Asymptotic bound on slow-roll parameter in stringy quintessence model

Min-Seok Seoa

aDepartment of Physics Education, Korea National University of Education,
Cheongju 28173, Republic of Korea

We study the late time behavior of the scalar part of the volume modulus and
the dilaton in stringy quintessence model, focusing on their contributions to
the Hubble slow-roll parameter $\epsilon$ which directly measures the
deviation of the spacetime geometry from de Sitter space. When only one of the
moduli is allowed to move, $\epsilon$ converges to the stable fixed point at
late time. The fixed point value is larger than $1$, thus the slow-roll cannot
be realized. Moreover, if the decay rate of the quintessence potential is
larger than some critical value, the positivity of the potential imposes that
the stable fixed point value is just given by $3$, independent of the details
of the moduli dynamics. Otherwise, the fixed point value coincides with the
potential slow-roll parameter. When both the volume modulus and the dilaton
roll down the potential simultaneously, we can find the relation between the
contributions of two moduli to $\epsilon$ satisfied at the fixed point. In
this case, the fixed point value is not in general the simple sum of fixed
point values in the single field case and cannot be larger than $3$.

##  1 Introduction

Construction of the model for the universe compatible with the cosmological
observations [2] has been one of challenges in string phenomenology. In
particular, whereas models like the KKLT [3] or the large volume scenario [4]
have been proposed to realize the metastable de Sitter (dS) vacuum which well
describes the observed almost constant vacuum energy density, the suspicion
has been raised that some unknown corrections may invalidate these models. It
comes from the fact that in string theory, the parametric control is achieved
in the asymptotic limits of the moduli space where the potential is dominated
by a few runaway terms, but the metastable dS vacuum requires that these terms
are significantly corrected [5]. Motivated by this, the ‘dS swampland
conjecture’ was proposed, which states that string theory does not admit dS
vacua in any parametrically controlled regime of the moduli space [6, 7] (see,
[8, 9, 10, 11, 12] for the refinement).

If the conjecture is true, the accelerated expansion of the universe driven by
an almost constant vacuum energy density may be explained by so-called the
‘quintessence model’, where the scalar field slowly rolls down the runaway
potential [13, 14, 15]. This possibility stimulated extensive studies on the
quintessence in the context of the string model building [16, 17, 18, 19] (see
also [20, 21, 22, 23] for earlier discussions), in particular focusing on the
behavior of the scalar fields in the asymptotic region of the moduli space
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. The remarkable claim made in recent
works is that stringy quintessence models in the asymptotic region suffer from
the no-go theorem : there is a lower bound of order one on the potential slow-
roll parameter which measures the slope of the potential in units of the
Hubble scale [24, 27]. This is linked to the fact that for the string length
and loop corrections to be suppressed, both the scalar part of the volume
modulus (denoted by $\sigma$) and the dilaton (denoted by $s$) take the large
values. In this case, the Kähler potential strongly restricts the slope of the
potential in the direction of $\sigma$ and $s$, each of which is too steep to
realize the slow-roll (see also [34, 35, 36, 37, 38] for earlier discussions
based on Type II string theory). In particular, it turns out that the leading
no-scale structure of the potential is salient to obtain an order one
contribution of $\sigma$ to the potential slow-roll parameter.

On the other hand, the time variation of the vacuum energy density can be more
directly measured by the Hubble slow-roll parameter defined by the rate of
change of the Hubble parameter $H$,

$\begin{split}\epsilon=-\frac{\dot{H}}{H^{2}}.\end{split}$ (1)

Since $1/H$ is interpreted as the horizon radius, it is clear that $\epsilon$
measures the deviation of the spacetime geometry from dS space. In this
regard, we claim that $\epsilon$ rather than the potential slow-roll parameter
is more appropriate parameter to describe the dS swampland conjecture, which
essentially deals with the instability of the dS geometry. 111We note that in
some models like the DBI inflationary mechanism, the effects of the non-
negligible higher-derivative terms can allow the sizeable potential slow-roll
parameter while the value of $\epsilon$ is kept smaller than $1$ [39, 40]. In
the slow-roll approximation where the condition

$\begin{split}1\gg\frac{\dot{u}}{Hu}\gg\frac{\ddot{u}}{H^{2}u}\end{split}$ (2)

is satisfied for any scalar field $u$, $\epsilon$ can be well approximated by
the potential slow-roll parameter. However, the bound on the potential slow-
roll parameter claimed by the no-go theorem is of order one, which indicates
that the slow-roll approximation does not hold, thus the potential slow-roll
parameter is not necessarily identified with $\epsilon$. Then it is natural to
ask if there is a bound on $\epsilon$ provided by the dynamics of $\sigma$ and
$s$ in the asymptotic region, and if so, what is the value of the bound. In
this article, we address this question by considering the specific model, the
compactification of Type IIB string theory.

To proceed, we use the fact that when the quantum corrections such as the non-
perturbative effects are negligible and the fluxes are turned off, the
superpotential is independent of both $\sigma$ and $s$. Then the F-term
potential $V$ satisfies $u_{i}\partial_{u_{i}}V=-\beta_{i}V$
($u_{i}=\sigma,s$) for some positive number $\beta_{i}$, which indeed is a
typical feature of the runaway potential in the quintessence model. If (for
some reason) the sizeable but controllable quantum corrections are allowed or
fluxes are turned on, one of $\sigma$ and $s$ can be stabilized, while another
still rolls down the potential. 222Of course, if both $\sigma$ and $s$ are
stabilized, they never contribute to $\epsilon$. Moreover, there is no
contribution of $\sigma$ to $\epsilon$ in the non-geometric compactifications
that have no Kähler moduli [28, 32].  We first consider this simple case in
Sec. 3, based on the generic properties of the single field quintessence model
summarized in Sec. 2. As we will see, the value of $\epsilon$ in this case
converges to some stable fixed point at late time. Comparing with the bound on
the potential slow-roll parameter in the no-go theorem, the fixed point value
is similar in size, but for $\sigma$, quite different in nature. More
concretely, when we redefine the field for the canonical kinetic term, the
potential decreases exponentially with respect to the redefined field. If the
decay rate of this potential is larger than some value, the positivity of the
potential imposes that the fixed point value is fixed to $3$, independent of
the decay rate. In our model, $\sigma$ corresponds to this case. This implies
that unlike the bound on the potential slow-roll parameter in the no-go
theorem, the fixed point value $3$ is not the result of the no-scale
structure. Meanwhile, when any quantum corrections are negligibly small and
the fluxes are turned off, both $\sigma$ and $s$ are allowed to roll down the
potential, which is visited in Sec. 4. In this case, the fixed point is
defined by the relation between contributions of $\sigma$ and $s$ to
$\epsilon$. Still, by the positivity of the potential, the fixed point value
of $\epsilon$ cannot be larger than $3$.

We also note that throughout the discussion, the 4-dimensional (reduced)
Planck scale $M_{\rm Pl}$ is fixed to the observed value $2.4\times
10^{18}$GeV : we do not consider the case in which the more fundamental string
scale is fixed and $M_{\rm Pl}$ induced from it is allowed to vary. More
precisely, $M_{\rm Pl}$ is determined by the string scale and the internal
volume ${\cal V}_{0}\ell_{s}^{6}$ through

$\begin{split}M_{\rm Pl}^{2}=\frac{{\cal
V}_{0}\ell_{s}^{6}}{\kappa_{10}^{2}},\end{split}$ (3)

where $\ell_{s}$ is the string length scale and
$\kappa_{10}^{2}=g_{s}^{2}\ell_{s}^{8}/(4\pi)$ is the gravitational coupling
in 10-dimensional supergravity ($g_{s}$ : the string coupling constant). This
also reads

$\begin{split}m_{s}=\frac{g_{s}}{\sqrt{4\pi{\cal V}_{0}}}M_{\rm
Pl}.\end{split}$ (4)

In Type IIB string theory, ${\cal V}_{0}$ and $g_{s}$ are given by
$\langle\sigma^{3/2}\rangle$ and $\langle s^{-1}\rangle$, respectively. This
shows that in the limit $s\to\infty$ ($\sigma\to\infty$), $m_{s}$ ($m_{s}$ as
well as the Kaluza-Klein mass scale) giving the fixed value of $M_{\rm Pl}$
becomes extremely low, as claimed by the distance conjecture [41].

##  2 Fixed point in single field quintessence model

We first consider the simplest quintessence model in which the single scalar
field $u$ rolls down the positive runaway potential. Assuming the spatial
homogeneity and isotropy, the metric can be written as

$\begin{split}ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}\end{split}$ (5)

and $u$ depends only on $t$, then the action is given in the form of

$\begin{split}S_{\rm quint}=\int d^{4}xa^{3}\Big{(}\frac{M_{\rm
Pl}^{2}}{2\alpha^{2}}\frac{\dot{u}^{2}}{u^{2}}-\frac{V_{0}}{u^{\beta}}\Big{)}.\end{split}$
(6)

We note that in terms of the canonically normalized field
$\varphi=\frac{M_{\rm Pl}}{\alpha}\log u$ the action can be rewritten as

$\begin{split}S_{\rm quint}=\int
d^{4}xa^{3}\Big{(}\frac{1}{2}{\dot{\varphi}}^{2}-V_{0}e^{-\alpha\beta\frac{\varphi}{M_{\rm
pl}}}\Big{)},\end{split}$ (7)

which shows that the potential decreases exponentially with respect to
$\varphi$ with the decay rate given by $\alpha\beta$. In the following, our
discussion will be made in terms of $u$ since (6) is a typical form of the
action for (the scalar part of) the modulus in the effective supergravity
description of string theory.

Taking the Einstein-Hilbert action into account in addition, we obtain
following equations of motion :

$\begin{split}&3M_{\rm Pl}^{2}H^{2}=\frac{M_{\rm
Pl}^{2}}{2\alpha^{2}}\frac{\dot{u}^{2}}{u^{2}}+\frac{V_{0}}{u^{\beta}},\\\
&3M_{\rm Pl}^{2}H^{2}+2M_{\rm Pl}^{2}\dot{H}=-\frac{M_{\rm
Pl}^{2}}{2\alpha^{2}}\frac{\dot{u}^{2}}{u^{2}}+\frac{V_{0}}{u^{\beta}},\\\
&\ddot{u}+3H\dot{u}-\frac{\dot{u}^{2}}{u}-\alpha^{2}\beta\frac{uV_{0}/u^{\beta}}{M_{\rm
Pl}^{2}}=0,\end{split}$ (8)

where $H=\dot{a}/a$ is the Hubble parameter. From the difference between the
first two equations, one finds that

$\begin{split}\epsilon=-\frac{\dot{H}}{H^{2}}=\frac{1}{2H^{2}M_{\rm
Pl}^{2}}\frac{M_{\rm
Pl}^{2}}{\alpha^{2}}\frac{\dot{u}^{2}}{u^{2}}=\frac{1}{2\alpha^{2}}\Big{(}\frac{\dot{u}}{Hu}\Big{)}^{2}.\end{split}$
(9)

Meanwhile, the first and the third equations can be rewritten as

$\begin{split}&1=\frac{1}{6\alpha^{2}}\Big{(}\frac{\dot{u}}{Hu}\Big{)}^{2}+\frac{V_{0}/u^{\beta}}{3H^{2}M_{\rm
Pl}^{2}},\\\
&\frac{\ddot{u}}{H^{2}u}+3\frac{\dot{u}}{Hu}-\Big{(}\frac{\dot{u}}{Hu}\Big{)}^{2}-\alpha^{2}\beta\frac{V_{0}/u^{\beta}}{H^{2}M_{\rm
Pl}^{2}}=0,\end{split}$ (10)

respectively, which shows that $\frac{\ddot{u}}{H^{2}u}$ and
$V=\frac{V_{0}}{u^{\beta}}$ can be written in terms of
$\sqrt{\epsilon}=\frac{1}{\sqrt{2}\alpha}\frac{\dot{u}}{Hu}$ :

$\begin{split}&\frac{V}{3H^{2}M_{\rm
Pl}^{2}}=\frac{V_{0}/u^{\beta}}{3H^{2}M_{\rm Pl}^{2}}=1-\frac{\epsilon}{3},\\\
&\frac{\ddot{u}}{H^{2}u}=3\alpha^{2}\beta-3\sqrt{2}\alpha\sqrt{\epsilon}+2\alpha^{2}\Big{(}1-\frac{\beta}{2}\Big{)}\epsilon.\end{split}$
(11)

Since $V\geq 0$, the first equation gives the bound $\epsilon\leq 3$. The
upper bound $\epsilon=3$ is saturated when $\frac{V}{3M_{\rm Pl}^{2}H^{2}}=0$.
If $u$ keeps rolling down the potential, one may naïvely expect that at late
time ($t\to\infty$), $V\to 0$, thus $\epsilon$ converges to $3$. However, as
we will see, this is not always the case.

Figure 1: The variation of the rate $\frac{d}{Hdt}\sqrt{\epsilon}$ as a
function of $\sqrt{\epsilon}$. Left : For $\alpha\beta>\sqrt{6}$, the
positivity of the potential imposes $\sqrt{\epsilon}<\sqrt{3}$, over which
$\frac{d}{Hdt}\sqrt{\epsilon}$ is always positive. Then regardless of the
initial condition, $\sqrt{\epsilon}$ increases in time and eventually
converges to $\sqrt{3}$. Right : For $\alpha\beta<\sqrt{6}$,
$\frac{d}{Hdt}\sqrt{\epsilon}$ is negative (positive) for
$\frac{\alpha\beta}{\sqrt{2}}<\sqrt{\epsilon}<\sqrt{3}$
($0<\sqrt{\epsilon}<\frac{\alpha\beta}{\sqrt{2}}$) in which case
$\sqrt{\epsilon}$ decreases (increases) in time until it reaches the stable
fixed point $\frac{\alpha\beta}{\sqrt{2}}$. The direction of the time
evolution of $\sqrt{\epsilon}$ is indicated by the arrow.

To see the time variation of $\epsilon$ in detail, consider the rate

$\begin{split}\frac{d}{Hdt}\sqrt{\epsilon}&=\frac{1}{\sqrt{2}\alpha}\frac{d}{Hdt}\Big{(}\frac{\dot{u}}{Hu}\Big{)}=\frac{1}{\sqrt{2}\alpha}\Big{[}\frac{\ddot{u}}{H^{2}u}-\Big{(}\frac{\dot{u}}{Hu}\Big{)}^{2}+\epsilon\frac{\dot{u}}{Hu}\Big{]}\\\
&=(\sqrt{\epsilon}+\sqrt{3})(\sqrt{\epsilon}-\sqrt{3})\Big{(}\sqrt{\epsilon}-\frac{\alpha\beta}{\sqrt{2}}\Big{)},\end{split}$
(12)

where (10) is used for the last equality. If $\alpha\beta>\sqrt{6}$,
$\frac{d}{Hdt}\sqrt{\epsilon}$ is positive for $0\leq\sqrt{\epsilon}<\sqrt{3}$
and negative for $\sqrt{3}<\sqrt{\epsilon}<\frac{\alpha\beta}{\sqrt{2}}$, but
the latter region is not physically meaningful due to the positivity of the
potential $V=\frac{V_{0}}{u^{\beta}}$ : see the first equation in (11). As
depicted in the left panel of Fig. 1, since $\frac{d}{Hdt}\sqrt{\epsilon}$ is
always positive, regardless of the initial value, $\epsilon$ increases in time
until it reaches $3$ at which the ratio $\frac{V}{3M_{\rm Pl}^{2}H^{2}}$
becomes $0$. Thus the upper bound on $\epsilon$ given by $3$ can be
interpreted as a stable fixed point. We note that this stable fixed point
value does not depend on both $\alpha$ and $\beta$. This reflects the fact
that for $\alpha\beta>\sqrt{6}$, the potential quickly decreases to $0$ : as
$u$ rolls down the potential, the ratio $\frac{V}{3M_{\rm Pl}^{2}H^{2}}$
converges to $0$ (where $\epsilon$ becomes $3$), while both $V$ and $H$
decrease to $0$. In fact, $\alpha\beta$ is nothing more than the decay rate of
the potential with respect to the canonically normalized field
$\varphi=\frac{M_{\rm Pl}}{\alpha}\log u$ (see (7)). We also note that at
$\epsilon=3$, $\frac{\ddot{u}}{H^{2}u}=-(3\sqrt{6}-6\alpha)\alpha$ is not
necessarily suppressed compared to $\frac{\dot{u}}{Hu}=\sqrt{6}\alpha$ thus
the slow-roll approximation is not guaranteed. More precisely,
$|\frac{\ddot{u}}{H^{2}u}|$ is larger than $\frac{\dot{u}}{Hu}$ for
$\alpha<\sqrt{\frac{2}{3}}$ and $\alpha>2\sqrt{\frac{2}{3}}$.

On the other hand, when $\alpha\beta=\sqrt{6}$, the value of
$\frac{\alpha\beta}{\sqrt{2}}$ at which $\frac{d}{Hdt}\sqrt{\epsilon}=0$
coincides with $\sqrt{3}$, thus $\frac{d}{Hdt}\sqrt{\epsilon}\geq 0$ for any
value of $\sqrt{\epsilon}$ in the physical region
$0\leq\sqrt{\epsilon}\leq\sqrt{3}$ and $\frac{d}{Hdt}\sqrt{\epsilon}=0$ only
at $\epsilon=3$. Then the qualitative feature of the time variation of
$\epsilon$ is the same as that in the case of $\alpha\beta>\sqrt{6}$.

Finally, when $\alpha\beta<\sqrt{6}$, $\frac{\alpha\beta}{\sqrt{2}}$ is
smaller than $\sqrt{3}$ hence belongs to the physical region. In this case, as
can be found in the right panel in Fig. 1, $\frac{d}{Hdt}\sqrt{\epsilon}$ is
positive for $0<\sqrt{\epsilon}<\frac{\alpha\beta}{\sqrt{2}}$ and negative for
$\frac{\alpha\beta}{\sqrt{2}}<\sqrt{\epsilon}<\sqrt{3}$. Then
$\epsilon=\frac{(\alpha\beta)^{2}}{2}$ corresponds to a stable fixed point :
if the initial value of $\epsilon$ is smaller (larger) than
$\frac{(\alpha\beta)^{2}}{2}$, $\epsilon$ increases (decreases) in time until
it becomes $\frac{(\alpha\beta)^{2}}{2}$. Indeed, at the stable fixed point,
the Hubble parameter,

$\begin{split}H(t)=\frac{H(0)}{1+\frac{(\alpha\beta)^{2}}{2}H(0)t},\end{split}$
(13)

behaves as $H(t)\sim\frac{2}{(\alpha\beta)^{2}}\frac{1}{t}\to 0$ at late time,
and at the same time, the potential

$\begin{split}V=\Big{(}1-\frac{\epsilon}{3}\Big{)}3H^{2}M_{\rm
Pl}^{2}=\Big{(}1-\frac{(\alpha\beta)^{2}}{6}\Big{)}3M_{\rm
Pl}^{2}\Big{(}\frac{H(0)}{1+\frac{(\alpha\beta)^{2}}{2}H(0)t}\Big{)}^{2}\end{split}$
(14)

also decreases to zero. That is, while $\sigma$ rolls down the potential, both
$V$ and $H^{2}$ decrease to zero with the same rate such that the ratio
$\frac{V}{3M_{\rm Pl}^{2}H^{2}}$ is kept constant. We also note that if
$\alpha\beta\ll\sqrt{2}$ the stable fixed point value
$\epsilon=\frac{(\alpha\beta)^{2}}{2}$ is much smaller than $1$. Then the
slow-roll condition is satisfied at late time provided
$\frac{\ddot{\sigma}}{H^{2}\sigma}=\alpha^{4}\beta^{2}\big{(}1-\frac{\beta}{2}\big{)}$
is much smaller than $\frac{\dot{\sigma}}{H\sigma}=\alpha^{2}\beta$, in which
case $V$ can be approximated by $3M_{\rm Pl}^{2}H(t)^{2}$.

We can compare $\epsilon$ with the ‘potential slow-roll parameter’ given by

$\begin{split}\epsilon_{V}=\frac{1}{2}M_{\rm
Pl}^{2}\Big{(}\frac{\alpha^{2}u^{2}}{M_{\rm
Pl}^{2}}\Big{)}\Big{(}\frac{dV/du}{V}\Big{)}^{2}=\frac{(\alpha\beta)^{2}}{2},\end{split}$
(15)

where $\frac{\alpha^{2}u^{2}}{M_{\rm Pl}^{2}}$ comes from the inverse of the
the Kähler metric. 333Obviously, the same value of $\epsilon_{V}$ can be
obtained in terms of the canonically normalized field $\varphi$ (see (7) for
the action) by using the well known expression $\epsilon_{V}=\frac{1}{2}M_{\rm
Pl}^{2}\Big{(}\frac{dV/d\varphi}{V}\Big{)}^{2}$, which is consistent with the
fact that $\alpha\beta$ is the decay rate of the potential. Then one finds
that if $\epsilon=\epsilon_{V}$, it is nothing more than the value of
$\epsilon$ at which $\frac{d}{Hdt}\sqrt{\epsilon}$ becomes $0$. Whether it is
a physical stable fixed point depends on the size of $\alpha\beta$. That is,
while we can approximate $\epsilon$ by $\epsilon_{V}$ when the slow-roll
condition is satisfied, the exact equality $\epsilon_{V}=\epsilon$ is
satisfied at the stable fixed point provided $\alpha\beta\leq\sqrt{6}$.

##  3 Volume modulus and dilaton runaways

We now move onto the contributions of (the scalar part of) the volume modulus
and the dilaton to $\epsilon$ in the asymptotic region when only one of them
rolls down the potential. For this purpose, we consider the compactification
of Type IIB string theory with a single Kähler modulus, the volume modulus
$\rho$. In this case, the Kähler potential is given by

$\begin{split}K/M_{\rm
Pl}^{2}=-3\log\big{[}-i(\rho-\overline{\rho})\big{]}-\log\big{[}-i(\tau-\overline{\tau})\big{]}+K_{0}/M_{\rm
Pl}^{2},\end{split}$ (16)

where $\tau$ is the axio-dilaton and $K_{0}$ depends on the complex structure
moduli. Meanwhile, the superpotential $W$ is independent of $\rho$ in the
absence of the quantum corrections such as the non-perturbative effects. Then
the action for the moduli $\phi^{I}$ (which include $\rho$ and $\tau$, as well
as the complex structure moduli) is

$\begin{split}S[\phi^{I}]=\int
d^{4}xa^{3}\big{(}K_{I{\overline{J}}}{\dot{\phi}}^{I}{\dot{\overline{\phi}}}^{\overline{J}}-V(\phi^{I})\big{)},\end{split}$
(17)

where the potential is given by

$\begin{split}V=e^{K/M_{\rm
Pl}^{2}}\Big{(}K^{I{\overline{J}}}D_{I}WD_{\overline{J}}\overline{W}-\frac{3}{M_{\rm
Pl}^{2}}|W|^{2}\Big{)},\end{split}$ (18)

with $D_{I}W=\partial_{I}W+\frac{1}{M_{\rm Pl}^{2}}\partial_{I}K$. Denoting
the scalar part of $\rho$ and $\tau$ by $\sigma$ and $s$, respectively (that
is, $\rho=i\sigma+\theta$ and $\tau=is+a$), the volume of the internal
manifold in string units is given by $\sigma^{3/2}$ and the string coupling
constant $g_{s}$ is identified with $s^{-1}$. Since
$K^{\rho{\overline{\rho}}}K_{\rho}K_{\overline{\rho}}=3M_{\rm Pl}^{2}$, in the
absence of the quantum corrections, the potential exhibits the no-scale
structure,

$\begin{split}V=\frac{e^{K_{0}/M_{\rm Pl}^{2}}}{2s\times
8\sigma^{3}}K^{i\overline{j}}D_{i}WD_{\overline{j}}\overline{W},\end{split}$
(19)

where $i,j$ run over moduli other than $\sigma$.

###  3.1 Volume modulus

We first assume that $s$ is stabilized, i.e., the value of $g_{s}$ is fixed,
and investigate the rolling of $\sigma$. While the potential can be written as
$V(\sigma)=V_{0}/\sigma^{3}$, for the comparison with the more suppressed
potential (which violates the no-scale structure), we consider the potential
given by $V=V_{0}/\sigma^{3+q}$, where $q\geq 0$. Then the action for $\sigma$
is

$\begin{split}S_{\sigma}=\int d^{4}xa^{3}\Big{(}\frac{3}{4}M_{\rm
Pl}^{2}\frac{\dot{\sigma}^{2}}{\sigma^{2}}-\frac{V_{0}}{\sigma^{3+q}}\Big{)},\end{split}$
(20)

which is the same form as (6) with $\alpha=\sqrt{\frac{2}{3}}$ and
$\beta=3+q$. In other words, the field redefinition for the canonical kinetic
term is given by $\varphi=\sqrt{\frac{3}{2}}M_{\rm Pl}\log\sigma$. The slow-
roll parameter in this case is given by

$\begin{split}\epsilon=\frac{2K_{\rho{\overline{\rho}}}\dot{\sigma^{2}}}{2H^{2}M_{\rm
Pl}^{2}}=\frac{3}{4}\Big{(}\frac{\dot{\sigma}}{H\sigma}\Big{)}^{2},\end{split}$
(21)

whereas the potential and $\frac{\ddot{\sigma}}{H^{2}\sigma}$ is written as

$\begin{split}&\frac{V}{3H^{2}M_{\rm
Pl}^{2}}=\frac{V_{0}/\sigma^{3+q}}{3H^{2}M_{\rm
Pl}^{2}}=1-\frac{\epsilon}{3},\\\
&\frac{\ddot{\sigma}}{H^{2}\sigma}=6\Big{(}1+\frac{q}{3}\Big{)}-2\sqrt{3}\sqrt{\epsilon}-\frac{2}{3}(1+q)\epsilon,\end{split}$
(22)

respectively.

The values of $\alpha$ and $\beta$ for $\sigma$ satisfy
$\alpha\beta=\sqrt{6}\big{(}1+\frac{q}{3}\big{)}$, thus according to the
discussion in Sec. 2, the rate of the change of $\sqrt{\epsilon}$ in time is
given by

$\begin{split}\frac{d}{Hdt}\sqrt{\epsilon}&=\frac{\sqrt{3}}{2}\frac{d}{Hdt}\Big{(}\frac{\dot{\sigma}}{H\sigma}\Big{)}=(\sqrt{\epsilon}+\sqrt{3})(\sqrt{\epsilon}-\sqrt{3})\Big{(}\sqrt{\epsilon}-\sqrt{3}\Big{(}1+\frac{q}{3}\Big{)}\Big{)},\end{split}$
(23)

and the positivity of $V$ imposes that the viable range of $\sqrt{\epsilon}$
is restricted to $0\leq\sqrt{\epsilon}\leq\sqrt{3}$. Since
$\frac{d}{Hdt}\sqrt{\epsilon}$ is always positive in this region, as time goes
on, $\epsilon$ increases toward the fixed point $\epsilon=3$, which is
satisfied at $V=0$, or equivalently, $\sigma=\infty$. Moreover, since the
fixed point value $\epsilon=3$ is larger than $1$ and the absolute value of
$\frac{\ddot{\sigma}}{H^{2}\sigma}=-2$ is the same as that of
$\frac{\dot{\sigma}}{H\sigma}=2$, the slow-roll condition is not satisfied.

The fact that the fixed point value $\epsilon=3$ is independent of the choice
of $q$ may be compared to the potential slow-roll parameter

$\begin{split}\epsilon_{V}=\frac{1}{2}M_{\rm
Pl}^{2}\frac{K^{\rho\overline{\rho}}}{2}\Big{(}\frac{dV/d\sigma}{V}\Big{)}^{2}=3\Big{(}1+\frac{q}{3}\Big{)}^{2}.\end{split}$
(24)

When $q>0$, $\epsilon_{V}$ cannot be the viable value of $\epsilon$ since
$\epsilon=\epsilon_{V}$ gives $\frac{V}{3H^{2}M_{\rm
Pl}^{2}}=-\frac{q}{3}\big{(}2+\frac{q}{3}\big{)}<0$ (see the first equation in
(22)), which is not compatible with the positivity of the potential.
Meanwhile, when $q=0$, $\epsilon_{V}$ coincides with the fixed point value
$\epsilon=3$. We note that $\epsilon_{V}$ is given by $3$ only if the Kähler
potential depends on $\sigma$ as $-3\log[2\sigma]$ (not, say,
$-(3+q)\log[2\sigma]$) and $W$ is independent of $\sigma$, or equivalently,
the potential exhibits the no-scale structure thus $q$ is exactly $0$. In
contrast, the fixed point value of $\epsilon$ is $3$ regardless of the value
of $q$, and it indeed originates from the more generic condition, the
positivity of the potential (given by the first equation in (11) or (22)), not
the no-scale structure.

###  3.2 Dilaton

We now consider the case in which the value of $\sigma$ is fixed but $s$ rolls
down the potential. When the fluxes are turned off, $W$ is independent of
$\tau$ (hence $s$) as well, then from
$K^{\tau{\overline{\tau}}}K_{\tau}K_{\overline{\tau}}=M_{\rm Pl}^{2}$ the
potential can be written as $V=V_{0}/s$. 444Of course, for $\sigma$ to be
stabilized, the no-scale structure is violated by the non-perturbative effects
or the supersymmetry breaking so (19) cannot be directly used. But here we
assume that the potential term deviates from (19) is suppressed.  From this,
the action for $s$ is given in the form of (6) with $\alpha=\sqrt{2}$ and
$\beta=1$,

$\begin{split}S_{s}=\int d^{4}xa^{3}\Big{(}\frac{1}{4}M_{\rm
Pl}^{2}\frac{\dot{s}^{2}}{s^{2}}-\frac{V_{0}}{s}\Big{)},\end{split}$ (25)

and the redefined field for the canonical kinetic term is
$\varphi=\sqrt{\frac{1}{2}}M_{\rm Pl}\log s$. Thus, the equations of motion
give the relations

$\begin{split}&\epsilon=\frac{2K_{\tau{\overline{\tau}}}\dot{s^{2}}}{2H^{2}M_{\rm
Pl}^{2}}=\frac{1}{4}\Big{(}\frac{\dot{s}}{Hs}\Big{)}^{2},\\\
&\frac{V}{3H^{2}M_{\rm Pl}^{2}}=\frac{V_{0}/s}{3H^{2}M_{\rm
Pl}^{2}}=1-\frac{\epsilon}{3},\\\
&\frac{\ddot{s}}{H^{2}s}=6-6\sqrt{\epsilon}+2\epsilon,\end{split}$ (26)

respectively. Since $\alpha\beta=\sqrt{2}<\sqrt{3}$, the rate of change of
$\sqrt{\epsilon}$ in time is

$\begin{split}\frac{d}{Hdt}\sqrt{\epsilon}=\frac{1}{2}\frac{d}{Hdt}\Big{(}\frac{\dot{s}}{Hs}\Big{)}=(\sqrt{\epsilon}+\sqrt{3})(\sqrt{\epsilon}-1)(\sqrt{\epsilon}-\sqrt{3}).\end{split}$
(27)

This shows that $\epsilon=1$ corresponds to the stable fixed point : at late
time $\epsilon$ converges to $1$ regardless of its initial value in the
physical region $0\leq\epsilon\leq 3$. From the equations of motion one finds
that $\frac{V}{3H^{2}M_{\rm Pl}^{2}}=\frac{1}{3}$ and
$\frac{\dot{s}}{Hs}=\frac{\ddot{s}}{H^{2}s}=2$ are satisfied at the fixed
point, which indicates that the slow-roll approximation is not valid. We also
note that the value of the potential slow-roll parameter

$\begin{split}\epsilon_{V}=\frac{1}{2}M_{\rm
Pl}^{2}\frac{K^{\tau\overline{\tau}}}{2}\Big{(}\frac{dV/ds}{V}\Big{)}^{2}=1\end{split}$
(28)

coincides with $\epsilon$ at the fixed point.

##  4 Fixed point in multifield quintessence model

So far we have considered the simplest quintessence model in which only a
single scalar field rolls down the runaway potential. When more than two
scalar fields $u_{i}$ ($i=1,\cdots,N$) simultaneously roll down the potential,
we need to consider the generalized equations of motion,

$\begin{split}&3M_{\rm Pl}^{2}H^{2}=\sum_{i}\frac{M_{\rm
Pl}^{2}}{2\alpha_{i}^{2}}\frac{\dot{u}_{i}^{2}}{u_{i}^{2}}+V,\\\ &3M_{\rm
Pl}^{2}H^{2}+2M_{\rm Pl}^{2}\dot{H}=-\sum_{i}\frac{M_{\rm
Pl}^{2}}{2\alpha_{i}^{2}}\frac{\dot{u}_{i}^{2}}{u_{i}^{2}}+V,\\\
&\ddot{u}_{i}+3H\dot{u}_{i}-\frac{\dot{u}_{i}^{2}}{u_{i}}+\sum_{i}\frac{\alpha_{i}^{2}}{M_{\rm
Pl}^{2}}u_{i}^{2}\frac{\partial V}{\partial u_{i}}=0,\end{split}$ (29)

where we assume that the kinetic mixing between fields is absent. 555In the
presence of the kinetic mixing, the kinetic term can be written as
$\frac{1}{2}G_{ij}\dot{u}^{i}\dot{u}^{i}$, which replaces
$\sum_{i}\frac{M_{\rm
Pl}^{2}}{2\alpha_{i}^{2}}\frac{\dot{u}_{i}^{2}}{u_{i}^{2}}$ in the first two
equations. Then the third equation becomes
$\begin{split}{\ddot{u}}^{i}+\Gamma^{i}_{~{}jk}\dot{u}^{j}\dot{u}^{k}+3H\dot{u}^{i}+G^{ij}\partial_{j}V=0,\end{split}$
(30) where
$\Gamma^{i}_{~{}jk}=\frac{1}{2}G^{il}[\partial_{j}G_{lk}+\partial_{k}G_{lj}-\partial_{l}G_{jk}]$.
Then they lead to

$\begin{split}&\epsilon=-\frac{\dot{H}}{H^{2}}=\sum_{i}\frac{1}{2\alpha_{i}^{2}}\Big{(}\frac{\dot{u}_{i}}{Hu_{i}}\Big{)}^{2}\equiv\sum_{i}\epsilon_{i},\\\
&\frac{V}{3H^{2}M_{\rm Pl}^{2}}=1-\frac{\epsilon}{3},\end{split}$ (31)

where
$\epsilon_{i}=\frac{1}{2\alpha_{i}^{2}}\big{(}\frac{\dot{u}_{i}}{Hu_{i}}\big{)}^{2}$.
Moreover, when the potential satisfies $u_{i}\partial_{i}V=-\beta_{i}V$, i.e.,
exhibits the runaway behavior with respect to each of $u_{i}$, we obtain the
relation

$\begin{split}&\frac{\ddot{u}_{i}}{H^{2}u_{i}}=3\alpha_{i}^{2}\beta_{i}-3\sqrt{2}\alpha_{i}\sqrt{\epsilon_{i}}+2\alpha_{i}^{2}\epsilon_{i}-\alpha_{i}^{2}\beta_{i}\epsilon.\end{split}$
(32)

Since $\epsilon$, as well as $\epsilon_{i}$, appears in the above equation
(see the last term), the dynamics of any one of moduli is affected by that of
others. Of course, in the slow roll approximation in which
$\frac{\ddot{u}_{i}}{H^{2}u_{i}}$, $\epsilon$, and $\epsilon_{i}$ are
suppressed compared to $\sqrt{\epsilon_{i}}$, each of $\epsilon_{i}$ is
decoupled from the rest, giving
$\epsilon_{i}\simeq\frac{(\alpha_{i}\beta_{i})^{2}}{2}$ provided
$\alpha_{i}\beta_{i}\ll 1$.

Figure 2: The behavior of the rate $\frac{d\epsilon}{Hdt}$ with respect to
$\epsilon_{\sigma}$ and $\epsilon_{s}$. The curve consists of points
satisfying
$\epsilon=\epsilon_{\sigma}+\epsilon_{s}=\sum_{i}\frac{\alpha_{i}\beta_{i}}{\sqrt{2}}\sqrt{\epsilon_{i}}$.
The blue region corresponds to $\epsilon=\epsilon_{\sigma}+\epsilon_{s}>3$,
which is excluded by the positivity of the potential. The yellow region
satisfies both $\epsilon<3$ and
$\epsilon<\sum_{i}\frac{\alpha_{i}\beta_{i}}{\sqrt{2}}\sqrt{\epsilon_{i}}$
(hence $\frac{d\epsilon}{Hdt}>0$). Then the part of the boundary of the yellow
region satisfying
$\epsilon=\sum_{i}\frac{\alpha_{i}\beta_{i}}{\sqrt{2}}\sqrt{\epsilon_{i}}$ or
$\epsilon=3$ consists of the stable fixed points.

In order to find out the fixed points, we consider

$\begin{split}\frac{d\epsilon}{Hdt}=\sum_{i}\frac{d\epsilon_{i}}{Hdt}&=\sum_{i}\frac{1}{\alpha_{i}^{2}}\Big{(}\frac{\dot{u}_{i}}{Hu_{i}}\Big{)}\Big{(}\frac{\ddot{u}_{i}}{H^{2}u_{i}}-\Big{(}\frac{\dot{u}_{i}}{Hu_{i}}\Big{)}^{2}+\epsilon\frac{\dot{u}_{i}}{Hu_{i}}\Big{)}\\\
&=2(\epsilon-3)\Big{(}\epsilon-\sum_{i}\frac{\alpha_{i}\beta_{i}}{\sqrt{2}}\sqrt{\epsilon_{i}}\Big{)},\end{split}$
(33)

which shows that the stable fixed points are given by

$\begin{split}\epsilon={\rm
min}\Big{[}3,\sum_{i}\frac{\alpha_{i}\beta_{i}}{\sqrt{2}}\sqrt{\epsilon_{i}}\Big{]}.\end{split}$
(34)

In particular, a set of values
$\epsilon_{i}=\frac{(\alpha_{i}\beta_{i})^{2}}{2}$ can be the stable fixed
point so far as $\epsilon=\sum_{i}\frac{(\alpha_{i}\beta_{i})^{2}}{2}$ is
smaller than $3$ (otherwise, it contradicts to the positivity of the
potential). In this case, each of $\epsilon_{i}$ coincides with the
contribution of $u_{i}$ to the potential slow roll parameter,

$\begin{split}\epsilon_{V,i}=\frac{1}{2}M_{\rm
Pl}^{2}\Big{(}\frac{\alpha_{i}^{2}u_{i}^{2}}{M_{\rm
Pl}^{2}}\Big{)}\Big{(}\frac{dV/du_{i}}{V}\Big{)}^{2}=\frac{(\alpha_{i}\beta_{i})^{2}}{2}.\end{split}$
(35)

We now investigate the contributions of $\sigma$ and $s$ to $\epsilon$. In
this case $(\alpha_{i},\beta_{i})$ ($i=\sigma,s$) are given by
$(\sqrt{\frac{2}{3}},3)$ for $\sigma$ and $(\sqrt{2},1)$ for $s$,
respectively. From this and $\epsilon=\epsilon_{\sigma}+\epsilon_{s}$ we
obtain

$\begin{split}\frac{d\epsilon}{Hdt}&=2(\epsilon-3)\Big{(}\epsilon-(\sqrt{3}\sqrt{\epsilon_{\sigma}}+\sqrt{\epsilon_{s}})\Big{)}\\\
&=2(\epsilon-3)\big{(}\sqrt{\epsilon_{\sigma}}(\sqrt{\epsilon_{\sigma}}-\sqrt{3})+\sqrt{\epsilon_{s}}(\sqrt{\epsilon_{s}}-1)\big{)}.\end{split}$
(36)

Since the positivity of the potential imposes $\epsilon\leq 3$,
$\frac{d\epsilon}{Hdt}$ is positive when both $\epsilon<3$ and
$\epsilon<\sqrt{3}\sqrt{\epsilon_{\sigma}}+\sqrt{\epsilon_{s}}$ are satisfied,
which corresponds to the yellow region in Fig. 2. Then a set of the stable
fixed points is nothing more than the part of the boundary of this region
satisfying (34). We note that two curves $\epsilon=3$ and
$\epsilon=\sqrt{3}\sqrt{\epsilon_{\sigma}}+\sqrt{\epsilon_{s}}$ intersect at
$(\epsilon_{\sigma},\epsilon_{s})=(\frac{3}{4},\frac{9}{4})$ and $(3,0)$, at
which $V=0$. Meanwhile, the point $(\epsilon_{\sigma},\epsilon_{s})=(3,1)$ at
which $\epsilon_{i}=\epsilon_{V,i}$ is satisfied is excluded by the positivity
of the potential. This also indicates that the fixed point value is not
necessarily the simple sum of the values obtained in the single field case.

##  5 Conclusions

In this short note, we consider the stringy quintessence model and investigate
the late time contributions of $\sigma$ (the scalar part of the volume
modulus) and $s$ (the dilaton) to the slow-roll parameter
$\epsilon=-\frac{\dot{H}}{H^{2}}$ which directly measures the deviation of the
geometry from dS space. We point out that as time goes on, each of these
contributions converges to the stable fixed point, at which the slow-roll
approximation does not hold. In particular, in the single field model, it
turns out that when the decay rate of the potential is larger than some
critical value, the positivity of the potential imposes that the fixed point
value is given by $3$, independent of the details of the dynamics. This is
somewhat different feature from the potential slow-roll parameter, and
$\sigma$ corresponds to this case. When both $\sigma$ and $s$ simultaneously
roll down the potential, we can find the curve consisting of the fixed points
in the plane of $\epsilon_{\sigma}$ and $\epsilon_{s}$, and as in the single
field case, there exists a direction in which the fixed point value of
$\epsilon$ is just given by $3$, the maximum value of $\epsilon$. Moreover,
unlike the potential slow-roll parameter, the fixed point value in the
multifield case is not necessarily given by the sum of fixed point values in
the single field case.

The no-go theorem we have investigated indicates that despite the
compatibility with the observations, both the metastable dS vacuum and the
quintessence suffer from the parametric control problem. It is also remarkable
that in order to resolve this problem, we need to understand the dynamics of
$\sigma$ and $s$ more clearly. Presumably, there may be an accidental fine-
tuning which prevents the instability of $\sigma$ and $s$ (for the instability
issue originating from the mixing between $\sigma$ and $s$, see, [42, 43]), or
unknown quantum gravity reason which recovers the parametric control.

### Acknowledgements

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# Classical and quantum harmonic oscillators
subject to a time dependent force

Henryk Gzyl,
Centro de Finanzas, IESA, Caracas.
<EMAIL_ADDRESS>
###### Abstract

In this work we address the problem of the quantization of a simple harmonic
oscillator that is perturbed by a time dependent force. The approach consists
of removing the perturbation by a canonical change of coordinates. Since the
quantization procedure uses the classical Hamiltonian formalism as staring
point, the change of variables is carried out using canonical transformations,
and to transform between the quantized systems the canonical transformation is
implemented as a unitary transformation mapping the states of the perturbed
and unperturbed system onto each other.

Keywords: Forced harmonic oscillator, canonical transformation, quantization
of the forced oscillator.

##  1 Introduction and Preliminaries

When confronted with a classical or quantum system subject to a perturbation,
one possible approach to describe it consists in removing the perturbation by
means of a change of coordinates. So the first problem to consider is to
determine the change of coordinates that simplifies the description of the
system. Here, the correct way is dictated by the fact that the time evolution
of a quantum system is described by a Hamiltonian operator which is obtained
from the classical (non quantum) Hamiltonian by an application of the Bohr-
Sommerfeld rules. Thus, if we want to remove the effect of a perturbation by
means of a change of variables, it is natural to do it first in the classical
version of the system and then transport the procedure to the quantized
version. It so happens that the canonical transformation that maps the
perturbed oscillator onto the simple oscillator can be implemented as a
unitary transformation between the corresponding quantized versions of the two
systems. It is the aim of this paper to work this out explicitly and thus
obtain the time evolution of the perturbed system in terms of that of the
unperturbed one.

The study of time dependent Hamiltonian systems in general and harmonic
oscillators with time dependent Hamiltonians is not new. Much a attention has
been devoted to systems with time dependent quadratic Hamiltonians. To cite
but a few references, consider [15] in which the quantum perturbed oscillator
appears as part of s system to detect gravitational waves. In [6] the
Schrödinger equation is solved directly. Time dependent quadratic
Hamiltonians, which include the cases of the damped harmonic oscillator and/or
the harmonic oscillator with a time dependent frequency, have been treated by
a variety of techniques. For example, [4] uses the Hamilton-Jacobi equation to
obtain a solution to the Schrödinger equation. The solution to the Hamilton-
Jacobi equation is the generating function of a canonical transformation that
brings the system to its initial state. Another approach is based on the
construction of invariants of motion (the Ermakov-Lewis invariant) using a
variety of techniques, including canonical transformations. See for example
[5], [7], [9], and [8] (in which references to applications are given). Other
approaches involve more advanced group theoretic or algebraic approaches using
symplectic geometry: [10], [11], [12], [14], [13] for a sample of works using
these approaches. Anyway, all of these differ from the approach developed
here, which consists of using a the machinery of canonical transformations and
their unitary representations to solve the quantum mechanical problem.

To establish notations, in the remainder of this section we solve the
equations of motion of the forced harmonic oscillator within the Hamiltonian
formalism, and then see how the solution can be described as the superposition
of the solution without the forcing term plus describing the motion of a
system under the action of the “external” forcing.

In Section 2 we explain how to relate the two descriptions by means of a
canonical transformation. This the first step to relate the quantized versions
on the forced and the non forced oscillators. The second is to implement the
canonical transformation as a unitary transformation between the corresponding
Hilbert spaces. We do that in Section 3, where we prove that the unitary
transformation maps the position, momentum and Hamiltonian operators as in the
classical case. On the one hand this means that the quantization rules behave
consistently under canonical transformations, and on the other, that we can
transform the solutions of the corresponding Schrödinger equations onto each
other. There we also indicate how we can compute transition probabilities
using the canonical mapping.

###  1.1 The Hamiltonian description of the forced harmonic oscillator

The harmonic oscillator subject to a time dependent force independent of its
position is a simple mechanical system. The Hamiltonian from which the
dynamics of the forced oscillator is obtained is
$H(x,p)=p^{2}/2m+m\omega^{2}x^{2}/2-xk(t).$ When $\omega=0$ the Hamiltonian
describes a particle in spatially constant but time dependent force, while
when $k(t)=k$ is time independent, the Hamiltonian describes a particle under
the action of a harmonic force plus a constant force.

In the Hamiltonian formalism the equations of motion are obtained from the
Hamiltonian $H(x,p)$ as follows:

$\frac{d}{dt}{x\atopwithdelims( )p}={\frac{\partial H}{\partial
p}\atopwithdelims( )-\frac{\partial H}{\partial x}}={p\atopwithdelims(
)-\omega^{2}x+k(t)}=\left(\begin{array}[]{cc}0&1/m\\\
-m\omega^{2}&0\end{array}\right){x\atopwithdelims( )p}+{0\atopwithdelims(
)k(t)}={\mathbb{H}}_{0}{x\atopwithdelims( )p}+{0\atopwithdelims( )k(t)}.$
(1.1)

The initial conditions are $x(0)=x_{0},p(0)=p_{0}.$ This is a common textbook
example. It appears as a mathematical model of many physical systems: from
electrical circuits to charged particles taken out of equilibrium by the
action an electric fields. See [1], [3] or [2] for example. The solution to
that system is simple to obtain and it is given by:

${x(t)\atopwithdelims( )p(t)}=U(t){x_{0}\atopwithdelims(
)p_{0}}+\int_{0}^{t}U(t-s){0\atopwithdelims( )k(s)}ds.$ (1.2)

Here it is clear that the time dependence of the forcing term is arbitrary,
subject to a mathematical requirement, namely, that the integrals on the right
hand side of (1.2) are defined. See the concluding remarks section for more on
this issue. In (1.2) we introduced the following notations:

$U(t)=\left(\begin{array}[]{cc}cos(\omega
t)&\frac{1}{\sqrt{m}\omega}sin(\omega t)\\\ -\omega\sqrt{m}sin(\omega
t)&cos(\omega t)\end{array}\right),$ (1.3)

Note as well that $U(t+s)=U(t)U(s)$ or $U(t-s)=U(t)U(-s)$ for all $s,t.$ To
simplify the notations in what comes below, let us denote the coordinates
$(x,v)$ by ${\boldsymbol{z}}$ (thought of as column vector), and denote by
${\boldsymbol{k}}(s)$ the transpose of $(0,k(s)).$ With those notations, write
the solution (1.2) to the system (1.1) as

${\boldsymbol{z}}(t)=U(t){\boldsymbol{z}}(0)+\int_{0}^{t}U(t-s){\boldsymbol{k}}(s)ds={\boldsymbol{z}}_{h}(t)+{\boldsymbol{z}}_{nh}(t).$
(1.4)

The subscript $h$ stands for homogeneous and $nh$ stands for non-homogeneous.
If we put $H_{0}(x,p)=\frac{1}{2}\big{(}m\omega^{2}x^{2}+p^{2}/m\big{)},$ then
${\boldsymbol{z}}_{h}(t)$ solves (1.1) with $H_{0}(x,p)$ instead of $H(x,p).$
Or, if you prefer, ${\boldsymbol{z}}_{nh}(t)$ is just the particular solution
to (1.1) with zero initial conditions. It is also easy to see that the matrix
${\mathbb{H}}_{0}$ introduced in (1.1) satisfies

$U^{\dagger}(t){\mathbb{H}}_{0}U(t)={\mathbb{H}}_{0}.$ (1.5)

Using this we have the following geometric way of visualizing the solutions is
as ellipses with center moving according to ${\boldsymbol{z}}_{nh}(t)$. This
follows from the fact that

$\langle\big{(}{\boldsymbol{z}}(t)-{\boldsymbol{z}}_{nh}(t)\big{)},{\mathbb{H}}_{0}\big{(}{\boldsymbol{z}}(t)-{\boldsymbol{z}}_{nh}(t)\big{)}\rangle=\mbox{constant}=\langle{\boldsymbol{z}}(0),{\mathbb{H}}_{0}{\boldsymbol{z}}(0)\rangle.$

That is, we might think of the two terms in the right hand side of (1.4) as
follows: Interpret ${\boldsymbol{z}}_{nh}(t)$ as the motion of the center of
coordinates of a “laboratory” that undergoes a non-uniform motion, and
interpret ${\boldsymbol{z}}_{h}(t)$ as the motion with respect to a system of
coordinates in which there is no external force, that is of the motion
described in a system of coordinates moving with the laboratory. To visualize
the motion of the laboratory system, consider the three ellipses displayed in
the panels of Figure 1. These correspond to ${\boldsymbol{z}}_{nh}(t)$ when
$k(t)=K$ is constant in time. In this case we have
$(\frac{K}{\omega})\big{(}\frac{1-cos(\omega t)}{\omega},sen(\omega
t)\big{)},$ which is an ellipse. Notice that it stretches out when $\omega\to
0$ and shrinks when $\omega\to\infty.$

(a) $\omega=2\pi/100$

(b) $\omega=2\pi$

(c) $\omega=200\pi$

Figure 1: Motion of the laboratory system

Notice that the scales in the time axis and in the coordinates axes are very
different. Had we plotted them on the same scale, the first ellipse (1a) would
be very elongated compared to (1b), while the third (1c) would be compressed
almost on top of the origin of coordinates.

##  2 The canonical transformations

To perform the change of frame of reference mentioned at the end of the
previous section in such a way that the Hamiltonian equations of motion are
preserved, we have to consider canonical transformations. We shall think of
the coordinates ${\boldsymbol{z}}=(x,p)^{t}$ as coordinates in the laboratory
system and the new coordinates ${\boldsymbol{\zeta}}=(\xi,\eta)$ as the
position and momentum in the moving coordinate system. Let us denote the
coordinates of ${\boldsymbol{z}}_{nh}(t)$ by $x_{nh}(t)$ and
$p_{nh}(t)=m\dot{x}_{nh}(t),$ where as customary, the dot stands for the
derivative with respect to time.

Let us write $H(x,p)=(p^{2}/m+m\omega^{2}x^{2})/2-xk(t)$ and
$K(\xi,\eta)=(\eta^{2}/m+m\omega^{2}\xi^{2})/2$ for the two Hamiltonian
functions of interest. The passage from one to the other by means of a
canonical transform can be achieved in several ways, but we shall consider the
following cases when implementing them as unitary transformations. We put

$\displaystyle\mbox{Either}\;\;\;F_{1}(x,\eta,t)=\big{(}x-x_{nh}(t)\big{)}\big{(}\eta+\dot{x}_{nh}(t)\big{)}+G(t)$
(2.1)
$\displaystyle\mbox{or}\;\;\;\;F_{2}(\xi,p,t)=\big{(}\xi+x_{nh}(t)\big{)}\big{(}p-\dot{x}_{nh}(t)\big{)}+G(t)$
(2.2)

depending on which coordinates we want to regard old or new. Which one we
consider will depend on which solution of the two possible Schrödinger
equations we want to transform onto which. The equations that relate the old
and new coordinates are (see [2] or [3]):

$\displaystyle\xi(t)=\frac{\partial F_{1}}{\partial\eta}$ (2.3) $\displaystyle
p(t)=\frac{\partial F_{1}}{\partial x}$ (2.4) $\displaystyle
K=H+\frac{\partial F_{1}}{\partial t}.$ (2.5)

Note that the transformation equations yield that $\xi(t)=x(t)-x_{nh}(t)$ and
$p=\eta+\dot{x}_{nh}(t).$ It is understood that in (2.5) the partial
derivation with respect to $t$ is carried out and the old coordinates are
substituted for the new after solving (2.3)-(2.4). To see that $H(x,p)$
becomes $K(\xi,\eta)$ when we apply (2.3)-(2.4)-(2.5) to (2.1) we have to make
use of the fact that

$\ddot{x}_{nh}(t)+m\omega^{2}x_{nh}(t)-k(t)=0$ (2.6)

and that $G(t)$ is to be chosen so that

$\dot{G}(t)-\dot{x}_{nh}(t)^{2}/2m+m\omega^{2}x_{nh}(t)^{2}/2-x_{nh}(t)k(t)=0.$
(2.7)

A similar procedure is followed for the passage from $K(\xi,\eta)$ to
$H(x,p).$

##  3 Representation of the canonical transformations and quantization

Our starting point consists of two classical systems, with phase spaces
labeled by $(x,p)$ and $(\xi,\eta),$ and with dynamics determined by the
Hamiltonians $H(x,p)$ and $K(\xi,\eta).$ We shall denote by $\Psi(x)$ and
$\Phi(\xi)$ the states in the Schrd̈inger representation, and we shall suppose
that they are at least twice continuously differentiable, square integrable,
complex valued functions such that all the integration by parts necessary to
verify the identities presented below are valid. Let us denote by
${\mathcal{H}}$ and ${\mathcal{K}}$ the corresponding state spaces provided
with the usual scalar product.

As the canonical transformations involve the momentum variables of one system
and coordinate variables of the other, the unitary transformations are defined
to act on the momentum representation of the states to yield states in the
coordinate representations. The description of the states in terms of the
momentum representation is given by taking the Fourier transforms, that is:

$\Psi(p)=\frac{1}{\sqrt{2\pi}}\int
e^{-ipx}\Psi(x)dx\;\;\;\;\mbox{and}\;\;\;\;\Phi(\eta)=\frac{1}{\sqrt{2\pi}}\int
e^{-i\eta\xi}\Phi(\xi)d\xi.$ (3.1)

Consider (2.1)-(2.2). In each case the transform depends on the “new” momentum
and the “old” coordinates. Thus the unitary transformations induced by $F_{1}$
and $F_{2}$ go in the opposite directions and are defined as follows.

$\displaystyle\tilde{\Phi}(x,t)=U_{F_{1}}(t)\Phi(x,t)=\frac{1}{\sqrt{2\pi}}\int
e^{iF_{1}(x,\eta)}\Phi(\eta,t)d\eta,$ (3.2)
$\displaystyle\tilde{\Psi}(\xi,t)=U_{F_{2}}(t)\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int
e^{iF_{2}(\xi,p)}\Psi(p,t)dp.$ (3.3)

The tilde is a notational reminder of the fact that the state is obtained by
applying the canonical transform. Keep in mind that the transformation is time
dependent and applied to the state dynamically. Note as well that at $t=0$ the
transforms reduce to the connection between the momentum and the coordinate
representations. The explicit computation of these transforms is simple:

$\displaystyle\widetilde{\Phi}(x,t)=e^{i\theta(x,t)}\Phi(x-x_{nh}(t),t),$
(3.4)
$\displaystyle\widetilde{\Psi}(\xi,t)=e^{i\theta^{\prime}(\xi,t)}\Psi(\xi+x_{nh}(t),t).$
(3.5)

Here we put

$\displaystyle\theta(x,t)=\big{(}x-x_{nh}(t)\big{)}\dot{x}_{nh}(t)+G(t)$ (3.6)
$\displaystyle\theta^{\prime}(\xi,t)=-\big{(}\xi+x_{nh}(t)\big{)}\dot{x}_{nh}(t)+G(t).$
(3.7)

Clearly the transformations are unitary. Let us now verify that the position
and momentum operators are transformed consistently with the correspondence
rules.

###  3.1 Transformation of the position and momentum operators

Let us begin by recalling that using (3.1) one verifies that

$\hat{\xi}\Phi(\eta)=-i\frac{\partial}{\partial\eta}\Phi(\eta)\;\;\;\mbox{and}\;\;\;\hat{\eta}\Phi(\eta)=\eta\Phi(\eta).$

Now, applying $U_{F_{1}}$ to $\Phi$ as in (3.2) and (3.3) we obtain that the
position operator transforms as:

$U_{F_{1}}\big{(}\hat{\xi}\Phi\big{)}(x,t)=\big{(}x-x_{nh}(t)\big{)}\widetilde{\Phi}(x,t)=\big{(}\hat{x}-x_{nh}(t)\big{)}\tilde{\Phi}(x,t),$
(3.8)

whereas the momentum operator transforms as:

$U_{F_{1}}\big{(}\hat{\eta}\Phi\big{)}(x,t)=\big{(}-i\frac{\partial}{\partial
x}-\dot{x}_{nh}(t)\big{)}\widetilde{\Phi}(x,t)=\big{(}\hat{p}-\dot{x}_{nh}(t)\big{)}\widetilde{\Phi}(x,t).$
(3.9)

To conclude, keep in mind that the Hamiltonian operators in the “laboratory”
system and in the “accelerated” system are given by

${\mathcal{H}}=-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{m\omega^{2}}{2}x^{2}-xk(t)\;\;\;\mbox{resp.}\;\;\;\;{\mathcal{K}}=-\frac{1}{2m}\frac{\partial^{2}}{\partial\xi^{2}}+\frac{m\omega^{2}}{2}\xi^{2}.$
(3.10)

###  3.2 The evolution in time transforms consistently

Let us now verity that the time evolution in with respect to one Hamiltonian
transforms into time evolution with respect to the other Hamiltonian. The
result drops out of the fact that

$i\frac{d}{dt}\tilde{\Psi}(\xi,t)=i\frac{d}{dt}\big{(}U_{F_{2}}\Psi\big{)}(\xi,t)=\big{(}i\frac{d}{dt}U_{F_{2}}\big{)}\Psi(\xi.t)+U_{F_{2}}\big{(}i\frac{d}{dt}\Psi\big{)}(\xi,t).$
(3.11)

Actually, a look at the definition in (3.2) and its effect on $\Psi$
explicitly shown in (3.5), shows that the claim will follow from the fact that

$\displaystyle
i\frac{d}{dt}\tilde{\Psi}(\xi,t)=i\frac{d}{dt}\big{(}e^{i\theta^{\prime}(\xi,t)}\Psi(\xi+x_{nh}(t),t)\big{)}\;\;\;\;\;\;\;\;\;$
$\displaystyle\
=i\frac{d}{dt}\big{(}e^{i\theta^{\prime}(\xi,t)}\big{)}\Psi(\xi+x_{nh}(t),t)+e^{i\theta^{\prime}(\xi,t)}\big{(}i\frac{d}{dt}\Psi(\xi+x_{nh}(t),t)\big{)}.$
(3.12)

Notice that

$\big{(}i\frac{d}{dt}\Psi(\xi+x_{nh}(t),t)\big{)}=i\dot{x}_{nh}(t)\frac{\partial}{\partial
x}\Psi(\xi+x_{nh}(t),t)+{\mathcal{H}}\Psi(\xi+x_{nh}(t),t).$

It is just a matter of substituting (3.8)-(3.9), making use of (2.3)-(2.5),
and the fact that neither $x_{nh}(t)$ nor $\dot{x}_{nh}(t)$ involves $x$ or
$p,$ to verify that all necessary cancellations take place to conclude that
when $\Psi(x,t)$ satisfies Schrödinger’s equation in the laboratory system,
then

$i\frac{d}{dt}\tilde{\Psi}(\xi,t)={\mathcal{K}}\tilde{\Psi}(\xi,t),\;\;\;\mbox{with}\;\;\;\tilde{\Psi}(\xi,0)=\Psi(\xi,0)=\Psi(x,0).$
(3.13)

That is, $\tilde{\Psi}(\xi,t)$ satisfies Schrödinger’s equation in the moving
system. An exactly analogous argument proves that if $\Phi(\xi,t)$ satisfies
Schrödinger’s equation with Hamiltonian operator ${\mathcal{K}},$ then
$\tilde{\Phi}(x,t)=\big{(}U_{F_{2}}\Phi\big{)}(x,t)$ satisfies ”Schrödinger’s
equation with Hamiltonian ${\mathcal{H}}.$ Next, we use these results to
compute the transition probabilities between the eigenstates of the oscillator
induced by the time dependent perturbation.

###  3.3 Computation of transition probabilities

Suppose that at $t=0$ the oscillator was in the $n$th eigenstate $\Psi_{n}$of
energy $E_{n}$ and the perturbation is turned on. We are interested in
computing the probability of finding the oscillator in some other eigenstate
$\Psi_{m}$ as time passes by. That is we want to compute

$P_{n,m}(t)=|\langle\Psi_{m},\Psi_{n}(t)\rangle|^{2}.$

From the closing comment in the previous section we know that the state at
current time $t$ that evolved from the eigenstate $\Psi_{n}(x)=\Phi_{n}(\xi)$
at time $0$ is, according to (3.4), given by
$\widetilde{\Phi_{n}}(x,t)=e^{-i\theta(x,t)}\Phi_{n}(x-x_{nh}(t),t)$ and that
$\Phi_{n}(x-x_{nh}(t),t)=\exp(-iE_{n}t)\Phi_{n}(x-x_{nh}(t)).$ From (3.6) we
obtain

$\langle\Psi_{m},\Psi_{n}(t)\rangle=e^{-iE_{n}t}e^{-iG(t)}\int\Psi_{m}(x)e^{-i(x-x_{nh}(t))\dot{x}_{nh}(t)}\Psi_{n}(x-x_{nh}(t))dx.$

The phase in front of the integral disappears when we consider the absolute
value, so we drop it. A change of variables in the integral yields

$|\langle\Psi_{m},\Psi_{n}(t)\rangle|=|\int\Psi_{m}(x+x_{nh}(t))\Psi_{n}(x)e^{-ix\dot{x}_{nh}(t)}dx|.$
(3.14)

To explicitly compute the scalar products we need to recall the following
facts: The eigenstate $\Psi_{n}(x)$ is expressed in terms of the Hermite
polynomials as

$\Psi_{n}(x)=\frac{1}{\sqrt{2^{n}n!}}\big{(}\frac{m\omega}{\pi}\big{)}^{1/4}H_{n}(\sqrt{m\omega}x)e^{-m\omega
x^{2}/2}.$ (3.15)

The constants are such that $\langle\Psi_{n},\Psi_{m}\rangle=\delta_{n,m}.$
The Hermite polynomials are obtained from their generating function as
follows:

$G(x,u)=e^{2xu-u^{2}}=\sum_{n=0}^{\infty}\frac{u^{n}}{n!}H_{n}(x).$ (3.16)

To complete the list, we need to keep in mind that for any complex $z$ we
have:

$\int_{{\mathbb{R}}}e^{zx}e^{-x^{2}}dx=\sqrt{\pi}e^{\frac{z^{2}}{4}}.$ (3.17)

. If we leave aside the factor $C(m,n)=\Big{(}2^{(m+n)}m!n!\pi\Big{)}^{-1/2},$
and make the change of variables $\sqrt{m\omega}x\to x,$ to calculate the
transition probabilities we need to compute

$\int_{{\mathbb{R}}}H_{m}(x+a)H_{n}(x)e^{-ixb}e^{-(x+a)^{2}/2}e^{-x^{2}/2}dx$
(3.18)

where we put $a=\sqrt{m\omega}x_{nh}(t)$ and
$b=\dot{x}_{nh}(t)/\sqrt{m\omega}.$ Instead of evaluation this integral for
each $m,n$ we make use of (3.16) and evaluate instead

$\int_{{\mathbb{R}}}G(x+a,v)G(x,u)e^{-ixb}e^{-(x+a)^{2}/2}e^{-x^{2}/2}dx.$
(3.19)

To obtain the desired integral, we compute $\partial^{m+n}/\partial
v^{m}\partial u^{n}$ at $v=u=0.$ The integrand in the last expression is an
exponential, which after collecting powers of $x$ looks like

$2av-a^{2}/2-u^{2}-v^{2}+[2(u+v)-(ib+a)]x-x^{2}.$

Now, make use of (3.17) to obtain

$\int_{{\mathbb{R}}}G(x+a,v)G(x,u)e^{-ixb}e^{-x^{2}}dx=\sqrt{\pi}e^{2av-a^{2}/2-u^{2}-v^{2}}e^{[2(u+v)-(ib+a)]^{2}/4}.$

At this point it is important to note that the square powers of $u$ and $v$
cancel out after the expansion and we are left with

$\displaystyle\int_{{\mathbb{R}}}G(x+a,v)G(x,u)e^{-ixb}e^{-x^{2}}dx=\sqrt{\pi}\exp\big{(}2uv+2av-(u+v)(a+ib)-\frac{(a-ib)^{2}}{4}\big{)}$
(3.20)
$\displaystyle=\sqrt{\pi}\exp\big{(}2uv+v(a-ib)-u(a+ib)-\frac{(a-ib)^{2}}{4}\big{)}.$

We can now replace back $a$ and $b$ with $\sqrt{\omega}x_{nh}(t)$ and
$b=\dot{x}_{nh}(t)/\sqrt{\omega}.$ As a simple example we can compute the
probability of finding the system in its ground state at time $t>0$ is given
by

$|\langle\Psi_{0},\Psi_{0}(t)\rangle|=\exp\big{(}-\frac{m\omega
x^{2}_{nh}(t)}{4}+\frac{x_{nh}(t)\dot{x}_{nh}(t}{2}-\frac{\dot{x}^{2}_{nh}(t)}{4m\omega}\big{)}.$
(3.21)

The term in the exponent can be written as $\big{(}\omega
L_{nh}-\mathcal{E}_{nh}\big{)}/2\omega,$ where $L_{nh}$ in the angular
momentum of the non-homogeneous solution and $\mathcal{E}_{nh}$ its total
energy.

We direct the interested reader to [16] for a different approach to computing
integrals of products of Hermite polynomials and exponentials. They use a
somewhat different scaling for the Hermite polynomials.

##  4 Closing comments

We mentioned at the beginning that the time dependence of the forcing term is
arbitrary, subject to the requirement that $\int_{0}^{t}|k(s)|ds$ be finite
for all $t>0.$ But actually, at the expense of complicating the mathematical
apparatus, we might consider impulsive forces (random or not), or white noise.

To finish and to repeat ourselves once more, sometimes a perturbed system can
be rendered unperturbed by a change of coordinates. It is therefore convenient
to consider a framework in which the equations of motion in both coordinate
systems are the same. This is the role of canonical transformations in the
Hamiltonian description of particle dynamics. The bonus is that in some cases
the canonical transformation can be explicitly implemented as a unitary
transformation between the quantized descriptions of the perturbed and
unperturbed systems.

Acknowledgments I want to thank the reviewers and the editors for their
thorough and careful reading of the manuscript and for their comments and
suggestions for improving it.

## References

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# A Comparative Study on Speaker-attributed Automatic Speech Recognition in
Multi-party Meetings

###### Abstract

In this paper, we conduct a comparative study on speaker-attributed automatic
speech recognition (SA-ASR) in the multi-party meeting scenario, a topic with
increasing attention in meeting rich transcription. Specifically, three
approaches are evaluated in this study. The first approach, FD-SOT, consists
of a frame-level diarization model to identify speakers and a multi-talker ASR
to recognize utterances. The speaker-attributed transcriptions are obtained by
aligning the diarization results and the recognized hypotheses. However, due
to the modular independence, such an alignment strategy may suffer from
erroneous timestamps which severely hinder the model performance. Therefore,
we propose the second approach, WD-SOT, to address alignment errors by
introducing a word-level diarization model, which can get rid of such
timestamp alignment dependency. To further mitigate the alignment issues, we
propose the third approach, TS-ASR, which trains a target-speaker separation
module and an ASR module jointly. By comparing various strategies for each SA-
ASR approach, experimental results on a real meeting scenario corpus,
AliMeeting, reveal that the WD-SOT approach achieves 10.7% relative reduction
on averaged speaker-dependent character error rate (SD-CER), compared with the
FD-SOT approach. In addition, the TS-ASR approach also outperforms the FD-SOT
approach and brings 16.5% relative average SD-CER reduction.

Index Terms: rich transcription, speaker-attributed, multi-speaker ASR,
AliMeeting

##  1 Introduction

Speaker-attributed automatic speech recognition (SA-ASR) is the major purpose
of rich transcription in real-world multi-party meetings [1, 2]. In general,
an SA-ASR aims at answering the question “who spoke what” [3, 4, 5, 6].
Compared with the multi-speaker ASR [7, 8, 9], SA-ASR not only focuses on
transcribing multi-speaker speech that may contain overlapped segments from
different speakers, but also assigns speaker labels to each recognized word.
Therefore, SA-ASR system needs to take more consideration of all involved
modules, such as speaker diarization [10, 11, 12] to count and identify
speakers, speech separation [13, 14, 15] to handle overlapping speech and ASR
[16, 17, 18] to recognize speech contents from the separated signals.

The accuracy of SA-ASR is affected by both transcript prediction and speaker
assignment. Recently, a lot of efforts have been made on designing an end-to-
end system that directly outputs multi-speaker transcriptions [7, 19, 20, 21,
9]. Speech separation and joint-training with multiple ASR decoders under
permutation invariant training (PIT) scheme is the typical approach [7, 19,
20, 21]. However, the maximum number of speakers that the model can handle is
constrained by the number of the decoders in the model. Besides, duplicated
hypotheses can be generated in different outputs, since the outputs are
independent of each another in PIT. To mitigate these issues, the serialized
output training (SOT) strategy [9] is proposed for multi-talker ASR, which
introduces a special symbol to represent the speaker change with only one
output layer. In this way, SOT-based models have no constraints on the maximum
number of speakers and avoid duplicated hypotheses naturally. In the recent
M2MeT challenge which focuses on multi-speaker ASR in the meeting scenario
[22, 23], SOT has been well applied and achieved remarkable performance.
Therefore, in this paper, we combine the multi-speaker hypotheses of the SOT-
based ASR model and the frame-level results of the speaker diarization model
(e.g, TS-VAD [24, 25]) to obtain the speaker-attributed transcriptions by
simply aligning the timestamps as our first approach, namely frame-level
diarization with SOT (FD-SOT).

However, due to the modular independence, such an alignment strategy may
suffer from erroneous timestamps which severely hinder the speaker assignment
performance. Considering that the lack of correlation between the ASR and
speaker diarization modules may cause potential alignment errors, the recently
updated revisions of SOT introduced speaker inventory and speaker encoder to
produce speaker labels for each ASR token [26, 27, 28]. Although they
significantly reduced the speaker-attributed word error rate, these approaches
cost a lot for changing ASR model structures and training pipelines, which is
not practical in real production systems. In order to retain the original ASR
network structure designed for single-speaker while addressing the alignment
errors, we propose the second approach named word-level diarization with SOT
(WD-SOT), which utilizes the recognized transcriptions from an SOT-based ASR
to model the diarization system, to get rid of such timestamp alignment
dependency. Meanwhile, we adopt self-attention to capture more contextual
information, which can further improve the diarization performance.

Both FD-SOT and WD-SOT depend on the output of the SOT model, and the errors
of SOT will seriously affect the performance of the overall framework.
Therefore, we turn to another solution of SA-ASR that uses a separation model
to handle overlapped speech while getting rid of the dependence on the multi-
talker ASR output [29, 30, 7, 19, 31, 32]. As the speech separation models are
usually trained based on the signal-level criterion, a joint training strategy
was proposed to mitigate the mismatch between the separation and the backend
ASR, leading to better recognition performance [7, 19, 31, 32]. However, most
of the prior works adopted blind source separation (BSS), which cannot
determine the specific speaker of the recognition transcriptions. In order to
fit the SA-ASR task, we use target speech separation (TSS) by adopting speaker
diarization and speaker extraction models to obtain the speaker embeddings
required by the TSS module. In this paper, we name this approach as joint
target-speaker separation and ASR (TS-ASR) approach.

Although most of the works described above show promising results in multi-
speaker ASR, they are mostly evaluated by simulated multi-speaker data.
Problems including unknown number of speakers, variable overlap rate, and
accurate speaker identity are still considered unsolved, especially for real-
world applications. SA-ASR for multi-speaker speech remains to be a very
difficult problem especially when speech utterances from multiple speakers
significantly overlap in monaural recordings. To the best of our knowledge, we
are the first to evaluate SA-ASR approaches on a real meeting corpus
AliMeeting to provide reasonable SA-ASR results and promote the research in
meeting rich transcription.

##  2 Evaluation Data

In this study, we use the AliMeeting corpus [22, 23] to evaluate various SA-
ASR systems. Collected in real meetings, the AliMeeting corpus contains 104.75
hours 111Hours are calculated in a single channel of audio. data for training
(Train), 4 hours for evaluation (Eval) and 10 hours for test (Test). Each set
contains several meeting sessions and each session consists of a 15 to
30-minute discussion with 2 to 4 participants. To highlight the speaker
overlap, the sessions with 4 participants account for 59%, 50% and 57%
sessions in Train, Eval and Test, respectively. Train and Eval sets contain
the 8-channel audios recorded from the microphone array (Ali-far) as well as
the near-field audio (Ali-near) from the participant’s headset microphone,
while the Test set only contains the far-field audios. Ali-far-bf is produced
by applying CDDMA Beamformer [33, 34]. In this paper, the model training and
evaluation are all based on the single-channel audio, namely Ali-near and Ali-
far-bf. The prefix Train-, Eval- and Test- are used to indicate different
sets, e.g., Train-Ali-far-bf denotes the one channel data outputted by the
beamformer which takes the AliMeeting 8-channel array Train data as input. We
use official scripts 222https://github.com/yufan-aslp/AliMeeting provided by
the M2MeT challenge to prepare the sentence segmentation timestamp. Meanwhile,
in order to improve the performance of the speech separation module used in
this paper, we simulate 50 hours mixed training data named Train-Ali-simu from
Train-Ali-near.

##  3 SA-ASR

###  3.1 SOT

The SOT method has an excellent ability to model the dependencies among
outputs for different speakers and no longer has a limitation on the maximum
number of speakers. To recognize multiple utterances, SOT serializes multiple
references into a single token sequence with a special token
$\langle\text{sc}\rangle$, which is used to concatenate the transcriptions of
the different utterances. In order to avoid the complex calculation of PIT for
all possible concatenation patterns, SOT sorts the reference labels by their
start times, which means “first-in, first-out” (FIFO). The experiments show
that the FIFO method achieves a better CER than the method of calculating all
permutations [9]. Meanwhile, considering the high overlap ratio and frequent
speaker turn on the AliMeeting corpus, we experimentally investigate the
speaker-based and utterance-based FIFO training schemes [35] in section 4.
Remarkably, we employ Transformer [16] model with Conformer encoder [17],
which includes 12 encoder layers and 6 decoder layers. The common parameters
of the encoder and decoder layers are: $d^{head}=4,d^{attn}=256,d^{ff}=2048$
for the number of attention heads, dimension of attention module, and
dimension of feedforward layer, respectively.

Figure 1:  Semantic diagram of the proposed world-level diarization method.

###  3.2 FD-SOT

As the top three teams all employ TS-VAD to find the overlap between speakers
in the M2MeT challenge speaker diarization track [23], we also re-implemented
it and achieved DERs of 4.20% and 5.42% on AliMeeting Eval and Test sets,
respectively. To further obtain the speaker-attributed transcriptions, we
combine the results of TS-VAD and SOT by aligning the timestamps. This
approach is named as frame-level diarization with SOT (FD-SOT). The detailed
process of FD-SOT is showing as follows:

  * 1)

We estimate the number of the utterances using the oracle sentence
segmentation, says $\hat{N}$, according to the diarization output of TS-VAD.

  * 2)

The utterance number of SOT output is defined as $N$. If $\hat{N}$ is equal to
$N$, no further effort is required.

  * 3)

If $\hat{N}$ is larger than $N$, we select $N$ out of the utterances that have
the longest duration in the TS-VAD diarization output, and discard other
utterances.

  * 4)

If $\hat{N}$ is smaller than $N$, we select $\hat{N}$ out of utterances that
have the longest text length in the SOT output, and discard other utterances.

  * 5)

Finally, we match utterances between TS-VAD and SOT in chronological order.

Figure 2:  Semantic diagram of compared approaches, including FD-SOT, WD-SOT
and TS-ASR.

###  3.3 WD-SOT

In the proposed world-level diarization (WD) method, as shown in Fig. 1, we
first use three individual encoders to encode the multi-talker hypotheses,
speech features and speaker embeddings. Given the encoded hypotheses
$H=\\{\mathrm{h}_{l}|l=1,\dots,L\\}$ and features
$X=\\{\mathrm{x}_{t}|t=1,\dots,T\\}$, a multi-head attention is used to
produce the aggregated feature representation $\mathrm{r}_{l}$ for each token:

$\mathrm{r}_{l}=\sum_{t=1}^{T}{a_{l,t}W_{v}\mathrm{x}_{t}}$ (1)

which is thought to include both acoustic and semantic information. $a_{l,t}$
is calculated by

$\begin{split}\alpha_{l,t}&=\text{dot}(W_{q}\mathrm{h}_{l},W_{k}\mathrm{x}_{t}),\\\
a_{l,t}&=\frac{\exp(\alpha_{l,t})}{\sum_{i=1}^{T}\exp(\alpha_{l,i})}.\end{split}$
(2)

$W_{q}$, $W_{k}$ and $W_{v}$ are trainable parameters in the attention layer.
Next, the context-independent (CI) score $S^{CI}$ is derived from the dot
product between the encoded speaker embeddings $\mathbf{v}_{n}$ and the
aggregated representations $\mathbf{r}_{l}$:

$S^{CI}_{l,n}=\text{dot}(\mathrm{r}_{l},\mathrm{v}_{n}).$ (3)

While CI scores only consider the representations of the current speaker, the
contextual information of different speakers is also useful to identify the
activated speaker from others. Therefore, we further design a context-
dependent (CD) score $S^{CD}_{l,n}$, which is defined as follows:

$S^{CD}_{l,n}=f(\mathbf{r}_{l},\mathbf{v}_{n};R,\Theta)$ (4)

where $f$ is a context-aware function, e.g., the self-attention based networks
(SAN) [16], and $\Theta$ are learnable parameters of $f$.
$R=\\{\mathbf{r}_{l}|l=1,\dots,L\\}$ contains all aggregated representations
in an utterance. Finally, the CI and CD scores are concatenated and fed to a
post-processing network to predict the corresponding speaker for each
character.

In this study, a four-layer self-attention based encoder is employed to encode
the recognized text with 8 attention heads and 256 hidden units in each layer.

###  3.4 TS-ASR

Target-speaker separation modules generate target-speaker representation from
multi-speaker signals by enrollment embedding for ASR. With the premise that
optimizing front-end and back-end separately will lead to sub-optimal
performance, joint modeling will make the whole system matching the final
metric. Meanwhile, we adopt TS-VAD described in Section 3.2 and d-vector
extraction network to obtain the speaker embeddings required by the target-
speaker separation module.

For the structure of target-speaker separation model , we compared the
performance between Conformer [17] and convolutional recurrent network (CRN)
[36]. Target-speaker separation Conformer network consists of 6 encoder layers
with 4 attention heads, 256 attention dimensions and 2048 dimensional feed-
forward network. CRN network consists of 2 bi-directional long short-term
memory (BLSTM) layers with 256 hidden dimensions and a convolutional encoder-
decoder (CED) with skip connections of corresponding encoder and decoder
layer. We employ a Res2Net-based d-vector extraction network trained on the
VoxCeleb corpus [37, 38] as the speaker embedding model. Noted that the
d-vector is combined with both front-end models using feature-wise linear
modulation (FILM) [39], which shows better performance on the multi-speaker
and noisy datasets. In order to control the TS-ASR model parameters to be
consistent with the ASR model of previous approaches, we adopt the ASR model
with 6 layers of encoder.

##  4 Experiments

###  4.1 Training details

In our work, the 80-dimensional log Mel-filter bank feature (Fbank) is used as
the input feature. The window size is 25 ms with a shift of 10 ms. We use 4950
Chinese characters extracted from the training transcriptions as the modeling
units. SOT models are trained directly based on Train-Ali-far-bf, Train-Ali-
near and Train-Ali-simu. For front-end and back-end joint models, we first
pre-train each module with Train-Ali-simu and Train-Ali-near, respectively.
Then we use the Train-Ali-far-bf data for fine-tuning and use ASR loss
function to update the parameters of the whole model for joint training. We
train the model for 100 epochs with the Adam optimizer.

###  4.2 Evaluation metric

We use two evaluation metrics in our experiments, referring as speaker
independent- (SI-) and speaker dependent- (SD-) character error rate (CER)
[40]. The SI-CER is designed to measure the performance of multi-speaker ASR
task (e.g, track 2 of M2Met challenge [22, 23]), which ignores the speaker
labels. On the other hand, the SD-CER, slightly different from cpWER [28], is
calculated by comparing the ASR hypothesis and the reference transcription of
the corresponding speaker instead of all possible speaker permutations. The
SD-CER is more rigorous in calculation due to the global calculation of the
entire meeting, which needs to determine the exact global speaker ID for each
utterance in the whole meeting.

###  4.3 Comparison of different SA-ASR approaches

As shown in Table 1, we evaluate our SA-ASR approaches on AliMeeting Eval and
Test sets. We first compare the speaker-based and utterance-based FIFO
training schemes of SOT models described in Section 3.1. Consistent with the
conclusion of [35], we find that the utterance-based FIFO training
significantly outperforms the speaker-based FIFO training with over 1%
absolute SI/SD-CER reduction for all SOT-based approaches, due to the high
overlap ratio and frequent speaker turns of AliMeeting. Based on this
conclusion, we employ the utterance-based FIFO scheme in the remaining
experiments.

For the SOT-based model, we regard the SI-CER result of SOT as the topline of
SD-CER results, assuming that each token matches the correct speaker. We can
see that our proposed WD-SOT approach outperforms the FD-SOT approach, leading
to 12.2% (41.0% $\to$ 36.0%) and 9.6% (41.2% $\to$ 37.1%) relative SD-CER
reduction on Eval and Test set, respectively. Compared with SOT-based SA-ASR
models, our proposed TS-ASR models achieve the lowest averaged SD-CER.
Specifically, TS-ASR (CRN) approach achieves SD-CERs of 32.5% and 35.1% on
Eval and Test sets, respectively.

Table 1: SA-ASR results for various modular approaches on Eval and Test sets (%). Approach | Metric | Eval | Test | Average
---|---|---|---|---
SOT [22, 23] | SI-CER | 29.7 | 30.9 | 30.6
FD-SOT | SD-CER | 41.0 | 41.2 | 41.2
WD-SOT | 36.0 | 37.1 | 36.8
TS-ASR (Confomer) | 34.8 | 34.7 | 34.7
TS-ASR (CRN) | 32.5 | 35.1 | 34.4

###  4.4 Comparison of various strategies for WD-SOT

We further compare the effect of various strategies for the WD-SOT approach,
and the results are shown in Table 2. The first-row result of WD-SOT is
trained using the ground truth transcripts of Train-Ali-far-bf only. To
increase the robustness of the model, we add the hypothetical transcriptions
to the training set, which dramatically decreases the average SD-CER from
39.1% to 37.9%. From the comparison of contextual information, we can conclude
that when the WD-SOT captures more context information through the self-
attention layer, the performance can be significantly improved, which achieves
2.9% (37.9% $\to$ 36.8%) relative average SD-CER reduction. Our WD-SOT
approach is based on the results of SOT, so the SOT performance will seriously
affect the performance of the whole framework. Considering the position offset
of the separator $\langle\text{sc}\rangle$, we intend to investigate the
impact of the separator prediction accuracy on the model performance. After
replacing the oracle separator, the overall performance has been improved from
36.8% to 36.3%, especially on Test sets with 2.5% relative SD-CER reduction.

Table 2: Comparison of the WD-SOT results of various strategies on Eval and Test sets (SD-CER %). Approach | Eval | Test | Average
---|---|---|---
WD-SOT | 40.6 | 38.5 | 39.1
\+ Hypothetical transcriptions | 37.8 | 38.0 | 37.9
\+ Contextual information | 36.0 | 37.1 | 36.8
\+ Oracle separator $\langle\text{sc}\rangle$ | 36.4 | 36.2 | 36.3

###  4.5 Impact of minimum time of diarization utterances for TS-ASR approach

In the TS-ASR approach, we need to determine the speakers within an oracle
sentence segment, which relies on two estimated information: the oracle
speaker labels and the speaker diarization results. Surprisingly, we obtain
better recognition performance by using speaker diarization results (0s) for
both TS-ASR approaches, comparing with the oracle speaker labels, leading to
7.0%/8.2% and 2.5%/3.4% relative SD-CER reduction on Eval and Test set for
Conformer and CRN TS-ASR approaches, as shown in Table 3. By analyzing the
decoding results, we find that some interfering speech is recognized when
target-speaker speech duration is short, resulting in a large number of
insertion errors. Compared with the insertion errors caused by oracle speaker
labels covering all speaker speech, the deletion errors caused by speaker
diarization results ignoring short speaker speech are fewer. Based on this
finding, we further investigate the impact of minimum time of diarization
utterances for TS-ASR approaches. From the table, we can see that both TS-ASR
models achieve the best results at minimum time equal to 0.5 s, which brings
absolute SD-CER reduction ranging from 0.5% to 0.8% on the Eval and Test sets
compared with the TS-ASR approaches without deleting short speaker speech.

Table 3: TS-ASR results of the different minimum time of diarization utterances on Eval and Test sets (SD-CER %). |  |  | Speaker diarization
---|---|---|---
Approach | Set | Oracle | 0s | 0.3s | 0.5s | 0.7s
Confomer | Eval | 37.4 | 34.8 | 34.4 |  34.3 | 34.3
CRN | 35.4 | 32.5 | 32.0 |  31.9 | 31.9
Confomer | Test | 35.6 | 34.7 | 34.3 |  34.2 | 34.3
CRN | 36.3 | 35.1 | 34.5 |  34.3 | 34.4

###  4.6 Effect of joint training for TS-ASR approach

The comparison results between Conformer and CRN TS-ASR approaches with
different optimization strategies are shown in Table 4. Here, the front-end
modules are pre-trained on Train-Ali-simu set and back-end modules are pre-
trained with Train-Ali-near data, respectively. The difference between the
separated and joint training strategies is whether we use ASR loss function to
update the front-end module when we use Train-Ali-far-bf data to fine-tune the
whole model. According to the Table 4, joint optimization for Conformer and
CRN TS-ASR approaches leads to 26.8% (47.4% $\to$ 34.7%) and 23.9% (45.1%
$\to$ 34.3%) relative average SD-CER reduction on Eval and Test set,
respectively. We conclude that the joint optimization can make the front-end
module more suitable and less distorted for the back-end ASR.

Table 4: The comparison of separated v.s. joint optimization for TS-ASR on Eval and Test sets (SD-CER %). Approach | Optimize strategy | Eval | Test | Average
---|---|---|---|---
Conformer | Separated | 46.0 | 48.0 | 47.4
Joint | 34.8 | 34.7 | 34.7
CRN | Separated | 43.3 | 45.8 | 45.1
Joint | 32.5 | 35.1 | 34.3

##  5 Conclusion

In this study, three SA-ASR approaches are evaluated on the AliMeeting corpus,
a challenging meeting dataset with multi-talker conversation. Compared with
the baseline approach, FD-SOT, the proposed WD-SOT approach addresses the
alignment errors by introducing a word-level diarization model and results in
10.7% relative average SD-CER reduction. To further get rid of the dependence
on multi-talker ASR output, the proposed TS-ASR approach trains a target-
speaker separation module and an ASR module jointly, which leads to 16.5%
relative average SD-CER reduction compared with FD-SOT. Moreover, ignoring
short diarization utterances can bring 0.8% absolute SD-CER reduction for the
TS-ASR task. In the future, we will investigate how to incorporate the single-
speaker ASR trained on large-scale data into our proposed approaches for real-
world applications.

##  6 Acknowledgement

This work was supported by Alibaba Group through Alibaba Research Intern
Program and Key R & D Projects of the Ministry of Science and Technology
(2020YFC0832500).

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