# Generalized Emergent Dark Energy Model and the Hubble Constant Tension

Weiqiang Yang<EMAIL_ADDRESS>Department of Physics, Liaoning Normal
University, Dalian, 116029, P. R. China  Eleonora Di Valentino  eleonora.di-
<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology,
Department of Physics, Durham University, Durham DH1 3LE, UK  Supriya Pan
<EMAIL_ADDRESS>Department of Mathematics, Presidency
University, 86/1 College Street, Kolkata 700073, India  Arman Shafieloo
<EMAIL_ADDRESS>Korea Astronomy and Space Science Institute, Daejeon
34055, Korea  University of Science and Technology, Yuseong-gu 217 Gajeong-ro,
Daejeon 34113, Korea  Xiaolei Li<EMAIL_ADDRESS>College of Physics,
Hebei Normal University, Shijiazhuang 050024, China

###### Abstract

We investigate a generalized form of the Phenomenologically Emergent Dark
Energy (PEDE) model, known as Generalized Emergent Dark Energy (GEDE),
introduced in Li and Shafieloo (2020) in light of a series of cosmological
probes and considering the evolution of the model at the level of linear
perturbations. This model introduces a free parameter $\Delta$ that can
discriminate between the $\Lambda$CDM (corresponds to $\Delta=0$) or the PEDE
(corresponds to $\Delta=1$) models, allowing us to determine which model is
preferred most by the fit of the observational datasets. We find a strong
evidence of the GEDE scenario for most of the employed observational datasets.
In particular, we find that $\Lambda$CDM model is disfavored at more than
$2\sigma$ CL for most of the observational datasets considered in this work
and PEDE is in agreement with Planck 2018+BAO+R19 combination within
$1\sigma$. Finally, a Bayesian Model comparison shows a strong evidence for
GEDE against $\Lambda$CDM for most of the dataset combinations.

##  I Introduction

With the growing sensitivity in the experimental data, cosmology is becoming
more rich than it was several years back. The observational estimates suggest
that the main ingredients of the universe are two dark fluids, one is
pressureless dark matter (DM) or cold dark matter (CDM) and one is some
hypothetical dark energy (DE). This DE fluid is solely responsible for the
observed accelerating expansion of the universe. This observational
description can be described in the framework of the $\Lambda$CDM cosmology
where the $\Lambda$ term - independent of the cosmic evolution, acts as the
so-called DE fluid. Even though the physics of DM and DE are not even clear,
$\Lambda$CDM has received greatest focus for its ability to fit most of the
observational data. If we even avoid the actual physics of DM and DE, however,
$\Lambda$CDM cannot explain some very serious issues. The tensions in the
Hubble constant $H_{0}$ Riess et al. (2019, 2020); Aghanim et al. (2020a);
Verde et al. (2019); Di Valentino (2020) and in the amplitude of the growth of
structure $\sigma_{8}$ (alternatively one also measures the parameter
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m0}/0.3}$) are the signals of the
limitations of the $\Lambda$CDM cosmology (see Refs. Di Valentino et al.
(2020a, b) for a recent overview, and references therein). Therefore, the
search for some alternative descriptions to the $\Lambda$CDM cosmology having
the explanations to the aforementioned tensions of these cosmological
parameters has been the heart of modern cosmology now. Usually, the
alleviation/solution of the tensions are achieved by extending the parameter
space of the six parameters $\Lambda$CDM model. This extension is performed in
many ways.

For example, early modifications of the expansion history are promising
because such modifications can increase the $H_{0}$ value and give a lower
sound horizon $r_{\mathrm{drag}}$ at the drag epoch Knox and Millea (2020);
Evslin et al. (2018), in agreement with the Baryon Acoustic Oscillations (BAO)
data. In this category we find the Early Dark Energy models Pettorino et al.
(2013); Poulin et al. (2019); Karwal and Kamionkowski (2016); Sakstein and
Trodden (2020); Niedermann and Sloth (2019); Akarsu et al. (2020); Gogoi et
al. (2020); Smith et al. (2020a); Hill et al. (2020); Murgia et al. (2020);
Lucca (2020); Haridasu et al. (2020); Smith et al. (2020b); D’Amico et al.
(2020); Chudaykin et al. (2020a); Kaloper (2019); Chudaykin et al. (2020b);
Berghaus and Karwal (2020); Alexander and McDonough (2019); Agrawal et al.
(2019a); Niedermann and Sloth (2020); Ye and Piao (2020); Lin et al. (2019,
2020); Yin (2020); Braglia et al. (2020)), and extra relativistic species at
recombination Vagnozzi (2020); Seto and Toda (2021); Carneiro et al. (2019);
Gelmini et al. (2019, 2020); Barenboim et al. (2017); D’Eramo et al. (2018);
Pandey et al. (2020); Xiao et al. (2020); Nygaard et al. (2020); Blinov et al.
(2020); Binder et al. (2018); Hryczuk and Jodłowski (2020); Vattis et al.
(2019); Haridasu and Viel (2020); Clark et al. (2020); Gu et al. (2020);
Alcaniz et al. (2021); Choi et al. (2020a); Escudero et al. (2019);
Anchordoqui (2020); Desai et al. (2020); Choi et al. (2020b); Di Valentino et
al. (2018a); Escudero and Witte (2020); Arias-Aragon et al. (2021); Blinov and
Marques-Tavares (2020); Buen-Abad et al. (2018); Gonzalez et al. (2020);
Flambaum and Samsonov (2019); Anchordoqui and Perez Bergliaffa (2019); Flores
and Kusenko (2021); Artymowski et al. (2020), but they can not solve the
tension with R19 below $3\sigma$ Arendse et al. (2020). Having features in the
form of the primordial power spectrum have been proposed as another possible
early universe solution for the Hubble tension problem Hazra et al. (2019);
Keeley et al. (2020) but these models still need to be tested against a
combination of various observations. One can also modify the physics at late
time to explain the Hubble constant issue. While the late time modifications
of the expansion history can solve the Hubble tension with R19 within
$1\sigma$, they do not modify the sound horizon, resulting in tension with the
BAO data. In this category, we have Phantom Dark Energy models Aghanim et al.
(2020a); Vagnozzi (2020); Martinelli and Tutusaus (2019); Alestas et al.
(2020a); Di Valentino (2020); Haridasu et al. (2020); Menci et al. (2020);
Kitazawa (2020); Di Valentino et al. (2020c); Yang et al. (2021a, 2019a); Di
Valentino et al. (2020d)), Phenomenologically Emergent Dark Energy (PEDE) Li
and Shafieloo (2019); Pan et al. (2020a); Rezaei et al. (2020); Liu and Miao
(2020); Li and Shafieloo (2020); Hernández-Almada et al. (2020); Yang et al.
(2021b), and additional Dark Energy proposals Li et al. (2019); Yang et al.
(2020a); Camarena and Marra (2021); Benevento et al. (2020); Alestas et al.
(2020b); Solà et al. (2017); Keeley et al. (2019); Dutta et al. (2020); da
Silva and Silva (2020a); Yang et al. (2019b); Guo et al. (2019); da Silva and
Silva (2020b); Agrawal et al. (2019b); Ó Colgáin et al. (2019); Colgáin and
Yavartanoo (2019); Banerjee et al. (2020); Di Valentino et al. (2019); Miao
and Huang (2018); Adler (2019). Other promising Hubble constant solutions are
models considering an interaction between the Dark Matter and the Dark Energy,
known as Interacting Dark Energy (IDE) models Kumar and Nunes (2017); Di
Valentino et al. (2020e); Kumar et al. (2019); Di Valentino and Mena (2020);
Lucca and Hooper (2020); Di Valentino (2020); Yang et al. (2020b); Martinelli
et al. (2019); Gómez-Valent et al. (2020); Di Valentino et al. (2017); Yang et
al. (2020c); Kumar and Nunes (2016); Yang et al. (2018); Pan et al. (2020b);
Yao and Meng (2020a, b); Pan et al. (2019a); Yang et al. (2019c); Pan et al.
(2019b); Yang et al. (2019d); Amirhashchi and Yadav (2020); Gao et al. (2021).
In this context, we refer to an exhaustive review on the Hubble constant
tension that appeared recently Di Valentino et al. (2021).

Effectively, in almost every work, an extension of the parameter space
(compared to the minimal $\Lambda$CDM scenario) is needed to release this
tension, however, both the approaches as described above have some
shortcomings. Even though a model with extended parameter space compared to
the minimal $\Lambda$CDM model can address the $H_{0}$ tension issue, at the
same time, due to the additional free parameters, the fitting of the model
gets worse compared to the excellent fitting that we witnessed in
$\Lambda$CDM. A natural question in this context therefore raised: whether one
could find some cosmological models having exactly six free parameters as in
the $\Lambda$CDM model, but at the same time alleviate/solve the $H_{0}$
tension. There are only a very few works where the $H_{0}$ tension can be
solved without adding any extra parameter from outside. One is the vacuum
metamorphosis model Di Valentino et al. (2018b, 2020f) and the other one is
the PEDE scenario, see for instance Refs. Li and Shafieloo (2019); Pan et al.
(2020a). Both the models are excellent to reconcile the $H_{0}$ tension but
the problem with the sound horizon remains. But interestingly, both the models
offer excellent fits to some of the observational datasets, not for all
though. So, the six-parameters based cosmological models can be considered to
be the potential contenders of other cosmological models.

Having such appealing possibilities associated with these six-parameters based
cosmological models, in the present article we focus only on an extended
version of the PEDE model known as Generalized Emergent Dark Energy (GEDE)
model Li and Shafieloo (2020) (also see Hernández-Almada et al. (2020)). The
GEDE model contains an extra free parameter $\Delta$ which can return both
$\Lambda$CDM ($\Delta=0$) and PEDE model ($\Delta=1$) as special cases. So,
this model allows a more wider cosmological picture between $\Lambda$CDM model
and PEDE model and any departure from $\Delta=0$ will strengthen the viability
of the new scenario. We investigated this scenario including the evolution of
the model at the level of linear perturbations in order to examine whether
this extended version of PEDE can equally release the $H_{0}$ tension when
almost all cosmological probes are taken into account. We quickly note here
that GEDE can reconcile $H_{0}$ and offer an excellent fit to most of the
observational data employed in this work, compared to the $\Lambda$CDM model.

The article has been organized in the following way. In section II we review
the GEDE model. After that in section III we describe the observational data
and the statistical methodology for the fitting of the model, and in the
following section IV we discuss the observational constraints and the $H_{0}$
problem in detail. Finally, in section V we summarize our results focusing on
the main ingredients of the model.

##  II Reintroducing Generalized Emergent Dark energy

We consider a spatially flat Friedmann-Lemaître-Robertson-Walker Universe
where radiation, pressureless matter (baryons plus cold dark matter) and a
dark energy fluid are present. All are assumed to be barotropic in nature and
none of them are interacting with each other. In such a background, one can
readily write down the Hubble expansion as

$\displaystyle
H^{2}(a)=H_{0}^{2}\Bigl{[}\Omega_{r0}a^{-4}+\Omega_{m0}a^{-3}+\widetilde{\Omega}_{\rm{DE}}(a)\Bigr{]}$
(1)

where $a$ being the scale factor of the FLRW Universe which is related with
the redshift $z$ as $1+z=a_{0}/a$ ($a_{0}$ being the present value of the
scale factor which we set to the unity); $\Omega_{r0}=\rho_{r0}/\rho_{\rm
crit,0}$, $\Omega_{m0}=\rho_{m0}/\rho_{\rm crit,0}$ are, respectively, the
present values of the density parameters for radiation and pressureless matter
and $\widetilde{\Omega}_{\rm{DE}}(a)=\rho_{\rm DE}(a)/\rho_{\rm crit,0}$ is
the energy density of the dark energy fluid with respect to the critical
energy density $\rho_{\rm{crit},0}(a)=3H^{2}(a)/8\pi G$ and
$\widetilde{\Omega}_{\rm{DE},0}=(1-\Omega_{m0}-\Omega_{r0})$. One can easily
notice that $\widetilde{\Omega}_{\rm{DE}}(a)$ is related to the density
parameter for DE ($\Omega_{\rm DE}(a)$) as follows:

$\displaystyle\widetilde{\Omega}_{\rm{DE}}(a)$
$\displaystyle\,=\,\frac{\rho_{\rm{DE}}(a)}{\rho_{\rm{crit,0}}}\,=\,\frac{\rho_{\rm{DE}}(a)}{\rho_{\rm{crit}}(a)}\times\frac{\rho_{\rm{crit}}(a)}{\rho_{\rm{crit,0}}}$
(2) $\displaystyle\,=\,{\Omega}_{\rm{DE}}(a)\times\frac{H^{2}(a)}{H_{0}^{2}}$
(3)

Therefore, once the evolution of $\widetilde{\Omega}_{\rm{DE}}(a)$ is known,
the dynamics of the Universe can in principle be determined either
analytically or numerically. In the present work we consider that
$\widetilde{\Omega}_{\rm{DE}}(a)$ evolves as

$\widetilde{\Omega}_{\rm{DE}}(z)\,=\,\Omega_{\rm{DE,0}}\frac{1-{\rm{tanh}}\left(\Delta\times{\rm{log}}_{10}(\frac{1+z}{1+z_{t}})\right)}{{1+{\rm{tanh}}\left(\Delta\times{\rm{log}}_{10}({1+z_{t}})\right)}}$
(4)

where $\Delta$ is the only free parameter which can take any values. The
symbol $z_{t}$ denotes the epoch where the matter energy density and the dark
energy density are equal, that means $\rho_{m}(z_{t})=\rho_{\rm
DE}(z_{t})\implies\Omega_{m0}(1+z_{t})^{3}=\widetilde{\Omega}_{\rm{DE}}(z_{t})$,
and hence, $z_{t}$ is a derived parameter, not a free parameter.

As we have a non-interacting scenario, therefore, using the conservation
equation for DE, namely, $\dot{\rho}_{\rm DE}+3H(1+w_{\rm DE}(z))\rho_{\rm
DE}=0$, where $w_{\rm DE}(z)$ is the barotropic equation of state for DE, can
be derived as

$w_{\rm
DE}(z)\,=\,\frac{1}{3}\frac{d\,{\rm{ln}}\,{\widetilde{\Omega}_{\rm{DE}}}}{dz}(1+z)-1$
(5)

which for the present model (4), $w_{\rm DE}$ turns out to be

$w_{\rm
DE}(z)=-1-\frac{\Delta}{3{\rm{ln}}\,10}\times\left({1+{\rm{tanh}}\left(\Delta\times{\rm{log}}_{10}(\frac{1+z}{1+z_{t}})\right)}\right).$
(6)

Thus, the cosmological scenario parametrized by $\tilde{\Omega}_{\rm DE}$ as
shown in eqn. (4) is boiled down to a parametrized dynamical dark energy
equation of state characterized by eqn. (5) or (6). With the expression for
$w_{\rm DE}$, the analysis of the model at the perturbation level becomes easy
and straightforward. Following Ma and Bertschinger (1995), one could derive
the velocity and density perturbations for the DE fluid characterized by the
dynamical equation-of-state parameter in eqn. (6).

In Figs. 1 and 2 we show how the parameter $\Delta$ affects the temperature
and the matter power spectra respectively, including the variation we have
with respect to the $\Lambda$CDM model corresponding to $\Delta=0$. We can
see, that increasing the value of $\Delta$ from $-2$ to $2$, we have a
suppression in the amplitude of the lower multipoles and a shift of the peaks
in the damping tail towards larger multipoles in the temperature power
spectrum, while we have an increase of the amplitude in the matter power
spectrum.

Figure 1: We show how the parameter $\Delta$ affects the temperature power
spectrum (top panel) and the variation we have with respect to the
$\Lambda$CDM model corresponding to $\Delta=0$ (bottom panel).

Figure 2: We show how the parameter $\Delta$ affects the matter power spectrum
(top panel) and the variation we have with respect to the $\Lambda$CDM model
corresponding to $\Delta=0$ (bottom panel).

##  III Observational data

Here we describe the main observational datasets and the statistical procedure
used to extract the observational constraints out of the present cosmological
scenario parametrized by the dark energy density with respect to the critical
energy density. In what follows we enlist the observational datasets:

  * •

CMB: We use the latest cosmic microwave background (CMB) observations from
Planck 2018 legacy release, i.e. plikTTTEEE+lowl+lowE Aghanim et al. (2020a,
b).

  * •

BAO: We use various measurements of the Baryon acoustic oscillation (BAO)
distance measurements from dFGS Beutler et al. (2011), SDSS MGS Ross et al.
(2015), and BOSS DR12 Alam et al. (2017).

  * •

$H_{0}$ from HST: We consider the Hubble constant estimation from the Hubble
Space Telescope measurements, using Cepheids as geometric distance
calibrators, which yields to the gaussian prior $H_{0}=74.03\pm 1.42$ km/s/Mpc
at $68\%$ CL Riess et al. (2019) (R19).

  * •

Supernovae Type Ia data: We also include the latest compilation from the
Pantheon catalog Scolnic et al. (2018), or alternatively, the Joint Light-
curve Analysis (JLA) Betoule et al. (2014) sample.

To extract the constraints of the GEDE scenario, we have modified the well
known CosmoMC package Lewis and Bridle (2002); Lewis et al. (2000). The
CosmoMC package is freely available from http://cosmologist.info/cosmomc/ and
this has a convergence diagnostic based on the Gelman-Rubin criterion Gelman
and Rubin (1992). Additionally, this also supports the Planck 2018 likelihood
Aghanim et al. (2020b). In Table 1, we have shown the flat priors imposed on
the free parameters of the present cosmological scenario during the
statistical analysis.

Parameter | prior
---|---
$\Omega_{\rm b}h^{2}$ | $[0.005,0.1]$
$\Omega_{\rm c}h^{2}$ | $[0.001,0.99]$
$100\theta_{MC}$ | $[0.5,10]$
$\tau$ | $[0.01,0.8]$
$n_{\mathrm{S}}$ | $[0.7,1.3]$
$\log[10^{10}A_{s}]$ | $[1.7,5.0]$
$\Delta$ | $[-10,10]$
Table 1: Flat priors on the free cosmological parameters of the GEDE scenario.  Parameters | Planck 2018 | Planck 2018+BAO | Planck 2018+R19 | Planck 2018+BAO+R19
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1198_{-0.0013-0.0027}^{+0.0013+0.0026}$ | $0.1198_{-0.0012-0.0025}^{+0.0012+0.0027}$ | $0.1201_{-0.0013-0.0027}^{+0.0013+0.0026}$ | $0.1203_{-0.0015-0.0029}^{+0.0015+0.0029}$
$\Omega_{b}h^{2}$ | $0.02240_{-0.00015-0.00028}^{+0.00014+0.00029}$ | $0.02238_{-0.00014-0.00028}^{+0.00014+0.00028}$ | $0.02238_{-0.00015-0.00029}^{+0.00015+0.00031}$ | $0.02239_{-0.00015-0.00029}^{+0.00015+0.00030}$
$100\theta_{MC}$ | $1.04095_{-0.00033-0.00060}^{+0.00032+0.00061}$ | $1.04096_{-0.00032-0.00061}^{+0.00031+0.00064}$ | $1.04094_{-0.00031-0.00062}^{+0.00033+0.00060}$ | $1.04090_{-0.00030-0.00059}^{+0.00031+0.00057}$
$\tau$ | $0.0530_{-0.0073-0.015}^{+0.0073+0.016}$ | $0.0546_{-0.0075-0.017}^{+0.0077+0.017}$ | $0.0541_{-0.0077-0.016}^{+0.0077+0.017}$ | $0.0544_{-0.0086-0.016}^{+0.0075+0.016}$
$n_{s}$ | $0.9655_{-0.0043-0.0088}^{+0.0046+0.0084}$ | $0.9659_{-0.0040-0.0087}^{+0.0045+0.0081}$ | $0.9651_{-0.0042-0.0084}^{+0.0042+0.0083}$ | $0.9644_{-0.0048-0.0086}^{+0.0044+0.0089}$
${\rm{ln}}(10^{10}A_{s})$ | $3.042_{-0.015-0.033}^{+0.015+0.033}$ | $3.045_{-0.016-0.034}^{+0.016+0.035}$ | $3.044_{-0.016-0.033}^{+0.016+0.033}$ | $3.046_{-0.017-0.029}^{+0.015+0.032}$
$\Delta$ | $>3.0\,>0.33$ | $0.26_{-0.40-0.73}^{+0.37+0.79}$ | $1.56_{-0.38-0.67}^{+0.35+0.72}$ | $0.85_{-0.41-0.79}^{+0.44+0.71}$
$\Omega_{m0}$ | $0.207_{-0.061-0.07}^{+0.021+0.10}$ | $0.304_{-0.012-0.023}^{+0.012+0.024}$ | $0.260_{-0.011-0.020}^{+0.010+0.020}$ | $0.285_{-0.0098-0.018}^{+0.0093+0.019}$
$\sigma_{8}$ | $0.958_{-0.043-0.148}^{+0.098+0.110}$ | $0.822_{-0.021-0.037}^{+0.018+0.038}$ | $0.876_{-0.017-0.034}^{+0.017+0.033}$ | $0.848_{-0.020-0.039}^{+0.023+0.036}$
$H_{0}$ | $85_{-6-17}^{+12+13}$ | $68.6_{-1.5-2.7}^{+1.3+2.9}$ | $74.3_{-1.4-2.7}^{+1.4+2.8}$ | $71.0_{-1.3-2.6}^{+1.4+2.5}$
$S_{8}$ | $0.785_{-0.035-0.050}^{+0.022+0.058}$ | $0.827_{-0.014-0.026}^{+0.013+0.028}$ | $0.814_{-0.015-0.029}^{+0.015+0.030}$ | $0.826_{-0.014-0.026}^{+0.014+0.027}$
$r_{\rm{drag}}$ | $147.10_{-0.30-0.58}^{+0.30+0.58}$ | $147.13_{-0.29-0.57}^{+0.30+0.57}$ | $147.06_{-0.29-0.56}^{+0.29+0.59}$ | $146.99_{-0.33-0.64}^{+0.32+0.64}$
Table 2: Observational constraints at 68% and 95% CL on the free and derived parameters of the GEDE scenario considering CMB from Planck 2018, and its combination with BAO and R19.  Parameters | Planck 2018+JLA | Planck 2018+Pantheon | Planck 2018+BAO+JLA+R19 | Planck 2018+BAO+Pantheon+R19
---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1202_{-0.0013-0.0026}^{+0.0013+0.0025}$ | $0.1203_{-0.0014-0.0026}^{+0.0014+0.0027}$ | $0.1201_{-0.0012-0.0024}^{+0.0012+0.0023}$ | $0.1199_{-0.0012-0.0024}^{+0.0012+0.0023}$
$\Omega_{b}h^{2}$ | $0.02235_{-0.00014-0.00029}^{+0.00015+0.00029}$ | $0.02236_{-0.00015-0.00029}^{+0.00015+0.00030}$ | $0.02238_{-0.00016-0.00028}^{+0.00014+0.00030}$ | $0.02240_{-0.00015-0.00029}^{+0.00014+0.00029}$
$100\theta_{MC}$ | $1.04090_{-0.00031-0.00062}^{+0.00031+0.00062}$ | $1.04092_{-0.00031-0.00059}^{+0.00031+0.00061}$ | $1.04096_{-0.00030-0.00060}^{+0.00031+0.00060}$ | $1.04095_{-0.00031-0.00061}^{+0.00031+0.00058}$
$\tau$ | $0.0544_{-0.0074-0.014}^{+0.0075+0.015}$ | $0.0541_{-0.0075-0.015}^{+0.0074+0.015}$ | $0.0543_{-0.0078-0.016}^{+0.0074+0.015}$ | $0.0550_{-0.0077-0.015}^{+0.0076+0.016}$
$n_{s}$ | $0.9647_{-0.0047-0.0084}^{+0.0043+0.0085}$ | $0.9647_{-0.0043-0.0089}^{+0.0047+0.0086}$ | $0.9649_{-0.0041-0.0079}^{+0.0041+0.0081}$ | $0.9656_{-0.0041-0.0080}^{+0.0041+0.0084}$
${\rm{ln}}(10^{10}A_{s})$ | $3.045_{-0.016-0.029}^{+0.015+0.031}$ | $3.045_{-0.016-0.031}^{+0.016+0.031}$ | $3.044_{-0.016-0.031}^{+0.015+0.032}$ | $3.046_{-0.016-0.030}^{+0.016+0.032}$
$\Delta$ | $0.30_{-0.40-0.73}^{+0.36+0.79}$ | $0.25_{-0.26-0.52}^{+0.26+0.51}$ | $0.69_{-0.25-0.48}^{+0.25+0.49}$ | $0.55_{-0.21-0.43}^{+0.20+0.42}$
$\Omega_{m0}$ | $0.305_{-0.015-0.029}^{+0.015+0.029}$ | $0.307_{-0.011-0.021}^{+0.011+0.021}$ | $0.289_{-0.0072-0.014}^{+0.0074+0.015}$ | $0.293_{-0.0067-0.013}^{+0.0065+0.013}$
$\sigma_{8}$ | $0.825_{-0.019-0.034}^{+0.017+0.037}$ | $0.823_{-0.013-0.028}^{+0.016+0.028}$ | $0.841_{-0.014-0.027}^{+0.014+0.027}$ | $0.834_{-0.012-0.025}^{+0.012+0.025}$
$H_{0}$ | $68.6_{-1.8-3.0}^{+1.5+3.3}$ | $68.3_{-1.1-2.1}^{+1.1+2.3}$ | $70.38_{-0.89-1.7}^{+0.87+1.8}$ | $69.86_{-0.74-1.4}^{+0.75+1.4}$
$S_{8}$ | $0.831_{-0.015-0.030}^{+0.015+0.031}$ | $0.832_{-0.015-0.029}^{+0.015+0.030}$ | $0.825_{-0.013-0.025}^{+0.013+0.026}$ | $0.824_{-0.013-0.025}^{+0.014+0.025}$
$r_{\rm{drag}}$ | $147.05_{-0.31-0.55}^{+0.28+0.58}$ | $147.03_{-0.30-0.57}^{+0.30+0.58}$ | $147.05_{-0.28-0.54}^{+0.27+0.53}$ | $147.10_{-0.27-0.54}^{+0.27+0.52}$
Table 3: Here we present the constraints on the same GEDE scenario but
focusing on the effects of JLA and Pantheon on the model parameters.  Figure
3: We show the triangular plot containing 1-dimensional marginalized posterior
distributions and the 2-dimensional joint contours for some important
parameters of the GEDE scenario considering the datasets Planck 2018, Planck
2018+BAO, Planck 2018+R19 and Planck 2018+BAO+R19.  Figure 4: Here we show the
triangular plot for the GEDE scenario for Planck 2018+Pantheon and Planck
2018+JLA datasets in order to predict mainly the individual effects of
Pantheon and JLA.

##  IV Observational Results

Let us summarize the main results of the GEDE model that we find using a
variety of cosmological probes. In Tables 2 and 3, we have shown the 68% and
95% CL constraints on the free and derived parameters of the GEDE scenario
using the CMB spectra from Planck 2018 and its combination with other external
cosmological probes.

In the second column of Table 2, we show the results from Planck 2018 alone
dataset. We find that the Hubble constant assumes a much larger mean value
than R19, but with relaxed error bars, solving the tension within $2\sigma$.
Due to the anti-correlation between $H_{0}$ and $\Omega_{m0}$, $\Omega_{m0}$
assumes a very small value ($\Omega_{m0}=0.207_{-0.061}^{+0.021}$ at 68% CL).
The only free parameter of this model, $\Delta$, has an upper bound
($\Delta>3$ at 68% CL and ($\Delta>0.33$ at 95% CL) and this upper bound
implies that the $\Lambda$CDM model (for which $\Delta=0$) is excluded at more
than 95% CL, while the PEDE model (for which $\Delta=1$) at more than 68% CL.
Thus, as we see, Planck 2018 data rule out the $\Lambda$CDM scenario. If we
compare this full Planck 2018 analysis with the previous work Li and Shafieloo
(2020) without perturbations, we can see that the $H_{0}$ constraint changes
significantly, and now it is more in agreement with R19. The very same thing
happens for $\Delta$, highlighting the importance of a robust data analysis.
This means that we are playing here with the $\Delta-H_{0}$ positive
degeneracy (see Fig. 3), and introducing perturbations, we allow for larger
values of both $\Delta$ and $H_{0}$ at the same time, as well as a lower value
of $S_{8}$ solving also the tension with the cosmic shear data.

When the BAO data are added to the CMB from Planck 2018 (see the third column
of Table 2), $H_{0}$ goes down and the existing tension with R19 is now
reduced to $2.8\sigma$. Recall that for Planck 2018 alone, the tension of
$H_{0}$ with R19 was alleviated within $2\sigma$. The free parameter $\Delta$
is well constrained and the scenario now becomes consistent with the
$\Lambda$CDM model within $1\sigma$ ($\Delta=0.26_{-0.40}^{+0.37}$ at 68% CL)
and with PEDE within $2\sigma$.

For Planck 2018+R19, $\Delta\neq 0$ at more than 95% CL, that means,
$\Lambda$CDM is ruled out at more than two standard deviations. However, we
are now in agreement with $\Delta=1$, the PEDE scenario, within $2\sigma$.
Interestingly, when we consider the full Planck 2018+BAO+R19 combination,
shown in the last column of Table 2, not only $\Lambda$CDM is ruled out at
more than 95% CL, but this model is in agreement within $1\sigma$ with
$\Delta=1$, this means that PEDE is the favored model by this combination of
data.

At this point, we can discuss the impacts on the constraints when JLA and
Pantheon datasets are separately added to the CMB. As expected we have similar
results for these combinations (see Fig. 4), with only marginal differences,
and these are shown in Table 3. For both Planck 2018+JLA and Planck
2018+Pantheon (see Fig. 4), $\Delta=0$ is consistent within $1\sigma$, and
thus recovering the $\Lambda$CDM model, but while Planck 2018+JLA is in
agreement with PEDE $\Delta=1$ within the 95% CL, Planck 2018+Pantheon is
ruling out this model at more than $2\sigma$. Similar considerations hold for
the Hubble constant tension: both the datasets combination give a similar mean
value for $H_{0}$, but Planck 2018+Pantheon has stronger error bars. This
translates in a $2.6\sigma$ tension for Planck 2018 + JLA, and a $3.2\sigma$
tension for Planck 2018+Pantheon with respect to R19.

For the two full dataset combinations, namely, Planck 2018+BAO+JLA+R19 and
Planck 2018+BAO+Pantheon+R19, $\Delta$ is found to be different from zero at
more than 95% CL. This result is mainly driven by the R19 dataset as one can
quickly realize by looking at various datasets in Tables 2 and 3. However, as
for the previous cases, Planck 2018+BAO+JLA+R19 is in agreement with PEDE
(corresponds to $\Delta=1$) within the 95% CL, but Planck
2018+BAO+Pantheon+R19 is ruling out PEDE too at more than $2\sigma$.

Finally, we compute the Bayesian evidence summarizing the results in Table 4,
to quantify which model performs better in the fit of the data between
$\Lambda$CDM and GEDE. We use the publicly available cosmological package
MCEvidence111github.com/yabebalFantaye/MCEvidence Heavens et al. (2017a, b).,
and compute the evidences in terms of the Bayes factor $\ln B_{ij}$ where its
negative (positive) values indicate a preference of the data for the
$\Lambda$CDM (GEDE) scenario. To interpret the numerical values of $\ln
B_{ij}$, we use instead the the revised Jeffreys scale by Kass and Raftery as
in Ref. Kass and Raftery (1995), for which if $0\leq|\ln B_{ij}|<1$ the model
has a weak evidence, if $1\leq|\ln B_{ij}|<3$ the model has a definite
evidence, if $3\leq|\ln B_{ij}|<5$ the model has a strong evidence, and if
$|\ln B_{ij}|\geq 5$ the model has a very strong evidence. Looking at the
Table 4 we can see that GEDE model is mostly preferred by the fitting of the
data, with the exception of the cases CMB+SNIa (Pantheon or JLA). In
particular, we find a Positive evidence for GEDE against $\Lambda$CDM for
Planck 2018, a weak evidence for Planck 2018+BAO, and a very strong evidence
for all the dataset combinations including R19.

Data | $\ln B_{ij}$
---|---
Planck 2018 | $2.9$
Planck 2018+BAO | $0.8$
Planck 2018+R19 | $12.1$
Planck 2018+BAO+R19 | $7.9$
Planck 2018+JLA | $-0.2$
Planck 2018+Pantheon | $-0.9$
Planck 2018+BAO+JLA+R19 | $6.1$
Planck 2018+BAO+Pantheon+R19 | $5.8$
Table 4: The table shows the values of $\ln B_{ij}$ calculated for the GEDE
model with respect to the $\Lambda$CDM as the reference model. The negative
value in $\ln B_{ij}$ indicates that $\Lambda$CDM is preferred over GEDE while
the positive sign denotes the opposite.

##  V Concluding remarks

No doubt, $\Lambda$CDM has received notable attention over the years for its
ability to fit most of the observational data, but the cosmological constant
problem has still remained unanswered. Even if we forget the cosmological
constant issue, some recent observations indicate that we cannot totally rely
on the $\Lambda$CDM driven cosmology – the measurements of some key
cosmological parameters, namely $H_{0}$ and $\sigma_{8}$, within this
paradigm, do not match with their measurements made by other astronomical
probes, and are many sigmas apart from one another. This is widely known as
the cosmological tension. These discrepancies at many standard deviations
cannot be avoided and, if not due to systematic errors in the experiments,
they actually hint for some limitations of the $\Lambda$CDM cosmology.
Therefore, modifications of the $\Lambda$CDM scenario are welcomed, since
there should not be any reason to favor it over others. If a new cosmological
model can solve those tensions, or at least one of the tensions, then the
model should be investigated robustly in order to understand how it fits
overall with the expansion history of the Universe. However, it is equally
true that the cosmological models beyond $\Lambda$CDM usually contain some
extra free parameters than six (number of free parameters in a flat
$\Lambda$CDM model), and for this reason, sometimes, the model comparison
disfavours them in comparison to the standard $\Lambda$CDM model. In this
context we recall PEDE Li and Shafieloo (2019) having exactly six free
parameters. The model can excellently solve the $H_{0}$ tension within
$1\sigma$ Pan et al. (2020a) and for some combination of the cosmological
probes, this model is preferred by the data over the $\Lambda$CDM model as
well. Having such excellent features, an extension of this model, called
generalized emergent dark energy model, was proposed and investigated Li and
Shafieloo (2020). The generalized emergent dark energy model introduces one
extra free parameter, $\Delta$, compared to the PEDE model, that can
discriminate between the $\Lambda$CDM model (recovered for $\Delta=0$) and the
PEDE model (recovered for $\Delta=1$). In this way we can determine which
model is preferred by the fit of the data.

We have constrained the present GEDE scenario considering its evolution at the
level of linear perturbations and with the use of latest cosmological probes
from Planck 2018, BAO, SNIa (both JLA and Pantheon samples) and R19. The
results are summarized in Table 2 and 3. We find that, while the $\Lambda$CDM
model is ruled out at more than 95% CL for most of the cases, with the
exception of Planck 2018+BAO or Planck 2018+SNIa, PEDE is in agreement with
the data for Planck 2018+BAO+R19 combination within $1\sigma$, and with Planck
+ R19, Planck 2018+JLA, and Planck 2018+JLA+BAO+R19 within $2\sigma$.

The model has been also assessed through the Bayesian Evidence analysis from
which we find a very strong evidence for the GEDE model against $\Lambda$CDM
for most of the datasets combinations (see Table 4). Therefore, the present
work clearly tells us that the cosmological models which can reconcile the
$H_{0}$ tension with an excellent fitting of the data are certainly appealing
even if they are motivated from the phenomenological ground. Last but not
least, we must say that there is no reason to exclude the phenomenology for
two reasons: firstly, the DE sector is totally unknown and secondly, since we
have a lot of observational data, hence, we have enough space to judge the
viability of the model arising from the phenomenological ground. A model can
be thrown at any time but before throwing it, one needs to replace it with
something good. Until we win this game, we have to be open minded.

###### Acknowledgements.

WY was supported by the National Natural Science Foundation of China under
Grants No. 11705079 and No. 11647153. EDV acknowledges the support of the
Addison-Wheeler Fellowship awarded by the Institute of Advanced Study at
Durham University. SP gratefully acknowledges the Mathematical Research
Impact-Centric Support Scheme (File No. MTR/2018/000940) funded by the Science
and Engineering Research Board, Govt. of India,. AS would like to acknowledge
the support of the Korea Institute for Advanced Study (KIAS) grant funded by
the Korea government. X. Li is supported by National Natural Science
Foundation of China under Grants Nos. 12003006, Hebei NSF under Grant No.
A202005002.

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# Relative Switching Point based Dynamic Positional Representation for Code-
Mixed Sentiment Analysis

Mohsin Ali1 Kandukuri Sai Teja 1 Sumanth Manduru1
Parth Patwa2 Amitava Das 3, 4

# PESTO: Switching Point based Dynamic and Relative Positional Encoding for
Code-Mixed Languages

Mohsin Ali1 Kandukuri Sai Teja 1 Sumanth Manduru1
Parth Patwa2 Amitava Das 3, 4

###### Abstract

NLP applications for code-mixed (CM) or mix-lingual text have gained a
significant momentum recently, the main reason being the prevalence of
language mixing in social media communications in multi-lingual societies like
India, Mexico, Europe, parts of USA etc. Word embeddings are basic building
blocks of any NLP system today, yet, word embedding for CM languages is an
unexplored territory. The major bottleneck for CM word embeddings is switching
points, where the language switches. These locations lack in contextually and
statistical systems fail to model this phenomena due to high variance in the
seen examples. In this paper we present our initial observations on applying
switching point based positional encoding techniques for CM language,
specifically Hinglish (Hindi - English). Results are only marginally better
than SOTA, but it is evident that positional encoding could be an effective
way to train position sensitive language models for CM text.

## Switching Points: The Bottleneck

Switching Points (SPs) are the positions in CM text, where the language
switches. Consider the text - aapHI seHI requestEN hainHI (request you to).
Here, when the language switches from Hindi to English (seHI requestEN) a HI-
EN (HIndi-ENglish) SP occurs. Correspondingly, a EN-HI SP occurs at requestEN
hainHI. In this work we to look at sentiment analysis of CM languages,
specifically Hinglish through the lens of language modeling. We propose PESTO
- a switching point based dynamic and relative positional encoding. PESTO
learns to emphasis on switching points in CM text. Our model marginally
outperforms the SOTA.

## Background - Dataset and Positional Encoding

Data and SOTA: The SentiMix task @ SemEval 2020 Patwa et al. (2020) released
20K Hinglish tweets, are annotated with word-level languages and sentence-
level sentiment i.e.  positive, negative, neutral. Liu et al. (2020a) achieved
the SOTA (75% f1 score) by fine-tuning a pre-trained XLM-R using adversarial
training.

Vaswani et al. (2017) introduced Positional Encoding (PE) for language
modeling. PE serves as an added feature along with the word embeddings,
providing both relative and absolute positional relations between a target
word and its context words.

### Absolute Positional Encoding (APE)

Sinusoidal PE: \- A predefined sinusoidal vector $p_{i}\in R^{d}$ is assigned
to each position i. This $p_{i}$ is added to the word embedding $w_{i}\in
R^{d}$ at position i, and $w_{i}+p_{i}$ is used as input to the model. In this
way, the Transformer can differentiate the words coming from different
positions and assign each token position-dependent attention Vaswani et al.
(2017). Sin/cos functions are used interchangeably to capture odd/even
numbered positional words in a sequence - equation 1

$\alpha_{ij}^{abs}=\frac{1}{\sqrt{d}}\left(\left(w_{i}+p_{i}\right)W^{Q,1}\right)\left(\left(w_{j}+p_{j}\right)W^{K,1}\right)^{T}$
(1)

Dynamic PE: \- Instead of using periodical functions like $sin/cos$, Liu et
al. (2020b), proposed to learn a dynamic function at every encoder layer that
can represent the positional info. A function $\theta(i)$ is introduced which
can learn positional info with gradient flow. - equation 2

$\alpha_{ij}=\frac{1}{\sqrt{d}}\left(\left(w_{i}+\theta\left(i\right)\right)W^{Q,1}\right)\left(\left(w_{j}+\theta\left(j\right)\right)W^{K,1}\right)^{T}$
(2)

### Relative Positional Encoding (RPE)

Shaw, Uszkoreit, and Vaswani (2018) introduced a learnable parameter
$a_{j-i}^{l}$ which learns the positional representation of the relative
position j-i at encoder layer l. This helps the model to capture relative word
orders explicitly - equation 3

$\alpha_{ij}^{rel}=\frac{1}{\sqrt{d}}\left(\left(w_{i}\right)^{l}W^{Q,l}\right)\left(\left(w_{i}\right)^{l}W^{K,l}+a_{j-i}^{l}\right)^{T}$
(3)

## Switching Point based Positional Encoding

We introduce a novel, switching point based PE. Consider the Hinglish sequence
- gaayeHI aurHI danceEN kareHI. SP based indices (SPI) - i) We set the index
to 0 whenever an SP occurs. Indexing would normally be = $\\{0,1,2,3\\}$, we
change it to $\\{0,1,0,0\\}$. ii) We consider Hindi as our base language and
English as the mixed language. We set the index to 0 only when the shift is
from base language (L1) to the mixed language (L2). So, the resultant index
would be $\\{0,1,0,1\\}$.

### Switching Point based Dynamic PE (SPDPE)

We introduce a function $S(l_{i})$, which takes the word level language labels
as input and returns SPI. Instead of passing an index directly as i to
$\theta$, we use $\theta(s(l_{i}))$ to dynamically learn the PE based on SPI -
equation 4

$\alpha_{ij}=\frac{1}{\sqrt{d}}\left(\left(w_{i}+\theta\left(S(l_{i})\right)\right)W^{Q,1}\right)\\\
\left(\left(w_{j}+\theta\left(S(l_{j})\right)\right)W^{K,1}\right)^{T}$

(4)

### PESTO - Switching Point based Dynamic and Relative PE (SPDRPE)

Here, in addition to the SPDPE, we use a learning parameter $a_{j-i}^{l}$,
which encodes the relative position j-i at the encoder layer l. This encoding
approach learns representations dynamically based on SPs along with the
embedding $a_{j-i}^{l}$ so that it can also capture relative word orders
(equation 5).

$\alpha_{ij}=\frac{1}{\sqrt{d}}\left(\left(w_{i}+\theta\left(S(l_{i})\right)\right)^{l}W^{Q,l}\right)\left(\left(w_{j}+\theta\left(S(l_{j})\right)\right)^{l}W^{K,l}+a_{j-1}^{l}\right)^{T}$

(5)

## Models

Baselines - Word2Vec, Multi Head Attention (MHA): We choose Word2Vec as the
baseline since it does not capture position info. We also choose attention
mechanism, which is widely used to capture relational dependencies, to see its
effects over SPs. We experiment with two lengths - i) Length 3 to capture the
local window of dependency, whereas, ii) 12 to see whether it can learn
anything from the whole sentence. 12 is the average length of sentences in our
corpus.

PESTO Overall Architecture: The local dependencies from skipgram Word2Vec
(trained from scratch) along with SPI obtained from SPDRPE are passed to a 12
headed transformer based encoder layer. On top of the transformer, a 1D CNN is
used to get the sentence level representation. We also obtain the sentence
embedding using tf-idf weighted average of Word2Vec embeddings. Finally, we
concatenate the representations of the CNN and the tf-idf sentence embedding
and pass it to a dense layer which applies softmax to predict the sentiment.
The architecture of PESTO is shown in Fig. 1. We train the entire model (2
encoder layers) from scratch, without using any pre-trained model.

## Results

PESTO achieves 75.56% F1 score and outperforms SOTA (Tab. 1). The main reason
for this is learning SP by aggregating both relative and dynamic PE with a
variable length MHA framework. PESTO is able to generate more thrust to the
switching point weatherEN achaaHI (Fig. 2). The experiments were conducted on
google Colab. The code is available at
https://github.com/mohammedmohsinali/PESTO.

Figure 1: PESTO - Proposed model for Relative Switching Point based Dynamic
Positional Representation for CM text.

Models | Positional Representation | F1 (%)
---|---|---
Sin/Cos | Index | SPI | Relative |
Word2Vec + LSTM | ✗ | ✗ | ✗ | ✗ | 56
ELMO | ✗ | ✗ | ✗ | ✗ | 55
BERT | ✓ | ✗ | ✗ | ✗ | 60
3HA with Sinusoidal PE | ✓ | ✗ | ✗ | ✗ | 65
3HA with Dynamic PE | ✗ | ✓ | ✗ | ✗ | 69.7
12HA with Dynamic PE+RPE | ✗ | ✓ | ✗ | ✓ | 73
12HA with RPE | ✗ | ✗ | ✗ | ✓ | 73.4
12HA with SPDPE | ✗ | ✗ | ✓ | ✗ | 73.52
SOTA (Liu et al. 2020a) | ✓ | ✗ | ✗ | ✗ | 75
PESTO (12HA with SPDRE) | ✗ | ✗ | ✓ | ✓ | 75.56

Table 1: Results of various position sensitive experiments on CM text. It is
evident from the results that positional encoding could be an effective way to
train position sensitive language models for CM text.

(a) Sinusoidal PE

(b) Dynamic PE

(c) PESTO

Figure 2: PESTO not only differentiates words coming from different positions,
but also pays high attention to the SPs like weatherEN and achaaHI.

## Conclusion

In this paper we report initial experiments on Hinglish sentiment analysis
problem through the lens of language modeling. We argued SPs are the major
bottleneck for CM. Our contribution could be seen as following - i) We
introduce the idea of switching-point based positional encoding. i) We propose
a relative switching point dynamic positional encoding technique named PESTO,
which yields better results than SOTA. iii) It is also noteworthy that our
model - PESTO achieves SOTA results without any pre-trained heavy language
model, whereas all the SOTA models in the SentiMix task used models like BERT,
or XLNet.

## References

  * Liu et al. (2020a) Liu, J.; et al. 2020a.  kk2018 at SemEval-2020 Task 9: Adversarial Training for Code-Mixing Sentiment Classification.  In _SemEval_.
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  * Patwa et al. (2020) Patwa, P.; et al. 2020.  SemEval-2020 Task 9: Overview of Sentiment Analysis of Code-Mixed Tweets.  In _SemEval 2020_.
  * Shaw, Uszkoreit, and Vaswani (2018) Shaw, P.; Uszkoreit, J.; and Vaswani, A. 2018.  Self-Attention with Relative Position Representations.  In _NAACL 2018_.
  * Vaswani et al. (2017) Vaswani, A.; et al. 2017.  Attention Is All You Need.  In _NeurIPS_.