# Dissipative Unified Dark Fluid: Observational Constraints

Esraa Ali Elkhateeb11footnotetext: Corresponding author. and Mahmoud Hashim

###### Abstract

We adopt a standard FRW cosmology with a unified scenario, where the usual
dark matter and dark energy sectors are replaced by a single dissipative
unified dark fluid (DUDF). The equation of state of such fluid can asymptote
between two power laws. As a result, it enables fluid to have a smooth
transition from dust at early times to dark energy at late times. The
dissipation is represented by a bulk viscosity with a constant coefficient,
whereas shear viscosity is excluded due to the isotropy of the universe. We
performed a Likelihood analysis using recent observational datasets from BAO,
CMB, Cosmic Chronometer measurements, and Type Ia supernovae to put
cosmological constraints on the model. We found that the viscosity coefficient
is not constrained by the data, whence the analysis is continued with this
coefficient fixed. The model yields a minimum $\chi^{2}$-value of $1357.91$
compared to $1363.34$ for the $\Lambda$CDM model. Using the Akaike Information
Criterion (AIC) we found that our model minimizes the AIC value with a
difference of -1.4 compared to the $\Lambda$CDM model, which indicates that
DUDF model is viable according to background observations. We studied the
evolution of the universe due to DUDF model by studying the evolution of
different cosmological parameters like the Hubble parameter, the distance
modulus, the deceleration parameter, the effective equation of state
parameter, and the density parameter. We show that DUDF model doesn’t deviate
from the standard $\Lambda$CDM model at early times, with the ability to play
the role of the cosmological constant by accelerating the universe in the late
times.

##  1 Introduction

The accelerating expansion of the universe was discovered in 1998 from
supernovae data and confirmed by baryon acoustic oscillation (BAO) data and
the analysis of galaxies clustering [1]-[4]. The phenomenon was unexpected due
to the gravitational attraction of normal matter in the universe and is
attributed to some peculiar energy, termed dark energy. One explanation is
through the cosmological constant, which was previously suggested by Einstein
[5]. This idea was the base for the standard model of cosmology, the
$\Lambda$CDM model, in which $\Lambda$ represents the cosmological constant
and plays the role of dark energy, while CDM is the Cold Dark Matter, another
sort of peculiar matter which is gravitating but non-baryonic. According to
recent cosmological observations, $95\%$ of the total energy budget of the
universe is composed of these dark sectors [6]. However, the cosmological
constant approach, although promising, suffers a conceptual problem [7], [8],
in addition to the famous coincidence problem [9]. Such problems have
motivated the construction of alternative scenarios to the cosmological
constant dark energy. In some of these scenarios, the dark energy is dynamic.
In other scenarios the dark matter and dark energy are interacting [10], [11],
while in some others they are non-interacting [12], [13]. There is also a
class of models that considers unification between the two sectors. In these
models, the two dark sectors are represented by a single dark fluid whereas
late transition from DM to DE can smoothly occur [14], [15]. Chaplygin gas
model and its generalizations are examples of these models [16], [17]. There
is also the new barotropic dark fluid model proposed by Elkhateeb [18].

As viscosity is a physical property of fluids, cosmological viscosity
attracted attention. Dissipative processes in cosmology have been discussing
since the work of Zel’dovich [19] and Misner [20] in 1967. In his work 1971,
Weinberg [21] claims that vanishing bulk viscosity for a general dissipative
fluid is just an exception rather than a rule. Many authors also discussed the
viscous bulk pressure effects relating the cosmological particle production,
e.g., [22]-[28]. Viscous pressure inducing cosmic expansion also brings
attention. A study made by Giovannini [29] declines the possibility of the
existence of an accelerated phase driven solely by viscosity, where the tensor
to scalar ratio exceeds the observational limit. He concluded that the viscous
modes are only tolerable as a subleading component of a dominant adiabatic
mode from a different physical origin. Some authors also discussed removing
instabilities in the fluid power spectrum by treating viscosity as the
physical origin for entropy perturbation, which in turn smoothing out
anisotropies in the universe [30]-[32]. Cosmological viscosity also included
studying early universe and entropy generation via cosmic inflation [33]-[39],
and bulk viscosity driven late universe acceleration [40]-[44]. Recently,
Brevik and Normann [45] inferred symmetry between the early-time and present-
day viscous cosmology by using a bulk viscosity function proportional to the
Hubble parameter $H$. Victor et al. [46] discussed the bulk viscosity effects
and matter production as an alternative for the description of the late times
behavior of the observable universe. Alleviation of the $H_{0}$-tension
between direct and indirect measurements is also a motivation for studying
viscous models [47]-[49]. Many authors also studied The viscosity of the
cosmic unified dark fluids [50]-[55]. Other aspects concerning dissipative
cosmology can be found in [56]-[61].

In this work, we aim to study the evolution of the universe under the effect
of bulk viscosity using the latest cosmological observations. We adopt the new
unified viscous dark fluid model introduced by Elkhateeb in a previous work
[52]. The dark fluid introduced has an equation of state with two power-law
asymptotes, consequently can interpolate between dust and dark energy at early
and late times. The added dissipative effects are in the form of bulk
viscosity with a constant viscosity coefficient. Shear was considered
negligible due to the isotropy of the universe. We implement recent
observational datasets including Baryon Acoustic Oscillations (BAO), Cosmic
Microwave Background (CMB), Cosmic Chronometer measurements (CC), and Type Ia
Supernovae SNe Ia to perform robust analysis. The paper is organized as
follows: Sec. 2 represents the theory of viscous fluid-based cosmology. Sec. 3
constrains some of the model parameters based on the asymptotic behavior of
the model. In sec. 4, recent datasets used to validate the analysis procedure
are presented. In sec. 5, we use the Akaike Information Criterion (AIC) to
compare our model with the $\Lambda$CDM model. In sec. 6, we study the
evolution of different cosmological parameters like the Hubble parameter, the
distance modulus, the deceleration parameter, the effective equation of state
parameter, and the density parameter. Sec. 7 for estimation of the present-day
viscosity coefficient of the cosmic fluid. We finally concluded in sec. 8.

##  2 Fluid Based Cosmological Model

Einstein’s equation for GR is given by

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^{4}}T_{\mu\nu},$ (2.1)

where $R_{\mu\nu}$ is the Ricci curvature tensor, $R$ is the scalar curvature,
$g_{\mu\nu}$ is the space metric and $T_{\mu\nu}$ is the energy-momentum
tensor given by

$T_{\mu\nu}=\rho U_{\mu}U_{\nu}+ph_{\mu\nu},$ (2.2)

where $\rho$ is the energy density of the cosmic fluid, $p$ is its pressure,
and $h_{\mu\nu}=U_{\mu}U_{\nu}-g_{\mu\nu}$ is the projection tensor to the
$3-$space orthogonal to the fluid element, where in comoving coordinates the
four-velocity $U_{\mu}={\delta^{0}}_{\mu}$.

Considering units with $c=1$, the metric in the standard FRW cosmology is
given by

$ds^{2}=dt^{2}-a^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}),$ (2.3)

where $a(t)$ represents the scale factor, $k$ is the curvature constant for
the space with the three distinct values $-1$, $0$, and $1$, represent
respectively a spatially open, flat, and a closed universe.
$d\Omega^{2}=d\theta^{2}+sin^{2}\theta d\phi^{2}$, with $\theta$ and $\phi$
are the usual polar and azimuthal angles of spherical coordinates. The
coordinates $(t,r,\theta,\phi)$ are the comoving coordinates. Considering our
universe is filled with several components: radiation, baryons, and dark
fluid, each with pressure $p_{i}$ and energy density $\rho_{i}$, Einstein’s
equation (2.1) together with the metric (2.3) will lead to the Friedmann
equations

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{3}\sum_{i}\rho_{i},$ (2.4)
$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{6}\sum_{i}(\rho_{i}+3p_{i}),$ (2.5)

where we consider flat geometry and units with $8\pi G=1$. The conservation
equation $T^{\mu}_{\nu(i);\mu}=0$ gives

$\dot{\rho_{i}}+3\frac{\dot{a}}{a}(\rho_{i}+p_{i})=0,$ (2.6)

where $T^{\mu}_{\nu(i)}$ is the energy momentum tensors for the $i^{th}$
component.

The effective pressure of dark fluid component $p_{\rm fld}$ is considered to
consist of an adiabatic part $p_{a}$ plus a viscous term

$p_{\rm fld}=p_{a}-\vartheta\zeta(\rho),$ (2.7)

where $\vartheta={U^{\mu}}_{;\mu}=3H$ is the expansion rate of the fluid with
$H=\frac{\dot{a}}{a}$ is the Hubble parameter, $\zeta=\zeta(\rho)$ is the
coefficient of bulk viscosity that arises in the fluid and is restricted to be
positive. For the adiabatic part, $p_{a}$, we adopt a barotropic pressure that
can asymptote between dust at the early time and DE at the late time [18]. It
has the form

$p_{a}=-\rho_{\rm fld}+\frac{\gamma\rho^{n}_{\rm fld}}{1+\delta\rho^{m}_{\rm
fld}},$ (2.8)

where $\gamma$, $\delta$, $n$, and $m$ are free parameters. As mentioned in
[18], this form of adiabatic pressure has the advantage that it enables
interpolation between different powers for the density, which allows for
smooth phase transitions during the universe evolution. It also has the
advantage of having a general equation of state (EoS) for DE that enables the
cosmological constant as a special case. The bulk viscosity coefficient is
considered to take the form

$\zeta(\rho_{\rm fld})=\zeta_{0}\rho^{\nu}_{\rm fld},$ (2.9)

where $\zeta_{0}$ and $\nu$ are constants. In this work we adopt a constant
bulk viscosity coefficient $\zeta(\rho)=\zeta_{0}$, so that $\nu=0$. This
assumption is attractive because of its mathematical simplicity while it is
physically acceptable. Accordingly, we consider a viscous dark fluid with an
effective equation of state

$p_{\rm fld}=-\rho_{\rm fld}+\frac{\gamma\rho_{\rm fld}^{n}}{1+\delta\rho_{\rm
fld}^{m}}-3\zeta_{0}H.$ (2.10)

##  3 The Model Parameters

The dark sector of the universe is supposed to consist of a viscous dark fluid
with an EoS asymptotes between two power laws characterizing dust at early
time and dark energy at late time. The asymptotic behavior of the model can
shed light on the relationship between its parameters, which reduces the
number of degrees of freedom of the model. In doing so, we define the
following dimensionless parameters,

$\tilde{\rho}_{\rm fld}=\delta^{\frac{1}{m}}\rho_{\rm fld},$ (3.1)

and

$\tau=\frac{\gamma}{{\delta}^{\frac{(2n-1)}{{2m}}}}t,$ (3.2)

so that
$\left[\delta^{\frac{1}{m}}\right]=\left[\rho^{-1}\right]=\left[H^{-2}\right]$.
Accordingly, the conservation equation for the fluid reduces to

$\frac{d\tilde{\rho}_{\rm fld}}{d\tau}=-\sqrt{3}\left(\frac{{\tilde{\rho}_{\rm
fld}}^{n+\frac{1}{2}}}{1+{\tilde{\rho}}^{m}_{\rm
fld}}-\tilde{\zeta_{0}}\tilde{\rho}_{\rm fld}\right),$ (3.3)

with

$\tilde{\zeta_{0}}=\frac{\sqrt{3}}{\gamma}\delta^{\frac{(2n-1)}{{2m}}}\zeta_{0},$
(3.4)

is also dimensionless, while $\left[\zeta_{0}\right]=\left[H\right]$.

DUDF has to mimic dust for large $\tilde{\rho}$ and DE for small
$\tilde{\rho}$. This means that $d\tilde{\rho}/d\tau$ has to be proportional
to $\tilde{\rho}$ for large $\tilde{\rho}$ and tends to zero for small
$\tilde{\rho}$. Accordingly, the parameter $n$ has to be positive. Restricting
$m$ also to be positive, relation (3.3) tends asymptotically to the following
form for large $\tilde{\rho}$

$\frac{d\tilde{\rho}}{d\tau}\sim-\sqrt{3}\left(\tilde{\rho}^{(n-m+\frac{1}{2})}-\tilde{\zeta_{0}}\tilde{\rho}\right).$
(3.5)

For small $\tilde{\zeta_{0}}$, the first term in the above equation dominates
for $(n-m)\geq 1$. Under this condition, Eq. (3.5) reduces to

$\frac{d\tilde{\rho}}{d\tau}\sim-\sqrt{3}\;\tilde{\rho}^{(n-m+\frac{1}{2})}.$
(3.6)

Comparing this with Friedmann equation for dust, namely

$\frac{d\tilde{\rho}}{d\tau}=-\sqrt{3}\;\frac{\delta^{\frac{(n-1)}{m}}}{\gamma}\;\tilde{\rho}^{\frac{3}{2}},$
(3.7)

we directly get

$n=m+1\;,\;\;\;\;\;\;\;\gamma=\delta.$ (3.8)

As a result

$\tilde{\zeta_{0}}={\sqrt{\frac{3}{\tilde{\delta}}}}\;\zeta_{0},$ (3.9)

with

$\tilde{\delta}=\delta^{-\frac{1}{m}},$ (3.10)

has the dimensions of $H^{2}$;
$\left[\tilde{\delta}\right]=\left[\rho\right]=\left[H^{2}\right]$.

In a previous work [52], we introduced the basic properties of DUDF model and
deduced its parameters using observations of different cosmological phenomena
such as today’s value of deceleration parameter, the redshift value of
deceleration-acceleration transition, and the age of the universe. In this
work, we seek to use the available cosmological data sets to constrain such
parameters robustly.

##  4 Data Analysis

###  4.1 Observational Data

We now aim to constrain DUDF model using observational data. We relied on
different data sets from Baryon Acoustic Oscillations (BAO), Cosmic Microwave
Background (CMB), Cosmic Chronometer (CC), and Type Ia Supernovae (SNe Ia).

In this analysis, the cosmological model is specified by the total relative
density,

$\Omega_{\rm tot}(z)=\Omega_{\rm r}(z)+\Omega_{\rm b}(z)+\Omega_{\rm fld}(z).$
(4.1)

The radiation term $\Omega_{\rm r}$ considers the standard thermal history
into account with today’s value $\Omega_{\rm r0}$ given by [62]

$\Omega_{\rm{r0}}(z)=\Omega_{\gamma,0}\left[1+\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{\rm
eff}\right].$ (4.2)

where $\Omega_{\gamma,0}=2.4728\times{10}^{-5}/h^{-2}$ is the current value of
photon density, $h$ is the dimensionless Hubble constant defined through
$H_{0}=100\;h\;km\;s^{-1}Mpc^{-1}$, and $N_{\rm eff}=3.046$ is the effective
number of neutrino species [63].

####  4.1.1 SNe Ia

Type Ia supernovae (SNe Ia) have a known brightness whence can serve as
standard candles and can be used for measuring cosmological distances.
Pantheon Sample [64] is the most complete updated compilation of SNe Ia.
Confirmed 276 SNe Ia from Pan-STARRS1 Medium Deep Survey at $0.03<z<0.65$ are
combined with previous available SNe Ia samples from other surveys to form a
sample of $1048$ SNe Ia spanning the redshift range $0.01<z<2.3$. Data for the
apparent magnitude $m(z)$ is given instead of the distance modulus $\mu$ as
the absolute magnitude $M$ of SNe Ia degenerates with $H_{0}$. Observations of
the apparent magnitude of SNe Ia, $m(z)$, are related to the distance modulus
$\mu(z)$ through the relation

$\mu(z)=m(z)-M=5log_{10}(\frac{D_{L}(z)}{10\;pc}),$ (4.3)

where the absolute magnitude $M$ is considered constant for all supernovae of
Type Ia due to the standard candle hypothesis, and $D_{L}(z)$ is the
luminosity distance given by

$D_{L}(z)=D_{H}(1+z)\int^{z}_{0}\frac{d{z^{\prime}}}{E({z^{\prime}})},$ (4.4)

with

$D_{H}=\frac{c}{H_{0}},$ (4.5)

is the Hubble distance, and the function $E(z)$ is given by

$E(z)=\frac{H(z)}{H_{0}}.$ (4.6)

####  4.1.2 OHD

In addition to the SNe Ia observations, we also considered observational
Hubble data (OHD) from CC and BAO measurements. CC techniques [65] allow
direct information about Hubble function as they are probe for direct measure
of the differential age, $\frac{\Delta z}{\Delta t}$, of passively evolving
galaxies, whence $H(z)=-\frac{1}{1+z}\frac{\Delta z}{\Delta t}$. Accordingly,
They allow for model-independent measurements of Hubble function. However, CC
is a good tool for Hubble function measure at redshifts $\leq 2$ [66]. In this
work, we considered the updated list compiled by Maga$\widetilde{n}$a et al.
[67] which contains 31 points from CC plus 20 points from BAO measurements.

####  4.1.3 $H_{0}$ Measurements

Additionally, we included in our analysis the local value of Hubble parameter
$H_{0}$ provided by Riess et al. 2019 results (R19) [68].

####  4.1.4 BAO

Anisotropic BAO analysis can measure the Hubble function $H(z)$ and comoving
angular diameter distance $D_{M}(z)$ due to the BAO shifts perpendicular and
parallel to the line of sight. Alam et al. [69] considered data provided by
BAO measurements from Sloan Digital Sky Survey (SDSS III). They constrained
three points for comoving angular diameter distance $D_{M}(z)$ at
$z=0.35,0.51,0.61$. The comoving angular diameter distance $D_{M}(z)$ for a
flat universe is defined to be

$D_{M}(z)=D_{H}\int^{z}_{0}\frac{d{z^{\prime}}}{E({z^{\prime}})}.$ (4.7)

Another important parameter one can extract from BAOs is the acoustic
parameter $A(z)$ introduced by Eisenstein et al. (2005) [70]. It describes the
distance-redshift relation [71]. $A(z)$ is defined as

$A(z)=\frac{D_{V}(z)\sqrt{\Omega_{m}H^{2}_{0}}}{cz},$ (4.8)

where the volume-average distance $D_{V}(z)$ is a combination of angular
diameter distance $D_{A}(z)$ and Hubble parameter, and is given by

$D_{V}(z)=\left[\left(1+z\right)^{2}D^{2}_{A}(z)\frac{cz}{H(z)}\right]^{1/3},$
(4.9)

with $D_{A}(z)$ is related to the comoving angular diameter distance
$D_{M}(z)$ through the relation $D_{A}(z)=\frac{D_{M}}{1+z}$. Using the SDSS
luminous red galaxy (LRG) sample, Eisenstein et al. constrained
$A(z)=0.469\left(\frac{n_{s}}{0.98}\right)^{-0.35}\pm 0.017$ at redshift
$z=0.35$, where $n_{s}=0.961$ is the spectral index [72].

####  4.1.5 CMB

We use the results of Planck collaboration for the CMB shift parameter
$\mathcal{R}$, which is given by

$\mathcal{R}=\frac{1}{c}D_{M}(z_{\ast})\sqrt{\Omega_{m}H^{2}_{0}},$ (4.10)

where $z_{\ast}$ is the redshift of decoupling epoch which can be calculated
as [73]

$z_{\ast}=1048\left[1+0.00124\left(\Omega_{b}h^{2}\right)^{-0.738}\right]\left[1+g_{1}\left(\Omega_{m}h^{2}\right)^{g_{2}}\right],$
(4.11)

where

$g_{1}=\frac{0.0783\left(\Omega_{b}h^{2}\right)^{-0.238}}{1+39.5\left(\Omega_{b}h^{2}\right)^{0.763}}\;,\;\;\;g_{2}=\frac{0.560}{1+21.1\left(\Omega_{b}h^{2}\right)^{1.81}}.$
(4.12)

We impose a Gaussian prior for $\mathcal{R}=1.7382\pm 0.0088$ [63], and a BBN
prior on $\Omega_{b}h^{2}=0.0222\pm 0.0005$ [6].

###  4.2 Parameters Estimation

The probability distribution function - posterior -
$p\left(\bm{\Theta|D}\right)$ for the model parameters $\bm{\Theta}$ given
different observational data sets $\bm{D}$ is given by Bayes’ theorem:

$p\left(\bm{\Theta|D}\right)=\frac{p\left(\bm{D|\Theta}\right)p\left(\bm{\Theta}\right)}{p(\bm{D})},$
(4.13)

where $p(\bm{\Theta})$ is the parameters prior, and $p(\bm{D})$, known as the
evidence, is the probability of observing the data. It is constant and can be
set equal to $1$ since, after all, we got the observations. The probability
$p\left(\bm{D|\Theta}\right)$ is the expected distribution of data given the
parameters - likelihood - and is given by:

$\mathcal{L}=\prod^{i}_{N}\frac{1}{\sqrt{2\pi\sigma^{2}_{i}}}\exp(-\frac{1}{2}\chi^{2}_{\rm
tot}),$ (4.14)

where

$\chi^{2}_{\rm tot}=\sum_{\rm d}\chi^{2}_{\rm d},$ (4.15)

with d represents each data group.

To constrain DUDF model we used the public Markov Chain Monte Carlo (MCMC)
emcee code [74]. The parameter space were first chosen to be seven-
dimensional; the three basic cosmological parameters: Hubble constant $H_{0}$,
today’s baryon density $\Omega_{b}h^{2}$, and today’s matter density
$\Omega_{m}$, plus the three model parameters: $m$, $\widetilde{\delta}$, and
$\zeta_{0}$, in addition to the nuisance parameter represents the absolute
magnitude $M$, i.e,
$\bm{\Theta}=\\{H_{0},\Omega_{b}h^{2},\Omega_{m},m,\widetilde{\delta},\zeta_{0},M\\}$.
The posterior is constructed from the likelihood and a flat prior on all
parameters. In table 1 we summarize the priors on the model parameters that
have been used during the statistical analysis.

Table 1: Prior of the model parameters for the seven-dimensional set analysis. $H_{0}$ | $\Omega_{b}h^{2}$ | $\Omega_{m}$ | m | $10^{-4}\tilde{\delta}$ | $\zeta_{0}$ | M
---|---|---|---|---|---|---
[65, 80] | [0.015, 0.028] | [0.2, 0.4] | [0.5, 4] | [0.8, 2.5] | [0, 0.01] | [-20, -18]
Figure 1: $68\%$ and $95\%$ full dataset contour plots between combinations of
some of the model parameters for different values of $\zeta_{0}$. The contours
represent the effect of variation of $\zeta_{0}$ on the model parameters.

As indicated from Fig.1, the contour plot of the viscosity parameter
$\zeta_{0}$ is flat, which means that the data couldn’t put a constraint on
DUDF viscous property. Consequently, we enforced to fix the viscosity
parameter $\zeta_{0}$ and analyse a six-dimensional parameter space,
$\bm{\Theta}=\\{H_{0},\Omega_{b}h^{2},\Omega_{m},m,\widetilde{\delta},M\\}$.
Analysis are made for $\zeta_{0}=0,0.001,0.1,0.2,1,4,8\;\&\;10$. In Fig.1 we
show the $68\%$ and $95\%$ full dataset combination contours between some of
the model parameters for $\zeta_{0}=0,0.001\;\&\;0.2$. Results indicate that
only $\tilde{\delta}$ is weakly affected by the variation of the viscosity
parameter $\zeta_{0}$Ṫhe other parameters are nearly unaffected. This
interprets the disability of data to restrict this parameter.

We constrained DUDF model parameters using five different datasets, namely,
SNe $+$ BBN $+$ $H_{0}$ as a Base dataset, Base $+$ BAO, base $+$ BAO $+$ OHD,
Base $+$ BAO $+$ CMB, and Base $+$ BAO $+$ OHD $+$ CMB. In Fig.2 we
illustrated the two-dimensional contour plots between the basic cosmological
parameters due to the different dataset combinations. The Fig indicates that
both $H_{0}$ and $\Omega_{m}$ are insensitive to Pantheon SNe data. Adding BAO
data to Pantheon lowers and tightens the constraint on each of these two
parameters. With the addition of the OHD measurements, these constraints are
strengthened. The same is figured out for the rest of the model parameters.
The values of the parameters due to different dataset combinations are
summarized in table 2. One can see from the table that the addition of the OHD
measurements decreases the error bars for $H_{0}$, while the addition of CMB
data tightens the constraint on $\omega_{b}h^{2}$. On the other hand, the best
constraints on $\Omega_{m}$ are achieved for the full dataset combinations.
The absolute magnitude parameter $M$ can be seen from this table to be
consistent with SNe Ia data.

Figure 2: $68\%$ and $95\%$ c.l. contour plots between various combinations of
the basic cosmological parameters using the observational data from different
sources. Table 2: $68\%$ CL parameters of DUDF model from different
observational datasets

Parameter | $\zeta_{0}$ | Base | Base $+$ BAO | Base $+$ BAO $+$ OHD | Base $+$ BAO $+$ CMB | Base $+$ BAO $+$ CMB $+$ OHD
---|---|---|---|---|---|---
| $0.0$ | $74.1\pm 1.4$ | $69.74\pm 0.61$ | $69.34\pm 0.54$ | $69.47\pm 0.62$ | $69.34\pm 0.54$
$H_{0}$ | $0.001$ | $74.1\pm 1.4$ | $69.74\pm 0.61$ | $69.34\pm 0.54$ | $69.47\pm 0.62$ | $69.34\pm 0.54$
| $0.2$ | $74.1\pm 1.4$ | $69.74\pm 0.61$ | $69.34\pm 0.54$ | $69.48\pm 0.62$ | $69.35\pm 0.54$
| $0.0$ | $0.0215\pm 0.0037$ | $0.0213^{+0.0030}_{-0.0062}$ | $0.0217^{+0.0062}_{-0.0037}$ | $0.02220\pm 0.00050$ | $0.02220\pm 0.00050$
$\Omega_{b}h^{2}$ | $0.001$ | $0.0215\pm 0.0038$ | $0.0213^{+0.0032}_{-0.0062}$ | $0.0217\pm 0.0038$ | $0.02220\pm 0.00050$ | $0.02220\pm 0.00050$
| $0.2$ | $0.0215\pm 0.0038$ | $0.0213^{+0.0032}_{-0.0062}$ | $0.0217\pm 0.0038$ | $0.02220\pm 0.00050$ | $0.02221\pm 0.00050$
| $0.0$ | $0.300\pm 0.058$ | $0.273\pm 0.020$ | $0.279\pm 0.020$ | $0.301\pm 0.019$ | $0.2798\pm 0.0088$
$\Omega_{m}$ | $0.001$ | $0.300\pm 0.058$ | $0.272\pm 0.020$ | $0.279\pm 0.020$ | $0.301\pm 0.019$ | $0.2798\pm 0.0088$
| $0.2$ | $0.300\pm 0.058$ | $0.272\pm 0.020$ | $0.279\pm 0.020$ | $0.301\pm 0.019$ | $0.2797\pm 0.0088$
| $0.0$ | $2.97^{+0.73}_{-0.50}$ | $3.26^{+0.73}_{-0.24}$ | $1.69^{+0.15}_{-0.17}$ | $2.19^{+0.29}_{-0.45}$ | $1.70^{+0.14}_{-0.17}$
$m$ | $0.001$ | $2.97^{+0.74}_{-0.50}$ | $3.25^{+0.73}_{-0.22}$ | $1.69^{+0.15}_{-0.18}$ | $2.19^{+0.29}_{-0.45}$ | $1.70^{+0.14}_{-0.17}$
| $0.2$ | $2.96^{+0.74}_{-0.50}$ | $3.25^{+0.73}_{-0.22}$ | $1.69^{+0.15}_{-0.17}$ | $2.20^{+0.29}_{-0.45}$ | $1.69^{+0.14}_{-0.17}$
| $0.0$ | $2.3^{+0.1}_{-0.14}$ | $2.018^{+0.051}_{-0.084}$ | $2.289\pm 0.07$ | $2.2^{+0.079}_{-0.01}$ | $2.29\pm 0.066$
$10^{-4}\tilde{\delta}$ | $0.001$ | $2.3^{+0.1}_{-0.14}$ | $2.018^{+0.051}_{-0.084}$ | $2.29\pm 0.07$ | $2.2^{+0.079}_{-0.01}$ | $2.29\pm 0.067$
| $0.2$ | $2.292^{+0.1}_{-0.14}$ | $2.01^{+0.05}_{-0.08}$ | $2.274\pm 0.07$ | $2.185^{+0.078}_{-0.1}$ | $2.275\pm 0.066$
| $0.0$ | $-19.235\pm 0.042$ | $-19.370\pm 0.017$ | $-19.369\pm 0.015$ | $-19.372\pm 0.017$ | $-19.369\pm 0.014$
$M$ | $0.001$ | $-19.235\pm 0.042$ | $-19.370\pm 0.017$ | $-19.369\pm 0.015$ | $-19.373\pm 0.017$ | $-19.369\pm 0.014$
| $0.2$ | $-19.235\pm 0.042$ | $-19.370\pm 0.017$ | $-19.369\pm 0.015$ | $-19.372\pm 0.018$ | $-19.369\pm 0.014$

In Figs.3 and 4, we plot the $68\%$ and $95\%$ two dimensional contours
between DUDF model parameters for $\zeta_{0}=0\;\&\;0.001$, together with the
$\Lambda$CDM model contours for comparison. In the Figs, we also show the one-
dimensional marginalized posterior distribution for the parameters of both
models. In Fig.5, we plot only the shared parameters between the two models.
From the Figs we can notice the correlation between different parameters. In
particular, there is the well-known anticorrelation between $H_{0}$ and
$\Omega_{m}$. There is also the positive correlation between $H_{0}$ and both
of the dark fluid parameters $m$ and $\tilde{\delta}$. This interprets
tightening the constraints on the dark fluid parameters $m$ and
$\tilde{\delta}$ once $H_{0}$ is well constrained. On the other hand, no
correlation exists between $m$ and $\tilde{\delta}$ themselves.

Figure 3: $68\%$ and $95\%$ c.l. contour plots between various combinations of
the model parameters for $\zeta_{0}=0$ using the full dataset combinations.
The Fig. also shows the $\Lambda$CDM contours for comparison and the one
dimensional marginalized posterior distributions for the model parameters.
Figure 4: $68\%$ and $95\%$ c.l. contour plots between various combinations of
the model parameters for $\zeta_{0}=0.001$ using the full observational data
set. The Fig also shows the $\Lambda$CDM contours for comparison and the one
dimensional marginalized posterior distributions for the model parameters.

Figure 5: DUDF model vs $\Lambda$CDM model cosmological parameters for
$\zeta_{0}=0$, left panel, and $\zeta_{0}=0.001$, right panel.

##  5 Model Comparison

The minimum $\chi^{2}$-value for DUDF model is obtained for $\zeta_{0}=0$ and
is found to be $1357.91$. This increases gradually as $\zeta_{0}$ increases to
reach $1363.22$ for $\zeta_{0}=10$. Table 3 shows $\chi^{2}_{\rm min}$ for
DUDF model for $\zeta_{0}=0,0.001$ and $0.2$ against that of the $\Lambda$CDM
model for the five dataset combinations used in the analysis.

Table 3: $\chi^{2}_{\rm min}$ of DUDF model for different values of
$\zeta_{0}$ vs $\Lambda$CDM model according to different data sets.

Data Set |  $\Lambda$CDM |  |  | Viscous Fluid |
---|---|---|---|---|---
|  |  | $\zeta_{0}=0.0$ | $\zeta_{0}=0.001$ | $\zeta_{0}=0.2$
Base | 1499.75 |  | 1501.04 | 1500.82 | 1501.82
Base $+$ BAO | 1484.9 |  | 1488.38 | 1486.79 | 1487.27
Base $+$ BAO $+$ OHD | 1353.4 |  | 1350.65 | 1351.24 | 1350.3
Base $+$ BAO $+$ CMB | 1496.43 |  | 1487.6 | 1395.33 | 1494.17
Base $+$ BAO $+$ CMB $+$ OHD | 1363.34 |  | 1357.91 | 1359.26 | 1359.9
$\Delta{AIC}=AIC_{fld}-AIC_{\Lambda CDM}$ | – |  | -1.43 | -0.08 | 0.56

The $\chi^{2}$-statistics enables the determination of the best-fit parameters
of a model. However, they are not sufficient for deciding which model is the
best, especially if the models have a different number of parameters. Many
model selection statistical methods are proposed in the context of cosmology
in the literature. The most commonly used is the Akaike information criteria
(AIC) [75]. It takes into consideration the number of parameters of each
model, and so it is more robust.

The AIC is defined for a given model as

$AIC=-2ln\mathcal{L}+2p,$ (5.1)

where $p$ is the number of free parameters of the model. The model which
minimizes the AIC is considered as the better model. As a result, the second
term in the criteria imposes a penalty against the extra number of parameters.
In general, the AIC criterion is inclined to select the model that better fits
the data [76]. However, the difference between AIC’s of different models is
that making sense. According to [77], if $\Delta_{i}=AIC_{i}-AIC_{\rm min}$,
then modeli which has $\Delta_{i}$ between 0 and 2 is considered as
substantial on the level of empirical support, while models with $\Delta_{i}$
between 4 and 7 has considerably less support. For $\Delta_{i}>10$, modeli may
be omitted in future consideration. In the last row of table 3 we include the
difference $\Delta{AIC}$ as computed with respect to the $\Lambda$CDM model
taken as a reference, so that $\Delta{AIC}=AIC_{\rm fld}-AIC_{\Lambda CDM}$.

We can see from the table that DUDF model minimizes the AIC value. It has a
difference of -1.43 to the $\Lambda$CDM for $\zeta_{0}=0$ and of $-0.08$ for
$\zeta_{0}=0.001$. Accordingly, observational data support DUDF model as a
viable framework for the description of our universe.

##  6 Cosmological Evolution of the Model

In this section, we study the cosmological evolution of the different physical
parameters of the standard cosmology and discuss the consistency of the
theoretical model with observations.

###  6.1 The Hubble Parameter

One of the most important physical parameters in cosmology is the Hubble
parameter. It describes the expansion history of the universe. In the upper
panel of Fig.6 we plot the evolution of the Hubble parameter due to DUDF model
with no viscosity, $\zeta_{0}=0$, and with constant viscosity parameter,
$\zeta_{0}=0.001$, against the OHD. We also displayed results from the
$\Lambda$CDM model for comparison. In the lower panel of the figure we present
the relative ratio $H(z)/H_{\Lambda CDM}$ at different values of $\zeta_{0}$.

Figure 6: Upper panel: evolution of the Hubble parameter due to DUDF model,
for $\zeta_{0}=0$ and $0.001$, vs the $\Lambda$CDM model. Lower panel: the
ratio between Hubble parameter as given by DUDF model to that of the
$\Lambda$CDM model.

The graph shows that the model represents the observational data very well. In
the upper panel of Fig.8 we show the behavior of the Hubble parameter due to
the model relative to the $\Lambda$CDM model at early and late times. The Fig.
shows that the fluid follows the history of the standard cosmology at early
times, while as time passes, it deviates slightly, in the range of $\approx
2\%$, from the $\Lambda$CDM model to follow the observational data, then
retains the standard behavior again at a later time.

###  6.2 Supernovae SNe Ia Observations

Distance measurements using SNe Ia lead to the discovery of the accelerating
expansion of the universe. This was first discovered in $1998$. SNe Ia are
referred to as standard candles, and their different characteristics extend to
their stellar age distribution. In the upper panel of Fig.7, we plot the
distance modulus calculated from DUDF model for the values of viscosity
coefficient $\zeta_{0}=0$ and $0.001$ against observations from Pantheon data.
Results from the $\Lambda$CDM model are also presented for comparison. In the
lower panel of Fig.7 we plot the ratio of the distance modulus from DUDF model
to that from the $\Lambda$CDM model.

Figure 7: Upper panel: distance modulus from viscous fluid model for
$\zeta_{0}=0$ and $0.001$ vs $\Lambda$CDM model. Lower panel: ratio between
the distance modulus due to the fluid model to that of the $\Lambda$CDM model.

This ratio is also plotted again up to early times in the lower panel of
Fig.8. We can see from the Figs that DUDF model excellently fits the SNe Ia
data. We can also see that DUDF model follows the standard behavior at early
times while slightly deviates as time passes before retaining the $\Lambda$CDM
behavior again at a later time.

Figure 8:  Upper panel: ratio between the distance modulus due to DUDF model
to that of the $\Lambda$CDM model. Lower Panel: the ratio between Hubble
parameter as given by DUDF model to that of the $\Lambda$CDM model.

###  6.3 Effective Equation of State parameter

In standard cosmology, the universe is isotropic and filled with a perfect
fluid. The equation of state (EoS) describing this fluid is characterized by a
constant parameter $\omega$ that is the ratio of fluid pressure to fluid
density. For the $\Lambda$CDM model, the universe is characterized mainly by
two components: dark energy with $\omega=-1$ and dark matter with $\omega=0$.

On the other hand, the effective EoS parameter of DUDF model is calculated
using relations (3.8) and (3.10) in (2.10) where we get

$p_{\rm fld}=-\rho_{\rm fld}+\frac{\rho_{\rm
fld}^{m+1}}{{\tilde{\delta}}^{m}+\rho^{m}_{\rm fld}}-3\zeta_{0}H.$ (6.1)

Accordingly

$\omega_{\rm fld}=-1+\frac{\rho_{\rm fld}^{m}}{{\tilde{\delta}}^{m}+\rho_{\rm
fld}^{m}}-3\zeta_{0}\frac{H}{\rho_{\rm fld}}.$ (6.2)

In Fig.9 we show the behavior of the EoS of DUDF model and how it is affected
with the viscosity parameter.

Figure 9: Behaviour of the EoS of DUDF model for different values of
$\zeta_{0}$.

The Fig shows the ability of the fluid for a smooth transition from dust at
early times, where $\omega(z)=0$, to dark energy domination at late time. It
also shows that increasing the viscosity coefficient slows the transition to
dark energy, which, in turn, slows the expansion of the universe.

###  6.4 Evolution of the Deceleration Parameter

Deceleration parameter is an important parameter in cosmology. Evolution of
the expansion of the universe can be discussed in terms of the this parameter.
Deceleration parameter is defined as

$q(z)=-a(z)\frac{\ddot{a}}{\dot{a}^{2}}.$ (6.3)

It was believed that the expansion of the universe is decelerating due to
self-gravitation. A study of the apparent magnitude of distant supernovae
reveals that the universe is instead accelerating. However, cosmological
observations showed that this acceleration is a recent phenomenon. That was
explained by the domination of dark energy in the late times. Fig.10 shows the
behavior of deceleration parameter due to DUDF model for different values of
$\zeta_{0}$. The Fig. also shows results due to the $\Lambda$CDM scenario for
comparison.

Figure 10: Behavior the deceleration parameter due to DUDF model for different
values of $\zeta_{0}$ compared to that results from the $\Lambda$CDM model.

One can see from the Fig that the deceleration parameter due to DUDF model
follows the standard evolution of the universe. The fluid can play the role of
the cosmological constant in the late times, and so, can switch the
deceleration expansion to an accelerating one. DUDF model expectations for
deceleration acceleration transition redshift depend on the value of the
viscosity parameter $\zeta_{0}$. The Fig shows again that increasing the
viscosity coefficient slowing the transition to the accelerating phase. The
transition redshift decreases from $z_{tr}\approx 0.84$ for $\zeta_{0}=0$ to
$z_{tr}\approx 0.78$ for $\zeta_{0}=8$, which is consistent with the results
of the observational analysis made in [78] and [79]. Similarly, the current
value of the deceleration parameter depends also on the value of $\zeta_{0}$.
Its value decreases from $q_{0}\approx-0.51$ for $\zeta_{0}=0$ to
$q_{0}\approx-0.53$ for $\zeta_{0}=8$, in agreement with [80].

###  6.5 Density of the Universe

Observations show that the universe is flat with a mean energy density equals
to its critical density. Friedmann equations for flat space enable the
determination of the energy budget of the universe at any time. For any
component $x$ of the universe’s fluid, the energy density is given by

$\Omega_{x}(z)=\frac{\rho_{x}(z)}{\rho_{crit}(z)},$ (6.4)

where $\rho_{x}$ is the density of the $x$-component, and $\rho_{crit}$ is the
critical density given by

$\rho_{crit}(z)=\frac{3cH^{2}(z)}{8\pi G}.$ (6.5)

In Fig.11 we show the density of different components of DUDF model vs the
$\Lambda$CDM model.

Figure 11: Evolution of density parameter for different components of the
universe fluid due to DUDF model for $\zeta_{0}=0$ and $0.001$. Shown also
results for $\Lambda$CDM model.

We can see from the Fig that the density parameter of DUDF model matches the
dust density due to the $\Lambda$CDM scenario at early times while gradually
follows the dark energy density to the future. It also shows that the
viscosity parameter does not much affect the behavior of the density
parameter.

##  7 Estimation of the Viscosity of the Cosmic Fluid

Different authors estimated the viscosity of the cosmic fluid found that its
present value is $\zeta_{0}\leq 10^{7}$ $Pa$ $s$. However, its value is highly
model-dependent. As we mentioned before, our observational data analysis is
not able to constrain the viscosity coefficient. However, we can rely on the
results of the Akaike criteria to estimate $\zeta_{0}$. DUDF model with
$\zeta_{0}=0$ minimizes the AIC value. Since the modeli with
$\Delta_{i}=AIC_{i}-AIC_{min}$ between 0 and 2 is considered as substantial on
the level of empirical support, we then can consider $\zeta_{0}\leq 0.2$ $km$
$s^{-1}Mpc^{-1}$ as acceptable for the value of viscosity coefficient as it
satisfies this condition. Multiplying this by $c^{2}/8\pi G$, we found that
the cosmic fluid is expected to have a present-day viscosity coefficient $\leq
3.48\times 10^{5}$ $Pa$ $s$.

##  8 Conclusions

We considered a standard FRW flat cosmology with several energy source
components. The two dark sectors of the cosmic fluid are replaced by a DUDF
model. The EoS of this fluid can asymptote between two power laws. As a
result, it enables fluid to have a smooth transition between early and late
times. The dissipative effects are described by a bulk viscosity with a
constant coefficient.

We have implemented data analysis for DUDF model using recent observational
datasets from BAO, CMB, CC, BBN, and SNe Ia. The model parameters are
constrained using five different datasets, namely, SNe $+$ BBN $+$ $H_{0}$ as
a base dataset, base $+$ BAO, base $+$ BAO $+$ OHD, base $+$ BAO $+$ CMB, and
base $+$ BAO $+$ OHD $+$ CMB. We found that the viscosity parameter of the
fluid, $\zeta_{0}$, is not constrained by the observational data, whence we
restricted our analysis to some chosen values of this parameter. We analyzed
for values of $\zeta_{0}$ from $0$ to $10$. We got a minimum $\chi^{2}$-value
of $1357.91$ at $\zeta_{0}=0$ compared to $1363.34$ for the $\Lambda$CDM
model. This increases to $1359.26$ at $\zeta_{0}=0.001$, and to $1359.9$ at
$\zeta_{0}=0.2$. It continues to increase gradually and reaches $1363.22$ at
$\zeta_{0}=10$. Such a result indicates that the observational data favors the
non-viscous barotropic fluid, at least for the constant-coefficient model. On
using the AIC for model comparison, we found that DUDF model minimizes the AIC
value for the $\chi^{2}$ of $\zeta_{0}=0$ and $0.001$, and has an AIC value
$<2$ for $\zeta_{0}=0.2$, compared to the $\Lambda$CDM model. Considering
this, we can estimate a present-day value for the viscosity coefficient to be
$\zeta_{0}\leq 3.48\times 10^{5}$ $\rm Pa.s$, which agrees with expectations
for this coefficient.

The study of the evolution of different cosmological parameters like the
Hubble parameter, the distance modulus, the effective equation of state of the
dark fluid, the cosmic deceleration parameter, and the cosmic density
parameter showed that DUDF model highly matches observations and follows the
standard scenario through the evolution history. Evolution of the deceleration
parameter proved to reflect the well-known behavior of such parameter, with
the present-day value of $q_{0}=-0.51$, and an acceleration-deceleration
transition redshift of $z_{tr}=0.84$, all calculated for $\zeta_{0}=0$, which
agrees with the results obtained by many previous authors. The evolution of
the EoS of the fluid manifests its ability for a smooth transition between
dark sectors at early and late times. The density parameter indicates again
that the fluid can smoothly evolve from one dark sector to the other, with a
dust density that matches $\Lambda$CDM model in the high redshift, and a dark
energy density dominates to the future.

We can then conclude that observational data analysis indicated that DUDF
model provides a viable framework for the description of our universe.

## Acknowledgments

The authors would like to thank A. Awad, A. El-zant, and W. El Hanafy for
valuable discussions.

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# Design Guidelines for Inclusive Speaker Verification Evaluation Datasets

###### Abstract

Speaker verification (SV) provides billions of voice-enabled devices with
access control, and ensures the security of voice-driven technologies. As a
type of biometrics, it is necessary that SV is unbiased, with consistent and
reliable performance across speakers irrespective of their demographic, social
and economic attributes. Current SV evaluation practices are insufficient for
evaluating bias: they are over-simplified and aggregate users, not
representative of usage scenarios encountered in deployment, and consequences
of errors are not accounted for. This paper proposes design guidelines for
constructing SV evaluation datasets that address these short-comings. We
propose a schema for grading the difficulty of utterance pairs, and present an
algorithm for generating inclusive SV datasets. We empirically validate our
proposed method in a set of experiments on the VoxCeleb1 dataset. Our results
confirm that the count of utterance pairs/speaker, and the difficulty grading
of utterance pairs have a significant effect on evaluation performance and
variability. Our work contributes to the development of SV evaluation
practices that are inclusive and fair.

Index Terms: speaker verification, voice biometrics, evaluation, audit, bias,
fairness, design guidelines

##  1 Introduction

Speaker verification (SV) technology, which determines the identity of a
speaker from their voice, is widely used across the world as a form of
biometric identification [1]. Applications range from devices like mobile
phones and smart speakers to transactional, investment and mobile banking,
call centers and proof-of-life verification of pensioners. While unobtrusive,
SV is extremely sensitive to the recording hardware, the deployment context
and environment [2]. Moreover, it is also sensitive to the demographic
attributes of speakers [3].

In machine learning (ML), bias has become a priority concern for fair and
inclusive development of ML technologies. A ML system is considered biased if
its predictive outputs favour or are prejudiced against individuals or groups
of people based on their demographic attributes [4]. An often-times unintended
consequence of bias is discrimination, which can lead to public mistrust [5]
and legal consequences [6]. It is thus important that potenatial bias in SV
systems is evaluated and mitigated during development. Studies of bias in SV,
and research to mitigate its effects, are currently limited. Recent research
has highlighted that existing SV evaluation practices contribute to
exacerbating bias in the development of SV systems [7]. In particular,
unrepresentative benchmarks make it impossible to evaluate bias, let alone
address it.

This paper presents design guidelines for the creation of more inclusive
evaluation datasets that are suitable for evaluating bias in SV. While
existing SV datasets consider demographic representation on a speaker level
[8, 9, 10], we show that it is also important to consider inclusion at the
more fine-grained level of utterance pairs. We address two key questions:

  1. 1.

How should utterance pairs be constructed to support an unbiased evaluation?

  2. 2.

How many utterance pairs are needed per speaker for a robust SV evaluation?

Using an iterative design approach, we propose an algorithm for constructing
evaluation datasets that are representative of realistic application
scenarios. We empirically validate the utility of the datasets. Our
experimental results present evidence that the difficulty of utterance pairs
impacts the evaluation outcome. We also show that randomized utterance
pairings can result in significant performance variation if the utterance pair
count per speaker is low. Based on these results we propose design guidelines
that consider inclusion on an utterance level.

In the next section we present background information on SV evaluation. Design
considerations for the evaluation dataset are discussed in Section 3. Section
4 outlines the experimental setup and results. We propose evaluation dataset
design guidelines and an outlook on future work in Section 5 and conclude in
Section 6.

##  2 Background

We now present a short overview of SV evaluation, and discuss shortcomings of
current practices. For a detailed overview of SV techniques and systems we
refer the reader to [1] for a historic, and to [11] for a recent survey of
deep learning methods.

###  2.1 Speaker Verification Evaluation

SV performance is determined by the false positive (FP) rate and false
negative (FN) rate at a threshold value to which the system has been
calibrated. It is accepted that the two error rates present a trade-off, and
that selecting an appropriate threshold is an application-specific design
decision [12]. The threshold value is determined by balancing the FP and FN
error rates for a particular objective, such as obtaining an _equal error
rate_ (EER) for FP and FN errors, or minimising a cost function. The US
National Institute of Standards and Technology (NIST), which has been
conducting speaker recognition evaluations for over two decades, advises
against the use of the _EER_ , as applications typically require either low FP
or FN [13]. The _detection cost function_ (DCF) is a weighted sum of FP and FN
rates across threshold values, with weights determined by the application
requirements [13]. To compare performance across models, systems are
frequently tuned to the threshold value at the minimum of the DCF (_minDCF_),
and the _minDCF_ is reported as metric. NIST recommends that SV systems are
evaluated with Detection Error Tradeoff (DET) curves to show SV performance
across a range of error rates.

###  2.2 Short-comings of Current Evaluation Practices

Alongside the NIST, independent SV competitions are a popular mechanism for
evaluating and driving advancement in SV techniques. Many competitions, such
as SITW [10], the VoxCeleb SRC [14] and the Far-Field SRC [15] have been
hosted at Interspeech. The competitions serve as benchmarks, promote
evaluation habits and implicitly guide the culture of evaluation in the
domain. To enable comparison across competition entries, they focus their
evaluations on one or two metrics, typically a version of the _minDCF_ or
_EER_.

Recent research on bias in SV has shown that evaluation outcomes are highly
dependent on the evaluation dataset and metrics used for evaluation [7]. The
paper found that current SV evaluation practices consider limited context,
environment and user variability in the design of evaluation datasets and
metrics. The authors argue that SV evaluation practices present an
oversimplified view of application scenarios, promote unrepresentative
benchmark populations and oversimplified metrics, do not consider the
consequences of errors and favour low FP rates while neglecting the
consequences of FN rates. This limits current evaluation practices in their
ability to detect bias in realistic application settings and adds to bias that
may already exist in SV technologies. We build on this work by studying
approaches to mitigating bias in the design of evaluation datasets to ensure
that evaluation practices are more inclusive and fair.

##  3 Dataset Design Considerations

In this section we consider consequences of speaker verification (SV) errors,
introduce a schema for grading utterance pairs, and propose an algorithm for
generating inclusive SV evaluation datasets.

We use the following terminology and notation. A _speaker_ ($S^{i}$) is a
unique individual with at least one speech _utterance_ ($u^{i}$) in the set of
all _evaluation utterances_ ($\mathcal{U}$). $\mathcal{U}^{i}$ is the subset
of evaluation utterances of speaker $S^{i}$ and $S=\\{S^{1}...S^{K}\\}$ is the
set of all speakers in $\mathcal{U}$. An _evaluation dataset_ ($\mathcal{D}$)
is constructed by generating utterance pairs $\\{u_{a},u_{b}\\}$ from the
evaluation utterances in $\mathcal{U}$. Conventionally $u_{a}$ is the
enrollment utterance, and $u_{b}$ is the test utterance. Practically, the
order of utterance pairs does not matter in SV evaluation.

A _same speaker_ pair $\\{u^{1}_{a},u^{1}_{b}\\}$ has the enrollment and test
utterances drawn from $\mathcal{U}^{1}$, while a _different speaker_ pair
$\\{u^{1}_{a},u^{2}_{a}\\}$ has the enrollment and test utterances drawn from
two different speakers, ie from $\mathcal{U}^{1}$ and $\mathcal{U}^{2}$. The
objective of creating an inclusive SV evaluation dataset, $\mathcal{D}$, is
then to:

  * •

offer an equivalent evaluation across all speakers in $S$

  * •

generate same speaker and different speaker utterance pairs that are
reflective of deployment usage scenarios

  * •

facilitate an evaluation that is robust to perturbations in utterance pairings
(e.g., if $\\{u^{1}_{a},u^{2}_{a}\\}$ is substituted with
$\\{u^{1}_{b},u^{3}_{f}\\}$ the false positive error rate should not change
significantly)

###  3.1 Difficulty Grading of Utterance Pairs

SV is a comparative task that compares an enrolled speaker utterance with a
test utterance based on the difference between embeddings of the voice
profiles. Naturally, system performance is affected by the utterance pairs
being compared. However, not all utterance pairs are born equal.

It is important that evaluation utterance pairs test likely scenarios that a
SV system will encounter when deployed, and that pairs of appropriate
difficulty are constructed to test the limits of the system. A qualitative
difficulty grading of utterance pairs has the advantage that it bares no
computational cost and that it can be interpreted in relation to the
application context. We thus created a schema for grading the difficulty of
utterance pairs across four categories: trivial (cat 1), easy (cat 2), medium
(cat 3) and hard (cat 4) for same and different speaker pairs. As a general
rule, evaluation of same speakers is easier if utterance pairs are more
similar, and harder if utterance pairs are more different. For different
speakers the opposite is true: similar utterance pairs are harder, and
different utterance pairs are easier to classify. Speaker, recording channel
and environmental attributes affect the similarity of utterance pairs. In
Table 1 we show how we use the schema to grade utterance pair difficulty based
on speaker gender, nationality, recording channel and background noise
(attributes chosen based on available metadata in the VoxCeleb1 dataset). Same
speaker pairs always have the same gender and nationality, while different
speaker pairs are never from the same recording. We assume that utterances
from the same recording have the same noise conditions.

Utterance
Pairs |  Difficulty |  Same
Gender |  Same
Nationality |  Same
Recording |  Same
Noise
---|---|---|---|---|---
Same |  cat 1 (trivial) |  Yes |  Yes |  Yes |  Yes
Speaker |  cat 3 (med.) |  Yes |  Yes |  No |  n.k.
|  cat 1 (trivial) |  No |  No |  No |  n.k.
Different |  cat 2 (easy) |  No |  Yes |  No |  n.k.
Speakers |  cat 3 (med.) |  Yes |  No |  No |  n.k.
|  cat 4 (hard) |  Yes |  Yes |  No |  n.k.
Table 1: Grading of utterance pairs (n.k. = not known)

###  3.2 Generating Evaluation Datasets

Taking these considerations and the dataset requirements into account, we
designed Algorithm 1 for generating an evaluation dataset, $\mathcal{D}$, from
a set of utterances, $\mathcal{U}$. Given available utterances, our intention
was to create an evaluation dataset in the most difficult category. To ensure
that the evaluation is robust to perturbations in utterance pairings, the
random seed for selecting same and different speaker utterance pairs can be
changed to generate similar datasets with different utterance pairings. SV
evaluation outcomes can then be compared across the datasets.

Algorithm 1 Evaluation Dataset Generation Algorithm

$S\leftarrow$ set of all speakers

$i\leftarrow$ unique speaker

$a,b\leftarrow$ two recording instances

$u\leftarrow$ utterance

$\mathcal{U}\leftarrow$ set of all utterances

$n\leftarrow$ count of _different speaker_ utterance pairs/speaker

$r\leftarrow$ random seed

$\mathcal{D}\leftarrow$ new evaluation dataset

for $S^{i}$ in $S$ do

$\mathcal{U}^{i}\leftarrow$ subset of $u^{i}$

$\mathcal{U}^{j}\leftarrow$ subset of $u^{j}$ $\triangleright$ $i\neq j$;
$S^{i},S^{j}$ same group

$\mathcal{D}^{i}_{same}\leftarrow$ new list of _same speaker_ utterance pairs

$\mathcal{D}^{i}_{diff}\leftarrow$ new list of _different speaker_ utterance
pairs

for $u^{i}_{a}$, $u^{i}_{b}$ in $\mathcal{U}^{i}$ do $\triangleright$ same
speaker pairs

if recordinga is not recordingb then $\triangleright$ difficulty: medium

$\mathcal{D}^{i}_{same}$ append ${\\{u^{i}_{a},u^{i}_{b}\\}}$

else

pass

end if

end for

$\mathcal{D}^{i}_{same}\leftarrow$ randomly select $n$ pairs from
$\mathcal{D}^{i}_{same}$ with seed $r$

$\mathcal{U}^{i}_{n}\leftarrow$ randomly select $n$ $u$ from $\mathcal{U}^{i}$
with seed $r$ (replace ok)

$\mathcal{U}^{j}_{n}\leftarrow$ randomly select $n$ $u$ from $\mathcal{U}^{j}$
with seed $r$

$x\leftarrow 0$

while $x<n$ do

for $u^{i}_{x}$, $u^{j}_{x}$ in $\mathcal{U}^{i}_{n}$, $\mathcal{U}^{j}_{n}$
do $\triangleright$ different speaker pairs

$\mathcal{D}^{i}_{diff}$ append ${\\{u^{i}_{x},u^{j}_{x}\\}}$

end for

end while

$\mathcal{D}$ append $\mathcal{D}^{i}_{same}$

$\mathcal{D}$ append $\mathcal{D}^{i}_{diff}$

end for

##  4 Experiments

We empirically validated our approach to generating SV evaluation datasets by
conducting experiments with the VoxCeleb1 dataset [16]. Due to space
limitations we only show experiments and results for speaker groups based on
nationality. Our experiments show that model performance deteriorates
significantly when evaluated on datasets with more difficult utterance pairs
of same and different speakers. Moreover, when the number of utterance pairs
per speaker is low, model performance has high variability. Evaluation
robustness improves as the number of utterance pairs increases. In this
section we describe the setup of the experiments and the baseline evaluation
dataset, and then present results to demonstrate the effect of utterance pair
grading and count.

###  4.1 Setup

We use a pre-trained end-to-end model based on a 34-layer ResNet trunk
architecture. The model is described in detail in [17] where it is referred to
as the performance optimized model. The model has been trained on the VoxCeleb
2 training set [8], with close to 1 million utterances of 5994 speakers. 61%
of speakers are male and 29% of speakers have a US nationality, which is the
most represented nationality. The feature input to the model are
64-dimensional log Mel-filterbanks.

We use the model in a SV inference pipeline to classify utterance pairs in an
evaluation dataset $\mathcal{D}$ as being from the same or from different
speakers. The only variable that we consider in the experiments is
$\mathcal{D}$. Evaluation datasets were generated from VoxCeleb1 with the
algorithm described in Section 3. Different speaker pairs were constructed
from speakers with the same gender and nationality. The evaluation sets are
characterised by speaker nationalities, the count of utterance pairs/speaker
($n$)111$n$ is the count of same _or_ different utterance pairs. As these two
counts are equal, the total number of utterance pairs per speaker is $2n$. and
the random seed with which the dataset was generated. The experiments are
detailed in Table 2.

Exp |  Nationalities
(speaker count) |  Pairs/speaker
($n$) |  Random
seed
---|---|---|---
1 |  Canada, India, USA, Ireland, Norway, UK, Australia, Germany |  520 |  12
2 |  Canada (54), India (26), UK (215) |  50, 100, 150, 225, 350, 450, 520 |  3, 6, 8, 12, 20
Table 2: Overview of SV evaluation experiments.

###  4.2 Baseline Evaluation Dataset

To benchmark our evaluation datasets, we compare them against a baseline
evaluation on VoxCeleb1-H. Of the three VoxCeleb1 evaluation sets, VoxCeleb1-H
is considered the ”hard” set. It includes only different speaker pairs of the
same gender and nationality, and requires groups of 5 or more speakers. Table
3 summarises the attributes of same speaker pairs in VoxCeleb1 H. As can be
seen from the table, the number of speakers, the total number of utterance
pairs, and the count of utterances pairs per speaker varies across speaker
groups based on nationality. Additionally, all groups contain trivial (cat 1)
same speaker pairs. However, the proportion of trivial pairs in relation to
all of the group’s same speaker pairs differs.

Nationality | Speakers | Pairs |  Pairs/
speaker |  cat 1
(trivial)
---|---|---|---|---
USA | 799 | 178122 |  222.9 |  12.9%
UK | 215 | 53111 |  247.0 |  10.3%
Canada | 54 | 10864 |  201.2 |  11.1%
India | 26 | 10053 |  386.7 |  10.6%
Australia | 37 | 8668 |  234.3 |  10.5%
Ireland | 18 | 4960 |  275.6 |  8.5%
Norway | 20 | 4906 |  245.3 |  10.0%
New Zealand | 6 | 1811 |  301.8 |  10.1%
Germany | 5 | 1256 |  251.2 |  17.0%
Mexico | 5 | 1130 |  226.0 |  10.2%
Italy | 5 | 571 |  114.2 |  17.0%
Table 3: VoxCeleb1-H Same Speaker Utterance Pairs.

###  4.3 Results

We now analyze the results of our experiments and show how utterance pair
grading and utterance pair count affect the evaluation outcome.

Figure 1: DET curves for a baseline and our inclusive dataset show significant
performance decline when the latter is used for evaluation. This implies that
the SV model performs worse under more realistic and inclusive evaluation
scenarios. Figure 2: DET curves show variability in evaluation outcomes for
evaluation sets with 50, 225 and 520 utterance pairs ($n$) for Canadian,
Indian and UK speakers. For each $n$ five datasets were generated with
different random seeds.

####  4.3.1 Effect of Utterance Pair Grading

Experiment 1 included speakers from 8 nationalities, with 520 utterance pairs
of different speakers (1040 total pairs) per speaker. We call this dataset
_VoxCeleb1-Inclusive_. The key differences between the VoxCeleb1-H baseline
and VoxCeleb1-Inclusive are that the latter contains an equal count of
pairs/speaker across individual speakers and speaker groups, that trivial same
speaker pairs have been excluded and that speakers were included if their
unique utterance pair count for different speakers was greater than our
selection value $n$. Due to our randomized utterance pair generation, the
different speaker pairs in the two evaluation sets are not the same.

The DET performance curves for the model evaluated on the two datasets is
shown in Figure 1. When evaluated on VoxCeleb1 H performance is reasonable: a
FP rate of 1% results in a FN rate of 5%. However, at the same FP rate the FN
rate triples to 15% when same speaker utterance pairs are exclusively of
medium difficulty grading.

####  4.3.2 Effect of Utterance Pair Count

In Experiment 2 we investigate the effect of the utterance pair count on the
robustness of the evaluation, given perturbations in utterance pairings. We
compare the evaluation outcome for Canadian, Indian and UK speakers across
evaluation datasets generated with 50, 100, 150, 225, 350, 450 and 520
distinct different speaker utterance pairs/speaker. For each utterance pair
count $n$ we generated five versions of the dataset with different random
seeds.

Figure 3: Variance in the minDCF metric for Canadian, Indian and UK speakers
for evaluation datasets with different counts of utterance pairs/speaker
($n$). Five versions of each dataset are compared. The evaluation is more
robust when $n$ is larger.

For the equal error rate (_EER_) and minimum Detection Cost Function
(_minDCF_) performance metrics we observe significant variability in
performance when $n$ is small. Results for the _minDCF_ are shown in Figure 3.
To illustrate, the _minDCF_ of the worst performing evaluation is 30% higher
than the best evaluation for Canadian speakers when $n=50$. This performance
difference can be attributed entirely to utterance pairings generated with
different random seeds. In general, we observe that variance is greater when
the count of unique speakers in a group is smaller: UK speakers experience the
lowest variance, while Indian speakers experience the greatest variance.

Variability is not limited to performance on the _EER_ and _minDCF_ metrics,
but exists across the DET performance curve, as shown in Figure 2. As $n$
increases, variability remains greatest at the end points of the curve where
FP and FN error rates are the greatest/smallest. Variability stablilizes first
in the center of the curve. This means that at very low FP rates the expected
FN rate will carry significant uncertainty (and vice versa), unless $n$ is
large enough to ensure stable performance.

##  5 Evaluation Dataset Design Guidelines

Based on our analysis and empirical validation, we propose that an inclusive
evaluation dataset for robust speaker verification evaluation should:

  1. 1.

have an equal number of same speaker and different speaker utterance pairs for
each speaker

  2. 2.

have at least 500 different speaker utterance pairs for each speaker

  3. 3.

have an equal number of utterance pairs per speaker

  4. 4.

have an equal proportion of utterance pairs of each difficulty grading per
speaker

  5. 5.

have utterance pairs with difficulty gradings that are representative of usage
scenarios of deployed applications

  6. 6.

be compared against datasets variations with randomly generated utterance
pairings to ensure robust evaluation outcomes

Limitations and Future Work: While we studied the effects of utterance pairs
on SV evaluation outcomes, we did not conduct a detailed study on the number
of unique speakers that are required for a robust evaluation. This is an
opportunity to extend the proposed design guidelines in future research. Our
classification of utterance pairs by difficulty grading was limited by the
metadata available for the VoxCeleb1 dataset. In actual usage scenarios hard
utterance pairs may be more challenging than the pairs we evaluated. For
example, smart device applications should consider different utterance pairs
from same gender family members or colleagues, which present the most likely
intruders to these systems. Future work should also consider speaker groups
along different dimensions, such as age, accent, health conditions and time.

##  6 Conclusion

In this paper we present design guidelines for generating inclusive evaluation
datasets towards robust speaker verification (SV) evaluation. During an
iterative development process we elaborate on design considerations that
account for consequences of SV errors, propose a taxonomy for grading the
difficulty of utterance pairs, and present an algorithm for generating
inclusive evaluation datasets. We empirically validate the evaluation datasets
generated with our proposed method in a set of experiments on the VoxCeleb1
dataset. Our results confirm that the count of utterance pairs/speaker, and
the difficulty grading of utterance pairs have a significant effect on
evaluation outcomes. We advocate that current speaker verification evaluation
practices can become more inclusive and robust by including equal counts of
utterance pairs with the same difficulty grading for all individual speakers
and groups.

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