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Ultra Light Bosonic Dark Matter and Cosmic Microwave Background The Astrophysical Journal, 721:1509–1514, 2010 October 1 doi:10.1088/0004-637X/721/2/1509 C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A. ULTRA LIGHT BOSONIC DARK MATTER AND COSMIC MICROWAVE BACKGROUND Ivan´ Rodr´ıguez-Montoya1,3, Juan Magana˜ 2,3, Tonatiuh Matos1,3, and Abdel Perez-Lorenzana´ 1,3 1 Departamento de F´ısica, Centro de Investigacion´ y de Estudios Avanzados del IPN, A.P. 14-740, 07000 Mexico City, Mexico; rrodriguez@fis.cinvestav.mx, tmatos@fis.cinvestav.mx, aplorenz@fis.cinvestav.mx 2 Instituto de Astronom´ıa, Universidad Nacional Autonoma´ de Mexico,´ Ciudad Universitaria, 04510 Mexico City, Mexico; [email protected] Received 2009 October 7; accepted 2010 July 22; published 2010 September 10 ABSTRACT In this paper, we consider the hypothesis in which a species of ultra light bosonic dark matter (ULBDM) −22 with mass mB ∼ 10 eV could be the dominant dark matter (DM) in the universe. As a first approach we work in the context of kinetic theory, where ULBDM is described by the phase space distribution function whose dynamics is dictated by the Boltzmann–Einstein equations. We investigate the effects that this kind of DM imprints in the acoustic peaks of the cosmic microwave background. We find that the effect of the Bose–Einstein statistics is small, albeit perceptible, and is equivalent to an increase of non- relativistic matter. It is stressed that in this approach, the mass-to-temperature ratio necessary for ULBDM to be a plausible DM candidate is about five orders of magnitude. We show that reionization is also necessary and we address a range of consistent values for this model. We find that the temperature of ULBDM is below the critical value implying that Bose–Einstein condensation is inherent to the ULBDM paradigm. Key words: cosmic background radiation – cosmology: theory – dark matter Online-only material: color figures 1. INTRODUCTION showed that the density profiles for SFDM halos are non-cuspy profiles, in accordance with the observations of LSB galaxies One of the most precise cosmological observations is the (see also Bohmer¨ & Harko 2007; Matos et al. 2009). More- measurement of the anisotropies in the cosmic microwave back- over, it is noticeable that in the relativistic regime scalar fields ground (CMB). The experimental data are useful for probing can form gravitationally-bound structures. These are called the dynamics and properties of many theoretical cosmological boson stars for complex scalar fields (Ruffini & Bonazzola models. Nowadays, the most successful model describing the 1969; Lee & Koh 1996; Guzman´ 2006), and oscillations for observed profiles of CMB anisotropies is the so-called cold dark real scalar fields (Seidel & Suen 1991;Urena-L˜ opez´ 2002; matter with a cosmological constant (ΛCDM). Nevertheless, the Alcubierre et al. 2003). There are also scalar field stable gravita- cold dark matter (CDM) model has some inconsistencies with tional structures described by the Schrodinger–Poisson¨ system observations on galactic and sub-galactic scales. For instance, (Guzman´ & Urena-L˜ opez´ 2003, 2006; Bernal & Guzman´ 2006). CDM predicts cusp central density profiles of dark halos in low One of the most promising and physically interesting features surface brightness (LSB) and dwarf galaxies; meanwhile, the of SFDM resides on the hypothesis that it describes cosmolog- measurements indicate a smooth distribution of matter. Also, ical Bose–Einstein condensates (BEC; see, for example, Woo CDM has some discrepancies between the number of predicted & Chiueh 2009;Urena˜ 2009). For that reason it is important satellite galaxies in high-resolution N-body simulations and ob- to provide a thermodynamic understanding of scalar particles, servations. In this sense, the possibility of alternative hypotheses putting aside for the moment the classical field description. on the nature of dark matter (DM) is open. In the SFDM model, the mass is constrained by phenomenol- In recent years, it has been argued that a real scalar field Φ, ogy to an extremely low value (∼10−23 eV). This ultra light minimally coupled to gravity, could be a plausible candidate scalar field mass fits the observed amount of substructure (Matos for DM. This alternative proposal (or similar ideas) is called &Urena˜ 2001), the critical mass of galaxies (Alcubierre et al. scalar field dark matter (SFDM; Ji & Sin 1994;Sin1994;Lee& 2003), the rotation curves of galaxies (Bohmer¨ & Harko 2007), Koh 1996;Huetal.2000; Matos & Guzman´ 2000; Matos et al. the central density profile of LSB galaxies (Bernal et al. 2008), 2000; Sahni & Wang 2000; Matos & Urena˜ 2001;Lee2009; the evolution of the cosmological densities (Matos et al. 2009), Garcia & Matos 2009). Several previous works have shown that etc. Furthermore, SFDM forms galaxies earlier than CDM; thus, a scalar field is able to reproduce the cosmological evolution of if SFDM is correct, we expect to see big galaxies at high red- the universe. To this end, the scalar field is endowed with a scalar shifts. potential V (Φ) of the form cosh(Φ)orΦ2 and obeys an equation If this scalar field could be considered as a system of indi- of state ωΦ ≡ pΦ/ρΦ that varies in time (−1 ωΦ 1; see, vidual light bosonic particles (with zero spin) and, moreover, if for example, Matos & Urena˜ 2001; Matos et al. 2009). Matos & there are some of these scalar particles in thermal equilibrium Urena˜ (2001) found that the SFDM model predicts a suppression forming an ideal gas, then they should obey the Bose–Einstein on the mass power spectrum for small scales. Thus, SFDM could statistics. From this perspective, ultra light bosonic dark mat- help to explain the excess of satellite galaxies. ter (ULBDM) seems to have some properties close to those of The SFDM paradigm has also been tested on galactic scales, neutrinos. In fact, neutrinos constitute a subdominant compo- showing interesting results. For instance, Bernal et al. (2008) nent of DM in the universe. For this reason, it is interesting to mention some of the most remarkable features of the neutrino 3 Part of the Instituto Avanzado de Cosmolog´ıa (IAC) collaboration http://www.iac.edu.mx/. cosmology. 1509 1510 RODRIGUEZ-MONTOYA´ ET AL. Vol. 721 At very early times of the universe, the neutrinos were in We stress that in our treatment ULBDM has a phase-space thermal equilibrium with the primeval fireball (see, for example, description, prescribed by the relativistic kinetic theory, i.e., the Dodelson 2003, p. 440). Due to its low mass compared with its evolution of ULBDM is dictated by the Boltzmann equation temperature in this epoch, they behaved exactly as radiation coupled to Einstein equations. This is a novel approach to the at the moment of its decoupling. This means that neutrinos scalar DM paradigm. Concretely, the object of treatment in our fall under the classification of hot dark matter (HDM).4 After scheme is neither a classical nor a quantum field, but rather the decoupling, neutrinos still keep the relativistic distribution, phase-space distribution function of an ideal gas of individual while they relax only with the expansion of the universe; this is noninteracting particles. The scalar particles are thought to be called the freeze out. Thus, the temperature of neutrinos evolves initially thermalized but decoupled from the rest of the universe. −1 simply as Tν ∝ a and eventually could reduce to values Even if a priori we do not restrict ourselves to the case in which lower than its mass. This epoch is known as the non-relativistic all the particles reside in a coherent phase, it is found that transition (NRT) of the neutrino. Since this epoch, gravitational Bose–Einstein condensation has a central role in the model. The attraction is sufficient to contribute to structure formation. BEC formation is assumed to take place before its decoupling Once decoupled and after electron–positron annihilations, the during the radiation epoch. The motivation to work in this temperature of neutrinos remains well determined in terms scheme is to explore the contribution to the CMB anisotropies 1/3 of the temperature of the photons as Tν = (4/11) Tγ ;this from possible thermal particles filling different energy states (0) ∼ −3 fixes the neutrino number density today at nν 100 cm .It in the ULBDM gas. This is precisely the reason why the name is now clear that if NRT occurs earlier, then neutrinos can form ULBDM rather than SFDM is more descriptive in this approach. more bounded structures and vice versa. However, in most of In the following, we consider a flat, homogenous, and the typical scenarios, this transition occurs too late, thus making isotropic universe. We take as fixed parameters the current the neutrino contribution subdominant (see an excellent review temperature of the CMB photons TCMB = 2.726 K, the current −1 −1 in Lesgourgues & Pastor 2006). Hubble’s constant H0 = 75.0kms Mpc , and the current Neutrinos and ULBDM are similar in that they are assumed baryon density parameter Ωbar = 0.04. Also, we assume, just for to be in thermal equilibrium but with negligible couplings with simplicity, that the dark energy in the universe is a cosmological other types of matter. The value of the mass also ensures that both constant Λ with a current density value ΩΛ = 0.74. We choose −5 decouple when still relativistic and also that their distribution units in which c = h¯ = kB = 1, then 1 K ≡ 8.617 × 10 eV. freezes out. They differ, however, in many aspects; an important This paper is organized as follows. Section 2 states the key intrinsic part of the nature of the neutrino is that it is a fermion equations of this calculation. First, we discuss the Bose–Einstein and therefore its density has an upper bound.
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