# Stress and Geometry for Isotropic Singularities

A. Rod Gover  Department of Mathematics, The University of Auckland, Private
Bag 92019, Auckland 1142, New Zealand<EMAIL_ADDRESS>Jarosław
Kopiński  Center for Theoretical Physics, Polish Academy of Sciences,
Al. Lotników 32/46, 02-668 Warsaw, Poland<EMAIL_ADDRESS>Center for
Quantum Mathematics and Physics (QMAP), Department of Mathematics, University
of California, Davis, CA95616, USA<EMAIL_ADDRESS>Andrew Waldron
Center for Quantum Mathematics and Physics (QMAP), Department of Mathematics,
University of California, Davis, CA95616, USA

###### Abstract

We develop the mathematics needed to treat the interaction of geometry and
stress at any isotropic spacetime singularity. This allows us to handle the
Einstein equations at the initial singularity and characterize allowed general
relativistic stress-energy tensors. Their leading behaviors are dictated by an
initial hypersurface conformal embedding. We also show that an isotropic Big
Bang determines a canonical non-singular metric on and about the initial
hypersurface as well as a cosmological time. This assigns a volume and energy
to the initial point singularity.

##  I Introduction

Is the causal structure of our universe singular at the Big Bang? This
question is of physical import since, as we shall show, a well-defined causal
structure at the initial Big Bang singularity imposes strong constraints on
the matter content of the early universe.

We perform our analysis in the context of isotropic singularities, meaning
that when multiplied by some power of a timelike coordinate, the observed
physical metric becomes non-singular on an initial hypersurface along which
the conformal structure is well-defined. Such spacetimes are physically
relevant in light of Penrose’s Weyl Curvature hypothesis [1] which asserts
that, at any physically relevant initial singularity, the Weyl tensor is
finite even when the Ricci curvature is singular [2]. Isotropic singularities
subject to various underlying matter model assumptions have been analyzed in
detail [7, 8, 9, 10, 11, 12, 13, 14, 15]. Our analysis applies to generic,
compatible stress.

Because the metric along an isotropic singularity is degenerate but the
conformal structure is still well-defined, early universe physics is dictated
by the mathematics of conformally embedded hypersurfaces. Conformal
submanifold embeddings are crucial to the theory of observables in the AdS/CFT
correspondence [16, 17]. This machinery can be fruitfully applied to
cosmology.

##  II Conformal Geometry

For simplicity we focus on generic, dimension 4 [18], causal structures given
by the data of a Lorentzian conformal geometry $(M,{\bm{g}})$ [19], where
${\bm{g}}$ denotes a conformal class of metrics $g$ with equivalence given by
rescalings $\Omega^{2}g\sim g$ for $0<\Omega\in C^{\infty}M$. Parallelism
determined by the Levi–Civita connection $\nabla$ is a central mathematical
construct of general relativity. Its conformal geometry generalization, known
as the tractor connection $\bm{\nabla}$, promotes the tangent bundle $TM$ to a
“tractor bundle” ${\bm{T}}M$ with dimension six fibers [20, 21]. Indeed
${\bm{g}}$ contains a metric solving the vacuum Einstein equations precisely
when there is a parallel tractor vector field $I\in{\bm{T}}M$ [21, 22], viz

${\bm{\nabla}}I=0\,.$ (1)

Given a choice of metric $g\in{\bm{g}}$, a tractor $I$ is a triple

$I\stackrel{{\scriptstyle
g}}{{=}}\scalebox{0.9}{{\hbox{\begin{pmatrix}\sigma\\\ n\\\
\rho\end{pmatrix}\stackrel{{\scriptstyle\Omega^{2}
g}}{{=}}\begin{pmatrix}\Omega\sigma\\\ \Omega(n+ \sigma d\log\Omega)\\\
\Omega^{-1}(\rho-{\mathcal{L}}_{n}\log\Omega-\tfrac{\sigma}{2}
|d\log\Omega|_{g}^{2}) \end{pmatrix}}}}\,,$ (2)

where $\sigma$, $\rho$ are scalars, ${\mathcal{L}}_{n}$ is the Lie derivative
along the vector $n$, and the above gauge transformation is valued in a
parabolic subgroup of the spacetime conformal group $SO(4,2)$. Tractors are
basic objects for theories incorporating local conformal transformations and
diffeomorphisms [23]. The tractor connection is given by

${\bm{\nabla}}_{a}I=\scalebox{0.9}{{\hbox{
\begin{pmatrix}\nabla_{a}{\bm{\sigma}} - n_{a} \\\ \nabla_{a} n^{b} +
\bm{\sigma} P_{a}{}^{b}+\rho{\hskip 0.85358pt}\delta_{a}{}^{b}\\\ \nabla_{a}
\rho- n_{c} P_{a}{}^{c}\end{pmatrix}}}}\,.$ (3)

In the above $\bm{\sigma}$ denotes a conformal density of weight $1$. A weight
$w$ conformal density [24] is a section of
$\big{[}(\wedge^{4}TM)^{2}]^{\frac{w}{8}}=:{\mathcal{E}}M[w]$. This is a power
of a tensor density, so the Levi-Civita connection is well-defined acting upon
it. A density $\bm{\sigma}$ may also be understood as an equivalence class of
metric-function pairs $(g,\sigma)\sim(\Omega^{2}g,\Omega\sigma)$. Indices are
raised and lowered using ${\bm{g}}$ and $\nabla$ is the Levi-Civita connection
(see [23]). Also $P$ denotes the Schouten tensor and $J$ its trace.

The standard tractor bundle ${\bm{T}}M$ comes equipped with a parallel
“tractor metric” $h$ and a canonical tractor vector field $X\in{\bm{T}}M[1]$
[21, 25, 23]. Indeed $h(I,X)=\bm{\sigma}\in{\mathcal{E}}M[1]$. Moreover, when
Eq. (1) holds, it follows that the vacuum cosmological constant $\Lambda_{\rm
vac}$ obeys

$-\tfrac{1}{3}\Lambda_{\rm
vac}=2\bm{\sigma}\rho+|n|_{\bm{g}}^{2}=h(I,I)=:{\hskip 0.85358pt}I\\!{\hskip
0.85358pt}\cdot\\!{\hskip 0.85358pt}I{\hskip 0.85358pt}=:I^{2}\,.$

To incorporate matter, we must couple the stress–energy tensor to the right
hand side of Eq. (1). On the other hand, when the function $\sigma$ is a good
coordinate for a hypersurface $\Sigma$, the local conformal embedding data
$\Sigma\hookrightarrow(M,{\bm{g}})$ is encoded by ${\bm{\nabla}}I\in
TM\otimes{\bm{T}}M$. The latter gives a conformal geometric analog of
extrinsic curvatures. It follows that there is a natural correspondence
between stress and local conformal embedding data.

##  III Isotropic Singularities

An isotropic singularity is a spacelike hypersurface $\Sigma$ in a spacetime
$M$ with a degenerate physical metric $\check{g}$ such that, for $\alpha<0$
and any defining function $\tau$ [26],

$g=\tau^{2\alpha}\check{g}$ (4)

extends to a smooth metric across $\Sigma$. The degree of metric singularity
for the (zero $\Lambda$) conformally flat Friedmann–Lemaître–Robertson–Walker
spacetime with perfect fluid pressure to density ratio $\kappa$ is

$\alpha_{\rm FLRW}=-\tfrac{2}{3\kappa+1}\,.$

Even this simplest of cosmological scenarios allows non-integer $\alpha$. This
parameter controls both smoothness of the physical metric $\check{g}$ and the
volume expansion rate.

Another bona fide metric $g^{\prime}=\Omega^{2}g$ corresponds to a rescaled
defining function $\tau^{\prime}=\Omega^{\frac{1}{\alpha}}\tau$. Thus we may
write (4) as

$\check{g}={\bm{\tau}}^{-2\alpha}{\bm{g}}\,,$

with ${\bm{\tau}}\in{\mathcal{E}}M[\frac{1}{\alpha}]$, where no particular
choice of metric in the conformal class ${\bm{g}}$ has been made. Indeed the
causal structure of ${\bm{g}}$ is well-defined across the initial singularity.

Our aim is to analyze the Einstein field equations

$\check{G}+\Lambda\check{g}=\check{T}\,,$ (5)

where $\check{T}$ is the stress–energy tensor of a universe with cosmological
constant $\Lambda$ and degenerate physical metric $\check{g}$. For this we use
maps from weight 1 scalar densities to weight 0 tractors and from tractor-
valued one-forms to weight 1 symmetric trace-free tensors [27, 23]

${\bm{\mu}}\\!\stackrel{{\scriptstyle
I}}{{\longmapsto}}\\!\\!\scalebox{0.8}{{\hbox{\begin{pmatrix}\bm{\mu}\\\
\nabla^{b} \bm{\mu}\\\ \\!\> -\tfrac{1}{4}(\square\\!+ \\!J){
\bm{\mu}}\\!{\hskip
0.85358pt}\end{pmatrix}}}}\,,\qquad\>\scalebox{0.8}{{\hbox{
\begin{pmatrix}0\\\ \mathring{x}_{a}{}^{b} \\\ \\! -\tfrac{1}{3} \nabla_{b}
\mathring{x}_{a}{}^{b} \end{pmatrix}}}}\stackrel{{\scriptstyle
q^{*}}}{{\longmapsto}}\\!\mathring{x}_{ab}\,.$ (6)

When $\bm{\mu}$ is non-vanishing almost everywhere, $I_{\bm{\mu}}$ is termed a
scale tractor. For any weight one density $\bm{\mu}$, the definitions of
$\bm{\nabla}$ and $I_{\bm{\mu}}$ in Equations (3) and (6) imply vanishing of
the top slot of $\bm{\nabla}I_{\bm{\mu}}$. So by virtue of Eq. (2) its middle
slot is a trace-free conformally covariant rank two tensor equaling
$q^{*}\bm{\nabla}I_{\bm{\mu}}$. Indeed,
$2\bm{\mu}^{-1}q^{*}\bm{\nabla}I_{\bm{\mu}}$ is precisely the trace-free
Einstein tensor for the metric $\bm{\mu}^{-2}{\bm{g}}$, wherever this is
defined, and Equations (5) now read

$\displaystyle q^{*}{\bm{\nabla}}I_{\bm{\tau}^{\alpha}}$ $\displaystyle=$
$\displaystyle\,\,\;\tfrac{{\bm{\tau}}^{\alpha}}{2}{\hskip
0.85358pt}\mathring{\check{T}}\,,$ (7) $\displaystyle
I^{2}_{\bm{\tau}^{\alpha}}\,\,\,$ $\displaystyle=$
$\displaystyle\tfrac{1}{12}\check{T}_{a}{}^{a}-\tfrac{1}{3}\Lambda\,.$ (8)

Scale tractors are potentials for Einstein’s equations. The trace of the
stress tensor for a spacetime with isotropic singularity cannot vanish along
$\Sigma$ unless $\alpha\\!=\\!-1$, as the leading behavior of
$I^{2}_{\bm{\tau}^{\alpha}}$ is
$\frac{\alpha(\alpha+1)}{2}\bm{\tau}^{2\alpha-2}|\nabla\bm{\tau}|^{2}_{{\bm{g}}}$.

##  IV Traversing the singularity

Spacetimes whose singularities are isotropic admit a global causal structure.
Hence, even though the Einstein tensor is singular across an isotropic
singularity, there are a number of well-defined geometric quantities,
invariant to the structure, that constrain matter. The Weyl tensor
$W_{ab}{}^{c}{}_{d}$ is defined independently of any choice of $g\in{\bm{g}}$
but, unlike the Bach tensor $B_{ab}$, it is not related to stress by a local
differential operator. Let ${\sf P}$ be the conformally covariant partially
massless wave-operator  defined, acting on a weight $1$ trace-free symmetric
tensor $\mathring{x}_{ab}$, by

${\sf P}{\hskip 0.85358pt}\mathring{x}_{ab}:=\square{\hskip
0.85358pt}\mathring{x}_{ab}\\!-\\!\nabla_{c}\nabla_{(a}\mathring{x}^{c}{}_{b)_{\circ}}\\!-\\!\tfrac{1}{3}\nabla_{(a}\nabla_{|c|}\mathring{x}^{c}{}_{b)_{\circ}}\\!+\\!W_{a}{}^{c}{}_{b}{}^{d}\mathring{x}_{cd}\,.$

Partially massless excitations measure Bach flat fluctuations around an
Einstein metric [28, 29]. For any non-vanishing weight one density $\bm{\mu}$
[30],

$\bm{\mu}{\hskip 0.85358pt}B:={\sf P}q^{*}\bm{\nabla}I_{\bm{\mu}}\,.$ (9)

Given a causal structure ${\bm{g}}$, the Bach tensor is non-singular so the
above implies that the physical stress obeys a d’Alembert-type equation [31]

$B=\tfrac{1}{\bm{\tau}^{\alpha}}{\hskip 0.85358pt}{\sf P}{\hskip
0.85358pt}\big{(}\tfrac{\bm{\tau}^{\alpha}}{2}\mathring{\check{T}}\big{)}\,.$
(10)

###  IV.1 Singularity Geometry

So far the conformal embedding data $\Sigma\hookrightarrow(M,{\bm{g}})$ has
not been used. This data determines uniquely the local asymptotics of a metric
$g_{+}$ whose scalar curvature obeys

$R^{g_{+}}=12+{\mathcal{O}}(\sigma^{4})\,,$ (11)

where $\sigma$ is any defining function for $\Sigma$. The initial hypersurface
$\Sigma$ is a conformal infinity of this singular Yamabe metric $g_{+}$. An
all order “singular Yamabe problem” [32, 33, 34, 35] solution amounts to
finding $\bm{\sigma}=[g,\sigma]\in{\mathcal{E}}M[1]$ such that
$I_{\bm{\sigma}}^{2}=-1$. The expansion coefficient of the $\bm{\sigma}^{4}$
term in $I_{\bm{\sigma}}^{2}+1$, along $\Sigma$, is a weight $-4$ conformal
hypersurface invariant [35, 36, 37] equaling the variation of an energy
functional $E_{\Sigma}$ [38, 39, 40]. This energy is the anomaly in the
renormalized volume of $(M,g_{+})$ and an invariant of the initial singularity

$E_{\Sigma}=\int_{\Sigma}\mathring{K}_{ab}\mathring{F}^{ab}\,dV_{g_{\Sigma}}\,.$

The above integral is over any metric in the conformal class of metrics
${\bm{g}}_{\Sigma}$ induced along $\Sigma$ by ${\bm{g}}$. It is invariantly
defined because the contraction of the trace-free extrinsic curvature
$\mathring{K}$ with the Fialkow tensor [41, 42],

$F_{ab}:=\hat{n}^{c}\hat{n}^{d}W_{cabd}-\mathring{K}_{ac}\mathring{K}_{b}{}^{c}+\tfrac{1}{4}\mathring{K}_{cd}\mathring{K}^{cd}\overline{g}_{ab}\in\odot^{2}T^{*}\Sigma\,,$

defines a conformal density of weight $-3$. The extrinsic curvatures $(K,F)$
have transverse orders $(1,2)$ and are termed second and third fundamental
forms [43]. They give the first two elements in a sequence of trace-free
conformal hypersurface invariants defined along $\Sigma$ and termed conformal
fundamental forms [44]. These are conformally invariant obstructions to the
problem of finding an asymptotically de Sitter (dS) metric with conformal
infinity $\Sigma$. They probe derivatives of ${\bm{g}}$ off of $\Sigma$ in the
direction of the (future-pointing) timelike unit normal $\hat{n}\in
TM[-1]|_{\Sigma}$.

The extrinsic curvature $K$ measures the difference between the Levi-Civita
connections $\nabla$ and $\bar{\nabla}$ of $M$ and $\Sigma$ respectively,
while the Fialkow tensor measures that of the respective tractor connections
$\bm{\nabla}$ and $\bar{\bm{\nabla}}$ of ${\bm{g}}$ and ${\bm{g}}_{\Sigma}$
[42].

The fourth conformal fundamental form [44, 45] takes three normal derivatives
of ${\bm{g}}$ [46],

$\mathring{L}_{ab}:=\left(\hat{n}^{c}C_{c(ab)}\right)^{\top}+H\hat{n}^{c}\hat{n}^{d}W_{acbd}\\\
-\bar{\nabla}^{c}(\hat{n}^{d}W_{d(ab)c})^{\\!\top}\\!\\!\in\\!\circledcirc^{2}T^{*}\Sigma[-1]\,.$

Here $C_{abc}:=2\nabla_{[a}P_{b]c}$ is the Cotton tensor and $\circledcirc$
denotes the trace-free symmetric product of one forms. The tensor
$\mathring{L}$ is distinguished in the context of dS4 metrics as it extracts
the second piece of boundary data (the first being ${\bm{g}}_{\Sigma}$).

The locally determined singular Yamabe asymptotics of (11) terminate at order
four, so the fourth conformal fundamental form $\mathring{L}$ is the last
tensor determined this way. Before using the geometric triple
$(\mathring{K},\mathring{F},\mathring{L})$ to constrain
$\mathring{\check{T}}$ , we study further (pseudo-)Riemannian data determined
by the isotropic singularity.

###  IV.2 The Big Bang Metric

Remarkably there is a canonical Riemannian metric along the big bang
hypersurface $\Sigma$. It is constructed from the solution $\bm{\sigma}$ to
the singular Yamabe problem: The weight 1 density $\bm{\gamma}$ defined by

$\bm{\gamma}^{\frac{1}{\alpha}-1}:=\frac{\bm{\tau}}{\bm{\sigma}}$ (12)

is nowhere vanishing by our smoothness assumptions and therefore defines a
Lorentzian metric

$\displaystyle g_{\bm{\gamma}}:=\bm{\gamma}^{-2}{\bm{g}}\,,$

and in particular a Riemannian metric $g_{\Sigma}$ on the spacelike isotropic
singularity $\Sigma$ [47]. Thus the Riemannian three manifold
$(\Sigma,g_{\Sigma})$ is an invariant of the Big Bang, as is its volume
$V_{\Sigma}=\int_{\Sigma}dV_{g_{\Sigma}}$.

Because the “Big Bang metric” $g_{\bm{\gamma}}$ is an everywhere smooth
element of the conformal class ${\bm{g}}$, it determines the triple of
conformal fundamental forms $(\mathring{K},\mathring{F},\mathring{L})$ by the
formulæ above. Importantly the metric $g_{\bm{\gamma}}$ also furnishes the
early universe with a canonical cosmological time coordinate

$t=\frac{\bm{\sigma}}{\bm{\gamma}}\,.$

##  V Big Bang Stress–Energy Tensor

The behavior of stress at an isotropic singularity can be studied using
potentials $I_{\bm{\tau}^{\alpha}}$, $I_{\bm{\sigma}}$ and $I_{\bm{\gamma}}$.
By virtue of Eq. (12), these must be related:

$I_{\bm{\tau}^{\alpha}}=t^{\alpha-2}\Big{[}-\tfrac{\alpha(\alpha-1)}{4\bm{\gamma}}I_{\bm{\sigma}}^{2}X+t\big{(}\alpha
I_{\bm{\sigma}}+\tfrac{\alpha(\alpha-1)}{2\bm{\gamma}}I_{\bm{\sigma}}.I_{\bm{\gamma}}{\hskip
0.85358pt}X\big{)}+t^{2}\big{(}(1-\alpha)I_{\bm{\gamma}}-\tfrac{\alpha(\alpha-1)}{4\bm{\gamma}}I_{\bm{\gamma}}^{2}{\hskip
0.85358pt}X\big{)}\Big{]}\,.$ (13)

###  V.1 Trace of Stress–Energy Tensor

The square of a scale tractor measures scalar curvature/trace of stress (see
Eq. (8)) so Eq. (13) implies

$\begin{split}\tfrac{1}{4}\check{T}_{a}{}^{a}\\!-\\!\Lambda&=\tfrac{3}{2}t^{2\left(\alpha-1\right)}\big{[}\\!-\\!\alpha\left(\alpha\\!+\\!1\right)+2t\alpha\left(\alpha\\!-\\!1\right)H^{\rm
ext}\\\
&-\tfrac{t^{2}}{12}\left(\alpha-1\right)\left(\alpha-2\right)R^{g_{\bm{\gamma}}}+{\mathcal{O}}(t^{4})\big{]}\,.\end{split}$
(14)

In the above $H^{\rm ext}:=-I_{\bm{\sigma}}\cdot I_{\bm{\gamma}}$ canonically
extends the mean curvature of $\Sigma\hookrightarrow(M,g_{\bm{\gamma}})$.

Given only the trace of stress and causal structure for a spacetime with
isotropic singularity, can we recover the physical metric $\check{g}{\hskip
0.85358pt}$? Remarkably there exists a “solution generating algebra” that
addresses this question: The operator $I$ acting on $\bm{\sigma}$ (of Section
III) is both conformally invariant and second order. It is an example of a
more general Thomas $D$-operator mapping tractors to tractors [21, 25]. Given
the data of a weight $w^{\prime}\neq 0,-1$ density $\bm{\mu}$, this yields a
conformally invariant “d’Alembert–Robin” operator [48]

${\bm{L}}_{\bm{\mu}}:=-w^{\prime}\bm{\mu}\left(\square+wJ\right)\\\
+2\left(w+1\right)\nabla_{a}\bm{\mu}\nabla^{a}-\tfrac{w(w+1)}{w^{\prime}+1}\left(\square\bm{\mu}+w^{\prime}J\bm{\mu}\right)\,,$
(15)

mapping weight $w$ densities to weight $w+w^{\prime}-2$ densities. When
$g_{\bm{\mu}}:={\bm{\mu}}^{-\frac{2}{w^{\prime}}}{\bm{g}}$ is a metric, this
gives a d’Alembert operator
$\square^{g_{\bm{\mu}}}+\tfrac{w(w+w^{\prime}+2)}{6(w^{\prime}+1)}R^{g_{\bm{\mu}}}$.
Specializing $\bm{\mu}$ to the singular Yamabe defining density $\bm{\sigma}$,
the operator ${\bm{L}}_{\bm{\sigma}}$ yields a conformally invariant, Robin-
type, boundary operator [51]

$\delta_{\rm R}\stackrel{{\scriptstyle\Sigma}}{{:=}}\nabla_{\hat{n}}-wH\,.$

The crucial point now is that, calling
${\mathcal{S}}_{\bm{\mu}}:={\bm{L}}_{\bm{\mu}}{\bm{\mu}}$, there is an
$\mathfrak{sl}(2)=\langle x,[x,y],y\rangle$ algebra generated by

$(x,y):=\big{(}\bm{\mu},-\tfrac{2(w^{\prime}+1)}{w^{\prime}{\mathcal{S}}_{\bm{\mu}}}{\bm{L}}_{\bm{\mu}}\big{)}\,.$

In fact Eq. (8) now becomes

${\bm{L}}_{\bm{\tau}^{\alpha}}{\bm{\tau}^{\alpha}}=\tfrac{1}{3}\check{T}_{a}{}^{a}-\tfrac{4}{3}\Lambda\
.$

The formal asymptotics for traced-Einstein equations can be solved iteratively
using the solution generating $\mathfrak{sl}(2)$ algebra, cf. [50].

###  V.2 Conformal Fundamental Forms and Stress–Energy Tensor

We now analyze the trace-free part of the matter coupled Einstein system in
terms of conformal embedding geometry. Eq. (7) implies that we must study the
tractor gradient of Eq. (13) relating the various scale tractors. Acting with
$q^{*}$ and multiplying by $t^{-\alpha}$ gives

$t^{-\alpha}q^{*}\bm{\nabla}I_{\bm{\tau}^{\alpha}}\\!=\tfrac{\alpha(\alpha-1)}{t^{2}}{\bm{\gamma}dt\circledcirc
dt}\\\
+\tfrac{\alpha}{t}q^{*}\bm{\nabla}I_{\bm{\sigma}}+(1-\alpha)q^{*}\bm{\nabla}I_{\bm{\gamma}}\,.$
(16)

Multiplying by an overall factor $2/\bm{\gamma}$, each (trace-free) term above
has a physical interpretation: The left hand side is the physical stress
tensor  $\mathring{\check{T}}$ . The first summand is the stress of a perfect
fluid. The second is $\alpha$ times the stress tensor of the singular Yamabe
metric. It captures the embedding data. The last is $1-\alpha$ times the
stress tensor of the Big Bang metric $g_{\bm{\gamma}}$. Hence we learn the
asymptotics of  $\mathring{\check{T}}$ :

$\mathring{\check{T}}=\frac{\alpha(\alpha-1)\mathring{T}_{\scriptscriptstyle\rm
fluid}}{t^{2}}+\frac{\alpha\mathring{T}_{\rm\scriptscriptstyle
Bach}}{t}-(\alpha-1)\,\mathring{T}_{\rm\scriptscriptstyle Big{\hskip
0.85358pt}Bang}\,,$ (17)

where $\mathring{T}_{\scriptscriptstyle\rm fluid}:=2dt\circledcirc dt$. Note
that Eq. (9) implies

${\sf
P}q^{*}\bm{\nabla}I_{\bm{\sigma}}=\bm{\sigma}B\stackrel{{\scriptstyle\Sigma}}{{=}}0\,,$
(18)

so the (transverse order 2) partially massless operator acting on
$\frac{\bm{\gamma}}{2}\mathring{T}_{\rm Bach}$ returns $\bm{\sigma}B$.

As advertized, Equation (17) characterizes allowed stress at an isotropic
singularity. As we next show, the coefficients of terms that diverge as $t\to
0$ are local invariants of the boundary. This is reminiscent of a similar
phenomenon in the AdS/CFT correspondence relating bulk geometry and boundary
renormalization group flows [52, 53, 54].

We want to study the first four orders of the early time ($t\sim 0$)
asymptotics of physical stress. Both the fluid and Big Bang terms in Eq. (17)
are completely determined to this order so we focus on the Bach term.
Conformally invariant transverse jets of $q^{*}\bm{\nabla}I_{\bm{\sigma}}$
generate the second and third but not fourth conformal fundamental forms (see
Eq. (18)). There is a notion of a fifth fundamental form, viz the projected
Bach tensor $B^{\top}|_{\Sigma}$. However the Bach-to-stress Equation (10)
determines the conformal structure ${\bm{g}}$ given initial data of the first
through fourth fundamental forms, so we focus on these.

First note the second fundamental form here obeys

$\mathring{K}=q^{*}\bm{\nabla}I_{\bm{\sigma}}|_{\Sigma}=\tfrac{\bm{\gamma}}{2}\,\mathring{T}_{\rm\scriptscriptstyle
Bach}\big{|}_{\Sigma}\,.$

To study the next order term, we use the tractor analog of the
d’Alembert–Robin operator ${\bm{L}}_{\bm{\sigma}}$ of Eq. (15) to make a
transverse order 1 operator [44], again called $\delta_{\rm R}$,

$\circledcirc T^{*}M[w]\ni\mathring{x}_{ab}\stackrel{{\scriptstyle\delta_{\rm
R}}}{{\longmapsto}}\big{[}(\nabla_{\hat{n}}+(2-w)H)\mathring{x}_{ab}\\\
\qquad\qquad+\tfrac{2}{w-3}\bar{\nabla}_{(a}\mathring{x}_{\hat{n}b)}^{\top}\big{]}^{\top,\circ}\in\circledcirc
T^{*}\Sigma[w-1]\,.$

The trace-free Fialkow tensor is then

$\mathring{F}=\delta_{\rm R}q^{*}\bm{\nabla}I_{\bm{\sigma}}=\delta_{\rm
R}\big{(}\tfrac{\bm{\gamma}}{2}\,\mathring{T}_{\rm\scriptscriptstyle
Bach}\big{)}\,.$

Because we cannot extract $\mathring{L}$ from a conformally invariant second
normal derivative of $q^{*}\bm{\nabla}I_{\bm{\sigma}}$ to relate the fourth
fundamental form to stress, we instead consider one normal derivative of Big
Bang stress $\mathring{T}_{\rm\scriptscriptstyle Big{\hskip 0.85358pt}Bang}$.
For this we employ the identity [55]

$\delta_{\rm R}{\hskip
0.85358pt}q^{*}{\bm{\nabla}I_{\bm{\gamma}}}=\bm{\gamma}\mathring{L}+\delta^{(2)}\bm{\gamma}\,.$

This yields the last line of Figure 1 summarizing the relations between
geometry and stress–energy tensor.

Geometry | Stress–Energy Tensor
---|---
$\hat{n}\circledcirc\hat{n}$ | $\tfrac{{\bm{\gamma}}^{2}}{2}\mathring{T}_{\rm fluid}\big{|}_{\Sigma}$
$\mathring{K}$ | $\tfrac{\bm{\gamma}}{2}\mathring{T}_{\rm\scriptscriptstyle Bach}\big{|}_{\Sigma}$
$\mathring{F}$ | $\delta_{\rm R}\big{(}\tfrac{\bm{\gamma}}{2}\,\mathring{T}_{\rm\scriptscriptstyle Bach}\big{)}$
$\mathring{L}$ | $\bm{\gamma}^{-1}\delta_{\rm R}\big{(}\tfrac{\bm{\gamma}}{2}\,\mathring{T}_{\rm\scriptscriptstyle Big{\hskip 0.85358pt}Bang}\big{)}-\bm{\gamma}^{-1}{\sf\delta^{(2)}}\bm{\gamma}$
Figure 1: Conformal fundamental forms related to stress.

##  VI Example: Poincaré–Einstein Conformal Cyclic Cosmology

Models where the present universe is seeded by pre-Big Bang data [57, 58, 59,
60], dovetail with the above results. One approach [61] employs an
asymptotically dS pre-Big Bang metric $\hat{g}$ and a physical metric
$\check{g}$ with isotropic singularity:

$\hat{g}=\frac{-d\hat{t}^{{\hskip 0.85358pt}2}+\hat{h}(\hat{t}{\hskip
0.85358pt})}{\hat{t}^{2}}\,,\quad\check{g}=\check{t}^{-2\alpha}\big{(}\\!-d\check{t}^{{\hskip
0.85358pt}2}+\check{h}(\check{t}{\hskip 0.85358pt})\big{)}\,.$

The conformal infinity/initial singularity hypersurface $\Sigma$ is at
$\hat{t}=0=\check{t}$. The pre-Big Bang spatial metric $\hat{h}$ is defined by
a Fefferman–Graham-type expansion [62] about the conformal infinity of
$\hat{g}$ obtained by solving Einstein’s equations with non-vanishing stress
for suitable late time $\hat{t}\to 0_{-}$ matter content. Conformal
fundamental forms are covariant analogs of Fefferman–Graham expansion
coefficients [44] and are determined by $\hat{h}(\hat{t}{\hskip 0.85358pt})$.
They can be matched [45] to those of the Big Bang model and thus its stress.
Schematically,

${\mathring{\hat{{\hskip 0.85358pt}T}}}{\hskip
0.85358pt}\mapsto\mbox{conformal fundamental
forms}\mapsto\mathring{\check{T}}\,.$

Just as for stellar models where interior and exterior solutions are matched
using fundamental forms [63, 64, 65], cyclic cosmological matching is via
conformal fundamental forms.

###### Acknowledgements.

We thank Paweł Nurowski useful discussions. A.R.G. and A.W. acknowledge
support from the Royal Society of New Zealand via Marsden Grant 19-UOA-008.
J.K. acknowledges funding received from the Norwegian Financial Mechanism
2014-2021, project registration number UMO-2019/34/H/ST1/00636. A.W. was also
supported by Simons Foundation Collaboration Grant for Mathematicians ID
686131. J.K. and A.W. thank the University of Auckland for warm hospitality.

## References

  * [1] R. Penrose, _Singularities and time-asymmetry_ , in: General Relativity and Gravitation: an Einstein centennial volume ed. by W. Israel and S.W. Hawking, CUP, Cambridge (1979).
  * [2] Conformally removable singularities yield a subset of cosmological solutions to the Einstein field equations; references [3, 4, 5, 6] discuss generic singularity formation.
  * [3] E. Ames, F. Beyer, J. Isenberg and T.A. Oliynyk, _Stability of AVTD Behavior Within the Polarized $\mathbb{T}{}^{2}$-Symmetric Vacuum Spacetimes_, Annales Henri Poincare 23, no.7, 2299-2343 (2022).
  * [4] G. Fournodavlos, I. Rodnianski and J. Speck, _Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime_ , J. Am. Math. Soc. 36, no.3, 827-916 (2023).
  * [5] H.O. Groeniger, O. Petersen and H. Ringström, _Formation of quiescent big bang singularities_ , arXiv:2309.11370 [gr-qc] (2023).
  * [6] G. Fournodavlos and J. Luk, _Asymptotically Kasner-like singularities_ , Am. J. Math. 145, no.4, 1183-1272 (2023).
  * [7] S.W. Goode and J. Wainwright, _Isotropic singularities in cosmological models_ , Class. Quant. Grav., 2, 99–115 (1985).
  * [8] K.P. Tod, _Isotropic singularities and the $\gamma=2$ equation of state_, Class. Quant. Grav., 7 L13 (1990).
  * [9] K.P. Tod, _Isotropic Singularities and the Polytropic Equation of State_ , Class. Quant. Grav., 8 L77 (1991).
  * [10] R.P.A.C. Newman, _On the structure of conformal singularities in classical general relativity_ , Proc. Roy. Soc. London Ser. A, 443, 473–492 (1993).
  * [11] K. Anguige and K.P. Tod, _Isotropic cosmological singularities 1: Polytropic perfect fluid spacetimes_ , Ann. Physics, 276, 257–293 (1999).
  * [12] K. Anguige and K.P. Tod, _Isotropic cosmological singularities 2: The Einstein–Vlasov system_ , Ann. Physics, 276, 294–320 (1999).
  * [13] K. Anguige, _Isotropic cosmological singularities 3: The Cauchy problem for the inhomogeneous conformal Einstein-Vlasov equations_ , Ann. Phys. (NY) 282 395–419, (2000).
  * [14] K.P. Tod, _Isotropic cosmological singularities: other matter models_ , Class. Quantum Grav., 20 521 (2003).
  * [15] K.P. Tod, _Isotropic cosmological singularities in spatially homogeneous models with a cosmological constant_ , Class. Quant. Grav., 24 2415 (2007).
  * [16] C.R. Graham and E. Witten, _Conformal anomaly of submanifold observables in AdS / CFT correspondence_ , Nucl. Phys. B 546, 52–64 (1999).
  * [17] S. Ryu and T. Takayanagi, _Aspects of Holographic Entanglement Entropy,_ JHEP 08, 045 (2006).
  * [18] We work in $d=4$ dimensions but all our considerations apply to dimension $d$ spacetimes $M$.
  * [19] A. García-Parrado and J.M.M. Senovilla, _Causal structures and causal boundaries_ , Class. Quant. Grav. 22, R1–R84 (2005).
  * [20] T.Y. Thomas, _On Conformal Geometry_ , Proc. Nat. Acad. Sci. 12, 352–359 (1926).
  * [21] T.N. Bailey, M.G. Eastwood and A.R. Gover, _Thomas’s structure bundle for conformal, projective and related structures_ , Rocky Mountain J. Math., 24, 1191–1217 (1994).
  * [22] A.R. Gover and P. Nurowski, _Obstructions to conformally Einstein metrics in $n$ dimensions_, J. Geom. Phys. 56, 450–484 (2006).
  * [23] S.N. Curry and A.R. Gover, _An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity_ , London Math. Soc. Lecture Note Ser., 443, 86–170 (2018).
  * [24] Note ${\mathcal{V}}M[w]:={\mathcal{V}}M\otimes{\mathcal{E}}M[w]$ so that ${\bm{g}}\in\odot^{2}T^{*}M[2]$.
  * [25] A.R. Gover, _Invariant theory and calculus for conformal geometries_ , Adv. Math. 163 206 (2001).
  * [26] A function $\tau$ is said to be defining for a hypersurface $\Sigma$ if $0\leq\tau\in C^{\infty}M$ and $\tau|_{\Sigma}=0\neq d\tau|_{p\in\Sigma}$ for all $p\in\Sigma$.
  * [27] M. Eastwood, _Notes on conformal differential geometry_ , Rend. Circ. Mat. Palermo (2) Suppl., 43, 57–76 (1996).
  * [28] J. Maldacena, _Einstein Gravity from Conformal Gravity_ , arXiv:1105.5632 [hep-th] (2011).
  * [29] S. Deser, E. Joung and A. Waldron, _Partial Masslessness and Conformal Gravity_ , J. Phys. A 46, 214019 (2013).
  * [30] A.R. Gover, P. Somberg and V. Souček, _Yang-Mills detour complexes and conformal geometry_ , Commun. Math. Phys. 278, 307–327 (2008).
  * [31] J. Kopiński and J.A. Valiente Kroon, _Bach equation and the matching of spacetimes in conformal cyclic cosmology models_ , Phys. Rev. D 106, no.8, 084034 (2022).
  * [32] C. Loewner and L. Nirenberg. _Partial Differential Equations Invariant under Conformal or Projective Transformations_ in: contributions to Analysis, Academic Press, New York, 1974.
  * [33] P. Aviles and R.C. McOwen, _Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds_ , Duke Math. J. 56, 395–398 (1988).
  * [34] R. Mazzeo, _Regularity for the singular Yamabe problem_ , Indiana Univ. Math. J. 40, 1277–1299 (1991).
  * [35] L. Andersson, P. Chruściel and H. Friedrich, _On the Regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einsteins field equations_ , Commun. Math. Phys. 149, 587–612 (1992).
  * [36] A.R. Gover and A. Waldron, _Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem_ , Comm. Anal. Geom., 29, 779–836 (2021).
  * [37] A.R. Gover and A. Waldron, _A calculus for conformal hypersurfaces and new higher Willmore energy functionals_ , Adv. Geom., 20, 29–60 (2020).
  * [38] C.R. Graham, _Volume renormalization for singular Yamabe metrics_ , Proc. Amer. Math. Soc., 145, 1781–1792 (2017).
  * [39] A.R. Gover and A. Waldron, _Renormalized Volume_ , Commun. Math. Phys. 354, 1205–1244 (2017).
  * [40] M. Glaros, A.R. Gover, M. Halbasch and A. Waldron, _Variational calculus for hypersurface functionals: Singular Yamabe problem Willmore energies_ , J. Geom. Phys. 138, 168–193 (2019).
  * [41] A. Fialkow, _Conformal differential geometry of a subspace_ , Trans. Amer. Math. Soc., 56 309–433 (1944).
  * [42] R. Stafford, _Tractor Calculus and Invariants for Conformal Sub-Manifolds_ , M.Sc. Thesis, University of Auckland (2005).
  * [43] Transverse order counts normal derivatives. Also, the first conformal fundamental form is the boundary conformal structure.
  * [44] S. Blitz, A.R. Gover and A. Waldron, _Conformal fundamental forms and the asymptotically Poincaré–Einstein condition_ Indiana to appear, arXiv:2107.10381 [math.DG] (2021).
  * [45] A.R. Gover and J. Kopiński, _Higher fundamental forms of the conformal boundary of asymptotically de Sitter spacetimes_ , Class. Quant. Grav. 40, 015001 (2023).
  * [46] The symbol $\top$ denotes projection along directions tangent to $\Sigma$. Also note there is no need to introduce new indices for hypersurface tensors if one identifies $T\Sigma$ with its image in $TM|_{\Sigma}$.
  * [47] Remarkably, there is another canonical metric $\big{(}|\nabla\bm{\tau}|_{\bm{g}}^{2}-\tfrac{\bm{\tau}}{\alpha+1}(\square\bm{\tau}+\tfrac{1}{\alpha}J^{\bm{g}}\bm{\tau})\big{)}^{\frac{\alpha}{\alpha-1}}\,{\bm{g}}$ defined to all orders in a collar neighborhood of spacelike $\Sigma$, which also induces the metric $g_{\Sigma}$. However, it is not directly related to conformal fundamental forms.
  * [48] This slightly generalizes the construction of [49, 50].
  * [49] A.R. Gover, _Almost Einstein and Poincaré–Einstein manifolds in Riemannian signature_ , J. Geometry and Physics, 60, 182–204 (2010).
  * [50] A.R. Gover and A. Waldron, _Boundary calculus for conformally compact manifolds_ , Indiana Univ. Math. J. 63, 119–163 (2014).
  * [51] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes. J. Funct. Anal., 57 154–206 (1984).
  * [52] J. de Boer, E.P. Verlinde and H.L. Verlinde, _On the holographic renormalization group_ , JHEP 08, 003 (2000).
  * [53] M. Henningson and K. Skenderis, _The Holographic Weyl anomaly_ , JHEP 07, 023 (1998).
  * [54] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, _Large N field theories, string theory and gravity_ , Phys. Rept. 323, 183-386 (2000).
  * [55] Here $\delta^{(2)}:=-\mathring{K}\delta_{\rm R}^{(2)}-\mathring{F}\delta_{\rm R}+\bar{\delta}^{(2)}$ is a conformally invariant, transverse order 2 operator on weight 1 densities. The operator $\delta_{\rm R}^{(2)}\stackrel{{\scriptstyle\Sigma}}{{=}}\hat{n}^{a}\hat{n}^{b}\nabla_{a}\nabla_{b}+\cdots$ is the second transverse order generalization of the Robin operator of [56] (see Lemma 3.10 of [44] for an explicit formula; our conventions differ by a factor $3$). The boundary operator $\bar{\delta}^{(2)}$ has transverse order 0 and is conformally invariant.
  * [56] A.R. Gover and L.J. Peterson, _Conformal boundary operators, t-curvatures, and conformal fractional Laplacians of odd order_ , Pac. J. Math. 311: 277-32 (2021).
  * [57] R. Penrose, _Cycles of Time: An Extraordinary New View of the Universe_ , Bodley Head (2010).
  * [58] K.P. Tod, _The equations of Conformal Cyclic Cosmology_ , Gen. Rel. Grav. 47, no.3, 17 (2015).
  * [59] M. Eckstein, _Conformal Cyclic Cosmology, gravitational entropy and quantum information_ , Gen. Rel. Grav. 55, no.2, 26 (2023).
  * [60] O. Markwell and C. Stevens, _Toward fixing a framework for conformal cyclic cosmology_ , Gen. Rel. Grav. 55, no.8, 93 (2023).
  * [61] P. Nurowski, _Poincaré–Einstein approach to Penrose’s conformal cyclic cosmology_ , Class. Quant. Grav. 38, no.14, 145004 (2021).
  * [62] C. Fefferman, and C.R. Graham, _Conformal invariants in: The mathematical heritage of Élie Cartan (Lyon, 1984)_. Astérisque 1985, Numero Hors Serie, 95–116.
  * [63] J. R. Oppenheimer and H. Snyder, On Continued Gravitational Contraction, Phys. Rev. 56, 455 (1939).
  * [64] G. Darmois, Les equations de la gravitation einsteinienne In Memorial de Sciences Mathematiques, Fascicule XXV; Gauthier-Villars et Cie: Paris, France, 1927.
  * [65] W. Israel, W. Singular Hypersurfaces and Thin Shells in General Relativity, Nuovo Cimento 10, 1–14 (1966).


# Laser-induced forced evaporative cooling of molecular anions below 4 Kelvin

Jonas Tauch  Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Im Neuenheimer Feld 226, 69120 Heidelberg, Germany  Saba Z. Hassan
Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer
Feld 226, 69120 Heidelberg, Germany  Markus Nötzold  Institut für Ionenphysik
und Angewandte Physik, Universität Innsbruck, Technikerstrasse 25, 6020
Innsbruck, Austria  Eric S. Endres  Institut für Ionenphysik und Angewandte
Physik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
Roland Wester  Institut für Ionenphysik und Angewandte Physik, Universität
Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria  Matthias Weidemüller
Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer
Feld 226, 69120 Heidelberg, Germany

The study of cold and controlled molecular ions is pivotal for fundamental
research in modern physics and chemistry [1]. Investigations into cooling
molecular anions, in particular, have proven to be of key consequence for the
production of cold antihydrogen [2], the creation and study of anionic Coulomb
crystals [3, 4] as well as in atmospheric research [5] and astrochemistry [6,
7, 8]. The commonly used anion cooling technique via collisions with a buffer
gas is limited by the temperature of the used cryogenic cooling medium. Here,
we demonstrate forced evaporative cooling of anions via a laser beam with
photon energies far above the photodetachment threshold of the anion [9, 10].
We reach runaway evaporative cooling of an anionic OH- ensemble from an
initial temperature of 370(12) Kelvin down to 2.2(8) Kelvin. This corresponds
to a three orders of magnitude increase in the ions’ phase space density
approaching the near-strong Coulomb coupling regime. A quantitative analysis
of the experimental results, via a full thermodynamic model including the role
of intrinsic collisional heating, represents the anion cooling dynamics
without any fitting parameters. This technique can be used to cool, in
principle, any anionic species below liquid helium temperature, providing a
novel tool to push the frontiers of anion cooling to much lower than the
state-of-the-art temperature regimes.

Recent technological developments in the hybrid trapping of ions and atoms
have provided key insights into the interaction dynamics of molecular ions
with atoms [1]. These studies involve investigations of the atom-ion reactive
collisions [11, 12, 13, 14, 15, 16, 17], as well as the elastic and inelastic
collisions, cooling the ions in their external (translational) and internal
(vibration and rotational) degrees of freedom respectively [18, 19, 20, 21,
22]. While most of these systems, primarily involve cationic ions, cooling
anions to the cold and even ultracold regime is of great interest in the study
of astrochemically relevant reaction dynamics, i.e. the growth of anionic
carbon chains detected in Titan’s atmosphere [6, 23, 24], associative
detachment reactions of hydrogen in the early universe [25, 26] or proton
exchange formation pathways of ammonia in interstellar media [27, 28]. The
outermost electron of a negative ion is bound by a short-range potential,
which results in a lack of multiple stable electronic states, and due to their
fragile nature, most anions cannot be cooled via traditional laser cooling
methods. Although few promising candidates exist [29, 30, 31], they still need
precooling techniques, and direct Doppler cooling of anions has been, so far,
experimentally elusive.

Building upon the success of evaporative cooling to create ultracold atoms,
forced evaporative cooling of anions was proposed three decades ago [9]. For a
static laser beam position, laser-induced evaporative cooling was demonstrated
in atomic anions, where O- anions were cooled from 1.15 eV to 0.33 eV [10].
Alternatively, evaporative cooling of trapped antiprotons to temperatures as
low as 9 $\mathrm{K}$ was achieved by altering the trap’s potential landscape
[32]. In our work, we employ this approach of laser-induced evaporative
cooling to molecular anions and thus demonstrate the generality of this
method. We reach temperatures below 4 $\mathrm{K}$, which sets the limit of
buffer-gas cooling using cryogenic helium. We remove the high-energetic tail
of a molecular anionic ensemble, stored in a radio-frequency (rf) ion trap,
via selective photodetachment. Subsequent rethermalization leads to efficient
cooling of the remaining ions, which can be further enhanced by sweeping the
detachment laser into the ion cloud (”forced evaporative cooling”). The
cooling process is well modeled by an ab-initio thermodynamic calculation,
revealing the role of radio-frequency heating induced by ion-ion collisions.
With this approach, we cool down an ensemble of OH- anions from 370(12) K to
2.2(8) K. We achieve a Coulomb coupling parameter of $\Gamma=0.06(5)$ in less
than 4 seconds along with an increase in the ensemble’s relative phase space
density by three orders of magnitude.

Figure 1: Experimental setup and thermalization dynamics under position-
dependent photodetachment. (a), Schematic representation of the octupole wire
trap with hollow-core endcaps for loading and extraction of the ions.
Perpendicular to the trap axis, we shine in a laser beam for photodetachment
tomography and energy selective evaporation. (b), Axial ion distribution in
the trap at 370(12) K. The solid points represent photodetachment rates, which
are proportional to the local column density. Each rate is determined by
fitting an exponential decay to a loss measurement depending on the
photodetachment time. The error bars show the $\pm 1\sigma$ uncertainty of
these fits. The solid curve is a fitted Gaussian function. Six different laser
beam positions and their width are illustrated by colored vertical lines. The
right panels show (c) the temporal evolution of temperature and (d) the
normalized ion number for the six laser beam positions. The open circles
represent the measured data and the solid lines show the numerical integration
of Equation 1 and 2.

In our experiment, we trap a cloud of OH- anions in a radio-frequency octupole
wire trap (see Figure 1 a). A description of the experimental details are
given in the Methods section. By addressing ions in a chosen axial interval,
i.e. via photodetachment with a focused laser beam we can manipulate the
ensemble’s mean energy. To perform evaporative cooling, ions with total energy
higher than the mean energy must be removed [9]. With this approach, one is
able to preserve the trapping potential landscape, compared to the forced
evaporative cooling of ions by lowering the trap depth [32]. We measured the
evolution of temperature and the number of ions for six laser beam positions.
The position and width of the laser beam compared to the ions’ initial
distribution for the six measurements are illustrated in Figure 1 b as colored
vertical bars.

The evolution of ensemble temperature and number of ions is measured as a
function of the interaction time with the photodetachment light, as shown in
Figure 1 c, 1 d respectively. For photodetachment laser beam position,
sufficiently far away from the center of the ion cloud, we observe a reduction
in the ion temperature by a factor of 2 within ten seconds of interaction time
with the laser beam. If the photodetachment beam is closer to the center, the
ions undergo heating. During this interaction, the number of ions are reduced
by roughly a factor of 10.

Based on ref. [9], the change in temperature (T) and ion number (N) in an
octupole trap is described by:

$\displaystyle\dot{T}=$
$\displaystyle\frac{3}{7}\Bigg{(}-\frac{\sigma_{\text{pd}}}{k_{\text{B}}}\int\rho_{\text{ion}}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})V(x,y,z)dxdydz+$
(1)
$\displaystyle\frac{5\sigma_{\text{pd}}}{6}T\int\rho_{\text{ion}}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})dxdydz\Bigg{)},$
$\displaystyle\dot{N}=$
$\displaystyle-\sigma_{\text{pd}}N\int\rho_{ion}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})dxdydz.$
(2)

$\sigma_{\text{pd}}$ is the absolute photodetachment cross section
($\sigma_{\text{OH${}^{-}$}}=8.5\cdot 10^{-22}$ m2 at 660 nm [33]),
$\rho_{\text{ion}}$ the normalized ion distribution and
$\Phi(x,z,x_{\text{L}},z_{\text{L}})$ the Gaussian distributed photon flux at
the position $(x_{\text{L}},z_{\text{L}})$. The total potential $V$ is defined
as the sum of the harmonic axial potential created by the static endcaps and
the time averaged rf potential in radial direction. In an octupole trap this
ponderomotive potential is proportional to $r^{6}$. The first term of Equation
1 corresponds to the mean potential energy of the ions addressed by the
photodetachment beam and the second term represents the mean potential energy
of the entire ensemble. The prefactors $\frac{3}{7}$ and $\frac{5}{6}$ stem
from applying the virial theorem. A detailed description of the model can be
found in the Methods section. For evaporative cooling, the ensemble’s
thermalization rate must be fast as compared to the induced loss rate. To
ensure the validity of this condition the laser power is adjusted accordingly
for different laser beam positions (see Supplementary Information, Section I).
Thus, as long as the ion-ion thermalization rate surpasses the ion loss rate,
the process of evaporative cooling (and heating) can be described via this
equilibrium picture.

Equations 1 and 2 are integrated numerically using the measured experimental
parameters without any free parameter. The results of this integration are
shown in Figures 1 c and d, respectively, as solid lines for each laser beam
configuration. It can be seen that the model accurately describes the
dynamics. The ensemble is cooled for sufficiently large distances of the laser
beam from the cloud center reaching temperatures around 200 K after 10 s,
while removing about 90% of the atoms.

Figure 2: Forced evaporative cooling via photodetachment. Evolution of (a) the
ion ensemble’s temperature, (b) normalized ion number and (c) the Coulomb
coupling constant, while moving the laser beam towards the trap center for 4
different beam velocities. The initial beam position is 4 mm off-center. For
each beam velocity, ions are extracted at the positions 4 mm, 3 mm, 2 mm, 1 mm
and 0.6 mm from the center. For 1 mm/s and 2 mm/s additional measurements at
0.5 mm, 0.4 mm and 0.3 mm where taken. The measured experimental results are
shown as solid points. The dashed lines represent the result of the numerical
integration of Equation 1 and 2. The solid lines include the heating rate of
Equation 3. (c) shows measured Coulomb coupling parameter as solid points and
the model including the heating rate as solid lines. The model is terminated
at the residual ion number of 1 % of the initial ion population.

Enhanced cooling can be achieved by moving the laser dynamically towards the
trap center. The resulting ion dynamics is shown for four different laser beam
velocities in Figures 2 a and b. Moving the laser beam sufficiently slowly to
the trap center the final ensemble temperature is 2.2(8) K with about 2 % of
the initially loaded ions remaining. This corresponds to an increase in phase
space density by three orders of magnitude.

The dashed lines shown in Figure 2 a and b are the numerical integration of
Equation 1 and 2 including a dynamically moving photodetachment laser beam.
Although the thermodynamic model describes the observed temperature evolution
very well, the measured number of ions deviates significantly from the
predicted numbers for lower beam velocities. This can be explained by the
influence of ion-ion collisions, which transfer the energy of the driving rf
field to the ions’ secular motion. The resulting heating rate can be
approximated by the product of the ion-ion collision rate, described by the
Chandrasekhar-Spitzer plasma self-collision rate $\nu_{ii}(T)$ [34], and the
relative energy transfer per collision $\bar{\epsilon}$:

$\displaystyle\dot{T}=\frac{3}{7}\nu_{\text{ii}}(T)\bar{\epsilon}\frac{1}{3}T=\frac{3}{7}\frac{ne^{4}\log\Lambda_{\text{ii}}}{12\epsilon_{0}^{2}\sqrt{\pi^{3}m(k_{\text{B}}T)^{3}}}\bar{\epsilon}\frac{1}{3}T.$
(3)

The mean rf motion in an octupole trap is given by $\langle
E_{\text{rf}}\rangle=\frac{1}{3}k_{\text{B}}T$ and the relative energy change
per collision $\bar{\epsilon}=0.045(6)$ is measured by a background heating
measurement (see Supplementary Information, Section II). $n$ is the ion
density and $\log\Lambda_{\text{ii}}$ the Coulomb logarithm, with
$\Lambda_{\text{ii}}$ being the ratio between the Debye length and the
classical distance of the closest approach.

Equation 1 and 2 are independent on the absolute ion number, but the ion-ion
collision rate is not. The detection efficiency of our setup is therefore
determined by an independent thermalization measurement (see Supplementary
Information, Section I). The zero fit-parameter models, including the heating
rate, are shown as the solid lines in Figure 2. The complete model describes
the change in both, the temperature and the ion number, very accurately.

The efficiency and pace of cooling our ensemble are limited by two timescales,
the evaporation rate and the thermalization rate. The evaporation rate is
proportional to the detachment rate, and thus photon flux. If we move the
laser beam faster than the removal of high-energy ions, the cooling efficiency
drops, as can be seen in Figure 2. The thermalization rate determines how fast
the energy is redistributed over the whole ensemble. If we move the laser beam
too fast, the evaporation rate exceeds the thermalization rate, and the ions
are only spilled without an increase in the ensembles’ phase space density. If
we move it too slowly, rf heating via ion-ion collisions becomes more
prominent and results in an additional ion loss. The thermalization rate of a
one-component plasma is proportional to the product of the Coulomb logarithm
and the ensembles’ phase space density, for which we see a tremendous increase
by three orders of magnitude (see Supplementary Information, Section III).

A one-component plasma is characterized by the dimensionless Coulomb coupling
constant $\Gamma$, which represents the nearest-neighbor Coulomb energy to the
thermal energy:

$\displaystyle\Gamma=\frac{e^{2}}{4\pi\epsilon_{0}k_{\text{B}}T\langle
r\rangle},$ (4)

where $\langle r\rangle$ is the mean inter-particle distance (Wigner-Seitz
radius). In a weakly coupled plasma, $\Gamma\ll 1$, Coulomb interaction can be
treated as two particle collisions perturbing the system, as we did in
Equation 3. For $\Gamma~{}1$, the assumption breaks down and reveals phenomena
as liquid-like behavior and a liquid-solid phase transition at $\Gamma=178$
[35][36]. Figure 2 c shows the Coulomb coupling constant corresponding to the
measurement shown in the two left panels. For the maximum forced evaporative
cooling efficiency (laser beam velocity of 1 mm/s), the coupling parameter
increases from initially $\Gamma=0.35(4)\cdot 10^{-3}$ to 0.06(5) resulting in
runaway evaporative cooling. Based on the model, the coupling constant could
be further increased by increasing the detachment rate beyond the rate
currently achievable in our experimental setting (see Supplementary
Information, Section IV).

Our experimental results and the model show that with forced evaporative
cooling via above-threshold photodetachment is an efficient tool to reduce the
thermal energy of, in principle, any anionic species by orders of magnitude.
This establishes a way to translationally cool anions below the temperature of
liquid helium without the need of a buffer gas, thus ensuring ultra-high
vacuum conditions. This would allow to enter the regime of low-energy
scattering resonances [37, 38, 39], as has recently been achieved with atomic
cations [40], eventually approaching the regime of strong Coulomb coupling. In
a multi-species anionic plasma, this laser-induced cooling provides the
advantage of addressing one anionic species, while sympathetically cooling the
other species, e.g. antiprotons, without the loss of the latter. The cooling
of the molecular anions can even be extended to address internal degrees of
freedom by tuning the laser light frequency close to the anion’s
photodetachment threshold [9].

## Methods

### Experimental details

Our molecular anion of interest, the hydroxide OH- anion, is created in a
pulsed plasma discharge source and subsequently trapped in an octupole wire
trap. This trap combines the advantage of reduced rf heating in multipole
traps and the use of wires to provide optical access to investigate atom-anion
interactions in a hybrid trap [41, 42, 43]. The measurements were performed on
trapped ions initially thermalized with a room temperature helium buffer gas,
which results in an ensemble with a temperature slightly above the buffer gas.
This increase in temperature is attributed due to rf heating [44]. We
determine the temperature of ions in the trap by extracting them through a
hollow-core endcap and subsequently detecting the time-of-flight (TOF) for
each particle on a micro-channel plate (MCP). With an optimized extraction
potential, one can linearly map the energy distribution onto the TOF to the
detector. A detailed description of the setup and the TOF thermometry can be
found in [44].

While the energy distribution is obtained from the TOF, the ions’ spatial
distribution in the trap can be measured via above-threshold photodetachment
tomography [45]. In our setup the laser light with a photon energy of 1.88 eV,
which is above the photodetachment threshold of OH- (1.83 eV [46]), is focused
down to a diameter of $180\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (1/e2) and
can be moved by a translational stage. One measurement of the ion distribution
along the trap axis, after initialization with the helium buffer gas pulse, is
shown in Figure 1 b as solid points. Since the axial potential is well-
described by a harmonic potential, a Gaussian distribution of the ion cloud
can be observed.

### Thermodynamic model

The approach of this rate model is based on [9] and it assumes that the
cooling process is slow enough, that the ions are at any time in thermal
equilibrium. This way the total energy can be described by the sum of the mean
kinetic energy and the mean potential energy:

$\displaystyle E_{\text{tot}}=\langle E_{\text{k}}\rangle_{v}+\langle
E_{\text{p}}\rangle_{r}=\frac{3}{2}k_{\text{B}}T+\frac{5}{6}k_{\text{B}}T=\frac{7}{3}k_{\text{B}}T.$
(5)

The prefactor $\frac{5}{6}$ is the result of the virial theorem for an
octupole trap. The change in ion number is described by the overlap integral.

$\displaystyle\dot{N}=-N\langle\langle\sigma\Phi\rangle_{r}\rangle_{v}=-N\int\sigma(\textbf{v})\Phi(\textbf{r})\rho_{\text{ion}}(\textbf{r},\textbf{v})d\textbf{r}d\textbf{v}$
(6)

$\sigma$ is the absolute photodetachment cross section, $\rho_{\text{ion}}$
the normalized ion density and $\Phi$ is the photon flux. The change of the
total energy is given by the lost ions’ energy:

$\displaystyle\dot{E}_{\text{tot}}$ $\displaystyle=-N\langle\langle\sigma\Phi
E\rangle_{r}\rangle_{v}$ (7)
$\displaystyle\Rightarrow\langle\langle\dot{E}\rangle_{r}\rangle_{v}$
$\displaystyle=-\langle\langle\sigma\Phi
E\rangle_{r}\rangle_{v}+\langle\langle\sigma\Phi\rangle_{r}\rangle_{v}\langle\langle
E\rangle_{r}\rangle_{v}$ (8)
$\displaystyle=-\langle\langle\sigma\Phi(E_{\text{k}}+E_{\text{p}})\rangle_{r}\rangle_{v}+\langle\langle\sigma\Phi\rangle_{r}\rangle_{v}\langle\langle(E_{\text{k}}+E_{\text{p}})\rangle_{r}\rangle_{v}$
(9)

Only the laser intensity and potential energy depends on r and the kinetic
energy on v. In the case of above-threshold photodetachment, the cross section
is independent of the ions’ velocity. Thus the kinetic energy terms cancel
out.

$\displaystyle\langle\langle\dot{E}\rangle_{r}\rangle_{v}=-$
$\displaystyle\sigma\langle\Phi
E_{\text{p}}\rangle_{r}+\sigma\langle\Phi\rangle_{r}\langle
E_{\text{p}}\rangle_{r}$ (10)
$\displaystyle\Rightarrow\frac{7}{3}k_{\text{B}}\dot{T}=-$
$\displaystyle\sigma\int\rho_{\text{ion}}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})V(x,y,z)dxdydz+$
(11)
$\displaystyle\frac{5}{6}k_{\text{B}}T\sigma\int\rho_{\text{ion}}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})dxdydz$
$\displaystyle\dot{N}=-$ $\displaystyle\sigma
N\int\rho_{\text{ion}}(x,y,z,T)\Phi(x,z,x_{\text{L}},z_{\text{L}})dxdydz-
k_{\text{bgr}}N$ (12)

The background loss rate has been measured to be $k_{\text{bgr}}=0.009(1)$
s-1. The potential is determined by photodetachment tomography measurements at
370(12) K in axial and radial direction assuming a quadratic and a
$r^{6}$-Potential:

$\displaystyle V(x,y,z)=2.04(9)\cdot
10^{-15}$\mathrm{J}\mathrm{/}\mathrm{m}^{2}$\cdot z^{2}+3.9(6)\cdot
10^{-3}$\mathrm{J}\mathrm{/}\mathrm{m}^{6}$\cdot\left(x^{2}+y^{2}\right)^{3}$
(13)

## References

  * [1] M. Tomza, K. Jachymski, R. Gerritsma, et al.  “Cold hybrid ion-atom systems”.  Rev. Mod. Phys. 91, 035001 (2019).
  * [2] A. Kellerbauer and J. Walz.  “A novel cooling scheme for antiprotons”.  New J. Phys. 8 (2006).
  * [3] M. Bonitz, P. Ludwig, H. Baumgartner, et al.  “Classical and quantum Coulomb crystals”.  Phys. Plasmas. 15(5) (2008).
  * [4] M. Drewsen, I. Jensen, J. Lindballe, et al.  “Ion Coulomb crystals: A tool for studying ion processes”.  Int. J. Mass Spectrom. 229(1-2), 83 (2003).
  * [5] V. Vuitton, P. Lavvas, R. V. Yelle, et al.  “Negative ion chemistry in Titan’s upper atmosphere”.  Planet. Space Sci. 57(13), 1558 (2009).
  * [6] M. C. McCarthy, C. A. Gottlieb, H. Gupta, and P. Thaddeus.  “ Laboratory and Astronomical Identification of the Negative Molecular Ion C6H-”.  ApJ 652(2), L141 (2006).
  * [7] M. Larsson, W. D. Geppert, and G. Nyman.  “Ion chemistry in space”.  Rep. Prog. Phys. 75(6) (2012).
  * [8] T. J. Millar, C. Walsh, and T. A. Field.  “Negative ions in space”.  Chem. Rev. 117(3), 1765 (2017).
  * [9] A. Crubellier.  “Theory of laser evaporative cooling of trapped negative ions. I. Harmonically bound ions and RF traps”.  J. Phys. B: At. Mol. Opt. Phys. 23(20), 3585 (1990).
  * [10] G. Cerchiari, P. Yzombard, and A. Kellerbauer.  “Laser-Assisted Evaporative Cooling of Anions”.  Phys. Rev. Lett. 123(10), 103201 (2019).
  * [11] P. Puri, M. Mills, C. Schneider, et al.  “Synthesis of mixed hypermetallic oxide BaOCa+ from laser-cooled reagents in an atom-ion hybrid trap”.  Science 357(6358), 1370 (2017).
  * [12] F. H. J. Hall and S. Willitsch.  “Millikelvin Reactive Collisions between Sympathetically Cooled Molecular Ions and Laser-Cooled Atoms in an Ion-Atom Hybrid Trap”.  Phys. Rev. Lett. 109, 233202 (2012).
  * [13] M. Germann, X. Tong, and S. Willitsch.  “Observation of electric-dipole-forbidden infrared transitions in cold molecular ions”.  Nat. Phys. 10(11), 820 (2014).
  * [14] A. Kilaj, H. Gao, D. Rösch, et al.  “Observation of different reactivities of para and ortho-water towards trapped diazenylium ions”.  Nat. Commun. 9(1), 2096 (2018).
  * [15] A. Härter, A. Krükow, M. Deiß, et al.  “Population distribution of product states following three-body recombination in an ultracold atomic gas”.  Nat. Phys. 9(8), 512 (2013).
  * [16] J. Wolf, M. Deiß, A. Krükow, et al.  “State-to-state chemistry for three-body recombination in an ultracold rubidium gas”.  Science 358(6365), 921 (2017).
  * [17] P. Puri, M. Mills, I. Simbotin, et al.  “Reaction blockading in a reaction between an excited atom and a charged molecule at low collision energy”.  Nat. Chem. 11(7), 615 (2019).
  * [18] E. R. Hudson.  “Sympathetic cooling of molecular ions with ultracold atoms”.  EPJ Techn. Instrum. 3(1), 8 (2016).
  * [19] S. Willitsch.  “Coulomb-crystallised molecular ions in traps: methods, applications, prospects”.  Int. Rev. Phys. Chem. 31(2), 175 (2012).
  * [20] W. G. Rellergert, S. T. Sullivan, S. J. Schowalter, et al.  “Evidence for sympathetic vibrational cooling of translationally cold molecules”.  Nature 495(7442), 490 (2013).
  * [21] T. Stoecklin, P. Halvick, M. A. Gannouni, et al.  “Explanation of efficient quenching of molecular ion vibrational motion by ultracold atoms”.  Nat. Commun. 7(1), 11234 (2016).
  * [22] D. Hauser, S. Lee, F. Carelli, et al.  “Rotational state-changing cold collisions of hydroxyl ions with helium”.  Nat. Phys. 11(6), 467 (2015).
  * [23] E. Herbst.  “Can negative molecular ions be detected in dense interstellar clouds?”  Nature 289(5799), 656 (1981).
  * [24] B. Bastian, T. Michaelsen, J. Meyer, and R. Wester.  “Anionic Carbon Chain Growth in Reactions of C${}_{2}^{-}$,C${}_{4}^{-}$,C${}_{6}^{-}$, C2H- , C4H- , and C6H- with C2H2”.  ApJ 878(2), 162 (2019).
  * [25] H. Kreckel, H. Bruhns, M. Čížek, et al.  “Experimental Results for H2 Formation from H- and H and Implications for First Star Formation”.  Science 329(5987), 69 LP  (2010).
  * [26] D. Gerlich, P. Jusko, Š. Roučka, et al.  “Ion trap studies of H- \+ H $\rightarrow$ H2 \+ e- Between 10 and 135 K”.  ApJ 749(1) (2012).
  * [27] R. Otto, J. Mikosch, S. Trippel, et al.  “Nonstandard behavior of a negative ion reaction at very low temperatures”.  Phys. Rev. Lett. 101(6), 1 (2008).
  * [28] F. A. Gianturco, E. Yurtsever, M. Satta, and R. Wester.  “Modeling Ionic Reactions at Interstellar Temperatures: The Case of NH${}_{2}^{-}$\+ H2 $\rightarrow$ NH3 \+ H-”.  J. Phys. Chem. A 123(46), 9905 (2019).
  * [29] E. Jordan, G. Cerchiari, S. Fritzsche, and A. Kellerbauer.  “High-Resolution Spectroscopy on the Laser-Cooling Candidate La-”.  Phys. Rev. Lett. 115(11), 1 (2015).
  * [30] S. Gerber, J. Fesel, M. Doser, and D. Comparat.  “Photodetachment and Doppler laser cooling of anionic molecules”.  New J. Phys. 20(2), 023024 (2018).
  * [31] M.-j. Wan, D.-h. Huang, Y. Yu, and Y.-g. Zhang.  “Laser cooling of the OH- molecular anion in a theoretical investigation”.  Phys. Chem. Chem. Phys. 19, 27360 (2017).
  * [32] G. B. Andresen, M. D. Ashkezari, M. Baquero-Ruiz, et al. (ALPHA Collaboration).  “Evaporative Cooling of Antiprotons to Cryogenic Temperatures”.  Phys. Rev. Lett. 105, 013003 (2010).
  * [33] P. Hlavenka, R. Otto, S. Trippel, et al.  “Absolute photodetachment cross section measurements of the O- and OH- anion”.  J. Chem. Phys. 130(6) (2009).
  * [34] L. Spitzer Physics of fully ionized gases 2nd ed. (Interscience Publ., New York, 1967).
  * [35] S. Ichimaru and S. Tanaka.  “Generalized viscoelastic theory of the glass transition for strongly coupled, classical, one-component plasmas”.  Phys. Rev. Lett. 56(26), 2815 (1986).
  * [36] S. L. Gilbert, J. J. Bollinger, and D. J. Wineland.  “Shell-structure phase of magnetically confined strongly coupled plasmas”.  Phys. Rev. Lett. 60(20), 2022 (1988).
  * [37] J. G. Lauderdale, C. W. McCurdy, and A. U. Hazi.  “Conversion of bound states to resonances with changing internuclear distance in molecular anions”.  J. Chem. Phys. 79(5), 2200 (1983).
  * [38] L. Sanche.  “Low-energy electron scattering from molecules on surfaces”.  J. Phys. B: At. Mol. Opt. Phys. 23(10), 1597 (1990).
  * [39] F. Engel, T. Dieterle, F. Hummel, et al.  “Precision Spectroscopy of Negative-Ion Resonances in Ultralong-Range Rydberg Molecules”.  Phys. Rev. Lett. 123, 073003 (2019).
  * [40] P. Weckesser, F. Thielemann, D. Wiater, et al.  “Observation of Feshbach resonances between a single ion and ultracold atoms”.  Nature 600(7889), 429 (2021).
  * [41] M. Kas, J. Loreau, J. Liévin, and N. Vaeck.  “Reactivity of Hydrated Hydroxide Anion Clusters with H and Rb: An ab Initio Study”.  J. Phys. Chem. A 123(41), 8893 (2019).
  * [42] S. Z. Hassan, J. Tauch, M. Kas, et al.  “Associative detachment in anion-atom reactions involving a dipole-bound electron”.  Nat. Commun. 13(1), 818 (2022).
  * [43] S. Z. Hassan, J. Tauch, M. Kas, et al.  “Quantum state-dependent anion–neutral detachment processes”.  J. Chem. Phys. 156(9), 094304 (2022).
  * [44] M. Nötzold, S. Z. Hassan, J. Tauch, et al.  “Thermometry in a Multipole Ion Trap”.  Appl. Sci. 10(15), 5264 (2020).
  * [45] S. Trippel, J. Mikosch, R. Berhane, et al.  “Photodetachment of cold OH- in a multipole ion trap”.  Phys. Rev. Lett. 97(19), 1 (2006).
  * [46] J. R. Smith, J. B. Kim, and W. C. Lineberger.  “High-resolution threshold photodetachment spectroscopy of OH-”.  Phys. Rev. A 55(3), 2036 (1997).