Modified Dust and the Small Scale Crisis in CDM
MModified Dust and the Small Scale Crisis in CDM
Fabio Capela a and Sabir Ramazanov b a DAMPT, Centre for Mathematical Sciences, Cambridge University,Wilberforce Road, Cambridge CB3 0WA, UK b Service de Physique Th´eorique, Universit´e Libre de Bruxelles (ULB),CP225 Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Abstract
At large scales and for sufficiently early times, dark matter is described as a pressurelessperfect fluid—dust—non-interacting with Standard Model fields. These features are capturedby a simple model with two scalars: a Lagrange multiplier and another playing the role ofthe velocity potential. That model arises naturally in some gravitational frameworks, e.g.,the mimetic dark matter scenario. We consider an extension of the model by means of higherderivative terms, such that the dust solutions are preserved at the background level, butthere is a non-zero sound speed at the linear level. We associate this
Modified Dust withdark matter, and study the linear evolution of cosmological perturbations in that picture.The most prominent effect is the suppression of their power spectrum for sufficiently largecosmological momenta. This can be relevant in view of the problems that cold dark matterfaces at sub-galactic scales, e.g., the missing satellites problem. At even shorter scales,however, perturbations of Modified Dust are enhanced compared to the predictions of morecommon particle dark matter scenarios. This is a peculiarity of their evolution in radiationdominated background. We also briefly discuss clustering of Modified Dust. We write thesystem of equations in the Newtonian limit, and sketch the possible mechanism which couldprevent the appearance of caustic singularities. The same mechanism may be relevant inlight of the core-cusp problem. a r X i v : . [ a s t r o - ph . C O ] A p r ontents K ( ∂ µ ϕ ) = 0 42.2 Generic case 6 γ ( (cid:3) ϕ ) A.1 Background level 17A.2 Linear level 17
B Term γ ∇ µ ∇ ν ϕ ∇ µ ∇ ν ϕ B.1 Background level 18B.2 Linear level 19
To date, particle physics provides us with a plethora of candidates for dark matter (DM).These include sterile neutrinos, supersymmetric partners of the Standard Model particles,light scalars (axions) and many others. Given that the new degrees of freedom are heavyenough and weakly interacting with the Standard Model constituents, one deals with colddark matter (CDM). The concept of CDM is of the uttermost importance in modern cos-mology. Namely, it is among the building blocks of the 6-parametric concordance modelestablished by the recent Planck and WMAP missions [1, 2]. In particular, the evolution oflinear perturbations developed in this framework is in excellent agreement with the picture ofthe Cosmic Microwave Background temperature anisotropies [3]. Furthermore, simulationsbased on CDM lead to correct predictions about the large scale structure distribution in theUniverse [4–6]. Finally, the concept of CDM agrees with the Bullet Cluster observations [7, 8]and provides an explanation for the flat galaxy rotation curves [9–11].Despite these successes, there is some tension between CDM predictions and astronom-ical data at the sub-galactic scales. This amounts to three problems. First, CDM predictsan overabundance of small structures, i.e., dwarf galaxies. Observations in the vicinity ofthe Milky Way, however, indicate a much smaller number. This states the “missing satellitesproblem” [12, 13]. Second, CDM leads to cuspy profiles of the DM halos [9, 10]. At the sametime, observations on the concentrations of dwarf galaxies rather prefer cored profiles [14, 15].Finally, N-body simulations result into a large central density of massive subhaloes in the1ilky Way—a fact, which is in conflict with observations of stellar dynamics in dwarf galaxieshosted by those subhaloes. This inconsistency is dubbed as the “too big to fail” problem [16].It is important to keep in mind that the aforementioned shortcomings of CDM may not berobust to the proper account of various astrophysical phenomena [17–25]. However, realistichigh resolution numeric simulations, which include baryonic processes, are challenging to im-plement at the moment. Meanwhile, it is interesting to speculate if part or all of the problemsare due to the peculiar nature of DM itself. For example, Warm Dark Matter (WDM) scenar-ios start from the proposal that the DM particles are still mildly relativistic at the freeze outtemperature. Then, the short wavelength perturbations get washed out due to free streamingprocesses [26, 27]. This provides a simple mechanism to suppress the number of small scalestructures, and thus can be relevant to alleviate the missing satellites problem [28]. However,WDM scenarios are perhaps too efficient in erasing the short wavelength perturbations, asindicated by the constraints following from the Lyman- α forest data [29]. Another approachto the problems at the sub-galactic scales is to go beyond the approximation of collisionlessmatter. Namely, allowing for sufficiently strong self-interactions of DM particles [30], onecan address the core-cusp and too big to fail problems [31]. On the other hand, constraintsobtained from the Bullet Cluster observations imply smaller self-interaction cross-sectionsthan what is required in view of the core-cusp problem [32].One can try to find a solution to the small scale crisis by switching to a paradigmdifferent from particle DM. Recently, an interesting proposal on the way to model darkmatter and dark energy has been made in Ref. [33]. There, the authors introduced a novelclass of theories, referred to as the Σ ϕ -fluid. The action for the Σ ϕ -fluid is given by S = (cid:90) d x √− g [Σ ( g µν ∂ µ ϕ∂ ν ϕ −
1) + K ( ϕ, ∂ µ ϕ )] . (1.1)In what follows, we assume the sign convention (+ , − , − , − ) for the metric. Here Σ is theLagrange multiplier; K ( ϕ, ∂ µ ϕ ) is some arbitrary function of the scalar ϕ and its derivatives.Varying the action (1.1) with respect to the field Σ enforces the constraint g µν ∂ µ ϕ∂ ν ϕ = 1,so that ∂ µ ϕ is a unit 4-vector. One can associate the field ϕ with the velocity potential ofthe Σ ϕ -fluid. The constraint equation tells us that the fluid elements follow geodesics, muchin the same manner as the dust particles. At the same time, given the non-trivial function K ( ϕ, ∂ µ ϕ ), one can allow for a non-zero effective pressure with a time-dependent equationof state. This opens up the possibility to construct a fluid, which may mimic dust at earlytimes and a positive cosmological constant Λ later on.In the present paper, however, we restrict the discussion to DM. A pressureless perfectfluid is obtained for the function K ( ϕ, ∂ µ ϕ ) identically equal to zero, i.e., K ( ϕ, ∂ µ ϕ ) = 0. Theenergy density of the dust is associated with the field Σ decaying as 1 /a with the scale factor a . It is thus tempting to view the construction with the Lagrange multiplier as the simplestmodel of DM: at the background and linear levels, it reproduces all the successes of morecommon particle scenarios. In the non-linear regime, however, the dust model has importantdrawbacks: it leads to caustic singularities and is unable to form stable DM halos [34].From now on, we should go beyond the approximation of a pressureless perfect fluid.In the theory described by the action (1.1), this is achieved by incorporating the function K ( ϕ, ∂ µ ϕ ). At the same time, we would like to keep the simple form of the background dustsolutions in the Universe dominated by the Σ ϕ -fluid: Σ ∝ /a . Interestingly, this is possiblewith the non-trivial choice of the function K ( ϕ, ∂ µ ϕ ), K ( ϕ, ∂ µ ϕ ) ≡ K ( ∂ µ ϕ ) = γ ∇ µ ∇ µ ϕ ∇ ν ∇ ν ϕ + γ ∇ µ ∇ ν ϕ ∇ µ ∇ ν ϕ . (1.2)2ere γ and γ are parameters with dimension of mass squared. Expressions (1.1) and (1.2)state the model, which we refer to as Modified Dust in what follows. We shortly give a briefsummary of our main results. Before that, let us comment on how the action (1.1) arises indifferent gravitational frameworks.In Ref. [35], the authors considered the standard Einstein’s metric g µν as the compositionof an auxilliary metric ˜ g µν and the first derivatives of the scalar field ϕ , g µν = ˜ g µν ˜ g αβ ∂ α ϕ∂ β ϕ. Unexpectedly, variation with respect to the fields ˜ g µν and ϕ results in a modification of theEinstein–Hilbert equations such that the traceless part is non-zero even in the absence ofmatter. This discrepancy with the standard equations of General Relativity is due to thepresence of an extra degree of freedom, which behaves as dust. The dust solution has beendubbed as mimetic dark matter in Ref. [35]. Soon afterwards, it has been realized that theproposed scenario is equivalent to adding a Lagrange multiplier into the Einstein–Hilbertaction [36]. This is the picture described by the action (1.1). Furthermore, in Ref. [37] it wasproved that the condition Σ > K ( ϕ, ∂ µ ϕ ) equals to zero in the mimetic dark matter scenario as it stands. Setting it byhands, however, opens up the possibility to mimic different types of cosmologies [40, 41].Constructions introducing the term with the Lagrange multiplier are known in thecontext of Einstein–Æther (EA) models [42]. In particular, scalar EA [43] has exactly theform given by Eqs. (1.1) and (1.2). This is interesting, as the scalar EA appears in the IR limitof the projectable version of Horava–Lifshitz model [44–46]— power counting renormalizabletheory of gravity [47]. It is thus not a surprise that DM has been identified in this setup [48].Contrary to the case of Modified Dust, however, it was suggested in [48, 49] to extend thepressureless perfect fluid by means of higher curvature terms inherent to Horava’s proposal.In the present paper we prefer to stay on a more phenomelogical side. Our main purposeis to study the linear evolution of cosmological perturbations of Modified Dust described byEqs. (1.1) and (1.2). We uncover several effects, which can be relevant in light of the smallscale crisis. The first is the suppression of perturbations with relatively large momenta. Aswe will see explicitly in Section 2, γ -terms result into a non-zero sound speed at the levelof perturbations [40]. Beyond the sound horizon, perturbations of Modified Dust behave asin the CDM picture: they grow linearly with the scale factor during the matter dominated(MD) stage. As they enter the sound horizon, their growth stabilizes. This places an impor-tant cutoff on the linear power spectrum in our model. In this regard, Modified Dust carriessimilarities with WDM. So, by setting γ i ∼ − M , where M pl is the Planck mass , one cansuppress perturbations with wavelengths below 100 kpc. Hence, Modified Dust is capable toaddress the missing satellites problem. There is, however, an important distinction from thecase of WDM scenarios. The difference is clearly seen from the evolution during the radia-tion dominated (RD) stage, when perturbations of Modified Dust of very small wavelengthsexperience a linear growth with the scale factor. We show this explicitly in Section 3. Asa result, corresponding perturbations get amplified compared to the predictions of WDM Hereafter, we define the Planck mass squared as the inverse of the Newton’s constant G , i.e., M = G − . We start with the case of the Universe dominated by Modified Dust. This is a good approx-imation to the Universe at relatively low redshifts, i.e., z (cid:28) , when the contribution ofthe radiation can be neglected. Non-trivial effects arising upon the inclusion of radiation willbe considered in the next Section. In the present paper, we will always neglect the contribu-tions from baryons and dark energy. The former is expected to change the behaviour of thegravitational potential at the percent level, while the latter becomes relevant only at verysmall redshifts, z (cid:46) K ( ∂ µ ϕ ) = 0In the simplest case, when the constants γ equal to zero, i.e., γ = γ = 0, we have for theenergy momentum tensor of the Σ ϕ -fluid, T µν = 2Σ ∂ µ ϕ∂ ν ϕ . Varying the action (1.1) with respect to the field Σ (Lagrange multiplier), one obtains ∂ µ ϕ∂ µ ϕ = 1 . (2.1)We see explicitly that the energy momentum tensor is the one of a pressureless perfect fluid,with 2Σ being the energy-density and ϕ the velocity potential. At the background level, theformer drops with the scale factor a as 2Σ = C/a . We fix the constant C in a way that 2Σcorresponds to the energy density of DM, i.e.,2Σ = ρ DM . (2.2)The background value of the field ϕ is given by ϕ = t ,
4p to an irrelevant constant of integration. The cosmological evolution of the Universe filledin with dust is then obtained from the ( ij )-component of Einstein’s equations,2 H (cid:48) + H = 0 . (2.3)Here H = a ( η ) H , where H denotes the Hubble parameter. From this point on we prefer towork in terms of the conformal time η . The prime denotes the derivative with respect to thelatter. From Eq. (2.3) we obtain H = 2 /η , which gives the standard solution for the scalefactor a ( η ) in the matter dominated Universe: a ( η ) ∝ η .Note one generic feature of the model, where the 4-velocity of the fluid u µ is the deriva-tive of the scalar field ϕ , i.e., u µ = ∂ µ ϕ . In that case, the conservation of the energy-momentum tensor implies just one equation. Indeed, ∇ µ T µν = ∇ µ (2Σ ∂ µ ϕ ) ∂ ν ϕ = 0 , where we took into account Eq. (2.1). Consistently, this equation can be obtained fromthe variation of the action (1.1) with respect to the field ϕ . Of course, this does not lead todegeneracy of solutions, as the constraint (2.1) itself plays the role of the missing equation [40].At the linear level, Eq. (2.1) reads δϕ (cid:48) = a Φ . (2.4)This is essentially the Euler equation linearized, where Φ is the scalar perturbation of the(00)-component of the metric. Hereafter, we choose to work in the Newtonian gauge. Asit follows from the form of the action (1.1), Eq. (2.4) remains unmodified even for the non-trivial choice of the function K ( ∂ µ ϕ ). In particular, the same equation is true for the case ofnon-zero coefficients γ and γ , to which we will turn soon.Before that, let us remind the picture of linear evolution in the case of the pure dust(CDM). In the Newtonian gauge, the perturbed Friedmann–Robertson–Walker metric hasthe form, ds = a ( η )[(1 + 2Φ) dη − (1 − d x ] . In the perfect fluid approximation, the simple relation holds between the potentials Φ andΨ: Φ = Ψ. We choose to work with the function Φ in what follows. The evolution of thepotential Φ can be easily inferred from the ( ij )-component of Einstein’s equations,Φ (cid:48)(cid:48) + 6 η Φ (cid:48) = 0 . Up to a negligible decaying mode, it has the constant solution Φ = const. In the presence ofthe constant gravitational potential, perturbations of DM grow linearly with the scale factor,as it immediately follows from the Poisson equation, − k Φ = 4 πGa ρ DM δ DM , (2.5)where G is the Newton’s constant. That is, during the MD stage, perturbations of DM aresubject to the Jeans instability independently of their momenta. Later on, they enter non-linear regime, and the clustering starts. This standard picture changes drastically upon theinclusion of γ -terms. 5 .2 Generic case Let us make two important comments before we dig into the details of calculations. With noloss of generality, we choose to work with the unique coefficient γ in the bulk of the paper,while setting the other one, γ , to zero. Besides considerations of simplicity, this is alsojustified, since both terms are expected to result into qualitatively the same phenomenology.We put the details regarding the case γ (cid:54) = 0 in the Appendix B. With this said, the energymomentum tensor has the form [40], T µν = 2Σ ∂ ν ϕ∂ µ ϕ + γ (cid:18) ∂ α ϕ∂ α (cid:3) ϕ + 12 ( (cid:3) ϕ ) (cid:19) δ µν − γ ( ∂ ν ϕ∂ µ (cid:3) ϕ + ∂ ν (cid:3) ϕ∂ µ ϕ ) , (2.6)where γ ≡ γ .For non-zero coefficients γ and γ the interpretation of the field Σ as the energy-density of DM may be misleading. Still, we prefer to stick to the simple convention (2.2)in what follows. Hopefully, this is not going to confuse the reader, as one is interested inthe behaviour of the gravitational potential in the end. Moreover, the difference between theformal energy density and the physical one, ρ phDM = T , remains small in most cases discussedin the present paper. This is true at both levels of background and linear perturbations. Theonly exceptional case occurs at very early times, deeply in radiation dominated era. We willcomment on that in due time.Let us write the background cosmological equations. The simplest is the one corre-sponding to the ( ij )-component of Einstein equations, (cid:0) H (cid:48) + H (cid:1) · (1 − πGγ ) = 0 . (2.7)Apart from the degenerate case 12 πGγ = 1, we have H = 2 /η , which corresponds to theUniverse driven by dust. Consistently, the conservation equation has the form, ρ (cid:48) DM + 3 H ρ DM = 32 γ (cid:18) H (cid:48) a + H a (cid:19) (cid:48) . (2.8)The Friedmann equation is given by3 H (1 − πγ G) = 8 πGa ρ DM Upon the formal change (1 − πγG ) − ρ DM → ρ DM , we get back to the standard set ofequations of the dust dominated Universe. In Appendix B, we show that the same conclusionholds for the γ -term in Eq. (1.2).Before turning to the study of linear perturbations, let us make one useful observation.We note that the linearized energy-momentum tensor is similar to that of a perfect fluidin a sense that δT ij ∝ δ ij . This means that one can impose the constraint on the scalarperturbation Ψ of the spatial part of the metric, Ψ = Φ . We choose to work with thepotential Φ in what follows.The simplest way to proceed is to write the (0 i )-component of Einstein’s equations.The reader can find the explicit expression in the Appendix A. Here we write it in theapproximation of the small parameter γ , i.e., γ (cid:28) M , δϕ (cid:48)(cid:48) + (cid:18) c s k − H (cid:19) δϕ = 0 , (2.9) The analogous conclusion is not applicable to the γ -term. However, the difference between potentials Φand Ψ is negligible and hence is irrelevant for phenomenology. See details in Appendix B. We thank A. Vikmanfor discussions on this point. − − − − − − − − − − − − g r a v i t a t i o n a l p o t e n t i a l η/η eq k = 55 Mpc − k = Mpc − k = Mpc − Figure 1 . The gravitational potential Φ is plotted as a function of conformal time η for differentcosmological wavenumbers k . Cases of Modified Dust and a pressureless perfect fluid/ CDM havebeen studied. The time η eq corresponds to the equilibrium between matter and radiation. For thewavenumber k = 55 Mpc − (green line) predictions of Modified Dust and CDM are indistinguishablefrom each other. Shorter wavelength perturbations are pictured with dark blue and red lines (CDM)and light blue and orange lines (Modified Dust). The parameter γ is set to γ = 10 − M . where c s is given by c s ≈ πγ G . Clearly, the second term in the equation mimics the sound speed . This explains the notation“ c s ” we use. Provided that the cosmological modes are beyond the speed horizon, i.e., c s k (cid:28) H , one obtains for the field ϕ perturbation δϕ ∝ η . Using Eq. (2.4), we getΦ = const. This is the standard solution for the gravitational potential in the MD Universe.Another story occurs after the modes enter the sound horizon, i.e., in the regime c s k (cid:29) H .Accordingly to Eq. (2.9), the rapid growth of the perturbations of the field ϕ stops and turnsinto oscillations, i.e., δϕ ∝ e ic s kη . As a result, the gravitational potential decreases with thescale factor.The behaviour of the DM energy density perturbations can be easily deduced from the(00)-component of Einstein’s equations, which takes the standard form (2.5) in the small γ approximation. As it follows, for modes with sufficiently short wavelengths and at relativelylate times, the linear growth of the energy density contrast δ DM stabilizes and turns intooscillations with a constant amplitude. Let us choose the constant γ in such a way that thegrowth stops at redshifts as small as z (cid:39)
10 for perturbations with the comoving wavelengthsof λ (cid:39)
100 kpc. These wavelengths roughly characterize the collection regions collapsing tothe halos of dwarf galaxies. The redshifts z (cid:39)
10 correspond to the times, when the modesof interest enter the non-linear regime (in the CDM picture). The estimate of the parameter γ reads γ ∼ − M , which is remarkably close to the Grand Unification Scale. This valueof the parameter γ is required to alleviate the missing satellites problem. For larger values, This observation was first made in the context of inflation [40]. γ (cid:28) − M are, however, plausible, as the proper account of baryonic processesmay eliminate the problem with the dwarf galaxies. In that case, however, the motivationfor Modified Dust is essentially lost. Therefore, we set γ ∼ − M in what follows, unlessthe opposite is stated.To put things on solid ground, we performed numerical simulations to obtain the grav-itational potential and the matter power spectrum. The results are presented in Figs. 1and 2, respectively. As it is clearly seen from Fig. 1, the gravitational potential deviatesfrom the standard behaviour predicted by the CDM scenarios for sufficiently large cosmo-logical momenta. Namely, at some point in MD stage it starts to oscillate with a decreasingamplitude and a frequency ω = c s k . The corresponding period of oscillations is very large:for wavelengths λ ∼
100 kpc it is comparable with the age of the Universe. The decrease ofthe gravitational potential with the scale factor translates into the suppression of the linearmatter power spectrum, which we plot in Fig. 2 for the redshift value z = 3. The choice ofthe redshift is dictated by simplicity considerations: for smaller values of z , we would need toincorporate the effects of the accelerated expansion of the Universe into the analysis. Notethat the plot in Fig. 2 represents the extrapolation of the linear evolution of Modified Dustto late times. This is by no means justified, as the cosmological modes of interest are deeplyin the non-linear regime at the redshift z = 3. Still, the plot is useful for the purpose ofcomparison with the particle DM scenarios. We use the conventional definition of the powerspectrum, P ( k ) = 2 π ∆ ( k ) k . Here ∆( k ) is the amplitude of the matter perturbations related to the two-point correlatorin the coordinate space by (cid:104) δ ( x ) (cid:105) = (cid:90) dkk ∆ ( k ) . The picture 2 is quite analogous to what one has in the case of WDM scenarios [50, 51].There are two important qualifications, however. While WDM models predict the exponentialsuppression of the small scale spectrum, we observe a more moderate power law drop. Thisdistinction might be relevant, since WDM does perhaps too good job with diluting sub-galactic structures. Second, we note the presence of slow oscillations in the high momentumtail of the Modified Dust spectrum. This could be a promising smoking gun of our scenario.The conclusions of the present Section work well for wavelengths in the kpc-range. Forperturbations with even smaller wavelengths another effect becomes prominent. That is,very small perturbations get enhanced during the evolution in the RD epoch (once again,compared to the predictions of CDM). We discuss this issue in the following Section.
In the presence of radiation, the evolution of Modified Dust changes considerably. In partic-ular, the energy density associated with the field Σ is now given by ρ DM = ρ + ˜ ρ r , ˜ ρ r = − γη a . (3.1)8 − − − − − − P ( k ) [ h − M p c ] k [h Mpc − ] Figure 2 . The linear matter power spectrum as a function of the momenta k . Black and orange linescorrespond to the cases of a pressureless perfect fluid/CDM and Modified Dust, respectively. Thechoice of the parameter γ = 10 − M is assumed. Here ρ is the energy-density corresponding to pure dust, i.e., ρ ∝ /a . The novel contri-bution ˜ ρ r mimics radiation. The solution (3.1) follows from Eq. (2.8), where we set H = 1 /η .The latter is the standard expression for the Hubble parameter during the RD stage. Theterm ˜ ρ r is somewhat worrisome, as it has a negative sign. It governs the evolution of theDM at redshifts as large as z (cid:29) z ∼ − . The value z corresponds to thepoint, when ρ DM ( z ) = 0. The temperature of the Universe at these early times reads T ∼
100 MeV − ρ DM < ρ phDM = T and remains positive. Moreover,due to the time-dependence of the term ˜ ρ r , it can be absorbed into the standard radiationwith no physical consequences at the background level.Let us discuss the linear perturbations around the background (3.1). At very early timescorresponding to the redshift values z (cid:29) z all the relevant modes are beyond the horizon.In the super-horizon regime, i.e., when k →
0, the conservation equation for the DM (A.9)takes the form, ( a δρ DM ) (cid:48) + 6 γa (cid:0) H (cid:48)(cid:48) − HH (cid:48) − H (cid:1) Φ i = 0 . The solution to this equation reads simply a δρ DM = C (cid:48) + 12 γa ( η ) η Φ i , (3.2)where C (cid:48) is the constant in time, which will be specified below. At very early times, formallyas η →
0, the second term on the r.h.s. is the most relevant. In the same limit ρ DM → At these early times the notion of“Modified Dust” appears to be misleading, as the Σ ϕ -fluid also mimicsradiation in that case. However, we choose to continue with our standard convention in what follows. γ/η a . Consequently, we obtain for the energy-density contrast at very early times δ DM = − i ,—the standard adiabatic initial condition for radiation. As a cross check of ourcalculations, we observe that this initial condition is consistent with the η → z (cid:28) z . In that case, Modified Dust takesthe standard form, i.e., ρ DM ∝ /a . The first term on the r.h.s. of Eq. (3.2) correspondsto the constant mode of the energy-density contrast δ DM , while the second term stands forthe decaying mode. Neglecting the latter, we get the initial condition for Modified Dust, δ DM = C (cid:48) /ρ DM , , where ρ DM , is the present energy density of DM. We fix the constant C (cid:48) insuch a way that we do not encounter problems with CMB observations. That is, we impose δ DM = − / i initially.A comment is in order before we proceed. Working in terms of the quantity δ DM canbe misleading at the redshift z , when ρ DM ( z ) = 0. At that point, we have δ DM ( z ) → ∞ .This, we believe, is just a formality, since the quantity δρ DM —more relevant one–remainsfinite at all times. To avoid the problem, we could always provide the calculations in termsof δρ DM and then convert the latter into δ DM . During the RD stage, cosmological perturbations follow a non-trivial evolution, which isparticularly prominent for short wavelength modes. At sufficiently early times, the maincontribution to the gravitational potential Φ follows from the perturbations of radiation. Inthat case, the potential Φ is given by [52]Φ = − i ( u s kη ) (cid:18) cos( u s kη ) − sin( u s kη ) u s kη (cid:19) , (3.3)where u s = 1 / √ δ DM = C − i ln( u s kη ) + 9 γk a ( η ) ρ DM , Φ i + decaying modes . (3.4)The first two terms on the r.h.s. represent the well-known constant and logarithmicallygrowing mode in CDM, while the third one is the novelty. It describes a mode linearly growingwith the scale factor. This originates from the term ∼ k in the conservation equation (A.9).While for long wavelength modes this term is negligble, it may start to dominate for shorterones at some point. This point η × in the linear evolution is defined from γk a ( η × ) ρ DM , ∼ ln( u s kη × ) . (3.5)Note that the effect takes place only for momenta larger than kH ∼ M pl √ γ √ z eq ; (3.6)10 − − − − − − − − − − − − − δ D M η/η eq wavelength λ =
10 pc A δ Figure 3 . The evolution of the energy density contrast for a mode with the wavelength λ = 10 pcas a function of conformal time η . Black and orange lines correspond to the cases of a pressurelessperfect fluid/CDM and Modified Dust, respectively. The choice of the parameter γ = 10 − M isassumed. otherwise, the condition (3.5) formally takes place at the MD stage , i.e., at η × > η eq , whenthe formula (3.3) and, consequently, the estimate (3.6) are not applicable. Here z eq and η eq denote the redshift value and the conformal time at the equilibrium between radiationand matter, respectively, z eq ∼ and η eq ∼
10 Mpc. For the value √ γ/ M pl ∼ − , wehave k/H (cid:38) , which corresponds to wavelengths λ (cid:46) z ∗ is defined from δ DM ∼ z ∗ z eq Φ i . (3.7)In this estimate, we took into account that perturbations of radiation do not grow, butexperience oscillations with the amplitude 6Φ i [53]. Since this point on, the formula (3.3)is not applicable anymore. To handle the situation, let us get back to the conservationequation (A.9), where we omit all the terms except for one proportional to k . Expressingthe energy density contrast δ DM through the perturbations of the velocity potential δϕ ,substituting δϕ into Eq. (A.6) and omitting the subleading terms, we obtain the secondorder equation for the density perturbations, δ (cid:48)(cid:48) DM + 4 πGγk δ DM = 0 . This equation has only oscillatory solutions. That is, the growth of perturbations δ DM stopsand we get back to the picture observed in the matter dominated Universe. To quantify theamplitude of the oscillations, we first estimate the redshift value z ∗ . By comparing Eq. (3.4)11ith the estimate (3.7), we obtain z ∗ ∼ √ z eq · √ γ M pl · kH . Substituting this back into Eq. (3.7), we have for the amplitude of the energy-density per-turbations, A δ ∼ √ z eq · √ γ M pl · kH Φ i . In Fig. 3, we plot the result of numerical simulations for the evolution of the energy densitycontrast δ DM . As it is clearly seen, it agrees well with our estimates. The associated gravi-tational potential also gets enhanced compared to the predictions of CDM scenarios, as it issourced by perturbations of Modified Dust already at redshifts z (cid:29) z eq . This enhancementgets compensated for the wavelengths λ (cid:38) λ (cid:46) λ fs drops as 1 /z with the redshift z . We are interested in the valueof λ fs at the times, when DM starts to dominate the linear evolution. This can be inferredfrom the value of λ fs at the recombination epoch, λ fs ( z ∗ ) ∼ z rec z ∗ λ fs ∼ λ H · . Here z rec (cid:39) λ (cid:28) − H − ∼
100 kpc, λ fs ( z ∗ ) (cid:28) λ .Hence, one can neglect the free-streaming effects for all interesting wavelengths. Finally, let us discuss the behaviour of Modified Dust in the non-linear regime, i.e., when δ DM (cid:38)
1. Generically, non-linear analysis is a very complicated task, involving non-trivialnumerical simulations. We leave this for the future work. What one can actually do at themoment is to write the relevant system of equations in the Newtonian limit, and discuss theirpossible physical consequences. 12efore that, let us remind the state of affairs with the collisionless particle DM. Inthat case, one normally runs the N-body simulations, which attempt to solve the system ofVlasov–Poisson equations. The problem can be paraphrased in terms of the momenta of theVlasov equation (collisionless Boltzmann equation). Formally, it leads to an infinite chain ofcoupled equations. Setting to zero all the momenta of the probability density starting fromthe velocity dispersion, results into the set of equations of the standard dust. That is, theconservation equation, ˙ ρ + 3 Hρ + 1 a ∇ · ( ρ v ) = 0 . and the Euler equation ˙ v + H v + 1 a ( v · ∇ ) v = − a ∇ Φ . (4.1)In the present Section, we omit the subscript “DM” in the notation of the energy densityof Modified Dust. This system coadded with the Poisson equation can be solved iterativelyusing Eulerian or Lagrangian perturbation schemes [54, 55]. In the mildly non-linear regimethe results are argued to be in a good agreement with N-body simulations. Furthermore,peaks of the energy density localized at the surface of particle crossing (caustics, or Zel’dovichpancake) give a qualitatively correct picture of the Cosmic Web. Since this point on, however,the dust approximation breaks down. That is, the divergence of the velocity and the energydensity become infinite at the caustics, i.e., ∇ · v → −∞ and ρ → ∞ . The singularityhas a clear physical meaning: it reflects the fact that the dust particles may pass unaffectedthrough each other. After particles cross, they fly away from the caustics, as there is nomechanism sticking them together. This leads to fast broadening of the Zel’dovich pancake,and eventually to diluting the structures [34]. The problem does not occur in the case of thereal particle DM: near the caustics high momenta of the Vlasov equation become relevant, andthey regularize the divergence. Qualitatively, this amounts to the appeareance of an effectivepressure (revealed in the non-zero velocity dispersion), which opposes gravity, preventingparticles to cross. Below we argue that a similar mechanism can be relevant in the case ofModified Dust.We restrict the discussion to the Universe filled in with Modified Dust. We writethe system of cosmological equations in the Newtonian limit. The analogue of the Poissonequation takes the form,∆Φ = 4 πG − πγG a (cid:20) δρ − Ha γ ∆ δϕ − γa ( ∇ · v ) (cid:21) . (4.2)The conservation equation reads now˙ ρ + 3 Hρ + 1 a ∇ · ( ρ v ) = − γa ∆( ∇ · v ) . (4.3)The Euler equation has the standard form (4.1), as it follows from the constraint equa-tion (2.1). Here v is the velocity defined with respect to the Euclidean space. It is relatedto the velocity potential ϕ by v = −∇ ϕ/a .One can show that the modification of the Poisson equation is irrelevant. Indeed, theextra terms in Eq. (4.2) result into small O (cid:16) γM Pl (cid:17) corrections to the terms already present Strictly speaking, this type of divergence is seen in the Zel’dovich approximation. It is argued, how-ever, that the latter becomes exact for particular initial conditions. Moreover, corrections to the Zel’dovichapproximation do not cure the problem.
13n the Euler equation (4.1). At the same time, the modification of the conservation equationis something new compared to the standard dust. Interestingly, deviations from the latterstart explicitly when the different trajectories come very close to each other, so that thequantity ∇ · v tends to blow up, i.e., ∇ · v → −∞ . In that case, the r.h.s. of Eq. (4.3)becomes sufficiently large. This gives rise to the flow of energy away from the region, whereone would expect the presence of singularities, to the outer regions. As nothing preventsthe energy density from becoming negative, one results with “anti-gravity”, i.e., a repulsiveforce between fluid elements. This is a necessary—though, not sufficient,—condition to curethe caustic singularities. Note that the appearence of the negative energy density does notimply catastrophic ghost instabilities in the model. Indeed, as it follows from Eq. (4.3), theintegral (cid:82) ρdV is conserved for any large physical volume V . Hence, the regions of negativeenergy do not swallow the entire space.To illustrate the picture described above, let us consider the toy example of the 1Dcollapse. Namely, we take some initial sufficiently smooth distribution of the energy density ρ ( t = 0 , x ) = A exp (cid:16) − x L (cid:17) with the initial velocity v ( t = 0 , x ) = 0. Here A is the constantamplitude and L is the characteristic size of the distribution. Let us focus on the case γ = 0.We assume in what follows that the Universe is empty, and the scale factor a = 1. Thesolution for the energy density reads in the approximation | x | (cid:28) L , ρ ( t, x ) = A − πGAt − Ax − πGAt ) L + O ( x ) . (4.4)In the same approximation, the solution for the velocity is given by v ( t, x ) = − πGAxt − πGAt + 2 πGAx t − πGAt ) L + O ( x ) . (4.5)While we are primarily interested in the behaviour at x = 0, where the appearance of thesingularity is expected, we write explicitly O ( x ) corrections for future purposes. As it isclearly seen, the energy density blows up at a finite time t s = 1 / √ πGA . The same happensto the divergence of the velocity, i.e., ∂ x v → −∞ . This is the caustic singularity. Now let usinclude the γ -term into the discussion. Still assuming that the solution (4.4) and (4.5) holds,we obtain for the r.h.s. of Eq. (4.3) at x = 0, − γ∂ x v = − πγGAt (1 − πGAt ) L . (4.6)As it follows, the higher derivative term becomes of the order of the standard dust termin Eq. (4.3) even for an arbitrarily small γ and large L at times sufficiently close to t s .Another important fact is that the γ -term has a negative sign. Therefore the growth ofthe energy density slows down. Since this point on, however, the solutions (4.4), (4.5) and,consequently, (4.6) are not valid anymore, as they were obtained in the assumption of anegligible γ -term. Assuming, however, that the higher derivative term becomes dominant,one observes the decrease of the energy density. As the positive values of the energy densityare not protected, ρ may become negative at some point. This is in agreement with thequalitative picture described above. One may worry that the energy density will continue todecrease infinitely. This, we expect, does not happen, since the case ρ < γ/AL (cid:38)
1. In a more interesting case γ/AL (cid:28)
1, however, the solution is thesubject of instabilities, which can be either due to some numerical artefacts or the presenceof caustic singularities. A study of this part of the parameter space is currently under way.Given that this mechanism indeed works, one deals with a novel way of curing causticsingularities. Recall that the common lore is to modify the Euler equation. Namely, oneadds viscous terms to the r.h.s. of Eq. (4.1) as in adhesive gravitational models [57, 58], or“quantum” pressure models [59] (see also [60] for the recent studies). These attempts amountto parametrizing the velocity dispersion present in the particle DM case. Therefore, we mayexpect that the Modified Dust scenario leads to qualitatively new results.In particular, the higher derivative term could be promising for solving the other long-standing problem of the CDM: the cuspy profile of the DM halos. To show this, let usestimate the distance from the centre of the dwarf spheroidal galaxies, at which the sourceterm on the r.h.s. of Eq. (4.3) becomes relevant: | γ ∆( ∇ · v ) ||∇ ( ρ · v ) | ∼ γρr (cid:38) , We estimate the energy density of DM in the central regions of the galaxies as ρ DM ∼
10 GeV / cm . We use the estimate for the parameter γ ∼ − M , as it is the most relevantfor the solution of the missing satellites problem. Then, the inequality above gives r (cid:46)
100 pc . (4.7)This roughly corresponds to the scales, at which the density profiles obtained in simulationsbegin to disagree with the observational data [61]. At distances smaller than about 100 pc,we expect a substantial flow of energy away from the centre. Hence, one has a chance toreduce the mass of DM in the central region.We reiterate that all the conclusions made in this Section are preliminary. By no means,they should be viewed as the proof of absence of caustic singularities or interpreted as thesuccessful solution to the core-cusp problem. This remains to be shown by making use ofnumerical simulations. We leave this for a future work. In this paper we showed that the Modified Dust scenario allows to address several cosmo-logical puzzles. For rather long wavelengths and relatively low values of the parameter γ ,cosmological perturbations behave as in the CDM picture. Below some wavelength, the powerspectrum gets suppressed compared to the standard predictions. This could be relevant foralleviating the missing satellites problem. In the previous Section, we also showed that Mod-ified Dust has some appealing features in the Newtonian limit. That is, unlike the standarddust, it may avoid developing singularities at the caustics.In this regard, the scenario which we considered in the present paper is a good modelfor DM. Before making that strong statement, however, several important issues must beaddressed: 15 Numerical simulations must be performed in the non-linear regime. In particular, itwould be interesting to see if caustic singularities are indeed absent. In the case of apositive answer, one can ask more sophisticated questions. For example, what is thestructure of DM halos formed by Modified Dust? • Lyman- α forest data is a powerful tool for descriminating between different DM frame-works. Therefore, it is important to test the predictions of the Modified Dust scenariousing these data, and possibly to deduce the constraints on the parameter γ . • Generically, suppression of sub-galactic scale structures implies the delayed formationof the first stars compared to CDM predictions. This may lead to some tension withthe CMB data that favors an early reionization of the Universe. The situation lookedhopeless with the first release of the WMAP data [62, 63], which reported large redshiftsvalues z re corresponding to the half-reionized Universe [64]: 11 < z re <
30 at 95%C.L.. However, the best-fit value of the redshift z re essentially decreased with the laterreleases of WMAP data and Planck data [1, 2]. Hence, one has a chance to avoidstringent constraints on the parameter γ . • Is the model under study consistent with the Bullet Cluster observations? This issueconcerns the non-linear dynamics of Modified Dust and thus remains obscure at themoment.From the theoretical point of view, there are the following issues: • The unified description of the dark matter and dark energy. Recall that this has beenthe original motivation of the Σ ϕ -fluid. One can try to address this issue following theguidelines of the paper [33]. • It would be certainly worth to search for a fundamental theory underlying ModifiedDust. In particular, higher derivative terms are naturally viewed as the part of an effec-tive theory associated with some broken global symmetry, with ϕ being the Goldstonefield. On the other hand, this analogy with the effective theory is complicated by thepresence of the constraint (2.1).Once a solution to these problems is found, it would be interesting to search for signa-tures of the Modified Dust scenario in the observational data. Note added.
At the final stage of this project, we became aware of the related work carriedout by L. Mirzagholi and A. Vikman. The discussion of Ref. [65] made available recentlyessentially extends that of the present paper.
Acknowledgments
We thank Dmitry Gorbunov, Michael Gustafsson, Mikhail Ivanov, Andrey Khmelnitsky,Maxim Pshirkov, Tiziana Scarna, Sergey Sibiryakov, Peter Tinyakov and Alexander Vikmanfor many useful comments and fruitful discussions. We are indebted to A. Vikman for sharingseveral ideas of the work [65] prior to its publication. This work is supported by the WienerAnspach Foundation (F. C.) and Belgian Science Policy IAP VII/37 (S. R.).16
Term γ ( (cid:3) ϕ ) In this Appendix, we write the system of cosmological equations for the Universe filled inwith Modified Dust and radiation. As in the bulk of the paper, we assume the parameter γ = 0. A.1 Background level
We start with background cosmological equations. In the presence of radiation and ModifiedDust, the Friedmann equation takes the form3 (cid:2) (1 − π G γ ) H + 8 π G γ H (cid:48) (cid:3) = 8 π G a ( ρ DM + ρ r ) . (A.1)Here ρ r is the energy density of radiation; we assume the simple identification 2Σ = ρ DM asin the bulk of the paper. The ( ij )-component of the Einstein’s equation is given by − − π G γ ) (cid:0) H (cid:48) + H (cid:1) = 8 π G a ρ r . (A.2)We supplement the system with conservation equations for radiation ρ (cid:48) r + 4 H ρ r = 0 (A.3)and Modified Dust a ρ (cid:48) DM + 3 H a ρ DM + 3 γ (cid:0) H + HH (cid:48) − H (cid:48)(cid:48) (cid:1) = 0 . (A.4)Taking present values for the energy density of radiation and DM, one can easily solve thissystem. In particular, neglecting the contribution for radiation, we get back to the systemof equations given in Section 2. A.2 Linear level
The simplest equation at the linear level follows from the constraint g µν ∂ µ ϕ∂ ν ϕ = 1, which,we remind, remains unmodified upon the inclusion of γ -terms. Namely, it is given by Eq. (2.4).We repeat it here for the sake of completeness, δϕ (cid:48) = a Φ . (A.5)The (00)-component of the Einstein’s equations reads in Fourier space, − k Φ (1 − πγ G) − (cid:0) H + 4 πγ G (cid:2) H (cid:48) − H (cid:3)(cid:1) − H Φ (cid:48) (1 − πγ G) − πγ GΦ (cid:48)(cid:48) = 4 π G (cid:16) a ρ r δ r + a ρ DM δ DM + H γ a k δϕ (cid:17) , (A.6)where δ r is the perturbation for the radiation energy density. The (0 i )-component is givenby: − − π G γ ) δϕ (cid:48)(cid:48) + 12 π G (cid:2) γ (cid:0) H − H (cid:48) (cid:1) + a ρ DM − γ k (cid:3) δϕ = − π G a ρ r v r , (A.7)where v r is the scalar velocity potential for radiation; here we exploited Eq. (A.5) to expressthe gravitational potential Φ through δϕ . Neglecting the energy density of the radiation in17he last equation, we get back to Eq. (2.9) from the main body of the paper with the soundspeed c s given by c s = 4 πGγ − πγ G . (A.8)The ( ij )-component of Einstein equations is given by3(1 − π G γ ) (cid:8) ( H − H (cid:48) ) δϕ (cid:48) − H δϕ (cid:48)(cid:48) − δϕ (cid:48)(cid:48)(cid:48) (cid:9) − π G γ k (cid:0) δϕ (cid:48) + H ϕ (cid:1) = − π G a δρ r . Note that the non-diagonal part of the same equation equals to zero identically. This, we re-mind, is a property of the γ -term, which preserves the simple form of the energy-momentumtensor δT ij ∝ δ ij . This similarity with the case of the perfect dust breaks down at the level ofthe γ -term, as we explain in details in the following Appendix.The system supplemented with the conservation equation for Modified Dust (cid:0) a δρ DM (cid:1) (cid:48) − γ (cid:0) H (cid:48)(cid:48) − HH (cid:48) − H (cid:1) δϕ (cid:48) + ρ DM k a δϕ = 3Φ (cid:48) a ρ DM + a γ (cid:0) (cid:3) ϕ (cid:1)(cid:12)(cid:12) linear (A.9)where (cid:0) (cid:3) ϕ (cid:1)(cid:12)(cid:12) linear = = a [( k − H (cid:48) k ) δϕ + (18 HH (cid:48) − H (cid:48)(cid:48) − H − H k ) δϕ (cid:48) +(12 H − H (cid:48) − k ) δϕ (cid:48)(cid:48) + 6 H δϕ (cid:48)(cid:48)(cid:48) − δϕ (cid:48)(cid:48)(cid:48)(cid:48) ] (A.10)and the standard conservation equation for the radiation can be solved numerically with theinitial conditions set deep in RD stage (see discussion in Section 3). B Term γ ∇ µ ∇ ν ϕ ∇ µ ∇ ν ϕ Now, let us consider the effects of the γ term on the cosmological evolution. That is, weset the parameter γ to zero. We restrict the discussion to the case of the matter dominatedUniverse filled in with Modified Dust. Namely, we neglect the contribution of the radiation. B.1 Background level
First, let us show that the background cosmological equations are the same as in the case of apressureless perfect fluid. This immediately follows from the ( ij ) component of the Einsteinequation, which has the standard form 2 H (cid:48) + H = 0. The Friedmann equation is given by3 (1 − π G γ ) H = 8 π G a ρ DM (B.1)and the conservation equation reads a ρ (cid:48) DM + 3 a H ρ DM + 3 γ H (cid:0) H (cid:48) + H (cid:1) = 0 . (B.2)We see that the quantity ρ DM ≡
2Σ drops as 1 /a with the scale factor. Hence, it can beidentified with the energy density of DM, as in the case of the γ -term.18 .2 Linear level Let us discuss the non-trivial effects for the linear cosmological perturbations due to thepresence of the γ -term. In that case, the simple relation δT ij ∝ δ ij does not hold anymore.Therefore, the gravitational potentials Φ and Ψ do not coincide. Still, the main conclu-sion of the paper holds: cosmological perturbations with sufficiently short wavelengths aresuppressed.To show this explicitly, we write down the (00) and ( ij )-components of the Einsteinequations,4 π G (cid:0) a ρ DM + 3 γ H − γ k (cid:1) δϕ − H (1 − π G γ ) δϕ (cid:48) − a (1 − π G γ ) Ψ (cid:48) = 0 , (B.3)and δ ji (cid:2) ak Ψ + 2 a (1 − π G γ )(2 H Ψ (cid:48) + Ψ (cid:48)(cid:48) ) + 4 H (cid:48) (1 − π G γ ) δϕ (cid:48) − k δϕ (cid:48) ++ 2 H (1 − π G γ ) δϕ (cid:48)(cid:48) ] + k i k j [8 π G γ H δϕ + (1 + 8 π G γ ) δϕ (cid:48) − a Ψ] = 0 , (B.4)respectively. The tensor structure of the last equation contains two parts: one proportionalto δ ij and the other proportional to k i k j . The latter gives the relation between the potentialsΦ and Ψ ( i (cid:54) = j case), a Ψ = 8 π G γ H δϕ + (1 + 8 π G γ ) δϕ (cid:48) . (B.5)Using this, we can express the derivatives of the potential Ψ in terms of the derivatives of thegravitational potential Φ and δϕ . Plugging the previous relation into the (00)-component ofthe Einstein’s equations, we finally obtain δϕ (cid:48)(cid:48) + (cid:18) π G γ π G γ − (8 π G γ ) k − H (cid:19) δϕ = 0 (B.6)Here, we made use of the background equation (B.1). We see that in the limit γ (cid:28) M , itreduces to Eq. (2.9) studied in the main body of the paper. We conclude that all the resultsstudied for the case of the γ -term in Section 2 are true for the case of the γ -term. References [1] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta,M. Halpern, R. S. Hill, N. Odegard, et al., Astrophys. J., Suppl. Ser. , 19 (2013), .[2] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud,M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, et al., ArXive-prints (2013), .[3] P. Ade et al. (Planck Collaboration) (2013), .[4] W. J. Percival, C. M. Baugh, J. Bland-Hawthorn, T. Bridges, R. Cannon, S. Cole, M. Colless,C. Collins, W. Couch, G. Dalton, et al., Mon. Not. R. Astron. Soc. , 1297 (2001), arXiv:astro-ph/0105252 .[5] D. J. Eisenstein et al. (SDSS Collaboration), Astrophys.J. , 560 (2005), astro-ph/0501171 .[6] M. Tegmark, M. R. Blanton, M. A. Strauss, F. Hoyle, D. Schlegel, R. Scoccimarro, M. S.Vogeley, D. H. Weinberg, I. Zehavi, A. Berlind, et al., Astrophys. J. , 702 (2004), arXiv:astro-ph/0310725 .
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