Physics Letters B 738 (2014) 418–423 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Small scale structures in coupled scalar field dark matter ∗ 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 dark matter 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 dark energy 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 cold dark matter and a cosmological constant issues, warm dark matter (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- V ( ) = c2 M4 cosh(λ /M) − 1 , (3) 2 χ χ 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.