# Late-time cosmology in a model of modified gravity with an exponential
function of the curvature

A. Oliveros<EMAIL_ADDRESS>Mario A. Acero
<EMAIL_ADDRESS>
###### Abstract

In this work, we analyse the late-time evolution of the Universe for a
particular $f(R)$ gravity model built from an exponential function of the
scalar curvature. Following the literature, we write the field equations in
terms of a suited statefinder function ($y_{H}(z)$) and considering well
motivated physical initial conditions, the resulting equations are solved
numerically. Also, the cosmological parameters $w_{\rm{DE}}$, $w_{\rm{eff}}$,
$\Omega_{\rm{DE}}$ and $H(z)$ and the statefinder quantities $q$, $j$, $s$ and
$Om(z)$ are explicitly expressed in terms of $y_{H}(z)$ and its derivatives.
Furthermore, setting an appropriate set of values for the model parameters,
the cosmological parameters as well as the statefinder quantities are plotted,
and their present values (at $z=0$), are shown to be compatible with Planck
2018 observations and the $\Lambda$CDM–model values. Considering updated
measurements from the dynamics of the expansion of the Universe, $H(z)$, we
perform an statistical analysis to constrain the free parameters of the model,
finding a particular set of values that fit the data well and predict
acceptable values for the cosmological and statefinder parameters at present
time. Therefore, the $f(R)$ gravity model is found to be consistent with the
considered observational data, and a viable alternative to explain the late-
time acceleration of the Universe.

###### keywords:

Modified gravity , Dark energy , $f(R)$ gravity.

###### PACS:

04.50.Kd , 98.80.-k

[1]organization=Programa de Física, Universidad del Atlántico,
addressline=Carrera 30 Número 8-49, city=Puerto Colombia, state=Atlántico,
country=Colombia

##  1 Introduction

Modified gravity theories are a set of proposals used in last years in order
to explain the observed late-time acceleration of the Universe, among other
cosmological and astrophysical phenomena. In these scenarios, it is not
necessary to consider dark energy (DE) or new forms of matter to explain the
late-time acceleration (for more details about this topic see Refs. [1, 2,
3]). Modified gravity theories include scalar-tensor theories, scalar-vector-
tensor theories, Einstein–aether, Bimetric theories, TeVeS, $f(R)$, general
higher-order theories, Hořava–Lifschitz gravity, Galileons, Ghost Condensates,
and models of extra dimensions, including Kaluza-Klein, Randall–Sundrum and
DGP (see e.g. Refs. [4, 5, 6] for a review). Among this wide spectrum of
proposals, $f(R)$ gravity is one of the most studied in the literature, [7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31] since in this framework, it is possible to analyse and unify
different cosmological eras (inflation and dark energy). In general, the
explicit form of the function $f(R)$ is given by some consistency requirements
and diverse constraints, which impose conditions for the cosmological
viability of $f(R)$ dark energy models [32], i.e., the function $f(R)$ must
satisfy both cosmological and solar-system tests in the small-field limit of
the parameter space. Particular $f(R)$ models satisfying such requirements
that have been studied intensively in the literature include: the Hu-Sawicki
model [33], Starobinsky’s model [34], Tsujikawa’s model [35], and the
Exponential model [36]. More recently, the Gogoi-Goswami model [37] has
attracted special attention. Usually in these models, the particle associated
with the cold dark matter component is not consider explicitly. In this sense,
the authors of Ref. [29] have considered an axion $f(R)$ gravity model, in
which the axion is the main component of cold dark matter in the Universe,
demonstrating that the such a model can unify the early-time with the late-
time acceleration (see Refs. [38, 39, 40, 41] for a more exhaustive exposition
about this topic).

In this work, we carry out a detailed study of the late-time evolution of the
Universe taking into account the $f(R)$ gravity model introduced in Ref. [42],
which is based on an exponential function of the scalar curvature. This model
behaves akin to $\Lambda$CDM at early times, and satisfies local and
cosmological constraints. The procedure that we follow here, consists in
writing the field equations in terms of a suitable statefinder function and,
using well motivated physical initial conditions, solving the resulting
equations numerically. This strategy has been exhaustively implemented in the
literature [33, 44, 45, 46, 47], but until now, it has not been applied for
the $f(R)$ gravity model under consideration. This study is mainly focused in
analysing the behavior of some cosmological parameters ($w_{\rm{DE}}$,
$w_{\rm{eff}}$, $\Omega_{\rm{DE}}$ and $H(z)$) together with the statefinder
quantities $q$, $j$, $s$ and $Om(z)$. Thereby, with this analysis we hope to
broaden the evidence which allows us to validate or to rule out this $f(R)$
gravity model. To add up to such scrutiny, we perform a statistical analysis,
using updated measurements from the dynamics of the expansion of the Universe,
$H(z)$, looking for establishing restrictions to the free parameter of the
model.

This paper is organized as follows: in section 2 we perform a brief review
about the $f(R)$ gravity. In section 3, we write the field equations in terms
of a suitable statefinder function, $y_{H}(z)$, and taking into account
physically motivated initial conditions, the resulting equations are solved
numerically. In this section we also explicitly present the cosmological
parameters $w_{\rm{DE}}$, $w_{\rm{eff}}$, $\Omega_{\rm{DE}}$, and $H(z)$, and
the statefinder quantities $q$, $j$, $s$, and $Om(z)$, in terms of $y_{H}(z)$
and its derivatives. Furthermore, considering appropriate values for the model
parameters, the evolution of the cosmological parameters and the statefinder
quantities is plotted with respect to the redshift $z$, carrying out a
complete cosmological analysis. Finally, in section 4 we consider a set of
observational data of $H(z)$, to constrain the distinct parameters of the
model under consideration. Our conclusions are exposed in section 5.

##  2 The $f(R)$ gravity: a brief review

In general, the action for an $f(R)$ gravity model in the presence of matter
components is given by

$S=\int{d^{4}x\sqrt{-g}\left(\frac{f(R)}{2\kappa^{2}}+\mathcal{L}_{M}\right)},$
(1)

where $g$ denotes the determinant of the metric tensor $g^{\mu\nu}$,
$\kappa^{2}=8\pi G=1/M_{p}^{2}$, with $G$ being the Newton’s constant and
$M_{p}$ the reduced Planck mass. $\mathcal{L}_{M}$ represents the Lagrangian
density for the matter components (relativistic and non-relativistic perfect
matter fluids). The term $f(R)$ is for now an arbitrary function of the Ricci
scalar $R$. Variation with respect to the metric gives the equation of motion

$f_{R}(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R)+(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})f_{R}(R)=\kappa^{2}T_{\mu\nu}^{(M)},$
(2)

where $f_{R}\equiv\frac{df}{dR}$, $\nabla_{\mu}$ is the covariant derivative
associated with the Levi-Civita connection of the metric, and
$\square\equiv\nabla^{\mu}\nabla_{\mu}$. Plus, $T_{\mu\nu}^{(M)}$ is the
matter energy–momentum tensor which is assumed to be a perfect fluid.
Considering the flat Friedman-Robertson- Walker (FRW) metric,

$ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j},$ (3)

with $a(t)$ representing the scale factor, the time and spatial components of
Eq. (2) are given, respectively, by

$3H^{2}f_{R}=\kappa^{2}\rho_{M}+\frac{1}{2}(Rf_{R}-f)-3H\dot{f}_{R},$ (4)

and

$-2\dot{H}f_{R}=\kappa^{2}(\rho_{M}+P_{M})+\ddot{f}_{R}-H\dot{f}_{R},$ (5)

where $\rho_{M}$ and $P_{M}$ denote the density and pressure, respectively, of
any perfect fluid of non-relativistic and relativistic matter. As usual in the
literature, it is possible to rewrite the field equations (4) and (5) in the
Einstein-Hilbert form:

$3H^{2}=\kappa^{2}\rho,$ (6) $-2\dot{H}^{2}=\kappa^{2}(\rho+P),$ (7)

where $\rho=\rho_{M}+\rho_{\rm{DE}}$ and $P=P_{M}+P_{\rm{DE}}$ correspond to
the total effective energy density and total effective pressure density of the
cosmological fluid. Note that $\rho_{M}=\rho_{m}+\rho_{r}$. In this case, the
dark energy component has a geometric origin, and after a some manipulation in
Eqs. (4) and (5), we obtain

$\rho_{\rm{DE}}=\frac{1}{\kappa^{2}}\left[\frac{Rf_{R}-f}{2}+3H^{2}(1-f_{R})-3H\dot{f}_{R}\right],$
(8)
$P_{\rm{DE}}=\frac{1}{\kappa^{2}}[\ddot{f}_{R}-H\dot{f}_{R}+2\dot{H}(f_{R}-1)-\kappa^{2}\rho_{\rm{DE}}].$
(9)

Following the notation used in Ref. [45], we shall assume that the perfect
fluids, composed of Cold Dark Matter (CDM) and radiation, can be written as

$\rho_{M}(t)=\rho_{m0}\frac{1}{a^{3}}\left(1+\chi\frac{1}{a}\right),$ (10)

where $\rho_{m0}$ is the energy density of the CDM at present time, and
$\chi=\rho_{r0}/\rho_{m0}=\Omega_{r0}/\Omega_{m0}\simeq 3.1\times 10^{-4}$,
with $\rho_{r0}$ being the current density of relativistic component.

##  3 Late-time cosmological evolution

In this section we analyse the late-time evolution of the Universe taking into
account a particular $f(R)$ gravity model. Since the set of equations obtained
here are very complicated, it will be necessary to solve them numerically. In
order to facilitate such implementation, we perform a change of variable,
i.e., we use the redshift, $z$, instead of the cosmic time $t$ as a dynamical
variable.

To begin with, we know that

$1+z=\frac{1}{a(t)},$ (11)

where $a(0)=1$ is assumed. Then, from Eq. (10), it is easy to derive the
following relation:

$\frac{d}{dt}=-H(1+z)\frac{d}{dz}.$ (12)

In this way, it is evident that

$\dot{H}=-H(z)(1+z)H^{\prime}(z),$ (13)

where prime denotes derivative with respect to the redshift $z$; in a similar
fashion, we can write $R$ and $\dot{f}_{R}$ as follows:

$R(z)=6(2H^{2}+\dot{H})=6[2H(z)^{2}-(1+z)H(z)H^{\prime}(z)],$ (14)

$\begin{split}\dot{f}_{R}(z)&=\dot{R}f_{RR}\\\
&=6H(z)(1+z)^{2}\left[H^{\prime}(z)^{2}+H(z)H^{\prime\prime}(z)-\frac{3H(z)H^{\prime}(z)}{1+z}\right]f_{RR}.\end{split}$
(15)

It is clear in this context that we only need to use Eq. (6) for our analysis,
but instead of solving the resulting equation for $H(z)$, we introduce a
suitable statefinder function $y_{H}(z)$, defined by [33, 44, 45, 46, 47]

$y_{H}(z)\equiv\frac{\rho_{\rm{DE}}}{\rho_{m0}}=\frac{H(z)^{2}}{m_{s}^{2}}-(1+z)^{3}-\chi(1+z)^{4},$
(16)

where $m_{s}^{2}=\frac{\kappa^{2}\rho_{m0}}{3}=\Omega_{m0}H_{0}^{2}\simeq
6.51108\times 10^{-67}\,\rm{eV}^{2}$.111This value is obtained using the
latest Planck data [48]: $\Omega_{m0}\simeq 0.315$ and $\rm{h}\simeq 0.674$,
and taking into account the conversion factors: $1\,\rm{s}=1.51927\times
10^{15}\,\rm{eV}^{-1}$ and $1\,\rm{m}=5.06765\times 10^{6}\,\rm{eV}^{-1}$,
which follows from considering the natural units, $\hbar=c=1$. Hence,
$H_{0}=100\,\rm{h}\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}=2.1331\,\rm{h}\times
10^{-42}\,\rm{GeV}=1.4377\times 10^{-33}\,\rm{eV}$. From Eq. (16), we obtain
the following useful expressions in terms of $y_{H}(z)$ and its derivatives:

$\displaystyle H(z)^{2}$ $\displaystyle=$ $\displaystyle
m_{s}^{2}[y_{H}(z)+(1+z)^{3}(1+(1+z)\chi)],$ $\displaystyle H(z)H^{\prime}(z)$
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{s}^{2}[y_{H}^{\prime}(z)+(1+z)^{2}(3+4(1+z)\chi)],$
(17) $\displaystyle H^{\prime}(z)^{2}+H(z)H^{\prime\prime}(z)$
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{s}^{2}[y_{H}^{\prime\prime}(z)+6(1+z)(1+2(1+z)\chi)].$

Therefore, Eqs. (14) and (15) take the following form:

$R(z)=3m_{s}^{2}[4y_{H}(z)-(1+z)y_{H}^{\prime}(z)+(1+z)^{3}],$ (18)
$\begin{split}\dot{f}_{R}(z)=&-3m_{s}^{3}(1+z)\sqrt{y_{H}(z)+(1+z)^{3}(1+(1+z)\chi)}\\\
&\times[3y_{H}^{\prime}(z)+(1+z)(3(1+z)-y_{H}^{\prime\prime}(z))]f_{RR},\end{split}$
(19)

and Eq. (10) in terms of the redshift is

$\rho_{M}(z)=\frac{3m_{s}^{2}}{\kappa^{2}}(1+z)^{3}[1+(1+z)\chi].$ (20)

Similarly, the cosmological parameters which will be used later, are given by:

$\Omega_{\rm{DE}}(z)=\frac{y_{H}(z)}{y_{H}(z)+(1+z)^{3}[1+\chi(1+z)]},$ (21)
$w_{\rm{DE}}(z)=-1+\frac{1}{3}(1+z)\frac{y^{\prime}_{H}(z)}{y_{H}(z)},$ (22)
$\begin{split}w_{\rm{eff}}(z)&=-1+\frac{2}{3}(1+z)\frac{H^{\prime}(z)}{H(z)}\\\
&=\frac{(1+z)y^{\prime}_{H}(z)-3y_{H}(z)+(1+z)^{4}\chi}{3\\{y_{H}(z)+(1+z)^{3}[1+\chi(1+z)]\\}}.\end{split}$
(23)

The statefinder parameters are also rewritten in the following form:

$\begin{split}q(z)&=-1+(1+z)\frac{H^{\prime}(z)}{H(z)}\\\
&=\frac{(1+z)y^{\prime}_{H}(z)-2y_{H}(z)+(1+z)^{3}[1+2\chi(1+z)]}{2\\{y_{H}(z)+(1+z)^{3}[1+\chi(1+z)]\\}},\end{split}$
(24) $\begin{split}j(z)&=-2-3q+\frac{\ddot{H}}{H^{3}}\\\
&=\frac{2y_{H}(z)+(1+z)\\{-2y^{\prime}_{H}(z)+(1+z)[2(1+z)(1+3(1+z)\chi)+y^{\prime\prime}_{H}(z)]\\}}{2\\{y_{H}(z)+(1+z)^{3}[1+\chi(1+z)]\\}},\end{split}$
(25)
$s(z)=\frac{j-1}{3(q-\frac{1}{2})}=\frac{(1+z)[(1+z)y^{\prime\prime}_{H}(z)-2y^{\prime}_{H}(z)+4(1+z)^{3}\chi]}{3(1+z)[(1+z)^{3}\chi+y^{\prime}_{H}(z)]-9y_{H}(z)},$
(26)
$\begin{split}Om(z)&=\frac{\left(\frac{H}{H_{0}}\right)^{2}-1}{(1+z)^{3}-1}\\\
&=\frac{y_{H}(z)+(1+z)^{3}[1+(1+z)\chi]-1-\chi-
y_{H}(0)}{[(1+z)^{3}-1][1+\chi+y_{H}(0)]},\end{split}$ (27)

where the expressions given in Eq. (3) were used systematically. Note that
these quantities depend explicitly only on the parameter $\chi$.

It is at this point that we introduce the $f(R)$ gravity model which plays the
central role in this work:

$f(R)=R-2\,\lambda\,\mu^{2}\,e^{-(\mu^{2}/R)^{\eta}},$ (28)

where $\lambda$ and $\eta$ are positive real dimensionless parameters, and
$\mu$ is a positive real parameter with dimension of $\rm{eV}$.

This model was introduced in Ref. [42], and it behaves very close to
$\Lambda$CDM at early times and satisfy local and cosmological constraints.
The model inserts an exponential function of the scalar curvature, which works
as a correction to the Einstein term and disappears in the limit $R\rightarrow
0$, containing flat space-time solutions and allowing the possibility of pure
geometrical explanation of the dark energy phenomenon. An extension of this
model (where an $R^{2}$ Starobinsky term is added) able to explain early time
inflation and late time accelerated expansion was studied in Ref. [49].
Although such an extension, i.e., including an extra $R^{2}$ term in Eq. (28),
might be of some interest (see [50] for a recent discussion about this
subject), we have gone through the numerical solution of the relevant
equations to demonstrate that its contribution is negligible at late-times
(dark energy era). Then, in Ref. [51], the author performs a generalization of
this model, introducing a general function of the scalar curvature in the
exponential term. In the literature, other authors have studied some $f(R)$
gravity models with exponential functions of the scalar curvature (see for
example Refs. [36] and [52]).

For the numerical analysis that follows, we consider well motivated physical
initial conditions at $z_{f}=10$, given by [44, 45, 46, 47, 53]

$y_{H}(z_{f})=\frac{\Lambda}{3m_{s}^{2}}\left(1+\frac{1+z_{f}}{1000}\right),\quad\left.\frac{dy_{H}(z)}{dz}\right|_{z=z_{f}}=\frac{1}{1000}\frac{\Lambda}{3m_{s}^{2}},$
(29)

where, for convenience, we choose $\Lambda\simeq 4.24\times
10^{-66}\,\rm{eV}^{2}$. Note that the second initial condition is always a
constant. The specific numerical values are: $y_{H}(10)\simeq 2.1945$ and
$y^{\prime}_{H}(10)\simeq 2.17\times 10^{-3}$, respectively. In addition, to
perform the numerical analysis presented in this section, it is necessary to
fix the values for the model parameters. Then, following the results of Ref.
[42], where bounds on these parameters are established, the values that we use
here are: $\lambda=9.72$, $\mu^{2}=10^{-66}\,\rm{eV}^{2}$ and $\eta=0.08$;
moreover $M_{p}=2.44\times 10^{27}\,\rm{eV}$. Taking into account the above
conditions, we solve Eq. (6) for $y_{H}(z)$ numerically, and from Eqs.
(21)–(27), we obtain the results displayed in Figs. 1–7.

Figure 1: Evolution of the statefinder function $y_{H}(z)$ (_left_) and the
deceleration parameter $q(z)$ (_right_), with the redshift, $z$.

First, in the left panel of Fig. 1 we plot the numerical solution for
$y_{H}(z)$ and we can see that this model predicts non negligible dark energy
oscillations for $z\gtrsim 4$. Further, its current value is predicted to be
$y_{H}(0)=2.24$, which is close to the inferred value using Eq. (16), i.e.,
$y_{H}(0)=\Omega_{\rm{DE}0}/\Omega_{m0}\simeq 2.18$. The observed dark energy
oscillations is more apparent in the evolution of the parameters $j$, $s$ and
$w_{\rm{DE}}$, shown in Figs. 2 and 3 (right panel).

Figure 2: The statefinder parameters $j(z)$ (_left_) and $s(z)$ (_right_) as a
function of the redshift, $z$.

Although the evolution of the parameters $q$, $Om$ and $w_{\rm{eff}}$ also
exhibit these oscillations, the corresponding amplitude is smaller (see Fig. 1
(right panel), Fig. 3 (left panel) and Fig. 4 (right panel), respectively). In
general, the amplitude of the oscillations is high at earlier epochs and
decreases as the evolution proceeds. This behavior is a generic feature of
viable $f(R)$ gravity models, in particular, exponential gravity as well as a
power form model (see Refs. [43] and [44] for an exhaustive analysis about
this issue).

Figure 3: Evolution of the statefinder parameter $Om(z)$ (_left_) and the dark
energy equation of state (EoS) parameter $w_{\rm{DE}}(z)$ (_right_).

From the left plot of Fig. 4 we can see that the parameter $\Omega_{\rm{DE}}$
is free of oscillations. These results are similar to those found in Ref. [46]
for a $f(R)$ Einstein-Gauss-Bonnet gravity model, and also in Ref. [54] for a
$f(R)$ Einstein-Gauss-Bonnet model with a non-minimal coupling between gravity
and the kinetic term of a scalar field. However, in Ref. [45] the authors have
demonstrated the absence of dark energy oscillations in the context of a
$k$-Essence $f(R)$ gravity model. In the right panel of Fig. 4, we have
plotted the effective EoS parameter for the exponential type $f(R)$ model
given by Eq. (28) and the $\Lambda$CDM model as functions of the redshift, and
from it, we see that the $f(R)$ model is almost indistinguishable from
$\Lambda$CDM.

Figure 4: Change of the dark energy density parameter $\Omega_{\rm{DE}}(z)$ in
terms of the redshift, $z$ (_left_). Comparison between the effective EoS
parameter for the exponential type $f(R)$ model (Eq. (28)) and the
$\Lambda$CDM model as functions of the redshift (_right_).

Another approach to compare the $f(R)$ model studied here against the
$\Lambda$CDM standard model is by using the dimensionless Hubble parameter,
$E(z)=H(z)/H_{0}$, and calculating

$\Delta E(z)=100\times\left[\frac{E(z)}{E(z)_{\Lambda\text{CDM}}}-1\right],$
(30)

which, by definition, is zero for the $\Lambda$CDM model (red-dashed line in
the right panel of Fig. 5). As it is well known, $H(z)$ for the $\Lambda$CDM
model is given by

$H(z)=H_{0}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{DE0}+\Omega_{r0}(1+z)^{4}},$
(31)

with $H_{0}$ the present value of the Hubble rate, while
$\Omega_{\rm{DE}0}\simeq 0.6852$, $\Omega_{m0}\simeq 0.3147$ and
$\Omega_{r0}\sim 10^{-4}$. Looking at the evolution of $\Delta E(z)$ in Fig. 5
(right panel), we can see that the $f(R)$ model gives $\Delta E(z)>0$,
indicating that $H(z)$ is larger than the $\Lambda$CDM prediction (see also
the left plot of Fig. 5). The largest deviation from $\Lambda$CDM occurs
around $z\simeq 0.84$, being $\Delta E(z\simeq 0.84)\simeq 2.3\%$. For
$z\simeq 0$, there is a difference of the order of $1\%$. For $z>4$, the
deviation decreases in an oscillating way, approaching to the $\Lambda$CDM
prediction.

Figure 5: (_Left_) Evolution of the Hubble parameter, $H(z)$, (in units of
$\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$) for the exponential type $f(R)$ model
given by Eq. (28) and the $\Lambda$CDM model as functions of the redshift.
(_Right_) Comparison of the $f(R)$ model against $\Lambda$CDM using the
dimensionless Hubble parameter through Eq. (30).

In Table 1 we summarize the present ($z=0$) values obtained for the
cosmological and statefinder parameters studied above. For comparison, we
include the corresponding values predicted by the $\Lambda$CDM model and the
measurements available for some of the parameters.

Table 1: Current values obtained for the cosmological and statefinder parameters using fixed and fitted (third column, BF = Best Fit) values for the free parameters of the $f(R)$ gravity model. The values for the $\Lambda$CDM model are obtained using Eq. (31) in Eqs. (23)-(27). The units for $H_{0}$ are $\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$.  Parameter | $f(R)$ | $f_{\rm{BF}}(R)$ | Planck 2018 or SNe Ia |  $\Lambda$CDM
---|---|---|---|---
$q_{0}$ | -0.504 | -0.538 |  $-0.53^{+0.17}_{-0.13}$ (SNe Ia) | -0.528
$j_{0}$ | 1.011 | 1.006 | - | 1.0002
$s_{0}$ | -0.004 | -0.002 | - | 0.002
$Om_{0}$ | 0.330 | 0.308 | - | 0.315
$w_{\rm{DE}0}$ | -0.968 | -0.999 | $-1.03\pm 0.03$ | -1
$\Omega_{\rm{DE}0}$ | 0.691 | 0.6926 | $0.6847\pm 0.0073$ | 0.6852
$w_{\rm{eff}0}$ | -0.684 | -0.692 | - | -0.685
$H_{0}$ | 68.11 | 68.24 | $67.4\pm 0.5$ | 67.4

An additional important test is evaluated by looking at $f_{R}(R)$ and
$f_{RR}(R)$ as a function of the redshift. This is shown in Fig. 6 from which
we can see that the considered $f(R)$ model satisfies the well known viability
criteria, i.e., $f_{R}>0$ and $f_{RR}>0$, also $f_{R}\rightarrow 1$ and
$f_{RR}\rightarrow 0$ at $R\rightarrow\infty$.

Figure 6: The terms $f_{R}(R)$ (_left_) and $f_{RR}(R)$ (_right_) as a
function of the redshift.

In general, the initial conditions used for the numerical calculations
described above were obtained from Eq. (29) at $z_{f}=10$. However, if $z_{f}$
is changed (keeping the same values for the free parameters), the late-time
evolution of the cosmological and statefinder parameters change as well. This
means that the divergence from the $\Lambda$CDM value depends strongly on the
choice of $z_{f}$. In order to illustrate the above, in Fig. 7 we show the
behavior for $y_{H}(z)$ and $w_{\rm{DE}}(z)$ using three different choices for
$z_{f}$. It is clear that the discrepancy among the three curves at low
redshift ($z<1$) is small, and in this case, the current values obtained for
the cosmological and statefinder parameters are similar to those illustrated
in Table 1. However, these values and the other plots are omitted by
simplicity. Let us point out that in Ref. [55], the authors perform a detailed
analysis about this issue using the Hu-Sawicki model.

Figure 7: Changes on the evolution of $y_{H}(z)$ (_left_) and $w_{\rm{DE}}(z)$
(_right_) when the initial conditions in Eq. 29 are modified. In both cases we
used three different choices for $z_{f}$: $z_{f}=5$ (blue), $z_{f}=10$ (red)
and $z_{f}=15$ (green). The black-dashed line in the right plot, corresponds
to the prediction of $\Lambda$CDM.

##  4 Parameter constraints from $H(z)$ observational data

Looking for properly characterizing the model studied here, we compare its
predictions for the Hubble parameter, $H_{\rm{th}}(z)$, against a set of
observational data. Let us stress that for the fit presented in this section,
we have considered $H_{0}$ as measured by Planck [48], which is known to be in
tension with the value estimated through the local probes [56] (see Ref. [57]
for a comprehensive review of this matter)222To avoid the tension, in Ref.
[58], the authors introduce a particular statistical procedure in which the
chi-square estimator is not affected by the value of the Hubble constant..

The statistical analysis presented in this section is performed by means of
the commonly used $\chi^{2}$ function

$\chi^{2}(z;\bm{\theta})=\sum_{i=1}^{N_{d}}\frac{\left(H_{\rm{th}}(z_{i};\bm{\theta})-H_{\rm{obs}}(z_{i})\right)^{2}}{\sigma_{i}^{2}},$
(32)

where $N_{d}=40$ is the number of observational data considered here (taken
from Ref. [59]), $\bm{\theta}=(\lambda,\mu,\eta)$ is the vector of free
parameters of the model (see Eq. (28)), and
$\left(H_{\rm{obs}}(z_{i}),\sigma_{i}\right)$ corresponds to the measured
value of the Hubble parameter at $z_{i}$ and its associated observational
uncertainty.

As mentioned earlier and developed in the preceding sections, given the
complexity of the $f(R)$ function, the equations were solved numerically.
Furthermore, it is noted that $H_{\rm{th}}(z;\bm{\theta})$ is extremely
sensible to changes of the parameters of the model, in particular to
variations of $\lambda$ and $\eta$. As a consequence, we find that
constraining these parameters is troublesome.

The result of this analysis if presented in Fig. 8, where an array of 2D- and
1D-plots are shown. The figures should be interpreted as follows: each 2D-plot
shows the allowed regions (contours) for the displayed pair of parameters at a
certain confidence level (C.L.), according to the line color. On the other
hand, the 1D-plots show the marginalized allowed interval for each individual
parameter, with the horizontal (colored) lines indicating the C.L.

From these plots it is evident that we obtain a well defined allowed interval
for the parameter $\mu$ up to a 99.73% C.L. and larger, while for $\eta$ there
is only an upper limit: $\eta\lesssim 9\times 10^{-2}$ at 95.45% C.L. In
addition, as a confirmation of what we explained above, $\lambda$ is not well
constrained by the set of observational data considered here. In particular,
the uppermost 1D-plot demonstrates that, since the black line does not
intersect any of the horizontal lines, every value of $\lambda$ would be
equally valid to fit the data. In fact, an exploration of a larger interval
than the one exhibited in the plot shows a similar low-valued oscillatory
behavior with a number of local minima, preventing us to set bounds on this
parameter. Clarifying this issue requires a deeper statistical analysis,
considering additional observational data, and such an approach would be
adrressed in a future work. In spite of this, a global minimum of the
$\chi^{2}$ function in Eq. (32) is found, and the best fit values obtained
from the analysis, together with the allowed intervals at three particular
levels of confidence are displayed in Table 2.

Figure 8: Allowed regions and intervals for the model parameters obtained from the $\chi^{2}$ analysis, at 68.27%, 90%, 95.45%, 99%, and 99.73% C.L. The ‘$+$’ sign in the 2D-plots locates the corresponding best fit. Table 2: Best fit and the corresponding allowed intervals for the parameters of the model. Note that the values of $\mu$ should be multiplied by $10^{-33}\,\rm{eV}$. Parameter | Best fit | 68.27% | 95.45% | 99.73%
---|---|---|---|---
$\lambda$ | 8.02 | - | - | -
$\eta$ | 0.001 | $<0.02$ | $<0.09$ | -
$\mu$ | 1.22 | (1.01, 1.30) | (0.90, 1.34) | (0.81, 1.36)

The specific values of the parameters that make our model fits better the
observational data (Table 2) can be used to compute the evolution of the
Hubble parameter against the redshift for $f(R)$ in Eq. (28). This is shown in
Fig. 9 (black-solid line), where the data are also included for comparison.
This figure allows us to see that the $f(R)$ model under consideration barely
marginally deviates from the $\Lambda$CDM predictions (red-dashed line), as it
is usually required. In fact, when these values are used to compute the
cosmological parameters and the statefinder quantities studied in the previous
section at $z=0$, we obtain the numbers recorded in the third column
($f_{\rm{BF}}(R)$) of Table 1, clearly close to those predicted by
$\Lambda$CDM and (for the existing cases) to the measured data. Plus, setting
the parameters ($\mu$ and $\eta$) at the 95.45% C.L. limit values (keeping
$\lambda$ at its BF), we obtain the gray-shadowed band around the best fit
which, as expected, covers the vast majority of the observational data (up to
their uncertainties). We have also confirmed that the best fit values of the
parameters in Table 2 satisfy the stability conditions as discussed in [42].

In the context of this kind of analysis, in Ref. [60], the authors inspect a
number of well-known $f(R)$ cosmological models and constrain the
corresponding parameters, namely the deviation parameter $b$ and the
cosmological parameters $\Omega_{m}$ and $h$. The authors found that for the
Hu-Sawicki and Starobinsky models, the optimized value of the deviation
parameter is much closer to zero than what has been reported in the
literature. In both cases, the zero value for $b$ falls safely within the
$1\sigma$ (equivalent to 68.27%) C.L., which indicates that these models
coincide observationally with the predictions of $\Lambda$CDM (see Figs. 3(a)
and 3(b) of [60]). This behavior is similar to that obtained above with
$\eta=10^{-3}$, since the limiting case of $\Lambda$CDM can be reached not
only at high curvature but also at $\eta\rightarrow 0$ with cosmological
constant $\Lambda\rightarrow e^{-1}\lambda\mu^{2}$. But our constraints imply
too small values of $m(r)$ at current or late times ($m\ll 10^{-6}$), making
it very difficult to detect measurable differences with the $\Lambda$CDM model
[42].

It is worth noting that, with this particular set of the model parameters (our
best fit, Table 2), the evolution with $z$ of the other cosmological and
statefinder quantities analysed in the previous section does not change
considerably and their corresponding plot remains essentially unaffected (and
the conclusions are unaffected), so we do not include new plots, for
simplicity.

Figure 9: Best-fit $f(R)$ model prediction for $H(z)$ (black-solid line)
compared against observational data (blue points –with vertical lines
indicating the uncertainty–) and the $\Lambda$CDM predictions (red-dashed
line). The gray-shadowed area shows the $f(R)$ model prediction when $\mu$ and
$\eta$ are varied inside their 95.45% C.L. allowed intervals.

##  5 Conclusions

In the last years, modified gravity theories have been established as
alternative explanation to the observed late-time acceleration of the
Universe, instead of the dark energy paradigm. In this context, among the wide
plethora of modified gravity theories, $f(R)$ gravity emerge as one of the
most studied in the literature. Following this idea, in this work we have
studied the late-time evolution of the Universe for a particular viable $f(R)$
gravity model built from an exponential function of the scalar curvature.
Inspired by the literature, the field equations were written in terms of a
suited statefinder function, $y_{H}(z)$, and implementing well motivated
physical initial conditions, the resulting equations were solved numerically.
Also, the cosmological parameters $w_{\rm{DE}}$, $w_{\rm{eff}}$,
$\Omega_{\rm{DE}}$ and $H(z)$ and the statefinder quantities $q$, $j$, $s$ and
$Om(z)$ were explicitly expressed in terms of $y_{H}(z)$ and its derivatives
(see Eqs. (21) and (27)).

Furthermore, setting an appropriate set of values for the model parameters,
the cosmological parameters as well as the statefinder quantities were
plotted, and from them, we achieved the following results: for $y_{H}(z)$ we
can see that this model predicts non negligible dark energy oscillations for
$z\geq 4$ (see Fig. 1 (left panel)). Further, its current value is
$y_{H}(0)=2.24$, which is very close to that inferred using Eq. (16), i.e.,
$y_{H}(0)=\Omega_{\rm{DE}0}/\Omega_{m0}\simeq 2.175$. The observed dark energy
oscillations is more appreciable in the evolution of the parameters $j$, $s$
and $w_{\rm{DE}}$ (see Figs. 2 and 3 (right panel)).

Although the parameters $q$, $Om$ and $w_{\rm{eff}}$ also present these
oscillations, those are with less amplitude (Fig. 1 (right panel), Fig. 3
(left panel) and Fig. 4 (right panel)). In general, the amplitude of the
oscillations is high at earlier epochs and decreases as the evolution
proceeds. For the parameter $\Omega_{\rm{DE}}$, is clear that it is free of
oscillations (see left panel of Fig. 4). From the plot of $w_{\rm{eff}}$ for
the given $f(R)$ gravity model and the $\Lambda$CDM model as functions of the
redshift in the right panel of Fig. 4, we saw that the $f(R)$ model is almost
indistinguishable from $\Lambda$CDM.

In general, the cosmological and statefinder parameters at $z=0$, are shown to
be compatible with Planck 2018 observations and the $\Lambda$CDM model values
as exhibited in Table 1. In addition, we have compared the $f(R)$ model
studied here against the $\Lambda$CDM standard model using the dimensionless
Hubble parameter $E(z)=H(z)/H_{0}$ and calculating $\Delta E(z)$ though Eq.
(30), from which it is observed that the $f(R)$ model gives $\Delta E(z)>0$,
indicating that $H(z)$ is larger than the $\Lambda$CDM prediction (see Fig. 5
(left panel)). The biggest deviation from $\Lambda$CDM takes place around
$z\simeq 0.84$, being $\Delta E(z\simeq 0.84)\simeq 2.3\%$. For $z\simeq 0$,
there is a difference of the order of $1\%$. For $z>4$, the deviation
decreases in an oscillating way, approaching to the $\Lambda$CDM prediction,
as depicted in Fig. 5 (right panel).

Additionally, $f_{R}(R)$ and $f_{RR}(R)$ were plotted as a function of the
redshift, and it was seen that the given $f(R)$ model satisfy the well known
viability criteria, ($f_{R}>0$ and $f_{RR}>0$), also $f_{R}\rightarrow 1$ and
$f_{RR}\rightarrow 0$ at $R\rightarrow\infty$ (see Fig. 6). We have also
explored the numerical calculations using other initial conditions, but
keeping the same values for the free parameters, and we observed that the
late-time evolution for the cosmological and statefinder parameters change,
making evident that the deviation from the $\Lambda$CDM value strongly depends
on the choice of $z_{f}$ (see Fig. 7). However, the deviation among the curves
at low redshifts ($z<1$) is small, and in this case, the current values
obtained for the cosmological and statefinder parameters are similar to those
illustrated in Table 1.

Finally, through familiar $\chi^{2}$ statistical analysis, we compared the
predictions of the selected $f(R)$ gravity model for the Hubble parameter,
$H(z)$, with a number of observational data. If was found that, because of the
complexity of the differential equations, as well as of the $f(R)$ model
itself, it is not easy to find a global minimum of the $\chi^{2}$ function in
Eq. (32) and that the dataset used in the analysis is not enough to
unequivocally constraint the whole set of free parameters. As presented in
Fig. 8 and Table 2, a well-defined allowed region is met for $\mu$ at a C.L.
larger than 99%, while only an upper limit is obtained for $\eta$ and no bound
is found for $\lambda$. In spite of this, when the best fit values are used to
compute the cosmological and statefinder parameters at present time ($z=0$),
the results are remarkably close to those predicted by the $\Lambda$CDM model
(see Table 1). Likewise, the $f(R)$–predicted evolution of the Hubble
parameter as a function of $z$ is only marginally distinguishable from that of
$\Lambda$CDM (Fig. 9), though this imply too small values of $m(r)$ at current
or late times ($m\ll 10^{-6}$), making it very difficult to detect measurable
differences with the $\Lambda$CDM model.

The $f(R)$ gravity model considered here predicts some of the cosmological
parameters to have an ill-oscillatory behavior which is an open problem for
this type of models, among others. Yet, as a result of the analyses presented
in this work, we can confidently say that the model is consistent with the
analyzed observations, but it would be necessary to use a more complete set of
observational data to obtain a best fit on the parameters of the model, e.g.,
using together with the measurements from the dynamics of the expansion of the
Universe, $H(z)$, the growth rate of cosmic structures, $[f\sigma_{8}](z)$, or
data from Supernovae (SNIa) Pantheon sample, among others, but that kind of
analysis is beyond the scope of this work and could be addressed later.

## Acknowledgments

A. O. is supported by Patrimonio Autónomo-Fondo Nacional de Financiamiento
para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas
(MINCIENCIAS-COLOMBIA) Grant No. 110685269447 RC-80740-465-2020, projects
69723 and 69553.

## References

  * [1] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559-606 (2003) [arXiv:astro-ph/0207347 [astro-ph]].
  * [2] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753-1936 (2006) [arXiv:hep-th/0603057 [hep-th]].
  * [3] K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Astrophys. Space Sci. 342, 155-228 (2012) [arXiv:1205.3421 [gr-qc]].
  * [4] S. Nojiri and S. D. Odintsov, Phys. Rept. 505, 59-144 (2011) [arXiv:1011.0544 [gr-qc]].
  * [5] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept. 513, 1-189 (2012) [arXiv:1106.2476 [astro-ph.CO]].
  * [6] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept. 692, 1-104 (2017) [arXiv:1705.11098 [gr-qc]].
  * [7] J. c. Hwang and H. Noh, Phys. Lett. B 506, 13-19 (2001) [arXiv:astro-ph/0102423 [astro-ph]].
  * [8] S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003) [arXiv:hep-th/0307288 [hep-th]].
  * [9] S. Capozziello, V. F. Cardone and A. Troisi, Phys. Rev. D 71, 043503 (2005) [arXiv:astro-ph/0501426 [astro-ph]].
  * [10] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, JCAP 02, 010 (2005) [arXiv:hep-th/0501096 [hep-th]].
  * [11] S. Capozziello, V. F. Cardone and M. Francaviglia, Gen. Rel. Grav. 38, 711-734 (2006) [arXiv:astro-ph/0410135 [astro-ph]].
  * [12] S. Nojiri and S. D. Odintsov, Phys. Rev. D 74, 086005 (2006) [arXiv:hep-th/0608008 [hep-th]].
  * [13] Y. S. Song, W. Hu and I. Sawicki, Phys. Rev. D 75, 044004 (2007) [arXiv:astro-ph/0610532 [astro-ph]].
  * [14] T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao, Phys. Rev. D 76, 063505 (2007) [arXiv:astro-ph/0612569 [astro-ph]].
  * [15] G. J. Olmo, Phys. Rev. D 75, 023511 (2007) [arXiv:gr-qc/0612047 [gr-qc]].
  * [16] I. Sawicki and W. Hu, Phys. Rev. D 75, 127502 (2007) [arXiv:astro-ph/0702278 [astro-ph]].
  * [17] V. Faraoni, Phys. Rev. D 75, 067302 (2007) [arXiv:gr-qc/0703044 [gr-qc]].
  * [18] R. Bean, D. Bernat, L. Pogosian, A. Silvestri and M. Trodden, Phys. Rev. D 75, 064020 (2007) [arXiv:astro-ph/0611321 [astro-ph]].
  * [19] S. Nojiri and S. D. Odintsov, Phys. Lett. B 657, 238-245 (2007) [arXiv:0707.1941 [hep-th]].
  * [20] S. Capozziello, A. Stabile and A. Troisi, Phys. Rev. D 76, 104019 (2007) [arXiv:0708.0723 [gr-qc]].
  * [21] N. Deruelle, M. Sasaki and Y. Sendouda, Prog. Theor. Phys. 119, 237-251 (2008) [arXiv:0711.1150 [gr-qc]].
  * [22] S. A. Appleby and R. A. Battye, JCAP 05, 019 (2008) [arXiv:0803.1081 [astro-ph]].
  * [23] S. Carloni, P. K. S. Dunsby and A. Troisi, Phys. Rev. D 77, 024024 (2008) [arXiv:0707.0106 [gr-qc]].
  * [24] S. Capozziello, V. F. Cardone and V. Salzano, Phys. Rev. D 78, 063504 (2008) [arXiv:0802.1583 [astro-ph]].
  * [25] P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. S. Gomez, Phys. Rev. D 82, 023519 (2010) [arXiv:1005.2205 [gr-qc]].
  * [26] S. Capozziello and M. De Laurentis, Annalen Phys. 524, 545-578 (2012)
  * [27] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 99, no.6, 064049 (2019) [arXiv:1901.05363 [gr-qc]].
  * [28] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 99, no.10, 104070 (2019) [arXiv:1905.03496 [gr-qc]].
  * [29] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 101, no.4, 044009 (2020) [arXiv:2001.06830 [gr-qc]].
  * [30] V. K. Oikonomou, Phys. Rev. D 103, no.12, 124028 (2021) [arXiv:2012.01312 [gr-qc]].
  * [31] V. K. Oikonomou, Phys. Rev. D 103, no.4, 044036 (2021) [arXiv:2012.00586 [astro-ph.CO]].
  * [32] L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, Phys. Rev. D 75, 083504 (2007) [arXiv:gr-qc/0612180 [gr-qc]].
  * [33] W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 (2007) [arXiv:0705.1158 [astro-ph]].
  * [34] A. A. Starobinsky, JETP Lett. 86, 157-163 (2007) [arXiv:0706.2041 [astro-ph]].
  * [35] S. Tsujikawa, Phys. Rev. D 77, 023507 (2008) [arXiv:0709.1391 [astro-ph]].
  * [36] E. V. Linder, Phys. Rev. D 80, 123528 (2009) [arXiv:0905.2962 [astro-ph.CO]].
  * [37] D. J. Gogoi and U. Dev Goswami, Eur. Phys. J. C 80, no.12, 1101 (2020) [arXiv:2006.04011 [gr-qc]].
  * [38] Y. Mambrini, S. Profumo and F. S. Queiroz, Phys. Lett. B 760, 807 (2016) [arXiv:1508.06635[hep-ph]].
  * [39] D. J. E. Marsh, Phys. Rept. 643, 1-79 (2016) [arXiv:1510.07633 [astro-ph.C]].
  * [40] R. T. Co, L. J. Hall and K. Harigaya, Phys. Rev. Lett. 124, no.25, 251802 (2020) [arXiv:1910.14152[hep-ph]].
  * [41] V. K. Oikonomou, Phys. Rev. D 106, no.4, 044041 (2022) [arXiv:2208.05544 [gr-qc]].
  * [42] L. N. Granda, Eur. Phys. J. C 80, no.6, 539 (2020) [arXiv:2003.09006 [gr-qc]].
  * [43] E. Elizalde, S. D. Odintsov, L. Sebastiani and S. Zerbini, Eur. Phys. J. C 72, 1843 (2012) [arXiv:1108.6184 [gr-qc]].
  * [44] K. Bamba, A. Lopez-Revelles, R. Myrzakulov, S. D. Odintsov and L. Sebastiani, Class. Quant. Grav. 30, 015008 (2013) [arXiv:1207.1009 [gr-qc]].
  * [45] S. D. Odintsov, V. K. Oikonomou and F. P. Fronimos, Phys. Dark Univ. 29, 100563 (2020) [arXiv:2004.08884 [gr-qc]].
  * [46] S. D. Odintsov, V. K. Oikonomou and F. P. Fronimos, Class. Quant. Grav. 38, no.7, 075009 (2021) [arXiv:2102.02239 [gr-qc]].
  * [47] V. K. Oikonomou and I. Giannakoudi, Int. J. Mod. Phys. D 31, no.09, 2250075 (2022) [arXiv:2205.08599 [gr-qc]].
  * [48] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].
  * [49] L. Granda, Symmetry 12, no.5, 794 (2020)
  * [50] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 104, no.12, 124065 (2021) [arXiv:2112.06269 [gr-qc]].
  * [51] L. N. Granda, JHEP 12, 205 (2021) [arXiv:2007.13956 [gr-qc]].
  * [52] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Phys. Rev. D 77, 046009 (2008) [arXiv:0712.4017 [hep-th]].
  * [53] S. D. Odintsov, V. K. Oikonomou, F. P. Fronimos and K. V. Fasoulakos, Phys. Rev. D 102, no.10, 104042 (2020) [arXiv:2010.13580 [gr-qc]].
  * [54] F. Fronimos, Eur. Phys. J. Plus 136, no.10, 1014 (2021) [arXiv:2110.00353 [gr-qc]].
  * [55] S. Nadkarni-Ghosh and S. Chowdhury, Mon. Not. Roy. Astron. Soc. 511, no.3, 3636-3656 (2022) [arXiv:2110.05121 [astro-ph.CO]].
  * [56] A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, L. Breuval and T. G. Brink, et al. Astrophys. J. Lett. 934, no.1, L7 (2022) [arXiv:2112.04510 [astro-ph.CO]].
  * [57] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk, Class. Quant. Grav. 38, no.15, 153001 (2021) [arXiv:2103.01183 [astro-ph.CO]].
  * [58] F. Anagnostopoulos and S. Basilakos, Phys. Rev. D 97, no.6, 063503 (2018) [arXiv:1709.02356 [astro-ph.CO]].
  * [59] S. Cao, T. J. Zhang, X. Wang and T. Zhang, Universe 7, no.3, 57 (2021) [arXiv:2103.03670 [astro-ph.CO]].
  * [60] J. Sultana, M. K. Yennapureddy, F. Melia and D. Kazanas, Mon. Not. Roy. Astron. Soc. 514, no.4, 5827-5839 (2022) [arXiv:2206.10761 [astro-ph.CO]].


finally voting on these as a group to establish the final 5 most important
scenes for each character. From each of these, a question was worded to ask if
the submitted video summary could be said to have answered that question. The
content questions for each character are specified below:

  * •

Janine

    1. 1.

What is causing Ryan to be sick in bed?

    2. 2.

How does Janine attempt to kill Ryan while in the hospital?

    3. 3.

What happens when Janine attempts to play recording of Stacey?

    4. 4.

Who stabbed Janine?

    5. 5.

Who gives Janine the recording of Stacey?

  * •

Ryan

    1. 1.

How does Janine attempt to kill Ryan in the hospital?

    2. 2.

What does Ryan do when Janine is lying in the hospital?

    3. 3.

Where is Ryan trapped?

    4. 4.

What does Ryan tell Phil he can do for him?

    5. 5.

Who is Ryan with when going to put his name on the baby’s birth cert?

  * •

Stacey

    1. 1.

Who climbs up to the roof to talk Stacey out of jumping off?

    2. 2.

What does Stacey reveal when in a cell with Janine, Kat, and Pat?

    3. 3.

What does Stacey admit to her mum in bedroom when mum is upset?

    4. 4.

Who confronts Stacey in restroom where Stacey finally admits to killing
Archie?

    5. 5.

Who calls to Stacey’s door to tell her to get her stuff and go, after Stacey’s
mum had called the police?

Assessors also marked video summaries on the subjective metrics of
tempo/rhythm, contextuality, and redundancy, on a 7-point Likert-scale, with
the following definitions. Tempo/Rhythm was defined as: How well do the video
shots flow together? Do shots cut mid-sentence (indicating poor tempo/rhythm)?
Do they flow together nicely so it wouldn’t be obvious that this is an
automatically generated summary (high tempo/rhythm)? (High is best).
Contextuality was defined as: Does the content provide the circumstances that
form the setting for an event, statement, or idea, and in terms of which it
can be fully understood and assessed? (High is best). Redundancy was defined
as: Does the video contain content considered to be unnecessary or
superfluous? (Low is best).

#### Metrics

Scores were calculated as a percentage using marks for the 5 content based
questions and the 3 subjective quality based questions. Base Likert-scale
scores for Tempo/Rhythm and Contextuality were taken as assessed by human
annotators. Scores for Redundancy, where a lower score is better, were
flipped. This gave a total of 21 possible marks available for subjective
quality scores. The remainder was calculated by taking the remaining 79
possible marks and dividing by the 5 content based questions, giving a total
of 15.8 possible marks for each correct content based question which was to be
rounded to the nearest integer. This would give a perfect summary 100 points.
A summary with no relevant content but all perfect scores for the other
factors would get 21 points. Overall this gave summaries a maximum score of
100 down to a minimum score of 3.

#### Results

Table 17 shows the individual results for each submission query and run on all
metrics and content questions. Team Me_MAD consistently achieves the best
results.

Figure 60 shows the average scores for each target query by team. Scores are
averaged across all runs. Ryan is seen to be the easiest character to
summarize, with Janine the most difficult. However the only differences
between Janine and Stacey are in the subjective measures of Tempo,
Contextuality and Redundancy.

Figure 61 shows the average scores for each target query by team. Scores are
averaged across all target queries. Run 3, containing 15 shots and up to a
maximum of 450 seconds in length scores higher than other runs. This is most
affected by target query Ryan who scored one extra content question by one
team for runs 3 and 4. Run 4 average scores were reduced by lost marks in
Tempo, Contextuality and Redundancy.

Figure 62 shows the average scores for each team. Scores are averaged across
all runs and target queries. Team Me_MAD achieves consistently higher marks,
mostly due to summaries containing at least one correct content question.

Figure 63 shows the individual scores for all teams, runs and target queries.
This chart visualizes the final results shown in table 17, from which it can
be seen that team Me_MAD scores higher for Ryan run 3 and Ryan run 4 than for
all other submitted summaries.

Table 17: Video Summarization Queries and Specifics Team_Run_Query | Tempo | Contextuality | Redundancy | Q1 | Q2 | Q3 | Q4 | Q5 | Score
---|---|---|---|---|---|---|---|---|---
Me_MAD_1_Janine | 6 | 4 | 5 | No | No | No | No | Yes | 29%
Me_MAD_2_Janine | 5 | 5 | 6 | No | No | No | No | Yes | 28%
Me_MAD_3_Janine | 5 | 5 | 6 | No | No | No | No | Yes | 28%
Me_MAD_4_Janine | 5 | 5 | 7 | No | No | No | No | Yes | 27%
NII_UIT_1_Janine | 5 | 3 | 7 | No | No | No | No | No | 9%
NII_UIT_2_Janine | 4 | 3 | 7 | No | No | No | No | No | 8%
NII_UIT_3_Janine | 4 | 3 | 7 | No | No | No | No | No | 8%
NII_UIT_4_Janine | 2 | 3 | 7 | No | No | No | No | No | 6%
Me_MAD_1_Ryan | 4 | 5 | 3 | No | No | No | No | Yes | 30%
Me_MAD_2_Ryan | 5 | 5 | 3 | No | No | No | No | Yes | 31%
Me_MAD_3_Ryan | 3 | 4 | 5 | No | No | No | Yes | Yes | 42%
Me_MAD_4_Ryan | 2 | 3 | 5 | No | No | No | Yes | Yes | 40%
NII_UIT_1_Ryan | 4 | 3 | 5 | No | No | No | No | No | 10%
NII_UIT_2_Ryan | 3 | 3 | 5 | No | No | No | No | No | 9%
NII_UIT_3_Ryan | 2 | 4 | 5 | No | No | No | No | No | 9%
NII_UIT_4_Ryan | 2 | 2 | 6 | No | No | No | No | No | 6%
Me_MAD_1_Stacey | 6 | 5 | 2 | No | Yes | No | No | No | 33%
Me_MAD_2_Stacey | 6 | 5 | 2 | No | Yes | No | No | No | 33%
Me_MAD_3_Stacey | 6 | 6 | 2 | No | Yes | No | No | No | 34%
Me_MAD_4_Stacey | 4 | 5 | 4 | No | Yes | No | No | No | 29%
NII_UIT_1_Stacey | 3 | 3 | 7 | No | No | No | No | No | 7%
NII_UIT_2_Stacey | 3 | 3 | 7 | No | No | No | No | No | 7%
NII_UIT_3_Stacey | 3 | 2 | 5 | No | No | No | No | No | 8%
NII_UIT_4_Stacey | 3 | 2 | 6 | No | No | No | No | No | 7%
Figure 60: VSUM: Average scores by Character Figure 61: VSUM: Average scores
for each run Figure 62: VSUM: Average scores by team Figure 63: VSUM:
Individual scores

#### Observations

This is the first year of the video summarization task. Due to this, the
decision was taken to require that teams submit results for 4 different runs,
specified by a maximum number of shots and maximum summary length in seconds.
While it was found that the conditions for run 3 scores higher, this is based
on too few submissions to make a reliable judgment. Hence for the next year of
the task these same conditions will apply. Following that, the plan is to
specify that runs differentiate between techniques applied and configurations
used. All submitted summaries used all of the maximum number of shots allowed,
however no submitted summaries were anywhere near as long as the specified
maximum length, thus the maximum length will be greatly reduced for the next
year of the task.

We should also note that this year a significant amount of time and effort was
spent trying to get the data agreement set with the donor (BBC) which may have
adversely affected the number of teams who did not get enough time to work on
and finish the task.

We now summarize the approaches taken by teams. Team MeMAD proposed a fan-
driven and character-centered approach to video summarization. In addition to
the BBC Eastenders data made available to teams participating in the task,
they also used fan-made content from the BBC Eastenders Fandom Wiki and
character images crawled from a search engine. In the first step of their
approach, they scraped synopsis found on the Eastenders Fandom Wiki with the
hypothesis that every sentence represents an important event to be included in
summaries. In parallel, they extracted shots from the series in which the
three target queries appear using Face Celebrity Recognition. This detects
faces using Multi-task Cascaded Convolutional Neural Networks (MTCNN)
algorithm 666https://github.com/ipazc/mtcnn, and the FaceNet
777https://github.com/davidsandberg/facenet model is applied to get face
embeddings.

Team NII_UIT proposed a framework to generate final summaries by combining the
score of a person’s facial detection scores and a self-attention based
network. They divided their network into three separate modules -
segmentation, score, selection. Videos were divided into short segments using
the shot time information on the BBC Eastenders data. They then calculated an
importance score for each shot by combining the person score of facial
detection similarity, and representation score. For person score, they use
their INS 2019 system which includes a face detector, a face descriptor, and a
face matching component. For calculating the representation score, they used
VASNet [Fajtl et al., 2018], an approach to sequence transformation for video
summarization based on soft, self-attention mechanism. The final importance
score of each shot was calculated by summing the person face and
representation score.

#### Conclusions

This was the first year of the Video Summarization task. Teams were asked to
produce summaries of the major life events of three target characters from
within a specified time frame of the BBC Eastenders series. The major
challenges of this task were to locate only shots for the target queries and
to identify those shots most likely to have been considered major life events.

There were a total of 2 finishing teams out of 12 participating teams in this
year’s task. All 2 finishing teams submitted notebook papers and presented
their approaches at the TRECVID workshop. It should also be noted that the
long delay in teams gaining access to the data set may have adversely affected
the number of teams who were able to complete the task.

##  4 Summing up and moving on

In this overview paper to TRECVID 2020, we provided basic information for all
tasks we run this year and particularly on the goals, data, evaluation
mechanisms, and metrics used. Further details about each particular group’s
approach and performance for each task can be found in that group’s site
report. The raw results for each submitted run can be found at the online
proceeding of the workshop [TV20Pubs, 2020]. Finally, we are looking forward
to continuing a new evaluation cycle in 2021 after refining the current tasks
and introducing any potential new tasks.

##  5 Authors’ note

TRECVID would not have happened in 2020 without support from the National
Institute of Standards and Technology (NIST). The research community is very
grateful for this. Beyond that, various individuals and groups deserve special
thanks:

  * •

Koichi Shinoda of the TokyoTech team agreed to host a copy of IACC.2 data.

  * •

Georges Quénot provided the master shot reference for the IACC.3 videos.

  * •

The LIMSI Spoken Language Processing Group and Vocapia Research provided ASR
for the IACC.3 videos.

  * •

Luca Rossetto of University of Basel for providing the V3C dataset collection.

  * •

Noel O’Connor and Kevin McGuinness at Dublin City University along with Robin
Aly at the University of Twente worked with NIST and Andy O’Dwyer plus William
Hayes at the BBC to make the BBC EastEnders video available for use in
TRECVID. Finally, Rob Cooper at BBC facilitated the copyright licence
agreement for the Eastenders data.

  * •

Jeffrey Liu and Andrew Weinert of MIT Lincoln Laboratory for supporting the
DSDI task by making the LADI dataset available and helping with the testing
dataset preparations.

Finally we want to thank all the participants and other contributors on the
mailing list for their energy and perseverance.

##  6 Acknowledgments

The ActEV NIST work was supported by the Intelligence Advanced Research
Projects Activity (IARPA), agreement IARPA-16002, order R18-774-0017. The
authors would like to thank Kitware, Inc. for annotating the dataset. The
Video-to-Text work has been partially supported by Science Foundation Ireland
(SFI) as a part of the Insight Centre at Dublin City University (12/RC/2289)
and grant number 13/RC/2106 (ADAPT Centre for Digital Content Technology,
www.adaptcentre.ie) at Trinity College Dublin. We would like to thank Tim
Finin and Lushan Han of University of Maryland, Baltimore County for providing
access to the semantic similarity metric. Finally, the TRECVID team at NIST
would like to thank all external coordinators for their efforts across the
different tasks they helped to coordinate.

## References

  * [Anderson et al., 2016] Anderson, P., Fernando, B., Johnson, M., and Gould, S. (2016).  Spice: Semantic propositional image caption evaluation.  In ECCV.
  * [Awad et al., 2016a] Awad, G., Fiscus, J., Joy, D., Michel, M., Kraaij, W., Smeaton, A. F., Quénot, G., Eskevich, M., Aly, R., Ordelman, R., Ritter, M., Jones, G. J., , Huet, B., and Larson, M. (2016a).  TRECVID 2016: Evaluating Video Search, Video Event Detection, Localization, and Hyperlinking.  In Proceedings of TRECVID 2016. NIST, USA.
  * [Awad et al., 2016b] Awad, G., Snoek, C. G., Smeaton, A. F., and Quénot, G. (2016b).  TRECVid Semantic Indexing of Video: A 6-year retrospective.  ITE Transactions on Media Technology and Applications, 4(3):187–208.
  * [Banerjee and Lavie, 2005] Banerjee, S. and Lavie, A. (2005).  Meteor: An Automatic Metric for MT Evaluation with Improved Correlation with Human Judgments.  In Proceedings of the acl workshop on intrinsic and extrinsic evaluation measures for machine translation and/or summarization, volume 29, pages 65–72.
  * [Barrault et al., 2020] Barrault, L., Biesialska, M., Bojar, O., Costa-jussà, M. R., Federmann, C., Graham, Y., Grundkiewicz, R., Haddow, B., Huck, M., Joanis, E., Kocmi, T., Koehn, P., Lo, C.-k., Ljubešić, N., Monz, C., Morishita, M., Nagata, M., Nakazawa, T., Pal, S., Post, M., and Zampieri, M. (2020).  Findings of the 2020 conference on machine translation (wmt20).  In Proceedings of the Fifth Conference on Machine Translation, pages 1–55, Online. Association for Computational Linguistics.
  * [Bojar et al., 2017] Bojar, O., Chatterjee, R., Federmann, C., Graham, Y., Haddow, B., Huang, S., Huck, M., Koehn, P., Liu, Q., Logacheva, V., Monz, C., Negri, M., Post, M., Rubino, R., Specia, L., and Turchi, M. (2017).  Findings of the 2017 conference on machine translation (wmt17).  In Proceedings of the Second Conference on Machine Translation, Volume 2: Shared Task Papers, pages 169–214, Copenhagen, Denmark. Association for Computational Linguistics.
  * [Fajtl et al., 2018] Fajtl, J., Sokeh, H. S., Argyriou, V., Monekosso, D., and Remagnino, P. (2018).  Summarizing videos with attention.  In Asian Conference on Computer Vision, pages 39–54. Springer.
  * [Gkountakos et al., 2019] Gkountakos, K., Galanopoulos, D., Mpakratsas, M., Touska, D., Moumtzidou, A., Ioannidis, K., Gialampoukidis, I., Vrochidis, S., Mezaris, V., and Kompatsiaris, I. (2019).  Iti-certh participation in trecvid 2020.
  * [Godil et al., 2020] Godil, A., Lee, Y., and Fiscus, J. (2020).  TRECVID 2020 actev evaluation plan.  https://actev.nist.gov/pub/TRECVID_2020_ActEV_EvaluationPlan.pdf.
  * [Graham et al., 2018] Graham, Y., Awad, G., and Smeaton, A. (2018).  Evaluation of automatic video captioning using direct assessment.  PloS one, 13(9):e0202789.
  * [Graham et al., 2016] Graham, Y., Baldwin, T., Moffat, A., and Zobel, J. (2016).  Can machine translation systems be evaluated by the crowd alone.  Natural Language Engineering, FirstView:1–28.
  * [Greenberg et al., 2020] Greenberg, C. S., Mason, L. P., Sadjadi, S. O., and Reynolds, D. A. (2020).  Two decades of speaker recognition evaluation at the national institute of standards and technology.  Computer Speech & Language, 60:101032.
  * [Han et al., 2013] Han, L., Kashyap, A., Finin, T., Mayfield, J., and Weese, J. (2013).  UMBC EBIQUITY-CORE: Semantic Textual Similarity Systems.  In Proceedings of the Second Joint Conference on Lexical and Computational Semantics, volume 1, pages 44–52.
  * [Karu and Jain, 1996] Karu, K. and Jain, A. K. (1996).  Fingerprint classification.  Pattern recognition, 29(3):389–404.
  * [Kitware, 2020] Kitware (2020).  MEVA Data Website.  https://www.mevadata.org.  Accessed: 2020-03-12.
  * [Le et al., 2014] Le, V.-B., Lamel, L., Messaoudi, A., Hartmann, W., Gauvain, J.-L., Woehrling, C., Despres, J., and Roy, A. (2014).  Developing stt and kws systems using limited language resources.  In Fifteenth Annual Conference of the International Speech Communication Association.
  * [Lee et al., 2018] Lee, Y., Godil, A., Joy, D., and Fiscus, J. (2018).  TRECVID 2019 actev evaluation plan.  https://actev.nist.gov/pub/Draft_ActEV_2018_EvaluationPlan.pdf.
  * [Li et al., ] Li, X., Zhou, F., and Chen, A.  Renmin university of china at trecvid 2020: Sentence encoder assembly for ad-hoc video search.
  * [Manly, 1997] Manly, B. F. J. (1997).  Randomization, Bootstrap, and Monte Carlo Methods in Biology.  Chapman & Hall, London, UK, 2nd edition.
  * [Martin et al., 1997] Martin, A., Doddington, G., Kamm, T., Ordowski, M., and Przybocki, M. (1997).  The DET curve in assessment of detection task performance.  In Proceedings, pages 1895–1898.
  * [Mille et al., 2020] Mille, S., Belz, A., Bohnet, B., Castro Ferreira, T., Graham, Y., and Wanner, L. (2020).  The third multilingual surface realisation shared task (SR’20): Overview and evaluation results.  In Proceedings of the Third Workshop on Multilingual Surface Realisation, pages 1–20, Barcelona, Spain (Online). Association for Computational Linguistics.
  * [NIST, 2020] NIST (2020).  ActEV Sequestered Data Leaderboard Website.  https://actev.nist.gov/sdl.  Accessed: 2020-03-12.
  * [Oh et al., 2011] Oh, S., Hoogs, A., Perera, A., Cuntoor, N., Chen, C.-C., Lee, J. T., Mukherjee, S., Aggarwal, J., Lee, H., Davis, L., et al. (2011).  A large-scale benchmark dataset for event recognition in surveillance video.  In Computer vision and pattern recognition (CVPR), 2011 IEEE conference on, pages 3153–3160. IEEE.
  * [Over et al., 2006] Over, P., Ianeva, T., Kraaij, W., and Smeaton, A. F. (2006).  TRECVID 2006 Overview.  www-nlpir.nist.gov/projects/tvpubs/tv6.papers/tv6overview.pdf.
  * [Papineni et al., 2002] Papineni, K., Roukos, S., Ward, T., and Zhu, W.-J. (2002).  BLEU: A Method for Automatic Evaluation of Machine Translation.  In Proceedings of the 40th annual meeting on association for computational linguistics, pages 311–318. Association for Computational Linguistics.
  * [Rossetto et al., 2019] Rossetto, L., Schuldt, H., Awad, G., and Butt, A. A. (2019).  V3C–a research video collection.  In International Conference on Multimedia Modeling, pages 349–360. Springer.
  * [TV20Pubs, 2020] TV20Pubs (2020).  http://www-nlpir.nist.gov/projects/tvpubs/tv.pubs.20.org.html.
  * [Ueki et al., 2019] Ueki, K., Hori, T., and Kobayashi, T. (2019).  Waseda_meisei_softbank at trecvid 2019: Ad-hoc video search.  In TRECVID.
  * [Vedantam et al., 2015] Vedantam, R., Lawrence Zitnick, C., and Parikh, D. (2015).  CIDEr: Consensus-based Image Description Evaluation.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4566–4575.
  * [Yilmaz and Aslam, 2006] Yilmaz, E. and Aslam, J. A. (2006).  Estimating Average Precision with Incomplete and Imperfect Judgments.  In Proceedings of the Fifteenth ACM International Conference on Information and Knowledge Management (CIKM), Arlington, VA, USA.
  * [Yilmaz et al., 2008] Yilmaz, E., Kanoulas, E., and Aslam, J. A. (2008).  A Simple and Efficient Sampling Method for Estimating AP and NDCG.  In SIGIR ’08: Proceedings of the 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 603–610, New York, NY, USA. ACM.
  * [Zhao et al., ] Zhao, Y., Song, Y., Chen, S., and Jin, Q.  Ruc_aim3 at trecvid 2020: Ad-hoc video search & video to text description.

##  Appendix A Ad-hoc query topics - 20 unique

641

Find shots showing an aerial view of buildings near water in the daytime

642

Find shots of a person paddling kayak in the water

643

Find shots of people dancing or singing while wearing costumes outdoors

644

Find shots of sailboats in the water

645

Find shots of a person wearing a necklace

646

Find shots of a woman sitting on the floor

647

Find shots of people or cars moving on a dirt road

648

Find shots of a man in blue jeans outdoors

649

Find shots of someone jumping while snowboarding

650

Find shots of one or more people drinking wine

651

Find shots of one or more people skydiving

652

Find shots of a little boy smiling

653

Find shots of group of people clapping

654

Find shots of one or more persons exercising in a gym

655

Find shots of one or more persons standing in a body of water

656

Find shots of a long haired man

657

Find shots of a woman with short hair indoors

658

Find shots of two or more people under a tree

659

Find shots of a church from the inside

660

Find shots of train tracks during the daytime

##  Appendix B Ad-hoc query topics - 20 progress topics

591

Find shots of a person holding an opened umbrella outdoors

592

Find shots of a person reading a paper including newspaper

593

Find shots of one or more women models on a catwalk demonstrating clothes

594

Find shots of people doing yoga

595

Find shots of a person sleeping

596

Find shots of fishermen fishing on a boat

597

Find shots of a shark swimming under the water

598

Find shots of a man in a clothing store

599

Find shots of a person in a bedroom

600

Find shots of a person’s shadow

601

Find shots of a person jumping with a motorcycle

602

Find shots of a person jumping with a bicycle

603

Find shots of people hiking

604

Find shots of bride and groom kissing

605

Find shots of a person skateboarding

606

Find shots of people queuing

607

Find shots of two people kissing who are not bride and groom

608

Find shots of two people talking to each other inside a moving car

609

Find shots of people walking across (not down) a street in a city

610

Find shots showing electrical power lines

##  Appendix C Instance search topics - 20 unique

9299

Find Ian sitting on couch

9300

Find Billy sitting on couch

9301

Find Ian Holding paper - including photos/envelope,notebooks, magazines, etc

9302

Find Bradley Holding paper - including photos/envelope,notebooks, magazines,
etc

9303

Find Billy Holding paper - including photos/envelope,notebooks, magazines, etc

9304

Find Max Drinking

9305

Find Dot Drinking

9306

Find Pat Holding cloth - including jackets, coats, kitchen towels, cleaning
towels, etc

9307

Find Heather Holding cloth - including jackets, coats, kitchen towels,
cleaning towels, etc

9308

Find Ian Crying

9309

Find Heather Crying

9310

Find Max smoking a cigarette - including holding a cigarette between fingers

9311

Find Dot smoking a cigarette - including holding a cigarette between fingers

9312

Find Pat smoking a cigarette - including holding a cigarette between fingers

9313

Find Stacey Laughing

9314

Find Pat Laughing

9315

Find Max Going up or down the stairs

9316

Find Bradley Going up or down the stairs

9317

Find Max holding a phone / handset - including talking on phone

9318

Find Stacey holding a phone / handset - including talking on phone

##  Appendix D Instance search topics - 20 progress topics

9279

Find Phil Sitting on a couch

9280

Find Heather Sitting on a couch

9281

Find Jack Holding phone

9282

Find Heather Holding phone

9283

Find Phil Drinking

9284

Find Shirley Drinking

9285

Find Jack Kissing

9286

Find Denise Kissing

9287

Find Phil Opening door and entering room / building

9288

Find Sean Opening door and entering room / building

9289

Find Shirley Shouting

9290

Find Sean Shouting

9291

Find Stacey Hugging

9292

Find Denise Hugging

9293

Find Max Opening door and leaving room / building

9294

Find Stacey Opening door and leaving room / building

9295

Find Max Standing and talking at door

9296

Find Dot Standing and talking at door

9297

Find Jack Closing door without leaving

9298

Find Dot Closing door without leaving


Rather, we focus on four countermeasures (see Section <ref>), without using any fallback to plain-text DNS.

Reviewer #1:  “Similarly in Section 6.1, you investigate various approaches to use secure versions of DNS to perform the lookups. Why misusing the DNS protocol to do something it is not designed for instead of using Restful APIs (as McAfee mentions in its response)?”

Response: Following the reviewer's comment, we revised the solution section to compare four proposed countermeasures (including REST APIs) on four categories of criteria.
Our review finds that REST APIs are in a secure countermeasure; they are also flexible, given their ability to support future extensions, because they are not limited to 255 characters, like the DNS protocol.
However, RESTful APIs have two disadvantages compared to the other countermeasures when considered as alternatives to existing DNSAML services, namely their performance and compatibility.
From the performance perspective, existing queries over the DNS protocol to DNSAML services outperform RESTful APIs, because they have native support for caching on DNS resolvers, and they are sent over the UDP transport layer therefore incurring the lowest response time overhead of all of the examined countermeasures.
From a compatibility perspective, both anti-malware solution providers and consumers must adapt their infrastructure and network policies to support a transition to RESTful APIs, an effort that is significantly greater than that associated with DoH or DoT.
We discuss these considerations regarding the RESTful APIs in Subsection <ref>:

“REST APIs <cit.> is a standard for a software architecture for interactive Web applications.
Compared to previous solutions, this solution proposes to deprecate the DNS protocol for exchanging signatures, and make use of HTTPS.
The main upside of REST APIs are its security and flexibility, since compared to DNS, it allows larger messages (i.e., up to 2MB with GET requests).
In contrast, it has the worst performance and compatibility issues compared to the other solutions, because providers are required to change their infrastructure to support HTTPS, and consumers are required to permit HTTPS."

Reviewer #2: “The countermeasures should also address shortcomings of the proposals, such as attacks against DNS encryption proposals.”

“In the proposed countermeasures, the authors propose using DNS encryption to ensure privacy. In the recent years a number of attacks against such solutions were demonstrated. Your should list those and explain how these impact your recommendation to use DNS encryption solutions. Would the techniques proposed in prior work for inferring the encrypted input enable your attacks despite encryption?”
“Add reference to attacks against DNS encryption proposals.”

Response: We updated the manuscript to discuss the security limitations of the proposed countermeasures, particularly encrypted DNS protocols.
The revised text appears in Subsection <ref>:

DNS over TLS (DoT) <cit.> is privacy-preserving protocol based on a combination of the TCP stack and TLS that is used to improve DNS security and reliability.
In DoT, DNS data is encrypted, which in turn reduces the impact of various DNS attacks (e.g., DNS hijacking) by establishing mutual connections.
DNS based on TLS is designed to provide greater privacy, support large payloads, and mitigate hijacking and reflection distributed denial-of-service attacks more effectively than existing UDP protocols.
DoT provides a security countermeasure against all of the mentioned threats, by design.
The main security limitation of this countermeasure is privacy leakage <cit.>, allowing an MITM attacker to learn whether a user visits a specific set of domains, which is arguably a lesser concern than learning which exact files are being scanned by a machine, but is still a concern.
From a performance perspective, the use of the TCP transport layer result in reduced performance compared to the DNS protocol over the UDP transport layer.
However, caching is still supported, similarly to DNS without TLS.
To support DoT in anti-malware solutions, consumers need to allow the DoT port (853), and solution providers must replace the DNS resolver with a DoT resolver.

DNS-over-HTTPS (DOH) <cit.> is a standard web protocol for sending DNS traffic over the privacy-preserving HTTPS.
DoH provides a security countermeasure against all of the attacks mentioned.
Its main security limitation is its susceptibility to downgrade attacks <cit.> that force an client (e.g., agent) to fall back to plain-text DNS.
From a performance perspective, the use of HTTP incurs more overhead than to plain-text DNS, and caching is only supported at the recursive DNS server level and not within internal servers, as in the case of plain-text DNS.
To support DoH in anti-malware solutions, consumers need to allow HTTPS traffic, and solution providers must replace the DNS resolver with a DoH resolver.

These limitations are also included in the summary of proposed countermeasures (see the security column in Table <ref>).

§.§ General concerns (Writing quality and minor comments)

Reviewer #2: “The paper is poorly written and the text is difficult to parse. For instance, “The results over time are presented over time in Figure 2, and their describing statistics appear in Table 2”.
“In our experiments which involved several scans, we witnessed file scans for which two specific IPv4 addresses were provide in response, resulting in a situation in which the file was not designated as malicious in the scan report;”.

Response: Before submitting the manuscript, it was reviewed by a native English speaker to improve the writing quality.

Reviewer #1: “The HTTPS protocol is typically spelled HTTPS and not HTTPS”

Response: We replaced all instances of the term “HTTPs” with “HTTPS.”