Unifying Dark Matter and Dark Energy with non-Canonical Scalars

Non-canonical scalar fields with the Lagrangian ${\cal L} = X^\alpha - V(\phi)$, possess the attractive property that the speed of sound, $c_s^{2} = (2\,\alpha - 1)^{-1}$, can be exceedingly small for large values of $\alpha$. This allows a non-canonical field to cluster and behave like warm/cold dark matter on small scales. We demonstrate that simple potentials including $V = V_0\coth^2{\phi}$ and a Starobinsky-type potential can unify dark matter and dark energy. Cascading dark energy, in which the potential cascades to lower values in a series of discrete steps, can also work as a unified model. In all of these models the kinetic term $X^\alpha$ plays the role of dark matter, while the potential term $V(\phi)$ plays the role of dark energy.


I. INTRODUCTION
A key feature of our universe is that 96% of its matter content is weakly interacting and non-baryonic. It is widely believed that this so-called dark sector consists of two distinct sub-components, the first of which, dark matter (DM), consists of a pressureless fluid which clusters, while the second, dark energy (DE), has large negative pressure and causes the universe to accelerate at late times.
Although numerous theoretical models have been advanced as to what may constitute dark matter, none so far has received unambiguous experimental support [1]. The same may also be said of dark energy. The simplest model of DE, the cosmological constant Λ, fits most observational data sets quite well [2]; see however [3,4]. Yet the fine tuning problem associated with Λ and the cosmic coincidence issue, have motivated the development of dynamical dark energy (DDE) models in which the DE density and equation of state (EOS) evolve with time [5,6].
In view of the largely unknown nature of the dark sector several attempts have been made to describe it within a unified setting [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Perhaps the earliest prescription for a unified model of dark matter and dark energy was made in the context of the Chaplygin gas (CG) [7,8,10]. CG possesses an equation of state which is pressureless (p ≃ 0) at early times and Λ-like (p ≃ −ρ) at late times. This led to the hope that CG may be able to describe both dark matter and dark energy through a unique Lagrangian. Unfortunately an analysis of density perturbations dashed these early hopes for unification [11]. In this paper we build on early attempts at unification and demonstrate that a compelling unified description of dark matter and dark energy can emerge from scalar fields with non-canonical kinetic terms.

The speed of sound [23]
can become quite small for large values of α since c s → 0 when α ≫ 1.
This latter property ensures that non-canonical scalars can, in principle, play the role of dark matter. In this paper we shall show that, for a suitable choice of the potential V (φ), non-canonical scalars can also unify dark matter with dark energy.
This paper works in the context of a spatially flat Friedmann-Robertson-Walker (FRW) universe for which the energy-momentum tensor has the form In the context of non-canonical scalars, the energy density, ρ φ , and pressure, p φ , can be written as It is easy to see that (6) reduces to the the canonical form The Friedmann equation which is solved in association with (3) is where ρ r is the radiation term and ρ b is the contribution from baryons. Note that we do not assume a separate contribution from dark matter or dark energy which are encoded in ρ φ .
As shown in [20], for sufficiently flat potentials with V ′ ≃ 0 the third term in (3) can be neglected, leading toφ Substituting (8) in which reduces to ρ X ∝ a −3 for α ≫ 1. Comparing (10) with ρ X ∝ a −3(1+w X ) one finds so that w X ≃ 0 for α ≫ 1. We therefore conclude that for flat potentials and large values of α, the kinetic term, ρ X , plays the role of dark matter while the potential term, V , plays the role of dark energy in (6).
The above argument is based on the requirement that the third term is much smaller than the first two terms in equation (3). In a recent paper Li and Scherrer [21] have made the interesting observation that the potential need not be flat in order that (8) be satisfied.
The analysis of Li and Scherrer suggests that the third term in (3) can be neglected under more general conditions than anticipated in [20] provided the potential is 'sufficiently rapidly decaying' [21]. Indeed it is easy to show that the requirement of the third term in (3) being much smaller than the second translates into the inequality which reduces to when α ≫ 1. Equation (13) can be recast as Equations (13) & (14) inform us that the variation in V can be quite large at early times when both H and ρ X are large. By contrast the same equations suggest that the potential should be quite flat at late times when H and ρ X are small. This latter property ensures that V (φ) can play the role of DE at late times and drive cosmic acceleration. The inequalities in (13), (14) can be satisfied by a number of potentials some of which are discussed below.
We first consider the inverse-power-law (IPL) family of potentials [26] V (φ) = V 0 (φ/m p ) p , p > 0. (15) Assuming that the background density falls off as (where m = 4, 3 for radiative and matter dominated epochs respectively), one can show that (8) is a late-time attractor provided the inequality is satisfied [21]. For the large values α ≫ 1 which interest us, (17) reduces to for which the late-time attractor is [21] V ∝ (ρ B ) From (19) one finds that for p > 2 the potential falls off faster than the background density ρ B . The case p = 2 is special since V ∝ ρ B , i.e. the potential scales exactly like the background density. This is illustrated in figure 1. In this case one finds (for α ≫ 1) where M is a free parameter in the non-canonical Lagrangian (1).
The above analysis suggests that the IPL potential V ∝ φ −p , p ≥ 2 cannot give rise to cosmic acceleration at late times. Thus while being able to account for dark matter (through ρ X ) this model is unable to provide a unified description of dark matter and dark energy.
However, as we show in the rest of this paper, a unified prescription for dark matter and dark energy is easily provided by any one of the following potentials 1 : It is interesting that all of these potentials belong to the α-attractor family [27,28] in the canonical case.
FIG. 1. This figure illustrates the evolution of the density in radiation (red curve), baryons (dotted green curve) and a non-canonical scalar field with the potential V ∝ φ −2 . The solid green curve shows the evolution of the kinetic term ρ X ∝ X α which behaves like dark matter, ρ X ∝ a −3 . The blue curve shows the evolution of the potential V (φ). Note that V scales like the background fluid so that V ∝ a −4 during the radiative regime and V ∝ a −3 during matter domination. (α = 10 5 has been assumed.)

A. Dark Matter and Dark Energy from
The potential [29] V (φ) = V 0 coth p φ m p , p > 0 can provide a compelling description of dark matter and dark energy on account of its two asymptotes: As discussed in the previous section, the IPL asymptote (22) ensures that the kinetic term behaves like dark matter ρ X ∝ a −3 , while V (φ) scales like the background density (for p = 2) or faster (for p > 2); see eqn. (19).
The late-time asymptote (23) demonstrates that the potential flattens to a constant value at late times. This feature allows V (φ) to play the role of dark energy. Indeed, a detailed numerical analysis of the coth potential, summarized in figures 2 and 3, demonstrates that (21) with p = 2 can provide a successful unified description of dark matter and dark energy in the non-canonical setting. (This is also true for p > 2. However for shallower potentials with p < 2 the third term in (3) cannot be neglected. This implies that for such potentials ρ X does not scale as a −3 and therefore cannot play the role of dark matter [21].) For non-canonical models the equation of state can be determined from (6), namely which simplifies to For the potential (21) ρ X ∝ a −3 whereas V (φ) approaches a constant value at late times.
The effective equation of state of the kinetic and potential components can be determined  Substituting for ρ X from (9) one finds which leads to after substitution for 3Hφ from (8). It is instructive to rewrite (29) as As noted in (13) the inequalityV ≪ 3Hρ X should be satisfied in order for ρ X to behave like dark matter. Since the densities in dark matter and dark energy are expected to be comparable at late times, one finds ρ X V ∼ O(1) at z ≤ 1. Substituting these results in (32) one concludes that the EOS of DE is expected to approach w V ≃ −1 at late times in unified models of the dark sector. (One should also note that since ρ X V can be fairly large at early times, w V is not restricted to being close to -1 at z ≫ 1.) Figure 5 compares the behaviour of w X and w V in the two potentials: coth 2 φ and φ −2 .
One notices that w V ≃ 1/3 at early times in both potentials, which is a reflection of the scaling behaviour V ∝ ρ B noted in (19). At late times w V in the coth potential drops to negative values causing the universe to accelerate. For V ∝ φ −2 on the other hand, w V always tracks the dominant background fluid which results in w V = 0 at late times. (Note that in this case the fluid which dominates at late times is the kinetic term, so that V ∝ ρ X .) For the IPL potential on the other hand (dashed green) the universe stays matter dominated at late times (q ≃ 0.5) and does not accelerate.

B. Dark Matter and Dark Energy from a Starobinsky-type potential
A unified model of dark matter and dark energy can also arise from the potential [27,28,30] which reduces to the Starobinsky potential in the Einstein frame [31] for λ = 2 3 . The potential in (33) is characterized by three asymptotic branches (see figure 6): where Before discussing the unification of dark matter and dark energy in a Starobinsky-type potential we briefly explore the dynamics of the scalar field as it rolls down the exponential branch (34).

Motion along the exponential branch
The exponential branch has been extensively studied in the canonical case (α = 1) for which the late time attractor is w φ = w B if λ 2 > 3(1 + w B ). The situation radically changes for non-canonical scalars (α = 1). As shown in [21], if the background density scales as ρ B ∝ a −m , then for (2α − 1)m > 6 the late time attractor iṡ Since ρ X ∝φ 2α one finds We therefore find that, as in the IPL case, for large values of the non-canonical parameter α the density of the kinetic term scales just like pressureless (dark) matter. From (38) one also finds thatφ ∼ constant when α ≫ 1. From this it is easy to show that for α ≫ 1 the amplitude of the scalar field grows as so that φ ∝ a 2 during the radiative regime and φ ∝ a 3/2 during matter domination. This behaviour is illustrated in figure 7. Substituting (41) in (34) one finds V ∝ exp [−2λφ] ∼ exp [−2λa m/2 ], which implies an exponentially rapid decline in the value of the potential as the universe expands and a(t) increases. One therefore concludes that like the IPL potential, an exponential potential too can never dominate the energy density of the universe and source cosmic acceleration.

Accelerating Cosmology from a Starobinsky-type potential
In the context of the Starobinsky-type potential in (33), the rapid growth of φ in (41) enables the scalar field to pass from the steep left wing to the flat right wing of V (φ). In other words the scalar field rolls from A to B in figure 6. Since V ≃ V 0 on the flat right wing, cosmological expansion in this model mimicks ΛCDM at late times. This is illustrated in figure 8. (Note that ρ X ∝ a −3 on both wings of the potential.)   (29) is an effective quantity hence its value can exceed unity.) By contrast the EOS of state of the kinetic term remains pegged at w X = 0. Figure 8 shows the behaviour of w X and w V as φ moves under the influence of the potential (33). A key feature to be noted is that w V encounters a pole as φ rolls from the steep left wing to the flat right wing of V (φ) (i.e. from A to B in figure 6).
A question occasionally directed towards dark energy is whether cosmic acceleration will continue forever (as in ΛCDM) or whether, like the earlier transient epochs (inflation, radiative/matter dominated) dark energy will also will be a fleeting phenomenon. In this section we investigate a transient model of DE in which the potential V (φ) is piece-wise flat and resembles a staircase; see figure 9. Such a potential might mimick a model in which an initially large vacuum energy cascades to lower values through a series of waterfallsdiscrete steps [32]. shows the evolution in the density of the kinetic term ρ X ∝ a −3 (solid green line) and the cascading potential V (φ) (blue). Also shown are the densities in radiation (red) and baryons (dotted green).
(Note that V initial = 10 25 × V final .) Locally the i-th step of this potential may be described by [20,33] where A + B = V i and A − B = V i+1 . If V i+1 ≃ 10 −47 GeV 4 then this potential could account for cosmic acceleration. (One might imagine yet another step at which V i+2 < 0. In this case the universe would stop expanding and begin to contract at some point in the future.) Motion along the staircase potential leads to a cascading model of dark energy. Remarkably, the inequality in (13) holds even as φ cascades from higher to lower values of V . This ensures that the kinetic term scales as ρ X ∝ a −3 and behaves like dark matter while V (φ) behaves like dark energy, as shown in fig. 9.
Note that the cascading DE model runs into trouble in the canonical context since the kinetic energy of a canonical scalar field moving along a flat potential declines as 1 2φ 2 ∝ a −6 .
This puts the brakes on φ(t) which soon approaches its asymptotic value φ * , resulting in inflation sourced by V (φ * ). By contrastφ ∼ constant in non-canonical models with α ≫ 1, see (38). This allows φ(t) to cross each successive step on the DE staircase in a finite amount of time, ∆t ≃ ∆φ φ , and drop to a lower value of V (φ); see left panel of fig. 9.

IV. DISCUSSION
In this paper we have demonstrated that a scalar field with a non-canonical kinetic term can play the dual role of dark matter and dark energy. The key criterion which must be satisfied by unified models of the dark sector is (13). This inequality ensures that the third term in the equation of motion (3) is small and can be neglected, resulting in ρ X ∝ a −3 and w X ≃ c S ≃ 0. In other words if (13) is satisfied the kinetic term behaves like dark matter with vanishing pressure and sound speed. Of equal importance is the fact that if (13) holds then eqn. (32) implies w V ≃ −1 at late times. This ensures that the potential V (φ) can dominate over ρ X and source cosmic acceleration at late times.
The following unified models of the dark sector have been discussed in this paper: values. It is interesting to note that for all of the above potentials 3 the kinetic term scales as ρ X ∝ a −3 throughout the expansion history of the universe, even as the shape of the potential continuously changes. This property allows the kinetic term to play the role of 2 Note that the width of each step is restricted by the fact that the universe does not appear to accelerate prior to z ∼ 1 (with the exclusion of an early inflationary epoch). 3 We note in passing that the asymptotically flat potential [27] V (φ) = V 0 tanh 2 φ also leads to a unified scenario of dark matter and dark energy although we do not discuss it in this paper. dark matter while the potential term V (φ) plays the role of dark energy and leads to cosmic acceleration at late times.
As shown in [20] the small (but non-vanishing) speed of sound in non-canonical models suppresses gravitational clustering on small scales. Non-canonical models with c s ≪ 1 can therefore have a macroscopic Jeans length which might help in resolving the cusp-core and substructure problems which afflict the standard cold dark matter scenario. In this context the dark matter content of our model shares similarities with warm dark matter [34,35] and fuzzy cold dark matter [30,[36][37][38] both of which are known to possess a large Jeans scale.
Finally it is interesting to note that all of the potentials discussed in this paper belong to the α-attractor family of potentials [27,28] and lead to interesting models of inflation, dark matter and dark energy [29,30] in the canonical case.