}$ on $\mathfrak{l}^{0}$ induces a
Minkowski norm $\overline{F}$ on $\mathfrak{l}^{0}/\mathfrak{l}^{1}$. We
denote by $\overline{g}_{\cdot}(\cdot,\cdot)$ the fundamental tensor of
$\overline{F}$. The fundamental tensor of $F|_{\mathfrak{l}^{0}}$ coincides
with that of for $F$, i.e., $g_{\cdot}(\cdot,\cdot)$, except that all three
inputs must be from $\mathfrak{l}^{0}$.

######  Lemma 4.3.

Choose any $y_{0}\in\mathfrak{g}\backslash\mathfrak{l}^{0}$, we have

$\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})\neq
0,\quad\forall\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}.$

Proof. We prove Lemma 4.3 by contradiction. Assume conversely that

$\displaystyle\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})=0\mbox{
for some
}\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}.$ (4.6)

Let $u$ be the horizonal lifting of $\overline{u}$, i.e.,
$u\in\mathfrak{l}^{0}\backslash\mathfrak{l}^{1}$ satisfies
$\mathrm{pr}_{1}(u)=\overline{u}$ and $g_{u}(u,\mathfrak{l}^{1})=0$. In
$[y_{0},u]+\mathfrak{l}^{1}$, there exists a unique $u^{\prime}$ satisfying
$g_{u}(u^{\prime},\mathfrak{l}^{1})=0$. So the assumption (4.6) implies

$\displaystyle
g_{u}(u,[y_{0},u])=g_{u}(u,[y_{0},u]+\mathfrak{l}^{1})=g_{u}(u,u^{\prime})=\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})=0,$
(4.7)

where we have applied Lemma 2.1 to get the third equal.

The condition $g_{u}(u,\mathfrak{l}^{1})=0$ implies

$\displaystyle g_{u}(u,[\mathfrak{l}^{0},u])=g_{u}(u,\mathfrak{l}^{1})=0.$
(4.8)

The first statement of (2), which has been proved, indicates that
$\mathfrak{g}=\mathfrak{l}^{0}+\mathbb{R}y_{0}$, so (4.7) and (4.8) can be
summarized as $g_{u}(u,[\mathfrak{g},u])=0$. To apply Lemma 2.4 to get the
contradiction to negative curvature, we just need to find
$v\in\mathfrak{g}\backslash\mathbb{R}u$ satisfying $[u,v]=0$. When
$\mathfrak{l}^{0}$ is $k$-step nilpotent with $k>1$, we can choose $v$ from
$\mathfrak{l}^{k-1}\backslash\\{0\\}\subset\mathfrak{l}^{1}$. When
$\mathfrak{l}^{0}$ is $1$-step nilpotent, i.e., $\mathfrak{l}^{0}$ is Abelian,
because $\dim_{\mathbb{R}}\mathfrak{l}^{0}=\dim_{\mathbb{R}}\mathfrak{g}-1\geq
2$, we can choose $v$ from $\mathfrak{l}^{0}\backslash\mathbb{R}u$. This ends
the proof of Lemma 4.3.

By Lemma 4.3, we can achieve
$\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})>0$
for some
$\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}$, by a
possible replacement of $y_{0}$ with $-y_{0}$. If
$\dim_{\mathbb{R}}\mathfrak{l}^{0}/\mathfrak{l}^{1}=1$,
$\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})$ has only one
eigenvalue, which is positive, so the second statement in (2) of Theorem 1.3
is proved in this case.

Now we consider the situation that
$\dim_{\mathbb{R}}\mathfrak{l}^{0}/\mathfrak{l}^{1}>1$. By the connectedness,
we have

$\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})>0,\quad\forall\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}.$

Then by the positive 2-homogeneity and the continuity, there is a constant

$c=\min_{\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}}\frac{\overline{g}_{\overline{u}}(\overline{u},\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})\overline{u})}{\overline{g}_{\overline{u}}(\overline{u},\overline{u})}>0.$

Fix any $\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}$,
we set
$\overline{u}(t)=e^{t\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})}(\overline{u})$
and $f(t)=\frac{1}{2}\overline{F}(\overline{u}(t))^{2}$. Then for each
$t\in\mathbb{R}$, $\overline{u}(t)\neq 0$ and $f(t)$ depend on $t$ smoothly.
Calculation shows

$\frac{d}{dt}f(t)=\overline{g}_{\overline{u}(t)}(\overline{u}(t),\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})(\overline{u}(t)))\geq
c\cdot\overline{g}_{\overline{u}(t)}(\overline{u}(t),\overline{u}(t))=cf(t),$

$f(t)\geq e^{ct}f(0)>0$ when $t\geq 0$. So we have
$\lim_{t\rightarrow+\infty}e^{t\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})}(\overline{u})=\infty$
for any $\overline{u}\in(\mathfrak{l}^{0}/\mathfrak{l}^{1})\backslash\\{0\\}$.
By (3)$\Rightarrow$(1) in Lemma 4.1,
$\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})$ only has eigenvalues
with positive real parts. This ends the proof of (1)$\Rightarrow$(2) in
Theorem 1.3.

###  4.3. Proof of (2)$\Rightarrow$(3) in Theorem 1.3

Let $y_{0}\in\mathfrak{g}$ be the vector given in (2) of Theorem 1.3. We will
prove that it meets the requirement in the second statement in (3) of Theorem
1.3.

######  Lemma 4.4.

For each $l\in\mathbb{N}$,
$\mathrm{ad}_{\mathfrak{l}^{l-1}/\mathfrak{l}^{l}}(y_{0})$ only has
eigenvalues with positive real parts.

Proof. When $l=1$, Lemma 4.4 is just the second statement in (2) of Theorem
1.3. Now we further assume that for $l\in\mathbb{N}$,
$\mathrm{ad}_{\mathfrak{l}^{l-1}/\mathfrak{l}^{l}}(y_{0})$ only has
eigenvalues with positive real parts.

Using the Jordan forms of
$\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})$ and
$\mathrm{ad}_{\mathfrak{l}^{l-1}/\mathfrak{l}^{l}}(y_{0})$, and similar
argument as in the proof of Lemma 4.1, we can get:

  1. (1)

For each $u\in\mathfrak{l}^{0}\otimes\mathbb{C}$, there exists
$p\in\mathbb{N}\cup\\{0\\}$, $n_{i}\in\mathbb{N}\cup\\{0\\}$,
$\lambda_{i}\in\mathbb{C}$ with $\mathrm{Re}\lambda_{i}>0$,
$u_{i}\in\mathfrak{l}^{0}\otimes\mathbb{C}$, $\forall 1\leq i\leq p$, such
that
$e^{t\mathrm{ad}_{\mathfrak{l}^{0}/\mathfrak{l}^{1}}(y_{0})}(\mathrm{pr}{}_{1}(u))=\sum_{i=1}^{p}t^{n_{i}}e^{\lambda_{i}t}\mathrm{pr{}_{1}}(u_{i})$,
i.e.,

$\displaystyle
e^{t\mathrm{ad}{}(y_{0})}(u)=\sum_{i=1}^{p}t^{n_{i}}e^{\lambda_{i}t}u_{i}\quad\mbox{(mod
}\mathfrak{l}^{1}\otimes\mathbb{C}\mbox{)};$ (4.9)

  2. (2)

For each $v\in\mathfrak{l}^{l-1}\otimes\mathbb{C}$, there exists
$q\in\mathbb{N}\cup\\{0\\}$, $m_{j}\in\mathbb{N}\cup\\{0\\}$,
$\mu_{j}\in\mathbb{C}$ with $\mathrm{Re}\mu_{j}>0$,
$v_{j}\in\mathfrak{l}^{l-1}\otimes\mathbb{C}$, $\forall 1\leq j\leq q$, such
that
$e^{t\mathrm{ad}_{\mathfrak{l}^{l-1}{}/\mathfrak{l}^{l}{}}(y_{0})}(\mathrm{pr}{}_{l}(v))=\sum_{j=1}^{q}t^{m_{j}}e^{\mu_{j}t}\mathrm{pr}{}_{l}(v_{j})$,
i.e.,

$\displaystyle
e^{t\mathrm{ad}{}(y_{0})}(v)=\sum_{j=1}^{q}t^{m_{j}}e^{\mu_{j}t}v_{j}\quad\mbox{(mod
}\mathfrak{l}^{l}\otimes\mathbb{C}\mbox{)}.$ (4.10)

Any vector $w\in\mathfrak{l}^{l}\otimes\mathbb{C}$, mod
$\mathfrak{l}^{l+1}\otimes\mathbb{C}$, is a complex linear combination of
vectors of the form $[u,v]$, with $u\in\mathfrak{l}^{0}\otimes\mathbb{C}$ and
$v\in\mathfrak{l}^{l-1}\otimes\mathbb{C}$. Since $e^{t\mathrm{ad}{}(y_{0})}$
is a complex Lie algebra automorphism for each $t\in\mathbb{R}$, for $u$ and
$v$ in (4.9) and (4.10) respectively, we have

$\displaystyle e^{t\mathrm{ad}{}(y_{0})}([u,v])$ $\displaystyle=$
$\displaystyle[e^{\mathrm{ad}{}(y_{0})}(u),e^{\mathrm{ad}{}(y_{0})}(v)]$
$\displaystyle=$
$\displaystyle[\sum_{i=1}^{p}t^{n_{i}}e^{\lambda_{i}t}u_{i},\sum_{j=1}^{q}t^{m_{j}}e^{\mu_{j}t}v_{j}]\quad(\mbox{mod
}\mathfrak{l}^{l+1}\otimes\mathbb{C})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{p}\sum_{j=1}^{q}t^{n_{i}+m_{j}}e^{(\lambda_{i}+\mu_{j})t}[u_{i},u_{j}]\quad(\mbox{mod
}\mathfrak{l}^{l+1}\otimes\mathbb{C}).$

Now we assume that $w$ is chosen from
$\mathfrak{l}^{l}\otimes\mathbb{C}\backslash\mathfrak{l}^{l+1}\otimes\mathbb{C}$,
then for each $t\in\mathbb{R}$,
$e^{t\mathrm{ad}(y_{0})}(w)\in\mathfrak{l}^{l}\otimes\mathbb{C}\backslash\mathfrak{l}^{l+1}\otimes\mathbb{C}$.
Above calculations and observations provide an integer $r>0$,
$k_{i}\in\mathbb{N}\cup\\{0\\}$, $\xi_{i}\in\mathbb{C}$ with
$\mathrm{Re}\xi_{i}>0$,
$w_{i}\in\mathfrak{l}^{l}\otimes\mathbb{C}\backslash\mathfrak{l}^{l+1}\otimes\mathbb{C}$,
$\forall 1\leq i\leq r$, such that the pairs in $\\{(k_{i},\xi_{i}),\forall
1\leq i\leq r\\}$ are all distinct and

$\displaystyle
e^{t\mathrm{ad}{}(y_{0})}(w)=\sum_{i=1}^{r}t^{k_{i}}e^{\xi_{i}t}w_{i},\quad\mbox{(mod
}\mathfrak{l}^{l+1}\otimes\mathbb{C}\mbox{)}.$ (4.11)

The equality (4.11) can equivalently presented as

$\displaystyle
e^{t\mathrm{ad}_{\mathfrak{l}^{l}{}/\mathfrak{l}^{l+1}{}}(y_{0})}(\mathrm{pr}_{l+1}{}(w))=\sum_{i=1}^{r}t^{k_{i}}e^{\xi_{i}t}\mathrm{pr}_{l+1}(w_{i}),$

where $\mathrm{pr}_{l+1}(w_{i})$ is nonzero in
$\mathfrak{l}^{l}\otimes\mathbb{C}/\mathfrak{l}^{l+1}\otimes\mathbb{C}$,
$\forall 1\leq i\leq r$. Lemma 4.2 provides a sequence $t_{n}\in\mathbb{R}$
satisfying

$\lim_{n\rightarrow+\infty}t_{n}=+\infty\quad\mbox{and}\quad\lim_{n\rightarrow+\infty}e^{t\mathrm{ad}_{\mathfrak{l}^{l}{}/\mathfrak{l}^{l+1}{}}(y_{0})}(\mathrm{pr}_{l+1}(w))=\lim_{n\rightarrow+\infty}\sum_{i=1}^{r}t_{n}^{k_{i}}e^{\xi_{i}t_{n}}\mathrm{pr}_{l+1}(w_{i})=\infty.$

By (3)$\Rightarrow$(1) in Lemma 4.1,
$\mathrm{ad}_{\mathfrak{l}^{l}/\mathfrak{l}^{l+1}}(y_{0})$ only has
eigenvalues with positive real parts. This ends the proof of Lemma 4.4 by
induction.

The spectrum (i.e., eigenvalue set, counting multiplicities) of
$\mathrm{ad}_{\mathfrak{l}^{0}}(y_{0}):\mathfrak{l}^{0}\rightarrow\mathfrak{l}^{0}$
is the union of those of
$\mathrm{ad}_{\mathfrak{l}^{l}/\mathfrak{l}^{l+1}}(y_{0}):\mathfrak{l}^{l}/\mathfrak{l}^{l+1}\rightarrow\mathfrak{l}^{l}/\mathfrak{l}^{l+1}$,
$\forall 0\leq l\leq k-1$. So the second statement in (3) of Theorem 4.4
follows after Lemma 4.4 immediately.

Finally, we remark that (3)$\Rightarrow$(1) in Theorem 1.3 is an immediate
corollary of Theorem 2.7. This ends the proof of Theorem 1.3.

Acknowledgement. This paper is supported by Beijing Natural Science Foundation
(No. 1222003), National Natural Science Foundation of China (No. 12131012, No.
11821101).

Declarations

Data Availability Not applicable.

Conflict of interest Not applicable.

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# Symmetry of Cosmological Observables, a Mirror World Dark Sector, and the
Hubble Constant

Francis-Yan Cyr-Racine  Department of Physics and Astronomy, University of New
Mexico, Albuquerque, New Mexico, USA 87106  Fei Ge  Department of Physics and
Astronomy, University of California, Davis, California, USA 95616  Lloyd Knox
Department of Physics and Astronomy, University of California, Davis,
California, USA 95616<EMAIL_ADDRESS>
###### Abstract

We find that a uniform scaling of the gravitational free-fall rates and
photon-electron scattering rate leaves most dimensionless cosmological
observables nearly invariant. This result opens up a new approach to
reconciling cosmic microwave background and large-scale structure observations
with high values of the Hubble constant $H_{0}$: Find a cosmological model in
which the scaling transformation can be realized without violating any
measurements of quantities not protected by the symmetry. A “mirror world”
dark sector allows for effective scaling of the gravitational free-fall rates
while respecting the measured mean photon density today. Further model
building might bring consistency with the two constraints not yet satisfied:
the inferred primordial abundances of deuterium and helium.

cosmic microwave background, light relics, neutrinos, cosmology, Hubble
constant

Introduction.—Different methodologies for determining the current rate of
expansion, the Hubble constant $H_{0}$, are leading to discrepant results. The
most precise of the cosmological-model-dependent methods uses the Planck
measurements of the cosmic microwave background (CMB). Assuming Lambda cold
dark matter $\Lambda\mathrm{CDM}$ the result is $H_{0}=(67.49\pm 0.53)$
km/sec/Mpc [1, 2, 3]. The most precise of the more direct methods that are
relatively independent of cosmological model assumptions comes from the SH0ES
team [4, 5, 6, 7]. Using Cepheid-calibrated supernovae they find
$H_{0}=(73.04\pm 1.04)$ km/sec/Mpc [8, hereafter R22], a 5$\sigma$ difference
from the above result.

The preference of $\Lambda\mathrm{CDM}$-dependent methods for a low $H_{0}$ is
remarkably robust to choice of datasets and details of applications; it
remains for both the “inverse distance-ladder” methods [9, 10, 11, 12, 13, 14,
15] that make minimal use of CMB data, and methods that rely on large-scale
structure observations instead of CMB data [16, 17, 18, 19, 20, 21].

Likewise, various averages of the more direct methodologies [22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34], including those that exclude Cepheid-
calibrated supernovae, also lead to $>4\sigma$ discrepancies with the CMB- and
$\Lambda\mathrm{CDM}$-inferred value of $H_{0}$ [35]. On the other hand, there
is no convergence of opinion yet among those using supernovae to measure
$H_{0}$. Most notably, the Carnegie-Chicago Hubble Program [36, 37, 38, 39,
40, 41, 42, 43, 44, 45] finds some inconsistencies in SH0ES estimates of
distances to nearby supernovae. For a review of the observational situation
see Refs. [30, 46].

A lot of recent theoretical work is inspired by the possibility that the
tension arises from a failure of $\Lambda\mathrm{CDM}$. For a wide-ranging
discussion of possible avenues for a solution, see Ref. [47], for a summary of
many of the proposed models, see Ref. [48], and for a ranking of their
performance with respect to a common dataset, see Ref. [49]. The search is
difficult due to the variety of cosmological measurements, their sensitivity
to the details of the cosmological models, their high precision, and their
high degree of consistency with $\Lambda\mathrm{CDM}$.

In this Letter, we provide new insight into the structure of cosmological
models to help theorists navigate this challenging environment, as they search
for solutions to the Hubble tension. In particular, we present a
transformation that leaves all distance ratios and the statistical properties
of fractional maps of CMB temperature anisotropy (i.e., $\Delta T/T$), CMB
polarization, and galaxy number overdensity invariant. This symmetry
transformation, under which all relevant length and timescales in the problem
are rescaled by a constant scaling factor $\lambda$ at all redshifts $z$
111Redshift is often used as a timelike variable in cosmology since the light
that arrives here from a given event on our past light cone is stretched by
the expansion by a factor of $1+z$, providing a monotonic relationship between
time of emission and redshift $z$ in a continuously expanding universe., has
its roots in the scale-free nature of primordial fluctuations.

A restricted version of this transformation was introduced in Ref. [51], in
which only the gravitational timescales $1/\sqrt{G\rho_{i}}$, where $\rho_{i}$
is the mean density of the $i$th component, were scaled. This leads to an
approximate symmetry that is severely broken at small angular scales and in
polarization on all scales. By extending their transformation to include a
scaling of the photon scattering rate, we have found a symmetry of the above-
mentioned observables that is exact in the limit of equilibrium recombination
and zero neutrino mass. Crucially, we find that symmetry-breaking effects of
real-world departures from these limits are mild.

We also report here on partial progress toward the use of this symmetry to
solve the $H_{0}$ tension. Significant observational constraints prevent us
from a straightforward scaling of gravitational rates by the scaling of mean
densities. To evade these constraints, and still reap the benefits of the
scaling transformation, we are led quite naturally to the addition of a mirror
world dark sector. Dark sector models that contain a mirror (or “twin”) sector
222Models have also been proposed that contain multiple copies of the SM, see
e.g. Refs. [170, 171, 172]. that has the same particle content and gauge
interactions as the standard model (SM), have been extensively studied in the
literature (see e.g. Refs. [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83])
in the context of the little hierarchy problem. To scale the photon scattering
rate, one can alter the primordial fraction of baryonic mass in helium $Y_{\rm
P}$, which is the method we use here. We show below that such models can, in
principle, exploit the scaling symmetry to accommodate a higher value of
$H_{0}$ with CMB observations.

A strict implementation of the scaling transformation, however, leads to a
conflict between the measurements of $Y_{\rm P}$ and the primordial deuterium
abundance with expectations from big bang nucleosynthesis (BBN) as well as the
$Y_{\rm P}$ necessary for scaling up the scattering rate. Thus, we have
essentially remapped the problem that has proven to be quite difficult, of
reconciling a high $H_{0}$ with phenomenologically complicated CMB data, to a
problem of reconciling a high $H_{0}$ with the observationally inferred
primordial abundances of helium and deuterium. Our work motivates the search
for models that can solve this reframed problem [84]. Finally, although we
focus here on inferences from CMB observations, all of the cosmological-model-
dependent inferences of $H_{0}$ we referred to above, including those in [9,
10, 11, 12, 13, 16, 14, 15, 16, 17, 18, 19, 20, 21], would be impacted
similarly; i.e., replacing $\Lambda\mathrm{CDM}$ with a model that solves our
reframed problem would also reconcile these inferences with a high $H_{0}$,
due to the nature of the symmetry.

The scaling transformation.— Let us assume, for now, that recombination
happens in equilibrium and neutrinos are massless. Then the only length scales
in the linear perturbation evolution equations in the $\Lambda\mathrm{CDM}$
model, written with the scale factor $a=1/(1+z)$ as the independent timelike
variable, are the gravitational timescales of each of the $i=1$ to $N$
components $1/\sqrt{G\rho_{i}(a)}$ and the photon mean free path between
electron scatters, $1/\left[\sigma_{T}n_{e}(a)\right]$. As a result, if we
consider the linear evolution of a single Fourier mode with wave number $k$,
any fractional perturbation, such as $\delta\rho_{\rm
m}({\mathbf{k}},a)/\rho_{\rm m}(a)$ satisfying the evolution equations will
also satisfy them when transformed by a uniform scaling of all relevant
(inverse) length scales (including $k$) by some factor $\lambda$. Since the
initial conditions in $\Lambda\mathrm{CDM}$ do not introduce a length scale
(the spectrum of initial perturbations is a power law), the statistical
properties of fractional perturbations are independent of $\lambda$ except for
an overall amplitude. Dependence on the amplitude can be removed [51] by
extending the scaling transformation to include $A_{s}\rightarrow
A_{s}/\lambda^{(n_{\rm s}-1)}$ where $A_{\rm s}$ is the amplitude of the
primordial power spectrum at some fiducial value of $k$, and $n_{\rm s}$ is
the spectral index of the power law power spectrum of initial density
perturbations.

In Ref. [51], this transformation was introduced but without the photon
scattering-rate scaling, which is critically important for our purposes.
Including it, the transformation leads to an exact symmetry in the limit of
equilibrium recombination 333We could drop this caveat by extending the
transformation further so that the atomic reaction rates scale appropriately
as well. Doing so would also extend the challenge (significantly) of finding a
model for which a direction in the parameter space corresponds to the scaling
transformation., massless neutrinos 444The very mild symmetry breaking from
non-zero neutrino masses could be removed by also scaling them by a factor of
$\lambda$., and linearized equations 555We include this caveat because we do
not have a proof that the symmetry holds in the full non-linear theory. It
might. We do know it holds to second order in perturbations; see our
Supplementary Material, which includes [173, 51, 174]. of distance ratios and
the statistical properties of maps made in projection on the sky of quantities
such as $\delta\rho/\rho$, and fractional CMB temperature and polarization
anisotropies. These include galaxy clustering power spectra, shear power
spectra, galaxy-shear cross-correlations, fractional CMB temperature and
polarization power spectra, the temperature-polarization cross spectrum, and
the CMB lensing spectrum. The invariance exists for the $\Lambda\mathrm{CDM}$
model, and any other model as long as additional length scales (if any, such
as those related to mean curvature or neutrino mass) are properly scaled as
well. Absent the introduction of new length scales, the full transformation
can be written as

$\begin{split}\sqrt{G\rho_{i}(a)}\rightarrow\lambda\sqrt{G\rho_{i}(a)},\ \ &\
\ \sigma_{\rm T}n_{e}(a)\rightarrow\lambda\sigma_{\rm T}n_{e}(a)\\\ {\rm and}\
\ A_{\rm s}\rightarrow A_{\rm s}/\lambda^{(n_{\rm s}-1)}.\end{split}$ (1)

Symmetry breaking.—The transformation given in Eq. (1) is severely constrained
by observations that are sensitive to _absolute_ densities of cosmological
components. Most importantly, we know very precisely the mean energy density
of the CMB today from measurements by FIRAS of its flux density across a broad
range of wavelengths [88, 89]. By anchoring $\rho_{\gamma}$, this measurement
severely limits our ability to exploit the scaling transformation to raise
$H_{0}$ 666 A similar point was made recently in Ref. [175].

Other important effects that break the above symmetry arise from departures
from thermodynamic equilibrium, as emphasized in Ref. [51]. Unlike periods of
equilibrium, during which we have no sensitivity to the rates of the reactions
that are maintaining equilibrium, periods in which equilibrium is lost provide
us with valuable sensitivity to the relevant reaction rates. If we then assume
that such microphysical rates are known, we can gain sensitivity to the
expansion rate. A prime example is BBN, where sensitivity of the yield of
helium and deuterium to nuclear reaction rates allows one to infer, from
measurements of helium and/or deuterium, the expansion rate during BBN, and
thus, through the Friedmann equation, the mass or energy density of the
Universe at that time. Similarly, hydrogen recombination is an out-of-
equilibrium process which is sensitive to atomic reaction rates, and thus
breaks the symmetry of the Eq. (1) transformation. We will see that the impact
of this latter symmetry breaking on our parameter constraints is mild.

A mirror world dark sector and free $Y_{\rm P}$.—Zahn and Zaldarriaga [51] did
not provide a physical mechanism by which one could realize their
transformation, other than by varying $G$, as their goal was only analytic
understanding. We now introduce a physical mechanism that, while not allowing
for the transformation as strictly written, permits a complete mimicry of its
effects, so that the same invariance is achieved.

By extending the $\Lambda\mathrm{CDM}$ model to include a dark copy of the
photons, baryons, and neutrinos (see, e.g., Refs. [91, 92, 93, 94, 95, 96, 97,
98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111]), all with
the same mean density ratios as in the visible sector, we can effectively
mimic the $\sqrt{G\rho_{i}}$ part of the scaling transformation while evading
the constraint from FIRAS. The dark photons (which have temperature $T_{\rm
D}$) are a replacement for the additional visible photons that would violate
the FIRAS constraint. The dark baryons [implemented as “atomic dark matter”
(ADM) [112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
126, 127, 128, 129, 130, 131, 132, 133, 134, 135]] allow us to scale up the
total baryonlike density without changing the well-constrained (visible
sector) baryon-to-photon ratio. The dark neutrinos allow us to scale up the
effective number of free-streaming neutrino species from its
$\Lambda\mathrm{CDM}$ value of $N_{\rm eff}^{\rm FS}=3.046$ 777We note that
using the more recent estimate of $N_{\rm eff}^{\rm fs}=3.044$ [176] would
have minimal impact on our results., preserving the well-constrained ratio of
free-streaming to tightly coupled relativistic particle densities [137, 138,
139].

In our implementation, the new mirror dark sector interacts purely
gravitationally with the visible sector and the CDM. Therefore, we can mimic a
$\Lambda\mathrm{CDM}$ model with scaled-up densities if the perturbations in
the mirror world evolve in the same way as what they replaced in the visible
sector and thus provide the same source to the metric perturbations. For this
to be the case, the dark photons must transition from tightly coupled to
freely streaming when the visible photons do. We thus ensure that the ADM
recombines at approximately the same time as regular hydrogen by keeping the
ratio $B_{\rm D}/T_{\rm D}$ fixed, where $B_{\rm D}$ is the binding energy of
the ADM, and by setting the dark fine structure constant and dark proton mass
equal to those in the light sector. For simplicity we assume there are no dark
versions of helium or heavier nuclei. We can keep the Thomson scattering rate
on the scaling trajectory by adjusting $Y_{\rm P}$. At fixed baryon density,
we have $n_{e}(z)\propto x_{e}(z)(1-Y_{\rm P})$ where $x_{e}(z)$ is the
fraction of free electrons. So to scale the scattering rate appropriately,
approximating $x_{e}(z)$ as fixed, we send $(1-Y_{\rm
P})\rightarrow\lambda(1-Y_{\rm P})$.

Figure 1: CMB temperature (TT) and polarization (EE) power spectra with (red
and green) and without (blue) scaling by $\lambda=1.1$ from the best-fit
$\Lambda\mathrm{CDM}$ model for dataset 1. Both $\lambda=1.1$ cases are scaled
by use of the mirror world dark sector, with (red) and without (green) scaling
of the photon scattering rate. Data points are from Planck 2018 [140].
Fractional differences with the best-fit $\Lambda\mathrm{CDM}$ model are shown
in the second and bottom panels.

In Fig. 1, we show the temperature (TT) and polarization (EE) power spectra
for the best-fit $\Lambda\mathrm{CDM}$ model given dataset 1 (defined below)
as well as spectra for the models scaled from that one with $\lambda=1.1$,
with and without the photon-scattering-rate scaling included. The scaled
spectra with no photon scattering scaling differ significantly from the
$\Lambda\mathrm{CDM}$ spectra. The small differences between the fully scaled
spectra and the $\Lambda\mathrm{CDM}$ spectra are primarily due to the
symmetry-breaking effects of the atomic reaction rates affecting hydrogen
recombination.

Label | Dataset
---|---
1 |  Planck TT, TE, EE, lensing + BAO
2 | Dataset 1 plus R21 ($H_{0}=73.2\pm 1.3$ km/s/Mpc)
Table 1: Definition of datasets 1 and 2. We use the Plik TT+TE+EE, LowlT, LowlE, and lensing Planck likelihoods described in Ref. [140]. Baryon acoustic oscillations (BAO) datasets are 6dFGS [141], SDSS MGS [142] and BOSS DR12 [143]. R21 is SH0ES measurement [7], of which R22 is a recent update. Label | Model space
---|---
A |  $\Lambda\mathrm{CDM}$ \+ $\lambda$ (scaling enforced), $x_{e}(z)$ fixed
B |  $\Lambda\mathrm{CDM}$ \+ $\lambda$ (scaling enforced), $x_{e}(z)$ calculated
C |  $\Lambda\mathrm{CDM}$ \+ $T_{\rm D}$ \+ $N_{\rm eff}^{\rm FS}$ \+ $f_{\rm ADM}$ \+ $Y_{\rm P}$
Table 2: Definition of model spaces A, B & C. For model space A the one
non-$\Lambda\mathrm{CDM}$ parameter is $\lambda$. We restrict the additional
components including $Y_{\rm P}$ to the scaling solution and (artificially)
hold $x_{e}(z)$ fixed to its $\Lambda\mathrm{CDM}$ best-fit value, for both
the dark and light sectors. We scale $Y_{\rm P}$ from its BBN-consistent
$\Lambda\mathrm{CDM}$ dataset 1 best-fit value of 0.2454. Model space B
differs only in that we calculate the visible sector $x_{e}(z)$ using the
atomic reaction rates (and the code RECFAST [144, 145]) and the dark sector
ionization evolution as in Ref. [122]. Model space C only differs from B in
that we allow $Y_{\rm P}$, the effective number of free-streaming neutrinos
$N_{\rm eff}^{\rm FS}$, and the fraction of mirror world (or “atomic”) dark
matter $f_{\rm ADM}$ to depart from their scaling values. For all model spaces
we adopt the uniform prior $1.00001<\lambda<1.3$, set the ratio of dark to
light photon temperatures to
$T_{D}/T_{\gamma}=\left(\lambda^{2}-1\right)^{1/4}$, and, although it
introduces a new length scale, we take one of the neutrino species to have a
mass of 0.06 eV [146]. We modified CAMB [147] to solve the relevant Einstein-
Boltzmann equations and used CosmoMC [148] to calculate parameter posterior
densities.

Results.—We now explore the parameter constraints in the framework just
described. We define datasets 1 and 2 and model spaces A, B, and C, in Tables
I and II. The results are shown in Fig. 2. In the upper left panel we see the
expected results from A1: The posterior is almost flat throughout the prior
region 888This prior is made necessary by our use of a mirror dark sector to
mimic a $\Lambda$CDM model with scaled-up densities. It ensures that the
energy densities of dark photons and dark atoms are always positive.. This is
the numerical manifestation of the exact symmetry we have presented.

For B1, the $H_{0}$ posterior remains quite broad. The soft symmetry-breaking
effects of nonequilibrium recombination, evident in Fig. 1, are sufficiently
degenerate with variation of other $\Lambda\mathrm{CDM}$ parameters to avoid
significant constraint on $H_{0}$. The tension with R21 has been completely
eliminated. The very slight preference for $H_{0}=73.2$ km/s/Mpc in B1 over
the $\Lambda\mathrm{CDM}$ value is a parameter-volume effect; the best-fit
$\Lambda\mathrm{CDM}$ model has a nearly identical $\chi^{2}$ value as the
best-fit B model constrained to $\lambda=1.08$, lower by $\Delta\chi^{2}=$
0.2.

Figure 2: Constraints on parameters from datasets 1 and 2 given models A, B
and C (see Tables I and II). Top left panel: the (unnormalized) posterior
probability density of $H_{0}$. Other panels: the 68% and 95% contours of
equal probability density in the $H_{0}-Y_{\rm P}$, $H_{0}-f_{\rm ADM}$, and
$H_{0}-T_{\rm D}/T_{\rm\gamma}$ planes. The gray band “R21” shows the
1$\sigma$ and 2$\sigma$ constraint on $H_{0}$ from R21. The purple band “A21”
shows the same for $Y_{\rm P}$ from [150, hereafter A21].

In the other three panels of Fig. 2 we show how well departures away from
scaling are constrained by the data, if they are not prevented by fiat. The B1
contours lie over the constraints given by model C; as expected, the scaling
direction is preferred by the data. We also see that the region of high
posterior probability density extends to parameter values far from the scaling
solution, an indication of some freedom that more detailed model building
could exploit. For C2, we find that $f_{\rm ADM}=0.027\pm 0.011$ and $T_{\rm
D}=(0.68\pm 0.06)T_{\gamma}$. Meanwhile, the value of $\sigma_{8}=0.808\pm
0.011$ is nearly unchanged (if not slightly lower) from its
$\Lambda\mathrm{CDM}$ value, as expected.

In Fig. 2 we also see a problem: The $Y_{\rm P}$ values consistent with R21
are inconsistent with inferences from spectral observations of hot “metal-
poor” gas such as the A21 finding of $Y_{\rm P}=0.2453\pm 0.0034$. From C2 we
have $Y_{\rm P}=0.170\pm 0.025$, a 3.0$\sigma$ difference. Compounding this
trouble, the additional light relics, if we do not otherwise alter the
standard thermal history, would increase the BBN-expected $Y_{\rm P}$ [151],
in C2, to $0.2614\pm 0.0038$.

Discussion.—We have found important constraints on the project of turning the
scaling transformation into a solution to the $H_{0}$ tension. The first of
these, from FIRAS, can be accommodated with a mirror world dark sector.
Constraints from light element abundances lead us to the articulation of two
additional targets for model building: (i) a new mechanism for increasing the
photon scattering rate and (ii) additional model features that would bring BBN
predictions for helium and deuterium abundances in line with observations.

Item (i) follows from the fact that the $Y_{\rm P}$ required for consistency
with R21 in models B and C is 3$\sigma$ too low compared to the inference from
observations in A21, hence strongly suggesting that modifying $Y_{\rm P}$ is
not a promising way to increase the photon scattering rate. Item (ii) follows
since the extra relativistic species in the mirror sector required by R21
alter predictions of $Y_{\rm P}$ and deuterium. We have for C2 $\Delta N_{\rm
eff}=1.3\pm 0.34$. On the other hand, BBN consistency with $Y_{\rm P}$ and
deuterium measurements leads to $\Delta N_{\rm eff}=-0.17\pm 0.28$ [84].

One idea worth exploring, to boost the photon scattering rate, is a heating of
the baryons by a spectral distortion in the Wien tail at frequencies beyond
FIRAS’s reach 999J. Chluba, private communication. A time-varying electron
mass [153, 154] is another possible solution that has had some
phenomenological success [153], and could even be well motivated by a
supersymmetric gravity sector [155, 156]. Successfully fitting the light
element abundances could be achieved, for instance, by reheating the dark
sector in the post-BBN era (see, e.g., Refs. [157]), or by possibly
introducing interactions between the light and dark sectors that could affect
the predicted yields; our inferred $T_{D}/T_{\gamma}$ even puts decoupling of
the dark and light sectors in the right ballpark for this to occur.

Our scenario requires a mirror sector that is phenomenologically close to the
SM, albeit with a lower temperature. In its simplest realization, such a
sector appears tightly constrained by particle collider data (see e.g. Ref.
[158]). Encouragingly however, preliminary work [159] indicates that such a
mirror sector could be successfully built. More generally, our work opens the
possibility of relaxing cosmological constraints (see e.g. Refs. [65, 69, 75,
160]) on such mirror scenarios, given mechanisms to adjust the photon
scattering rate and light-element abundances.

We note that having $\sim 3\%$ of dark matter in ADM can lead to interesting
astrophysical phenomenology such as exotic compact objects [161, 162], dark
stars [163, 164], exotic gamma-bright point sources [165], and dark disks
within galaxies [125, 126, 127, 128, 166, 167, 168, 169, 80].

Conclusions.—We have generalized a scaling transformation so that it now
allows for large changes in the cosmological model while preserving the
precisely measured and feature-rich CMB temperature and polarization spectra,
as well as many other cosmological observables. Implementing this
transformation, while evading constraints from FIRAS, leads us directly to a
mirror world dark sector, a type of dark sector that has been proposed before
for independent reasons.

While the scaling symmetry does not, by itself, provide an end-to-end solution
to the $H_{0}$ tension, we have used it to single out the photon scattering
rate and light-element abundances as players with important roles in the
discrepancy. We have thus provided clear model-building targets for the
community to explore.

###### Acknowledgements.

Acknowledgments: F.G. and L.K. were partially supported by the U.S. Department
of Energy Office of Science. F.-Y. C.-R is supported by the National Science
Foundation (NSF) under Grant No. AST-2008696. We thank B. Fields, D. Green, D.
Liu, and M. Luty for useful conversations and the UNM Center for Advanced
Research Computing, supported in part by the NSF, for providing some of the
computing resources used in this work. Part of this work was performed at the
Aspen Center for Physics, which is supported by NSF Grant No. PHY-1607611.

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##  Appendix A Supplemental Material: Symmetry of the Equations of Motion

In this supplementary material, we examine how the equations of motion
governing the evolution of cosmological fluctuations are invariant under the
scaling transformation introduced in the main text. We first consider the
standard first-order equations, and then examine how the scaling
transformation generalizes to higher order in perturbation theory.

###  A.1 First-order equations

As a starting point, let us first examine the Boltzmann equations governing
the evolution of photons and baryons fluctuations. Using the scale factor $a$
as our time variable, these take the form [173]

$\displaystyle\frac{\partial F_{\gamma 0}}{\partial a}$
$\displaystyle=-\frac{k}{a^{2}H}F_{\gamma 1}+4\frac{\partial\phi}{\partial
a},$ (2) $\displaystyle a^{2}H\frac{\partial F_{\gamma 1}}{\partial a}$
$\displaystyle=\frac{k}{3}(F_{\gamma 0}-2F_{\gamma
2})+\frac{4k}{3}\psi+\dot{\kappa}(\frac{4}{3}v_{\rm b}-F_{\gamma 1}),$
$\displaystyle a^{2}H\frac{\partial F_{\gamma 2}}{\partial a}$
$\displaystyle=\frac{k}{5}(2F_{\gamma 1}-3F_{\gamma
3})-\frac{9}{10}\dot{\kappa}F_{\gamma 2},$ $\displaystyle a^{2}H\frac{\partial
F_{\gamma l}}{\partial a}$
$\displaystyle=\frac{k}{2l+1}\left[lF_{\gamma(l-1)}-(l+1)F_{\gamma(l+1)}\right]-\dot{\kappa}F_{\gamma
l},$ $\displaystyle\frac{\partial\delta_{\rm b}}{\partial a}$
$\displaystyle=-\frac{k}{a^{2}H}v_{\rm b}+3\frac{\partial\phi}{\partial a},$
$\displaystyle a^{2}H\frac{\partial v_{\rm b}}{\partial a}$
$\displaystyle=-aHv_{\rm b}+c_{\rm s}^{2}k\delta_{\rm
b}+\frac{\rho_{\gamma}}{\rho_{\rm b}}\dot{\kappa}(F_{\gamma
1}-\frac{4}{3}v_{b}),$

where $F_{\gamma l}$ are the multipole moments of the photon temperature
perturbation, $k$ is the Fourier wavenumber, $\dot{\kappa}=an_{\rm
e}\sigma_{\rm T}$ is the Thomson opacity, $\delta_{\rm b}$ is the baryon
density perturbation, $v_{\rm b}$ is the baryonic bulk velocity, $c_{\rm s}$
is the baryonic sound speed, and $\phi$ and $\psi$ are the two gravitational
potentials in conformal Newtonian gauge. Note that we have used the
relationship

$\frac{d}{d\eta}=a^{2}H\frac{d}{da}$ (3)

to convert between conformal time ($\eta$) derivatives and scale-factor
derivatives. It is straightforward to see that these equations are invariant
under the transformation

$H\to\lambda H,\,k\to\lambda k,\,\dot{\kappa}\to\lambda\dot{\kappa}.$ (4)

These transformations correspond to equally rescaling _all_ length scales
appearing in the Boltzmann equations: the Hubble horizon, the wavelength of
fluctuations, and the photon mean free path. To close this system of
equations, we need the perturbed Einstein equations for the $\phi$ and $\psi$
potentials. We use here the Poisson and shear equations

$\displaystyle k^{2}\phi+3aH\left(a^{2}H\frac{d\phi}{da}+aH\psi\right)$
$\displaystyle=-4\pi Ga^{2}\sum_{i}\rho_{i}\delta_{i},$ (5) $\displaystyle
k^{2}(\phi-\psi)$ $\displaystyle=12\pi
Ga^{2}\sum_{i}(\rho_{i}+P_{i})\sigma_{i},$

where $\delta_{i}$ and $\sigma_{i}$ are the energy density perturbation and
anisotropic stress of species $i$, respectively. These equations are invariant
under the transformation given in Eq. (4), provided that the energy density of
each component is individually rescaled, that is,
$\sqrt{G\rho_{i}}\to\lambda\sqrt{G\rho_{i}}$. Massless neutrinos and dark
matter follow collisionless versions of the equation given in Eqs. (2),
implying that they too are invariant under the transformation $H\to\lambda H$
and $k\to\lambda k$. We note that the evolution of massive neutrinos
perturbations is also invariant under this transformation, once their masses
are also properly scaled.

We thus see that the linear evolution equations of _all_ components present in
the Universe are invariant under the transformation

$\sqrt{G\rho_{i}}\to\lambda\sqrt{G\rho_{i}},\,k\to\lambda
k,\,\dot{\kappa}\to\lambda\dot{\kappa}.$ (6)

This means that we can express the solution $\tilde{\Phi}$ to the perturbation
equations in the presence of this scaling symmetry in terms of the original
solution $\Phi$ in the absence of scaling (i.e. $\lambda=1$) as

$\tilde{\Phi}(k,\dot{\kappa},a,\lambda)=\Phi(k/\lambda,\dot{\kappa}/\lambda,a,\lambda=1),$
(7)

where $\Phi$ here stands for any of the dimensionless perturbation variables
(e.g. $\delta,v,F_{\gamma l}$, etc.). Such a relation was first presented in
Ref. [51] in the context of the tight-coupling approximation (
$\dot{\kappa}\gg H$), but we see here that it applies in a broader context
once the Thomson opacity is also rescaled.

###  A.2 Second-order equations

Equations (2) and (5) are invariant under the scaling transformation given in
Eq. (6) essentially on dimensional ground. By consistency, all terms in a
single differential equation need to have the same mass (or length)
dimensions. Since the scaling transformation uniformly rescale _all_ length
scales in the problem, then these equations are invariant as all terms are
multiplied by the same scaling factor. The same logic applies to higher-order
equations. For instance, the second-order equations for photons, baryons, cold
dark matter and neutrinos are presented in Ref. [174]. As in the first-order
case, all terms in the relevant differential equations (when written with the
scale factor $a$ as the time variable) either involve a factor of $H$, a
factor of $k$, or a factor of $\dot{\kappa}$, each multiplied by dimensionless
perturbation variables. Such equations are thus invariant under the scaling
transformation given in Eq. (6).