11institutetext:  Université Paris-Saclay, CNRS, Institut d’Astrophysique
Spatiale, Bâtiment 121, 91405 Orsay, France 22institutetext: Kavli IPMU (WPI),
UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
22email<EMAIL_ADDRESS>
# Can decaying dark matter models be a solution to the $S_{8}$ tension from
the thermal Sunyaev-Zel’dovich effect?

Hideki Tanimura 1122 Marian Douspis 11 Nabila Aghanim 11 Joseph Kuruvilla 11

Considering possible solutions to the $S_{8}$ tension between the Planck
cosmic microwave background (CMB) measurement and low-redshift probes, we
extended the standard $\Lambda$CDM cosmological model by including decay of
dark matter (DDM). We first tested the DDM model in which dark matter decays
into a form of non-interacting dark radiation. Under this DDM model, we
investigated the DDM’s impacts on the Sunyaev Zel’dovich effect by varying the
decay lifetime, $\Gamma^{-1}$, including the background evolution in cosmology
and non-linear prescription in the halo mass function. We performed a
cosmological analysis under the assumption of this extended cosmological model
by combining the latest high-redshift Planck CMB measurement and low-redshift
SZ measurement. Our result shows a preference for $\Gamma^{-1}\sim 200$ Gyr
with a lower bound on the decay lifetime of $\sim$ 77 Gyr at 95% confidence
level. However, the CMB data do not prefer this model, and the $S_{8}$ tension
still remains. Additionally, we tested the other DDM model in which dark
matter decays into warm dark matter and dark radiation. This model supports
$\Gamma^{-1}\sim 204$ Gyr to resolve the $S_{8}$ tension with a lower bound on
the decay lifetime of $\sim$ 47 Gyr at 95% confidence level.

###### Key Words.:

galaxies: clusters: general - intracluster medium - Cosmology: large-scale
structure of Universe - dark matter - cosmic background radiation

##  1 Introduction

The current standard cosmological model, called the $\Lambda$ Cold Dark Matter
($\Lambda$CDM) model, has been supported by many observational results at
different epochs, such as Cosmic Microwave Background (CMB), Big Bang
Nucleosynthesis (BBN), and luminosity distances to SuperNovae of type Ia
(SNIa). However, as the measurement precision has increased over the past few
years, the validity of the $\Lambda$CDM model is reexamined, particularly due
to a discrepancy in the $S_{8}(\equiv\sigma_{8}(\Omega_{m}/0.3)^{0.5}$
parameter: the amplitude of matter density fluctuations, $\sigma_{8}$, scaled
by the square root of the matter density, $\Omega_{m}$. For example, Planck
measured $S_{8}=0.830\pm 0.013$ with the CMB anisotropies ($z\sim 1100$)
(Planck Collaboration, 2020a), but low-redshift ($z\sim 0-1$) cosmological
probes, such as the abundance of galaxy clusters by Planck (Planck
Collaboration, 2014, 2016b) and gravitational lensing by Kilo Degree Survey
(KiDS) (Heymans et al., 2021) and Dark Energy Survey (DES) (DES Collaboration,
2021), show a preference for a lower value of $S_{8}\sim 0.76-0.78$,
representing up to a $\sim 2-3\sigma$ tension with the CMB constraints, this
is the so-called $S_{8}$ tension. This tension indicates that the growth rate
of the cosmic structure is less than expected from the CMB measurements and
may demand modifications to the standard model.

Many attempts have been made to solve the $S_{8}$ tension by breaking
assumptions in the standard cosmological model, mainly on the nature of
neutrinos (e.g., Salvati et al. 2018; Tröster et al. 2021), dark energy (e.g.,
Bolliet et al. 2018; Salvati et al. 2018; Tröster et al. 2021; Lambiase et al.
2019), and modified gravity (e.g., Ilić et al. 2019). However, none of the
proposed solutions have gained wide acceptance so far. One possible solution
is the decaying dark matter (DDM) model (e.g., Aoyama et al. 2014; Audren et
al. 2014; Enqvist et al. 2015; Poulin et al. 2016; Enqvist et al. 2020;
Abellán et al. 2020, 2021; Chen et al. 2021; Hubert et al. 2021; Fuß & Garny
2022; Brinch Holm et al. 2022; McCarthy & Hill 2022). The DDM model has two
features that reduce the cosmic structure growth and could naturally explain
the $S_{8}$ tension. First, the cosmic structure growth is reduced by the
decay of massive dark matter into lighter or massless daughter particles.
Second, the DDM model can only change the structure formation scenario at low
redshift, probed by clusters of galaxies, lensing, etc., without modifying the
one at high redshift, probed by CMB, before dark matter decays.

While several scenarios can be considered for the decay modes, we consider two
scenarios: one scenario in which dark matter (DM) decays into a form of non-
interacting dark radiation (DR) (hereafter $\Lambda$DDM1 model) (e.g.,, Audren
et al. 2014; Enqvist et al. 2015; Poulin et al. 2016; Enqvist et al. 2020) and
the other scenario where DM decays into one massless DR component and one
massive Warm DM (WDM) particle (hereafter $\Lambda$DDM2 model) that interacts
only through gravity with the standard model particles (e.g.,, Aoyama et al.
2014; Abellán et al. 2020, 2021). The $\Lambda$DDM1 model can be parameterized
with the decay rate, $\Gamma$ ($\Gamma^{-1}$ represents the DM’s lifetime), in
addition to the cosmological parameters under the standard $\Lambda$CDM model.
The DDM2 model can be parameterized with two parameters: $\Gamma$ and
$\varepsilon$ that represents the mass-energy fraction transferred to the
massless component of DR (Aoyama et al., 2014; Abellán et al., 2020, 2021).
Many studies were performed for the simplest scenario of $\Lambda$DDM1 model
and tight constraints were put on the DM particle’s lifetime of $\Gamma^{-1}$
¿ 160 Gyr (e.g., Audren et al. 2014; Enqvist et al. 2015; Poulin et al. 2016;
Enqvist et al. 2020), which is much longer than the current age of the
Universe. Extended DDM models was considered in fewer studies, such as
$\Lambda$DDM2 scenario. For example, Abellán et al. (2020) showed that the
$\Lambda$DDM2 can fully explain the low-$S_{8}$ measurement with a shorter
lifetime of $\Gamma^{-1}\simeq 55\,(\varepsilon/0.007)^{1.4}$ Gyr using the
Planck CMB measurement (Planck Collaboration, 2020a), combined with the
Baryonic Acoustic Oscillations (BAO) measurements from 6-degree Field Galaxy
Survey (6dF) (Beutler et al., 2011), Sloan Digital Sky Survey (SDSS) (Ross et
al., 2015), Baryon Oscillation Spectroscopic Survey (BOSS) (Alam et al.,
2017), and Extended Baryon Oscillation Spectroscopic Survey (eBOSS) (Blomqvist
et al., 2019; de Sainte Agathe et al., 2019), including the SNIa measurement
from Pantheon (Scolnic et al., 2018). Further study was performed in Abellán
et al. (2021), including KiDS and DES weak lensing data (Abbott et al., 2018;
Joudaki et al., 2020; Heymans et al., 2021), and the authors concluded a
preference for the $\Lambda$DDM2 model than the standard $\Lambda$CDM model
when KiDS and DES weak lensing data were included, without degrading the fit
to other cosmological datasets such as CMB, BAO and SNIa measurements.

In this paper, we use the thermal Sunyaev-Zel’dovich (SZ) effect (Sunyaev &
Zeldovich, 1970, 1972) as a probe for the $S_{8}$ tension. To investigate the
$S_{8}$ parameter, the SZ effect, caused by the inverse Compton scattering of
CMB photons by hot electrons along the line of sight, is a useful probe
because the SZ power spectrum, $C_{\ell}^{\rm SZ}$, is sensitive to the
$\sigma_{8}$ and $\Omega_{\rm m}$ cosmological parameters as $C_{\ell}^{\rm
SZ}\propto\sigma_{8}^{8}\Omega_{\rm m}^{3}$ (Planck Collaboration, 2016a;
Salvati et al., 2018). The $S_{8}$ tension was first exhibited by cosmological
analysis of the SZ cluster number counts from Planck (Planck Collaboration,
2014). Recent SZ results (Planck Collaboration, 2016a; Tanimura et al., 2022)
showed a similar amplitude of $S_{8}$ value to the weak-lensing and cluster-
count observations, confirming a tension with the Planck’s CMB result. As a
solution for the $S_{8}$ tension, we assume that the origin of the tension is
caused by the decay of dark matter and constrain the cosmological parameters
and the DDM models using the latest SZ power spectrum measurement in Tanimura
et al. (2022) (hereafter T22). The present paper is structured as follows.
Section 2 describes the model of the SZ power spectrum, including the DDM
model. Section 3 presents the constraint on the DDM model through our
cosmological analysis. We finally end with the conclusions in Sect. 4.

##  2 Theoretical background

In this section, we first describe the theoretical prescription of the SZ
effect, in particular the SZ angular power spectrum under the $\Lambda$CDM
model by following T22, and then modify it including the DDM models.

###  2.1 Compton $y$ parameter

The Compton $y$ parameter is proportional to the line-of-sight integral of
electron pressure, $P_{\rm e}=n_{\rm e}k_{\rm B}T_{\rm e}$, where $n_{\rm e}$
is the physical electron number density, $k_{\rm B}$ is the Boltzmann
constant, and $T_{\rm e}$ is the electron temperature. In an angular
direction, $\bm{\hat{n}}$, it is expressed by

$y(\bm{\hat{n}})=\frac{\sigma_{\rm T}}{m_{\rm e}{\it c}^{2}}\int P_{\rm
e}(\bm{\hat{n}})\,{\rm d}l\ ,$ (1)

where $\sigma_{\rm T}$ is the Thomson cross section, $m_{\rm e}$ is the mass
of electron, $c$ is the speed of light, and $l$ is the physical distance. The
change to the CMB temperature by the SZ effect, $\Delta T$, at frequency $\nu$
is given by

$\frac{\Delta T}{T_{\rm CMB}}(\nu,\bm{\hat{n}})=f(x)\,y(\bm{\hat{n}}),$ (2)

where $T_{\rm CMB}$ is the CMB temperature. The frequency dependence of the SZ
effect is included in the pre-factor $f(x)$ as

$f(x)=x\
\mathrm{coth}\left(\frac{x}{2}\right)-4\quad\left(x=\frac{h\nu}{k_{\rm
B}T_{\rm CMB}}\right)$ (3)

in the thermodynamic temperature unit, where $h$ is the Planck constant.

###  2.2 SZ angular power spectrum

The SZ power spectrum can be modeled with a halo model (Cooray & Sheth, 2002).
The halo model consists of ‘one-halo term’, which accounts for the correlation
arising within an individual halo, and ‘two-halo term’, which accounts for the
correlation arising due to the environment surrounding a halo (Komatsu et al.,
2002; Cooray & Sheth, 2002). In this paper, we consider only the one-halo term
because the two-halo term’s contribution is minor to the total SZ power
spectrum (Komatsu & Kitayama, 1999) at scales we consider ($\ell>60$). Then,
the SZ power spectrum is given by

$C^{SZ}_{\ell}=\int{\rm d}z\frac{{\rm d}^{2}V}{{\rm d}z{\rm d}\Omega}\int{\rm
d}M\frac{{\rm d}n(M,z)}{{\rm d}M}\,|\tilde{y}_{\ell}(M,z)|^{2},$ (4)

where ${\rm d}^{2}V/{\rm d}z{\rm d}\Omega$ is the comoving volume element per
redshift per steradian, $n(M,z)$ is the comoving number density of halos of
mass $M$ and redshift $z$, called the halo mass function (HMF), and
$\tilde{y}_{\ell}(M,z)$ is the 2D Fourier transform of the $y$-profile of a
halo, given by

$\tilde{y}_{\ell}(M,z)=\frac{\sigma_{\rm T}}{m_{\rm e}{\it c}^{2}}\frac{4\pi
r_{\rm s}}{\ell^{2}_{\rm s}}\int{\rm d}x_{\rm r}x_{\rm r}^{2}\frac{\sin(\ell
x_{\rm r}/\ell_{\rm s})}{\ell x_{\rm r}/\ell_{\rm s}}P_{\rm e}(x,M,z)\ ,$ (5)

where $r_{\rm s}$ is the characteristic scale radius of the pressure profile
and $x_{\rm r}=r/r_{\rm s}$ is the dimensionless radial scale. $d_{\rm A}$ is
the angular diameter distance and $\ell_{\rm s}=d_{\rm A}/r_{\rm s}$ is the
associated multipole moment. We integrate the contribution of halos in the
redshift range from 0 to 3 and the mass range from $10^{13}$ ${\rm M}_{\odot}$
to $5\times 10^{15}$ ${\rm M}_{\odot}$ as well as the scaled radial distance,
$x_{\rm r}$, in the range from 0 to 5, following T22.

###  2.3 Halo mass function

We use the mass function from Tinker et al. (2008), as in T22, in which the
number of halos per unit volume is given by

$\frac{{\rm d}n}{{\rm d}M}=f(\sigma)\frac{\rho_{m,0}}{M}\frac{{\rm d}{\rm
ln}\sigma^{-1}}{{\rm d}M},$ (6)

where $\rho_{m,0}$ is the matter density at $z=0$ and $f(\sigma)$ is given by

$f(\sigma)=A\left[1+\left(\frac{\sigma}{b}\right)^{-a}\right]{\rm
exp}\left(-\frac{c}{\sigma^{2}}\right),$ (7)

where $A,a,b,c$ are constants calibrated with simulations in Tinker et al.
(2008). $\sigma$ is the standard deviation of density perturbations in a
sphere of radius $R=(3M/4\pi\rho_{m,0})^{1/3}$, and it is given by

$\sigma^{2}=\frac{1}{2\pi^{2}}\int{\rm d}kk^{2}P_{\rm m}(k,z)|W(kR)|^{2},$ (8)

where $W(kR)$ is the window function of a spherical top hat of radius $R$.

###  2.4 Universal pressure profile

For the electron pressure profile, we use the model from Planck Collaboration
(2013), used as a fiducial model in T22. It adopts the ‘universal’ pressure
profile (UPP; Nagai et al., 2007), which is a form of generalized Navarro,
Frenk, & White (NFW; 1997) profile (gNFW),

$\mathbb{P}(x_{\rm 500})=\frac{P_{0}}{(c_{500}x_{\rm
500})^{\gamma}[1+(c_{500}x_{\rm 500})^{\alpha}]^{(\beta-\alpha)/\gamma}}\ .$
(9)

Here, $x_{\rm 500}=r/R_{500}$ and we remind the reader that $R_{500}$ relates
to $500$ times the critical density. The model is defined by the following
parameters: $P_{0}$, normalization; $c_{500}$, concentration parameter defined
at a characteristic radius $R_{500}$; and the slopes in the central $(x_{\rm
500}\ll 1/c_{500})$, intermediate $(x_{\rm 500}\sim 1/c_{500})$ and outer
regions $(x_{\rm 500}\gg 1/c_{500})$, given by $\gamma$, $\alpha$ and $\beta$,
respectively. The scaled pressure profile for a halo with $M_{500}$ and $z$ is

$\frac{P(r)}{P_{500}}=\mathbb{P}(x_{\rm 500}),$ (10)

with

$\displaystyle P_{500}$ $\displaystyle=1.65\times
10^{-3}\left[\frac{H(z)}{H_{0}}\right]^{8/3}$ (11)
$\displaystyle\times\left[\frac{(1-b)\,M_{500}}{3\times
10^{14}\,(h/0.7)^{-1}{\rm M}_{\odot}}\right]^{2/3+\alpha_{\rm
p}}\left(\frac{h}{0.7}\right)^{2}\rm\ keV\ cm^{-3},$

where $H(z)$ is the Hubble parameter at redshift $z$ and
$H_{0}=100h\,\mathrm{kms^{-1}Mpc^{-1}}$ is the present value. $P_{500}$ is the
characteristic pressure reflecting the mass variation expected in a self-
similar model of pressure evolution when $\alpha_{\rm p}=0$, purely based on
gravitation (Arnaud et al., 2010). Deviation from the self-similarity appears
in a variation of the scaled pressure profile, given by

$\frac{P(r)}{P_{500}}=\mathbb{P}(x)\left[\frac{(1-b)\,M_{500}}{3\times
10^{14}\,(h/0.7)^{-1}{\rm M}_{\odot}}\right]^{\alpha_{\rm p}},$ (12)

expressed as a function of $M_{500}$. $b$ is the hydrostatic mass bias. Note
that we use the pressure model in Planck Collaboration (2013) in which
$M_{500}$ actually corresponds to the hydrostatic mass, $(1-b)\,M_{500}$. For
the mass bias, we adopt $(1-b)=0.780\pm 0.092$ derived from the Canadian
Cluster Comparison Project (Hoekstra et al. 2015; CCCP), which is consistent
with most of the results from hydrodynamic simulations (Gianfagna et al.,
2021). For the parameters of the generalised NFW electron pressure profile, we
adopt the best-fit values of
$[P_{0},c_{500},\gamma,\alpha,\beta]=[6.41,1.81,0.31,1.33,4.13]$, estimated
from 62 massive nearby clusters ($10^{14.4}<M_{500}<10^{15.3}{\rm M}_{\odot}$)
using the Planck SZ and XMM-Newton X-ray data in Planck Collaboration (2013).
We also adopt $\alpha_{\rm p}=0.12$ from the measurement in Arnaud et al.
(2010).

###  2.5 Decaying dark matter model implementation

We include the DDM model based on the public code provided by Abellán et al.
(2021)111https://github.com/PoulinV/class_majoron (hereafter G21), in which
the DDM model is implemented in the modified version of the Boltzmann code,
CLASS (Blas et al., 2011; Lesgourgues & Tram, 2011). In this code, the 1-body
($\Lambda$DDM1 model) and 2-body ($\Lambda$DDM2 model) decaying DM models are
implemented by including two additional parameters with respect to
$\Lambda$CDM; one is the decay rate, $\Gamma$, and the other is the mass-
energy fraction transferred to DR, $\varepsilon=(1/2)(1-m_{\rm WDM}^{2}/m_{\rm
DDM}^{2})$, where $0\leq\varepsilon\leq 1/2$. In the two extreme cases,
$\varepsilon$ = 0 corresponds to the standard $\Lambda$CDM case with no DM
decay and $\varepsilon$ = 1/2 to the $\Lambda$DDM1 case. The values in between
correspond to the $\Lambda$DDM2 case. Abellán et al. (2021) included the
background evolution and, in particular, density perturbations of daughter
particles, which was not included in previous studies (Vattis et al., 2019;
Haridasu & Viel, 2020; Clark et al., 2021).

In the $\Lambda$DDM1 model (DM $\rightarrow$ DR), the DM decay reduces the
matter content in the Universe, causing the reduction in the expansion rate
compared to the $\Lambda$CDM model, as shown in Fig. 1 in G21. It reduces the
growth of matter density fluctuations at small scales compared to the
$\Lambda$CDM model, as shown in Fig. 4 in G21. The amount of matter power
suppression increases as the DM lifetime becomes shorter.

In the $\Lambda$DDM2 model (DM $\rightarrow$ WDM+DR), the WDM component
partially contributes to the matter density, leaving the expansion rate almost
unchanged compared to the $\Lambda$CDM model, as shown in Fig. 1 in G21.
However, it suppresses the growth of matter density fluctuations due to the
free-streaming of WDM at small scales compared to the $\Lambda$CDM case, as
shown in Fig. 4 in G21. The amount of matter power suppression increases as
the DM lifetime becomes shorter, and the scale of the power suppression is
determined by the free-streaming length of WDM, similarly to that induced by
massive neutrinos.

In order to include these DDM models in the model of SZ power spectrum, we
modified the comoving volume, HMF, and halo’s pressure profile in Eq. 4.
First, the comoving volume at a given redshift depends on the background
evolution of assumed cosmological models, and the one including the DDM can be
directly computed with G21. Second, we consider the modification of the HMF.
The theoretical prescription of the SZ power spectrum, including the DDM
model, was studied in Takahashi et al. (2004). The authors included the DDM’s
impact on the HMF by modifying the Press–Schechter formalism (Press &
Schechter, 1974). We instead use the HMF from Tinker et al. (2008) but replace
the matter power spectrum in Eq. 8 and $\rho_{m,0}$ in Eq. 6 with ones
including the DDM model. Third, we consider the modification of the halo’s
pressure profile. Currently, the model of the halo’s pressure profile is
constrained by the pressure measurements in galaxy clusters with X-ray or the
SZ effect. For example, in Planck Collaboration (2013) and Pointecouteau et
al. (2021), the SZ profiles of galaxy clusters were stacked and fitted with
the gNFW model with five parameters in Eq. 9. We include the impact from the
DDM model on the pressure profile model in Planck Collaboration (2013) by
modifying the time evolution in Eq. 11 and replacing the $H(z)$ with the one
including the DDM model. The halo’s pressure profile is also given as a
function of mass, and this relation might be modified when the DDM model is
included. However, we assume that it is not the case. This assumption would
not be valid if the DM decay depends on their density, however, the DDM model
we consider here does not have such a dependency.

The SZ power spectra, including these modifications, are shown in Figure. 1
for the $\Lambda$DDM1 model with different decay lifetime of $\Gamma^{-1}$ =
30, 100, 300 Gyr, which are compared with the $\Lambda$CDM model with stable
cold dark matter under the Planck 2018 cosmology of Table 1 in Planck
Collaboration (2020a). As expected, the DM lifetime $\Gamma^{-1}$ determines
the depth of the suppression, and the amount of power suppression increases
for smaller lifetimes.

We also show the SZ power spectra for the $\Lambda$DDM2 model in Figure. 2
with different decay lifetime of $\Gamma^{-1}$ = 30, 100, 300 Gyr and also
with different mass-energy fraction transferred to DR with $\varepsilon$ = 0.1
and 0.01. As in the case of the DDM1 model, the lifetime $\Gamma^{-1}$
determines the depth of the suppression, which increase for smaller lifetimes.
$\varepsilon$ determines the free-streaming scale of WDM and the suppression
scale of matter power spectrum, reducing the number of halos in the HMF.
Because this reduction is more dramatic for massive halos, the SZ power
spectrum is more suppressed at large scales as shown in Figure. 3.

Figure 1: SZ power spectra with the $\Lambda$DDM1 model (DM $\rightarrow$ DR)
with different decay lifetime of $\Gamma^{-1}$ = 30 (dotted line), 100 (dash-
dot line), 300 (dashed line) Gyr. “$\Lambda$CDM” represents the $\Lambda$CDM
model with stable cold dark matter (solid line). These all assume the Planck
2018 cosmological parameters.

Figure 2: SZ power spectra with the $\Lambda$DDM2 model (DM $\rightarrow$
WDM+DR) with different decay lifetime of $\Gamma^{-1}$ = 30 (dotted line), 100
(dash-dot line), 300 (dashed line) Gyr. “$\Lambda$CDM” represents the
$\Lambda$CDM model with stable cold dark matter (solid line). The mass-energy
fractions transferred to DR are $\varepsilon$ = 0.1 (blue) in the left panel
and 0.01 (green) in the right panel. These all assume the Planck 2018
cosmological parameters.  Figure 3: SZ power spectra with the $\Lambda$DDM2
model (DM $\rightarrow$ WDM+DR) for the lifetime $\Gamma^{-1}=200$ Gyr with
different values of $\varepsilon$ = 0.1 (dotted line), 0.01 (dash-dot line),
0.001 (dashed line). “$\Lambda$CDM” represents the $\Lambda$CDM model with
stable cold dark matter (solid line). These all assume the Planck 2018
cosmological parameters.

###  2.6 Emulator of SZ power spectrum

We compute the $C_{\ell}^{\rm SZ}$ in our cosmological analysis with the Monte
Carlo Markov chain (MCMC) in Sect. 3.2, and thus the $C_{\ell}^{\rm SZ}$’s
computation time has to be minimized. To achieve this, we adopted the machine
learning technique of Random Forest algorithm, developed in Douspis et al.
(2022) and used in Gorce et al. (2022), and modified it to build an emulator
adapted to the DDM models. We built $\sim$50,000 SZ power spectra with a
random sampling of cosmological parameters within the ranges listed in Table.
1 at 50 different multipoles between 2 and 3072 and used 60%, 20% and 20% of
the sample for the training, validation, and test, respectively. We optimized
the Random-Forest’s hyperparameters, such as numbers of trees and depths to be
40 and 30, to maximize the final score of the prediction. This resulted in the
accuracy of the constructed model to be better than 1% in our analysis range
of 60 ¡ $\ell$ ¡ 1411, which is well below the uncertainty in the current SZ
measurement of $\sim$ 30%, used in our analysis.

##  3 Cosmological analysis

We use the measurement of the SZ angular power spectrum in T22 to explain the
$S_{8}$ tension and constrain the DDM model. In the analysis, the SZ
measurement is compared to the theoretical prescription, including the DDM
model explained in Sect. 2.

###  3.1 Compton $y$-map

A new all-sky $y$ map was reconstructed in T22 from the Planck 2020 data
release 4 (Planck Collaboration, 2020b)222https://pla.esac.esa.int in
HEALpix333http://healpix.sourceforge.net/ format (Górski et al., 2005) with a
pixel resolution of $N_{\rm side}$ = 2048 ($\sim 1.7$ arcmin). The
reconstruction was performed based on the modified internal linear combination
algorithm (MILCA) (Hurier et al., 2013) by combining six Planck HFI frequency
maps between 100 and 857 GHz so that the spectral response of the Compton $y$
parameter is unity.

We use sky masks suitable for the analysis of the $y$ maps provided in the
2015 Planck data release (Planck Collaboration, 2016a). These masks cover
point sources and regions around the Galactic plane, excluding about 58% of
the sky. However, the residual extragalactic foreground emissions may remain
in the $y$ map. Thus, we model them including radio and IR point sources with
the Planck sky model (Delabrouille et al., 2013) and CIB fluctuations from
Maniyar et al. (2021).

Our cosmological analysis uses two $y$-maps reconstructed from the first and
last half-ring Planck frequency maps and computes their cross-power spectrum
to avoid the bias induced by the noise in the auto-power spectrum. We also
limit our analysis to the multipole range of 60 ¡ $\ell$ in order to minimize
the contribution from the foreground and to $\ell$ ¡ 1411 in order to minimize
the contribution from the noise.

###  3.2 Maximum likelihood analysis

Cosmological constraints can be obtained by fitting the SZ power spectrum
measurement with the SZ and foreground models simultaneously. In our model, we
consider four components: SZ, CIB, radio point sources, and infrared point
sources. We also include an instrumental noise, CN, in the model. Finally, the
observed SZ power spectrum, $C_{\ell}^{\rm obs}$, is modeled by

$C_{\ell}^{\rm obs}=C_{\ell}^{\rm SZ}(\Theta,1-b)+A_{\rm CIB}\,C_{\ell}^{\rm
CIB}+A_{\rm IR}\,C_{\ell}^{\rm IR}+A_{\rm rad}\,C_{\ell}^{\rm rad}+A_{\rm
CN}\,C_{\ell}^{\rm CN},$ (13)

where $C_{\ell}^{\rm SZ}(\Theta,1-b)$ is the SZ power spectrum, $\Theta$ is
the set of free cosmological parameters shown below in Eq. 14, $1-b$ is the
hydrostatic mass bias, $C_{\ell}^{\rm CIB}$ is the CIB power spectrum,
$C_{\ell}^{\rm IR}$ and $C_{\ell}^{\rm rad}$ are the infrared and radio source
power spectra and $C_{\ell}^{\rm CN}$ is the empirical model for the
instrumental noise. For the cosmological parameters, we vary

$\Theta_{\rm\Lambda DDM1}=\\{\Omega_{\rm b}h^{2},\ln{(10^{10}A_{\rm
s})},n_{s},\tau_{\rm reio},H_{0},\Omega_{\rm dcdm}^{\rm
ini}h^{2},\log_{10}\Gamma\\}$ (14)

in the $\Lambda$DDM1 model and

$\Theta_{\rm\Lambda DDM2}=\\{\Omega_{\rm b}h^{2},\ln{(10^{10}A_{\rm
s})},n_{s},\tau_{\rm reio},H_{0},\Omega_{\rm dcdm}^{\rm
ini}h^{2},\log_{10}\Gamma,\log_{10}\varepsilon\\}$ (15)

in the $\Lambda$DDM2 model, where $\Omega_{\rm b}h^{2}$ is the baryon density,
$\ln{(10^{10}A_{\rm s})}$ is the primordial density perturbation amplitude,
$n_{s}$ is the primordial density perturbation spectral index,
$\rm\tau_{reio}$ is the Reionization optical depth, $H_{0}$ is the Hubble
constant, $\Omega_{\rm dcdm}^{\rm ini}h^{2}$ is the initial matter density of
decaying dark matter, $\log_{10}\Gamma$ is the decay rate in logarithmic
scale, and $\log_{10}\varepsilon$ is the fractional mass energy transferred to
DR in logarithmic scale.

We performed the MCMC likelihood analysis using the MONTEPYTHON-v3 (Audren et
al., 2013; Brinckmann & Lesgourgues, 2019). We first performed the
cosmological analysis only with the high-redshift Planck CMB data from
high-$\ell$ TT,TE,EE lite and low-$\ell$ TT, EE (Planck Collaboration, 2020a)
under the $\Lambda$DDM1 and $\Lambda$DDM2 models, respectively, without the SZ
data. We then combined the CMB data with the low-redshift SZ power spectrum
from T22 and compared the results. We used flat priors for the cosmological
parameters and Gaussian priors for nuisance parameters. The sampling
parameters and priors used in our cosmological analysis are summarized in
Table. 1.

Table 1: Sampling parameters and priors.

Parameter | Symbol | Prior
---|---|---
Baryon density | $\Omega_{\rm b}h^{2}$ | [0.0212 - 0.0234]
Primordial density perturbation amplitude | $\ln{(10^{10}A_{\rm s})}$ | [2.94 - 3.16]
Primordial density perturbation spectral index | $n_{s}$ | [0.92 - 1.00]
Reionization optical depth | $\rm\tau_{reio}$ | [0.01 - 0.10]
Hubble constant | $H_{0}$ | [64 - 71]
Initial matter density of decaying dark matter | $\Omega_{\rm dcdm}^{\rm ini}h^{2}$ | [0.11 - 0.13]
Inverse of decay lifetime | $\log_{10}\Gamma$ | [-4 - 1]
Fractional mass energy transferred to DR | $\log_{10}\varepsilon$ | [-4 - $\log_{10}(0.5)$]
CIB contamination | $A_{\rm CIB}$ |  $N$(1, 0.5)
IR-source contamination | $A_{\rm IR}$ |  $N$(1, 0.5)
Radio-source contamination | $A_{\rm rad}$ |  $N$(1, 0.5)
Noise | $A_{\rm CN}$ |  $N$(1, 0.5)
Mass bias | $1-b$ |  $N$(0.78, 0.092)

Sampling cosmological parameters and nuisance parameters are listed. Nuisance
parameters include $A_{\rm CIB}$, $A_{\rm IR}$, $A_{\rm rad}$, $A_{\rm CN}$,
and $1-b$. [min - max] corresponds to a flat prior with minimum and maximum
values. $N$(mean, variance) corresponds to a Gaussian prior with given mean
and variance.

###  3.3 Results

Figure 4 shows the result of our cosmological analysis with the $\Lambda$DDM1
model. In this model, DM decays into DR and $\Omega_{\rm m}$ decreases
compared to the one in the $\Lambda$CDM. Thus, the lower $S_{8}$ value is
mainly induced by the lower value of $\Omega_{\rm m}$, not of $\sigma_{8}$, as
shown in Figure 4. The figure shows the posterior distributions of the
cosmological parameters, $\log_{10}\Gamma$, $S_{8}$, $\sigma_{8}$, and
$\Omega_{m}$ with 68% and 95% confidence interval contours, when only the
Planck CMB data is used in gray and the Planck CMB data is combined with the
SZ data from T22 in red. The posterior distributions of other cosmological
parameters are summarized in Table. 2 with 68% confidence interval, and they
are consistent between the two cases. The CMB data can constrain the DDM model
through the integrated Sachs-Wolfe (ISW) effect as shown in Figure 5 of Aoyama
et al. (2014) and put a tight constraint of $\Gamma^{-1}$ ¿ 160 Gyr on the
$\Lambda$DDM1 model (e.g., Audren et al. 2014; Enqvist et al. 2015; Poulin et
al. 2016; Enqvist et al. 2020). A combination of the high-redshift CMB data
and low-redshift SZ data provides the best-fit value on the DM decay lifetime
of

$\Gamma^{-1}=200.3_{-61.8}^{+141.0}\;{\rm[Gyr]}\quad({\rm\Lambda DDM1}),$ (16)

showing a preference of $\Gamma^{-1}\sim 200$ Gyr to resolve the $S_{8}$
tension between the Planck CMB measurement and the SZ measurement from T22.
However, the CMB data do not prefer this model, and the tension still remains.
Given the uncertainty of our measurement, we also constrained a lower bound on
the DM decay lifetime to be $\sim$ 77 Gyr at 95% confidence level but did not
obtain a tighter constraint than the one from the CMB data.

Figure 4: Posterior distributions of the cosmological parameters under the
$\Lambda$DDM1 model, $\log_{10}\Gamma$, $S_{8}$, $\sigma_{8}$, and
$\Omega_{m}$ with 68% and 95% confidence interval contours obtained from our
cosmological analysis, when only the Planck CMB data is used in gray and the
Planck CMB data is combined with the SZ data from T22 in red.

Figure 5 shows the result of our cosmological analysis with the $\Lambda$DDM2
model. In this model, DM decays into WDM, but $\Omega_{\rm m}$ does not
decrease unlike the case in the $\Lambda$DDM1 model. Rather, the $\sigma_{8}$
is lowered by the matter power suppression at small scales induced by the
free-streaming length of WDM, similar to that induced by massive neutrinos.
Under the $\Lambda$DDM2 model, the lower $S_{8}$ value is thus mainly induced
by the lower value of $\sigma_{8}$, not of $\Omega_{\rm m}$, as shown in
Figure 5. The figure shows the posterior distributions of the cosmological
parameters, $\log_{10}\Gamma$, $S_{8}$, $\log_{10}\varepsilon$, $\sigma_{8}$,
and $\Omega_{m}$ with 68% and 95% confidence interval contours, when only the
Planck CMB data is used in gray and the Planck CMB data is combined with the
SZ data from T22 in red. The posterior distributions of other cosmological
parameters are summarized in Table. 2 with 68% confidence interval, and they
are consistent between the two cases. Similarly to the case with the
$\Lambda$DDM1 model, A combination of the high-redshift CMB data and low-
redshift SZ data provides the best-fit value on the DM decay lifetime of

$\Gamma^{-1}=204.6_{-78.9}^{+132.9}\;{\rm[Gyr]}\quad({\rm\Lambda DDM2}),$ (17)

providing a lower bound on the DM’s decay lifetime to be $\sim$ 47 Gyr at 95%
confidence level. However, $\varepsilon$ was not well constrained in our
analysis. To constrain this parameter, precise measurements of the shape of
the SZ power spectrum are required as shown in Figure. 3. However, the
uncertainty in the current SZ measurement is $\sim$ 30%, and it does not seem
to allow determining the value of this parameter. For this purpose, more
sensitive data at small scales will be useful, such as AdvACT Henderson et al.
(2016) and SPT-3G Benson et al. (2014) and upcoming Simons Observatory Ade et
al. (2019) and CMB-S4 Abazajian et al. (2019).

Figure 5: Posterior distributions of the cosmological parameters under the
$\Lambda$DDM2 model, $\log_{10}\Gamma$, $\log_{10}\varepsilon$, $S_{8}$,
$\sigma_{8}$, and $\Omega_{m}$ with 68% and 95% confidence interval contours
obtained from our cosmological analysis, when only the Planck CMB data is used
in gray and the Planck CMB data is combined with the SZ data from T22 in red.
Table 2: Sampling parameters and priors.

Parameter |  $\Lambda$DDM1 |  $\Lambda$DDM2
---|---|---
|  Planck CMB | +SZ |  Planck CMB | +SZ
100 $\Omega_{\rm b}h^{2}$ | 0.0223 $\pm$ 0.0002 | 0.0223 $\pm$ 0.0002 | 0.0223 $\pm$ 0.0002 | 0.0224 $\pm$ 0.0002
$\ln{(10^{10}A_{\rm s})}$ | $3.045_{-0.016}^{+0.017}$ | $3.042_{-0.013}^{+0.014}$ | $3.045_{-0.015}^{+0.016}$ | $3.046_{-0.014}^{+0.013}$
$n_{s}$ | 0.9642 $\pm$ 0.0045 | $0.9636_{-0.0034}^{+0.0047}$ | $0.9641_{-0.0043}^{+0.0044}$ | $0.9653^{+0.0034}_{-0.0039}$
$\rm\tau_{reio}$ | $0.0542_{-0.0078}^{+0.0081}$ | $0.0533_{-0.0069}^{+0.0071}$ | $0.0542_{-0.0072}^{+0.0078}$ | $0.0542_{-0.0063}^{+0.0060}$
$H_{0}$ | 67.2 $\pm$ 0.6 | $67.6_{-0.5}^{+0.7}$ | 67.2 $\pm$ 0.6 | 67.6 $\pm$ 0.6
$\Omega_{\rm dcdm}^{\rm ini}h^{2}$ | 0.1202 $\pm$ 0.0014 | $0.1204_{-0.0017}^{+0.0012}$ | 0.1202 $\pm$ 0.0014 | $0.1195_{-0.0010}^{+0.0009}$
$\log_{10}\Gamma$ | Not preferred | $-2.30_{-0.23}^{+0.16}$ | Not constrained | $-2.31_{-0.22}^{+0.20}$
$\log_{10}\varepsilon$ | - | - | Not constrained | $-1.92_{-1.12}^{+0.49}$
$\Omega_{\rm m}$ | $0.3168_{-0.0087}^{+0.0089}$ | $0.296_{-0.011}^{+0.010}$ | $0.3167_{-0.0084}^{+0.0087}$ | 0.3144 $\pm$ 0.0055
$\sigma_{8}$ | $0.8116_{-0.0076}^{+0.0078}$ | $0.8053_{-0.0072}^{+0.0051}$ | $0.8108_{-0.0085}^{+0.0080}$ | $0.7843_{-0.0158}^{+0.0226}$
$S_{8}$ | 0.834 $\pm$ 0.017 | $0.799_{-0.019}^{+0.018}$ | 0.833 $\pm$ 0.017 | $0.799_{-0.019}^{+0.022}$

Sampling cosmological parameters and nuisance parameters are listed. Nuisance
parameters include $A_{\rm CIB}$, $A_{\rm IR}$, $A_{\rm rad}$, $A_{\rm CN}$,
and $1-b$. [min - max] corresponds to a flat prior with minimum and maximum
values. $N$(mean, variance) corresponds to a Gaussian prior with given mean
and variance.

###  3.4 Systematic effects

Finally, we consider systematic uncertainties in our analysis. As discussed in
Planck Collaboration (2016a) and T22, the mass bias and the pressure profile
model are not well constrained, affecting the cosmological analysis with the
SZ power spectrum. We summarize the results in Table. 4 and Table. 4 for the
$\Lambda$DDM1 and $\Lambda$DDM2 models.

First, in order to investigate the impact on our DDM’s constraints from the
mass bias, we replaced the CCCP’s mass bias prior with two other ones: one
from the Weighting the Giants weak lensing measurements (von der Linden et al.
2014; WtG) and the other from cosmological hydrodynamical simulations (Biffi
et al. 2016; BIFFI). Table. 4 shows their comparison in $\Gamma_{-1}$ and its
lower bound at 95% confidence level. The result shows that the $\Gamma^{-1}$
value increases as the mass bias increases (or $1-b$ decreases). The reason is
that the amplitude of the SZ power spectrum model shifts downwards when the
mass bias increases, as in Eq. 11, which reduces the $S_{8}$ discrepancy
between the Planck CMB and SZ results and thus requires a smaller amount of
dark matter decay, in other words, longer decay lifetime.

In addition, we investigated the impact on our DDM’s constraints from the
pressure profile model by replacing P13 with two other models: one based on
the combination of XMM-Newton measurements and numerical simulations (Arnaud
et al. 2010; A10) and the other based on the analysis with combined data from
Planck and ACT (Pointecouteau et al. 2021; PACT21). Table. 4 shows their
comparison in $\Gamma^{-1}$ and its lower band at 95% confidence level. The
result shows that the $\Gamma^{-1}$ value of PACT21 is higher than the others.
The reason is that the amplitude of the PACT21 pressure profile is slightly
lower than the others, as shown in Fig. 5 in Pointecouteau et al. (2021). It
shifts the amplitude of the SZ power spectrum model downwards, reduces the
$S_{8}$ discrepancy between the Planck CMB and SZ results, and requires less
amount of dark matter decay.

Table 3: $\Gamma^{-1}$ constraints obtained with different mass bias priors
and pressure profile models under ${\rm\Lambda DDM1}$.

Mass bias prior | $1-b$ |  $\Gamma^{-1}\rm[Gyr]$ (lower bound)
---|---|---
CCCP (fiducial) | $0.780\pm 0.092$ |  $200.3_{-61.8}^{+141.0}$ (¿ 76.7)
WtG | $0.688\pm 0.072$ |  $231.7_{-73.5}^{+108.4}$ (¿ 84.6)
BIFFI | $0.877\pm 0.015$ |  $141.8_{-34.6}^{+75.0}$ (¿ 72.7)
Pressure model |  |  $\Gamma^{-1}\rm[Gyr]$ (lower bound)
P13 (fiducial) |  |  $200.3_{-61.8}^{+141.0}$ (¿ 76.7)
PACT21 |  |  $250.8_{-95.6}^{+129.8}$ (¿ 59.7)
A10 |  |  $190.1_{-49.5}^{+100.3}$ (¿ 91.0)

Table 4: $\Gamma^{-1}$ constraints obtained with different mass bias priors
and pressure profile models under ${\rm\Lambda DDM2}$.

Mass bias prior | $1-b$ |  $\Gamma^{-1}\rm[Gyr]$ (lower bound)
---|---|---
CCCP (fiducial) | $0.780\pm 0.092$ |  $204.3_{-78.9}^{+132.9}$ (¿ 46.5)
WtG | $0.688\pm 0.072$ |  $282.9_{-113.3}^{+118.6}$ (¿ 56.3)
BIFFI | $0.877\pm 0.015$ |  $152.1_{-52.2}^{+100.4}$ (¿ 47.7)
Pressure model |  |  $\Gamma^{-1}\rm[Gyr]$ (lower bound)
P13 (fiducial) |  |  $204.3_{-78.9}^{+132.9}$ (¿ 46.5)
PACT21 |  |  $270.7_{-122.4}^{+149.5}$ (¿ 25.8)
A10 |  |  $209.9_{-76.5}^{+123.7}$ (¿ 56.9)

##  4 Summary and conclusion

To solve the $S_{8}$ tension between the Planck CMB measurement and low-
redshift probes, we extend the $\Lambda$CDM model, including a decaying dark
matter model. Two DDM models are tested in our study: one DDM model
($\Lambda$DDM1) that dark matter decays into a form of non-interacting dark
radiation, which is parameterized with the decay rate, $\Gamma$, and the other
model ($\Lambda$DDM2) that the dark matter decays into warm dark matter and
dark radiation, which is parameterized with the decay rate, $\Gamma$, and the
mass-energy fraction transferred to the massless component, $\varepsilon$. For
the low-redshift probe, we use the Sunyaev Zel’dovich effect and compute the
DDM’s impact on the SZ power spectrum by varying the DDM parameters, including
the background evolution in cosmology and non-linear prescription in the halo
mass function. The result shows the suppression of the SZ power spectrum
relative to the $\Lambda$CDM model due to the dark matter decay. As one
expects, the suppression becomes more remarkable as the decay time becomes
shorter.

We perform a cosmological analysis and compare the results when only the
Planck CMB data are used and when the Planck CMB data are combined with the SZ
data from T22. The result shows a preference of
$\Gamma^{-1}=200.3_{-61.8}^{+141.0}$ Gyr with the $\Lambda$DDM1 model to
resolve the tension between the Planck CMB and SZ measurements on the $S_{8}$
parameter, in agreement with literatures supporting $\Gamma^{-1}$ ¿ 160 Gyr.
Given the uncertainty of our measurement, we also set a lower bound on the
DM’s decay lifetime to be $\sim$ 77 Gyr at 95% confidence level. However, the
CMB data do not prefer this model, and the tension still remains. We also
tested for the $\Lambda$DDM2 model using the Planck CMB and SZ measurements.
The result shows a preference of $\Gamma^{-1}=204.3_{-78.9}^{+132.9}$ Gyr to
resolve the $S_{8}$ tension. This result provides a lower bound on the DM’s
decay lifetime to be $\sim$ 47 Gyr at 95% confidence level.

We also checked the systematic uncertainty in our analysis with the SZ power
spectrum, which includes the impact of the mass bias and pressure profile
model. The lower bound changes depending on the choice of these models. It
indicates that the DDM model can be more accurately constrained only if the
systematic uncertainties originating from the internal structures in galaxy
clusters, such as contributions of non-thermal pressure and baryonic feedback
effects, are well understood and modeled.

Finally, current studies based on weak lensing and galaxy clustering as low-
redshift probes indicate that a lower $S_{8}$ value may be caused by a lower
$\sigma_{8}$, not $\Omega_{m}$ (Di Valentino et al., 2021). To investigate
this using SZ signal, additional high-resolution and high-sensitivity data
will be needed, such as those of AdvACT and SPT-3G and upcoming Simons
Observatory and CMB-S4.

###### Acknowledgements.

This research has been supported by the funding for the Baryon Picture of the
Cosmos (ByoPiC) project from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme grant
agreement ERC-2015-AdG 695561. The authors acknowledge fruitful discussions
with the members of the ByoPiC project (https://byopic.eu/team). This work is
supported by World Premier International Research Center Initiative (WPI),
MEXT, Japan. Kavli IPMU was established by World Premier International
Research Center Initiative (WPI), MEXT, Japan. Kavli IPMU is supported by
World Premier International Research Center Initiative (WPI), MEXT, Japan.

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# On the Diophantine equations
$X^{3}+Y^{3}+Z^{3}+aU^{k}=\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}$ ; $k=3,4$

Farzali Izadi  Farzali Izadi
Department of Mathematics
Faculty of Science
Urmia University
Urmia 165-57153, Iran<EMAIL_ADDRESS>and  Mehdi Baghalaghdam  Mehdi
Baghalaghdam
Department of Mathematics
Faculty of Science
Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran<EMAIL_ADDRESS>
###### Abstract.

In this paper, elliptic curves theory is used for solving the Diophantine
equations $X^{3}+Y^{3}+Z^{3}+aU^{k}=\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}$,
$k=3,4$, where $n,t_{i}\in\mathbb{N}\cup\\{0\\}$, and $a\neq 0$, $a_{i}$, are
fixed arbitrary rational numbers. We try to transform each case of the above
Diophantine equations to a cubic elliptic curve of positive rank, then get
infinitely many integer solutions for each case. We also solve these
Diophantine equations for some values of $n$, $a$, $a_{i}$, $t_{i}$, and
obtain infinitely many solutions for each case, and show among the other
things that how sums of four, five, or more cubics can be written as sums of
four, five, or more biquadrates as well as sums of 5th powers, 6th powers and
so on.

###### Key words and phrases:

Diophantine equations, High power Diophantine equations, Elliptic curves

###### 2010 Mathematics Subject Classification:

11D45, 11D72, 11D25, 11G05 and 14H52

##  1\. Introduction

The authors in two different papers, used elliptic curves to solve two
Diophantine equations

(1.1) $X^{4}+Y^{4}=2U^{4}+\sum_{i=1}^{n}T_{i}U_{i}^{\alpha_{i}},$

where $n,\alpha_{i}\in\mathbb{N}$, and $T_{i}$, are appropriate rational
numbers, and

(1.2)
$\sum_{i=1}^{n}a_{i}x_{i}^{6}+\sum_{i=1}^{m}b_{i}y_{i}^{3}=\sum_{i=1}^{n}a_{i}X_{i}^{6}\pm\sum_{i=1}^{m}b_{i}Y_{i}^{3},$

where $n,m\in\mathbb{N}$ and $a_{i},b_{i}\neq 0\in\mathbb{Q}$. (see [1], [2])

In this paper, we are interested in the study of the Diophantine equations:

(1.3) $X^{3}+Y^{3}+Z^{3}+aU^{k}=\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}};k=3,4$

where $n,t_{i}\in\mathbb{N}\cup\\{0\\}$, and $a\neq 0$, $a_{i}$, are fixed
arbitrary rational numbers.
This shows that how sums of four, five, or more cubics can be written as sums
of four, five, or more biquadrates as well as sums of 5th powers, 6th powers
and so on.

We conclude this introduction with a standard fact which is needed in section
$3$. (see [4])

######  Lemma 1.1.

Let K be a field of characteristic not equal to $2$. Consider the equation

$v^{2}=au^{4}+bu^{3}+cu^{2}+du+q^{2}$, with $a$, $b$, $c$, $d$ $\in K$.

Let $x=\frac{2q(v+q)+du}{u^{2}}$,
$y=\frac{4q^{2}(v+q)+2q(du+cu^{2})-(\frac{d^{2}u^{2}}{2q})}{u^{3}}$.

Define $a_{1}=\frac{d}{q}$, $a_{2}=c-(\frac{d^{2}}{4q^{2}})$, $a_{3}=2qb$,
$a_{4}=-4q^{2}a$, $a_{6}=a_{2}a_{4}$.

Then $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$.

The inverse transformation is $u=\frac{2q(x+c)-(\frac{d^{2}}{2q})}{y}$,
$v=-q+\frac{u(ux-d)}{2q}$.

The point $(u,v)=(0,q)$ corresponds to the point $(x,y)=\infty$ and

$(u,v)=(0,-q)$ corresponds to $(x,y)=(-a_{2},a_{1}a_{2}-a_{3})$.

##  2\. The Diophantine equation (DE)
$X^{3}+Y^{3}+Z^{3}+aU^{3}=\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}$

######  Main Theorem 2.1.

Consider the DE (1.3) for the case $k=3$, where
$n,t_{i}\in\mathbb{N}\cup\\{0\\}$, $a\neq 0$, $a_{i}$, are fixed arbitrary
rational numbers.

We try to transform this DE to a cubic elliptic curve of positive rank.
Let $Y^{2}=X^{3}+FX^{2}+GX+H$, be an elliptic curve in which the coefficients
$F$, $G$, and $H$, are all functions of $a$, $a_{i}$, $t_{i}$ and the other
rational parameters $P_{i}$, $t$, $s$, yet to be found later. If the elliptic
curve has positive rank, depending on the values of $P_{i}$, $s$, the DE (1.3)
has infinitely many integer solutions.

######  Proof 2.1.

Firstly, it is clear that if we find rational solutions for each case of the
Diophantine equations (1.3), then by canceling the denominators of $X$, $Y$,
$Z$, $U$, $U_{i}$, and by multiplying both sides of these Diophantine
equations by appropriate number $M$, we may obtain integer solutions for each
case.

Let $X=-Z+t$, $Y=-Z-t$, where $Z$, $t$ are rational variables. By substituting
these variables in the DE (1.3), and some simplifications, we get:

(2.1)
$t^{2}=(\frac{a}{6Z})U^{3}+(\frac{-Z^{2}}{6}-\frac{\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}}{6Z}).$

By multiplying both sides of the Eq. (2.1), by $(\frac{a}{6Z})^{2}$, and
letting

(2.2) $Y^{\prime}=t.(\frac{-a}{6Z}),$

and

(2.3) $X^{\prime}=U.(\frac{a}{6Z}),$

we get:

(2.4) $Y^{\prime 2}=X^{\prime
3}+(-\frac{a^{2}}{216}-(\frac{a^{2}}{216Z^{3}}).\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}).$

Now by choosing appropriate values for $Z$ (equal to $s$), and $U_{i}$ (equal
to $P_{i}$), such that the rank of the elliptic curve (2.4) to be positive,
and by calculating $t$, $U$, $X$, $Y$, from the relations (2.2), (2.3),
$X=-Z+t$ and $Y=-Z-t$, some simplifications and canceling the denominators of
$X,Y,Z,U,U_{i}$, we obtain infinitely many integer solutions for the DE (1.3).
The proof is completed.

Now we work out some examples.

###  2.1. Application to examples

######  Example 1.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{3}+U_{1}^{3}+U_{2}^{3}$
i.e., the sum of $4$ cubics can be written as the sum of $3$ cubics.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{3}+U_{1}^{3}+U_{2}^{3}}{216Z^{3}})$.
By taking $Z=5$, $U_{0}=1$, $U_{1}=2$, $U_{2}=3$, the above elliptic curve
becomes
$Y^{\prime 2}=X^{\prime 3}-\frac{161}{27000}$.
Rank=1.
Generator: $P=(X^{\prime},Y^{\prime})=(\frac{643}{90},\frac{2578}{135})$.
$(t,U)=(\frac{-5156}{9},\frac{643}{3})$.

Solution:

$9^{3}+18^{3}+27^{3}+5201^{3}=45^{3}+1929^{3}+5111^{3}$.

######  Example 2.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{4}$
i.e., fourth power of an integer is written as a sum of four cubics.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{4}}{216Z^{3}})$.
Now letting $Z=-2$, and $U_{0}=7$, yields
$Y^{\prime 2}=X^{\prime 3}+\frac{2393}{1728}$.
Rank=2.
Generators: $P_{1}=(X^{\prime},Y^{\prime})=(\frac{115}{48},\frac{249}{64})$,
$P_{2}=(X^{\prime\prime},Y^{\prime\prime})=(\frac{1320481}{291600},\frac{1528666129}{157464000})$.
$(t,U)=(\frac{747}{16},\frac{-115}{4})$, for the point $P_{1}$,
$(t^{\prime},U^{\prime})=(\frac{1528666129}{13122000},-\frac{1320481}{24300})$,
for the point $P_{2}$.

Solutions:

$779^{3}+(-715)^{3}+(-32)^{3}+(-460)^{3}=56^{4}$.

$7774550645^{3}+(-7512110645)^{3}+(-131220000)^{3}+(-3565298700)^{3}=5103000^{4}$.

######  Example 3.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{4}+U_{1}^{4}+U_{2}^{4}+U_{3}^{4}$
i.e., the sum of $4$ cubics is written as the sum of $4$ biquadrates.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{4}+U_{1}^{4}+U_{2}^{4}+U_{3}^{4}}{216Z^{3}})$.
Now letting $Z=-5$, $U_{0}=1$, $U_{1}=2$, $U_{2}=3$, $U_{3}=4$, yields
$Y^{\prime 2}=X^{\prime 3}+\frac{229}{27000}$.
Rank=1.
Generator:
$P=(X^{\prime},Y^{\prime})=(\frac{367}{7260},\frac{14821}{159720})$.
$(t,U)=(\frac{14821}{5324},-\frac{367}{242})$.

Solution:

$1823404^{3}+519156^{3}+(-1171280)^{3}+(-355256)^{3}=10648^{4}+21296^{4}+31944^{4}+42592^{4}$.

######  Example 4.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{5}+U_{1}^{5}+U_{2}^{5}+U_{3}^{5}$
i.e., the sum of $4$ cubics is written as the sum of $4$ fifth powers.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{5}+U_{1}^{5}+U_{2}^{5}+U_{3}^{5}}{216Z^{3}})$.
Now letting $Z=-6$, $U_{0}=1$, $U_{1}=2$, $U_{2}=3$, $U_{3}=4$, yields
$Y^{\prime 2}=X^{\prime 3}+\frac{271}{11664}$.
Rank=1.
Generator: $P=(X^{\prime},Y^{\prime})=(\frac{37}{324},\frac{917}{5832})$.
$(t,U)=(\frac{917}{162},-\frac{37}{9})$.

Solution:

$90672^{3}+2640^{3}+(-46656)^{3}+(-31968)^{3}=216^{5}+432^{5}+648^{5}+864^{5}$.

######  Example 5.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{6}+U_{1}^{6}+U_{2}^{6}+U_{3}^{6}$
i.e., the sum of $4$ cubics is written as the sum of $4$ sixth powers.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{6}+U_{1}^{6}+U_{2}^{6}+U_{3}^{6}}{216Z^{3}})$.
Now letting $Z=-7$, $U_{0}=1$, $U_{1}=2$, $U_{2}=3$, $U_{3}=4$, yields
$Y^{\prime 2}=X^{\prime 3}+\frac{4547}{74088}$.
Rank=2.
Generators:
$P_{1}=(X^{\prime},Y^{\prime})=(\frac{284747}{318402},\frac{111956735}{127042398})$,
$P_{2}=(X^{\prime\prime},Y^{\prime\prime})=(\frac{-162932615}{499254336},\frac{1819882008883}{11155338883584})$.
$(t,U)=(\frac{111956735}{3024819},-\frac{284747}{7581})$,
$(t^{\prime},U^{\prime})=(\frac{1819882008883}{265603306752},\frac{162932615}{11887008})$.

Solutions:

$2529478892^{3}+(-1724877038)^{3}+(-402300927)^{3}+(-2158667007)^{3}=\\\
7581^{6}+15162^{6}+22743^{6}+30324^{6}$.

$489320985767551^{3}+5232371404673^{3}+484195324491480^{3}+\\\
(-247276678586112)^{3}=$ $5943504^{6}+11887008^{6}+17830512^{6}+23774016^{6}$.

If we take $Z=-7$, $U_{0}=U_{1}=3$, $U_{2}=U_{3}=4$, the above elliptic curve
becomes
$Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{3^{6}+3^{6}+4^{6}+4^{6}}{216.(-7)^{3}})=X^{3}+\frac{9307}{74088}$.
Rank=2.
Generators:
$P_{1}=(X^{\prime},Y^{\prime})=(\frac{1654081}{3572100},\frac{3201764021}{6751269000})$,
$P_{2}=(X^{\prime\prime},Y^{\prime\prime})=(\frac{1526057}{71442},\frac{-666521155}{6751269})$.
$(t,U)=(\frac{3201764021}{160744500},-\frac{1654081}{85050})$,
$(t^{\prime},U^{\prime})=(\frac{-1333042310}{321489},-\frac{1526057}{1701})$.

Solutions: (for the DE $X^{3}+Y^{3}+Z^{3}+U^{3}=2U_{0}^{6}+2U_{2}^{6}$):

$21634877605^{3}+(-10382762605)^{3}+(-5626057500)^{3}+(-15631065450)^{3}=\\\ $
$2.(85050)^{6}+2.(113400)^{6}$.

$(-1330791887)^{3}+1335292733^{3}+(-2250423)^{3}+(-288424773)^{3}=$
$2.(1701)^{6}+2.(2268)^{6}$.

######  Example 6.

$X^{3}+Y^{3}+Z^{3}+U^{3}=U_{0}^{7}+U_{1}^{7}+U_{2}^{7}+U_{3}^{7}$
i.e., the sum of $4$ cubics is written as the sum of $4$ seventh powers.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{U_{0}^{7}+U_{1}^{7}+U_{2}^{7}+U_{3}^{7}}{216Z^{3}})$.
Now letting $Z=2$, $U_{0}=1$, $U_{1}=2$, $U_{2}=3$, $U_{3}=4$, yields
$Y^{\prime 2}=X^{\prime 3}-\frac{1559}{144}$.
Rank=1.
Generator:
$P=(X^{\prime},Y^{\prime})=(\frac{55825}{1521},\frac{-52754017}{237276})$.
$(t,U)=(\frac{52754017}{19773},\frac{223300}{507})$.

Solution:

$365855455514133^{3}+(-366404379540849)^{3}+274462013358^{3}$ $+\\\
60441190910100^{3}=59319^{7}+118638^{7}+177957^{7}+237276^{7}$.

######  Example 7.

$X^{3}+Y^{3}+Z^{3}+U^{3}+U_{0}^{3}=U_{1}^{8}+U_{2}^{8}+U_{3}^{8}+U_{4}^{8}+U_{5}^{8}$
i.e., the sum of $5$ cubics is written as the sum of $5$ eighth powers.

The cubic elliptic curve (2.4) is $Y^{\prime 2}=X^{\prime
3}-\frac{1}{216}-(\frac{-U_{0}^{3}+U_{1}^{8}+U_{2}^{8}+U_{3}^{8}+U_{4}^{8}+U_{5}^{8}}{216Z^{3}})$.
Now letting $Z=2$, $U_{0}=1$, $U_{1}=\frac{2}{3}$, $U_{2}=2$,
$U_{3}=\frac{1}{3}$, $U_{4}=3$, $U_{5}=\frac{4}{3}$, yields
$Y^{\prime 2}=X^{\prime 3}-\frac{14946019}{3779136}$.
Rank=1.
Generator:
$P(X^{\prime},Y^{\prime})=(\frac{4905677718694537513}{2497113793896353424},\frac{-7515135371775048887228789899}{3946003812454638191475172032})$.

$(t,U)=(\frac{7515135371775048887228789899}{328833651037886515956264336},\frac{4905677718694537513}{208092816158029452})$.

Solution:

$X=\frac{6857468069699275855316261227}{328833651037886515956264336}$,

$Y=\frac{-8172802673850821919141318571}{328833651037886515956264336},$

$U=\frac{4905677718694537513}{208092816158029452}$,

$Z=2$, $U_{0}=1$, $U_{1}=\frac{2}{3}$, $U_{2}=2$, $U_{3}=\frac{1}{3}$,
$U_{4}=3$, $U_{5}=\frac{4}{3}$.

######  Remark 2.2.

By choosing the other points on the above elliptic curves such as nP ($n=3,4$,
$\cdots$,), P is one of the elliptic curves generators, we obtain infinitely
many solutions for each case of the above Diophantine equations.

##  3\. The DE $X^{3}+Y^{3}+Z^{3}+aU^{4}=\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}$

######  Main Theorem 3.1.

Consider the DE (1.3) for the case $k=4$, where
$n,t_{i}\in\mathbb{N}\cup\\{0\\}$, and $a\neq 0$, $a_{i}$, are fixed arbitrary
rational numbers.

Let $Y^{2}=X^{3}+FX^{2}+GX+H$, be an elliptic curve in which the coefficients
$F$, $G$, and $H$, are all functions of $a$, $a_{i}$, $t_{i}$ and the other
rational parameters $P_{i}$, $t$, $s$, yet to be found later. If the elliptic
curve has positive rank, depending on the values of $P_{i}$, $s$, the DE (1.3)
has infinitely many integer solutions.

######  Proof 3.1.

Let $X=-Z+t$, $Y=-Z-t$, where $Z$, $t$ are rational variables. By substituting
these variables in the DE (1.3), and some simplifications, we get:

(3.1)
$t^{2}=(\frac{a}{6Z})U^{4}+(\frac{-Z^{2}}{6}-\frac{\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}}{6Z}).$

Now by choosing appropriate values for $U_{i}$ (equal to $P_{i}$), and $Z$
(equal to $s$), such that the rank of the quartic elliptic curve (3.1) to be
positive (Then we get infinitely many rational solutions for $U$, $t$.), by
calculating $U$, $t$, $X$, $Y$, $U_{i}$, $Z$, from the relations (3.1),
$X=-Z+t$, $Y=-Z-t$, $U_{i}=P_{i}$, $Z=s$, after some simplifications and
canceling the denominators of the values obtained for variables, we obtain
infinitely many integer solutions for the DE (1.3). The proof is completed.

######  Remark 3.2.

(If in the quartic elliptic curve (3.1),

(3.2) $Q:=(\frac{-Z^{2}}{6}-\frac{\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}}{6Z}),$

to be square (It is done by choosing appropriate values for $Z$, and
$U_{i}$.), say $q^{2}$, we may use the lemma $1.1$ for transforming this
quartic to a cubic elliptic curve of the form
$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$, where
$a_{i}\in\mathbb{Q}$.
Then we solve the cubic elliptic curve just obtained of the rank$\geq 1$, and
get infinitely many solutions for the above DE.

Generally, it is not essential that $Q$ to be square, because we may transform
the quartic (3.1) to a quartic that its constant number is square if the rank
of the quartic (3.1) is positive. The only important thing is that the rank of
the quartic elliptic curve (3.1), to be positive for getting infinitely many
nontrivial solutions for the above DE. See the example $8$.)

Now let us take

(3.3) $q^{2}=(\frac{-Z^{2}}{6}-\frac{\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}}}{6Z}).$

Then for the quaratic elliptic curve (3.1), we have

(3.4) $t^{2}=(\frac{a}{6Z})U^{4}+q^{2}.$

With the inverse transformation

(3.5) $U=\frac{2qX^{\prime}}{Y^{\prime}},$

and

(3.6) $t=-q+\frac{U^{2}X^{\prime}}{2q},$

the corresponding cubic elliptic curve is

(3.7) $Y^{\prime 2}=X^{\prime
3}+(\frac{aZ}{9}+\frac{a.(\sum_{i=0}^{n}a_{i}U_{i}^{t_{i}})}{9Z^{2}})X^{\prime}.$

Then if the elliptic curve (3.7) has positive rank ( This is done by choosing
appropriate values for $U_{i}$, and $Z$.), by calculating $U$, $t$, $X$, $Y$,
from the relations (3.5), (3.6), $X=-Z+t$, and $Y=-Z-t$, after some
simplifications and canceling the denominators of values obtained for
variables, we obtain infinitely many integer solutions for the DE (1.3). Thus
we conclude that we must choose appropriate values for $U_{i}$, and $Z$, such
that the rank of the elliptic curve (3.7) to be positive, then obtain
infinitely many solutions for the DE (1.3).

###  3.1. Application to examples

######  Example 8.

$X^{3}+Y^{3}+Z^{3}=U^{4}+a_{0}U_{0}^{t_{0}}+a_{1}U_{1}^{t_{1}}$

First of all, we solve the DE
$X^{3}+Y^{3}+Z^{3}=U^{4}+a_{0}U_{0}^{t_{0}}+a_{1}$.
Let $X=-Z+t$ and $Y=-Z-t$. By substituting these variables in the above DE,
and some simplifications, we get

(3.8)
$t^{2}=(\frac{-1}{6Z})U^{4}+(\frac{-Z^{2}}{6}-\frac{a_{0}U_{0}^{t_{0}}+a_{1}}{6Z}).$

Let us transform the quaratic (3.8) to a cubic elliptic curve.

Let

(3.9) $q^{2}=(\frac{-Z^{2}}{6}-\frac{a_{0}U_{0}^{t_{0}}+a_{1}}{6Z}).$

Now with the inverse transformation

(3.10) $U=\frac{2qX^{\prime}}{Y^{\prime}},$

and

(3.11) $t=-q+\frac{U^{2}X^{\prime}}{2q},$

the quaratic (3.8), maps to the cubic elliptic curve

(3.12) $Y^{\prime 2}=X^{\prime 3}+(\frac{2q^{2}}{3Z})X^{\prime}.$

Then if the above elliptic curve has positive rank (This is done by choosing
appropriate arbitrary values for $q$, and $Z$, such as the rank of the above
elliptic curve to be $\geq 1$.), we obtain infinitely many rational solutions
for the DE $X^{3}+Y^{3}+Z^{3}=U^{4}+a_{0}U_{0}^{t_{0}}+a_{1}$.
After multiplying both sides of the above DE by appropriate number $M$, we
obtain integer solutions for the main DE
$X^{3}+Y^{3}+Z^{3}=U^{4}+a_{0}U_{0}^{t_{0}}+a_{1}U_{1}^{t_{1}}$.

As an example, if we take $Z=-2$, $q=7$, $a_{0}=1$, $U_{0}=2$,
$t_{0}=t_{1}=4$, ($a_{1}=580$),
the elliptic curve (3.12) becomes

$Y^{\prime 2}=X^{\prime 3}-\frac{49}{3}X^{\prime}$.
Rank=1.
Generator: $P=(X^{\prime},Y^{\prime})=(\frac{-300}{361},\frac{24730}{6859})$.
$(t,U)=(\frac{-46590103}{6115729},\frac{-7980}{2473})$.

Solution:

$(-210128161627205)^{3}+359736726432969^{3}+(-74804282402882)^{3}=\\\
48803517420^{4}+580.(15124197817)^{4}+30248395634^{4}$.

By taking $Z=-1$, $q=12$, $a_{0}=1$, $U_{0}=3$, $t_{0}=t_{1}=6$,
($a_{1}=136$), the elliptic curve (3.12) becomes $Y^{\prime 2}=X^{\prime
3}-96X^{\prime}$.
Rank=1.
Generator: $P=(X^{\prime},Y^{\prime})=(-8,16)$.
$2P=(25,-115)$.
$(t,U)=(\frac{8652}{529},\frac{-120}{23})$, (for the point 2P).

Solution:

$4856749^{3}+(-4297067)^{3}+(-279841)^{3}=63480^{4}+1587^{6}+136.(529)^{6}$.

Letting $Z=-1$, $q=9$, $a_{0}=1$, $U_{0}=2$, $t_{0}=t_{1}=8$, ($a_{1}=231$),
in the elliptic curve (3.12), yields
$Y^{\prime 2}=X^{\prime 3}-54X^{\prime}$.
Rank=1.
Generator: $P=(X^{\prime},Y^{\prime})=(-2,10)$.
$(t,U)=(\frac{-261}{25},\frac{-18}{5})$.

Solution:

$(-3687500)^{3}+4468750^{3}+(-390625)^{3}=56250^{4}+250^{8}+231.(125)^{8}$.

######  Example 9.

$Y_{1}^{3}+Y_{2}^{3}+Y_{3}^{3}=X_{1}^{5}+X_{2}^{5}+X_{3}^{5}$ ($a=0$)
i.e., the sum of $3$ cubics can be written as the sum of $3$ fifth powers.

We solve this DE with another new method.
Let: $Y_{1}=t+v$, $Y_{2}=t-v$, $Y_{3}=\beta t$, $X_{1}=t+x_{1}$,
$X_{2}=t-x_{1}$, $X_{3}=\alpha t$.

By substituting these variables in the above DE, we get

(3.13)
$(2t^{3}+6tv^{2})+\beta^{3}t^{3}=(2t^{5}+10x_{1}^{4}t+20x_{1}^{2}t^{3})+\alpha^{5}t^{5}.$

Then after some simplifications and clearing the case of $t=0$, we obtain

(3.14)
$v^{2}=\frac{2+\alpha^{5}}{6}t^{4}+\frac{20x_{1}^{2}-2-\beta^{3}}{6}t^{2}+(\frac{5}{3})x_{1}^{4}.$

Note that $(\frac{5}{3})x_{1}^{4}$, is not a square. Then we may not use from
the lemma $1.1$ for transforming this quartic to a cubic elliptic curve, but
we do this work by another method. Let us take $x_{1}=1$, $\alpha=\beta=2$.
Then the quartic (3.14) becomes

(3.15) $v^{2}=\frac{17}{3}t^{4}+\frac{5}{3}t^{2}+\frac{5}{3}.$

By searching, we see that the above quartic has two rational points
$P_{1}=(1,3)$, and $P_{2}=(7,117)$, among others. Let us put $T=t-1$. Then we
get

(3.16) $v^{2}=\frac{17}{3}T^{4}+\frac{68}{3}T^{3}+\frac{107}{3}T^{2}+26T+9.$

Now with the inverse transformation

(3.17) $T=\frac{6(X+\frac{107}{3})-\frac{26^{2}}{6}}{Y},$

and

(3.18) $v=-3+\frac{T(XT-26)}{6},$

the quaratic (3.16), maps to the cubic elliptic curve

(3.19)
$Y^{2}+\frac{26}{3}XY+136Y=X^{3}+\frac{152}{9}X^{2}-204X-\frac{10336}{3}.$

Rank=2.
Generators: $P_{1}=(X^{\prime},Y^{\prime})=(\frac{-152}{9},\frac{280}{27})$,
$P_{2}=(X^{\prime\prime},Y^{\prime\prime})=(\frac{-44}{3},\frac{20}{9})$.

To square the left hand of (3.19), let us put $M=Y+\frac{13}{3}X+68$. Then the
cubic elliptic curve (3.19) transforms to the Weierstrass form

(3.20) $M^{2}=X^{3}+\frac{107}{3}X^{2}+\frac{1156}{3}X+\frac{3536}{3}.$

Rank=2.
Generators: $G_{1}=(X^{\prime},M^{\prime})=(\frac{-44}{3},\frac{20}{3})$,
$G_{2}=(X^{\prime\prime},M^{\prime\prime})=(\frac{-152}{9},\frac{140}{27})$.

Thus we conclude that we could transform the main quartic (3.15) to the cubic
elliptic curve (3.20) of the rank equal to $2$.

Because of this, the above cubic elliptic curve has infinitely many rational
points and we may obtain infinitely many solutions for the above DE too.
Since $G_{1}=(X^{\prime},M^{\prime})=(\frac{-44}{3},\frac{20}{3})$, we get
$(t,v)=(7,-117)$, that is on the (3.15), by calculating $X_{i}$, $Y_{i}$, from
the above relations and after some simplifications and canceling the
denominators of $X_{i}$, $Y_{i}$, we obtain a solution for the above DE:

$(-110)^{3}+124^{3}+14^{3}=8^{5}+6^{5}+14^{5}$.

It is iteresting to see that $(-110)+124+14=8+6+14$, too.

Also we have $2G_{2}=(\frac{373}{36},\frac{-21721}{216})$.

By using this new point $2G_{2}=(\frac{373}{36},\frac{-21721}{216})$, we get
$(t,v)=(\frac{11}{47},\frac{2943}{2209})$.

Solution:

$359227580^{3}+(-251874598)^{3}+107352982^{3}=128122^{5}+(-79524)^{5}+48598^{5}$.

The Sage software has been used for calculating the rank of the elliptic
curves. (see [3])

## References

  * [1] F. Izadi and M. Baghalaghdam, On the Diophantine equation $X^{4}+Y^{4}=2U^{4}+\sum_{i=1}^{n}T_{i}U_{i}^{\alpha_{i}}$, submitted, $(2016)$.
  * [2] F. Izadi and M. Baghalaghdam, On the six-three power Diophantine equations, submitted, $(2016)$.
  * [3] SAGE software, available from http://sagemath.org.
  * [4] L. C. Washington, Elliptic Curves: Number Theory and Cryptography, Chapman-Hall, $(2008)$.