We consider theories which break the invariance under diffeomorphisms Diff down to transverse diffeomorphisms TDiff in the matter sector consisting of multiple scalar fields. In particular we regard shiftsymmetric models with two free TDiff scalar fields in a flat RobertsonWalker spacetime and use the perfect fluid approach to study their dynamics. As a consequence of the symmetry breaking an effective interaction between the fields is induced from the conservation of the total energymomentum tensor without the necessity to introduce any explicit interacting term in the Lagrangian. We study the different singlefield domination regimes and analyze the energy exchange between the fields. Thereupon we introduce an application of these models for the description of interactions in the dark sector and compare the theoretical predictions of our model to observational data from Type Ia supernovae.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']
Multifield TDiff theories for cosmology
{'Multi-field TDi ff theories for cosmology': "Diego Tessainer a, ∗ , Antonio L. Maroto a , Prado Mart'ın-Moruno a  \na Departamento de F'ısica Te'orica and Instituto de F'ısica de Part'ıculas y del Cosmos (IPARCOS-UCM), Universidad Complutense de Madrid, 28040, Madrid, Spain.", 'Abstract': 'We consider theories which break the invariance under di ff eomorphisms (Di ff ) down to transverse di ff eomorphisms (TDi ff ) in the matter sector, consisting of multiple scalar fields. In particular, we regard shift-symmetric models with two free TDi ff scalar fields in a flat Robertson-Walker spacetime and use the perfect fluid approach to study their dynamics. As a consequence of the symmetry breaking, an e ff ective interaction between the fields is induced from the conservation of the total energy-momentum tensor, without the necessity to introduce any explicit interacting term in the Lagrangian. We study the di ff erent single-field domination regimes and analyze the energy exchange between the fields. Thereupon, we introduce an application of these models for the description of interactions in the dark sector, and compare the theoretical predictions of our model to observational data from Type Ia supernovae.  \nKeywords: cosmology, scalar fields, shift-symmetry, transverse di ff eomorphisms, dark energy, interacting dark sector.', '1. Introduction': 'It is widely known, as observational data indicate, that our Universe currently exhibits an accelerated expansion [1]. Many models explain this as the consequence of a dark energy component dominating the cosmic expansion, taken to be a cosmological constant in the standard model. However, there are other alternatives, such as quintessence , involving a canonical scalar field with a dynamical equation of state determined by its potential [2], and k-essence [3], which also display dynamical dark energy and can avoid fine-tuning problems, but include non-canonical kinetic terms in the action. Additionally, observational data also indicate that most of the matter composition of our Universe is dark matter [4]. Furthermore, it is nowadays recognized that there exists a tension in the Hubble parameter H 0 measurements [5], which could be alleviated by models involving dark sector interactions [6] or phantom models [7], in which the energy density of the dark energy component increases with the expansion. In addition, it has been proven that in order to ease both the H 0 and S 8 cosmological tensions simultaneously by taking into account new physics that is relevant only at late cosmic times, a dark energy component crossing the phantom limit is necessary [8]. On the other hand, as the nature of the dark sector is unknown, possible modifications of gravity at cosmological scales are often considered [9]. Regarding this possibility, multiple modified gravity theories extending upon General Relativity (GR) have been explored [10].  \nEven if GR provides a very powerful tool for studying gravity and cosmology, theories breaking invariance under di ff eomorphisms (Di ff ) have been recently gaining popularity, with one  \nof the most prominent ones being Unimodular Gravity (UG) [11, 12, 13]. In UG, the metric determinant is taken to be a fixed non-dynamical field and the Di ff invariance is broken down to transverse di ff eomorphisms (TDi ff ) and Weyl rescalings. UG theories could provide a solution to the problem of vacuumenergy which does not gravitate in this type of theories [14]. Nevertheless, in this work we will focus on theories that are only invariant under TDi ff , which have lately started to be studied more deeply. Thus for instance, TDi ff models beyond UG have been studied in references [15, 16, 17, 18]. The cosmological evolution in TDi ff -invariant theories propagating a scalar graviton mode was analyzed in reference [17]. On the other hand, TDi ff invariant models with broken di ff eomorphisms in the matter sector have been analyzed in references [19, 20] for single scalar fields. There it is shown that even though on small scales such theories behave as standard Di ff models, on superHubble scales the behaviour can be drastically di ff erent, thus opening up a wide range of possibilities for cosmological model building. Thus, in particular, a simple TDi ff model for dark matter based on a free scalar field was proposed in references [19, 20]. A unified TDi ff model for the dark sector has been considered in reference [21]. A general classification of singlefield TDi ff models based on their speed of sound and equation of state was performed in reference [22]. TDi ff models for single abelian gauge fields can be found in reference [23] and their phenomenological implications for cosmic magnetic field evolution in reference [24].  \nIn this work we will extend the previous works and consider multi-scalar TDi ff invariant models in the matter sector in flat Robertson-Walker (RW) spacetimes. We will specifically regard shift-symmetric models, which are invariant under shift transformations of the field, i.e., ϕ → ϕ + C , where C is a constant. Thus, we will only consider the exact kinetic domination regime for each field. The motivation behind this approach  \nlies in the fact that in this way we can avoid fine-tuning problems depending on the specific choice for the potential term in the action. On the other hand, not considering any mass or potential-like terms in the action also results in the EinsteinHilbert action only receiving higher-order radiative corrections, which also motivates our choice to only break the Di ff symmetry in the matter sector.  \nUnlike the single-field case, the energy-momentum tensor (EMT) conservation will entail an e ff ective interaction between the fields as a consequence of the symmetry breaking even without introducing any interaction terms in the Lagrangian. This fact opens up a wide range of phenomenological implications for multi-field models. Particularly, we will apply this e ff ect to describe an interacting dark sector, comparing its predictions with observational data.  \nThe work is organized as follows. In section 2 we briefly review the TDi ff formalism, focusing on shift-symmetric theories and lay the groundwork for our particular models. Section 3 is devoted to explain the theoretical framework for multi-scalar TDi ff models. In section 4 we perform a numerical analysis for our model, applying it to the dark sector. Results will be compared both with observations and w CDM, and physical predictions for our TDi ff model will be obtained. Finally, in section 5 we will discuss the conclusions.', '2. Single-field shift-symmetric TDi ff theories': 'In this section we will briefly recap the main results obtained for shift-symmetric TDi ff theories involving one scalar field.', '2.1. Transverse di ff eomorphisms and matter action': "Let us first consider a general infinitesimal coordinate transformation x µ 7→ x ' µ = x µ + ξ µ ( x ) given by the vector field ξ . As it is well known, the variation of metric tensor g µν ( x ) will be given by its Lie derivative, i.e.,  \nδ g µν = L ξ ( g µν ) = -∇ νξµ - ∇ µξν, (1)  \nand thus it follows that the metric determinant ( g : = | det( g µν ) | ) will transform according to  \nδ g = gg µν δ g µν = -2 g ∇ µξ µ . (2)  \nLet us now write down our action. This is  \nS = S EH[ g µν ] + S mat[ g µν, ϕ ] , (3)  \nwhere S mat denotes the matter part of the action involving a single scalar field ϕ . Since we will only break the Di ff symmetry in the matter action, the geometrical part will just be the usual Einstein-Hilbert action  \nS EH[ g µν ] = -1 16 π G Z d 4 x √ g R . (4)  \nOn the other hand, the matter part will read  \nS mat[ g µν, ϕ ] = Z d 4 x f ( g ) L ( g µν ( x ) , ϕ ( x ) , ∂µϕ ( x )) , (5)  \nwhere L denotes the corresponding scalar under Di ff Lagrangian density and f ( g ) an arbitrary function of the metric determinant. Recalling (1) and (2) we can compute δξ S , which, after integration by parts and assuming that the fields vanish at infinity, reads [19]  \nδξ S = Z d 4 x ∂µξ µ [ f ( g ) -2 gf ' ( g )] L . (6)  \nThus, we see that the action is invariant under any infinitesimal coordinate transformation (Di ff invariant) only when f ( g ) -2 gf ' ( g ) = 0, i.e. f ( g ) ∝ √ g . However, the action is also invariant for any form of f ( g ) if the transformations satisfy ∂µξ µ = 0. This corresponds to a smaller subgroup of symmetry, the transverse di ff eomorphisms (TDi ff ).", '2.2. Single scalar-field models in the kinetic regime': 'Let us first consider the matter part of the action with a simple kinetic term [19, 20]:  \nS mat = Z d 4 x 1 2 f ( g ) ∂µϕ∂ µ ϕ, (7)  \nwhere f ( g ) is a positive coupling function of the metric determinant. The corresponding equation of motion reads  \n∂µ GLYPH<0> f ( g ) ∂ µ ϕ GLYPH<1> = 0 , (8)  \nand the EMT will be defined as usual:  \nT µν : = -2 √ g δ S mat δ g µν , (9)  \nwhich in this case reads  \nT µν = f ( g ) √ g GLYPH<16> ∂µϕ∂νϕ -F ( g ) g µν ✷ ϕ GLYPH<17> , (10)  \nwhere we have defined F ( g ) : = d ln f ( g ) / d ln g . Since we are not modifying the Einstein-Hilbert action, the Bianchi identities are preserved and thus the local conservation of the EMT will still hold [19, 20] under solutions of Einstein equations.  \nIn relation to the background geometry, we will consider a spatially flat FLRW metric. Since we have less gauge freedom than in the Di ff case, we will not generally be able to perform a coordinate change that fixes the lapse function to one and we will have more physical degrees of freedom than in the Di ff case. Thus, our spacetime can be described by the following line element [25]:  \nds 2 = b 2 ( τ ) d τ 2 -a 2 ( τ ) dx 2 , (11)  \nwhere a ( τ ) and b ( τ ) are the independent components that will act as the scale factor and lapse function, respectively; and τ denotes the time-coordinate. Both must be computed from Einstein equations 1 .  \nLet us now apply the perfect fluid approach. It is worth recalling that, when ∂µϕ is a time-like vector, the EMT (10) takes the form [20]  \nT µν = ( ρ + p ) u µ u ν -p g µν, (12)  \nwhere ρ = T 0 0 denotes the energy density, p = -T i j δ j i / 3 the pressure, and u µ is the four-velocity of the fluid, a time-like unit vector. Recalling (10) and using (11) we get  \nρ = f ( g ) b 2 √ g GLYPH<2> 1 -F ( g ) GLYPH<3> ( ϕ \' ) 2 , (13)  \np = f ( g ) b 2 √ g F ( g ) ( ϕ \' ) 2 , (14)  \nwhere we have considered a homogeneous field ϕ = ϕ ( τ ). It is straightforward to see from equations (13) and (14) that  \nw ϕ : = p ρ = F ( g ) 1 -F ( g ) ; (15)  \nwhich will generally depend on τ and, thus, the equation of state parameter w ϕ will generally evolve throughout time. One particular case of interest takes place when the coupling function is a power-law, i.e., f ( g ) = kg α , where k and α are constants. In this case we obtain for w ϕ the following result:  \nw ϕ = α 1 -α = const . (16)  \nNotice how this requires α < 1 in order for the weak energy condition to be satisfied [20]. In addition, the zeroth component of the EMT conservation equation ∇ ν T µν = 0 yields the usual result [19]:  \nρ \' + 3 a \' a ( ρ + p ) = 0 . (17)  \nOn the other hand, the equation for the G 00 component of the Einstein tensor yields [25]  \na \' a ! 2 = 8 π G 3 ρ b 2 , (18)  \nwhich is the usual Friedmann equation in time τ . Notice that it recovers its original form under the coordinate transformation d t = b ( τ ) d τ , where t is the cosmological time. We will denote \' = d / d τ and · = d / d t .  \nFinally, let us write the equation of motion of ϕ ( τ ) in this space-time (8):  \nϕ \'\' ( τ ) + ϕ \' ( τ ) L \' ( τ ) L ( τ ) = 0 , (19)  \nwhere L ( τ ) ≡ f ( g ( τ )) / b 2 ( τ ). This equation of motion implies that  \nϕ \' ( τ ) = C ϕ L ( τ ) , (20)  \nwith C ϕ a constant parameter. Substituting (20) in equations (13) and (14); factoring out ρ + p in equation (17) and recalling g = b 2 a 6 , the conservation law (17) reads  \nd d τ ln GLYPH<16> a 6 GLYPH<17> = g \' ( τ ) d d g " ln (1 -2 F ( g )) g f ( g ) !# , (21)  \nwhich provides the geometrical constraint that allows the conservation law (17) to be satisfied. This is [19]:  \ng f ( g ) (1 -2 F ( g )) = Cga 6 , (22)  \nwhere Cg is a constant. This geometrical constrain on the metric determinant g allows us to obtain the relation between b and a for any given coupling. For instance, if f ( g ) ∝ g α , equation (22) implies that  \nb ∝ a 3 α/ (1 -α ) . (23)  \nNotice that only when we take α = 1 / 2 (Di ff limit), we recover the standard sti ff -fluid behaviour ρ ( a ) ∝ a -6 of a kinetically dominated scalar field [26, 27]. In conclusion, TDi ff symmetry allows for a much wider phenomenology for simple kinetically driven scalar fields.', '3. Shift-symmetric multi-field TDi ff models': "In this section we will extend the previous results to the case of two free shift-symmetric TDi ff homogeneous scalar fields in the matter action with di ff erent coupling functions. Since both fields will be kinetically driven, our action will read  \nS mat = Z d 4 x 1 2 2 X i = 1 GLYPH<16> fi ( g ) ∂µϕ i ∂ µ ϕ i GLYPH<17> . (24)  \nNotice that we did not consider an interaction potential between both fields. As we will see, the energy exchange and the rich phenomenology will arise from geometrical constrains coming from the conservation of the total EMT, since the individual EMTs of each field will not be conserved as a consequence of the symmetry breaking. In fact, since our fields are free, the total EMT will simply be the sum of the individual EMTs of each field:  \nT µν = T (1) µν + T (2) µν = ( ρ 1 + p 1) u µ u ν -p 1 g µν + ( ρ 2 + p 2) u µ u ν -p 2 g µν (25)  \nFor homogeneous fields in a Robertson-Walker background both fields share a common velocity u µ and  \nρ i = fi ( g ) b 2 √ g GLYPH<2> 1 -Fi ( g ) GLYPH<3> ( ϕ ' i ) 2 , i = 1 , 2 (26)  \npi = fi ( g ) b 2 √ g Fi ( g ) ( ϕ ' i ) 2 , i = 1 , 2 (27)  \nwhere very much as in the single-field case, we have defined Fi ( g ) : = d ln fi ( g ) / d ln g , so that the corresponding equations of state read  \nwi : = pi ρ i = Fi ( g ) 1 -Fi ( g ) , i = 1 , 2 (28)  \nThe conservation of the total energy-momentum tensor implies  \n∇ µ T µν = ∇ µ T (1) µν + ∇ µ T (2) µν = 0 , (29)  \nwhich in the Robertson-Walker background reads  \nρ ' 1 + 3 a ' a ( ρ 1 + p 1) + ρ ' 2 + 3 a ' a ( ρ 2 + p 2) = 0 . (30)  \nNotice that the previous expression does not imply the energy conservation for individual fields, but in general we will have  \nρ ' 1 + 3 a ' a ( ρ 1 + p 1) = Q , (31)  \nρ ' 2 + 3 a ' a ( ρ 2 + p 2) = -Q , (32)  \nwhere Q is commonly referred to as the interacting kernel in the literature [28].  \nOn the other hand, the fields equations of motion read  \nϕ '' i ( τ ) + ϕ ' i ( τ ) L ' i ( τ ) Li ( τ ) = 0 , i = 1 , 2 (33)  \nwith Li ( τ ) ≡ fi ( g ( τ )) / b 2 ( τ ), so that very much as in the singlefield case, we can write  \nϕ ' i ( τ ) = C ϕ i Li ( τ ) , i = 1 , 2 (34)  \nwith C ϕ i constants.  \nSubstituting these expressions into the respective pressures and energy densities (26) and (27), recalling the conservation equation (30) and proceeding analogously to the single-field case, calculations yield the following geometrical constrain:  \nC 2 ϕ 1 g | 2 F 1 -1 | f 1 + C 2 ϕ 2 g | 2 F 2 -1 | f 2 = Cga 6 , (35)  \nIn the case in which the coupling functions are simple power laws  \nfi ( g ) = λ ig α i , i = 1 , 2; (36)  \nwith λ i , α i constants, the conservation equation (35) implies  \nC 1 g 1 -α 1 | 2 α 1 -1 | + C 2 g 1 -α 2 | 2 α 2 -1 | = Cga 6 ; (37)  \nwhere C 1 = C 2 ϕ 1 /λ 1 and C 2 = C 2 ϕ 2 /λ 2. This is a very illuminating result, since as we observe from equation (37) it does not require the individual EMT conservation of each field and thus it will involve a geometrical-like interaction between the two components caused by the symmetry breaking. Unlike the single-field case, an explicit solution of this equation cannot be obtained even for simple power-law functions.  \nLastly, here we include the expression for the energy density in the power-law coupling case, which will be of valuable use throughout the rest of the work:  \nρ i ( a , b ) = Ci (1 -α i ) b 1 -2 α i a 6 α i + 3 , i = 1 , 2 (38)  \nwhich is straightforwardly obtained from equation (26) using (34). Notice how the e ff ective interactions will be reflected on the particular form of b ( a ) obtained through the EMT conservation law (37).", '3.1. Approximate results: single-field domination': 'Let us first consider the case in which one of the fields, for example ϕ 1, dominates over the other, ϕ 2. We can thus neglect the contribution of ϕ 2 in (37), so  \nC 1 g 1 -α 1 | 2 α 1 -1 | ≃ Cga 6 , (39)  \nwhich can be solved as  \nb ∝ a 3 w 1 (40)  \nw 1 = α 1 1 -α 1 , (41)  \nw 2 = α 2 1 -α 2 , (42)  \nas we can see from (28). This is the same geometrical constrain one would obtain if ϕ 1 was the only field. Notice that this is just an approximation that provides the leading order of b ( a ), but it gives us valuable information concerning the evolution of the energy densities in the di ff erent domination regimes. Recalling (38) and using (40) yields  \nρ 1( a ) ∝ a -3(1 + w 1) and ρ 2( a ) ∝ a -3(1 + w e ff ) , (43)  \nand thus ϕ 1 decays as expected from its equation of state, but the subdominant field ϕ 2 will exhibit a decay as if it were a perfect fluid with constant equation of state parameter w e ff , w 2, where  \nw e ff = 2 w 2 -w 1 + w 1 w 2 1 + w 2 . (44)  \nIt is worth noting that the individual equation of state parameters wi will then depict the asymptotic decay behavior of each component when it is dominant. Fig.1 summarizes the wide range of phenomenological possibilities for the subdominant component.  \nFigure 1: E ff ective equation of state parameter w e ff of the subdominant field ϕ 2 under ϕ 1 domination in terms of the individual equation of state parameters w 1 and w 2.  \n<!-- image -->  \nwhere  \nThis result happens to be physically illuminating with regards to cosmological contexts. As we can see above, the induced interactions between perfect TDi ff fluids with di ff erent equation of state parameters allow for a wide range of possible evolutions for the subdominant component. In particular, all of the possible dark energy behaviors are plausible for the subdominant field, including phantom dark energy [29] ( w e ff < -1, where its energy density increases over time) and quintessence. We emphasize that these behaviours can be obtained without the addition of non-canonical kinetic terms [3], they are a result of breaking the Di ff symmetry down to TDi ff . Interestingly, although wi < -1 is not allowed for each individual field, in accordance to the weak energy condition [20], the dominance regimes allow for subdominant phantom behavior without violating the energy conditions. As a result, this provides a vast range of possibilities to describe an interacting dark matter-dark energy sector ( w 1 = 0, w 2 < -1 / 3) with an evolving dark energy decay given by a function w e ff ( a ) stemming from the broken Di ff invariance, exhibiting phantom decay at early times during the matter epoch. This will allow for phantom-crossing, as it will later be discussed.', '3.2. Energy exchange': "We will now analyze the exchange of energy between the fields induced by the e ff ective interaction, and its evolution through the several field domination regimes by studying the interacting kernel Q . Let us consider two kinetically-driven scalar fields ϕ 1 and ϕ 2, with constant equation of state parameters w 1 and w 2, respectively. Let us also assume that ϕ 1 dominates over ϕ 2. Using (38) on equation (32) and recalling (44) we obtain the following expression  \nQ = 3 C 2(1 -α 2) a ' a a -3(1 + w e ff ) ( w e ff -w 2) . (45)  \nwhich can be rewritten as  \nQ = 3 ρ 2 H ( w 2 -w 1)(1 -w 2) 1 + w 2 . (46)  \nwhere H = a ' / a denotes the Hubble parameter in time coordinate τ .  \nLet us now study the sign of Q during the single-field domination regimes. Firstly, we observe from (46) that when ϕ 1 dominates, Q has two zeros, those being at w e ff = w 2, i.e., w 2 = w 1 and w 2 = 1. On the other hand, the analysis in the ϕ 2 domination regime is fully akin to the previous one, but we have to perform the change w 1 7→ w 2 and change the sign of Q (remember we defined Q with respect to the conservation law for ϕ 1). We show in Fig. 2 the sign of the interaction kernel in both cases ( ϕ 1 and ϕ 2 domination).  \nIn light of this analysis, we distinguish three scenarios. In the first case, in which both equation of state parameters are smaller than one ( w 1, w 2 < 1), the sign of Q does not change between both domination regimes and thus the direction of the energy exchange will not be altered over time. More clearly, if we assume w 1 > w 2 we see from Fig. 2 that when ϕ 1 dominates Q < 0 and ϕ 1 loses energy in favor of ϕ 2, with the same happening as well when ϕ 2 is dominant. The same reasoning can  \nbe applied to the case in which w 1 < w 2 (although in this case Q > 0), allowing us to conclude that in this case it is the field with the greater equation of state parameter who always loses energy.  \nSecondly, we also have the case in which both fields are beyond sti ff fluids ( w 1, w 2 > 1). We can immediately check (see Fig. 2), similarly to how we proceeded in the previous case, that the direction of the energy exchange will not change during the interaction and it will always be the field with the larger equation of state parameter which gains energy from the other component.  \nLastly, there is the case in which one of the fields is beyond a sti ff fluid and the other is not ( w 1 > 1, w 2 < 1 and vice versa). As opposed to the previous scenarios, we see from Fig. 2 that the direction of the energy flux changes between both domination regimes. For instance, if w 1 > 1 and w 2 < 1, Q will be smaller than zero under ϕ 1 domination and thus ϕ 2 will be gaining energy from ϕ 1. However, when ϕ 2 is dominant, since w 1 > 1 we can see that Q > 0 and thus it is ϕ 1 which gains energy from ϕ 2 now (the analysis is analogous if w 1 < 1, w 2 > 1).  \nRegarding the potential applications for the description of the dark sector, notice that in (44) w e ff = -1 / 3 when w 2 = ( -1 + 3 w 1) / (7 + 3 w 1) ≡ A < w 1. This separates the region of w 2 values in which the subdominant field, taken to be ϕ 2 for this example, starts decaying as dark energy. Similarly, w e ff = -1 occurs at w 2 = ( -1 + w 1) / (3 + w 1) ≡ B and it corresponds to the phantom behavior boundary for ϕ 2. Hence, if w 1 < 1 we can see that if w 2 ∈ ( -1 , B ) the subdominant component will exhibit phantom dark energy behavior and the dominant field will lose energy in favor of this; and if w 2 ∈ ( B , A ) it will also gain energy from the dominant component ϕ 1, but not enough to display phantom nature.  \nMore specifically, if we consider a dark sector model consisting of dark matter (DM) with w 1 = 0 and dark energy (DE), with w 2 < -1 / 3, we can observe from (44) that DE will always be phantom during the matter domination epoch due to the energy flux from DM ( Q < 0). The energy exchange will occur in the same direction when DE dominates, although it will now not be enough to keep the phantom behavior, and the DE decay will gradually transition to resemble its asymptotic value for the equation of state parameter w 2. On the other hand, DM will slowly start to exhibit a di ff erent decay than the typical a -3 as DE becomes more dominant.  \nLastly, before we go on with our analysis let us briefly comment about the existence of tracking solutions in this model. Recalling (43) we see that the condition that must be satisfied in order for both fields to exhibit the same decay would be  \n-3(1 + w 1) = 3 w 1 -9 w 2 -3 w 1 w 2 -3 1 + w 2 , (47)  \nwhich cannot be accomplished unless we are in the trivial case in which both components are indeed the same, i.e., w 1 = w 2, and there would be no interaction. Thus, there will not be tracking solutions in this particular TDi ff model.  \n<!-- image -->  \nFigure 2: Sign of the interaction kernel Q when the ϕ 1 fluid dominates (left) and for ϕ 2 domination (right)  \n<!-- image -->", '3.3. Analytical model': 'Solving the general constrain (37) is not a simple task, and it usually requires numerical treatment. However, there is a particular dark sector model for which equation (37) can be analytically solved, consisting of DM with w 1 = 0 ( α 1 = 0) and DE with w 2 = -1 / 2 ( α 2 = -1). Despite not being the best fitting model, as we will later see, being analytical provides us with a wide insight to further understand the physics behind shiftsymmetric multi-field TDi ff models. The constrain (37) then reads  \nC 1 g + 3 C 2 g 2 = Cga 6 , (48)  \nwhich is quadratic in g and can be easily solved as  \ng = -C 1 6 C 2 + q C 2 1 + 12 C 2 Cga 6 6 C 2 , (49)  \nwhere we have taken into account that Ci = C 2 ϕ i /λ i should be positive to avoid ghosts instabilities, so that only the positiveroot solution of equation (48) is physically sensible. This solution allows us to explicitly obtain the relation b ( a ):  \nb ( a ) = r C 1 6 C 2         a -6         s 1 + 12 C 2 Cg C 2 1 a 6 -1                 1 / 2 , (50)  \nvalid for all values of a . As we will later see, the remote past a ≪ 1 will correspond to the matter era, and in the distant future a ≫ 1 DE will be dominant, as expected.  \nFor a ≪ 1, expanding (50) in powers of a yields  \nb ( a ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> a ≪ 1 ≃ p CgC 1       1 -3 2 C 2 Cg C 2 1 a 6       , (51)  \nfrom which we can obtain the respective energy densities:  \nρ 1( a ) ≃ p C 1 Cg       a -3 -3 C 2 Cg 2 C 2 1 a 3       , (52)  \nρ 2( a ) ≃ 2 Cg C 1 ! 3 / 2 C 2 a 3 . (53)  \nNotice how the DM ( ρ 1) decay is governed by the a -3 term, which corresponds to the expected behavior according to w 1 = 0. Consequently, DM is dominant at early times. Besides, DE ( ρ 2) evolves with a 3 , exhibiting the phantom nature we previously discussed (in particular, w e ff = -2) as a result of it gaining energy from DM. This can be illustrated writing the conservation equations for each component in terms of the energy density of the other, which read:  \nρ \' 1 + 3 a \' a ρ 1 ≃ -9 2 H ρ 2 , (54)  \nρ \' 2 + 3 a \' a ( ρ 2 + p 2) ≃ + 9 C 2 g C 2 C 1 H 1 ρ 1 ; (55)  \nwhere the phantom nature is exposed in (55) as a result of ρ 1 appearing in the denominator.  \nOn the other hand, for a ≫ 1, expanding (50) yields  \nb ( a ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> a ≫ 1 ≃ r C 1 6 C 2 √ Aa -3 / 2 -1 √ 2 A a -9 / 2 ! , (56)  \nwith A ≡ q 12 C 2 Cg / C 2 1 . The energy densities thus read  \nρ 1( a ) ≃ C 3 / 2 1 √ 6 C 2 √ Aa -9 / 2 -1 2 √ A a -15 / 2 ! , (57)  \nρ 2( a ) ≃ C 3 / 2 1 6 3 / 2 C 1 / 2 2 ( -3 a -9 / 2 √ A + 2 a -3 / 2 A 3 / 2 ) . (58)  \nThe leading order for large values of a in ρ 2 indicates that now our DE will decay as expected from its equation of state ( w 2 = -1 / 2), and DM decays faster than a -3 . This implies that at later times it is DE who becomes dominant, and from the reasoning of the previous subsection, we can see that DM  \nis still giving energy to DE, but not enough to keep the phantom behavior as DM starts becoming subdominant. One could write analogous expressions to (54) and (55), but they are not as physically enlightening due to the lack of phantom nature under dark energy domination.  \nWe will now write the exact expressions for both energy densities in order to discuss the whole evolution. From the previous analysis we know that both energy densities become equal at a certain time: a = a eq, ρ i = ρ eq. We can thus write ρ 1 and ρ 2 substituting (50) in (38) and equating them. We obtain  \nC 1 = 5 2 ρ 2 eq a 6 eq C -1 g , C 2 = 125 16 ρ 4 eq a 6 eq 1 C 3 g ; (59)  \nwhich allow us to write down the energy densities in terms of these parameters, which are easier to physically interpret than the integration constants Ci . Thus, we have:  \nρ 1( z ) = 1 √ 3 ρ eq 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) , (60)  \nρ 2( z ) = 1 √ 27 ρ eq 1 + z 1 + z eq ! 6 Θ 3 / 2 ( z ); (61)  \nwhere we defined Θ ( z ) ≡ q 1 + 15[(1 + z eq) / (1 + z )] 6 -1, and where z = 1 / a -1 denotes the redshift.  \nWe will now obtain some physical results and compare this model to Λ CDM before analyzing the general case. Firstly, recalling (38) and the conservation law (17), we can parameterize the decay of each component with a function w e ff , i( z ) which satisfies the individual conservation law  \nρ \' i + 3 a \' a [1 + w e ff , i( z )] ρ i = 0; (62)  \nwhich yields the following result when recalling (38):  \nw e ff , i( z ) = -1 3 " -1 + z b ( z ) (1 -2 α i ) d b d z -6 α i -3 # -1 , (63)  \nwhere α i denote the exponents of the coupling functions of each component. This recovers the previously studied constant results when considering the respective field domination regimes. The functions w e ff , i( z ) can easily be simplified for the analytical case since we know b ( z ) explicitly.  \nUsing the exact expression for b ( z ) (50) yields the result in Fig.3, for two di ff erent values of the free parameter z eq. Notice how in reality we only have z eq as our free parameter, since the cosmic sum rule ( ρ 1 + ρ 2) GLYPH<12> GLYPH<12> GLYPH<12> t = t 0 = (1 -Ω B) ρ c must be satisfied and, thus, it enforces an extra relation between the parameters that allows to remove the dependence on ρ eq. As a reminder, Ω B depicts the baryonic matter component of the universe, ρ c is the critical density and we are ignoring the radiation component at late times. It is worth noting that in this work we assumed that only the dark sector breaks the Di ff invariance, hence we will treat baryons as ordinary Di ff matter. As we can see from Fig.3, the DE decay behavior starts being phantom-like at early times, as expected, and then evolves in time until it reaches the  \nFigure 3: w e ff ( z ) for DM and DE for various z eq. Dark energy transitions from being phantom during the matter era to decaying as usual dark energy with equation of state parameter w 2, with there being phantom crossing at recent times. Dark matter starts to decay faster than expected from w 1 = 0 as dark energy starts dominating.  \n<!-- image -->  \nasymptotic value reflected in the equation of state w 2 > -1 in the future, with there being phantom-crossing near the present. The parameter z eq only changes slightly the behavior in the intermediate regimes, without altering the main physical behavior. On the other hand, DM will exhibit its usual a -3 decay at early times but it will decay faster when DE starts to dominate. In light of this we can see that TDi ff models can provide a very rich phenomenology involving di ff erent time evolutions for the dark sector. This could lead to new models for interactions in the dark sector without the introduction of non-canonical terms or ghost instabilities in the action.  \nLastly, we will analyze this model from the perspective of the density parameters to further understand shift-symmetric TDi ff dark sector models. We will denote the density parameters for DM and DE, respectively, as Ω DMT and Ω DET. We will also use the standard notation for the Λ CDM parameters: Ω M = Ω DM + Ω B for matter and ΩΛ for the cosmological constant. Recalling the Friedmann equation (18) and using cosmological time d t = b ( τ )d τ yields  \nH 2 = 8 π G 3 ( ρ B + ρ 1 + ρ 2) . (64)  \nMultiplying and dividing this expression by the Hubble parameter at t = t 0 (today) H 2 0 , and recalling that ρ c = 8 π G / (3 H 2 0 ) it is straightforward to obtain  \nΩ DMT( z ) = 1 √ 3 ρ eq ρ c 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) E 2 ( z ) , (65)  \nΩ DET( z ) = 1 3 √ 3 ρ eq ρ c 1 + z 1 + z eq ! 6 Θ 3 / 2 ( z ) E 2 ( z ) , (66)  \nwhere we defined  \nE 2 ( z ) ≡ Ω B(1 + z ) 3 + ρ eq √ 3 ρ c 1 + z 1 + z eq ! 6 Θ 1 / 2 ( z ) " 1 + Θ ( z ) 3 # . (67)  \nThis allows us to obtain the time evolution for each density parameter. This will obviously depend on the specific value of z eq, but the general behaviour will be similar. For the sake of simplicity, we included the case z eq = 1 . 1 in the Fig.4 to show qualitative results. This results in a higher DE abundance and a  \nFigure 4: Evolution of the density parameters: Λ CDM (continuous lines) vs analytical TDi ff case (dashed lines), for z eq = 1 . 10. In shift-symmetric TDi ff models dark energy would be more dominant as a consequence of it being phantom at early times, gaining energy from dark matter.  \n<!-- image -->  \nlower DM one at t = t 0 than those from Λ CDM, which can be interpreted as a consequence of the phantom era during the DM domination regime. As we previously discussed, DM transfers part of its energy to DE, which translates into its phantom behavior and thus contributes to obtaining higher values of Ω DET. It is worth mentioning, however, that we shall not directly compare these parameters to those from Λ CDM, as Ω DMT and Ω DET may not be regarded as true DM and DE density parameters, since, as opposed to Λ CDM, this model presents an interacting dark sector and thus there may be contributions from both components to each parameter.', '4. A TDi ff model for dark sector interactions': 'We will now consider a more general case with w 1 = 0, which could play the role of DM; and arbitrary w 2 < -1 / 3, which could play the role of DE. We will then contrast the predictions of this simple model to observations to get a glimpse on the viability of shift-symmetric TDi ff models for describing the dark sector. Recalling the geometrical constrain (37) arisen from the conservation of the EMT, using α 1 = 0 and dividing by C 2 yields  \nλ (1 -α 2) g + g 1 -α 2 | 2 α 2 -1 | = a 6 ( λ (1 -α 2) + | 2 α 2 -1 | ) . (68)  \nwhere we have used  \nρ 1( t 0) ρ 2( t 0) = C 1 C 2 (1 -α 2) -1 = Ω DMT Ω DET ≡ λ. (69)  \nand normalized 1 a ( t 0) = 1, and also g ( t 0) = 1, which leads to b ( t 0) = 1. Therefore, we only have two free parameters, those  \nbeing the exponent of the power-law coupling function of the DE component, α 2, and λ . However, we will use another physical parameter instead of λ in order to obtain a more direct analysis and an easier comparison to observations. In fact, recalling Friedman equation (64) in cosmological time, using (38) and noting that ( ρ 1 + ρ 2) GLYPH<12> GLYPH<12> GLYPH<12> t 0 = (1 -Ω B) ρ c as a consequence of the cosmic sum rule yields the following expression for H 2 ( z ):  \nH 2 ( z ) = H 2 0        Ω B(1 + z ) 3 + (1 -Ω B) 1 + 1 λ ! -1 b ( z )(1 + z ) 3 + (1 -Ω B) 1 λ 1 + 1 λ ! -1 b ( z ) 1 -2 α 2 (1 + z ) 6 α 2 + 3        , (70)  \nwhere we neglected radiation, as the purpose of this model is to study the DM and DE domination epochs. Otherwise, we should have included the corresponding Ω R(1 + z ) 4 contribution from radiation, assuming it is a Di ff component. Notice that at early times, when the ϕ 1 fluid dominates over ϕ 2, we can neglect the last term in (70). In addition, b ( a ) ∝ a 3 w 1 takes a constant value at early times b ( z ) ≃ bearly since w 1 = 0. This allows us to define the following e ff ective density parameter for total matter at high redshift  \nΩ e ff M ≡ Ω B + (1 -Ω B) 1 + 1 λ ! -1 b early , (71)  \nWe will use this parameter Ω e ff M instead of λ , since both are trivially related through (71). Acknowledge that b early can be directly computed from the conservation equation (68) taking into consideration that DM dominates at this time and radiation does not contribute to the geometrical constrain, since we are treating it as a Di ff component and thus its EMT is automatically conserved. Thus, using (69) we obtain  \nb early = s λ (1 -α 2) + | 2 α 2 -1 | λ (1 -α 2) . (72)  \nIf we express H -1 0 as 2997 . 9 h -1 Mpc, with h being the reduced Hubble constant, this will allow us to fit our parameters ( w 2 , Ω e ff M ) to observations and obtain physical predictions for this model. Notice that we are using w 2 instead of α 2 as the model parameter since the are trivially related by (42). We will consider the baryon density parameter obtained from the abundance of light elements Ω B h 2 = 0 . 02240 ± 0 . 00069 [30], as it is independent of the particular choice for the cosmological model, and we will use the central value of the Planck measurement for H 0 = 67 . 66 km s -1 Mpc -1 [31]. In particular, we developed a code in Python that solves the conservation law (68) for any given pair of these two parameters. Hence, we can obtain b ( z ) and H ( z ) through (70). We will then regard the Union2-database observational data coming from type Ia Supernovae [32, 33] consisting of 557 data for 0 . 015 < z < 1 . 030  \nand compare them to the theoretical distance moduli µ ( z ) predictions of our model. We will study the agreement between theory and observations using the χ 2 s statistical estimator [34]:  \nχ 2 s = X i ( µ obs ( zi ) -µ th ( zi )) 2 E 2 i , (73)  \nwhere the theoretical distance modulus is given by  \nµ th ( z ) = 5 log 10 d L( z ) 1 Mpc ! + M = ˆ µ ( z ) + M , (74)  \nwith d L( z ) the luminosity distance computed from  \nd L( z ) = (1 + z ) Z z 0 d z H ( z ) , (75)  \nfor flat spatial sections and M being the absolute magnitude, which we marginalized the following way:  \nM = X i       1 σ µ obs ( zi ) -ˆ µ ( zi ) E 2 i       . (76)  \nwhere Ei denotes the error in the µ i measurement at redshift zi and σ = P i E -2 i . Numerical integration will allow us to perform the analysis in the subsequents sections.', '4.1. One-parameter fit': 'We will first start by studying which w 2 value for the dark energy equation of state parameter fits best to the observational data, when fixing Ω e ff M to the central value measured by Planck CMB observations ( Ω M = 0 . 315 ± 0 . 007) [31]. This will allow us to get a glimpse of the viability of this model before tackling the two parameter case, in which we will regard Ω e ff M as another free parameter (notice from (71) that fixing Ω e ff M leaves us with w 2 as our only free parameter). We have considered a set of values of w 2 lying in the interval ( -0 . 99 , -0 . 33) and computed the distance moduli using (74) for the respective redshift values present in Union2 data. This allows us to study the agreement between our models and observations by using the χ 2 s estimator defined above. For the sake of comparison, we will also include the results obtained from the respective one-parameter w CDM fit, for which the distance moduli are computed from (74), the Hubble parameter being the only di ff erence:  \nH ( z ) w CDM = H 0 p Ω M(1 + z ) 3 + ΩΛ (1 + z ) 3(1 + w ) , (77)  \nfor the DM and DE domination eras, where w denotes the constant DE equation of state. This allows us to compare our oneparameter TDi ff fit to the analogous w CDM fit when considering w as the only free parameter by fixing Ω M to the CMB measurement (we used values of w in (-2,0)). This yields the result in Tab.1.  \nTable 1: One-parameter fit: TDi ff vs w CDM results. Both models fit very well to observational data. In the TDi ff case, dark energy would be phantom in the past and then it would slowly transition to decay as quintessence with equation of state parameter w 2 in the future, there being phantom crossing. The 1σ regions for each parameter are also included.  \n.  \nWe can thus infer that both models provide excellent concordance with observations, with the minimal χ 2 di ff ering just a 0.10% between them. As we know, in Λ CDM, dark energy is a cosmological constant with a non-dynamical equation of state. However, in our ideal TDi ff model, it would be a TDi ff component with p = -0 . 736 ρ but a dynamical decay behavior given by an e ff ective parameter w e ff ( z ), through (63). Using (63) for the best TDi ff fit allows us to obtain the DE decay behavior for all redshifts. Thus, the DE component would have been phantom in the past under DM domination and, afterwards, it would slowly transition until reaching the asymptotic decay reflected by w 2 (quintessence dark energy). Numerically we obtained that w e ff ≃ -0 . 8 today.', '4.2. Two-parameter fit': 'We will lastly analyze our TDi ff model without constraining Ω e ff M , which will allow us to conclude if the actual best TDi ff fit is compatible with type Ia supernovae observations and the CMB measurement for Ω M. On the grounds of this, we will consider a set of values for w 2 in the interval ( -0 . 993 , -0 . 60) and values of Ω e ff M in the interval (0 . 18 , 0 . 56). For each possible pair of values, we calculated the distance moduli using (74) and then computed the respective χ 2 s estimator for each case using (73). In the following analysis we will also include the direct w CDM analogue of this two-parameter fit, in which both Ω M and w are fitted (using values for w in ( -2 , 0) and Ω M in (0 . 01 , 0 . 60)), in order to compare both models. Numerical analysis thus yields the results in Tab.2.  \nTable 2: Two-parameter fit: TDi ff vs w CDM. Both models are in good agreement with observations. The 1σ intervals for each parameter have also been included.  \nThese results indicate that both models fit extremely well the observational data, with the TDi ff model even achieving this in a subtly better way. Therefore, we will focus on the TDi ff case from now on and present the contour plot for both parameters up to the 4σ region in Fig.5. We can infer that, not only does our TDi ff model provide excellent agreement with observational data, but it is also compatible with the Ω M CMB measurement in the 1σ region, indicating that TDi ff models provide good compatibility when it comes to type Ia supernovae observations.  \nThe respective 1σ intervals for each of the parameters were obtained by marginalizing the joint likelihood  \nL ( w 2 , Ω e ff M ) = N e -χ 2 s / 2 , (78)  \nwhich can be done for each variable by performing the integration with respect to the other. This yields:  \nL m ( w 2) = N 1 Z e -χ 2 s / 2 d Ω e ff M , L m ( Ω e ff M ) = N 2 Z e -χ 2 s / 2 d w 2; (79)  \nFigure 5: Two-parameter fit: contour plot for χ 2 s up to the 4σ region using Union2 data. The 1σ region is compatible with the CMB measurement for Ω M. The red dot indicates the best fit. Notice that the 68% contour region di ff ers a bit from the marginalized 1σ intervals, which is a consequence of the non-gaussianity of the distributions.  \n<!-- image -->  \nwhere N 1 and N 2 are normalization constants (the way of proceeding for the w CDMcase is fully analogous). The maximization of these marginalized likelihood distributions L m allowed us to obtain the 1σ regions for both of the parameters, which has been done making use of the GetDist package for Python. 1  \nNotice how, although the model is compatible with a cosmological constant behavior in the 2σ region (see Fig.5), the best fit area lies in the range of w 2 in the interval ( -0 . 873 , -0 . 711). This indicates that, in light of observational data, the TDi ff model would favor a non-cosmological constant behavior in the asymptotic future for the dark energy component, with it being phantom in the matter era and there being phantom crossing, as we studied from the single-field dominance regimes. The 1σ region for the e ff ective matter parameter Ω e ff M is compatible with the Planck measurement from CMB data.  \nFig.6 summarizes the results of the best fitting model and its comparison to w CDM, displaying excellent agreement with observations, and also to w CDM, although there start being minor di ff erences between both models at higher redshift values.  \nLastly, we include the evolution of the e ff ective equation of state parameter for the DE and DM components for the best fitting TDi ff model in Fig.7. We see that today DE evolves with an e ff ective equation of state w e ff , 2( t 0) ≃ -0 . 9. As a result, TDi ff models favor the presence of a dynamical DE, starting from phantom at early times and slowly transitioning to usual quintessence DE, with an asymptotic quintessence decay dictated by w 2), roughly being a cosmological constant at current times. Similarly, DM will exhibit a faster decay than that expected from w 1 = 0 at recent times as a consequence of the symmetry breaking, without the usual a -3 decay being altered during the matter era.  \nThe results obtained throughout this section indicate that this TDi ff model should be further explored in the future. Particularly, the Hubble tension problem should be taken into con-  \nFigure 6: Best fit: comparison to w CDMandobservations. Both models exhibit great accordance with observational data from type Ia supernovae and do not di ff er much from each other. Minor di ff erences start appearing between the models at higher redshift values.  \n<!-- image -->  \nFigure 7: w e ff ( z ) for the two-parameter best fit. DE behaves as a phantom component under DM domination and its dynamical decay transitions to depict its quintessence w 2 behavior in the future. There is phantom crossing taking place near the current time, when DE approximately behaves as a cosmological constant.  \n<!-- image -->  \nsideration, as other models involving phantom DE have been proven to be favored by observations [7].  \nIt is worth remarking that this time-evolving DE behavior involving phantom-quintessence transitions was obtained without enforcing any type of interaction potential in the Lagrangian, and without the addition of non-canonical or ghost terms in the matter action.', '5. Conclusions and future work': 'In this work, we have considered shift-symmetric theories with two kinetically-driven scalar fields breaking the Di ff symmetry down to TDi ff and studied their cosmological consequences. We have analyzed the geometrical condition imposed by the conservation of the total EMT, which still holds as a consequence of the Bianchi identities. When working in a flat FLRWbackground, this conservation allows us to obtain a geometrical constraint that leads to a particular shape for the lapse  \nfunction, which cannot be freely chosen now due to the symmetry breaking.  \nThis geometrical constraint enforces an exchange of energy between both fields, as their individual EMTs are not conserved. In light of this fact, we have proposed a dark sector model involving two TDi ff scalar fields coupled to gravity through power-law functions of the metric determinant, with one field describing a DM fluid and the other DE. We have regarded the di ff erent field domination regimes and showed that, although the equation of state parameters of both fluids are constant, both components will exhibit a di ff erent dynamical decay as that corresponding to Di ff models with the same constant parameters. Particularly, when imposing that DM decays as a -3 at early times, we show that the DE component will present phantom behavior during the matter era for it to slowly transition into quintessential behavior in the future, even if its equation of state parameter takes a constant value larger than minus one. It should be emphasized that this interaction of the dark sector is obtained without including interacting potentials. Moreover, ghost or non-canonical kinetic terms have not been considered to obtain phantom behaviour. In this framework one naturally obtains an interacting dark sector with a dark energy component that crosses the phantom regime. In addition, the shift symmetry of the fields allows us to describe a dynamical interacting dark sector avoiding fine-tuning problems depending on the specific choice for the potential.  \nWe have also studied the evolution of the energy exchange between the fluids and shown how in these models it is always DE which gains energy from DM. On the other hand, we have also considered a particular analytical model to understand the physics involved in this TDi ff dark sector framework. For that model, we have analysed the form of the interaction kernel, and investigated the decay of the fluids, parameterized through w e ff , i ( z ).  \nBeyond the simplest dark sector model, we have studied this interacting dark matter-dark energy framework in deeper detail using numerical techniques. We have considered its parameters: w 2 (the equation of state parameter for the DE field, linked to the exponent of the coupling function) and Ω e ff M (an e ff ective density parameter at high redshift values, which can be easily compared to the CMB measurement for Ω M). Moreover, we have used the Union2 data for Ia supernovae and fitted our two parameters to these observations to get a first glance regarding the viability of these theories. Our results shows exceptional compatibility with those data and a goodness of the fit similar to that of w CDM. In addition, the best fitting region for our parameters at 1σ is compatible with the CMB measurement of the matter density parameter. It is worth mentioning that these results have been obtained considering two parameters ( w 2 and Ω e ff M ), but further and more complete results should be obtained in future work when extending to the full observational analysis using more data sets and more parameters (namely the Hubble constant and the baryonic abundance, as well as the supernovae absolute magnitude). That is, after having introduced this shiftsymmetric multi-field TDi ff model for the first time in this work and studied it from a more theoretical point of view, these first positive results definitely indicate that the model deserves fur-  \nalyses.  \nFinally, it is worthy to emphasize that the present work has started a new line of research based on studying multi-field TDi ff theories. Future projects include to investigate the stability under cosmological perturbations of these theories, in order to study structure formation from the TDi ff perturbation formalism perspective; the covariantized approach, studied in detail in reference [22], could be of special relevance for this purpose. Moreover, one could also analyse the models resulting from considering more general coupling functions or non-homogeneous fields, as well as going beyond the shiftsymmetric case and / or breaking the Di ff symmetry also in the Einstein-Hilbert action. From an observational point of view, future work will also be done regarding a deeper likelihood analysis using additional data sets, such as (Pantheon + SH0Es [32, 33], CMB, BAO and H ( z )) and an extended parameter space, together with the possible impact on the Hubble tension problem.', 'Acknowledgements': "The authors would like to thank Dar'ıo Jaramillo Garrido and Alfredo Delgado Miravet for useful comments and discussions regarding TDi ff theories. DTB also acknowledges financial help from the Ayudas de M'aster IPARCOSUCM / 2023. This work has been supported by the MICIN (Spain) Project No. PID2022-138263NB-I00 funded by MICIU / AEI / 10.13039 / 501100011033 and by ERDF / EU.", 'References': "- [1] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009-1038. arXiv:astro-ph/9805201 , doi:10.1086/300499 .\n- [2] S. Tsujikawa, Quintessence: A Review, Class. Quant. 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