Anisotropic Dark Energy from String Compactifications

We explore the cosmological dynamics of a minimalistic yet generic string-inspired model for multifield dark energy. Adopting a supergravity four-dimensional viewpoint, we motivate the model's structure arising from superstring compactifications involving a chiral superfield and a pure $U(1)$ gauge sector. The chiral sector gives rise to a pair of scalar fields, such as the axio-dilaton, which are kinetically coupled. However, the scalar potential depends on only one of them, further entwined with the vector field through the gauge kinetic function. The model has two anisotropic attractor solutions that, despite a steep potential and thanks to multifield dynamics, could explain the current accelerated expansion of the Universe while satisfying observational constraints on the late-times cosmological anisotropy. Nevertheless, justifying the parameter space allowing for slow roll dynamics together with the correct cosmological parameters, would be challenging within the landscape of string theory. Intriguingly, we find that the vector field, particularly at one of the studied fixed points, plays a crucial role in enabling geodesic trajectories in the scalar field space while realizing slow-roll dynamics with a steep potential. This observation opens a new avenue for exploring multifield dark energy models within the superstring landscape.


Introduction
For over two decades, there has been strong evidence of a present accelerated epoch in the universe [1,2], supported by recent observations of the Cosmic Microwave Background (CMB) [3][4][5] (also see [6]).The simplest explanation for this behaviour is a vacuum energy or cosmological constant Λ [7,8], which is accommodated in the so-called ΛCDM (Cold Dark Matter) or concordance model.Interestingly, this model started to gain traction as a strong candidate for describing cosmological data even before such observations were made [9].
There are several reasons to explore alternative approaches.Firstly, the vacuum energy interpretation suffers from the lack of a solid theoretical prediction, with a notable concern being the discrepancy of 120 orders of magnitude [10].Secondly, recent results have led to cosmological tensions that cannot, at least easily, be adequately accommodated within this framework [11].Among these challenges, the most significant is the Hubble constant discrepancy [12][13][14].(For a comprehensive overview and the current status, please refer to [15].)Furthermore, exploring alternative models enables us to address issues such as the coincidence problem.All this is happening against the backdrop of the dawn of precision cosmological experiments (e.g., Refs.[16][17][18]), which hold the promise of refining our understanding of the early and late universe.
The goal is to replicate an accelerating late phase of cosmological evolution by emulating the effects of the cosmological constant through field dynamics [19,20].This strategy aligns with the concept of Dark Energy (DE) and seeks to elucidate the origin of the fields and their couplings, potentially arising from an ultraviolet completion like string theory.In fact, in a broader context, any ultraviolet completion will extend this foundational model by introducing additional fields, consequently leading to non-standard cosmologies.
Despite the potential conceptual challenges surrounding its compatibility with accelerated universes [21,22], superstring theory provides a top-down approach that serves as both a playground for model construction and a guiding principle in ensuring consistency within possible ultraviolet completions that incorporate quantum gravity.Remarkably, the phenomenology of string theory has made significant strides over recent decades, offering a wealth of insights into potential fields and their interactions.Although precise and exact outcomes remain elusive due to the intricacies of transitioning from the 10-dimensional theory to the 4-dimensional effective description, guidelines have surfaced, exemplified by the swampland conjectures [23][24][25][26][27][28].These conjectures shed light on what can be anticipated from theories aiming for a coherent ultraviolet completion that encompasses gravity. 1calar fields are pervasive in effective 4-dimensional theories derived from string theory.They describe various couplings (e.g., dilaton modulus), internal dimensions' size and shape (e.g., Kähler modulus), brane locations, and more.Thus, the simplest scenarios involve single scalar field models.Encouragingly, several positive results have already been achieved.For instance, there are potential UV descriptions of generalizations of the Chaplygin gas derived from D-branes [30][31][32], stringy axions [33][34][35], and early dark energy scenarios like the one presented in Ref. [36].
Additionally, there are multifield scenarios, as seen in references such as [37][38][39][40][41][42].In many cases, multifield scenarios address issues present in the single field case and include elements like the axionic superpartner for scalar fields.These multifield situations are not only more natural due to the expectation of many scalar fields in UV completions but also align with the swampland conjectures.These conjectures demand very steep scalar potentials, often in conflict with working scenarios of quintessence.Interestingly, a non-trivial curvature in the scalar field manifold can evade such constraints, as shown in Ref. [43] (see also Ref. [44]), and permit behaviours resembling inflation [34,41,[45][46][47].An interesting situation of multifield dynamics can simultaneously address the puzzle of dark matter, as shown for example in Ref. [48] and, recently in Ref. [47,49,50].The former work builds on a string inspired model of the dilaton that couples with a hidden sector, but does no regard its superpartner.This is done in Ref. [51] (also in Ref. [52] with more involved kinetic mixings) where, however, no any other energy components are included.This system, together with a barotropic perfect fluid is instead studied in Refs.[53,54] and with a direct universal coupling in Ref. [55].
Vector fields are, indeed, a more intrinsic component of nature than scalar fields and are anticipated to be inherent in any extension of the standard model.The issue of consistency becomes less straightforward in this context, owing to the potential emergence of ghost fields.Nevertheless, substantial progress has been made in comprehending a bottomup approach for constructing effective Lagrangians involving these fields, as discussed in references such as [56][57][58][59][60][61][62], facilitating a broad spectrum of cosmological investigations.
In the line of the present study, dynamical dark energy scenarios have been formulated, wherein vector fields play a pivotal role.For example, Refs.[63][64][65] introduces vector fields associated with an SO(3) symmetry, forming what is termed the cosmic triad, first proposed in [66].This scenario allows for isotropic solutions, while also accommodating situations with a preferred direction, as explored in Refs.[67,68], considering a Bianchi I spacetime as the background.A simpler scenario involves a U (1) gauge symmetry, where even general scalar-vector-tensor analyses have been conducted, for example in Refs.[69,70].Motivated by UV considerations, studies have delved into specific constructions, including the work of Thorsrud et al. in Ref. [71], who investigated the cosmological implications of a scalar field with an exponentially field-dependent gauge kinetic function inspired by dilaton field couplings.A detailed analysis of this model was subsequently undertaken in Ref. [72], exploring alternative kinetic term structures for the scalar field.
The primary objective of our current investigation is to explore the simplest yet least constrained model inspired by string theory compactifications.Specifically, we consider two scalar fields associated with a complex modulus and a U (1) vector field, where the gauge kinetic function is dependent on the modulus.The analysis leads to study a dynamical system dependent on three parameters, resulting in four fixed points.One of these fixed points involves only one scalar field, reproducing models initially studied in this type of scenario [19] but which is in tension with the swampland constraints on the steepness of the potential.Two other fixed points are inherited from previous models [53,72], which involve only a pair of fields in the model.Additionally, a completely novel fixed point is discovered in which all three fields play a role.We pay special attention to the cases where the vector field has a non-trivial role.We find that it is possible to describe the observed cosmology, albeit in a parameter space challenging to justify from superstring compactifications.
The paper has the following structure.The second section provides an overview of the construction of the scalar sector from a four-dimensional N = 1 Supergravity (SUGRA) theory, resulting in a generic superstring compactification.These results are then utilised to re-examine the dynamics of a cosmological system involving the modulus and its superpartner.Moving forward, the third section introduces vector fields within a SUGRA theory, proposing a contribution to the Lagrangian in the case of a U (1) symmetry inspired by Heterotic and type II D-brane constructions.This model is subjected to detailed scrutiny in subsequent analyses.Then, we present our concluding remarks.The appendix shows some details about the construction of 4D effective actions from string compactifications.

Scalar Multifield model from string compatifications
Much in the spirit of inflationary models, our main concern will be with the scalar sector whose dynamics are expected to lead to a dynamical equation of state that hopefully presents a late-time accelerated expansion epoch.The dynamics of the scalar sector are mainly encoded in the scalar potential, which in supergravity can be split into two components: the F -term and D-term contributions.The former has the following general structure [73], with M P the reduced Planck mass, where K = K( ΦĪ , Φ I ) is the Kähler potential and W = W (Φ I ) is the superpotential.Here, I and J range over the complete set of chiral superfields, and the bar symbol representing the conjugate quantities; therefore, the superpotential is a holomorphic function for the chiral superfields, while the Kähler potential is a real one.Additionally, using the shorthand notation ∂ I f ≡ ∂f ∂Φ I , we define K I J ≡ ∂ 2 K ∂Φ I ∂ Φ J , and its inverse The covariant derivative is defined as On the other hand, the D-term contribution has to do with the gauge symmetries, with f AB the inverse of f AB , the gauge kinetic function (more on it below), while where X I A are the, in general, field-dependent Killing vectors associated with the gauge symmetry A. In other words, a transformation associated with real parameters λ A induces one on the chiral superfields given by δϕ i = λ A X i A , such that D A is a gauge-invariant combination of charged fields.
While string compactifications exhibit a diverse array of scalar fields, the most distinctive aspect of four-dimensional effective supersymmetric theories is the presence of moduli fields, characterizing flat directions within the field space.This peculiarity motivates their incorporation into string-inspired model constructions for slow-roll inflationary and quintessence scenarios (for a review and specific instances, see [74] and references therein.)The underlying rationale for this behaviour lies in the moduli fields serving as superpartners to compactified p-forms potentials with gauge freedom, represented by C p → C p + dA p−1 , which manifests as shift symmetries.Denoting the scalar component of our chiral modulus superfield as Ψ = s+iσ with s and σ being real, the axionic symmetry σ → σ +c is reflected in the Kähler potential as a dependency in the form with M ranging over all chiral fields but the modulus, while the superpotential should be modulus-independent, i.e., W (Φ I ) = W 0 (Φ M ) .
In light of these considerations, the F -term scalar potential adopts the following expression: We further simplify by considering that the structure of the Kähler potential is of the form Consequently, the scalar potential takes the following general form: with modulus-independent functions and A typical form for the modulus Kähler potential is given by a logarithmic dependency with p a real constant, such that and with V 0,F a s-independent function.Given that V 0,F depends on all other fields, we examine the scenario wherein all fields except s have previously been determined and stabilised through dynamics investigated over the last two decades, as illustrated for instance in [75][76][77][78][79][80][81] (for a recent review and a more complete set of references see [82]).In this context, we treat V 0,F as a non-vanishing constant.Now, let us explore the D-term contribution.For it we restrict to the simplest case of a single U (1) sector beyond the Standard Model symmetries and examine two cases.In the first case, we assume that the Killing vectors are independent of moduli and X Ψ = 0, indicating that Ψ is neutral, resulting in where the numerator is s-independent and its value is fixed by the stabilization of all other fields.Here the dependency on s will appear in the gauge kinetic function, as discussed below.
A second possibility is one of a pseudo-anomalous U (1) sector, for which the field appearing as the gauge kinetic function develops a non-linear transformation with a constant Killing vector X Ψ = iδ [83][84][85].This induces a field-dependent Fayet-Ilioupoulus term of the form using the logarithmic dependency introduced above.Regarding the case on which the component iX N K N = 0 once the rest of the fields are fixed2 and the D-term scalar potential reduces to In order to be explicit we use a linear dependency for the gauge kinetic function, namely f (Ψ) = Ψ/M P , leading to the following general structure for the D-term scalar potential with n = 1 and 3 respectively in the two cases and V 0,D = 1 2M 4 P D 2 in the former case and in the pseudo-anomalous one.The kinetic part is dictated solely by the Kähler potential and is given by thus K I J is the metric of the field space so the consideration done in (2.5) means that the scalar manifold in the moduli sector factorises.Using the considerations above we get the following explicit kinetic term for the scalar fields Or, defining a canonical normalised field ϕ 1 = p 2 ln(s/M P )M P and renaming ϕ 2 = σ, In terms of these fields the scalar potential takes the form This result, motivates the study of a general two-field model of the form with exponential functions, as was done in [53].In the following section, we will study the cosmological dynamics encoded in this model.

Cosmological setup
In a flat, homogeneous, and isotropic Friedman-Lemaître-Robertson-Walker (FLRW) spacetime, described by the metric where a(t) is the cosmic time-dependent scale factor, the Lagrangian (2.17) regarding homogeneous fields, yields the following equations of motion: where the dot symbol denotes time derivatives, H ≡ ȧ/a, and the subscript ϕ 1 denotes derivatives with respect to this field.To investigate the resulting cosmology, these equations must be coupled with the Einstein equation of motion M 2 P G µν = T µν , where G µν is the Einstein tensor, and T µν is the energy-momentum tensor.The contribution of these scalar fields to the energy-momentum tensor is expressed as For homogeneous fields, the density and pressure are specifically given by However, for a more realistic scenario, additional components must be considered in the universe's energy budget.These contributions are represented by two perfect fluids, each following radiation and matter equations of state.Consequently, the complete energymomentum tensor is formulated as and the dynamics for these two sectors adhere to the conservation of their energy-momentum tensors, i.e., ∇ µ T (r) In light of these considerations, the Friedman equations from the Einstein field equations take the form As proposed in the preceding section and elucidated in appendix A, superstring theory likely necessitates the coupling function and potential to conform to the expressions given in (2.15) and (2.16),3 explicitly represented as where f 0 and V 0 are constants, and the parameters λ = √ 2p and ν = 2/p are determined by the integer p. 4 Nevertheless, in the subsequent analysis, we treat λ and ν as arbitrary and examine the parameter space of the model yielding to viable cosmologies.Subsequently, we comment on the discrete parameter space favoured by string theory.

Dynamical Analysis
Unveiling the cosmological dynamics of the model requires solving Eqs.(2.19), (2.23) and (2.24) given some parameters and initial conditions.However, we can extract information of the asymptotic behaviour of the model encoded in its fixed points [87].To do so, we recast these equations in terms of the following dimensionless variables 26) The first Friedman equation (2.24) reduces to the constraint equation and the continuity equations in Eq. (2.23) and the equations of motion for the scalar fields in Eq. (2.19) are replaced by the following closed autonomous system ) where a prime denotes a derivative with respect to the number of e-folds, which is related to the cosmic time by dN ≡ Hdt.The deceleration parameter, q ≡ −aä/ ȧ2 , is written in terms of these variables as The fixed points of this system are computed by solving the algebraic system resulting from x ′ 1 = 0, x ′ 2 = 0, y ′ = 0, and Ω ′ r = 0. We characterise the physical meaning of these fixed points by computing the effective equation of state and the equation of state of dark energy where ρ tot ≡ ρ ϕ + ρ m + ρ r and p tot = p ϕ + ρ r /3.Here, we focus on accelerated solutions driven by dark energy, i.e., fixed points where w eff < −1/3 and w DE < −1/3.Additionally, notice that the autonomous system exhibits the following symmetries: (2.34) Therefore, without loss of generality we restrict ourselves to solutions with x 2 , y ≥ 0 and consider non-negative values of the parameters λ and ν while leaving free x 1 .Under these conditions we obtained just two possible accelerated solutions: N G : (2.36) These points were called "geodesic" (G) and "non-geodesic" (N G) solutions in Ref. [53], since they are related to the "curvature" in the field space of the scalar fields.
In the left panel of figure 1, we show the existence region of the points (G) and (N G), defined as the parameter space regions {ν, λ} where the variables in the fixed points [Eqs.(2.35) and (2.36)] attain real values.On the other hand, the stability of a fixed point can be known by analysing the behaviour of small perturbations of the autonomous system around the given fixed point.At the linear level, this amounts to evaluate the real part of the eigenvalues of the Jacobian matrix of the autonomous system in the corresponding fixed points.If all eigenvalues are negative, it is said that the point is an attractor.If there is a mix of negative and positive eigenvalues, then we have a saddle fixed point, and if all eigenvalues are positive, this point is a source [87].In the right panel of figure 1, we show the parameter space where these points emerge as attractors of the system, i.e., their corresponding regions of attraction.We particularly emphasize the subregions associated with an accelerated epoch, meeting the condition −1 ≤ w eff ≤ −0.98.Recalling that λ = √ 2p and ν = 2/p are demanded by string theory, we depict the points for some values of p, observing that none of them describes a scenario of late-time accelerated expansion in agreement with current observations.This clearly indicates that the current state of the universe is challenging to explain within the framework of this model, as derived from string compactifications.To grasp this, observe that as p increases, the steepness of the potential for ϕ 1 also intensifies, necessitating a more pronounced interplay with ϕ 2 , given by a larger ν parameter, as depicted in the figure.However, ν decreases with increasing p, creating a tension between these two requirements.This possibility was identified in Ref. [53] as a regime of "large curvature" in the scalar fields space.However, as illustrated in Figure 1, the parameter values required by string theory lie outside the regions where the model exhibits viable cosmological attractors.
3 Scalar multifield coupled to a vector field While currently challenging to justify the existence of accelerated solutions in the multifield model that align with observations, alternative mechanisms may potentially broaden the model's parameter space.For example, in Refs.[44,64,67,72,88], it was demonstrated that steep potentials, such as Higgs-like or exponential potentials resembling that in Eq. (2.25) with λ 2 > 2, can yield stable accelerated solutions when supported by the presence of other fields, including non-abelian SU (2) vector fields, abelian U (1) vector fields, or 2-form fields.In this section, we explore how this possibility holds true for the scalar multifield model.

Cosmological setup
Here, inspired by Ref. [72] where it was shown that a vector field coupled to a scalar field yields anisotropic accelerated attractors even for a steep exponential potential, we explore a scenario involving a coupling between the scalar field responsible for the accelerated expansion, denoted as ϕ 1 , and a vector field represented by A µ .The Lagrangian of this extended model reads where F µν ≡ ∇ µ A ν − ∇ ν A µ is the strength tensor associated to the vector field, and h 2 (ϕ 1 ) is the field dependent coupling to the gauge kinetic function.First of all, the energymomentum tensor in Eq. (2.22) is extended by a contribution from the gauge sector which reads In a FLRW spacetime, all the components of F µν are zero since only a time-dependent timelike vector field keeps the isotropy of the background [89].Nonetheless, there is a recent increase in the amount of evidence suggesting that the Universe could not be as isotropic as thought [11,[90][91][92][93]. Therefore, as in Ref. [72], we adopt an anisotropic Bianchi-I metric with a residual symmetry in the y-z plane 5 and a spatial-like vector field with only one component aligned with the x axis.Explicitly where a(t) in this case represents an overall scale factor, and the anisotropy is encoded in the geometric shear σ(t).Using this metric and this vector field profile, we distinguish the vector field contributions to the density and pressure From the Einstein equations, besides the Friedman equations which will take into account the contribution of the vector field to the stress-energy tensor, we have the corresponding equation for the evolution of the geometric shear 6 (3.7) In the last equation, we note that in the case ρ A = 0, the shear decays as σ ∝ a −3 , i.e., the anisotropy of the Universe is erased by the expansion.This resembles the well-known no-cosmic hair theorem valid for ΛCDM [95].Instead, in our case, the density of the vector field sources the shear preventing this fast decay.However, for a phenomenologically viable scenario, the predicted values must conform to the current observational constraints [96,97].
Since the vector field is not coupled to ϕ 2 , the equation of motion of this scalar field is not modified [see Eq. (2. 19)].In turn, the equation of motion for ϕ 1 and A µ are given by In the following section we perform the analysis of this enriched dynamics.

Dynamical Analysis
To investigate the asymptotic evolution of the fully-fledged model, we introduce two extra dimensionless variables to the set outlined in Eq. (2.26).Furthermore, we adopt an exponential functional form for the gauge kinetic function, a choice motivated by results from string compactifications, as elucidated later on.Thus we define 6 An equation of motion for the shear can be derived from the difference between the "11" and "22" components of the Einstein equations, i.e., Table 1.Values of the variables in the anisotropic accelerated solutions (DE1) and (DE2).We have defined A ≡ (λ + 2µ), B ≡ λ − 6µ, C ≡ λ + 6µ to keep the presentation simple.
where h 0 and µ are constants.The first Friedman equation in Eq. (3.5) becomes the constraint Concerning the autonomous system in Eqs.(2.28)-(2.31),we observe that only the equation for the variable x 1 undergoes a modification in its structure.Additionally, two more equations must be incorporated to account for the introduction of new variables, namely z and Σ.We have that the complete set is ) ) where we used the equation of motion for the vector field, which we have conveniently recast in terms of the density ρ A as The deceleration parameter 1 + q ≡ − Ḣ/H 2 is given by In this case, the effective equation of state undergoes corrections due to the presence of shear.It can be computed as Together with the scalar and vector fields, it is convenient to include the shear in the definition of the dark energy fluid, as follows: such that the equation of state of dark energy is (3.22) The autonomous system exhibits the following symmetries: Hence, without loss of generality, we confine our exploration to solutions with positive values for x 2 , y, and z, along with non-negative parameters λ and ν, while leaving x 1 and µ unrestricted.Under these conditions, we obtained the isotropic solutions (G) and (N G), as discussed in section 2.3.Additionally, we identified two potential anisotropic accelerated solutions, named (DE1) and (DE2).The variable values at these points are presented in Table 1.Notably, point (DE1) aligns precisely with the anisotropic accelerated solution discovered in Ref. [72] for the quintessence model.This correspondence is expected since x 2 = 0 is an automatic solution to (3.13), reducing the system to a case where ϕ 2 is absent.On the other hand, point (DE2) corresponds to a clear generalisation of point (N G), considering anisotropy.

Fixed Point (DE1)
In left panel of Figure 2, we plot the existence region for (DE1) considering that where the bounds for Σ are within the constraints given in Refs.[96,97].We also plot three lines denoting the values of λ for p = 1, 2, and 3. We see that in general, the existence of anisotropic accelerated solutions in agreement with observations require λ ≪ µ.Under this condition, we see that Also, it is possible to estimate the eigenvalues of the Jacobian matrix in this point.We get From the latter expressions, we can estimate the attraction region of (DE1), i.e., the region where η 1,...,6 < 0. We get two branches depending on the sign of µ: However, when we consider the parameter space for viable cosmological solutions [Eq.(3.24)], the positive branch is dynamically selected and µ ≫ λ implies that the cosmological interesting attraction region is delimited by µ > 2ν/3. (3.28) Notably, when λ = √ 2p and ν = 2/p, we observe in Figure 2 that (DE1) serves as an accelerated attractor of the system, for some values of p within the range expected from string theory and large values of µ.For instance, for p = 1, the viable cosmological attractor require µ ≳ 496.This can be physically interpreted as the vector field interaction, controlled by µ, playing a crucial role in inducing a slow-roll behavior in the multifield configuration evolution, despite the presence of a steep potential.

Fixed Point (DE2)
From the expressions in Table 1, we see that for (DE2) to be a cosmologically viable point, i.e., w DE ∼ −1 and Σ small [see Eq. (3.24)], we require ν ≫ λ.Under this condition, we get Hence, the admissible solutions for any µ dismiss the values λ = √ 2p and ν = 2/p stemming from string theory.Nevertheless, in the subsequent analysis, we extensively survey the parameter space of this novel solution, allowing for arbitrary values of these parameters.
Concerning (DE2), the eigenvalues of the Jacobian matrix manifest as highly intricate expressions, rendering an analytical approach challenging to implement.Consequently, we  All the (DE2) attractors are near the plane µ = 2ν/3 but not necessarily into this plane as it seems in Figure 3.We corroborated that the overlap between blue and red points is merely due to the projection onto λ, and that a proper bifurcation manifold between (DE1) and (DE2) exists in the complete parameter space {λ, ν, µ}.
opt for a numerical assessment of the real part of the eigenvalues of the Jacobian matrix at these points, employing a methodology akin to that proposed in Ref. [98].
The numerical methodology can be succinctly encapsulated in the following manner.A large number of random points is sampled within the parameter space {λ, ν, µ}, and the fixed points are assessed at these locations.Subsequently, only the parameter sets corresponding to real-valued fixed points are retained.The eigenvalues of the Jacobian matrix are then computed for these specific parameter sets, and those associated with attractor solutions are selected.In figures 3 and 4, we show the result of this procedure when applied to the points (DE1) and (DE2).
In Figure 3, we explored the parameter space by varying λ from 0 to 3, and {ν, µ} from 0 to 500.The left panel of the figure illustrates that all the blue points, corresponding to locations where all eigenvalues evaluated at (DE1) are negative, are allocated above the plane µ = 2ν/3.Conversely, the red points, indicating positions where all eigenvalues at (DE2) are negative, align with the plane described by µ = 2ν/3.This observation is further supported by analysing the projection of the stability regions onto the {ν, µ} plane, as depicted in the right panel of Figure 3.
To assess scenarios where all parameters are of the same order, we conducted a more focused sampling within the parameter space, specifically {λ, ν, µ} ∈ [0, 10].The outcomes are presented in Figure 4.In this case, the projection onto the {ν, µ} plane reveals a mixture of attractor locations between (DE1) and (DE2).However, this should be expected since we are considering a projection while there is a mild dependency on λ.We numerically corroborate that a proper bifurcation manifold exists in the complete parameter space {λ, ν, µ}.We also notice that (DE2) attractors (red points) are situated near the plane µ = 2ν/3, but not necessarily into this plane as it seems in Figure 3.
Intriguingly, the attractor nature of (DE2) emerges at points where the anisotropy is naturally suppressed, i.e., µ ≈ 2ν/3, while, simultaneously, exhibits a late-time accelerated phase, implying λ ≪ ν, as evident from equation (3.29).However, this observation also implies that the parameter µ, which governs the gauge dynamics, should be finely tuned in relation to scalar couplings controlled by ν, while also resulting in a subdominant contribution of the vector field to the dark energy budget.
This final observation leads us to the conclusion that, although both (DE1) and (DE2) can portray a scenario explaining the current observations, (DE2) is much less likely to occur compared to (DE1) when the complete model is taken into account.This is evident in Figures 3 and 4, where the parameter space for (DE1) is significantly larger than that for (DE2).

Cosmological Dynamics
In this section, we numerically solve the autonomous set in Eqs.(3.12)-(3.17)for a particular set of parameters.The chosen parameters are tailored to make (DE1) the attractor point, with a subsequent discussion on the situation for (DE2).Specifically, we opt for λ = √ 2p, ν = 2/p, where p = 2, and µ = 10 3 .The initial conditions are defined as follows: These conditions represent a radiation-dominated era, initiated at z r,i = 6.5 × 10 7 , corresponding to N = − log (1 + z) = −18.Given our interest in late-time mechanisms for generating an anisotropic expansion, we assume an initially smooth Universe, i.e., Σ i = 0.
In the left panel of Figure 5, we plot the evolution of Ω r , Ω m , and Ω DE ≡ ρ DE /(3M 2 P H 2 ), and the effective equation of state.In particular it shows that the model evolves following the standard expansion history of the Universe: after the radiation dominated era (Ω r ≈ 1 and w eff ≈ 1/3) there is a radiation-matter transition around z r = 3000.The matterdominated epoch (Ω m ≈ 1 and w eff ≈ 0) then extends until z r ∼ 0.3 when the matter-dark energy transition takes place.Importantly, during the radiation and matter-dominated We see that the model can reproduce a viable expansion history, i.e., radiation dominance at early times (red dotted line), then a matter dominance (light brown dashed line), and dark energy dominance at late times (black solid line) characterized by w eff ≃ −1 (blued dot-dashed line).Note that a early times, w eff behave as a stiff fluid and then as a ultra-relativistic fluid without relevant contributions to the energy budget.(Right) Evolution of the variables associated to the kinetic energies of the scalar field ϕ 1 and the vector field A. The equation of state of dark energy w DE inherits the oscillations of these fields.
epochs, the contribution of dark energy is subdominant.Indeed, we verified that Ω DE follows the BBN constraint, Ω DE < 0.045, and the CMB constraint, Ω DE < 0.02 at z r = 50.Dark energy dominance (Ω DE ≈ 1 and w eff < −1/3 for z → −1) is characterised by a quasi-de Sitter evolution where w DE ≈ −1 and small oscillations at late times.As shown in Ref. [72], this oscillations are supported by the presence of the vector field.In the right panel of Figure 5, we can see that, at late times, the kinetic energy of the scalar field, driving the accelerated expansion, and the kinetic energy of the vector field are of the same order, i.e., x 1 ∼ z, and both oscillate.These oscillations directly influence the equation of state of dark energy.However, they remain relatively small and are likely beyond the observational scope of current and forthcoming surveys.See Ref. [72] for further details about the cosmology of this fixed point.We verified that indeed (DE1) is the attractor of the system by computing the values of the variables in the future, i.e., when z r → −1 which we take as N = 50.These are given by which are consistent with the values computed from Table 1.In summary, the cosmological trajectory of the model compatible with the requirements from string theory, i.e., λ = √ 2p and ν = 2/p, is Radiation → Matter → Anisotropic dark energy (DE1).
Note that although x 2 = 0 in the fixed point, the scalar field ϕ 2 could be relevant to the cosmological dynamics well before the attractor is reached.However, we confirmed that the dynamics of the field ϕ 2 is many orders of magnitude subdominant during the whole expansion history.
Attempting to implement a numerical solution for (DE2) poses a challenging task.The constrained parameter space necessary for it to function as an attractor point results in an exceedingly narrow region in phase space with trajectories rapidly converging to this point.To be more precise, the coexistence with the significantly more favoured (DE1) point implies that, despite the instability of (DE1) under the chosen parameters, it behaves akin to a pseudo-stable fixed point and trajectories spend virtually infinite time in (DE1) before transitioning to (DE2).
Another way to understand the unlikely nature of (DE2) comes directly from the solution of the equations of motion which for ϕ 2 and A µ read Indeed, for homogeneous fields in the context of the Bianchi I spacetime and a vector field as given in Eq. (3.3), yields the solutions: with c ϕ 2 and c A integration constants.Consequently, the corresponding density contributions of these fields to the first Friedman equation are given by: . (3.34) At late time the systems approaches the fixed point and φ1 ≈ 0, thus the scalar field ϕ 1 reaches its asymptotic value and the coupling functions f (ϕ 1 ) and h(ϕ 1 ) also stabilise at constant values.Therefore, it can be concluded that ρ ϕ 2 ∝ 1/a 6 while ρ A ∝ 1/a 4 .Considering the rapid decay of ρ ϕ 2 , two main observations can be made: i) scalar fields can exhibit stiff matter behaviour, dominating the Universe's energy budget at very early times before the radiation-dominated epoch as shown in the left panel of Figure 5, and ii) at late times the contribution of the vector field to the total density consistently supersedes that of the scalar ϕ 2 .Hence, (DE1) describes the most natural asymptotic evolution of the system, while (DE2) corresponds only to a marginal option needing some degree of fine-tuning for its viability.

Vector field from supergravity
We have already mentioned the possibility of including some dynamics from gauge symmetries when contemplating a potential contribution to the scalar dynamics from the D-term scalar potential.Indeed, it is expected to have such, and to be general enough, we might also include the dynamics of the vector fields themselves.As before, we consider only the simplest case of a U (1) sector, different from the one already included for the D-term scalar potential, as a non-vanishing D-term contribution implies a gauge symmetry breaking for the corresponding sector.
The minimal setup will include a pure gauge U (1), which can be justified by the absence of charged particles in the effective theory; that is, the charged sector turns out to be heavy compared with the energy scale involved in our studies.The general pure gauge sector in SUGRA is given by [73] where the gauge kinetic function f AB is a holomorphic function of the chiral superfields, and F µν and Fµν are the usual field strength and its dual.The main outshot extracted from this structure is the possibility of having field-dependent gauge couplings, once the gauge kinetic function is not trivial, and a clear connection between the kinetics term and the theta term, i.e., the gauge coupling and the axion are superpartners.As before, we consider a diagonal metric in the gauge sector, f AB ∝ δ AB .Moreover, for homogenous field configurations, like the ones we are dealing with, the pseudoscalar term vanishes.Then we end up with a Lagrangian of the form As advertised when dealing with the D-term scalar dynamics, gauge kinetic functions in superstring compactifications are naturally linear in the moduli fields, i.e., f = Ψ/M P .Thus, once the canonical normalization is done, s = M P e Therefore, an exponential dependence on the scalar field ϕ 1 is inherent in an effective field theory originating from string theory, providing a rationale for its inclusion in the preceding analysis.However, aligning the parameter µ with the microscopic result yields µ = − 2/p = −ν.Unfortunately, this falls beyond the attractor region for both fixed points, (DE1) and (DE2): firstly, as this requires positive values for µ; additionally, all parameters are anticipated to be of the same order, whereas an accelerated expansion scenario necessitates substantially small values of λ compared to µ and ν in each case; finally, achieving the relation µ ≈ 2ν/3 appears challenging given the precise expression from string compactifications.Unfortunately, this non-isotropic backgrounds have not been the primary focus of investigations in superstring constructions.However, during the 1990s, several works emerged that delved into the direct solution of the equations of motion of the superstring sigma model with background fields [99,100].These endeavors sought to address the question of what types of solutions are consistent with non-zero background fields in the vacuum.Probably the first exploration of such a possibility was the work by Kaloper [101], where anisotropic spaces with a constant dilaton were discovered, albeit exhibiting a recollapse cosmology.Subsequently, Batakis [102][103][104][105] classified possible homogeneous solutions that included a non-trivial 3-form field (H field).These identified a specific class representing spaces with 4 non-compact dimensions akin to Bianchi spaces.Similar studies and results can be found in [106,107], and [108].Notably, the latter work, in particular, incorporated a background with B field, H = dB, and no H as a background.Interestingly, it found that under these circumstances, an expanding universe is only possible in the anisotropic case.The inclusion of non-trivial moduli also presents solutions with this characteristic, as found in [109,110].Non-isotropic solutions to the beta functions, now including 2-loop α ′ corrections and cases with central charge deficit, were also found recently in [111,112], confirming the existence of non-isotropic solutions.

Conclusions
Our study delved into the cosmological implications of a simple yet versatile string-inspired model for dynamical dark energy.The model, rooted in a chiral modulus, exhibits common features in superstring compactifications, including a logarithmic Kähler potential, a modulus-independent superpotential, and a modulus-dependent gauge kinetic function in a pure gauge sector.This formulation results in a 4D model featuring two scalar fields and an abelian U (1) vector field, coupled through non-trivial kinetic functions, and governed by a scalar potential -all displaying an exponential dependence on only one of the fields.These choices prompted a comprehensive three-parameter dynamical analysis.The study reviews previous results in which one of the fields plays a spectator role, and explore a novel situation where the interplay of all three fields is in the game.
In the scenario where the vector field assumes the role of a spectator, we observe a reduction to the case studied in Ref. [53].Here, the interplay of two fields with non-geodesic trajectories in the scalar manifold gives rise to a quintessence attractor point, characterized by a steep potential.However, achieving correct values for cosmological parameters necessitates the coupling parameters to lie outside the preferred values derived from superstring string theory constructions.
The introduction of a non-trivial contribution from the vector field inherently results in a preferred spatial direction, prompting the consideration of a Bianchi-I background spacetime.Although such a structural choice for non-compact dimensions is unconventional in string constructions, positive outcomes from direct examinations of string sigma models suggest not only its potential relevance but also its potential natural appearance in superstring compactifications.In this context, the model manifests a quintessence attractor fixed point, that we called (DE1), where the axionic scalar component assumes a trivial value-a scenario previously investigated by two of the authors in Ref. [72].In this particular instance, achieving alignment with the observed values and bounds for the equation of state parameters and shear necessitates a coupling with the vector field significantly stronger than the dynamics originating from the scalar potential.
This intriguing result aligns seamlessly with the swampland condition, as the dynamics from the vector field facilitate a slow-roll dynamics along a geodesic trajectory for the scalar field, despite its steep potential.In a broader context, this outcome extends the wellknown results from non-geodesic trajectories in the scalar manifold, as formally discussed in Refs.[43,44], and has already found application in various multifield scenarios.
Under these circumstances we analyse the cosmic evolution, which can be summarized as follows.An initially isotropic radiation domination epoch is followed by a matter domination period.Then, late accelerated expansion is driven by dark energy which also source anisotropy within current observational bounds [96,97].Another key feature of our dark energy model is the ocillating behavior of its equation of state.This feature has been found in other models [88] and thoroughly explained in [72].
A second and novel fixed point (DE2) is also under consideration, where all three fields actively contribute to the dynamics.Remarkably, the system naturally accommodates this configuration, and the attractor quintessence point is realised for parameter values that are concurrently consistent with a small anisotropy.Intriguingly, the region hosting the attractor dynamics, marked also by accelerated behaviour, resides within a parameter space where the dynamics of the scalar fields are finely tuned to that of the vector field.However, this fine-tuning implies this attractor is significantly more constrained compared to the parameter space for the fixed point (DE1).In other words, it represents a much less probable scenario compared to the fixed point discussed above as can be seen in Figure 3.We numerically corroborated that cosmological dynamics stay around (DE1), even in the case when the parameters λ, ν, and µ are choice in the attraction region of (DE2), which is all around the plane µ = 2ν/3.
Unfortunately, in all the scenarios presented by the system, the parameter space required, either from stability requirements or cosmological observations, falls outside the suggested range from superstring compactifications.Specifically, in all four cases (two isotropic and two anisotropic attractors), a mandatory hierarchy for the coupling parameters is essential to obtain the correct value for the effective equation of state parameter and demonstrate a late accelerated epoch.In contrast, string compactifications naturally yield values of the same order.Additionally, for (DE1) and (DE2), achieving an attractor nature is only possible if the exponentials related to the kinetic coupling have the same sign in their exponents, whereas string compactifications are likely to produce exponents with opposite signs.Therefore, our findings intensify concerns about a top-down approach to constructing cosmological models for dark energy [113,114].The simplest realisations struggle to replicate a viable expansion history, and even more complex models, though anticipated from a UV completion theory, might face similar challenges.However, it could be worth exploring configurations involving non-abelian gauge sectors, where input from the vector field dynamics might aid in realizing an accelerated late epoch.
To assess the generalizability of our findings, it is imperative to consider various corrections.These encompass quantum corrections to the Kähler potential (e.g., [115,116]), nonperturbative deformations of the superpotential (e.g., [117][118][119]), quantum corrections to the effective Lagrangian following Supersymmetry breaking [120], and thermal corrections (e.g., [121][122][123][124]).These factors have the potential to significantly influence the dynamics of the system.Despite the prospect of these modifications, we maintain our anticipation that our primary concern regarding the improbable region in parameter space holds true on general grounds.appear in the gauge kinetic function in type IIB string compactifications, exhibiting a dependency similar to the one employed in the work.For more detailed descriptions and involved constructions with analogous results see for instance [131][132][133][134][135][136][137].Now, to ensure decoupling from the visible sector, it is imperative that the D-brane carrying our U (1) field is positioned away from the branes supporting the standard model.

Figure 1 .
Figure 1.(Left) Existence (real valued variables) regions of (G) and (N G) in the parameter space {ν, λ}.(Right) Attraction regions of these points.The darker regions correspond to the regions where w DE ∼ −1, which in general require ν ≫ λ for both points, although (G) can be a suitable accelerated attractor for any ν given that λ ≪ 1 as in the usual quintessence scenario.We can see that the values of λ and ν predicted by string theory happen to fall outside the region associated with stable and viable accelerated attractors.

Figure 2 .
Figure 2. (Left) Existence region of the point (DE1).The dashed lines represent λ = √ 2p for some values of p. Viable solutions [w DE ∼ −1 and Σ small as in Eq. (3.24)] within the string theory requirements (λ = √ 2p) imply µ ≫ λ.Note that the value of ν is irrelevant for the existence of this fixed point since x 2 = 0 in (DE1).(Right) Stability region of (DE1) considering µ ≫ λ, λ = √ 2p and ν = 2/p in the approximated expresions for the eigenvalues η 1,...,6 in Eq. (3.26).We can see that this point can provide viable anisotropic accelerated solutions adjusting the values of µ to the given discrete value of p.

Figure 4 .
Figure 4. Similar to Figure 3 but considering a smaller parameter region to perform the random sampling, namely, {λ, ν, µ} ∈ [0, 10].All the (DE2) attractors are near the plane µ = 2ν/3 but not necessarily into this plane as it seems in Figure3.We corroborated that the overlap between blue and red points is merely due to the projection onto λ, and that a proper bifurcation manifold between (DE1) and (DE2) exists in the complete parameter space {λ, ν, µ}.

Figure 5 .
Figure 5. (Left) Evolution of several cosmological parameters.The initial conditions were chosen deep in the radiation era.We see that the model can reproduce a viable expansion history, i.e., radiation dominance at early times (red dotted line), then a matter dominance (light brown dashed line), and dark energy dominance at late times (black solid line) characterized by w eff ≃ −1 (blued dot-dashed line).Note that a early times, w eff behave as a stiff fluid and then as a ultra-relativistic fluid without relevant contributions to the energy budget.(Right) Evolution of the variables associated to the kinetic energies of the scalar field ϕ 1 and the vector field A. The equation of state of dark energy w DE inherits the oscillations of these fields.