© ESO 2018

1 Introduction

The quantum systems of interacting bosons are interesting in relation to the fundamental question of stability of matter (Lieb & Seiringer 2009), in astrophysics as potential candidates of dark matter (Jetzer 1992), and also because they can simulate condensates of the Bose–Einstein type (Griffin et al. 1996). Ruffini & Bonazzola (1969) studied the properties of a self-gravitating Newtonian sphere of bosons by means of variational method and established an upper bound for the ground state energy of this system, namely: E₀ ≤−0.1626M³G²m²ℏ⁻². Here, M is the mass of the sphere, G is the gravitational constant, m, the boson mass, and ℏ, the Planck’s constant. This result was in conflict with the lower bound for the same system established by Lévy-Leblond (1969): E₀ ≥−0.125M³G²m²ℏ⁻². This conflict worsened when Basdevant et al. (1990) significantly improved the lower bound; they obtained E₀ ≥−0.0625M³G²m²ℏ⁻². In 1989, we studied the same system in the Hartree approximation, and obtained an upper bound E₀ ≤−0.05426M³G²m²ℏ⁻² (Membrado et al. 1989), which brought peace with the mentioned lower bounds: the correction of a factor three had the answer.

Recently, Hui et al. (2017) have solved the time-independent Schrödinger–Poisson equation for its lowest eigenstates, assuming spherical symmetry. The energy per particle of the ground eigenstate that they obtained is the same we derived in Membrado et al. (1989). Our upper bound coincides with the total energy of the ground state they obtained when the phase of the wave function is position independent.

Using a simple method, Membrado & Pacheco (2012) estimated a limit for the existence of those Newtonian boson clusters in the presence of dark energy. We imposed that the critical distance where the attractive gravitational effect is balanced by the repulsion exerted by the cosmological constant must be greater than the cluster length scale.

Scalar fields fulfilling different potentials and masses ranging from 10⁻²⁶ eV up to 10⁻²³ eV have been proposed to model dark matter (see, for example: Lee & Koh 1996; Sahni & Wang 2000; Matos & Ureña-López 2000; Matos & Arturo Ureña-López 2001). We should also mention the work by Amendola & Barbieri (2006), who proposed that part of dark matter may be an ultra-light pseudo-Goldstone-boson with a mass m ≥ 10⁻²³ eV, arising fromthe spontaneous breaking at a scale close to the Planck’s mass of an extended approximate symmetry.

It so happens that the standard cosmological constant and cold dark matter model (ΛCDM), at scale of individual galaxies smaller than 10 kpc, produces an excess of dwarf galaxies and singular galactic cores not supported by observations(see, e.g. Weinberg et al. 2015). Hu et al. (2000) showed that these problems could be solved by introducing fuzzy dark matter (FDM) instead CDM. This FDM would be constituted by ultra-light scalar particles with a mass in the order of 10⁻²² eV, initially in a Bose–Einstein condensate. For a thorough list of references on this subject, see Chavanis (2011). The physics that support this new model and its astrophysical consequences are reviewed and analysed by Hui et al. (2017).

In Membrado & Pacheco (2013), we applied a dark halo made of collisionless Newtonian bosons to the ultra-faint Milky Way satellite galaxy Segue 1 (Simon et al. 2011; Martinez et al. 2011). We saw that when taking its half-light radius, r_1∕2, as the radius that covers 99% of the dark assembly mass, the Segue 1 mass at r_1∕2 can be reproduced. Thus, we derived an upper bound for the boson mass, m < 10⁻²⁰ eV.

Chen et al. (2017) applied Jeans analysis to the kinematic data of eight classical dwarf spheroidal (dSph) galaxies, assuming a soliton core profile (Schive et al. 2014) connecting to a Navarro–Frenk–White profile (Navarro et al. 1997) at larger radii. They obtain masses between $m=8_{-3}^{+5}\times {10}^{-23}$ eV (for Draco) and $m=6_{-2}^{+7}\times {10}^{-22}$ eV (for Sextans).

Calabrese & Spergel (2016) applied the same model than Chen et al. (2017) to Draco II and Triangulum II, but using two observational limits: the half-light mass, M_1∕2 (Wolf et al. 2010), and the maximum halo mass, M_max ~ 2 × 10¹⁰ M_⊙, based on the mass function of the Milky Way dwarf satellite galaxies (Giocoli et al. 2008). They find that if the stellar component is within the core, the particle mass would be about m ~ 3.7–5.6 ×10⁻²² eV.

In this paper, we have retaken and extended the calculation made in Membrado & Pacheco (2013) to 19 dSph galaxies. We have dealt with a sphere of self-gravitating bosons as in Membrado et al. (1989), but have assumed the presence of dark energy. The aim of this paper is: (1) to determine the effects of the cosmological constant; and (2) to estimate the boson mass.

The paper is organized as follows. In Sect. 2, we present a Hartree-like method which allows us to obtain the energy per unit mass of the ground state of a self-gravitating assembly of identical bosons in the presence of the cosmological constant. In this section, the Hartree method is stated as a variational problem in the particle density. The solutions for the particle number density are shown in Sect. 3. The virial theorem for these systems is treated in Sect. 4. In Sect. 5, we use clusters of self-gravitating collisionless Newtonian bosons to model dark haloes of dSph galaxies. Finally, our main conclusions are summarized in Sect. 6.

2 The single-boson energy equation in boson clusters embedded in dark energy

Let us consider an assembly of N self-gravitating identical bosons with mass m and dark energy. Assuming Newtonian gravity for matter and the cosmological constant model for dark energy, the Hamiltonian of the system is $$\widehat{H}=-\frac{{\hslash }^2}{2m}{\sum }_{i=1}^N{\nabla }_i^2-G{m}^2{\sum }_{i>j=1}^N\frac{1}{\mid r_i-r_j\mid }-\frac{m\Lambda c^2}{6}{\sum }_{i=1}^N{r}_i^2.$$(1)

In Eq. (1), Λ is the cosmological constant; for the cosmological constant term, see, for example, Membrado & Pacheco (2012).

For the ground state of the system, ∣ψ⟩, its energy, E, is $$E=\langle \psi \mid \widehat{H}\mid \psi \rangle .$$(2)

In the Hartree approximation, the particle density, n(r), at the point r is given by $$n(r)=N{f}^{\star }(r)f(r),$$(3)

where $f={[n/N]}^{1/2}$ is the single-particle wave function (we note that f is real because corresponds to a bound state). Thus $$\begin{array}{lll}E\hfill & =\hfill & -\frac{\hslash }{2m}{\displaystyle \int \text{d}}r{n}^{1/2}(r){\nabla }^2{n}^{1/2}(r)\hfill \\ \hfill & \hfill & -\frac{G{m}^2}{2}\left(\frac{N-1}{N}\right){\displaystyle \int \text{d}}r{\displaystyle \int \text{d}}{r}^{\prime }\frac{n(r)n{(r)}^{\prime }}{\mid r-r^{\prime }\mid }\hfill \\ \hfill & \hfill & -\frac{m\Lambda c^2}{6}{\displaystyle \int \text{d}}rn(r)r^2.\hfill \end{array}$$(4)

Henceforth, we assume that N ≫ 1.

The minimum energy solution can be expressed as a variational problem. Thus, E, given by Eq. (4), must be minimized as a function of n(r) with the constraint $$N={\displaystyle \int \text{d}}rn(r).$$(5)

Hence, we deal with $$\delta (E[n]+\lambda N[n])=0,$$(6)

where λ is the Lagrange multiplier. Assuming spherical symmetry and radius R, the variational problem given by Eq. (6) is of the kind $$\delta {\displaystyle {\int }_0^{R}F}[n]\text{d}r=0,$$(7)

where $F=F(r,n,\dot{n},\ddot{n})$, a dot meaning differentiation with respect to r. Explicitly, $$F=4\pi r^{2}n\left[e_{\text{kin}}+\frac{1}{2}m{\varphi }_M+m{\varphi }_{\Lambda }+\lambda \right].$$(8)

In Eq. (8), e_kin, ϕ_M, and ϕ_Λ are the kinetic energy per particle, the gravitational potential due to matter, and the gravitational potential due to dark energy, respectively. They are given by $$\begin{array}{lll}\hfill & \hfill & e_{\text{kin}}=-\frac{{\hslash }^2}{2m}\left[\frac{\dot{n}}{nr}-\frac{{\dot{n}}^2}{4n^2}+\frac{\ddot{n}}{2n}\right],\hfill \\ \hfill & \hfill & {\varphi }_M=-Gm\left[\frac{1}{r}{\displaystyle {\int }_0^{r}n}{r^{\prime }}^2\text{d}{r}^{\prime }+{\displaystyle {\int }_r^{R}n}{r}^{\prime }\text{d}{r}^{\prime }\right],\hfill \\ \hfill & \hfill & {\varphi }_{\Lambda }=-\frac{1}{6}\Lambda c^2{r}^2.\hfill \end{array}$$

The solution of the variational problem is shown in Appendix A. Thus, using Eq. (8) in (A.9), the boson number density, n(r), must fulfil $$e=-\lambda =e_{\text{kin}}(r)+m{\varphi }_M(r)+m{\varphi }_{\Lambda }(r),$$(12)

It should be noticed that e in Eq. (12) is the energy of the single-particle wave function f; that is, $$\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+m{\varphi }_M+m{\varphi }_{\Lambda }\right]f=ef.$$(13)

We take $$b=\frac{{\hslash }^2}{2GM{m}^2}$$(14)

as the length scale of a cluster of mass M that contains N identical bosons of mass m (i.e. M = Nm). This quantity allows us to define dimensionless variables η and x as $$\begin{array}{lll}\hfill & \hfill & n=\frac{N}{4\pi b^3}\eta ,\hfill \\ \hfill & \hfill & r=bx.\hfill \end{array}$$

As the mass enclosed within a sphere of radius r is $$M(r)=4\pi m{\displaystyle {\int }_0^{r}n}(r^{\prime }){r^{\prime }}^2\text{d}{r}^{\prime },$$(17)

using Eqs. (15) and (16) we define the dimensionless mass $$\mu (x)={\displaystyle {\int }_0^x\eta }{x^{\prime }}^2\text{d}{x}^{\prime },$$(18)

fulfilling $$M(r)=M\mu (r/b).$$(19)

Hence, Eq. (5) reads as $$1=\mu (R/b).$$(20)

We also introduce the dimensionless parameters ϵ and ξ given by $$\begin{array}{lll}\hfill & \hfill & e=\frac{GMm}{b}ϵ.\hfill \\ \hfill & \hfill & \xi =\frac{c^2\Lambda b^3}{GM}.\hfill \end{array}$$

In Eq. (21), ϵ is the dimensionless energy of the single-particle wave function. The parameter ξ, defined in Eq. (22), is a measure of the influence of the cosmological constant repulsive effect with respect to the attractive effect due to matter. From Eqs. (14) and (22), the mass of the cluster can be expressed as a function of m and ξ by $$M=\left(\frac{\hslash c}{2Gm}\right){\left(\frac{\hslash }{m{c}^2}\right)}^{1/2}{\left(\frac{2\Lambda c^2}{\xi }\right)}^{1/4}.$$(23)

Using Eqs. (14)–(16) and (21)–(22), Eq. (12) reads as $$\begin{array}{lll}\hfill & \hfill & ϵ=\kappa (x)+{\nu }_M(x)+{\nu }_{\Lambda }(x),\hfill \\ \hfill & \hfill & \kappa (x)=-\left[\frac{{\eta }^{\prime }}{\eta x}-\frac{{{\eta }^{\prime }}^2}{4{\eta }^2}+\frac{{\eta }^{″}}{2\eta }\right],\hfill \\ \hfill & \hfill & {\nu }_M(x)=-\frac{\mu (x)}{x}-{\displaystyle {\int }_x^{(R/b)}n}(y)y\text{d}y,\hfill \\ \hfill & \hfill & {\nu }_{\Lambda }(x)=-\frac{\xi }{6}{x}^2,\hfill \end{array}$$

In Eq. (24), κ(x) is dimensionless kinetic energy per particle (an apostrophe means differentation with respect to x); ν_M (x) and ν_Λ (x) are dimensionless gravitational energies per particle due to matter and dark energy, respectively.

When the cosmological constant contribution is neglected, the solution of Eq. (24) must not have any node (1s state), so η(x →∞) → 0 (see Fig. 1 of Membrado et al. 1989). Such a solution has a classical turning point at x = x₁, where k(x₁) = 0 (see Fig. 1). From the single-particle wave function, f, the probability of finding the particle inside the volume of dimensionless radius x is given by μ(x) (see Eq. (18)). As μ(x₁) = 0.83, the probability of finding it beyond the classical turning point cannot be considered as negligible.

When ξ≠0, the cosmological constant term of the gravitational energy makes the kinetic term be also zero at some x = x₂, beyond the classical turning point (see right hand panels of Fig. 3). This means that there could be a tunnelling process through the potential barrier from x = x₁ up to x = x₂. The values of the wave function, f, at x = x₁ and x = x₂ will allow to estimate the mean life of the system.

In the absence of dark energy, the mass density decays as x⁻⁴; that is, it vanishes at an infinite distance from the origin. Considering the virial theorem of this system at x = ∞, the result is the canonical form 2T = −V, where T is kinetic energy, and V, gravitational energy (see Membrado et al. 1989). In the case of not using the endpoint at infinity, the canonical virial relation should have to be corrected by a surface term.

When the cosmological constant is taken into account, we see in Sect. 4 that 2T = −V + 2V_Λ (the final term corresponds to the cosmological constant contribution) is fulfilled at x₂. This means that surface terms are null there and indicates that x₂ is the dimensionless radius of the system.

From Eqs. (24)–(27) together with Eq. (20), the dimensionless distance x₂ and the classical turning point, x₁, fulfil $$\begin{array}{lll}\hfill & \hfill & ϵ=-\frac{1}{x_2}-\frac{\xi }{6}{x}_2^2.\hfill \\ \hfill & \hfill & ϵ=-\frac{\mu (x_1)}{x_1}-{\displaystyle {\int }_{x_1}^{x_2}n}(x)x\text{d}x-\frac{\xi }{6}{x}_1^2.\hfill \end{array}$$

Finally, the distance x₀, where the attractive gravitational effect is balanced by the repulsion (i.e. where $\text{d}({\nu }_M+{\nu }_{\Lambda })/\text{d}x{\mid }_{x_0}=0$), fulfils $$0=\frac{\mu (x_0)}{x_0^2}-\frac{\xi }{3}{x}_0.$$(30)

If we assume that for x₀ ≤ x ≤ x₂, the contribution of η(x) to the particle number is very small (i.e. μ(x₀) ≈ 1), and that x₀ ≫ 1, $$x_0={(3/\xi )}^{1/3};$$(31)

in other words $$r_0=b{x}_0={\left(\frac{3GM}{\Lambda c^2}\right)}^{1/3}.$$(32)

This distance r₀ is the same as that derived assuming a point mass M (see, for example: Membrado & Pacheco 2012). In Membrado & Pacheco (2012), we concluded that for r > r₀, the stability of any self-gravitational cluster of matter would be affected by the cosmological constant. In this work, we see that at x₀, the potential energy per particle, v_M + v_Λ, shows a maximum. This should cause the system to lose particles by tunnelling. Hence, the instability would be manifested in this quantum system by the loss of bosons. The ratio between the mean life of the system and the Hubble time will inform us of the importance of this instability (see Sect. 3).

Fig. 1 For ξ = 0, gravitational energy, νM and single-particle energy, ϵ, as a functionof the dimensionless distance x. The classicalturning point x1 is marked.

Fig. 2 Choice of a0 for different values of ξ (see text).

Fig. 3 Left panels: dimensionless particle number density, η, as a functionof x, for three values of ξ. The bottom panel corresponds to the critical value ξc. Right panels: dimensionless energies per particle for the same values of ξ shown in left panels (gravitational energy due to matter, νM; total gravitational energy, νM + νΛ; single-particle energy, ϵ), as a functionof the dimensionless distance x. The distances x1, x0 and x2 for ξc are marked in the bottom right panel.

3 Solutions for the particle number density

Equation (24) is an integro-differential equation. Instead of solving it, we apply the dimensionless Laplacian operator, b² ∇², to Eq. (24) to obtain the following four-order differential equation $$\begin{array}{lll}{{\eta }^{‴}}^{\prime }\hfill & =\hfill & 2{\eta }^2\left[1-\frac{\xi }{\eta }\right]-\frac{4{\eta }^{‴}}{x}+\frac{10{\eta }^{\prime }{\eta }^{″}}{x\eta }-\frac{6{{\eta }^{\prime }}^3}{x\eta }+\frac{3{\eta }^{\prime }{\eta }^{‴}}{n}\hfill \\ \hfill & \hfill & +\frac{2{{\eta }^{″}}^2}{\eta }-\frac{7{{\eta }^{\prime }}^2{\eta }^{″}}{{\eta }^2}+\frac{3{{\eta }^{\prime }}^4}{{\eta }^3}.\hfill \end{array}$$(33)

It can be seen that η(x → 0) = ∑ a_nxⁿ; therefore, the solution of Eq. (33) requires knowledge of a₀, a₁, a₂, and a₃. From Eqs. (24) and (33), we obtain a₁ = 0 and $a_3=(a_1/a_0)[(5/6)a_2-(1/4)(a_1^2/a_0)]$, respectively; hence a₃ = 0, too. Thus, the problem is reduced to finding a₀ and a₂. For this subject, we chose a value for a₀ and seek its corresponding a₂ leading to a decreasing η(x) without any node up to x₂. Finally, a₀ is determined by imposing that N particles are contained up to x = x₂.

In Fig. 2, we show, for several values of ξ, the quantity μ(x₂) (see Eq. (18)) as a function of a₀ (the reader should remember that a₂ is adequately fixed for each a₀). In the figure, the line μ(x₂) = 1 is also represented. Hence, the chosen value for a₀ will be that fulfilling μ(x₂) = 1. As can be seen from Fig. 2, for each ξ < ξ_c = 1.65 × 10⁻⁴, there are two values of a₀ leading to μ(x₂) = 1; the greatest a₀, corresponding to the smallest ϵ, is chosen. As an example, let us consider the case ξ = 10⁻⁵. For this ξ, a₀ = 6.811 × 10⁻³ and a₀ = 3.827 × 10⁻⁵ fulfil μ(x₂) = 1 for a₂ = −1.7186 × 10⁻⁴ and a₂ = −4.9123 × 10⁻⁸, respectively; these values lead to the dimensionless energies of the single-particle wave function, ϵ = −8.124 × 10⁻² and ϵ = −2.255 × 10⁻². As we are looking for minimum energy solution which was expressed as a variational problem in this work, the greatest value of a₀ would lead to the ground state solution (the variational problem is just designed to determine the ground state, but not to excited states). In the case ξ_c = 1.65 × 10⁻⁴, only one a₀ is allowed. For ξ > ξ_c, μ(x₂) > 1 for any a₀. This means that the cosmological constant imposes a limit for the existence of self-gravitating boson clusters. Thus, when the scale of the gravitational repulsive force due to the cosmological constant (Λc²bm), is greater than 1.65× 10⁻⁴ times the scale of the gravitational attraction force (GMm∕b²), self-gravitating boson assemblies can not be created. The condition ξ ≤ ξ_c imposes a constraint for the minimum mass of the cluster. Thus, from Eq. (23), $$M\ge \left(\frac{\hslash c}{2Gm}\right){\left(\frac{\hslash }{m{c}^2}\right)}^{1/2}{\left(\frac{2\Lambda c^2}{{\xi }_c}\right)}^{1/4}.$$(34)

In this work, we have assumed that bosons are in Newtonian gravity. This means that (2GM∕b) ≪ c². Hence, using Eq. (14), the mass of the cluster must fulfil $$M\ll \frac{\hslash c}{2Gm}.$$(35)

This means that $$m{c}^2\gg \hslash {\left(\frac{2\Lambda c^2}{{\xi }_c}\right)}^{1/2}=2.47\times {10}^{-31}\text{\hspace{0.17em}eV}.$$(36)

In Eq. (36), we have used Λ = 1.29 × 10⁻⁵⁶ cm⁻². This value is calculated from the energy density of dark energy, ρ_Λ, by Λ = (8πGρ_Λ∕c⁴). We have taken into account that ρ_Λ = ρ₀Ω_Λ, where ${\rho }_0=(3c^2{H}_0^2/8\pi G)$ is the background energy density, H₀ being the Hubble constant. Values for Ω_Λ and H₀ can be taken from Spergel et al. (2003); in this work we assumed Ω_Λ = 0.73 and H₀ = 71 km s⁻¹ Mpc⁻¹.

Profiles of η(x) for three values of ξ are presented in the left panels of Fig. 3. Values of a₀ and a₂ for the models are presented in Table 1. We have only shown density profiles for η(x) ≥ 10⁻⁸. For ξ_c, η(x₂) = 4.7 × 10⁻⁷; and, for ξ = 10⁻⁵ and ξ = 10⁻⁶, dimensionless densities smaller than 10⁻⁸ do not already contribute to the bound particle number. The behaviour of η(x) in the neigbourhood of x₂ can be seen in the bottom left panel (ξ = ξ_c). This behaviour is of the kind $$\begin{array}{lll}\hfill & \hfill & \eta (x\to x_2)\approx \eta (x_2)\left[1+\alpha {(x_2-x)}^3\right],\hfill \\ \hfill & \hfill & \alpha =\frac{\xi }{9}{x}_2-\frac{1}{3x_2^2}>0.\hfill \end{array}$$

In the panel devoted to ξ_c, the profile of η(x) is finished at x = x₂, where the potential barrier ends (the kinetic energy per particle is zero).

Once the particle density is known, ν_M(x) can be calculated from Eq. (26), and ϵ, from Eqs. (24)–(27). Then, x₁ and x₀ are obtained from Eqs. (29) and (30). In Table 2, x₁, x₀ and x₂ are shown for different values of ξ. From this table, it can be seen that the greatest difference between x₀ and that calculated from Eq. (31), appears for ξ_c, being smaller than 0.8%. This means that beyond x₀, the contribution of the particle density to the number of bound particles can be considered as negligible.

In the right hand panels of Fig. 3, we present ν_M (x), ν_M (x) + ν_Λ(x), and ϵ for the same values of ξ treated in the left hand panels. Their profiles end at x = x₂. In the figures, x₁ and x₀ can be observed, together with the potential barrier and the effect of the cosmological constant in the dimensionless gravitational energy per particle.

From the potential barrier, we were able to estimate the mean life of these systems, τ_c. This can be calculated from a timescale for the cluster, t_c, and the transmission factor, T (see, for example, Matthews 1963, p. 96f). If τ_c is smaller than the age of the Universe, t_u = 1.37 × 10¹⁰ years, these clusters should not exist at present. A timescale, t_c, for the cluster is $$t_c=\frac{m{b}^2{x}_1^2}{\hslash }.$$(39)

In Eq. (39), we have assumed non-relativistic bosons, and we have used the classical turning radius. Thus, using the transmission factor T = η(x₂)∕η(x₁), the mean life reads as $${\tau }_c=\frac{t_c}{T}=\text{(}\frac{m{b}^2}{\hslash }\text{)}\text{(}\frac{x_1^2\eta (x_1)}{\eta (x_2)}\text{)}.$$(40)

Hence, using Eq. (14) in (40), bound boson clusters can exist if $$M\ll \text{(}\frac{\hslash c}{2Gm}\text{)}{\text{(}\frac{\hslash }{m{c}^2}\text{)}}^{1/2}{\text{(}\frac{x_1^2\eta (x_1)}{t_u\eta (x_2)}\text{)}}^{1/2.}$$(41)

When Eq. (23) is taken into account in Eq. (41), the condition $$\frac{{\xi }^{1/2}{x}_1^2\eta (x_1)}{\eta (x_2)}\gg t_u{(2\Lambda c^2)}^{1/2}=2.08$$(42)

is derived. The minimum value of the left hand term of Eq. (42) happens for ξ_c (it is a decreasing function of ξ). For ξ = ξ_c = 1.65 × 10⁻⁴, x₁ = 13.9, η(x₁) = 2.38 × 10⁻⁴ and η(x₂) = 4.69 × 10⁻⁷; thus, $${\frac{{\xi }^{1/2}{x}_1^2\eta (x_1)}{\eta (x_2)}]}_{\xi ={\xi }_c}=1.26\times {10}^3.$$(43)

Hence, the present existence of Newtonian boson clusters embedded in dark energy is not threatened by the age of the Universe.

4 The virial theorem

Taking the differential of Eq. (12) with respect to the radial co-ordinate, and after multiplying this result by the particle density, $$0=n\frac{\text{d}{e}_{\text{kin}}}{\text{d}r}+mn\frac{\text{d}{\varphi }_M}{\text{d}r}+mn\frac{\text{d}{\varphi }_{\Lambda }}{\text{d}r}$$(44)

is obtained. Equation (44) is the equation of radial motion which shows the balance of strength per unit volume.

If spherical symmetry is assumed, the virial theorem is derived by first multiplying the radial motion equation by the radial co-ordinate and then integrating it over a volume. Thus, if the volume is that of a sphere of radius R, the virial theorem reads as $$\begin{array}{lll}0\hfill & =\hfill & 4\pi {\displaystyle {\int }_0^R{r}^3}n\frac{\text{d}{e}_{\text{kin}}}{\text{d}r}\text{d}r\hfill \\ \hfill & \hfill & +4\pi m{\displaystyle {\int }_0^R{r}^3}n\frac{\text{d}{\varphi }_M}{\text{d}r}\text{d}r+4\pi m{\displaystyle {\int }_0^R{r}^3}n\frac{\text{d}{\varphi }_{\Lambda }}{\text{d}r}\text{d}r.\hfill \end{array}$$(45)

By taking R = bx₂ as the radius of the system, the first term of Eq. (45) is equal to minus two times the kinetic energy of the system, T (see Appendix B; we note that, from Eq. (37), $\dot{n}(R)=\ddot{n}(R)=0$); the second term is minus the gravitational energy due to matter, V_M, calculated from the trace of the energy potential tensor of Chandrasekar (see, for example, Binney & Tremaine 1987, p. 67); the third term is two times the gravitational energy due to dark energy, V_Λ (from using Eq. (11)). Thus, $$0=-2T-V_M+2V_{\Lambda }.$$(46)

where $$\begin{array}{lll}\hfill & \hfill & T=4\pi {\displaystyle {\int }_0^R{r}^2}n{e}_{\text{kin}}\text{d}r,\hfill \\ \hfill & \hfill & V_M=\frac{4\pi m}{2}{\displaystyle {\int }_0^R{r}^2}n{\varphi }_M\text{d}r,\hfill \\ \hfill & \hfill & V_{\Lambda }=4\pi m{\displaystyle {\int }_0^R{r}^2}n{\varphi }_{\Lambda }\text{d}r.\hfill \end{array}$$

Equation (46) can be expressed as a function of dimensionless magnitudes, $\tilde{T}$, Ṽ_M and Ṽ_Λ given by $$\begin{array}{lll}\hfill & \hfill & \tilde{T}=-{\displaystyle {\int }_0^{x_2}{x}^2}\left[\frac{{\eta }^{\prime }}{x}-\frac{{{\eta }^{\prime }}^2}{4\eta }+\frac{{\eta }^{″}}{2}\right]\text{d}x,\hfill \\ \hfill & \hfill & {\tilde{V}}_M=-{\displaystyle {\int }_0^{x_2}\eta }\mu x\text{d}x,\hfill \\ \hfill & \hfill & {\tilde{V}}_{\Lambda }=-\frac{\xi }{6}{\displaystyle {\int }_0^{x_2}\eta }{x}^4\text{d}x.\hfill \end{array}$$

They are related with T, V_M and V_Λ by $$\begin{array}{lll}\hfill & \hfill & T=\frac{G{M}^2}{b}\tilde{T},\hfill \\ \hfill & \hfill & V_M=\frac{G{M}^2}{b}{\tilde{V}}_M,\hfill \\ \hfill & \hfill & V_{\Lambda }=\frac{G{M}^2}{b}{\tilde{V}}_{\Lambda }.\hfill \end{array}$$

Equation (50) is obtained from Eq. (47) using Eqs. (9), (14)–(16) and (53). Equation (51) is obtained from the second term on the right hand side of Eq. (45); there, Eqs. (10), (14)–(16), (19), and (54) are used. Finally, Eq. (52) derives from Eq. (49), taking into account Eqs. (11), (14)–(16), (22), and (55). Thus, $$0=-2\tilde{T}-{\tilde{V}}_M+2{\tilde{V}}_{\Lambda }.$$(56)

Values of $\tilde{T}$, Ṽ_M and Ṽ_Λ for three ξ-models are presented in Table 3.

Dimensionless total energies $$\tilde{E}=\tilde{T}+{\tilde{V}}_M+{\tilde{V}}_{\Lambda }$$(57)

are also shown in Table 3. The total energy E, is then given by (see Eqs. (53)–(55)) $$E=\frac{G{M}^2}{b}\tilde{E}.$$(58)

The energy, e, of the singleparticle wave function can be also obtained by multiplying Eq. (12) by the number density, n(r), and integrating it over the volume of the system. Thus, $$eN=T+2V_M+V_{\Lambda }.$$(59)

Then, using Eqs. (21) and (53)–(55), the dimensionless energy, ϵ, of the single particle wave function is $$ϵ=\tilde{T}+2{\tilde{V}}_M+{\tilde{V}}_{\Lambda }.$$(60)

Values of ϵ are also presented in Table 3. Finally, from Eqs. (56), (57) and (60), it can be seen that $$\tilde{E}=\frac{ϵ}{3}+\frac{4}{3}{\tilde{V}}_{\Lambda }.$$(61)

5 Application to dark halos of dSph galaxies

Dwarf spheroidal galaxies are among the smallest and faintest galactic systems. However, their velocity dispersions suggest large mass-to-light ratios, being some of the most dark matter dominated objects in the Universe (e.g. Gilmore et al. 2007). Dwarf spheroidal galaxies have been found inside the virial radius of groups and clusters. These galaxies appear in halos of massive galaxies where tidal effects are present, so their star formation history is strongly dependent on their environment. It is generally considered that these gas-poor galaxies with old stellar population come from irregular dwarfs which have suffered processes of gas stripping and strangulation that limit further star formation.

Observational data from dSph galaxies reveal discrepancies with respect to predictions of ΛCDM models. High-resolution simulations in the standard ΛCDM cosmology (see, for example: Klypin et al. 1999; Moore et al. 1999; Diemand et al. 2007; Springel et al. 2008) predict a number ofsub-halos within the Local Group which is about two orders of magnitude higher than the total number of observed satellite galaxies (Kauffmann et al. 1993; Moore et al. 1999; Klypin et al. 1999). This result is known as the “missing satellites problem”. Simulations also lead to power-law density profiles for sub-halos (e.g. Moore et al. 1999; Navarro et al. 1997, 2004), against the more flattened profiles derived from observations (e.g. Burkert 1995; de Blok & Bosma 2002; Gentile et al. 2005; de Blok 2005). It has even been shown that tidal interactions are not able to give rise to the dSph galaxies of the Milky Way group from the most massive sub-halos in a ΛCDM Universe (Kazantzidis et al. 2004).

Moreover, dSph galaxies have also been found in isolated environments, far away from any massive galaxy (Makarov et al. 2012; Karachentsev et al. 2015). The evolution of these galaxies seems to be regulated by their own star formation rather than by their environment. It has been proposed that these objects were formed in the early Universe, before the reionization in small haloes M < 2 × 10⁸ M_⊙, where an active star formation would deplete gas resources (see: Bovill & Ricotti 2009; Ricotti & Gnedin 2005). It should also be said that, as happen with sub-halos, standard ΛCDM model predicts a factor of ten more dwarf haloes in the field than the number of observed dwarf galaxies (Tikhonov & Klypin 2009).

Hu et al. (2000) showed that these problems might be solved, maintaining the advantages of the ΛCDM models, if the dark matter particle is an ultra-light scalar of m ~ 10⁻²² eV. This kind of dark matter is often called fuzzy dark matter. Initially, this particle would be in a cold Bose–Einstein condensate, similar to axion dark matter models (e.g. Marsh 2016). The wave properties of the dark matter would stabilize the gravitational collapse, giving rise to smoother cores and would suppress small-scale linear power. For a review of the FDM model see Suárez et al. (2014). Recently, Hui et al. (2017) describe the arguments that motivate FDM, review previous works and analyse several aspects of its behaviour.

In 2013, we dealt with dark haloes composed of degenerate fermions to reproduce rotation velocities of galaxies (Membrado & Pacheco 2013), and tried to use them to describe haloes of dispersion-supported dwarf stellar system. We chose the ultra-faint Milky Way satellite galaxy Segue 1 (Simon et al. 2011; Martinez et al. 2011). For this dSph galaxy, Simon et al. (2011) derived that the mass within its half-light radius, r_1∕2 = 38 pc, was M_1∕2 ≈ 6 × 10⁵ M_⊙. We concluded that to adequately model the dark halo of Segue 1, it was necessary to have more compact selfgravitating spheres than those provided by degenerate fermions. We then thought on self-gravitating collisionless Newtonian boson cluster in which all bosons occupy the lowest lying one-particle Hartree orbital. Then, taking r_1∕2 as the radius that covers 99% in mass of the dark assembly, we derived an upper bound for the boson mass, m < 10⁻²⁰ eV.

The goal of this section is to estimate the mass of a boson which would be able to describe dark halos of dSph galaxies. For this purpose, we used characteristic quantities of these galaxies: the edge of the galaxy, r_lim; the 3D deprojected half-light radius, r_1∕2 (i.e. the radius at which luminosity is half total luminosity); the mass M_1∕2 enclosed within a sphere of radius r_1∕2; the luminosity-weighted average of the square of the line-of-sight velocity dispersion, $\langle {\sigma }_{\text{los}}^2\rangle$; and errors in M_1∕2 and ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$. For the stellar component, we assumed constant mass-to-light ratio and used a truncated King profile with a core radius r_c calculated from r_1∕2 and r_lim. For simplicity, star mass contribution to system mass is neglected in any point. Two expressions are used to estimate m from the above quantities. One of themrelates M_1∕2 and r_1∕2 with M and m. The second one is the virial theorem for the star component which relates r_lim and $\langle {\sigma }_{\text{los}}^2\rangle$ to M and m.

5.1 Formulae used to estimate the boson mass

From Eqs. (16) and (14), $$r_{1/2}=\frac{{\hslash }^2}{2GM{m}^2}{x}_{1/2};$$(62)

and, from Eq. (19), $$M_{1/2}=M\mu (x_{1/2}).$$(63)

Hence, $$\left(\frac{M_{1/2}}{{10}^7{M}_{\odot }}\right)=0.427x_{1/2}\mu (x_{1/2}){\left(\frac{m{c}^2}{{10}^{-21}\text{\hspace{0.17em}eV}}\right)}^{-2}{\left(\frac{r_{1/2}}{100\text{\hspace{0.17em}pc}}\right)}^{-1}.$$(64)

In Eq. (64), r_1∕2 is expressed in hundreds of parsecs, and M_1∕2 in units of 10⁷ M_⊙, as those deduced from observations of dSph galaxies (e.g. Wolf et al. 2010, who derived an accurate mass estimator for dispersion-supported stellar systems). In that equation, boson mass appears in units of 10⁻²¹ eV in order the numerical coefficient to be close to unity.

ShowingM in the same units that M_1∕2, the length scale b reads as $$b=4.27\times {10}^1{\left(\frac{M}{{10}^7{M}_{\odot }}\right)}^{-1}{\left(\frac{m{c}^2}{{10}^{-21}\text{\hspace{0.17em}eV}}\right)}^{-2}\text{\hspace{0.17em}pc}.$$(65)

Thus, usingEq. (65) in (22), the dimensionless parameter ξ can be expressed as $$\xi =2.00\times {10}^{-8}{\left(\frac{M}{{10}^7{M}_{\odot }}\right)}^{-4}{\left(\frac{m{c}^2}{{10}^{-21}\text{\hspace{0.17em}eV}}\right)}^{-6}.$$(66)

Hence, the repulsive effect of the cosmological constant imposes the constraint (from ξ ≤ ξ_c = 1.65 × 10⁻⁴) $$\left(\frac{M}{{10}^7{M}_{\odot }}\right)\ge 1.05\times {10}^{-1}{\left(\frac{m{c}^2}{{10}^{-21}\text{\hspace{0.17em}eV}}\right)}^{-3/2}.$$(67)

The numerical coefficient in Eq. (67) is five times greater than that derived in Membrado & Pacheco (2012); there, it was imposed that the distance r₀ (see Eq. (32)) where the gravitational effect is balanced by the repulsion must be greater than the length scale, b (see Eq. (14)).

Hui et al. (2017) found similar results to those given by Eq. (67) by imposing two different arguments. One argues that dark halo mean density inside half-mass radius must be greater than two hundred times the critical density of the Universe; the other is based on the Jeans length. Within the considerable uncertainties in their results, they give a minimum mass halo of (1–2) ×${10}^7{M}_{\odot }{(m{c}^2/{10}^{-22}\text{\hspace{0.17em}eV})}^{-3/2}$. This bound is 0.3–0.6 times lower than that of Eq. (67). According to this result, the cosmological constant is what actually imposes the lower bound for the mass of these systems.

The virial theorem that we used for the stellar component (assuming spherical symmetry and steady and static conditions) comes from multiplying the radial Jeans equation (see, for example, Binney & Tremaine 1987, p. 198) by 4πr³ and integrating from r = 0 up to r = r_lim, where surface terms are neglected. Thus $${\langle {\sigma }^2\rangle }_{M_{\star }}=G{\langle \frac{M(r)}{r}\rangle }_{M_{\star }}.$$(68)

In Eq. (68), σ is the total velocity dispersion of stars; M(r) is the mass at distance r from the system centre; M_⋆, the stellar mass enclosed within a sphere of radius r_lim; and, for any quantity Q, $${\langle Q\rangle }_{M_{\star }}=\frac{{\displaystyle {\int }_0^{r_{\text{lim}}}{\rho }_{\star }}(r)Q(r)r^2\text{d}r}{{\displaystyle {\int }_0^{r_{\text{lim}}}{\rho }_{\star }}(r)r^2\text{d}r},$$(69)

ρ_⋆ (r) being stellar mass density at r.

In our study, ρ_⋆(r) is approximated by a truncated King profile (King 1962, Eqs. (27) and (29)). Its core radius, r_c, is chosen to fulfil $$\frac{{\displaystyle {\int }_0^{r_{\text{lim}}}F}(r)r^2\text{d}r}{{\displaystyle {\int }_0^{r_{1/2}}F}(r)r^2\text{d}r}=2,$$(70)

where $$\begin{array}{lll}\hfill & \hfill & F(r)=\frac{1}{z^2}\left[\frac{{\cos }^{-1}(z)}{z}-{(1-z^2)}^{1/2}\right],\hfill \\ \hfill & \hfill & z={\left[\frac{1+{(r/r_c)}^2}{1+{(r_{\text{lim}}/r_c)}^2}\right]}^{1/2}.\hfill \end{array}$$

For ${\langle {\sigma }_{\text{los}}^2\rangle }_{M_{\star }}$, we take three times the luminosity-weighted average of the square of the line-of-sight velocity dispersion, $\langle {\sigma }_{\text{los}}^2\rangle$. Thus, using Eqs. (14), (16), (19) and (18) and the truncated King profile in (68), $$\left(\frac{{\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}}{10\text{\hspace{0.17em}km\hspace{0.17em}}{\text{s}}^{-1}}\right)=1.832S(x_{\text{lim}})\left(\frac{M}{{10}^7{M}_{\odot }}\right)\left(\frac{m{c}^2}{{10}^{-21}\text{\hspace{0.17em}eV}}\right),$$(73)

where $$x_{\text{lim}}=r_{\text{lim}}/b,$$(74) $$S(x_{\text{lim}})={\text{[}\frac{{\int }_0^{x_{\text{lim}}}F(bx)\mu (x)x\text{d}x}{{\int }_0^{x_{\text{lim}}}F(bx)x^2\text{d}x}\text{]}}^{1/2.}$$(75)

In order Eq. (73) to have a numerical coefficient close to unity, ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ is presented in tens of km s⁻¹.

5.2 Procedure followed to estimate m

Now, let us suppose that we are dealing with N_g galaxies. Then, N_d = 2 N_g is the number of data, and N_p = N_g + 1, the number of unknown parameters. In the following, for the galaxy i, we denote data by M_1∕2;i and ${\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}$, and dark halo mass parameter by M_i.

For a particle mass parameter m, we calculated the best-fitting dark halo parameters. Our procedure is as follows:

• 1.

  For each galaxy i, numerical values from our model (M) at r_1∕2;i of the dark halo mass ${[M_{1/2;i}]}^{\text{M}}$ and of the velocity dispersion ${[{\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}]}^{\text{M}}$ are calculated from m and M_i: particle mass and dark halo mass fix the length scale b_i and ξ_i from Eqs. (65) and (66), respectively; the dimensionless density profile, η_i(x), is then obtained by solving Eq. (33); this allows us to calculate μ_i(x_1∕2;i) from Eq. (18), with x_1∕2;i = r_1∕2;i∕b_i. ${[M_{1/2;i}]}^{\text{M}}$ is then determined from Eq. (64), and ${[{\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}]}^{\text{M}}$ from Eq. (73), with x_lim;i = r_lim;i∕b_i.

• 2.

  χ² is calculated as $${\chi }^2(m,M_1,\dots ,M_{N_g})={\sum }_{i=1}^{N_g}{\chi }_i^2(m,M_i),$$(76)

  with$${\chi }_i^2(m,M_i)={\text{(}\frac{{[M_{1/2;i}]}^{\text{M}}-M_{1/2;i}}{{\Delta }_{M_{1/2;i}}^{\pm }}\text{)}}^2+{\text{(}\frac{{[{\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}]}^{\text{M}}-{\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}}{{\Delta }_{{\langle {\sigma }_{\text{los};i}^2\rangle }^{1/2}}^{\pm }}\text{)}}^2.$$(77)

  For any quantity Q in Eq. (77), ${\Delta }_Q^{\pm }$ are errors in Q. If ${[Q]}^{\text{M}}>Q$, ${\Delta }_Q^{+}$ is taken; otherwise, ${\Delta }_Q^{-}$.

• 3.

  The best-fitting dark halo parameters are those which minimize Eq. (76); that is, those which minimize${\chi }_i^2(m,M_i)$ given by Eq. (77). We denote the minimum value of ${\chi }_i^2(m,M_i)$ by ${\chi }_i^2(m)$, and the dark halo parameter which leads to it by M_0;i(m); that is, ${\chi }_i^2(m)\equiv {\chi }_i^2(m,M_{0;i}(m))$. The minimum value of ${\chi }^2(m,M_1,\dots ,M_{N_g})$ is denoted by χ²(m), which appears for $M_{0;1}(m),\dots ,M_{0;N_g}(m)$; that is, ${\chi }^2(m)\equiv {\chi }^2(m,M_{0;1}(m),\dots ,M_{0;N_g}(m))$. Thus, $${\chi }^2(m)={\sum }_{i=1}^{N_g}{\chi }_i^2(m).$$(78)

  The best estimate of m is the particle mass for which χ²(m) takes its minimum value. It happens for m₀; that is, ${\chi }_{\text{min}}^2\equiv {\chi }^2(m_0)$.

  We can also derive a range of values of m in which there is a 68% chance of finding the true mass in it. This range of values are determined by the condition $${\chi }^2(m)\le {\chi }_{\text{min}}^2+\Delta {\chi }_{1\sigma ,N_d-N_p}^2,$$(79)

  $\Delta {\chi }_{1\sigma ,n}^2$ being solution of $$\frac{\gamma (n/2,\Delta {\chi }_{1\sigma ,n}^2/2)}{\Gamma (n/2)}=0.6827$$(80)

  (Γ(x) is the Gamma function and γ(x, y), the incomplete gamma function).

5.3 Boson mass estimation from dSph galaxy data

In this work we have dealt with 19 Milky Way (MW) dSph galaxies (eight classical and eleven post-Sloan Digital Sky Survey (SDSS)). Data for r_1∕2, r_lim, M_1∕2, ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ and errors in M_1∕2 and ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ have been taken from the work by Wolf et al. (2010). The core radius, r_c, of the truncated King profile for the stellar component of each galaxy is obtained by imposing Eq. (70).

In Fig. 4, we show χ²(m)∕(N_d − N_p) obtained from data of the eight classical MW dSph, and that obtained from including those of eleven post-SDSS MW dSph as well. In both cases, the minimum of χ² appears for m₀c² = 3.5 × 10⁻²² eV. In the first case, ${\chi }_{\text{min}}^2/7=11.2/7$, while in the second case, ${\chi }_{\text{min}}^2/18=17.8/18$. This value m₀ is in agreement with the result by Hui et al. (2017) who state that the most significant observational consequences occur if the candidate mass is in the range 1–10 ×10⁻²² eV.

This estimation of m₀ used in Eq. (67) leads to the lower bound (LB) for the dark halo mass $$M_{\text{LB}}={5.1}_{-2.8}^{+2.2}\times {10}^6{M}_{\odot }.$$(81)

The model for m = m₀ and M = M_LB presents a length scale b = 0.69 kpc. The classical turning point is found at x₁ = 13.9, that is, at r₁ = 9.6 kpc, containing 79% total dark halo mass, M(r₁) = 4.0 × 10⁶ M_⊙. The sphere containing 99% halo mass has a dimensionless radius x₉₉ = 30.6, which represents r₉₉ = 21.1 kpc. The dark halo ends at x₂ = 42.7; that is r₂ = 29.4 kpc. It should be expected that such large halo would suffer tidal effects in the Milky Way group, but it could be found as an isolated galaxy in regions far away from large galaxies.

For N_g = 8, $\Delta {\chi }_{1\sigma ,7}^2=8.2$; then, applying Eq. (79) to the results shown in Fig. 4 from classical dSph, $m_0{c}^2={3.5}_{-1.0}^{+1.3}\times {10}^{-22}$ eV. When post-SDSS dSph are also considered, N_g = 19 and $\Delta {\chi }_{1\sigma ,18}^2=20.3$; so, Eq. (79) together with Fig. 4 lead to $m_0{c}^2={3.5}_{-2.5}^{+13.4}\times {10}^{-22}$ eV. Differences between both mass ranges appear to be due to the large errors in M_1∕2 given by Wolf et al. (2010) for the post-SDSS MW dSph galaxies. We could say that the data from the classical dSph are those which roughly set the best estimate of m; when more dSph data with larger errors are adding, the estimate does not significantly change but errors increase. In this sense, we note that when the χ² fit is only done with eleven post-SDSS dSph, the best estimate is m₀c² = 1.7 × 10⁻²¹ eV, with ${\chi }_{\text{min}}^2/10=2.1/10$, but the range of values of m with a 68% chance of finding the true mass extends up to m₀c² = 2 × 10⁻²³ eV (for m₀c² = 3.5 × 10⁻²² eV, χ² ∕10 = 6.6∕10).

We should also say that Lyman-α forest, favours mc² ≥ 10–20 ×10⁻²² eV, though it is discussed in the literature whether more sophisticated models of recognization could go in favour of smaller masses (Hui et al. 2017). Nevertheless, it must be borne in mind that these bounds are not derived from FDM simulations; they come from CDM simulations in which a cut-off in the linear power spectrum would account for quantum effects. To derive reliable bounds for Lyman-α forest, it would require a self-consistent fuzzy dark matter simulations.

In Table 4, we show several quantities obtained from the model that minimize χ², for the galaxies treated in this work. From this table:

• 1.

  All dSph galaxies present ξ ≪ ξ_c. Hence, the cosmological constant does not affect to the structure of their dark haloes. It should be noticed that though dark halo masses are between nine and twenty times the lower bound given by Eq. (81), a factor 10 in M applied to Eq. (66) makes ξ to decrease 10⁴ times.

• 2.

  Dark halo masses, M₀, that minimize χ² range from 4.7 × 10⁷ M_⊙ (Hercules) up to 1.1 × 10⁸ M_⊙ (Willman I). As one notes, there is only a factor of 2.3 between these two masses.

• 3.

  With respect to length scale b, it ranges from 31.1 pc (Willman I) up to 74.8 pc (Hercules).

• 4.

  It is actually surprising that the ultra-faint dSph galaxies Segue I and Willman I show the more massive dark haloes. But we note that there is no relation between galaxy luminosity and dark halo mass. In fact, although there are not great differences between dark haloes of the dSph galaxies treated (as much, a factor of 2.4 in mass or length scale), the stellar component differs appreciably from one to another galaxy.

• 5.

  Galaxies present great differences with respect to values of the dimensionless half-light radius, r_1∕2∕b, and the dimensionless limit radius of galaxy luminosity, r_lim∕b (and hence, with respect to the dimensionless core radius, r_c∕b (see Eq. (70)).

• 6.

  There are five dSph galaxies with r_1∕2∕b much smaller than the dimensionless classical turning point of the dark component, x₁ = 11.9 (Willman, Segue I, Canes Ventici II, Leo IV and Coma Berenices show r_1∕2∕b < 2.2). And, only three galaxies show r_1∕2∕b > x₁ (Ursa Minor, Sextans and Formax that shows r_1∕2∕b = 21). The mean value of r_1∕2∕b is 6.2, close to half x₁.

• 7.

  For seven galaxies, their edge of luminosity does not reach x₁ (Willman, Segue I, Coma Berenices, Canes Ventici II, Ursa Major II, Leo IV and Leo II). However, there are other five (Sculptor, Ursa Minor, Canes Venatici I, Sextans and Fornax) that have it further than the sphere containing 99% dark halo mass, x₉₉ = 19.9 (Membrado et al. 1989). In any case, the mean value of r_lim∕b is close to that value, ⟨r_lim∕b⟩ = 21.9.

• 8.

  The quotient M_1∕2∕M₀ gives the percentage of dark halo mass up to half-light radius, r_1∕2. It presents great differences among the galaxies studied. They range from 0.3% for Willman I up to 99.4% for Fornax, showing a mean value of 31.1%.

• 9.

  Mass-to-light ratios also differ appreciably (for luminosity values, L, see Wolf et al. 2010). At half-light radius, the quotient M_1∕2∕(L∕2), ranges from 9.0 (Fornax) up to 3140 (Ursa major II), with ⟨M_1∕2∕(L∕2)⟩ = 712.7.

• 10.

  With respect to values of M_1∕2 and ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ derived from the model minimizing χ², we can see that they are in agreement with data presented by Wolf et al. (2010), except for two dSph galaxies. Sculptor exceeds 12% in M_1∕2 while the upper error represents 7%; and Sextans, 24% and 16%, respectively. Sculptor is also 3% down in ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ while error is 2%.

Now, we could estimate tidal force effects exerted by the Milky Way group on dSph galaxies. A rough estimation of a tidal radius, r_T, can be derived from the equation (see e.g. Membrado & Pacheco 2013) $$\begin{array}{lll}\hfill & \hfill & \left[\frac{G{M}_{\text{MW}}[R]}{R^2}-\frac{\Lambda c^{2}R}{3}\right]+\left[\frac{G{M}_{DS}[r_T]}{r_T^2}-\frac{\Lambda c^2{r}_T}{3}\right]\hfill \\ \hfill & \hfill & \approx \left[\frac{G{M}_{\text{MW}}[R-r_T]}{{(R-r_T)}^2}-\frac{\Lambda c^2(R-r_T)}{3}\right].\hfill \end{array}$$(82)

This equation is the balance of strengths per unit mass in the direction that joins the centres of the dSph and the MW. The first term in brackets on the left hand side denotes the rotation of the dSph around the MW; M_MW[R] stands for the mass of the MW group at a distance R (dark halo and galaxy masses are included) where the dSph galaxy is located. The other terms in brackets are gravitational strengths per unit mass due: (1) to the dSph mass up to r_T, M_DS[r_T], and to the cosmological constant at a distance r_T from the centre of the dSph; and (2) to the MW group mass and to the cosmological constant at a distance R − r_T from the centre of the MW.

For the mass of the Milky Way group as a function of the distance from the MW centre, we have used that proposed by Membrado & Pacheco (2016): $$M_{\text{MW}}[R]=M_{\text{MW}}[R_s]{\left(R/R_s\right)}^p,$$(83)

with R_s = 514 kpc, p = 0.631, and M_MW[R_s] = 3.8 × 10¹¹ M_⊙. In Membrado & Pacheco (2016), we proposed two equations (discrete model) containing surface terms to estimate galaxy sample masses. When the surface terms are neglected, these equations provide the so-called virial and projected masses. In that work, the galaxies studied by Karachentsev (2005) were used. The parameters for M_MW[R] were chosen to reproduce the harmonic radius of the Milky Way group and the mean separation between galaxies and the MW, derived from the discrete model (199 kpc and 214 kpc, respectively).

We haveestimated r_T for the eight classical dSph. The distances to those galaxies given by Karachentsev (2005) have been used. For their masses, we use Eq. (19), that is M_DS[r] = M₀ μ(r∕b), with the M₀’s presented in Table 4. We have seen that for the classical dSph galaxies, r_T ∕b ranges from 75.4 (Sextans) up to 236.8 (Leo I). These dimensionless distances are beyond the dimensionless distance enclosing 99% of the mass (x₉₉ = 19.9). From these results, we conclude that tidal effects on the eight classical dSph galaxies would be negligible.

It can also be estimated the mass, M_m[R], of a dSph galaxy that orbiting around the MW and located at a distance R fulfils r_T = r₉₉. DSph galaxies with masses smaller than M_m[R] could suffer disruption by tidal forces. A value for its crossing time can be obtained by $$t_{\text{cross}}[R]\approx R{\left[\frac{G{M}_{\text{MW}}[R]}{R}-\frac{\Lambda c^2{R}^2}{3}\right]}^{-1/2}.$$(84)

As examples: M_m[200 kpc] = 1.07 × 10⁷ M_⊙, H₀ t_cross[200 kpc] = 0.22, where H₀ is the Hubble function at present; M_m[400 kpc] = 7.4 × 10⁶ M_⊙, H₀ t_cross[400 kpc] = 0.53. Hence, a dSph with the lower limit mass, M_LB = 5.1 × 10⁶ M_⊙, could not be orbiting around the MW. If these galaxies exist, they should be isolated, far away from any massive galaxy.

We have also compared cosmological constant contributions to the gravitational force in the host halo and in the classical dSph galaxies. In the dark halo of the Milky Way group, the nearest dSph galaxy, Ursa Minor, is located at 63 kpc, while the farthest one, Leo I, is found at 250 kpc. At these distances, the repulsive contributions represent 0.4% and 5.5% the attractive contributions. At the tidal radii of Ursa Minor (4.9 kpc) and of Leo I (13.6 kpc), Λ contribution represents 0.1% and 3.5% the mass contribution to the gravitational strength. Hence, it could be said that for the classical dSph galaxies, repulsive to attractive forces ratio is even a bit smaller at their tidal radii than at their locations in the Milky Way Group halo.

As has been seen, MW dSph galaxies show different luminosity component profiles which do not depend on dark halo mass. Despite this variety, by simplicity, we could address our attention to a profile family leading to the same M_1∕2∕M, M_1∕2∕(L∕2) and r_lim ∕b, in order to see how the luminosity component changes for different ξ. We have chosen their mean values from the sample of MW dSph galaxies; that is, M_1∕2∕M = 0.311, M_1∕2∕(L∕2) = 713 and r_lim ∕b = 21.9. Results are presented in Table 5.

The quantities shown in Table 5 are derived as follows. After assuming m = 3.5 × 10⁻²² eV, Eq. (66) gives the dark halo mass, M, for each ξ; then, Eq. (65) provides the length scale b. The value chosen for M_1∕2∕M, allows us to fix M_1∕2. The use of Eq. (63), together with (18), the dimensionless densities η given in Fig. 3, and Eq. (16) determine r_1∕2. The edge of the luminosity, r_lim comes from assuming the value of r_lim∕b. The core radius r_c is calculated from Eq. (70). The velocity dispersion ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ is then obtained using Eq. (73). Finally, luminosity comes from assuming the value of M_1∕2∕(L∕2). The radius enclosing 99% of the dark halo mass, r₉₉, is also presented in this table.

From this table, we can see that, from ξ = 10⁻⁶ up to ξ_c, the edge of the luminosity component, r_lim, ranges from 4 to 15 kpc. This large values, together with small numbers of stars, would make the identification of dSph galaxy members difficult.

Fig. 4 Normalized χ2 from the eight classical MW dSph, and from 19 dSph galaxies which include the classical and the post-SDSS MW dSph, as a function of the boson mass, m.

6 Conclusions

• 1.

  We have included dark energy of the cosmological background in the structure of a cluster composed by self-gravitating collisionless Newtonian bosons. Dark energy is assumed to be the cosmological constant, and bosons are treated in their ground state. The structure of the system is derived by solving a variational problem in the particle number density. The model is used to describe dark haloes of dSph galaxies, and is tested in 19 Milky Way dSph galaxies.

• 2.

  The magnitude of the effects of the cosmological constant on the structure of these systems only depends on the value of one parameter. This parameter, ξ, is the quotient between two gravitational force scales: one, repulsive, due to the cosmological constant, and the another, attractive, due to matter.

• 3.

  The influence of the cosmological constant on bound structures is only appreciable for 10⁻⁵ < ξ ≤ ξ_c = 1.65 × 10⁻⁴. For ξ > ξ_c, bound assemblies can not be created due to the repulsion exerted by the cosmological constant. Hence, ξ_c imposes a lower bound for the system mass which is function of the particle mass. This lower limit is approximately five times greater than that estimated by us in Membrado & Pacheco (2012), where a more simple argument were used. From ξ_c, it is also derived that assemblies are Newtonian if the particle mass is much greater than 2.5 × 10⁻³¹ eV.

• 4.

  The solution for ξ = 0 has a classical turning point where the kinetic energy per particle is null. However, when 0 < ξ ≤ ξ_c, the cosmological constant makes the kinetic energy to have a second zero beyond the classical turning point. This means that there is tunnelling through a potential barrier. We have estimated the mean life of these systems, concluding that their existence is not affected by the age of the Universe.

• 5.

  We have developed the virial theorem for these systems to check the solutions of the variational problem. We have found that the energy of the system is not a third of the product of the energy per particle and the number of particles, as happens when cosmological constant is neglected: now, four thirds of the potential energy due to the cosmological constant must be added.

• 6.

  The model is tested by using four characteristic data from 19 Milky Way dSph galaxies (eight classical and eleven post-Sloan Digital Sky Survey), taken from the work by Wolf et al. (2010): galaxy radius, r_lim; 3D deprojected half-light radius, r_1∕2; luminosity-weighted averages of the square of line-of-sight velocity dispersions, ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$; and mass M_1∕2 enclosed up to r_1∕2. Mass distribution from our model allows us to relate r_1∕2 and M_1∕2 with the halo total mass, M, and the particle mass, m; and, the virial theorem for the stellar component, r_lim and ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$ with M and m. The stellar component is approximated by a truncated King profile.

• 7.

  We have calculated χ² taking into account errors in M_1∕2 and in ${\langle {\sigma }_{\text{los}}^2\rangle }^{1/2}$. From χ², we find that when classical dSph are considered, the range of values of m in which there is a 68% chance of finding the true mass, is ${3.5}_{-1.0}^{+1.3}\times {10}^{-22}$ eV. These values are in the range 1–10 ×10⁻²² eV proposed by Hui et al. (2017) if the dark matter is composed by fuzzy dark matter (ultralight bosons). When post-SDSS galaxies are included, $m={3.5}_{-2.5}^{+13.4}\times {10}^{-22}$ eV are obtained. We should say that discrepancies between both ranges are due to large M_1∕2 errors inpost-SDSS dwarfs.

• 8.

  The best estimate of particle mass, m₀ = 3.5 × 10⁻²² eV, leads to a lower limit for bound dark haloes of M_LB = 5.1 × 10⁶ M_⊙. Hui et al. (2017) have also derived a lower bound, but based on arguments about the mass density at half-light radius (greater than two hundred times the critical density) and on the Jeans length; within the considerable uncertainties and assuming m₀, their value would be about 1.5–3 ×10⁶ M_⊙. These results would make us conclude that the repulsion of the cosmological constant could impose the minimum mass for these dark haloes. In our calculations, surface terms of the virial theorem at x₂ are null; that is, quantum pressure at x₂ is zero. This is due to the fact that at x₂, n′ = n^″ = 0. This result is for isolated dSph galaxies and without taking into account their environments. Nevertheless, it should be said that simulations indicate that quantum pressure would affect the survivability of the smallest halos (see, for example: Mocz et al. 2017). Hence, the minimum mass of dSph galaxies would be also limited by quantum pressure.

• 9.

  The 19 dSph galaxies show values of ξ from 1.6 × 10⁻⁸ (Carina) up to from 6.9 × 10⁻¹⁰ (Willman I). This means that the cosmological constant effects are negligible in the structure of these galaxies. However, their total halo masses range from 4.7 × 10⁷ M_⊙ (Hercules) up to 1.1 × 10⁸ M_⊙ (Willman I); that is ~10–20 times the lower bound for dark halo masses. This is because ξ α M⁻⁴. The influence of the cosmological constant begins to be relevant for dark halo masses smaller than 10⁷ M_⊙ (ξ > 10⁻⁵) up to the lower limit imposed by the cosmological constant repulsion (5.1 × 10⁶ M_⊙). However, we note once again that for these low halo masses there are other mechanisms that could dominate over the dark energy repulsion regarding the disruption of halos, such as quantum pressure or dynamical friction. In this respect, we have estimated that the minimun mass of a dSph galaxy in the Milky Way group that does not suffer tidal effects would be about 7 × 10⁶ M_⊙. Less massive dSph galaxies would have to be isolated, and far away from any massive galaxy.

• 10.

  Assuming M∕M_1∕2 = 0.311, M_1∕2∕(L∕2) = 713, and r_lim∕b = 21.9, corresponding to mean values of those quantities from the 19 galaxies, 10⁻⁵ < ξ ≤ ξ_c dark haloes would contain stellar components inside ~8–15 kpc with luminosities ~9–4 ×10³ L_⊙. With these characteristics, their detection could be rather difficult.

Appendix A: The variational problem

We are dealing with a variational problem of the kind given by Eq. (7); that is, with $${\displaystyle {\int }_0^R\delta }F[n]\text{d}r=0,$$(A.1)

where $F=F(r,n,\dot{n},\ddot{n})$, and R, the radius of the system. Assuming a profile for n(r), we increase a quantity δn(r). Thus, $$\begin{array}{lll}\hfill & \hfill & n(r)\to n(r)+\delta n(r),\hfill \\ \hfill & \hfill & \dot{n}(r)\to \dot{n}(r)+\delta \dot{n}(r),\hfill \\ \hfill & \hfill & \ddot{n}(r)\to \ddot{n}(r)+\delta \ddot{n}(r).\hfill \end{array}$$

Hence, at each point, F[n] will be modified by an amount δF[n] given by $$\delta F=\frac{\text{d}F}{\text{d}n}\delta n=\frac{\partial F}{\partial n}\delta n+\frac{\partial F}{\partial \dot{n}}\delta \dot{n}+\frac{\partial F}{\partial \ddot{n}}\delta \ddot{n}.$$(A.5)

And, taking into account that $$\begin{array}{lll}\hfill & \hfill & \delta \dot{n}=\frac{\text{d}\delta n}{\text{d}r},\hfill \\ \hfill & \hfill & \delta \ddot{n}=\frac{\text{d}\delta \dot{n}}{\text{d}r},\hfill \end{array}$$

Eq. (A.5) reads as $$\delta F=\delta n\text{[}\frac{\partial F}{\partial n}-\frac{\text{d}}{\text{d}r}\text{(}\frac{\partial F}{\partial \dot{n}}\text{)}+\frac{{\text{d}}^2}{\text{d}{r}^2}\text{(}\frac{\partial F}{\partial \ddot{n}}\text{)}\text{]}+\frac{\text{d}}{\text{d}r}\left\{\delta n\left[\frac{\partial F}{\partial \dot{n}}-\frac{\text{d}}{\text{d}r}\left(\frac{\partial F}{\partial \ddot{n}}\right)\right]+\frac{\text{d}\delta n}{\text{d}r}\left[\frac{\partial F}{\partial \ddot{n}}\right]\right\}.$$(A.8)

Thus, using Eq. (A.8) in Eq. (A.1), and assuming δn(0) = δn(R) = 0, and δṅ (0) = δṅ(R) = 0, the condition $$\frac{\partial F}{\partial n}-\frac{\text{d}}{\text{d}r}\text{(}\frac{\partial F}{\partial \dot{n}}\text{)}+\frac{{\text{d}}^2}{\text{d}{r}^2}\text{(}\frac{\partial F}{\partial \ddot{n}}\text{)}=0,$$(A.9)

must be fulfilled at any point.

Appendix B: Kinetic contribution to the virial theorem

The first term of Eq. (45), which we call T₁, can be expressed as $$T_1=4\pi {\displaystyle {\int }_0^R{r}^3}n\frac{\text{d}}{\text{d}r}\left[\frac{n{r}^2{e}_{\text{kin}}}{n{r}^2}\right]\text{d}r.$$(B.1)

Hence, $$T_1=4\pi {\displaystyle {\int }_0^R\left[r\frac{\text{d}(n{r}^2{e}_{\text{kin}})}{\text{d}r}-r^3{e}_{\text{kin}}\frac{\text{d}n}{\text{d}r}-2n{r}^2{e}_{\text{kin}}\right]}\text{d}r,$$(B.2)

and using the expression of e_kin given by Eq. (9) in the first and second term of the integral (B.2), $$T_1=4\pi \left\{-\frac{{\hslash }^2}{4m}{\left[r^2\dot{n}+r^3\ddot{n}-r^3\frac{{\dot{n}}^2}{n}\right]}_0^R-2{\displaystyle {\int }_0^{R}n}{r}^2{e}_{\text{kin}}\text{d}r\right\}.$$(B.3)

The first term on the right hand side of Eq. (B.3) is a surface term. When ṅ (R) = 0 and $\ddot{n}(R)=0$, $$T_1\equiv 4\pi {\displaystyle {\int }_0^R{r}^3}n\frac{\text{d}{e}_{\text{kin}}}{\text{d}r}\text{d}r=-8\pi {\displaystyle {\int }_0^{R}n}{r}^2{e}_{\text{kin}}\text{d}r\equiv -2T,$$(B.4)

T being the kinetic energy of the system.

References

All Tables

All Figures

	Fig. 1 For ξ = 0, gravitational energy, νM and single-particle energy, ϵ, as a functionof the dimensionless distance x. The classicalturning point x1 is marked.
In the text

	Fig. 2 Choice of a0 for different values of ξ (see text).
In the text

Fig. 3 Left panels: dimensionless particle number density, η, as a functionof x, for three values of ξ. The bottom panel corresponds to the critical value ξc. Right panels: dimensionless energies per particle for the same values of ξ shown in left panels (gravitational energy due to matter, νM; total gravitational energy, νM + νΛ; single-particle energy, ϵ), as a functionof the dimensionless distance x. The distances x1, x0 and x2 for ξc are marked in the bottom right panel.

In the text