Dynamical Dark sectors and Neutrino masses and abundances
Weiqiang Yang Eleonora Di Valentino Olga Mena Supriya Pan
astro-ph.CO gr-qc







d11102004@163.com
Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. China


eleonora.divalentino@manchester.ac.uk
Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK


omena@ific.uv.es
IFIC, Universidad de Valencia-CSIC, 46071, Valencia, Spain


supriya.maths@presiuniv.ac.in
Department of Mathematics, Presidency University, 86/1 College Street,  Kolkata 700073, India 




We investigate generalized interacting dark matter-dark energy scenarios with a time-dependent coupling parameter, allowing also for freedom in the neutrino sector. The models are tested in the phantom and quintessence regimes, characterized by an equation of state w_x<-1 and w_x>-1, respectively. Our analyses show that for some of the scenarios the existing tensions on the Hubble constant H_0 and on the clustering parameter S_8 can be significantly alleviated. The relief is either due to (a) a dark energy component which lies within the phantom region; or  (b) the presence of a dynamical coupling in quintessence scenarios. The inclusion of massive neutrinos into the interaction schemes does not affect neither the constraints on the cosmological parameters nor the bounds on the total number or relativistic degrees of freedom N_ eff, which are found to be extremely robust and, in general, strongly consistent with the canonical prediction N_ eff=3.045. The most stringent bound on the total neutrino mass M_ν is M_ν<0.116 eV and it is obtained within a quintessence scenario in which the matter mass-energy density is only mildly affected by the presence of a dynamical dark sector coupling.





Dynamical Dark sectors and Neutrino masses and abundances
    Supriya Pan
    December 30, 2023
=========================================================





§ INTRODUCTION


Cosmological models where a non-gravitational interaction between the dark fluids of the universe, dark matter and dark energy, are still a very appealing and  interesting solution to the so-called why now? problem. Early models were based on coupled quintessence scenarios <cit.>, while more recent phenomenological approaches have adopted a number of possible parametrizations of the  energy exchange rate, see e.g. <cit.>. Following our pioneering previous work <cit.> we shall consider here a time-dependent coupling in non-minimal cosmologies. Given the fact that neutrinos can play a non-standard role within non-minimal dark energy scenarios <cit.>, we extend our previous analyses by inspecting the impact of neutrino properties within  interacting cosmologies  with a time-dependent coupling. We also generalize the work of Ref. <cit.> with the inclusion of a constant dark energy state parameter that may freely vary in a certain region. This picture also entails the case of a coupling parameter that remains constant in cosmic time. For our analyses we have assumed that our  universe is homogeneous and isotropic, that is, its geometry is well described by the Friedmann-Lemaître-Robertson-Walker line element. In order to perform robust statistical analyses, we shall make use of various cosmological datasets such as the Cosmic Microwave Background radiation, Baryon Acoustic Oscillation distance measurements, and, finally, a local measurement of the Hubble constant from the Hubble Space Telescope. 



The manuscript has been organized as follows: In Sec.  <ref> we briefly introduce the gravitational equations for the two interacting dark fluids. Section <ref> describes the observational data, methodology and the priors imposed on the cosmological parameters. Section <ref> presents the current observational constraints on the interacting cosmic scenarios considered here. Section <ref> contains our main conclusions. 


  


§ INTERACTING DARK SECTORS: GRAVITATIONAL EQUATIONS


Observations suggest that at large scales, our universe is homogeneous and isotropic and therefore well described by the Friedmann-Lemaître-Robertson-Walker (FLRW) line element 


    ds^2 =  - dt^2 + a^2 (t) [dr^2/1-κ r^2 + r^2 (dθ ^2 + sin^2 θ d ϕ^2 ) ] ,

where a(t) is the expansion scale factor of the universe and (t, r, θ, ϕ) are the co-moving coordinates. Having specified the metric of the underlying geometry of our universe, we assume in the following that the gravitational sector of the universe is described by General Relativity, the matter sector is minimally coupled to gravity, and, finally, that there is a non-gravitational interaction between the dark sectors of the universe, namely, between the pressureless dark matter (DM) and the dark energy (DE) fluids:


    ρ̇_c + 3 H ρ_c = - Q ,
          ρ̇_x + 3 H (1+w_x) ρ  =  Q ,

where H ≡ȧ/a is the Hubble rate of the FLRW universe; ρ_c (p_c), ρ_x (p_x) are the energy density (pressure) for DM and DE respectively (albeit the DM fluid, being pressureless here, has p_c  = 0), w_x  = p_x/ρ_x denotes the barotropic DE equation of state parameter (assumed here to be constant) and, finally, Q determines the interaction rate between DM and DE. In general, when a specific form of the interaction rate is given, one can solve either analytically or numerically the background evolution for ρ_c and ρ_x. We shall explore here the (time-dependent) interaction models of Ref. <cit.>:

    IDE1:    Q = 3 ξ (a) H ρ_x, 
       IDE2:    Q= 3 ξ (a) H ρ_c ρ_x/ρ_c+ ρ_x,

where ξ (a) is a 
time-dependent dimensionless coupling parameter. Similar to our earlier work <cit.>, we keep the parametrization of ξ (a) as follows 

    ξ (a) = ξ_0 + ξ_a   (1-a) ,

where ξ_0 and ξ_a are real constants. 
Finally, based on the stability criteria of the perturbation evolution <cit.>, we shall classify the models as
 


    IDE1p:   w_x< -1, ξ_0 < 0, ξ_a < 0, 
     IDE1q:   w_x> -1, ξ_0 > 0, ξ_a > 0,

for the IDE1 case, and, equivalently,
  
  
    IDE2p:   w_x< -1, ξ_0 < 0, ξ_a < 0, 
     IDE2q:   w_x> -1, ξ_0 > 0, ξ_a > 0,


for the IDE2 model, where p and q in IDEp and IDEq stand for phantom and quintessence regimes, respectively.






§ OBSERVATIONAL DATA AND METHODOLOGY



In the following we briefly describe the cosmological data sets used in this work. 


    
  * Cosmic Microwave Background (CMB): our  default data set is the one containing the latest CMB temperature and polarization measurements  in both the high and low multipole regions, i.e. Plik TT,TE,EE + lowl + lowE, from the final 2018 Planck legacy release <cit.>.
    
    
  * Baryon Acoustic Oscillations (BAO): we make use of several BAO measurements from different cosmological observations, as considered by the Planck collaboration <cit.>: 6dFGS <cit.>, SDSS-MGS <cit.>, and BOSS DR12 <cit.> surveys. 
    
    
    
  * Hubble constant Gaussian prior (R19): we assume a Gaussian prior on the Hubble constant, in agreement with that obtained by the SH0ES collaboration in 2019, i.e. H_0 = 74.03 ± 1.42 km/s/Mpc at 68% CL <cit.>.
    
    



For the analysis of the cosmological data, we adopt a fiducial model described by nine cosmological parameters. In particular, we vary the six parameters of the standard ΛCDM model, i.e. the baryon energy density Ω_ bh^2, the cold dark matter energy density Ω_ ch^2, the ratio between the sound horizon and the angular diameter distance at decoupling 100θ_MC, the reionization optical depth 
τ, the spectral index n_s and the amplitude of the scalar primordial power spectrum A_s. In addition we vary the three parameters of the dark sector physics considered here, i.e. the DE equation of state w_x and the strength of the coupling, parametrized by ξ_0 and ξ_a, see Eq. (<ref>). The parameter space will therefore be described by: 


    𝒫≡{Ω_bh^2, Ω_ch^2, 100θ_MC, τ, n_s, log[10^10A_s], 
    ξ_0, ξ_a, w_x} .

As aforementioned, the stability of the perturbation evolution restricts the IDE scenarios to two phantom cases (w_x<-1) (IDE1p, Eq.(<ref>) and IDE2p, Eq.(<ref>)) with ξ_0<0 and ξ_a<0 and two quintessence regimes (w_x>-1) (IDE1q, Eq.(<ref>) and IDE2q, Eq.(<ref>)) with ξ_0>0 and ξ_a>0. Table <ref> lists the priors on all the parameters considered in this work. 

We shall also consider an enlarged cosmological scenario with eleven parameters, allowing the sum of the neutrino masses M_ν and the number or relativistic degrees of freedom N_ eff to freely vary (IDE + M_ν + N_ eff): 


    𝒫≡{Ω_bh^2, Ω_ch^2, 100θ_MC, τ, n_s, log[10^10A_s], 
    ξ_0, ξ_a, w_x, M_ν, N_ eff} ,


and also in this case we will have four cases, depending on the scenario of IDE considered and on the phantom or quintessence regime, i.e. IDE1p, IDE1q, IDE2p and IDE2q, respectively.

To derive the constraints on the cosmological parameters we shall use a modified version with models IDE1 and IDE2 implemented of the publicly available Markov Chain Monte Carlo code  <cit.> package. This version supports the new 2018 Planck likelihood <cit.> and uses a convergence diagnostic following the Gelman-Rubin criteria <cit.>. 









§ RESULTS






 §.§ IDE1

In the following we shall show the results obtained for the IDE1 scenario presented in Eq. (<ref>), both in the phantom and in the quintessence regimes, and with and without varying the neutrino sector.




  §.§.§ IDE1p


The results for the IDE1 model in the phantom regime, i.e. with w_x<-1, ξ_0 < 0 and ξ_a < 0 are reported in Tab.  <ref> and Fig. <ref>.

For an interacting dark energy with a phantom-like equation of state, the CDM energy density Ω_ch^2 is larger than in the ΛCDM model, provided the energy transfer is from the DE to the DM sector <cit.>. Furthermore, due  to the strong degeneracy between w_x and H_0, see Fig. <ref>, the Hubble constant is almost unconstrained for CMB only data. The well known H_0 tension is strongly alleviated within this model. While ξ_0 has only lower limit for all the combinations of data  considered here, being therefore consistent with a vanishing interaction at present, we find ξ_a different from zero at one standard deviation for the CMB only (ξ_a=-0.077^+0.064_-0.032 at 68% CL) and for the CMB+R19 (ξ_a=-0.077^+0.059_-0.037 at 68% CL) cases. A very interesting  feature of this model is the strong evidence for a phantom-like equation of state w_x<-1  for all the data combinations, with a statistical significance increasing from 1σ for the CMB only case (w_x=-1.80^+0.49_-0.39 at 68% CL), to about 2σ for CMB+BAO.
Finally, the S_8 parameter moves towards lower values for the CMB only case, enough to bring it in agreement with the cosmic shear experiments DES <cit.>, KiDS-450 <cit.>, CFHTLenS <cit.>, or the combination of KiDS+VIKING-450 and DES-Y1 <cit.> , i.e.  S_8=0.789±0.037 at 68% CL. However, when the BAO or the R19 priors are added to the CMB, the S_8 values are increased, restoring the tension at more than 3 standard deviations. 

Finally, in Table <ref>, we show the χ^2 values for this model, as well as other models considered in this work, for all the observational datasets employed here. In the same Table <ref>, we have also shown the χ^2 values for the non-interacting scenario wCDM model as the reference model.   
From Table <ref> we can see that the χ^2 values obtained for this scenario (i.e., IDE1p)  are improved with respect to the wCDM model of about 2 (for Planck 2018+BAO) and 4.5 (Planck 2018+R19), even if in our case we have two more degrees of freedom compared to the wCDM model. 





                                         
                                        




  §.§.§ IDE1p + M_ν + N_ eff


The results for the IDE1 model in the phantom regime with the addition of the neutrino parameters, i.e. M_ν and N_ eff, are shown in Tab. <ref> and Fig. <ref>.

The constraints from the previous section on the cosmological parameters and their correlations (IDE1p) are barely affected by allowing M_ν and N_ eff to freely vary simultaneously.
In particular, Ω_ch^2 is larger than in ΛCDM model and the Hubble constant tension with R19 is solved within 3σ even when BAO data are included.
Also in this case ξ_0 has just a lower limit and is consistent with zero, while ξ_a is different from zero at one standard deviation for the CMB only (ξ_a=-0.081^+0.060_-0.037 at 68% CL) and CMB+R19 (ξ_a=-0.087^+0.055_-0.048 at 68% CL) cases, but consistent with zero when BAO data are included. 

The indication for a phantom equation of state w_x<-1 is instead present for all the dataset combinations with a statistical significance always larger two standard  deviations, even for the CMB only case.
The neutrino sector parameters M_ν and N_ eff are mostly uncorrelated with the other cosmological parameters, with the exception of w_x that strongly anti-correlates with the total neutrino mass,  M_ν. The existence of anti-correlation between w_x and M_ν is not new, in fact, in the usual non-interacting w (z)CDM cosmology, this has been already pointed out  <cit.>, however, the interesting observation in this case that we find, even if the presence scenario allows an interaction in the dark sector, 
this anti-correlation does not get affected due to such interaction. The preference for w_x<-1 is therefore the reason for the much weaker upper limits on M_ν with respect to the same combinations of data within a ΛCDM model <cit.>. 
The most stringent limit we find on the sum of the neutrino masses is when adding BAO data to the CMB, i.e. M_ν<0.162 eV at 95% CL. 

Regarding the constraints on the effective number of relativistic degrees of freedom N_ eff, these are completely unaffected by the inclusion of the interaction ξ(a): in this scenario N_ eff is always consistent with its expected value of  3.045 <cit.>.

In Table <ref> we can see that the χ^2 values for this scenario (i.e., IDE1p + M_ν + N_ eff) are always below compared to the wCDM + M_ν + N_ eff model, up to 4.2 for Planck 2018+R19. We note that the model IDE1p + M_ν + N_ eff has two more degrees of freedom compared to the wCDM + M_ν + N_ eff model.



                                                                                
                                         




  §.§.§ IDE1q


The results for the IDE1 model in the quintessence regime, Eq. (<ref>), are reported in Tab. <ref> and Fig. <ref>.

For an interacting dark energy with a quintessence-like equation of state, the CDM energy density Ω_ch^2 is always smaller than in a ΛCDM model: indeed,  only an upper limit for this cosmological parameter is found 
<cit.>.
The most interesting feature of this IDE1q scenario is that, even if the well-known anti-correlation between w_x and H_0 is present, see Fig. <ref>, the positive correlation between ξ_0 and H_0 shifts the Hubble constant towards higher values, solving the H_0 tension within 1σ for the CMB only case (H_0=70.2^+4.1_-3.1 km/s/Mpc at 68% CL).

Contrarily to the IDE1p case, in this IDE1q scenario the value of ξ_0, i.e. the interaction today, is found to be different from zero at low (high) significance for the CMB (CMB+R19) data. While for the CMB only and  the CMB+R19 cases only an upper limit on w_x  is found, an indication at 1σ for w_x>-1 appears for CMB+BAO (w_x=-0.895^+0.040_-0.093 at 68% CL). In this scenario, the S_8 parameter moves towards larger values, however the error bars are very large, enabling an agreement with cosmic shear experiments.

Finally, in Table <ref> we can see that the χ^2 for this scenario (i.e., IDE1q) is systematically higher than the wCDM model, therefore it is disfavoured by the fit of the data. 





                                                                                
                                         




  §.§.§ IDE1q + M_ν + N_ eff


The results for the IDE1 model in the quintessence regime  extended to include the neutrino parameters are shown in Tab. <ref> and Fig. <ref>.

Similarly to the phantom case, both the constraints on the cosmological parameters and the correlations presented above are robust and are not affected by the introduction of the neutrino parameters M_ν and N_ eff. As in the previous section, ξ_0 is found to be different from zero at one standard deviation for the CMB only dataset (ξ_0=0.137^+0.087_-0.089 at 68% CL), at several standard deviations for CMB+R19, and it has just an upper limit for the CMB+BAO case. In this extended scenario ξ_a is always consistent with zero, as well as w_x is consistent with -1 at 95% CL for all the data combinations.

Also in this case the only important correlation between the neutrino sector and the remaining cosmological parameters is the one present between M_ν and w_x. However, in this quintessence regime, the CMB only upper limit on M_ν is stronger than the one found in the phantom regime (see Ref. <cit.>), and including the R19 prior this upper bound becomes even stronger (M_ν<0.221 eV at 95% CL). We note here that similar to the w(z)CDM case explored in  <cit.> the anti-correlation between M_ν and w_x remains unaltered in presence of the interaction between these dark sectors. This is an important point which clarifies that the anti-correlation between M_ν and w_x seems to be independent of the coupling in the dark sector. The most stringent limit in this case we find on the sum of the neutrino masses is when adding BAO data to the CMB, i.e. M_ν<0.189 eV at 95% CL.

Finally, in this extended scenario (as in the phantom one), the constraints on the effective number of relativistic degrees of freedom N_ eff are completely consistent with its canonical value N_ eff=3.045 for all the data combinations.

In Table <ref> we can see that the χ^2 values for this scenario (i.e., IDE1q + M_ν + N_ eff) are always larger than the wCDM + M_ν + N_ eff model. Therefore, this case is also disfavoured by the data.



                                                                                
                                         



 §.§ IDE2

In the  following we shall show the bounds on the cosmological parameters obtained for the IDE2 scenario, see Eq. (<ref>), both in the phantom and in the quintessence regimes, and with and without varying the neutrino sector.




  §.§.§ IDE2p


The results for the IDE2 model in the phantom regime are reported in Tab. <ref> and Fig. <ref>.

In the IDE2 model the interaction rate depends on both the cold dark matter density and the dark energy density. For this reason the flux of energy in the dark sector, from DE to DM and vice versa, can change with time. In this scenario, the bound on the cold dark matter energy density Ω_ch^2 is in perfect agreement with that obtained within a ΛCDM model, as we can notice from Tab. <ref>.
The well-known negative correlation present between w_x and H_0 when w_x is in the phantom regime (see Fig. <ref>) shifts the Hubble constant towards very larger values. The H_0 tension is then reduced within three standard deviations for all the combination of data sets considered in this work.

Both the interaction parameters ξ_0 and ξ_a have only a lower limit for all the dataset combinations at 68% CL and are consistent with zero, i.e. consistent  with a model without interaction, as we notice from Tab. <ref>. A strong evidence for a phantom equation of state w_x<-1 is present at more than 2σ for the CMB only case, and at many standard deviations for the CMB+R19 combination. However, this is not the case for CMB+BAO data. In this scenario IDE2p, the S_8 value shifts down enough to solve the tension with the cosmic shear experiments for all the data combinations considered here.

Finally, for this IDE2p scenario, we have that the χ^2 values are systematically higher than the wCDM model, as we can see in Table <ref>, showing that this is disfavoured by the fit of the data.



                                                                                
                                         





  §.§.§ IDE2p + M_ν + N_ eff


The results for the IDE2 model within the phantom regime with the addition of the neutrino parameters, i.e. M_ν and N_ eff, are shown in Tab. <ref> and Fig. <ref>.


As in the IDE1 model, the results from the previous section are not modified significantly with the introduction of M_ν and N_ eff as extra parameters. Indeed, in this scenario the bound on Ω_ch^2 is really robust, shifted only one standard deviation towards lower values with respect to the case in which the neutrino parameters are fixed, but still in agreement with what obtained in a ΛCDM model, see e.g. Tabs. <ref> and <ref>. Also here the Hubble constant is almost unconstrained when the CMB data only is considered, due to the negative correlation with w_x, see Fig. <ref>. For the very same reason, the H_0 tension is reduced within 2.5σ even after including BAO data in the analysis.

The neutrino sector parameters M_ν and N_ eff do not show any strong correlation with the other cosmological parameters, with the exception of w_x, that is anti-correlated with the total neutrino mass M_ν. As already pointed out, this anti-correlation between w_x and M_ν is independent of the coupling in the dark sector. 
The preference for w_x<-1 is the reason of the softening of the M_ν upper limit. The most stringent bound we find on the sum of the neutrino masses is when adding BAO data to the CMB, i.e. M_ν<0.181 eV at 95% CL. The mean values of the effective number of relativistic degrees of freedom N_ eff are lower than in a model without interaction ξ(a), even if always highly consistent with its expected value N_ eff=3.045.

In Table <ref> we can see that the χ^2 value for Planck 2018 data for this scenario (i.e., IDE2p + M_ν + N_ eff) is larger than the corresponding χ^2 value obtained for the wCDM + M_ν + N_ eff model, but concerning the other two datasets, the χ^2 values for IDE2p + M_ν + N_ eff are lower than the wCDM + M_ν + N_ eff model. However, these lower values are consistent with the introduction of two more degrees, so do not correspond to an actual improvement of the fit. Therefore, these cases are almost equivalent.



                                                                                
                                         





  §.§.§ IDE2q


The results for the IDE2 model in the quintessence regime, Eq. (<ref>), are presented in Tab. <ref> and Fig. <ref>.

In the IDE2q scenario, the well known anti-correlation present between w_x and H_0, shifts the Hubble constant towards lower values, see Fig. <ref>, exacerbating the H_0 tension at more than 4σ with respect to previous models.

Both the interaction parameters ξ_0 and ξ_a are constrained by an upper limit for all the dataset combinations and are uncorrelated with the other cosmological parameters, as can be noticed from Fig. <ref>.
Only an upper limit is present also for the equation of state in the quintessence regime w_x>-1, and the S_8 tension with the cosmic shear experiments is restored.

Finally, even this IDE2q scenario is disfavoured by the fit of the data as showed in Table <ref>.



                                                                                
                                         




  §.§.§ IDE2q + M_ν + N_ eff


The results for the IDE2 model in the quintessence regime with  M_ν plus N_ eff as additional parameters are shown in Tab. <ref> and Fig. <ref>.
However, for this model IDE2q, the neutrino parameters M_ν and N_ eff are  correlated with other cosmological parameters. In particular, we notice an important correlation with the Hubble constant H_0. This degeneracy is responsible, when the R19 prior is included in the data, i.e. for the combination CMB+R19, of the shift of N_ eff towards higher values. The value N_ eff = 3.43 ^+0.15_-0.16 at 68% CL deviates from the canonical expectation more than 2 standard deviations. In this IDE2q scenario we  obtain our strongest limit on the M_ν, M_ν<0.116 eV at 95% CL, as expected in quintessential non-interacting scenarios <cit.> where an anti-correlation between w_x and M_ν exists similar to this coupled case. 

In Table <ref> we can see that the χ^2 values for IDE2q + M_ν + N_ eff are larger than the χ^2 values obtained in the wCDM + M_ν + N_ eff model for Planck 2018 alone and Planck 2018 + R19, but lower for Planck 2018 + BAO. However, this improvement quantified through Δχ^2 ∼ 2 is consistent with the the fact that in the interacting scenario we have two extra degrees of freedom, so it does not correspond to an actual improvement of the fit. 



                                                                                
                                         







                                         
                                        





§ SUMMARY AND CONCLUSIONS



In this manuscript we further investigate the presence of a exchange rate Q between dark matter (DM) and dark energy (DE) allowing for a time-dependent coupling <cit.>. We add new ingredients in the models, such as (i) freely varying neutrino parameters and (ii) a DE with a constant, freely varying equation-of-state, rather than vacuum dark energy. 
We restrict ourselves to the natural form of the coupling parameter ξ (a) = ξ_0 + (1-a)ξ_a and consider two interacting models, namely, IDE1 (Q = 3 H [ξ_0 + ξ_a (1-a)] ρ_x) and IDE2 (Q = 3 H [ξ_0 + ξ_a (1-a)] ρ_c ρ_x/ρ_c+ρ_x). In order to avoid instabilities in the perturbation evolution, we consider the  regions (A) w_x< -1, ξ_0 < 0, ξ_a < 0, and (B) w_x> -1, ξ_0 > 0, ξ_a > 0, and investigate the interacting scenarios with and without the presence of neutrinos. The scenario with phantom DE equation-of-state (w_x< -1) is labeled as IDEp and the scenario where DE has a quintessence-like equation-of-state (w_x> -1) is labeled as IDEq. 
Let us summarize the main observational results that we find for all these scenarios:



    
  * IDE1: We have explored this interaction model for both regimes, namely, w_x< - and w_x> -1 with and without the presence of neutrinos. We have therefore investigated four different scenarios: IDE1p, IDE1p + M_ν +  N_ eff, IDE1q, and IDE1q +  M_ν +  N_ eff.
    
    We find that for both IDE1p and IDE1p + M_ν+ N_ eff, Ω_ch^2 is larger than within the ΛCDM cosmology and w_x prefers a phantom nature with high significance. The parameter ξ_0 determining the current value of the DM-DE interaction is consistent with a null value, while ξ_a prefers a value different from zero (albeit only mildly). We also notice that within these two phantom frameworks, the tension on H_0 is alleviated satisfactorily for all the data combinations considered here (CMB, CMB+BAO and CMB+R19). Concerning the S_8 parameter, its tension is significantly reduced only for the case of CMB data alone. The inclusion of M_ν and N_ eff to IDE1p does not change the constraints on other parameters.  The most stringent bound on M_ν is obtained for the CMB+BAO case and is M_ν< 0.162 eV  at  95% CL.  
    
    As regards the remaining two scenarios IDE1q and IDE1q + M_ν + N_ eff, similarly to the phantom case, the inclusion of the neutrinos does not affect the constraints on the remaining cosmological parameters. The tightest bound on M_ν appears for CMB+BAO case (M_ν< 0.189 eV at 95% CL) which is slightly larger than the one obtained within the ΛCDM framework for the same data combination. Contrarily to the previous two cases, the value of  Ω_c h^2 is much smaller. The parameter  ξ_0 is found to be non-zero for all the cases. However, ξ_a is consistent with zero for all the datasets exploited in this work. The H_0 tension is solved for the CMB case (H_0 = 70.2^+4.1_-3.1 km/s/Mpc) and due to the very large error bars on the S_8 parameter, the S_8 tension is mildly alleviated. 
    
    
    
  * IDE2: Using the very same observational data than for IDE1, we have investigated four scenarios, namely, IDE2p, IDE2p + M_ν + N_ eff, IDE2q, and IDE2q +  M_ν +  N_ eff.  
    
    The scenario IDE2p is very interesting because both the H_0 and S_8 tensions are alleviated for all the data combinations used in this analysis. The dark energy equation of state shows a strong preference for  a  phantom nature. When neutrinos are considered into this picture (IDE2p + M_ν+ N_ eff) no significant changes are obtained, apart from  the large anti-correlation between w_x and M_ν. The DE equation of state still prefers w_x< -1 with high significance. Finally, for both IDE2p and IDE2p + M_ν + N_ eff models we find that ξ_0 and ξ_a are consistent with zero, leading to a negligible preference for an interacting scenario. 
    
    The scenario IDE2q is quite different from the previous cases. Within this interaction scheme we find that none of the tensions (H_0,  S_8) are alleviated. We do not find any evidence for an interaction among the dark sectors, since both the parameters ξ_0 and ξ_a, quantifying the interaction, are consistent with zero. An interesting outcome of this scenario is that it provides the most stringent bound on M_ν found in this study (M_ν< 0.116 eV at 95% CL), which is obtained for the combination of CMB+R19.   
    
    

Finally, to conclude, 
the bounds on the effective number of neutrino species N_ eff as we see in almost all of the scenarios above are extremely robust and consistent with the standard value of N_ eff = 3.046 and are, therefore, completely unaffected by the dynamics of the dark sectors. However, the H_0 tension scenario which for some cases in this work is alleviated, needs further investigations in light of other cosmological datasets. The excess of lensing in the CMB damping tail (see for instance <cit.>) might be an appealing investigation in this context.



The authors thank the referee for some valuable comments aiming to improve the quality of the manuscript. 
WY acknowledges the support from the National Natural Science Foundation of China under Grants No.  
11705079 and No.  11647153. EDV was supported from the European Research Council in the form of a Consolidator Grant 
with number 681431. OM is supported  by  the  Spanish  grants FPA2017-85985-P and SEV-2014-0398 of the MINECO, by PROMETEO/2019/083 and by the European Union Horizon  2020  research  and  innovation  program  (grant  agreements No.  690575 and 67489). SP was supported by the the Science and Engineering Research Board (SERB), Govt. of India through the Mathematical Research Impact-Centric Support Scheme (MATRICS), File No. MTR/2018/000940. 


  

  
  Carroll:1998zi 
  S. M. Carroll,
  Quintessence and the rest of the world,
  Phys. Rev. Lett.  81, 3067 (1998)
  
  [astro-ph/9806099].
  
  
Wetterich:1994bg 
  C. Wetterich,
  The Cosmon model for an asymptotically vanishing time dependent cosmological 'constant',
  Astron. Astrophys.  301, 321 (1995)
  [hep-th/9408025].
  
  
Amendola-ide1 L. Amendola, Coupled Quintessence,
Phys. Rev. D 62, 043511 (2000) 
[arXiv:astro-ph/9908023].

Amendola-ide2 L. Amendola and C. Quercellini, Tracking and coupled dark energy as seen by WMAP, 
Phys. Rev. D 68, 023514 (2003) 
[arXiv:astro-ph/0303228].

Pavon:2005yx 
D. Pavón and W. Zimdahl, Holographic dark energy and cosmic coincidence,
Phys. Lett. B 628, 206 (2005)
[arXiv:gr-qc/0505020]. 

delCampo:2008sr 
S. del Campo, R. Herrera and D. Pavón, Toward a solution of the coincidence problem,
Phys. Rev. D 78, 021302 (2008)
[arXiv:0806.2116 [astro-ph]].

delCampo:2008jx 
S. del Campo, R. Herrera and D. Pavón, Interacting models may be key to solve the cosmic coincidence problem,
J. Cosmol. Astropart. Phys. 0901, 020 (2009)
[arXiv:0812.2210 [gr-qc]].

Billyard:2000bh
  A. P. Billyard and A. A. Coley,
  Interactions in scalar field cosmology,
  Phys. Rev. D 61, 083503 (2000)
  [astro-ph/9908224].
  
  Barrow:2006hia 
  J. D. Barrow and T. Clifton,
  Cosmologies with energy exchange,
  Phys. Rev. D 73, 103520 (2006)
  [gr-qc/0604063].
  
  Amendola:2006dg
  L. Amendola, G. Camargo Campos and R. Rosenfeld,
  Consequences of dark matter-dark energy interaction on cosmological parameters derived from SNIa data,
  Phys. Rev. D 75, 083506 (2007)
  [astro-ph/0610806].
  
 
He:2008tn
  J. H. He and B. Wang,
  Effects of the interaction between dark energy and dark matter on cosmological parameters,
  JCAP 0806, 010 (2008)
  [arXiv:0801.4233 [astro-ph]].
  
   Valiviita:2008iv 
  J. Väliviita, E. Majerotto and R. Maartens,
  Instability in interacting dark energy and dark matter fluids,
  JCAP 0807, 020 (2008)
  
  [arXiv:0804.0232 [astro-ph]].
  
  
 Gavela:2009cy 
  M. B. Gavela, D. Hernandez, L. Lopez Honorez, O. Mena and S. Rigolin,
  Dark coupling,
  JCAP 0907, 034 (2009)
  
  [arXiv:0901.1611 [astro-ph.CO]].
  
   Majerotto:2009np 
  E. Majerotto, J. Valiviita and R. Maartens,
  Adiabatic initial conditions for perturbations in interacting dark energy models,
  Mon. Not. Roy. Astron. Soc.  402, 2344 (2010)
  
  [arXiv:0907.4981 [astro-ph.CO]].
  
  Gavela:2010tm 
  M. B. Gavela, L. Lopez Honorez, O. Mena and S. Rigolin,
  Dark Coupling and Gauge Invariance,
  JCAP 1011, 044 (2010)
  
  [arXiv:1005.0295 [astro-ph.CO]].
  
  Clemson:2011an 
  T. Clemson, K. Koyama, G. B. Zhao, R. Maartens and J. Valiviita,
  Interacting Dark Energy – constraints and degeneracies,
  Phys. Rev. D 85, 043007 (2012)
  
  [arXiv:1109.6234 [astro-ph.CO]]. 
  
  Pan:2012ki
  S. Pan, S. Bhattacharya and S. Chakraborty,
  An analytic model for interacting dark energy and its observational constraints,
  Mon. Not. Roy. Astron. Soc.  452, no.3,  3038 (2015)
  [arXiv:1210.0396 [gr-qc]].
  
Pan:2013rha 
  S. Pan and S. Chakraborty,
  Will there be again a transition from acceleration to deceleration in course of the dark energy evolution of the universe?,
  Eur. Phys. J. C 73, 2575 (2013)
  [arXiv:1303.5602 [gr-qc]].
  
  Yang:2014vza
  W. Yang and L. Xu,
  Testing coupled dark energy with large scale structure observation,
  JCAP 1408, 034 (2014)
  [arXiv:1401.5177 [astro-ph.CO]].
  
Yang:2014gza
  W. Yang and L. Xu,
  Cosmological constraints on interacting dark energy with redshift-space distortion after Planck data,
  Phys. Rev. D 89, no.8,  083517 (2014)
  [arXiv:1401.1286 [astro-ph.CO]].
  
 Nunes:2014qoa 
  R. C. Nunes and E. M. Barboza,
  Dark matter-dark energy interaction for a time-dependent EoS parameter,
  Gen. Rel. Grav.  46, 1820 (2014)
  
  [arXiv:1404.1620 [astro-ph.CO]]. 
  
 
vandeBruck:2015ida
C. van de Bruck and J. Morrice,
Disformal couplings and the dark sector of the universe,
JCAP 04, 036 (2015)

[arXiv:1501.03073 [gr-qc]].
  
  Nunes:2016dlj 
  R. C. Nunes, S. Pan and E. N. Saridakis,
  New constraints on interacting dark energy from cosmic chronometers,
  Phys. Rev. D 94, no. 2, 023508 (2016)
  
  [arXiv:1605.01712 [astro-ph.CO]].
  
  Kumar:2016zpg 
  S. Kumar and R. C. Nunes,
  Probing the interaction between dark matter and dark energy in the presence of massive neutrinos,
  Phys. Rev. D 94, no. 12, 123511 (2016)
  
  [arXiv:1608.02454 [astro-ph.CO]].
  
Pan:2016ngu 
  S. Pan and G. S. Sharov,
  A model with interaction of dark components and recent observational data,
  Mon. Not. Roy. Astron. Soc.  472, no. 4, 4736 (2017)
  [arXiv:1609.02287 [gr-qc]].
  
  vandeBruck:2016hpz
C. van de Bruck, J. Mifsud and J. Morrice,
Testing coupled dark energy models with their cosmological background evolution,
Phys. Rev. D 95, no.4, 043513 (2017)

[arXiv:1609.09855 [astro-ph.CO]].
  
  Mukherjee:2016shl 
  A. Mukherjee and N. Banerjee,
  In search of the dark matter dark energy interaction: a kinematic approach,
  Class. Quant. Grav.  34, no. 3, 035016 (2017)
  
  [arXiv:1610.04419 [astro-ph.CO]].
  
  Sharov:2017iue
  G. S. Sharov, S. Bhattacharya, S. Pan, R. C. Nunes and S. Chakraborty,
  A new interacting two fluid model and its consequences,
  Mon. Not. Roy. Astron. Soc.  466, 3497 (2017)
  [arXiv:1701.00780 [gr-qc]].
  
  Kumar:2017dnp 
  S. Kumar and R. C. Nunes,
  Echo of interactions in the dark sector,
  Phys. Rev. D 96, no. 10, 103511 (2017)
  
  [arXiv:1702.02143 [astro-ph.CO]].
  
  Yang:2017yme 
  W. Yang, N. Banerjee and S. Pan,
  Constraining a dark matter and dark energy interaction scenario with a dynamical equation of state,
  Phys. Rev. D 95, no. 12, 123527 (2017)
  
  
  [arXiv:1705.09278 [astro-ph.CO]].
 
 
 Yang:2017zjs 
  W. Yang, S. Pan and J. D. Barrow,
  Large-scale Stability and Astronomical Constraints for Coupled Dark-Energy Models,
  Phys. Rev. D 97, no. 4, 043529 (2018)
  
  [arXiv:1706.04953 [astro-ph.CO]].
  
 
 Mifsud:2017fsy
J. Mifsud and C. Van De Bruck,
Probing the imprints of generalized interacting dark energy on the growth of perturbations,
JCAP 11, 001 (2017)

[arXiv:1707.07667 [astro-ph.CO]]. 
  
  vandeBruck:2017idm
C. Van De Bruck and J. Mifsud,
Searching for dark matter - dark energy interactions: going beyond the conformal case,
Phys. Rev. D 97, no.2, 023506 (2018)

[arXiv:1709.04882 [astro-ph.CO]].
  
  Kumar:2017bpv 
  S. Kumar and R. C. Nunes,
  Observational constraints on dark matter–dark energy scattering cross section,
  Eur. Phys. J. C 77, no. 11, 734 (2017)
  
  [arXiv:1709.02384 [astro-ph.CO]].
 
 Yang:2017ccc 
  W. Yang, S. Pan and D. F. Mota,
  Novel approach toward the large-scale stable interacting dark-energy models and their astronomical bounds,
  Phys. Rev. D 96, no. 12, 123508 (2017)
  
  [arXiv:1709.00006 [astro-ph.CO]].
  
  
  
  Pan:2017ent
  S. Pan, A. Mukherjee and N. Banerjee,
  Astronomical bounds on a cosmological model allowing a general interaction in the dark sector,
  Mon. Not. Roy. Astron. Soc.  477, 1189 (2018)
  [arXiv:1710.03725 [astro-ph.CO]].
  
  Yang:2018pej
  W. Yang, S. Pan and A. Paliathanasis,
  Cosmological constraints on an exponential interaction in the dark sector,
  Mon. Not. Roy. Astron. Soc.  482, no.1,  1007 (2019)
  [arXiv:1804.08558 [gr-qc]].
  

Yang:2018ubt 
  W. Yang, S. Pan, L. Xu and D. F. Mota,
  Effects of anisotropic stress in interacting dark matter – dark energy scenarios,
  Mon. Not. Roy. Astron. Soc.  482, no. 2, 1858 (2019)
  
  [arXiv:1804.08455 [astro-ph.CO]].

Yang:2018xlt
  W. Yang, S. Pan, R. Herrera and S. Chakraborty,
  Large-scale (in) stability analysis of an exactly solved coupled dark-energy model,
  Phys. Rev. D 98, no.4,  043517 (2018)
  [arXiv:1808.01669 [gr-qc]].
  
  Yang:2018qec 
  W. Yang, N. Banerjee, A. Paliathanasis and S. Pan,
  Reconstructing the dark matter and dark energy interaction scenarios from observations,
  Phys. Dark Univ.  26, 100383 (2019)
  
  [arXiv:1812.06854 [astro-ph.CO]].
  
 Martinelli:2019dau 
  M. Martinelli, N. B. Hogg, S. Peirone, M. Bruni and D. Wands,
  Constraints on the interacting vacuum–geodesic CDM scenario,
  Mon. Not. Roy. Astron. Soc.  488, no. 3, 3423 (2019)
  
  [arXiv:1902.10694 [astro-ph.CO]].

 Paliathanasis:2019hbi 
  A. Paliathanasis, S. Pan and W. Yang,
  Dynamics of nonlinear interacting dark energy models,
  Int. J. Mod. Phys. D 28, no. 12, 1950161 (2019)
  
  [arXiv:1903.02370 [gr-qc]].
  
  Pan:2019jqh 
  S. Pan, W. Yang, C. Singha and E. N. Saridakis,
  Observational constraints on sign-changeable interaction models and alleviation of the H_0 tension,
  Phys. Rev. D 100, no. 8, 083539 (2019)
  
  [arXiv:1903.10969 [astro-ph.CO]].
  
  Kumar:2019wfs 
  S. Kumar, R. C. Nunes and S. K. Yadav,
  Dark sector interaction: a remedy of the tensions between CMB and LSS data,
  Eur. Phys. J. C 79, no. 7, 576 (2019)
  
  [arXiv:1903.04865 [astro-ph.CO]].
  
  Yang:2019bpr
  W. Yang, S. Pan, E. Di Valentino, B. Wang and A. Wang,
  Forecasting Interacting Vacuum-Energy Models using Gravitational Waves,
  arXiv:1904.11980 [astro-ph.CO].
  
  
  Yang:2019vni 
  W. Yang, S. Vagnozzi, E. Di Valentino, R. C. Nunes, S. Pan and D. F. Mota,
  Listening to the sound of dark sector interactions with gravitational wave standard sirens,
  JCAP 1907, 037 (2019)
  
  [arXiv:1905.08286 [astro-ph.CO]].
  
  Barrow:2019jlm 
  J. D. Barrow and G. Kittou,
  Non-linear interactions in cosmologies with energy exchange,
  Eur. Phys. J. C 80, no. 2, 120 (2020)
  
  [arXiv:1907.06410 [gr-qc]].
  
  Pan:2019gop 
  S. Pan, W. Yang, E. Di Valentino, E. N. Saridakis and S. Chakraborty,
  Interacting scenarios with dynamical dark energy: Observational constraints and alleviation of the H_0 tension,
  Phys. Rev. D 100, no. 10, 103520 (2019)
  
  [arXiv:1907.07540 [astro-ph.CO]].
  
  Papagiannopoulos:2019kar 
  G. Papagiannopoulos, P. Tsiapi, S. Basilakos and A. Paliathanasis,
  Dynamics and cosmological evolution in Λ-varying cosmology,
  Eur. Phys. J. C 80, no. 1, 55 (2020)
  
  [arXiv:1911.12431 [gr-qc]].
  
  Yang:2020zuk 
  W. Yang, E. Di Valentino, S. Pan, S. Basilakos and A. Paliathanasis,
  Metastable dark energy models in light of Planck 2018: Alleviating the H_0 tension,
  arXiv:2001.04307 [astro-ph.CO].
  
  Pan:2020zza 
  S. Pan, G. S. Sharov and W. Yang,
  Field theoretic interpretations of interacting dark energy scenarios and recent observations,
  arXiv:2001.03120 [astro-ph.CO].
  
  
  
  Pan:2020mst 
  S. Pan, J. de Haro, W. Yang and J. Amorós,
  Understanding the phenomenology of interacting dark energy scenarios and their theoretical bounds,
  arXiv:2001.09885 [gr-qc].
  
  Pan:2020bur 
  S. Pan, W. Yang and A. Paliathanasis,
  Nonlinear interacting cosmological models after Planck 2018 legacy release and the H_0 tension,
  Mon. Not. Roy. Astron. Soc.  493, 3114 (2020)
  arXiv:2002.03408 [astro-ph.CO].
  
  
  Lucca:2020zjb 
  M. Lucca and D. C. Hooper,
  Tensions in the dark: shedding light on Dark Matter-Dark Energy interactions,
  arXiv:2002.06127 [astro-ph.CO].
  
Yang:2019uzo 
  W. Yang, O. Mena, S. Pan and E. Di Valentino,
  Dark sectors with dynamical coupling,
  Phys. Rev. D 100, no. 8, 083509 (2019)
  
  [arXiv:1906.11697 [astro-ph.CO]].
  
  
  
  Giusarma:2016phn 
  E. Giusarma, M. Gerbino, O. Mena, S. Vagnozzi, S. Ho and K. Freese,
  Improvement of cosmological neutrino mass bounds,
  Phys. Rev. D 94, no. 8, 083522 (2016)
  
  [arXiv:1605.04320 [astro-ph.CO]].
  
  Gerbino:2016sgw 
  M. Gerbino, K. Freese, S. Vagnozzi, M. Lattanzi, O. Mena, E. Giusarma and S. Ho,
  Impact of neutrino properties on the estimation of inflationary parameters from current and future observations,
  Phys. Rev. D 95, no. 4, 043512 (2017)
  
  [arXiv:1610.08830 [astro-ph.CO]].
  
  
  Vagnozzi:2017ovm 
  S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho and M. Lattanzi,
  Unveiling ν secrets with cosmological data: neutrino masses and mass hierarchy,
  Phys. Rev. D 96, no. 12, 123503 (2017)
  
  [arXiv:1701.08172 [astro-ph.CO]].
  
  Vagnozzi:2018jhn 
  S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goobar and O. Mena,
  Constraints on the sum of the neutrino masses in dynamical dark energy models with w(z) ≥ -1 are tighter than those obtained in ΛCDM,
  Phys. Rev. D 98, no. 8, 083501 (2018)
  
  [arXiv:1801.08553 [astro-ph.CO]].
  
  Giusarma:2018jei 
  E. Giusarma, S. Vagnozzi, S. Ho, S. Ferraro, K. Freese, R. Kamen-Rubio and K. B. Luk,
  Scale-dependent galaxy bias, CMB lensing-galaxy cross-correlation, and neutrino masses,
  Phys. Rev. D 98, no. 12, 123526 (2018)
  
  [arXiv:1802.08694 [astro-ph.CO]].
  
  Choudhury:2018byy 
  S. Roy Choudhury and S. Choubey,
  Updated Bounds on Sum of Neutrino Masses in Various Cosmological Scenarios,
  JCAP 1809, 017 (2018)
  
  [arXiv:1806.10832 [astro-ph.CO]].
  
  Vagnozzi:2018pwo 
  S. Vagnozzi, T. Brinckmann, M. Archidiacono, K. Freese, M. Gerbino, J. Lesgourgues and T. Sprenger,
  Bias due to neutrinos must not uncorrect'd go,
  JCAP 1809, 001 (2018)
  
  [arXiv:1807.04672 [astro-ph.CO]].
  
  Vagnozzi:2019utt 
  S. Vagnozzi,
  Cosmological searches for the neutrino mass scale and mass ordering,
  arXiv:1907.08010 [astro-ph.CO].

  Yang:2019uog 
  W. Yang, S. Pan, R. C. Nunes and D. F. Mota,
  Dark calling Dark: Interaction in the dark sector in presence of neutrino properties after Planck CMB final release,
  to appear in JCAP,
  arXiv:1910.08821 [astro-ph.CO].
  
  Yang:2020uga 
  W. Yang, E. Di Valentino, O. Mena, S. Pan and R. C. Nunes,
  All-inclusive interacting dark sector cosmologies,
  to appear in  Phys. Rev. D,
  arXiv:2001.10852 [astro-ph.CO].
  
  
  Hagstotz:2020ukm 
  S. Hagstotz, P. F. de Salas, S. Gariazzo, M. Gerbino, M. Lattanzi, S. Vagnozzi, K. Freese and S. Pastor,
  Bounds on light sterile neutrino mass and mixing from cosmology and laboratory searches,
  arXiv:2003.02289 [astro-ph.CO].
  
  
Aghanim:2018eyx 
  N. Aghanim et al. [Planck Collaboration],
 Planck 2018 results. VI. Cosmological parameters,
  arXiv:1807.06209 [astro-ph.CO].
  
    Aghanim:2018oex 
  N. Aghanim et al. [Planck Collaboration],
  Planck 2018 results. VIII. Gravitational lensing,
  arXiv:1807.06210 [astro-ph.CO].
  
  Aghanim:2019ame 
  N. Aghanim et al. [Planck Collaboration],
  Planck 2018 results. V. CMB power spectra and likelihoods,
  arXiv:1907.12875 [astro-ph.CO].
  
  Beutler:2011hx 
  F. Beutler et al.,
  The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant,
  Mon. Not. Roy. Astron. Soc.  416, 3017 (2011)
  
  [arXiv:1106.3366 [astro-ph.CO]].
 
  Ross:2014qpa 
  A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, The 
clustering of the SDSS DR7 main Galaxy sample – I. A 4 per cent distance measure at z = 
0.15,
  Mon. Not. Roy. Astron. Soc.  449, no. 1, 835 (2015)
  
  [arXiv:1409.3242 [astro-ph.CO]].
  
  Alam:2016hwk 
  S. Alam et al. [BOSS Collaboration],
  The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample,
  Mon. Not. Roy. Astron. Soc.  470, no. 3, 2617 (2017)
  
  [arXiv:1607.03155 [astro-ph.CO]].
  
  
  Riess:2019cxk 
  A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic,
  Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond ΛCDM,
  Astrophys. J.  876, no. 1, 85 (2019)
  
  [arXiv:1903.07603 [astro-ph.CO]].
  
  
 
  
    Lewis:2002ah 
  A. Lewis and S. Bridle,
  Cosmological parameters from CMB and other data: A Monte Carlo approach,
  Phys. Rev. D 66, 103511 (2002)
  [astro-ph/0205436].

  Lewis:1999bs 
  A. Lewis, A. Challinor and A. Lasenby,
  Efficient computation of CMB anisotropies in closed FRW models,
  Astrophys. J.  538, 473 (2000)
  
  [astro-ph/9911177].

 Gelman-Rubin
  A. Gelman and D. Rubin, 
  Inference from iterative simulation using multiple sequences, 
  Statistical Science 7, 457 (1992). 
  
  

  
  DiValentino:2019jae 
  E. Di Valentino, A. Melchiorri, O. Mena and S. Vagnozzi,
  Nonminimal dark sector physics and cosmological tensions,
  Phys. Rev. D 101, no. 6, 063502 (2020)
  
  [arXiv:1910.09853 [astro-ph.CO]].
  
  
DiValentino:2019ffd 
  E. Di Valentino, A. Melchiorri, O. Mena and S. Vagnozzi,
  Interacting dark energy after the latest Planck, DES, and H_0 measurements: an excellent solution to the H_0 and cosmic shear tensions,
  arXiv:1908.04281 [astro-ph.CO].
  

Abbott:2017wau 
  T. M. C. Abbott et al. [DES Collaboration],
  Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,
  arXiv:1708.01530 [astro-ph.CO].
  
  

Troxel:2017xyo 
  M. A. Troxel et al. [DES Collaboration],
  Dark Energy Survey Year 1 Results: Cosmological Constraints from Cosmic Shear,
  arXiv:1708.01538 [astro-ph.CO].
  
  

Hildebrandt:2016iqg 
  H. Hildebrandt et al.,
  KiDS-450: Cosmological parameter constraints from tomographic weak gravitational lensing,
  arXiv:1606.05338 [astro-ph.CO].
  

Kuijken:2015vca 
  K. Kuijken et al.,
  Gravitational Lensing Analysis of the Kilo Degree Survey,
  Mon. Not. Roy. Astron. Soc.  454, no. 4, 3500 (2015)
  
  [arXiv:1507.00738 [astro-ph.CO]].
  
  
Conti:2016gav 
  I. Fenech Conti, R. Herbonnet, H. Hoekstra, J. Merten, L. Miller and M. Viola,
  Calibration of weak-lensing shear in the Kilo-Degree Survey,
  Mon. Not. Roy. Astron. Soc.  467, no. 2, 1627 (2017)
  
  [arXiv:1606.05337 [astro-ph.CO]].
  
  

Heymans:2012gg 
  C. Heymans et al.,
  CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey,
  Mon. Not. Roy. Astron. Soc.  427, 146 (2012)
  
  [arXiv:1210.0032 [astro-ph.CO]].


Erben:2012zw 
  T. Erben et al.,
  CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey - Imaging Data and Catalogue Products,
  Mon. Not. Roy. Astron. Soc.  433, 2545 (2013)
  
  [arXiv:1210.8156 [astro-ph.CO]].
  
  
  
Joudaki:2016mvz 
  S. Joudaki et al.,
  CFHTLenS revisited: assessing concordance with Planck including astrophysical systematics,
  Mon. Not. Roy. Astron. Soc.  465, no. 2, 2033 (2017)
  
  [arXiv:1601.05786 [astro-ph.CO]].
  
  Asgari:2019fkq 
  M. Asgari et al.,
  KiDS+VIKING-450 and DES-Y1 combined: Mitigating baryon feedback uncertainty with COSEBIs,
  Astron. Astrophys.  634, A127 (2020)
  
  [arXiv:1910.05336 [astro-ph.CO]].
  
  
  
    Mangano:2005cc 
  G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti and P. D. Serpico,
  Relic neutrino decoupling including flavor oscillations,
  Nucl. Phys. B 729, 221 (2005)
  
  [hep-ph/0506164].
  
  
deSalas:2016ztq 
  P. F. de Salas and S. Pastor,
  Relic neutrino decoupling with flavour oscillations revisited,
  JCAP 1607, 051 (2016)
  
  [arXiv:1606.06986 [hep-ph]].
  
  
Motloch:2018pjy
P. Motloch and W. Hu,
Tensions between direct measurements of the lens power spectrum from Planck data,
Phys. Rev. D 97, no.10, 103536 (2018)

[arXiv:1803.11526 [astro-ph.CO]].




