11institutetext: School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham NG7 2RD, United Kingdom22institutetext: The
Nottingham Centre of Gravity, University of Nottingham, Nottingham NG7 2RD, UK

# On de Sitter vacua in O(d,d) invariant cosmology

Yang Liu, Antonio Padilla, Paul M. Saffin and Robert G. C. Smith
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a thorough analysis of de Sitter vacua in $\textrm{O}(d,d)$
invariant cosmologies. Starting with a homogeneous and isotropic framework we
examine conditions for the existence of such vacua, non-perturbative in
$\alpha^{\prime}$ in both the string frame and the Einstein frame. We
elucidate the nature of the instability in the string frame vacuum. For the
Einstein frame, we demonstrate that the de Sitter vacuum cannot be eternal. We
then extend our analysis to include Bianchi I universes where the
$\textrm{O}(d,d)$ symmetry includes scale factor exchange as well as scale
factor duality. We show how the theory can be extended to the anisotropic case
so that it admits de Sitter vacua, noting the crucial role played by the
$\textrm{O}(d,d)$ symmetry in satisfying any additional constraints.

##  1 Introduction

Cosmological observations cmb ; SN1 ; SN2  indicate that our universe has
undergone two phases of accelerated expansion, at early Baumann  and at late
times Ed . At early times we have inflation, consistent with the evolution of
a scalar field in slow roll. At late times we have dark energy, which may
correspond to our universe approaching a de Sitter vacuum, or else
quintessence, which is a low scale version of early universe inflation. Data
recently released by the DESI collaboration shows some preference for dark
energy with an evolving equation of state DESI:2024mwx , consistent with
quintessence. Whether it takes the form of quintessence or de Sitter,
obtaining an accelerated expansion from string theory is an ongoing challenge.
Although we can construct stable de Sitter vacua using background flux and
brane uplifts (see eg. KKLT ; Tbranes  and for reviews Grana ; STcos ) the
consistency of these constructions is a matter of considerable debate (see eg.
nods ; dSVQ ). This has led to the so-called de Sitter conjecture which
implies that fully consistent and stabilised de Sitter vacua are absent from
the spectrum of weakly coupled string theory at or near the boundary of moduli
space ds1 ; ds2 ; ds3 . If the conjecture is correct and these vacua are
indeed absent, then quintessence is also expected to be ruled out quin1 ; us1
; us2 ; arthur .

String theory is a consistent quantum theory of gravity, with a well defined
low energy description in terms of higher dimensional supergravities. For this
to be relevant to our universe, it must be able to admit a compactification
down to four dimensions, where gravity is well described by General Relativity
and matter by a suitable extension of the Standard Model of Particle Physics.
It should also be able to accommodate an accelerated cosmological expansion.
Of course, it is still possible that the concerns surrounding de Sitter vacua
in perturbative string theory will go away and we will be able to find
consistent accelerated cosmological solutions that everyone is happy with.
However, if this is not the case, we will need a non-perturbative
understanding of the theory to probe the full vacuum structure and establish
the existence of de Sitter vacua far from the boundary of moduli space. This
will likely require a third string revolution - a step change in our current
approach to string theory/M theory. We can also consider the possibility that
de Sitter is a resonant or excited state of string theory res1 ; res2 ;
excitedstate .

We can try and get some hints into what can happen non-perturbatively by
focussing on the effective field theory for classical strings in the
homogeneous limit. The Neveu-Schwarz sector describing the metric, the Kalb-
Ramond field and the dilaton, are known to possess an $\textrm{O}(d,d)$
symmetry to leading order Veneziano1 ; Meissner1 ; Meissner2 ; Sen1 ;
Gasperini:1991ak . This symmetry is expected to be preserved to all orders in
the slope parameter $\alpha^{\prime}$, motivating corrections to the leading
order Neveu-Schwarz theory using higher order $\textrm{O}(d,d)$ invariant
operators Meissner3 . Note that $\textrm{O}(d,d)$ invariance can be made
manifest using double field theory, albeit at the expense of manifest
diffeomorphism invariance DFT ; DFT2 ; Hohm:2013jaa ; Hohm:2014xsa . Duality
invariant cosmology has been used to tackle the problem of the initial
singularity (see eg. Gasperini:2002bn ; Wu ; Brandenberger:2018xwl ; Bran2018
; Quintin ).

Remarkably, for homogeneous fields, Hohm and Zwiebach have been able to
provide a complete classicfication of all higher order terms invariant under
$\textrm{O}(d,d)$ Hohm:2015doa ; Hohm:2019ccp ; Hohm:2019jgu . With field
redefinitions, the system is reduced to first order time derivatives allowing
them to write down the most general duality invariant theory to all orders in
$\alpha^{\prime}$ for the metric, Kalb-Ramond field and dilaton. To leading
order in the expansion in $\alpha^{\prime}$ the theory coincides with the
Neveu-Schwarz sector of the supergravity theories describing the effective
theory of classical strings, in the homogeneous limit. In the limit of
vanishing Kalb-Ramond field and an isotropic spatially flat metric, the theory
exhibits scale factor duality $a\to 1/a$ as a remnant of the underlying
$\textrm{O}(d,d)$ symmetry. The full theory is now tractable at all orders in
$\alpha^{\prime}$, enabling the authors of Hohm:2019ccp ; Hohm:2019jgu  to
identify string frame de Sitter vacua that are non-perturbative in
$\alpha^{\prime}$. The dynamics of this new class of theories has been studied
in a series of subsequent papers EFdS ; Bernardo:2019bkz ; Bernardo:2020zlc ;
Bernardo:2020nol ; Bernardo:2021xtr ; Quintin:2021eup ; Bieniek:2022mrv ;
Bernardo:2022nex ; Bieniek:2023ubx ; Codina:2023fhy ; Gasperini:2023tus ;
Conzinu:2023fth . In particular, one can extend the system to include
$\textrm{O}(d,d)$ invariant matter fields Gasperini:1991ak ; Bernardo:2019bkz
, or find theories that admit de Sitter solutions even in Einstein frame EFdS
; Bernardo:2019bkz . Instabilities for homogeneous and isotropic perturbations
were studied in detail in Bernardo:2020zlc ; Bieniek:2022mrv ;
Bernardo:2022nex  with subsequent extensions to anisotropic perturbations
Bieniek:2023ubx . Bianchi I universes were studied in Codina:2023fhy ;
Gasperini:2023tus ; Conzinu:2023fth , along with two-dimensional black hole
solutions Codina:2023fhy  and bouncing cosmologies Gasperini:2023tus ;
Conzinu:2023fth .

In this paper, we revisit the dynamics of $\textrm{O}(d,d)$ invariant
cosmologies, starting with isotropic and spatially flat metrics before moving
on to the anisotropic case. Our focus is on the existence of de Sitter vacua
in both the string frame and the Einstein frame. Of course, the isotropic case
is simpler and much more extensively studied. Within the isotropic framework,
string frame de Sitter vacua are known to exist but have been shown to be
unstable Bernardo:2020zlc ; Bieniek:2022mrv ; Bernardo:2022nex , something we
are able to verify using slightly different methods. de Sitter solutions in
the Einstein frame have been obtained in the presence of a source
Bernardo:2019bkz  and in vacuum EFdS . The latter only exist when the theory
takes on a very bespoke form at all orders in $\alpha^{\prime}$. As we will
show, these Einstein frame de Sitter solutions are the only solutions to the
vacuum field equations for these particular theories for a finite time period,
ensuring their local stability under homogeneous fluctuations. However, the
solutions are not eternal in both past and future. This is because the dilaton
evolves. After a finite time (towards either the past or future), the solution
is kicked into a new dynamical phase where Einstein frame de Sitter vacua are
no longer possible. This resonates (!) with the idea of de Sitter as a
resonance res1 ; res2 .

These solutions enjoy a natural extension to anisotropic universes thanks to
the $\textrm{O}(d,d)$ symmetry. Here we consider a Bianchi I universe, with a
number of different scale factors with corresponding Hubble parameters. In a
notable departure from the isotropic framework, the $\textrm{O}(d,d)$ symmetry
now implies an invariance under the exchange of any two of those Hubble
parameters, placing important restrictions on the structure of the underlying
theory. Although isotropy is no longer assumed at the level of the theory, we
are interested in isotropic solutions: namely, the de Sitter vacua in both
string frame and Einstein frame. In string frame, de Sitter solutions are
possible, provided the anisotropic theory obeys properties analogous to the
isotropic case. These solutions are once again unstable. In the Einstein
frame, we find conditions for the de Sitter vacuum to exist. This mirrors the
isotropic case discussed in the previous paragraph, with additional conditions
pointing along the anisotropic directions. Remarkably, these extra conditions
hold automatically thanks to the $\textrm{O}(d,d)$ symmetry.

The rest of this paper is organized as follows. In section 2 we give a brief
review of $O(d,d)$-invariant cosmology. In section 3, we review the structure
of the underlying theory for an isotropic and spatially flat universe,
including sources. In section 3.1 we review the derivation and instability of
de Sitter vacua in the string frame, providing some added insight into the
source of the instability. In section 3.2 we demonstrate how de Sitter
solutions can also be found in the Einstein frame. As discussed above, these
solutions exist for a finite time period, during which time they are stable
against homogeneous fluctuations. However, the solutions are not external, and
will eventually give way to a new cosmological phase. In section 4, we change
gears and extend our analysis to include Bianchi I universes. In section 4.1
we establish the appropriate conditions for string-frame de Sitter vacua,
commenting on the stability. In section 4.2 we do the same for de Sitter vacua
in the Einstein frame, noting the crucical role played by the
$\textrm{O}(d,d)$ symmetry. In section 5, we summarise our conclusions, and
extend the results to non-critical dimensions.

##  2 Review of O$(d,d)$-invariant cosmology

We begin with a review of all orders in $\alpha^{\prime}$ duality invariant
cosmology, as originally proposed by Hohm and Zwiebach Hohm:2019ccp ;
Hohm:2019jgu  building on earlier pioneering work Veneziano1 ; Meissner1 ;
Meissner2 ; Sen1 ; Gasperini:1991ak . Consider the leading order theory
describing the universal massless sector of closed strings in $d+1$
dimensions. This corresponds to the well known Neveu-Schwarz action depending
on the string frame metric, $\mathbb{g}_{\mu\nu}$, the antisymmetric Kalb-
Ramond field, $\mathbb{b}_{\mu\nu}$, and the scalar dilaton $\phi$,

$I_{0}=\frac{1}{2\l_{s}^{d-1}}\int\mathrm{d}^{1+d}x\sqrt{-\mathbb{g}}e^{-2\phi}\left[R+4(\partial\phi)^{2}-\frac{1}{12}H^{\mu\nu\rho}H_{\mu\nu\rho}-2\Lambda\right],$
(1)

where the string length $l_{s}=\sqrt{\alpha^{\prime}}$,
$H_{\mu\nu\rho}=3\partial_{[\mu}\mathbb{b}_{\nu\rho]}$ is the Kalb-Ramond
field strength and $R$ is the Ricci scalar for the string frame metric. The
cosmological term, $\Lambda$, is present for non-critical strings and scales
as $l_{s}^{-2}$. For critical strings it vanishes, so we shall neglect it in
the following. We make a cosmological ansatz for the metric, $b$-field and
dilaton,

$\displaystyle\mathbb{g}_{\mu\nu}$
$\displaystyle=\begin{bmatrix}-n^{2}(t)&0\\\ 0&g_{ij}(t)\end{bmatrix},$ (2)
$\displaystyle\mathbb{b}_{\mu\nu}$ $\displaystyle=\begin{bmatrix}0&0\\\
0&b_{ij}(t)\end{bmatrix},$ (3) $\displaystyle\phi$ $\displaystyle=\phi(t),$
(4)

where $n(t)$ is the lapse and $g_{ij}(t)$ are the components of the spatial
metric. This breaks diffeomorphisms down to the subgroup $t\to f(t),x^{i}\to
x^{i}$, with $\phi,~{}g_{ij}$ and $b_{ij}$ transforming as scalars, and the
lapse as a vector, $n\to n\dot{f}$. In this limit, the corresponding field
equations also exhibit an $\textrm{O}(d,d)$ symmetry. Indeed, the full system
can be written in terms of $\textrm{O}(d,d)$ covariant objects: a generalised
dilaton,

$\Phi=2\phi-\ln\sqrt{\det g},$ (5)

and a generalised metric

$\mathcal{H}=\begin{bmatrix}g^{-1}&-g^{-1}b\\\
bg^{-1}&g-bg^{-1}b\end{bmatrix}.$ (6)

The latter transforms as a tensor,
$\mathcal{H}\to\Omega^{T}\mathcal{H}\Omega$, under $\Omega\in\textrm{O}(d,d)$
where

$\Omega^{T}\eta\Omega=\eta,\qquad\eta=\begin{bmatrix}0&\mathbf{1}\\\
\mathbf{1}&0\end{bmatrix}.$ (7)

In contrast, the generalised dilaton and the lapse function are
$\textrm{O}(d,d)$ scalars, transforming trivially as $\Phi\to\Phi$ and $n\to
n.$

The spatial contribution to the action can now be integrated out, resulting in
a one-dimensional, two-derivative action that can be written in a form that is
manifestly invariant under the residual diffeomorphisms as well as the
underlying $\textrm{O}(d,d)$ symmetry Meissner1 ; Gasperini:1991ak ,

$I_{0}=\kappa^{2}\int\mathrm{d}t\;ne^{-\Phi}\left[-(\partial_{n}\Phi)^{2}-\frac{1}{8}\mathrm{tr}[(\partial_{n}S)^{2}]\right].$
(8)

where $\kappa^{2}>0$ is some dimensionful constant that plays no role in what
follows. Note that we have introduced the following notation for the covariant
time derivative

$\partial_{n}=\frac{1}{n(t)}\frac{\mathrm{d}}{\mathrm{d}t},$ (9)

and further defined

$S=\eta\mathcal{H}=\begin{bmatrix}bg^{-1}&\;\;g-bg^{-1}b\\\
g^{-1}&\;\;-g^{-1}b\end{bmatrix}.\\\ $ (10)

Note also that both $\eta$ and $S$ are involutory matrices, squaring to the
identity, $S^{2}=\eta^{2}=1$. The action is now seen to be manifestly
invariant under $\textrm{O}(d,d)$ since $S$ transforms as111To see this, note
that $S=\eta\mathcal{H}\to
S^{\prime}=\eta\Omega^{T}\mathcal{H}\Omega=\eta\Omega^{T}\eta S\Omega$, where
we have used the fact that $\eta^{2}=1$. It then follows from (7) that
$\eta\Omega^{T}\eta=\Omega^{-1}$ and so $S^{\prime}=\Omega^{-1}S\Omega$, as
stated in (11). In Hohm:2019ccp ; Hohm:2019jgu , these transformations are
expressed in terms of $h=\Omega^{-1}$ where (7) along with the fact that
$\eta^{2}=1$ now implies that $h\eta h^{T}=\eta$.

$S\to S^{\prime}=\Omega^{-1}S\Omega,\\\ $ (11)

We now consider the residual diffeomorphism and $\textrm{O}(d,d)$ invariant
$\alpha^{\prime}$ corrections to $I_{0}$. These terms are constructed out of
$\textrm{O}(d,d)$ scalars such as $\Phi$ and its covariant derivatives,
$\partial_{n}\Phi,\partial^{2}_{n}\Phi,\ldots$ along with traces of
$\textrm{O}(d,d)$ covariant terms such as $\partial_{n}S$ and higher covariant
derivatives $\partial_{n}^{2}S,\partial_{n}^{3}S,\ldots$. Through a judicious
use of equations of motion, field redefinitions, integration by parts, and
useful relations of the form

$\displaystyle\mathrm{tr}(S)=\mathrm{tr}(\partial_{n}S)=\mathrm{tr}(\partial_{n}^{2}S)=...$
$\displaystyle=0,$ (12) $\displaystyle\mathrm{tr}((\partial_{n}S)^{2k+1})$
$\displaystyle=0\qquad\text{for}\qquad k=0,1,...,$ (13)
$\displaystyle\mathrm{tr}(S(\partial_{n}S)^{k})$
$\displaystyle=0\qquad\text{for}\qquad k=0,1,...,$ (14)

we can eliminate all higher order derivatives in the system. Indeed, Holm and
Zwiebach Hohm:2019jgu  showed that the most general action at all order in
$\alpha^{\prime}$ must take the following form

$I=\kappa^{2}\int dt\
ne^{-\Phi}\left[-(\partial_{n}\Phi)^{2}+\sum_{k=1}^{\infty}c_{k}\alpha^{\prime
k-1}\mathrm{tr}[(\partial_{n}S)^{2k}]+\textrm{multitrace}\right]$ (15)

where $c_{1}=-\frac{1}{8}$ to match the leading order Neveu-Schwarz action
(8). For any particular string theory, there ought to be a tower of
dimensionless coefficients of higher order operators given by $c_{k}$, along
with a tower of coefficients for the multitrace operators.

##  3 Isotropic cosmologies

In the case of isotropic and spatially flat cosmology,
$g_{ij}=a^{2}(t)\delta_{ij}$, with a vanishing $b$ field, the system
simplifies considerably and the action (15) can be written in the following
compact form Hohm:2019jgu :

$I=\kappa^{2}\int\mathrm{d}t\;ne^{-\Phi}\left[-\Phi^{2}_{n}+\frac{1}{\alpha^{\prime}}f(\sqrt{\alpha^{\prime}}H)\right],$
(16)

where we have introduced the shorthand
$\Phi_{n}=\partial_{n}\Phi,\Phi_{nn}=\partial_{n}^{2}\Phi,\ldots$ and define
the string frame Hubble factor as $H=\partial_{n}\ln a$. Matching with (15) we
note that the function $f$ is analytic in a neigbourhood of the origin where
it takes the following form

$\displaystyle f(\sqrt{\alpha^{\prime}}H)$
$\displaystyle=\sum^{\infty}_{k=1}d_{k}\alpha^{\prime k}H^{2k},\qquad
d_{1}={d}.$ (17)

and the dimensionless coefficients $d_{k}$ can, in principle, be determined
from the $c_{k}$ and the coefficients of the multitrace operators. Note that
$f$ is a manifestly even function in $H$. This follows from the
$\textrm{O}(d,d)$ symmetry which takes the form of a simple scale factor
duality, $a\to 1/a$ and so $H\to-H$, in the homogeneous and isotropic limit.

In the remainder of the paper, we choose units where $\alpha^{\prime}=1$. It
will also be convenient to represent the scale factor through its logarithm,
$q=\ln a$. The equations of motion for the full system can be obtained from
varying the action with respect to the lapse, the dilaton and the scale
factor. This yields the equations $E_{a}=0$ where $a=n,\Phi,q$ and

$\displaystyle E_{n}$ $\displaystyle=\frac{\delta I}{\delta
n}=\kappa^{2}e^{-\Phi}\left[\Phi^{2}_{n}-L_{f}(H)\right],$ (18) $\displaystyle
E_{\Phi}$ $\displaystyle=\frac{\delta
I}{\delta\Phi}=\kappa^{2}ne^{-\Phi}\left[2\Phi_{nn}-\Phi^{2}_{n}-f(H)\right],$
(19) $\displaystyle E_{q}$ $\displaystyle=\frac{\delta I}{\delta
q}=-\kappa^{2}\frac{\mathrm{d}}{\mathrm{d}t}\left[e^{-\Phi}f^{\prime}(H)\right].$
(20)

Here, $L_{f}(H)=Hf^{\prime}(H)-f(H)$ denotes the Legendre transform of $f$ and
$f^{\prime}(H)=\mathrm{d}f/\mathrm{d}H$. Thanks to the residual
diffeomorphisms, these equations are not independent and satisfy the following
identity

$\frac{\mathrm{d}}{\mathrm{d}t}E_{n}=\Phi_{n}E_{\Phi}+HE_{q}.$ (21)

We can include matter in an $\textrm{O}(d,d)$ invariant way by including a
matter action of the form $S_{m}[\Phi,\mathcal{S},\chi_{I}]$, where
$\chi_{I}(t)$ are $\textrm{O}(d,d)$ invariant matter fields Gasperini:1991ak ;
Bernardo:2019bkz . For a homogeneous and isotropic cosmology, this is
equivalent to $S_{m}[\Phi,q,\chi_{I}]$. The vacuum equations of motion are now
sourced as follows

$E_{n}=\bar{\rho},\quad E_{q}=-dn\bar{p},\quad
E_{\Phi}=\frac{1}{2}n\bar{\sigma}$ (22)

where the $\textrm{O}(d,d)$ covariant energy density, pressure and dilaton
charge are respectively given by

$\bar{\rho}=-\frac{\delta S_{m}}{\delta
n}\Big{|}_{\Phi,q},\quad\bar{p}=\frac{1}{dn}\frac{\delta S_{m}}{\delta
q}\Big{|}_{n,\Phi},\quad\bar{\sigma}=-\frac{2}{n}\frac{\delta
S_{m}}{\delta\Phi}\Big{|}_{n,q}.$ (23)

Note that the $\textrm{O}(d,d)$ covariant sources differ from their
diffeomorphism invariant counterparts by a factor of $\sqrt{-g}=a^{d}$,
namely,
$\rho=\bar{\rho}/a^{d},~{}p=\bar{p}/a^{d},~{}\sigma=\bar{\sigma}/a^{d}$. For
the pressure, we should also note that $p$ is now defined by holding
$\Phi=2\phi-dq$ fixed when we vary the matter action. This differs from the
usual definition of pressure given by

$p_{\text{standard}}=\frac{1}{dna^{d}}\frac{\delta S_{m}}{\delta
q}\Big{|}_{n,\phi},$

where $\phi$ is held fixed instead. It is easy enough to show that the two
pressures can be related using the dilaton charge,
$p_{\text{standard}}=p+\frac{1}{2}\sigma$ Quintin:2021eup . Note also that
diffeomorphism invariance in the matter sector implies that

$\partial_{n}\rho+dH(\rho+p)=\frac{1}{2}\Phi_{n}\sigma.$ (24)

Although we have included the matter sources for completeness, in the
remainder of this paper we shall mostly neglect them, since our focus will be
on finding consistent solutions to the vacuum field equation. In particular,
we will seek conditions for consistent de Sitter vacua in both the string and
the Einstein frame. We refer the reader to Gasperini:1991ak ; Bernardo:2019bkz
for more details on matter couplings.

###  3.1 dS solutions in the string frame

It was already shown in Hohm:2019ccp ; Hohm:2019jgu  that string frame de
Sitter solutions are possible in this class of $\alpha^{\prime}$ complete
cosmologies for suitable choices of the function $f(H)$. These solutions are
necessarily non-perturbative in $\alpha^{\prime}$, and could not have been
found using standard techniques in perturbative string theory. However, it has
also been shown that these solutions are unstable under vacuum fluctuations
Bernardo:2020zlc ; Bieniek:2022mrv ; Bernardo:2022nex . We will review the
derivation of these solutions and add some additional insight into the nature
of the instability.

We know that in a neighbourhood of the origin, the function $f(H)$ is even and
admits the following Taylor expansion in $H^{2}$

$f(H)={d}H^{2}+d_{2}H^{4}+\ldots$ (25)

Although we cannot assume that the function remains analytic for all values of
$H$, we shall assume that it is twice differentiable in order to have well
defined field equations. We shall also assume it to be an even function in
order to preserve the $\textrm{O}(d,d)$ symmetry.

Here we are interested in the properties of the function in a neighbourhood of
a string frame de Sitter solution with $H=H_{c}>0$, for some constant $H_{c}$.
Whilst it is tempting to motivate this with the current phase of accelerated
expansion, it is important to note that we are working in $d+1$ dimensions.
For the critical dimension ($d+1=10,26$), we must perform compactification
down to four dimensions before making contact with observation. Therefore, we
remain agnostic as to the value of $H_{c}$.

Setting $E_{a}=0$ for $a=n,\Phi,q$, we immediately see that $H=H_{c}>0$, for
some constant $H_{c}$, if and only if

$\Phi^{2}_{n}=f(H_{c})=L_{f}(H_{c}),\quad f(H_{c})f^{\prime}(H_{c})=0,$ (26)

which is equivalent to $\Phi_{n}=f(H_{c})=f^{\prime}(H_{c})=0$. This means
that $f(H)$ must have a double zero at $H=H_{c}$. Thanks to the
$\textrm{O}(d,d)$ symmetry, it must also have a double zero at $H=-H_{c}$.
This suggests the following general form

$f(H)=\frac{g(H)}{H_{c}^{4}}(H^{2}-H_{c}^{2})^{2}$ (27)

for some even function $g(H)$ that admits a Taylor expansion in
$H^{2}-H_{c}^{2}$

$g(H)=g_{0}+g_{1}(H^{2}-H_{c}^{2})+\ldots.$ (28)

For the form of $f(H)$ given in (27) to overlap with the form close to the
origin (25), we obtain the following constraints

$g(0)=g^{\prime}(0)=0,\qquad g^{\prime\prime}(0)=2{d}.$ (29)

It is also important to establish the stability of these de Sitter vacua. Of
course, a full analysis of cosmological perturbations requires knowledge of
the underlying theory beyond the homogeneous limit under consideration.
Although there has been some interesting recent work in this direction
Codina:2023fhy , it is well known that $\textrm{O}(d,d)$ invariance in $d+1$
dimensions is at odds (!) with manifest diffeomorphism invariance, as
evidenced through double field theory Hohm:2013jaa ; Hohm:2014xsa  (see also
Brustein:1998kq  for a related duality observed in cosmological
perturbations). Faced with these limitations, we can only consider the
stability of solutions under homogeneous perturbations captured by the general
action (15). Instabilities for homogeneous and isotropic perturbations of
string frame de Sitter vacua were identified in Bernardo:2020zlc ;
Bieniek:2022mrv ; Bernardo:2022nex  with subsequent extensions to anisotropic
perturbations Bieniek:2023ubx . Here we take a subtlely different approach to
those works but draw the same conclusions.

We consider only homogeneous and isotropic perturbations, so that the system
of equations $E_{i}=0$ remains unchanged. Assuming $f^{\prime\prime}(H)\neq
0$, these can be written in the following form

$f^{\prime}(H)=ce^{\Phi},\qquad\partial_{n}H=s\frac{\sqrt{L_{f}(H)}f^{\prime}(H)}{f^{\prime\prime}(H)},$
(30)

where $c$ is a constant of integration. Here $s=\pm 1$ corresponds to the sign
of $\dot{\Phi}=d\Phi/dt$, appearing when we take the square root in (18). This
ability to take either sign was identified as a potential source of error in
the stability analysis, as discussed in Bernardo:2022nex . Next we will show
how the choice of sign plays a crucial role in opening up the unstable
channel.

We begin by making a choice of the time co-ordinate by setting the lapse
function to unity, $n=1$, and solve for the Hubble parameter in a
neighbourhood of the vacuum, giving

$H=H_{c}+\delta H$ (31)

where $\delta H$ satisfies

$\dot{\delta H}\approx s\sqrt{H_{c}f^{\prime\prime}(H_{c})}\delta H^{3/2}.$
(32)

This gives a consistent solution

$\delta H\approx\frac{4}{H_{c}f^{\prime\prime}(H_{c})(t-t_{c})^{2}}$ (33)

with $s=-\text{sgn}(t-t_{c})$. We immediately see the unstable channel if and
only if the initial time $t_{i}<t_{c}$, or equivalently $s=+1$. This means the
instability is excited if and only if $\dot{\Phi}(t_{i})>0$.

###  3.2 dS solutions in Einstein frame

We now turn our attention to the Einstein frame. Einstein frame de Sitter
solutions with constant $\dot{\Phi}$ were obtained in Bernardo:2019bkz  in the
presence of a source with $\bar{p}=-\bar{\rho}$. Here we shall present
Einstein frame de Sitter vacua as solutions to the vacuum equations of motion
for bespoke choices of the function $f(H)$. Although the solutions are stable,
we will also show that the required form of $f(H)$ cannot be smoothly
connected to the leading order Neveu-Schwarz theory in a neighbourhood of
$H=0$. This agrees with the conclusions drawn in EFdS .

The string frame and the Einstein frame are related by a conformal
transformation. Labelling the Einstein frame with the letter “E”, we write
$\mathbb{g}_{\mu\nu}=\mathcal{A}^{2}\mathbb{g}_{\mu\nu}^{E}$ and so
$n=\mathcal{A}n_{E}$, $a_{S}=\mathcal{A}a_{E}$, where $\mathcal{A}$ is the
conformal factor to be derived presently. Focussing on the leading order
curvature term in the action, we note that

$\int\mathrm{d}^{1+d}x\sqrt{-\mathbb{g}}e^{-2\phi}R=\int\mathrm{d}^{1+d}x\sqrt{-\mathbb{g}_{E}}e^{-2\phi}\mathcal{A}^{d-1}\left[R_{E}+\ldots\right]$
(34)

where the ellipsis include derivatives of $\mathcal{A}$ and $R_{E}$ is the
Ricci scalar for the transformed metric. For the latter to coincide with the
Einstein frame metric, we must choose $\mathcal{A}=e^{\frac{2\phi}{d-1}}$.

Setting $E_{i}=0$ for $i=n,\Phi,q$, we see that the vacuum field equations are
equivalent to

$f^{\prime}(H)=ce^{\Phi},\qquad\Phi_{n}=s\sqrt{L_{f}(H)}$ (35)

where we recall that $c$ is a constant of integration and $s=\pm 1$. We wish
to recast these equations in terms of the Hubble parameter defined in the
Einstein frame

$H_{E}=\partial_{n_{E}}q_{E}$ (36)

where $\partial_{n_{E}}=\frac{1}{n_{E}}\frac{d}{dt}$ and $q_{E}=\ln a_{E}$.
From the definition of the generalised dilaton $\Phi=2\phi-dq$ and the
conformal mapping between frames $q-q_{E}=\ln(n/n_{E})=\frac{2\phi}{d-1}$, we
can derive the following useful relations

$\Phi=-q-(d-1)q_{E}\implies\Phi_{n}=-H-(d-1)\frac{n_{E}}{n}H_{E}$ (37)

and

$\phi=-\frac{(d-1)}{2}(\Phi+dq_{E})\implies\frac{n}{n_{E}}=\frac{e^{-\Phi}}{\mathcal{V}_{E}}$
(38)

where $\mathcal{V}_{E}=a_{E}^{d}$ is volume scale factor in Einstein frame.
After some manipulations, the system of equations (35), (37) and (38) can be
expressed as follows

$e^{\Phi}=\frac{f^{\prime}(H)}{c},\qquad\frac{n}{n_{E}}=\frac{c}{f^{\prime}(H)\mathcal{V}_{E}},\qquad\partial_{n_{E}}\mathcal{V}_{E}=c\mathcal{F}(H),\qquad\partial_{n_{E}}H=\frac{c\mathcal{G}(H)}{\mathcal{V}_{E}}$
(39)

where

$\mathcal{F}(H)=\frac{d}{1-d}\left[\frac{H+s\sqrt{L}}{f^{\prime}(H)}\right],\qquad\mathcal{G}(H)=\frac{s\sqrt{L}}{f^{\prime\prime}(H)}$
(40)

and we have assumed $f^{\prime}(H)f^{\prime\prime}(H)\neq 0$. To explore the
possibility of a de Sitter vacuum in the Einstein frame, it is convenient to
construct

$\partial_{n_{E}}H_{E}=dH_{E}^{2}\left[\frac{\mathcal{V}_{E}\partial_{n_{E}}^{2}\mathcal{V}_{E}}{(\partial_{n_{E}}\mathcal{V}_{E})^{2}}-1\right]=dH_{E}^{2}\left[\frac{\mathcal{F}^{\prime}(H)\mathcal{G}(H)}{\mathcal{F}(H)^{2}}-1\right],\qquad
c\mathcal{F}(H)\neq 0.$ (41)

If we choose the function $f(H)$ such that the right hand of this equation
vanishes identically, the Hubble parameter in Einstein frame is guaranteed to
be constant, $H_{E}=H_{\star}$, for a solution to the vacuum field equations.
Thus, the existence of the de Sitter vacuum in Einstein frame requires that

$\left(\frac{1}{\mathcal{F}}\right)^{\prime}+\frac{1}{\mathcal{G}}=0,$ (42)

or in other words, that $f(H)$ satisfies the following differential equation,

$\frac{d+1}{d-1}sH\sqrt{Hf^{\prime}-f}f^{\prime\prime}+\frac{1}{d-1}(Hf^{\prime}-f)f^{\prime\prime}+\frac{d}{d-1}H^{2}f^{\prime\prime}+s\sqrt{Hf^{\prime}-f}f^{\prime}+\frac{1}{2}Hf^{\prime}f^{\prime\prime}=0.$
(43)

Note further that we can use the third equation in (39) to show that the
Hubble parameters in the Einstein and string frame are related according to
the following formula

$H_{E}=\frac{c\mathcal{F}(H)}{d\mathcal{V}_{E}}.$ (44)

For a suitable choice of $f(H)$ satisfying (43), the corresponding constant
curvature solution is stable under vacuum perturbations, as it is the only
solution to (41).

Finding a suitable choice of $f(H)$ is not easy as the differential equation
(43) is challenging to solve in full generality. However, if we assume that
$H>0$, it does admit the following analytic solution when $s=-1$,

$\bar{f}(H)=(H+H_{m})^{2}-H^{2}_{m},$ (45)

where $H_{m}$ is a constant.

With $f=\bar{f}$ and taking the negative ($s=-1$) root we find that
$\mathcal{F}=0$ and $\mathcal{G}=-H/2$. The vanishing of $\mathcal{F}$
suggests this solution lies outside of the original regime of validility for
which (43) was derived. Nevertheless, it is interesting to explore the
solutions to the system of equations (39) in this particular case. It turns
out that they imply a constant but vanishing Hubble rate in Einstein frame

$H_{\star}=0,$ (46)

consistent with (44), along with the following solutions for the string frame
Hubble parameter, the dilaton and the lapse,

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle
H_{c}\exp\left(-\frac{ct}{2a_{0}^{d}}\right),$ (47) $\displaystyle\Phi(t)$
$\displaystyle=$
$\displaystyle\ln\left[\frac{2}{c}\left(H_{c}\exp\left(-\frac{ct}{2a_{0}^{d}}\right)+H_{m}\right)\right],$
(48) $\displaystyle n(t)$ $\displaystyle=$
$\displaystyle\frac{c}{2a_{0}^{d}}\left[H_{c}\exp\left(-\frac{ct}{2a_{0}^{d}}\right)+H_{m}\right]^{-1},$
(49)

where $H_{c}$ is a dimensionful constant of integration and we have assumed
$d\neq 1$ and $c\neq 0$. Clearly the solution runs into a singularity as
$H\to-H_{m}$ although this can be trivially avoided by taking $H_{m}>0$.

We have not been able to find an exact solution to (43) that gives non-
vanishing $\mathcal{F}$ and, correspondingly, $H_{\star}>0$. However,
numerical solutions are presented in figure 1, with the corresponding value of
$H_{E}$ shown in figure 2, where we recall that we have set units such that
$\alpha^{\prime}=1$.

Figure 1: Examples of numerically computed $f(H)$, with the central black
curve corresponding to $\bar{f}(H)$.

Figure 2: Time evolution of the Einstein-frame Hubble parameter for the $f(H)$
given in Fig. 1

.

We can also seek a perturbative solution to (43). To this end, we consider

$f(H)=\bar{f}(H)\left[1+\epsilon(H)\right],$ (50)

where we assume $|\epsilon|\ll 1$. Plugging this into (43) (with $s=-1$) and
keeping terms up to linear order in $\epsilon$, we arrive at the linear
differential equation

$(H_{m}-H)(2H_{m}-H)H\epsilon^{\prime\prime}(H)+(2H_{m}^{2}-3HH_{m}+2H^{2})\epsilon^{\prime}(H)-H_{m}\epsilon(H)=0$
(51)

with the following solution

$\epsilon(H)=\frac{C_{1}H_{m}+C_{2}H+h_{m}C_{2}\ln\left|\frac{H}{H_{m}}\right|}{H+2h_{m}},$
(52)

where $C_{1}$ and $C_{2}$ are constants of integration. Note that $C_{1}$ can
be removed by redefining $H_{m}\to H_{m}\left(1-C_{1}/2\right)$ and working to
linear order in $C_{1}$. Thus we can set $C_{1}=0$ without loss of generality
and express (52) as a simple function of the ratio $\delta=H/H_{m}$,

$\epsilon(H)=C_{2}\left[\frac{\delta+\ln|\delta|}{\delta+2}\right].$ (53)

Recall that this expression is valid as long as $|\epsilon(H)|\ll 1$. If we
assume $|C_{2}|\ll 1$ then this condition is violated if and only if $\delta$
strays too close to the singularities at $\delta=0,-2$. More precisely, we
have $\epsilon(H)\ll 1$ whenever $|C_{2}|\ll 1$, provided

$|\delta|\gg e^{-2/|C_{2}|}\ ,\quad\Big{|}\delta+2\Big{|}\gg|C_{2}|\left(2-\ln
2\right).$

We now revisit the field equations (39) and (42) using the perturbed form of
$f$ given by (53). To leading order in $|C_{2}|\ll 1$ (42) gives the following
expression for Hubble rate in Einstein frame

$H_{E}\approx\frac{C_{2}c}{4(d-1)\mathcal{V}_{E}}.$ (54)

If we assume $H_{E}=H_{\star}$ for some constant $H_{\star}$ and set
$n_{E}=1$, it follows that that we have
$\mathcal{V}_{E}=a_{0}^{d}e^{dH_{\star}t}$. We can then solve (54) explicitly,
giving

$H_{\star}\approx\frac{1}{dt}W\left(\frac{dC_{2}ct}{4(d-1)a_{0}^{d}}\right)\approx\frac{C_{2}c}{4(d-1)a_{0}^{d}}$
(55)

where we have used the fact that the Lambert W function $W(x)\approx x$ for
small $x$. We see that we have a constant and non-vanishing Hubble rate in the
Einstein frame, with the approximation valid as long as $|t|\ll
4(d-1)a_{0}^{d}/d|c||C_{2}|$. Since $H_{\star}$ is required to be positive in
an expanding universe, we assume that $C_{2}c>0$. The remaining fields are
dominated by their non-vanishing zeroth order (in $\epsilon$) solutions
(47)-(49). Recall that in a neighbourhood of the origin, perturbative string
theory suggests that the function $f(H)$ should admit a Taylor expansion of
the form (25). We will now show that any solution to the differential equation
(43) valid for $H>0$ cannot be analytically continued towards a solution of
the form (25) in a neighbourhood of $H=0$.

To see this, we begin by assuming that $f(H)$ admits a Taylor expansions of
the form (25) in a neighbourhood of the origin

$f(H)={d}H^{2}+\mathcal{O}(H^{4}).$ (56)

In this neighbourhood, it immediately follows that

$\frac{\mathcal{F}^{\prime}(H)\mathcal{G}(H)}{\mathcal{F}(H)^{2}}=\mathcal{O}(H^{2}).$
(57)

In order to satisfy (43), the right hand side of this expression should be
equal to unity, which is clearly not possible in a neighbourhood of $H=0$,
regardless of the sign of $s=\pm 1$. Thus, it is not possible to smoothly
connect a solution to (43) to the desired leading order Neveu-Schwarz theory
at low curvatures. However, perhaps this is too demanding. As noted earlier,
for the field equations to be well defined, we only need the form of $f(H)$ to
be twice differentiable. For any finite $N$, it is certainly possible to match
the first $N$ derivatives of a solution to (43) to a function of the form
(56), at some value of $H=H_{\text{match}}>0$.

It is, perhaps, optimistic to expect such a matching to emerge from a non-
perturbative understanding of string theory. Nevertheless, we could imagine a
discontinuity in the higher order derivatives of $f(H)$ to be a remnant of the
truncation to the massless sector of the theory. Indeed, if it emerges from
string oscillators with string scale masses, we might expect
$H_{\text{match}}\sim 1$ (in units of $\alpha^{\prime}=1$).

Even if this desired form of $f(H)$ can be justified, it is clear that we can
only trust our de Sitter vacuum in the regime where $H>H_{\text{match}}$.
However, recall that $H$ is the Hubble parameter in the string frame and this
will change over time. If the system evolves into a regime where
$H<H_{\text{match}}$, the form of $f(H)$ will no longer admit de Sitter
solutions and the Einstein frame de Sitter vacuum is destabilised.

To avoid this instability, we need to stop $H$ evolving into the unstable
region with $H<H_{\text{match}}$ in either time direction. In other words, we
need $H$ to have a turning point in its evolution at some
$H_{\text{turn}}>H_{\text{match}}$. At such a turning point, the last equation
in (39) tells us that $\mathcal{G}(H_{\text{turn}})=0$. Since
$H_{\text{turn}}>H_{\text{match}}$ by assumption, the differential equation
(42) ought to be satisfied at $H=H_{\text{turn}}$, further implying that
$\mathcal{F}(H_{\text{turn}})=0$ must have a zero at the same point. Since we
assume that $f(H)$ is twice differentiable,
$\mathcal{F}(H_{\text{turn}})=\mathcal{G}(H_{\text{turn}})=0$ if and only if
$H_{\text{turn}}=0$. This contradicts the underlying assumption that
$H_{\text{turn}}>H_{\text{match}}>0$. It follows that our de Sitter vacua in
Einstein frame cannot be eternal.

##  4 Anisotropic cosmologies

We now turn our attention to anisotropic cosmologies in $d+1$ spacetime and
consider a string frame line element of the form

$ds^{2}=-n(t)^{2}dt^{2}+\sum_{i=1}^{d}a_{i}(t)^{2}dx_{i}^{2},$ (58)

where $a_{i}(t)$ is the scale factor along the $i$th spatial direction. A
detailed derivation of the $\textrm{O}(d,d)$ invariant action requires a
careful computation of all the terms in (15), including the multitrace
operators. Instead of computing that explicitly, we note that the action will
take the form

$I=\kappa^{2}\int\mathrm{d}t\;ne^{-\Phi}\left[-\Phi^{2}_{n}+f(H_{i})\right],$
(59)

where we have assumed units with $\alpha^{\prime}=1$. The Hubble parameter
along the $i$th direction is given by $H_{i}=\partial_{n}q_{i}$ for $q_{i}=\ln
a_{i}$, while the generalised dilaton is given in terms of the standard
dilaton via the following relation

$\Phi=2\phi-\sum_{i}q_{i}.$ (60)

To match with the leading order Neveu-Schwarz action (1) at low curvatures, we
expect that $f$ admits a Taylor expansion near the origin such that

$f(H_{i})=\sum_{i}H_{i}^{2}+\mathcal{O}(H^{4}).$ (61)

The $\textrm{O}(d,d)$ symmetry now includes scale factor duality along each
direction, $a_{i}\to 1/a_{i}$ and scale factor exchange $a_{i}\leftrightarrow
a_{j}$ for each $i$ and $j$. This suggests that the function $f(H_{i})$ can
also be expressed as

$f(H_{i})=F\left(\hat{e}^{d}_{i}(H_{j}^{2})\right),$ (62)

where we define the normalised elementary symmetric polynomials

$\hat{e}_{i}^{d}(X_{j})=\frac{1}{\binom{d}{i}}\sum_{1\leq j_{1}<\ldots
j_{i}\leq d}X_{j_{1}}\ldots X_{j_{i}}$ (63)

for $i=1,\ldots d$.

Varying the action (59) with respect to the lapse, the dilaton and each of the
scale factors, we obtain $E_{a}=0$ for $a=n,\Phi,q_{i}$, where

$\displaystyle E_{n}$ $\displaystyle=\frac{\delta I}{\delta
n}=\kappa^{2}e^{-\Phi}\left[\Phi^{2}_{n}-L_{f}(H_{i})\right],$ (64)
$\displaystyle E_{\Phi}$ $\displaystyle=\frac{\delta
I}{\delta\Phi}=\kappa^{2}ne^{-\Phi}\left[2\Phi_{nn}-\Phi^{2}_{n}-f(H_{i})\right],$
(65) $\displaystyle E_{q_{i}}$ $\displaystyle=\frac{\delta I}{\delta
q_{i}}=-\kappa^{2}\frac{\mathrm{d}}{\mathrm{d}t}\left[e^{-\Phi}f_{i}\right],$
(66)

where we denote partial derivatives $f_{i}=\partial f/\partial H_{i}$ and the
Legendre transform $L_{f}(H_{i})=\sum_{i}H_{i}f_{i}-f(H_{i})$. It is also
convenient to define the average Hubble parameter in string frame as
$H=\frac{1}{d}\sum_{i}H_{i}$.

###  4.1 dS solutions in the string frame

We now seek solutions to the anisotropic field equations corresponding to the
string frame de Sitter vacua. Using similar arguments to the isotropic case in
section 3.1, we see that there is an isotropic solution to (64) to (66) with
$H_{i}=H_{c}>0$ for all $i$ and some constant $H_{c}$ if and only if

$\Phi_{n}=f(H_{c},\ldots,H_{c})=f_{i}(H_{c},\ldots,H_{c})=0.$ (67)

Given the form of $f$ dictated by the $\textrm{O}(d,d)$ symmetry (62), this
suggests that

$f(H_{i})=\sum_{j}\frac{g_{j}(H_{l})}{H_{c}^{4j}}(\hat{e}_{j}^{d}(H_{l}^{2})-H_{c}^{2j})^{2}$
(68)

where the functions $g_{j}(H_{k})$ take the $\textrm{O}(d,d)$ symmetric form
(62) and admit a Taylor expansion in each of
$\hat{e}_{j}^{d}(H_{l}^{2})-H_{c}^{2j}$,

$g_{j}(H_{k})=g_{j0}+\sum_{k}g_{jk}(\hat{e}_{k}^{d}(H_{l}^{2})-H_{c}^{2k})+\ldots.$
(69)

Close to the origin, the form of $f$ should coincide with (61) to recover the
leading order Neveu-Schwarz theory at small curvatures. This implies that

$\sum_{j}g_{j}(0,\ldots,0)=\sum_{j}\partial_{H_{k}}g_{j}(0,\ldots,0)=0,\qquad\sum_{j}\partial_{H_{k}}\partial_{H_{l}}g_{j}(0,\ldots,0)=2\delta_{kl}.$
(70)

On the isotropic line where $H_{i}=H$ for all $i$, the form of $f$ is
compatible with the isotropic solutions (27), as expected

$f(H_{i})|_{\text{iso}}=\sum_{j}\frac{g_{j}(H,\ldots,H)}{H_{c}^{4j}}(H^{2j}-H_{c}^{2j})^{2}.$
(71)

It immediately follows that the corresponding de Sitter vacua will once again
be unstable to isotropic fluctuations.

###  4.2 dS solutions in the Einstein frame

We now switch our attention to the Einstein frame to see if the family of de
Sitter solutions found in the isotropic framework EFdS  carries over to the
anisotropic framework. As usual, we perform a conformal transformation to
switch to Einstein frame, with $n=\mathcal{A}n_{E}$ and
$a_{i}=\mathcal{A}a_{i}^{E}$ where $\mathcal{A}=e^{\frac{2\phi}{d-1}}$. We
therefore have $\ln(n/n_{E})=q_{i}-q_{i}^{E}=\frac{2\phi}{d-1}$ for
$q_{i}^{E}=\ln a_{i}^{E}$. This allows us to extract the following useful
relation

$\phi=-\frac{d-1}{2}\left(\Phi+\sum_{i}q_{i}^{E}\right),$ (72)

implying

$\frac{n}{n_{E}}=\frac{e^{-\Phi}}{\mathcal{V}_{E}},\qquad
q_{i}-q_{i}^{E}=-\Phi-\ln\mathcal{V}_{E}$ (73)

where $\mathcal{V}_{E}=\prod_{i}(a_{i}^{E})^{k_{i}}$ is the Einstein frame
volume scale factor.

In the Einstein frame, we define Hubble parameters
$H_{i}^{E}=\partial_{n_{E}}q_{i}^{E}$ and the corresponding average
$H^{E}=\frac{1}{d}\sum_{i}H_{i}^{E}$. These are easily related to their string
frame counterparts via the relation

$H_{i}=\frac{n_{E}}{n}\left(H_{i}^{E}-dH^{E}\right)-\Phi_{n}.$ (74)

From the field equations (64) to (66) we can derive expressions for the
generalised velocities in Einstein frame

$\partial_{n_{E}}\Phi=\frac{e^{-\Phi}}{\mathcal{V}_{E}}s\sqrt{L_{f}},\qquad
H_{i}^{E}-H^{E}=\frac{e^{-\Phi}}{\mathcal{V}_{E}}\left(H_{i}-H\right),\qquad
H^{E}=\frac{e^{-\Phi}}{\mathcal{V}_{E}}\frac{\Sigma}{d}\mathcal{F}$ (75)

where

$\mathcal{F}=\frac{d}{1-d}\left(\frac{H+s\sqrt{L_{f}}}{\Sigma}\right),\qquad\Sigma=\sqrt{\sum_{j}f_{j}^{2}}$
(76)

and $s=\pm 1$, depending on the sign of $\Phi_{n}$. From equation (66), we
have that

$e^{-\Phi}f_{i}=c_{i}\implies
e^{-\Phi}\Sigma=|\underline{c}|\equiv\sqrt{\sum_{j}c_{j}^{2}}$ (77)

for some constants $c_{i}$. We can now derive the following constraints on the
dynamics of the system, analogous to the last two equations in (39)

$\qquad\partial_{n_{E}}\mathcal{V}_{E}=|\underline{c}|\mathcal{F},\qquad\partial_{n_{E}}H_{i}=\frac{|\underline{c}|\mathcal{G}_{i}}{\mathcal{V}_{E}}$
(78)

where

$\mathcal{G}_{i}=\frac{s\sqrt{L_{f}}\sum_{j}f^{ij}f_{j}}{\Sigma}$ (79)

and $f^{ij}$ denotes the inverse of the Hessian of the function $f$, assumed
to be well defined. We proceed in a similar way to the isotropic case, and
compute the rate of change of the Einstein frame Hubble parameters. We find
that

$\partial_{n_{E}}H_{i}^{E}=d(H^{E})^{2}\left[(1+\xi_{i})(\mathcal{O}\ln\mathcal{F}-1)+\mathcal{O}\xi_{i}\right],$
(80)

where

$\displaystyle\mathcal{O}$
$\displaystyle=\frac{1}{\mathcal{F}}\sum_{j}\mathcal{G}_{j}\partial_{H_{j}},$
(81) $\displaystyle\xi_{i}$
$\displaystyle=\frac{d(H_{i}-H)}{\Sigma\mathcal{F}}.$ (82)

Since $\sum_{j}\xi_{j}=0$, it follows that the rate of change of the average
Hubble factor is given by a somewhat simpler formula

$\partial_{n_{E}}H^{E}=d(H^{E})^{2}(\mathcal{O}\ln\mathcal{F}-1),$ (83)

which is strongly reminiscent of the corresponding isotropic equation (41), as
expected. We may alternatively express (80) as

$\partial_{n_{E}}H_{i}^{E}-(1+\xi_{i})\partial_{n_{E}}H^{E}=d(H^{E})^{2}\mathcal{O}\xi_{i}.$
(84)

Let us now evaluate our system on the isotropic line $H_{i}=H$, along which we
have $\xi_{i}=0$ from (82). For consistency, it must follow that

$(\partial_{n_{E}}H^{E})|_{\text{iso}}=d(H^{E})^{2}(\mathcal{O}\ln\mathcal{F}-1)|_{\text{iso}},\qquad(\mathcal{O}\xi_{i})|_{\text{iso}}=0$
(85)

where $|_{\text{iso}}$ indicates evaluation on the isotropic line. Requiring
the vacuum to be de Sitter in the Einstein frame on the isotropic line, i.e.
$H_{i}^{E}=H^{E}=const$, now yields the following constraints

$\mathcal{O}(\ln\mathcal{F})|_{\text{iso}}=1,\qquad(\mathcal{O}\xi_{i})|_{\text{iso}}=0.$
(86)

Defining $\mathcal{G}=\sum_{i}\mathcal{G}_{i}/d$, we note that

$\displaystyle\mathcal{O}\left(\ln\mathcal{F}\right)=-\sum_{i}\mathcal{G}_{i}\partial_{H_{i}}\left(\frac{1}{\mathcal{F}}\right),$
(87)
$\displaystyle\mathcal{O}(\xi_{i})=\frac{1}{\Sigma\mathcal{F}^{2}}(\mathcal{G}_{i}-\mathcal{G})-\xi_{i}\mathcal{O}\ln(\Sigma\mathcal{F}),$
(88)

and impose (86), giving

$\left[\sum_{i}\partial_{H_{i}}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{\mathcal{G}}\right]_{\text{iso}}=0,\qquad\mathcal{G}_{i}|_{\text{iso}}=\mathcal{G}|_{\text{iso}}\neq
0.$ (89)

To explore the implications of these results, consider the $N$ dimensional
Euclidean space spanned by the Hubble parameters, $(H_{1},\ldots,H_{d})$. It
is convenient to introduce a new set of coordinates $(H,v_{\alpha})$, adapted
to the isotropic line, defined in terms of the average Hubble parameter
$H=\sum_{i}H_{i}/d$ and $d-1$ orthogonal directions

$\displaystyle v_{\alpha}$ $\displaystyle=\sum_{i}(H_{i}-H)n_{i}^{\alpha},$
(90)

so that

$H_{i}=H+\sum_{\alpha}n^{\alpha}_{i}v_{\alpha}.$ (91)

Here we have introduced an orthonormal set of vectors in Hubble space,
$n^{\alpha}_{i}$, orthogonal to the tangent vector, $u_{i}=1/d$, on the
isotropic line

$\sum_{i}u_{i}n_{i}^{\alpha}=\frac{1}{d}\sum_{i}n_{i}^{\alpha}=0,\qquad\sum_{i}n_{i}^{\alpha}n_{i}^{\beta}=\delta^{\alpha\beta}.$
(92)

We can now think of the isotropic line as being parametrized by the average
Hubble parameter, $H$, with each of the remaining coordinates vanishing there,
$v_{\alpha}|_{\text{iso}}=0$. At this point, we recall that the
$\textrm{O}(d,d)$ symmetry implies that $f(H_{i})$ is given in terms of the
normalised elementary symmetric polynomials, as in (62). In a neighbourhood of
the isotropic line, we can show that

$\hat{e}^{d}_{i}(H_{j}^{2})=H^{2i}+\frac{i(d+1-2i)}{d(d-1)}H^{2(i-1)}\sum_{\alpha,\beta}\delta^{\alpha\beta}v_{\alpha}v_{\beta}+\mathcal{O}(v^{3}).$
(93)

Correspondingly we find that the function $f$ takes the form

$f(H_{i})=f_{0}(H)+g_{0}(H)\sum_{\alpha,\beta}\delta^{\alpha\beta}v_{\alpha}v_{\beta}+\mathcal{O}(v^{3}),$
(94)

where

$f_{0}(H)=F(H^{2},\ldots,H^{2d}),\qquad
g_{0}(H)=\sum_{i=1}^{d}\frac{i(d+1-2i)}{d(d-1)}H^{2(i-1)}\frac{\partial
F}{\partial\hat{e}^{d}_{i}}\Bigg{|}_{\hat{e}^{d}_{j}=H^{2j}}.$ (95)

For the de Sitter vacuum to exist, we require the conditions (89) to hold. The
second of these is equivalent to

$\left[\sum
u_{i}\mathcal{G}_{i}\right]_{\text{iso}}=\mathcal{G}_{\text{iso}}\neq
0,\qquad\left[\sum n_{i}^{\alpha}\mathcal{G}_{i}\right]_{\text{iso}}=0.$ (96)

Making use of the chain rule

$\frac{\partial}{\partial H}=\sum_{i}\frac{\partial}{\partial
H_{i}},\qquad\frac{\partial}{\partial
v_{\alpha}}=\sum_{i}n_{i}^{(\alpha)}\frac{\partial}{\partial H_{i}},$ (97)

and assuming $g_{0}(H)\neq 0$, we immediately see that the condition (96)
holds automatically. Meanwhile, the first condition in (89) yields

$\left[\partial_{H}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{\mathcal{G}}\right]_{\text{iso}}=0,$
(98)

which is equivalent to (43) for $f(H)=f_{0}(H)$. It follows that the Einstein
frame de Sitter vacua found in the isotropic framework extend naturally to the
anisotropic framework. This is because the underlying $\textrm{O}(d,d)$
symmetry ensures that deviations from isotropy only enter the theory at
$\mathcal{O}(v^{2})\sim\mathcal{O}((H_{i}-H_{j\neq i})^{2})$.

##  5 Conclusions

Can we obtain de Sitter solutions in string theory? Although perturbative
methods provide some rich phenomenology, they have not been able to uncover de
Sitter solutions that are universally accepted as consistent and fully under
control. This motivates attempts to probe the non-perturbative structure of
string theory. Of course, we do not have the tools to do this in full
generality; but progress can be made by restricting access to the Neveu-
Schwarz fields in the homogeneous limit. This is because they admit an
$\textrm{O}(d,d)$ symmetry that can be extended to higher order operators,
extending to all orders in $\alpha^{\prime}$.

The underlying structure of these theories is easy to write down whenever we
have a homogeneous and isotropic framework, where the $\textrm{O}(d,d)$
symmetry is equivalent to scale factor dulaity $a\to 1/a$. For a suitable non-
linear structure, de Sitter solutions can be found in both the string frame
and the Einstein frame. As shown previously Bernardo:2020zlc ; Bieniek:2022mrv
; Bernardo:2022nex , homogeneous and isotropic fluctuations about the string
frame solutions can trigger an instability, although we have now shown that
this is excited if and only if the generalised dilaton is growing at the
initial time $\dot{\Phi}(t_{i})>0$. The situation in the Einstein frame is
even more delicate. de Sitter solutions can and do exist in the Einstein
frame, for particular non-perturbative extensions of the Neveu-Schwarz theory.
However, these extensions do not connect smoothly to the leading order Neveu-
Schwarz theory. This means the Einstein frame de Sitter solutions cannot be
eternal to both past and future. In other words, a de Sitter solution can
survive for a finite period of time, or even a semi-infinite time, but at some
point towards either the future or the past, the solution breaks down. The
solution enters a new phase of the theory in a neighbourhood of small string
frame curvature, where Einstein frame de Sitter solutions are no longer
possible. Perhaps this should have been expected, as the breakdown occurs in
the perturbative regime where de Sitter vacua are conjectured to be absent ds1
; ds2 ; ds3 .

All of these results can be extended to include Bianchi I metrics. The
$\textrm{O}(d,d)$ symmetry now plays a more interesting role than in the
isotropic case, where scale factor duality is accompanied by scale factor
exchange $a_{i}\leftrightarrow a_{j}$. This is crucial, especially for the
existence of de Sitter solutions in the Einstein frame where the additional
constraints required for existence in an anisotropic setting are satisfied
automatically thanks to $\textrm{O}(d,d)$ symmetry.

Although we have focused our analysis on the case of classical strings in a
critical dimension, for which the constant $\Lambda=0$ in (1), the results are
easily extended to non-critical dimensions. This is because a non-vanishing
$\Lambda\propto D-D_{c}$ is easily absorbed into a redefinition of the
function $f$ that defines theory to all orders in $\alpha^{\prime}$,

$f\to f-2\Lambda.$ (99)

This changes the structure of the theory in neighbourhood of the origin, but
little else. For the string frame de Sitter solutions in the isotropic
framework, this will affect the boundary conditions (29), so that we now have
$g(0)=-2\Lambda$. Similarly, in the anisotropic case, we take (70) with
$\sum_{i}g_{j}(0,\ldots,0)=-2\Lambda$. The stability analysis is unchanged.
For the Einstein frame, we can revisit the question of whether or not the
bespoke choice of $f(H)$ can be smoothly connected to the desired form in a
neighbourhood of $H=0$. Focusing on the isotropic case, we set
$f(H)=-2\Lambda+{d}H^{2}+\mathcal{O}(H^{4})$ and compute

$\frac{\mathcal{F}^{\prime}(H)\mathcal{G}(H)}{\mathcal{F}(H)^{2}}=\frac{d-1}{d}+\mathcal{O}(H).$
(100)

Clearly this cannot be set equal to unity, as required for Einstein frame de
Sitter solutions. Thus, we conclude that the situation is qualitatively
unchanged from the critical case, $D=D_{c}$, and one cannot have eternal de
Sitter solutions in the Einstein frame.

#### Acknowledgements

We would like to thank Heliudson Bernardo and Guilherme Franzmann for useful
discussions. AP and PMS were supported by STFC consolidated grant number
ST/T000732/1 and YL by an STFC studentship. RGCS was supported by a Bell-
Burnell studentship. This paper was drafted during the period of the UCU
marking and assessment boycott, demonstrating work carried out by AP and PMS
as employees of the University of Nottingham. For the purpose of open access,
the authors have applied a CC BY public copyright licence to any Author
Accepted Manuscript version arising.

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# In the Danger Zone: U-Net Driven Quantile Regression can Predict High-risk
SARS-CoV-2 Regions via Pollutant Particulate Matter and Satellite Imagery

Jacquelyn A. Shelton  Przemyslaw Polewski  Wei Yao

###### Abstract

Since the outbreak of COVID-19 policy makers have been relying upon non-
pharmacological interventions to control the outbreak. With air pollution as a
potential transmission vector there is need to include it in intervention
strategies. We propose a U-net driven quantile regression model to predict
$PM_{2.5}$ air pollution based on easily obtainable satellite imagery. We
demonstrate that our approach can reconstruct $PM_{2.5}\,$concentrations on
ground-truth data and predict reasonable $PM_{2.5}\,$values with their spatial
distribution, even for locations where pollution data is unavailable. Such
predictions of $PM_{2.5}\,$characteristics could crucially advise public
policy strategies geared to reduce the transmission of and lethality of
COVID-19.

COVID19, SARS-CoV-2, Pollutant Particulate Matter, Satellite Imagery, U-net,
Remote Sensing, Convolutional Neural Networks

##  1 Introduction

Since the outbreak of the Severe Acute Respiratory Syndrome Corona Virus 2
(SARS-CoV-2), popularly referred to as _COVID-19_ , many questions have been
asked around the disease. Of particular interest are the questions of
transmission methods, and characteristics that identify vulnerable
populations. Studies into these properties are made more complex by the lack
of information around the behavior of the virus, E.g. the percent of cases
that are asymptomatic. However, as indicated by the inclusion in the CDC’s
vulnerable Population index, asthma and chronic lung disease seem to be a
major factor in the severity of the cases. For example, in (Conticini et al.,
2020) the connections between the lethality rate in Lombardy and Emilia
Romagna, areas with high level of atmospheric pollution, are explored. In
particular, the paper studies the correlation between pollution, which is a
known instigator of chronic lung disease even in young and other wise healthy
subjects, and the lethality of SARS-CoV-2. A similar result for the lethality
in the United States (Wu et al., 2020) using county level fatality rate, and
county level long term air pollution, shows that, after adjusting for other
known factors, that there is a strong correlation between the concentration of
particulate matter $2.5$ micrometers or less in diameter, or $PM_{2.5}$, and
the county level lethality. Specifically, a 1 $\frac{\mu g}{m^{3}}$ increase
in $PM_{2.5}$ corresponds to an $8\%$ increase in the fatality rate.

However, as observed in (Coccia, 2020), there is also a correlation between
particulate matter ($PM$) air pollution and the number of reported cases. This
suggests that pollution-to-human may serve as another transmission dynamic for
SARS-CoV-2. These two results suggest a two factor vulnerability to SARS-CoV-2
caused by increased particulate matter in the air: On one hand it increases
the likelihood of having a more sever reaction to infection, and the other it
serves a transmission vector.

Figure 1: Network structure for predicting $PM_{2.5}\,$concentrations from
multispectral imagery, based on the classic U-Net (Ronneberger et al., 2015)
architecture up to the top-level upsampling layer, where 3 parallel sequences
of convolutional filters and activations branch off the upsampled features,
corresponding to the 3 types of outputs predicted by the network (lower/upper
bound, median).

Traditionally, $PM_{2.5}$ concentration data can be obtained from ground
sensors and measurement stations. However, the spatial resolution of these
measurements is greatly limited by the sparsity of sensor networks. Thus,
detailed pollution maps have been developed that integrate heterogeneous data
sources (see e.g. van Donkelaar et al. (2019)), to create a database of
estimated monthly pollutant concentrations over several countries. This data
was used by Wu et al. (2020) linking $PM_{2.5}$ concentrations to COVID-19
mortality rates. Although this historical data helped establish this causality
relationship, up-to-date/live pollution information is still needed to monitor
pollutant $PM_{2.5}$.

By knowing where pollution is we can better understand the impacts of social
segregation policy on subpopulations in order to e.g. allocate medical funds
to the most vulnerable populaces. The health needs for a population often
require very granular data about their circumstances, and air pollution is one
such measure that is only sporadically known in poorer areas and thus is
insufficiently represented in modeling the health situation. In order to
properly control COVID-19 hotspots there is a need to predict the spread and
intensity of COVID-19. While contact tracing and non-pharmacological
interventions (NPIs) have shown value, there is mounting evidence that there
are other transmission vectors, including via pollution particles, as
discussed earlier. Therefore there is mounting need to understand local
pollution dynamics in order to correctly deal with the pandemic. In order to
properly understand the effectiveness of a given NPI it is essential we
understand the main transmission vectors. It is, therefore, vital we gain
understanding of the strength of air pollution as a transmission vector in
order to understand and design NPIs efficiently.

The goal of the present work is to model air pollution
$PM_{2.5}\,$concentrations using readily available satellite imagery. The aim
is to aid in any planning for efficacious COVID-19 geared strategies. The
paper is organized as follows: Sec. 2 introduces the proposed U-net model,
Sec. 3 presents the data, Sec. 4 describes the experiments and results, and
finally Sec. 5 provides a summary and outlook.

##  2 U-net model for pollutant particulate matter

We build upon the well known U-net architecture (Ronneberger et al., 2015) to
predict dense (per-pixel) $PM_{2.5}$ concentrations from multispectral
satellite imagery. Since its introduction in 2015, the U-net has been
successfully applied to various semantic segmentation tasks, E.g. in medical
imaging (Dong et al., 2017) and astronomy(Akeret et al., 2017). The U-net
consists of two symmetrical parts. The encoder path downsamples the original
image into meaningful features by means of convolutional filters and pooling
operations, whereas the upsampling path aims at decoding these features into a
full-sized output map using transposed convolution operations, driven by an
appropriate loss function. Moreover, upsampling layers are augmented with
feature maps from the downsampling path at the corresponding resolution, to
provide more context information. The original U-net was meant for
classification and featured a softmax layer after the top-level upsampling
layer’s output, trained using a cross-entropy objective with discrete ground-
truth labels. Yet the U-net has also been used for dense regression by
removing the softmax layer and optimizing the squared difference between the
upsampled output and a continuous target variable (Yao et al., 2018; Payer et
al., 2016).

It is well known that least squares regression estimates the conditional mean
of the response variable given the predictors, and is therefore sensitive to
outliers. Alternatively, Quantile regression (Koenker, 2005) is more robust to
outliers. Let $S=\\{(x_{i},y_{i})\\},1\leq i\leq N$ denote a set of predictor
variable vectors $x$ and matching continuous response variables $y$ sampled
from their respective distributions $\mathcal{X},\mathcal{Y}$. Let
$f(x|\theta)$ indicate the prediction function parameterized by $\theta$, and
let $0<q<1$ refer to the quantile level. In this work, we consider losses of
the form:

$\small\begin{split}L_{q}(\theta)&=\frac{1}{|S|}\sum_{(x,y)\in
S}\rho_{q}(y-f(x|\theta))\\\ \rho_{q}(r)&=r[r\geq 0]-(1-q)r\end{split}$ (1)

In the above, $[a]$ denotes the indicator function for event $a$, whereas the
term $\rho_{q}(r)$ is the _check function_ (Koenker, 2005). To better quantify
the approximated distribution, we learn 3 quantiles $q_{l}<q_{m}=0.5<q_{u}$
simultaneously, which allows us to obtain both a point-wise estimate from the
median $q_{m}$ and a $(q_{u}-q_{l})-$confidence interval for the value. We
cast the aggregate loss function as a linear combination of partial $L_{q}$
terms with coefficients $\gamma_{l},\gamma_{u}$, in a simple multi-task
learning setting (Gong et al., 2019):

$L_{aggr}(\theta)=\gamma_{l}L_{q_{l}}(\theta)+L_{0.5}(\theta)+\gamma_{u}L_{q_{u}}(\theta)$
(2)

To mirror the structure of the loss function, the U-net architecture was
extended to include a separate sequence of convolutional filters at the top
level of the upsampling path per quantile loss term (see Fig. 1). One
disadvantage of using the loss $L_{q}$ (Eq. 1) is the fact that the check
function $\rho_{q}$ is not differentiable at zero. Moreover, the derivative is
piece-wise constant on $R^{+},R^{-}$. This might pose a challenge to gradient-
based optimization schemes (E.g. back propagation in neural networks) because
the gradient norm does not get smaller as the optimization converges to a
local minimum. To alleviate that, some approximations of the check function
have been proposed. The Huber loss (Hastie et al., 2001) combines quadratic
behavior within a $\delta-$neighborhood of 0 with linear behavior on
$R\setminus[-\delta;\delta]$. This can be used to approximate $\rho_{0.5}$.
Recently, Gupta et al. (2020) introduced an asymmetric version of the Huber
loss:

$H(r|\delta_{l},\delta_{u})=r^{2}-(r-\delta_{l})_{+}^{2}-(-r-\delta_{u})_{+}^{2}$
(3)

Figure 2: Comparison of loss functions: the differentiable asymmetric Huber
function in Eq. (3) for various check function $\rho$ approximations versus
the standard squared loss.

This loss function is differentiable everywhere. The parameters
$\delta_{l},\delta_{u}$ control both the slope of the linear functions on
their respective sides of the real axis, and the locations where the function
starts to show quadratic behavior. We propose to approximate $\rho_{q}$ with
$R^{\alpha}_{q}(r)=\alpha H(r|q/2\alpha,(1-q)/2\alpha)$, where $\alpha$ is a
parameter which controls the location of the change from linear to quadratic
characteristics. See Fig. 2 for a comparison of loss functions.

##  3 Data

In order to address this problem, we use satellite imagery as a source of
predictor variables, and approximate $PM_{2.5}$ concentrations published by
van Donkelaar et al. (2019) for the time period of 2000-2018 in the role of
ground-truth.

Satellite data. Our work used Landsat 8 satellite imagery published by the
United States Geological Survey (Lan, 2016). Landsat 8 is the latest in a
series of Earth observation missions containing a total of $11$ spectral
bands, ranging in wavelengths from $0.435\mu m$ to $12.51\mu m$, with spatial
resolution between $15$ to $100m$ depending on the band.

Pollution data. We downloaded monthly $PM_{2.5}$ concentration maps for North
America from (Don, ). Readily available, these maps contain $0.01$ degrees per
pixel and use a standard WGS84 coordinate reference system. We used the data
made available immediately after Landsat 8 mission launched, namely from March
2013 to December 2018.

Preprocessing. The Landsat imagery was first reprojected to the WGS84
coordinate system, and downsampled to the ground-truth resolution of $0.01$
degrees (the panchromatic band was dropped). To match the temporal resolution
of the ground-truth, we computed per-band average images grouped by month of
acquisition. Next, we derived a per-pixel mask of regions within the image
covered by cirrus clouds and hence not suitable for analysis, based on the
estimated cloud cover percentage (Foga et al., 2017). Finally, we combined the
cloud cover mask with the data availability mask from the ground-truth
$PM_{2.5}$ maps. Also, pixels corresponding to the top and bottom $1\%$ of
ground-truth values were masked out as outliers. All input imagery bands were
normalized to the interval $[0;1]$ individually per band.

##  4 Experiments

We selected an initial number $N=133$ Landsat images, spanning March 2013 to
December 2018, of $24$ major cities from representative regions of the United
States (see Fig. 3, green boxes) and preprocessed them as described in Sec. 3.
Next, we defined our ground-truth data as these images paired with
corresponding $PM_{2.5}$ images from which we use an 80:20$\%$ random split to
create training/testing set of 106:27 images. A U-net based on adapting the
implementation by Akeret et al. (2017) was trained using an ADAM optimized
over $1000$ epochs with minibatch size of $15$ and $100$ internal iterations,
with the following parameterization: dropout ratio was $0.5$, learning rate
$0.00005$, regression quantiles $q_{l}=0.1,q_{r}=0.9$. For the Huber loss
function (3), all aggregates (2) contributed in equal proportion and the
parameter controlling the function shape was $\alpha=2$.

Figure 3: Locations of US cities chosen as a basis for our study. Rectangles
indicate bounding boxes of the downloaded Landsat images. Locations marked
with green boxes were used for training and validation, whereas blue boxes
represent regions used exclusively for testing. The background color map
represents average concentration of $PM_{2.5}\,$in 2018.

4.1: Sanity check with ground-truth. The network converged to a state which
produced a mean absolute error of $\approx 1\mu g/m^{3}$ between predicted and
ground-truth values on the training set (see Fig. 4). The median width of the
predicted confidence intervals (upper bound - lower bound) for training data
was $1.94\mu g/m^{3}$, and $70\%$ of ground-truth values fell into the
predicted interval. The number of ground-truth values above the lower bound
and below the upper bound was respectively $88\%$ and $82\%$. This
demonstrates that the model was able to approximate the $0.1$ and $0.9$
quantiles well, and also to provide quality point-estimates. Correspondingly,
our approach can predict $PM_{2.5}\,$pollution maps that match the structure
and approximate concentrations of ground-truth reconstructions. See Fig. 5 for
some examples.

Figure 4: The error of the U-net trained in Exp. 4.1 shown with the mean
absolute error (MAE). Red curve: error predicting $PM_{2.5}\,$concentrations
of data with ground-truth. Blue curve: error predicting $PM_{2.5}\,$of
validation data in Exp. 4.2, where the U-net was generalized to cities at
times not present in the training data but locations it has been trained on.
Error with the GT data converges to $MAE\approx 1$ and the error with
validation data converges to $MAE\approx 2$. This shows that the U-net can
generalize well to temporally novel data.

Figure 5: The U-net trained on ground-truth satellite images was able to
successfully reconstruct the $PM_{2.5}\,$concentrations, corresponding to the
low error shown in Fig. 4. Shown here are a few examples.

4.2: Generalizability to temporally novel data. To assess the ability of our
model to generalize in the temporal domain, we utilized $27$ previously unseen
images that overlapped with the training set spatially but not temporally. The
test set contained images from $15$ of the $24$ cities. The mean absolute
validation error followed the same trend as Exp. 4.1 with GT training data,
however it converged to about $2\mu g/m^{3}$ (see Fig. 4). The performance of
the predicted confidence intervals degraded to 40% of contained ground-truth
values, whereas the median interval width remained low at $2.06\mu g/m^{3}$.
This indicates a degree of overfitting, however the predicted
$PM_{2.5}\,$concentrations are still within reasonable distance of the ground-
truth.

4.3: Cities with similar pollution profile. We selected $28$ new images at
$20$ additional locations within the United States, showing visually similar
distributions of ground-truth $PM_{2.5}$ within the Landsat image frames to
the original training cities, in order to evaluate the performance of
predicting the $PM_{2.5}\,$concentrations at locations unseen during training.
The obtained mean absolute error was $2.81\mu g/m^{3}$, and the predicted
confidence intervals contained $31\%$ of ground-truth values. This shows that
it is harder to generalize in the spatial than in the temporal domain.

4.4: Before and after SARS-CoV-2-induced lock-down. The goal of the fourth
experiment is to verify that our model can predict expected
$PM_{2.5}\,$concentration trends at time points before and after the
government mandated lock-down in March 2020 intended to hinder the spread of
SARS-CoV-2 (which consequently e.g. drastically reduced industrial emissions).
We applied the U-net learned in Exp. 4.3 to satellite images of Los Angeles,
CA from 2018, 2019, and early 2020. The results, shown in Fig. 6, illustrate
that the U-net was able to learn $PM_{2.5}\,$concentrations consistent with
world events at the time. Namely, the pollution is inferred to be notably
higher before the lock-down than after the lock-down, shown by the
considerable shift of the predicted $PM_{2.5}\,$distribution’s $0.9$ quantile
between October 2019 and April 2020 – the quantile shifted from $13.2$ to
$9.7$, respectively. Additionally, our approach was able to successfully
isolate regions in LA that are the most densely populated by predicting higher
pollutant $PM_{2.5}\,$values. This suggests that our approach can generalize
to new data and reliably predict pollutant $PM_{2.5}\,$concentrations and
their spatial structure, which informs on the presence and lethality of SARS-
CoV-2.

Figure 6: Density of predicted $PM_{2.5}\,$concentrations in Los Angeles for
time periods before and during the COVID-19 outbreak (top left), corresponding
Landsat images with their predicted $PM_{2.5}\,$pollution maps (bottom left).
As expected, we see a significant reduction in the $0.9$-quantile of the
predicted $PM_{2.5}\,$pollution distributions between April 2020 (immediately
after the lock-down began) and the previous years. The predicted
$PM_{2.5}\,$maps match this trend, showing relatively less pollution in 2020
(darker pixels). Interestingly, yet unsurprisingly, the most densely populated
counties, Los Angeles and Orange County, (encircled in red on the map) are
predicted to have higher $PM_{2.5}\,$concentrations for all of the years, as
shown in the $PM_{2.5}\,$pollution maps. The 2010 population map shows this
dense region had a joint population of $12.8$ million (top right), and between
October 2019 and April 2020, this region showed the greatest shift in
predicted $PM_{2.5}\,$concentrations (bottom right). The scatter plot of a
sample of predictions from October 2019 versus April 2020 (teal line
represents identity mapping) from 2010 (bottom right) illustrates this
$PM_{2.5}\,$shift, where points are color-coded by originating county (black
points correspond to the ocean). The most significant drop in predicted
$PM_{2.5}\,$concentration occurs in Los Angeles (red points) and Orange County
(peach).

##  5 Discussion

SARS-CoV-2 lethality has been concretely linked to the concentration of
pollutant particulate matter, where a slight increase in $PM_{2.5}\,$can
drastically enhance the morbidity rate. We have proposed a means of learning
the structure, spatial distribution, and physical concentration of pollutant
particulate matter $PM_{2.5}\,$based solely on global satellite imagery and
monthly $PM_{2.5}\,$values from the years 2013-2018. Our approach proposes a
U-net convolutional network with quantile regression loss to learn dense per-
pixel $PM_{2.5}\,$prediction maps. We have demonstrated not only that our
approach can successfully reconstruct $PM_{2.5}\,$concentrations with low
error, but it can also generalize to completely unseen locations for which
pollutant data is not available. As pollution has been shown to be a main
vector of SARS-CoV-2 transmission, this ability to identify regions where it
may be particularly transmissible and lethal can provide critical advice to
implementable public health strategies, such as the ability to monitor,
understand, and design NPIs efficiently.

Given that our approach passes several sanity checks by demonstrating that its
$PM_{2.5}\,$concentration predictions reflect ground-truth
values/measurements, it would follow that it can make meaningful predictions
for locations for which there has never been recordings of the particulate
matter (only satellite imagery available). Additionally, although we currently
do not have rigorous methodology to quantify what makes satellite imagery of
different cities ‘similar’, we could address this e.g. with dependence tests
such as kernel two-sample tests (Gretton et al., 2012). Also, data from
multiple satellites could be included to provide greater temporal resolution,
such as from the Sentinel-2 mission (Drusch et al., 2012). With larger
satellite datasets we can expect an increase in the accuracy of our
predictions.

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