Cosmological constraints on Bose-Einstein-condensed scalar field dark matter

Bohua Li, Tanja Rindler-Daller, and Paul R. Shapiro

Phys. Rev. D 89, 083536 – Published 30 April 2014

Abstract

Despite the great successes of the cold dark matter (CDM) model in explaining a wide range of observations of the global evolution and the formation of galaxies and large-scale structure in the Universe, the origin and microscopic nature of dark matter is still unknown. The most common form of CDM considered to date is that of weakly interacting massive particles (WIMPs), but, so far, attempts to detect WIMPs directly or indirectly have not yet succeeded, and the allowed range of particle parameters has been significantly restricted. Some of the cosmological predictions for this kind of CDM are even in apparent conflict with observations (e.g., cuspy-cored halos and an overabundance of satellite dwarf galaxies). For these reasons, it is important to consider the consequences of different forms of CDM. We focus here on the hypothesis that the dark matter is comprised, instead, of ultralight bosons that form a Bose–Einstein condensate, described by a complex scalar field, for which particle number per unit comoving volume is conserved. We start from the Klein–Gordon and Einstein field equations to describe the evolution of the Friedmann–Robertson–Walker universe in the presence of this kind of dark matter. We find that, in addition to the radiation-, matter-, and $Λ$ -dominated phases familiar from the standard CDM model, there is an earlier phase of scalar-field domination, which is special to this model. In addition, while WIMP CDM is nonrelativistic at all times after it decouples, the equation of state of Bose–Einstein condensed scalar field dark matter (SFDM) is found to be relativistic at early times, evolving from stiff ( $\bar{p} = \bar{ρ}$ ) to radiationlike ( $\bar{p} = \bar{ρ} / 3$ ), before it becomes nonrelativistic and CDM-like at late times ( $\bar{p} = 0$ ). The timing of the transitions between these phases and regimes is shown to yield fundamental constraints on the SFDM model parameters, particle mass $m$ , and self-interaction coupling strength $λ$ . We show that SFDM is compatible with observations of the evolving background universe, by deriving the range of particle parameters required to match observations of the cosmic microwave background (CMB) and the abundances of the light elements produced by big bang nucleosynthesis (BBN), including $N_{eff}$ , the effective number of neutrino species, and the epoch of matter-radiation equality $z_{eq}$ . This yields $m \geq 2.4 \times 1 0^{- 21} eV / c^{2}$ and $9.5 \times 10^{- 9} {eV}^{- 1} {cm}^{3} \leq λ / ({mc}^{2})^{2} \leq 4 \times 10^{- 17} {eV}^{- 1} {cm}^{3}$ . Indeed, our model can accommodate current observations in which $N_{eff}$ is higher at the BBN epoch than at $z_{eq}$ , probed by the CMB, which is otherwise unexplained by the standard CDM model involving WIMPs. We also show that SFDM without self-interaction (also called “fuzzy dark matter”) is not able to comply with the current constraints from BBN within 68% confidence and is therefore disfavored.

Received 22 October 2013

DOI:https://doi.org/10.1103/PhysRevD.89.083536

Authors & Affiliations

Bohua Li^*, Tanja Rindler-Daller^†, and Paul R. Shapiro^‡

Department of Astronomy and Texas Cosmology Center, The University of Texas at Austin, 2515 Speedway C1400, Austin, Texas 78712, USA

^*bohuali@astro.as.utexas.edu
^†Present address: Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, MI 48109, USA. daller@umich.edu
^‡shapiro@astro.as.utexas.edu

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Vol. 89, Iss. 8 — 15 April 2014

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Figure 1
Left-hand plot: Evolution of the SFDM energy density ${\bar{ρ}}_{SFDM}$ vs scale factor $a$ . The SFDM parameters are $m = 3 \times 1 0^{- 21} eV / c^{2}$ and $λ / (m c^{2})^{2} = 2 \times 1 0^{- 18} {eV}^{- 1} {cm}^{3}$ (fiducial model). The vertical solid line depicts the epoch of matter-radiation equality $a_{eq}$ from Table 1, while the cross indicates the point after which SFDM is well described as fully nonrelativistic matter (CDM-like). Right-hand plot: Evolution of the equation of state $\bar{w} = {\bar{p}}_{SFDM} / {\bar{ρ}}_{SFDM}$ . The solid curve corresponds to the fiducial model plotted in the left panel. The other curves represent models with the same mass $m$ but different ratios of $λ / (m c^{2})^{2}$ in unit of ${eV}^{- 1} {cm}^{3}$ , as seen in the legend. The vertical dotted lines depict the epoch of neutron-proton freeze-out $a_{n/p}$ and the epoch of light-element production $a_{nuc}$ , respectively (see Sec. 4b). The larger the value of $λ / (m c^{2})^{2}$ , the longer lasts the radiationlike phase of SFDM; this provides constraints on this ratio from CMB observations of $a_{eq}$ and $N_{eff}$ during BBN; see Secs. 4a and 4b.
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Figure 2
Left-hand plot: Hubble parameter $H (a)$ vs scale factor $a$ for our fiducial SFDM model with $m = 3 \times 1 0^{- 21} eV / c^{2}$ and $λ / (m c^{2})^{2} = 2 \times 1 0^{- 18} {eV}^{- 1} {cm}^{3}$ . Right-hand plot: Evolution of the ratio of the oscillation angular frequency and Hubble parameter, $ω / H$ , for that same model. The vertical solid line depicts the epoch of matter-radiation equality $a_{eq}$ from Table 1. The vertical dotted lines depict the beginning of the neutron-proton ratio freeze-out $a_{n/p}$ and the epoch of light-element production $a_{nuc}$ , respectively (see Sec. 4b).
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Figure 3
Evolution of the fractions $Ω_{i}$ of the energy density of each cosmic component $i$ with SFDM of mass $m = 3 \times 1 0^{- 21} eV / c^{2}$ and self-interaction $λ / (m c^{2})^{2} = 2 \times 1 0^{- 18} {eV}^{- 1} {cm}^{3}$ (fiducial model) represented by the thick curves. Different components are depicted with different line styles, as labeled in the legend. The solid vertical line corresponds to $a_{eq}$ . On the lower left part of the figure, the thin curves represent the constraint from BBN. The solid one refers to a universe with a constant $N_{eff}$ of the central value in Eq. (40), and the two dash-dotted ones refer to such universes with $N_{eff}$ of the $1 σ$ limits there. The dotted vertical lines indicate the beginning of the neutron-proton ratio freeze-out $a_{n/p}$ and the epoch of light-element production $a_{nuc}$ , respectively.
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Figure 4
Upper plots: Evolution of the normalized Hubble expansion rate $H (a) / H_{Λ CDM} (a)$ vs scale factor $a$ . Lower plots: Evolution of the effective number of neutrino species, $N_{eff}$ , vs scale factor $a$ . The thick curves represent the evolution of $Λ SFDM$ models with various particle parameters. In the left-hand plots, $m$ is fixed. In the right-hand plots, $λ / (m c^{2})^{2}$ is fixed. The solid ones again correspond to our fiducial model with SFDM parameters $m = 3 \times 1 0^{- 21} eV / c^{2}$ and $λ / (m c^{2})^{2} = 2 \times 1 0^{- 18} eV {cm}^{3}$ ; see the legends for the corresponding values of $(λ / (m c^{2})^{2}, m)$ of each thick curve, in units of ( ${eV}^{- 1} {cm}^{3}$ , $eV / c^{2}$ ). Among the thin curves, the solid (dashed-dotted) ones refer to universes with constant $N_{eff}$ at the central value (68% confidence limits) of the measured $N_{eff}$ (40). The error bar in the lower plots is from the result of CMB measurements, $N_{eff} = 3.36 \pm 0.34$ [6].
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Figure 5
Parameter space of SFDM ( $λ / (m c^{2})^{2}$ , $m$ ). The solid vertical line represent the upper bound on $λ / (m c^{2})^{2}$ , which makes SFDM complete its transition from radiationlike to CDM-like (i.e., $⟨ \bar{w} ⟩ = 0.001$ ) just before the observed $z_{eq}$ . The arrow indicates that the region on the left side of the solid vertical line is allowed by this constraint from $z_{eq}$ . The two shaded bands are the allowed regions derived from the constraints that $N_{eff}$ be within the $1 σ$ interval of the value measured by BBN, at $a_{n/p}$ and $a_{nuc}$ , as labeled respectively. For each band, the thick solid (dashed) boundary curve corresponds to the upper (lower) $1 σ$ limit of the measured value of $N_{eff}$ in Eq. (40). The final allowed region is crosshatched, after combining all constraints. Our fiducial model, indicated by the star at $m = 3 \times 1 0^{- 21} eV / c^{2}$ , lies on the dotted vertical line at $λ / (m c^{2})^{2} = 2 \times 1 0^{- 18} {eV}^{- 1} {cm}^{3}$ , which corresponds to a radius of an equilibrium halo around 1 kpc.
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Figure 6
Radius of a virialized, polytropic SFDM halo, which forms during the matter-dominated era, as a function of $N_{eff}$ during the radiation-dominated era. The relation is shown by the solid curve, on which the star represents our fiducial model. The polytrope radius is considered as the minimum length scale of structures. The two dashed-dotted vertical lines indicate the $1 σ$ limits of $N_{eff}$ from BBN measurements, while the dashed vertical line indicates the central value of $N_{eff}$ from CMB measurements (the latter is the same as in Fig. 4, lower plots). The solid vertical line denotes the upper bound of $N_{eff}$ during the plateau so as to fulfill the constraint from fixed $z_{eq}$ .
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