# Reassessing constant-roll Warm Inflation

Sandip Biswas<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Kanpur: 208016, Uttar Pradesh, India  Kaushik Bhattacharya
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology,
Kanpur: 208016, Uttar Pradesh, India  Suratna Das<EMAIL_ADDRESS>Department of Physics, Ashoka University, Rajiv Gandhi Education City, Rai,
Sonipat: 131029, Haryana, India

###### Abstract

Departing from standard slow-roll conditions is one way of putting the
inflationary paradigm to test, and constraining the dynamics of the inflaton
field with a constant-rate of roll of the inflaton field, a.k.a. the constant-
roll scenario, is one way of exploring such deviation from the standard slow-
roll dynamics. In this manuscript we explore such a possibility in a variant
inflationary scenario, known as Warm Inflation. We construct and derive the
conditions for having constant-roll WI models where inflation lasts at least
for 60 $e-$folds, gracefully exits the constant-roll inflation phase, and
maintains near thermal equilibrium of the system which is an essential feature
of WI in the slow-roll regime. We show that while certain models of WI (the
ones with dissipative coefficient as a function of temperature alone) can
accommodate constant-roll dynamics, others (with dissipative coefficient as a
function of temperature and the inflaton field both) fail to maintain thermal
equilibrium once the constant-roll condition is imposed and hence cannot
produce a constant-roll WI phase.

##  I Introduction

On the one hand, cosmic inflation Kazanas:1980tx ; Guth:1980zm ; Sato:1981ds ;
Sato:1980yn ; Linde:1981mu ; Albrecht:1982wi  solves the fine-tuning problems
of the Big Bang cosmology, i.e. the horizon problem, the flatness problem and
the monopole problem. On the other hand, it helps generate inhomogeneties in
an otherwise homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker
(FLRW) background, which eventually give rise to the large scale structures we
see around us today as well as the fluctuations in the temperature of the
Cosmic Microwave Background Radiation (CMBR) which have been measured to a
very good accuracy in recent times. The dynamics of the inflationary phase is
often portrayed as the dynamics of a single scalar field which is slowly
rolling down its approximately flat potential. The slow-roll dynamics of the
inflaton field is often invoked in any standard inflationary dynamics as it
helps gracefully exit the inflationary phase as well as generate nearly scale-
invariant spectra of primordial perturbations (both scalar and tensor), and
thus makes definite predictions which can be tested through the observations
of the CMBR and the large scale structures Riotto:2002yw ; Baumann:2009ds ;
Mishra:2024axb . This simple vanilla model of single field slow-roll inflation
is in good agreement with the current data Planck:2018jri . Significant
departure from the slow-roll dynamics, as it happens during an ultra-slow-roll
phase Dimopoulos:2017ged  for example, results in departure from the nearly
scale-invariant spectra that slow-roll generates. It has been shown that a
transient phase of Ultra-slow-roll yields large primordial perturbations which
eventually collapse to form Primordial Black Holes Motohashi:2017kbs , a
coveted candidate for Dark Matter Carr:2016drx .

Despite being in accordance with the current observations, inflation is still
a paradigm which allows one to explore many different ways to depart from the
standard single field slow-roll dynamics of inflation while still being in
tune with the available data. One such attempt is to consider constant-roll
inflation Martin:2012pe ; Motohashi:2014ppa  which, depending on the value of
the constant-roll parameter $\beta$, can be significantly different from the
standard slow-roll inflation. The concept of constant-roll inflation was first
introduced in Martin:2012pe , but the dynamics of constant-roll was first
clearly penned down in Motohashi:2014ppa . The term “constant-roll” was also
first coined in Motohashi:2014ppa . As has been indicated in Motohashi:2014ppa
, the name “constant-roll” signifies a constant rate of roll of the inflaton
field, an additional constraint imposed on the dynamics of the inflaton field.
The constraint can be written as follows Martin:2012pe :

$\displaystyle\ddot{\phi}+3\beta H\dot{\phi}=0,$ (1)

where the constant-roll parameter $\beta$ is a constant and can take both
positive and negative values. Note that the constant-roll parameter $n$ in
Martin:2012pe  is equal to $-3\beta$ in our case. The above equation can be
rewritten as $d\ln\dot{\phi}/dN=-3\beta$, where $N$ is the number of
$e-$folds. This clearly shows that $\dot{\phi}$ changes with a constant rate
($-3\beta$) with $e-$folds, and hence the name. If $|\beta|\ll 1$, then one
can ignore $\ddot{\phi}$ with respect to the $3H\dot{\phi}$ term, and in such
cases the dynamics boils down to the familiar slow-roll dynamics. On the other
hand, $\beta=1$ corresponds to the ultra-slow-roll dynamics for a phase when
$\partial V/\partial\phi=0$, where $V$ represents the inflaton potential
Dimopoulos:2017ged . One can get an estimate of the allowed values of $\beta$
in two different ways. First of all, it has been shown in Morse:2018kda ;
Lin:2019fcz  that only small-$\beta$ constant-roll models ($\beta<0.5$) can be
universal attractors, whereas large-$\beta$ constant-roll ($\beta>0.5$) should
be treated as an initial transient phase followed by an evolution dominated by
slow-roll dynamics. The constant-roll parameter $\bar{\eta}$ in Morse:2018kda
; Lin:2019fcz  is equal to $3\beta$ in our case. Secondly, the perturbations
generated during a constant-roll inflationary phase (not the initial transient
phase followed by slow-roll dynamics) are compatible with current data only
when $\beta$ is negative and of the order $10^{-3}$ Motohashi:2017aob . Note
that the constant roll parameter $\beta$ in Motohashi:2017aob  is equal to
$-3\beta$ in our case. However, it is shown in Gao:2019sbz  that $\beta\sim 1$
is also allowed observationally, but such constant-roll inflation models fails
to gracefully exit an inflationary phase. Note that the constant-roll
parameter $\eta_{H}$ in Gao:2019sbz  is equal to $3\beta$ in our case. This
indicates that despite imposing the constant-roll condition, given in Eq. (1),
on a single-field inflationary dynamics, to be an attractor solution as well
as to remain viable by the data (in addition, to be able to gracefully exit an
inflationary phase), the dynamics should not differ much from a slow-roll
evolution.

In this article we will reassess the constant-roll dynamics in a variant
inflationary scenario, known as Warm Inflation Berera:1995ie . Unlike in the
standard single-field inflationary scenario described above, where the
couplings of the inflaton field is generally ignored during an inflationary
phase, in Warm Inflation (WI) these couplings play a significant role in
dissipating inflaton’s energy to maintain a constant radiation bath throughout
inflation (for more recent reviews on WI, see Kamali:2023lzq ; Berera:2023liv
). In doing so, WI can gracefully exit to a radiation dominated era, whereas
the more standard inflationary scenario, which we will refer to as Cold
Inflation (CI) henceforth, requires a subsequent reheating phase, dynamics of
which is still largely unknown. Also, the requirement of reheating post
inflation in CI scenarios restricts the classes of inflationary potentials to
the ones with a minimum. Otherwise, one requires other mechanism to reheat the
universe, such as gravitational reheating Ford:1986sy ; Chun:2009yu  for
example, apart from standard reheating dynamics via the inflaton’s oscillation
at the minima of the potential Mishra:2024axb . As WI doesn’t require a
subsequent reheating phase, such scenarios can accommodate larger classes of
inflaton potentials. For example, generalised runaway potentials, which do not
have minima, have been explored in Das:2020xmh  and was shown that the WI
dynamics smoothly transits to a radiation dominated era. However, graceful
exit in WI is a much more complex process than in CI as has been pointed out
in Das:2020lut . Some of the attractive features of WI include (a) generation
of smaller tensor-to-scalar ratio which helps accommodate potentials, like
quartic and quadratic, which are otherwise ruled out in CI for generating way
too much tensor perturbations compared to scalar perturbations Bartrum:2013fia
, (b) generation of Primordial Black Holes without departing from the slow-
roll dynamics Arya:2019wck ; Bastero-Gil:2021fac ; Correa:2022ngq ;
Arya:2023pod , (c) accommodating the de Sitter Swampland Conjecture
Ooguri:2018wrx ; Garg:2018reu  naturally Das:2018hqy ; Motaharfar:2018zyb ;
Das:2018rpg ; Das:2019hto ; Das:2019acf  which otherwise disfavours the
standard CI dynamics Kinney:2018nny , to name of few.

An important feature of WI is that the radiation bath produced during WI is
treated to be in near thermal equilibrium due to which a temperature $T$ can
be assigned to the bath. The inflaton field, on the other hand, may or may not
thermalize with the radiation bath. The radiation bath, produced during WI is
sub-dominant (the potential energy density of the inflaton field dominates the
energy density during WI just like it happens in CI), but not negligible. One
measure to differentiate between CI and WI is that if the radiation energy
density $\rho_{r}$ is greater (less) than the Hubble parameter $H$ during an
inflationary phase, then it is to be treated as WI (CI) Kamali:2023lzq . Upon
thermalization of the radiation bath, the WI condition, $\rho_{r}^{1/4}>H$,
translates into the thermalization condition $T>H$, which is to be maintained
throughout WI. The near thermal equilibrium condition also plays a major role
in determining the cosmological perturbations produced during WI. Unlike CI,
where the primordial perturbations are quantum in nature, WI produces thermal
fluctuations as well, which are classical and determines the form of the
scalar power spectrum Kamali:2023lzq ; Berera:2023liv . It has been noted
earlier that it becomes a daunting task to maintain near thermal equilibrium
of the radiation bath during WI when the dynamics veers away from slow-roll
dynamics, such as in the case of ultra-slow-roll Biswas:2023jcd . It has been
shown in Biswas:2023jcd  that the thermalization condition, which is an
essential feature of WI, can be maintained only for a few $e-$foldings and
that too in a very restrictive class of WI models where the dissipative
coefficient is a function of both temperature and the inflaton field.

Constant-roll dynamics in WI has been explored a few times in the literature
before Kamali:2019wdh ; Mun:2021kzb ; Setare:2021iei ; AlHallak:2021hwb ;
Saleem:2023aof . The latter three works Setare:2021iei ; AlHallak:2021hwb ;
Saleem:2023aof  deal with constant-roll WI in modified gravity theories, and
thus of no interest for the present discussion. The first analysis
Kamali:2019wdh  dealt with toy models of WI where the parameter $Q$ (ratio of
the two frictional terms in the equation of motion of the inflaton field in
WI) is kept constant. The second analysis Mun:2021kzb  dealt with some
specific form of $Q$ depending on the inflaton field $\phi$ and the
temperature $T$, though they did not consider the standard forms of
dissipative coefficients appearing in various WI models Kamali:2023lzq ;
Berera:2023liv . Their parametrization of $Q$ as a function of $\phi$ and $T$
are rather ad hoc. More importantly, both these papers do not take into
account the thermalization condition which requires to be maintained even in a
constant-roll dynamics of WI (simply to call it a WI scenario). Above all,
both these works do not comment on graceful exit conditions in constant-roll
WI. We will comment more on these works in a latter section.

What we want in this paper is to obtain constant-roll WI models where (a)
constant-roll defines the full dynamics of an inflationary phase (at least 60
$e-$folds of it), and not just a transient phase of it, (b) graceful exit from
the constant-roll phase is ensured, and (c) near thermal equilibrium condition
($T>H$) is maintained throughout the constant roll phase so that we can
logically call it a constant-roll WI model. In doing so, we will explore all
possible kinds of WI scenarios, and will not restrict ourselves to only
constant-$Q$ toy models of WI. We will explore both weak dissipative $(Q\ll
1)$ and strong dissipative $(Q\gg 1)$ scenarios of WI as well. We have
furnished the rest of the paper as follows. In Sec. (II) we revisit the
constant-roll CI scenario and determine the graceful exit conditions, which,
to the best of our knowledge, has not been addressed so far in the literature.
In doing so, we will show that the graceful exit itself constrain the
constant-roll parameter $\beta$. In Sec (III) we then determine the graceful
exit conditions for any WI model. In Sec. (IV) we will determine the
conditions to maintain near thermal equilibrium during a constant-roll WI
scenario. We will show that to maintain near thermal equilibrium condition in
constant-roll WI scenario, one cannot differ much from a slow-roll evolution.
This is a condition which was also obtained for constant-roll CI models to be
compatible with the current data. However, it comes from the perturbation
analysis of the constant-roll CI dynamics. Here, in the case of WI, even the
background evolution constrains the dynamics to evolve closely to a slow-roll
dynamics. We will also show that WI models where the dissipative coefficient
is a function of temperature and the inflaton field, cannot maintain the
thermalization condition during a constant-roll phase, and thus constant-roll
dynamics cannot be realised in the class of such WI models. In Sec.(V), we
will discuss the main results obtained in this paper and will conclude with
some future outlooks.

##  II A fresh look at constant-roll Cold Inflation: Graceful exit condition

In standard CI scenario, the background geometry evolves according to the
first and second Friedmann equations, whereas the dynamics of the scalar
inflaton field $\phi$ is governed by the Klein-Gordon equation. These three
equations together determine the evolution of the universe during CI, which
can be written as

$\displaystyle 3M_{\rm Pl}^{2}H^{2}=\frac{\dot{\phi}^{2}}{2}+V(\phi),$ (2)
$\displaystyle 2M_{\rm Pl}^{2}\dot{H}=-\dot{\phi}^{2},$ (3)
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V,_{\phi}=0,$ (4)

respectively. Here the overdot denotes derivative with respect to the cosmic
time $t$, $V,_{\phi}\equiv dV(\phi)/d\phi$, $H\equiv\dot{a}/a$ is the Hubble
parameter determining the rate of change of the scale factor $a$ of the
Friedmann-Lemaître-Robertson-Walker background metric, and $M_{\rm Pl}$ is the
reduced Planck mass. The evolution of the background is often quantified in
terms of the Hubble slow-roll parameters defined as Martin:2012pe

$\displaystyle\epsilon_{i+1}=\frac{d\ln\epsilon_{i}}{dN},$ (5)

where $N\equiv\ln a$ is the number of $e-$folds. The first two Hubble slow-
roll parameters are of interest to us, which starting from $\epsilon_{0}=1/H$
in the above relation, can be written, as

$\displaystyle\epsilon_{1}=-\frac{\dot{H}}{H^{2}},\quad\quad\epsilon_{2}=\frac{\dot{\epsilon}_{1}}{\epsilon_{1}H}.$
(6)

One requires $\epsilon_{1}<1$ to ensure an inflationary phase, and
$\epsilon_{1}\sim 1$ designates end of inflation. The standard slow-roll of
the inflaton field requires the subsequent Hubble slow-roll parameters to be
smaller than unity $(|\epsilon_{i}|\leq 1)$. However, the subsequent slow-roll
parameters can become of order unity or larger when the dynamics deviates from
slow-roll. Even if the dynamics deviates from slow-roll, one requires the
first Hubble slow-roll parameter to be smaller than unity $(\epsilon_{1}<1)$
to ensure inflation. Thus, to end an inflationary phase, $\epsilon_{1}$ should
grow from smaller than unity values to unity. Therefore, graceful exit demands
a growing $\epsilon_{1}$ during any inflationary phase, slow-roll or
otherwise, which can be written as

$\displaystyle\frac{d\epsilon_{1}}{dN}=\frac{\dot{\epsilon}_{1}}{H}>0.$ (7)

As $H$ is always positive in an expanding Universe, we thus require
$\dot{\epsilon}_{1}>0$ to gracefully exit any inflationary phase.

Imposing the constant-roll constraint given in Eq. (1), the Klein-Gordon
equation in Eq. (4) would become

$\displaystyle 3(1-\beta)H\dot{\phi}=-V,_{\phi}.$ (8)

We would first like to investigate for what values of $\beta$ constant-roll CI
can make a graceful exit. From Eqs. (6), we note that

$\displaystyle\dot{\epsilon}_{1}=-\frac{1}{H^{2}}\left(\ddot{H}-\frac{2\dot{H}^{2}}{H}\right).$
(9)

In constant-roll inflation, we get

$\displaystyle M_{\rm Pl}^{2}\ddot{H}=3\beta H\dot{\phi}^{2},$ (10)

whereas, $\dot{H}$ can be read from Eq. (3). Therefore, we obtain

$\displaystyle\ddot{H}-\frac{2\dot{H}^{2}}{H}=\frac{3H\dot{\phi}^{2}}{M_{\rm
Pl}^{2}}\left(\beta-\frac{\dot{\phi}^{2}}{6M_{\rm Pl}^{2}H^{2}}\right).$ (11)

It is clear from this expression that if $\beta<0$, then the above expression
is always negative, yielding $\dot{\epsilon}_{1}>0$ from Eq. (9). The
constant-roll CI models which are compatible with the observations also
restricts the values of $\beta$ in the negative domain Motohashi:2017aob .
Hence, graceful exit is also ensured in such observationally viable constant-
roll CI models.

However, positive $\beta$ values with

$\displaystyle\beta<\frac{\dot{\phi}^{2}}{6M_{\rm Pl}^{2}H^{2}}$ (12)

can also yield $\dot{\epsilon}_{1}>0$, and hence, can end constant-roll CI.
Inserting Eq. (8) into Eq. (2) and solving for $H^{2}$ one gets

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{V}{6M_{\rm
Pl}^{2}}\left(1+\sqrt{1+\frac{2}{3}\frac{M_{\rm
Pl}^{2}}{(1-\beta)^{2}}\frac{V,_{\phi}^{2}}{V^{2}}}\right)$ (13)
$\displaystyle=$ $\displaystyle\frac{V}{6M_{\rm
Pl}^{2}}\left(1+\sqrt{1+\frac{4\epsilon_{V}}{3(1-\beta)^{2}}}\right),$

which yields

$\displaystyle\beta<\frac{4\epsilon_{V}}{3(1-\beta)^{2}}\left(1+\sqrt{1+\frac{4\epsilon_{V}}{3(1-\beta)^{2}}}\right)^{-2},$
(14)

and further assuming $\beta<1$, the above condition simplifies to
$\beta<\epsilon_{V}/3$. In the above equations, $\epsilon_{V}$ is a potential
slow-roll parameter defined as $\epsilon_{V}\equiv(M_{\rm
Pl}^{2}/2)(V,_{\phi}/V)^{2}.$ Thus one can afford to have positive $\beta$
which ends constant-roll CI, if $\beta$ satisfies the above conditions. To
note, this condition clearly depends on the form of the potential. However,
observationally, $\beta>0$ models are not favoured in CI because such models
yield anomalous super-Hubble evolution of curvature perturbations
Motohashi:2014ppa  111Note that the constant-roll parameter $\alpha$ defined
in Motohashi:2014ppa  is related to our constant-roll parameter $\beta$ as
$\beta=1+\alpha/3$..

##  III Graceful exit condition for constant-roll Warm Inflation

In Warm Inflation (WI), the inflaton field dissipates its energy to a
subdominant, yet non-negligible radiation bath throughout inflation. This
feature not only modifies the inflaton dynamics, but also affects the
background evolution. The Friedmann equations, the Klein-Gordon equation and
the evolution of the energy density of the subdominant radiation bath
$(\rho_{r})$ in WI can be written as

$\displaystyle 3M_{\rm
Pl}^{2}H^{2}=\frac{\dot{\phi}^{2}}{2}+V(\phi)+\rho_{r},$ (15) $\displaystyle
2M_{\rm Pl}^{2}\dot{H}=-\left(\dot{\phi}^{2}+\frac{4}{3}\rho_{r}\right),$ (16)
$\displaystyle\ddot{\phi}+3H(1+Q)\dot{\phi}+V,_{\phi}=0,$ (17)
$\displaystyle\dot{\rho}_{r}+4H\rho_{r}=\Upsilon\dot{\phi}^{2},$ (18)

where $\Upsilon$ designates the rate at which the inflaton dissipates its
energy to the radiation bath, and the parameter $Q$ in the Klein-Gordon
equation is the ratio of the two frictional terms present in the inflaton
dynamics, the friction due to dissipation and the friction due to the
expansion of the background universe, and is defined as $Q=\Upsilon/3H$. In
general, the dissipative term $\Upsilon$ is a function of the inflaton field
($\phi$) and the temperature of the radiation bath ($T$), assuming the system
evolves in a near thermal-equilibrium condition.

For constant-roll WI, we again impose the constant-roll condition, given in
Eq. (1), on the Warm inflationary inflaton dynamics. We need a growing
$\epsilon_{1}$ in order to gracefully exit WI, and from Eq. (9) we see that
this condition can be achieved in two ways:

  1. 1.

$\ddot{H}<0$ is a sufficient condition to achieve the condition. Given the
above set of equations, one gets

$\displaystyle M_{\rm Pl}^{2}\ddot{H}$ $\displaystyle=$
$\displaystyle-\left(-3H\beta\dot{\phi}^{2}+\frac{2}{3}\dot{\rho}_{r}\right)$
(19) $\displaystyle=$ $\displaystyle
3H\left(\beta-\frac{2}{3}Q\right)\dot{\phi}^{2}+\frac{8}{3}H\rho_{r}.$

Therefore, the graceful exit condition for constant-roll WI boils down to
having

$\displaystyle\beta<\frac{2}{3}Q-\frac{8}{9}\frac{\rho_{r}}{\dot{\phi}^{2}}.$
(20)

Therefore, as $Q$ is always positive, it is evident that not only negative
values of $\beta$, but also positive values of $\beta$ are allowed in WI as
far as graceful exit is concerned. However, as the rhs of the above inequality
evolves with time, this condition should be maintained throughout WI, and
needs to be checked numerically.

  2. 2.

If $\ddot{H}>0$ but $\ddot{H}<2\dot{H}^{2}/H$, then also one gets growing
$\epsilon_{1}$. However, as this condition doesn’t lead to any simplified
condition, we leave it at this form. This condition needs to be fulfilled
throughout inflation.

##  IV Maintaining thermal equilibrium during constant-roll Warm Inflation

During standard slow-roll WI a constant radiation bath is maintained
throughout inflation by the dissipation mechanism. In such a case, one can
assume $\dot{\rho}_{r}\approx 0$ in Eq. (18). This radiation bath is also
assumed be to evolving near thermal equilibrium, and hence a temperature $T$
can be associated with this radiation bath (the inflaton field, on the other
hand, may or may not thermalize with this radiation bath). In such a case, the
radiation energy density can be written as

$\displaystyle\rho_{r}=\frac{\pi^{2}}{30}g_{*}T^{4},$ (21)

where $g_{*}$ is the relativistic degrees of freedom of the radiation bath.
Furthermore, $\dot{\rho}_{r}\approx 0$ also suggests $\dot{T}\approx 0$
implying that the radiation bath evolves near thermal equilibrium. These
features of slow-roll WI suggest that to maintain thermal equilibrium during
constant-roll WI $\dot{\rho}_{r}\approx 0$ is a valid assumption. Hence, we
will proceed with this condition to analyze the thermal equilibrium during
constant-roll WI. With this assumption Eq. (18) can be written as

$\displaystyle\rho_{r}\approx\frac{3}{4}Q\dot{\phi}^{2},$ (22)

and plugging this information into Eq. (15) and Eq. (16) yields

$\displaystyle 3M_{\rm
Pl}^{2}H^{2}\approx\left(1+\frac{3}{2}Q\right)\frac{\dot{\phi}^{2}}{2}+V(\phi),$
(23) $\displaystyle 2M_{\rm Pl}^{2}\dot{H}\approx-(1+Q)\dot{\phi}^{2}.$ (24)

We will proceed with these approximated expressions to determine the
conditions for thermal stabilization during constant-roll WI.

###  IV.1 The constant $Q$ case with $\beta>0$

We will first assume that $\beta$ is positive. Integrating Eq. (1) with
respect to $N$, we see that

$\displaystyle\dot{\phi}(N)=\dot{\phi}_{0}e^{-3\beta N},$ (25)

where $\dot{\phi}_{0}$ is the initial value of $\dot{\phi}$. In such a case,
as inflation progresses the kinetic energy of the inflaton field keeps
decreasing. Integrating Eq. (24), with $\dot{\phi}$ as given above, we get

$\displaystyle 3M_{\rm
Pl}^{2}H^{2}=\frac{(1+Q)}{2\beta}\dot{\phi}_{0}^{2}e^{-6\beta N}+V_{0},$ (26)

where $V_{0}$ is the integration constant. This constant can be determined
from the fact that $\epsilon_{1}$ becomes 1 at the end of inflation
($N=N_{f}$, say). Therefore, using Eq. (24) and Eq. (26) to construct
$\epsilon_{1}$ and then setting it to unity at the end of inflation yields

$\displaystyle
V_{0}=\left(\frac{3\beta-1}{2\beta}\right)(1+Q)\dot{\phi}_{0}^{2}e^{-6\beta
N_{f}}.$ (27)

We note here that $\beta>1/3$ yields positive $V_{0}$ whereas $\beta<1/3$
yields negative $V_{0}$. Moreover, to be consistent, Eq. (23) and Eq. (26)
should be the same which can only happen for a specific form of the potential
$V(\phi)$, such as

$\displaystyle
V(\phi)=\left(\frac{1+Q}{\beta}-1-\frac{3}{2}Q\right)\frac{\dot{\phi}_{0}^{2}}{2}e^{-6\beta
N}+V_{0}.$ (28)

We will later determine the form of the potential in terms of $\phi$.

We will now determine what values of $\beta$ can yield temperature
stabilization during constant-roll WI. Assuming a constant radiation bath is
maintained during constant-roll WI, we determine the evolution of the
temperature $T$ of the radiation bath as

$\displaystyle\frac{d\ln
T}{dN}=\frac{1}{4-p}\left(\frac{\Upsilon,_{\phi}}{H\Upsilon}\dot{\phi}+\epsilon_{1}-6(1+Q)-2\frac{V,_{\phi}}{H\dot{\phi}}\right),$
(29)

where $p\equiv(\Upsilon,_{T}T)/\Upsilon$. We have used Eq. (21), Eq. (22) and
Eq. (17) to determine the above expression. For constant $Q$ we see that

$\displaystyle\frac{\Upsilon,_{\phi}}{H\Upsilon}\dot{\phi}=\frac{1}{3QH^{2}}\frac{d}{dt}(3QH)\frac{dt}{d\phi}\dot{\phi}=-\epsilon_{1}.$
(30)

In addition to that, the potential given in Eq. (28) yields

$\displaystyle
V,_{\phi}=3H\dot{\phi}\left[\beta\left(1+\frac{3}{2}Q\right)-(1+Q)\right].$
(31)

Plugging the above two expressions in Eq. (29), we get

$\displaystyle\frac{d\ln
T}{dN}=-\frac{6\beta}{4-p}\left(1+\frac{3}{2}Q\right).$ (32)

Now, if WI is taking place in a weak dissipative regime $(Q\ll 1)$ then one
requires $\beta\ll 1$ to maintain thermal equilibrium during constant-roll.
However, if WI is taking place in strong dissipative require $(Q\gg 1)$, then
thermal stabilization during constant-roll demands $Q\beta\ll 1$. In both the
scenarios $V_{0}$ will become negative because one requires $\beta\ll 1$, as
can be seen from Eq. (27).

Furthermore, we note that plugging the constant-roll condition Eq. (1) in the
WI inflaton equation of motion, given in Eq. (17), one obtains

$\displaystyle V,_{\phi}=3H\dot{\phi}[\beta-(1+Q)].$ (33)

It is required that the potential given in Eq. (28) should yield the same
equation of motion of the inflaton field during constant roll. Hence, Eq. (31)
should resemble the above equation. In the weak dissipative regime, if $Q\ll
2/3$, then these two expressions are equivalent. This imposes another
condition on weak dissipative constant-roll WI models with constant $Q$.
However, in the strong dissipative regime, these two expressions become
equivalent when $\beta\ll 2/3$, which is consistent with the thermal
stabilization condition.

To summarize, thermal stabilization can be maintained during constant-roll WI
(in models where $Q$ is constant) if

  * •

in weak dissipative regime $(Q\ll 1)$ one has $\beta\ll 1$ and $Q\ll 2/3$

  * •

in strong dissipative regime $(Q\gg 1)$ one has $Q\beta\ll 1$ and $\beta\ll
2/3$.

We will now determine the form of the potential, given in Eq. (28), in terms
of $\phi$ which gives rise to constant-roll WI. Integrating Eq. (25) again
with respect to $N$ (while keeping in mind that $V_{0}$ is negative) we get

$\displaystyle\phi(N)=\phi_{0}-\frac{M_{\rm
Pl}}{3}\sqrt{\frac{6}{\beta(1+Q)}}\cosh^{-1}\left(\sqrt{\frac{1+Q}{2\beta|V_{0}|}}\dot{\phi}_{0}e^{-3\beta
N}\right)\,,$

where $|V_{0}|$ is the absolute value of $V_{0}$ and $\phi_{0}$ is the
integration constant. Here $\phi_{0}$ may not be equal to $\phi(N=0)$.
Rearranging Eq. (IV.1) we find

$\displaystyle\dot{\phi}_{0}e^{-3\beta
N}=\sqrt{\frac{2\beta|V_{0}|}{1+Q}}\cosh\left(\frac{3}{M_{\rm
Pl}}\sqrt{\frac{\beta(1+Q)}{6}}(\phi_{0}-\phi)\right).$

Plugging this expression into Eq. (28) we find the form of the potential
giving rise to constant-roll WI as

$\displaystyle
V(\phi)=|V_{0}|\left[\left(1-\frac{\beta(2-3Q)}{2(1+Q)}\right)\cosh^{2}\left(\frac{3}{M_{\rm
Pl}}\sqrt{\frac{\beta(1+Q)}{6}}(\phi_{0}-\phi)\right)-1\right].$ (36)

Fig (1) depicts the form of this potential.

Figure 1: The form of the potential given in Eq. (36), with $\beta=10^{-3}$,
$Q=10^{-2}$ and $\phi_{0}=15$.

This potential corresponds to the Hubble parameter

$\displaystyle H=\sqrt{\frac{|V_{0}|}{3M_{\rm
Pl}^{2}}}\sinh\left(\sqrt{\frac{3\beta(1+Q)}{2}}\frac{(\phi-\phi_{0})}{M_{\rm
Pl}}\right).$ (37)

The above expression can be obtained by substituting Eq. (IV.1) into Eq. (26).
One can tally these results with the constant-roll Cold Inflation case by
setting $Q=0$ (and replacing $\beta$ with $1+\alpha/3$ to compare with
Motohashi:2014ppa ), which yields the Hubble parameter given in Eq. (2.13) of
Motohashi:2014ppa  and its corresponding potential given in Eq. (2.26).
However, unlike the constant-roll WI case we are discussing here, the
corresponding constant-roll CI case doesn’t give rise to an inflationary phase
because it yields $\ddot{a}(t)<0$, as has been pointed out in
Motohashi:2014ppa . 222The constant-roll WI scenarios corresponding to the
other two constant-roll CI scenarios discussed in Motohashi:2014ppa  with
Hubble parameters given in Eq. (2.11) and Eq. (2.13) along with their
corresponding potentials, given in Eq. (2.14) and Eq. (2.18), can be found by
considering $V_{0}=0$ and $V_{0}>0$ in Eq. (26), respectively, and finding
their corresponding potentials. However, these two constant-roll WI scenarios
cannot be realised while maintaining near thermal equilibrium during
inflation.

####  IV.1.1 Numerical analysis of constant-roll WI models in weak dissipative
regime

We evolve a constant-roll Warm Inflation scenario, with the dynamical
equations given in Eq. (15), Eq. (33)and Eq. (18), fully numerically with the
choice of parameter values as $\beta=10^{-3}$, $Q=10^{-2}$, $g_{*}=106.75$,
$\phi_{0}=15$ and initial conditions as $T(N=0)=2.79\times 10^{-6}\,M_{\rm
Pl}$ and $\dot{\phi}(N=0)\equiv\dot{\phi}_{0}=5.33\times 10^{-10}\,M_{\rm
Pl}^{2}$. Once the parameters are fixed we also know the values of $V_{0}$
from Eq. (27) by setting $N_{f}=60$. We get, with the above choice of
parameters, $|V_{0}|\approx 10^{-16}\,M_{\rm Pl}^{4}$ and
$\phi(N=0)=-1\,M_{\rm Pl}$. We also note that our choice of parameters are
such that $\beta\ll 1$ and $Q\ll 2/3$, which are the theoretically determined
conditions to be met to have a constant-roll WI in weak dissipative regime
with constant $Q$ models which can evolve near thermal equilibrium throughout
inflation.

The results of this numerical analysis is depicted in Fig. (2). As in this
model $\ddot{H}>0$ throughout, thus to have $\dot{\epsilon}_{1}>0$ one needs
to maintain $\ddot{H}<2\dot{H}^{2}/H$ throughout inflation in order to
gracefully exit inflation. Sub-figure (a) confirms that this condition has
been maintained. Sub-figures (b) and (c) show the evolution of $\epsilon_{1}$
and $\epsilon_{2}$ of this model, and we see that both the slow-roll
parameters remain less than unity throughout the evolution. The temperature of
the radiation bath doesn’t evolve much during the inflationary phase as has
been shown in sub-figure (d), along with the thermalization condition $T>H$
maintained throughout, indicating near thermal evolution of the system.

(a) Graceful exit condition (b) Evolution of $\epsilon_{1}$ (c) Evolution of
$\epsilon_{2}$

(d) Evolution of temperature $T$ (e) Thermalization condition $T>H$

Figure 2: Results of numerical evolution of a constant-roll WI scenario with
positive $\beta$ and constant $Q$ in weak dissipative regime.

####  IV.1.2 Numerical analysis of constant-roll WI models in strong
dissipative regime

We again evolve Eq. (15), Eq. (33)and Eq. (18) fully numerically, now with the
choice of parameter values as $\beta=10^{-4}$, $Q=2.8\times 10^{2}$,
$g_{*}=106.75$, $\phi_{0}=0$ and initial conditions as $T(N=0)=0.6\times
10^{-6}\,M_{\rm Pl}$ and $\dot{\phi}(N=0)\equiv\dot{\phi}_{0}=8.59\times
10^{-14}\,M_{\rm Pl}^{2}$. Once the parameters are fixed we get
$|V_{0}|\approx 10^{-20}\,M_{\rm Pl}^{4}$ and $\phi(N=0)=-1\,M_{\rm Pl}$. We
also note that our choice of parameters are such that $\beta\ll 2/3$ and
$Q\beta\ll 1$, which are the theoretically determined conditions to be met to
have a constant-roll WI in strong dissipative regime with constant $Q$ models
which can evolve near thermal equilibrium throughout inflation.

The results of this numerical analysis is depicted in Fig. (3). As in this
model too $\ddot{H}$ remains positive throughout, sub-figure (a) confirms that
the condition, $\ddot{H}<2\dot{H}^{2}/H$, has been maintained throughout and
thus guarantees graceful exit. Sub-figures (b) and (c) show the evolution of
$\epsilon_{1}$ and $\epsilon_{2}$ of this model, and we see that though
$\epsilon_{1}$ remains less than unity throughout the evolution,
$\epsilon_{2}$ starts with a value larger than unity, but quickly becomes less
than unity and remains so for the rest of the evolution till inflation lasts.
The temperature of the radiation bath doesn’t evolve much during the
inflationary phase as has been shown in sub-figure (d). The thermalization
condition $T>H$ maintained throughout, indicating near thermal evolution of
the system.

(a) Graceful exit condition (b) Evolution of $\epsilon_{1}$ (c) Evolution of
$\epsilon_{2}$

(d) Evolution of temperature $T$ (e) Thermalization condition $T>H$

Figure 3: Results of numerical evolution of a constant-roll WI scenario with
positive $\beta$ and constant $Q$ in strong dissipative regime.

####  IV.1.3 Some comments on the analysis presented in Ref. Kamali:2019wdh
and Ref. Mun:2021kzb

The first paper which made an attempt to formulate constant-roll WI is
Kamali:2019wdh . The authors in this paper formulated the background evolution
of the inflationary universe undergoing constant-roll WI and presented the
results related to the cosmological perturbations during such a constant-roll
WI. A few features of this analysis which we think are required to be
reconsidered are as follows:

  1. 1.

The paper doesn’t specify whether their model is suitable for sustaining at
least 60 $e-$folds of constant-roll inflation or whether it is related to a
transient constant-roll phase. This issue stems from the fact the authors did
not comment on graceful exit from a constant-roll phase.

  2. 2.

The analysis is done for positive $\beta$ values only. Also, the weak
dissipative regime is not explored in the paper. As we have shown here,
constant-roll WI can be realized in both weak and strong dissipative regimes
with both positive and negative values of $\beta$.

  3. 3.

The analysis is done only for toy models of WI where $Q$ is treated as
constant. In realistic models of WI, $Q$ becomes a function of both $\phi$ and
$T$.

  4. 4.

The background analysis done in this paper indicates that the dynamics doesn’t
evolve in a near thermal equilibrium condition. Yet, when the perturbation
calculations are done, it appears that the temperature parameter $T$ is taken
to be a constant. The references used by the previous authors to write the
Langevin equation, (Eq. (4.5) in Ref. Kamali:2019wdh ) explicitly requires the
system to be have a nearly constant temperature $T$. Thus the background
analysis doesn’t provide the environment for the perturbation analysis
presented in the paper.

  5. 5.

The authors mention that the constant-roll WI dynamics appears to be an
interplay between both cold and warm inflation. The challenge one might face
while considering CI and WI simultaneously is that the perturbation analysis
in both these scenarios are vastly different. In CI, the generated
perturbations determining the scalar power spectrum are quantum in nature,
while in WI they are classical being thermal fluctuations. Thus, treating CI
and WI simultaneously, if at all called for, requires more serious
consideration.

The second paper Mun:2021kzb  which also addresses the issue of constant-roll
WI defines the constant-roll condition for WI in a rather different way:

$\displaystyle\ddot{\phi}=-3H(1+Q)\lambda\dot{\phi},$ (38)

where $\lambda$ is a dimensionless constant. This stems form the fact that in
the inflaton’s equation of motion of WI, not only the Hubble friction term has
a $\dot{\phi}$ factor, but the dissipative term also has one. This raises the
question whether the constant rate of roll of the inflaton field in a
constant-roll WI scenario should depend on both these terms in the inflaton’s
equation of motion. We think that as $Q$ is not a constant in realistic models
of WI, the above way of defining the constant-roll condition would not lead to
constant rate of roll of the inflaton field in WI. Thus we stick to the
original constraint (Eq. (1)) as has been defined in cases of CI. The analysis
presented in this paper, too, does not address the issues of graceful exit or
thermalization condition (thermalization is assumed, without showing whether
the thermalization condition can be maintained throughout). Above all the
parametrization of $Q$ as a function of $\phi$ and $T$ considered in this
paper are rather ad hoc and doesn’t stem from any realistic model of WI. This
point has already been mentioned in the introduction of the paper.

###  IV.2 The constant $Q$ case with $\beta<0$

We will do a similar analysis as above, but now with a negative $\beta$. We
define $\tilde{\beta}=|\beta|$. We first note that with the constant-roll
constraint equation, Eq. (1), $\dot{\phi}$ will now be an increasing function
of $e-$folds:

$\displaystyle\dot{\phi}(N)=\dot{\phi}_{0}e^{3\tilde{\beta}N},$ (39)

which then yields using Eq. (24)

$\displaystyle 3M_{\rm
Pl}^{2}H^{2}=-\frac{(1+Q)}{2\tilde{\beta}}\dot{\phi}_{0}^{2}e^{6\tilde{\beta}N}+V_{0}.$
(40)

It is to note here that, unlike the previous case with $\beta>0$, here $V_{0}$
has to be positive as the first term on rhs of the above equation is always
negative throughout inflation, and it is required that $V_{0}$ always remains
greater than the first term so that $H$ remains real. One can determine
$V_{0}$ as before by setting $\epsilon_{1}=1$ at $N=N_{f}$ and get

$\displaystyle
V_{0}=\left(\frac{1+3\tilde{\beta}}{2\tilde{\beta}}\right)(1+Q)\dot{\phi}_{0}^{2}e^{6\tilde{\beta}N_{f}},$
(41)

which turns out to be positive. Also, the form of the potential which is
consistent with both Eq. (23) and Eq. (40) is

$\displaystyle
V(\phi)=V_{0}-\left(1+\frac{3}{2}Q+\frac{1+Q}{\beta}\right)\frac{\dot{\phi}_{0}^{2}}{2}e^{6\tilde{\beta}N}.$
(42)

Proceeding with the same analysis as before, we find the temperature evolution
in such cases can be written as

$\displaystyle\frac{d\ln
T}{dN}=\frac{6\tilde{\beta}}{4-p}\left(1+\frac{3}{2}Q\right).$ (43)

Therefore, as before, temperature stabilizes in weak dissipative regime $(Q\ll
1)$ when $\tilde{\beta}\ll 1$ and in the strong dissipative regime $(Q\gg 1)$
when $Q\tilde{\beta}\ll 1$. Also, comparing the Klein-Gordon equation of WI
including the constant-roll constraint,

$\displaystyle V,_{\phi}=-3H\dot{\phi}[\tilde{\beta}+(1+Q)],$ (44)

with the derivative of the potential obtained in Eq. (42) as

$\displaystyle
V,_{\phi}=-3H\dot{\phi}\left[\tilde{\beta}\left(1+\frac{3}{2}Q\right)+(1+Q)\right],$
(45)

we see that one also requires $Q\ll 2/3$ in the weak dissipative regime and
$\tilde{\beta}\ll 2/3$ in the strong dissipative regime to have a consistent
picture. Thus, we obtained similar conditions for near thermal equilibrium as
before in the weak and strong dissipative regime.

Finally to determine the form of the potential as a function of $\phi$ we
proceed as before: first integrate Eq. (39) to obtain $\phi(N)$ as

$\displaystyle\phi(N)=\frac{M_{\rm
Pl}}{3}\sqrt{\frac{6}{\tilde{\beta}(1+Q)}}\sin^{-1}\left(\sqrt{\frac{1+Q}{2V_{0}\tilde{\beta}}}\dot{\phi}_{0}e^{3\tilde{\beta}N}\right)-\phi_{0},$
(46)

rearranging the expression to obtain $\dot{\phi}_{0}e^{3\tilde{\beta}N}$, and
then insert it into Eq. (42). We get

$\displaystyle
V(\phi)=V_{0}\left[1-\left(1+\frac{\tilde{\beta}(2+3Q)}{2(1+Q)}\right)\sin^{2}\left(\frac{3}{M_{\rm
Pl}}\sqrt{\frac{\tilde{\beta}(1+Q)}{6}}(\phi_{0}+\phi)\right)\right].$ (47)

Fig (4) depicts the form of this potential.

Figure 4: The form of the potential given in Eq. (47), with $\beta=-10^{-2}$,
$Q=10^{-2}$ and $\phi_{0}=0$.

This potential corresponds to the Hubble parameter

$\displaystyle H=\sqrt{\frac{V_{0}}{3M_{\rm
Pl}^{2}}}\cos\left(\sqrt{\frac{3\tilde{\beta}(1+Q)}{2}}\frac{(\phi+\phi_{0})}{M_{\rm
Pl}}\right).$ (48)

As $V_{0}>0$ in this case, it corresponds to the Hubble parameter of constant-
roll CI as given in Eq. (2.12) of Motohashi:2014ppa  and the above potential
corresponds to the potential given in Eq. (2.22) of Motohashi:2014ppa  in the
$Q\rightarrow 0$ limit.

####  IV.2.1 Numerical analysis of constant-roll WI models in weak dissipative
regime

We evolve Eq. (15), Eq. (44)and Eq. (18) fully numerically, with the choice of
parameter values as $\beta=-10^{-2}$, $Q=10^{-2}$, $g_{*}=106.75$,
$\phi_{0}=0$ and initial conditions as $T(N=0)=1.56\times 10^{-6}\,M_{\rm Pl}$
and $\dot{\phi}(N=0)\equiv\dot{\phi}_{0}=1.7\times 10^{-10}\,M_{\rm Pl}^{2}$.
Once the parameters are fixed we get $V_{0}\approx 10^{-16}\,M_{\rm Pl}^{4}$
(from Eq. (41)) and $\phi(N=0)=1\,M_{\rm Pl}$ (from Eq. (46)). We also note
that our choice of parameters are such that $\tilde{\beta}\ll 1$ and $Q\ll
2/3$, which are the theoretically determined conditions to be met to have a
constant-roll WI in weak dissipative regime with constant $Q$ models to evolve
near thermal equilibrium throughout inflation.

The results of this numerical analysis is depicted in Fig. (5). As in this
model $\ddot{H}$ remains negative throughout, as can be seen from sub-figure
(a), graceful exit is always guaranteed. Sub-figures (b) and (c) show the
evolution of $\epsilon_{1}$ and $\epsilon_{2}$ of this model, and we see that
both $\epsilon_{1}$ and $\epsilon_{2}$ remain less than unity throughout the
evolution. The temperature of the radiation bath doesn’t evolve much during
the inflationary phase as has been shown in sub-figure (d), along with the
thermalization condition $T>H$ maintained throughout, indicating near thermal
evolution of the system.

(a) Graceful exit condition (b) Evolution of $\epsilon_{1}$ (c) Evolution of
$\epsilon_{2}$

(d) Evolution of temperature $T$ (e) Thermalization condition $T>H$

Figure 5: Results of numerical evolution of a constant-roll WI scenario with
negative $\beta$ and constant $Q$ in weak dissipative regime.

####  IV.2.2 Numerical analysis of constant-roll WI models in strong
dissipative regime

We again evolve Eq. (15), Eq. (44)and Eq. (18) fully numerically, now with the
choice of parameter values as $\beta=-10^{-4}$, $Q=2.8\times 10^{2}$,
$g_{*}=106.75$, $\phi_{0}=7.65$ and initial conditions as $T(N=0)=1.56\times
10^{-6}\,M_{\rm Pl}$ and $\dot{\phi}(N=0)\equiv\dot{\phi}_{0}=8.28\times
10^{-12}\,M_{\rm Pl}^{2}$. Once the parameters are fixed we get $V_{0}\approx
10^{-16}\,M_{\rm Pl}^{4}$ (from Eq. (41)) and $\phi(N=0)=-1\,M_{\rm Pl}$ (from
Eq. (46)). We also note that our choice of parameters are such that
$\tilde{\beta}\ll 2/3$ and $Q\tilde{\beta}\ll 1$, which are the theoretically
determined conditions to be met to have a constant-roll WI in strong
dissipative regime with constant $Q$ models to evolve near thermal equilibrium
throughout inflation.

The results of this numerical analysis is depicted in Fig. (6). Like in the
previous case, $\ddot{H}$ remains negative throughout, as can be seen from
sub-figure (a), and thus graceful exit is always guaranteed. Sub-figures (b)
and (c) show the evolution of $\epsilon_{1}$ and $\epsilon_{2}$ of this model,
and we see that both $\epsilon_{1}$ and $\epsilon_{2}$ remain less than unity
throughout the evolution. The temperature of the radiation bath doesn’t evolve
much during the inflationary phase as has been shown in sub-figure (d), along
with the thermalization condition $T>H$ maintained throughout, indicating near
thermal evolution of the system.

(a) Graceful exit condition (b) Evolution of $\epsilon_{1}$ (c) Evolution of
$\epsilon_{2}$

(d) Evolution of temperature $T$ (e) Thermalization condition $T>H$

Figure 6: Results of numerical evolution of a constant-roll WI scenario with
negative $\beta$ and constant $Q$ in weak dissipative regime.

###  IV.3 The case with $\Upsilon$ as a function of $T$ alone

This case resembles the analysis of the constant $Q$ case. Let us first
discuss a subclass of such models with $\beta>0$. We see from Eq. (29) that if
$\Upsilon$ is a function of $T$ alone, then $\Upsilon,_{\phi}=0$. In such a
case the evolution of temperature with $e-$folds can be written as

$\displaystyle\frac{d\ln
T}{dN}=\frac{1}{4-p}\left[\epsilon_{1}-6\beta\left(1+\frac{3}{2}Q\right)\right].$
(49)

As $\epsilon_{1}$ remains much smaller than unity throughout constant-roll WI,
the thermal stabilization condition remains the same as has been pointed out
in the case of constant $Q$ in Subsection IV.1. As we can see that the rest of
the calculations of the constant $Q$ case don’t depend on the form of
$\Upsilon$, the outcome of the case where $\Upsilon$ is a function of $T$
alone is identical to the constant $Q$ case, but now with $p\neq 0$. A similar
argument can also be presented for the subclass of such models with negative
$\beta$.

There are several models of Warm Inflation where the dissipative coefficient
$\Upsilon$ is a function of $T$ alone. For example, the Minimal Warm Inflation
model ($\Upsilon\propto T^{3}$, and thus $p=3$) Berghaus:2019whh  and the
model presented in Bastero-Gil:2019gao  ($\Upsilon\propto T^{-1}$, and thus
$p=-1$) are models of WI where WI is realized in the strong dissipative
regime, whereas the Warm Little inflaton model ($\Upsilon\propto T$, and thus
$p=1$) is an example where WI is realized in the weak dissipative regime. All
these models can be realised with constant-roll dynamics as well in which the
system will evolve in a near thermal equilibrium condition.

###  IV.4 The case with $\Upsilon$ as a function of $\phi$

The situation drastically changes when we deal with WI models where the
dissipative coefficient $\Upsilon$ is a function of $\phi$. A general form of
the dissipative coefficients generally appearing in WI models can be written
as

$\displaystyle\Upsilon(\phi,T)=C_{\Upsilon}T^{p}\phi^{c}M^{1-p-c},$ (50)

where the power of $T$ and $\phi$, i.e. $p$ and $c$ respectively, can take
integer values (positive or negative), $C_{\Upsilon}$ is a dimensionless
quantity which depends on the microscopic details of the WI model under
consideration and $M$ is some appropriate mass scale so that the
dimensionality of $\Upsilon$ is kept at $[\Upsilon]=[{\rm mass}]$.

If we want such models to evolve with constant-roll dynamics in a near thermal
equilibrium condition, we can treat $T$ to be a constant. Also, both Eq. (21)
and Eq. (22) should hold. Equating these two equations with the above form of
$\Upsilon(\phi,T)$ one gets

$\displaystyle
H=\frac{C_{\Upsilon}M^{1-p-c}}{4C_{r}}T^{p-4}\phi^{c}\dot{\phi}^{2},$ (51)

where we have defined $C_{r}=(\pi^{2}/30)g_{*}$. As in the constant-roll
dynamics $\dot{\phi}=\dot{\phi}_{0}e^{-3\beta N}$ (assuming $\beta$ to be
positive for the time being), we can write the above equation as

$\displaystyle
H=\left(\frac{C_{\Upsilon}M^{1-p-c}}{4C_{r}}T^{p-4}\dot{\phi}_{0}^{2}\right)\phi^{c}e^{-6\beta
N}.$ (52)

This situation is quite peculiar because now $H$ has become an explicit
function of $\phi$. In a general situation, either slow-roll or constant-roll
with constant $Q$ and where $\Upsilon$ is a function of $T$ alone, the $\phi$
dependence of $H$ comes from the potential of the inflaton field appearing in
the first Friedmann equation (see Eq. (15)). However, in the present case, the
dependence of $H$ is not due to the inflaton potential but solely due to the
constant-roll condition imposed on the constant radiation bath with nearly
constant temperature. Thus, the evolution of $H$ according to the first
Friedmann equation and according to the above equation will not be the same.
Proceeding with the above form of $H$ and knowing that
$H(d\phi/dN)=\dot{\phi}$, we can replace the form of $H$ from the above
equation to obtain

$\displaystyle\frac{d\phi}{dN}=\left(\frac{4C_{r}T^{4-p}}{C_{\Upsilon}M^{1-p-c}}\right)\frac{1}{\phi^{c}\dot{\phi}_{0}}e^{3\beta
N}.$ (53)

One can integrate this equation assuming $T$ to be a constant to obtain $\phi$
as a function of $N$ as

$\displaystyle\phi^{c+1}=\bar{\phi}^{c+1}+\left(\frac{4(c+1)}{3\beta}\frac{C_{r}T^{4-p}}{C_{\Upsilon}M^{1-p-c}\dot{\phi}_{0}}\right)\left(e^{3\beta
N}-1\right),$

where $\bar{\phi}$ is the value of $\phi$ at $N=0$. Again, as this form of
$\phi$ is not produced from the background Friedmann equations, it will not
satisfy the WI inflaton equation of motion with constant-roll condition
imposed on it (see Eq. (33)). Thus the whole dynamics becomes inconsistent as
soon as we impose the thermal stability condition, through Eq. (21) and Eq.
(22), on such constant-roll WI models where $\Upsilon$ is a function of
$\phi$. This indicates that such constant-roll WI models cannot be treated
near thermal equilibrium condition. The same argument holds for the negative
$\beta$ cases as well.

A quite common dissipative coefficient used in WI models is of the form
$\Upsilon\propto T^{3}/\phi^{2}$ Berera:2008ar ; Bastero-Gil:2010dgy ;
Bastero-Gil:2012akf , where WI takes place in weak dissipative regime.
According to our analysis, such a model cannot be treated with constant-roll
dynamics while maintaining thermal equilibrium of the system.

##  V Discussion and Conclusion

In this paper we derived the conditions to obtain constant-roll WI models
where (a) constant-roll dynamics drives at least 60 $e-$folds of inflation,
(b) gracefully exits a constant-roll phase, and (c) the thermalization
condition ($T>H$) is maintained throughout the constant-roll phase as it
happens during standard slow-roll dynamics of WI. In doing so, we showed that
both positive and negative values of $\beta$ are allowed in constant-roll WI.
However, the thermalization condition forces one to restrict to values of
$|\beta|$ much smaller than unity, which signifies that the dynamics of the
constant-roll phase is not significantly different from a slow-roll evolution.
In constant-roll CI, too, one requires to consider such small values of
$\beta$ so that the predictions of the theory are in tune with the
observations Motohashi:2017aob , and thus the dynamics does not evolve into
non-slow-roll regimes there as well. These conditions on $\beta$ in constant-
roll CI is obtained from perturbation analysis. However, in constant-roll WI,
the background evolution itself restricts the dynamics to evolve near slow-
roll regime. We first consider constant-$Q$ toy models of WI to derive the
dynamics of constant-roll WI. We numerically checked the models for both
positive and negative values of $\beta$ in both weak and strong dissipative
regimes. We then showed that the constant-$Q$ toy model analysis can be
naturally extended to more realistic models of WI where the dissipative
coefficient $\Upsilon$ is a function of $T$ alone. However, the most important
result of our analysis is to show that WI models with dissipative coefficient
$\Upsilon$ as a function of both $T$ and $\phi$ is untenable in thermal
equilibrium once the constant-roll conditions are imposed. Therefore, such WI
models cannot be realized with constant-roll conditions. More importantly,
this analysis clearly indicates that the constant-roll dynamics can in certain
circumstances be significantly different from slow-roll dynamics in WI. We
further want to point out that in our previous analysis exploring a transient
ultra-slow-roll phase in WI Biswas:2023jcd , we noted that models with
$\Upsilon$ as a function of $T$ alone cannot maintain thermal equilibrium even
for a few $e-$foldings, whereas models with $\Upsilon$ as a function of $T$
and $\phi$ both can successfully have a transient ultra-slow-roll phase of a
few $e-$foldings while maintaining near thermal equilibrium condition.
However, what we have noticed in this paper is that, unlike in the ultra-slow-
roll case, models with $\Upsilon(T)$ are consistent with constant-roll whereas
models having $\Upsilon(\phi,T)$ are not suitable for constant-roll WI when
thermal equilibrium is maintained. This also indicates that, unlike CI, the
constant-roll dynamics is not a generalization of the ultra-slow-roll dynamics
in WI.

From the above discussion some key features of constant-roll WI emerges which
we state below:

  1. 1.

We showed that constant-roll WI occurring in thermal equilibrium is similar to
a slow-roll WI phase if we purely want to describe an inflationary phase by
the behaviour of the Hubble slow-roll parameters (both $\epsilon_{1}$ and
$\epsilon_{2}$ being smaller than unity throughout the evolution), but this
similarity is not deep rooted. Thermalization of constant-roll WI yields
restrictions on the system (by constraining the value of $|\beta|$), whereas
in standard slow-roll WI the thermalization of the radiation bath occurs
naturally.

  2. 2.

The constant-roll dynamics also differes from the slow-roll dynamics in the
sense that the potential for the constant-roll is determined by the dynamics,
unlike in the case of slow-roll where one is free to choose among any flat
potentials which yield observarionally viable models. This is true for both CI
and WI constant-roll models.

  3. 3.

Lastly, the fact that constant-roll WI must happen maintaining thermal
equilibrium yields further important restrictions on the theory by
constraining the form of the dissipative coefficient. Thermal equilibrium
rules out cases of constant-roll WI where the dissipation coefficient is a
general function of $T$ and $\phi$.

It would be interesting to check whether such constant-roll WI models are
universal attractors or not. Also, analysis of cosmological perturbations for
such constant-roll WI models is not done in this paper. We defer both these
issues for future projects in the hope that such analysis can help figure out
whether the positive or negative $\beta$ is preferred.

###### Acknowledgements.

The work of S.D. is supported by the Start-up Research Grant (SRG) awarded by
Science and Engineering Research Board (SERB) [Grant No: SRG/2023/000101/PMS].
S.D. is also thankful to Axis Bank and acknowledges the financial support
obtained from them which partially supports this research. S.D. thanks Rudnei
Ramos for many useful discussions on Warm Inflation on various occasions.

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# Origin of the uncurling metric formalism from an Inverse Problems
regularization method

Fred Greensite

###### Abstract.

As we have previously shown, recognition of an algebra’s space of uncurling
metrics leads to a generalization of the usual algebra norm and a novel set of
isomorphism invariants. It is herein detailed how this arose in the context of
a particular Inverse Problems regularization method.

##  1\. Introduction

As introduced in [1], a member of a space of uncurling metrics associated with
unital associative algebra whose vector space of elements is $\mathbb{R}^{n}$
is a self-adjoint linear transformation which, with respect to the standard
basis, corresponds to a real symmetric matrix $L$ satisfying
$d\left((ds)^{\prime}Ls^{-1}\right)=d\left([Ls^{-1}]\cdot ds\right)=0$ in a
neighborhood of the multiplicative identity $\mathbf{1}$ of the algebra (i.e.,
the linear transformation “uncurls” the algebra’s multiplicative inverse
operation). If in addition we also have $s^{\prime}Ls^{-1}=\|\mathbf{1}\|^{2}$
(interpreted to mean that multiplicative inverses $s,s^{-1}$ also behave
inversely with respect to “inner product” $L$), then an (incomplete) unital
norm is given by
$\mathfrak{u}(s)=\text{exp}\left(\frac{1}{\|\mathbf{1}\|^{2}}\int_{\mathbf{1}}^{s}[Lt^{-1}]\cdot
dt\right)$. The expression $\|\mathbf{1}\|^{2}$ is the number of independent
entries on the main diagonal of the matrices comprising the left regular
representation of the algebra. The essence of this program is that, assuming a
norm should be a degree-1 positive homogeneous function on the vector space of
algebra elements, and its selection should be influenced by the existence of
an algebra’s inverse operation and little else, then the above unital norm
choices from a family of uncurling metrics are determined by the property that
$s^{-1}$ and $\frac{\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}$ behave the same
way when $s$ is replaced by $\alpha s$, $\alpha>0$ (this property follows from
the Euler Homogeneous Function Theorem). The role of the latter property in
the context of algebras is strongly analogous to the way norms (and inner
products) with respect to vector spaces can be seen to emerge [2]. The feature
that $L$ uncurls $s^{-1}$ is necessary so as to allow us to equate $Ls^{-1}$
to $\frac{\nabla\mathfrak{u}(s)}{\mathfrak{u}(s)}$ (a gradient) and then
integrate to find $\mathfrak{u}(s)$ subject to the above minimum of conditions
the norm should satisfy. A real finite-dimensional unital associative algebra
is always associated with a non-empty unital norm family, as well as a space
of uncurling metrics that is at least one-dimensional. Also, the unital norm
family typically includes a member essentially equivalent to the algebra’s
“usual norm” (the latter being the determinant of the element’s image under
the left regular representation). This formalism leads to a set of novel
algebra isomorphism invariants along with a useful functor [1].

As will be made explicit by the end of Section 2, the above program had its
origins in the field of Inverse Problems, where a particular regularization
method has an interpretation which forwards the equations in the first two
sentences of this Introduction as they relate to a particular algebra - which
suggested the interest of applying the relevant structures to algebras more
generally. This regularization method and its interpretation in the unital
norm context will be presented here.

##  2\. The finite-dimensional case

Consider the linear problem described by,

(2.1) $y^{\delta}=Fx+\nu,$

where one is given a data vector $y^{\delta}$ that is a version of an unknown
source vector $x$ as transformed by the known linear operator
$F:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$, but where the data vector has
been corrupted by a “noise” vector $\nu$ that is only characterized by

(2.2) $\|y^{\delta}-Fx\|^{2}\leq\delta^{2},$

where $\delta$ is given and $\|\cdot\|$ is the Euclidean norm.

If $F$ is an ill-conditioned matrix (e.g., a discretization of a linear
compact operator), a solution estimate $F^{-1}y^{\delta}$ or
$F^{\dagger}y^{\delta}$ is usually worthless. That is, consider the singular
system for $F$ as $\\{u_{i},\sigma_{i},v_{i}\\}$, where $\\{u_{i}\\}$ is the
set of eigenvectors of $FF^{\prime}$, $\\{v_{i}\\}$ is the set of eigenvectors
of $F^{\prime}F$, and $\\{\sigma_{i}^{2}\\}$ is the set of eigenvalues of
$F^{\prime}F$ (i.e., $\\{\sigma_{i}\\}$ is a set of singular values). The
least squares solution estimate of smallest norm is then,

(2.3) $x_{\rm poor}=F^{\dagger}y^{\delta}=\sum_{\\{i:\sigma_{i}\neq
0\\}}\frac{u_{i}^{\prime}y^{\delta}}{\sigma_{i}}v_{i},$

where an expression like $a^{\prime}b$ indicates the Euclidean inner product
of vectors $a$ and $b$. It is evident that the existence of small singular
values leads to potentially extreme noise amplification, so that the above is
a poor estimate of $x$ as regards the at least possibly approachable
$x^{\dagger}\equiv F^{\dagger}Fx$. Instead, a common treatment (upon which
zero-order Tikhonov regularization with the “Discrepancy Principle” is based
[3, 4]) is to supply the solution estimate of minimum norm that is consistent
with the modification of (2.2) as

(2.4) $\|y^{\delta}-Fx\|^{2}=\delta^{2}.$

That is, the solution estimate is the point of smallest Euclidean norm on the
“discrepancy ellipsoid”,

(2.5)
$\partial\Omega_{\delta}\equiv\\{x\in\mathcal{D}(F):\|y^{\delta}-Fx\|^{2}=\delta^{2}\\},$

where $\mathcal{D}(F)$ is the domain of $F$. We use $\mathcal{D}(F)$ here
instead of $\mathbb{R}^{m}$ because we will want (2.5) to also be relevant
when $F$ is a compact operator (the general Inverse Problems setting [4]). A
Lagrange multiplier argument then gives the solution estimate as,

(2.6) $x^{\delta}_{D}=(F^{\prime}F+\gamma I)^{-1}F^{\prime}y^{\delta},$

where $I$ is the identity operator, and regularization parameter $\gamma$ is
the reciprocal of the Lagrange multiplier as determined by the requirement
that (2.4) be satisfied (the subscript “D” indicates the estimate follows from
a “deterministic” treatment).

Alternatively, one may consider both $x$ and $\nu$ to be zero-mean Gaussian
random vectors (“signal” and “noise”, respectively), with the covariance
matrix of $x$ being $C_{x}$ (assumed for the moment to be nonsingular), and
the covariance matrix of noise vector $\nu$ being $\epsilon^{2}I$
($\delta^{2}$ in (2.4) might now be viewed as an estimate of the sum of the
autocovariances of the components of $\nu$). The maximum a posteriori
(Bayesian) estimate is then,

(2.7)
$x^{\delta}_{S}=(F^{\prime}F+\epsilon^{2}C_{x}^{-1})^{-1}F^{\prime}y^{\delta},$

where the subscript “S” indicates that the estimate follows from a
“statistical” treatment [3].

We can thus associate the deterministic solution estimate (2.6) with a
statistical interpretation (2.7), as resulting from assigned signal and noise
covariance matrices each proportional to the identity matrix, with $\gamma$ as
the inverse of the square of the “signal-to-noise” ratio (this inverse square
being the ratio of the respective noise and signal component variances). Thus,
the diagonal entries of the noise covariance matrix are assumed to be
identical, and these represent the noise variance $\epsilon^{2}$ in each of
the components of the data vector (i.e., the variances of each component of
random vector $\nu$). So the full noise covariance matrix is assumed known a
priori, perhaps based on knowledge of $\delta^{2}$ under the supposition that
$\delta^{2}\approx n\epsilon^{2}$ in (2.4), where $n$ is the dimension of the
range of $F$. The factor multiplying the identity matrix to give $C_{x}$, so
that this factor is the (assumed uniform) variance of each component of the
signal random vector $x$, then follows once $\gamma$ is computed. Hence, from
a statistical standpoint, (2.4), (2.6) can be re-interpreted as implying a
method resulting in simultaneous estimation of $x$ and the presumably uniform
signal component variances $\epsilon^{2}/\gamma$.

But as long as one is simultaneously estimating the solution and a covariance
matrix feature, why stop there? Instead of assuming that the signal covariance
matrix is proportional to the identity and estimating the proportionality
constant simultaneously with the signal, one might only assume that the signal
covariance matrix is diagonal in the basis defined by the eigenvectors of
$F^{\prime}F$, and seek to estimate the diagonal elements simultaneously with
the signal (i.e., one assumes that the matrix whose columns are the right
singular vectors of $F$ diagonalizes the signal covariance matrix). The latter
basis, $\\{v_{i}\\}$, is the natural one to be used in the context of this
type of assumption since, considering the underlying setting where $F$ is a
compact operator, one thing we know is that for the inverse problem to make
sense the signal must be in the domain of $F$ (the problem is defined by $F$
operating on the signal). That is, we already know that the sum of the squares
of the signal components in this basis is finite - and (absent other prior
information) this is the only basis for which we know that to be true. So, the
assumption is simply that the components of random vector $x$ in this basis
are independent, rather than additionally assuming that they all have the same
variance (which would correspond to the assumption that the signal covariance
matrix is proportional to $I$).

Proceeding in this way, one notes that the value of the $i$-th eigenvalue of
$C_{x}$ is then the autocovariance of the $i$-th component of the signal
random vector in the $\\{v_{i}\\}$-basis, i.e., the expectation of the square
of the $i$-th component of random vector $x$ in that basis (which is the
expectation of $(v_{i}^{\prime}x)^{2}$). Thus, an alternative approach to the
statistical treatment of (2.1), (2.4) (which we do not claim to be optimal in
the solitary estimation of $x$) would be to require that the eigenvalues of
$C_{x}$ be the squares of the components of the solution estimate vector -
inspired by the actual meaning of those covariance matrix values as
expectations of the squares of the signal component random variables in the
$\\{v_{i}\\}$-basis. Such a solution estimate would then be a fixed point of
the dynamical system on $\mathbb{R}^{m}$ defined by iteration of the operator,

(2.8) $A_{\epsilon}[w]\equiv\lim_{\alpha\rightarrow
0}\left(F^{\prime}F+\epsilon^{2}\,V\mbox{\rm
diag}\\{(v_{i}^{\prime}w)^{2}+\alpha^{2}\\}^{-1}V^{\prime}\right)^{-1}F^{\prime}y^{\delta},$

where $V$ is the matrix whose $i$-th column is $v_{i}$ for each $i$. The role
of $\alpha$ and $\lim_{\alpha\rightarrow 0}$ is to account for the possibility
that one of the $v_{i}^{\prime}w$ is zero. To get the iterations started, a
solution estimate candidate $w$ (even chosen arbitrarily) is associated with a
signal covariance matrix candidate $V\mbox{\rm
diag}\\{(v_{i}^{\prime}w)^{2}\\}V^{\prime}$. According to (2.7),
$A_{\epsilon}[w]$ is then an updated solution estimate using the latter signal
covariance matrix. For the next iteration, the squares of the components of
this updated solution estimate $A_{\epsilon}[w]$ in the $\\{v_{i}\\}$-basis
are used to define the entries of an updated version of the diagonal matrix on
the right-hand-side of (2.8) (i.e., updating the signal covariance matrix),
which then supplies a successive updated solution estimate as
$A_{\epsilon}[A_{\epsilon}[w]]$, etc. This iteration process can be repeated
indefinitely. A fixed point of the dynamical system evidently supplies a
covariance matrix whose eigenvalues are the squares of the solution estimate
components in the $\\{v_{i}\\}$-basis. The fixed point $x=0$ is associated
with a signal covariance matrix as the matrix of zeros. Thus, one would not
select $w=0$ as an initial guess for the solution estimate to begin the
iteration process (also, $w=0$ is very far from satisfying (2.4)).

But it is helpful that one can easily identify the fixed points of
$A_{\epsilon}$. Inserting the previously noted singular system associated with
$F$ into the right-hand-side of (2.8), we obtain

(2.9)
$A_{\epsilon}[w]=\sum_{i}\frac{\sigma_{i}(u_{i}^{\prime}y^{\delta})}{\sigma_{i}^{2}+\epsilon^{2}/(v_{i}^{\prime}w)^{2}}v_{i}.$

A fixed point $p$ must satisfy

(2.10)
$A_{\epsilon}[p]=\sum_{i}\frac{\sigma_{i}(u_{i}^{\prime}y^{\delta})}{\sigma_{i}^{2}+\epsilon^{2}/(v_{i}^{\prime}p)^{2}}v_{i}=p=\sum_{i}(v_{i}^{\prime}p)v_{i}.$

Since the coefficients of $v_{i}$ in the two sums in (2.10) are necessarily
equal, for $v_{i}^{\prime}p\neq 0$ we obtain

(2.11)
$\sigma_{i}v_{i}^{\prime}p=\frac{u_{i}^{\prime}y^{\delta}\pm\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon^{2}}}{2}.$

In the first place, this means that the series for a fixed point will only
contain terms $i$ such that $(u_{i}^{\prime}y^{\delta})^{2}>4\epsilon^{2}$.
Second, using Taylor series, it is easy to show that if the sign in front of
the radical on the right-hand-side above is not the same as the sign of
$u_{i}^{\prime}y^{\delta}$, then
$|\sigma_{i}v_{i}^{\prime}p|=O(\epsilon^{2}/|u_{i}^{\prime}y^{\delta}|)$.
Indeed, whatever signal was contained in $u_{i}^{\prime}y^{\delta}$ will have
then been mostly removed from the right-hand-side of (2.11) (at least until
$|u_{i}^{\prime}y^{\delta}|$ is on the order of $\epsilon$, at which point in
practical settings the data is mostly noise-dominated). Thus, any fixed point
of interest will be such that the sign in front of the radical is the same as
the sign of $u_{i}^{\prime}y^{\delta}$ . The solution estimate that takes
advantage of the greatest number of data subspaces available to this method is
then,

(2.12)
$p=\frac{1}{2}\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|>2\epsilon\\}}\left(u_{i}^{\prime}y^{\delta}+{\rm
sgn}(u_{i}^{\prime}y^{\delta})\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon^{2}}\right)\frac{v_{i}}{\sigma_{i}}.$

Note that the fixed points of of $A_{\epsilon}$ are truncations - a feature
this solution estimate shares with the (quite different) commonly used
Truncated Singular Value Decomposition (TSVD) solution estimate [3].

The following result answers an obvious question of interest.

######  Lemma 2.1.

Consider the solution estimate (2.12) but with removal of the terms of the sum
on the right-hand-side for which $|u_{i}^{\prime}y^{\delta}|<4\epsilon$. This
is an attracting fixed point of the dynamical system resulting from repeated
iterations of $A_{\epsilon}$.

###### Proof.

For fixed point $p$ to be attracting, it is sufficient that $A_{\epsilon}$ be
continuously differentiable, with $\|\mbox{\bf
D}\big{[}A_{\epsilon}[p]\big{]}\|_{2}<1$, where D is the operator derivative
and $\|\cdot\|_{2}$ denotes the operator norm subject to the Euclidean norm on
the domain of $F$. Using (2.9), we obtain

(2.13) $\mbox{\bf
D}\big{[}A_{\epsilon}[w]\big{]}=\sum_{i}\left(\frac{2\epsilon^{2}\,(\sigma_{i}v_{i}^{\prime}w)(u_{i}^{\prime}y^{\delta})}{(\sigma_{i}^{2}(v_{i}^{\prime}w)^{2}+\epsilon^{2})^{2}}\right)v_{i}\,v_{i}^{\prime},$

where $v_{i}v_{i}^{\prime}$ indicates the outer product of $v_{i}$ with
itself. Inspection of the terms of the finite sum on the right-hand-side of
(2.13) indicates that $A_{\epsilon}$ is continuously differentiable. Thus, for
$p$ to be attracting it is sufficient that for each $i$,

(2.14)
$2\epsilon^{2}\,|(\sigma_{i}v_{i}^{\prime}p)(u_{i}^{\prime}y^{\delta})|<(\sigma_{i}^{2}(v_{i}^{\prime}p)^{2}+\epsilon^{2})^{2}.$

From (2.11),

(2.15) $\frac{1}{2}|u_{i}^{\prime}y^{\delta}|<|\sigma_{i}v_{i}^{\prime}p|,$

based on the requirement that the sign of the radical match the sign of
$u_{i}^{\prime}y^{\delta}$. Thus, it is sufficient that the inequality (2.14)
hold when on the left-hand-side $u_{i}^{\prime}y^{\delta}$ is replaced by
$2\sigma_{i}v_{i}^{\prime}p$, and on the right-hand-side $\epsilon^{2}$ is
replaced by $0$. One then obtains
$4\epsilon^{2}<(\sigma_{i}v_{i}^{\prime}p)^{2}$ as a sufficient condition for
$p$ to be attracting. Applying (2.15) again, one notes that it is sufficient
if $4\epsilon^{2}<(1/4)(u_{i}^{\prime}y^{\delta})^{2}$, i.e.,
$|u_{i}^{\prime}y^{\delta})|>4\epsilon$. ∎

For the purposes of this paper, the point in presenting all this is that the
solution estimate $p$ given by (2.12) is a critical point of the geometric
mean of the solution estimate component absolute values on an implied
ellipsoid derived from the discrepancy ellipsoid on which $p$ lies - in
contrast to the most commonly used solution estimate, which is a critical
point of the Euclidean norm, i.e., a critical point of the quadratic mean, on
the discrepancy ellipsoid on which it lies. That is, the geometric mean has
both a statistical and deterministic interpretation in the context of the
regularization of inverse problems, just as the Euclidean norm does.

######  Theorem 2.2.

Consider

(2.1) $y^{\delta}=Fx+\nu,$

where $y^{\delta}$ is known, $F:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a
given nonsingular linear transformation associated with singular value
decomposition matrix $U\Sigma V^{\prime}$, and $\\{u_{i}\\}$ and $\\{v_{i}\\}$
are the sets of columns of $U$ and $V$, respectively. Further consider the
dynamical system on $\mathbb{R}^{k}$ defined by iteration of the operator,

(2.8) $A_{\epsilon}[w]\equiv\lim_{\alpha\rightarrow
0}\left(F^{\prime}F+\epsilon^{2}\,V\mbox{\rm
diag}\\{(v_{i}^{\prime}w)^{2}+\alpha^{2}\\}^{-1}V^{\prime}\right)^{-1}F^{\prime}y^{\delta}.$

As a solution estimate for $x$, the fixed point $p_{\epsilon}$ of largest
Euclidean norm of this dynamical system has the following interpretations:

  1. (1)

Statistical: If $\epsilon^{2}$ is the variance of zero-mean white Gaussian
noise random vector $\nu$, then $p_{\epsilon}$ is the maximum a posteriori
estimate for zero-mean Gaussian signal random vector $x$ of greatest Euclidean
norm that results from a signal covariance matrix whose eigenvectors are the
members of the basis $\\{v_{i}\\}$ and with the additional feature that its
eigenvalues are the squares of the components of the solution estimate with
respect to those basis vectors.

  2. (2)

Deterministic: Denote the discrepancy of $p_{\epsilon}$ as
$\delta_{\epsilon}\equiv\|y^{\delta}-Fp_{\epsilon}\|$, which may be adjusted
by varying $\epsilon$ so as to allow for the influence of prior knowledge
concerning the magnitude of unknown noise vector $\nu$. Let
$\mathbb{V}_{\epsilon}\subset\mathbb{R}^{k}$ be the space spanned by the
subset of eigenvectors $\\{v_{i}:|u_{i}^{\prime}y^{\delta}|>2\epsilon\\}$.
Consider the ellipsoid

$\partial\Omega\equiv\\{z\in\mathbb{V}_{\epsilon}:\|y^{\delta}-Fz\|^{2}=\delta_{\epsilon}^{2}\\}.$

Then $p_{\epsilon}\in\mathbb{V}_{\epsilon}$ is a critical point of the
geometric mean of the absolute values of the coordinates of the points of
$\partial\Omega$ with respect to the above eigenvector subset basis.

###### Proof.

The statistical interpretation follows from the prior observations related to
the form of the right-hand-side of (2.7) and inspection of the form of (2.8)
as regards a fixed point.

We now consider the deterministic interpretation. To simplify our notation,
without loss we assume an arrangement of the sequence of right singular
vectors of $F$ such that $\\{v_{i}\\}_{i=1}^{n}$ is the set
$\\{v_{i}:|u_{i}^{\prime}y^{\delta}|>2\epsilon\\}$, and we will write
$p_{\epsilon}$ as simply $p$. We observe that $p$ is a member of
$\mathbb{V}_{\epsilon}$, since we have previously shown that $p$ is given by
(2.12). Define $\hat{p}\equiv(v_{1}^{\prime}p,\dots,v_{n}^{\prime}p)$, i.e.,
$\hat{p}$ is the coordinate expression of $p$ as a member of
$\partial\Omega\subset\mathbb{V}_{e}$ (thus, $\hat{p}$ is $p$ with suppression
of the latter’s null-valued coordinates with respect to the full basis of
$\mathbb{R}^{k}$ given by $\\{v_{i}\\}$). Define
$C_{p}\equiv\mbox{diag}\\{(v_{i}^{\prime}p)^{2}\\}$ and
$C_{\hat{p}}\equiv\mbox{diag}\\{(v_{i}^{\prime}p)^{2}\\}_{i=1}^{n}$. Since $p$
is a fixed point of $A_{\epsilon}$, (2.8) implies

(2.16) $p=\lim_{\alpha\rightarrow
0}\left(F^{\prime}F+\epsilon^{2}\,\left[C_{p}+\mbox{diag}\\{\alpha^{2}\\}\right]^{-1}\right)^{-1}F^{\prime}y^{\delta}.$

However, the null coordinates of $p$ correspond to the null diagonal values of
$C_{p}$. There is no intrinsic loss in suppressing these null entries. Hence,
the right-hand-side of (2.16), can be cast as the solution to the variational
problem of finding the point minimizing $z^{\prime}C^{-1}_{\hat{p}}z$ over
$z\in\partial\Omega\subset\mathbb{R}^{n}$, which is $\hat{p}$. Specifically,
(2.16) implies, and is replaced by,

(2.17)
$\hat{p}=\left(\hat{F}^{\prime}\hat{F}+\epsilon^{2}\,C_{\hat{p}}^{-1}\right)^{-1}\hat{F}^{\prime}y^{\delta},$

where $\hat{F}$ is the operator resulting from the restriction of the domain
of $F$ to $\mathbb{V}_{\epsilon}$. According to the principle of Lagrange
multipliers, the point minimizing $z^{\prime}C^{-1}_{\hat{p}}z$ on

$\partial\Omega=\\{z\in\mathbb{V}_{\epsilon}:\|\hat{F}z-y^{\delta}\|^{2}=\delta_{\epsilon}^{2}\\}=\\{z\in\mathbb{V}_{\epsilon}:(z^{\prime}\hat{F}^{\prime}-(y^{\delta})^{\prime})(\hat{F}z-y^{\delta})=\delta_{\epsilon}^{2}\\},$

is the critical point of,

$\mathcal{L}(z,\lambda)\equiv\lambda\left[(z^{\prime}\hat{F}^{\prime}-(y^{\delta})^{\prime})(\hat{F}z-y^{\delta})-\delta_{\epsilon}^{2}\right]+z^{\prime}C^{-1}_{\hat{p}}z.$

Setting to zero all of the partial derivatives of $\mathcal{L}$ with respect
to the components of $z$ leads to the solution being given as the right-hand-
side of (2.17), where the factor $\epsilon^{2}$ is the reciprocal of the
Lagrange multiplier. The latter is determined by the final equation
$\frac{\partial\mathcal{L}}{\partial\lambda}=0$, which places the solution on
$\partial\Omega$. This solution is indeed $\hat{p}$ since we are given (2.17)
(it is implied by (2.16)) and we know that $\hat{p}$ lies on $\partial\Omega$.

Now define the function $h:\mathbb{V}_{\epsilon}\rightarrow\mathbb{R}$ with
$h(z)\equiv z^{\prime}C^{-1}_{\hat{p}}z$. According to the prior paragraph,
$\hat{p}$ minimizes $h(z)$ on $\partial\Omega$. Consider the level set of $h$
given by the ellipsoid,

$\mathcal{H}\equiv\\{z\in\mathbb{V}_{\epsilon}:z^{\prime}C_{\hat{p}}^{-1}z=n\\}=\left\\{z\in\mathbb{V}_{\epsilon}:\sum_{i=1}^{n}\frac{z_{i}^{2}}{p_{i}^{2}}=n\right\\},$

where $z_{j}\equiv v_{j}^{\prime}z$ and $p_{j}\equiv v_{j}^{\prime}p$ for
$v_{j}\in\\{v_{i}\\}_{i=1}^{n}$. The point $\hat{p}$ is evidently a member of
$\mathcal{H}$. The tangent space of ellipsoid $\mathcal{H}$ at $\hat{p}$ has
normal vectors proportional to the gradient of $h$ at $\hat{p}$, and with
respect to the $\\{v_{i}\\}_{i=1}^{n}$-basis,

(2.18) $\nabla
h(z)\Big{|}_{z=\hat{p}}=\,\,2\left(\frac{1}{p_{1}},\dots,\frac{1}{p_{n}}\right).$

But as we have noted above, $\hat{p}$ is the solution estimate minimizing
$z^{\prime}C^{-1}_{\hat{p}}z$ on $\partial\Omega$. Thus, $\mathcal{H}$ is
tangent to $\partial\Omega$ at $\hat{p}$.

We next introduce $g:\mathbb{V}_{\epsilon}\rightarrow\mathbb{R}$ with
$g(z)=\prod_{i=1}^{n}|z_{i}|$ (the $n$-th power of the geometric mean of the
absolute values of a point’s components with respect to the
$\\{v_{i}\\}_{i=1}^{n}$-basis). The level set of $g$ that contains $\hat{p}$
is,

$\mathcal{G}\equiv\left\\{z\in\mathbb{V}_{\epsilon}:\prod_{i=1}^{n}|z_{i}|=g(\hat{p})\right\\}.$

We observe that at any point of this level set,

(2.19) $\frac{\partial g}{\partial z_{j}}={\rm sgn}(z_{j})\prod_{i\neq
j}|z_{i}|=\frac{g(\hat{p})}{z_{j}}.$

The tangent space of $\mathcal{G}$ at $\hat{p}$ has normal vectors
proportional to the gradient of $g(z)$ at $\hat{p}$ and, with respect to the
$\\{v_{i}\\}_{i=1}^{n}$-basis, (2.19) implies

(2.20) $\nabla
g(z)\Big{|}_{z=\hat{p}}=\,\,g(\hat{p})\left(\frac{1}{p_{1}},\dots,\frac{1}{p_{n}}\right),$

which according to (2.18) is proportional to $\nabla h(x)\big{|}_{z=\hat{p}}$.
Hence, $\mathcal{H}$ and $\mathcal{G}$ both contain $\hat{p}$, and share the
same tangent space at $\hat{p}$, i.e., $\mathcal{G}$ and $\mathcal{H}$ are
tangent at $\hat{p}$. But we have seen that $\mathcal{H}$ is tangent to
$\partial\Omega$ at $\hat{p}$. Hence $\mathcal{G}$ is tangent to
$\partial\Omega$ at $\hat{p}$, and $\hat{p}$ is thereby a critical point of
$g$ on $\partial\Omega$. The $n$-th root of $g$, which is the geometric mean,
then also has a critical point on $\partial\Omega$ at $\hat{p}$, which is
equivalent to the final statement in the theorem. ∎

Thus, the squared Euclidean norm/quadratic mean and the geometric mean have
kindred statistical interpretations (as regards associated implied covariance
matrices) in the application to Inverse Problems.

With a little imagination, we can write the key feature exploited in proof of
the “Deterministic” portion of Theorem 2.2 as $s^{-1}=\frac{\nabla
g(s)}{g(s)}$, i.e., (2.19), (2.20), where “$s^{-1}$” is the $n$-tuple of
inverses of the components of $s$. The fact that we have introduced the
inverse of a point $s=(s_{1},\dots,s_{n})$ as
$s^{-1}\equiv\left(\frac{1}{s_{1}},\dots,\frac{1}{s_{n}}\right)$ suggests that
the points $s$ are members of the algebra defined as a direct sum of algebras
$\bigoplus_{i=1}^{n}\mathbb{R}$. The multiplicative identity on this algebra
is $\mathbf{1}=(1,1,\dots,1)$. Recalling the Introduction (or the equations of
Definition 2.1 in [1]), it is a simple matter to verify that the geometric
mean $\bar{g}(s)\equiv(g(s))^{\frac{1}{n}}$ is then an incomplete unital norm
resulting from the uncurling metric $L$ as the identity. The notion that the
geometric mean is a type of “norm” comes from Theorem 2.2, in that it (or
equivalently, its $n$-th power, $g(s)$) is being used to regularize an inverse
problem (see the prior paragraph) - something typically accomplished with
(semi)norms. So this Inverse Problems application, concerning a “norm” on
algebra $\bigoplus_{i=1}^{n}\mathbb{R}$, was the origin of uncurling metrics
and unital norms as ultimately applied to real finite-dimensional unital
associative algebras in general [1].

##  3\. Extension to compact operators

The following theorem establishes that the above “geometric mean methodology”
is indeed a regularization method, i.e., convergent for suitable selection of
a regularization parameter based on knowledge of the noise magnitude. But we
hasten to add that although the geometric mean application is associated with
a regularization method that works well in practice, the latter is not
asserted to have efficacy comparable to that of e.g., zero-order Tikhonov
regularization. In fact, derivation of convergence rates for typical subsets
of the domain of the relevant compact operator is problematic.

######  Theorem 3.1.

Let $F$ be a given linear compact operator with singular value expansion
$F=\sum_{i}\sigma_{i}u_{i}v_{i}^{\prime}$, and consider,

(2.1) $y^{\delta}=Fx+\nu,$

where $y^{\delta}$ in the range of $F$ is known, but $x$ in the domain of $F$
and $\nu$ in the range of $F$ are unknown. Suppose also that,

(2.2) $\|Fx-y^{\delta}\|^{2}\leq\delta^{2},$

where $\delta$ is known. On the domain of $F$, the greatest-magnitude fixed
point of the operator,

(3.1) $A_{\epsilon}[w]\equiv\lim_{\alpha\rightarrow
0}\left(F^{\prime}F+\epsilon^{2}\,\sum_{i}\frac{v_{i}v_{i}^{\prime}}{(v_{i}^{\prime}w)^{2}+\alpha^{2}}\right)^{-1}F^{\prime}y^{\delta},$

is given by

(3.2)
$x^{\delta}=\frac{1}{2}\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|>2\epsilon\\}}\left(u_{i}^{\prime}y^{\delta}+{\rm
sgn}(u_{i}^{\prime}y^{\delta})\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon^{2}}\right)\frac{v_{i}}{\sigma_{i}}.$

If we specify $\epsilon=\epsilon(\delta)=\delta$, then for $x^{\dagger}\equiv
F^{\dagger}Fx$ we have $\lim_{\epsilon(\delta)\rightarrow
0}\|x^{\dagger}-x^{\delta}\|=0$.

###### Proof.

In the following argument we will assume that there are no vanishing singular
values of $F$, so that $x^{\dagger}=x$. If there are vanishing singular
values, the argument is easily modified by appropriately exchanging $x$ for
$x^{\dagger}$ as necesary.

The prior derivation of (2.12) holds regardless of whether $F$ is a finite-
dimensional matrix or a linear compact operator, so (3.2) follows with $p$
replaced by $x^{\delta}$. The error of the solution estimate is thus,

(3.3) $\displaystyle x-x^{\delta}$ $\displaystyle=$
$\displaystyle\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|\leq
2\epsilon(\delta)\\}}(v_{i}^{\prime}x)v_{i}\,\,+\sum_{\\{i:|u_{i}^{\prime}y^{\delta})|>2\epsilon(\delta)\\}}\frac{(\sigma_{i}v_{i}^{\prime}x-u_{i}^{\prime}y^{\delta})}{\sigma_{i}}v_{i}$
$\displaystyle+\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|>2\epsilon(\delta)\\}}\frac{1}{2}\bigg{(}u_{i}^{\prime}y^{\delta}-{\rm
sgn}(u_{i}^{\prime}y^{\delta})\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon(\delta)^{2}}\bigg{)}\frac{1}{\sigma_{i}}v_{i},$

where the first sum above is the “truncation error” and the last two sums are
the “stability error”. It is now simply a matter of evaluating the norms of
the three sums on the right-hand-side above, as the norm of the solution
estimate error will be less than the sum of those norms.

The square of the norm of the first sum on the right-hand-side of (3.3) is,

$\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|\leq
2\epsilon(\delta)\\}}(v_{i}^{\prime}x)^{2}\,\,=\,\,\sum_{i=1}^{\infty}(v_{i}^{\prime}x)^{2}H\big{(}2\epsilon(\delta)-|u_{i}^{\prime}y^{\delta}|\big{)}\,\,\leq\,\,\|x\|^{2},$

where $H(\cdot)$ is the Heaviside function. The Dominated Convergence Theorem
can then be invoked so that application of $\lim_{\epsilon(\delta)\rightarrow
0}$ to the sum on the left-hand-side above is seen to be equal to the sum over
the individual limits of the terms of the sum on the right-hand-side of the
first equality. Since the limit of each of those terms is zero, the norm of
the first sum on the right-hand-side of (3.3) vanishes in the limit as
$\epsilon(\delta)=\delta\rightarrow 0$.

Now, both the second and third sums on the right-hand-side of (3.3) are over
terms with $i$ such that $|u_{i}^{\prime}y^{\delta})|>2\epsilon(\delta)$. But
since $\epsilon(\delta)=\delta$, we have,

$2\delta=2\epsilon(\delta)<|u_{i}^{\prime}y^{\delta}|=|\sigma_{i}v_{i}^{\prime}x+u_{i}^{\prime}\nu|\leq|\sigma_{i}v_{i}^{\prime}x|+\delta,$

with the final inequality above following from (2.2). Consequently, for both
the second and third sums on the right-hand-side of (3.3),

(3.4) $\frac{\delta}{\sigma_{i}}<|v_{i}^{\prime}x|.$

So we can apply this to the second sum on the right-hand-side of (3.3) to
obtain,

(3.5)
$\sum_{\\{i:\,|u_{i}^{\prime}y^{\delta}|>2\epsilon(\delta)\\}}\frac{(\sigma_{i}v_{i}^{\prime}x-u_{i}^{\prime}y^{\delta})^{2}}{\sigma_{i}^{2}}<\frac{\delta^{2}}{\sigma_{m-1}^{2}}<(v_{m-1}^{\prime}x)^{2},$

where $\sigma_{m-1}$ is the smallest singular value appearing in the sum. The
first inequality follows from (2.2) (i.e., (2.2) implies that the above sum
over just the numerators of the included terms is $\leq\delta^{2}$). As
$\epsilon(\delta)=\delta\rightarrow 0$, it follows that
$\sigma_{m-1}\rightarrow 0$ (since $F$ is compact), i.e.,
$m\rightarrow\infty$. But
$\lim_{m\rightarrow\infty}(v_{m-1}^{\prime}x)^{2}=0$. Equation (3.5) then
implies that the norm of the second sum on the right-hand-side of (3.3) tends
to zero as $\epsilon(\delta)=\delta\rightarrow 0$.

So now we consider the third sum on the right-hand-side of (3.3). Note that
$0<1-\sqrt{1-z^{2}}<z^{2}$ for $z\in(-1,1)$. Since each term of the sum is
such that $|u_{i}^{\prime}y^{\delta})|>2\epsilon(\delta)$, it follows that,

(3.6) $\left|u_{i}^{\prime}y^{\delta}-{\rm
sgn}(u_{i}^{\prime}y^{\delta})\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon(\delta)^{2}}\right|<\frac{[2\epsilon(\delta)]^{2}}{|u_{i}^{\prime}y^{\delta}|}<2\epsilon(\delta).$

Applying this to the square of the norm of the third sum on the right-hand-
side of (3.3), we obtain that squared norm as

$\displaystyle\sum_{i=1}^{\infty}\frac{1}{4\sigma_{i}^{2}}\left(u_{i}^{\prime}y^{\delta}-{\rm
sgn}(u_{i}^{\prime}y^{\delta})\sqrt{(u_{i}^{\prime}y^{\delta})^{2}-4\epsilon(\delta)^{2}}\right)^{2}\Big{(}1-H\big{(}2\epsilon(\delta)-|u_{i}^{\prime}y^{\delta}|\big{)}\Big{)}$
(3.7)
$\displaystyle<\,\,\,\,\sum_{i=1}^{\infty}\left(\frac{\epsilon(\delta)}{\sigma_{i}}\right)^{2}\Big{(}1-H\big{(}2\epsilon(\delta)-|u_{i}^{\prime}y^{\delta}|\big{)}\Big{)}\,\,\,<\,\,\,\,\sum_{i=1}^{\infty}(v_{i}^{\prime}x)^{2}=\|x\|^{2}.$

The second inequality in (3.7) follows from (3.4), since
$\epsilon(\delta)=\delta$. The Dominated Convergence Theorem then implies that
$\lim_{\epsilon(\delta)\rightarrow 0}$ can be taken inside the sum on the
left-hand-side of (3.7) in evaluation of the limit of that sum. The limit of
each of the terms of the sum is zero, indicating that the limit of the norm of
the third sum on the right-hand-side of (3.3) is zero as
$\epsilon(\delta)=\delta\rightarrow 0$.

Since the limit of the norm of each of the sums in (3.3) is zero as
$\epsilon(\delta)=\delta\rightarrow 0$, it follows that
$\lim_{\epsilon(\delta)\rightarrow 0}\|x-x^{\delta}\|=0$. ∎

##  4\. An example

The following figure shows the results of a numerical simulation, computed in
MatLab.

The source “$x$” is a discetized time series of spikes in 100 data points,
shown at the (1,1) position in the $(3\times 3)$-matix of tracings in the
figure. The source is smoothed by convolution with a Gaussian density and then
some simulated white noise is added, giving the measurements
$``y+\text{noise}"$ time series at the (2,1) position of the figure. The zero-
order Tikhonov regularized signal estimate corresponding to the L-curve corner
regularization parameter selection method [5] is shown at the figure’s (1,2)
position. The alternative method presented in this paper, also using the
L-curve corner regularization parameter selection method (for selection of
$\epsilon$ in Theorem 2.2), is at the (1,3) position [in the caption, “orthog”
relates to the method’s use of an orthogonal operator rather than a seminorm
as in Tikhonov regularization; “zero-order” refers to the use of the
eigenvector matrix of $F^{\prime}F$ for this purpose (where $F$ is the
transfer matrix as in Theorem 2.2), rather than another choice based on some
degree of prior knowledge of the source features]. The L-curve for the zero-
order Tikhonov method and the method of Theorem 2.2 are as indicated in the
(2,2) and (2,3) positions, respectively. The L-curves are generated over 30
orders of magnitude of regularization parameter value, with the solution
estimate relative errors for each value of the regularization parameter shown
at the respective (3,2) and (3,3) positions. The latter depict the relative
error of the methods for each value of regularization parameter, where
relative error is defined by $\frac{\|\text{source - source
estimate}\|}{\|\text{source}\|}$, and $\|\cdot\|$ is the Euclidean norm on
$\mathbb{R}^{n}$ where $n$ is the number of data points comprising $x$ (i.e.,
$n=100$ here). The plot at the (3,1) position is a scatter plot of (abscissa,
ordinate) of the L-curve corner, for the L-curve generated at each iteration
step of the new method (so, the plot at the (2,3) position is the full L-curve
for the final iteration step, and its corner is one of the points of the
scatter plot at the (3,1) position). The points associated with successive
iterations are connected by a dotted line.

The quality of the new method’s solution estimate is not quite as good as the
estimate provided by zero-order Tihonov regularization, as is to be expected
since zero-order Tikhonov regularization is essentially optimal under
conditions of this simulation. Nevertheless, the new method’s solution
estimate is respectable and, as indicated by Theorem 3.1, it represents a
convergent method.

## References

  * [1] Greensite F (2023) Novel geometrical features of real finite-dimensional unital algebras, arXiv:2207.11358
  * [2] Greensite, FS (2022), A new proof of the Pythagorean Theorem, generalization of the usual algebra norm, and novel algebra isomorphism invariants, arXiv:2209.14119
  * [3] Vogel C (2002) Computational Methods for Inverse Problems. Philadelphia: SIAM.
  * [4] Engl HW, Hanke M, Neubauer A (2000) Regularization of Inverse Problems, Klewer, Boston.
  * [5] Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34:563-580.