EXACT SOLUTION TO FINITE TEMPERATURE SFDM: NATURAL CORES WITHOUT FEEDBACK

The idea that DM is a scalar field was first considered by Sin (1994) and independently introduced by Guzmán & Matos (2000). In the SFDM model, the main hypothesis is that DM is an auto-interacting real SF that condensates forming Bose–Einstein Condensate (BEC) "drops" (Magaña & Matos 2012). We interpret these BEC drops to be the halos of galaxies, such that their wave properties and the Heisenberg uncertainty principle stop the DM phase-space density from growing indefinitely, and thus, it avoids cuspy halos and reduces the number of small satellites (Hu et al. 2000).

In the SFDM model, the scenario of galaxy formation proposes that galactic halos form from the condensation of an SF with an ultra-light mass of the order of m ∼ 10⁻²² eV (units where the speed of light c = 1). From this mass, it follows that the critical temperature of the condensation in the SF is T_crit ∼ m^−5/3 ∼ TeV, which is very high. Thus, BEC drops are formed very early in the universe. It has been proposed that these drops are the halos of galaxies (Matos & Ureña 2001), i.e., that halos are gigantic clumps of SF.

The DM halos of galaxies can be described in the non-relativistic regime, where they can be seen as Newtonian gas. When the SF has self-interaction, we need to add a quartic term to the SF potential, and in the Newtonian limit, the equation of state of the SF is that of a polytrope of index 1 (Suárez & Matos 2011; Harko 2011). Some studies of the stability of these SF configurations have shown that stable large-scale configurations are not preferred (Colpi et al. 1986; Balakrishna et al. 1998; Valdez et al. 2011). Though the critical mass for stability depends on which parameter values were used, all reach the same conclusion, that very large configurations, like those of cluster scales (masses of M ⩾ 10¹³ M_☉), are usually unstable. Therefore, these structures were most likely to form just as in the CDM model—by hierarchy (Matos & Ureña 2001; Suárez & Matos 2011), i.e., by mergers of smaller halos. The concept behind the SFDM model is as follows: after inflation big structures start to grow hierarchically, as in the CDM model, and its growth is then boosted by the SB mechanism. Inside these structures, galactic sized halos will form by condensation. Thus, all predictions of the CDM model at large scales are reproduced by SFDM (Colpi et al. 1986; Gleiser 1988; Matos & Ureña 2001; Chavanis 2011). Although the model is quite successful, there are at least two reasons why the model must be complemented in addition to finding an explanation to the two discrepancies discussed in the Introduction.

The first discrepancy is found when we consider the fully condensed system at temperature T = 0. The fits to the rotation curves (RCs) of the LSBs (Robles & Matos 2012; Böhmer & Harko 2007) show deviations from the observed data at large radii because the DM halos are modeled by only considering the ground state (complete condensation), the DM density of which and velocity profile are given by (Böhmer & Harko 2007)

$$\begin{eqnarray} \rho ^0(r) & = & \rho ^0_{0} \frac{\sin (\pi r/ \hat{R})}{\pi r/ \hat{R}}, \end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray} V^2_{0}(r) & = & \frac{4 \pi G \rho ^0_{0}}{K^{2}} \biggl (\frac{\sin (Kr)}{Kr} - \cos (Kr) \biggr), \end{eqnarray} \tag{ 1b }$$

where $K= \pi / \hat{R}$, ρ⁰₀ = ρ⁰(0) is the central density, and the halo radius determined by $\rho (\hat{R})=0$ is

$$\begin{equation} \hat{R}= \pi \sqrt{\frac{\hbar ^{2} b}{G m^{3}}}. \end{equation} \tag{ 2 }$$

Here, ℏ is Planck's constant divided by 2π, m is the mass of the DM particle, G is the gravitational constant, and b is the scattering length. The latter is related to the coupling constant λ' by λ' = 4πℏ²b/m. The density profile depends on two fitting parameters, the central density and a length scale $\hat{R}$. However, $\hat{R}$ depends only on fixed parameters, the interaction parameter λ' and the mass of the SF particle, which implies that it should not vary from galaxy to galaxy. However, when fitting the rotation curves of galaxies, as $\hat{R}$ is a fitting parameter, it has different values for each galaxy (Robles & Matos 2012; Böhmer & Harko 2007). This is somewhat contradictory and represents a problem for the simplest SFDM model.

The second problem lies in the rapid decrease of the velocity profile (Equation (1b)) after its maximum value. Such a decrease causes a disagreement between the fits and the observed data in large galaxies because the data usually remain "flat" until the outermost regions (Swaters et al. 2000). In addition to this, and for large galaxies, when we aim for the best fit to the velocity maximum, we obtain a worse fit in the outer regions, i.e., the better the fit to the velocity maximum in the RC, the worse the fit becomes in the outer regions and vice versa.

One approach to solving those problems was considered by Matos & Ureña (2007) and later by Harko & Madarassy (2011). It consisted of including the finite temperature of the DM in the DM halos. Harko & Madarassy (2011) show that by including a small correction to the pure condensed state, albeit in a different way from the one in this paper, it is possible to solve the first discrepancy and to partially solve the second one. The inclusion of temperature T mainly does two things: (1) it makes the halo radius temperature dependent, thus it is no longer fixed for all galaxies and (2) it lifts the RC fit in the outer region and keeps it flat until the last value. However, the consequence of (2) is negligible when the halo temperature is T < 0.5 T_BEC, where T_BEC is the critical temperature of the Bose–Einstein condensation. If T_BEC ∼ TeV (c = 1), then we would expect that the present halos would be well approximated by Equation (1b) and hence, we would be unable to simultaneously obtain a good fit to the RC maximum and the RC outer regions. Though this last problem is not readily visible in galaxies with a small radius (outer radius of ⩽10 kpc), it is conspicuous in large galaxies, therefore problem (2) is only partially solved.

Thus, an alternative approach to solving these two SFDM discrepancies considers the non-condensed SF configurations at T = 0, i.e., SF configurations in excited states (Balakrishna et al. 1998; Ureña & Bernal 2010; Bernal et al. 2010). These configurations fit RCs up to the last data point and can even reproduce the wiggles seen at large radii in high-resolution observations (Sin 1994; Colpi et al. 1986). The problem found in this kind of solution was that the configurations required for good fits (4-5 exited states) were unstable and decayed to the ground state in a short time (Siddhartha & Ureña 2003). Therefore, we will expect to see that most DM halos are in their ground state, which means that the disagreement at large radii would remain.

Motivated to solve all these issues, we consider the following scenario that includes the temperature of the DM and the excited states of the SF.

The idea is that the DM is a spin 0 scalar field Φ, with a repulsive interaction embedded in a thermal bath of temperature T; we also consider the finite temperature corrections up to one loop in the perturbations. This system is described by the potential (Kolb & Turner 1987; Dalfovo et al. 1999)

$$\begin{equation} V (\Phi) = -\frac{1}{2} \frac{\hat{m}^2 c^2}{\hbar ^2} \Phi ^2+\frac{\hat{\lambda }}{4}\Phi ^4 + \frac{\hat{\lambda }}{8}k_\mathrm{B}^2T^2\Phi ^2 -\frac{\pi ^2 k_\mathrm{B}^4T^4}{90\, \hbar ^2 c^2}, \end{equation} \tag{ 3 }$$

for the case when k_BT ≫ $\hat{m} c^2$. Here, k_B is Boltzmann's constant, $\hat{\lambda }= \lambda /(\hbar ^2 c^2)$ is the parameter describing the interaction, $\hat{\mu }^2$ := $\hat{m}^2 c^2$/ℏ² is a parameter, and T is the temperature of the thermal bath. The first term in V(Φ) relates to the mass term, the second to the repulsive self-interaction, the third to the interaction of the field with the thermal bath, and the last to the thermal bath alone.

At some high-enough temperature in the early universe, the SF interacts with the rest of the matter; however, due to the expansion of the universe, its temperature will keep decreasing. Eventually, when the temperature is sufficiently small, the SF decouples from the interaction with the rest of the matter and follows its own thermodynamic history, while the background keeps cooling down due to the expansion. Moreover, as the temperature continues decreasing, the field will reach the minimum of the potential Φ ≈0. This initial minimum of the potential eventually becomes a local maximum and after this moment, the initial Z₂ symmetry of the potential V(Φ) will be broken. The latter happens at a critical temperature T_C given by

$$\begin{equation} k_\mathrm{B}T_C=\frac{2\hat{m}c^2}{\sqrt{\lambda }}. \end{equation} \tag{ 4 }$$

We will see that the critical temperature determines the moment at which the DM fluctuations can start growing. They grow from the moment when T <T_C until they reach a stable equilibrium point, for example, in Φ²_min = k²_B(T²_C − T²)/4 (see Section 2.3).

The perturbed system of a scalar field with a quartic repulsive interaction and with temperature of zero has been studied before (Colpi et al. 1986; Ureña & Bernal 2010). Following the same procedure, we study the evolution of the SF in an FRW universe. We write the metric tensor as $\mathbf {g}= \mathbf {g}^0 + \delta \mathbf {g}$, where $\mathbf {g}^0$ is the unperturbed FRW background metric and $\delta \mathbf {g}$ is the perturbation. The perturbed line element in conformal time η is (we take c = 1 in this subsection)

$$\begin{eqnarray} ds^2 &=& a(\eta)^2(-(1+2\psi)d\eta ^2+2B,_id\eta dx^i \nonumber \\ &&+ a(\eta)^2[(1-2\phi)\delta _{ij}+2E,_{ij}]dx^idx^j, \end{eqnarray} \tag{ 5 }$$

with a being the scale factor, ψ the lapse function, ϕ gravitational potential, B the shift, and E the anisotropic potential. We separate the energy–momentum tensor and the field as $\mathbf {T}=\mathbf {T}_0+\delta \mathbf {T}$ and Φ(x^μ) = Φ₀(η) + δΦ(x^μ), respectively. As we are studying the linear regime δΦ(x^μ) ≪ Φ₀(η), we can approximate V(Φ) ≈ V(Φ₀). We work in the Newtonian gauge where the metric tensor $\mathbf {g}$ becomes diagonal and as a result, in the trace of Einstein's equations, the scalar potentials ψ and ϕ are identical, therefore; ψ relates to the gravitational potential.

Changing to the cosmological time t when using the relation (d/dη) = a(d/dt), the perturbed Einstein's equations δGⁱ_j = 8πGδTⁱ_j to first order for a scalar field in the Newtonian gauge (E = 0 = B) are

$$\begin{eqnarray} {-}8 \pi G \delta \rho _{\Phi }&=&6H(\dot{\phi }+H\phi)-\frac{2}{a^2}\nabla ^2\phi, \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} 8\pi G \dot{\Phi }_0\delta \Phi ,_i&=&2(\dot{\phi }+H\phi),_i, \end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray} 8\pi G\delta p_{\Phi }&=&2[\ddot{\phi }+3H\dot{\phi }+(2\dot{H}+H^2)\phi] \end{eqnarray} \tag{ 6c }$$

with $\dot{}=\partial /\partial t$ and H = d(ln a)/dt: The perturbed density δρ_Φ and the perturbed pressure δp_Φ are defined in terms of the perturbed energy momentum tensor as

$$\begin{eqnarray} \delta T^0_0&=&{-}\delta \rho _{\Phi }= {-} \big(\dot{\Phi }_0\dot{\delta \Phi }-\dot{\Phi }_0^2\psi +V,_{\Phi _{0}}\delta \Phi \big), \nonumber \\ \delta T^0_i&=& {-}\frac{1}{a}(\dot{\Phi }_0\delta \Phi ,_i), \nonumber \\ \delta T^i_j&=&\delta p_{\Phi }= \big(\dot{\Phi }_0\dot{\delta \Phi }-\dot{\Phi }_0^2\psi -V,_{\Phi _{0}}\delta \Phi \big)\delta ^i_j. \end{eqnarray} \tag{ 7 }$$

The systems of equations (6) and (7) describe the evolution of the scalar perturbations. To study the evolution of the SF perturbations, we use the perturbed Klein–Gordon equation

$$\begin{eqnarray} \ddot{\delta \Phi }&+& 3 H \dot{\delta \Phi }-\frac{1}{a^{2}} \mathbf {\nabla }^2\delta \Phi + V,_{\Phi _0\Phi _0}\delta \Phi \nonumber \\ &+&2V,_{\Phi _0}\phi -4\dot{\Phi }_0\dot{\phi }=0. \end{eqnarray} \tag{ 8 }$$

Equation (8) can be rewritten as

$$\begin{equation} \square \delta \Phi +\left.\frac{d^2V}{d\Phi ^2}\right|_{\Phi _0}\delta \Phi +\left.2\frac{dV}{d\Phi }\right|_{\Phi _0}\phi -4\dot{\Phi }_0\dot{\phi }=0, \end{equation} \tag{ 9 }$$

where the D'Alambertian operator is defined as

$$\begin{equation} \square : = \frac{\partial ^2}{\partial t^2}+3H\frac{\partial }{\partial t}- \frac{1}{a^2} \mathbf {\nabla }^2. \end{equation} \tag{ 10 }$$

Essentially Equation (8) represents a harmonic oscillator with a damping of $3H\dot{\delta \Phi }$ and an extra force of $-2\phi V,_{\Phi _0}$. Equation (8) has oscillating solutions if $(V_{,\Phi \Phi } -({1}/{a^{2}})\mathbf {\nabla }^2)\delta \Phi$ is positive. This equation contains growing solutions if this term is negative, that is, if V_{, ΦΦ} is negative enough. From here, we see an important feature. These perturbations grow only if V has a maximum, even if it is a local one. Here, the potential is unstable and during the time when the scalar field remains at the maximum, the scalar field fluctuations grow until they reach a new stable point. If we use Equation (3) the transition from a local minimum to a local maximum happens when T = T_C, thus, we see why T_C determines the moment in which the DM fluctuations can start growing. This implies that the galactic scale halos could have formed within this period and with similar features. Finally for the background field equation we assume that ϕ ≈ 0, therefore its equation reads

$$\begin{equation} \ddot{\Phi }_0 + 3H \dot{\Phi }_0 + V,_\Phi (\Phi _0) =0. \end{equation} \tag{ 11 }$$

Φ₀ depends only on time.

We now suppose that the temperature is sufficiently small, such that the interaction between the SF and the rest of matter has decoupled—after this moment the field stops interacting with the rest of the particles. We also assume that the symmetry break (SB) took place in the radiation-dominated era in a flat universe. We mentioned that after the SB, the perturbations can grow until they reach their new minimum, thus, each perturbation has a temperature at which it forms and separates from the background field following its own evolution. We denote this temperature of formation by $T_\Phi$: = T(t_form), where t_form is the time in which the halo forms. Under these assumptions the equation for an SF perturbation which is formed at $T_\Phi$ reads

$$\begin{eqnarray} \square \delta \Phi &+&\frac{\hat{\lambda }}{4 } \big[k_\mathrm{B}^2 (T^2_{\Phi }-T^2_C) + 12\Phi _0^2 \big]\delta \Phi -4\dot{\Phi }_0\dot{\phi } \nonumber \\ &+&\frac{\hat{\lambda }}{2} \big[k_\mathrm{B}^2 (T^2_{\Phi }-T^2_C)+4\Phi _0^2 \big]\Phi _0\phi =0. \end{eqnarray} \tag{ 12 }$$

In Figure 1, we show the behavior of the potential for different temperatures, and we see that the SB takes place at T = T_C.

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In the SFDM model, the initial fluctuations come from inflation as in the standard CDM paradigm. Later on the field decouples from the rest of the matter and goes through an SB which can increase the fluctuation's amplitude, forming the initial structures of the universe.

In this work, as one of our main interests' is to find an exact analytical solution to Equation (12). We will not pursue the task of solving it numerically here. However, the numerical work done in Magaña & Matos (2012) has confirmed that the behavior of the SF perturbation just after the SB is what we had expected from our analysis of Equation (9). They have analyzed in some detail the evolution of a perturbation with a wavelength of 2 Mpc and a density contrast of δ = 1 × 10⁻⁷ after the SB. They took a = 10⁻⁶ as an initial condition and evolved it until a = 10⁻³. They also analyzed the case where T ∼ T_C and showed that as the temperature decreases and goes below T_C, Φ₀ falls rapidly to a new minimum, where it remains oscillating. Meanwhile, the SF fluctuation grows quickly as Φ₀ approaches the new minimum. It is only before the SB that the SF remains homogeneous.

From here we see that in the SFDM model, the primordial DM halos form almost at the same time due to the phase transition produced by the SB. In the nonlinear regime, these halos can merge with other halos, forming larger structures just as in the CDM model. Therefore, one of the main differences with the CDM model lies in the initial formation of the DM halos, where they are formed very rapidly and almost at the same time. From this we expect that they possess similar features. This difference between the SFDM and CDM models can be tested by observing well-formed high-redshift galaxies and also by comparing the characteristic parameters of several DM dominated systems, for example, by observing that indeed, dwarf or LSB galaxies possess cores even at high redshift, especially since the CDM simulations of dwarf galaxies by Governato et al. (2012) suggest that their DM density profiles were initially cuspy, but later turn into core profiles due to feedback processes.

In this work, we are interested in the galactic scale of DM halos after their formation. Thus, we constrain ourselves to solve the Newtonian limit of Equation (12), when Φ is near the minimum of the potential and after the SB, where we expect it to be stable. In this limit, we also expect the gravitational potential to be locally very homogeneous in the beginning of its collapse, thus, $\dot{\phi }\approx 0$. The stability of these halos will be shown elsewhere. For clarity, from here on we stop using units in which c = 1. We now find an exact spherically symmetric solution for the SF when it is near the minimum of the potential ($V^{\prime }(\Phi)|_{\Phi _0}\approx 0$), i.e., when Φ²₀ = Φ²_min = k²_B(T²_C −T²_Φ)/4 and T_Φ < T_C. For the Newtonian case H = 0, $\dot{\Phi }_ {0} \approx 0$, and Equation (12) reads

$$\begin{equation} \ddot{\delta \Phi }-\nabla ^2\delta \Phi + \frac{\lambda k^2_\mathrm{B}}{2 \hbar ^2} \big(T^2_C-T^2_{\Phi } \big)\delta \Phi =0. \end{equation} \tag{ 13 }$$

Equation (13) describes the evolution of the SF fluctuation, which formed at T_Φ after the SB. Moreover, this equation is linear. In general, halos can evolve with different histories depending on their environment, star formation, gas accretion, and other secular processes that occur in the hosted galaxy. Thus, in order to study these systems in detail, we will need the full nonlinear regime. However, we can still regard Equation (13) as a first approximation for those halos of isolated galaxies or of galaxies that are not heavily influenced by their neighbors, for example, isolated galaxies or galaxies that are in low-mass density regions can be well described by this equation. This is mainly because their halo profiles are not expected to be affected substantially by their initial profiles during their evolution. Furthermore, we expect that the galaxies remain in the Newtonian gravitational regime, thus we can consider that the linear approximation is sufficient for describing a galaxy. Therefore, we interpret a solution of this system as the temperature-corrected density profile of early DM halos.

We find that the ansatz

$$\begin{equation} \delta \Phi = \delta \Phi _0 \frac{\sin (k r)}{kr} \cos (\omega t) \end{equation} \tag{ 14 }$$

is an exact solution to Equation (13) provided that

$$\begin{equation} \omega ^2 = k^2 c^2 + \frac{\lambda k^2_\mathrm{B}}{2 \hbar ^2} \big(T^2_C-T^2_{\Phi } \big). \end{equation} \tag{ 15 }$$

Here, δΦ₀ is the amplitude of the fluctuation. From Equation (15) we note that now k = k(T_Φ). For an easier comparison with the observations, we use the standard definition of number density $n(\mathbf {\mathit {x}},t)$ = κ(δΦ)², where κ is a constant that gives us the necessary units so that we can interpret $n(\mathbf {\mathit {x}},t)$ as the number density of DM particles, as Φ has energy units. With this in mind, we can define an effective mass density of the SF fluctuation as ρ = mn and a central density by ρ₀ = mκ(δΦ₀)². It is important to note that while δΦ₀ is not obtained directly from observations, the value of ρ₀ is a direct consequence of the RC fit, and for this reason, it is preferable to work with ρ₀ instead of δΦ₀.

Combining Equation (14) and the definition of n we obtain a finite temperature density profile

$$\begin{equation} \rho (r) = \rho _{0} \frac{\sin ^{2} (kr)}{(kr)^{2}}, \end{equation} \tag{ 16 }$$

provided that

$$\begin{eqnarray} \quad\quad\, k^2_\mathrm{B} T^2_{\Phi }& =& k^2_\mathrm{B} T^2_C - 4\Phi _0^2, \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \Phi ^2_0 &=& \Phi ^2_{\mathrm{min}}. \end{eqnarray} \tag{ 18 }$$

Here, k(T_Φ) and ρ₀ = ρ₀(T_Φ) are fitting parameters while λ, T_C (or m), and κ are free parameters to be constrained by observations.

For galaxies the Newtonian approximation provides a good description; therefore, from Equation (16), we obtain the mass and rotation curve velocity profiles

$$\begin{eqnarray} M(r) &=& \frac{4 \pi \rho _0}{k^2} \frac{r}{2} \biggl (1-\frac{\sin (2 k r)}{2 k r} \biggr), \end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray} V^2(r) &=& \frac{4 \pi G \rho _0}{2 k^2} \left(1-\frac{\sin (2 k r)}{2 k r} \right), \end{eqnarray} \tag{ 19b }$$

respectively. For comparison, we give the Einasto rotation curve profile, which lately seems to give a better description of DM halos in CDM simulations (Navarro et al. 2010; Merrit et al. 2006; Graham et al. 2006),

$$\begin{equation} V^2_\mathrm{E} = 4 \pi G \rho _{-2} \frac{r^3_{-2}}{r} \biggl [ \frac{e^{2/ \alpha }}{\alpha } \biggl (\frac{\alpha }{2}\biggr)^{(3/\alpha)} \gamma \left(\frac{3}{\alpha },x ^{\prime } \right) \biggr ], \end{equation} \tag{ 20 }$$

with γ being the incomplete gamma function given by

$$\begin{equation*} \gamma \left(\frac{3}{\alpha },x^{\prime } \right)= \int ^{x^{\prime }}_0 e^{-\tau } \ \tau ^{(3/\alpha)-1} d\tau \end{equation*}$$

where x' = (2/α)(r/r₋₂)^α, r₋₂ is the radius in which the logarithmic slope of the density is −2, ρ₋₂ is the density at the radius r₋₂, and α is the parameter that describes the degree of curvature of the profile (Merrit et al. 2006; Graham et al. 2006).

We now define the radius R of the SFDM distribution using the condition ρ(R) = 0. This fixes the relation

$$\begin{equation} k_j \mathrm{R} = j \pi , \ \ \ j=1,2,3,..., \end{equation} \tag{ 21 }$$

where j is the number of the excited states required to fit a galaxy RC up to the last measured point. From Equations (13), (14), and (21) we find that the halo allows excited states, i.e., the excited states are also solutions of Equation (13). As this equation is linear, there can be halos in a combination of excited states: For these cases, the total density ρ_tot is the sum of the densities in the different states

$$\begin{equation} \rho _\mathrm{tot}=\sum _j \rho ^j_0 \frac{\sin ^{2} (j \pi r/\mathrm{R})}{(j \pi r/\mathrm{R})^{2}}, \end{equation} \tag{ 22 }$$

with ρ^j₀ being the central density of the state j. It is important to note that there is only one halo formation temperature, $T_\Phi$, and that the number of states that compose one DM halo depends on each halo—one reason for this is that the initial conditions change from one to another. Finally, we note that for each state j, there exists both ω_j and k_j, which satisfy their corresponding Equation (15).

An additional feature of Equation (16) is the presence of "wiggles." These oscillations are characteristic of SF configurations in excited states, which were also seen in Sin (1994). Also, we define the distance where the first peak (maximum) in the RC is reached as r¹_max. This determines the first local maximum of the RC velocity, which can be obtained from Equation (23)

$$\begin{equation} \frac{\cos (2 \pi j y)}{2(\pi j)^2 y} \left[ \frac{\tan (2 \pi j y)}{2 \pi j y} -1\right] = 0, \end{equation} \tag{ 23 }$$

where we used Equation (21) and y := (r¹_max/R).