# Primordial black holes and secondary gravitational waves from the inflation
potential with a tiny bump

Wei<EMAIL_ADDRESS>Yu-Xuan<EMAIL_ADDRESS>Arshad
<EMAIL_ADDRESS>Tao-Tao<EMAIL_ADDRESS>Chen-Hao
<EMAIL_ADDRESS>Ya-Peng<EMAIL_ADDRESS>corresponding
author aCollege of Physics, Nanjing University of Aeronautics and
Astronautics, Nanjing, 211106, China
bSchool of Physics Sciences, University of Science and Technology of China,
Hefei 230026, China
cInstitute for Advanced Study & School of Physical Science and Technology,
Soochow University, Suzhou 215006, P.R. China
dKey Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT,
Nanjing 211106, China

###### Abstract

This paper explores the generation of primordial black holes (PBHs) and
scalar-induced gravitational waves (SIGWs) from the inflation potential with a
tiny bump. We propose a Lorentz function that makes a tiny bump characteristic
of the inflation model potential. This property makes the scalar field move
locally ultra slowly, which not only makes the primordial curvature power
spectrum have $\mathcal{O}(10^{-2})$ peaks at a small scale but also satisfies
observational constraints of the cosmic microwave background (CMB) on a large
scale. Specifically, we calculate the abundances of PBHs for different mass
ranges in this model, where PBHs with mass $10^{-12}M_{\odot}$ can make up
almost all dark matter and PBHs with mass $10^{-5}M_{\odot}$ can explain OGLE
ultrashort-timescale microlensing events. Moreover, we find that SIGWs
accompanying the PBHs can be tested by the Square Kilometre Array (SKA),
TianQin, Taiji, Laser Interferometer Space Antenna (LISA), and DECIGO. As for
the parameter set I, the consequent SIGWs can explain the NANOGrav 12.5yrs
signal.

Primordial black holes; Inflationary cosmology; Dark matter; Starobinsky
inflation model; Scalar induced gravitational waves.

##  I Introduction

The investigation of gravitational waves (GWs) has played a pivotal role in
cosmology and astrophysics since the formulation of the theory of general
relativity in the 20th century. The Laser Interferometer Gravitational-Wave
Observatory (LIGO) achieved a historic milestone by successfully detecting GW
signals originating from binary black hole mergers Abbott  _et al._ (2016).
Subsequently, the LIGO/Virgo collaborations extended this achievement to
encompass mergers involving black holes and neutron stars, ushering in a new
era of multi-messenger astronomy Abbott  _et al._ (2019). Notably, the
emergence of GW signals from such events has spurred heightened interest in
primordial black holes (PBHs) as they are posited to explain binary black hole
mergers Bird  _et al._ (2016); Sasaki  _et al._ (2016).

Several astrophysical observations provide credible evidence that dark matter
(DM) is the non-negligible element of our universe Rubin  _et al._ (1980);
Persic  _et al._ (1996); Bertone and Hooper (2018). As viable candidates of
cold DM, PBHs have also attracted lots of attention Frampton  _et al._ (2010);
Clesse and García-Bellido (2015); Inomata  _et al._ (2017); García-Bellido
(2017); Kovetz (2017); Carr  _et al._ (2016, 2021); Carr and Kuhnel (2020). A
large number of different types of cosmological and astronomical observations
have been made on the abundance of primordial black holes as dark matter in
many mass ranges Limit is givenAli-Haïmoud and Kamionkowski (2017); Poulin
_et al._ (2017); Carr  _et al._ (2010); Graham  _et al._ (2015); Niikura  _et
al._ (2019); Griest  _et al._ (2013); Tisserand  _et al._ (2007); Vaskonen and
Veermäe (2020). However, the abundances of PBHs at the mass windows of
$10^{-17}-10^{-15}M_{\odot}$ and $10^{-14}-10^{-12}M_{\odot}$ are largely
unconstrained by observations and specific assumptions, and can be explained
almost all elements of the DM Green and Kavanagh (2021), and PBHs with
planetary masses can explain the ultrashort-timescale microlensing events in
the OGLE dataNiikura  _et al._ (2019).

There are many PBH formation mechanisms, the most popular one is the overdense
region gravitational collapse inside a Hubble horizon, where the density
exceeds the threshold value Hawking (1971). Many situations can form the
overdense region, such as enhancement of curvature perturbation Garcia-Bellido
and Ruiz Morales (2017); Di and Gong (2018); Sasaki  _et al._ (2018),
accumulation of topological defect Garriga and Vilenkin (1993); Polnarev and
Zembowicz (1991); Deng  _et al._ (2017); Eroshenko (2021), and postponed false
vacuum decay during first-order transition Lewicki  _et al._ (2023); Liu  _et
al._ (2022). Beyond PBHs, these perturbations also contribute to the
production of scalar-induced gravitational waves (SIGWs)Saito and Yokoyama
(2009, 2010); Orlofsky  _et al._ (2017); Wang  _et al._ (2018); Cai  _et al._
(2019); Yi (2023), which can be tested by space-based GW detectors, such as
the Laser Interferometer Space Antenna (LISA) Amaro-Seoane  _et al._ (2017),
Taiji Hu and Wu (2017), TianQin Luo  _et al._ (2016), and Deci-hertz
Interferometer Gravitational-Wave Observatory (DECIGO) Kawamura  _et al._
(2006) in the future. To produce a sufficient abundance of PBHs as DM and
detectable SIGWs, we need a sufficient large-density perturbation at a small
scale.

The traditional single-field and multi-field inflation models can enhance
curvature perturbations by fine-tuning the parameters Xu  _et al._ (2020);
Anguelova (2021); Zhao  _et al._ (2023); Chen  _et al._ (2023); Yi  _et al._
(2021). The form of perturbation of the potential function is not uniqueMishra
and Sahni (2020); Fu and Chen (2023). Some people use Gaussian and Lorentz
functions to construct antisymmetric perturbationsWang  _et al._ (2021); Zhao
_et al._ (2023). In this paper, we propose a tiny bump in the form of a
Lorentz function as a local correction of the basic inflation potential
$V_{b}(\phi)$, which can create a perfect platform in the inflation potential
function to bring inflation into the ultra-slow roll (USR) phase, and it
exhibits good symmetry in the USR region for $\phi_{0}$. Nevertheless, we find
that this property does not affect the inflation potential to satisfy the
cosmic microwave background (CMB) observation constraints on a large scale,
but also leads to peaks in the curvature power spectrum. We use three sets of
parameters to numerically calculate the corresponding peaks of the power
spectrum of the primordial curvature perturbations, and find that they all
reach $\mathcal{O}(10^{-2})$ on a small scale. It is worth noting that the
enhancement of the curvature perturbations corresponding to different scales
also promotes the generation of PBHs with different masses and SIGWs with
different frequencies.

This paper is organized as follows. In section II, we propose a bump function
and calculate the slow-roll parameters In section III, we numerically
calculate the power spectrum of the primordial curvature perturbations,
$n_{s}$, $r$ and the abundance of PBH. In section IV, we study the energy
density spectrum of SIGWs. Finally, we draw our conclusion and discussion in
section V.

##  II A tiny bump in potential and slow roll inflation

In this paper, we study single-field inflation, and its corresponding action
is

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi-V(\phi)\right].$
(1)

Our model of a small local perturbation forming PBH is based on potential with
the general form Mishra and Sahni (2020)

$V(\phi)=V_{b}(1+f(\phi)).$ (2)

We have considered this general form of the potential as a phenomenological
model. Where $V_{b}$ is the base inflation potential responsible for
generating quantum fluctuations compatible with the CMB constraints, Here
$f(\phi)$ represents a tiny local bump, so here it’s a small quantity much
smaller than 1.

Of the numerous inflation models, the Starobinsky inflation model is almost
the most well-known one and it is well-aligned with current CMB measurements
Starobinsky (1980); Mishra  _et al._ (2018). Thus, we choose the base
potential $V_{b}$ in Eq.(2) is associated with this inflation model

$V_{b}=\frac{3}{4}m^{2}\left(1-e^{-\sqrt{\frac{2}{3}}\phi}\right)^{2},$ (3)

where $m=1.13\times 10^{-5}m_{p}$, denotes the mass parameter. Here, we adopt
the reduced Planck mass as $m_{p}=\frac{1}{\sqrt{8\pi G}}=1$. We suggest a new
function form of this local bump as

$f(\phi)=bL(\sqrt{c}(\phi-\phi_{0}))=\frac{b}{1+c(\phi-\phi_{0})^{2}},$ (4)

where $L(\sqrt{c}(\phi-\phi_{0}))$ is a Lorentz function, which Lorentz
function acts as a local bump in the potential function, and to ensure that
the potential function can enter a relatively flat platform, the value of $c$
should be much larger than 1 to ensure that the width of the peak is
relatively large. $\phi_{0}$ controls the bump position of the potential
function, which is also a field value in inflation, and $b$ represents the
amplitude, and the nature of the small perturbations causes $b$ to be limited
to a range of $0<b\ll 1$. Notably, when $b=0$, the model reduces to the
Starobinsky inflation model for the Eq.(2). This feature of the local bump
will cause the inflation potential to enter the USR phase near the $\phi_{0}$,
and we will introduce slow rolling inflation below.

In the context of a spatially flat homogeneous and isotropic universe with
FLRW metric $ds^{2}=-dt^{2}+a(t)\delta_{ij}dx^{i}dx^{j}$, the Friedmann
equation and dynamical equations of the inflation field $\phi$ are as follows
Garcia-Bellido and Ruiz Morales (2017)

$\displaystyle 3H^{2}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$
$\displaystyle\dot{H}=-\frac{1}{2}\dot{\phi}^{2},$
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{\phi}(\phi)=0.$ (5)

Here, $a(t)$ is the scale factor, and $H=\dot{a}/a$ represents the Hubble
parameter. For the homogeneous background, the inflation field is only time-
dependent, $\phi=\phi(t)$. During the inflation epoch, the scalar field’s
movement is characterized by extreme slowness, with the potential energy being
the dominant contributor to the scalar field’s energy. The scalar field
adheres to the slow-roll condition, which entails
$\frac{1}{2}\dot{\phi}^{2}\ll V(\phi)$ and $|\ddot{\phi}|\ll 3H|\dot{\phi}|$.
The dynamics equations to approximate Yi  _et al._ (2021); Di and Gong (2018)

$\displaystyle 3H^{2}\simeq V(\phi),~{}~{}~{}~{}3H\dot{\phi}\simeq-
V_{\phi}(\phi).$ (6)

The slow-roll parameters are defined as

$\displaystyle\epsilon_{H}=-\frac{\dot{H}}{H^{2}},~{}~{}~{}~{}~{}~{}\eta_{H}=-\frac{\ddot{\phi}}{H\dot{\phi}}.$
(7)

The $\epsilon_{H}\ll 1,|\eta_{H}|\ll 1$ during inflation, the end of inflation
condition is $\epsilon_{H}=1$. In this paper, we choose the initial value time
of inflation $\phi_{*}=5.42$, which makes $\epsilon_{H}(\phi_{*})\approx
10^{-4}$ satisfies the slow-roll condition, and we can obtain
$\phi_{\text{end}}=0.614$ by $\epsilon_{H(\phi_{\text{end}})}=1$. Therefore,
the parameter $\phi_{0}$ of the bump function has a range value of
$0.614<\phi_{0}<5.42$. According to the above constraints of $b$ and $c$, we
choose an optimal set of parameters, $b=4\times 10^{-4}$, $c=10013.8$,
$\phi_{0}=5.1$, and they are brought into Eq.(4) to satisfy $f(\phi)\ll 1$.
The evolution of the potential function containing a bump for $\phi$ as shown
in Fig.1, It is not difficult to see that the potential function changes very
little near $\phi_{0}$, which will also lead to The number of e-folds is
superimposed during inflation, according to the formula of number of e-folds

$\displaystyle
N=\int_{\phi_{*}}^{\phi_{in}}\frac{1}{\sqrt{2\epsilon_{H}}}d\phi,$ (8)

where $\phi_{in}$ is scalar field values in inflation. Fig.2(a) visually
represents that the value change of the inflation field is very small and the
number of e-folds increases around $\Delta{N}\approx 10$ e-folds near the
$\phi_{0}\approx 5.1$. However, it will not make the total number of e-folds
by the end of inflation too large, when the $\phi_{in}$ closer to
$\phi_{\text{end}}$ can keep the total number of e-folds N around 60, that
satisfy the CMB constraints. This implies that the slow-roll parameter
$\epsilon_{H}$ will have a minimum value in the corresponding region, as shown
in Fig.2(b). The $\epsilon_{H}$ satisfies the inflation condition
$\epsilon_{H}\ll 1$ during inflation and $\epsilon_{H\text{min}}\sim
10^{-10}$, that provides favorable conditions for the calculation of the
primordial curvature power spectrum.

Figure 1: The evolution of the potential function containing a bump for
$\phi$. The small figures show the behavior of the potential around the
$\phi_{0}$, and the parameters are $b=4\times 10^{-4}$, $c=10013.8$,
$\phi_{0}=5.1$.

Figure 2: (a) shows the evolutions of e-folds number $N$. (b) shows the
evolutions of the first slow-roll parameter $\epsilon_{H}$. (c) shows the
evolutions of the second slow-roll parameter $\eta_{H}$.

The well-known slow-roll approximation formula for the power spectrum of
primordial scalar perturbation is given by Garcia-Bellido and Ruiz Morales
(2017); Di and Gong (2018)

$\displaystyle P_{\zeta}=\frac{H^{2}}{8\pi^{2}\epsilon_{H}}.$ (9)

It is inversely proportional to the slow-roll parameter $\epsilon_{H}$, and
due to the flattening of the potential energy near $\phi_{0}$, the Hubble
parameter $H$ can be approximated as constant from Eq.(6). The
$\epsilon_{H\text{min}}\sim 10^{-10}$ near $\phi_{0}$, signifying a reduction
of six orders of magnitude compared to the initial value of
$\epsilon_{H}(\phi_{*})$. Thus, we obtain the approximated results of the
power spectrum is $\mathcal{O}(10^{-3})$, which does not reach
$\mathcal{O}(10^{-2})$. In addition, although when entering an ultra-slow roll
period, the $\epsilon_{H}\ll 1$ implies inflation does not end. But the
$|\eta_{H}|$ is greater than 1 for a short period which seems to violate the
slow roll condition as shown in Fig.2(c). Therefore, Eq.(9) cannot be used to
calculate the power spectrum Yi  _et al._ (2021); Di and Gong (2018), we will
calculate the power spectrum numerically in the next section.

##  III The power spectrum and primordial black holes

In this section, we first investigate the behavior of the power spectrum for
primordial curvature perturbation to ensure it satisfies the constraints of
CMB observations. By numerically solving the Mukhanov-Sasaki equation Mukhanov
(1985); Sasaki (1986)

$\displaystyle\frac{d^{2}v_{k}}{d\eta^{2}}+\left(k^{2}-\frac{1}{z}\frac{d^{2}z}{d\eta^{2}}\right)v_{k}=0,$
(10)

where the conformal time $d\eta=dt/a$, the quantum canonical field
$v_{k}=z\zeta_{k}$ and $z=a\dot{\phi}/H$, we can obtain the scalar power
spectrum

$\displaystyle P_{\zeta}=\frac{k^{3}}{2\pi^{2}}\frac{|v_{k}|^{2}}{z^{2}}.$
(11)

Table 1: Model parameters and the numerical results $set$ | $b(10^{-4})$ | $c$ | $\phi_{0}$ | N | $n_{s}$ | r | $k_{\text{peak}}/\text{Mpc}^{-1}$ | $P_{\zeta(peak)}$ | $\frac{M_{\text{peak}}}{M_{\odot}}$ | $Y_{\text{PBH}}^{\text{peak}}$ | $f_{c}/\text{Hz}$
---|---|---|---|---|---|---|---|---|---|---|---
$I$ | 4 | 10013.8 | 5.1 | 67 | 0.967906 | 0.003023 | $3.13\times{10^{5}}$ | 0.018 | 37.51 | $0.00059$ | $4.84\times{10^{-10}}$
$II$ | 4.1 | 16824 | 4.89 | 62 | 0.967879 | 0.003024 | $5.57\times 10^{8}$ | 0.015 | $1.186\times 10^{-5}$ | 0.026 | $8.61\times{10^{-7}}$
$III$ | 4.4 | 34131.8 | 4.63 | 61 | 0.967876 | 0.003024 | $1.42\times 10^{12}$ | 0.013 | $1.83\times 10^{-12}$ | 0.862 | $2.19\times{10^{-3}}$

In this work, we use three suitable sets of parameters I, II, and III in Table
1, and numerically calculate the power spectrum. We find that three sets of
parameters can not only eliminate the contingency of the single set of
parameters model but also broaden the peak scale $k_{peak}$ range
corresponding to the peak of the power spectrum $P_{peak}$. In Fig.3, we show
the power spectrum results of numerical calculation for models I, II, and III
with blue, red, and black lines, respectively. We find the peak values of the
power spectrum to reach the order of $\mathcal{O}(10^{-2})$ at the scales
$k_{peak}\sim 10^{5}Mpc^{-1}$, $k_{peak}\sim 10^{8}Mpc^{-1}$ and $k_{peak}\sim
10^{12}Mpc^{-1}$ as shown in Table 1.

The local bump of the potential function causes the power spectrum to have a
peak, and the effect on large-scale cosmic microwave background (CMB) is
discussed below. Compared with Planck 2018 CMB observation constraints with a
pivot scale of $k=0.05\,\text{Mpc}^{-1}$, which provide constraints such as
the scalar spectral index $n_{s}=0.9649\pm 0.0042$ (68% CL) and tensor-to-
scalar ratio $r_{0.05}<0.06$ (95% CL) at large scale Akrami  _et al._ (2020);
Ade  _et al._ (2018). The relevant CMB parameters, $n_{s}$ and $r$ are Yi  _et
al._ (2021)

$\displaystyle n_{s}$ $\displaystyle=1+\frac{d\ln{P_{\zeta}}}{d\ln{k}}\simeq
1+2\eta_{H}-4\epsilon_{H},$ $\displaystyle r$
$\displaystyle=\frac{P_{T}}{P_{\mathcal{\zeta}}}\simeq 16\epsilon_{H}.$ (12)

We numerically calculate the scalar spectral index $n_{s}$ and the tensor-to-
scalar ratio $r$ as shown in Table 1, which satisfy the constraints of Planck
2018 CMB observations. The significance and richness of the primordial
curvature perturbation power spectrum peak results indicate that the dark
matter abundance of primordial black holes should have corresponding
significant results.

Figure 3: The scalar power spectrums in the models, are enhanced to
$\mathcal{O}(10^{-2})$ on small scales. The dashed line represents the power
spectrum, and the shaded regions represent the observation constraint Akrami
_et al._ (2020); Inomata and Nakama (2019); Inomata  _et al._ (2016); Fixsen
_et al._ (1996).

Next, we explore the abundance of PBHs within the context of the model. PBHs
could form due to the amplification of small-scale curvature perturbations
during the inflation process, and they can emerge if the density perturbations
exceed a certain threshold. We have ignored non-Gaussian effects in our
analysis for simplicity. As previously mentioned, the comoving curvature
perturbation in our model is assumed to follow the Gaussian distribution. The
most simple method to estimate $\beta$ is to use the Press-Schechter
formalism, the fractional energy density of PBHs of the universe is Press and
Schechter (1974); Di and Gong (2018)

$\displaystyle\beta$
$\displaystyle=\frac{\rho_{\text{PBH}}}{\rho_{tot}}\approx\it{erfc}\left(\frac{9\delta_{c}}{4\sqrt{2P_{\zeta}}}\right)\approx{\sqrt{\frac{2}{\pi}}\frac{\sqrt{P_{\zeta}}}{\mu_{c}}e^{\left(-\frac{\mu_{c}^{2}}{2P_{\zeta}}\right)}},$
(13)

where $\rho_{tot}$ and $\rho_{\text{PBH}}$ represent the energy density of the
cosmic background and PBHs, respectively. The $\it{erfc}(x)$ is the
complementary error function and $\mu_{c}=9\delta_{c}/4$ in the radiation-
dominated period Di and Gong (2018). The threshold of density perturbation for
the formation of the PBHs can be $\delta_{c}=0.4$ Harada  _et al._ (2013).

The PBHs abundance is given as Carr  _et al._ (2016); Di and Gong (2018)

$\displaystyle Y_{\text{PBH}}=$ $\displaystyle\frac{\beta}{3.94\times
10^{-9}}\left(\frac{\gamma}{0.2}\right)^{\frac{1}{2}}\left(\frac{g_{*}}{10.75}\right)^{-\frac{1}{4}}$
$\displaystyle\times\left(\frac{0.12}{{\Omega}_{\text{DM}}h^{2}}\right)\left(\frac{M_{\text{PBH}}}{M_{\odot}}\right)^{-\frac{1}{2}}.$
(14)

In this equation, the numerical factor $\gamma=0.2$, the effective degrees of
freedom $g_{*}=107.5$ for $T>300\,\text{GeV}$ and $g_{*}=10.75$ for
$0.5\,\text{MeV}<T<300\,\text{GeV}$ Carr (1975), Planck measurements provide
the following value for the dark matter abundance
$\Omega_{\text{DM}}h^{2}=0.12$ Aghanim  _et al._ (2020),
$h=H/100=0.6727km\cdot s^{-1}\cdot Mpc^{-1}$, which measures the Hubble rate
in units of 100, and $M_{\text{PBH}}$ and $M_{\odot}$ are the mass of PBHs and
solar mass, respectively.

The relationship between mass and scale of PBHs is given by

$\displaystyle
M_{\text{PBH}}=3.68\left(\frac{\gamma}{0.2}\right)\left(\frac{g_{*}}{10.75}\right)^{-\frac{1}{6}}\left(\frac{k}{10^{6}Mpc^{-1}}\right)^{-2}M_{\odot}.$
(15)

Combine the numerical results of the power spectrum and Eq.(13) and Eq.(15),
we can obtain the peak mass of PBHs, as shown in Fig.4 and Table 1. We find
that the three sets of parameters we choose make the peak scale $k_{peak}$
cover more ground, and also have the wider mass range of the PBHs. In other
words, PBHs with different masses correspond to different peak scales
$k_{peak}$, which will give our results a wider scope of application to
explain the PBHs theory. Specifically speaking, for set I, PBHs with the
stellar mass of approximately $30M_{\odot}$ can be generated. These PBHs might
be explained to the LIGO events Bird  _et al._ (2016); Sasaki  _et al._
(2016). For set II, our model produces PBHs with the mass $M_{\text{PBH}}\sim
10^{-5}M_{\odot}$, which can explain OGLE ultrashort-timescale microlensing
eventsNiikura  _et al._ (2019)and be interpreted as Planet 9 that a
hypothetical outer solar system object used to explain the anomalous orbits of
trans-Neptunian objects Scholtz and Unwin (2020). For set III, PBHs with
masses around $10^{-13}-10^{-11}M_{\odot}$ make up almost all Dark Matter, and
the peak abundance is approximately $Y_{\text{PBH}}\approx 1$.

Figure 4: The abundances of PBH produced by models. The dashed line represents
the PBH abundance, and the shaded regions represent the constraints on PBHs
from various observationsAli-Haïmoud and Kamionkowski (2017); Poulin  _et al._
(2017); Carr  _et al._ (2010); Graham  _et al._ (2015); Niikura  _et al._
(2019); Griest  _et al._ (2013); Tisserand  _et al._ (2007); Vaskonen and
Veermäe (2020); Niikura  _et al._ (2019); Carr  _et al._ (2021)

##  IV Scalar-induced gravitational waves

In this section, to investigate the energy density of SIGWs, we shed light on
the production of SIGWs. The primordial scalar perturbation starts to evolve
after re-entering the event horizon, and its transverse traceless component
induces a stochastic gravitational wave background that can be observed by
different detectors when the small-scale density perturbation is large enough.
Here, we consider the perturbed metric in Newtonian gauge in the cosmological
background which reads Kohri and Terada (2018)

$\displaystyle ds^{2}$ $\displaystyle=-a^{2}(\eta)(1+2\mathbf{\Phi})d\eta^{2}$
$\displaystyle\quad+a^{2}(\eta)\left[(1-2\mathbf{\Phi})\delta_{ij}+\frac{1}{2}h_{ij}\right]dx^{i}dx^{j},$
(16)

where $a(\eta)$ is the scale factor of the universe. The scalar perturbation
$\mathbf{\Phi}$ is of the first order, and the transverse traceless part
$h_{ij}$ is the second-order tensor mode. In the following, in the evaluations
of SIGWs, first, we briefly analyze the well-known results on $h_{ij}$. The
Fourier component equation of $h_{ij}$ can be expressed as

$\displaystyle h_{ij}(\eta,\mathbf{x})$
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3/2}}\left[e^{+}_{ij}(\mathbf{k})h^{+}_{\mathbf{k}}(\eta)+e^{\times}_{ij}(\mathbf{k})h^{\times}_{\mathbf{k}}(\eta)\right]e^{i\mathbf{k}\cdot\mathbf{x}},$
(17)

where the polarization tensors are defined as Kohri and Terada (2018); Yi  _et
al._ (2021)

$\displaystyle
e^{+}_{ij}(\mathbf{k})=\frac{1}{\sqrt{2}}[e_{i}(\mathbf{k})e_{j}(\mathbf{k})-\bar{e_{i}}(\mathbf{k})\bar{e_{j}}(\mathbf{k})],$
(18) $\displaystyle
e^{+}_{ij}(\mathbf{k})=\frac{1}{\sqrt{2}}[e_{i}(\mathbf{k})\bar{e_{j}}(\mathbf{k})+\bar{e_{i}}(\mathbf{k}){e_{j}}(\mathbf{k})],$
(19)

with $e_{i}(\mathbf{k})$ and $\bar{e_{i}}(\mathbf{k})$ being normalized
vectors orthogonal to each other and to $k$. Satisfying
$e_{i}(\mathbf{k})\cdot\bar{e_{i}}(\mathbf{k})=e_{i}(\mathbf{k})\cdot\mathbf{k}=\bar{e_{i}}(\mathbf{k})\cdot\mathbf{k}$.
From perturbation of the Einstein’s field equations
$G_{\mu\nu}=8{\pi}GT_{\mu\nu}$ to the second order, the Fourier component
equation of the second order tensor perturbation $h_{k}$ is Kohri and Terada
(2018); Ananda  _et al._ (2007); Baumann  _et al._ (2007)

$\displaystyle
h_{\mathbf{k}}^{\prime\prime}(\eta)+2\mathcal{H}h_{\mathbf{k}}^{\prime}(\eta)+k^{2}h_{\mathbf{k}}(\eta)=4S_{\mathbf{k}}(\eta),$
(20)

where, ′ denotes $\frac{d}{d\eta}$, $\eta$ is conformal time $\eta=\int
dt/a(t)$, and the conformal Hubble parameter is $\mathcal{H}=a^{\prime}/a$,
and the source term is Kohri and Terada (2018)

$\displaystyle S_{\mathbf{k}}$
$\displaystyle=\int\frac{d^{3}q}{(2\pi)^{3/2}}e_{ij}(\mathbf{k})q_{i}q_{j}\Big{[}2\mathbf{\Phi}_{\mathbf{q}}\mathbf{\Phi}_{\mathbf{k-q}}+$
$\displaystyle\quad\left(\mathcal{H}^{-1}\mathbf{\Phi}^{\prime}_{\mathbf{q}}+\mathbf{\Phi}_{\mathbf{q}}\right)\left(\mathcal{H}^{-1}\mathbf{\Phi}^{\prime}_{\mathbf{k-q}}+\mathbf{\Phi}_{\mathbf{k-q}}\right)\Big{]}.$
(21)

During the radiation-dominated era, the scalar perturbation satisfies the
following equation of motion in Fourier space to be expressed as

$\displaystyle\Phi_{k}^{\prime\prime}+4\mathcal{H}\Phi^{\prime}_{k}-\frac{k^{3}}{3}\Phi_{k}=0,$
(22)

where $\mathbf{\Phi_{k}}$ is the Fourier component of the gravitational
potential, and it can be related to its primordial value $\phi_{k}$ by the
transfer function

$\displaystyle\mathbf{\Phi}_{\mathbf{k}}=\mathbf{\Phi}(k\eta)\phi_{\mathbf{k}}.$
(23)

The transfer function for the radiation-dominated period is given by

$\displaystyle\mathbf{\Phi}(k\eta)=\frac{9}{(k\eta)^{2}}\left(\frac{\sin(k\eta/\sqrt{3})}{k\eta/\sqrt{3}}-\cos(k\eta/\sqrt{3})\right),$
(24)

and its two-point correlation function is determined by the power spectrum of
the curvature perturbation

$\displaystyle\langle\phi_{\mathbf{k}}\phi_{\mathbf{k^{\prime}}}\rangle=\delta(\mathbf{k}+\mathbf{k^{\prime}})\frac{2\pi^{2}}{k^{3}}\left(\frac{3+3w}{5+3w}\right)^{2}P_{\zeta}(k).$
(25)

The power spectrum of induced gravitational waves can be written as

$\displaystyle\langle
h^{\lambda}_{\mathbf{k}}(\eta)h^{\lambda^{\prime}}_{\mathbf{k}^{\prime}}(\eta)\rangle=\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}+\mathbf{k}^{\prime})\frac{2\pi^{2}}{k^{3}}P_{h}(k,\eta),$
(26)

where $\lambda,\lambda^{\prime}=+,\times$ represents the polarization index,
which we omit in the following. We solve the tensor Eq.(20) by employing the
Green’s function method, we obtain $h_{\mathbf{k}}(\eta)$ is expressed as

$\displaystyle
h_{\mathbf{k}}(\eta)=\frac{4}{a(\eta)}\int^{\eta}d\bar{\eta}G_{k}(\eta,\bar{\eta})a(\bar{\eta})S_{\mathbf{k}}(\bar{\eta}),$
(27)

during the radiation-dominated period, the Green’s function of GW is

$\displaystyle G_{k}(\eta,\bar{\eta})=\frac{\sin[k(\eta-\bar{\eta})]}{k}.$
(28)

The fraction of the GW energy density is Kohri and Terada (2018); Ali  _et
al._ (2023)

$\displaystyle\Omega_{\text{GW}}(k,\eta)=\frac{1}{24}\left(\frac{k}{a(\eta)H(\eta)}\right)^{2}\overline{P_{h}(k,\eta)}.$
(29)

Substituting the solution Eq.(27) into Eq.(26), Then the calculated the power
spectrum for the SIGWs $P_{h}$ is brought into Eq.(29) to obtain the fraction
of energy density Kohri and Terada (2018); Lu  _et al._ (2020)

$\displaystyle\Omega_{\text{GW}}(k)=$
$\displaystyle\frac{1}{6}\left(\frac{k}{aH}\right)^{2}\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du$
(30)
$\displaystyle\left\\{\left[\frac{4v^{2}-(1-u^{2}+v^{2})^{2}}{4uv}\right]^{2}\right.$
$\displaystyle\left.\times
I_{R}^{2}(u,v,x\rightarrow\infty)\mathcal{P_{\zeta}}(kv)\mathcal{P_{\zeta}}(ku)\right\\},$

where $u=|\mathbf{k-\widetilde{k}}|$, $v=\widetilde{k}/k$, $x=k\eta$, and
$I_{R}$ is kernel function. The current frequency $f$ and wave number $k$ of
the induced gravitational wave satisfy the following equation Fu  _et al._
(2020)

$\displaystyle f=1.546\times 10^{-15}\frac{k}{1Mpc^{-1}}Hz.$ (31)

The fractional energy density of SIGWs today

$\Omega_{\text{GW,0}}(k)=\frac{\Omega_{\text{r,0}}(k)}{\Omega_{\text{r}}(k)}\Omega_{\text{GW}}(k)$
(32)

where $\Omega_{\text{r,0}}(k)=9.17\times 10^{-5}$ is the current fractional
energy density of radiation, and we choose $\Omega_{\text{r}}(k)=1$ during the
radiation dominationYi  _et al._ (2021); Zhang (2022). By substituting the
numerical results of the power spectrum into Eq.(32), we obtain the energy
density $\Omega_{\text{GW,0}}$ of the SIGWs for the corresponding model and
the peak frequencies $f_{c}$ of these SIGWs are shown in Table 1, we see that
the peak frequencies of the SIGWs are nHz, $\mu$Hz and mHz, respectively.

Figure 5: The energy density spectrum of SIGWs. The curves show the
sensitivity of current and future GW observatories. The black dashed curve,
the green dot-dashed curve, and gray dotted curve denotes the EPTA limit Hobbs
_et al._ (2010); McLaughlin (2013); Hobbs (2013); Ferdman  _et al._ (2010);
Lentati  _et al._ (2015), PPTA Shannon  _et al._ (2015) and the SKA limitMoore
_et al._ (2015), respectively. The other limits from the TianQin Luo  _et al._
(2016), Taiji Hu and Wu (2017), the DECIGOKawamura  _et al._ (2006), the LISA
Amaro-Seoane  _et al._ (2017) and the aLIGO Abbott  _et al._ (2016, 2019),
respectively. The blue region is the observational result from the North
American Nanohertz Observatory for Gravitational Wave (NANOGrav) 12.5-year
dataArzoumanian  _et al._ (2020)

In this part, we show the predictions of the energy density spectrum of GWs
for the above-mentioned parameters in Fig.5 and compare them with the
constraints and future sensitivity curves for the GW observatories. As for the
parameter set I, the consequent SIGWs (blue) can explain the NANOGrav 12.5yrs
signal and can be tested by SKA. The model with parameter set III can be
tested by LISA/Taiji/TianQin/ DECIGO.

##  V Conclusion and discussion

In this paper, we focus on the generation of PBHs and SIGWs within an
inflation potential bump. We propose a simple Lorentz function form of bump as
a local correction of the base inflation potential, which can show good
symmetry about $\phi_{0}$ in the USR region. At the same time, Our numerical
results for the scalar spectral index $n_{s}$ and tensor-to-scalar ratio $r$
predicted by this model align with the Planck 2018 results. Furthermore, our
inflation mechanism enhances the power spectrum of the primordial curvature
perturbation to $\mathcal{O}(10^{-2})$ at the scales $k_{peak}\sim
10^{5}Mpc^{-1}$, $k_{peak}\sim 10^{8}Mpc^{-1}$ and $k_{peak}\sim
10^{12}Mpc^{-1}$, that satisfies the prerequisites for generating PBHs.

Subsequently, we derive that the abundances of PBH with stellar mass,
planetary mass, and $10^{-12}M_{\odot}$ are $\mathcal{O}(10^{-4})$,
$\mathcal{O}(10^{-2})$, and approximately $1$, respectively. The PBHs with
mass $10^{-12}M_{\odot}$ can make up almost all dark matter, and PBHs with
mass $10^{-5}M_{\odot}$ can explain OGLE ultrashort-timescale microlensing
events and can be explained to Planet 9. As is widely recognized, PBHs
originate from matter collapse triggered by high-density perturbations in the
early universe. It is important to note that the process of PBH formation is
concomitant with the generation of SIGWs. Leveraging the PBHs mass
distribution and the primordial curvature power spectrum, we have extracted
the characteristics of the gravitational wave signal, focusing on the
frequency and amplitude. Promisingly, we get these SIGWs that can be tested by
gravitational wave observatories like SKA, Tianqin, Taiji, LISA, and DECIGO.
As for the parameter set I, the consequent SIGWs (blue) can explain the
NANOGrav 12.5yrs signal.

We give a new function form of the local correction term of the inflation
model, this function does not depend on the concrete form of the model, and we
can try to apply it in more models. In addition, we have ignored non-
Gaussianity and quantum stochastic effects in our analysis for simplicity,
which is to be examined in the future. Future studies could include these
effects to obtain more precise results for PBH formation with our bump
potentialDe Luca  _et al._ (2019); Matsubara and Sasaki (2022); Cai  _et al._
(2018); Franciolini  _et al._ (2018). The different calculation methods of PBH
mass fraction $\beta$ often lead to different final results. We have used the
Press-Schechter (PS) theory to calculate in our present study. Hence, it will
be challenging and meaningful work to use other theories such as peak theory
Bardeen  _et al._ (1986); Green  _et al._ (2004) to calculate $\beta$ in this
model and compare it with the results of the PS theory. These questions
deserve further investigation in future research.

## Acknowledgements

We would like to thank Dr. Zhu Yi and our group members for their assistance.
This work is supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 12175105, No. 11575083, No. 11565017, No. 12147175, Ya-
Peng Hu is supported by the Top-notch Academic Programs Project of Jiangsu
Higher Education Institutions (TAPP).

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# Improving Contextual Coherence in Variational Personalized and Empathetic
Dialogue Agents

###### Abstract

In recent years, latent variable models, such as the Conditional Variational
Auto Encoder (CVAE), have been applied to both personalized and empathetic
dialogue generation. Prior work have largely focused on generating diverse
dialogue responses that exhibit persona consistency and empathy. However, when
it comes to the contextual coherence of the generated responses, there is
still room for improvement. Hence, to improve the contextual coherence, we
propose a novel Uncertainty Aware CVAE (UA-CVAE) framework. The UA-CVAE
framework involves approximating and incorporating the aleatoric uncertainty
during response generation. We apply our framework to both personalized and
empathetic dialogue generation. Empirical results show that our framework
significantly improves the contextual coherence of the generated response.
Additionally, we introduce a novel automatic metric for measuring contextual
coherence, which was found to correlate positively with human judgement.

Index Terms—  Dialogue Generation, Latent Variables, Uncertainty Estimation,
Artificial Intelligence

##  1 Introduction

The ultimate goal of open-domain dialogue agents is to engage humans in
seamless, natural conversation. To achieve this goal, dialogue agents are
expected to generate responses which demonstrate empathy and persona
consistency. Hence, the tasks of personalized [1, 2, 3] and empathetic (or
emotional) dialogue generation [4, 5, 6, 7] has received significant research
attention in recent years. Traditionally, neural personalized or empathetic
open-domain dialogue agents demonstrate a tendency to generate generic and
repetitive responses. To address the lack of diversity in the generated
responses, numerous approaches have explored the application of latent
variable models, particularly the Conditional Variational Auto Encoder (CVAE),
to personalized and empathetic dialogue generation. Empirical results from
prior works [8, 9, 10, 11, 12, 6] have shown that CVAE-based models have
largely succeeded in generating diverse dialogue responses. Despite the gains
in response diversity, there is still much room for improvement, in
particular, when it comes to contextual coherence.

To improve the contextual coherence of the responses generated by CVAE-based
agents, we propose approximating and incorporating uncertainty during the
generation process. There are two types of uncertainty: aleatoric and
epistemic uncertainty [13]. While aleatoric uncertainty captures the
uncertainty in the input data, epistemic uncertainty quantifies the
uncertainty due to the model. Typically, aleatoric uncertainty is predicted as
a model output, and the epistemic uncertainty is obtained via ensemble methods
[14] or monte carlo simulations [15]. Aleatoric uncertainty can be further
categorized as either homoscedastic or heteroscedastic. Homoscedastic, or
task-dependant, uncertainty refers to input invariant noise which differs from
task to task. On the other hand, heteroscedastic, or data-dependant,
uncertainty quantifies the uncertainty (or noise) inherent in each input and
is thus unique to each input.

Hence, we introduce an Uncertainty-Aware CVAE (UA-CVAE) framework, which
involves approximating and incorporating the heteroscedastic aleatoric
uncertainty via an uncertainty aware latent variable. We also propose an
Utterance Entailment (UE) score that quantifies the contextual coherence of an
agent’s responses. Experimental results show that our framework significantly
improves overall response coherence. The remainder of this paper is organized
as follows: An overview of aleatoric uncertainty approximation in CVAE-based
dialogue agents is provided in Section 2. In Section 3, the UA-CVAE framework
and UE score are described in detail. Experimental results and the conclusion
are provided in sections 4 and 5 respectively.

##  2 Uncertainty in Dialogue Generation

For our discussion, let $X=\\{x_{0,0},...,x_{M-1,N-1}\\}$ refers to the
dialogue context ($M$ refers to the number of utterances in the context, and
$N$ denotes the number tokens in a given utterance), and
$Y=\\{y_{0},...,y_{L-1}\\}$ refers to the generated dialogue response ($L$
represents the number of tokens). Similarly, $\bar{Y}$ will refer to the
reference dialogue response. Additionally, we use $z$ and $c$ to denote the
sampled latent variable and any external information (persona description or
emotion label) respectively.

###  2.1 CVAE-based Dialogue Generation

In a CVAE-based dialogue agent, the generated response $Y$ is conditioned on
the the dialogue context $X$, the external information $c$ (persona
description or emotion label), as well as latent variable $z$, which is
randomly sampled from latent Gaussian distribution $p(z|X,c)$. An overview is
provided in Fig 1. The stochasticity introduced by $z$ improves the diversity
of the generated dialogue.

Fig. 1: Overview of a CVAE-based dialogue agent.

###  2.2 Aleatoric Uncertainty Approximation

In this section, we show that for a CVAE-based dialogue agent, the variance
$\sigma^{2}$ of the latent Gaussian distribution is an approximation of the
_aleatoric uncertainty_ of the generated dialogue response. For a CVAE-based
dialogue agent, the latent variable $z$ is assumed to follow an isotropic
multivariate Gaussian distribution:

$z\sim\mathcal{N}(\mu,\sigma^{2}\textbf{I})$ (1)

where $\mu$ and $\sigma^{2}$ refer to the mean and variance vectors of the
latent distribution. The CVAE-based dialogue generation process can be
represented by the following conditional distribution:

$p(Y,z|X,c)=p(Y|z,X,c)p(z|X,c)$ (2)

In a CVAE-based dialogue agent, both $p(Y|z,X,c)$ and $p(z|X,c)$ will be
defined by neural networks parameterized by $\pi$ and $\theta$ respectively.
In particular, as $z$ is a Gaussian random variable and
$p(z|X,c)=\mathcal{N}(\mu,\sigma^{2}\textbf{I})$, the network
$p_{\theta}(z|X,c)$ will be tasked with generating a mean $\mu$ and a variance
$\sigma^{2}$. Hence, since the variance $\sigma^{2}$ predicted by the neural
network $p_{\theta}(z|X,c)$ is dependant on the input data, $\sigma^{2}$ would
capture the uncertainty inherent in the input $X$ and $c$ i.e., the
heteroscedastic aleatoric or data-dependant uncertainty. This is similar to
the uncertainty estimation approach introduced in [15], where a data dependant
function of the uncertainty is learned. From (2), we observe that the response
generation process $p(Y,z|X,c)$ and $p(z|X,c)$ are both conditioned only on
$X$ and $c$. Hence, we conclude that the aleatoric uncertainty of the
generated dialogue $p(Y,z|X,c)$ can be approximated by the variance
$\sigma^{2}$, which captures the aleatoric uncertainty of $p(z|X,c)$.

Fig. 2: (a) Overview of the combination network in UA-CVAE(C). (b) Overview of
the combination network in UA-CVAE(M). $\oplus$ indicates element-wise
addition. Both networks output $z_{\text{u}}$. The embedding dimension
(embed_dim), intermediate dimension (inter_dim) and the latent dimension
(latent_dim) are fixed at 768, 384 and 256 respectively.

##  3 Uncertainty-Aware CVAE

Fig. 3: Overview of the UA-CVAE framework. The dashed and dotted connection
occurs in only the training and inference stages respectively.

As mentioned, the conditional distributions $p(Y|z,X,c)$ and $p(z|X,c)$ are
defined by neural networks parametrized by $\pi$ and $\theta$ respectively. A
GPT-2 pretrained model [16] is used to obtain the sequence embeddings of $X$,
$c$ and $\bar{Y}$, denoted by $X^{\prime}$, $c^{\prime}$ and
$\bar{Y}^{\prime}$ respectively. The prior network $p_{\theta}(z|X,c)$ is
composed of a single-layer MLP tasked with generating the mean
$\mu_{\text{p}}$ and the log-variance $log(\sigma^{2}_{\text{p}})$ of the
latent Gaussian distribution:

$\begin{bmatrix}\mu_{\text{p}}\\\
log(\sigma^{2}_{\text{p}})\end{bmatrix}=W_{\text{p}}\begin{bmatrix}c^{\prime}\\\
X^{\prime}\end{bmatrix}+b_{\text{p}}$ (3)

where $W_{\text{p}}$ and $b_{\text{p}}$ refer to the weights and bias of the
prior network. Before being fed into the prior network, the sequence embedding
of the dialogue context and the external information are obtained via a GPT-2
pretrained model. The variance, is attained by computing the exponential of
the log-variance $log(\sigma^{2}_{\text{p}})$. To obtain $z$, the latent
Gaussian distribution is randomly sampled via the reparametrization trick to
ensure differentiability. During training, the true posterior distribution
$q(z|X,c,\bar{Y})$ is approximated by another MLP parameterized by $\phi$.
This network, termed the recognition network $q_{\phi}(z|X,c,\bar{Y})$, also
consists of a single layer MLP:

$\begin{bmatrix}\mu_{\text{r}}\\\
log(\sigma^{2}_{\text{r}})\end{bmatrix}=W_{\text{r}}\begin{bmatrix}c^{\prime}\\\
X^{\prime}\\\ \bar{Y}^{\prime}\end{bmatrix}+b_{\text{r}}$ (4)

where $W_{\text{r}}$ and $b_{\text{r}}$ refer to the weights and bias of the
recognition network. Similarly, Before being fed into the prior network, the
sequence embedding of the dialogue context and the external information are
obtained via a GPT-2 pretrained model [16]. The KL divergence between the
Gaussian distributions generated by the prior and recognition network will be
incorporated in the final loss function. Then, the latent variable $z$ and the
variance vector $\sigma^{2}_{\text{p}}$ (inference) or
$\sigma^{2}_{\text{r}}$(training), an approximation of the _aleatoric
uncertainty_ of the generated dialogue response, is fed to the combination
network which outputs the uncertainty aware latent variable $z_{\text{u}}$. By
conditioning the decoder on the uncertainty aware latent variable
$z_{\text{u}}$, the decoder would possess an awareness of the uncertainty or
noise inherent in $X$ and $c$. This allows the decoder to recognize and adapt
to uncertain or noisy dialogue contexts and persona description/emotion
labels, reducing the number of contextually incoherent responses. We implement
2 variants of the combination network (Fig 2), resulting in 2 variants of the
UA-CVAE framework: UA-CVAE(M) and UA-CVAE(C). While the combination network in
UA-CVAE(M) constitutes a several linear layers, the combination network in UA-
CVAE(C) involves a 1D convolution layer. Then, $z_{\text{u}}$ is added to the
positional encoding and token-level embedding at each decoding step, and fed
to the response decoder $p_{\pi}(Y|z,X,c)$ which constitutes a GPT-2
pretrained language model [16].

Since training a CVAE would require an intractable marginalization over $z$,
Stochastic Gradient Variational Bayes (SGVB) is used to train the UA-CVAE.
Additionally, due to the vanishing latent variable problem, we include the
Bag-of-Words (BoW) loss in the loss function [12]. The final loss function is
as follows:

$\begin{split}L(X,&Y,c,z,z_{\text{u}})=-KL(q_{\phi}(z|X,c,Y)||p_{\theta}(z|X,c))\\\
&+E_{q_{\phi}(z|X,c,Y)}[log\>p(Y|X,c,z_{\text{u}})]\\\
&+E_{q_{\phi}(z|X,c,Y)}[log\>p(Y_{bow}|X,c,z_{\text{u}})]\end{split}$ (5)

where $Y_{bow}$ refers to the generated dialogue response BoW. A overview is
provided in Fig 3.

###  3.1 Utterance Entailment Score

Prior work have shown that the task of Natural Language Inference, which
involves evaluating if a given premise entails or contradicts a predefined
hypothesis, correlates with human judgement with regard to contextual
coherence. Our metric, termed the Utterance Entailment (UE) score, requires
finetuning a BERT model on the SNLI corpus and applying the resultant model to
the dialogue context and generated response. Specifically, given a single
utterance in the dialogue context $x_{i}$ and a generated response $Y$, the
model judges if the premise $x_{i}$ and the hypothesis $Y$ are contradictory,
entailing or neutral. Then, a rating of 1,-1 or 0 is assigned when the premise
and hypothesis are entailing, contradictory or neutral respectively. For each
response, the final Utterance Entailment score is computed by summing all
assigned ratings. Hence, for each dialogue example, the UE score can be
computed as follows:

$UE(X,Y)=\sum_{i=1}^{M}BERT_{NLI}(x_{i},Y)$ (6)

where $X$ and $Y$ refer to the dialogue context and generated response
respectively. The final UE-score for a given test set is obtained by averaging
the UE-scores of each dialogue example.

##  4 Experiments & Results

Table 1: Results of automatic evaluation.

| Model | PPL | Avg Length | Rouge-l (F1) | Rouge-l (R) | Rouge-l (P) | METEOR | Distinct-1 | Distinct-2 | Distinct-3 | UE-score
---|---|---|---|---|---|---|---|---|---|---|---
ConvAI2 | seq2seq | 42.5615 | 10.3481 | 0.0115 | 0.0672 | 0.0064 | 0.3109 | 0.3927 | 0.6870 | 0.7509 | 0.0002
HRED | 33.0022 | 10.3594 | 0.0126 | 0.0732 | 0.0069 | 0.3023 | 0.4134 | 0.7531 | 0.8204 | 0.0006
VHRED | 39.6422 | 10.5012 | 0.0110 | 0.0636 | 0.0061 | 0.3154 | 0.4339 | 0.8142 | 0.8817 | 0.0008
VHCR | 32.1812 | 10.8626 | 0.0114 | 0.0660 | 0.0063 | 0.3099 | 0.4364 | 0.7641 | 0.8183 | 0.0004
GPT-2 | 15.5766 | 9.5820 | 0.1434 | 0.1846 | 0.1236 | 0.2294 | 0.4496 | 0.8160 | 0.8705 | 0.0540
CVAE | 14.2457 | 11.0069 | 0.1172 | 0.1406 | 0.1073 | 0.2631 | 0.4773 | 0.8729 | 0.9245 | 0.0406
UA-CVAE(M) | 14.5423 | 11.7421 | 0.1291 | 0.1517 | 0.1200 | 0.2576 | 0.4562 | 0.8554 | 0.9149 | 0.0648
| UA-CVAE(C) | 14.1253 | 11.3658 | 0.1169 | 0.1475 | 0.1245 | 0.2507 | 0.4981 | 0.8648 | 0.9106 | 0.0906
Empathetic Dialogues | seq2seq | 52.9023 | 10.9731 | 0.0054 | 0.0355 | 0.0030 | 0.3243 | 0.4468 | 0.7561 | 0.8130 | 0.0103
HRED | 68.2843 | 11.0345 | 0.0053 | 0.0353 | 0.0029 | 0.3311 | 0.4444 | 0.8333 | 0.8988 | 0.0095
VHRED | 55.4275 | 11.0139 | 0.0045 | 0.0297 | 0.0025 | 0.3305 | 0.4396 | 0.8149 | 0.8798 | 0.0089
VHCR | 63.4752 | 11.2758 | 0.0034 | 0.0416 | 0.0034 | 0.3017 | 0.4884 | 0.7668 | 0.8112 | 0.0090
GPT-2 | 16.6456 | 7.3657 | 0.0390 | 0.0676 | 0.0299 | 0.2525 | 0.4848 | 0.7902 | 0.8315 | 0.1081
CVAE | 16.3177 | 8.0514 | 0.0745 | 0.1087 | 0.0651 | 0.2333 | 0.5139 | 0.8715 | 0.9328 | 0.1074
UA-CVAE(M) | 15.2021 | 9.8180 | 0.0867 | 0.1132 | 0.0817 | 0.2913 | 0.5526 | 0.9149 | 0.9370 | 0.1198
| UA-CVAE(C) | 17.1932 | 12.1111 | 0.0859 | 0.1045 | 0.0825 | 0.2794 | 0.4922 | 0.8877 | 0.9239 | 0.1257

Table 2: Results of human evaluation. Kappa scores generally range from 0.5 to
0.6, indicating moderate agreement.

| Model | UA-CVAE(M) | UA-CVAE(C)
---|---|---|---
Win | Tie | Loss | Kappa | Win | Tie | Loss | Kappa
ConvAI2 | seq2seq | 53% | 45% | 2% | 0.54 | 55% | 39% | 6% | 0.59
HRED | 49% | 42% | 9% | 0.49 | 58% | 38% | 4% | 0.46
VHRED | 43% | 41% | 16% | 0.52 | 43% | 51% | 6% | 0.50
VHCR | 42% | 38% | 20% | 0.64 | 51% | 41% | 8% | 0.59
GPT-2 | 37% | 58% | 5% | 0.57 | 41% | 45% | 14% | 0.51
CVAE | 34% | 49% | 17% | 0.43 | 44% | 45% | 11% | 0.47
UA-CVAE(M) | - | - | - | - | 29% | 48% | 23% | 0.63
UA-CVAE(C) | 23% | 48% | 29% | 0.63 | - | - | - | -
Empathetic Dialogues | seq2seq | 50% | 42% | 8% | 0.42 | 58% | 32% | 10% | 0.66
HRED | 53% | 43% | 4% | 0.49 | 64% | 29% | 7% | 0.58
VHRED | 57% | 38% | 5% | 0.53 | 57% | 38% | 5% | 0.55
VHCR | 59% | 33% | 8% | 0.54 | 61% | 37% | 2% | 0.60
GPT-2 | 33% | 52% | 15% | 0.64 | 52% | 29% | 19% | 0.49
CVAE | 39% | 47% | 14% | 0.56 | 34% | 53% | 13% | 0.54
UA-CVAE(M) | - | - | - | - | 31% | 56% | 13% | 0.64
UA-CVAE(C) | 13% | 56% | 31% | 0.64 | - | - | - | -

###  4.1 Corpora

We benchmark our framework on the ConvAI2 [17] and EMPATHETICDIALOGS [18]
corpora. For EMPATHETICDIALOGS, the human speaker describes a personal
situation and the dialogue agent is expected to generate an empathetic
response. Each dialogue example is provided an emotion label (e.g.,
‘excited’). ConvAI2 consists of dialogues where two interlocutors are trying
to get to know each other. Each interlocutor is assigned a persona description
consisting of multiple persona statements (e.g., ‘I love dogs’, ‘I like
eggs’).

###  4.2 Implementation & Baselines

In our implementation of UA-CVAE(M) and UA-CVAE(C), the response decoder
consists of a GPT-2 pretrained language model [16] which is finetuned during
training. We utilize the GPT-2 base model from HuggingFace consisting of 12
layers, 12 attention heads as well as a hidden dimension of 768. The size of
the latent variable $z$ and $z_{\text{u}}$ was fixed at 256. The Adam
optimizer (learning rate = 0.0001, batch size = 16) was used. The maximum
utterance length and maximum number of turns were fixed at 40 and 4
respectively.

In addition to UA-CVAE(C) and UA-CVAE(M), we implement the following
baselines: Seq2seq with attention applied to the GRU decoder, Hierarchical
Recurrent Encoder-Decoder (HRED) [19], Variational VHRED (VHRED) [20],
Variational Hierarchical Conversation RNN (VHCR) [21], GPT-2, and the CVAE
with a GPT-2 decoder.

##  5 Results & Discussion

###  5.1 Results

We evaluate the generated dialogue with the following automatic metrics (Table
1): PPL, ROUGE [22], METEOR [23], intra-response Distinct-n (1,2,3) [24], and
the UE-Score. The average response length is also provided. For human
evaluation (Table 2), we engage 4 graduates to evaluate the contextual
coherence of the generated responses. Specifically, each graduate was told to
compare a response generated by a UA-CVAE variant with a response generated by
the baselines, and select the more contextually coherent response.

###  5.2 Discussion

Based on the human evaluation results and the automatic metrics, we can
conclude that there is a positive correlation between the proposed UE-score
and human judgement. UA-CVAE(M)/(C) generally achieve a lower percentage of
wins and higher percentage of losses against baselines with higher UE-scores
such as GPT-2 and CVAE.

It is also apparent that UA-CVAE(M) and UA-CVAE(C) outperformed all
implemented baselines in terms of contextual coherence. Furthermore, the
Distinct-n, Rouge and METEOR scores achieved by UA-CVAE(M) and UA-CVAE(C) were
comparable to that of the CVAE baseline, suggesting comparable response
diversity and fluency. In terms of UE-score, compared to the CVAE baseline,
UA-CVAE(M) and UA-CVAE(C) achieved a 60% and 121% improvement respectively on
ConvAI2. However, on EMPATHETICDIALOGS, UA-CVAE(M) and UA-CVAE(C) achieved a
11% and 17% gain. The difference in gains could be attributed to the
informative nature and shorter length of the dialogue utterances in ConvAI2,
which would result in more positive NLI predictions (‘entailment’) when
computing the UE-score. Also, we notice that UA-CVAE(C) has a slight edge over
UA-CVAE(M) in terms of coherence. In addition to attaining higher UE-scores,
the UA-CVAE(C) model achieved a larger percentage of wins on both corpora.
This shows that a 1D convolution layer allows for a slightly more effective
combination of the variance, or uncertainty, vector and latent variable.

##  6 Conclusion

We have proposed two UA-CVAE frameworks: UA-CVAE(M) and UA-CVAE(C), which
improved coherence on both ConvAI2 (60% and 121%) and EMPATHETICDIALOGS (11%
and 17%). Additionally, we introduced the UE-score, which correlated
positively with human judgement in terms of contextual coherence. Future work
could involve exploring the possibility of enhancing the dialogue quality by
leveraging the epistemic or homoscedastic aleatoric uncertainty.

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