Eur. Phys. J. C (2017) 77:225 DOI 10.1140/epjc/s10052-017-4796-7

Regular Article - Theoretical Physics

A new recipe for CDM

Varun Sahni1,a, Anjan A. Sen2,b 1 Inter-University Centre for Astronomy and , Pune, India 2 Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India

Received: 12 May 2016 / Accepted: 28 March 2017 © The Author(s) 2017. This article is an open access publication

Abstract It is well known that a canonical scalar field is Despite its enormous success in explaining observations, the able to describe either dark or but not origin of  is not known. It may simply be a residual vacuum both. We demonstrate that a non-canonical scalar field can fluctuation, although quantum field theory usually predicts describe both and dark energy within a unified much larger values, and the cosmological constant problem setting. We consider the simplest extension of the canoni- remains unresolved. As concerns dark matter, mainstream cal Lagrangian L ∝ X α − V (φ) where α ≥ 1 and V is thinking usually assumes it to be a non-baryonic relic of the a sufficiently flat potential. In this case the kinetic term in big bang [14Ð18] but other explanations can also be found the Lagrangian behaves just like a perfect fluid, whereas the in the literature [19Ð28]. Furthermore, since 96% of the con- potential term mimicks dark energy. For very large values, tent of the universe is of unknown origin, attempts have been α  1, the equation of state of the kinetic term drops to made to describe both dark matter and dark energy within a zero and the universe expands as if filled with a mixture of unified setting. The Chaplygin gas (and its subsequent gener- dark matter and dark energy. The velocity of sound in this alisation) belongs to this category of models, since its equa- model and the associated gravitational clustering are sensi- tion of state (EOS) behaves like pressureless dust at early tive to the value of α. For very large values of α the clustering times and like a -term at late times [29Ð31]; also see [32]. properties of our model resemble those of Unfortunately the Chaplygin gas has problems with gravi- (CDM). But for smaller values of α, gravitational clustering tational clustering and so falls short of describing the real on small scales is suppressed, and our model has properties universe [33Ð39]. resembling those of (WDM). Therefore In this paper we show that a unified description of dark our non-canonical model has an interesting new property: matter and dark energy can emerge from non-canonical scalar its expansion history resembles CDM, while its clustering fields; see [40Ð42] for earlier work in this direction. These properties are akin to those of either cold or warm dark matter. fields possess an additional degree of freedom (encoded in the parameter α) which allows a scalar field rolling along a flat potential to behave like a two component fluid consisting of 1 Introduction an almost pressureless kinetic component (dark matter) and a cosmological constant. For large values of α the equation of Ever since the discovery that high type Ia supernovae state of the kinetic component drops to zero and the expansion supported an accelerating universe, concordance cosmology of the universe is similar to CDM. Non-canonical scalars or CDM, has come to dominate popular thinking. Although cluster on small scales, thereby providing us with a realistic issues relating to the smallness of  have given rise to several model of an accelerating universe consisting of dark matter rival models of cosmic acceleration [1Ð8] there is no doubt and dark energy. For very large values of α the kinetic com- that, despite some recent evidence to the contrary [9Ð12], ponent clusters like cold dark matter, whereas for smaller α CDM agrees well with a large set of cosmological obser- values, clustering in our model resembles warm dark matter. vations [13]. As its name suggests, CDM consists of two components:  the cosmological constant, , and cold dark matter (CDM). 2 Non-canonical scalars and CDM a e-mail: [email protected] Perhaps the simplest generalisation of the canonical scalar b e-mail: [email protected] field Lagrangian density 123 225 Page 2 of 8 Eur. Phys. J. C (2017) 77:225

1 φ˙ ∝ −3 L(X,φ)= X − V (φ), X = φ˙2 (1) and which reduces to the canonical result, a , when 2 α = 1. Substituting for X ≡ φ˙2/2from(12)into(7) one which preserves the second order nature of the field equations readily finds is the non-canonical Lagrangian [41,43Ð47] ρφ = ρX + V (φ) (13)   α−1 X pφ = pX − V (φ), (14) L(X,φ)= X − V (φ), 4 (2) M ρ = ( α − ) X( X )α−1 ≡ ρ a−3(1+w) p = with X 2 1 M4 0X and X where M has dimensions of mass while α is dimensionless. wρ X , where When α = 1 the k-essence Lagrangian (2) reduces to (1). p 1 w = X = , (15) We shall be working in the spatially flat FriedmannÐ ρX 2α − 1 RobertsonÐWalker (FRW) universe is the equation of state (EOS) of the kinetic component of ds2 = dt2 − a2(t) [dx2 + dy2 + dz2], (3) the scalar field. From (15) one notes that w ≥ 0forα ≥ 1, therefore models based solely on the kinetic term cannot for which the energyÐmomentum tensor has the form describe cosmic acceleration, including the phantom regime μ recently reviewed in [48]. T = diag(ρφ, −pφ, −pφ, −pφ), (4) ν From (13) and (14) it follows that the non-canonical scalar where the energy density, ρφ, and pressure, pφ,aregivenby field behaves like a mixture of two non-interacting perfect   fluids: ρX and V (φ), where the equation of state of ρX is ∂L given by (15). Assuming for simplicity that V (φ) = /8πG ρφ = (2 X) − L, (5) ∂ X one finds (after setting 8πG = 1) pφ = L. (6) ρφ = ρX +  (16) L Substituting for from (2)into(5) and (6) one gets pφ = pX − , (17)  α− X 1 substituting (16)into(8)gives ρφ = (2α − 1) X + V (φ),   4 1/2 M 3(1+w)  α− (7) H(z) = H0 0X (1 + z) +  (18) X 1 pφ = X − V (φ), π ρ 4 = 8 G 0X w M where 0X 2 and is described by (15). 3H0 which reduces to the canonical form ρφ = X + V , pφ = From (15) and (18) we find that the expansion history is X − V when α = 1. The two Friedmann equations are very sensitive to the value of the non-canonical parameter   α. For the canonical value α = 1 the scalar field behaves  α− 8πG X 1 like a mixture of ‘ + stiff matter’. However, for α = 2the H 2 = (2α − 1) X + V (φ) , (8) 3 M4 expansion history mimicks ‘ + radiation’. For the physi-   −3  α− cally interesting case α  1, w → 0 and ρX ∝ a , conse- a¨ 8πG X 1 =− (α + 1) X − V (φ) , (9) quently (18) describes CDM in this limit: a 3 M4 3 1/2 H(z)  H0[ 0m(1 + z) + ] , where 0m≡ 0X . φ( ) where t satisfies the equation of motion (19)    α−1 φ˙ (φ) 4 The equation of state of the scalar field, wφ, is given by φ¨ + 3 H + V 2 M = , 0 (10)  2α − 1 α(2α − 1) φ˙2 −1  1 + wφ(z) = (1 + w) 1 + (20) ¨ ˙ 3(1+w) which reduces to the standard canonical form φ + 3 Hφ + 0X (1 + z) (φ) = α = V 0 when 1. where w is described by (15). We find that wφ  w when z  Consider the equation of motion (10) for the simplest case  −1 1, while its current value is wφ,0 = (1 + w)[1 + ] − 1.    0X when V is small and can be neglected in (10). Setting V 0 From (15) and (20) one finds that, for α  1, wφ  0at in (10) one finds z  1, and wφ,0 −  at z = 0. Thus for large values of φ˙ α, the EOS of the scalar field smoothly interpolates between ¨ 3 H φ =− , (11) dust-like behaviour at high and a negative value at 2α − 1 present.1 In Fig. 1 we plot the fractional difference between which is easily integrated to give 1 In the limit when α →∞our model has properties resembling those − 3 φ˙ ∝ a 2α−1 , (12) of [41]. 123 Eur. Phys. J. C (2017) 77:225 Page 3 of 8 225

2 2 0.07 φ = x + y = 1, (22) 2 0.06 γ = 1 + wφ = x (1 + wk).

0.05 For α  1, wk ∼ 0 and one can safely approximate 0.04 H γ ∼ x2. Using the formulation prescribed in De-Santiago 0.03 et al. [49], we can now form an autonomous system of equations for γ and σ : 0.02

 √ 0.01 γ = 3σ(1 − γ) γ − 3γ(1 − γ), 0.00 √ √  2 3 0.0 0.2 0.4 0.6 0.8 1.0 σ =−3σ γ( − 1) + σ(1 − σ(1 − γ)/ γ). a 2 (23) Fig. 1 H is plotted against the expansion factor, a,for α = 50, 100, 200, 1000 (top to bottom). Here H = (H − Here ‘prime’ denotes derivative w.r.t. log a and = H )/H and a = 1 corresponds to the present epoch (φ) CDM CDM VV = 1 V = 1 m2φ2 V (φ)2 , so that 2 for 2 . In order to solve this autonomous system, one requires initial condi- the Hubble parameter H(z) in our model from CDM for tions for γ and σ . We set these at decoupling, a ∼ 10−3, different values of the parameter α (but identical values of the assuming that initially the scalar field kinetic energy dom- matter density). One can see that for α ≥ 103 the deviation 1 inates over its potential energy, so that wφ ∼ α− . With α 2 1 is less than 1%. Hence for such large values of our model α  1, we have γi ∼ 1. Similarly one finds σi ∼ 0at  −6 will be virtually indistinguishable from CDM model by decoupling. We set σi = 10 for our subsequent cal- observables measuring background cosmology alone. culations. (One should note that wφ is not exactly equal We have assumed thus far that V (φ) is a constant. This to zero initially, due to the large but finite value of α.) however need not necessarily be the case. It is important to With these initial conditions, the autonomous system of note that CDM-like expansion can also arise in the case of equations in (20) is evolved from decoupling until today, other potentials which are flat. The reason for this is simple. and the resultant behaviour of the equation of state for Our treatment above was based on the assumption that the last the scalar field, wφ, is shown in Fig. 2 (left panel). We term in (10) was negligibly small compared to the remaining find that the behaviour of wφ in our model is quite similar two terms, allowing the former to be neglected. This feature is to that described by (18)forCDM. Nevertheless, the shared by several flat potentials some of which are described two models are by no means identical. Indeed, the expan- below. sion history in both models can easily be distinguished • (φ) = 1 2φ2 V 2 m . In order to study this power-law poten- by means of the Om diagnostic. The Om diagnostic [50] tial, we first form an autonomous system of equations for L = ( ) − our model. For Lagrangians of the form F X (H/H )2 − 1 (φ) Om(a) = 0 (24) V , De-Santiago et al. [49] have already constructed a−3 − 1 an autonomous system of equations involving the dimen- sionless variables has the interesting property that its value stays pegged to √ the current value of the density parameter, i.e. Om(z) = 2XFX − F x = √ , 0m, only in CDM. In other models one expects 3m H ( ) = √ pl Om z 0m. V The right panel of Fig. 2 shows the behaviour of Om = √ , y in a CDM model described by Eq. (18), with w  0 3m pl H (21) (dashed line). The same figure also shows the Om diag- F wk = , nostic for a dark matter-dark energy model described by 2XF − F X the Lagrangian (2) with V = 1 m2φ2. Figure 2 clearly m dlogV 2 σ =−√ pl . shows that the two potentials V =  and V = 1 m2φ2 |ρ | dt 2 3 k can be distinguished by their expansion histories, as encoded in the Om diagnostic. Thus the m2φ2 model Applying these variables to our Lagrangian given by (2), has a time-dependent value of Om which is lower than we get that in CDM. The Omdiagnostic therefore emerges as a 1 useful means of distinguishing between rival dark matter- wk = , α − 1 dark energy models based on non-canonical scalars. One 123 225 Page 4 of 8 Eur. Phys. J. C (2017) 77:225

0.0 0.40

0.2 0.35

0.4 0.30 Om

0.6 0.25

0.20 0.001 0.005 0.010 0.050 0.100 0.500 1.000 0.2 0.4 0.6 0.8 1.0 a a

−6 Fig. 2 Left The equation of state for the scalar field wφ as a function of σi = 10 . Right The behaviour of the Om parameter. The solid and scale factor for V (φ) ∼ φ2 (solid line). The dashed line is for equation dashed lines represent the same models as in the left panel of state described by (18)for  = 0.7and 0X = 0.3. We have set

   might add that future measurements of the expansion his-  z  2 2 R + 2 R + c k Rk = 0, (26) tory are likely to determine Om to great precision, help- k z k s ing break near-degeneracies between rival dark energy where models [51,52]. / • a (ρφ + pφ)1 2 Finally, an interesting example of a piece-wise flat poten- z ≡ , (27) tial is the step potential cs H R is the curvature perturbation V (φ) = A + B tanh βφ (25)   H R ≡ ψ + δφ, (28) φ˙ where A + B = Vinitial and A − B = Vfinal.For 64 4 −47 4 ψ δφ Vinitial  10 GeV , Vfinal  10 GeV this potential and , correspond to the metric perturbation and the scalar would interpolate between inflation at early times, and field perturbation, respectively. The derivative in (26) is taken dark energy at late times, and therefore might describe a with respect to the conformal time, η = dt/a(t) and cs is model of quintessential inflation. We shall examine this the effective sound speed of perturbations in the scalar field possibility in greater detail in a companion paper. [56]   To summarise, we have demonstrated that the two com- (∂L/∂ ) 2 ≡ X . ρ cs (29) ponents of the non-canonical scalar field density, namely X (∂L/∂ X) + (2 X) ∂2L/∂ X 2 and V (φ) in (13), can play the dual role of dark matter and dark energy viz-a-viz the expansion history of the universe. Rewriting (26) in terms of the MukhanovÐSasaki variable In Fig. 2 we have shown how different non-canonical mod- uk ≡ z Rk, one gets   els of dark energy can be distinguished by means of the Om   2 2 z diagnostic. In order to deepen the parallel between a non- u + c k − uk = 0. (30) k s z canonical scalar field and a unified treatment of dark mat- ter and dark energy we also need to demonstrate that the The key to our understanding of gravitational clustering is field φ can cluster. In order to do this we first note that lin- provided by the sound speed. Substituting (2)into(29)we earised scalar perturbations in a spatially flat FRW universe get are described by the line element [53Ð55] 2 = 1 . cs (31) 2 2 i 2 α − 1 ds = (1 + 2 A) dt − 2 a(t)(∂i B) dt dx 2 i j We therefore find that the sound speed is a constant, and that, − a (t) [(1 − 2 ψ) δij + 2 (∂i ∂ j E)] dx dx . for α  1, c2 → 0. In other words, when the value of the μ μ S The linearised Einstein equation δG ν = κδT ν together non-canonical parameter α is large, the sound speed vanishes, with the perturbation equation for the scalar field gives and the scalar field begins to behave like a pressureless fluid. 123 Eur. Phys. J. C (2017) 77:225 Page 5 of 8 225

k 0.01 h Mpc k 0.1 h Mpc 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a a

Fig. 3 The scale-dependence of linear gravitational clustering is illus- for the non-canonical scalar field with α = 5 × 104, 105, 5 × 105 and trated for a non-canonical model with V (φ) = /8πG. The linear also for CDM (bottom to top with CDM at the top). Two scales are density contrast, δ, is shown as a function of the expansion factor, a, considered: k = 0.01 h/Mpc (left)andk = 0.1h/Mpc (right)

An important property of our model follows from (19) and and gravitational clustering. Another possibility provided by (31), namely, when α  1, the background universe expands our model is that the non-canonical scalar comes as an add- like CDM, while its clustering properties could resemble on to dark matter (instead of replacing it). This is the usual those of cold dark matter or even warm dark matter. The procedure adopted by models such as quintessence, in which non-canonical scalar therefore provides a unified prescription the matter part of the Lagrangian remains unchanged while for dark matter and dark energy since both components are dark energy is sourced by a potential such as V ∝ φ−α. sourced by the same non-canonical scalar field. We elaborate It is easy to see that the expansion history in such a model on this issue below. (consisting of conventional dark matter and a non-canonical The evolution equation for the linear density contrast of scalar) is ρX −¯ρX theX-fluidin(13), namely δ = ρ¯ , evaluated on sub- X ( ) = [ ( + )3 + ( + )3(1+w) + ]1/2 horizon scales (|k|H/c), is given by [36] H z H0 0m 1 z 0X 1 z 

  (33) δ =−[2 + A − 3(2w − c2)]δ k s k   2 where the last two terms are sourced by the scalar field and 3 2 2 kcs + X (1 − 6c + 8w − 3w )δk − δk (32) w 2 s aH is described by (15). The clustering properties of the non-canonical scalar are where w, the equation of state of the X-fluid, is described by once more given by (31). The new model (33) therefore ( 2) (15), = 8πGρ /3H 2, A = H and  ≡ d .We describes a universe filled with the cosmological constant X X 2H 2 dloga evolve this equation from the decoupling epoch (a = 10−3) and two kinds of dark matter: the first being the usual dark δ w ≡ 2 when it is reasonable to assume δ ∼ a and d k ∼ 1. matter whereas, depending upon the value of cs , the sec- k da Our results are shown in Fig. 3 for two different scales, ond component, 0X , can behave like a hot, warm or cold k = 0.01 h/Mpc, 0.1h/Mpc in the context of a non-canonical dark matter component. This could have interesting cosmo- model with V (φ) = /8πG. We find that gravitational clus- logical consequences. For instance, as recently demonstrated  tering in this model is scale-dependent. On very large scales in [57], a model with 0X 0m could help alleviate the  k ≤ 0.01 h/Mpc, scalar field models with large values of tension faced by CDM in simultaneously fitting CMB and α ≥ 104 display clustering identical to CDM. However, weak lensing data. A subdominant component of dark mat- on smaller scales k ≥ 0.1h/Mpc, the density contrast in our ter, like the one discussed in this paper, could also seed early model is suppressed relative to CDM even for α values as black hole formation, as discussed in [58]. large as 105, for which the background expansion is indis- tinguishable from CDM, as demonstrated in Fig. 1.(The reader might like to note that similar results may be obtained 3 Discussion if one models dark matter by a perfect fluid with p = kρ, 0 ≤ k  1, and adds it to the cosmological constant in the In this paper we have demonstrated that a single non- so-called standard model.) canonical scalar field can play the dual role of describing both We have thus demonstrated that our model is capable dark matter and dark energy. To summarise, a non-canonical of mimicking the behaviour of a dark matter + vacuum scalar field rolling along a flat potential has a kinetic energy energy model both with respect to cosmological expansion which decreases rapidly with time and a potential energy 123 225 Page 6 of 8 Eur. Phys. J. C (2017) 77:225 which decreases much more slowly. For large values of the 1.0 non-canonical parameter α in (2), the kinetic energy can play the role of dark matter while the potential energy behaves like 0.8 dark energy. If V (φ) = /8πG then the expansion history of this model mimicks CDM. Other (suitably flat) potentials 0.6 give rise to a slightly different expansion history. On its own this result, while surprising, is not unique. It 0.4 is well known that, for a given expansion history, a(t),it is always possible to reconstruct the canonical scalar field 0.2 potential V (φ) which will reproduce the expansion history precisely [59,60]. Therefore, in principle, it is possible to 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 obtain a potential which reproduces the CDM expansion k in h Mpc ( ) ∝ ( 3  )2/3 rate a t sinh 2 3 t . However, the fact that (non- Fig. 4 Perturbations, δk , in the non-canonical scalar field model are oscillating) canonical scalar fields do not cluster on sub- 7 shown at the present epoch. δk is plotted against k for α = 2×10 , 7× horizon scales, prevents this potential from providing one 107, 5 × 108 (bottom to top, solid lines). Perturbations in warm dark with a realistic portrayal of CDM; also see [37]. matter consisting of a with mass = 0.5, 1, 3 KeV are also The big advantage of non-canonical scalars arises from shown (bottom to top, dashed lines). The top most solid line corresponds  the fact that, for large values of the non-canonical parameter to CDM α, the sound speed in (31) drops to zero. Therefore the non- canonical scalar field can cluster, in contrast to canonical Fig. 3. It is interesting that a similar situation arises when models in which clustering is absent. dark matter is sourced by an oscillating massive canonical It is necessary to point out that properties similar to those scalar field with mass m [21Ð27]. In this case, as shown in possessed by our model have also appeared in other discus- [24,28], the Jeans length depends upon the scalar field mass −1/4 −1/2 sions of unification (of dark matter and dark energy). For as λJ ∼ (Gρ) m . Ultra-light scalars are therefore instance in [40] Scherrer proposed a non-canonical model able to suppress clustering on small scales, thereby providing which had an expansion rate exactly like CDM. Our model one with a resolution to the substructure and cuspy core prob- differs from [40] in two main respects: lems which plague standard cold dark matter.2 One expects that a similar mechanism will operate in our model as well, (i) The purely kinetic Lagrangian L(X) in [40] possesses with α playing the role of m. (A key distinction between the an extremum in X about which it is expanded in a Taylor two models is that whereas the canonical scalar field needs to series. Our Lagrangian, on the other hand, has a power- oscillate in order to describe dark matter, the non-canonical law kinetic term with no extremum. field does not oscillate but simply rolls along its flat poten- (ii) The sound velocity in [40] drops off as a−3 whereas in tial.) our model the sound velocity is a constant and is given A useful analogy can also be drawn between clustering by (31). in our model and that in particle dark matter. Consider the case when a relic particle of mass m (such as a We therefore conclude that the whereas the expansion his- or a sterile neutrino) plays the role of dark matter. In this tory in our model and in [40] is identical (corresponding to case perturbations on scales smaller than the free-streaming CDM), the nature of gravitational clustering in these two distance [14]   models is rather different. Indeed, gravitational clustering in m −1 λ ∼ 40 Mpc (35) our model is scale-dependent, and is sensitive to the choice fs 30 eV of α. are effectively erased during the relativistic motion of the In this context one should note that the value of α can particle. never be infinitely large. Consider two models characterised A larger value of m in (35) leads to a smaller value of λ . by α and α where 1  α  α . Since the Jeans length in fs 1 2 1 2 Comparing (35) with (34) we find that the role of mass,in our model is particle dark matter models, is played by the parameter α in  1 λ ∼ c / Gρ, where c = √ , (34) J s s α − 2 1 2 The substructure problem relates to the observation that CDM predicts it follows that the clustering properties of our field will be an order of magnitude more faint galaxies than are observed. The cuspy sensitive to the value of α. Clearly gravitational clustering core problem refers to the tension between simulations of CDM, which ρ ∼ /r α predict a density profile steeper than 1 for dark matter halos, and in the model with 1 will be inhibited on small scales rela- the much shallower ‘cored’ profiles observed in individual galaxies; see tive to the model with α2. This property was illustrated by [19] and references therein. 123 Eur. Phys. J. C (2017) 77:225 Page 7 of 8 225 our model. In other words, whereas very large values of α ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, will make clustering in our model resemble cold dark matter, and reproduction in any medium, provided you give appropriate credit smaller values of α will make our model closer to warm dark to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. matter. We demonstrate the similarity of our model with the Funded by SCOAP3. sterile neutrino model for warm dark matter in Fig. 4.In this figure we show δk for our model, obtained by solving Eq. (25), and compare it with the density contrast in the warm dark matter model, as described in [61]; also see [62Ð65]. References This figure clearly demonstrates that, for suitable values of α, clustering in our model can be like cold or warm dark 1. V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000) matter, even as its expansion history mimicks CDM. 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