Physics Letters B 738 (2014) 418–423

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Physics Letters B

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Small scale structures in coupled scalar field dark ∗ J. Beyer , C. Wetterich

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany a r t i c l e i n f o a b s t r a c t

Article history: We investigate structure formation for ultra-light scalar field coupled to quintessence, in Received 10 July 2014 particular the cosmon–bolon system. The linear power spectrum is computed by a numerical solution of Received in revised form 10 September the coupled field equations. We infer the substructure abundance within a Milky Way-like halo. Estimates 2014 of dark halo abundances from recent galaxy surveys imply a lower bound on the bolon mass of about Accepted 6 October 2014 − 9 ×10 22 eV. This seems to exclude a possible detection of scalar field dark matter through time variation Available online 13 October 2014 Editor: M. Trodden in pulsar timing signals in the near future. © 2014 Published by Elsevier B.V. This is an open access article under the CC BY license Keywords: (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. Scalar field dark matter Coupled Small scale structure

1. Introduction in the Milky way galaxy (the missing satellite problem [7,8]) and the cusp-like density profiles of halos which could be inconsistent with The cosmological standard model (or CDM model) has pro- observed velocity dispersions both in galaxies (the cusp-core prob- vided a solid foundation for modern cosmology for a number of lem [9,10]) and dwarf-galaxies (the too big to fail problem [11–13]). years by now. Still, the nature of two of its key components, dark While some of these issues, in particular the missing satellite prob- matter and dark energy, remains unknown. So far both these com- lem, might just be a result of our lack of understanding of the ponents have eluded direct detection and can be seen only through baryonic physics of galaxy formation [14–17], they may still be a gravitational effects. hint towards possible modifications of dark sector physics. In the CDM scenario the dark sector consists of a pressure- Amongst the many proposals that have emerged to solve these less fluid modeling and a cosmological constant issues, (WDM) is probably the most popular making up dark energy. While the particle physics models gener- one. If the dark matter particle is comparatively light (of the or- ating a cold dark matter component are plentiful, the value of the der of 1–4 keV) and is produced thermally in the early universe, it cosmological constant Λ is so tiny that is seems to contradict com- has a non-negligible velocity dispersion, thus suppressing the for- mon expectations from quantum field theory. This is known as the mation of structure on the relevant scales. This can solve some of cosmological constant problem, which has prompted many investiga- the small scale problems of CDM individually, as has been shown tions over the past years. A possible solution to this problem lies in several recent works [18–21]. However, constructing a consis- in dynamical theories of dark energy, most notably quintessence tent model obeying all current observational constraints seems to models, where a cosmological scalar field is used to describe dark be more difficult. In Ref. [22] Schneider et al. argued that a WDM energy [1–6]. model consistent with all current observational constraints does Despite its relative simplicity (and potential theoretical issues) not provide a significant improvement over cold dark matter pre- the cosmological standard model has been very successful in ex- dictions on small scales, at least not in the case of the simplest plaining the vast majority of observed cosmological phenomena. models of a single, thermally produced dark matter particle. One Several predictions of the CDM scenario for structure formation may therefore have to resort to more complicated scenarios of on small scales, however, have been claimed to be in conflict with warm dark matter generation, or look for alternatives elsewhere. increasingly precise cosmological observations. Most notable are Recently, we have proposed a unified picture of the dark sec- probably the apparent predicted overabundance of dwarf galaxies tor, in which both dark energy and dark matter are modeled by scalar fields which couple through their common potential [23]. The mass of the scalar field responsible for dark matter was found * Corresponding author. to be somewhat larger than the inverse size of galaxies. In the E-mail address: [email protected] (J. Beyer). present letter we show that the effects on structure formation are http://dx.doi.org/10.1016/j.physletb.2014.10.012 0370-2693/© 2014 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. J. Beyer, C. Wetterich / Physics Letters B 738 (2014) 418–423 419

similar to WDM, thus establishing such a model as an interest- matter.) Typical values for χ0 are somewhat below the reduced ing alternative explanation if small scale structures should indeed Planck mass M. The bolon evolution towards the end of the radi- turn out to behave differently from the CDM expectations. Our ation dominated epoch, as well as for the subsequent epochs, is model belongs to a general class of scalar field dark matter mod- therefore governed by three parameters, m0, χ0 (or respectively λ) els which have been investigated in various incarnations [24–29]. and β. One parameter has to be adapted in order to obtain the Besides its phenomenological interest it has the benefit of address- correct present matter density. For χ0 ≈ M one finds a typical ing the cosmological constant problem, as we will briefly discuss present bolon mass of the order of the inverse galactic radius, at the end. Furthermore, it provides for a natural explanation of a while smaller χ0 yield somewhat larger masses [23],   possible coupling between dark energy and dark matter (“coupled 4 − χ0 quintessence” [5,30]) which is often postulated somewhat ad hoc. m 1 ≈ 10 kpc. (7) χ M 2. Class of models Modifications of the CDM scenario on subgalactic scales therefore arise rather naturally in our setting. We consider two scalar fields ϕ and χ with canonical kinetic terms and a common potential of the form 3. Linear perturbations − V (ϕ, χ) = V (ϕ) + e 2βϕ/M V (χ), (1) 1 2 The evolution of linear perturbations around such a cosmolog- with M = 2.44 × 1018 GeV the reduced Planck mass. We adopt the ical solution is rather intricate, as the oscillations present in the common name cosmon for the quintessence field ϕ. The field χ background interfere with oscillations in the perturbative quan- is responsible for dark matter and dubbed bolon, following earlier tities. We have addressed this issue by means of an analytical work [23]. The potential (1) has been motivated by an investigation time-averaging procedure, which connects the scalar field descrip- of possible consequences of approximate scale symmetry in higher tion for H  mχ with an effective fluid description for H  mχ . dimensions [31–33]. The dimensionless parameter β will turn out For this purpose we start by expanding all dynamical k-space to be the effective coupling between dark energy and dark matter. quantities (for both the background and the perturbations) in a =  The potential V 1 can in principle be any quintessence potential. Taylor series in μ H/mχ 1, using the scalar field perturbations For definiteness we use here an exponential potential directly. In a second step we then expand the Taylor coefficients at each order in μ in a Fourier series for multiples of a ‘base fre- 4 −αϕ/M = V 1 = M e . (2) quency’ x mχ dt. The coefficients of this expansion evolve adi- abatically (i.e. are almost constant) during one oscillation period. On the other hand, V 2 is restricted to an effectively quadratic Plugging the resulting ‘double-expansion’ into the linear perturba- shape at least in the late universe, where the field χ is supposed tion equations and comparing coefficients tells us which Fourier to act like dark matter [24]. During the early stages of the cosmic frequencies are present at which order in μ for each dynamical evolution, V 2 can look very different indeed, and in fact a much quantity. From the k-modes of the scalar field perturbations we steeper potential may be natural and desirable to ensure both in- construct the k-modes of the perturbed energy momentum ten- sensitivity of the cosmic evolution on the precise initial conditions sor. Finally we map the scalar energy momentum tensor of the and stability of the adiabatic perturbation mode, which can be an bolon to the perfect fluid form, resulting in first order equations issue in such coupled models [34,35]. A shape very suitable for our for the bolon density contrast and its velocity potential. After in- purposes is the one proposed in Ref. [28] serting the combined Taylor–Fourier expansion we can integrate   over one oscillation period and recover a system of averaged effec- = 2 4 − V 2(χ) c M cosh(λχ/M) 1 , (3) tive equations. which effectively matches an exponential to a quadratic potential The details of these calculation go beyond the scope of this let- and satisfies both criteria. ter, we refer the interested reader to Ref. [36]. This procedure can in principle be applied to any order in , but it is not a priori During the later stages of its evolution, when V 2 is effectively μ quadratic, χ follows a Klein–Gordon equation clear that the ansatz works, i.e. that one can actually find a set of Fourier expansions that solve the linear perturbation equations to D Dμ + 2 = each order. To second order in , however, it works, and we have μ χ mχ (ϕ)χ 0, (4) μ performed this calculation, which results in an effective description complicated by the fact that the mass is time-dependent as ϕ in- which could have been guessed from the results of earlier works creases with t, [25,28]: The averaged bolon perturbations behave like cold dark matter coupled to quintessence, but with a small (non-adiabatic) = −βϕ/M = mχ (ϕ) m0e , m0 cMλ. (5) sound-speed present for large wavenumbers, which is given by

In a FLRW cosmological setting under rather generic assump- k2 tions, χ will oscillate quickly around its potential minimum once c2 = . (8) s,χ 4m2 (ϕ)a2 mχ  H. The energy density scales as coupled dark matter χ This sound-speed effectively suppresses the growth of modes with ∝ −3 −βϕ/M ρχ a e . (6) wavenumbers The pressure pχ on the other hand is highly oscillatory, but k2 vanishes when averaged over a suitable timescale. At the back-  1, (9) a2 Hm (ϕ) ground level such a coupled scalar field model results in a coupled χ quintessence cosmology [5,30] for late enough times. which behave similar to photons. Here H denotes the Hubble pa- For a wide range of initial conditions the value of χ at the rameter. later stages of the radiation dominated epoch depends only on the For a given k an effective growth sets in the matter dominated parameter λ (and weakly on β), χ = χ0(λ, β), but not on the de- epoch only once the scale factor a exceeds k/ Hmχ (ϕ). The small- tails of the initial conditions [23]. (This differs from dark est wavenumber which is suppressed as compared to CDM can be 420 J. Beyer, C. Wetterich / Physics Letters B 738 (2014) 418–423

2 2 roughly estimated by evaluating k = a Hmχ (ϕ) at the time when oscillations start (a similar scenario is described in Ref. [28]). The corresponding Jeans wavenumber can be used to define a Jeans- mass, which is given by

 ∗  2 −2βϕ /M −3/4 4π m e ρr,0 M = 0 ρ , (10) J 2 − ∗ − ∗ χ,0 3 3M (1 Ωϕ Ωχ ) ∗ ∗ ∗ where ϕ , Ωϕ and Ωχ are evaluated when oscillations start. (Sometimes a cutoff mass is defined by the wavenumber at which the linear power spectrum is suppressed by a factor of 2 compared to CDM. We choose to use a different notation for this cutoff mass, for which we find numerically Mc ≈ 3.3 × M J , which is very simi- lar to the result for WDM models discussed in [37].) We have computed the perturbation equations for the coupled cosmon–bolon system and solved them numerically, also including Fig. 1. Power spectrum for a cosmon–bolon cosmology. We show results for differ- photons, neutrinos and . Our code is a typical Boltzmann- ent values of the coupling β. The solid green line stands for a bolon power spectrum with β = 0and λ = 65, the dashed (blue) and dotted (orange) lines are obtained for code which uses manifestly gauge-invariant quantities and em- the same λ but with β = 0.05 and β =−0.1 respectively. These curves represent − ploys the same approximations as used in the CLASS-code [38, (in the order described) present inverse bolon masses of (4.9, 4.2, 6.4) × 10 3 pc − 39] for photons, neutrinos and baryons, adapted to fit the gauge- or (7.7, 6.5, 10) × 1020 eV 1. For comparison, the solid black line represents a CDM invariant setting. The treatment of the scalar field sector requires power spectrum and the dotted-dashed line (gray) a WDM modification for a ther- mally produced WDM particle of mass m = 2.284 keV. The three gridlines to the a reliable map between the description by an oscillatory scalar wdm right correspond to the inverse length scale one would assign to a virialized spher- field and the effective fluid description which is needed once the ical dark halo (with density contrast of δv = 200) with radii of 8, 10 and 12 kpc via = 3 −1/3 scalar oscillations become very rapid on the relevant time scales. kr (δv r ) . The single gridline on the left corresponds to a radius of 100 kpc, We use the exact equations for the scalar field perturbations for roughly the size of the Milky way . (For interpretation of the refer- ences to color in this figure legend, the reader is referred to the web version of this both the bolon and the cosmon in the early universe and switch article.) to the effective fluid description for the bolon given above for later times. The initial values for the fluid description are obtained by 4. Small scale structures explicit numerical integration over one oscillation period in the field description. In case of the cosmon we have extended the ap- We are interested in how this kind of power-spectrum mod- proximation called radiation streaming approximation in Ref. [38] to ification changes structure formation on small scales. A full in- the quintessence field for sufficiently late times. vestigation of structure formation would require high resolution We have checked the accuracy of the effective fluid description runs of N-body codes (see Ref. [43] for first successes of such runs by varying the time of transition from the field to the fluid de- in a similar model). They would necessarily be based on effective scription. We have confirmed that the matter-power spectrum in a (averaged) equations for the full non-linear perturbations. A more CDM cosmology obtained from our code agrees with the CLASS- direct (and numerically much less involved) approach can be found result to excellent accuracy. More details can be found in Ref. [36], by employing the extended Press–Schechter excursion set formal- where we present the analytical averaging procedure leading to ism (ePS) [44–46]. In this scenario hierarchical structure formation the effective fluid description in detail and discuss the numerical is modeled as a random walk of trajectories in density contrast treatment further. Our results for the power spectrum of adiabatic space δ(S), where S denotes the variance calculated from the lin- scalar fluctuations are shown in Fig. 1 for different values of β and ear power spectrum. The formation of a halo in this scenario is fixed λ. represented by the absorption of a trajectory by an absorbing bar- As is common in models of scalar field dark matter, the cutoff rier. The shape of the barrier has originally been assumed to be in the power spectrum is somewhat sharper than in WDM-models. a constant, derived from the spherical collapse model. For some Notably, a positive coupling β leads to a shift of the cutoff to larger years now it has been known however, that a barrier modification wavenumbers. This effect originates from our adjustment the cur- motivated by an elliptical collapse is necessary to match the re- 2 2 rent bolon energy density ρχ,0 to the fixed value 3M H0Ωc,0, sults of high-resolution N-body simulations [47,48]. Furthermore, which effectively increases the bolon mass at the time when os- in warm dark matter models, Barkana et al. showed that the bar- −βϕ/M cillations start for positive β despite the e -dependence. rier obtained from spherical collapse needs to be adjusted [37]. In addition to the position of the power spectrum cutoff, the The correct barrier exhibits a sharp upturn near the Jeans mass, coupling between the cosmon and the bolon leads to a number a fitting formula is given by Eq. (7) in [49]. Whether a similar up- of effects which are present in other coupled CDM models as well turn in the barrier is present in coupled scalar field models of dark and not connected to the scalar nature of bolon dark matter. The matter as well remains an open question for now, but it was pro- coupling β induces a ‘fifth force’ acting on dark matter, which can posed to be for a model of mixed axion dark matter in Ref. [50], be interpreted as a changed effective Newton’s constant Geff for which bears some resemblance to ours. We will merely illustrate 2 the bolon: Geff/G = 1 + 2β [5,40,41]. Since there is no coupling to the results such an effect might have on the predicted number of baryons, this could be seen as a violation of the equivalence prin- Milky Way subhalos. ciple between dark matter and baryons. Furthermore it leads to a The ePS-formalism does not directly yield predictions for num- change in the linear growth function on subhorizon scales [40,41], bers of subhalos within a given halo. It does however provide the which affects halo number counts [42]. This effect, as well as mod- conditional mass function f (M1, ω(δsc,1, S)|M2, ω(δsc,2, S)), which ifications of the growth rate by early dark energy is fully included describes the fraction of mass of a halo of mass M2 at in our numerical code. In Fig. 1 this effect leads to different high k z2 corresponding to the barrier ω(δsc,2, S), which was contained slopes of the power spectrum for different couplings. We keep the in halos of mass M1 < M2 at z1 > z2 corresponding to the bar- cosmon perturbations explicitly at all times. rier ω(δsc,1, S) > ω(δsc,2, S). As is common in ePS-analyses, we do J. Beyer, C. Wetterich / Physics Letters B 738 (2014) 418–423 421

Fig. 3. Allowed parameter range for the cosmon–bolon model. The colored contours Fig. 2. Cumulative number of Milky Way subhaloes as a function of halo mass Mh . −22 show different current bolon masses mχ (t0) in units of 10 eV. The solid red The solid black line represents a CDM power spectrum and the gray lines a WDM line displays the boundary of parameters which yield more than 66 subhalos in the modification for a thermally produced WDM particle of mass m = 2.284 keV. wdm Milky way if the modified barrier is used, the dashed red line shows the same ex- The green lines stands for a bolon power spectrum with β = 0andλ = 65, the clusion curve for the standard elliptical barrier. (For interpretation of the references blue and orange lines obtain for the same λ but with β = 0.05 and β =−0.1re- to color in this figure legend, the reader is referred to the web version of this arti- spectively. For all WDM and bolon models, the dashed lines are results calculated cle.) without an upturn of the barrier near the Jeans mass, whereas we included such an upturn for the solid lines. As an additional orientation we added the dashed–dotted gridline at the WDM Jeans mass. (For interpretation of the references to color in the elliptical barrier modification has been calculated in this way, this figure legend, the reader is referred to the web version of this article.) too, and deviating from it would introduce additional uncertain- ties. not modify the power-spectrum when going to higher , The resulting cumulative number counts for several models are 12 but put all the time-dependence in the spherical collapse barrier shown in Fig. 2 for a halo of mass M2 = 1.8 × 10 M at z = 0. instead, i.e. To gauge the accuracy of our calculations, one should compare the WDM-model with the results given in Fig. 11 in Ref. [52]. Up to = δsc(z) δsc,0/D(z), (11) masses slightly above the Jeans mass our scenario seems to fit where D(z) is the linear growth function. Furthermore we denote the N-body results rather well, but for smaller masses our curve the elliptical barrier adjustment by the function stagnates whereas the N-body results continue to rise for a while    longer. We seem to underestimate the asymptotic total number c √ S of subhalos by a factor of about 2. The reasons for this could be ω(δ , S) = Aδ 1 + b , (12) sc sc 2 twofold: First, structure formation is not strictly hierarchical, there Aδsc are violent mergers and disruption processes present in N-body = = = with A 0.707, b 0.5 and c 0.6. simulations, which cannot be represented in the strictly hierarchi- Following [51], we now calculate the current number of subha- cal ePS-scenario. Such processes can generate halos even below the los by a simple integration in barrier space, i.e. Jeans mass which cannot form hierarchically. This might explain ∞ the underprediction of low mass halos we see in our approach.   dn M2 Second, this is precisely the regime where spurious halos start to = f M1, ω(δsc,1) M2, ω(δsc,2) dδsc,1. (13) dm m play a role in N-body codes, and uncertainties may arise in the δ0 identification of such halos. When employing this procedure one loses the overall normaliza- One can use these results to put constraints on the allowed tion, and we have to normalize the resulting number counts to parameter range for our coupled cosmon–bolon model, simply by N-body simulations. We used the CDM simulation in [52] to adjust demanding that the number of subhalos should not fall below the the CDM curve, and employed the resulting normalization for all number of dwarf galaxies estimated from observations. Estimates other models. for this number range from 66 [52] to several hundred [58]. For The calculation of these first crossing rates needs to be done a bound on the bolon mass we choose the lower value of 66. We numerically for such complicated barriers, we used the recipe de- want to point out that galaxy formation for such small masses ap- scribed in the appendix of [49]. This approach treats the random pears to be a highly stochastic process [59], potentially leaving a walks as Markovian (i.e. uncorrelated), an assumption which is large number of halos void of stars, and the bounds we set here strictly speaking only true if one uses a sharp-k filter to obtain are therefore very conservative. the variance function S(M). However, as is well known, there is The masses of the ultra-faint dwarf-galaxies appear to be uni- no unique way to assign a mass to a filtering radius for this choice versally around 107 M , which is where we set our cut. At these of filter. As a result, huge uncertainties get introduced when one masses, our method already underestimates the WDM N-body re- choses this filter, as we discuss in some detail in Ref. [36]. We sults already by a factor of roughly 1.5(we have checked this for therefore stick to a spatial tophat-filter, where this issue is not all four models given in [52]), so we artificially raise our obtained present. The price to pay is that the assumption of an uncorrelated number counts by this factor when we use the modified spherical random walk is incorrect in this case. One could calculate cor- collapse barrier in order to remain extra cautious. The results can rections resulting from the non-Markovian nature of the random be seen in Fig. 3. walks [53–57], but based on the results of these investigations we Clearly larger couplings β allow for smaller current bolon expect them to be rather small compared to other effects neglected masses. Allowed values for the coupling strength are however con- in this approach and do not do so here. Furthermore, the fitting of strained by CMB observations [60,61] to roughly |β|  0.1. From 422 J. Beyer, C. Wetterich / Physics Letters B 738 (2014) 418–423 this constraint we can derive an upper bound for current bolon [22]. At first glance, our model predicts a scenario similar to WDM mass, which we estimate by evaluating the boundaries presented models. The onset of suppression of the power spectrum and the in Fig. 3 for β = 0.05: associated predicted number counts are almost the same for our model and WDM for a suitable choice of parameters. This similar- −22 mχ (t0)  9.2(4.1) × 10 eV (14) ity between scalar field dark matter and WDM at the linear level needs not extend to the non-linear evolution of perturbations. It for the modified (ellitpical) barrier. The typical scale at which we is as of yet unclear how effective non-linear equations would look expect the formation of structures to be suppressed are however ∗ like for our model, and bounds derived for WDM from the Lyman- linked to the mass at a and not today. Any mass can be related to alpha forest [62] might have to be recalculated for our scenario. a length scale via If one looks at the internal structure of dark matter halos, the   −1 bolon model looks rather different from WDM. The scalar field os- −1 mχ m ≈ 0.64 × pc. (15) cillations are expected to translate to the gravitational potential χ 10−23 eV in non-linear structures (similar to oscillations or boson stars [63, Our bound lies within the typical range of ultra-light scalar field 64]), and such effects could in principle be detected. In a recent dark matter masses, but at the larger end, with important conse- study Khmelnitsky and Rubakov investigated which mass range of quences for observational signatures. a dark matter scalar field could lead to detection of such a signal through time variation in pulsar signals [65]. Our considerations 5. Discussion of structure formation exclude this mass range by more than one order of magnitude (compare with Fig. 1 in Ref. [65]). Even with In summary, we have presented an analysis of structure for- conservative estimates, the mass of the scalar dark matter particle mation in the cosmon–bolon scenario of scalar field dark matter is probably too large to detect the scalar field oscillations in the coupled to quintessence. We find that our scenario constitutes a foreseeable future with pulsar timing signals. valid alternative to standard cold dark matter. An interesting mo- tivation of our model originates from higher dimensional theories Acknowledgements of gravity. 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