Unifying Dark Matter and Dark Energy with non-Canonical Scalars
Swagat S. Mishra Varun Sahni















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    m̃_PΩ_0m...a0.9ex< ∼ 0.9ex> ∼ Astroph. J. Mon. Not. Roy. Ast. Soc. Astron. Astrophys. Astron. J. Phys. Rev. Lett. Phys. Rev. D Nucl. Phys. Nature Phys. Lett. B Class.  Quant.  Grav.Mod. Phys.  Lett. A JETP Lett.  et al.Inter-University Centre for Astronomy and Astrophysics,
        Post Bag 4, Ganeshkhind, Pune 411 007, India
        
        
        empty
        
        
        Non-canonical scalar fields with the Lagrangian L = X^α - V(ϕ),
        possess the attractive property that the speed of sound, c_s^2 = (2 α - 1)^-1,
        can be exceedingly small for large values of α.
        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_0^2ϕ 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^α plays the role of dark matter,
        while the potential term V(ϕ) plays the role of dark energy.
        
        Unifying Dark Matter and Dark Energy with non-Canonical Scalars
            Swagat S. Mishra and Varun Sahni
            
        ===============================================================
        
        
        
        § 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 <cit.>.
        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
        <cit.>; see however <cit.>. 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 <cit.>.
        
        In view of the largely unknown nature of the dark sector several
         attempts have been made to describe it within
        a unified setting <cit.>. Perhaps the earliest prescription for a unified model of dark matter and dark energy
        was made in the context of the Chaplygin gas (CG) <cit.>. 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 <cit.>.
        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.
        
        
        
        
        § NON-CANONICAL SCALAR FIELDS
        
        
        The non-canonical scalar field Lagrangian
        <cit.> L(X,ϕ) = X(X/M^4)^α-1 -  V(ϕ),
        
        presents a simple generalization of the canonical scalar field Lagrangian
         L(X,ϕ) = X -  V(ϕ),     X = 1/2ϕ̇^2
        
        to which (<ref>) reduces when α=1.
        
        Two properties of non-canonical scalars make them attractive for the study of 
        cosmology:
        
          *  Their equation of motion
        ϕ̈+ 3  Hϕ̇2α -1 + (V'(ϕ)α(2α -1))(2 M^4ϕ̇^2)^α - 1 =  0,
        
         is of second order, as in the canonical case.
        Indeed, (<ref>) reduces to the standard canonical form
        ϕ̈+ 3  H ϕ̇ + V'(ϕ) = 0 when α =1.
        
        
          *  The speed of sound <cit.>
        c_s^2 = 12 α - 1
        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
        
        T^μ_   ν = diag(ρ__ϕ, -p__ϕ, - p__ϕ, - p__ϕ)̊.
        
        In the context of non-canonical scalars,
         the energy density, ρ__ϕ, and pressure, p__ϕ, can be written as
        ρ__ϕ   =   ł(2α-1)̊Xł(X/M^4)̊^α-1 +   V(ϕ),
        
        p__ϕ   =    Xł(X/M^4)̊^α-1 -  V(ϕ).
        
        It is easy to see that (<ref>) reduces to the the canonical form
        ρ__ϕ = X + V,  p__ϕ = X - V when α = 1.
        
        The Friedmann equation which is solved in association with (<ref>) is
        
        H^2 = 8 π G/3ł(ρ__ϕ + ρ__r +ρ__b)̊
        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 <cit.>, for sufficiently flat potentials with V' ≃ 0
        the third term in (<ref>) can be neglected, leading to
        ϕ̈≃ -3  Hϕ̇2α -1   ⇒  ϕ̇∝ a^-3/2α - 1 .
        
        Substituting (<ref>) in 
        ρ__X = ł(2α-1)̊Xł(X/M^4)̊^α-1,    X = 1/2ϕ̇^2
        
        one finds
        ρ__X∝ a^-6α/2α - 1
        which reduces to ρ__X∝ a^-3 for α≫ 1.
        Comparing (<ref>) with ρ__X∝ a^-3(1+w__X) one finds
        
        w__X = 1/2α-1 ,
        
        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 (<ref>).
        
        The above argument is based on the requirement that the third term
        is much smaller than the first two terms in equation (<ref>).
        In a recent paper Li and Scherrer <cit.> have made the interesting
        observation that the potential need not be flat in order that
        (<ref>) be satisfied.
        
        The analysis of Li and Scherrer suggests that the third term
        in (<ref>) can be neglected under more general conditions than
        anticipated in <cit.>
        provided the potential is `sufficiently rapidly decaying' <cit.>.
        Indeed it is easy to show that the requirement of the third term in (<ref>) being much
        smaller than the second translates into the inequality
        
        V' ≪ (2α - 1/2α )3Hρ__X/ϕ̇ ,
        
        
        which reduces to
        
        V' ≪3Hρ__X/ϕ̇  ⇒  V̇≪ 3Hρ__X,
        
        when α≫ 1.
        Equation (<ref>) can be recast as
        |dV/dz|≪3ρ__X/1+z .
        
        Equations (<ref>)&(<ref>) 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 (<ref>), (<ref>) 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 <cit.>
        V(ϕ) = V_0/(ϕ/m_p)^p,  p>0.
        
        Assuming that the background density falls off as
        ρ__ B∝ a^-m
        (where m=4, 3 for radiative and matter dominated epochs respectively),
        one can show that (<ref>)
        
        
        is a late-time attractor provided the inequality
        
        p ≥12α/(2α - 1)m - 6
        is satisfied <cit.>.
        For the large values α≫ 1 which interest us,
        (<ref>) reduces to
        
        p ≥6/m
        for which the late-time attractor is <cit.>
        V ∝ł(ρ__ B)̊^p/2    and   ρ__X∝ a^-3 .
        
        
        
        
        
        From (<ref>) 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, 
         the potential scales exactly like the background density.
        This is illustrated in figure <ref>.
        In this case one finds (for α≫ 1)
        V(ϕ)/ρ__ B≃V_0/ρ__0 B (H_0 m_p/M^2 )^2
        
        where M is a free parameter in the non-canonical Lagrangian
        (<ref>).
        
        
        
        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[Perhaps the simplest potential for unification is
        V(ϕ) = V_0 which results in a ΛCDM cosmology <cit.>.]: (i) V(ϕ) = V_0 ^2ϕ, 
        (ii) the Starobinsky-type potential V(ϕ) = V_0  ( 1 - e^-ϕ )^2, (iii) 
        the step-like potential
        V(ϕ) = A + Btanhβϕ.
        It is interesting that all of these potentials 
        belong to the α-attractor family <cit.>
        in the canonical case.
        
        
        
        
        
        § UNIFIED MODELS OF DARK MATTER AND DARK ENERGY
        
        
         §.§ Dark Matter and Dark Energy from V = V_0^2ϕ
        
        
        The potential <cit.>
        V(ϕ)    =    V_0 ^p(ϕ/m_p),   p>0 
        
           ≡    V_0 ł (1 + e^-2ϕ/m_p/1 - e^-2ϕ/m_p)̊^p
        
        can provide a compelling description of dark matter and dark energy on account of
        its two asymptotes:
        
        V(ϕ)    ≃   V_0/(ϕ/m_p)^p ,        for  λϕ≪ m_p
        
        V(ϕ)    ≃    V_0  ,                 for  λϕ≫ m_p .
        
        As discussed in the previous section,
        the IPL asymptote (<ref>) 
        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. (<ref>).
        
        The late-time asymptote (<ref>) 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  potential, 
        summarized in figures <ref> and <ref>, demonstrates that
        (<ref>) 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 (<ref>)
         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 <cit.>.)
        
        
        
        For non-canonical models the equation of state can be determined from (<ref>), namely
        
        w_ϕ= p__ϕ/ρ__ϕ = -1 + (2α/2α - 1)(ρ__X/ρ__X+V(ϕ)) ,
        
        which simplifies to
        
        w_ϕ= -1 + ρ__X/ρ__X+V(ϕ) ,     for  α≫ 1 .
        
        For the potential (<ref>) ρ__X∝ a^-3 whereas V(ϕ) approaches a constant value 
        at late times.
        Consequently one gets 
         
        w_ϕ(z=0)≃ -1 + Ω_0X/Ω_0X+Ω_0V ,
        
        where Ω_0V = V_0/ρ_cr,0.
         Assuming Ω_0V=0.7, Ω_0b=0.04, one finds Ω_0X=0.26 so that
         
         w_ϕ(z=0)≃ -0.7292.
         
        Our results for the deceleration parameter q = -ä/aH^2 and w_ϕ
        are shown in figure <ref>
        for the IPL model (<ref>) and the  potential (<ref>), both with p=2.
        
        
        
        The effective equation of state of the kinetic and potential components can be determined from
        
        
        w__X   =    - ρ̇_̇_̇Ẋ/3Hρ__X-1 ,
        
        w__V   =    -V̇/3HV-1  .  
        Substituting for  ρ__X from (<ref>) one finds 
        
        w__X = -1-2αϕ̈/3Hϕ̇ .
        
        which leads to 
        
        w__X = c__S^2 =(2α - 1)^-1
        after substitution for 3Hϕ̇ from (<ref>).
        
        It is instructive to rewrite (<ref>) as
        
        1+w__V = -V̇/3Hρ__X(ρ__X/V ) .
        
        As noted in (<ref>) the inequality V̇≪ 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 (<ref>)
        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 <ref> compares  the behaviour of w__X and w__V in the
        two potentials: ^2ϕ 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 (<ref>).
         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.)
        
        
        
         §.§ 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 <cit.>
        V(ϕ) = V_0  ( 1 - e^-λϕ/m_p )^2 ,  λ>0 ,
        
        which reduces to the Starobinsky potential in the Einstein frame <cit.> for λ=√(2/3).
        
        
        The potential in (<ref>) is characterized by three asymptotic branches (see
        figure <ref>):
             V(ϕ)    ≃    V_0  e^-2λϕ/m_p ,   ϕ < 0  ,   λ|ϕ| ≫ m_p  ,
        
           V(ϕ)    ≃    V_0  ,   λϕ≫ m_p   ,
        
           V(ϕ)    ≃   1/2μ^2ϕ^2  ,   λ|ϕ| ≪ m_p   , 
        where
        μ^2 = 2V_0λ^2/m_p^2.
        
        
        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 (<ref>).
        
        
        
          §.§.§ 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 <cit.>, if the background density scales as ρ__ B∝ a^-m, then for 
        (2α - 1) m > 6
        the late time attractor is
        ϕ̇^2α - 1∝ t^-6/m   ⇒  ϕ̇∝ a^-3/2α - 1 .
        
        Since ρ__X∝ϕ̇^2α one finds
        
        ρ__X∝ a^-6α/2α - 1
        which reduces to
        ρ__X∝ a^-3    for  α≫ 1.
        
        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 (<ref>) one also finds that ϕ̇∼ constant when 
        α≫ 1. From this it 
        is easy to show that for α≫ 1 the amplitude of the scalar field grows as 
        ϕ/m_p∝ a^m/2 ,
        
        so that ϕ∝ a^2 during the radiative regime and ϕ∝ a^3/2 during matter domination.
        This behaviour is illustrated in figure <ref>.
        Substituting (<ref>) in  (<ref>) one finds 
        V ∝exp[-2λϕ]∼exp[-2λ a^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 (<ref>),
        the rapid growth of ϕ in (<ref>) 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 <ref>.
        Since V ≃ V_0 on the flat right wing, 
        cosmological expansion in this model mimicks ΛCDM at late times.
        This is illustrated in figure <ref>.
        (Note that ρ__X∝ a^-3 on both wings of the potential.)
        
        
        
        
        Figure <ref> shows the behaviour of w__X and w__V 
        as ϕ moves under the influence of 
        the potential 
        (<ref>). 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) ( from A to B in figure <ref>).
        
        
        
         §.§ Cascading Dark Energy
        
        
        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 <ref>.
        Such a potential might mimick a model in which an initially large vacuum energy cascades to lower values through
        a series of waterfalls – discrete steps <cit.>.
        
        
        
        
        Locally the i-th step of this potential may be described by <cit.>
        V(ϕ) = A + Btanhβϕ
        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 (<ref>)
         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. <ref>. 
        
        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 1/2ϕ̇^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 contrast ϕ̇∼ constant in non-canonical models with α≫ 1,
        see (<ref>).
        This allows ϕ(t) to cross each successive step on the DE staircase in a finite amount of time,
        Δ t ≃Δϕ/ϕ̇, and drop to a lower value of V(ϕ); see left panel
        of fig. <ref>.
        
        
        
        § 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 (<ref>). This inequality ensures that the third term in the equation of motion 
        (<ref>)
        is small and can be neglected, resulting in ρ__X∝ a^-3 and w__X≃ c__S≃ 0.
        In other words if (<ref>) is satisfied the kinetic term behaves like dark matter with vanishing
        pressure and sound speed. Of equal importance is the fact that if
         (<ref>) holds then eqn. (<ref>) 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:
        
        
        
        (i) Models with exactly flat potentials V' = 0,  V=V_0.
        As shown in <cit.> the entire expansion history of this model resembles ΛCDM.
        (ii) Successful unification of the dark sector can also arise from
        potentials which are steep at early
        times and flatten out at late times. 
        Both the  potential (<ref>) and the Starobinsky-type potential (<ref>)
         provide us with examples of this category.
        (iii) The step-like potential (<ref>) also leads to unification
        [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).].
        In this case the motion of ϕ resembles a series of waterfalls as V(ϕ) cascades to lower
        and lower values.
        It is interesting to note that
        for all of the above potentials [We note in passing that the
        asymptotically flat potential <cit.>V(ϕ) = V_0 tanh^2ϕ
        also leads to a unified scenario of dark matter and dark energy although we do not discuss it in this paper.]
         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 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 <cit.> 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 <cit.> and fuzzy cold dark matter <cit.>
        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 <cit.>
        and lead to interesting models of inflation, dark matter and dark energy <cit.>
        in the canonical case.
        
        
        
        § ACKNOWLEDGEMENTS
        
        S.S.M thanks the Council of Scientific and Industrial Research (CSIR), India, for
        financial support as senior research fellow. S.S.M also acknowledges the hospitality of the
          Yukawa Institute for Theoretical Physics (YITP) at Kyoto University, where a substantial part of this research work was conducted during the workshop "Gravity and Cosmology 2018".
        
        
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