to encompass the description of living systems from an elementary and
fundamental viewpoint coming from first principles, us theoretical physicists
need to extend our toolkit to include phenomena arising through combinatorial
innovation. One of these cases is the theory of the adjacent possible, which
we described extensively.

We must highlight the difference between the ergodic and non-ergodic regimes
as crucial and key to the derivation of these conclusions. While we are in the
ergodic regime, objects can be repeatedly made simply because the conditions
to make them occur repeatedly. For instance, in the early stages of molecular
chemistry the same small molecules are independently created many times
through collisions of their constituents — the Universe has no need to learn
and remember how to make those molecules. The present non-ergodic Universe is
radically different, because to create biologically (or technologically)
complex items repeatedly requires the prescription for their creation to be
learned, stored, and then executed. It is this latter regime which lies
outside the realm of the conventional physics description, and for which TAP
provides a phenomenological model.

###  11.2 The conclusions

Some of the important conclusions drawn from this merger of cosmology and
biology can be summarised as follows:

  * •

In a theory that is well described by an underlying fundamental model, like
the Standard Model of particle physics, the global space of states, the
Hilbert space, does not expand. The states that the Universe goes through are
vector states which were present in the model from the beginning, and do not
appear unpredictably as the Universe undergoes different phase transitions.
Whether or not those states are available to the Universe at a given time
depends on the temperature regime the Universe is in. All states in the
evolution of the Universe along with their effective field theory can be
derived from the Standard Model (or its appropriate extension).

  * •

In a system that has no standard model, such as biology, the configuration
space genuinely and unpredictably expands in real time. As the system evolves,
new microstates are (combinatorially) found and tested by the system’s basic
constituents. Those states are genuinely novel, in that they could not have
been derived a priori by any underlying theory. Therein lies the crucial
distinction between physics and biology.

The central question for cosmology that we are addressing here is: Does the
Universe create large amounts of information as it evolves from early times to
late times? We can compare the probable answers to this question as given by
two paradigms for cosmological theory. These are:

  1. 1.

The Newtonian paradigm (defined above), which asserts that there is a fixed
space of states on which fixed laws operate.

  2. 2.

An evolving-laws paradigm, which denies that there are either a fixed set of
states or fixed laws.

Different authors, including the present ones, formulate evolving-law
paradigms differently, using various frameworks. But we all agree on the
denial of the Newtonian paradigm, which is sufficient to reach a firm
conclusion on the forementioned question. That is, our differences matter much
less than what we agree on.

We can all agree on the following. Both paradigms have well-defined quantities
called $\\{\rho\\}_{{\rm initial\;lawful}}$, the set of ‘lawful’ possible
initial states of the Universe of which one, $\rho_{\rm initial}$, was the
actual initial state of the Universe, and $\\{\rho\\}_{\rm late\;lawful}$, a
set of ‘lawful’ final or late states of the Universe of which one, $\rho_{\rm
final}$, will be the actual final or late state of the Universe. Different
theories may possess either a final state or a late state, the latter being
one after everything interesting has happened.

There are various options for speaking of measures on sets of states. Some,
like the Boltzmann or von Neumann entropies, depend on the theory and only
make sense in the first of the two paradigms we are comparing. But Bateson
information, defined informally as ‘the difference that makes a difference’
[82], is universal enough to let us compare the two paradigms.

Let $I_{\rm initial}^{\rm NP}$ be the amount of Bateson information needed to
select the actual initial state of the universe from the possible lawful
initial states in the Newtonian paradigm. Let $Q^{\rm NP}(\rho_{\rm
initial}^{\rm NP}\rightarrow\rho_{\rm final}^{\rm NP})$ be the information
needed to predict the exact final state given the exact initial state in the
Newtonian paradigm. The same quantities labeled with ‘EL’ arise in the
evolving-laws paradigm.

Our key question is, does the Universe learn during its evolution? In other
words is $Q$ order unity or huge? Most cosmologists focus on the $I$, which is
called the entropy of the initial state, or just the entropy of the Universe.
We instead are interested in the $Q$, which tell us how much information the
Universe learns during its evolution. This is information that is not present
in the initial conditions and also is not a consequence of the actual laws.

If you believe in the Newtonian paradigm, you must believe $Q^{\rm NP}=1$, as
all the information needed to precisely determine the final state is present
in the initial state. But an evolving-laws framework makes possible a creative
universe in which $Q^{\rm EL}\gg 1$. In other words, all evolving-law
cosmologists agree that the Universe creates information. We call $Q$ the
creative potential of the Universe. A biological universe can create more
information than was present at the Big Bang by conventional accounts.

###  11.3 The outcome

To summarise, we have taken the first steps in bringing the biosphere into
cosmology by proposing a tool for estimating the number of classical
microscopic states contained in the biosphere configuration space. Within the
context of an emergent and non-reductionist view of complex systems, we argued
that it is not self-evident that the contribution of the biosphere is sub-
dominant to the enormous numbers coming from gravitational physics. Emergent
complexity may dramatically expand the available volume of configuration
space. As a demonstration, we have adapted equations emerging from the Theory
of the Adjacent Possible [9], which show an explosive growth that can be
super-combinatorial and hence in principle able to overcome the traditional
permutational accounting of particle configuration space. Living biospheres
may be the dominant source of information in the Universe! How and whether the
potential associated divergence of complexity can be tamed (either in
equations or in reality itself) remains to be seen. An exciting speculation is
that cosmic acceleration may have a role to play in this.

Has our new result worsened the problem of cosmological initial conditions?
Surprisingly, probably not. The vast numbers generated via the biospheres are
due to the growth in state space enabled by the emergent complexity of life.
Those states simply did not exist until life came along to make them. The
newborn Universe had no opportunity to occupy them, nor pays any cost in
failing to do so.

To end we note that one should not regard TAP as describing actual physical
collisions of two or more objects to make a single new one; it operates at a
much more abstract and elevated level. It quantifies the remembered ability of
a system to make and remake an object. The word remembered is crucial; it is
not enough to make something once, we must be able to do so again and again.
Its existence must be encoded in some way; embedded in RNA perhaps, or passed
on through folklore, or written down in a stored patent application. This
requirement can become incredibly complex. To make one new leopard we need
both the actual parent leopards and the information encoded in their genomes
(which comes conveniently packaged within the leopards themselves). But to
keep making new leopards requires the entire sustainable ecosystem in which
they are embedded. Consider John Muir’s famous quote [83]: “When we try to
pick out anything by itself we find that it is bound fast by a thousand
invisible cords that cannot be broken, to everything in the universe.”

## Acknowledgments

We would first and foremost like to acknowledge Barbara Drossel, without whom
several of the connections in this work would never have been made. We thank
Niayesh Afshordi, Priyal Bordia, Latham Boyle, Hal Haggard, Wim Hordijk, Jaron
Lanier, Roberto Mangabeira Unger, Pierre Martin-Dussaud, Mark Neyrinck, Roger
Penrose, James Rosindell, and Carlo Rovelli for discussions.

This research was supported in part by Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported by the Government of
Canada through Industry Canada and by the Province of Ontario through the
Ministry of Research and Innovation. This research was also partly supported
by grants from NSERC and FQXi. This work was supported by the Fundação para a
Ciência e a Tecnologia (FCT) through the research grants UIDB/04434/2020 and
UIDP/04434/2020. M.C. acknowledges support from the FCT through grant
SFRH/BPD/111010/2015 and the Investigador FCT Contract No. CEECIND/02581/2018
and POPH/FSE (EC). A.R.L. acknowledges support from the FCT through the
Investigador FCT Contract No. CEECIND/02854/2017 and POPH/FSE (EC). M.C. and
A.R.L. are supported by the FCT through the research project EXPL/FIS-
AST/1418/2021. We are especially thankful to the John Templeton Foundation for
their generous support of this project.

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# Gravitational particle production of superheavy massive particles in
Quintessential Inflation II: $\alpha$-attractors

Llibert Aresté Saló<EMAIL_ADDRESS>School of Mathematical Sciences,
Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom
Jaume de Haro<EMAIL_ADDRESS>Departament de Matemàtiques, Universitat
Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

###### Abstract

We compute the gravitational production of conformally coupled superheavy
particles during the phase transition from the end of inflation to the
beginning of kination for $\alpha$-attractors potentials in the context of
Quintessential Inflation ($\alpha$-QI), showing that the maximum value of the
reheating temperature, independently of the value of the parameter $\alpha$,
is near $10^{9}$ GeV. This result, which contradicts the usual belief that the
reheating via the production of superheavy massive particles leads to an
inefficient reheating temperature, is due to the fact that in our numerical
calculations we take into account the contribution of the large wavelength
modes to the reheating temperature, which never happens in analytical
calculations where only ultraviolet modes are considered.

Gravitational Particle production; Quintessential Inflation;
$\alpha$-attractors; Reheating Temperature.

###### pacs:

04.20.-q, 98.80.Jk, 98.80.Bp

##  I Introduction

The so-called gravitational particle production Parker ; gmm ; ford ;
Zeldovich  of superheavy particles conformally coupled with gravity, which was
applied to standard inflation (potentials with a deep well) in kolb ; kolb1 ;
Birrell1 ; ema , is one of the mechanisms used to reheat the universe in
scenarios containing a period of inflation. This also happens in
Quintessential Inflation (QI) to match the inflationary period with the usual
hot Big Bang universe guth . However, the gravitational reheating in QI is
normally applied to very light fields Spokoiny ; pv ; A  and only in few
papers, which deal with toy non-smooth models as the Peebles-Vilenkin one, it
is applied to massive particles H ; ha ; hap18 ; J ; hashiba ; Hashiba . On
the other hand, regarding smooth QI potentials, particle creation has to be
analytically studied using the complex WKB approximation hashiba1 , whose
effective application is limited to the creation of particles by parabolic
potentials kofman .

In the present work we continue with our study of particle production by
smooth potentials started in ah , now focusing on a smooth exponential
potential coming from $\alpha$-attractors in Quintessential Inflation
($\alpha$-QI) vardayan ; K ; benisty3 . Since the $\alpha$-attractors come
from supergravity theories containing particles with only gravitational
interactions, the late-time decay of these relics may jeopardize the success
of the standard BBN lindley . To solve this problem one has to consider
sufficiently low reheating temperature (of the order of $10^{9}$ GeV or less)
eln . On the contrary, a lower bound for the reheating temperature comes from
the fact that the radiation-dominated era occurs before the Big Bang
Nucleosynthesis (BBN) epoch, which takes place in the $1$ MeV regime gkr .

Coming back to the gravitational production of superheavy particles, we will
use the well-known Hamiltonian diagonalization method (see gmmbook  for a
review), based on the computation of the time dependent $\beta$-Bogoliubov
coefficient which encodes the polarization effects and also the real particles
created during the phase transition. Fortunately, these polarization effects
disappear when the universe evolves adiabatically, which happens soon after
the beginning of kination, allowing its numerical calculation. Thus, in order
to calculate the energy density of the produced particles, one can safely use
the square modulus of the $\beta$-Bogoliubov coefficient after the beginning
of kination, whose numerical value is, for the relevant modes that contribute
to the reheating, of the order of $10^{-9}-10^{-10}$ depending on the
superheavy masses which in our simulations are of the order of
$10^{15}-10^{17}$ GeV.

Finally, once one has the value of the $\beta$-Bogoliubov coefficients, one
can calculate the value of the energy density of the superheavy particles,
which must decay into lighter ones before or after the end of the kination
phase to form a relativistic plasma. In the former case the reheating
temperature is greater than $10^{6}$ GeV and in the second one its maximum
value is around $10^{9}$ GeV, which shows that the gravitational production of
superheavy particles is a very efficient mechanism to reheat our universe.

Throughout the manuscript we use natural units, i.e., $\hbar=c=k_{B}=1$ and
the reduced Planck’s mass is denoted by $M_{pl}\equiv\frac{1}{\sqrt{8\pi
G}}\cong 2.44\times 10^{18}$ GeV.

##  II Particle creation of superheavy particles conformally coupled to
gravity

We consider a superheavy quantum field $\chi$ conformally coupled with
gravity. In order that the polarization effects due to this quantum field do
not affect the dynamics of the scalar field responsible for Quintessential
Inflation, the mass of the quantum field, namely $m_{\chi}$, must be greater
than the Hubble rate, in fact the condition $H\ll m_{\chi}$ has to be
satisfied (see Felder  to take a look at the problems associated to quantum
light fields). Therefore, since the most accepted idea is that inflation
starts at GUT scales, where the temperature is around $10^{16}$ GeV, then
using the Stefan-Boltzmann law $\rho=\frac{\pi^{2}}{30}g_{*}T^{4}$, where the
effective number of degrees of freedom for the Standard model is $106.75$, for
a flat FLRW spacetime at the beginning of inflation one has $H\cong 2\times
10^{13}$ GeV. This is the reason why we will chose $m_{\chi}\sim 10^{15}$ GeV
or greater.

To calculate the energy density of the produced particles, we will use the
well-known diagonalization method, based on the Bogoliubov coefficients, which
in the conformally coupled case must satisfy the first order system of
differential equations (see gmmbook  for a detailed discussion)

$\displaystyle\left\\{\begin{array}[]{ccc}\alpha_{k}^{\prime}(\tau)&=&\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}e^{2i\int^{\tau}\omega_{k}(\bar{\tau})d\bar{\tau}}\beta_{k}(\tau)\\\
\beta_{k}^{\prime}(\tau)&=&\frac{\omega_{k}^{\prime}(\tau)}{2\omega_{k}(\tau)}e^{-2i\int^{\tau}\omega_{k}(\bar{\tau})d\bar{\tau}}\alpha_{k}(\tau),\end{array}\right.$
(3)

where the time dependent frequency is denoted by
$\omega_{k}(\tau)=\sqrt{k^{2}+m_{\chi}^{2}a^{2}(\tau)}$ and $\tau$ is the
conformal time.

Finally, in terms of the $\beta$-Bogoliubov coefficient, the vacuum energy
density of the $\chi$-field is given by ah

$\displaystyle\rho_{\chi}(\tau)=\frac{1}{2\pi^{2}a^{4}(\tau)}\int_{0}^{\infty}k^{2}\omega_{k}(\tau)|\beta_{k}(\tau)|^{2}dk.$
(4)

###  II.1  Gravitational particle creation by $\alpha$-attractors in the
context of Quintessential Inflation

In the present work, the potential that we will consider is an exponential
$\alpha$-attractor in Quintessential Inflation, plotted in Figure 1 and given
by

$\displaystyle V(\varphi)=\lambda
M_{pl}^{4}e^{-n\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)},$ (5)

where $\lambda$, $\alpha$ and $n$ are dimensionless parameters, whose relation
is in order to match with the current observation data the following one (see
benisty3  for details):

$\displaystyle\frac{\lambda}{\alpha}e^{n}\sim
10^{-10}\qquad\mbox{and}\qquad\lambda e^{-n}\sim 10^{-120}.$ (6)

Figure 1: Plot of the Exponential $\alpha$-attractor potential, for
$\alpha\sim 10^{-2}$, $n\sim 10^{2}$ and $\lambda\sim 10^{-66}$.

First of all, in order to calculate the energy density of the produced
particles one has to integrate numerically the conservation equation for the
inflaton field, namely

$\displaystyle\ddot{\varphi}+3H\dot{\varphi}+V_{\varphi}=0,$ (7)

where
$H=\frac{1}{\sqrt{3}M_{pl}}\sqrt{\frac{\dot{\varphi}^{2}}{2}+V(\varphi)}$,
with initial conditions (the value of the scalar field and its first
derivative) during inflation. Since the slow-roll regime is an attractor, one
only has to take initial conditions in the basin of attraction of the slow-
roll solution, for example, $\varphi=\varphi_{*}$ and
$\dot{\varphi}=-\frac{V_{\varphi}(\varphi_{*})}{3H_{*}}$, where the “star”
denotes that the quantities are evaluated at the horizon crossing.

Once one has obtained the evolution of the scalar field and in particular the
evolution of the Hubble rate, one can compute the evolution of the scalar
factor, whose value at the horizon crossing we have chosen to be equal to $1$.

From the evolution of the scale factor, we can see on the left-hand side of
Figure 2 that a spike appears in the plot of the quantity
$\omega_{k}^{\prime}/\omega_{k}^{2}$ during the phase transition from the end
of inflation to the beginning of kination, that is, at that moment when the
adiabatic evolution is broken and particles are gravitationally produced (see
the right-hand side of Figure 2).

Figure 2: Adiabatic evolution (left) and evolution of the $\beta$-Bogoliubov
coefficient (right) for a heavy field with mass $m_{\chi}\sim 10^{16}$ GeV,
when $\alpha\sim 10^{-1}$. Here we have used the value $k=a_{kin}H_{kin}$,
which is in the range ${k}\lesssim a_{kin}m_{\chi}$, where we have observed
that particles are produced.

Then, we have numerically solved equation (3), with initial conditions
$\alpha_{k}(\tau_{*})=1$ and $\beta_{k}(\tau_{*})=0$ at the horizon crossing
(there were neither particles nor polarization effects at that moment because
during the slow-roll regime the derivatives of the Hubble rate are negligible
compared with the powers of $H$, i.e., the system is in the adiabatic regime).

For the value $k=a_{kin}H_{kin}$, we obtain in Figure 2 that
$|\beta_{k}(\tau)|^{2}$ stabilizes soon to a non-zero value after the
beginning of kination, containing only particle production effects. We have
numerically done the calculations for masses $m_{\chi}\cong 10^{15}-10^{17}$
GeV and for a large range of values of $\alpha$. We have obtained that the
relevant modes that contribute significantly to the particle production are in
the range $0\lesssim k\lesssim a_{kin}m_{\chi}$ (see Figure 3), leading to
values of $|\beta_{k}|^{2}$ of order $10^{-9}$ for $m_{\chi}\sim 10^{15}$ GeV
and values of $|\beta_{k}|^{2}$ of order $10^{-10}$ for $m_{\chi}\sim
10^{16}-10^{17}$ GeV, as one can see in Figure 4.

Figure 3: Plot of the logarithm of $|\beta_{k}|^{2}$, as a function of $k$,
for a heavy field with mass $m_{\chi}\sim 10^{15}$ GeV and $\alpha=0.1$.

Figure 4: Plot of the square modulus of the $\beta$-Bogoliubov coefficient, as
a function of $\alpha$, for a heavy field with masses $m_{\chi}\sim 10^{15}$,
$10^{16}$ and $10^{17}$ GeV.

Next, introducing these values of the $\beta$-Bogoliubov coefficient in the
energy density (4) and taking into account that the modes that contribute to
the energy density satisfy $0\lesssim k\lesssim a_{kin}m_{\chi}$ and lead
practically to the same value of the $\beta$-Bogoliubov coefficient, one can
safely do the approximation that after the beginning of kination
$\omega_{k}(\tau)\cong m_{\chi}a(\tau)$, obtaining

$\displaystyle\rho_{\chi}(\tau)\cong\frac{m_{\chi}^{4}}{6\pi^{2}}|\beta_{k}|^{2}\left(\frac{a_{kin}}{a(\tau)}\right)^{3}\sim\left\\{\begin{array}[]{ccc}10^{-11}m_{\chi}^{4}\left(\frac{a_{kin}}{a(\tau)}\right)^{3}&\mbox{when}&m_{\chi}\sim
10^{15}\mbox{ GeV}\\\ &&\\\
10^{-12}m_{\chi}^{4}\left(\frac{a_{kin}}{a(\tau)}\right)^{3}&\mbox{when}&m_{\chi}\sim
10^{16}-10^{17}\mbox{ GeV},\end{array}\right.$ (11)

for $\tau>\tau_{kin}$.

A final remark is in order: In hashiba ; ema , the authors infer through
numerical calculations using toy models that, for the relevant modes, the
square of the $\beta$-Bogoliubov coefficient must decay as $e^{-\kappa
m_{\chi}/H_{inf}}$, where $H_{inf}$ is the scale of inflation and $\kappa$ a
dimensionless factor of order 1. However, throughout all the calculation the
mass of the superheavy field is of the same order or less than the scale of
inflation. So, in practice, for the relevant models the authors obtain in all
numerical simulations $|\beta_{k}|^{2}\geq 10^{-7}$.

In our case we have chosen masses which are many orders greater than the scale
of inflation. Effectively, since the power spectrum of scalar perturbations
$P_{\zeta}=\frac{H_{*}^{2}}{8\pi^{2}\epsilon_{*}M_{pl}^{2}}\sim 2\times
10^{-9}$ (here, once again, the star denotes that the quantities are evaluated
at the horizon crossing) leads to $H_{*}\sim 4\times
10^{-4}\sqrt{\epsilon_{*}}M_{pl}$ -with
$\epsilon_{*}=\frac{3\alpha}{16}(1-n_{s})^{2}$ for the case of
$\alpha$-attractors, choosing for the spectral index its central value
$n_{s}=0.9649$ (see planck )-, one gets the following scale of inflation,

$\displaystyle H_{*}\sim 1.5\sqrt{\alpha}\times 10^{13}\mbox{ GeV},$ (12)

which is minimum two orders less than $10^{15}$ GeV.

In summary, there is gravitational production of superheavy particles, with
the square of the $\beta$-Bogoliubov, for the relevant modes, of the order of
$10^{-9}-10^{-10}$, but for superheavy masses the production is not suppressed
by an abnormally small factor as $e^{-\kappa m_{\chi}/H_{inf}}$.

##  III The reheating process

After the production of the superheavy particles, they have to decay into
lighter ones which after the thermalization process form a relativistic plasma
that depicts our hot universe. Then, this decay could occur before the end of
kination (when the energy density of the inflaton becomes of the same order
than the one of the $\chi$-field), or after its end.

###  III.1 Decay before the end of kination

In this case, the energy density of the background, i.e. the one of the
inflaton field, and the one of the relativistic plasma, when the decay is
finished, that is when ${\Gamma}\sim
H_{dec}=H_{kin}\left(\frac{{a}_{kin}}{a_{dec}}\right)^{3}$, will be

$\displaystyle\rho_{\varphi,dec}=3{\Gamma}^{2}M_{pl}^{2}\qquad\mbox{and}\qquad\rho_{\chi,dec}=\rho_{\chi,kin}\left(\frac{{a}_{kin}}{a_{dec}}\right)^{3}\cong\frac{m_{\chi}^{4}}{6\pi^{2}}|\beta_{k}|^{2}\frac{\Gamma}{H_{kin}}.$
(13)

Imposing that the end of the decay precedes the end of kination, which means
$\rho_{\chi,dec}\leq\rho_{\varphi,dec}$, and taking into account that it is
after the beginning of the kination, i.e., $\Gamma\leq H_{kin}\cong 6\times
10^{-7}M_{pl}$, one gets

$\displaystyle\frac{1}{18\pi^{2}}|\beta_{k}|^{2}\frac{m_{\chi}^{4}}{H_{kin}M_{pl}^{2}}\leq\Gamma\leq
H_{kin}.$ (14)

Finally, the reheating temperature, i.e., the temperature of the universe when
the relativistic plasma in thermal equilibrium starts to dominate, which
happens when $\rho_{\varphi,reh}\sim\rho_{\chi,reh}$, can be calculated as
follows: Since after the decay the evolution of the respective energy
densities are given by

$\displaystyle\rho_{\varphi,reh}=\rho_{\varphi,dec}\left(\frac{a_{dec}}{a_{reh}}\right)^{6}\qquad\mbox{and}\qquad\rho_{\chi,reh}=\rho_{\chi,dec}\left(\frac{a_{dec}}{a_{reh}}\right)^{4},$
(15)

we will have
$\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}=\left(\frac{a_{dec}}{a_{reh}}\right)^{2},$
and thus, from the Stefan-Boltzmann law
$\rho_{reh}=\frac{\pi^{2}}{30}g_{reh}T_{reh}^{4}$, where $g_{reh}=106.75$ is
the effective number of degrees of freedom for the Standard Model, the
reheating temperature will be

$\displaystyle
T_{reh}=\left(\frac{30}{\pi^{2}g_{reh}}\right)^{1/4}\rho_{\chi,reh}^{\frac{1}{4}}=\left(\frac{30}{\pi^{2}g_{reh}}\right)^{1/4}\rho_{\chi,dec}^{\frac{1}{4}}\sqrt{\frac{\rho_{\chi,dec}}{\rho_{\varphi,dec}}}$
$\displaystyle\cong\frac{1}{3\sqrt{2}\pi^{3/2}}\left(\frac{30}{\pi^{2}g_{reh}}\right)^{1/4}|\beta_{k}|^{3/2}\left(\frac{H_{kin}}{6\Gamma}\right)^{1/4}\frac{m_{\chi}^{3}}{M_{pl}^{2}H_{kin}}M_{pl}.$
(16)

So, taking into account the bound (14), the reheating temperature ranges
between

$\displaystyle\frac{1}{3\sqrt{2}\pi^{2}}\left(\frac{5}{g_{reh}}\right)^{1/4}|\beta_{k}|^{3/2}\frac{m_{\chi}^{3}}{M_{pl}^{2}H_{kin}}M_{pl}\leq
T_{reh}\leq$
$\displaystyle\frac{1}{3\sqrt{2}\pi^{3/2}}\left(\frac{90}{g_{reh}}\right)^{1/4}|\beta_{k}|\left(\frac{m_{\chi}}{M_{pl}}\right)^{2}\sqrt{\frac{M_{pl}}{H_{kin}}}M_{pl},$
(17)

which shows that $6\times 10^{5}$ GeV $\lesssim T_{reh}\lesssim 6\times
10^{8}$ GeV for $m_{\chi}\sim 10^{15}$ GeV and $T_{reh}\gtrsim 10^{8}$ GeV for
$m_{\chi}\sim 10^{16}$ GeV. However, for masses of order $10^{17}$ GeV, the
reheating temperature is some orders greater than $10^{9}$ GeV. This fact, as
we have already explained in the Introduction, supposes a great problem in
order to ensure the BBN success.

Therefore, to have a viable reheating temperature, we conclude that only
superheavy particles with masses of the order $10^{15}-10^{16}$ GeV could
decay before the end of kination.

###  III.2 Decay after the end of kination

In the case that the decay of the $\chi$-field is after the end of kination
(recall that kination ends when $\rho_{\varphi}\sim\rho_{\chi}$), one has to
impose ${\Gamma}\leq H(\tau_{end})\equiv H_{end}$, where we have denoted by
$\tau_{end}$ the time at which kination ends. Taking this into account, one
has

$\displaystyle H^{2}_{end}=\frac{2\rho_{\varphi,end}}{3M_{pl}^{2}}$ (18)

and

$\displaystyle\rho_{\varphi,end}={\rho}_{\varphi,kin}\left(\frac{{a}_{kin}}{a_{end}}\right)^{6}=\frac{{\rho}_{\chi,kin}^{2}}{{\rho}_{\varphi,kin}},$
(19)

where we have used that the kination ends when
${{\rho}_{\chi,end}}={{\rho}_{\varphi,end}}$, meaning
$\left({a}_{kin}/a_{end}\right)^{3}=\frac{{\rho}_{\chi,kin}}{{\rho}_{\varphi,kin}}$.
So, the condition ${\Gamma}\leq H_{end}$ leads to the bound

$\displaystyle\Gamma\leq\sqrt{\frac{2}{3}}\frac{\rho_{\chi,kin}}{M_{pl}\sqrt{\rho_{\varphi,kin}}}\cong\frac{\sqrt{2}}{18\pi^{2}}|\beta_{k}|^{2}\frac{m_{\chi}^{4}}{H_{kin}M_{pl}^{2}}.$
(20)

On the other hand, assuming once again instantaneous thermalization, the
reheating temperature (i.e., the temperature of the universe when the
thermalized plasma starts to dominate) will be obtained when all the
superheavy particles decay, i.e. when $H\sim\Gamma$, obtaining

$\displaystyle
T_{reh}=\left(\frac{30}{\pi^{2}g_{reh}}\right)^{1/4}\rho_{\chi,dec}^{1/4}=\left(\frac{90}{\pi^{2}g_{reh}}\right)^{1/4}\sqrt{{\Gamma}M_{pl}},$
(21)

where we have used that, after the end of the kination regime, the energy
density of the produced particles dominates the one of the inflaton field.

Consequently, since the BBN epoch occurs at the $1$ MeV regime, one can find
that, in that case, the reheating temperature is bounded by

$\displaystyle 1\mbox{ MeV}\leq
T_{reh}\leq\frac{1}{3\pi}\left(\frac{45}{\pi^{2}g_{reh}}\right)^{1/4}|\beta_{k}|\left(\frac{m_{\chi}}{M_{pl}}\right)^{2}\sqrt{\frac{M_{pl}}{H_{kin}}}M_{pl},$
(22)

which for masses $m_{\chi}\sim 10^{15}-10^{17}$ GeV leads to a reheating
temperature in all the range of viable values $1$ MeV $\leq T_{reh}\leq
10^{9}$ GeV.

##  IV  A simple model containing the Cosmological Constant

A very simple model comes from a lineal potential with the Cosmological
Constant denoted by $\Lambda$ vardayan ,
$V(\phi)=\lambda(\phi+\sqrt{6\alpha})+\Lambda M_{pl}^{2}$, which in terms of
the canonically unitary field $\varphi$, defined as
$\phi=\sqrt{6\alpha}\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)$,
becomes

$\displaystyle
V(\varphi)=\lambda\sqrt{6\alpha}\left(\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)+1\right)+\Lambda
M_{pl}^{2}.$ (23)

To obtain the values of the parameters, first of all we calculate the main
slow roll parameter

$\displaystyle\epsilon\cong\frac{1}{12\alpha\cosh^{4}\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)\left(\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)+1\right)^{2}},$
(24)

thus, in order that inflation ends, which happens when $\epsilon=1$, one needs
that $\frac{1}{12\alpha}>1$. So, we will take for example $\alpha\sim
10^{-2}$.

To get the value of $\lambda$ we use the formula (12) and the fact that at the
horizon crossing

$\displaystyle
H_{*}^{2}\cong\frac{V(\varphi_{*})}{3M_{pl}^{2}}\cong\frac{2\lambda\sqrt{6\alpha}}{3M^{2}_{pl}}.$
(25)

Combining both results one gets $\frac{\lambda}{\sqrt{\alpha}}\sim 2\times
10^{-11}M_{pl}^{4}$.

Finally, it is well-known that, to match with the current observational data,
the Cosmological Constant $\Lambda$ must be of the order $10^{-120}M_{pl}^{2}$
because at the present time the current observational data state that
$\Omega_{\Lambda}=\frac{\Lambda M_{pl}^{2}}{3H_{0}^{2}M_{pl}^{2}}\cong 0.7$.

###### Remark IV.1

The potential (23) does not belong to the class of Quintessential Inflation
potentials because to deal with the current cosmic acceleration it needs a
Cosmological Constant. The difference with the potential (5) is that this one
contains two parameters, namely $\lambda$ and $n$, needed to unify the early
and late-time acceleration. However, the potential
$\lambda\sqrt{6\alpha}\left(\tanh\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)+1\right)$
only contains the parameter $\lambda$ which is determined by the power
spectrum of scalar perturbation. Then, to correctly depict the late-time
acceleration one needs another parameter, in this case the Cosmological
Constant.

###### Remark IV.2

Note also that for the potential (5) the value of $\alpha$ is not restricted
to be small as for the potential (23), because for that potential the main
slow roll parameter satisfies
$\epsilon=\frac{n^{2}}{12\alpha\cosh^{4}\left(\frac{\varphi}{\sqrt{6\alpha}M_{pl}}\right)}$,
so in order that inflation ends one needs $\frac{n^{2}}{12\alpha}>1$, and
since $n$ is of the order $10^{2}$ (see the equation (6) and also benisty3
for a detailed discussion) this allows a large range of values of $\alpha$.

The important point is that this model leads to the same results for particle
production as the exponential model studied previously in detail, and thus, it
leads to the same bounds for the reheating temperature.

##  V Conclusions

In the present work we have numerically studied the gravitational particle
production of superheavy particles conformally coupled to gravity in
$\alpha$-Quintessential Inflation. To calculate the energy density of the
produced particles we have used the well-known diagonalisation method, where
the key point is the calculation of the time-dependent $\beta$-Bogoliubov
coefficient.

This coefficient encodes all the polarization effects and the produced
superheavy particles during the phase transition from the end of inflation to
the beginning of kination. Fortunately, the polarization effects disappear
soon after the beginning of kination, which enables us to extract from it only
the part which has to do with particle production. In fact, from the relevant
modes that contribute to the particle production the square modulus of the
$\beta$-Bogoliubov coefficient is of the order $10^{-9}-10^{-10}$, depending
on the mass of superheavy particles.

Once these superheavy particles have been created, they must decay into
lighter ones to form a relativistic plasma which eventually becomes dominant
and matches with the hot Big Bang universe. Then, two different situations
arise, namely when the decay occurs before the end of the kination regime and
when the decay occurs after the end of the kination regime. We have shown that
for both situations the maximum value of the reheating temperature is quite
big, more or less around $10^{9}$ GeV, which demystifies the belief that heavy
masses suppress the particle production, thus leading to an abnormally low
reheating temperature. What really happens is that the main contribution of
particle production is in a long wavelength regime, which without a numerical
calculation is impossible to quantify. This is the reason why in many papers
the production of superheavy particles is simply disregarded because
analytically only ultraviolet effects can be calculated, but, as it is well-
known, the ultraviolet modes do not contribute significantly to the particle
production.

## Acknowledgments

JdH is supported by grant MTM2017-84214-C2-1-P funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, and also
in part by the Catalan Government 2017-SGR-247. L.A.S thanks the School of
Mathematical Sciences (Queen Mary University of London) for the support
provided.

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# Lower Bound for the Simplicial Volume of Closed Manifolds Covered by
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$

Jixiang Fu  School of Mathematical Sciences, Fudan University, Shanghai
200433, China<EMAIL_ADDRESS>and  Xiaofeng Meng  School of Mathematical
Sciences, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
(Date: 2021/3/14)

###### Abstract.

We estimate the upper bound for the $\ell^{\infty}$-norm of the volume form on
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$ seen as a class in
$H_{c}^{6}(\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R};\mathbb{R})$.
This gives the lower bound for the simplicial volume of closed Riemennian
manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$.
The proof of these facts yields an algorithm to compute the lower bound of
closed Riemannian manifolds covered by $\big{(}\mathbb{H}^{2}\big{)}^{n}$.

###### 1991 Mathematics Subject Classification:

Primary 53C20; Secondary 55N10

##  1\. Introduction

The simplicial volume is a topological invariant of manifolds introduced by
Gromov ([4]) and Thurston ([7]). For an oriented closed connected
$n$-dimensional manifold $M$, $H_{n}(M,\mathbb{Z})$ is generated by the
fundamental class $[M]_{\mathbb{Z}}$ of $M$. Denote the image of
$[M]_{\mathbb{Z}}$ via the change of coefficients map
$H_{n}(M,\mathbb{Z})\hookrightarrow H_{n}(M,\mathbb{R})$ by $[M]\in
H_{n}(M,\mathbb{R})$. Then the simplicial volume of $M$ is defined by

$\|M\|=\mathrm{inf}\left\\{\sum_{i}|a_{i}|\ \middle|\
[\sum_{i}a_{i}\sigma_{i}]=[M]\in H_{n}(M,\mathbb{R})\right\\}.$

Moreover, simplicial volume can also be defined for a non-orientable closed
connected manifold $M$. Let $\widehat{M}$ be the oriented double covering of
$M$. Then the simplicial volume of $M$ is defined by
$\|M\|=\|\widehat{M}\|/2$.

Although several vanishing and non-vanishing results for the simplicial volume
are presented now, the exact value of non-vanishing simplicial volume has only
been calculated for a few cases. Those include closed hyperbolic manifolds
([4], [7]), Hilbert modular surfaces ([5]) and closed manifolds covered by
$\mathbb{H}^{2}\times\mathbb{H}^{2}$ ([2]).

For the simplicial volume of products, we have the following result by [4] for
closed manifolds $M$ and $N$ with dimension $m$ and $n$, respectively.

$\|M\|\cdot\|N\|\leq\|M\times N\|\leq\binom{n+m}{n}\|M\|\cdot\|N\|.$ (1)

When $M$ and $N$ are closed surfaces, it has been proved in [2] that

$\|M\times N\|=\frac{3}{2}\|M\|\cdot\|N\|.$ (2)

Combining (1) and (2), we get

$\dfrac{3}{2}\prod_{i=1}^{3}\|M_{i}\|\leq\Bigg{\|}\prod_{i=1}^{3}M_{i}\Bigg{\|}\leq\dfrac{45}{2}\prod_{i=1}^{3}\|M_{i}\|,$

for closed surfaces $M_{i}$, $i=1,2,3$. In this paper we improve the above
estimation of the lower bound to the following one and furthermore, offer an
algorithm to compute the lower bound of closed Riemannian manifolds covered by
$(\mathbb{H}^{2})^{n}$.

######  Theorem 1.

Let $M_{i}$ be a closed surface, where $i=1,2,3$. Then

$\Bigg{\|}\prod_{i=1}^{3}M_{i}\Bigg{\|}\geq\frac{45}{11}\prod_{i=1}^{3}\|M_{i}\|.$

Recall the proportionality principle by Gromov in [4]. Let $M$ be an
$n$-dimensional closed Riemannian manifold. Then

$\|M\|=\dfrac{\mathrm{Vol(M)}}{\|[\omega_{\widetilde{M}}]_{c}^{G}\|_{\infty}},$
(3)

where $\widetilde{M}$ is the universal covering of $M$, $G$ is the group of
orientation-preserving isometries of $\widetilde{M}$, $\omega_{\widetilde{M}}$
is the volume form of $\widetilde{M}$ and $[\omega_{\widetilde{M}}]_{c}^{G}$
is the volume class of $\widetilde{M}$ viewed as a class in
$H^{n}(C_{c}^{*}(\widetilde{M})^{G})$. According to [7], the constant
$\|[\mathrm{Vol}_{\widetilde{M}}]_{c}^{G}\|_{\infty}$ is $\pi$ for a closed
oriented surface $M$ supporting a hyperbolic structure. According to [4], the
simplicial volume of a closed connected manifold admitting a self-map of non-
trivial degree (i.e., not equal to $-1$, $0$, or $1$) vanishes. This implies
that the simplicial volume of a closed surface covered by $2$-sphere or torus
vanishes. Therefore, Theorem 1 naturally holds when one of the universal
coverings of $M_{1}$, $M_{2}$ and $M_{3}$ is $2$-sphere or Euclidean plane. To
prove Theorem 1, we only need to consider the case where the universal
covering of $M_{1}\times M_{2}\times M_{3}$ is
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$. Hence the following
Theorem proves Theorem 1.

######  Theorem 2.

Let $M$ be a closed Riemannian manifold whose universal covering is
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$. Then

$\|M\|\geq\frac{45}{11\pi^{3}}\mathrm{Vol}(M).$

To prove Theorem 1 and 2, we recall a way to compute the proportionality
constant $\frac{\|M\|}{\mathrm{Vol}(M)}$ provided by Bucher-Karlsson in [2]
for closed locally symmetric spaces of noncompact type. Let $M$ be an
$n$-dimensional closed locally symmetric space of non-compact type. We use the
same notations as in (3). Let $\mathcal{J}$ be the Van Est isomorphism mapping
from $A^{n}(\widetilde{M})^{G}$ to $H_{c}^{n}(G,\mathbb{R})$. Then

$\|M\|=\dfrac{\mathrm{Vol}(M)}{\|\mathcal{J}\big{(}\omega_{\widetilde{M}}\big{)}\|_{\infty}}.$

We establish the desired inequality as following.

######  Theorem 3.

Let $\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}}\in
A^{6}(\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2},\mathbb{R})$ be
the Riemannian volume form on
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$. Let $\mathcal{J}$ be
the Van Est isomorphism mapping from
$A^{6}(\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2},\mathbb{R})$ to
$H_{c}^{6}(\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R},\mathbb{R})$.
Then

$\|\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})\|_{\infty}\leq\frac{11}{45}\pi^{3}.$

We briefly outline the content of each section. To estimate the upper bound
for
$\|\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})\|_{\infty}$,
we need an explicit cocycle in class
$\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})$.
In Section 2 we recall the definition of continuous (bounded) cohomology of
Lie groups and select a cocycle $\pi^{3}\cdot\Theta_{\theta}$ representing
$\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})$.
In Section 3 we recall another bounded cohomology
$H_{\delta}^{k}\big{(}C_{b}^{*}(\mathbb{T}^{3},\mathbb{R})^{(PSL_{2}\mathbb{R})^{3}}\big{)}$
which is isometrically isomorphic to the continuous bounded cohomology of
$(PSL_{2}\mathbb{R})^{3}$. This allows us to estimate the upper bound of
$\|[\Theta_{\theta}]\|_{\infty}$ by calculating the $l^{\infty}$-norm of a
selected cocycle $\Theta$ in
$C_{b}^{6}(\mathbb{T}^{3},\mathbb{R})^{(PSL_{2}\mathbb{R})^{3}}$. We show that
$\|\Theta\|_{\infty}$ can be computed by reducing it to a programming problem.
This proves Theorem 3. Finally, in Section 4 we give several conjectures
related to the simplicial volume of closed manifolds covered by
$\big{(}\mathbb{H}^{2}\big{)}^{n}$.

##  2\. The volume form on
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$

Before giving a cocycle representing
$\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})$,
we recall some definitions first.

###  2.1. The continuous (bounded) cohomology of Lie groups

Let $G$ be a connected Lie group. For every $k\in\mathbb{N}$, we define the
complex $C^{k}(G,\mathbb{R})$ by

$C^{k}(G,\mathbb{R})=\left\\{f:G^{k+1}\rightarrow\mathbb{R}\ \middle|\ \text{f
is continuous}\right\\}.$

Define the coboundary $d^{k}:C^{k}(G,\mathbb{R})\rightarrow
C^{k+1}(G,\mathbb{R})$ by

$(d^{k}f)(g_{0},...,g_{k+1})=\sum_{i=0}^{k+1}(-1)^{i}f(g_{0},...,\hat{g}_{i},...,g_{k+1}),$

for all $f$ in $C^{k}(G,\mathbb{R})$ and all $(g_{0},...,g_{k+1})$ in
$G^{k+2}$. Define the $G$-action on $C^{k}(G,\mathbb{R})$ by

$(h\cdot f)(g_{0},...,g_{k})=f(h^{-1}g_{0},...,h^{-1}g_{k}),$

for all $f$ in $C^{k}(G,\mathbb{R})$, all $h\in G$ and all
$(g_{0},...,g_{k})\in G^{k+1}$. Let $C^{k}(G,\mathbb{R})^{G}$ be the
$G$-invariant elements in $C^{k}(G,\mathbb{R})$. Set $C_{b}^{k}(G,\mathbb{R})$
to be the subspace of $C^{k}(G,\mathbb{R})$ consisting all bounded functions
and $d_{b}^{k}=d^{k}|_{C_{b}^{k}(G,\mathbb{R})}$. Restrict the $G$-action to
$C_{b}^{k}(G,\mathbb{R})$. Then $C_{b}^{k}(G,\mathbb{R})^{G}\coloneqq
C_{b}^{k}(G,\mathbb{R})\cap C^{k}(G,\mathbb{R})^{G}$ is the subspace of
$C_{b}^{k}(G,\mathbb{R})$ with $G$-invariant elements. Therefore
$\big{(}C^{k}(G,\mathbb{R})^{G},d\big{)}$ and
$\big{(}C_{b}^{k}(G,\mathbb{R})^{G},d_{b}\big{)}$ induce continuous cohomology
$H_{c}^{k}(G,\mathbb{R})$ of $G$ and the continuous bounded cohomology
$H_{cb}^{k}(G,\mathbb{R})$ of $G$, respectively. For all $f\in
C_{b}^{k}(G,\mathbb{R})$, let $\|f\|_{\infty}$ be the $l^{\infty}$-norm of
$f$. This induces the semi-norms of $H_{cb}^{k}(G,\mathbb{R})$ and
$H_{c}^{k}(G,\mathbb{R})$, both of which we still denote by
$\|\cdot\|_{\infty}$.

Here we use a form of Van Est isomorphism introduced in [3]. Let $G$ be a Lie
group, $K<G$ be a maximal compact subgroup of $G$ and $X=G/K$ be the
associated symmetric space. For $k\in\mathbb{N}$, define
$\mathrm{A}^{k}(X,\mathbb{R})$ to be the set of differential $k$-forms. The
Lie group $G$ acts on $\mathrm{A}^{k}(X,\mathbb{R})$ by the pullbacks. Let
$\mathrm{A}^{k}(X,\mathbb{R})^{G}$ be the subspace of
$\mathrm{A}^{k}(X,\mathbb{R})$ with $G$-invariant elements. Denote the Van Est
isomorphism by
$\mathcal{J}:\mathrm{A}^{k}(X,\mathbb{R})^{G}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H_{c}^{k}(G,\mathbb{R})$.

###  2.2. The $2$-form representing $\mathcal{J}(\omega_{\mathbb{H}^{2}})$

Let $\omega_{\mathbb{H}^{2}}\in A^{2}(\mathbb{H}^{2},\mathbb{R})$ be the
volume form on $\mathbb{H}^{2}$. View $\mathbb{H}^{2}$ as the upper half-plane
$\big{\\{}(x,y)\in\mathbb{R}^{2}|y>0\big{\\}}\subset\mathbb{C}$. Then
$PSL_{2}\mathbb{R}$ acts on $\mathbb{R}^{2}$ by the Möbius transformations.
This action can be restricted to $\big{\\{}(x,0)\in\mathbb{R}^{2}\big{\\}}$.
Notice that through $z\mapsto\dfrac{z-i}{z+i}$ the upper half-plane is
identified with the unit disc $\\{|z|<1\\}\subset\mathbb{C}$ and the
$PSL_{2}\mathbb{R}$-action on $\big{\\{}(x,0)\in\mathbb{R}^{2}\big{\\}}$ can
induce a $PSL_{2}\mathbb{R}$-action on $\mathbb{S}^{1}$. Define a function
$Or$ by

$\begin{array}[]{cccl}\mathrm{Or}:&(\mathbb{S}^{1})^{3}&\longrightarrow&\mathbb{R}\\\
&(\theta_{0},\theta_{1},\theta_{2})&\longmapsto&\begin{cases}+1&\text{if
}\theta_{0},\theta_{1},\theta_{2}\text{ are distinct and positively
oriented,}\\\ -1&\text{if }\theta_{0},\theta_{1},\theta_{2}\text{ are distinct
and negatively oriented,}\\\ 0&\text{if
}\theta_{0},\theta_{1},\theta_{2}\text{ are not
distinct}.\end{cases}\end{array}$

We fix a point $\theta$ in $\mathbb{S}^{1}$. Define a cocycle
$\mathrm{Or}_{\theta}$ in
$C_{b}^{2}(PSL_{2}\mathbb{R},\mathbb{R})^{PSL_{2}\mathbb{R}}$ by

$\mathrm{Or}_{\theta}(g_{0},g_{1},g_{2})=\mathrm{Or}(g_{0}\theta,g_{1}\theta,g_{2}\theta)$

for all $(g_{0},g_{1},g_{2})$ in $(PSL_{2}\mathbb{R})^{3}$. Then it is easy to
check that $\mathcal{J}(\omega_{\mathbb{H}^{2}})=\pi\cdot[Or_{\theta}]$.

###  2.3. The $6$-form representing
$\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})$

Recall the definition of cup product. Take a $p$-cochain $c^{p}$ in
$C^{p}\big{(}(PSL_{2}\mathbb{R})^{3},\mathbb{R}\big{)}$ and a $q$-cochain
$c^{q}$ in $C^{q}\big{(}(PSL_{2}\mathbb{R})^{3},\mathbb{R}\big{)}$. The cup
product $c^{p}\cup c^{q}$ is defined by

$(c^{p}\cup c^{q})(g_{0},...,g_{p+q})=c^{p}(g_{0},...,g_{p})\cdot
c^{q}(g_{p},...,g_{p+q}),$

for all $g_{i}=(g_{i}^{1},g_{i}^{2},g_{i}^{3})$ in $(PSL_{2}\mathbb{R})^{3}$,
$i=0,...,p+q$. This induces cup product for classes.

Recall that the alternation of a $p$-cochain $c^{p}$ in
$C^{p}\big{(}(PSL_{2}\mathbb{R})^{3},\mathbb{R}\big{)}$ is

$\mathrm{Alt}(c^{p})(g_{0},...,g_{p})=\dfrac{1}{(p+1)!}\sum_{\sigma\in
Sym(p+1)}sign(\sigma)c^{p}(g_{\sigma(0)},...,g_{\sigma(p)})$

for all $g_{i}$ in $(PSL_{2}\mathbb{R})^{3}$, $i=0,...,p$. Note that for a
$p$-cocycle $f$ we have $[\mathrm{Alt}(f)]=[f]$ and
$\|\mathrm{Alt}(f)\|_{\infty}\leq\|f\|_{\infty}$.

Note Let $p^{\mathbb{H}}_{i}$ be the $i$-th projection from
$\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$ to $\mathbb{H}^{2}$
for $i=1,2,3$. Let $p^{PSL_{2}\mathbb{R}}_{i}$ be the $i$-th projection from
$PSL_{2}\mathbb{R}\times PSL_{2}\mathbb{R}\times PSL_{2}\mathbb{R}$ to
$PSL_{2}\mathbb{R}$ for $i=1,2,3$. Let $p^{\mathbb{T}}_{i}$ be the $i$-th
projection from $\mathbb{T}^{3}\times\mathbb{T}^{3}\times\mathbb{T}^{3}$ to
$\mathbb{T}^{3}$ for $i=1,2,3$, where $\mathbb{T}^{3}$ is
$\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$.

Define a function $\Theta:(\mathbb{T}^{3})^{7}\rightarrow\mathbb{R}$ by

$\begin{split}\Theta(\theta_{0},...,\theta_{6})&=\mathrm{Alt}\big{(}(p_{1}^{\mathbb{T}})^{*}(Or)\cup(p_{2}^{\mathbb{T}})^{*}(Or)\cup(p_{3}^{\mathbb{T}})^{*}(Or)\big{)}(\theta_{0},...,\theta_{6})\\\
&=\dfrac{1}{7!}\sum_{\sigma\in
Sym(7)}sign(\sigma)\prod_{i=1}^{3}Or(\theta_{2i-2}^{i},\theta_{2i-1}^{i},\theta_{2i}^{i})\end{split}$
(4)

for all $\theta_{i}=(\theta_{i}^{1},\theta_{i}^{2},\theta_{i}^{3})$ in
$\mathbb{T}^{3}$, $i=0,...,6$. Let $(PSL_{2}\mathbb{R})^{3}$ acts on
$\mathbb{T}^{3}$ diagonally. Fix a point
$\theta=(\theta^{1},\theta^{2},\theta^{3})$ in $\mathbb{T}^{3}$. Define a
$6$-form $\Theta_{\theta}$ in
$C^{6}\big{(}(PSL_{2}\mathbb{R})^{3},\mathbb{R}\big{)}$ by

$\Theta_{\theta}(g_{0},...,g_{6})=\Theta(g_{0}\theta,...,g_{6}\theta)$

for all $g_{i}=(g_{i}^{1},g_{i}^{2},g_{i}^{3})$ in $(PSL_{2}\mathbb{R})^{3}$,
$i=0,...,6$.

######  Proposition 4.

$\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})=[\pi^{3}\cdot\Theta_{\theta}]$.

###### Proof.

By (4), we have

$\begin{split}[\pi^{3}\cdot\Theta_{\theta}]&=\pi^{3}\cdot\Big{[}Alt\big{(}(p^{PSL_{2}\mathbb{R}}_{1})^{*}(Or_{\theta^{1}})\cup(p^{PSL_{2}\mathbb{R}}_{2})^{*}(Or_{\theta^{2}})\cup(p^{PSL_{2}\mathbb{R}}_{3})^{*}(Or_{\theta^{3}})\big{)}\Big{]}\\\
&=\pi^{3}\cdot\Big{[}(p^{PSL_{2}\mathbb{R}}_{1})^{*}(Or_{\theta^{1}})\cup(p^{PSL_{2}\mathbb{R}}_{2})^{*}(Or_{\theta^{2}})\cup(p^{PSL_{2}\mathbb{R}}_{3})^{*}(Or_{\theta^{3}})\Big{]}\\\
&=\Big{[}\pi\cdot(p^{PSL_{2}\mathbb{R}}_{1})^{*}(Or_{\theta^{1}})\Big{]}\cup\Big{[}\pi\cdot(p^{PSL_{2}\mathbb{R}}_{2})^{*}(Or_{\theta^{2}})\Big{]}\cup\Big{[}\pi\cdot(p^{PSL_{2}\mathbb{R}}_{3})^{*}(Or_{\theta^{3}})\Big{]}\\\
&=\mathcal{J}\big{(}(p^{\mathbb{H}}_{1})^{*}(\omega_{\mathbb{H}^{2}})\big{)}\cup\mathcal{J}\big{(}(p^{\mathbb{H}}_{2})^{*}(\omega_{\mathbb{H}^{2}})\big{)}\cup\mathcal{J}\big{(}(p^{\mathbb{H}}_{3})^{*}(\omega_{\mathbb{H}^{2}})\big{)}\\\
&=\mathcal{J}\big{(}(p^{\mathbb{H}}_{1})^{*}(\omega_{\mathbb{H}^{2}})\wedge(p^{\mathbb{H}}_{2})^{*}(\omega_{\mathbb{H}^{2}})\wedge(p^{\mathbb{H}}_{3})^{*}(\omega_{\mathbb{H}^{2}})\big{)}\\\
&=\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}}).\end{split}$

Therefore this proposition is proved.

∎

##  3\. Upper bound of
$\|\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}}\|_{\infty}$

Denote $(PSL_{2}\mathbb{R})^{3}$ by $G$. For the convenience of calculation,
we introduce another complex
$\big{(}C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})^{G},\delta\big{)}$. For every
$k\in\mathbb{N}$, define $C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})$ by

$C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})=\left\\{f:(\mathbb{T}^{3})^{k+1}\rightarrow\mathbb{R}\
\middle|\ \text{f is continous, measurable and bounded}\right\\}.$

The coboundary $\delta:C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})\rightarrow
C_{b}^{k+1}(\mathbb{T}^{3},\mathbb{R})$ is defined by

$(\delta
f)(\theta_{0},...,\theta_{k+1})=\sum_{i=0}^{k+1}(-1)^{i}f(\theta_{0},...,\hat{\theta}_{i},...,\theta_{k+1})$

for all $f$ in $C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})$ and all $\theta_{i}$ in
$\mathbb{T}^{3}$, $i=0,...,k+1$. Define the action of $G$ on
$C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})$ by

$(g\cdot
f)(\theta_{0},...,\theta_{k})=f(g^{-1}\theta_{0},...,g^{-1}\theta_{k}).$

Denote the $G$-invariant elements in $C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})$ by
$C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})^{G}$. Then the complex
$\big{(}C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})^{G},\delta\big{)}$ induces a
cohomology
$H_{\delta}^{k}\big{(}C_{b}^{*}(\mathbb{T}^{3},\mathbb{R})^{G}\big{)}$. Let
$\|f\|_{\infty}$ be the $l^{\infty}$-norm of $f$ in
$C_{b}^{k}(\mathbb{T}^{3},\mathbb{R})$. This induces the semi-norm of
$H_{\delta}^{k}\big{(}C_{b}^{*}(\mathbb{T}^{3},\mathbb{R})^{G}\big{)}$, which
we still denote by $\|\cdot\|_{\infty}$.

Let $P$ be the subgroup of $PSL_{2}\mathbb{R}$ defined as below.

$P=\left\\{\begin{pmatrix}a&0\\\ c&a^{-1}\end{pmatrix}\in SL_{2}\mathbb{R}\
\middle|\ a\in\mathbb{R}\setminus\\{0\\},\
c\in\mathbb{R}\right\\}\bigg{/}\\{\pm 1\\}.$

Notice that $\mathbb{T}^{3}=(PSL_{2}\mathbb{R})^{3}\big{/}P^{3}$ and $P$ is
amenable. Using [6, Corollary 7.5.9.], $\big{(}H_{cb}^{k}(G,\mathbb{R}),\|\
\|_{\infty}\big{)}$ is isometrically isomorphic to
$\big{(}H_{\delta}^{k}(C_{b}^{*}(\mathbb{T}^{3},\mathbb{R})^{G}),\|\
\|_{\infty}\big{)}$. Note that this isomorphism maps $[\Theta_{\theta}]$ to
$[\Theta]$. Hence

$\|\mathcal{J}(\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}})\|_{\infty}=\inf\left\\{\|f\|_{\infty}\
\middle|\ f\in[\Theta_{\theta}]\right\\}\leq\|\Theta\|_{\infty}.$

As a conclusion, to estimate the upper bound of
$\|\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}}\|_{\infty}$
we need to calculate $\|\Theta\|_{\infty}$.

######  Proposition 5.

$\|\Theta\|_{\infty}=\dfrac{11}{45}$.

Before using algorithm to compute the exact value of $\|\Theta\|_{\infty}$ we
need to simplify the formula (4). Otherwise the computation progress using
algorithm will take forever. For all $(\theta_{0},...,\theta_{6})$ in
$(\mathbb{T}^{3})^{7}$, we have

$\begin{split}\Theta(\theta_{0},...,\theta_{6})&=\dfrac{1}{7!}\sum_{\sigma\in
Sym(7)}sign(\sigma)Or\Big{(}\theta_{\sigma(0)}^{1},\theta_{\sigma(1)}^{1},\theta_{\sigma(2)}^{1}\Big{)}\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\Big{(}\theta_{\sigma(2)}^{2},\theta_{\sigma(3)}^{2},\theta_{\sigma(4)}^{2}\Big{)}Or\Big{(}\theta_{\sigma(4)}^{3},\theta_{\sigma(5)}^{3},\theta_{\sigma(6)}^{3}\Big{)}\\\
&=\dfrac{1}{7!}\sum_{\sigma\in
Sym(7)}sign(\sigma)Or\Big{(}\theta_{\sigma(0)}^{1},\theta_{\sigma(1)}^{1},\theta_{\sigma(2)}^{1}\Big{)}\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\Big{(}\theta_{\sigma(0)}^{2},\theta_{\sigma(3)}^{2},\theta_{\sigma(4)}^{2}\Big{)}Or\Big{(}\theta_{\sigma(4)}^{3},\theta_{\sigma(5)}^{3},\theta_{\sigma(6)}^{3}\Big{)}\\\
&=\dfrac{4}{7!}\sum_{\begin{subarray}{c}\sigma\in Sym(7)\\\
\sigma(1)<\sigma(2),\
\sigma(5)<\sigma(6)\end{subarray}}sign(\sigma)Or\Big{(}\theta_{\sigma(0)}^{1},\theta_{\sigma(1)}^{1},\theta_{\sigma(2)}^{1}\Big{)}\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\Big{(}\theta_{\sigma(0)}^{2},\theta_{\sigma(3)}^{2},\theta_{\sigma(4)}^{2}\Big{)}Or\Big{(}\theta_{\sigma(4)}^{3},\theta_{\sigma(5)}^{3},\theta_{\sigma(6)}^{3}\Big{)}.\end{split}$
(5)

View $\mathbb{S}^{1}$ as the quotient space $[0,1]/\sim_{f}$, where
$f(x)=e^{2\pi ix}$. Denote $\theta_{i}$ by
$\big{(}[x_{i}],[y_{i}],[z_{i}]\big{)}$, where $x_{i},y_{i},z_{i}\in[0,1]$,
for $i=0,...,6$. To prove Proposition 5, we need the following lemmas.

######  Lemma 6.

If the elements in $\\{\theta_{j}^{i}\\}_{j=0}^{6}$ are pairwise distinct for
$i=1,2,3$,

$|\Theta(\theta_{0},...,\theta_{6})|\leq\dfrac{11}{45}.$

###### Proof.

We can assume that $x_{0}<...<x_{6}<1$ and $\Theta$ is alternating. We notice
that for all $\theta_{i}=[x_{i}]\in\mathbb{S}^{1}$, where $i=0,1,2$,

$Or(\theta_{0},\theta_{1},\theta_{2})=sign(x_{0},x_{1},x_{2}).$

Here we abuse the notation $sign$. If the elements of $\\{x_{i}\\}_{i=0}^{k}$
are pairwise distinct, assuming $x_{0}<...<x_{k}$, we define
$sign(x_{i_{0}},...,x_{i_{k}})$ to be $sign(i_{0},...,i_{k})$. If the elements
of $\\{x_{i}\\}_{i=0}^{k}$ are not pairwise distinct, we define
$sign(x_{i_{0}},...,x_{i_{k}})$ to be $0$. Therefore we only need to consider
the order of $\\{y_{i}\\}$ and $\\{z_{i}\\}$. Moreover, notice that take
$g=(g_{1},g_{2},g_{3})$ in $(PSL_{2}\mathbb{R})^{3}$ with $g_{i}$ being
rotation or reflection of $\mathbb{S}^{1}$ for $i=1,2,3$. Then

$|(g\cdot\Theta)(\theta_{0},...,\theta_{6})|=|\Theta(\theta_{0},...,\theta_{6})|$
(6)

for all $\theta_{i}\in\mathbb{T}^{3}$, $i=0,...,6$. Thus we can assume
$y_{0}=z_{0}=0$, $y_{1}<y_{2}$ and $z_{1}<z_{2}$. Therefore we only need to
consider $360$ possible orders of $\\{y_{i}\\}_{i=0}^{6}$, as well as of
$\\{z_{i}\\}_{i=0}^{6}$. We denote these possible orders by a $7\times 360$
matrix $P$, where each column represents a possible order. For example, a
column $(2,3,5,1,4,6,7)^{t}$ represents
$y_{3}<y_{0}<y_{1}<y_{4}<y_{2}<y_{5}<y_{6}$ or
$z_{3}<z_{0}<z_{1}<z_{4}<z_{2}<z_{5}<z_{6}$. Let $S$ be a $7\times 1260$
matrix, where each column represents a permutation $\sigma$ in $Sym(1,...,7)$
satisfying $\sigma(2)<\sigma(3)$ and $\sigma(6)<\sigma(7)$. To further
simplify the computation progress, we define two $1260\times 360$ matrices
$P_{y360}$ and $P_{z360}$ by

$(P_{y360})_{i,j}=sign\big{(}P_{(S_{1,i}),j},P_{(S_{4,i}),j},P_{(S_{5,i}),j}\big{)}$

and

$(P_{z360})_{i,j}=sign\big{(}P_{(S_{5,i}),j},P_{(S_{6,i}),j},P_{(S_{7,i}),j}\big{)}.$

We define a $1260\times 1$ matrix $P_{sign}$ by

$(P_{sign})_{i,1}=sign(S_{1,i},...,S_{7,i})\cdot
sign(S_{1,i},S_{2,i},S_{3,i}).$

We apply the input values $P_{y}=P_{y360}$, $P_{z}=P_{z360}$ and
$P_{sign}=P_{sign}$ to Algorithm 3 and denote the corresponding output by
$O_{360}$. Then we get all $360\times 360$ possible values (possibly repeated)
$O_{360}$ of $\Theta$.

Algorithm 1 possible values of $\Theta(\theta_{0},...,\theta_{6})$.

0: Matrices $P_{y}$, $P_{z}$ and $P_{sign}$;

0: Matrix $O$;

1: Set p and q to be the number of columns of $P_{y}$ and $P_{z}$
respectively;

2: Set $O$ to be a $p\times q$ zero matrix;

3: for each $i=1,...,p$ do

4: for each $j=1,...,q$ do

5: for each $l=1,...,1260$ do

6:
$(O)_{i,j}=(O)_{i,j}+\dfrac{1}{1260}(P_{sign})_{l,1}\cdot(P_{y360})_{l,i}\cdot(P_{z360})_{l,j}$;

7: end for

8: end for

9: end for

Hence in this case $|\Theta(\theta_{0},...,\theta_{6})|\leq\dfrac{11}{45}$.
For $(\theta_{0},...,\theta_{6})$ defined as in Figure 1, $\Theta$ attains its
maximal volume $\dfrac{11}{45}$.

$[x_{0}]$$[x_{1}]$$[x_{2}]$$[x_{3}]$$[x_{4}]$$[x_{5}]$$[x_{6}]$$[y_{0}]$$[y_{1}]$$[y_{2}]$$[y_{3}]$$[y_{4}]$$[y_{5}]$$[y_{6}]$$[z_{0}]$$[z_{1}]$$[z_{2}]$$[z_{3}]$$[z_{4}]$$[z_{5}]$$[z_{6}]$
Figure 1.

∎

######  Lemma 7.

If $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ has exactly $3$ distinct points in
$\mathbb{S}^{1}$,

$|\Theta(\theta_{0},...,\theta_{6})|<\dfrac{11}{45}.$

###### Proof.

By (5), we have

$\begin{split}\Theta(\theta_{0},...,\theta_{6})&=\dfrac{1}{7!}\sum_{\sigma\in
Sym(7)}sign(\sigma)Or\big{(}[x_{\sigma(0)}],[x_{\sigma(1)}],[x_{\sigma(2)}]\big{)}\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{\sigma(0)}],[y_{\sigma(3)}],[y_{\sigma(4)}]\big{)}Or\big{(}[z_{\sigma(4)}],[z_{\sigma(5)}],[z_{\sigma(6)}]\big{)}\\\
&=\dfrac{1}{7!}\sum_{\begin{subarray}{c}a<b<c\\\
\\{a,b,c\\}\in\\{0,...,6\\}\end{subarray}}2\cdot
Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\\{i,j,k,l\\}=\\{0,...,6\\}\backslash\\{a,b,c\\}\\\
k<l\end{subarray}}\\\ &\ \ \ \ 2\cdot
sign(a,b,c,i,j,k,l)\Big{[}Or\big{(}[y_{a}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}+\\\
&\ \ \ \
Or\big{(}[y_{b}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}+\\\
&\ \ \ \
Or\big{(}[y_{c}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}\Big{]}.\end{split}$

######  Claim 1.

For three fixed element $a,b,c\in\\{0,...,6\\}$ satisfying $a<b<c$ there are
at least $\dfrac{1}{3}$ of the items in

$\begin{split}&Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\\{i,j,k,l\\}=\\{0,...,6\\}\backslash\\{a,b,c\\}\\\
k<l\end{subarray}}2\cdot sign(a,b,c,i,j,k,l)\\\ &\ \ \ \ \ \ \ \ \ \
\Big{[}Or\big{(}[y_{a}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}+Or\big{(}[y_{b}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}+\\\
&\ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{c}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}\Big{]}\end{split}$

that vanish or cancel with each other.

######  Proof of claim 1.

Fix $\\{a,b,c\\}$ in $\\{0,...,6\\}$, and denote
$\\{0,...,6\\}\backslash\\{a,b,c\\}$ by $\\{i_{0},...,i_{3}\\}$. We can assume
that $z_{0}\leq z_{1}\leq...\leq z_{6}$ as $\Theta$ is alternating. Assuming
$i_{0}<...<i_{3}$, we have $z_{i_{0}}\leq...\leq z_{i_{3}}$.

If $\theta_{i_{0}}^{3}$, $\theta_{i_{1}}^{3}$, $\theta_{i_{2}}^{3}$ and
$\theta_{i_{3}}^{3}$ are four pairwise distinct points in $\mathbb{S}^{1}$, we
have

$\begin{split}&\ \ \
Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\\{i,j,k,l\\}=\\{0,...,6\\}\backslash\\{a,b,c\\}\\\
k<l\end{subarray}}2\cdot sign(a,b,c,i,j,k,l)\\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{a}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}\\\
&=Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\tau\in
Sym(i_{0},...,i_{3})\\\ \tau(i_{2})<\tau(i_{3})\end{subarray}}2\cdot
sign(a,b,c,\tau)sign(\tau(i_{1}),\tau(i_{2}),\tau(i_{3}))\\\ &\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{a}],[y_{\tau(i_{0})}],[y_{\tau(i_{1})}]\big{)}\\\
&=Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\tau\in
Sym(i_{0},...,i_{3})\\\
\tau(i_{2})<\tau(i_{3})\end{subarray}}2\cdot(-1)^{a+b+c+\alpha(i_{0})}Or\big{(}[y_{a}],[y_{\tau(i_{0})}],[y_{\tau(i_{1})}]\big{)},\end{split}$
(7)

where for $\sigma(i_{0})=i_{p}$, $\alpha\big{(}\sigma(i_{0})\big{)}=p$. Notice
that the items corresponding to $\sigma=(a,b,c,i_{3},i_{1},i_{0},i_{2})$
cancels the items corresponding to $\sigma=(a,b,c,i_{1},i_{3},i_{0},i_{2})$
and the items corresponding to $\sigma=(a,b,c,i_{2},i_{0},i_{1},i_{3})$
cancels the items corresponding to $\sigma=(a,b,c,i_{0},i_{2},i_{1},i_{3})$.
Therefore at least $\dfrac{1}{3}$ of the items in (7) cancel with each other.

If $\theta_{i_{0}}^{3}$, $\theta_{i_{1}}^{3}$, $\theta_{i_{2}}^{3}$ and
$\theta_{i_{3}}^{3}$ have at least two points coincide with each other, we
have $\dfrac{\binom{1}{2}}{\binom{3}{4}}=\dfrac{1}{2}$ of the items in (7)
vanish. The same goes for

$\begin{split}&\ \ \
Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\\{i,j,k,l\\}=\\{0,...,6\\}\backslash\\{a,b,c\\}\\\
k<l\end{subarray}}2\cdot sign(a,b,c,i,j,k,l)\\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{b}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}\end{split}$

and

$\begin{split}&\ \ \
Or\big{(}[x_{a}],[x_{b}],[x_{c}]\big{)}\sum_{\begin{subarray}{c}\\{i,j,k,l\\}=\\{0,...,6\\}\backslash\\{a,b,c\\}\\\
k<l\end{subarray}}2\cdot sign(a,b,c,i,j,k,l)\\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
Or\big{(}[y_{c}],[y_{i}],[y_{j}]\big{)}Or\big{(}[z_{j}],[z_{k}],[z_{l}]\big{)}.\end{split}$

Hence we get the required result.

Now that we have these facts, we can prove this lemma in 4 cases.

######  Case 1.

There are $1$, $1$ and $5$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ that
respectively have the same values in $\mathbb{S}^{1}$.

We notice that $Or\big{(}[x_{0}],[x_{1}],[x_{2}]\big{)}=0$ when there exists
$i\neq j$ such that $[x_{i}]=[x_{j}]$. Hence there are only
$\dfrac{\binom{1}{5}}{\binom{3}{7}}=\dfrac{1}{7}<\dfrac{11}{45}$ of the items
in (4) which may not vanish. Therefore this case stands.

######  Case 2.

There are $1$, $2$ and $4$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ that
respectively have the same values in $\mathbb{S}^{1}$.

Similarly to Case 1, there are only
$\dfrac{\binom{1}{2}\cdot\binom{1}{4}}{\binom{3}{7}}=\dfrac{8}{35}<\dfrac{11}{45}$
of the items in (4) which may not vanish. Therefore this case stands.

######  Case 3.

There are $1$, $3$ and $3$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ that
respectively have the same values in $\mathbb{S}^{1}$.

According to Claim 1, there are only
$\dfrac{\binom{1}{3}\cdot\binom{1}{3}}{\binom{3}{7}}\times\dfrac{2}{3}=\dfrac{6}{35}<\dfrac{11}{45}$
of the items in (4) which may not vanish or cancel with each other. Therefore
this case stands.

######  Case 4.

There are $2$, $2$ and $3$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ that
respectively have the same values in $\mathbb{S}^{1}$.

Similarly to Case 3, there are only
$\dfrac{\binom{1}{2}\cdot\binom{1}{2}\cdot\binom{1}{3}}{\binom{3}{7}}\times\dfrac{2}{3}=\dfrac{8}{35}<\dfrac{11}{45}$
of the items in (4) which may not vanish or cancel with each other. Therefore
this case stands.

In conclusion, this lemma is proved.

∎

######  Lemma 8.

If $\\{\theta_{j}^{1}\\}_{j=0}^{6}$ has at least $4$ distinct points in
$\mathbb{S}^{1}$,

$|\Theta(\theta_{0},...,\theta_{6})|<\dfrac{11}{45}.$

###### Proof.

We prove this lemma in $3$ cases.

######  Case 5.

There are $1$, $1$, $1$ and $4$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$
that respectively have the same values in $\mathbb{S}^{1}$.

When there are $2$ points in $\\{\theta_{j}^{2}\\}_{j=0}^{6}$ that have the
same value in $\mathbb{S}^{1}$, assuming that
$x_{0}=...=x_{3}<x_{4}<x_{5}<x_{6}<1$, the worst case is $y_{4}=y_{5}$. Taking
a closer look at the proof of Claim 1, we have that there are at most

$\dfrac{(1+4)\times\frac{2}{3}+(2+4)\times\frac{2}{3}\times\frac{8+2}{12}}{\binom{3}{7}}=\dfrac{2}{9}<\dfrac{11}{45}$

of the items in (4) that may not vanish or cancel with each other. Therefore
we can assume that $\\{\theta_{j}^{i}\\}_{j=0}^{6}$ are pairwise distinct for
$i=2$ and $3$, respectively. We use Algorithm 3 to prove this. Let
$(x_{1},...,x_{7})$ be $(1,1,1,1,2,3,4)$. Define a $1260\times 1$ matrix
$P_{sign1114}$ by

$(P_{sign1114})_{i,1}=sign(S_{1,i},...,S_{7,i})\cdot
sign(x_{\scriptscriptstyle S_{1,i}},x_{\scriptscriptstyle
S_{2,i}},x_{\scriptscriptstyle S_{3,i}}).$

Here we input $P_{y}=P_{y360}$, $P_{z}=P_{z360}$ and $P_{sign}=P_{sign1114}$.
Denote the corresponding output by $O$. The maximal value of entries of $O$ is
$\dfrac{2}{15}$ which is smaller than $\dfrac{11}{45}$. Therefore this case
stands.

######  Case 6.

There are $1$, $1$, $2$ and $3$ points in $\\{\theta_{j}^{1}\\}_{j=0}^{6}$
that respectively have the same values in $\mathbb{S}^{1}$.

Let $P_{2}$ be a $7\times\dfrac{7!}{2!}$ (i.e., $7\times 2520$) matrix which
each column represents a permutation of $(1,1,2,3,4,5,6)$. Let $P_{3}$ be a
$7\times\dfrac{7!}{2!\times 2!}$ (i.e., $7\times 1260$) matrix which each
column represents a permutation of $(1,1,2,2,3,4,5)$. Let $P_{4}$ be a
$7\times\dfrac{7!}{2!\times 2!}$ (i.e., $7\times 1260$) matrix which each
column represents a permutation of $(1,1,2,3,3,4,5)$. Let $P_{5}$ be a
$7\times\dfrac{7!}{3!}$ (i.e., $7\times 840$) matrix which each column
represents a permutation of $(1,1,1,2,3,4,5)$. Let $P_{6}$ be a
$7\times\dfrac{7!}{2!\times 2!\times 2!}$ (i.e., $7\times 630$) matrix which
each column represents a permutation of $(1,1,2,2,3,3,4)$. Let $P_{7}$ be a
$7\times\dfrac{7!}{3!\times 2!}$ (i.e., $7\times 420$) matrix which each
column represents a permutation of $(1,1,1,2,2,3,4)$. Let $P_{8}$ be a
$7\times\dfrac{7!}{2!\times 2!}$ (i.e., $7\times 420$) matrix which each
column represents a permutation of $(1,1,1,2,3,3,4)$. Define a $7\times 7710$
matrix $P_{r}$ by stacking $P$, $P_{2}$,…,$P_{7}$ and $P_{8}$ vertically. Let
$P_{yr}$ be

$(P_{yr})_{k,l}=sign\big{(}(P_{r})_{(S_{1,k}),l},(P_{r})_{(S_{4,k}),l},(P_{r})_{(S_{5,k}),l}\big{)}.$

Let $P_{zr}$ be

$(P_{zr})_{k,l}=sign\big{(}(P_{r})_{(S_{5,k}),l},(P_{r})_{(S_{6,k}),l},(P_{r})_{(S_{7,k}),l}\big{)}.$

Let $(x_{1},...,x_{7})$ be $(1,1,1,2,2,3,4)$. Define a $1260\times 1$ matrix
$P_{sign1123}$ by

$(P_{sign1123})_{i,1}=sign(S_{1,i},...,S_{7,i})\cdot
sign(x_{S_{1,i}},x_{S_{2,i}},x_{S_{3,i}}).$

Input $P_{y}=P_{yr}$, $P_{z}=P_{zr}$ and $P_{sign}=P_{sign1123}$ to get the
corresponding output $O1123$. Define $(x_{1},...,x_{7})=(1,1,1,2,3,3,4)$.
Define a $1260\times 1$ matrix $P_{sign1213}$ by

$(P_{sign1213})_{i,1}=sign(S_{1,i},...,S_{7,i})\cdot
sign(x_{S_{1,i}},x_{S_{2,i}},x_{S_{3,i}}).$

Input $P_{y}=P_{yr}$, $P_{z}=P_{zr}$ and $P_{sign}=P_{sign1213}$ to get the
corresponding output $O1213$. Notice that the maximal values of entries of
$O1123$ and $O1213$ are both $\dfrac{7}{45}$. Notice that

$\begin{split}\Theta\Big{(}\big{(}[x_{0}],[y_{0}],[z_{0}]\big{)},...,\big{(}[x_{6}],[y_{6}],[z_{6}]\big{)}\Big{)}&=\Theta\Big{(}\big{(}[y_{0}],[x_{0}],[z_{0}]\big{)},...,\big{(}[y_{6}],[x_{6}],[z_{6}]\big{)}\Big{)}\\\
&=\Theta\Big{(}\big{(}[x_{0}],[z_{0}],[y_{0}]\big{)},...,\big{(}[x_{6}],[z_{6}],[y_{6}]\big{)}\Big{)}.\end{split}$
(8)

Therefore this case stands.

######  Case 7.

All other cases which are not include above.

Using equation (6) and (8), we only need to apply Algorithm 3 as below. Let
$(x^{i}_{1},...,x^{i}_{7})$ be $(1,1,2,2,3,3,4)$, $(1,1,1,2,3,4,5)$,
$(1,1,2,2,3,4,5)$ and $(1,1,2,3,4,5,6)$ respectively for $i=1,...,4$. Define
four $1260\times 1$ matrices $\\{P_{sign}^{i}\\}_{i=1}^{4}$ by

$(P_{sign}^{i})_{j,1}=sign(S_{1,j},...,S_{7,j})\cdot
sign\big{(}x^{i}_{S_{1,j}},x^{i}_{S_{2,j}},x^{i}_{S_{3,j}}\big{)}$

respectively for $i=1,...,4$. Input $P_{y}=P_{yr}$, $P_{z}=P_{zr}$ and
$P_{sign}=P_{sign}^{i}$ to get the outputs $O^{i}=O$ respectively for
$i=1,...,4$. The maximal values of entries of $O^{i}$ are $\dfrac{8}{45}$,
$\dfrac{8}{45}$, $\dfrac{1}{5}$ and $\dfrac{2}{9}$ for $i=1,...,4$,
respectively. Therefore this case stands. ∎

Consequently, Proposition 5 is proved. Therefore, we get the desired
estimation
$\|\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}}\|_{\infty}\leq\dfrac{11\pi^{3}}{45}$.
As a conclusion, Theorem 3 is proved. Furthermore, Algorithm 3 can easily be
generalized to provide an algorithm to compute the lower bound of closed
Riemannian manifolds covered by $\big{(}\mathbb{H}^{2}\big{)}^{n}$.

##  4\. Conjectures

Denote the $4$-cocycle representing
$\omega_{\mathbb{H}^{2}\times\mathbb{H}^{2}}$ in [2, Proposition 7] by
$\Theta^{\prime}$. Notice that when taking value
$(\theta_{0},...,\theta_{4})=\Big{(}\big{(}[x_{0}],[y_{0}]\big{)},...,\big{(}[x_{4}],[y_{4}]\big{)}\Big{)}$
as in Figure 2, $\Theta^{\prime}$ attains its maximal volume $\dfrac{2}{3}$.

$[x_{0}]$$[x_{1}]$$[x_{2}]$$[x_{3}]$$[x_{4}]$$[y_{0}]$$[y_{1}]$$[y_{2}]$$[y_{3}]$$[y_{4}]$
Figure 2.

We notice a pattern comparing Figure 1 with 2 as following.

######  Conjecture 9.

Define a $2n$-form $\Theta_{n}$ in $C_{b}^{2n}(\mathbb{T}^{n},\mathbb{R})$ as

$\Theta_{n}(\theta_{0},...,\theta_{2n})=\dfrac{\pi^{n}}{(2n+1)!}\sum_{\sigma\in
Sym(2n+1)}sign(\sigma)\prod_{i=1}^{n}Or\big{(}\theta_{2i-2}^{i},\theta_{2i-1}^{i},\theta_{2i}^{i}\big{)}$

for all $\theta_{i}=(\theta_{i}^{1},...,\theta_{i}^{n})$ in $\mathbb{T}^{n}$,
$i=0,...,2n$. Then
$\|\Theta_{n}\|_{\infty}=\Theta_{n}(\theta_{0},...,\theta_{2n})$, where
$\theta^{k}_{i}=e^{\frac{2(ki)\pi\sqrt{-1}}{2n+1}}$ for $k=1,...,n$ and
$i=0,...,2n$.

We already have
$\|\omega_{(\mathbb{H}^{2})^{n}}\|_{\infty}\leq\|\Theta_{n}\|_{\infty}$ and
the proportionality principle for locally symmetric spaces. We can ask further
that whether this inequality is actually equality.

######  Conjecture 10.

Let $M$ be a closed oriented manifold covered by
$\big{(}\mathbb{H}^{2}\big{)}^{n}$. Then the simplicial volume
$\|M\|=\dfrac{\mathrm{Vol}(M)}{\|\Theta_{n}\|_{\infty}}$.

If these two conjecture are true, we can get the exact value of simplicial
volume of all closed oriented manifolds covered by
$\big{(}\mathbb{H}^{2}\big{)}^{n}$.

## References

  * [1] M. Bucher-Karlsson: The proportionality constant for the simplicial volume of locally symmetric spaces. Colloq. Math. 111, 183–198 (2008)
  * [2] M. Bucher-Karlsson: The simplicial volume of closed manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}$. J. Topol. 1(3), 584-–602 (2008)
  * [3] J. L. Dupont: Simplicial de Rham cohomology and characteristic classes of flat bundles. Topology 15(3), 233–245 (1976)
  * [4] M. Gromov: Volume and bounded cohomology. Publ. Math. Inst. Hautes Études Sci. 56, 5–99 (1982)
  * [5] C. Löh, R. Sauer: Simplicial volume of Hilbert modular varieties. Comment. Math. Helv. 84(3), 457–470 (2009)
  * [6] N. Monod: Continuous bounded cohomology of locally compact groups. Lecture Notes in Math. 1758, Springer-Verlag, New York (2001)
  * [7] W. Thurston: The geometry and topology of 3-manifolds. Lecture notes, Princeton (1978)


# Integrating Elastic Bands to Enhance Performance for Textile Robotics

Cem Suulker1, Sophie Skach1, and Kaspar Althoefer1 1All authors are with the
Centre for Advanced Robotics, at the School of Engineering and Materials
Science, Queen Mary University of London, United Kingdom<EMAIL_ADDRESS>
##  I Introduction

The field of soft robotics is growing rapidly, and innovative elastic
materials are replacing heavy metallic links while soft inflatable actuators
are taking the place of electromechanical rotational motors. The use of
flexible textile materials in human-robot interaction has also been shown to
offer attractive design options due to their safe nature [1, 2].

The creation of soft robots and actuators often involves the use of various
materials and methods from the clothing industry. One crucial variable is the
stretch quality of the fabric material. Knitted fabrics are commonly used for
their stretchiness, but they often stretch in all directions, which is usually
not desirable. Woven fabrics, are normally known as non-stretchy. But
integrating elastane yarn into the weft, makes them more stretch in one
direction than the other. This quality makes them more suitable for creating
actuators. They are also more durable than knitted ones. While using coating
can make these structures airtight, it also eliminates the material’s ability
to stretch.

Another approach to creating stretch in soft robotic structures involves
increasing the material density using methods such as pleats and ruffles. The
use of pleats in soft robotics has been extensively researched and is
considered an effective method [1]. However, the use of ruffles has not
received as much attention. Ruffles are often used in the neck area of
clothing to gather the fabric material in a dense way. From a soft robotics
perspective, this gathered dense material can be inflated to create high
elongation.

Various types of elastic bands can be used to create ruffles, but in this
extended abstract, we will focus on braided elastic bands. These bands are
used for storing energy in soft robotics. However, their full potential is
realized when they are integrated into the fabric structure using the ruffles
method. In this extended abstract, we will showcase two soft robotic
applications that utilize elastic bands to enhance the system’s performance.

Figure 1: Before and after the integration of elastic bands to an actuator.

##  II Application: Bending Actuator for Wearables

To create bending of an inflatable textile actuator, an imbalance between two
layers of fabric must be achieved. This imbalance is created by using pleating
techniques [1], excess material of one layer is folded and stitched onto the
other, and unfolds when inflated. This approach, however, has potential
drawbacks, e.g. when the space between the layers is too small for the pleats
to unfold (which, for finger sized designs, can become imminent). Another
technique to create such imbalance is the use of different types of fabric
with varying elasticity. This way, one layer stretches more when actuated and
the structure will bend towards the less elastic one. Integrating braided
elastic band to the stretch fabric option with the ruffles technique is proven
more efficient in terms of blocking force and bending angle capabilities [3].

###  II-1 Materials

To create this textile actuator the two different textile properties layers
are selected. Bottom: a plain cotton weave for the (light fabric in Fig. 1).
Top: a cotton mix with elastane yarn integrated to the weft of the fabric
(dark fabric in Fig. 1).

To enhance a fabric’s stretch behavior and create the desired material
imbalance between layers, an additional support material is used that is
integrated when assembling the actuators: a braided elastic band, also called
elastics, see Figure 1, commonly used in clothing to create ruffles or elastic
waistbands. It consists of braided polyester and a small part of thin rubber,
making it durable and extremely stretchy.

###  II-2 Fabrication

First, the mono-directional stretch black fabric was cut 80% longer than the
cotton bottom layer fabric. The elastic band was integrated on the side seam
between the top and bottom layer, first stitched onto the top layer while
being stretched, and then sewn onto the bottom layer in a relaxed state. This
enabled the top layer to reach an excess length of 180% of the bottom layer.
The actuator is equipped with a latex bladder to ensure air tightness.

Figure 2: Blocking force output and flexion angle versus pressure graphs for
stretch fabric actuator, and elastic band integrated actuator. Integration of
the elastic band significantly boosts the performance.

###  II-3 Results

Two important parameters for soft bending actuators for wearables are flexion
angle and blocking force [4] In example, for rehabilitation or assistive hand
exoskeletons it is imperative that each finger is unrestricted in relation to
its maximum angle, and actuators should apply 10-15 N blocking force to the
fingers[5].

In Fig. 2 the elastic band integration boosts both of these critical
parameters for the actuator. The force capability increases approximately from
10 N to 25 N, and maximum bending angle increases from approximately 180
degrees to 360 degrees.

##  III Application: Soft Cap for Eversion Robots

Growing robots based on the eversion principle are known for their ability to
extend from within rapidly, along their longitudinal axis, and in doing so,
reach deep into hard-to-access, remote spaces. Because of their unique
movement principle maintaining a payload at the tip is a major challenge.
Various tip mechanisms have been proposed, including complex, rigid designs
that may not be compatible with functional hardware. To address these
shortcomings, we proposed a soft, entirely fabric-based cylindrical cap that
can be easily slipped onto the tip of eversion robots [6].

We created a series of caps of different sizes and materials and conducted an
experimental study to evaluate their effectiveness in maintaining their
position and transporting payloads, such as a camera, across long distances.
We also assessed the caps’ ability to navigate through narrow openings and
found that our soft, flexible cap does not significantly hinder the robot’s
flexibility or overall maneuverability. Our design offers a solution to the
challenge of maintaining sensory payloads at the tip of eversion robots
without compromising their performance or flexibility.

Figure 3: a) Pattern for stretch fabric cap b) Pattern for elastic band
integrated cap.

###  III-1 Materials & Fabrication

To keep the cap attached to the tip of the eversion body, the main concept is
based on squeezing the cap against the body. We have devised two different
designs to achieve this squeezing motion: one using a stretch fabric and the
other utilizing ruffles with elastic bands. Please see Fig. 3 for more
information on the patterns of the caps.

###  III-2 Results

In order to assess the performance of the caps, we subjected them to a series
of challenges that evaluated their ability to adapt to varying layer
thicknesses, protruding objects from the robot body, navigability, and
squeezability. The performance of the caps was measured using a percentage
index that took into account the number of challenges that the caps
successfully completed.

The stretch fabric caps achieved a performance rating of 84-86%, while the
elastic band integrated caps achieved a rating of 90-96% (Fig. 4). This
improvement can be attributed to the smaller contact area between the two
fabrics, with the stretch fabric caps having a contact area of 15 cm and the
elastic band integrated caps having a contact area of only 1 cm.

Figure 4: The eversion robot, with a stretchy cap, with an elastic band
integrated cap. Integration of elastic band boosts the performance of the
robot.

##  IV Conclusions

The elastic bands integrated using the ruffles technique proved to be
effective in enhancing the performance of the soft robotic structures. In the
actuator application, the elastic bands greatly increased the bending
capability and force capability of the structure, while in the eversion robot
cap application, the elastic bands improved the performance slightly by
maintaining the sensory payload at the tip without restricting the eversion
process. These findings demonstrate the potential of using elastic bands and
textile techniques in soft robotics to create more efficient and adaptable
structures.

## References

  * [1] L. Cappello, J. T. Meyer, K. C. Galloway, J. D. Peisner, R. Granberry, D. A. Wagner, S. Engelhardt, S. Paganoni, and C. J. Walsh, “Assisting hand function after spinal cord injury with a fabric-based soft robotic glove,” _Journal of neuroengineering and rehabilitation_ , vol. 15, no. 1, pp. 1–10, 2018.
  * [2] C. Suulker, A. Hassan, S. Skach, and K. Althoefer, “A comparison of silicone and fabric inflatable actuators for soft hand exoskeletons,” in _2022 IEEE 5th International Conference on Soft Robotics (RoboSoft)_. IEEE, 2022, pp. 735–740.
  * [3] C. Suulker, S. Skach, and K. Althoefer, “Soft robotic fabric actuator with elastic bands for high force and bending performance in hand exoskeletons,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 4, pp. 10 621–10 627, 2022.
  * [4] C. Suulker, S. Skach, and K. Althoefer, “A fabric soft robotic exoskeleton with novel elastic band integrated actuators for hand rehabilitation,” _arXiv preprint arXiv:2212.07206_ , 2022.
  * [5] C. D. Takahashi, L. Der-Yeghiaian, V. Le, R. R. Motiwala, and S. C. Cramer, “Robot-based hand motor therapy after stroke,” _Brain_ , vol. 131, no. 2, pp. 425–437, 2008.
  * [6] C. Suulker, S. Skach, D. Kaleel, T. Abrar, Z. Murtaza, D. Suulker, and K. Althoefer, “Soft cap for eversion robots,” _arXiv preprint arXiv:2301.12862_ , 2023.