# A natural model for curved inflation

Quentin Vigneron<EMAIL_ADDRESS>Institute of Astronomy, Faculty of
Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5, Toruń, 87-100, Poland  ENS de Lyon, CRAL UMR5574, Université
Claude Bernard Lyon 1, CNRS, Lyon, F-69007, France  Julien Larena
<EMAIL_ADDRESS>Laboratoire Univers & Particules de Montpellier,
Université de Montpellier, CNRS, Montpellier, France

###### Abstract

Inflationary models with a non-zero background curvature are ill-defined in
general relativity because scalar modes cannot be canonically quantized.
Therefore, there is no consensus on the primordial power spectrum that should
be considered at large scales in a curved Universe. In this letter, we propose
a model of curved inflation where canonical quantization is possible for any
curvature, and we unambiguously obtain the resulting primordial power spectra.
The framework is a recently proposed modification of general relativity in
which a non-dynamical topological term is added to the Einstein equation. The
main strength of this model is that no additional degree of freedom compared
to the standard model of cosmology is needed, giving a natural solution to the
problem of constructing curved inflation, and at the same time providing an
additional argument for this topological modification of general relativity.

††preprint: APS/123-QED

Cosmological inflation successfully explains the Cosmic Microwave Background
(CMB) anisotropies, and provides at the same time a solution to the flatness
and horizon problems. The success of the model lies in its simplicity: single
field slow-roll inflation with a Bunch-Davies vacuum condition allowing for a
satisfactory fit of the CMB data (Planck Collaboration, 2020). However, while
inflation in general relativity (GR) predicts a negligible late-time spatial
curvature, it is a priori not necessarily negligible at the start of
inflation, and can even be required to be non-zero by the topology of the
Universe. Therefore, for the consistency of the theory, one should be able to
easily construct an inflationary scenario regardless of the initial curvature.
There is however no consensus on the construction of such a scenario in GR,
and in particular on the shape of the power spectrum that should result from
it (e.g. Lyth and Woszczyna, 1995; Garriga  _et al._ , 1998; Lasenby and
Doran, 2005; Bonga  _et al._ , 2016; Ratra, 2017; Handley, 2019; Thavanesan
_et al._ , 2021; Renevey  _et al._ , 2021; Hergt  _et al._ , 2022; Renevey
_et al._ , 2022; Letey  _et al._ , 2022; D’Agostino  _et al._ , 2023). The
difficulty is mainly due to the fact that, for non-zero background curvature
$K$, the evolution equation for the curvature perturbation $\mathcal{R}$ is no
longer the solution of a wave equation, and features a non-local operator
$\mathcal{Z}$ (Handley, 2019):

$\displaystyle
v_{K}^{\prime\prime}-\left(\frac{\mathcal{Z}^{\prime\prime}}{\mathcal{Z}}+2K+\frac{2K\mathcal{Z}^{\prime}}{\mathcal{H}\mathcal{Z}}+\Delta\right)v_{K}=0,$
(1) $\displaystyle{\rm with}\ \
v_{K}\coloneqq\mathcal{Z}\mathcal{R}\quad;\quad\mathcal{Z}\coloneqq
z\sqrt{\frac{\mathcal{D}^{2}}{\mathcal{D}^{2}-K\mathcal{E}}},$

where $\mathcal{H}$ is the conformal Hubble rate,
$\mathcal{D}^{2}\coloneqq\Delta+3K$ and
$\mathcal{E}\coloneqq\frac{\varphi^{\prime 2}}{2\mathcal{H}^{2}}$ with
$\varphi$ the background inflaton. This makes it impossible to use standard
quantization procedures to define initial conditions and predict the
primordial power spectrum, leading to the use of non-standard initial
conditions (e.g. kinetic dominance Handley  _et al._ (2014); Hergt  _et al._
(2022); Letey  _et al._ (2022)). Additionally, $\mathcal{R}$ can only be
conserved on super-Hubble scales during inflation if the curvature scale is
also super-Hubble, as shown by

$\displaystyle\frac{1}{\mathcal{H}}\mathcal{R}^{\prime}$
$\displaystyle=\frac{1}{\mathcal{E}}\left\\{\frac{-k^{2}+3K}{\mathcal{H}^{2}}\Psi+\frac{K}{\mathcal{H}^{2}}\left[\frac{\Psi^{\prime}}{\mathcal{H}}+\Psi\right]\right\\},$
(2)

where $\Psi$ is the Bardeen potential and $k$ the wave number. Without this
hypothesis, numerical resolution is necessary to track the evolution of the
power spectrum (Handley, 2019; Renevey  _et al._ , 2022). Clearly, it is quite
unsavory to require a priori that spatial curvature is super-Hubble simply to
ensure conservation of $\mathcal{R}$, since this is supposed to be one of the
main achievements of inflation. It is also possible to work with a variable
solution of a wave equation without non-local operator, as shown by Letey  _et
al._ (2022). However, there is still not a natural initial condition that can
be considered, the super-Hubble conservation is not fulfilled, and a numerical
resolution is necessary.

Additionally, at the level of the background, changes in the evolution of the
scale factor induced by the presence of curvature (e.g. the bounce present in
the spherical case, or the upper bound on the Hubble radius in the hyperbolic
case) further complicate the case for curved inflation in GR. In particular,
additional fine-tuning on the number of e-folds is required (Uzan  _et al._ ,
2003; Renevey  _et al._ , 2021; Hergt  _et al._ , 2022), and, for $K>0$,
different power spectra are obtained if the initial conditions are taken
before or after the bounce (Renevey  _et al._ , 2022).

Overall, the procedure to obtain a power spectrum resulting from an
inflationary phase in a curved universe in GR is considerably more difficult,
and quite ambiguous, than that of the flat case. In this sense, with a
background curvature, we lose the simplicity and naturalness of inflation
theory that made its success.

The goal of this letter is to propose a curved inflationary scenario bypassing
the problems previously mentioned, as simple as flat inflation, and without
introducing any additional parameters. The philosophy of our approach lies in
a slight modification of the Einstein equation that was proposed by (Vigneron,
2024), rather than within GR. This slight, albeit fundamental, deviation is
amply compensated by the resulting simplicity of the model and the fact that
the initial motivation for the modification is not linked at all to inflation.

We first proceed with the presentation of this modification of GR, which we
name topo-GR, along with its perturbation equations that were derived in
Vigneron and Poulin (2023). All the related equations from this theory are
presented and compared with the ones from GR in Table 1. We then show how
curved inflation is constructed in this theory, in particular through the use
of a modified curvature perturbation variable, and give the resulting power
spectra. Details of the derivations can be found in the Supplementary
Materials. Throughout this letter, we use Greek letters for spacetime indices,
and Roman letters for spatial indices.

Topo-GR
The original motivation for the modification of GR proposed in Vigneron (2024)
is not related to inflation, but is to obtain a relativistic field equation
which admits a non-relativistic limit in any topology, something not possible
in GR (Vigneron, 2024). This modification does not add any degrees of freedom
compared to GR, and the equivalence principle is preserved. Physical
differences only appear at scales comparable to the topology scale. In this
theory, the spacetime Ricci curvature $R_{\mu\nu}$ in the Einstein equation is
replaced by the difference between that tensor and a reference Ricci curvature
$\bar{R}_{\mu\nu}$ that is fixed and depends on the spatial covering space of
the spatial topology, i.e. $R_{\mu\nu}\rightarrow
R_{\mu\nu}-\bar{R}_{\mu\nu}$, leading to equation (1), in Table 1. We direct
the reader to Lachieze-Rey and Luminet (1995); Vigneron (2024) for the precise
definitions of the topological notions involved in the definition of the topo-
GR theory, involving in particular the Thurston classification of closed
3-manifolds. The additional term appearing in the Einstein equation is
conserved by the second equation (1).

The precise way $\bar{R}_{\mu\nu}$ is chosen as function of the topology is
described in Sec. 5.2.2 of Vigneron (2024). In the present paper, where we
consider perturbations around a homogeneous and isotropic background,
$\bar{R}_{\mu\nu}$ takes the form (Vigneron and Poulin, 2023)

$\displaystyle\bar{R}_{\mu\nu}=2K\left(\gamma_{\mu\nu}+\mathcal{L}_{X}\gamma_{\mu\nu}\right),$
(3)

where $X^{\mu}$ is a first order vector field, and $\gamma_{\mu\nu}$ is the
background comoving spatial metric. For $K$ = 0, we have $\bar{R}_{\mu\nu}=0$,
and topo-GR is equivalent to GR.

Figure 1: Qualitative difference on the role of matter between GR (left) and
topo-GR (right). In GR, spacetime is Ricci flat in vacuum, i.e.
$R_{\mu\nu}=0$, and matter induces a deviation from flatness. In topo-GR, in
vacuum, spacetime is curved by topology, i.e. $R_{\mu\nu}=\bar{R}_{\mu\nu}$,
and matter induces a deviation from that ‘topological curvature’.

Qualitatively, within topo-GR, matter does not directly curve spacetime, as in
GR, but only induces deviations (not necessarily small) from the curvature
“imposed” by topology, i.e. deviations from $\bar{R}_{\mu\nu}$. This is seen
through the modified Einstein equation (1) from which, in the absence of
matter, $R_{\mu\nu}$ reduces to the reference (topological) curvature
$\bar{R}_{\mu\nu}$ (see also Fig. 1).

Table 1: Comparison between GR and topo-GR for the homogeneous and isotropic solution and its first order perturbations (longitudinal gauge) Vigneron and Poulin (2023). Terms in bold blue are present in GR but missing (or different) in topo-GR. Terms in bold red are present in topo-GR but missing (or different) in GR. For $K=0$, these two systems are equivalent. We defined $\kappa\coloneqq 8\pi G$. General relativity |  topo-GR
---|---
Field equations
$R_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{T_{\alpha\beta}g^{\alpha\beta}}{2}\,g_{\mu\nu}\right)$ |  $R_{\mu\nu}-{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{\bar{R}_{\mu\nu}}}}=\kappa\left(T_{\mu\nu}-\frac{T_{\alpha\beta}g^{\alpha\beta}}{2}\,g_{\mu\nu}\right)$ (4)
$\emptyset$ |  ${\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{\nabla^{\nu}\left(\bar{R}_{\mu\nu}-\frac{\bar{R}_{\alpha\beta}g^{\alpha\beta}}{2}g_{\mu\nu}\right)=0}}}$ (5)
Homogeneous and isotropic solution
$\begin{aligned} &\mathcal{H}^{2}=a^{2}\frac{\kappa}{3}\rho-{\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}{\bm{K}}}\\\ &\mathcal{H}^{\prime}=-a^{2}\frac{\kappa}{6}\left(\rho+3p\right)\end{aligned}$ |  $\begin{aligned} &\mathcal{H}^{2}=a^{2}\frac{\kappa}{3}\rho\\\ &\mathcal{H}^{\prime}=-a^{2}\frac{\kappa}{6}\left(\rho+3p\right)\end{aligned}$ $\quad\quad{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{\forall K}}}$ (6)
First order perturbations
Scalar modes
$(\Delta+{\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}{\bm{3K}}})\Psi=a^{2}\frac{\kappa}{2}\delta\rho_{l}+3\mathcal{H}\left(\Psi^{\prime}+\mathcal{H}\Phi\right)$ |  $\Delta\Psi=a^{2}\frac{\kappa}{2}\delta\rho_{l}+3\mathcal{H}\left(\Psi^{\prime}+\mathcal{H}\Phi\right)+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{K\Delta\mathcal{C}}}}$ (7)
$\Psi^{\prime}+\mathcal{H}\Phi=-a^{2}\frac{\kappa}{2}(\rho+p)\mathcal{V}$ |  $\Psi^{\prime}+\mathcal{H}\Phi=-a^{2}\frac{\kappa}{2}(\rho+p)\mathcal{V}+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{K\mathcal{C}^{\prime}}}}$ (8)
$\Psi-\Phi=a^{2}\kappa\Pi$ |  $\Psi-\Phi=a^{2}\kappa\Pi+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{4K\mathcal{C}}}}$ (9)
$\begin{aligned} &\Psi^{\prime\prime}+2\mathcal{H}\Psi^{\prime}+\mathcal{H}\Phi^{\prime}+(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\Phi-{\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}{\bm{K\Psi}}}=\\\ &\qquad a^{2}\frac{\kappa}{2}\left(c_{\rm s}^{2}\delta\rho_{l}+\delta p_{\not=\rm ad}+\frac{2}{3}\Delta\Pi\right)\\\ \end{aligned}$ |  $\begin{aligned} &\Psi^{\prime\prime}+2\mathcal{H}\Psi^{\prime}+\mathcal{H}\Phi^{\prime}+(2\mathcal{H}^{\prime}+\mathcal{H}^{2})\Phi=&\\\ &\qquad a^{2}\frac{\kappa}{2}\left(c_{\rm s}^{2}\delta\rho_{l}+\delta p_{\not=\rm ad}+\frac{2}{3}\Delta\Pi\right)+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{K\Delta\mathcal{C}}}}\end{aligned}$ (10)
Vector modes
$\left(\Delta+{\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}{\bm{2K}}}\right)Q_{i}=-a^{2}2\kappa(\rho+p)(\mathcal{V}_{i}-Q_{i})$ |  $\left(\Delta-{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{2K}}}\right)Q_{i}=-a^{2}2\kappa(\rho+p)\left(\mathcal{V}_{i}-Q_{i}\right)+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{4K\mathcal{C}_{i}^{\prime}}}}$ (11)
$Q_{i}^{\prime}+2\mathcal{H}Q_{i}=a^{2}\kappa\Pi_{i}$ |  $Q_{i}^{\prime}+2\mathcal{H}Q_{i}=a^{2}\kappa\Pi_{i}+{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{4K\mathcal{C}_{i}}}}$ (12)
Tensor mode
$h_{ij}^{\prime\prime}+2\mathcal{H}h_{ij}^{\prime}+\left({{\color[rgb]{0,0,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.8}{\bm{2K}}}-\Delta}\right)h_{ij}=2a^{2}\kappa\Pi_{ij}$ |  $h_{ij}^{\prime\prime}+2\mathcal{H}h_{ij}^{\prime}+\left({{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{6K}}}-\Delta}\right)h_{ij}=2a^{2}\kappa\Pi_{ij}$ (13)
Conservation of
$\bar{R}_{\alpha\beta}-\frac{\bar{R}_{\mu\nu}g^{\mu\nu}}{2}g_{\alpha\beta}$
$\emptyset$ |  ${\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{\mathcal{C}^{\prime\prime}+2\mathcal{H}\mathcal{C}^{\prime}-\Delta\mathcal{C}}}}{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{=-a^{2}\kappa\Pi}}}$ (14)
$\emptyset$ |  ${\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{\mathcal{C}_{i}^{\prime\prime}+2\mathcal{H}\mathcal{C}_{i}^{\prime}+\left(2K-\Delta\right)\mathcal{C}_{i}}}}{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}{\bm{=-a^{2}\kappa\Pi_{i}}}}$ (15)

Perturbation equations
Table 1 summarises the perturbation equations around the homogeneous and
isotropic solution of topo-GR that were derived in (Vigneron and Poulin, 2023)
(right column), and compare them with GR (left column), using the longitudinal
gauge, i.e.

$\displaystyle g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$
$\displaystyle=a^{2}\left\\{-(1+2\Phi)\textrm{d}\tau^{2}-Q_{i}\textrm{d}x^{i}\textrm{d}\tau\right.$
$\displaystyle\qquad\left.+\left[(1-2\Psi)\gamma_{ij}+h_{ij}\right]\textrm{d}x^{i}\textrm{d}x^{j}\right\\}.$
(16)

The two systems of equations are equivalent for $K=0$, and differ only for
$K\not=0$. Consequently, flat inflation is equivalent in GR and topo-GR. For
$K\not=0$, three main differences of topo-GR compared to GR are particularly
relevant to the design of inflationary scenarios.

First, the background equations (1) are the flat Friedmann equations for any
curvature, i.e. the scale factor evolution is independent of the value of $K$.
Especially, during a de Sitter phase, we have

$\displaystyle a(t)\propto-\frac{1}{\tau}\quad\quad{\rm
for}\quad\quad\tau<0,\quad\quad\forall\,K.$ (17)

This is a radical difference with respect to GR since there is no more bounce
or upper-bound for the Hubble radius. Consequently, we will be able to use the
limit $\tau\rightarrow-\infty$ to obtain an exact Bunch-Davies vacuum for any
curvature.

Second, the scalar mode equations (1)–(1) no longer feature $K\Psi$. That term
is one of the reasons a non-local operator appears in GR, because it requires
inverting the Laplacian in the Poisson equation to eliminate $\Psi$ when
deriving the equation for $\mathcal{R}$. Its absence in the topo-GR equations
will allow us to obtain a wave equation without non-local operator.
Third, additional gauge invariant variables are present: a scalar mode
$\mathcal{C}$ and a vector mode $\mathcal{C}_{i}$, both being solutions of
wave equations sourced by the matter anisotropic stress [equations (1) and
(1)]. These variables come from the first order contribution to
$\bar{R}_{\mu\nu}$ in the gauge (16), and can be related to the tilt of the
reference observer induced by $\bar{R}_{\mu\nu}$ in that gauge Vigneron and
Poulin (2023). For a vanishing stress, equation (1) implies that $\mathcal{C}$
decays during radiation and matter dominated areas on sub-Hubble scales, but
is conserved on super-Hubble scales. Hence if generated during inflation,
$\mathcal{C}$ could have an impact on the power spectrum at large scales. For
this reason, we will consider $\mathcal{C}\not=0$. The same can be said for
$\mathcal{C}_{i}$ with equation (1). However, while this variable can be non-
zero, even if it sources the metric vector mode $Q_{i}$ through equations (1)
and (1), these imply that $Q_{i}$ always decays on all scales for vanishing
stress, and therefore can be neglected. Since observational imprints of vector
modes come from $Q_{i}$, we can also neglect $\mathcal{C}_{i}$. This shows
that vector modes are not important in topo-GR, as in GR. Consequently, from
now on we will not consider these modes anymore.

Curvature deviation perturbation
We define the curvature deviation perturbation as

$\displaystyle\tilde{\mathcal{R}}\coloneqq\mathcal{R}-K\mathcal{C}.$ (18)

Physically, $\tilde{\mathcal{R}}$ is related to the difference between the
first order spatial scalar curvature $\delta\left({{}^{(3)}R}\right)$ and the
first order spatial reference curvature
$\delta\left(\gamma^{ij}\bar{R}_{ij}\right)$ in the comoving gauge (with
subscript c):

$\displaystyle\left[\delta\left({\mathcal{R}^{(3)}}\right)-\delta\left(g^{ij}\bar{R}_{ij}\right)\right]_{\rm
c}$ $\displaystyle=\frac{4}{a^{2}}\Delta\tilde{\mathcal{R}}.$ (19)

For a flat background, $\tilde{\mathcal{R}}$ reduces to $\mathcal{R}$. But
while in GR, $\mathcal{R}$ is the natural variable to use to study inflation
for $K=0$, in topo-GR this becomes $\tilde{\mathcal{R}}$ for any $K$. There
are three main reasons for this:

  1. 1.

$\tilde{\mathcal{R}}$ is conserved on super-Hubble scales for shear-free
adiabatic perturbations or single field inflation, without assuming
additionally the curvature scale to be super-Hubble. This results from

$\displaystyle\mathcal{H}\tilde{\mathcal{R}}^{\prime}=\frac{a^{2}\kappa}{3(1+w)}\left[c_{\rm
s}^{2}\delta\rho_{\rm c}+\delta
p_{\not=ad}+\frac{2}{3}\left(\Delta+3K\right)\Pi\right].$ (20)

For adiabatic perturbations, that formula also shows that the density
perturbation after inflation is directly generated by $\tilde{\mathcal{R}}$
for any $K$, as with $\mathcal{R}$ in GR for $K=0$.

  2. 2.

With a single scalar field as a source, $\tilde{\mathcal{R}}$ is solution of
the Mukhanov-Sasaki equation for any $K$:

$\displaystyle\tilde{\mathcal{R}}^{\prime\prime}+\frac{2z^{\prime}}{z}\tilde{\mathcal{R}}^{\prime}-\Delta\tilde{\mathcal{R}}=0\qquad\forall
K,$ (21)

where $z\coloneqq\frac{a\varphi^{\prime}}{\mathcal{H}}$. This equation for
$\tilde{\mathcal{R}}$ is formally equivalent to that of $\mathcal{R}$ in GR
for a flat background, but with the Laplacian depending on spatial curvature.

  3. 3.

$\tilde{\mathcal{R}}$ is one of the two free scalar fields to appear in the
action at second order (see next section).

The fact that $\tilde{\mathcal{R}}$ is the relevant variable to consider in
topo-GR rather than $\mathcal{R}$ is consistent with the main idea of the
theory: what matters for the dynamics is not the (spatial/spacetime)
curvature, but the difference between that curvature and the reference
curvature.

Second order action
To summarize, single field inflation in topo-GR features three independent
variables: two scalar modes $\tilde{\mathcal{R}}$ and $\mathcal{C}$, and a
tensor mode $h_{ij}$, solutions of the wave equations (21), (1) and (1),
respectively. Additionally, the scale factor evolves as $a\propto-1/\tau$ for
any curvature during an exact de Sitter phase. Therefore, apart from the
additional field $\mathcal{C}$, the situation is formally equivalent to that
of GR with a flat background. Thus, a canonical quantization with a Bunch-
Davies vacuum choice can be implemented. For this, the second order action is
necessary, allowing us to define the canonical variables.

In the presence of the reference topological curvature $\bar{R}_{\mu\nu}$, the
action is (Vigneron, 2024):

$\displaystyle S=\int\textrm{d}x^{4}\sqrt{|g|}\left[\frac{m_{\rm
p}^{2}}{2}\left(R_{\mu\nu}-\bar{R}_{\mu\nu}\right)g^{\mu\nu}+\mathcal{L}_{\rm\varphi}\right],$
(22)

where $m_{\rm p}$ is the Planck mass. The second order expansion of this
action can be expressed as

$\displaystyle S$
$\displaystyle=\frac{1}{2}\int\textrm{d}\tau\textrm{d}^{3}x\sqrt{\gamma}\left\\{\left[\left(\tilde{v}^{\prime}\right)^{2}+\frac{z^{\prime\prime}}{z}\tilde{v}^{2}-\left(D_{i}\tilde{v}\right)^{2}\right]\right.$
$\displaystyle\quad-\left[C^{\prime}\frac{\mathcal{D}^{2}}{K}C^{\prime}+\frac{a^{\prime\prime}}{a}C\frac{\mathcal{D}^{2}}{K}C-D_{i}CD^{i}\frac{\mathcal{D}^{2}}{K}C\right]$
(23)
$\displaystyle\quad\left.+\left[\left(\mathfrak{h}^{\prime}_{ij}\right)^{2}+\frac{a^{\prime\prime}}{a}\left(\mathfrak{h}_{ij}\right)^{2}-\left(D_{k}\mathfrak{h}_{ij}\right)^{2}-6K\left(\mathfrak{h}_{ij}\right)^{2}\right]\right\\},$

where we introduced $\tilde{v}\coloneqq am_{\rm
p}\sqrt{2\epsilon}\,\tilde{\mathcal{R}}$, $C\coloneqq am_{\rm
p}K\sqrt{2}\,\mathcal{C}$ and $\mathfrak{h}_{ij}\coloneqq\frac{am_{\rm
p}}{2}h_{ij}$, and with $\epsilon\coloneqq
1-\mathcal{H}^{\prime}/\mathcal{H}$. This action can be compared to the much
more complicated second order action in GR which, for $K\not=0$, features a
non-local operator, as derived in (Handley, 2019). This further emphasizes the
simplicity of the equations obtained when topo-GR is considered over GR once
$K\not=0$.

The Lagrangian for $C$ is fourth order in the spatial derivatives due to the
presence of the operator ${\mathcal{D}^{2}\coloneqq\Delta+3K}$. However, the
resulting field equation is still the wave equation (1), but with that
operator applied to it. Additionally, the presence of $\mathcal{D}^{2}$
implies that oscillating modes for $C$ such that
${\mathcal{D}^{2}=-k^{2}+3K>0}$ have a negative kinetic term and $C$ is a
ghost field for those modes. However, due to the discretization of $k$ for
$K>0$, only $k=0$ is permitted among those modes, which is pure gauge and not
physical. The second discrete mode $k^{2}=3K$ is also not physical as it
disappears from the action. This can also be seen from the perturbation
equations which can be rewritten to depend only on
$\mathcal{D}^{2}\mathcal{C}$ (Sec. 4.4 in Vigneron and Poulin (2023)). Hence,
the physical modes of $C$ always have $-k^{2}+3K<0$ and $C$ is a canonical
field. Consequently, the presence of $\mathcal{D}^{2}$ in the action will only
affect the amplitude of the primordial power spectrum of $K\mathcal{C}$.

Choice of initial conditions
For a de Sitter phase, the background evolution (17) implies that, for any
$K$, we have $a^{\prime\prime}/a\rightarrow 0$ and
$z^{\prime\prime}/z\rightarrow 0$ in the limit $\tau\rightarrow-\infty$, and
the variables $\tilde{v}$, $C$ and $\mathfrak{h}_{ij}$ become harmonic
oscillators in a static background (almost for $C$). That background can be
described as a “curved Minkowski spacetime”, i.e. homogeneous and isotropic
without expansion, but spatially curved. Consequently, in the limit
$\tau\rightarrow-\infty$, a canonical quantization can be applied on these
three variables along with a Bunch-Davies choice, leading to

$\displaystyle\tilde{v}(\tau,k)$
$\displaystyle\overset{\tau\rightarrow-\infty}{\sim}\frac{e^{-ik\tau}}{\sqrt{2k}},$
(24) $\displaystyle C(\tau,k)$
$\displaystyle\overset{\tau\rightarrow-\infty}{\sim}{\frac{K}{k^{2}-3K}}\frac{e^{-ik\tau}}{\sqrt{2k}}$
$\displaystyle\ \ k^{2}>3K,$ (25) $\displaystyle\mathfrak{h}_{ij}(\tau,k)$
$\displaystyle\overset{\tau\rightarrow-\infty}{\sim}\frac{e^{-i\sqrt{k^{2}+6K}\tau}}{\sqrt{2\sqrt{k^{2}+6K}}}$
$\displaystyle\ \ k^{2}+6K>0.$ (26)

The frequency $\sqrt{k^{2}+6K}$, instead of $k$, appearing for the tensor mode
results from its effective mass being $k^{2}+6K$. Additionally, the extra
factor of $K/(k^{2}-3K)$ appearing in the normalisation of $C$ comes from its
canonical momentum being $\pi_{C}=-\frac{\mathcal{D}^{2}}{K}C^{\prime}$, i.e.
different from the usual formula $\pi_{C}=C^{\prime}$. This normalisation is
well-defined since physical modes for $C$ have $k^{2}>3K$, as explained
before.

Power spectra from slow-roll inflation
Using the wave equations (1), (1) and (21), the scale factor evolution (17)
and the initial conditions (24)–(26), we can derive the power spectra of
$\tilde{\mathcal{R}}$, $\mathcal{C}$ and $h_{ij}$ from slow-roll inflation in
the same way it is done in GR with a flat background. The only difference is
that the wave number $k$ refers to harmonic functions adapted to the given
spatial curvature.

Defining the power spectrum as $\langle\Psi_{\bf k}(t)\Psi_{\bf
k^{\prime}}(t)\rangle=(2\pi)^{3}\delta(\bf k+k^{\prime})\mathcal{P}_{\Psi}$,
and $\mathcal{P}_{\Psi}\coloneqq|\Psi_{k}(\tau)|^{2}$, the primordial power
spectra resulting from a slow-roll inflationary phase are

$\displaystyle\mathcal{P}_{\tilde{\mathcal{R}}}$
$\displaystyle=\frac{1}{\epsilon}\frac{H^{2}}{2m_{\rm
p}^{2}}2^{2\nu-4}\left(\frac{\Gamma(\nu)}{\Gamma(\frac{3}{2})}\right)^{2}\,\frac{k^{-6\epsilon+2\eta}}{k^{3}},$
(27) $\displaystyle\mathcal{P}_{K\mathcal{C}}$
$\displaystyle=\frac{H^{2}}{2m_{\rm
p}^{2}}2^{2\mu-4}\left(\frac{\Gamma(\mu)}{\Gamma(\frac{3}{2})}\right)^{2}\,\left(\frac{K}{k^{2}-3K}\right)^{2}\,\frac{k^{-2\epsilon}}{k^{3}},$
(28) $\displaystyle\mathcal{P}_{h}$ $\displaystyle=16\frac{H^{2}}{2m_{\rm
p}^{2}}2^{2\mu-4}\left(\frac{\Gamma(\mu)}{\Gamma(\frac{3}{2})}\right)^{2}\,\frac{\left(\sqrt{k^{2}+6K}\right)^{-2\epsilon}}{\left(\sqrt{k^{2}+6K}\right)^{3}},$
(29)

where $H=\mathcal{H}/a$ is the Hubble rate during inflation, and we have
defined the usual slow-roll parameters $\epsilon\coloneqq-\dot{H}/H^{2}$ and
$\eta\coloneqq\epsilon-\ddot{\varphi}/(H\dot{\varphi})$; moreover, we set
$\mu\coloneqq 3/2+\epsilon$ and $\nu\coloneqq 3/2+3\epsilon-\eta$.

This is is an unambiguous way of defining the scaling and amplitude of a
spectrum resulting from a single field slow-roll inflation in any curved
space, something out of reach in GR. The power spectrum of the additional
scalar mode $\mathcal{C}$ is fully fixed by the slow-roll parameters and the
curvature, and no additional parameters are needed. Initial conditions on the
density perturbation are obtained from $\mathcal{P}_{\tilde{\mathcal{R}}}$
using relation (20).

At first order in the slow-roll parameters, we have the following ratios

$\displaystyle\mathcal{P}_{K\mathcal{C}}/\mathcal{P}_{\tilde{\mathcal{R}}}$
$\displaystyle=\epsilon{\left(\frac{K}{k^{2}-3K}\right)^{2}},$ (30)
$\displaystyle\mathcal{P}_{h}/\mathcal{P}_{\tilde{\mathcal{R}}}$
$\displaystyle=16\epsilon\left(\frac{k}{\sqrt{k^{2}+6K}}\right)^{3}.$ (31)

The tensor-to-scalar ratio $r$ features a scale dependence for scales
approaching the curvature scale, with an enhancement for $K<0$, similarly to
the GR case D’Agostino  _et al._ (2023). While there is a divergence for
$k^{2}=-6K$, it might be avoided for specific closed hyperbolic topologies for
which the minimal wave number of tensor modes would be $k_{\rm min}^{2}>-6K$
(e.g. Cornish and Spergel (1999)). Additionally, the consistency relation
between $r$ and the scaling index $n_{h}=-2\epsilon$ of tensor modes features
a dependence on the wave number for $K\not=0$. This prediction would be
observable only for a current curvature scale comparable to the CMB scale.

But most importantly, while topo-GR features one additional gauge invariant
variable $\mathcal{C}$, the amplitude of its power spectrum is negligible
compared to that of the curvature deviation perturbation. Additionally,
$\mathcal{C}$ decays during radiation and matter dominations on sub-Hubble
scales. Therefore, its effects on scalar mode fluctuations can be neglected
from the start of inflation to the late time Universe, and considering
$\mathcal{C}=0$ is a good approximation. In this case $\tilde{\mathcal{R}}$
exactly corresponds to the usual curvature perturbation $\mathcal{R}$, and the
situation is even closer to that of GR for $K=0$. Especially, the scale
dependence and the amplitude of $\mathcal{R}$ is equivalent for any $K$.

Conclusion
Curved inflationary scenarios are essential to the coherence of the inflation
hypothesis, but cannot be easily constructed within general relativity. In
this letter, we have shown that a canonical quantization of the inflaton is
possible for any background spatial curvature once a non-dynamical topological
term is added to the Einstein equation. The procedure is equivalent to that of
general relativity for the flat case, but applicable to any curvature. In
particular, (i) no bounce and no upper bound on the Hubble radius exists
during inflation and a Bunch-Davis vacuum condition is well-defined for any
$K$; (ii) there exists a variable $\tilde{\mathcal{R}}$, reducing to the
curvature perturbation for $K=0$, that is conserved on super-Hubble scales and
is solution of the Mukhanov-Sasaki equation for any $K$; (iii) a unique well-
defined primordial power spectrum is obtained for any $K$, with the same scale
dependence for scalar modes in each case.

This model of curved inflation has the same number of degrees of freedom as in
the standard model, which gives a natural answer to how curved inflation
should be built. We advocate that this is also a strong argument for
considering a non-dynamical topological term in the Einstein equation in the
form proposed by the topo-GR theory, when non-asymptotically flat spacetimes
and non-Euclidean topologies are considered.

QV is supported by the Centre of Excellence in Astrophysics and Astrochemistry
of Nicolaus Copernicus University in Toruń, and by the Polish National Science
Centre under grant No. SONATINA 2022/44/C/ST9/00078. We thank Sébastien
Renaux-Petel and Elliot Lynch for valuable discussions, and Guillaume Laibe
for his comments on the manuscript.

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##  Appendix A Vector modes during inflation in topo-GR

The equations constraining the vector modes $Q_{i}$ and $\mathcal{C}_{i}$
during single field inflation are

$\displaystyle\left(\Delta-2K\right)Q_{i}$
$\displaystyle=4K\mathcal{C}_{i}^{\prime}\,,$ (A1) $\displaystyle
Q_{i}^{\prime}+2\mathcal{H}Q_{i}$ $\displaystyle=4K\mathcal{C}_{i}.$ (A2)

The wave equation (1) for $\mathcal{C}_{i}$ follows from these two equations.
In topo-GR, $Q_{i}$ is sourced by $\mathcal{C}_{i}$, which seems to indicate
that vector modes might be non-negligible, for $K\not=0$. However, during a de
Sitter phase, characterised by $a(\tau)\propto 1/\tau$ for any $K$, the
general solution of this system of equations is

$\displaystyle\mathcal{C}_{i}$
$\displaystyle=\mathcal{C}_{i,1}\left[e^{-i\tau\sqrt{k^{2}+2K}}\left(1+i\tau\sqrt{k^{2}+2K}\right)+ie^{i\tau\sqrt{k^{2}+2K}}\left(1-i\tau\sqrt{k^{2}+2K}\right)\right],$
(A3) $\displaystyle Q_{i}$
$\displaystyle=-4K\mathcal{C}_{i,1}\,\tau\left[e^{-i\tau\sqrt{k^{2}+2K}}+ie^{i\tau\sqrt{k^{2}+2K}}\right],$
(A4)

where $\mathcal{C}_{i,1}$ is a constant. Therefore, the metric vector mode
$Q_{i}$ decays on all scales and can be neglected.

##  Appendix B Conservation of $\tilde{\mathcal{R}}$

The combination

$\displaystyle\left\\{\mathcal{H}\eqref{eq_blind_pert_S_2}-\frac{2\left[\eqref{eq_blind_pert_S_2}^{\prime}-\eqref{eq_blind_pert_S_4}+2\mathcal{H}\eqref{eq_blind_pert_S_2}+\eqref{eq_blind_pert_C_1}\right]}{3(1+w)}\right\\}$

of the first order equations of topo-GR leads to

$\displaystyle\mathcal{H}\tilde{\mathcal{R}}^{\prime}=\frac{a^{2}\kappa}{3(1+w)}\left[c_{\rm
s}^{2}\delta\rho_{\rm c}+\delta
p_{\not=ad}+\frac{2}{3}\left(\Delta+3K\right)\Pi\right],$ (B1)

where $\delta\rho_{\rm
C}\coloneqq\delta\rho_{\sigma}-3\mathcal{H}(1+w)\mathcal{V}$ is the density
perturbation in the comoving gauge. Apart for the term $3K\Pi$, that equation
is equivalent to the one obtained in GR for the curvature perturbation
$\mathcal{R}$ for $K=0$. Then, using the Poisson equation (1), equation (B1)
leads to

$\displaystyle\frac{1}{\mathcal{H}}\tilde{\mathcal{R}}^{\prime}$
$\displaystyle\underset{\delta p_{\not=\rm
ad}=0}{\overset{\Pi=0}{=}}\quad\frac{2c_{\rm
s}^{2}}{3(1+w)}\frac{a^{2}\kappa}{2\mathcal{H}^{2}}\delta\rho_{\rm c}$
$\displaystyle\quad=\frac{2c_{\rm
s}^{2}}{3(1+w)}\left[\frac{-k^{2}}{\mathcal{H}^{2}}\left(\Psi-K\mathcal{C}\right)-3K\frac{\mathcal{C}^{\prime}}{\mathcal{H}}\right],$
(B2)

for shear-free adiabatic perturbations, and to

$\displaystyle\frac{1}{\mathcal{H}}\tilde{\mathcal{R}}^{\prime}$
$\displaystyle\overset{\varphi}{=}\frac{2}{3(1+w)}\frac{a^{2}\kappa}{2\mathcal{H}^{2}}\delta\rho_{\rm
c}$
$\displaystyle=\frac{1}{\epsilon}\left[\frac{-k^{2}}{\mathcal{H}^{2}}\left(\Psi-K\mathcal{C}\right)-3K\frac{\mathcal{C}^{\prime}}{\mathcal{H}}\right],$
(B3)

for single field inflation.

The wave equation (1) implies that, for shear-free matter, $\mathcal{C}$ is
conserved on super-Hubble scales during a de Sitter phase, radiation or matter
dominated eras. Therefore, the right-hand-sides of equations (B2) and (B3)
vanish on super-Hubble scales, and $\tilde{\mathcal{R}}$ is conserved on these
scales.

The uniform energy density curvature perturbation
$\zeta\coloneqq-\mathcal{R}-\frac{\mathcal{H}}{\rho^{\prime}}\delta\rho_{\rm
c}$ is often used in GR to study inflation, since it is also constant on
super-Hubble scales for adiabatic matters, or single field inflation, in the
flat case. Similarly to $\tilde{\mathcal{R}}$, we can define a uniform energy
density curvature deviation perturbation $\tilde{\zeta}$ as

$\displaystyle\tilde{\zeta}$ $\displaystyle\coloneqq\zeta+K\mathcal{C},$ (B4)

which is also conserved on super-Hubble scales when $\tilde{\mathcal{R}}$ is,
for any $K$.

##  Appendix C Wave equation for $\tilde{\mathcal{R}}$

Consider a scalar field $\phi$ with energy-momentum tensor

$\displaystyle
T^{\mu}{}_{\nu}=\nabla^{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}\nabla^{\alpha}\phi\nabla_{\alpha}\phi\,\delta^{\mu}_{\nu}-V\,\delta^{\mu}_{\nu}.$
(C1)

We define the perturbation of the scalar field in the longitudinal gauge as
$\phi=\varphi+\delta\varphi$. The first order matter variables from the scalar
field are

$\displaystyle\delta\rho_{l}=\frac{1}{a^{2}}\left(\varphi^{\prime}\delta\varphi^{\prime}-\varphi^{\prime
2}\Phi\right)+\partial_{\varphi}V\delta\varphi\quad;\quad\mathcal{V}=-\frac{\delta\varphi}{\varphi^{\prime}}\quad;\quad\Pi=\Pi^{i}=\Pi^{ij}=0\
;\ \mathcal{V}_{i}=Q_{i}\quad;\quad\delta p_{\not=\rm ad}=\left(1-c_{\rm
s}^{2}\right)\delta\rho_{\rm C}.$

The first order field equations (1)–(1) of topo-GR become in this case

$\displaystyle\Delta\Psi-3\mathcal{H}\Psi^{\prime}-a^{2}V\Psi-\frac{1}{2}\left(\varphi^{\prime}\delta\varphi^{\prime}+a^{2}\partial_{\varphi}V\delta\varphi\right)=K\Delta\mathcal{C}-4KV\mathcal{C},$
(C2)
$\displaystyle\Psi^{\prime}+\mathcal{H}\Psi-\frac{1}{2}\varphi^{\prime}\delta\varphi=4K\mathcal{H}\mathcal{C}+K\mathcal{C}^{\prime},$
(C3) $\displaystyle\Phi-\Psi=-4K\mathcal{C}.$ (C4)

The derivative of equation (C3) is

$\displaystyle\Psi^{\prime\prime}+\mathcal{H}\Psi^{\prime}+\mathcal{H}^{\prime}\Psi-\frac{1}{2}\left[\varphi^{\prime}\delta\varphi^{\prime}-\left(2\mathcal{H}\varphi^{\prime}+a^{2}\partial_{\varphi}V\right)\delta\varphi\right]=K\left(\mathcal{C}^{\prime\prime}+4\mathcal{H}\mathcal{C}^{\prime}+4\mathcal{H}^{\prime}\mathcal{C}\right).$
(C5)

Finally, the time component of the conservation of the energy-momentum tensor
is, using equation (C4),

$\displaystyle\delta\varphi^{\prime\prime}+2\mathcal{H}\delta\varphi^{\prime}+a^{2}\partial_{\varphi}^{2}V\delta\varphi-\Delta\varphi-4\varphi^{\prime}\Psi^{\prime}+2a^{2}\partial_{\varphi}V\Psi=4K\left(-\varphi^{\prime}\mathcal{C}^{\prime}+2a^{2}\partial_{\varphi}V\mathcal{C}\right).$
(C6)

The left-hand-sides of equations (C2)–(C6) are equivalent to the first order
equations obtained with the Einstein equation in the flat case, but here for
any curvature. On the RHS, additional terms are present that depend on
$\mathcal{C}$. The combination of equations

$\displaystyle\left\\{\eqref{eq_3}+\frac{\mathcal{H}}{\varphi^{\prime}}\eqref{eq_4}+\frac{2z^{\prime}}{z}\eqref{eq_2}-\eqref{eq_1}\right\\},$

with the use of
${\frac{2z^{\prime}}{z}=-2\mathcal{H}-\frac{2\mathcal{H}^{\prime}}{\mathcal{H}}-2a^{2}\frac{\partial_{\varphi}V}{\varphi^{\prime}}}$
and $z\coloneqq\frac{a\varphi^{\prime}}{\mathcal{H}}$, leads to

$\displaystyle\mathcal{R}^{\prime\prime}+\frac{2z^{\prime}}{z}\mathcal{R}^{\prime}-\Delta\mathcal{R}=K\mathcal{C}^{\prime\prime}+K\frac{2z^{\prime}}{z}\mathcal{C}^{\prime}-K\Delta\mathcal{C}.$
(C7)

This is a wave equation for the curvature perturbation, with a source term
depending on $\mathcal{C}$. This equation can be rewritten as,

$\displaystyle\tilde{\mathcal{R}}^{\prime\prime}+\frac{2z^{\prime}}{z}\tilde{\mathcal{R}}^{\prime}-\Delta\tilde{\mathcal{R}}=0\qquad\qquad\forall\,K,$
(C8)

with $\tilde{\mathcal{R}}=\mathcal{R}-K\mathcal{C}$.

##  Appendix D Second order action

The action at second order of topo-GR can be calculated following the same
approach as in GR, within the ADM formalism Maldacena (2003). From the action
(22) of topo-GR, we obtain

$\displaystyle S=\frac{1}{2}\int\textrm{d}\tau\textrm{d}x^{3}\sqrt{s}\Big{[}$
$\displaystyle
N\left(\mathcal{R}^{(3)}_{ij}-\bar{R}_{ij}\right)s^{ij}+\frac{1}{N}\left(E^{ij}E_{ij}-E^{2}\right)+\frac{1}{N}\left(\phi^{\prime}+N^{i}\partial_{i}\phi\right)^{2}+ND_{i}\phi
D^{i}\phi-2NV$ $\displaystyle\
+\frac{1}{N}N^{i}N^{j}\bar{R}_{ij}-\frac{2}{N}N^{i}\bar{R}_{i0}+\frac{1}{N}\bar{R}_{00}\Big{]},$
(D1)

where $s_{ij}$ is the spatial metric with $\mathcal{R}^{(3)}_{ij}$ its Ricci
curvature and $\sqrt{s}\coloneqq\sqrt{{\rm det}\,s_{ij}}$, $N$ is the lapse,
$N^{i}$ is the shift, and

$\displaystyle
E_{ij}\coloneqq\frac{1}{2}\gamma^{\prime}_{ij}-D_{(i}N_{j)},\quad E\coloneqq
E^{i}{}_{i}.$ (D2)

We first consider only scalar modes. The goal is to develop the action (D1) up
to second order in the first order variables $\mathcal{R}$ and $\mathcal{C}$.
For this, the first and second orders of the reference curvature as functions
of these variables are needed. Because the reference curvature is a fixed
tensor, its perturbation orders are pure gauge. This means that in a general
gauge, up to second order, $\bar{R}_{\mu\nu}$ is of the form Vigneron and
Poulin (2023)

$\displaystyle\bar{R}_{\mu\nu}=\overset{(0)}{R}_{\mu\nu}+\left[\mathcal{L}_{X}\overset{(0)}{R}_{\mu\nu}\right]+\left[\frac{1}{2}\mathcal{L}_{X}^{2}\overset{(0)}{R}_{\mu\nu}+\mathcal{L}_{Y}\overset{(0)}{R}_{\mu\nu}\right],$
(D3)

where $X^{\mu}$ and $Y^{\mu}$ are respectively first and second order vectors,
and $\overset{(0)}{R}_{\mu\nu}=2K\gamma_{ij}$. Since all second order
variables will vanish in the second order action by using the background
equations, we are only interested in determining $X^{\mu}$ in the gauge we are
choosing. The time component $X^{0}$ is not present in the first order of
$\bar{R}_{\mu\nu}$, and in the second order is only present in a spatial
boundary term. Hence, it can be assumed to be zero in the calculation of the
second order action, leading to $X^{\mu}=\delta^{\mu}_{i}X^{i}$.

In the comoving gauge, for which $\delta\varphi=0$, the spatial metric have
the general form

$\displaystyle
s_{ij}=a^{2}\left[\left(1-2\mathcal{R}\right)\gamma_{ij}+2D_{i}D_{j}E\right],$
(D4)

where the choice of $E$ changes the shift but not the foliation, hence the
gauge remains comoving. The choice $E=-\mathcal{C}$ leads to $X^{i}=0$ because
the gauge invariant variable $\mathcal{C}$ is defined as
$\mathcal{C}\coloneqq\mathcal{X}-E$, where $X^{i}\eqqcolon D^{i}\mathcal{X}$
Vigneron and Poulin (2023). The choice $E=0$ is the standard one, and the
easiest in which calculating the second order action. For this choice, we have
$X^{\mu}=\delta^{\mu}_{i}D^{i}\mathcal{C}$, and the first and second orders of
the reference curvature take the form

$\displaystyle\overset{(1)}{\bar{R}}_{\mu\nu}$
$\displaystyle=4K\delta_{(\mu}^{0}\delta_{\nu)}^{i}\partial_{i}\mathcal{C}^{\prime}+4K\delta_{\mu}^{i}\delta_{\nu}^{j}\,D_{i}D_{j}\mathcal{C},$
(D5) $\displaystyle\overset{(2)}{\bar{R}}_{\mu\nu}$
$\displaystyle=2K\delta^{0}_{\mu}\delta^{0}_{\nu}\,D_{k}\mathcal{C}^{\prime}D^{k}\mathcal{C}^{\prime}+2K\left(D^{k}\mathcal{C}D_{k}D_{i}D_{j}\mathcal{C}+2D_{k}D_{i}\mathcal{C}D^{k}D_{j}\mathcal{C}\right)\delta^{i}_{\mu}\delta^{j}_{\nu}+2\overset{(2)}{\bar{R}}_{0i}\delta^{0}_{(\mu}\delta^{i}_{\nu)},$
(D6)

where $\overset{(2)}{\bar{R}}_{0i}$ is not needed for the calculation of the
second order action.

We define the perturbations of the lapse and shift respectively as

$\displaystyle N=a\left(1+\alpha\right)\quad;\quad
N_{i}=a^{2}\partial_{i}\beta.$ (D7)

The variation of the first order action (D1) with respect to $\alpha$ and
$\beta$ gives the constraints equations, respectively as

$\displaystyle\alpha=-\frac{\tilde{\mathcal{R}}^{\prime}}{\mathcal{H}},\quad\quad\Delta\left(\beta+\mathcal{C}^{\prime}\right)=-3K\mathcal{C}^{\prime}+\frac{1}{\mathcal{H}}\Delta\tilde{\mathcal{R}}-\epsilon\tilde{\mathcal{R}}^{\prime},$
(D8)

which also correspond, respectively, to equations (1) and (1). From there, the
goal is to calculate the action (D1) at second order in $\tilde{\mathcal{R}}$
and $\mathcal{C}$. The gauge choice (D4), with $E=0$, implies

$\displaystyle a^{2}s^{ij}$
$\displaystyle=\gamma^{ij}+\left[2\mathcal{R}\gamma^{ij}\right]+\left[4\mathcal{R}^{2}\gamma^{ij}\right],$
(D9) $\displaystyle\overset{(1)}{\Gamma}{}^{a}_{bc}$
$\displaystyle=-2\delta^{a}_{(b}D_{c)}\mathcal{R}+\gamma_{bc}D^{a}\mathcal{R},$
(D10) $\displaystyle\frac{1}{a^{3}}\sqrt{{\rm det}\,s_{ij}}$
$\displaystyle=1+\left[-3\mathcal{R}\right]+\left[\frac{3}{2}\mathcal{R}^{2}\right].$
(D11)

Using $\alpha=-\frac{\tilde{\mathcal{R}}^{\prime}}{\mathcal{H}}$, the
background expansion laws and several integration by parts, we obtain

$\displaystyle\int\textrm{d}\tau\textrm{d}x^{3}\sqrt{s}\left\\{\frac{1}{N}\left(E^{i}{}_{j}E^{j}{}_{i}-E^{2}\right)+\left[\frac{1}{N}\left(\phi^{\prime}+N^{i}D_{i}\phi\right)^{2}+ND_{i}\phi
D^{i}\phi-2NV\right]+\frac{1}{N}N^{i}N^{j}\bar{R}_{ij}-\frac{2}{N}N^{i}\bar{R}_{i0}-\frac{1}{N}\bar{R}_{00}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\
\;\,=\int\textrm{d}\tau\textrm{d}x^{3}a^{2}\sqrt{\gamma}\left[2\epsilon\left(\tilde{\mathcal{R}}^{\prime}\right)^{2}-6\left(K\mathcal{C}^{\prime}\right)^{2}+2KD_{k}\mathcal{C}^{\prime}D^{k}\mathcal{C}^{\prime}\right],$
(D12)
$\displaystyle\int\textrm{d}\tau\textrm{d}x^{3}\sqrt{s}\,N\left(\mathcal{R}^{(3)}_{ij}-\bar{R}_{ij}\right)s^{ij}=\int\textrm{d}\tau\textrm{d}x^{3}a^{2}\sqrt{\gamma}\left[-2\epsilon
D_{k}\tilde{\mathcal{R}}D^{k}\tilde{\mathcal{R}}+6K^{2}D_{k}\mathcal{C}D^{k}\mathcal{C}-2K\Delta\mathcal{C}\Delta\mathcal{C}\right].$
(D13)

Finally, the second order action for scalar modes takes the form

$\displaystyle S[\tilde{\mathcal{R}},\mathcal{C}]$
$\displaystyle=\frac{1}{2}\int\textrm{d}\tau\textrm{d}x^{3}2a^{2}m_{\rm
p}^{2}\sqrt{\gamma}\left\\{\epsilon\left[\left(\tilde{\mathcal{R}}^{\prime}\right)^{2}-\left(D_{k}\tilde{\mathcal{R}}\right)^{2}\right]-\left[3\left(K\mathcal{C}^{\prime}\right)^{2}-3\left(KD_{k}\mathcal{C}\right)^{2}+KD_{k}\mathcal{C}^{\prime}D^{k}\mathcal{C}^{\prime}-K\Delta\mathcal{C}\Delta\mathcal{C}\right]\right\\},$

which, with some additional integration by parts, can be rewritten as

$\displaystyle
S[\tilde{\mathcal{R}},\mathcal{C}]=\frac{1}{2}\int\textrm{d}\tau\textrm{d}x^{3}\sqrt{\gamma}\left(am_{\rm
p}\sqrt{2}\right)^{2}\left\\{\epsilon\left[\left(\tilde{\mathcal{R}}^{\prime}\right)^{2}-\left(D_{k}\tilde{\mathcal{R}}\right)^{2}\right]-K^{2}\left[\mathcal{C}^{\prime}\left(\Delta/K+3\right)\mathcal{C}^{\prime}+\mathcal{C}\Delta\left(\Delta/K+3\right)\mathcal{C}\right]\right\\}.$
(D14)

Introducing $\tilde{v}\coloneqq am_{\rm
p}\sqrt{2\epsilon}\,\tilde{\mathcal{R}}$ and $C\coloneqq am_{\rm
p}K\sqrt{2}\,C$, we obtain the first two terms in the action (A natural model
for curved inflation). Interestingly, when deriving (D12), the relation for
$\Delta(\beta+\mathcal{C}^{\prime})$ is not needed as this term disappears
during the calculation. On the contrary, in GR for $K\not=0$, $\beta$ remains
present in the action, even without a Laplacian. This requires inverting the
equation for $\Delta\beta$, leading to the presence of a non-local operator in
the second order action and the field equation for $\mathcal{R}$, as shown in
Handley (2019). The fact that this inversion of $\Delta\beta$ is not required
in topo-GR is the reason why no non-local operator appears in the second order
action.

For tensor modes, the derivation in the framework of topo-GR is equivalent to
the one in GR because the reference curvature up to second order does not
feature tensor modes. The only differences come from the expansion laws not
featuring the background curvature and the presence of
$\overset{(0)}{R}_{\mu\nu}=2K\gamma_{ij}$. The consequence is the replacement
of $2K$ with $6K$ in the action, leading to

$\displaystyle S[h_{ij}]$
$\displaystyle=\frac{1}{2}\int\textrm{d}\tau\textrm{d}x^{3}\sqrt{\gamma}\left(\frac{am_{\rm
p}}{2}\right)^{2}\left[\left(h^{\prime}_{ij}\right)^{2}-\left(D_{k}h_{ij}\right)^{2}-6K\left(h_{ij}\right)^{2}\right].$
(D15)

Introducing $\mathfrak{h}_{ij}\coloneqq am_{\rm p}\frac{1}{2}h_{ij}$, we
obtain the third term in the action (A natural model for curved inflation).

##  Appendix E Canonical quantization of $C$

The Lagrangian of $C$ features spatial derivatives of the velocity field
$C^{\prime}$ which cannot be removed with integration by parts. Therefore, to
calculate the conjugate momentum of the field $C$, we need to come back to the
general definition of the canonical momentum.

We consider a general Lagrangian depending on a field $q$, its time derivative
$\dot{q}$ and the spatial derivatives (first order or higher) of $q$ and
$\dot{q}$, i.e.
$\mathcal{L}\left(q,D_{i}q,D_{i}D_{j}q,...,\dot{q},D_{i}\dot{q},D_{i}D_{j}\dot{q},...\right)$.
The conjugate momentum $\pi$ of $q$ is defined as the functional derivative of
$\mathcal{L}$ with respect to $\dot{q}$, i.e. for a trial function $\phi$ we
have

$\displaystyle\int\textrm{d}x^{3}\pi\,\phi$
$\displaystyle\coloneqq\int\textrm{d}x^{3}\frac{\delta\mathcal{L}}{\delta\dot{q}}\,\phi$
$\displaystyle:=\int\textrm{d}x^{3}\left[\partial_{\dot{q}}\mathcal{L}+\frac{\partial\mathcal{L}}{D_{i}\dot{q}}D_{i}+\frac{\partial\mathcal{L}}{D_{i}D_{j}\dot{q}}D_{i}D_{j}+...\right]\,\phi,$
(E1)

which becomes, after integration by parts,

$\displaystyle\int\textrm{d}x^{3}\,\pi\,\phi$
$\displaystyle=\int\textrm{d}x^{3}\left[\partial_{\dot{q}}\mathcal{L}-D_{i}\frac{\partial\mathcal{L}}{D_{i}\dot{q}}+D_{i}D_{j}\frac{\partial\mathcal{L}}{D_{i}D_{j}\dot{q}}+...\right]\phi.$
(E2)

Up to second spatial derivatives, this yields

$\displaystyle\pi\coloneqq\frac{\partial\mathcal{L}}{\partial\dot{q}}-D_{i}\frac{\partial\mathcal{L}}{\partial{D_{i}\dot{q}}}+D_{i}D_{j}\frac{\partial\mathcal{L}}{\partial{D_{i}D_{j}\dot{q}}}+...$
(E3)

Due to the dependence of $\mathcal{L}$ in $D_{i}\dot{q}$, we get extra terms
in the definition of $\pi$, compared to the usual formula
$\pi=\partial\mathcal{L}/\partial{\dot{q}}$. Eq (E) also shows that adding a
boundary term in the Lagrangian, e.g.
$\mathcal{L}\rightarrow\mathcal{L}+\Delta\dot{q}^{2}$, does not change the
expression for the momentum.

Now using Eq (E3) for the Lagrangian of $C$, in the limit
$\tau\rightarrow-\infty$, so that we neglect $a^{\prime\prime}/a$, i.e.

$\displaystyle\mathcal{L}_{C}\coloneqq-\frac{1}{2}\left(3C^{\prime
2}+\frac{1}{K}C^{\prime}\Delta C^{\prime}+3C\Delta C+\frac{1}{K}C\Delta\Delta
C\right),$ (E4)

the conjugate momentum of $C$ is

$\displaystyle\pi_{C}=-\left(\Delta/K+3\right)C^{\prime}.$ (E5)

Performing a canonical quantization of $C$ and considering the minimal energy
state of the Hamiltonian as the initial condition, we constrain $C$ to a
single initial oscillating mode

$\displaystyle C\ \overset{\tau\rightarrow-\infty}{\propto}\ e^{\pm
ik\tau},\qquad k^{2}\not=3K.$ (E6)

From the canonical commutation rule $[\hat{C},\hat{\pi}_{C}]=i\hbar$, we
select the mode $e^{-ik\tau}$ and constrain the amplitude such that

$\displaystyle
C\overset{\tau\rightarrow-\infty}{\sim}\left(\frac{K}{k^{2}-3K}\right)\frac{1}{\sqrt{2k}}e^{-ik\tau},\quad
k^{2}>3K.$ (E7)

The mode $k^{2}=3K$ is not physical since $C$ disappears from the action for
that mode. Additionally, for $K>0$, the wave-number $k$ is discrete and given
by ${k^{2}=n(n+2)K}$ for $n\in\mathbb{N}$. Consequently, the first physical
mode is $k^{2}\geq 8K$, implying that the initial condition (E7) is well-
defined for all physical modes and the amplitude of $C$ is bounded from above
by a finite value, for $K$ positive or negative.


# Snowmass White Paper: prospects for measurements of the bottom quark mass

Javier Aparisi  Instituto de Física Corpuscular, CSIC-University of Valencia,
Valencia, Spain  Juan Fuster  Instituto de Física Corpuscular, CSIC-University
of Valencia, Valencia, Spain  André Hoang  University of Vienna, Austria
Adrián Irles  Instituto de Física Corpuscular, CSIC-University of Valencia,
Valencia, Spain  Christopher Lepenik  University of Vienna, Austria  Vicent
Mateu  Departamento de Física Fundamental e IUFFyM, Universidad de Salamanca,
Salamanca, Spain  Germán Rodrigo  European Research Council Executive Agency,
European Commission, BE-1049 Brussels, Belgium. Disclaimer: the views
expressed in this article are strictly those of the author and may not in any
circumstance be regarded as stating an official position of the European
Commission.  Instituto de Física Corpuscular, CSIC-University of Valencia,
Valencia, Spain  Michael Spira  PSI Villigen, Switzerland  Seidai Tairafune
Tohoku University, Sendai, Japan  Junping Tian  University of Tokyo, Tokyo,
Japan  Marcel Vos<EMAIL_ADDRESS>Instituto de Física Corpuscular,
CSIC-University of Valencia, Valencia, Spain  Hitoshi Yamamoto  On leave from
Tohoku University, Sendai, Japan  Instituto de Física Corpuscular, CSIC-
University of Valencia, Valencia, Spain  Ryo Yonamine  Tohoku University,
Sendai, Japan

###### Abstract

In this white paper for the Snowmass ’21 community planning exercise we
provide quantitative prospects for bottom quark mass measurements in high-
energy collisions at future colliders that can provide a precise test of the
scale evolution, or “running” of quark masses predicted by QCD.

bottom quark mass, scale evolution, Higgs boson

###### pacs:

##  I Introduction

The masses of quarks are free parameters in the Standard Model of particle
physics, whose values must be determined experimentally. Precise measurements
are performed through the comparison of measurements of physical observables
sensitive to the mass to SM predictions in Quantum Chromodynamics (QCD) beyond
leading order accuracy. Like the strong coupling constant, quark masses depend
on the renormalization scheme and are renormalization-scale-dependent
parameters or “running constants”. In this study, we adopt the most popular
renormalization scheme, the modified minimal subtraction scheme or
$\overline{\rm MS}$ scheme, where the strong coupling $\alpha_{s}(\mu)$ and
the quark masses $m_{q}(\mu)$ depend on the dimensionful renormalization scale
$\mu$ that is identified with the energy scale of the scattering process
111Throughout this White Paper we define the $\overline{\rm MS}$ bottom quark
mass in the $n_{f}=5$ flavor scheme.. The scale evolution is predicted by QCD.
Given a measurement at one scale, the value at any other scale is predicted by
the renormalization group equation (RGE). RGE calculations have reached 5-loop
(${\cal O}(\alpha_{s}^{5})$) accuracy Vermaseren  _et al._ (1997); Chetyrkin
(1997); Baikov  _et al._ (2014) and software packages such as RunDec Herren
and Steinhauser (2018) and REvolver Hoang  _et al._ (2021) provide access to
state-of-the-art renormalization evolution and scheme conversions. The
predicted evolution can be tested experimentally by performing measurements at
different scales. The evolution of the strong coupling has been tested over a
broad range of energies Zyla  _et al._ (2020) and experiments have studied the
“running” of $\overline{\mathrm{MS}}$ quark masses for the the charm quark at
HERA Gizhko  _et al._ (2017) and the top quark at the LHC CMS collaboration
(2020). We discuss the case of the bottom quark mass in this White Paper.

The most precise extractions of the bottom quark mass Narison (2020); Peset
_et al._ (2018); Kiyo  _et al._ (2016); Penin and Zerf (2014); Alberti  _et
al._ (2015); Beneke  _et al._ (2015); Dehnadi  _et al._ (2015); Lucha  _et
al._ (2013); Bodenstein  _et al._ (2012); Laschka  _et al._ (2011); Chetyrkin
_et al._ (2009) rely on the measurement of the mass of bottomonium bound
states and the $e^{+}e^{-}\rightarrow$ hadrons cross section as experimental
input, in combination with QCD sum rules and perturbative QCD calculations.
Several lattice QCD groups have also published results, the most recent of
which reaches a precision of approximately 0.3% Bazavov  _et al._ (2018);
Colquhoun  _et al._ (2015); Bernardoni  _et al._ (2014); Lee  _et al._ (2013);
Dimopoulos  _et al._ (2012) (see also the FLAG report Aoki  _et al._ (2020)).
The world average provided by the Particle Data Group (PDG) Zyla  _et al._
(2020) also includes inputs from HERA Abramowicz  _et al._ (2018) and the
BaBar and Belle experiments at the B-factories Schwanda  _et al._ (2008);
Aubert  _et al._ (2010). This world average of low-scale bottom-quark mass
measurements has a relative precision better than 1%:

$m_{b}(m_{b})=4.18^{+0.03}_{-0.02}~{}{\mathrm{Ge\kern-1.00006ptV}}{},$ (1)

where the reference value of the bottom mass is quoted in the
$\overline{\mathrm{MS}}$ scheme, at a scale given by the mass itself.

Bottom quark mass measurements at a much higher scale became possible at LEP
and SLC, where jet rates and event shapes are sensitive to subleading mass
effects. A practical method to extract the bottom-quark mass from $Z$-pole
data was proposed in Ref. Bilenky  _et al._ (1995). Three independent groups
completed the necessary next-to-leading order (NLO) theoretical calculation of
the three-jet rate for massive quarks Rodrigo  _et al._ (1997); Bilenky  _et
al._ (1999); Rodrigo  _et al._ (1999); Bernreuther  _et al._ (1997);
Brandenburg and Uwer (1998); Nason and Oleari (1997, 1998) (an NNLO
calculation for the three-jet rate in $e^{+}e^{-}$ collisions, without bottom-
quark-mass effects, is available in Ref. Gehrmann-De Ridder  _et al._ (2008)).
The first measurement of this type was performed by the DELPHI collaboration
Abreu  _et al._ (1998) using the LEP $Z$-pole data. Similar measurements were
also performed with SLD Brandenburg  _et al._ (1999); Abe  _et al._ (1999)
data, and by ALEPH Barate  _et al._ (2000), OPAL Abbiendi  _et al._ (2001) and
DELPHI Abdallah  _et al._ (2006, 2008). The combination of the most precise
determinations from three-jet rates of each experiment yields the following
average for $m_{b}(m_{Z})$:

$m_{b}(m_{Z})=2.82\pm 0.28~{}{\mathrm{Ge\kern-1.00006ptV}}{}.$ (2)

This value is in good agreement with the average of Ref. Kluth (2006) that is
based on a slightly different sub-set of measurements.

Recently, a new measurement was published of the bottom quark mass at the
scale of the Higgs boson mass Aparisi  _et al._ (2022). The value of
$m_{b}(m_{H})$ is inferred from measurements by the ATLAS ATLAS Collaboration
(2020) and CMS CMS collaboration (2019) experiments of the bottom quark decay
width to bottom quarks $\Gamma(H\rightarrow b\bar{b}$). The width is
normalized to $\Gamma(H\rightarrow ZZ)$, the decay width for the $ZZ$ decay
mode. The average of the $m_{b}(m_{H})$ results obtained from both
measurements yields:

$m_{b}(m_{H})=2.60^{+0.36}_{-0.30}~{}{\mathrm{Ge\kern-1.00006ptV}}{}.$ (3)

This result reinforces the experimental evidence for the “running” of the
$\overline{\mathrm{MS}}$ bottom quark mass, definitively excluding the no-
running scenario with a statistical significance greater than 7 standard
deviations.

Figure 1:  The scale evolution of the bottom quark $\overline{\mathrm{MS}}$
mass. The measurements include the PDG world average for $m_{b}(m_{b})$ from
low-scale measurements, the measurements of $m_{b}(m_{Z})$ from jet rates at
the $Z$-pole at LEP and SLC and the measurement of $m_{b}(m_{H})$ from Higgs
boson branching fractions. The prediction of the evolution of the mass is
calculated at five-loop precision with REvolver Hoang  _et al._ (2021). The
inner dark grey error band includes the effect of missing higher orders and
the parametric uncertainties from $m_{b}(m_{b})$ and $\alpha_{s}$ from the PDG
averages. The outer band with a lighter shading includes additionally the
effect of a $\pm$ 0.004 variation of $\alpha_{s}(m_{Z})$. Figure from Ref.
Aparisi  _et al._ (2022).

The three sets of measurements, of $m_{b}(m_{b})$, $m_{b}(m_{Z})$ and
$m_{b}(m_{H})$, are shown in Fig. 1 and compared to the evolution of the PDG
world average from $m_{b}(m_{b})$ to a higher scale using the RGE calculation
included in the REvolver code Hoang  _et al._ (2021) at five-loop precision.

We argue that in the next decade the study of the “running” of the bottom
quark mass will turn into a precise test of QCD. Precise measurements at
several energy scales can be used to rule out or confirm the presence of
massive new coloured states that may contribute to the quark mass evolution.
New collider facilities can further enhance this potential in several ways. In
this White Paper we provide a quantitative assessment of the potential of the
High-Luminosity phase of the LHC (HL-LHC) and a future “Higgs factory”
electron-positron collider to test the scale evolution of the bottom quark
mass predicted by QCD.

##  II The bottom-quark mass from low-energy measurements

Both from the theoretical and experimental points of view, the ideal
observable to determine $m_{b}$ is the bottomonium spectrum. The masses of
$b\bar{b}$ bound states are very sensitive to the bottom quark mass and have
nearly vanishing experimental uncertainties. On the other hand, we can compute
the masses of the low-lying narrow bottominum resonances only through
perturbative expansions supplemented with non-perturbative corrections. The
typical dynamical scale of these narrow bottominum masses are of the order of
the inverse Bohr radius $\sim C_{F}\alpha_{s}m_{b}\gg\Lambda_{\rm QCD}$, such
that perturbative expansions can be used reliably and non-perturbative
corrections remain small. Quarkonium masses have been computed to N3LO for
arbitrary quantum numbers and ultrasoft resummation is known to N2LL. The most
up to date analyses have been carried out in Refs. Peset  _et al._ (2018) and
Mateu and Ortega (2018) (earlier analyses use the $\overline{\rm MS}$ mass and
are not discussed). They share some features, like employing low-scale short-
distance masses (RS and MSR, respectively), using a 3-flavor scheme plus
finite charm mass corrections and varying two renormalization scales, whose
variation range is inferred from the perturbative logarithms. There are small
differences as well: while Peset  _et al._ (2018) determines the mass from a
single bound state, Mateu and Ortega (2018) performs global fits with
correlated scale variation using a $\chi^{2}$ function. In this latter
analysis non-perturbative effects are estimated by comparing the $m_{b}$
results for different sets of quarkonia states. The analyses in Ref. Peset
_et al._ (2018) and Mateu and Ortega (2018) both use PDG Zyla  _et al._ (2020)
data and find compatible results. Ref. Peset  _et al._ (2018) finds:

$m_{b}(m_{b})=4.186\pm 0.037\,\rm GeV,$ (4)

while Ref. Mateu and Ortega (2018) finds:

$m_{b}(m_{b})=4.216\pm 0.039\,\rm GeV.$ (5)

Since experimental data is already extremely accurate, more precision in the
future could only come from the theory side. If a new perturbative order
becomes available the theoretical uncertainty (that clearly dominates) could
go down to $0.026\,$GeV. This estimate is obtained with a quadratic
extrapolation (assuming that the error from N3LO to N4LO goes down by the same
factor as it does for going from N2LO to N3LO yields a slightly smaller
error). This should be taken as a rough estimate only. On the other hand, if
N3LL resummation becomes available for all bound states (so far it is only
known for P-wave states), an additional error reduction should be expected as
well.

Weighted averages of the bottom-tagged hadronic R-ratio in $e^{+}e^{-}$
annihilation are alternative physical observables to measure $m_{b}$. They are
very sensitive to the bottom quark mass, and precise theoretical calculations
can be achieved. This method goes under the name of QCD sum rules, and it can
take different forms depending on the specific weight function and on whether
the integration is cut off at some finite energy or not. For relativistic sum
rules it was shown in Refs. Dehnadi  _et al._ (2013, 2015) that in order to
properly estimate perturbative uncertainties it is mandatory to vary the
renormalization scales of the strong coupling $\alpha_{s}$ and bottom quark
${\overline{\rm MS}}$ mass independently, in order to avoid a biased
dependence on the expansion prescription. On the experimental side one has to
account for contributions from narrow resonances (whose masses and electronic
widths can be taken from the PDG Zyla  _et al._ (2020)) and continuum data (so
far only BaBar Aubert  _et al._ (2009) has released data), which only exists
for energies below $11.21$ GeV. This limits the experimental precision of this
method if sum rules without an energy cut are considered (relativistic sum
rules). These are very clean, theoretically, since their intrinsic dynamical
scale is of order $m_{b}$. Alternatively, one can consider sum rules with an
energy cut at the last experimental data point (finite-energy sum rules), or
use moments with specific weight functions that strongly suppress the
contributions from high-energies (non-relativistic sum rules). These
approaches, in turn, have additional theoretical drawbacks, related to either
a higher sensitivity to non-perturbative effects or to somewhat lower
renormalization scales (as is also the case for the bottomonium mass method
mentioned above).

We show results from two representative analyses for relativistic sum rules.
Ref. Chetyrkin  _et al._ (2009) uses correlated scale variation for the bottom
quark mass and $\alpha_{s}$ and finds:

$m_{b}(m_{b})=4.163\pm 0.016\,\rm GeV.$ (6)

Ref. Dehnadi  _et al._ (2015) with the independent variation mentioned above,
finding:

$m_{b}(m_{b})=4.176\pm 0.023\,\rm GeV.$ (7)

where the experimental systematic uncertainty by itself amounts to $19$ MeV
and the theoretical uncertainty to $9$ MeV for the more conservative analysis
of Ref. Dehnadi  _et al._ (2015). This uncertainty can be reduced
substantially if additional data in the continuum region around and above
$11.2$ GeV becomes available. This would potentially open up the possibility
for using the $n=1$ moment, which is theoretically the cleanest and most
precise, but has the strongest sensitivity to high-energy data. Assuming that
this new data decreases the uncertainty on the continuum by a factor of $2$,
the new experimental error on $m_{b}$ — obtained from the $n=2$ moment — would
be reduced to $13\,$MeV, making for a total uncertainty of $16\,$MeV if the
theoretical error is estimated as in Ref. Dehnadi  _et al._ (2015). If the
next perturbative order becomes available, the theoretical uncertainty may be
reduced by $30\%$ independent of the way how the perturbative truncation error
is estimated.

Concerning non-relativistic sum rules (for which non-perturbative effects are
smaller than in quarkonium masses) we discuss the following two analyses: Ref.
Beneke  _et al._ (2015) in which (potential) pNRQCD is used (which provided
N3LO fixed-order non-relativistic calculations) together with the PS short-
distance mass, and Ref. Hoang  _et al._ (2012) which uses (velocity) vNRQCD
(which provided NNLL renormalization group improved non-relativistic
calculations) and employs the 1S short-distance mass. The result from Ref.
Beneke  _et al._ (2015) is:

$m_{b}(m_{b})=4.193_{-0.035}^{+0.022}\,\rm GeV,$ (8)

and Ref. Hoang  _et al._ (2012) yields:

$m_{b}(m_{b})=4.235\pm 0.055\,\rm GeV.$ (9)

where the theoretical uncertainties are an order of magnitude larger than the
experimental ones. Improvements on this kind of bottom mass measurements can
only come from the theoretical side: additional perturbative orders may become
available (either through fixed-order corrections or anomalous dimensions for
the summation of logarithms) or an improved determination on the effects of
the finite charm quark mass could be reached. In either case an error
reduction will require a significant theoretical effort.

In the light of the available results as of 2022 one can envisage that the
most precise determination in the future will come from relativistic sum rules
(although it is not inconceivable that on the time scale of some of the future
collider facilities discussed in the Snowmass study lattice simulations with
dynamical bottom quarks are feasible). Assuming new data becomes available,
together with one extra perturbative coefficient, and hopefully new
theoretical strategies based on a better understanding of the perturbative
series (see e.g. Ref. Boito  _et al._ (2021a), where optimized moments were
tentatively introduced), it is not unrealistic to assume that the uncertainty
on $m_{b}(m_{b})$ is reduced below $10\,$MeV.

##  III The bottom-quark mass from three-jet rates

Future $e^{+}e^{-}$ colliders can improve the precision of the $m_{b}(m_{Z})$
measurement. A dedicated high-luminosity run at the $Z$-pole, i.e. the “GigaZ”
programme of a linear collider or the “TeraZ” run at the circular colliders,
yields a sample of $Z$-bosons that exceeds that of the LEP experiments and SLD
by orders of magnitude. We adopt the extrapolation of LEP/SLD results in Ref.
Fuster  _et al._ that assumes that the extraction of $m_{b}(m_{Z})$ from the
three-jet rates will be limited by the theory uncertainty and hadronization
uncertainties. Both sources of uncertainty are assumed to be reduced by a
factor 2. This requires fixed-order calculations at NNLO accuracy, with full
consideration of mass effects, which is available for Higgs decays Bernreuther
_et al._ (2018).

The Higgs factory program itself, with several inverse attobarn at a center-
of-mass energy of 240-250 GeV, can take advantage of radiative-return events.
The Lorentz-boost of the $Z$-bosons complicates the selection, reconstruction
and interpretation. A dedicated full-simulation study is therefore required to
provide a reliable, quantitative projection. However, it is clear that the
radiative-return data has the potential to significantly improve the precision
of existing LEP/SLC analysis.

Finally, a high-energy electron-positron collider operated at a center-of-mass
energy of 250 GeV or above can extend the analysis to higher energies and thus
probe the effect of coloured states with masses heavier than that the Higgs
boson on the running of the bottom quark mass. The potential of the three-jet
rate measurement to determine $m_{b}(\mu)$ for $\mu=$ 250 GeV has been studied
in Ref. Fuster  _et al._ . The mass dependence of the observable is found to
drop rapidly with increasing $\mu$, since the bottom quark mass dependence is
a power-suppressed correction. A measurement with a precision of 1 GeV is
feasible for $\mu=$ 250 GeV.

##  IV The bottom-quark mass from $Z-$boson decay

The bottom quark mass at the scale of the $Z$-boson mass can also be inferred
from the $Z\rightarrow b\bar{b}$ decay width. Currently, this method does not
offer a competitive precision. Using $R_{0,b}=\Gamma(Z\rightarrow
b\bar{b})/\Gamma_{\rm total}=0.21582\pm 0.00066$, as reported by the LEP/SLC
Electro-weak Working Group Schael  _et al._ (2006), Ref. Kluth (2022) finds an
uncertainty greater than 1 GeV.

A future high-statistics $Z$-pole run, together with theory improvements, can
significantly enhance the potential of this approach. Following the FCCee
Conceptual Design Report Abada  _et al._ (2019a, b), that predicts a ten-fold
increase of the precision of $R_{0,b}$, one can expect a precision of 140 MeV
(5%) on $m_{b}(m_{Z})$ after the “TeraZ” program. This requires considerable
improvements in the modelling of B- and D-hadron decays, compared to the
reference analysis performed by SLC that forms the basis for the extrapolation
by the FCCee study.

##  V The bottom-quark mass from Higgs decay

The measurement of $m_{b}(m_{H})$ from the Higgs decay width to a bottom-
antibottom quark pair is expected to increase rapidly in precision as the
precision of Higgs coupling measurement improves. The method of Ref. Aparisi
_et al._ (2022) provides a very clean theoretical basis that allows for steady
progress as the experimental precision improves. The key aspect of this method
is that the Higgs boson is a color-less spin-0 state with a relatively small
decay width, such that the analysis is essentially insensitive to the
theoretical knowledge of the Higgs production rate. For the same reason very
precise theoretical predictions can be made for the Higgs partial width into a
bottom-antibottom quark pair.

At the relevant dynamical scale, $m_{H}$, the QCD corrections are very well
under control using $\mu\sim m_{H}$ as the renormalization scales of
$\alpha_{s}$ and $m_{b}$. The partial width $\Gamma(H\rightarrow b\bar{b}$ is
proportional to the squared of the bottom quark mass. This dependence arises
because the decay is governed the bottom Yukawa coupling. For these reasons we
expect the determination of $m_{b}(m_{H})$ in $H\rightarrow b\bar{b}$ decay to
become the “golden” measurement among the high-energy determinations.

The current theory uncertainty from missing higher orders and parametric
uncertainties from $\alpha_{s}$ and $m_{H}$ is estimated to be 60 MeV Aparisi
_et al._ (2022), well below the current experimental precision. The theory
uncertainty is dominated by the parametric uncertainty from the Higgs boson
mass. The current uncertainty on the Higgs mass of 240 MeV leads to an
uncertainty of $\sim$40 MeV on $m_{b}(m_{H})$ and is expected to come down
considerably as more precise determinations of $m_{H}$ appear. Future
prospects for Higgs mass measurements are summarized in Ref. de Blas  _et al._
(2020). Both the HL-LHC Cepeda  _et al._ (2019) and the Higgs factory Yan  _et
al._ (2016) are expected to provide a measurement of the Higgs boson mass to
10-20 MeV precision, which is sufficient to reduce the impact of this source
of uncertainty on $m_{b}(m_{H})$ to below 10 MeV.

The effect of the strong coupling $\alpha_{s}$, which amounts to an 0.2%
uncertainty in the ratio of branching ratios for an uncertainty of $0.001$ in
the value of $\alpha_{s}(m_{Z})$, is relatively small. A much larger
parametric $\alpha_{s}$ uncertainty was reported in Ref. de Florian  _et al._
(2016), which stems mainly from the parametric dependence of the evolution of
the bottom quark mass from $\mu=m_{b}$ to $\mu=m_{H}$ on the value of
$\alpha_{s}$. This source of uncertainty does not exist in the measurement of
$m_{b}(m_{H})$. The parametric uncertainty from the value of $\alpha_{s}$ is
expected to remain sub-dominant even with only a very modest improvement of
its world average.

The dominant theory uncertainty is due to missing higher order electroweak
corrections for the branching ratios, that are currently known with NLO
precision. These electroweak uncertainties are about a factor of two larger
than the current uncertainties from QCD. Thus the knowledge of the leading
NNLO EW correction is expected to be sufficient to take full advantage of the
power of the Higgs factory data.

Ref. Cepeda  _et al._ (2019) provides the projections for the LHC and its
luminosity upgrade, extrapolating the partial run 2 results under the
following assumptions: both statistical and systematic uncertainties are
envisaged to scale with integrated luminosity $L$ as $1/\sqrt{L}$ up to
certain limits, while theory uncertainties are expected to improve by a factor
two. This “S2 scenario” leads to a projected uncertainty on the Higgs
branching ratio to bottom quarks of 4.4% (1.5% stat., 1.3% exp., 4.0% theo.)
and on $\lambda_{bz}=\mu^{bb}/\mu^{ZZ}$ of 3.1% (1.3% stat., 1.3% syst., 2.6%
theo.), an improvement by nearly a factor of ten with respect to the first
measurement in Ref. Aparisi  _et al._ (2022).

The Higgs boson couplings will be measured to even higher precision at future
$e^{+}e^{-}$ Higgs factories de Blas  _et al._ (2020). The recoil analysis
yields a precisely measured production cross section, and normalization of the
total width, while the branching fractions can be inferred from a simple
counting experiment on the selected events. Sub-% precision is expected for
the $Hb\bar{b}$ coupling, based on detailed prospect studies of Ref. Barklow
_et al._ (2018); Abramowicz  _et al._ (2017).

For a numerical estimate of the potential of the Higgs factory projects to
improve $m_{b}(m_{H})$ we adopt the projections for the ratio $BR(H\rightarrow
b\bar{b})/BR(H\rightarrow W^{+}W^{-}$ from the ILC project. This ratio can be
more precisely measured than the ratio $BR(H\rightarrow
b\bar{b})/BR(H\rightarrow ZZ)$, due to the better statistical precision in the
$H\rightarrow W^{+}W^{-}$ channel. The 250 GeV stage of the International
Linear Collider (ILC) can measure the former ratio to 0.86% precision Fujii
_et al._ (2019); Bambade  _et al._ (2019). The corresponding uncertainty on
$m_{b}(m_{H})$ is $\pm$12 MeV. The complete ILC programme with stages at 250
GeV and 500 GeV is expected to improve the precision further, to 0.47% on
$BR(H\rightarrow b\bar{b})/BR(H\rightarrow W^{+}W^{-})$, and yields an
experimental uncertainty of 6 MeV on $m_{b}(m_{H})$. Other Higgs factory
projects are expected to reach similar precision.

At this stage, the experimental precision of $m_{b}(m_{H})$ is expected to
reach a relative precision of 0.2%, a factor two better than the current world
average for $m_{b}(m_{b})$. At this point, additional theoretical care will be
needed to assess the uncertainties in the theoretical predictions that are
used for extraction of $m_{b}(m_{H})$ and in relating it to $m_{b}(m_{b})$.

##  VI The strong coupling

A thorough and precise test of the scale evolution of the bottom quark mass
requires precise values for the strong coupling, both at the scale of the
bottom quark mass and at the electro-weak scale $\mu\sim m_{Z}\sim m_{H}$. In
the traditional approach to the determination of quark masses, one usually
assumes that the strong coupling is an external parameter that leads only to
parametric uncertainties. In this context, it is deemed sufficient to consider
the PDG world average for $\alpha_{s}(m_{Z})$ (or at any other renormalization
scale $\mu$) and use the Standard Model evolution equation to determine the
strong coupling at the scale needed for the theoretical calculations. This is
a reasonable approach for the determination of the bottom quark mass at a
given scale, but for a precise test of the QCD evolution one must apply a more
conservative view. The running of the strong coupling may be affected by the
same new physics effects that alter the running of the quark masses and
therefore cannot be assumed to follow the Standard Model RGE evolution. The
analysis of the scale evolution of the bottom mass for scales between $m_{b}$
and $m_{Z}$ or $m_{H}$ therefore also requires precise measurements of the
strong coupling $\alpha_{s}(\mu)$ over the interval of scales considered.

Lattice determinations of the strong coupling have achieved sub-% precision
Aoki  _et al._ (2021). The Flavour Lattice Averaging Group (FLAG) working
group on the strong coupling constant, provides the following projection in
Ref. d’Enterria  _et al._ (2022): “a total error clearly below half a percent
for $\alpha_{s}(m_{Z})$ seems achievable within the next few years by pushing
the step-scaling method further, possibly in combination with the decoupling
strategy.”

An independent low-energy determination of the strong coupling comes from
determinations from $\tau$-decays d’Enterria  _et al._ (2022). Today, these
yield a value of $\alpha_{s}(m_{\tau})=0.3077\pm 0.0065(exp.)\pm
0.0038(theory)$ Boito  _et al._ (2021b). The dominant experimental
uncertainties are expected to be reduced strongly using Belle II data Fujikawa
_et al._ (2008) and eventually new $Z$-pole data at a future electron-positron
collider Dam (2021), potentially achieving sub-% precision.

The determination of the strong coupling at the electro-weak scale is
currently dominated by LEP measurements. In Ref. Aparisi  _et al._ (2022) an
uncertainty of 0.004 was assigned to $\alpha_{s}(m_{Z})$ based solely on
electroweak-scale measurements. A future electron-positron collider running at
the $Z$ pole provides an ideal environment to reduce this uncertainty. Ref.
d’Enterria and Jacobsen (2020) claims the strong coupling can be measured to
better than 0.1% exploiting $Z$-boson hadronic pseudo-observables, provided
the theoretical uncertainties are reduced by incorporating missing higher-
order QCD and mixed QCD+EW corrections Freitas  _et al._ (2019).

Based on these qualititative prospects, we assign an uncertainty of 0.5% to
the strong coupling over the range from $m_{b}$ to $m_{H}$.

##  VII Projection for the test of the scale evolution

The projections and extrapolations discussed in the previous sections have
been included in Fig. 2. The markers are centered on the current central
values for $m_{b}(m_{Z})$ and $m_{b}(m_{H})$ and the error bars indicate the
projected precision. The solid line indicates the evolution of the PDG world
average from $m_{b}(m_{b})$ to a higher scale using the RGE calculation
included in the REvolver code Hoang  _et al._ (2021) at five-loop precision.
The uncertainty band includes the projected uncertainty of 10 MeV on
$m_{b}(m_{b})$ (dark grey) and an 0.5% uncertainty on $\alpha_{s}(m_{Z})$.

Figure 2:  Prospects for measurements of the scale evolution of the bottom
quark $\overline{\mathrm{MS}}$ mass at future colliders. The markers are
projections for $m_{b}(m_{Z})$ from three-jet rates at the $Z$-pole and for
$m_{b}(m_{H})$ from Higgs boson branching fractions. The RGE evolution of the
mass is calculated at five-loop precision with REvolver Hoang  _et al._
(2021).

The independent determinations of the bottom quark mass at different energies
yield a precision test of the scale evolution of the bottom quark mass. High-
scale determinations can be used to search for the impact of new massive
coloured states on the scale evolution, using a similar strategy to studies of
$\alpha_{s}$ Llorente and Nachman (2018); Jezabek and Kuhn (1993), and
possibly incorporating the analysis of $\alpha_{s}$ and $m_{b}$ in a combined
fit. The implementation of this programme, and a precise estimate of its
sensitivity, is left for future work.

##  VIII Summary

In the next decades, with the completion of the high luminosity programme of
the LHC and the construction of a new “Higgs factory” electron-positron
collider, rapid progress is envisaged in the measurement of Higgs coupling
measurement. These precise measurements will enable an extraction of the
$\overline{\mathrm{MS}}$ bottom quark mass $m_{b}(\mu)$ at the scale given by
the Higgs boson mass, $m_{b}(m_{H})$, with a precision of the order of 10 MeV.
With a relative precision of 2 per mille, the high-scale measurement can reach
a similar precision as $m_{b}(m_{b})$ based on low-energy measurements.

Together with improved measurements of $m_{b}(m_{b})$ from low-energy data,
$m_{b}(m_{Z})$ from three-jet rates in $e^{+}e^{-}$ collisions (and possibly
new measurements at scales smaller than $m_{Z}$ and larger than $m_{H}$), one
can expect to map out the scale evolution of the bottom quark mass from
$m_{b}$ to $m_{H}$ with a precision at the few per mille level. At the same
time, improved measurements of the strong coupling at each of these scales
reduce the uncertainty in the evolution between the two energies. When all
these elements are brought together, they form a powerful test of the
“running” of quark masses predicted by the Standard Model and allow for
stringent limits on coloured states with mass below the electroweak scale.

###### Acknowledgements.

The authors than Stefan Sint for help understanding the lattice prospects and
acknowledge support from projects FPA2015-65652-C4-3-R, PID2020-114473GB-I00,
PID2019-105439GB-C22 and PGC2018-094856-B-100
(MICIN/AEI/10.13039/501100011033), support from the U. Valencia and CSIC for
H. Yamamoto, PROMETEO/2021/071, PROMETEO-2018/060 and CIDEGENT/2020/21
(Generalitat Valenciana), iLINK grant LINKB20065 (CSIC), the FWF Austrian
Science Fund Project No. P28535-N27 and Doctoral Program No. W1252-N27; the EU
STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement
No. 824093 and the COST Action CA16201 PARTICLEFACE.

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