Cosmological solutions from 4D $N=4$ matter-coupled supergravity

From four-dimensional $N=4$ matter-coupled gauged supergravity, we study smooth time-dependent cosmological solutions interpolating between a $dS_2\times \Sigma_2$ spacetime, with $\Sigma_2 = S^2$ and $H^2$, in the infinite past and a $dS_4$ spacetime in the infinite future. The solutions were obtained by solving the second-order equations of motion from all the ten gauged theories known to admit $dS_4$ solutions, of which there are two types. Type I $dS$ gauged theories can admit both $dS$ solutions as well as supersymmetric $AdS$ solutions while type II $dS$ gauged theories only admit $dS$ solutions. We also study the extent to which the first-order equations that solve the aforementioned second-order field equations fail to admit the $dS_4$ vacua and their associated cosmological solutions.


Introduction
Cosmological solutions connecting a dS 4 spacetime with a very large Hubble constant in the past to another dS 4 spacetime with a very small Hubble constant (both of the right order of magnitude) in the future can provide a good description of our universe. In the context of the dS/CFT holography, as formulated in [1], [2], these and the more general cosmological solutions connecting dS spacetimes of different dimensions, such as dS D−d × Σ d and dS D , are fundamental building blocks on the gravity side. These are dual to Euclidean conformal field theories with different dimensions which are connected by time evolution since the holographic dimension is time itself. Despite much effort, a metastable dS vacuum remains elusive from string theory [9]. Furthermore, there exist many outstanding issues regarding the quantum nature of dS space itself [6]. Because of these difficulties, dS/CFT holography based on the correspondence between a dS D+1 solution of (D +1)-dimensional supergravity and a Euclidean CFT in D dimensions, analogous to the now routinely-applied AdS/CFT correspondence between an AdS (D+1) spacetime and a CFT D theory, remains very poorly understood. To date, arguably the most concrete form of dS/CFT is found in higher spin theories [3], [4], [5] instead of ordinary supergravity.
A radically different way to obtain dS solutions is from unconventional gauged dS supergravities that arise from the compactification of the exotic 11D M* and 10D type IIB* theories [14], which are themselves obtained from the timelike T-dualities of the original M/string theories [11], [12], [13]. This framework, originally proposed in [10], naturally gives dS space the priviledged position occupied by AdS space in the conventional framework of M/string theories. Consequently, dS/CFT holography, in the sense proposed by [10], naturally arises in the context of these *-theories in the same manner that AdS/CFT holography does in conventional M/string theories. It is then unsurprising that many cosmological solutions can be found in this framework, see for examples [16], [18].
On the other hand, only a few examples involving cosmological solutions in conventional gauged supergravities are known. These include [15] and [17]. The cosmological solution in [15] interpolates between a dS spacetime and a singularity, while the cosmological solution in [17], interpolating between a dS 2 × S D−2 spacetime in the past and a dS D spacetime in the future, is an analytic continuation of the domain wall solution interpolating between an AdS 2 × S D−2 spacetime and an AdS D spacetime. In [7], cosmological solutions in 5D N = 4 supergravity interpolating between a dS 5−d × H d (with d = 2, 3) spacetime in the infinite past and a dS 5 spacetime in the infinite future were presented in all the gauged theories capable of admitting dS 5 solutions. Since the dS 5 vacua of 5D N = 4 supergravity are unstable, these and their associated cosmological vacua studied in [7] are not relevant in the context of dS holography. Nonetheless, independently of holography, it was instructive to derive and study the cosmological solutions in 5D N = 4 supergravity in their own right. The main motivation for the study in [7] was straightforward -we would like to know whether there exist fixed point solutions dS 5−d × Σ d , with Σ d = S d , H d where d = 2, 3, and interpolating solutions connecting these in the infinite past to the dS 5 solutions in the infinite future. It is our goal in this work to carry out the same analysis in the four-dimensional, N = 4 supergravity that is structurally similar to, but more complex than the five-dimensional theory studied in [7]. More specifically, our primary goal will be to study cosmological solutions interpolating between a dS 2 × Σ 2 spacetime, with Σ 2 = S 2 , H 2 , at early times and a dS 4 spacetime at late times from 4D N = 4 matter-coupled gauged supergravities. The solutions will be derived by solving the second-order field equations from the ten gauged theories known to admit dS 4 solutions. These gauged theories can be classified into two types, depending on whether the gauge group under consideration allows for an AdS solution in addition to a dS solution, or just dS solutions [30]. Secondly, similar to the 5D case in [7], our secondary goal will be to study the possibility of obtaining these dS 4 and cosmological solutions from some system of first-order equations that solve the second-order field equations. Although the fact that all dS 4 vacua of 4D N = 4 supergravity are unstable precludes the existence of pseudosupersymmetry which guarantees the existence of first-order pseudo-BPS equations capable of giving rise to these solutions, we will nevertheless perform our analysis, analogous to the 5D case, to pinpoint the exact failure characterizing the lack of first-order systems.
This rest of this paper is organized as follows. In section 2, we review the theory of 4D N = 4 gauged supergravity coupled to vector multiplets in the embedding tensor formalism. In section 3, we review the dS 4 solutions derived within the framework of the embedding tensor formalism and their classifications into type I and type II. In section 4, the ansatze for the dS 2 ×Σ 2 solutions are specified and the associated equations of motion are derived. In section 5 and 6, we present the cosmological solutions from the type I and type II dS gauged theories, respectively. In section 7, we investigate how the cosmological solutions found in sections 5,6 fail to arise from the first-order equations that solve the second-order field equations. Section 8 concludes the paper.

4D N = 4 matter-coupled supergravity
To make this paper self-contained, we highlight some relevant details of four-dimensional N = 4 matter-coupled gauged supergravity. The detailed construction of theory can be found in [24] and the references therein.
The field content of the theory includes an N = 4 supergravity multiplet and an arbitrary number n of vector multiplets. The supergravity multiplet contains the graviton eμ µ , four gravitini ψ µi , i = 1, ..., 4, six vectors A m µ , m = 1, ..., 6, four spin-1 2 fermions λ i , and a complex scalar τ . The vector multiplet contains a vector field A µ , four gaugini λ i and six scalars φ m . Spacetime and tangent space indices will be denoted by µ, ν, . . . = 0, 1, 2, 3 andμ,ν, . . . = 0, 1, 2, 3, respectively. Indices m, n, . . . = 1, 2, . . . , 6 label vector representation of SO(6) ∼ SU(4) R-symmetry while i, j, . . . denote chiral spinor of SO(6) or fundamental representation of SU(4). The n vector multiplets are labeled by indices a, b, . . . = 1, ..., n. Collectively, the fied content of the n vector multiplets can be written as Altogether, there are (6n + 2) scalars from both the gravity and vector multiplets. These scalars span the coset manifold The first factor in (1) is parameterized by a complex scalar τ consisting of the dilaton φ and the axion χ from the gravity multiplet, where The second factor in (1) is parameterized by the 6n scalars from the vector multiplets. Both factors of (1) will be described in terms of coset representatives. For SL(2)/SO (2), this coset representative is with an index α = (+, −) denoting the SL(2) fundamental representation. In the next section, we will also be using the notation α = (e, m). V α satisfies the relation in which M αβ is a symmetric matrix with unit determinant, while ǫ αβ is anti-symmetric with ǫ +− = ǫ +− = 1. The coset manifold SO(6, n)/SO(6)×SO(n) will be described by a coset representative V M A transforming under the global SO(6, n) and local SO(6) × SO(n) symmetry by left and right multiplications, respectively. The local index A can be split as A = (m, a) with m = 1, 2, . . . , 6 and a = 1, 2, . . . , n denoting vector representations of SO(6) × SO(n). Accordingly, V M A can be written as With V M A being an SO(6, n) matrix, the following relation is satisfied where η M N = diag (−1, −1, −1, −1, −1, −1, 1, . . . , 1) is the SO(6, n) invariant tensor. SO(6, n) indices M, N, . . . are lowered and raised using η M N and its inverse η M N . Equivalently, the symmetric matrix which is manifestly SO(6) × SO(n) invariant, is used to describe the SO(6, n)/SO(6) × SO(n) coset. Gaugings of the matter-coupled N = 4 supergravity are performed using the embedding tensor formalism [24] that provides an embedding of a gauge group G 0 in the global symmetry group SL(2, R) × SO(6, n). N = 4 supersymmetry restricts the embedding tensor to include only the components ξ αM and f αM N P = f α[M N P ] . A closed subalgebra of SL(2, R) × SO(6, n) satisfying the required commutation relations is ensured by the following quadratic constraints on the components of the embedding tensor The gauge group generators are constructed from these components of the embedding tensor as follows.
In this work, we are only interested in solutions involving the metric, scalars and some Abelian gauge fields. The bosonic Lagrangian reads where e is the vielbein determinant, and L top is the topological term whose explicit form will not be of relevance to our purpose because this term will always vanish for the solutions considered here. Note that in the Lagrangian above, only covariant electric gauge field strengths H M + µν are present while magnetic gauge fields H M − µν enter the equations of motion. The covariant derivatives acting on the scalars 1 read where In general, the covariant field strengths H M α µν contain the auxiliary two-form fields B and B αβ µν due to their complexity and the fact that these two-forms can be consistently truncated out for the specific embedding tensor components corresponding to all the gauge groups studied in this work. With The scalar potential V reads 1 Note that there are two equivalent ways to write the kinetic term for the supergravity scalar τ where M M N and M αβ are the inverse matrices of M M N and M αβ , respectively. M M N P QRS is obtained by raising indices of M M N P QRS defined by Alternatively, the potential V can be written in terms of the fermion-shift matrices A ij 1 , A ij 2 and A 2ai j that appear in the fermion mass-like terms and supersymmetry transformations of fermions as follows.
In terms of the scalar coset representatives, the fermion shift-matrices are given by V M ij is obtained from V M m by converting the SO(6) vector index to an anti-symmetric pair of SU(4) fundamental indices using the chiral SO(6) gamma matrices. The explicit gamma matrices used are given in the Appendix of [30].

dS 4 solutions from 4D N = 4 supergravity
In this section we summarize the known results on dS 4 solutions from [30]. dS 4 solutions of 4D N = 4 gauged supergravity coupled to n vector multiplets were originally derived in [20], [21], and rederived within the framework of the embedding tensor formalism in [30] using the following two ansatze for the fermionic shift matrices (18,19,20) where V 0 is the extremized value of the potential, and denotes the enclosed quantities being evaluated in the considered background. We recall that all solutions from [20] and [21] were recovered in [30] and in addition, a new solution with the gauge group SO(2, 1) × SO(4, 1) was found in [30]. While the ansatze (21,22) ensure the positivity of the scalar potential, they must also be subject to the quadratic constraints (8), and the extremization condition on the scalar potential. For more details, see [30]. Solutions to either (21) or (22) consist of the set of embedding tensor components (ξ α M , f α M N P ) from which the gauge group generators can be constructed using (9). As shown in [30], there exist solutions to both (21) and (22) which form the two types of gaugings that give rise to two different types of dS solutions in gauged N = 4 four dimensional supergravity.
The common features of these two types are as follows. Firstly, all gaugings with dS 4 solutions must have ξ αM = 0 which simplifies the formula for the gauge generators (9) Secondly, the gauge groups G 0 must be dyonic, comprising a product of at least one electric G e and one magnetic G m gauge factor under the SL(2, R) duality group 2 .
The differences between these two types of gaugings are as follows.
• Type I dS 4 solutions arise from gaugings of the first type, which can admit both dS and AdS solutions. The compulsory components of the 4D embedding tensor in this case are Other allowed (but not required) components of the embedding tensors are The gauge group is of the form where G e,− , G m,− , . . . are in general non-compact. The compact parts of (27) are always SO(3) e × SO(3) m , generated by f α mnp 3 , and lie entirely in the R-symmetry group SO(6) ⊂ SO(6, n) of (1). The non-compact parts of (27), generated by f α mab , lie completely in the matter-symmetry directions SO(n) ⊂ SO(6, n) of (1). Accordingly, the − subscript in the G e(m) factors in (27) is used to denote the embedding of the compact parts of (27) in the R-symmetry directions. Additionally, f α abc generate a purely compact gauge factor, embedded in the matter symmetry directions SO(n), that is optional and does not contribute anything to the scalar potential. When f α mab vanish, we can have the completely compact gauging SO(3) e,− × SO(3) m,− which can exist in the pure supergravity theory without any coupled matter.
• Type II dS 4 solutions arise from gaugings of the second type, which can only admit dS solutions with no AdS solutions possible. The compulsory components of the 4D embedding tensor in this case are Other allowed (but not required) components of the embedding tensors are The gauge group is always noncompact and is of the product form The non-compact parts of (30) are generated by the components f α amn and lie entirely along the R-symmetry directions SO(6) ⊂ SO(6, n) of (1). The compact parts of (30) are generated by f α abc and lie fully in the matter symmetry directions SO(n) ⊂ SO(6, n) of (1). Accordingly, the + subscript in the G e(m) factors of (30) is used to denote the embedding of the compact parts of (30) in the matter symmetry directions. This is in direct constrast to those gaugings of the first type above. When f α abc = 0, the only gauging possible is SO(2, 1) e,+ × SO(2, 1) m,+ . Just like in 5D [31], gaugings of this second type cannot exist in the pure theory, but only in the matter-coupled theories.
We emphasize that the +/− subscripts in (27) and (30) are not to be mistaken with the earlier notation of α = (+, −) used in section 2. Here the α subscript is explicitly labeled as α = (e, m), while the + or − subcripts are used to denote whether the compact parts of the gauge group are embedded in the matter or R-symmetry directions. The embedding tensor components for all 4D gaugings with dS 4 solutions are listed in table 1 where we have dropped the e(m) subscripts, but retained the (+/−) subscripts in the gauge factors. Because of the SL(2, R) duality, either G + or G − can be electric or magnetic and vice versa -this is reflected from the embedding tensor components f αM N P where α can assume either the value of e or m.
While the full form of all gauge groups is given as in Table 1, we must point out that not all gauge factors in these gauge groups contribute to the scalar potentials, as mentioned above. Only the compulsory components (25), (28) are needed for each type of gaugings. Consequently, to simplify the analysis, we can reduce some of the gauge groups to the bare forms needed to admit dS 4 solutions without any loss of generality. For later convenience, we summarize the explicit scalar potentials corresponding to the simplified versions of those gauge groups listed in Table 1 in Table 2. These scalar potentials are constructed from the explicit components of the embedding tensor given in Table 1 using (15) or (17). For concreteness, g 1 and g 2 are used to denote the electric factor G e and magnetic factor G m , respectively, of the gauge group G 0 . From here on, to lighten the notations, the explicit +/− subscripts in the gauge groups will also be dropped since their distinction is made clear from their assignment to either type I or type II gaugings. It is important to note that while the scalar potentials for the type II gaugings are all different, there is only a single scalar potential resulting from the type I gaugings.
Gauge group G 0 Embedding tensor f αM N P Type I: Table 1: Embedding tensor components and gauge groups for the two types of gaugings that yield dS 4 vacua as given in [30]. Some of the coupling constants have been rescaled compared to the original ones used in [30].
Gauge group Scalar potential V (φ, χ) and g 1 /g 2 scaling for dS 4 Table 2: Scalar potentials constructed from the embedding tensor f αM N P given in Table  1 using (15) or (17) for all gauge groups. For concreteness, we use g 2 for magnetic gauge factor G m and g 1 for electric factor G e . Note that the magnetic gauge coupling always appears with χ 2 .
To obtain cosmological solutions interpolating between a dS 2 × Σ 2 solution and a dS 4 solution in each of the gauge groups listed in Table 2 above, we need to turn on an Abelian U(1) gauge field, together with the metric and supergravity scalars φ, χ. All other fields are truncated out. Specifically, Accordingly, the full Lagrangian (10) reduces to the following general form However, this Lagrangian (32) will not be the final one that we will work with. In particular, for the dS 2 × Σ 2 solutions that are of interest to us, we will turn on the gauge Since all our gaugings are dyonic, we will dualize the magnetic gauge factor of each gauging into an electric one using the procedure outlined in [19], so that the SL(2, R) frame under consideration is purely electric. This is necessary for us in order to use the Lagrangian (32) in which only SL(2, R) electric field strengths are present. Effectively, this means that instead of G e × G m we will have G e ×G e whereG e is the dualized G m . The dualization in this case is simply the following SL(2, R) transformation acting on τ as 4 so that the part of the Lagrangian (32), involving the two gauge groups G e ×G e , takes the following specific form with F M ,F M being the field strengths corresponding to U(1) ⊂ G e and (1) ⊂G e , respectively 4 A general SL(2, R) transformation acting on τ has the form Taking into account (34), the Lagrangian (32) becomes 5 Next, we will truncate the axion χ in (36). This axion truncation is consistent as long as the following terms in (34) which source χ vanish This is the case because of the purely magnetic gauge ansatz that we will use, so χ can be safely truncated out. Consequently, setting χ = 0 in (36) gives us the following Lagrangian that we will work with The explicit scalar potentials V (φ) for all gauge groups are given by those specified in Table 2 with χ = 0.
Having established the Lagrangian (38), we now move on to specify the various ansatze for the dS 2 × Σ 2 solutions. For the metric, the ansatz is where dΩ 2 2 is the line element for S 2 or H 2 , The ansatz for the Abelian The exact gauge field ansatz with the specified values 6 for M will be given in subsequent sections for each gauge group. 7 For type I gaugings, we need to turn on the gauge fields corresponding to the U(1) × U(1) subgroup of the SO(3) × SO(3) compact subgroups that lie entirely along R-symmetry directions, so M = m. For type II gaugings, the U(1) × U(1) have to be the subgroup of the compact parts that are embedded in the matter symmetry directions, so M = a. The corresponding gauge field strengths to (41) read The equations of motion for dS 2 × Σ 2 solutions resulting from using the ansatze (39, 41) in the Lagrangian (38) are A dS 2 × Σ 2 fixed-point solution of the equations (43) is given by The full solution described by (39) and φ = φ(t) of (43) is a cosmological solution interpolating between the above dS 2 × Σ 2 fixed point at early times t → −∞ and a dS 4 fixed point at late times t → +∞. Before moving on, we note that real solutions can only be obtained if the product a 2 g 2 1 of the gauge flux a and gauge coupling g 1 is negative. In particular, we will impose the following constraint This situation resembles the case of cosmological solutions in those dS supergravities, arising from dimensionally reducing the exotic ⋆-theories, with the wrong sign for the gauge field strengths [16], and is similar to the result obtained from the 5D analyses in 5D N = 4 supergravity done in [7].

Type I dS gauged theories
There are four gauge groups in the first type of dS gaugings, namely SO ( SL(3, R). These four gauged theories can give rise to both dS 4 and AdS 4 solutions. The SU(2) × SU(2) common subgroup of the four gaugings are generated by X 1 , X 2 , X 3 and X 4 , X 5 , X 6 , as can be seen from the embedding tensor components given in Table 1. When the field content is truncated to only the metric and the scalars φ, χ from the supergravity multiplet, all four gauged theories produce the same scalar potential that admits the following vacua at φ = χ = 0 (48)

SO(3) × SO(3)
This gauge group can be embedded entirely in the R-symmetry group SO(6) without the need for any coupled matter. It is the simplest and only fully compact gauging for both dS and AdS solutions. To get dS 4 solution, we will set g 1 = g 2 in (47) and also χ = 0 so that the scalar potential of the type I dS gauged theory is with the following dS 4 solution As mentioned above, the SO(3)×SO(3) group is generated by X 1 , X 2 , X 3 and X 4 , X 5 , X 6 . Therefore, the U(1) × U(1) gauge fields are given by (41) with M = 3 for the electric part and M = 6 for the magnetic part, corresponding to the generators X 3 and X 6 , respectively. Eqs. (43) together with the potential (49) yield the following dS 2 × Σ 2 solutions with κ = +1 for Σ 2 = S 2 and κ = −1 for Σ 2 = H 2 . After imposing (46), the solutions become The cosmological solutions interpolating between the dS 2 × Σ 2 fixed points (52) and the dS 4 solution (50) are plotted in Fig.1.   Consequently, we will not repeat these analyses but note only that dS 2 × Σ 2 fixed point solutions and cosmological solutions in these gauged theories are given by (51, 52) and Fig. 1, respectively.

Type II dS gauged theories
For this type of gaugings, the U(1) × U(1) gauge fields are given by (41) with M assuming the values along the matter symmetry group SO(n) ⊂ SO(6, n) in 1 for both the electric and the magnetic parts.

SO(2, 1) × SO(2, 2)
From the embedding tensor given in Table 1, the compact part SO(2) × SO(2) × SO(2) ⊂ SO(2, 1) × SO(2, 2) is generated by X 9 , X 8 and X 7 , respectively for each of the three SO(2)'s. Correspondingly, we can turn on the U(1) × U(1) gauge fields (41) with M = 7 or M = 8 for the electric part and M = 9 for the magnetic part. The scalar potential of this theory is with a dS 4 critical point at The equations (43) with the potential V (57) admit the following dS 2 × Σ 2 fixed points which become with κ = 1 for Σ 2 = S 2 and κ = −1 for Σ 2 = H 2 , after imposing (46). The cosmological solutions interpolating between the dS 2 × Σ 2 solutions (60) and the dS 4 solution (58) are plotted in Fig. 3.    1) is generated by X 7 , X 8 , X 9 and X 10 , X 11 , X 12 (see Table 1), so we can turn on the U(1) × U(1) gauge fields (41) with M = 9 for the electric part and M = 12 for the magnetic part. The scalar potential for this gauge group is with a dS 4 vacuum at The equations of motion (43) together with the potential (61) yield the following dS 2 × Σ 2 fixed point solution where κ = 1 for Σ 2 = S 2 and κ = −1 for Σ 2 = H 2 . After imposing the condition (46), (63) become κ = 1 : The cosmological solutions interpolating between (64) at early times and (62) at late times are numerically solved for and plotted in Fig.4.

Summary of all solutions
In this section, we summarize all solutions of the type II dS gauged theories. For all gauge groups, excluding SO(3, 1) × SO(3, 1), the fixed-point solutions can be rewritten in a common form using the g 1 /g 2 ratios given in Table 2. We list all the rewritten solutions in Table 3. Furthermore, we collect all cosmological solutions given Figs. 2, 3, 4, 5, 6 in a single plot Fig. 7.

First-order systems
In this section, we will derive the first-order equations, containing some relevant pseudosuperpotential W [25], [26], [27], that can solve the second order equations of motion 43. Next, we check whether these first-order equations admit the dS 4 vacua and their associated cosmological solutions found in sections 5 and 6. Although first-order equations are most often linked to the supersymmetric AdS case in which domain walls or holographic RG flow solutions can arise as solutions of some first-order BPS equations obtained by setting to zero the supersymmetry transformations of the fermionic fields, it must be noted that first-order equations can and do arise completely independently from supersymmetric systems in the so-called fake supersymmetric case. In this scenario, the Hamilton-Jacobi (HJ) formalism has been shown to produce first-order equations containing some fake superpotential obtained from the factorization of the HJ characteristic function [29]. More details on the relation between the HJ formalism and fake supersymmetry can be found in [28], [29]. As mentioned in section 1, the fact that dS 4 vacua of 4D N = 4 supergravity are unstable is a strong indication of the lack of pseudosuperymmetry that ensures the existence of relevant pseudosuperpotentials and corresponding first-order equations. It is then the objective of this section to characterize the extent to which there fail to exist relevant pseudosuperpotentials for the type I and type II dS theories. We will proceed as follows. For the purpose of deriving the first-order equations in the dS gauged theories, our analysis will be based on the AdS case since their equations of motion are almost identical up to various signs. The form of the first-order equations for the dS case will be inferred from that of the BPS equations in the AdS case. To derive the field equations for either AdS 4 and dS 4 solutions, we will work with the minimally required field content consisting only of the metric and dilaton φ. The axion χ can be truncated out since it vanishes in either AdS 4 or dS 4 vacuum. The Lagrangian reads In the cases of the AdS 2 × Σ 2 and dS 2 × Σ 2 , we will use (38). The ansatz for AdS 4 metric is The field equations resulting from using (85) in (77) are The equations (79) are solved by the following set of first-order equations subject to the following condition on the scalar potential and the superpotential Recall that there are four gaugings which admit fully supersymmetric AdS 4 solutions. These are the exact ones given in the type I dS gaugings (see Table 1). With the vector multiplet scalars truncated out, the scalar potentials from these four gaugings are identical and can be obtained from the ones in Table 2 by setting χ = 0.
To obtain AdS 4 solutions we need to set g 1 = −g 2 so that the potential (82) becomes There exists the following superpotential W such that the potential (83) can be written in terms of W as in the required relation (81). Note that the equations (80) can be derived by setting to zero the supersymmetry transformations of the gravitino and dilatino fields, δψ µ i = 0, δχ i = 0, as were done in [23] 8 . Before moving on, we remark that the BPS equations (80) with the superpotential W (84) and the equations of motion (79) with the scalar potential V (83) admit the same AdS 4 vacuum as should be the case.

First-order equations for AdS 2 × Σ 2
The metric ansatz for AdS 2 × Σ 2 is with dΩ 2 2 being the line element for Σ 2 = S 2 , H 2 . The gauge field ansatz is given by (41), with A M e(m) φ being the U(1) × U(1) gauge fields in the R-symmetry directions where M = 3 for α = e and M = 6 for α = m. This is exactly the same as the gauge ansatz for the type I dS gauged theories. The equations of motion for AdS 2 × Σ 2 resulting from using the ansatze (86, 41) in (38) are with λ = +1 for Σ 2 = S 2 and λ = −1 for Σ 2 = H 2 . It is instructive to compare the field equations (87) in this case to those from the dS case (43). The two sets are almost identical except for the opposite signs for the non-derivative terms (V, λe −2g , and gauge field strength terms). The equations of motion for the AdS 2 × S 2 (87) case are solved by the following set of first-order system subject to the condition (81) on the scalar potential and the superpotential, and the following condition on the gauge flux a and gauge coupling g 1 For the scalar potential V given by (83), the relation (81) was shown to be satisfied with the superpotential given by (84). The BPS equations (88) subject to (89) with W given by (84) admit the same AdS 2 × Σ 2 solutions as the equations of motion (87) with the potential (83). However, not all solutions are real and thus physically acceptable. In particular, only in the case λ = −1, there exists the following real AdS 2 × H 2 solution to both (88) and (87) The domain wall solution interpolating between the AdS 2 × H 2 solution (90) and the AdS 4 solution (85) can be obtained by either solving (87) or (88) numerically. Finally, we note that the first-order equations (88) with W given by (84) and the twist condition (89) are essentially identical to the BPS equations that are given in [22] for the 4D N = 4 AdS 2 × Σ 2 holographic RG flow solutions. The metric ansatz for dS 4 is Using this ansatz in (77) gives the following set of equations of motion These equations are almost identical to the ones for AdS 4 (79), save for the opposite signs in front of the terms involving the scalar potential and its derivative. Eqs. (92) can be solved by the same set of first-order equations (80) as in the AdS 4 casė but with the relation (81) replaced by where V has an opposite sign to (81).

First-order equations for dS
The equations of motion (43) are solved by the same first-order equations (88) as in the where λ = 1 for Σ 2 = S 2 and −1 for Σ 2 = H 2 , subject to the following constraint between the gauge flux a and gauge coupling constant g 1 and the relation (94) Note that unless an explicit pseudo-superpotential is substituted in the first-order equations (95), the relation (94) and the constraint (89) are not enough to solve the equations of motion (43).
Having established the first-order equations that solve the second-order field equations for both the dS 4 and dS 2 × Σ 2 cases, we now move on to check whether there exists any pseudo-superpotential W that satisfies the required relation (94) for both the type I and type II gauged theories.

Type I gauged theories
With the vector multiplet scalars truncated out, the scalar potential from the four type I gaugings that admit dS 4 solutions is given in (49), This scalar potential appears in the equations of motion for both the cases of dS 4 (92) and dS 2 ×Σ 2 (43). The pseudo-superpotential that satisfies the relation (94) with V given by (49) reads The first-order equations (93) and (95) with W given in (97) solve the equations of motion (92) and (43), respectively. Although this is the case, these first-order equations do not give rise to either the dS 4 solution (50) nor the comoslogical solutions interpolating between this dS 4 and the dS 2 × Σ 2 fixed-point solutions (52). We elaborate more on this below.
• dS 4 solution. The dS 4 solution (50) admitted by V (49) is not a critical point of the pseudo-superpotential (97). This is because given (94), the extremization of V , V ′ = 0, implies either one of the two following conditions At the dS 4 point φ 0 = 0, although W ′ = 0, leading to V ′ = 0. As such, the dS 4 solution (50) cannot arise from the first-order equations (93). It was pointed out in [26] that in general, for a cosmological solution, if a scalar potential can be written in terms of a pseudo-superpotential as where α is a constant, then the critical point at which V ′ = 0 can arise from either When the former condition is satisfied then the solution is pseudosupersymmetric and W is the corresponding pseudo-superpotential. When the latter condition is satisfied then the solution is not pseudo-supersymmetric. These conditions are tied to the Breitenlohner-Freedman (BF) bound of the solutions, which, in the case of cosmology, reads where D is the spacetime dimension and L = 1/f 0 is the radius of dS space. Solutions admitted by the pseudo-superpotential W (corresponding to the case W ′ = 0) do not violate the BF bound, while those not admitted by the pseudo-superpotential W (corresponding to the case W ′′ − α 2 W = 0) do. This argument follows from the work of [26]. In this case, for D = 4 and L = 3 V 0 = 1/f 0 , it can be verified from [21] that dS 4 solutions of the four gaugings in the type I theories do violate the BF bound. We explicitly list the mass values violating this BF bound for each of the four type I gauged theories in Table 4.

Type II gauged theories
For the scalar potentials given in Table 2, no suitable W can be found such that (94) is satisfied. Instead, we found the following W that satisfies the following relation Consequently, there do not exist any systems of first-order equations that solve the secondorder equations (92) for the dS 4 case or (43) for the dS 2 × Σ 2 case for the type II dS solutions. It is worth recalling that since the type II dS gauged theories in 4D are directly related to 5D dS gauged theories via dimensional reduction as shown in [30], this situation agrees with the result from the five-dimensional analysis [7] which shows there do not exist any suitable pseudo-superpotentials (and systems of first-order equations) for the dS 5 and dS 2,3 × Σ 3,2 cosmological solutions. We also note that the pseudo-superpotentials W as given in (103) do not admit the dS 4 solutions at φ 0 = χ 0 = 0 that are admitted by the scalar potentials V given in (2). The reason for this is the same as the type I case above. Given the relation (104), the extremization of V , V ′ = 0, can be obtained from either At the dS 4 point φ 0 = 0, W ′ = 0, but rather It can again be verified from [21] and [30] that dS 4 solutions of the six gaugings in the type II theories violate the BF bound (102). The mass values violating the BF bound (102) for each of the six type II gauged theories are listed in Table 4.

BF bounds for dS 4 solutions
For D = 4, the BF bound (102) reads Given that the dS 4 length L is related to the extremized value V 0 of V as the bound (107) can also be written in terms of V 0 The mass spectra of dS 4 solutions in all dS gauged theories, excluding SO(2, 1)×SO(4, 1), can be found in [21]. The mass spectrum of the dS 4 solution of the SO(2, 1) × SO(4, 1) gauged theory can be found in [30]. In Table 4, we list V 0 and the mass values violating the BF bound as given in [21] and [30]. Only one mass value for each gauged theory is sufficient to show that the BF bound is violated. Due to the different notations used in [21] and [30], we will use (109) for [21] and (107) for [30]. Some comments regarding these notations are in order to avoid any potential confusion. In [21], V 0 is given in terms of a ij that is defined as where indices i, j label the various individual gauge factors G 1 × G 2 × . . . constituting the gauge group G 0 , g i , g j are the corresponding gauge couplings, and α i are the SU(1, 1) ∼ = SL(2, R) angles. Although there can be more than two gauge factors in G 0 , all cases were shown to be reduced to just two factors (see Table 2). The mass spectra are also given in terms of a ij , making it convenient to check the BF bound using (109). In [30], V 0 is given in terms of g 2 1 after applying the scaling ratio g 2 /g 1 to bring the dS 4 critical point φ 0 = 0 to the origin of the scalar manifold at φ 0 = 0, while the mass spectrum is given in units of m 2 L 2 . Accordingly, it is convenient to use (107).

Summary
For both types of dS gauged theories, we checked whether the first-order systems of equations, to which the second-order field equations reduce, can give rise to the dS 4 and cosmological solutions of sections 5, 6. Following the domain wall/cosmology correspondence established in [26], this analysis was performed using the AdS case as a reference, since the equations of motion for the AdS case are almost identical to those of the dS case, except for a few opposite signs in front of some non-derivative terms. The first-order equations for the dS 4 and dS 2 × Σ 2 solutions, if they exist, should be identical in form to the BPS equations for AdS 4 and AdS 2 ×Σ 2 solutions, respectively. The reducibility of the second-order field equations to the first-order equations hinges on there being a required relation V (W ) between V and W such that V can be written in terms of the squares of W and its derivatives. This V (W ) relation in the dS case should be identical, save for an overall opposite sign, to the V (W ) relation in the AdS case. While supersymmetry automatically ensures the existence of the superpotential W in the AdS case, there is no guarantee that a suitable pseudo-superpotential W can be found in the dS case. Thus, while the general form of the first-order equations in the dS case is settled, its validity is only confirmed if a suitable W exists. The suitability of W is decided by two conditions, the first one being whether it satisfies the required V (W ) relation, and the second one being whether it admits the same dS critical point as V .
For the type I dS gauged theories, we did find a pseudo-superpotential W satisfying the required V (W ) relation, but this W does not admit the dS 4 solution that is admitted by the scalar potential V . Hence, the first suitability condition is satisfied but the second is not. This eliminates the possibility of the type I dS 4 solution and its associated dS 2 × Σ 2 cosmological solutions arising from the first-order equations that have the same form as the BPS equations in the AdS case. For the type II dS gauged theories, on the other hand, there does not exist any W that satisfies the required V (W ) relation. Thus, even the first suitability condition fails. This is almost the same as the result of a similar analysis in the five-dimensional case [7] where there is only one type of dS gaugings that corresponds to type II gaugings in four dimensions. In five dimensions, although the first suitability condition is not fulfilled in an exact manner as in the type II theories in four dimensions, the second one is. Regardless of the exact way the pseudo-superpotential fails to be suitable in either the 4D type I or type II theories, and even in the 5D dS gauged theories, this failure can be traced to the fact that firstly these dS solutions are unstable, and secondly they violate the BF bound that serves as a means to guarantee pseudo-supersymmetry.

Concluding remarks
In this work, we have studied cosmological solutions interpolating between a dS 2 × Σ 2 spacetime, with Σ 2 = S 2 and H 2 , and a dS 4 spacetime from N = 4 four-dimensional gauged supergravity. We emphasize that our motivation for the study of these solutions is completely decoupled from any holographic contexts. Instead, we are solely motivated by the question of whether cosmological solutions exist in 4D N = 4 supergravity, given the existence of dS 4 vacua [30]. Consequently, the cosmological solutions found were obtained by solving the second-order field equations in theories with gauge groups capable of admitting dS 4 solutions. Although the methodology used in this work is the same as that used in the 5D work [7], there is an important distinction between the two. In 5D there is only one type of gaugings that can admit dS vacua. In 4D there are two types of gaugings. Type I dS gauged theories consist of four theories with gauge groups Type II dS gauged theories consist of six theories with gauge groups SO(2, 1) × SO(2, 1), SO(2, 1) × SO(2, 2), SO(2, 1) × SO(3, 1), SO(3, 1) × SO(3, 1), SO(2, 1) × SO(4, 1) and SO(2, 1)×SU(2, 1) whose compact subgroups are entirely embedded in the SO(n) matter symmetry directions. These type II theories can only admit dS 4 solutions without the possibility of admitting AdS 4 solutions. To obtain cosmological solutions in the type II dS theories, we needed to turn on an Abelian gauge field corresponding to the diagonal of the U(1) × U(1) subgroup of the compact part along the SO(n) matter directions. Only 4D type II gaugings can be derived from 5D dS gaugings, while 4D type I gaugings can be derived from 5D AdS gaugings [30]. This crucial difference between 5D and 4D theories is an important motivation to explore the 4D theories given the results of the 5D case [7].
Furthermore, we also characterized the extent to which these nonsupersymmetric dS 4 and their associated cosmological solutions fail to arise from the relevant first-order equations that solve the second-order field equations by studying the lack of suitable pseudosuperpotentials in the type I and II dS gauged theories. Finally, we note that cosmological solutions arising from either type I or type II theories require the square of the product of the gauge flux a and gauge coupling g 1 to be negative in order to be real. This feature is already encountered in the five-dimensional cosmological solutions [7] and resembles the situation in the dS supergravities, arising from the dimensional reduction of M⋆/IIB⋆theories, with the wrong sign for the gauge kinetic terms. Instead of regarding these solutions as pathological due to this particular feature, we interpret this as an additional characterization of the dS vacuum structure of half-maximal supergravity. The implications of this remain to be understood and we hope that more work will elucidate this matter further.