# Constraining über gravity with recent observations and elucidating the
$H_{0}$ problem

Gustavo A. Concha Valdez1<EMAIL_ADDRESS>Claudia Quintanilla2
<EMAIL_ADDRESS>Miguel A. García-Aspeitia3<EMAIL_ADDRESS>A.
Hernández-Almada4<EMAIL_ADDRESS>V. Motta5<EMAIL_ADDRESS>1Facultad de
Ciencias Físicas, Universidad Nacional Mayor de San Marcos, Ciudad
Universitaria, Cercado de Lima 15081, Perú.  2Facultad de Ciencias Naturales,
Universidad Nacional de San Agustín, Arequipa 04000, Perú.  3Depto. de Física
y Matemáticas, Universidad Iberoamericana Ciudad de México, Prolongación Paseo
de la Reforma 880, México D. F. 01219, México.  4Facultad de Ingeniería,
Universidad Autónoma de Querétaro, Centro Universitario Cerro de las Campanas,
76010, Santiago de Querétaro, México,  5Instituto de Física y Astronomía,
Universidad de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile.

###### Abstract

This paper studies both $\Lambda$CDM and CDM models under the über gravity
theory, named ü$\Lambda$CDM and üCDM respectively. We report bounds over their
parameter phase-space using several cosmological data, in particular, the
recent Pantheon+ sample. Based on the joint analysis, the best fit value of
the über characteristic parameter is $z_{\oplus}=0.028^{+0.036}_{-0.020}$ and
$z_{\oplus}=0.960^{+0.031}_{-0.030}$ at 68% confidence level for ü$\Lambda$CDM
and üCDM respectively. Furthermore, we find that the $\mathbb{H}0(z)$
diagnostic suggests the $H_{0}$ tension is not alleviated. Finally, both
models are statistically compared with $\Lambda$CDM through the Akaike and
Bayesian information criteria, which suggest that there is a modest evidence
against for ü$\Lambda$CDM, a strongest evidence for üCDM, against for the
joint analysis, but both über gravity models and $\Lambda$CDM are equally
preferred for most of the single samples.

###### pacs:

##  I Introduction

The $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model is one of the most
successful theories that explain with great precision the beginning and the
evolution of our Universe at its current stages. However, among its many
conundrums, the dark matter and energy components are the most intriguing due
to the extremely weak interactions (or possible negligible) with the standard
fields, being only the gravitational field the sole probe of its existence.
The dark energy (DE) is related to the current Universe acceleration first
observed by the supernova teams lead by Riess, Perlmutter among others Riess
_et al._ (1998); Perlmutter  _et al._ (1999) and later sustained by the Planck
Satellite Aghanim  _et al._ (2018a). Clearly an acceleration of the space-time
is not expected, thus it requires an unnatural dynamics: the addition of an
extra component into the Friedmann equations which is able to create an
accelerated expansion of the Universe and produce a late de Sitter scale
factor. Without doubts, the cosmological constant is the cheapest way to
tackle this problem, being an essential part of the $\Lambda$CDM paradigm,
having also an exquisite capability to simulate the late Universe Carroll
(2001). However, it is important to emphasize that the cosmological constant
is afflicted with several problems like those related to the vacuum energy
density and the coincidence problem Zeldovich (1968); Weinberg (1989). Is in
this vein that the community is exploring diverse alternatives to the
cosmological constant, being the problem not yet settled (see Motta  _et al._
(2021) for a compilation).

Additionally, cosmologists concentrate most of their efforts in tackling the
case of the current expansion rate of the Universe, the so-called $H_{0}$
tension. The reduction of uncertainties has led to significant discrepancies
between the Planck Aghanim  _et al._ (2018a) and SH0ES Riess  _et al._ (2021)
collaborations, whose disagreements are approximately at $\sim 5.0\sigma$.
Whether these discrepancies are due to systematic errors or mistaken
theoretical assumptions, the $H_{0}$ tension could represent a window into new
physics. Although modifying the standard cosmological model, in order to
address the tension, without compromising its success in other areas has
proven to be a very difficult task Abdalla  _et al._ (2022), the model-
dependence of early $H_{0}$ measurements may suggest that it is worth
exploring models beyond $\Lambda$CDM. The proposals include early dark energy
Karwal and Kamionkowski (2016); Poulin  _et al._ (2019); Smith  _et al._
(2020), extra-relativistic degrees of freedom Aloni  _et al._ (2021);
Aboubrahim  _et al._ (2022); Anchordoqui and Goldberg (2012), new early dark
energy Niedermann and Sloth (2021, 2020), diffusion models Haba  _et al._
(2016); Koutsoumbas  _et al._ (2018), holographic dark energy Li (2004); Huang
and Li (2005); Zhang  _et al._ (2014), to mention some of them.

In the present study we focus on two proposals: the ü$\Lambda$CDM and üCDM
models, based on the über-gravitational model Khosravi (2018), which
understands the $H_{0}$ tension as a transition in the context of a
gravitational theory Khosravi  _et al._ (2019). Über-gravity sets the idea of
taking an ensemble average over all possible $f(R)$ theories Khosravi (2016),
a process inspired by statistical mechanics. Although the criteria for
assigning probabilities to each theory are not clear, the general properties
of ü$\Lambda$CDM and üCDM can be studied and it shows promise in alleviating
the $H_{0}$ tension. Indeed, the model contains two branches where the Hubble
parameter evolves in different ways, with a threshold region determined by the
free parameter $z_{\oplus}$, which is the redshift of transition to the über
gravity. In this sense, it is also possible to determine the deceleration and
jerk parameters in order to elucidate the characteristics of the component
responsible for the acceleration. As stated, ü$\Lambda$CDM contains an extra
parameter in comparison with üCDM, which contains the same number as
$\Lambda$CDM. Therefore, in the case of ü$\Lambda$CDM, besides introducing a
statistical penalization (e.g. AIC or BIC criteria) due to the extra
parameter, it also keep the unsolved problem associated to the cosmological
constant.

Additionally, we constrain the free parameters through a Bayesian analysis,
using recent observations of Observational Hubble Parameters, Type Ia
Supernovae, Baryon Acoustic Oscillations, HII Galaxies and a joint analysis.
Moreover, we implement a $H_{0}$ diagnostic technique to study a possible
alleviation to the $H_{0}$ tension using the über-gravity paradigm.

The paper is organized as follows: in Sec. II we review the mathematical
background of ü$\Lambda$CDM and üCDM models, emphasizing its main properties
and defining the parameters to be analyzed. In Sec. III the data and
methodology are described, and the results are shown in Sec. IV. Finally, in V
we summarize the conclusions and give some outlooks on the subject. We
henceforth use units in which $\hbar=k_{B}=c=1$.

##  II Mathematical Background

The idea of taking an ensemble average over gravity theories is condensed in
the following Lagrangian

$\mathcal{L}=\left(\sum_{i=1}^{N}\mathcal{L}_{i}e^{-\beta\mathcal{L}_{i}}\right)\left(\sum_{i=1}^{N}e^{-\beta\mathcal{L}_{i}}\right)^{-1},$
(1)

where the index $i$ corresponds to the $i$-th theory considered through the
Lagrangian associated $\mathcal{L}$, being N the dimension of the
gravitational models space over the manifold $\mathbb{M}$. The free parameter
$\beta$ assigns probabilities to each model. For greater rigor, the ensemble
average is taken over all analytic models of gravity, described by the $f(R)$
family, where the $R$ stands for the Ricci scalar. However, in Khosravi  _et
al._ (2019) is shown that the Lagrangian, after a change of basis, can be
written as

$\mathcal{L}_{\mathrm{uber}}=\left(\sum_{n=1}^{\infty}(\bar{R}^{n}-2\Lambda)e^{-\beta(\bar{R}^{n}-2\Lambda)}\right)\left(\sum_{n=1}^{\infty}e^{-\beta(\bar{R}^{n}-2\Lambda)}\right)^{-1},$
(2)

where $\bar{R}\equiv R/R_{0}$ contains a new cosmological parameter $R_{0}$,
$n$ is an integer, and $\Lambda$ is the cosmological constant. Next, we will
show two models associated with ü$\Lambda$CDM and üCDM.

###  II.1 ü$\Lambda$CDM model

In background cosmology it is possible to assume the homogenity and isotropy
in the line element, through the Friedmann-Lemaitre-Robertson-Walker (FLRW)
equations as $ds^{2}=-dt^{2}+a(t)^{2}[dr^{2}+r^{2}d\Omega^{2}]$, where
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ and $a(t)$ is the scale
factor. Thus, the Friedmann equation in this scenario can be written as
Khosravi (2018)

$E^{2}(z)_{\mathrm{\ddot{u}\Lambda
CDM}}=\left\\{\begin{array}[]{cc}\Omega_{0m}(z+1)^{3}+\Omega_{0\Lambda},&z>z_{\oplus}\\\
\\\
\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4},&z<z_{\oplus}\\\
\end{array}\right.$ (3)

where $E(z)\equiv H(z)/H_{0}$, $H_{0}$ is the Hubble parameter, $\Omega_{0m}$
is the matter density parameter, both at $z=0$ and $\bar{R}_{0}\equiv
R_{0}/6H_{0}^{2}$. In this case, $\Omega_{0\Lambda}$ plays a role at
$z>z_{\oplus}$, being $z_{\oplus}$ the region of transition to über gravity.

Using the continuity for $E(z)$ and $E^{\prime}(z)$, where the prime denotes a
derivative with respect to $z$, $\bar{R}_{0}$ takes the form

$\frac{\bar{R}_{0}}{2}=\frac{(z_{\oplus}+1)^{4}+3\Omega_{0\Lambda}}{3+(z_{\oplus}+1)^{4}}.$
(4)

One of the magnitudes of interest for the present study is the deceleration
parameter, written in terms of the redshift, which is given by

$q(z)_{\mathrm{\ddot{u}\Lambda
CDM}}=\left\\{\begin{array}[]{cc}\frac{3\Omega_{0m}(z+1)^{3}}{2[\Omega_{0m}(z+1)^{3}+\Omega_{0\Lambda}]}-1,&z>z_{\oplus}\\\
\\\
\frac{2(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}{\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}-1,&z<z_{\oplus}\\\
\end{array}\right.$ (5)

Moreover, the jerk parameter is given by the expression

$j(z)_{\mathrm{\ddot{u}\Lambda
CDM}}=\left\\{\begin{array}[]{cc}1,&z>z_{\oplus}\\\ \\\
\frac{2(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}{\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}+1,&z<z_{\oplus}\\\
\end{array}\right.$ (6)

which behaves like a cosmological constant in the region $z>z_{\oplus}$, while
for $z<z_{\oplus}$ the über gravity dominates its evolution.

Finally, the transition redshift $z_{T}$, which determines the beginning of
the accelerated expansion of the universe $(q(z_{T})=0)$, was calculated for
both models. In the case of ü$\Lambda$CDM this redshift takes the form

$z_{T}=\left\\{\begin{array}[]{cc}\sqrt[3]{\frac{\Omega_{0\Lambda}(3+(1+z_{\oplus})^{4})}{2(1-\Omega_{0\Lambda})(1+z_{\oplus})}}-1,&z_{T}>z_{\oplus}\\\
\\\
\sqrt[4]{\frac{(z_{\oplus}+1)^{4}+3\Omega_{0\Lambda}}{1-\Omega_{0\Lambda}}}-1,&z_{T}<z_{\oplus}.\\\
\end{array}\right.$ (7)

The transition in this model occurs for $z_{T}>z_{\oplus}$, so in this study
only the first expression is taken into account.

###  II.2 üCDM model

In this case, we propose that the Universe acceleration is only driven by über
contributions, thus we consider $\Omega_{\Lambda}=0$, reducing the free
parameters to the same number as in the standard model and assuming FLRW
cosmology. Therefore, we have

$E^{2}(z)_{\mathrm{\ddot{u}CDM}}=\left\\{\begin{array}[]{cc}\Omega_{0m}(z+1)^{3},&z>z_{\oplus}\\\
\\\
\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4},&z<z_{\oplus}\\\
\end{array}\right.$ (8)

being

$\bar{R}_{0}=\frac{2(z_{\oplus}+1)^{4}}{3+(z_{\oplus}+1)^{4}},\;\;\Omega_{0m}=\frac{4(z_{\oplus}+1)}{3+(z_{\oplus}+1)^{4}}.$
(9)

For the deceleration parameter we have

$q(z)_{\mathrm{\ddot{u}CDM}}=\left\\{\begin{array}[]{cc}\frac{1}{2},&z>z_{\oplus}\\\
\\\
\frac{2(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}{\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}-1,&z<z_{\oplus}\\\
\end{array}\right.$ (10)

and the jerk parameter reads

$j(z)_{\mathrm{\ddot{u}CDM}}=\left\\{\begin{array}[]{cc}\frac{3}{4},&z>z_{\oplus}\\\
\\\
\frac{2(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}{\frac{1}{2}\bar{R}_{0}+(1-\frac{1}{2}\bar{R}_{0})(z+1)^{4}}+1,&z<z_{\oplus}\\\
\end{array}\right.$ (11)

being the first region $z>z_{\oplus}$ (for $q$ and $j$) a matter domination
and the later one a domination of the über effects that drives the
acceleration.

Finally, in üCDM the transition is not allowed in the region
$z_{T}>z_{\oplus}$, giving, as a result, the following analytic expression,
valid for $z_{T}<z_{\oplus}$

$z_{T}=\frac{1+z_{\oplus}}{\sqrt[4]{3}}-1.$ (12)

Note that $z_{T}$ shows a dependence only on $z_{\oplus}$.

##  III Data and methodology

Both ü$\Lambda$CDM and üCDM cosmologies are confronted using CC, HIIG, SnIa,
BAO and joint data through a Bayesian Markov Chain Monte Carlo (MCMC)
analysis. To bound their free parameter phase-space, ($h$,
$\Omega_{0\Lambda}$, $z_{\oplus}$) for ü$\Lambda$CDM and ($h$, $z_{\oplus}$)
for üCDM, we use the emcee package Foreman-Mackey  _et al._ (2013) under
Python language. We establish a configuration to achieve the convergence of
the chains using the autocorrelation function, and generate a set of 3000
chains with 250 steps. Additionally, we use a Gaussian prior over $h$ as
$h=0.7403\pm 0.0142$ Riess  _et al._ (2019) and flat priors over
$\Omega_{0\Lambda}$ and $z_{\oplus}$ in the region [0,1] and [0,2]
respectively. Thus, the $\chi^{2}$-function is given by

$\chi_{\rm Joint}^{2}=\chi_{\rm CC}^{2}+\chi_{\rm HIIG}^{2}+\chi_{\rm
SnIa}^{2}+\chi_{\rm BAO}^{2}\,,$ (13)

where each term corresponds to the $\chi^{2}$ function per sample.

###  III.1 Cosmic chronometers

We use a sample of 31 measurements of the Hubble parameter using differential
age method Moresco  _et al._ (2016) (see also Magaña  _et al._ (2018)). Due
these points are cosmological model independent, they are useful to test
alternative cosmologies to $\Lambda$CDM. The $\chi^{2}$ function can be built
as

$\chi^{2}_{{\rm
OHD}}=\sum_{i=1}^{31}\left(\frac{H_{th}(z_{i})-H_{obs}(z_{i})}{\sigma^{i}_{obs}}\right)^{2},$
(14)

where $H_{th}(z_{i})$ is the theoretical Hubble parameter using Eq. (3), and
$H_{obs}(z_{i})\pm\sigma_{obs}^{i}$ is the observational counterpart with its
uncertainty at the redshift $z_{i}$.

###  III.2 Type Ia Supernovae (Pantheon$+$)

Recently, a sample of 1701 measurements of the luminosity modulus coming from
SNIa, namely Pantheon$+$ sample, is reported by Scolnic  _et al._ (2018);
Brout  _et al._ (2022). This represents the largest sample and covers a region
$0.001<z<2.26$. Considering that this sample is extracted from 1550 distinct
SNIa, we build the $\chi^{2}$ function as

$\chi_{\rm SnIa}^{2}=a+\log\left(\frac{e}{2\pi}\right)-\frac{b^{2}}{e},$ (15)

where

$\displaystyle a$ $\displaystyle=$
$\displaystyle\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot\mathbf{Cov_{P}^{-1}}\cdot\Delta\boldsymbol{\tilde{\mu}},$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\Delta\boldsymbol{\tilde{\mu}}^{T}\cdot\mathbf{Cov_{P}^{-1}}\cdot\Delta\mathbf{1},$
(16) $\displaystyle e$ $\displaystyle=$
$\displaystyle\Delta\mathbf{1}^{T}\cdot\mathbf{Cov_{P}^{-1}}\cdot\Delta\mathbf{1},$

and $\Delta\boldsymbol{\tilde{\mu}}$ is the vector of the difference between
the theoretical distance modulus and the observed one,
$\Delta\mathbf{1}=(1,1,\dots,1)^{T}$, $\mathbf{Cov_{P}}$ is the covariance
matrix formed by adding the systematic and statistic uncertainties. The
transpose of the vectors are denoted with the super-index $T$ on the previous
expressions.

The theoretical counterpart of the distance modulus is estimated by

$m_{th}=\mathcal{M}+5\log_{10}\left[\frac{d_{L}(z)}{10\,{\rm pc}}\right],$
(17)

where $\mathcal{M}$ is a nuisance parameter which has been marginalized by Eq.
(15). The luminosity distance, denoted as $d_{L}(z)$, is computed through

$d_{L}(z)=(1+z)c\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})},$ (18)

where $c$ is the speed of light.

###  III.3 Baryon Acoustic Oscillations

The Baryon Acoustic Oscillations signature is an standard ruler useful to
constrain cosmological model parameters and is the result of the interactions
between baryons and photons in the recombination era. We use 6 correlated
points obtained by Percival  _et al._ (2010); Blake  _et al._ (2011); Beutler
_et al._ (2011) and collected by Giostri  _et al._ (2012). We confront them
with the cosmological models by building the $\chi^{2}$ function as

$\chi^{2}_{\rm BAO}=\vec{X}^{T}{\rm C}_{BAO}^{-1}\vec{X}$ (19)

where $\vec{X}^{T}$ is the residual vector between observational measurements
and theoretical values of the ratio $d_{A}(z_{*})/D_{V}(z)$, where
$d_{A}(z_{d})$ is the comoving angular diameter distance at the the photon
decoupling epoch ($z_{*}$) and the dilation scale is given by Eisenstein  _et
al._ (2005)

$D_{V}(z)=\left[d_{A}^{2}(z)cz/H(z)\right]^{1/3}$ (20)

where $c$ is again the speed of light. For this work we use $z_{*}=1089.80\pm
0.21$ Aghanim  _et al._ (2018a).

###  III.4 HII Galaxies

A sample of 181 measurements coming from Hydrogen II galaxies (HIIG), with
their luminosity dominated by young massive burst of star formation, is
reported by González-Morán  _et al._ (2019, 2021). This sample which covers a
region $0.01<z<2.6$ and is useful to establish bounds over cosmological
parameters due the correlation between the measured luminosity $L$ of the
galaxies and the inferred velocity dispersion $\sigma$ of their ionized gas
Chávez  _et al._ (2012, 2014); Terlevich  _et al._ (2015); Chávez  _et al._
(2016). The $\chi^{2}$-function is built as

$\chi^{2}_{{\rm
HIIG}}=\sum_{i}^{181}\frac{[\mu_{th}(z_{i},{\Theta})-\mu_{obs}(z_{i})]^{2}}{\epsilon_{i}^{2}},$
(21)

where $\epsilon_{i}$ is the observational uncertainty measured at $z_{i}$
having $68\%$ of confidence level. Additionally, the observational distance
modulus ($\mu_{obs}$) is expressed

$\mu_{obs}=2.5(\alpha+\beta\log\sigma-\log f-40.08)\,.$ (22)

Here, $\alpha$ and $\beta$ are the intercept and slope of the $L$-$\sigma$
relation and $f$ is the measured flux. The theoretical estimate is given as

$\mu_{th}(z,\Theta)=5\log_{10}\left[\frac{d_{L}(z,\Theta)}{1\,{\rm
Mpc}}\right]+25,$ (23)

where $d_{L}$ is the luminosity distance measured in Mpc (see Eq. (18)).

##  IV Results

We perform the über gravity in the context of ü$\Lambda$CDM and üCDM models to
understand the nature of the current Universe acceleration and help us to
interpret the origin of the $H_{0}$ tension. In Figs. 1 and 2 we present the
2D parameter likelihood contours at $68\%$ (1$\sigma$) and $99.7\%$
(3$\sigma$) confidence level (CL) respectively for both models. Moreover Table
1 shows the mean values of the parameters and their uncertainties at
1$\sigma$.

Additionally to this, we statistically compare these results with $\Lambda$CDM
model, applying the corrected Akaike information criterion (AICc) Akaike
(1974) and the Bayesian information criterion (BIC) Schwarz (1978). Both
criteria penalize according to the size of the data sample ($N$) and the
number of degrees of freedom ($k$) defined as ${\rm AICc}=\chi^{2}_{\rm
min}+2k+2(k^{2}+k)/(N-k-1)$ and ${\rm BIC}=\chi^{2}_{\rm min}+k\log(N)$,
respectively, where $\chi^{2}_{\rm min}$ is the minimum of $\chi^{2}$.
Therefore, a model with a lower value of AICc (BIC) is preferred by the data.

For the $\Delta{\rm AICc}$ we have the following conditions: if $|\Delta{\rm
AICc}|<4$ both models are statistically equivalent, if $4<{\rm AICc}<10$ the
data still support the given model (über gravity) but less than the preferred
one (concordance model) and finally if ${\rm AICc}>10$ indicates that the data
do not support the given model. In addition, for $\Delta{\rm BIC}$ the
interpretation is as follows: if $\Delta{\rm BIC}<2$ there is no evidence
against the new model which in this case is über gravity, if $2<\Delta{\rm
BIC}<6$ there is modest evidence against the new model, and finally if
$\Delta{\rm BIC}>10$, gives the strongest evidence against it.

We start by showing the AICc and BIC for the $\Lambda$CDM model: for AICc
values we have 22.07 (CC), 440.73 (HIIG), 2002.41 (SnIa), 12.82 (BAO) and
2478.32 (Joint); as for BIC it has 24.51 (CC), 447.06 (HIIG), 2013.28 (SnIa),
6.04 (BAO) and 2489.43 (Joint). Thus, Table 1 shows the above comparison being
$\Lambda$CDM the reference scenario. According to AICc we find that both
ü$\Lambda$CDM and $\Lambda$CDM are preferred equally for CC, HIIG and Joint
and the ü$\Lambda$CDM is not supported for SnIa and BAO. Regarding to BIC, we
find the strongest evidence against ü$\Lambda$CDM for SnIa and a modest
evidence against this model for the Joint analysis. On the other hand, we
observe that üCDM and $\Lambda$CDM are equally preferred by CC, HIIG and BAO
while the strongest evidence against üCDM is provided by SnIa and Joint
analysis. It is worth to mention that $\Delta$AICc and $\Delta$BIC are reduced
to $\Delta\chi^{2}$ because both models have same number of free parameters.

On the other hand, notice that the ü$\Lambda$CDM scenario produces a late
acceleration, constraining the free parameter of über gravity according to the
joint analysis as $z_{\oplus}\simeq 0.028$, which also coincides with the
epoch of transition to über gravity. The parameter $\Omega_{0\Lambda}\simeq
0.685$ marks a cosmological constant domination, however the unsolved
questions about its characteristics remains. The redshift transition is
compatible with $\Lambda$CDM model which is at $0.634$, as estimated from the
joint analysis. The transition from GR to über gravity is perfectly observed
through the jerk factor shown in Fig. 3, where a discontinuity in the $j(z)$
function is observed. According to the $\mathbb{H}_{0}$ diagnostic presented
in Fig. 4, the tension in $H_{0}$ is not alleviated because it keeps the trend
towards values consistent with supernovae results.

On the other hand, we explore the üCDM model which is not studied in the
literature. In this case, the über component $z_{\oplus}$ acts like a
cosmological constant, having the same free parameters as $\Lambda$CDM. For
the joint analysis we conclude that the value of the über parameter us
$z_{\oplus}\simeq 0.960$ and the transition to an accelerated Universe happens
earlier in its evolution $z_{T}\simeq 0.489$. From Fig. 3, in particular with
$q(z)$ and $j(z)$, we observe that the transition to über gravity domination
is more extreme in comparison with the ü$\Lambda$CDM model. Additionally to
this, in this figure we also observe a trend to values that are consistent
with supernova results instead of those obtained from Planck, concluding that
the $H_{0}$ tension persist despite the über gravity presence.

Figure 1: 2D contours at $1\sigma$ (inner region) and $3\sigma$ (outermost
region) CL for the ü$\Lambda$CDM. Figure 2: 2D contours at $1\sigma$ (inner
region) and $3\sigma$ (outermost region) CL for the üCDM.

Sample | $\chi^{2}_{\rm min}$ | $h$ | $\Omega_{0\Lambda}$ | $z_{\oplus}$ | $z_{T}$ | AICc | $\Delta{\rm AICc}$ | BIC | $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
ü$\Lambda$CDM
CC | 17.42 | $0.730^{+0.013}_{-0.013}$ | $0.689^{+0.006}_{-0.006}$ | $0.293^{+0.124}_{-0.135}$ | $0.706^{+0.059}_{-0.043}$ | 24.31 | 2.24 | 27.72 | 3.21
HIIG | 436.67 | $0.732^{+0.011}_{-0.011}$ | $0.689^{+0.006}_{-0.006}$ | $0.407^{+0.146}_{-0.162}$ | $0.760^{+0.086}_{-0.071}$ | 442.81 | 2.08 | 452.27 | 5.21
SnIa | 2012.73 | $0.740^{+0.014}_{-0.014}$ | $0.682^{+0.005}_{-0.005}$ | $0.019^{+0.028}_{-0.014}$ | $0.627^{+0.013}_{-0.013}$ | 2018.74 | 16.33 | 2035.05 | 21.77
BAO | 2.91 | $0.740^{+0.014}_{-0.014}$ | $0.688^{+0.005}_{-0.005}$ | $0.107^{+0.085}_{-0.071}$ | $0.654^{+0.020}_{-0.017}$ | 32.91 | 20.09 | 7.74 | 1.70
Joint | 2476.65 | $0.711^{+0.008}_{-0.009}$ | $0.685^{+0.005}_{-0.005}$ | $0.028^{+0.036}_{-0.020}$ | $0.634^{+0.013}_{-0.013}$ | 2482.66 | 4.34 | 2499.33 | 9.90
üCDM
CC | 15.66 | $0.735^{+0.013}_{-0.013}$ | — | $1.219^{+0.086}_{-0.080}$ | $0.686^{+0.065}_{-0.061}$ | 20.09 | -1.98 | 22.53 | -1.98
HIIG | 436.11 | $0.734^{+0.011}_{-0.011}$ | — | $1.309^{+0.125}_{-0.118}$ | $0.754^{+0.095}_{-0.090}$ | 440.18 | -0.55 | 446.51 | -0.55
SnIa | 2017.91 | $0.740^{+0.014}_{-0.014}$ | — | $0.791^{+0.032}_{-0.030}$ | $0.361^{+0.024}_{-0.023}$ | 2021.92 | 19.51 | 2032.79 | 19.51
BAO | 2.56 | $0.740^{+0.014}_{-0.014}$ | — | $1.312^{+0.063}_{-0.064}$ | $0.757^{+0.048}_{-0.049}$ | 12.56 | -0.26 | 5.78 | -0.26
Joint | 2541.55 | $0.706^{+0.009}_{-0.009}$ | — | $0.960^{+0.031}_{-0.030}$ | $0.489^{+0.024}_{-0.023}$ | 2545.56 | 67.24 | 2556.67 | 67.24

Table 1: Bestfit values and their uncertainties at $1\sigma$ of the free
parameters for both ü$\Lambda$CDM and üCDM models. Additionally, we show
values for the AICc, BIC, $\Delta{\rm AICc}\equiv{\rm AICc}-{\rm
AICc}^{\Lambda\rm CDM}$ and $\Delta{\rm BIC}\equiv{\rm BIC}-{\rm
BIC}^{\Lambda\rm CDM}$ for both ü$\Lambda$CDM and üCDM models.

Figure 3: Reconstruction of $H(z)$ (left panel), $q(z)$ (middle panel) and
$j(z)$ (right panel) for ü$\Lambda$CDM and üCDM, for each dataset. Circle
marker represents the best value of $z_{\oplus}$ for each sample.

Figure 4: The $\mathbb{H}0(z)$ diagnostic for über gravity, showing üCDM and
ü$\Lambda$CDM models. Here we present the behavior of $\mathbb{H}0(z)$ under
the constraints of the four data-samples and the joint analysis. The
continuous red line represents the results for $\Lambda$CDM cosmology assuming
$h=0.6766$ and $\Omega_{0m}=0.3111$ according to Aghanim  _et al._ (2018b) .

##  V Conclusions and Outlooks

This paper presents an exhaustive revision of the über gravity for the
ü$\Lambda$CDM and üCDM models, obtaining stringent constrictions with
different cosmological data samples, in particular the recent Pantheon$+$
sample. We statistically compare both models with $\Lambda$CDM by using AICc
and BIC criteria. We find that ü$\Lambda$CDM shows a modest evidence against
and üCDM a strongest evidence against for the joint analysis. However, uber
gravity and $\Lambda$CDM are equally preferred for most of the single samples
(being SNIa sample the exception).

First, the ü$\Lambda$CDM contains one extra free parameter in comparison with
the standard paradigm, maintaining the same open questions related to the
understanding of the cosmological constant. Our results for the joint analysis
point out that $\Omega_{0\Lambda}\simeq 0.685$, consistent with the standard
cosmological model, but with a subdominant value for the über parameter
$z_{\oplus}\simeq 0.028$. It is important to remind that the über parameter
$z_{\oplus}$ is the point where the über gravity starts to dominate over the
standard GR, thus, the presence of the $\Lambda$CDM delays the apparition of
the über gravity as observed in Figs. 3 for the $H(z)$, $q(z)$ and $j(z)$
parameters. The transition redshift is also in consistency with the standard
cosmological model (see Table 1). Regarding $\mathbb{H}0(z)$ diagnostic, Fig.
4 reveals that, according to the joint analysis, the $H_{0}$ value is more
consistent with the supernova data than with the Planck ones at $z=0$, unable
to reduce the tension under this scenario. This could be the result of
$\Lambda$CDM being behind the dynamics while über gravity start its domination
at $z_{\oplus}\simeq 0.028$. Additionally to this, the new sample of Pantheon+
could generate a tendency to a value greater than the one predicted by Planck.

Furthermore, we explore the üCDM, where the über parameter plays the role of
the cosmological constant, thus having the same parameters as in the standard
cosmology. In this case, über parameter generates the late acceleration of the
Universe and its presence start earlier, specifically at $z_{\oplus}\simeq
0.960$ according to the joint analysis presented in Table 1. The transition
also happens earlier than in the standard model, at $z_{T}\simeq 0.489$ (see
Table 1). The evolution of $H(z)$, $q(z)$ and $j(z)$ is presented in Fig. 3
where we can see that the behavior is more abrupt, mainly in the $q(z)$ and
$j(z)$ behavior. However, über gravity eventually mimics the cosmological
constant because $j=1$ when $z=-1$. The $\mathbb{H}0(z)$ diagnostic plot
presented in Fig. 4 shows that at $z=0$ the value for $H_{0}$ does not
coincide with the Planck result for $\Lambda$CDM, being in concordance with
supernova results, which is also a sign that the über parameter acting like a
cosmological constant is inadequate to alleviate the tension in $H_{0}$.

In summary, we observe that the über gravity is an alternative to study the
late Universe acceleration by mimicking the cosmological constant behavior
through the über parameter $z_{\oplus}$. However, the mystery of the Hubble
tension remains unsolvable and gives values that are compatibles with
supernova results. Nevertheless, we consider that a deeper exploration of the
über Lagrangian for other values of $n$ is necessary in order to elucidate any
extra dynamics that could help us understand the reason for the Hubble
tension. Such exploration will be presented elsewhere.

###### Acknowledgements.

G.A.C.V. thanks the support from the Grupo de Física Teórica of the
Universidad Nacional Mayor de San Marcos and Dr. Teófilo Vargas Auccalla for
his invaluable mentoring. C.Q. thanks the support of Universidad Nacional de
San Agustin and Mg. Rolando Perca Gonzáles. M.A.G-A acknowledges support from
cátedra Marcos Moshinsky (MM) and Universidad Iberoamericana for the support
with the National Research System (SNI) grant. The numerical analysis was
carried out by the Numerical Integration for Cosmological Theory and
Experiments in High-energy Astrophysics ”Nicte Ha” cluster at IBERO
University, acquired through cátedra MM support. A.H.A. thanks to the support
from Luis Aguilar, Alejandro de León, Carlos Flores, and Jair García of the
Laboratorio Nacional de Visualización Científica Avanzada. V.M. acknowledges
partial support from Centro de Astrofísica de Valparaíso.

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