Interacting dark energy: clarifying the cosmological implications and viability conditions

In this study, cosmological models are considered, where dark matter and dark energy are coupled and may exchange energy through non-gravitational interactions with one other. These interacting dark energy (IDE) models have previously been introduced to address problems with the standard $\Lambda$CDM model of cosmology (which include the coincidence problem, Hubble tension and $S_8$ discrepancy). However, conditions ensuring positive energy densities have often been overlooked. Assuming two different linear dark energy couplings, $Q = \delta H \rho_{\rm{de}}$ and $Q = \delta H \rho_{\rm{dm}}$, we find that negative energy densities are inevitable if energy flows from dark matter to dark energy (iDMDE regime) and that consequently, we should only seriously consider models where energy flows from dark energy to dark matter (iDEDM regime). To additionally ensure that these models are free from early time instabilities, we need to require that dark energy is in the `phantom' ($\omega<-1$) regime. This has the consequence that model $Q=\delta H \rho_{\rm{dm}}$ will end with a future big rip singularity, while $Q = \delta H \rho_{\rm{de}}$ may avoid this fate with the right choice of cosmological parameters.


Introduction
The expansion of the universe has thus far been well described by the ΛCDM model, where the energy budget of the universe is divided between ≈ 5% baryonic matter (standard model particles), ≈ 25% non-baryonic cold dark matter (which keeps galaxies from flying apart) and ≈ 70% dark energy in the form of the cosmological constant Λ (which explains late-time accelerated expansion).This model has proven to be a very successful fit for astrophysical observations [1][2][3][4][5], but problems with the ΛCDM model remain, which include: • The Cosmological Constant Problem or vacuum catastrophe, refers to the measured energy density of the vacuum being over 120 orders of magnitude smaller than the theoretical prediction.This has been referred to as the worst prediction in the history of physics and casts doubt on dark energy being a cosmological constant [6,7].
• The Cosmic Coincidence Problem, which alludes to the dark matter and dark energy densities having the same order of magnitude at the present moment while differing with many orders of magnitude in the past and predicted future [8][9][10][11][12].The initial conditions of dark matter and dark energy should be fine-tuned to about 95 orders of magnitude to produce a universe where the two densities nearly coincide today, approximately 14 billion years later [13].
• The S 8 discrepancy, which concerns the 3σ level difference between measurements made from the CMB against weak lensing measurements and redshift surveys of the parameter S 8 , which quantifies the amplitude of late-time matter fluctuations and structure growth.[24][25][26][27][28][29].
These problems motivate research into new physics beyond the ΛCDM model.A popular approach to solving these problems has been to investigate cosmological models in which there are non-gravitational interactions between the dark sectors of the universe.This allows the two dark sectors to exchange energy (and/or momentum) while dark matter (DM) and dark energy (DE) are not separately conserved, but the energy (and momentum) of the total dark sector is conserved instead.These models are broadly known as interacting dark energy (IDE) models.In these models, we assume that both radiation and baryonic matter is uncoupled and separately conserved since there are strong 'fifth-force' observational constraints on baryonic matter [30] and any new significant interactions with photons would probably cause deviations from photons following a geodesic path [11].We, therefore, have the following conservation equations for interacting dark energy models: where H is the Hubble parameter, ρ is the energy density, and the subscripts denote radiation (r), baryonic matter (bm), dark matter (dm) and dark energy (de).Here we still assume pressure-less dark matter (P dm = 0 → ω dm = 0), and note that baryonic matter ω bm = 0 and radiation ω r = 1/3 are uncoupled Q = 0, as in the ΛCDM model.The dark energy equation of state (ω de = ω from here onwards) is left as a free variable since the dark energy may be either vacuum energy (ω = −1), in the quintessence (−1 < ω < −1/3) or phantom (ω < −1) regime.Here Q is an arbitrary coupling function whose sign determines how energy (or momentum) is transferred between dark energy and dark matter.If Q > 0, then the energy (or momentum) is transferred from dark energy to dark matter and vice versa for Q < 0, such that [11,12,20,27,[31][32][33][34][35][36][37][38]: > 0 Dark Energy → Dark Matter (iDEDM regime) < 0 Dark Matter → Dark Energy (iDMDE regime) = 0 No interaction. (1.2) Here we have denoted the interacting case where energy flows from dark energy to dark matter (Q > 0) as the interacting Dark Energy Dark Matter regime (iDEDM), and vice versa as the interacting Dark Matter Dark Energy regime (iDMDE) [27].Since there is currently no fundamental theory for the coupling equation Q, the coupling in most works is purely phenomenologically motivated; and must be tested against observations [11,12,20,27,[31][32][33][34][35][36][37][38].
The coupling is thus freely chosen, but we will only consider models where the coupling function Q is either proportional to the dark matter or the dark energy density, which could have a strong field theoretical ground [39].The core publications we considered for these models are [32][33][34].For recent developments and observational constraints, see [19,20,[25][26][27][28][35][36][37][38][39]; and for comprehensive review articles on interacting dark energy, see [11,12].IDE models were first introduced to address the coincidence problem [8][9][10][11][12], but this approach has recently become less popular due to observational constraints on the interaction strength needed to solve this problem significantly.Instead, in recent years these models have received more attention as possible candidates to alleviate the Hubble tension [19][20][21][22][23][24][25][26] and have most recently been shown also to alleviate the S 8 discrepancy while making an excellent fit to the latest cosmological data available [24][25][26][27][28].
Even though IDE models have proven to be popular candidates to address the biggest problems in cosmology, we believe that the parameter space in the most popular model is often not well understood as conditions to ensure positive energy densities and to avoid a future big rip are often ignored.We would like to clarify this.New results that we obtained will be shown in italics to clearly differentiate from previously known results.In this article: • We clarify the general cosmological implications of IDE models for any interaction Q (summarised in table 1).
• We derive a new equation (2.17) that may be used to easily obtain phase portraits for the evolution of dark matter and dark energy densities (figure 1), which applies to any interaction Q without the need to solve the conservation equation (1.1), which may be excessively difficult for most interaction functions.
• We analyze the most popular IDE model where the interaction is proportional to the dark energy density Q = δHρ de .For this model, we will derive the viability condition 0 < δ < −3ω/(1 + 1 r 0 ) (3.18) to avoid negative energy densities, which is the most important result of this paper, as this is nearly always ignored in the literature and shows that negative energy densities are inevitable if energy flows from dark matter to dark energy (iDMDE regime) and that consequently, we should only seriously consider models where energy flows from dark energy to dark matter (iDEDM regime).Most of the rest of the paper is dedicated to clarifying previous results of this model while showing how this negative energy problem is always present for the iDMDE regime's future expansion.
• For this model, we show that the iDEDM regime may alleviate the coincidence problem for the past expansion while solving the coincidence problem for the future expansion, as seen in 3.27 and figure 2. Conversely, the iDMDE regime is shown only to worsen the coincidence problem.This result is well known, but as an application of the previous point, we included newly extended plots for the future expansion which clearly show negative values of r, indicating negative energy densities.The way we have related the effective equations of state ω eff dm and ω eff de to various parameters in this section and throughout the rest of the paper are not new results, but rather a newly suggested formulation for analyzing familiar problems and results more clearly.
• We clarify the additional cosmological consequences of the iDEDM and iDMDE regime for this model, using imposed parameters for the sake of clarity.We show how the coupling influences the evolution of the energy densities (figure 3 and 4), the total fluid effective equation of state (figure 5), deceleration parameter (figure 6), the expansion rate (figure 7), and the age of the universe (figure 8).Most of these results have been shown before, but we included newly extended plots for the future expansion clearly showing how negative dark matter energy densities plague all these previously known results for the iDMDE regime.
• We derived new analytical expressions for the redshift where the radiation-matter z (r=dm+bm) (3.28) and matter-dark energy z (dm+bm=de) (3.29) equalities occur as well the transition redshift z t (3.31)where accelerated expansion starts, known as the cosmic jerk.These results, with the imposed parameters, are summarised in tables 3, 4 and 5.
• We show that to ensure that this model is not only free from negative energy densities but also free from early time instabilities, we need to require that dark energy is in the 'phantom' (ω < −1) regime, as shown in table 6.This has the consequence that these universe models will end with a future big rip singularity (figure 9) unless the effective equation of state is ω eff de = ω + δ 3 > −1.The results from table 6 and equation (3.40) give the most useful theoretical constraints on parameters for other researchers who want to constrain these models with new observational data.A new equation (3.38), which gives the exact time of this big rip, has also been derived.
• We briefly summarise the same results above for the interaction function Q = δHρ dm .
Here the positive energy density condition 0 < δ < − 3ω (1+r 0 ) (3.45) and table 8 summarises the most useful theoretical constraints for this model.The negative energy densities and the inevitable big rip for this model are well known, thus all results in this section may be considered as only new formulations of known results.Equation (3.49) giving the time of this big rip is the only new result.
2 Properties of interacting dark energy models

Background cosmology and its implications
In IDE cosmology, the standard assumptions of isotropy and homogeneity of the universe, as characterised by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, still hold.At the same time, only the conservation equations are modified, changing the evolution of the DM and DE density profiles.This implies that the standard ΛCDM equations may be used, which include those for the Friedmann equation, the deceleration parameter and the total effective equation of state, respectively: ) where ω dm = ω bm = 0 and ω r = 1/3.The crucial difference in the behaviour of IDE models may be understood by how the interaction affects the effective equations of state of both dark matter ω eff dm and dark energy ω eff de , relative to the uncoupled background equations (Q = 0) in (1.1) such that [12,32,33]: Thus, the effects of an interaction may be understood to imply that if [12,32,33]: Dark matter redshifts slower than a −3 (less DM in past), ω eff de > ω de Dark energy has less accelerating pressure, (2.5) Dark matter redshifts faster than a −3 (more DM in past), ω eff de < ω de Dark energy has more accelerating pressure.
(2.6)This implies that even if ω de = −1, when Q < 0 or Q > 0, then the dark energy may behave like either uncoupled quintessence ω eff de > −1 or uncoupled phantom ω eff de < −1 dark energy respectively.If there is no interaction between dark matter and dark energy (Q = 0), the effective equations of state reduce back to the uncoupled model, such that ω eff dm = ω dm = 0 and ω eff de = ω de .These effective equations of state allow us to make predictions regarding the consequences of a dark coupling; and why it was initially introduced to address the cosmic coincidence problem.This can be seen by considering the ratio r IDE of ρ dm to ρ de from (1.1) for interacting dark energy models: with ζ indicating the magnitude of the coincidence problem.Thus, from (2.7), we see that the smaller the difference between ω eff dm and ω eff de the more the coincidence problem will be alleviated while being solved if ζ = 0, which happens when ω eff dm = ω eff de .This can be achieved if dark matter redshifts slower ω eff dm < ω dm and dark energy redshifts faster ω eff de > ω de , which coincides with the iDEDM (Q > 0) scenario.The opposite holds for the iDMDE (Q < 0) scenario.From (2.5),(2.6)and (2.7), while noting that ζ ΛCDM = 3, we may conclude: (2.8)Besides addressing the coincidence problem, IDE models have other far-reaching cosmological consequences.Since ω eff dm < 0 for iDEDM, DM redshifts slower, which leads to less DM in the past and the radiation-matter equality happening later [33], which in turns causes suppression in the matter power spectrum, alleviating the S 8 discrepancy [27] (see [24][25][26][27][28] for how IDE models address the S 8 tension).Similarly, ω eff de > ω de , such that DE redshifts faster, causing more DE in the past.Less DM and more DE in the past have the consequence that both the cosmic jerk and the matter-dark energy equality happen earlier in cosmic history.From the Friedmann equation (2.1), we can see that this overall suppression of dark matter density causes a lower value of the Hubble parameter at late times.This lower value of H 0 worsens the Hubble tension with regard to late time probes [27] (see [20][21][22][23][24][25][26] for how IDE models address the Hubble tension).Since the Hubble parameter, and therefore the expansion rate, is lower throughout most of the expansion, the universe must have taken longer to reach its current size.Since more time was needed to reach current conditions, the universe should also be older.The opposite holds for the iDMDE scenario.These general consequences of a dark sector coupling (if all other parameters are kept constant), are summarised in table 1.
These implications will only hold if the IDE model is viable.Any cosmological model may be considered unviable due to theoretical concerns, such as internal inconsistencies, instabilities or negative energy densities.A model free of these problems can only be deemed viable if it meets observational constraints, such as predicting an expansion history that coincides with the most recent cosmological data.This paper will only consider theoretical constraints, while we refer readers to [19][20][21][22][23][24][25][26][27][28][35][36][37][38][39] for observational constraints.

Instabilities and the doom factor
The coupling between the dark sectors will influence the evolution of dark matter and dark energy perturbations.A complete perturbation analysis of the models considered in the paper is found in [32] and [33].For our purposes, we only want to know what combination of parameters may be used to avoid instabilities.This can be found in [33], by introducing the so-called doom factor d: This is called the doom factor since this factor is proportional to the coupling function Q and may induce non-adiabatic instabilities in the evolution of the dark energy perturbations [33].
The sign of d will determine if there is an early time instability.It was shown that if the doom factor is positive and large d > 1; the dark energy perturbations will become dominated by the terms which are dependent on the coupling function Q, leading to a runaway; unstable growth regime [33].As long as d < 0, the model should be free of non-adiabatic instabilities at large scales.This doom factor can therefore provide the range of parameters that will give a priori stable universe, as is often done in literature [20-24, 26, 27, 33, 37].

Evolution of energy densities and phase portraits
Since the coupling function, Q, is phenomenologically motivated, many different interaction functions exist, which could either be simple linear or complex non-linear interactions [11,12].This often leads to difficulties when trying to solve the coupled conservation equations (1.1) to obtain analytical expressions for how ρ dm and ρ de evolve.It is, therefore, informative to consider how the derivatives of the density parameters Ωx evolve for any arbitrary coupling Q.This can be used to obtain phase portraits with flow lines in the (Ω dm , Ω de )-plane that has attractor and repulsor points.These attractor and repulsor points can tell us how the ratio of DM to DE evolves and indicate whether the coupling solves the coincidence problem.Furthermore, these phase portraits can also tell us if the DM or DE energy densities become negative at any point, which indicates that the interaction Q is unphysical.
For this analysis, we will consider models which contain radiation Ω r , baryonic matter Ω bm , dark matter Ω dm and dark energy Ω de .This may be done by first considering how the density parameters evolve with time, which is done by taking the derivative of Ω x = 8πG 3H 2 ρ x , (where x can be either r, bm, dm or de), giving: From the conservation equations for interacting dark energy models (1.1), we have: where ± = + for x =dm and ± = − for x =de.Substituting (2.11) into (2.10)gives: Where we also have that: (2.13) Substituting (2.13) and ρ x = 3H 2 8πG Ω x into (2.12)gives: Substituting in the expression for the deceleration parameter q (2.2) gives: This relation holds for either dark matter or dark energy with any coupling function Q.If Q = 0, this reduces back to the same expression for the uncoupled case and may be used not only for dark matter and dark energy but for radiation (ω r = 1/3) and baryonic matter (ω bm = 0) as well.For the different components, one therefore has: (2.16) Equation (2.15) reduces back to the ΛCDM case if Q = 0 and ω de = −1, which can be found in [7].For our purposes, we are interested in the parameter space of how dark matter and dark energy evolve with regard to each other.This can be obtained by dividing corresponding dark matter Ωdm and dark energy Ωde evolution equations (2.16) by each other, such that: This can be used to obtain a set of trajectories or flow lines in the (Ω dm , Ω de )-plane, which in turn have stable attractor and unstable repulsor points.These will be used to see if the ratio of dark matter to dark energy becomes fixed in the past or present, thus addressing the model's potential to solve the coincidence problem.Before considering any IDE models, we will first show how these phase portraits work for the ΛCDM model, as this will be the standard model to which we will compare our later results.

Two interacting dark energy case studies
Now that the general properties of IDE models have been discussed, we will move on to two case studies.This will show that the properties from table 1 hold in general for IDE models of any function Q while highlighting significant differences between the couplings.First, we will consider two of the most common IDE models in the literature, where there is a linear coupling function Q proportional to either the dark matter or dark energy density.These couplings have a strong field theoretical ground [39] and have the form: where H is the Hubble parameter (this dependence on H may naturally arise from first principles) and δ is a dimensionless coupling constant that determines the strength of the interaction between dark matter and dark energy [12,33,34].It should be noted that the coupling constant δ is often indicated by α [31,32] (which has an opposite sign to δ) or ξ in the literature [11,19,20,27,[35][36][37]. For these models, we assume that δ < −3ω (so that the coupling strength |δ| is not too strong [32]).This condition implies (δ < −3ω) → (δ+3ω < 0).Furthermore, since we require that H > 0 ; ρ dm > 0 ; ρ de > 0 the sign of δ will determine the direction of energy flow.Therefore, δ > 0 → Q > 0 corresponds to the iDEDM regime and δ < 0 → Q < 0 to the iDMDE regime.The greatest qualitative difference between the two coupling functions is that Q 1 ∝ ρ dm and Q 2 ∝ ρ de , which implies that the effect of the coupling will be either most prominent during early dark matter domination, or later dark energy domination respectively.Similarly to what was done in [33], we will use the cosmological parameters of the ΛCDM model from [1] for all subsequent figures and calculations, with δ = 0.25 for the iDEDM and δ = −0.25 for iDMDE regimes respectively, while temporarily choosing ω de = ω = −1, so that the coupling constant δ is the only variable that differs from the ΛCDM model, thus easing comparisons between the ΛCDM and IDE models.These chosen parameters are purely illustrative.For the latest cosmological parameters obtained from observations for these IDE models, see [26,28].A preliminary analysis done by us, obtaining cosmological parameters for these IDE models from only supernovae data, can be found in [42,43].

Interaction model: Q = δHρ de
The model Q = δHρ de is one of the most common interaction functions in the literature and is more popular than the model Q = δHρ dm , which we briefly discuss in section 3.2.A possible explanation for this is that in the iDMDE regime for Q = δHρ dm model, ρ de < 0 in the past, as was pointed out in [33].These authors then advocated for a coupling Q ∝ ρ de , since all energy densities remain positive throughout the past universe history, even in the iDMDE (δ < 0) regime [33].This result has often been taken at face value in the literature.However, we would like to focus attention on the fact that in the iDMDE (δ < 0) regime, these models will always suffer from negative dark matter energy densities (ρ dm < 0) during future expansion.This can be seen immediately from the interaction function's proportionality to the dark energy density, Q ∝ ρ de .This has an immediate effect on the iDMDE regime.During future expansion, the dark matter density will decrease, and energy will be transferred away from DM to DE until the DM density eventually reaches zero density.However, as there is no mechanism to stop this energy transfer (since energy transfer is only proportional to dark energy density), the energy transfer will still continue, inevitably leading to negative dark matter densities (ρ dm < 0) in the future.This observation should render these models less favourable and should have been noted by many recent papers that have neglected to mention this problem [20,[26][27][28][35][36][37].The exact conditions to ensure positive energy densities throughout the past and future expansion is calculated in section 3.1.3.However, this result can immediately be seen from the phase portraits of these models.

Phase portraits
Assuming the coupling Q = δHρ de = δH 3H 2 8πG Ω de , equation (2.16) becomes: where we have used the fact that 8πG 3H 2 ρ de = Ω de .Since this coupling is not proportional to the dark matter density, it may be seen from (2.16) that the evolution of baryonic matter and dark matter may be grouped together.It should be noted that baryonic matter is still separately conserved here and experiences no new interaction.Thus, if dark matter is grouped with baryonic matter and radiation is negligible (Ω r = 0), (3.2) becomes: Using (3.3), the evolution of matter and dark energy may now be expressed with a phase portrait in the (Ω m , Ω de )-plane, as seen in figure 1.In figure 1, every point on the plane defines a unique trajectory (as indicated by the blue arrows).However, for convenience, we have specified different trajectories (red lines) which pass through specific values for the present matter Ω (m,0) = 0.3 and dark energy densities Ω (de,0) = 0.1, 0.2, ..., 1.1, as was done in [7] for the ΛCDM model.In figure 1, the left panel shows the phase portrait of a positive δ (iDEDM), while the right panel shows a negative δ (iDMDE).The equilibrium points are calculated by setting Ωm = 0 and Ωde = 0 in equation (3.3) and solving for Ω m and Ω de , yielding: Each of the trajectories starts at (1, 0), which is an unstable repulsor point from which all the trajectories diverge.Finally, these paths all converge again at the stable point which is known as an attractor [7].It can be seen that the repulsor point is the same in both cases but that the attractor point is instead shifted by the dark coupling.This highlights the point that the effect of the coupling is more dominant during later dark energy dominance for Q ∝ ρ de .These equilibrium points also highlight the coincidence problem, as the ratio of their coordinates indicates which value r tends to in the past r − or the future r + .The ΛCDM model has r − → ∞ in the past, whilst approaching r + → 0 in the future.IDE models that can find a constant non-zero or non-infinite value for either r − or r + should solve the coincidence problem in either the past or the future, respectively.For this model, we have: Therefore, this model will not solve the coincidence problem in the past as r − but will stabilise r in the future r + , thereby solving the coincidence problem for future expansion.Furthermore, positive δ > 0 (iDEDM) solves the coincidence problem, but δ < 0 (iDMDE) causes negative energy densities.This brings us to the often-overlooked problem of the iDMDE regime for this model.It is clear from (3.4) that Ω dm,+ ≈ Ω m,+ = − δ 3ω , alongside (ω < 0), must imply that δ < (iDMDE) leads to a negative energy attractor solution for Ω dm .We should also note that baryonic matter is grouped with dark matter.However, in the distant future, it dilutes as in the ΛCDM model, and its contribution should become negligible, validating the approximation Ω dm,+ ≈ Ω m,+ .

Background analytical equations
To obtain analytical solutions for how the dark matter ρ dm and dark energy ρ de densities evolve, we need to solve the conservation equations (1.1) with Q = δHρ de , which yields expressions for ρ dm and ρ de : ) Here (3.6) and (3.7) matching the energy densities found in [12,33,35,37].The effective equation of states for this model can be obtained by substituting the coupling equation Q = δHρ de into (2.4).The dark matter effective equation of state is then: Similarly, for dark energy, we have the effective equation of state: which matches with [12,28,33,36,37].It can be seen that ω eff dm is dynamical with a dependence on r, while, in contrast ω eff de is constant.Equations (3.6), (3.7), (3.8) and (3.9) also reduce back to the ΛCDM model when δ = 0 and ω = −1.Using the relations ρ (x,0) = 3H 2 0 8πG Ω (x,0) and Ω x = 8πG 3H 2 ρ x , as well as the scalefactor redshift relation a = (1 + z) −1 , we obtain useful equations for the density parameters Ω dm and Ω de from (3.6) and (3.7) that can be added to the standard ΛCDM model density parameters for baryonic matter Ω bm and radiation Ω r : ) ) ) (3.13)

Positive energy density conditions
Positive energy conditions for these models were obtained in [38] from a dynamical systems analysis, where it was claimed that no viable scenarios exist.However, we want to show that viable conditions exist using a similar approach to what was done in [32].For this model, it can be seen that ρ de (3.7) is always positive (since a −(δ+3ω+3) > 0 for all values of δ), while ρ dm (3.6) has multiple terms which could become negative.We now derive the exact conditions to ensure that the ρ dm is always positive.To do this, we need to find out where the dark matter energy density crosses the zero energy density boundary and becomes negative so that conditions may be chosen to avoid this zero crossing.This is found when we set the dark matter energy density (3.6) equal to zero and solve for a: Using (3.14), we can find solutions where the dark matter energy density crosses zero and becomes negative (ρ dm < 0).From (3.14) and the relation a = (1 + z) −1 , this zero crossing (ρ dm = 0) happens at exactly the redshift z (dm=0) : (3.15) Using (3.14), we may explore four scenarios, (A -(D), where the energy density may possibly cross zero and become negative.These scenarios will be either the iDMDE (δ < 0) or iDEDM (δ > 0) scenarios for either the past or the future.This leads to: Here we can immediately see that for both (A) and (D) the L.H.S. and R.H.S. will never cross, which means that there will be no solution for (3.16) and thus the ρ dm will never cross zero and become negative.Therefore, ρ dm will always remain positive for scenerio's (A) (Past expansion with δ < 0) and (D) (future expansion with δ > 0).Furthermore, scenario (B) will always have a solution, and therefore, the dark energy density will always become negative in the future, as shown by the attractor point in figure 1.
Thus, if condition (3.17) is met, then scenario (C) (Past expansion with δ > 0) will always have positive energy densities.Therefore, since both (C) and (D) will always have positive energy densities, the positive coupling δ > 0 (with condition (3.17) met) may be seen as physical.Since the condition (3.17) holds, it implies that the condition δ < −3ω must necessarily hold as well.Taking the conditions (δ > 0) ; (δ < −3ω) and δ < − 3ω (1+1/r 0 ) together, a general condition is obtained to ensure positive energy densities for this model: The energy densities for all these conditions may be encapsulated in table 2 below.
Conditions ρ dm (Past) ρ dm (Future) ρ de (Past) ρ de (Future) Physical In table 2, (+) means that the energy densities will always remain positive, (−) means that the energy densities will always become negative somewhere in the cosmic evolution.Any scenario leading to negative energy densities should be considered unphysical.Thus, only systems that abide by the condition (3.18) may be considered physical.From this condition (3.18), it may be concluded that only IDE models where energy flows from dark energy to dark matter iDEDM (δ > 0) should be seriously considered as couplings where energy flows from dark matter to dark energy iDMDE (δ < 0) will always lead to either negative energies in the past or the future at redshift z (dm=0) (3.15).This only holds for the coupling Q = δHρ de , and may not be the case for other coupling models.

Cosmic coincidence problem
For this model, the coincidence problem is not solved in the past but instead in the future.This can be seen from the repulsor point r − = ∞ (3.5) being the same as in the ΛCDM model.Conversely, the attractor point r + in (3.5) shows that this model should solve the coincidence problem in the future, at least for the iDEDM regime.We will now reproduce these results from the analytical expression for ρ dm (3.6) and ρ de (3.7), which we may use to obtain the following simplified expression for r in terms of redshift z: From (3.19), it can be seen that r has the proportionality, such that: For the ΛCDM model ζ ΛCDM = 3, and for a general uncoupled model ζ = −3ω, thus from (3.20) it can be seen that: This behaviour coincides with the original analysis in (2.8).Furthermore, this effect becomes more extreme in both the distant past (at large redshifts (1 + z) → ∞) and the distant future (at redshifts (1+z) → 0).This can be seen by considering these limits for (3.44), while noting the condition δ + 3ω < 0, thus: These results match what was found from the phase portrait in figure 1, with the repulsor point r − and attractor point r + (3.5) being the same as the (1 + z) → ∞ and (1 + z) → 0 redshift limits found for r in (3.22), respectively.Furthermore, in the distant future, r has the proportionality: Since r is constant and ζ (Q,+) = 0, this model solves the coincidence problem for future expansion.This only holds for the δ > 0 (iDEDM) regime, since δ < 0 (iDMDE) will lead to a negative constant r + due to ρ dm which becomes negative at z (dm=0) (3.15), as shown in table 2, which is unphysical.Thus, for (1 + z) → 0 in the future, we have: (3.24) To understand this behaviour, we can consider how the effective equations of state ω eff for this model evolve.To do this, we first need the explicit relation for ω eff dm , which is obtained by substituting in r from (3.19) into (3.8): In the distant past the ratio r − → ∞, while in the distant future r + → − δ δ+3ω , as was independently shown in both (3.5) and (3.22).Noting that ω eff de = ω + δ 3 from (3.9), we can see how the dynamical effective equation of state ω eff dm behaves in both the distant past and future: .
The effective equations of state for dark matter and dark energy are, therefore, the same in the distant future (ω eff dm = ω eff de ).This shows that dark matter and dark energy redshift and dilute at the same rate in the future, effectively solving the coincidence problem by keeping the ratio of dark matter to dark energy constant.This shows that whenever r = +constant → ζ = 0, we also have ω eff dm = ω eff de .Furthermore, we can also see that in the distant past, ω eff dm = ω dm .The effect of the coupling on dark matter will thus become negligible for past expansion, effectively mimicking the behaviour of uncoupled dark matter.These predictions agree with what was found by [8,11] and is confirmed by plotting r (3.19) alongside both ω eff dm (3.25) and ω eff de (3.9) in figure 2. From the left panel in figure 2, it can be seen that for the coupled model with δ > 0 (iDEDM), r differs with many orders of magnitude in the past but converges to a constant value in the future r → r + (indicated by the dashed green line), as predicted by (3.5) and (3.22), making the present value less coincidental.The coincidence problem is thus solved for the future expansion history.This coincides with the right panel where ω eff dm = ω eff de , as shown in (3.26).The coincidence problem is also alleviated for the past expansion since the slope of r is smaller (as predicted by (3.21)), which coincides with ω eff de > ω de from (1.1), causing a smaller difference in (ω eff dm − ω eff de ).Conversely, for δ < 0 (iDEDM), we have ω eff de < ω de , which worsens the coincidence problem for the past expansion history (since the slope is greater than the case δ = 0).For future expansion, it can also be seen that r becomes zero, while ω eff de diverges at the same point.This is due to the dark matter density ρ dm which becomes zero in the future at redshift z (dm=0) (red dotted line) from (3.15), and then stays negative for the rest of the future expansion, indicating the unviability of the iDMDE regime.Thus, the results from (3.22), (3.21), (3.24) and (3.26) can clearly be seen in figure 2 and may be summarised as: Past expansion: δ < 0 (iDMDE) Past expansion: worsens coincidence problem Future expansion: ω eff dm = ω eff de (ρ de < 0) negative energy densities.

Evolution of energy densities and cosmic equalities
For this model, we saw that the coincidence problem is solved for the future (3.27).This can be clearly seen by plotting ρ dm (3.6) and ρ de (3.7) against redshift z in figure 3.In figure 3, we see that for δ > 0 (iDEDM), dark matter receives energy from dark energy, causing ρ dm to redshift slower ω eff dm < ω dm (smaller slope), while ρ de redshifts faster (greater slope).This behaviour alleviates the coincidence problem in the past.In the future the slope at which ρ dm and ρ de redshift becomes the same, coinciding with ω eff dm = ω eff de (3.26) and the coincidence problem being solved, while ρ dm dilutes similar to the ΛCDM model in the past where ω eff dm = ω dm (3.26).All these observations coincide with (3.27).It may also be noted that if δ > 0 (iDEDM), ρ de decreases over time; while if δ < 0 (iDMDE), ρ de increases over time.The dark energy, therefore, effectively behaves like either quintessence or phantom dark energy, respectively, with an equation of state ω eff de = ω + δ 3 .Since this effect continues indefinitely, it may cause a big rip singularity in the future at the time (3.38).We can now show that other implications of a dark coupling from table 1 hold for this coupling function.This is done by plotting the density parameters of dark matter Ω dm (3.10), dark energy Ω de (3.11), radiation Ω r (3.13) and baryonic matter Ω bm (3.12) in figure 4, as was done in figure 2 of [33] for only the past expansion, but we include the crucial future expansion as well.For δ > 0 (iDEDM), there will also be a time in the future when ρ dm = 0 at redshift z (dm=0) (3.15), after which ρ dm < 0 for the rest of expansion, which is unphysical.The predicted value for z (dm=0) is indicated by the red marker in figure 4. From figures 3 and 4, it is seen that for δ > 0 (iDEDM), there is less dark matter and more dark energy in the past, and vice versa for δ < 0 (iDMDE).For δ > 0 the matter-radiation equality happens later and the matter-dark energy equality earlier in cosmic history, with the opposite holding for δ < 0 [33].Analytical expressions giving the exact redshift where the radiation-matter z (r=dm+bm) (3.28) equality occurs may be calculated for this model by setting Ω (bm,0) + Ω (dm,0) = Ω (r,0) from equations (3.10), (3.12) and (3.13) and solving for z (using an approximation neglecting a small term), giving: Ω (bm,0) + Ω (dm,0) + Ω (de,0) The matter dark energy equality z (dm+bm=de) occurs when Ω (bm,0) + Ω (dm,0) = Ω (de,0) from equations (3.10), (3.11) and (3.12) and solving for z, giving: Ω (bm,0) +Ω (dm,0) Ω (de,0) In figure 3 it may also be seen there is complete matter domination (Ω dm , Ω de ) − = (1, 0), as in the ΛCDM case.Conversely, dark energy never completely dominates in the future, but instead, dark matter and dark energy have the density parameters (Ω dm+bm , Ω de ) + = − δ 3ω , 1 + δ 3ω from the attractor point (3.4).

Evolution of deceleration parameter
For this model, we have seen that the density parameters mostly deviate from the ΛCDM model during dark energy domination.We, therefore, expect this coupling function to change the behaviour of the deceleration parameter q and the effective equation of state for the fluid ω eff most dramatically in the future expansion.The expressions for both q (equation 2.  From figures 5 and 6, we can see that the past behaviour for the coupled models is almost identical to that of the ΛCDM model, with initial deceleration followed by acceleration from the cosmic jerk onwards.This cosmic jerk occurs at the transition redshift z t , for which an analytical expression (3.31) can be derived by setting q = 0 in equation (2.2), giving: Ω (de,0) The transition redshift for all three values of δ is calculated from (3.31) and indicated by the marker in figure 5 and 6, while the exact redshift for each can be found in tables 4 and 5.

Hubble parameter and age of the universe
The interaction Q will affect the age of the universe, which can be seen from the evolution of the Friedmann equation from (2.1) with the energy densities ρ r = ρ (r,0) a −4 , ρ bm = ρ (bm,0) a −3 , ρ dm from (3.6), ρ de from (3.7) and k = 0. Since both the deceleration parameter and the total effective equation of state deviate mostly during dark energy domination, we expect the expansion rate to change mostly for future expansion.This may be seen by plotting the Hubble parameter (2.1), relative to the non-interacting case (H/H δ=0 ), against redshift.The evolution of the scale factor against time is also plotted, and the universe's age is calculated by numerically integrating (2.1).This yields figures 7 and 8.In figure 7, H/H δ=0 < 1 for δ > 0 (iDEDM) throughout most of the expansion history, indicating a slower expansion rate.This is again due to the overall suppression of dark matter seen in figure 4, which causes a lower value for q and ω eff and therefore a slower expansion rate.Due to δ > 0 (iDEDM) having a slower expansion rate, more time is needed for the universe to evolve from a singularity (a = 0) to its current size (a = 1), causing an older age for the universe as seen in figure 8.The opposite of this holds for δ < 0 (iDEDM).We therefore confirm the following result from table 1: Age of universe (t 0 ) δ > 0 (iDEDM): t (0,IDE) > t (0,ΛCDM) Older universe than ΛCDM, δ < 0 (iDMDE): t (0,IDE) < t (0,ΛCDM) Younger universe than ΛCDM.
The stability of this model will once again be dependent on the doom factor d (2.9).This condition d < 0 guarantees an a priori stable universe as discussed in section 2.2.Thus, for Q = δHρ de we have the doom factor (2.9) [33]: In figure 9, we can see that the predicted time of the big rip singularity t rip (3.38) coincides with the time at which both a → ∞ and ρ de → ∞.The time t rip is significantly affected by the coupling, such that: Big rip δ > 0 (iDEDM): t (rip,IDE) > t (rip,δ=0) Later big rip than δ = 0, δ < 0 (iDMDE): t (rip,IDE) < t (rip,δ=0) Earlier big rip than δ = 0.
These models can still be viable (ω < −1 in the δ > 0 (iDEDM) regime) and avoid a big rip, as long as condition (3.37) is not met, such that ω eff de = ω + δ 3 > −1, while quintessence models with ω > −1 may also have a big rip if ω eff de < −1.Thus, if we want to avoid a big rip at t rip (3.38), we require the condition ω eff de = ω + δ 3 > −1 (3.37), which may be rewritten as δ > −3(ω + 1).This result may be combined with the positive energy condition 0 < δ < −3ω/(1 + 1 r 0 ) (3.18) to get the following condition:  We will now consider an interaction model where the interaction is proportional to the dark matter density Q ∝ ρ dm .This model is less popular in the literature than the previous model where Q = δHρ de , which may be due to the iDMDE regime in this model having negative dark energy densities ρ de < 0 in the past, as was pointed out in [33]).The two models may be analysed in the same manner due to the similarity of the two interaction functions.For the sake of brevity and to avoid needless repetition, we will only briefly summarise the equivalent main results of this model.A full analysis of this model with all equivalent figures

Conclusions
In this paper, we clarified the cosmological consequences of IDE models for any generic interaction Q as summarised in table 1.We also derived equation (2.17) which may be used to obtain phase portraits for the evolution of dark matter and dark energy densities for any generic interaction Q without the need to solve the conservation equation (1.1).We then considered two case studies of linear dark energy couplings, Q = δHρ de and Q = δHρ dm .For these models, we derived the often neglected positive energy conditions 0 < δ < −3ω/(1 + 1 r 0 ) (3.18) and 0 < δ < −3ω/ (1 + r 0 ) (3.45) respectively, from which we note the important fact that the δ < 0 (iDMDE) regime will always lead to ρ dm < 0 in the future for Q = δHρ de and ρ de < 0 in the past for Q = δHρ dm .This implies that the δ < 0 (iDMDE) regime should not be taken seriously as a potential dark energy candidate for these models.For the more viable δ > 0 (iDEDM) regime, we saw that the model Q = δHρ de could solve the coincidence problem in the future whilst alleviating the problem for the past (3.27).Conversely, the model Q = δHρ dm can solve the coincidence problem in the past and alleviate the problem for the future (3.47).Furthermore, the iDEDM regime for both models predicts a later radiationmatter equality, while both the matter-dark energy equality (3.30) and cosmic jerk will occur earlier (3.32).The age of these universe models will also be older (3.34).The opposite holds for δ < 0 (iDMDE).From tables 6 and 8, we see that the only viable regime for both these models, which avoid both negative energy densities and gravitational instabilities, is phantom dark energy ω < −1 in the δ > 0 (iDEDM) regime.This has the consequence that model Q = δHρ dm will always end with a future big rip singularity at the derived time t rip (3.49), while Q = δHρ de may avoid this fate with the right choice of cosmological parameters.The model Q = δHρ de will only experience a big rip future singularity at the derived time t rip (3.38) if the condition ω eff de = ω + δ 3 < −1 (3.37) is met.This big rip may be avoided if the conditions 3(ω + 1) < δ < −3ω 1 + 1 r 0 , alongside ω de < −1(3.40) is also obeyed.
In conclusion, we are not advocating these IDE models as superior alternatives to the ΛCDM model.However, we instead want to emphasise the importance of choosing the correct parameter space (iDEDM regime, with phantom dark energy) to avoid both negative energies and instabilities (as summarised in tables 6 and 8).We hope other researchers will use these theoretical constraints on the parameter space when further investigating the viability of these IDE models to address the current problems in cosmology with the latest observational data.

Figure 2 .
Figure 2. Coincidence problem and effective equations of state (Q = δHρ de )

Figure 9 .
Figure 9. Evolution of energy density, scale factor and the big rip for phantom (ω = −1.15)IDE models -(Q = δHρ de ) 40) describes the iDEDM regime with phantom dark energy.It ensures that all energies are positive throughout the future and past universe evolution and that early-time gravitational instabilities and late-time future big rip singularities are avoided.

3. 2 A
brief summary of results for interaction model Q = δHρ dm

Table 2 .
Conditions for positive energy densities throughout cosmic evolution (Q = δHρ de ) IDE < z ΛCDM happens later than ΛCDM Matter-dark energy equality: z IDE > z ΛCDM happens earlier than ΛCDM, IDE > z ΛCDM happens earlier than ΛCDM Matter-dark energy equality: z IDE < z ΛCDM happens later than ΛCDM,