# Bach Flow of Simply Connected Nilmanifolds

Adam Thompson

###### Abstract.

The Bach flow is a fourth order geometric flow defined on four manifolds. For
a compact manifold, it is a conformally modified gradient flow for the
$L^{2}$-norm of the Weyl curvature. In this paper we study the Bach flow on
four-dimensional simply connected nilmanifolds whose Lie algebra is
indecomposable. We show that the Bach flow beginning at an arbitrary left
invariant metric exists for all positive times and after rescaling converges
in the pointed Cheeger-Gromov sense to an expanding Bach soliton which is non-
gradient. Combining our results with previous results of Helliwell gives a
complete description of the Bach flow on simply connected nilmanifolds.

##  1\. Introduction

The success of the Ricci flow and other second order geometric flows has led
to a desire to better understand higher order geometric flows. In particular,
the study of fourth order flows is a natural progression from second order
flows. Fourth order flows can be more difficult to since we no longer have
access to maximum principles. An important class of fourth order flows are the
gradient flows of quadratic curvature functionals.

In [BahuaudHelliwellSTEFSHOGF] Bahuad and Helliwell introduced a family of
higher order flows called the _ambient obstruction flow_ which involves the
ambient obstruction tensor of Fefferman and Graham introduced in
[FeffermanGrahamCI]. In dimension 4, the ambient obstruction tensor coincides
with the _Bach tensor_ and the corresponding flow is the _Bach flow_ :

(1) $\frac{\partial}{\partial t}g=\operatorname{B}(g)+\frac{\Delta
s}{12}g,\qquad g(0)=g_{0}.$

Here $\operatorname{B}(g)$ is the Bach tensor defined in (2) below, $s=s(g)$
is the scalar curvature and $\Delta$ is the Laplace-Beltrami operator. This
flow corresponds to a conformally modified gradient flow of the $L^{2}$-norm
of the Weyl curvature, and so is fourth order (see the introduction in
[StreetsTLTBOFOCF]). While this might not be the first choice of a Bach flow
due to the $\frac{1}{12}(\Delta s)g$ term, this is needed to counteract the
invariance of the Bach tensor under the conformal group [LopezAOF, Section
1.2]. In any case, if a solution $g(t)$ to (1) has constant scalar curvature
at each point in time, as is the case for homogeneous solutions, then $\Delta
s\equiv 0$. Short time existence and uniqueness for the flow on compact
manifolds was proved by Bahaud and Helliwell in [BahuaudHelliwellSTEFSHOGF,
BahuaudHelliwellUFSHOGF].

The Bach flow was studied by Helliwell in [HelliwellBFOHP] on homogeneous
manifolds of the form $M^{4}=S^{1}\times N^{3}$ where $N^{3}$ is a closed,
homogeneous 3-manifold and $M$ is equipped with the product metric. By lifting
to the universal cover, this reduces to the study of the Bach flow on 9 simply
connected homogeneous spaces of the form
$\widetilde{M}=\mathbb{R}\times\widetilde{N}$. In [GriffinGAOSOHM], Griffin
uses the computations of the Bach tensor from [HelliwellBFOHP] and a theorem
of Petersen and Wylie [PetersenWylieROHGSMARE, Theorem 3.6] to study gradient
Bach solitons on the product manifolds from Helliwell’s paper.

The Bach tensor of a left-invariant metric is determined by its value at the
identity, and therefore can be computed explicitly using metric coefficients
and structure constants of the of the Lie algebra. In practice, however, this
is difficult due to the fourth order nature of the tensor. For Riemannian
products this formula greatly simplifies [HelliwellBFOHP, Section 2.1]. In
order to make the problem more tractable in the non-product case we have
chosen to study the flow on Lie group whose Lie algebra has a large number of
structure constants which vanish.

In this article we study the Bach flow the simply connected nilpotent Lie
group, $N^{4}$, whose Lie algebra is the unique four-dimensional
indecomposable nilpotent Lie algebra. By indecomposable, we mean that the Lie
algebra cannot be written as a direct product of two lower dimensional Lie
algebras. Our main result is the following theorem.

######  Theorem 1.

Let $N^{4}$ be the simply connected nilpotent Lie group with Lie algebra
$\operatorname{Lie}(N)=\mathfrak{n}_{4}$, the unique four dimensional,
indecomposoable, nilpotent Lie algebra and let $g$ be a left invariant metric
on $N$. Then, there exists a unique solution $(g(t))_{t\in(-\varepsilon,T)}$
to the Bach flow, defined on a maximal interval of existence
$(-\varepsilon,T)$, which is $N^{4}$-invariant. This satisfies

  1. (1)

$T=\infty$ and the metrics $g(t)$ converges in the pointed Cheeger-Gromov
sense to the Euclidean metric in $\mathbb{R}^{4}$ as $t\to\infty$.

  2. (2)

If $s_{t}=s(g(t))$ denotes the scalar curvature of $g(t)$ then the rescaled
metrics

$\tilde{g}(t)=|s_{t}|g(t)$

converge in the pointed Cheeger-Gromov sense to an expanding Bach soliton
$g_{\infty}$ on $N^{4}$ as $t\to\infty$.

A _Bach soliton_ is a solution to (1) which is self similar. That is, the
solution $g(t)$ to (1) beginning at a soliton $g$ can be written
$g(t)=c(t)\varphi(t)^{*}g$ for some positive function $c$ and time dependent
diffeomorphisms $\varphi(t)$. We say a soliton is expanding, steady or
shrinking if the scaling constant is increasing, constant or decreasing
respectively.

The Bach soliton $g_{\infty}$ is a non-trivial algebraic Bach soliton (see
[LauretRSHN, LauretGFATSOHS] for literature on algebraic Ricci solitons). That
is, if $B:\mathfrak{n}_{4}\to\mathfrak{n}_{4}$ is the endomorphism defined by
$\operatorname{B}(g_{\infty})_{e}=g(B\cdot,\cdot)$ then,

$B=\lambda I+D,\qquad\lambda\in\mathbb{R},$

where $D\in\operatorname{Der}(\mathfrak{n}_{4})$ is a non-zero derivation. To
prove Theorem 1 we use the bracket flow technique introduced by Lauret to
study the Ricci flow in homogeneous manifolds [LauretRFOHM] (Lauret extends
the bracket flow technique to more general geometric structures in
[LauretGFATSOHS]). The bracket Bach flow is an ODE in the variety of Nilpotent
Lie brackets
$\mathcal{N}^{4}\subset\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$
which is equivalent to the Bach flow. Algebraic solitons correspond to
solutions of the bracket flow which evolve only by scaling. These are easier
to detect than general solitons since we have removed the action of the
diffeomorphism group. Our analysis follows Lauret’s analysis of the Ricci flow
on simply-connected nilmanifolds in [LauretTRFFSCN].

We also remark that the soliton $g_{\infty}$ is not of gradient type due to a
Theorem of Petersen and Wylie [PetersenWylieROHGSMARE, Theorem 3.6]. To our
knowledge, this is the first example of a non-gradient Bach soliton.

The paper is structured as follows. In §2 we introduce the Bach tensor and its
basic properties. In §3 review some group actions on the variety of nilpotent
Lie algebras and recall the classification of four-dimensional Lie algebras.
In §4 we review the structure of four-dimensional simply connected
nilmanifolds. In particular, we show that up to isometry, left-invariant
metrics on $N^{4}$ depend on 3 real parameters. By suitably gauging the Bach
flow (Proposition 10), we obtain an ODE in three variables, $a,b,c$, which is
equivalent to the bracket Bach flow the simply connected Lie group $N^{4}$. By
studying the long time behaviour of the quantities $a,b,c$ we are able to
understand the long time behaviour of the Bach flow. Self similar solutions of
the Bach flow are studied in §6. In terms of the variables $a,b,c$ the soliton
condition becomes a set of algebraic equations. Finally, we considered the
normalised flow in §7.

##  2\. Background on The Bach Flow

Let $M^{4}$ be a four dimensional Riemannian manifold. The _Bach tensor_ is
the tensor given in local coordinates by

(2)
$\operatorname{B}(g)_{ij}=\nabla^{k}\nabla^{l}W_{kijl}+\frac{1}{2}R^{kl}W_{ikjl}.$

Here, $W$ is the Weyl tensor,
$R^{kl}=g^{ka}g^{lb}R_{ab}=g^{ka}g^{lb}(\operatorname{Rc})_{ab}$ are the
components of the Ricci tensor with both indices raised.

The Bach tensor is conformally invariant ($\operatorname{B}(\rho
g)=\rho^{-1}\operatorname{B}(g)$ for any positive function $\rho$) and arises
in the study of manifolds which are conformally Einstein
[KuhnelRademacherCTOPRM, Section 6]. In particular, Theorem 6.6 in
[KuhnelRademacherCTOPRM] says $\operatorname{B}(g)\equiv 0$ is a necessary
condition for $(M^{4},g)$ to be conformal to an Einstein manifold. The
converse holds if $g$ is conformal to a metric with harmonic Weyl tensor
[KuhnelRademacherCTOPRM, Corollary 6.8]. The Bach tensor is trace and
divergence free [GriffinGAOSOHM].

On a compact manifold the Bach tensor is $-1/4$ times the gradient of the
Riemannian functional

$g\longmapsto\int_{M}|W|_{g}^{2}\,dv(g)$

[BesseEM, 4.76] (note that the definition given here is only agrees with the
definition in [BesseEM] up to a factor of $-4$).

The Bach flow has been studied on homogeneous $4$-manifolds whose universal
cover has a product structure (that is, $(\widetilde{M}^{4},g)=(N_{1}\times
N_{2},g_{1}\oplus g_{2})$ where $(N_{i},g_{i})$ are simply connected
homogeneous manifolds of lower dimension which admit compact quotients) by
Helliwell in [HelliwellBFOHP]. When $\dim N_{2}=3$ we must have that $N_{2}$
is a unimodular Lie group and that $N_{1}=\mathbb{R}$. In this case, Helliwell
uses Milnor frames to diagonalise the Ricci and Bach tensor and then studies
the resulting ODE’s. In particular, this includes the study of the Bach flow
on $(\mathbb{R}^{4},\bar{g})$ and $(\mathbb{R}\times H^{3},\bar{g}\oplus g)$
where $\bar{g}$ denotes the Euclidean metric on $\mathbb{R}^{n}$ and $g$ is a
left invariant metric on the three-dimensional Heisenberg group $H^{3}$. For
$\dim N_{1}=\dim N_{2}=2$ the only possibilities are $M(c_{1})^{2}\times
M(c_{2})$ where $M(c)^{2}$ denotes the simply connected space form with
constant curvature $c\in\mathbb{R}$. In this case the flow remains a product
of two space forms and the flow is equivalent to a system of ODE’s for the
curvature of the factors.

##  3\. Four-Dimensional Nilpotent Lie algebras

Any bilinear skew-symmetric map
$\mu:\mathbb{R}^{4}\times\mathbb{R}^{4}\to\mathbb{R}^{4}$ which satisfies the
_Jacobi identity_ defines a Lie algebra structure structure on
$\mathbb{R}^{4}$. There is a natural ‘change of basis’ action of
$\operatorname{GL}_{4}(\mathbb{R})$ on
$\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$, the vector space of
bilinear skew-symmetric maps, given by

(3) $h\cdot\mu=h\mu(h^{-1}\cdot,h^{-1}\cdot),\qquad\forall
h\in\operatorname{GL}_{4}(\mathbb{R}).$

Two Lie brackets $\mu_{1}$ and $\mu_{2}$ define isomorphic Lie algebras if and
only if there is a $h\in\operatorname{GL}_{4}(\mathbb{R})$ such that
$h\cdot\mu_{1}=\mu_{2}$.

Moreover, any inner product $\left\langle\cdot,\cdot\right\rangle$ on
$\mathbb{R}^{4}$ induces an inner product on
$\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$ [LauretTRFFSCN]:

(4)
$\left\langle\mu,\lambda\right\rangle=\sum_{i,j=1}^{4}\left\langle\mu(e_{i},e_{j}),\lambda(e_{i},e_{j})\right\rangle,\,\forall\mu,\lambda\in\Lambda^{2}(\mathbb{R}^{n})^{*}\otimes\mathbb{R}^{n}$

where $\\{e_{i}\\}_{i=1}^{4}$ is any orthonormal basis.

If the map $(\operatorname{ad}_{\mu}x)$, defined by
$(\operatorname{ad}_{\mu}x)y:=\mu(x,y)$ for all $y\in\mathbb{R}^{4}$, is a
nilpotent endomorphism for all $x\in\mathbb{R}^{4}$, we say that $\mu$ is
_nilpotent_. A bracket $\mu$ which satisfies the Jacobi identity and is
nilpotent defines a nilpotent Lie algebra [VaradarajanLGLAATR, Theorem 3.5.4].
In this way, the set

(5)
$\mathcal{N}_{4}=\\{\mu\in\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}:\mu\text{
is nilpotent and satisfies the Jacobi Identity.}\\}$

parametrises structure constants of $4$-dimensional, nilpotent Lie algebras.

For any element $\mu\in\mathcal{N}_{4}$, $(\mathbb{R}^{4},\mu)$ is isomorphic
to one of the three following Lie algebras [deGraafCOSLA, Section 5.5]:

  1. (1)

$\mathbb{R}^{4}$: with the abelien Lie bracket $\mu(x,y)=0$ for all
$x,y\in\mathbb{R}^{4}$.

  2. (2)

$\mathbb{R}\oplus\mathfrak{h}_{3}$: This is the product of the 3 dimensional
Heisenberg algebra with $\mathbb{R}$. The non-trivial bracket relations are
$\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{3}$.

  3. (3)

$\mathfrak{n}_{4}$: This is not a product Lie algebra. The non-trivial bracket
relations are

$\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{3},\quad\mu(e_{1},e_{3})=-\mu(e_{3},e_{1})=e_{4}.$

Note that 1. and 2. are decomposable Lie algebras.

##  4\. Four-Dimensional Nilpotent Lie Groups

In this paper, we are concerned with simply connected Lie groups, $N^{4}$,
such that $\operatorname{Lie}(N)\simeq\mathfrak{n}_{4}$. Any such simply
connected Lie group can be identified with $\mathbb{R}^{4}$ under the
operation

$x\cdot_{\mu}y:=x+y+\frac{1}{2}\mu(x,y)+\frac{1}{12}\mu(\mu(x,y),y)-\frac{1}{12}\mu(\mu(x,y),x)\qquad\forall
x,y\in\mathbb{R}^{n}.$

See [VaradarajanLGLAATR, Section 3].

Let $\left\langle\cdot,\cdot\right\rangle$ be the standard inner product on
$\mathbb{R}^{n}$. Then any other inner product on $\mathbb{R}^{n}$ can be
written as

$(\cdot,\cdot)=\left\langle h\cdot,h\cdot\right\rangle$

for some $h\in\operatorname{GL}_{n}(\mathbb{R})$. If $(\cdot,\cdot)$ is an
inner product on $\mathbb{R}^{n}$, we denote by $g_{\mu,(\cdot,\cdot)}$ the
left-invariant metric on $(\mathbb{R}^{n},\cdot_{\mu})$ which agrees with
$(\cdot,\cdot)$ on $T_{0}\mathbb{R}^{n}$ [LauretTRFFSCN]. When $(\cdot,\cdot)$
is the standard inner product on $\mathbb{R}^{n}$, we will write $g_{\mu}$
instead of $g_{\mu,(\cdot,\cdot)}$. We then have the following proposition.

######  Proposition 1 ([LauretTRFFSCN], Theorem 4.1).

For $\mu,\lambda\in\mathcal{N}_{n}$ and an inner product
$(\cdot,\cdot)=\left\langle h\cdot,h\cdot\right\rangle$ on $\mathbb{R}^{n}$,
the metrics $g_{\mu}$ and $g_{\lambda,(\cdot,\cdot)}$ are isometric if and
only if $\lambda=h\cdot\mu$. In particular, $g_{\mu}$ and $g_{\lambda}$ are
isometric if and only if $\lambda\in O(n)\cdot\mu$.

Note that by the previous proposition, up to isometry we may assume that a
left invariant metric agrees with the standard inner product on
$T_{0}\mathbb{R}^{n}$.

Suppose now that $\mu\in\mathcal{N}_{4}$ is a Lie bracket such that
$(\mathbb{R}^{4},\mu)\simeq\mathfrak{n}_{4}$ and
$\left\langle\cdot,\cdot\right\rangle$ is the standard inner product on
$\mathbb{R}^{4}$. Proposition 2 below gives a convenient basis of
$\mathbb{R}^{4}$ for us to work in.

######  Proposition 2 ([VanThuongMO4DULG],Theorem 4.2).

Let $(\mathbb{R}^{4},g_{\tilde{\mu},(\cdot,\cdot)})$ be a four dimensional,
simply connected Nilpotent Lie group equipped with a left-invariant metric
$g_{\tilde{\mu},(\cdot,\cdot)}$ such that
$(\mathbb{R}^{4},\tilde{\mu})\simeq\mathfrak{n}_{4}$. Then
$(\mathbb{R}^{4},g_{\tilde{\mu},(\cdot,\cdot)})$ is isometric to
$(\mathbb{R}^{4},g_{\mu})$ where the bracket $\mu$ has the structure constants

(6) $\mu(e_{1},e_{2})=ae_{3}+be_{4},\qquad\mu(e_{1},e_{3})=ce_{4},\qquad
a,b,c\in\mathbb{R},a,c>0,$

and $\\{e_{i}\\}$ is the standard basis of $\mathbb{R}^{4}$.

If $\mu\in\mathcal{N}_{4}$ has the structure coefficients (6) with respect to
the standard basis of $\mathbb{R}^{4}$ then we will write $\mu=\mu_{a,b,c}$.

Let us define

(7) $\mathcal{O}=\\{\mu\in\mathcal{N}_{4}:\mu=\mu_{a,b,c}\text{ with respect
to the standard basis of }\mathbb{R}^{4}\\}.$

It follows from the above discussion that if $(\mathbb{R}^{n},g_{\mu})$ is a
simply connected Nilpotent Lie group with left-invariant metric $g_{\mu}$,
then up to isometry we may assume that $\mu\in\mathcal{O}$. Clearly we can
identify $\mathcal{O}$ with the set

$\\{(a,b,c)\in\mathbb{R}^{3}:0<a,c\\}.$

We will use this in §5 in order to reduce our study of the bracket flow, to be
introduced later on. Of course, there is no guarantee that a solution to the
bracket flow (9) will remain in $\mathcal{O}$. However, we will show in §5
that we can gauge our flow so that the solution does remain within
$\mathcal{O}$. In particular, our problem reduces to the study of an ODE in an
open subset of $\mathbb{R}^{3}$.

If $\mu=\mu_{a,b,c}\in\mathcal{O}$, then the norm of $\mu$ induced from the
inner product in (4) reduces to

$\lVert\mu\rVert^{2}=a^{2}+b^{2}+c^{2}.$

It will be useful to have an explicit description of an arbitrary derivation
$D\in\operatorname{Der}(\mu)$ of a bracket $\mu\in\mathcal{O}$. We can obtain
this by differentiating the description of an automorphism of
$\mathfrak{n}_{4}$ given in [VanThuongMO4DULG].

######  Lemma 1.

Let $\mu=\mu_{a,b,c}\in\mathcal{O}$. Then, any derivation
$D\in\operatorname{Der}(\mu)$ has the form

$D=\begin{pmatrix}\alpha&0&0&0\\\ *&\beta&0&0\\\
*&a\gamma+b\beta&\alpha+\beta&0\\\
*&*&c\gamma&2\alpha+\beta\end{pmatrix},\qquad\alpha,\beta,\gamma\in\mathbb{R},$

where each $*$ is an arbitrary real number.

##  5\. The Bach Flow on a Nilpotent Lie group

Suppose now that $(M^{4},g)=(\mathbb{R}^{4},g_{\mu})$ is a Nilpotent Lie group
with left-inavriant metric and that we have a
$(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution $(g(t))_{t\in(a,b)}$ to the
Bach flow with $g(0)=g_{\mu}$ (a solution $(g(t))_{t\in(a,b)}$ is
$(\mathbb{R}^{4},\cdot_{\mu})$-invariant if it is for all $t\in(a,b)$). Since
the scaler curvature is constant, we find that $\operatorname{B}(g(t))(0)$
satisfies the ODE

(8)
$\frac{d}{dt}\left\langle\cdot,\cdot\right\rangle_{t}=\operatorname{B}(\left\langle\cdot,\cdot\right\rangle_{t}),\qquad\left\langle\cdot,\cdot\right\rangle_{0}=g_{\mu}(0).$

Here, we have written
$\operatorname{B}(\left\langle\cdot,\cdot\right\rangle_{t})=\operatorname{B}(g(t))(0)$.
Conversely, given a solution $(\cdot,\cdot)_{t}$ to the ODE (8) we obtain a
$(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution $g(t)$ to the Bach flow by
defining $g(t)=g_{\mu,(\cdot,\cdot)_{t}}$ for all $t$. By the usual existence
and uniqueness of ODE’s, we are guaranteed a unique
$(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution with a maximal interval of
existence [LauretGFATSOHS]. The need for this reasoning is that uniqueness of
the Bach flow is an open problem on a general manifold.

Let us fix once and for all initial metric $g_{0}=g_{\mu_{0}}$ which is
invariant under the Nilpotent Lie group $(\mathbb{R}^{4},\cdot_{\mu_{0}})$.
Note that we have we have tacitly assumed that the initial inner product on
$T_{0}\mathbb{R}^{n}$ is the standard inner product, but this is not an issue
since up to isometry this is always the case (see §3).

By the above discussion, there is a unique curve of left-invariant metrics of
inner products $\left\langle\cdot,\cdot\right\rangle_{t}$ on $\mathbb{R}^{4}$
satisfying (8) which corresponds to the unique
$(\mathbb{R}^{4},\cdot_{\mu_{0}})$-invariant solution of the Bach flow
beginning at $g_{\mu_{0}}$. It follows that for each $t$ there is a
$h(t)\in\operatorname{GL}_{4}(\mathbb{R})$ such that

$\left\langle\cdot,\cdot\right\rangle_{t}=\left\langle
h(t)\cdot,h(t)\cdot\right\rangle.$

One can show that the one parameter family of matrices $h(t)$ can be chosen to
be a smooth curve (see [LauretGFATSOHS, Section 4.1] or Proposition 3 below).
The corresponding curve of brackets is given by $\mu(t)=h(t)\cdot\mu_{0}$. The
bracket flow, introduced by Lauret to study Ricci flow on homogeneous
manifolds in [LauretRFOHM], is motivated by considering what equation the
curve $\mu(t)\in\operatorname{GL}_{4}(\mathbb{R})\cdot\mu_{0}$ satisfies. For
more examples of the bracket flow technique see [LauretTRFFSCN,
BohmLafuenteIHRF, StanfieldPHCF].

######  Definition 1.

Let $(\mathbb{R}^{4},g_{\mu_{0}})$ as above. The bracket Bach flow is the ODE

(9) $\frac{d}{dt}\mu=\frac{1}{2}\pi(B_{\mu})\mu,\qquad\mu(0)=\mu_{0},$

where $B_{\mu}$ is defined by
$\operatorname{B}(g_{\mu(t)})(0)=g(0)(B_{\mu}\cdot,\cdot)$ is the Bach tensor
determined by $\mu$ and $\pi$ is the representation defined by

$\pi(A)\mu:=A\mu-\mu(A\cdot,\cdot)-\mu(\cdot,A\cdot),\qquad\forall
A\in\mathfrak{gl}_{n}.$

Note that $\pi$ is the derivative of the $\operatorname{GL}_{n}(\mathbb{R})$
representation on $\Lambda^{2}(\mathbb{R}^{n})^{*}\otimes\mathbb{R}^{n}$
defined in (3). Moreover, the solution $\mu(t)$ of (9) remains in the orbit
$\operatorname{GL}_{4}(\mathbb{R})\cdot\mu_{0}$ since
$\frac{1}{2}\pi(B_{\mu})\in
T_{\mu}(\operatorname{GL}_{4}(\mathbb{R})\cdot\mu)$ (see [LauretRFOHM, Lemma
3.2]).

Beginning at the inital metric $g_{\mu_{0}}$ we now have two families of
Riemannian manifolds

$(\mathbb{R}^{4},g(t))\qquad(\mathbb{R}^{4},g_{\mu(t)})$

where $g(t)$ is the unique $(\mathbb{R}^{n},\cdot_{\mu_{0}})$-invariant
solution of the Bach flow and $\mu(t)$ is the solution of the bracket flow
(9). The next proposition shows that these are equivalent in a precise way.

######  Proposition 3 ([LauretTRFFSCN], Theorem 5.1).

Let $(\mathbb{R}^{4},g(t))$, $(\mathbb{R}^{4},g_{\mu(t)})$ be solutions of the
homonegeous Bach flow and the bracket flow respectively. Then, there exists a
family of isomorphisms
$\varphi(t):(\mathbb{R}^{4},\cdot_{\mu_{0}})\to(\mathbb{R}^{4},\cdot_{\mu(t)})$
such that

$g(t)=\varphi(t)^{*}g_{\mu(t)}\quad\forall t.$

Moreover, $h(t)=d\varphi(t)$ satisfies

  1. (1)

$\left\langle\cdot,\cdot\right\rangle_{t}=\left\langle
h\cdot,h\cdot\right\rangle$

  2. (2)

$\mu(t)=h\mu_{0}(h^{-1}\cdot,h^{-1}\cdot)$.

For a proof of Proposition 3, one should consult [LauretTRFFSCN, Theorem 5.1]
(note that the proof in [LauretTRFFSCN] is for the Ricci flow however only
symmetry of the Ricci tensor is used). In particular, Proposition 3 shows that
the solutions of (1) and (9) have the same maximal interval of existence and
the same curvature (see the Remark after Theorem 3.3 in [LauretRFOHM]).

Recall from §3 that if $\lambda\in\operatorname{O}(4)\cdot\mu$ then the
metrics $g_{\mu}$ and $g_{\lambda}$ are isometric. This is due to the fact
that if $h\in\operatorname{GL}_{4}(\mathbb{R})$ gives rise to the inner
product $(\cdot,\cdot)=\left\langle h\cdot,h\cdot\right\rangle$ and
$k\in\operatorname{O}(4)$ then $hk\in\operatorname{GL}_{4}(\mathbb{R})$ gives
rise to the same inner product. It will be useful to exploit this
$\operatorname{O}(4)$ equivariance when studying the bracket flow (9). Böhm
and Lafuente describe this as a refinement of Uhlenbeck’s trick of moving
frames (see Sections 2 and 3 in [BohmLafuenteIHRF]).

######  Proposition 4 ([BohmLafuenteIHRF], Proposition 3.1).

Let $R:\operatorname{GL}_{4}(\mathbb{R})\cdot\mu_{0}\to\mathfrak{so}(4)$ be a
smooth map and let $\mu(t),\bar{\mu}(t)$ denote respectively solutions to the
bracket flow (9) and to the modified bracket flow

(10)
$\frac{d\bar{\mu}}{dt}=\frac{1}{2}\pi(B_{\mu}-R_{\mu})\bar{\mu},\qquad\bar{\mu}(0)=\mu_{0}.$

Then, there is a smooth curve $(k(t))\subset\operatorname{O}(4)$ such that
$\bar{\mu}=k\cdot\mu$.

In particular, the solutions $\mu,\bar{\mu}$ to the bracket flow (9) and the
gauged bracket flow (10) have the same maximal interval of existence and the
same curvature.

We now want to study the bracket flow for $(\mathbb{R}^{4},g_{\mu})$ when
$(\mathbb{R}^{4},\mu)\simeq\mathfrak{n}_{4}$.

######  Proposition 5.

Let $\mu=\mu_{a,b,c}\in\mathcal{O}$. Then, the endomorphism
$B_{\mu}:\mathbb{R}^{4}\to\mathbb{R}^{4}$ defined by
$\operatorname{B}(g_{\mu})(0)=g(B_{\mu}\cdot,\cdot)$ is given by the following
matrix

(11) $B_{\mu}=\begin{pmatrix}b_{1}&0&0&0\\\ 0&b_{2}&b_{5}&0\\\
0&b_{5}&b_{3}&b_{6}\\\ 0&0&b_{6}&b_{4}\end{pmatrix},$

where

$\displaystyle b_{1}$
$\displaystyle=\frac{1}{8}\left(4a^{4}+8a^{2}b^{2}-a^{2}c^{2}+4b^{4}+8b^{2}c^{2}+4c^{4}\right),$
$\displaystyle b_{2}$
$\displaystyle=\frac{1}{24}\left(12a^{4}+24a^{2}b^{2}-a^{2}c^{2}+12b^{4}+8b^{2}c^{2}-4c^{4}\right),$
$\displaystyle b_{3}$
$\displaystyle=\frac{-1}{24}\left(20a^{4}-a^{2}c^{2}+24a^{2}b^{2}+4b^{4}-8b^{2}c^{2}-12c^{4}\right),$
$\displaystyle b_{4}$
$\displaystyle=\frac{1}{24}\left(-4a^{4}+3a^{2}c^{2}-8a^{2}b^{2}-20\left(b^{2}+c^{2}\right)^{2}\right),$
$\displaystyle b_{5}$
$\displaystyle=\frac{2}{3}bc\left(a^{2}+b^{2}+c^{2}\right),$ $\displaystyle
b_{6}$ $\displaystyle=-\frac{2}{3}ab\left(a^{2}+b^{2}+c^{2}\right),$

######  Remark.

From the equations in Proposition 5 we can see the following:

  1. (1)

We can see explicitly the rescaling formula for the Bach tensor
$B_{c\cdot\mu}=c^{4}B_{\mu}$.

  2. (2)

The expressions for $b_{5}$ and $b_{6}$ show that $B_{\mu}$ is diagonal if and
only if $b=0$.

  3. (3)

It is interesting to note that

$b_{1}=\lvert W_{\mu}\rvert^{2}\geq 0.$

It is not clear to us why this is the case.

We have noted in §3 that the solution $\mu(t)$ to the bracket flow (9)
beginning at $\mu_{0}\in\mathcal{O}$ may not remain in $\mathcal{O}$. However,
we have also seen in §3 that for any $\mu\in\mathcal{N}_{4}$, the orbit
$\operatorname{O}(4)\cdot\mu$ intersects $\mathcal{O}$. That is to say, for
each $t$ we can find a $k(t)\in\operatorname{O}(4)$ such that
$k(t)\cdot\mu(t)\in\mathcal{O}$. Since $\mu$ and $k\cdot\mu$ determine
isometric Riemannian manifolds for $k\in\operatorname{O}(4)$, we may use
$\operatorname{O}(4)$ to gauge our flow, readjusting at each point in time to
ensure that the solution remains in $\mathcal{O}$. We formalise this in
Proposition 6 below by appealing to Proposition 10 in §5.

######  Proposition 6.

Let $\mu_{0}=\mu_{a_{0},b_{0},c_{0}}\in\mathcal{O}$ and let $B_{\mu(t)}$ be
the Bach endomorphism along the solution $\mu(t)$ of (9). Then, the solution
$\bar{\mu}(t)$ of the gauged bracket flow (10) remains in $\mathcal{O}$ where
the gauging, $R_{\mu}$, is given by

$R_{\mu}=\begin{pmatrix}0&0&0&0\\\ 0&0&b_{5}&0\\\ 0&-b_{5}&0&b_{6}\\\
0&0&-b_{6}&0\end{pmatrix}.$

###### Proof.

To show that the symmetries are preserved, it suffices to show that
$\dot{\mu}_{ij}^{k}=0$ whenever $i<j$ and
$(i,j,k)\notin\\{(1,2,3),(1,2,4),(1,3,4)\\}$ since then $\mu_{ij}^{k}$ will
solve the system $\dot{u}=0,u(0)=0$ and hence be $u\equiv 0$ by uniqueness.
Here a dot denotes a derivative with respect to time (i.e $\dot{}:=d/dt)$.

The effect of gauging is that $B_{\mu}-R_{\mu}=L_{\mu}$ is lower triangular
for all $t$. With respect to the basis, $\\{e_{i}\\}$, (10) is

(12)
$\dot{\mu}_{ij}^{k}=\frac{d}{dt}\left\langle\mu(e_{i},e_{j}),e_{k}\right\rangle=\left\langle
L_{\mu}\mu(e_{i},e_{j})-\mu(L_{\mu}e_{i},e_{j})-\mu(e_{i},L_{\mu}e_{j}),e_{k}\right\rangle\\\
=\sum_{l=1}^{4}\big{(}\mu_{ij}^{l}L_{l}^{k}-L_{i}^{l}\mu_{lj}^{k}-L_{j}^{l}\mu_{il}^{k}\big{)}=\sum_{i<j\leq
l\leq
k}\big{(}\mu_{ij}^{l}L_{l}^{k}-L_{i}^{l}\mu_{lj}^{k}-L_{j}^{l}\mu_{il}^{k}\big{)}.$

(Note that we are only summing over $l$). The last equality follows since
$B_{l}^{k}=0$ for $l>k$ since it is lower triangular and
$\mu_{il}^{k}=-\mu_{li}^{k}=0$ for $l>k$ by our choice of structure constants.
For $k=1,2$ the right hand side is zero since each term will have a factor of
$\mu_{ij}^{1}$ or $\mu_{ij}^{2}$, all of which are equal to $0$. If $k=3$,
then the only triples $(i,j,k)$ we need to check are $(i,3,3)$ for $i=1,2$.
But

$\dot{\mu}_{i3}^{3}=\sum_{i<3\leq l\leq
3}\big{(}\mu_{ij}^{l}L_{l}^{k}-L_{i}^{l}\mu_{lj}^{k}-L_{j}^{l}\mu_{il}^{k}\big{)}=\mu_{i3}^{3}L_{3}^{3}-L_{i}^{3}\mu_{33}^{3}-L_{3}^{3}\mu_{i3}^{3}=0.$

If $k=4$ then we must check $(2,j,4)$ for $j=3,4$. This is

$\dot{\mu}_{2j}^{4}=\sum_{2<j\leq l\leq
4}\big{(}\mu_{ij}^{l}L_{l}^{k}-L_{i}^{l}\mu_{lj}^{k}-L_{j}^{l}\mu_{il}^{k}\big{)}=\mu_{2j}^{l}L_{l}^{4}-L_{2}^{l}\mu_{lj}^{4}-L_{j}^{l}\mu_{2l}^{4}=0$

since $\mu_{2j}^{k}=0$ for $j>2$ and $\mu_{jl}^{4}=0$ for $j,l>2$. ∎

Therefore, the bracket flow (9) is equivalent to the following ODE for
$\mu=\mu_{a,b,c}\in\mathcal{O}$:

$\displaystyle a^{\prime}$
$\displaystyle=\frac{-a}{48}(44a^{4}+72a^{2}b^{2}-5a^{2}c^{2}+28b^{4}+24b^{2}c^{4}-4c^{4})$
(13) $\displaystyle b^{\prime}$
$\displaystyle=\frac{-b}{48}(60a^{4}+104a^{2}b^{2}+57a^{2}c^{2}+44b^{4}+104b^{2}c^{2}+60c^{4})$
$\displaystyle c^{\prime}$
$\displaystyle=\frac{-c}{48}(-4a^{4}+24a^{2}b^{2}-5a^{2}c^{2}+28b^{4}+72b^{2}c^{2}+44c^{4})$

With these in hand we are in a position to study the long time behaviour of
the flow.

######  Lemma 2.

If $\mu_{0}\in\mathcal{O}$, the following evolutions hold along the gauged
bracket flow:

(14)
$\displaystyle\frac{d}{dt}\log\frac{a}{c}=(c^{2}-a^{2})\lVert\mu\rVert^{2},$
(15)
$\displaystyle\frac{d}{dt}\frac{b^{2}}{a^{2}}\leq\frac{-2}{3}\frac{b^{2}}{a^{2}}\lVert\mu\rVert^{4},$
(16)
$\displaystyle\frac{d}{dt}\lVert\mu\rVert^{2}\leq\frac{-1}{12}\lVert\mu\rVert^{6}.$

###### Proof.

  1. (1)

Follows from a direct calculation using (5)

$\frac{d}{dt}\log\frac{a}{c}=\frac{d}{dt}\log a-\frac{d}{dt}\log
c=\frac{a^{\prime}}{a}-\frac{c^{\prime}}{c}=(c^{2}-a^{2})\lVert\mu\rVert^{2}.$

  2. (2)

Taking the derivative gives

$\frac{d}{dt}\frac{b^{2}}{a^{2}}=2\frac{b}{a}\frac{ab^{\prime}-a^{\prime}b}{b^{2}}=2\frac{b^{2}}{a^{2}}\bigg{(}\frac{b^{\prime}}{b}-\frac{a^{\prime}}{a}\bigg{)}.$

The claim then follows since

$\bigg{(}\frac{a^{\prime}}{a}-\frac{b^{\prime}}{b}\bigg{)}=\frac{-1}{24}\big{(}8\lVert\mu\rVert^{4}+3c^{2}(5a^{2}+b^{2}+c^{2})\big{)}\leq\frac{-1}{3}\lVert\mu\rVert^{4}.$

  3. (3)

Similarly to 2,

$48\frac{d}{dt}\lVert\mu\rVert^{2}=-44\lVert\mu\rVert^{6}+3a^{2}c^{2}(47a^{2}+53b^{2}+47c^{2}).$

Now, we observe that by the multinomial theorem

$\lVert\mu\rVert^{6}=(a^{2}+b^{2}+c^{2})^{3}\geq
a^{6}+c^{6}+3a^{4}c^{2}+3a^{2}c^{4}+6a^{2}b^{2}c^{2}.$

Hence,

$\displaystyle-40\lVert\mu\rVert^{6}+3a^{2}c^{2}(47a^{2}+53b^{2}+47c^{2})$
$\displaystyle\leq-40a^{6}+21a^{4}c^{2}+21a^{2}c^{4}-40c^{6}$
$\displaystyle=(a^{2}+c^{2})(-40a^{4}+61a^{2}c^{2}-40c^{4})$
$\displaystyle\leq(a^{2}+c^{2})(-40a^{4}+80a^{2}c^{2}-40c^{4})=-40(a^{2}+c^{2})(a^{2}-c^{2})^{2}\leq
0.$

Therefore,

$\frac{d}{dt}\lVert\mu\rVert^{2}\leq\frac{-1}{48}\cdot
4\lVert\mu\rVert^{6}=\frac{-1}{12}\lVert\mu\rVert^{6}.$

∎

Lemma 2 gives the following important corollaries.

######  Corollary 1.

The Bach flow on a four dimensional simply connected Nilpotent Lie group is
immortal, that is, the maximal interval of existence contains $(0,\infty)$.

###### Proof.

Since $\lVert\mu\rVert^{2}$ decreases along the flow, it remains within the
closed ball of radius $\lVert\mu_{0}\rVert$ which is a compact set. ∎

######  Corollary 2.

Let $\mu(t)$ be a solution to (9) in $\mathcal{N}_{4}$. Then,

$\lVert\mu(t)\rVert^{2}\leq\frac{\sqrt{6}}{\sqrt{t}}.$

###### Proof.

This follows since $\lVert\mu(t)\rVert^{2}$ is a sub-solution of

$x^{\prime}=\frac{-1}{12}x^{3},\qquad x(0)=\lVert\mu_{0}\rVert^{2}$

the solution of which is easily seen to be
$x(t)=\sqrt{6}(t+A)^{\frac{-1}{2}}\leq t^{-1/2}\sqrt{6}$. ∎

Corollary 2 shows that the bracket flow beginning at an arbitrary bracket
$\mu_{0}\in\mathcal{O}$ converges to the Euclidean bracket as $t\to\infty$.

##  6\. Self Similar Solutions to the Bach Flow

In this section we study _solitons_ of the Bach flow. A Bach soliton is a
Riemannian manifold $(M^{4},g)$ such that

(17) $\operatorname{B}(g)+\frac{\Delta s}{12}=\lambda g+\mathcal{L}_{X}g,$

for a constant $\lambda\in\mathbb{R}$, and a complete vector field,
$X\in\mathfrak{X}(M)$ ($\mathcal{L}_{X}$ is the Lie derivative in the
direction $X$). The soliton is called expanding, steady or shrinking if
$\lambda>0$, $\lambda=0$ or $\lambda<0$ respectively. If the vector field
arises as the gradient of a potential function $u\in C^{\infty}(M)$, then (17)
becomes

(18) $\operatorname{B}(g)+\frac{\Delta s}{12}=\lambda
g+2\operatorname{Hess}u.$

In this case, we say the soliton is a gradient soliton. For homogeneous
solitons, the Laplacian term in (17) vanishes since the scalar curvature is
constant. Homogeneous gradient Bach solitons were studied by Griffin in
[GriffinGAOSOHM]. Griffin applies Theorem 3.6 from [PetersenWylieROHGSMARE] to
reduce the study to Riemannian products of manifolds of dimension less than
$4$. Proposition 7 below is a simple corollary of [PetersenWylieROHGSMARE,
Theorem 3.6] which we will use to deduce that any non-product solitons we find
cannot be gradient.

######  Proposition 7 ([PetersenWylieROHGSMARE], [GriffinGAOSOHM]).

Let $(M^{4},g)$ be a homogeneous Riemannian manifold. If there is a non-
constant function $u\in C^{\infty}(M)$ such that

$\operatorname{B}(g)=\lambda
g+2\operatorname{Hess}u,\qquad\lambda\in\mathbb{R},$

then $M$ splits isometrically as a product
$(M,g)=(M^{\prime}\times\mathbb{R}^{k},g^{\prime}\oplus\overline{g})$ where
$\overline{g}$ is the Euclidean metric and $u$ is a function on the Euclidean
factor.

Bach solitons correspond to solutions of the Bach flow which evolve by scaling
and pullback by diffeomorphisms [LauretGFATSOHS, Theorem 4.10].

If $(M^{4},g)=(\mathbb{R}^{n},g_{\mu})$ is a simply connected Nilpotent Lie
group, then an analogous condition to the metric evolving self similarly under
the Bach flow is that the solution $\mu(t)$ to the bracket flow evolves only
by scaling (see the discussion before Theorem 6 in [LauretGFATSOHS]). If this
is the case, the bracket $\mu=\mu(0)$ satisfies

(19) $B_{\mu}=\lambda
I+D,\qquad\lambda\in\mathbb{R},D\in\operatorname{Der}(\mu).$

A simply connected Nilpotent Lie group $(\mathbb{R}^{n},g_{\mu})$ whose Bach
endomorphism satisfies (19) is called an _algebraic Bach soliton_. Algebraic
solitons were introduced by Lauret to study Ricci Nilsolitons (Ricci solitons
on Nilpotent Lie groups) in [LauretRSHN]. In [LauretGFATSOHS] Lauret
generalises the the bracket flow technique and algebraic solitons to a large
class of geometric structures. An important observation made by Lauret is that
an algebraic soliton is indeed a soliton in the sense of (17). For
convenience, we have summarised this for the Bach tensor in Proposition 8
below.

######  Proposition 8 (Theorem 6, [LauretGFATSOHS]).

For a simply connected Nilpotent Lie group $(\mathbb{R}^{n},g_{\mu_{0}})$ the
following are equivalent:

  1. (1)

The solution to the bracket flow (9) beginning at $\mu_{0}$ is given by

$\mu(t)=\lambda(t)\cdot\mu_{0},\quad\text{for some
}\lambda(t)>0,\,\lambda(0)=1.$

  2. (2)

The operator $B_{\mu}$ associated to the Bach tensor satisfies (19).

Moreover, whenever either of these conditions hold, the Riemannian manifold
$(\mathbb{R}^{n},g_{\mu_{0}})$ is a Bach soliton.

######  Remark.

If $\lambda<0$ in (19) then the corresponding bracket flow from Proposition 8
will shrink homothetically towards $0$. Since
$g_{\lambda\cdot\mu}=\lambda^{-2}g_{\mu}$ this corresponds to an expanding
Bach soliton.

Note that in general, not all solitons are algebraic solitons. Since we have
an explicit description of what a derivation $D\in\operatorname{Der}(\mu)$
looks like for a bracket $\mu\in\mathcal{O}$, the notion of an algebraic
soliton reduces our search for a soliton to simply solving a system of
equations in terms of $a,b,c,$ and the components of $D$ (note that since
$B_{\mu}$ is trace free, $\lambda=-\operatorname{tr}D/4$). In fact, we can
reduce the difficulty of this system further.

######  Lemma 3.

If $\mu=\mu_{a,b,c}\in\mathcal{O}$ is an algebraic soliton, then $B_{\mu}$ is
diagonal.

###### Proof.

Since $D=B_{\mu}-\lambda I$, the derivation $D\in\operatorname{Der}(\mu)$ must
be symmetric as the difference of two symmetric matrices. But by Lemma 1, $D$
is lower triangular. Hence, $D$ must be diagonal and so $B_{\mu}=\lambda I+D$
must also be diagonal. ∎

Since $B_{\mu}$ is diagonal if and only if $b=0$, this allows us to set $b=0$
when searching for algebraic solitons. By Lemma 1, a diagonal derivation has
eigenvalues $\alpha,\beta,\alpha+\beta,2\alpha+\beta$ for some
$\alpha,\beta\in\mathbb{R}$. Therefore, existence and uniqueness of algebraic
solitons reduces to existence and uniqueness of solutions to a set of
polynomials in $a,b,\alpha,\beta$.

######  Theorem 2.

The bracket $\mu\in\mathcal{O}\subset\mathcal{N}_{4}$ given by

$\mu(e_{1},e_{2})=e_{3}\quad\mu(e_{1},e_{3})=e_{4}$

is a Bach soliton. Moreover, this soliton is a non-gradient expanding soliton
and is the unique algebraic Bach soliton up to isometry and scaling within the
orbit $\operatorname{GL}_{4}(\mathbb{R})\cdot\mu\subset\mathcal{N}_{4}$.

###### Proof.

By setting $a=c=1$ and $b=0$ in Proposition 5 it is not difficult to check
that $\alpha=-7/12$ and $\beta=-7/6$ solves (19). This gives
$\lambda=-21/16<0$ so the soliton is expanding. The soliton is not of gradient
type due to Proposition 7.

To see uniqueness we assume that (19) holds for
$a,c,\alpha,\beta\in\mathbb{R}$ such that $\mu_{a,0,c}\in\mathcal{O}$ and
$a\neq c$. We then show that this leads to a contradiction. ∎

######  Remark.

Up to isometry any left-invariant metric on $\mathbb{R}\times H^{3}$ can be
described by a bracket $\mu=\mu_{a,b,c}$ with $b=c=0$ and $a>0$. By
Proposition 8 these must be solitons since they trivially evolve by scaling.
Moreover, by [GriffinGAOSOHM, Proposition 4.14] these are also non-gradient.

##  7\. Normalised Bach Flow and Convergence

Similarly to the Ricci flow case [LauretTRFFSCN, Section 7], one may consider
a normalised Bach flow:

(20) $\frac{\partial}{\partial t}g(t)=\operatorname{B}(g(t))+r(t)g(t),\quad
g(0)=g_{0},$

for some normalisation function $r:[0,T)\to\mathbb{R}$.

In the varying brackets perspective the analogue of (20) is the following
$r-$normalised bracket flow equation.

######  Definition 2.

An $r$-normalised bracket Bach flow for a normalisation function
$r:[0,T)\to\mathbb{R}$ is a curve $(\mu^{r}(t))\subset\mathcal{N}_{4}$ such
that

(21)
$\frac{d}{dt}\mu^{r}=\frac{1}{2}\pi(B_{\mu^{r}}-rI){\mu^{r}},\quad\mu(0)=\mu_{0}.$

######  Remark.

There is an equivalence between (20) and (21) which is analogous to the case
of un-normalised flows given in Proposition 3 [LauretTRFFSCN, Section 7].

The usefulness of (20) and (21) is that the addition of the $r$ term allows us
to keep a geometric quantity fixed along the flow. Moreover, the next
proposition shows that solutions of the normalised flows only differ from the
solutions of the original flows by a scaling and reparametrisation of time.

######  Proposition 9.

Let $\mu(t)$ and $\mu^{r}(t)$ be solutions of the bracket flow (9) and the
$r-$normalised bracket flow (21) respectively. Then, there are functions
$\tau:[0,T)\to[0,T)$, $\lambda:[0,T)\to\mathbb{R}$ such that

(22) $\mu^{r}(t)=\lambda(t)\mu(\tau(t))\qquad\forall t\in[0,T).$

The fucntions $\tau$ and $\lambda$ are the solutions of the ODE’s

(23)
$\tau^{\prime}=\lambda^{4},\quad\tau(0)=0\qquad\qquad\lambda^{\prime}=\frac{1}{2}r\lambda,\quad\lambda(0)=1.$

###### Proof.

Let $\mu(t)$ be a solution to (9) and define

$\mu^{r}(t)=\lambda(t)\mu(\tau(t))$

where $\tau,\lambda$ are the solutions of (23). Clearly $\mu^{r}(0)=\mu_{0}$.
Differentiating gives

$\frac{d}{dt}\mu^{r}(t)=\frac{d}{dt}(\lambda(t)\mu(\tau(t)))=\lambda^{\prime}\mu(\tau(t))+\lambda\tau^{\prime}\frac{d}{dt}\bigg{|}_{\tau(t)}\mu=\frac{1}{2}r\lambda\mu(\tau(t))+\frac{1}{2}\lambda^{5}\pi(B_{\mu})\mu\\\
=\frac{1}{2}r\mu^{r}+\frac{1}{2}\pi(B_{\mu^{r}})\mu^{r}=\frac{1}{2}\pi(B_{\mu^{r}}-rI)\mu^{r}$

where we have used that the Bach tensor scales by
$B_{\lambda\cdot\mu}=\lambda^{4}B_{\mu}.$ ∎

######  Corollary 3.

Let $u:\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}\to\mathbb{R}$ be
scale invariant (i.e. $u(\lambda\mu)=u(\mu)$ for any
$\lambda\in\mathbb{R}^{*}$) and smooth away from $0$. Then $u$ increases
(resp. decreases) along a solution of the bracket flow if and only if it $u$
increases (resp. decreases) along a solution if the normalised bracket flow.

###### Proof.

Let $\mu,\mu^{r}$ be solutions to the bracket flow and $r$-normalised bracket
flow respectively. If $\tilde{u},u^{r}$ denote the restrictions of $u$ to
$\mu,\mu^{r}$ then Proposition 9 and scale invariance of $u$ imply that
$u^{r}(t)=\tilde{u}(\tau(t))$. Since $\tau$ is increasing, the claim follows.
∎

In particular, if $\lVert\mu_{0}\rVert=2$ and

$r=-\frac{1}{4}\left\langle\pi(B_{\mu})\mu,\mu\right\rangle,$

then $\lVert\mu(t)\rVert\equiv 2$ since

$\frac{d}{dt}\lVert\mu\rVert^{2}=2\left\langle\frac{1}{2}\pi(B_{\mu}-rI)\mu,\mu\right\rangle=\left\langle\pi(B_{\mu})\mu,\mu\right\rangle-\frac{\left\langle\pi(B_{\mu})\mu,\mu\right\rangle\lVert\mu\rVert^{2}}{\lVert\mu\rVert^{2}}=0.$

The scalar curvature is also constant since $s_{\mu}=-\lVert\mu\rVert^{2}/4$
on Nilpotent Lie groups (c.f. 7.39 in [BesseEM]).

Since the quantities $a/c$ and $b^{2}/a^{2}$ are scale invariant, we can use
Lemma 2 and Corollary 3 to determine the behaviour of the normalised flow.

######  Theorem 3.

Let $\mu_{0}\in\mathcal{O}$ with $\lVert\mu_{0}\rVert=2$. If $\mu(t)$ is the
solution to the $r$-normalised Bach flow (21) for
$r=-\frac{1}{4}\left\langle\pi(B_{\mu})\mu,\mu\right\rangle$ then
$g_{\mu(t)}\to g_{\mu_{\infty}}$ uniformly on compact subsets of
$\mathbb{R}^{4}$ as $t\to\infty$ where $\mu_{\infty}$ is (up to scaling) the
algebraic Bach soliton given in Theorem 2.

###### Proof.

Let $\mu_{0}=\mu_{a_{0},b_{0},c_{0}}\in\mathcal{O}$ with
$a_{0}^{2}+b_{0}^{2}+c_{0}^{2}=4$. We show that $\mu(t)\to\mu_{\infty}$,
[LauretTRFFSCN, Proposition 2.1] then implies $g_{\mu(t)}\to g_{\mu_{\infty}}$
uniformly on compact subsets of $\mathbb{R}^{4}$ as $t\to\infty$. Showing
$\mu(t)\to\mu_{\infty}$ amounts to showing that $a,c\to 2$ and $b\to 0$ as
$t\to\infty$.

We first show that $b\to 0$ as $t\to\infty$. By Lemma 2 and Corollary 3, we
know that

$\frac{d}{dt}\frac{b^{2}}{a^{2}}\leq\frac{-2}{3}\frac{b^{2}}{a^{2}}\tau^{\prime}<0.$

Therefore, $b^{2}/a^{2}$ is monotone decreasing and is bounded below by $0$ so
it must converge. But this implies the derivative of $b^{2}/a^{2}$ must
apporach $0$, which implies $b^{2}/a^{2}$ converges to $0$ since
$\tau^{\prime}=\lambda^{4}\geq 1$ for $t$ sufficiently large. Since $a\leq 2$
this implies $b\to 0$ as $t\to\infty$.

Next, we have that

$\frac{d}{dt}\log\frac{a}{c}=\tau^{\prime}(c^{4}-a^{4})\lVert\tilde{\mu}(\tau(t))\rVert=2\lambda^{3}(c^{4}-a^{4}).$

Here $\tilde{\mu}$ denotes the un-normalised flow and we have used that
$2=\lVert\mu\rVert=\lambda\lVert\tilde{\mu}\rVert$. Observe that if
$a(t_{0})=c(t_{0})$ for some $t_{0}\geq 0$, then $a\equiv c$ for all $t>t_{0}$
by uniqueness of ODE solutions. Therefore, if $a(0)>c(0)$ (resp. $a(0)<c(0)$)
then we can assume $a(t)>c(t)$ for all $t$ (resp. $a(t)<c(t)$). In this case,
$\log\frac{a}{c}$ is a monotone and bounded, and hence must be convergent. But
then $d/dt\log\frac{a}{c}\to 0$ as $t\to\infty$ so $c^{4}-a^{4}\to 0$. Since
$a^{2}+c^{2}\to 1$ as $t\to\infty$, it must hold that

$\lim_{t\to\infty}a=\lim_{t\to\infty}c=2.$

∎

Note that Theorem 3 implies that the Bach flow $g(t)=\varphi(t)^{*}g_{\mu(t)}$
converges in the pointed Cheeger-Gromov sense to
$g_{\infty}:=g_{\mu_{\infty}}$ which is a non-trivial, non-gradient algebraic
Bach soliton by Theorem 2.

######  Acknowledgements.

I am grateful to Dr Ramiro Lafuente for his support and encouragement
throughout this research.

## References


#  RUNHETC-2024-13, UT-WI-04-2024
Holographic Inflation, Primordial Black Holes and Early Structure Formation
Submitted to 2024 Gravitation Research Foundation Essay Contest May 13, 2024

Tom Banks
NHETC and Department of Physics
Rutgers University, Piscataway, NJ 08854-8019
E-mail<EMAIL_ADDRESS>

Willy Fischler (corresponding author)
Department of Physics and Texas Cosmology Center
Weinberg Institute, Center for Theory
University of Texas, Austin, TX 78712
E-mail<EMAIL_ADDRESS>
###### Abstract

Evidence has accumulated that there are supermassive black holes (SMBHs) in
the centers of most galaxies, and that these were formed in the very early
universe by some as yet unknown process. In particular, there is evidence[15]
that at least some galaxies formed as early as $10^{8}-10^{9}$ years after the
Big Bang host SMBHs. We suggest that the holographic model of inflation, whose
dark matter candidates are primordial black holes carrying a discrete gauge
charge, which originated as a small subset of the inflationary horizon volumes
in the very early universe, can provide the seeds for this early structure
formation. Aspects of the model pointed out long ago suggested an early era of
structure formation, with structures dominated by dark matter. The additional
assumption that the dark matter consists of discretely charged black holes
implies black hole dominance of early structures, which seems to be implied by
JWST data.

##  1 A Speculation About the Seeds For Early Galaxy Formation

The holographic model of inflationary cosmology[1] began as an attempt to
construct a finite quantum model of the earliest moments of the universe,
matching on the flat FRW universe with equation of state $p=\rho$ in the
hydrodynamic limit. Later, we realized that it could incorporate a smooth
termination in a model of de Sitter (dS) space with a finite dimensional
Hilbert space. The additional insight that localized objects in dS space are
constrained subspaces of the dS Hilbert space, with the property that
subsystems in those subspaces remain independent for long periods of
time111This may seem somewhat obscure when stated abstractly, but the explicit
matrix models of[2] show exactly how it is possible., leads to a formulation
of inflationary cosmology that resolves the Trans-Planckian problem of
conventional inflation.

We can explain this, without too much detail, by pointing out that our
model[3] of the flat FRW universe with

$a(t)=\sinh^{1/3}(3t/R_{I}),$ (1.1)

ended up with a system whose density matrix was

$\rho=\frac{e^{-L_{0}}}{{\rm Tr}\ e^{-L_{0}}},$ (1.2)

where $L_{0}$ is the Virasoro generator of a (cut-off) $1+1$ dimensional CFT,
with central charge proportional to the area of the inflationary horizon. This
is exactly the same as the Carlip[4]-Solodukhin[5] ansatz for the density
matrix of a black hole of the same area. In the cosmological context this is
the density matrix of a causal diamond along any geodesic in the FRW universe,
for times $>R_{I}$. We remind the reader that

  * •

In Holographic Space Time Models, the background space-time is viewed,
following Jacobson[7] as a hydrodynamic approximation to the underlying
quantum theory. Homogeneity and isotropy are built into the model by choosing
the same time evolution operator $U(t)$ along each geodesic in the background,
and by having the variables transform under the rotation group that preserves
a geodesic, and the time dependent Hamiltonian be invariant.

  * •

Flatness follows from the dynamics of this particular model at all times, as a
consequence of insisting that the covariant entropy bound is saturated.

In order to build a realistic model of cosmology, containing localized
objects, we allow the horizon to expand after some number of inflationary
e-folds, but constrain the initial conditions such that individual horizon
volumes do not interact for a long time. The mechanism for this involves two
separate features. The first is that the interaction between localized
subsystems is mediated by exciting degrees of freedom that are set to zero by
the initial conditions. A model for this is a matrix model with single trace
interactions in which localized degrees of freedom are blocks in a matrix that
has been set to be block diagonal by the initial conditions. The second
feature is the Milne redshift. The interactions which turn the frozen degrees
of freedom on and off are all located on the stretched apparent horizon and
have a natural time scale which is the horizon radius. If the horizon expands
more rapidly than the time it takes to equilibrate the frozen variables, then
the localized objects will survive. Since their internal dynamics is identical
to that of black holes, we argued that the slow roll era ends with a dilute
gas of black holes. In the explicit quantum model the Milne redshift is
implemented by writing the time dependent Hamiltonian at any time as a sum of
terms corresponding to contributions from different time-like trajectories in
the conformally rescaled Casini-Huerta-Myers[8] coordinates defined below,
with red shift factors read from that metric. Excitations closer to the
geodesic are defined as those that evolve more rapidly in the quantum
Hamiltonian.

Given a flat FRW metric, Einstein’s equations define a pressure and energy
density $p,\rho$, and if $p+\rho\geq 0$ we can invent an ”inflaton” field such
that the metric is sourced by the inflaton. This field has no relation to any
field in low energy effective field theory around a Minkowski vacuum of
string/M-theory, so none of the constraints of[9] apply to it. The only
constraint on the slow roll metric is that the expansion rate during slow roll
is faster than the scrambling rate which would equilibrate the localized black
holes with the rest of the degrees of freedom on the apparent horizon. This is

$\epsilon\equiv\frac{\dot{H}}{H^{2}}>C{\rm ln}[(M_{P}/H)]^{-1}.$ (1.3)

$C$ is a constant characteristic of a particular fast scrambling system and is
not easy to determine from the formula for the Hamiltonian. Combined with the
observational constraint from the CMB that

$\epsilon^{-1}(H/M_{P})\sim 10^{-5},$ (1.4)

this leads to the estimate

$\epsilon\sim 0.1\ if\ C\ is\ o(1).$ (1.5)

As we’ll see, this is consistent with data, because our prediction for the
tensor to scalar ratio differs from that in conventional inflation models.

There is no semi-classical picture of the slow roll era in terms of particles
or a quantum field theory. It is defined by a quantum mechanical model in
which causality is imposed by making the Hamiltonian time dependent so that
degrees of freedom close to a geodesic only couple to those far from it after
a sufficient proper time has elapsed. Geometrically, one is working in
coordinates[8] where, at a coarse grained level,

$ds^{2}=a^{2}(t)\frac{T}{2}(\cosh t+\cosh
x)^{-2}(-dt^{2}+dx^{2}+\sinh^{2}xd\Omega^{2}).$ (1.6)

Here $T$ is the proper time since the beginning of the universe and $a(t)$ the
scale factor. We call this coarse grained because after the horizon begins to
expand, and we have distinct non-interacting quantum systems, each of which
behaves like an inflationary horizon volume, we cannot place these systems on
geodesics that are a Planck length apart for all time, as we did during the
earliest moments of the universe. When the slow roll era ends we have, on FRW
time slices, a collection of black holes that must be separated by more than
their Schwarzschild radii, in order to be consistent with treating them as
approximately non-interacting. So ”homogeneity” exists only on this larger
length scale, and even here it is violated by the statistical fluctuations in
the black hole entropy/mass. We can calculate the size of those fluctuations
from the Carlip-Solodukhin ansatz, and the tensor to scalar ratio $r$ is
determined[10] by the entropy formula for Kerr black holes. One finds that $r$
scales like $\epsilon^{2}$ rather than $\epsilon$. The reason for this is
that, according to the C-S ansatz $\frac{\delta H}{H}$ is independent of
$\epsilon$, whereas in field theoretic inflation it scales like
$\sqrt{\epsilon}$. The fluctuation calculation is done by assuming that the
classical Einstein equations, linearized around some slow roll metric, give a
coarse grained hydrodynamic description of our quantum model, with stochastic
initial conditions set by our assumption that we have a dilute gas of black
holes, with relation between the fluctuations in $\frac{\delta M}{M}$ and
$\delta{L}{L}$ given by the Kerr formula. The mass distribution of the black
hole fluctuation is smoothed to a horizon sized Gaussian and the spacing
between black holes is left as an input parameter in the fit, as is, of
course, the slow roll metric.

At a given proper time $T$ since the beginning of the universe, the variables
of our quantum model are two dimensional quantum fermion fields (with a UV
cutoff), labelled by $T\times T+1$ matrix indices. The initial conditions are
such that the square $T\times T$ matrices bilinear in these are block
diagonal, with a number $K$ of $R_{I}\times R_{I}$ blocks and one block of
size $T-KR_{I}$. The Hamiltonian is a single trace in matrix bilinears.
According to[11] the bilinear term is just a sum of free Dirac Hamiltonians,
while the quartic term is a certain abelian Thirring interaction. One term in
the Hamiltonian involves the full $T\times T$ matrix and evolves the system
with a characteristic time scale $T$. In addition to this there are terms of
the same form involving only a single $R_{I}\times R_{I}$ block matrix. Each
of those is associated with a particular $x$ value in the CHM coordinates of
the diamond at time $T$. The one associated with the geodesic at $x=0$ evolves
in Planck units, while each other value of $x$ is assigned the appropriate red
shift factor. We will not go into details of the quantum model of the slow
roll era here. It is unlikely that they will lead to testable predictions, and
we mention the model only to demonstrate that an explicit system incorporating
the principles we’ve outlined, exists.

The slow roll era is followed by an era dominated by a marginal dilute gas of
black holes. We use the word marginal because any state with localized
excitations is constrained, and is a priori a less probable initial condition
than one which leads to the maximal entropy universe with
$a(t)=\sinh^{1/3}(3t/R_{final}),$ for some choice of the final dS radius.
Thus, the most probable state with localized excitations is one where the post
inflationary gas of black holes is just dilute enough to escape this fate.
This means that the black hole number density on a surface of fixed FRW time
is

$n_{0}=\frac{C_{0}}{R_{IBH}^{3}},$ (1.7)

where $C_{0}$ is something like $10^{-2}-10^{-3}$ and $R_{IBH}$ is the size of
the apparent horizon at the end of inflation. From this point on, we can
follow our cosmology semi-classically. The dilute black hole gas is a matter
dominated universe and density fluctuations grow. The initial magnitude of the
fluctuations is determined[10] by our identification of the early inflation
phase with a system whose modular Hamiltonian is the cutoff $L_{0}$ generator
of a CFT. They are of order222Note the different power of the slow roll
parameter, compared to field theoretic inflation models.
$\epsilon^{-1}(R_{IBH}/L_{P})^{-1}$ . Fluctuations go non-linear when

$t\sim(\epsilon R_{IBH}/L_{P})^{3/2}L_{P},$ (1.8)

, which is much shorter than the black hole decay time

$t_{decay}\sim g^{-1}(R_{IBH}/L_{P})^{3}L_{P}.$ (1.9)

$g$ is the number of particle degrees of freedom with mass below the Hawking
temperature of the decaying black holes. Primordial structures thus begin to
develop in the dilute black hole gas (DBHG) before black hole decay sets off
the Hot Big Bang (HBB). Crude estimates[1] put the reheat temperature at
$10^{10}$ GeV.

We have also postulated[12] that a fraction $f$ of the inflationary horizon
volumes carry a discrete $Z_{N}$ gauge charge, which implies that their decay
will leave over stable remnants of at least Planck mass. We now note two
further complications. It is likely that each of these remnants is part of a
bound cluster, possibly containing other remnants, and certainly containing
other decaying black holes. Furthermore, if there are superpartners in the
world, half of the decay products of black holes will actually be massive
particles, with masses above $10^{3}$ GeV. Many of them will also have gauge
interactions that are long range on the scale of the bound clusters. Thus, the
fate of the bound clusters as the black holes decay is a complicated dynamical
problem. Rather than try to solve it we will simply assume that the result is
a distribution of primordial black holes, with masses ranging from the Planck
mass to black holes of the size of the horizon at the end of the black hole
dominated era. We estimate this to be $\sim 10^{14}-10^{15}L_{P}$. The smaller
black holes quickly decay down to Planck scale remnants, leaving no trace but
those near the horizon size have Hawking temperatures well below the reheat
temperature and are meta-stable for relatively long periods.

Given this conjecture, the radiation dominated era begins with a sprinkling of
near horizon size black holes, along with a collection of stable Planck mass
black hole remnants. Novikov and Zeldovich[13] observed long ago that the
evolution equation for black holes in an FRW universe with a single component
equation of state had a scaling solution in which the black hole size tracks
the size of the apparent horizon. Black holes much smaller than the horizon
grow slowly, but in the homogeneous approximation seem to evolve to the
scaling solution. Carr and Hawking[14] argued that inhomogeneous corrections
would prevent the evolution of black holes much smaller than the horizon size
to the scaling branch of the equation, and that no black hole solution that is
asymptotically FRW can exactly follow the scaling solution. Of course, once
Hawking radiation is taken into account, these holes will also have a
probability to decay. Although there is still an exact scaling solution it is
unstable, and generic initial conditions lead to eventual black hole decay,
even in the homogeneous approximation. However, for our estimate of the
horizon size at the beginning of the radiation dominated era, the decay does
not begin until the temperature has fallen below at least $10$ GeV. During
this time, lots of interesting things can happen.

The oldest black holes that have been observed[15] by the James Webb Space
Telescope (JWST) have a mass of about $10^{44}M_{P}$ and their Schwarzschild
radius was the horizon size when the age of the universe was about a second.
So in order to explain black holes of this magnitude we need only hypothesize
that some near horizon size black holes, which formed during the black hole
dominated era of our cosmology, followed the NZ scaling solution for this
fairly short period during the radiation dominated era. We have no other
probes of cosmology at these early times, so this does not seem implausible. A
crucial question, to which we have not yet found an answer, is what determines
$10^{6}$ solar masses as the scale where Carr-Hawking instabilities halt the
growth of black holes following the NZ scaling solution. The determination of
the scale at which the NZ scaling solution is halted is the crucial test of
our model. If a natural explanation of the observed scale of galactic center
black holes comes out of this mechanism, we would find it hard to believe that
it was incorrect. A more detailed numerical investigation of the Carr-Hawking
equations, within the parameters of our model might lead to an understanding
of this length scale.

There is also some evidence that the surprising early galaxies seen by JWST
have unusual elongated ”banana” shapes[16]. We would like to propose a crude
model for such elongated shapes. Our model of the radiation dominated era
consists of a gas of ”radiation” with an initial temperature between
$10^{9}-10^{10}$ GeV. Half of the radiative species are really massive
superpartners of standard model particles with masses in the range
$10^{3}-10^{5}$ GeV. In addition there is a population of stable Planck mass
primordial black holes, and another population of large black holes whose mass
grows from somewhere around $10^{14}M_{P}$ to around $10^{44}M_{P}$ in the
first second of the radiation dominated era. Together the two populations of
black holes make up the dark matter. Most of the dark matter consists of
stable Planck mass black hole remnants.

By construction, each of the black holes originated as an inflationary horizon
volume moving on a geodesic of the coarse grained FRW model that describes the
black hole dominated era of cosmology before fluctuations go non-linear. It is
plausible that after the brief eras of very early structure formation and
black hole decay, the resulting swarm of remnant black holes are not moving
rapidly in the rest frame of the gas of radiation. If we now consider a pair
of large black holes, the Newtonian escape velocity of the pair is

$v_{esc}\sim\frac{M_{1}M_{2}n_{L}^{1/3}}{M_{P}^{3}},$ (1.10)

where $n_{L}$ is the large black hole number density. Whenever this is larger
than the Hubble velocity $H/M_{P}$, we expect a black hole bound state. If it
has a reasonable angular momentum it will be meta-stable because gravitational
radiation is a slow process.

We want to emphasize that this non-relativistic description of bound state
formation is not consistent with causality, but could be a consequence of
causal processes. In HST models[17] Newton’s law arises causally from
processes in second order perturbation theory, in which frozen q-bits linking
two localized objects in the same causal diamond are excited and de-excited.
This leads to a phase in the transition amplitude for the two objects to enter
and exit the diamond at different points on its holographic screen, which can
be interpreted semi-classically as arising from a non-relativistic
instantaneous potential on FRW time slices. We are using this classical non-
relativistic language as shorthand for something that is quite complex to
describe in a proper causal and quantum mechanical manner.

The large black hole bound state is of course immersed in a dense gas
consisting of radiation and small black holes, and these will be attracted to
the black holes. But gas on the axis between the two will be pulled in
opposite directions, so we should expect a shape with a preferred axis,
determined by the initial angular momentum in the large black holes. The
rotation of the black holes will produce a Coriolis force on the matter
between them, which will tend to swirl it into a disk shape, but since the
motion of the heavy black holes takes place on a much longer time scale, there
will be a period where the structures have a large dipole moment. At these
early times, it’s obvious that the mass of these structures is mostly
dominated by the black holes at the ends. There are indications from the JWST
data that very early galaxies have a higher ratio of black hole to normal
mass, and this might be the beginning of the explanation for that fact. Of
course, the black holes at the end of the bananas will be absorbing matter and
growing all the time, and there are surely frictional processes which could
slow the rotation and cause the structure to collapse and the black holes to
merge.

It is also probable that many of the large black holes do not pair up into
binaries, so these may lead to a qualitatively different type of galaxy.
Perhaps there is some reason that galaxies based on single large black holes
take longer to begin star formation, so that they are too dim to be seen in
the current JWST observations. Star formation takes place mostly in regions of
high dark matter density. Near single black holes this will occur mostly near
the black hole itself and the dark matter may be absorbed before stars can
form. On the other hand, in the elongated systems, dense clusters of dark
matter in the center could light up and will remain stable against the
competing pull of opposing black holes. It’s important to estimate the time
scale for collapse of elongated systems into more spherical ones in this
binary model of early galaxies.

The observations of very early galaxies go back to a time hundreds of millions
of years after the Hot Big Bang, whereas our theoretical considerations are at
most valid over the first few seconds. There is a lot more detailed modeling
to be done before one can say that our models have successfully explained the
surprising features of the youngest galaxies in the universe. They do however
provide a solid basis for expecting to find a very early universe populated by
a collection of large black holes, as well as Planck mass primordial black
hole dark matter.

To summarize: the HST model of inflationary cosmology leads one to expect an
early era of structure formation that ends with a Hot Big Bang set off by the
decay of most of the Inflationary Black Holes. We’ve also postulated that a
certain fraction of the black holes leave over Planck mass remnants, whose
stability is guaranteed by a discrete $Z_{N}$ gauge charge. The temperature of
the ”radiation” at the beginning of the radiation dominated era is
approximately $10^{10}$ GeV. Our ignorance about particle physics above the
scale of $1-2$ TeV leaves open the possibility that much of the ”radiation”
actually consists of massive particles, and certainly much of it experiences
long range forces333Long range always means of order the apparent horizon.
stronger than gravitation. So it’s entirely plausible that some of the stable
black holes have evolved to horizon sized structures at the beginning of the
radiation dominated era. The horizon size at this time is only about
$10^{14}L_{P}$.

Such horizon sized black holes will follow the NZ scaling solution until Carr-
Hawking inhomogeneities stop their growth. Since the Schwarzschild radius of
even the largest super massive black hole known is miniscule compared to the
size of the apparent horizon at any era to which we have direct observational
access, we conjecture that the origin of all SMBHs is in fact in this very
early era. We do not as yet understand why the scale of $10^{6}$ solar masses
is the scale around which black hole growth following the NZ solution stops.
Bound binaries of such primordial SMBHs will attract the Planck mass black
hole dark matter, as well as the baryons and radiation in the universe, and
form elongated shapes in the early universe. It’s important to calculate how
long these elongated shapes last before collapsing into more spherical
profiles. One would not however expect that most of the early SMBHs formed
binaries, so one would expect another class of early galaxies, centered around
single SMBHs. The present authors do not understand enough about star
formation to speculate about why the elongated ones are more easily visible,
or whether failure to see a second population of early galaxies rules out a
model like this. One possible explanation relies on the fact that star
formation tends to occur in overdensities of dark matter. For a galaxy formed
around a single SMBH overdensities will occur preferably near the central
black hole and may be absorbed by it before they can form stars. For black
hole binaries, overdensities can occur on the axis between the two black holes
and survive long enough to form stars. We note that there are alternative
models of early prolate galaxies[18] based on LCDM simulations without SMBHs.
As far as we know there has not yet been any confirmation that the observed
elongated early galaxies harbor SMBHs, or that the early galaxies containing
SMBHs are elongated. Thus, our conjecture must survive many observational
tests before it can be confirmed.

The holographic model of inflation, combined with the assumption of a discrete
gauge symmetry that stabilizes a certain fraction of black hole remnants, is a
cosmological model that can account for most of the know facts about the very
early universe in terms of a very small number of parameters. It is based on
some general ideas about quantum gravity, is completely finite and does not
suffer from ”Transplanckian” paradoxes. It predicts that dark matter takes the
form of primordial black hole remnants, which are very hard to detect by
anything but their gravitational effects. In this paper we have shown that it
may also provide us with an intriguing explanation of the properties of the
very earliest galaxies, which are beginning to be revealed by the JWST.

Acknowledgments

The work of T.B. was supported by the Department of Energy under grant DE-
SC0010008. Rutgers Project 833012. The work of W. F. was supported by the
National Science Foundation under grant PHY-2210562.

## References

  * [1] T. Banks and W. Fischler,“Holographic Inflation Revised,”
doi:10.1017/9781316535783.013 [arXiv:1501.01686 [hep-th]].

  * [2] T. Banks, B. Fiol and A. Morisse,“Towards a quantum theory of de Sitter space,” JHEP 12, 004 (2006) doi:10.1088/1126-6708/2006/12/004 [arXiv:hep-th/0609062 [hep-th]].
  * [3] T. Banks, W. Fischler and L. Mannelli, “Microscopic quantum mechanics of the p = rho universe,” Phys. Rev. D 71, 123514 (2005) doi:10.1103/PhysRevD.71.123514 [arXiv:hep-th/0408076 [hep-th]].
  * [4] S. Carlip, “Black hole entropy from conformal field theory in any dimension,” Phys. Rev. Lett. 82, 2828-2831 (1999) doi:10.1103/PhysRevLett.82.2828 [arXiv:hep-th/9812013 [hep-th]].
  * [5] S. N. Solodukhin, “Conformal description of horizon’s states,” Phys. Lett. B 454, 213-222 (1999) doi:10.1016/S0370-2693(99)00398-6 [arXiv:hep-th/9812056 [hep-th]].
  * [6] T. Banks and K. M. Zurek, “Conformal description of near-horizon vacuum states,” Phys. Rev. D 104, no.12, 126026 (2021) doi:10.1103/PhysRevD.104.126026 [arXiv:2108.04806 [hep-th]].
  * [7] T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. 75, 1260-1263 (1995) doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [gr-qc]].
  * [8] H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 05, 036 (2011) doi:10.1007/JHEP05(2011)036 [arXiv:1102.0440 [hep-th]].
  * [9] A. Bedroya and C. Vafa,“Trans-Planckian Censorship and the Swampland,” JHEP 09, 123 (2020) doi:10.1007/JHEP09(2020)123 [arXiv:1909.11063 [hep-th]]; N. B. Agmon, A. Bedroya, M. J. Kang and C. Vafa, “Lectures on the string landscape and the Swampland,” [arXiv:2212.06187 [hep-th]], and references therein.
  * [10] S. A, T. Banks, ”Predictions for the Cosmic Microwave Background from the Holographic Space-time Model of Inflation”, manuscript in preparation.
  * [11] T. Banks, “Hilbert Bundles and Holographic Space-time Models,”[arXiv:2306.07038 [hep-th]].
  * [12] T. Banks and W. Fischler, “Discretely Charged Dark Matter in Inflation Models Based on Holographic Space-Time,” Universe 8, no.11, 600 (2022) doi:10.3390/universe8110600 [arXiv:2209.08361 [hep-th]].
  * [13] Y. B. Zel’dovich and I. D. Novikov, “The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model,” Sov. Astron. 10, 602 (1967)
  * [14] B. J. Carr and S. W. Hawking,“Black holes in the early Universe,” Mon. Not. Roy. Astron. Soc. 168, 399-415 (1974) doi:10.1093/mnras/168.2.399
  * [15] J. Silk, M. C. Begelman, C. Norman, A. Nusser and R. F. G. Wyse, “Which Came First: Supermassive Black Holes or Galaxies? Insights from JWST,” Astrophys. J. Lett. 961, no.2, L39 (2024) doi:10.3847/2041-8213/ad1bf0 [arXiv:2401.02482 [astro-ph.GA]]; A. Bogdan, A. D. Goulding, P. Natarajan, O. E. Kovács, G. R. Tremblay, U. Chadayammuri, M. Volonteri, R. P. Kraft, W. R. Forman and C. Jones, et al. “Evidence for heavy-seed origin of early supermassive black holes from a z $\approx$ 10 X-ray quasar,” Nature Astron. 8, no.1, 126-133 (2024) doi:10.1038/s41550-023-02111-9 [arXiv:2305.15458 [astro-ph.GA]]. See also: S. Y. Guo, M. Khlopov, X. Liu, L. Wu, Y. Wu and B. Zhu, [arXiv:2306.17022 [hep-ph]].
  * [16] Viraj Pandya and Haowen Zhang and Marc Huertas-Company and Kartheik G. Iyer and Elizabeth McGrath and Guillermo Barro and Steven L. Finkelstein and Martin Kuemmel and William G. Hartley and Henry C. Ferguson and Jeyhan S. Kartaltepe and Joel Primack and Avishai Dekel and Sandra M. Faber and David C. Koo and Greg L. Bryan and Rachel S. Somerville and Ricardo O. Amorin and Pablo Arrabal Haro and Micaela B. Bagley and Eric F. Bell and Emmanuel Bertin and Luca Costantin and Romeel Dave and Mark Dickinson and Robert Feldmann and Adriano Fontana and Raphael Gavazzi and Mauro Giavalisco and Andrea Grazian and Norman A. Grogin and Yuchen Guo and ChangHoon Hahn and Benne W. Holwerda and Lisa J. Kewley and Allison Kirkpatrick and Anton M. Koekemoer and Jennifer M. Lotz and Ray A. Lucas and Laura Pentericci and Pablo G. Perez-Gonzalez and Nor Pirzkal and Dale D. Kocevski and Casey Papovich and Swara Ravindranath and Caitlin Rose and Marc Schefer and Raymond C. Simons and Amber N. Straughn and Sandro Tacchella and Jonathan R. Trump and Alexander de la Vega and Stephen M. Wilkins and Stijn Wuyts and Guang Yang and L. Y. Aaron Yung, “Galaxies Going Bananas: Inferring the 3D Geometry of High-Redshift Galaxies with JWST-CEERS,” [arXiv: 2310.15232 [astro-ph.GA]]
  * [17] T. Banks and W. Fischler, “Holographic Space-time, Newton‘s Law, and the Dynamics of Horizons,” [arXiv:2003.03637 [hep-th]].
  * [18] D. Ceverino, J. Primack and A. Dekel, “Formation of elongated galaxies with low masses at high redshift,” Mon. Not. Roy. Astron. Soc. 453, no.1, 408-413 (2015) doi:10.1093/mnras/stv1603 [arXiv:1504.04988 [astro-ph.GA]]; M. Huertas-Company and J. R. Primack and A. Dekel and D. C. Koo and S. Lapiner and D. Ceverino and R. C. Simons and G. F. Snyder and M. Bernardi and Z. Chen and H. Domínguez-Sánchez and C. T. Lee and B. Margalef-Bentabol and D. Tuccillo “Deep Learning Identifies High-z Galaxies in a Central Blue Nugget Phase in a Characteristic Mass Range”, The Astrophysical Journal, 858, 114 (2018), doi10.3847/1538-4357/aabfed [arXiv: 1804.07307 [astro-ph.GA]]