We present a comprehensive reassessment of the state of interacting dark energy DE cosmology namely models featuring a nongravitational interaction between dark matter and DE. To achieve high generality we extend the dark sector physics by considering two different scenarios a nondynamical DE equation of state inlineformulammlmath displayinlinemmlmrowmmlmsubmmlmrowmmlmiwmmlmimmlmrowmmlmrowmmlmn0mmlmnmmlmrowmmlmsubmmlmommlmommlmommlmommlmn1mmlmnmmlmrowmmlmathinlineformula and a dynamical inlineformulammlmath displayinlinemmlmrowmmlmiwmmlmimmlmo stretchyfalsemmlmommlmiammlmimmlmo stretchyfalsemmlmommlmommlmommlmsubmmlmrowmmlmiwmmlmimmlmrowmmlmrowmmlmn0mmlmnmmlmrowmmlmsubmmlmommlmommlmsubmmlmrowmmlmiwmmlmimmlmrowmmlmrowmmlmiammlmimmlmrowmmlmsubmmlmo stretchyfalsemmlmommlmn1mmlmnmmlmommlmommlmiammlmimmlmo stretchyfalsemmlmommlmrowmmlmathinlineformula. In both cases we distinguish two different physical regimes resulting from a phantom or quintessence equation of state. To circumvent early time superhorizon instabilities the energymomentum transfer should occur in opposing directions within the two regimes resulting in distinct phenomenological outcomes. We study quintessence and phantom nondynamical and dynamical models in light of two independent cosmic microwave background CMB experimentsthe Planck satellite and the Atacama Cosmology Telescope. We analyze CMB data both independently and in combination with supernovae distance moduli measurements from the PantheonPlus catalog and baryon acoustic oscillations from the SDSSIV eBOSS survey. Our results update and extend the stateoftheart analyses significantly narrowing the parameter space allowed for these models and limiting their overall ability to reconcile cosmological tensions. Although considering different combinations of data leaves some freedom to increase inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula towards the value measured by the SH0ES collaboration our most constraining dataset inlineformulammlmath displayinlinemmlmrowmmlmiCMBmmlmimmlmommlmommlmibaryonmmlmimmlmrowmmlmathinlineformula acoustic inlineformulammlmath displayinlinemmlmrowmmlmioscillationsmmlmimmlmommlmommlmisupernovaemmlmimmlmrowmmlmathinlineformula indicates that fully reconciling the tension solely within the framework of interacting DE remains challenging.
['Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'High Energy Physics - Phenomenology']
Tightening the reins on nonminimal dark sector physics Interacting dark energy with dynamical and nondynamical equation of state
{'Tightening the reins on non-minimal dark sector physics: Interacting Dark Energy with dynamical and non-dynamical equation of state': 'William Giarè, 1, ∗ Yuejia Zhai, 1, † Supriya Pan, 2, 3, ‡ Eleonora Di 1, § 4, 5, ¶ 1,  \nValentino, Rafael C. Nunes, and Carsten van de Bruck ∗∗ 1 School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom 3 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa 5 Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais,  \n2 Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India  \n4 Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre RS, Brazil  \nAvenida dos Astronautas 1758, São José dos Campos, 12227-010, São Paulo, Brazil  \n̸  \nWe present a comprehensive reassessment of the state of Interacting Dark Energy (IDE) cosmology, namely models featuring a non-gravitational interaction between Dark Matter (DM) and Dark Energy (DE). To achieve high generality, we extend the dark sector physics by considering two different scenarios: a non-dynamical DE equation of state w 0 = -1 , and a dynamical w ( a ) = w 0 + w a (1 -a ) . In both cases, we distinguish two different physical regimes resulting from a phantom or quintessence equation of state. To circumvent early-time superhorizon instabilities, the energymomentum transfer should occur in opposing directions within the two regimes, resulting in distinct phenomenological outcomes. We study quintessence and phantom non-dynamical and dynamical models in light of two independent Cosmic Microwave Background (CMB) experiments - the Planck satellite and the Atacama Cosmology Telescope. We analyze CMB data both independently and in combination with Supernovae (SN) distance moduli measurements from the Pantheon-Plus catalog and Baryon Acoustic Oscillations (BAO) from the SDSS-IV eBOSS survey. Our results update and extend the state-of-the-art analyses, significantly narrowing the parameter space allowed for these models and limiting their overall ability to reconcile cosmological tensions. Although considering different combinations of data leaves some freedom to increase H 0 towards the value measured by the SH0ES collaboration, our most constraining dataset (CMB+BAO+SN) indicates that fully reconciling the tension solely within the framework of IDE remains challenging.', 'I. INTRODUCTION': 'In spite of the successes accumulated in past decades, the standard Λ CDM model of cosmology seems to be struggling to provide an exhaustive description of the most recent observations. As experimental precision has improved, various anomalies and tensions between experiments have come to light [1-4]. Among them, one in particular seriously calls into question the validity of our best-working model of the Universe: the so-called Hubble tension [5-8].  \nThe Hubble tension refers to a ∼ 5 σ disagreement between the value of the Hubble constant as inferred by CMB data from the Planck collaboration [9] assuming a Λ CDM cosmology ( H 0 = 67 . 4 ± 0 . 5 km/s/Mpc) and the value of the same parameter as directly obtained by local distance ladder measurements from Type Ia supernovae from the SH0ES collaboration [10, 11] ( H 0 = 73 ± 1 km/s/Mpc). Barring any possible systematic origin of  \nthe tension, 1 an exciting possibility to consider is that this problem could represent an indication of new physics beyond the standard cosmological paradigm.  \nIn this regard, we note that a somewhat surprising outcome of modern cosmology is that merely 5% of the total energy density of the Universe comprises baryonic matter, whose physical properties are fairly well understood in the framework of the Standard Model of particle physics (SM). In stark contrast, the remaining 95% is attributed to enigmatic entities - Dark Energy (DE) and Dark Matter (DM) - whose origins remain an enigma for modern cosmology and high-energy physics. Within the Λ CDM model, we opt for a quite minimal parameterization, describing DM as a perfect fluid made of cold non-relativistic particles with low momenta that do not interact with the other SM particles except through gravity. In addition, we assume DE to be a cosmological constant ( Λ ) in the Einstein Equation. However, it seems natural to speculate that, given our limited understanding of DM and DE, a part of the aforementioned tensions  \ncould originate from an oversimplification of the theoretical parameterization adopted for the dark sector of the cosmological model. For this reason, several intriguing alternatives involving additional interactions or couplings in the dark sector have been considered.  \nA model among many that has gained some research attention is the so-called Interacting Dark Energy (IDE) cosmology [12-74]. At its core, the model postulates a non-gravitational interaction between DM and DE, allowing an exchange of energy-momentum between the two, see also Refs. [19, 75] for reviews.  \nIt is worth noting that, from a purely theoretical point of view, there is no fundamental symmetry in nature for which non-gravitational DM-DE interactions are forbidden. Cosmological models featuring an interacting dark sector (in part motivated by the idea of coupled quintessence [30, 76-83] ) have been largely explored and discussed in the literature, see, e.g., [12, 13, 21, 31, 36, 84129]. Furthermore, many have speculated that a coupled dark sector could help address the coincidence (or "why now?") problem [130-134].  \nOn the other hand, from a more observational standpoint, IDE cosmology has emerged as a possible solution to cosmological tensions [15, 20, 24, 25, 29, 3338, 69, 135-149]. Allowing an exchange of energymomentum from DM to DE, can increase the value of H 0 inferred from CMB observations, possibly restoring the agreement with the direct measurement provided by the SH0ES collaboration. In addition, as recently shown by some of us in Ref. [69], IDE seems to be supported by different independent CMB experiments, leading to an overall consistency of view concerning the amount of energy-momentum transferred in the dark sector. Having said that, whether or not this model could provide a successful solution for the Hubble trouble is still a matter of debate. The model suffers from the typical problem faced by any late-time solution (i.e., solutions that are aimed to solve the Hubble tension by introducing new physics post-recombination). Namely, Baryon Acoustic Oscillation (BAO) data and distance moduli measurements for Supernovae (SN) are very constraining on local distances, leaving us with little freedom to introduce any deviation from a basic Λ CDM late-time cosmology [150, 151]. As a result, when considering low-redshift ( z ) probes, the ability of IDE to increase the present-day expansion rate of the Universe is strongly reduced, if not completely lost. 2  \nTo better understand the extent to which IDE (and their relatives) can provide an actual solution to the H 0 -tension, in this work, we focus more closely on the role  \nplayed by the DE equation of state (EoS) in IDE cosmology. First and foremost, we note that the nature of the DE EoS acquires primary importance in the model. We will be more precise on this in the next sections (specifically in section II and section III), but we anticipate that to avoid early-time superhorizon instabilities with cosmological perturbations [152-154], the EoS must be theoretically confined to either the quintessence or phantom regime, depending on the direction in which energymomentum is transferred between DM and DE. In the scenario we will consider in this work, a quintessential DE EoS ( w 0 > -1 ) implies an energy-momentum flow from DM to DE. Conversely, a phantom DE EoS ( w 0 < -1 ) implies a transfer of energy-momentum from DE to DM.  \n̸  \nAn implicit assumption underlying a large portion of the results mentioned thus far is fixing the DE EoS to a very tiny quintessence value, w 0 ≃ -0 . 999 ≃ -1 , essentially resembling a cosmological constant. Nevertheless, a few scattered analyses (largely conducted by some of us), have already considered the possibility of leaving the EoS w 0 a free parameter of the model [38, 53, 64, 155]. We refer to scenarios featuring a non-dynamical w 0 = -1 EoS as w 0 IDE. For these models, many important aspects remain unclear, and important questions are pending. For example, in Ref. [38], non-dynamical models were examined in the context of Planck-2018 data along with Baryon Acoustic Oscillations (BAO) and (SH0ES calibrated) Supernovae (SN) data. This analysis revealed a significant preference for quintessence IDE, establishing the model as a highly promising solution to the Hubble tension. However, as far as we know, the state-of-theart constraints on w 0 IDE remain largely unchanged since 2019, and such preferences have not been tested subsequently with CMB data other than Planck or against the latest low-redshift probes. Therefore, as a first step, we undertake a comprehensive reassessment of the constraints on the non-dynamical w 0 IDE scenario, updating the constraints on the w 0 IDE model (for both the quintessence and phantom regime), and incorporating the latest BAO, SN, and CMB data in the analysis. In this regard, we aim to clarify the following important aspects:  \n- · We place special emphasis on the constraints arising from (updated) local distance measurements, in the form of SN and BAO measurements to examine if any leeway remains to address cosmological tensions through DM-DE energy-momentum transfer (in either the quintessence or phantom regimes). In this context, interesting aspects to clarify are whether the latest BAO and SN data independently validate or dismiss the w 0 IDE scenario as a viable solution to the Hubble tension and shed light on the role of the SH0ES calibration for SN.\n- · As we already mentioned, in Ref [69], some of us pointed out that different CMB experiments share a consistent view on IDE when w 0 ≃ -1 is fixed. Here we extend the analysis of small-scale CMB measurements  \nreleased by the Atacama Cosmology Telescope (ACT) to the case where w 0 is let free to vary. In this regard, a particularly relevant aspect to clarify is whether the well-documented Planck preference for a phantom DE EoS [9, 156] (not confirmed by ACT [157-159]) could play any effect on the amount of energy-momentum transfer supported by data and, more broadly, if independent CMB experiments validate or dismiss the w 0 IDE scenario as a viable solution to the Hubble tension.  \nWe then move to consider more exotic models where the DE EoS w ( z ) is dynamical and changes with cosmic expansion. We refer to this scenario as w ( z ) IDE. Notice that the interaction between DM and DE via dynamical DE EoS has been investigated in various other contexts, see e.g., Refs. [16, 64, 98]. Here, we describe the dynamical evolution of w ( z ) by adopting a simple ChevallierPolarski-Linder (CPL) parametrization [160, 161] and provide a comprehensive overview of the most recent observational constraints on the dynamical w ( z ) IDE cosmology. We are fueled by the following motivations:  \n- · To the best of our knowledge, a proper updated analysis aimed to understand whether the dynamical w ( z ) IDE model could represent a solution to the Hubble tension (either in the quintessence regime or in the phantom regime) is missing. Therefore, we believe it is intrinsically interesting to understand how a possible dynamic behavior of the DE EoS could impact the constraints derived for the non-dynamical case.\n- · From a more practical point of view, the questions we seek to answer are not different from what we already pointed out: interesting aspects to clarify are whether the latest BAO and SN data independently validate or dismiss the w ( z ) IDE scenario as a viable solution to the H 0 tension. Additionally, we want to assess whether ACT and Planck always share a consistent view on w ( z ) IDE.  \nThe paper is organized as follows. In section II, we introduce the theoretical framework that underpins our study. In section III, we outline the methodology and the updated datasets used to establish observational constraints within the models considered in this work. Moving further, section IV and section V delve into our primary findings for the non-dynamical and dynamical cases, respectively. Finally, in section VI, we summarize our conclusions and offer insights into future perspectives. As usual, a subindex zero attached to any quantity means that it must be evaluated at present time.', 'II. INTERACTING DARK ENERGY COSMOLOGY': 'In this section, we review in a nutshell the wellestablished aspects of background evolution and linear  \nperturbations that govern the coupling between two dark fluids. We consider that the gravitational sector of the universe is described by the Einstein\'s General Relativity and a flat Friedmann-Lemaître-Robertson-Walker (FLRW) line element in the synchronous gauge ds 2 = a 2 ( η ) [ -dη 2 +( δ ij + h ij ) dx i dx j ] , where a ( η ) is the scale factor as a function of the conformal time η ; δ ij , h ij are respectively the unperturbed and perturbed metric tensors. The conservation equations of DM and DE in the presence of a dark interaction, characterized by a coupling function Q ( t ) , can be expressed as follows:  \n∇ ν T µν DM = Qu µ a ( η ) , (1)  \n∇ ν T µν DE = -Qu µ a ( η ) , (2)  \nwhere T µν DM and T µν DE are respectively the energymomentum tensor for DM and DE; u µ is the four-velocity vector of DM which in the synchronous gauge can be defined as u µ = a ( η )( -1 , u i ) where µ = 0 , 1 , 2 , 3 , i = 1 , 2 , 3 and u i is the proper velocity of the DM fluid. Now, to understand the evolution of the dark fluids, at the background and perturbation levels, one needs to prescribe the nature of the dark fluids and also the interaction function. We assume that DM is pressureless, with ρ c denoting its energy density, while DE enjoys a (dynamical or non-dynamical) barotropic equation of state, represented by w x = p x /ρ x , where p x and ρ x are respectively the pressure and energy density of the DE fluid. By examining the sign of Q ( t ) , we can determine the direction of energy transfer between the dark components. When Q > 0 , energy transfers from DE to cold dark matter (CDM), while Q < 0 indicates the reverse situation, signifying energy transfer from CDM to DE.  \nGiven the complexity of describing both dark species, at this stage, to proceed, an exact phenomenological approach that quantifies the coupling must be assumed. Several proposals in this regard have been put forward in the literature in recent times. In this article, we consider a widely studied model of the interaction function, which has received considerable attention in recent years. The interaction rate that we employ is as follows:  \nQ = ξ H ρ x , (3)  \nwhere ξ represents the dimensionless coupling constant which is independent of cosmic time. Here, H denotes the conformal Hubble parameter, following the standard definition as in General Relativity: 3 H 2 = 8 πG a 2 ( η ) ∑ ρ i , where ρ i represents the energy density of the i -th fluid. In addition to CDM and DE, we have also accounted for the presence of baryons, radiation, and neutrinos, including one massive and two massless species. Hence, consistent with the sign convention of Q ( t ) , ξ > 0 ( < 0 ) denotes the transfer of energy from DE to CDM (from CDM to DE),  \nrespectively. 3  \nNow, considering the linear perturbations and to prevent any potential unphysical scenarios related to the DE equation of state and c 2 s,x , we set the DE sound speed c 2 s,x = 1 . This allows us to express the evolution of density perturbations in terms of δ c,x and velocity perturbations ( θ c,x ) as follows:  \nδ \' x = -(1 + w x ) ( θ x + h \' 2 ) -ξ ( kv T 3 + h \' 6 ) -3 H (1 -w x ) [ δ x + H θ x k 2 (3(1 + w x ) + ξ ) ] ,  \nθ \' x = 2 H θ x + k 2 δ x +2 H ξ θ x -ξ H θ c ,  \nδ \' c = -θ c -1 2 h \' + ξ H ρ x ρ c ( δ x -δ c ) + ξ ρ x ρ c ( kv T 3 + h 6 ) ,  \nθ \' c = -H θ c ,  \n(4) 1 + w x 1 + w x 1 + w x (5) \' (6) (7)  \nwhere h is the trace of the scalar metric perturbation h ij ; k is the Fourier-space wave number; the prime attached to any quantity denotes its derivative with respect to the conformal time and v T refers to the center of mass velocity of the total fluid [154].  \nEnsuring the stability of linear perturbations over time is crucial within the dynamical scenarios considered for the dark coupling in this study. As demonstrated in [13], the parameter known as the "doom factor", denoted as d = Q/ (3 H (1 + w x )) , plays a pivotal role in determining the stability of the scalar modes. When d < 0 , indicating stability, it implies that for Q > 0 , the equation of state parameter w x must be less than -1 . Conversely, for Q < 0 , w x needs to be greater than -1 to maintain stability. In the following section, we will delve into a detailed description of the parameter space that ensures an absence of instabilities.', 'A. Markov Chain Monte Carlo Analysis': 'We perform Markov Chain Monte Carlo (MCMC) analyses using the publicly available sampler COBAYA [163]. The code explores the posterior distributions of a given parameter space using the MCMC sampler developed for CosmoMC [164] and tailored for parameter spaces with speed hierarchy, implementing the "fast dragging" procedure detailed in Ref. [165]. We compute the theoretical model by means of the Cosmic Linear Anisotropy Solving System code, CLASS [166],  \nTable I. List of the uniform parameter priors for the phantom and quintessence regimes. When considering the ACT CMB data, we assume a Gaussian prior τ = 0 . 065 ± 0 . 015 with a width much smaller than the uniform prior reported in this table.  \nmodified to introduce the possibility of interactions between dark energy and dark matter.  \nOur sampling parameters are the usual six Λ CDM parameters, namely the baryon ω b . = Ω b h 2 and cold dark matter ω c . = Ω c h 2 energy densities, the angular size of the horizon at the last scattering surface θ MC , the optical depth τ , the amplitude of primordial scalar perturbation log(10 10 A s ) , and the scalar spectral index n s . In addition, we consider the coupling parameter ξ defined in Eq. (3) and the set of parameters describing the DE EoS. In this regard, we distinguish two different cases:  \n- · Non-Dynamical DE EoS In this case w 0 becomes an additional free parameter that we vary in two different regimes: the quintessence regime where w 0 > -1 and the phantom regime where w 0 < -1 . To avoid instabilities in primordial perturbations, the coupling parameter ξ can vary within the following priors: ξ < 0 when w 0 > -1 and ξ > 0 when w 0 < -1 . 4 Therefore the behavior of the DE EoS fixes the direction of energy-momentum transfer between DM and DE. In summary:  \ni) w 0 > -1 ⇔ ξ < 0 ⇔ DM to DE,  \nii) w 0 < -1 ⇔ ξ > 0 ⇔ DE to DM.  \n- · Dynamical DE EoS In this case, we adopt a CPL parameterization:  \nw ( a ) = w 0 + w a (1 -a ) , (8)  \nwhere w 0 represents the present value w ( a = 1) , and w a is another free parameter such that dw d ln(1+ z ) ∣ ∣ ∣ ∣ z =1 = w a 2 . As usual, we distinguish two different regimes:  \n- i) The quintessence regime where w ( z ) > -1 for any z . In the MCMC analysis, we sample over these two parameters, ensuring that for every randomly sampled pair of values w 0 -w a , the condition w ( z ) > -1 holds true at any z . 5 If this condition is not met, the point is rejected. The test is performed dynamically during the MCMC run, without assuming any restrictive prior on the parameter w a controlling the dynamical evolution of w ( z ) . In contrast, we assume a prior w 0 > -1 that automatically follows from requiring w ( z ) > -1 at z = 0 . Our methodology ensures proper sampling of the parameter space and convergence of the chains.\n- ii) The phantom regime where w ( z ) < -1 for any z . We ensure that for every sampled pair of values w 0 -w a during the MCMC run, the condition w ( z ) < -1 is satisfied at any z . If this condition is not met, the point is rejected. Also in this case the test is performed dynamically without assuming any restrictive prior on w a and imposing a prior w 0 < -1 that automatically follows from requiring w ( z ) < -1 at z = 0 .  \nAs for the coupling parameter ξ , to avoid instabilities in primordial perturbations, in the quintessence regime we need ξ < 0 , while in the phantom case ξ > 0 . Therefore, also in the dynamical case, the DE EoS regime determines the direction of the energy and momentum flow between DM and DE. In summary:  \ni) w ( z ) > -1 ∀ z ⇔ ξ < 0 ⇔ DM to DE  \nii) w ( z ) < -1 ∀ z ⇔ ξ > 0 ⇔ DE to DM  \nA summary of the uniform prior distributions adopted for all the cosmological parameters considered in the analysis is given in Table I (except for the optical depth at reionization τ for which we adopt a prior distribution that depends on the specific CMB dataset, as discussed below). We test the convergence of the chains obtained using this approach by means of the Gelman-Rubin criterion. We establish a threshold for chain convergence of R -1 ≲ 0 . 02 .', 'B. Cosmological Data': 'Our reference datasets for both the dynamical and nondynamical IDE scenarios are the following:  \n- · The Planck 2018 temperature and polarization (TT TE EE) likelihood, which also includes low multipole data ( ℓ < 30 ) [9, 167, 168] and the Planck 2018 lensing likelihood [169], constructed from measurements of the power spectrum of the lensing potential. We refer to this dataset as P18 .\n- · Atacama Cosmology Telescope temperature and polarization anisotrpy DR4 likelihood in combination with the gravitational lensing DR6 likelihood covering 9400 deg 2 reconstructed from CMB measurements made by the Atacama Cosmology Telescope from 2017 to 2021 [170, 171]. In our analysis for the lensing spectrum we include only the conservative range of lensing multipoles 40 < ℓ < 763 . We consider a Gaussian prior on τ = 0 . 065 ± 0 . 015 , as done in [157]. We refer to this dataset as ACT .\n- · Baryon Acoustic Oscillation data from the finalized SDSS-IV eBOSS survey. These data encompass both isotropic and anisotropic distance and expansion rate measurements, as outlined in Table 3 of Reference [172]. We refer to this dataset as BAO .\n- · Distance modulus measurements of Type Ia supernovae obtained from the Pantheon-Plus sample [173]. This dataset comprises 1701 light curves representing 1550 unique Type Ia supernovae, spanning a redshift range from 0.001 to 2.26. In all our analyses, we consider two distinct possibilities. On the one hand, we consider the uncalibrated Pantheon-Plus SNe Ia sample that we will henceforth refer to as SN . On the other hand, we will consider the SH0ES Cepheid host distances to calibrate the SNe Ia sample [10]. We refer to the SH0ES-calibrated SN dataset as SN+SH0ES .', 'IV. RESULTS FOR NON-DYNAMICAL EOS': 'In this section, we present the results obtained considering a non-Dynamical DE EoS. We divide the section into two different subsections. In section IV A, we focus on quintessence models characterized by w 0 > -1 and a negative DM-DE coupling ξ < 0 . These models feature a flow of energy-momentum from the DM sector to the DE sector of the theory. Conversely, in section IV B, we study phantom models with w 0 < -1 and ξ > 0 . In this case the energy momentum is transferred from DE to DM.', 'A. Quintessence EoS': 'The results obtained imposing a quintessence dynamical DE EoS are provided in Table VI for the combinations of data involving P18 and in Table VII for those involving ACT.', '1. CMB-only': 'As usual, we start by considering only CMB temperature, polarization and lensing observations. Comparing the results with those derived for the non-dynamical case, we note that considering a dynamical phantom EoS allows a much larger fraction of energy-momentum flow from DE to DM. This is because within a CPL parametrization, we have more parameters to constrain. Taking the results at face value, from P18 we obtain ξ < 0 . 522 , while for ACT we obtain ξ < 0 . 499 . These large values of ξ result in a preference for a larger matter fraction (fed by DE) and consequently into a shift in the Hubble parameter towards values smaller than the Λ CDM one ( H 0 = 65 . 1 +1 . 9 -1 . 7 km/s/Mpc for P18 and H 0 = 65 . 1 ± 1 . 7 km/s/Mpc for ACT). Therefore, dynamical phantom models may actually prove to be less effective in potentially addressing the Hubble constant tension compared to their non-dynamical counterpart. Having said that, as usual, CMB-only data leaves us with  \nvery large error bars, making the inclusion of additional local universe probes a necessary step to take before deriving any definitive conclusions.', '2. CMB and SN': 'Combining CMB and SN together, we enhance the constraining power, narrowing down the upper limit on the coupling parameter ξ . Specifically, using uncalibrated SN, for P18+SN we obtain ξ < 0 . 224 , while for ACT+SN, we get ξ < 0 . 269 . Further tightening occurs when considering the SH0ES calibration. For P18+SN+SH0ES and ACT+SN+SH0ES, we find ξ < 0 . 0647 and ξ < 0 . 103 , respectively. However, uncalibrated SN data yield lower values H 0 = 66 . 31 ± 0 . 72 km/s/Mpc for P18+SN and H 0 = 66 . 14 ± 0 . 89 km/s/Mpc for ACT+SN. These results are in clear tension with SH0ES.  \nOverall, SN confirm that dynamical phantom models struggle to represent a possible solution to the H 0 -tension, in line with what we anticipated from the CMBonly analysis.', '3. CMB and BAO': 'When it comes to considering BAO data, we find that the coupling parameter is constrained to ξ < 0 . 0583 for P18+BAO and ξ < 0 . 0748 for ACT+BAO. Both datasets predict values of H 0 that are essentially the same obtained for Λ CDM.', '4. Joint Analyses': 'Finally, we come to the most constraining CMB+SN+BAO dataset. The correlations among the most relevant parameters are shown in Figure 4.  \nTable VIII. Constraints at 68% (95%) CL and upper limits at 95% CL on the parameters of the dynamical w ( z ) IDE model with w ( z ) < -1 at any z . The results are obtained by different combinations of P18, BAO and SN (with and without the SH0ES calibration) measurements.  \nTable IX. Constraints at 68% (95%) CL and upper limits at 95% CL on the parameters of the dynamical w ( z ) IDE model with w ( z ) < -1 at any z . The results are obtained by different combinations of ACT, BAO and SN (with and without the SH0ES calibration) measurements.  \nAs evident from the figure, the joint analysis confirms the overall trend observed consistently in the phantom dynamical case. The amount of energy and momentum that can be transferred from DE to DM is very constrained for both P18+BAO+SN ( ξ < 0 . 0642 ) and ACT+BAO+SN ( ξ < 0 . 0864 ). Everything is in line with a late-time Λ CDM cosmology, including the value inferred for the Hubble parameter.', 'B. Phantom EoS': 'We conclude our explorative study of IDE cosmology by considering models featuring a phantom dynamical EoS w ( z ) < -1 . The results involving P18 are given in Table VIII, those involving ACT are given in Table IX.', 'V. RESULTS FOR DYNAMICAL EOS': 'In this section, we discuss the results for a Dynamical DE EoS given by the CPL parameterization in Eq. (8). We divide the section into two different subsections. In section V A, we focus on quintessence models characterized by an EoS w ( z ) > -1 at any z and a DM-to-DE energy-momentum flow (i.e., ξ < 0 ). Instead, in section V B, we study phantom models with w ( z ) < -1 at any z and a DE-to-DM energy-momentum transfer ( ξ > 0 ).', 'VI. CONCLUSIONS': 'In this paper, we undertake a comprehensive reassessment of the constraints on IDE cosmology, namely cosmological models featuring energy-momentum flow between DM and DE.  \nOur model is detailed in section II. On top of this model, we expand the dark sector physics, allowing for  \nFigure 4. Joint marginalized contours at 68% and 95% CL illustrating the correlation between the coupling parameter ξ , the dynamical EoS parameters w 0 and w a of the CPL parameterization (obtained by imposing a phantom EoS w ( z ) < -1 for any z ), and the Hubble parameter H 0 for P18+BAO+SN and ACT+BAO+SN.  \n<!-- image -->  \n̸  \nmore freedom in the DE sector by not restricting the EoS to being that of a cosmological constant. We review, update and extend the state-of-the-art analyses performed in earlier similar studies by considering two distinct physical scenarios: IDE cosmology with a non-dynamical EoS w 0 = -1 , and IDE models with a dynamical EoS w ( z ) . For the latter, we adopt a simple CPL parameterization given by Eq. (8).  \nAvoiding early-time superhorizon instabilities in the dynamics of cosmological perturbations imposes stability conditions on the DM-DE coupling ξ and the DE EoS, forcing the fraction ξ/ (1 + w ) to be positive. Both in the dynamical and non-dynamical case, we have carefully taken into account stability conditions, studying separately two physical regimes represented by phantom and quintessence EoS. In the quintessence and phantom regimes, the energy-momentum transfer is forced to flow in different directions (from DM to DE and from DE to DM, respectively), producing a quite different phenomenology both in terms of perturbations and background dynamics.  \nAimed to conclusively assess whether IDE models featuring dynamical and/or non-dynamical DE EoS can represent a possible solution to the Hubble constant tension, we systematically study all the possibilities deriving updated observational constraints from the latest cosmological and astrophysical observations. Specifically, we consider two different independent CMB experiments: the Planck-2018 temperature polarization and lensing data as well as small-scale Atacama Cosmology Telescope CMB measurements. CMB experiments are considered on their own as well as in different combinations involving low-redshift probes, such as Supernovae distance moduli measurements from the Pantheon-Plus catalog and the most recent Baryon Acoustic Oscillations from the SDSS-IV eBOSS survey.  \nOur updated and extended analysis reveals significant differences compared to the state-of-the-art results, significantly restricting the parameter space allowed to IDE  \nmodels with dynamical and non-dynamical EoS, as well as limiting their overall ability to reconcile cosmological tensions. Notably, all our most important findings are always independently corroborated by the two different CMB experiments (that share a consistency of view on IDE even allowing the dark sector physics), making the conclusions of our analysis robust. The most important takeaway results read as follows:  \n- · IDE models featuring a non-dynamical quintessence EoS ( w 0 > -1 ) produce larger values of the presentday expansion rate H 0 when analyzed in terms of CMB data. This is due to the fact that the total amount of energy-momentum flow allowed from the DM to the DE is poorly constrained, leaving enough freedom to obtain large negative ξ and higher H 0 . However, including low-redshift probes, this preference is essentially lost. While using SN measurements calibrated with SH0ES, we can still obtain a value of H 0 large enough to reduce the Hubble tension down to 2 . 5 -2 . 7 σ . Considering the uncalibrated SN dataset or BAO distance measurements (both separately and in conjunction), we become very constrained on the coupling ξ , and no room is left to increase H 0 towards local distance ladder values anymore. The most constraining cosmological bounds on these scenarios are summarized in Figure 1.\n- · IDE models featuring a non-dynamical phantom EoS ( w 0 < -1 ) predict an energy-momentum transfer from the DE to DM. Interestingly, when this model is analyzed with CMB and CMB+BAO, and CMB+SN+SH0ES data, we get larger values of the Hubble constant, primarily due to a phantom DE EoS. Therefore, this model can, in principle, help with the Hubble tension as well. However, considering uncalibrated SN (both in combination CMB+SN and CMB+BAO+SN), the preference for larger H 0 is strongly reduced. Also, in this case, the joint analysis of CMB, SN, and BAO data (whose results are shown in Figure 2) strongly limits the ability of the model to  \nrepresent a solution to the Hubble tension.  \n- · IDE models featuring a dynamical quintessence EoS ( w ( z ) > -1 at any z ) perform better in attempting to increase the value of H 0 compared to the respective non-dynamical case. However, even allowing for a dynamical w ( z ) , when considering the joint analysis of CMB, BAO, and SN data, we are not able to significantly increase H 0 to solve the tension, as clear from Figure 3.\n- · IDE models featuring a dynamical phantom EoS perform worse than the non-dynamical case. We experience the same pattern noticed in the other scenarios: local universe observations rule out the model as a possible solution to the Hubble tension. This becomes pretty much evident when considering CMB, BAO, and SN data altogether, see also Figure 4.  \nOverall, our comprehensive reanalysis shows that updated BAO and SN data appear to constrain the possibility that w 0 IDE or w ( z ) IDE alone can conclusively resolve the Hubble tension for the proposed interaction model. Using the SH0ES calibration for SN (which is a well-motivated choice given the larger values of H 0 produced by the CMB-only analysis), we still have room to mitigate, although not fully solve, the Hubble trouble. 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Overall, it is worth investigating whether some new ingredients could be added to the IDE cosmology, which can overcome the SN and BAO issues pointed out in this article.', 'ACKNOWLEDGMENTS': "We thank the referee for many insightful comments which helped us to improve the quality of the manuscript. SP acknowledges the financial support from the Department of Science and Technology (DST), Govt. of India under the Scheme 'Fund for Improvement of S&T Infrastructure (FIST)' (File No. SR/FST/MS-I/2019/41). EDV is supported by a Royal Society Dorothy Hodgkin Research Fellowship. RCN thanks the financial support from CNPq under the project No. 304306/2022-3, and the FAPERGS for partial financial support under the project No. 23/2551-0000848-3. CvdB is supported (in part) by the Lancaster-Sheffield Consortium for Fundamental Physics under STFC grant: ST/X000621/1. 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