© The Authors 2024

1. Introduction

According to the Planck 2018 mission (Planck Collaboration VI 2020), dark matter (DM) constitutes 26.8% of the total mass-energy density of the Universe, which is almost 5.5 times more than the amount of ordinary matter (OM). Of the theories proposed to explain DM (e.g., Rees 2003; Freeman & McNamara 2006; Frieman et al. 2008; Overduin & Wesson 2004; Barbier et al. 2005; Sugita et al. 2008; Arkani-Hamed et al. 2009; Komatsu et al. 2011; Bartone & Hooper 2018; Giagu 2019; Watson & Musielak 2020; Hui 2021; Oks 2021; Musielak 2021, 2022; Chadha-Day & Ellis 2022), some predict the existence of weakly interacting massive particles (WIMPs); however, all attempts to detect WIMPs have failed so far (e.g., Ackermann et al. 2011; Ibarra et al. 2013; Marrodán Undagoitia & Rauch 2016; Hochberg et al. 2022). Recently, Rogers et al. (2023) demonstrated that the large-scale distribution of matter may imply that DM is composed of ultralight axions with masses in the range of 10⁻²⁸ − 10⁻²⁵ eV. A number of experiments (e.g., Crisosto et al. 2020; Jiang et al. 2021; Adair et al. 2022; Chadha-Day 2022) have failed to find any trace of axions.

In order to understand the nature and origin of DM, different relativistic and nonrelativistic theoretical models have been constructed. In this paper, only the latter are considered and discussed. A model of galactic DM halos that is based on the Schrödinger equation, which is known for its correct description of the microscopic structure of OM (e.g., Merzbacher 1998), was proposed by Sin (1994), who suggested that DM particles are extremely light bosons with masses of about 10⁻²⁴ eV. It was an interesting idea because it allowed us to solve the Schrödinger equation on the galactic scale because these particles have a very long Compton wavelength. In this model, the gravitational potential added to the equation is calculated by solving the Poisson equation with the DM density as the forcing term. Hu et al. (2000) considered DM particles with masses of about 10⁻²² eV and improved the previous models. However, more detailed numerical studies by Spivey et al. (2013, 2015) revealed that the models require the DM particles to have different masses for different galactic halos. This conclusion is difficult to justify physically.

Since the masses of ultralight axions proposed by Rogers et al. (2023) are even lower than the masses of extremely light bosons considered by Sin (1994), Hu et al. (2000), and Spivey et al. (2013, 2015), we might expect that numerical solutions to the Schrödinger equation for ultralight axions would also reveal that these particles must have different masses for different galactic DM halos. However, this extrapolation may not be valid because a correct description of ultralight axions in galactic DM halos may require a fully relativistic theory of DM, instead of using its nonrelativistic approximation based on the Schrödinger equation with the gravitational potential term.

The previous attempts to formulate nonrelativistic theories of DM using the Schrödinger equation have failed. Therefore, a different governing equation may be needed to formulate a new DM theory. An equation like this was recently derived (Musielak 2021) by using the irreducible representations (irreps) of the so-called extended Galilean group of the metric, which includes rotations, translations, and boost in Galilean space and time (Lévy-Leblond 1967, 1969; Kim & Noz 1986). The new equation is nonrelativistic and asymmetric in space and time derivatives, and it was recently used to describe the propagation of classical waves (Musielak 2023a) and to construct a quantum theory of DM (Musielak 2022, 2023b). It is known (e.g., Musielak & Fry 2009) that the Schrödinger equation can also be derived from the same irreps, which means that both equations are complementary to each other.

The main physical implication of this complementarity is that the basic quantum rules (e.g., Merzbacher 1998) can be applied to them, and each equation can be used to formulate a quantum theory. However, while the Schrödinger equation correctly describes the atomic structure and its unitary evolution, the new equation naturally accounts for its interactions with macroscopic measuring devices. In other words, a quantum theory based on the new equation can be used to describe the quantum measurement problem (Musielak 2024), which violates the unitary evolution represented by the Schrödinger equation (e.g., Merzbacher 1998).

In this paper, the new equation is used to develop a quantum model of a galactic cold dark matter halo with a given Navarro–Frenk–White (NFW) density profile (Navarro et al. 1996, 2010). The model predicts a quantum structure of the halo that is significantly different than that proposed by any previous DM model (e.g., Sin 1994; Hu et al. 2000; Spivey et al. 2013, 2015; Schive et al. 2014; Zhang et al. 2017). The model also naturally accounts for DM being collisionless, and it predicts that DM can emit radiation of only one frequency, whose values are computed for DM particles of different masses. The paper suggests potential observational verification of the theory.

This paper is organized as follows: The basic equations and a method for constructing galactic halo models are presented in Sect. 2. A quantum model of a dark matter halo is described in Sect. 3. The developed quantum model is applied to a halo with an NFW density profile in Sect. 4. The dark matter characteristic frequency and its observational detection is discussed in Sect. 5. Section 6 is finally devoted to our conclusions.

2. Basic equations and galactic halo models

In Galilean Relativity, space and time are separated and represented by two different metrics that are invariant with respect to all translations, rotations, and boosts that form the Galilean group of the metric (Lévy-Leblond 1967, 1969; Kim & Noz 1986). According to Inönu & Wigner (1952) and Bargmann (1954), the irreducible representations (irreps) of the Galilean group are known, and they can be used to derive the basic equations of physics that are allowed to exist in Galilean space and time. A scalar wavefunction Φ(t, x) transforms as one of the irreps if the following eigenvalue equations i∂_tΦ = ωΦ and −i∇Φ = kΦ are satisfied, where ∂_t = ∂/∂t, and ω and k are the eigenvalues that label the irreps (Musielak & Fry 2009).

By using the de Broglie relation (e.g., Merzbacher 1998), the eigenvalues can be determined (Musielak 2021). Then, the eigenvalue equations give the following two equations that are asymmetric in space and time derivatives:

$$\begin{array}{c}\hfill [i\frac{\partial }{\partial t}+\frac{ħ}{2m}{\mathrm{\nabla }}^2]\mathrm{\Psi }(t,\mathit{x})=0,\end{array}$$(1)

which is the Schrödinger equation (e.g., Merzbacher 1998), and a new asymmetric equation (Musielak 2021),

$$\begin{array}{c}\hfill [\frac{i}{\omega }\frac{{\partial }^2}{\partial t^2}+\frac{ħ}{2m}\mathit{k}·\mathrm{\nabla }]\mathrm{\Phi }(t,\mathit{x})=0,\end{array}$$(2)

which is with x = (x, y, z), and Ψ(t, x) and Φ(t, x) denoting the wavefunctions satisfying these equations. The differences in time and space derivatives between these equations and the explicit presence of both eigenvalues in Eq. (2) makes the equations complementary to each other.

The equations describe free (noninteracting) particles of mass m and zero spin. To construct models of galactic DM halos, a potential V(x) must be included in the equations. In previous work (e.g., Sin 1994; Hu et al. 2000), the Schrödinger equation was used with the potential V(r), where r is the spherical coordinate, determined from the Poisson equation ∇²V(r) = 4πGρ_DM, with ρ_DM = mM_o|Ψ|², and M_o being a mass parameter. The Schrödinger and Poisson equations were solved numerically, and the resulting radial probability densities were computed and compared to the Einasto DM density profiles (Einasto & Haud 1989). Despite reasonably good agreement, it was concluded that the constructed models were physical incorrect because they required DM particles of different masses for different galaxies Spivey et al. (2013, 2015).

Since the models of galactic DM halos based on the Schrödinger equation have failed, attempts were made to construct halo models by using the new asymmetric equation (Musielak 2021, 2022). Following a different approach, the models were constructed based on full analytical solutions to the new asymmetric equation (Musielak 2023b). The main advantage of having these analytical solutions is that relations between the physical parameters of the model are obtained and are used to make predictions that can be verified by observations.

Let a DM particle of mass m be located at point P inside a spherical DM halo of radius R_h and total mass M_h, and let the distance between P and the halo center be r. Then, according to Newtonian dynamics, the force acting on the particle is F_g(r) = GM(r)m/r², and this force is balanced by the centrifugal force $F_c(r)=mr{\mathrm{\Omega }}_c^2(r)$, which gives the circular orbital frequency of the particle,

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_c^2(r)=\frac{GM(r)}{r^3}·\end{array}$$(3)

where $M(r)=4\pi {\displaystyle {\int }_0^r}\rho (\stackrel{\sim }{r}){\stackrel{\sim }{r}}^2\mathrm{d}\stackrel{\sim }{r}$, with ρ(r) denoting the density of DM inside the halo, which must be specified.

Equation (2) can be modified by including ${\mathrm{\Omega }}_c^2(r)$ and identifying the eigenvalues ω and k with the characteristic frequency, ω_o, and wavevector, k_o, of DM by taking ω = ω_o and k = k_o. The result is

$$\begin{array}{c}\hfill [\frac{{\partial }^2}{\partial t^2}-i\frac{{\epsilon }_o}{2m}|{\mathit{k}}_o·\widehat{\mathit{r}}|\frac{\partial }{\partial r}+{\mathrm{\Omega }}_c^2(r)]\mathrm{\Phi }(t,r)=0,\end{array}$$(4)

where ε_o = ℏω_o represents quanta of energy, and $\widehat{\mathit{r}}=\mathit{r}/r$; there is no dependence of the angle on θ nor on ϕ. This is the governing equation used herein to develop a quantum theory of a galactic DM halo.

3. Quantum model of a dark matter halo

Solutions of Eq. (4) were found by separating the variables Φ(t, r) = χ(t)η(r), which gives the following time-independent equation for η(r):

$$\begin{array}{c}\hfill \frac{\mathrm{d}\eta }{\eta }=i\frac{2m}{{\epsilon }_o|{\mathit{k}}_o·\widehat{\mathit{r}}|}[{\mu }^2-{\mathrm{\Omega }}_c^2(r)]\mathrm{d}r,\end{array}$$(5)

where μ² is the separation constant to be determined. In general, the solutions of this equation are complex (Musielak 2023b). Their real part can be written as

$$\begin{array}{c}\hfill \eta (r)={\eta }_{o}cos\left(\frac{2G{m}^2[{\mu }^{2}r-I_c(r)]}{{\epsilon }_o^2|{\widehat{\mathit{k}}}_o·\widehat{\mathit{r}}|}\right),\end{array}$$(6)

where $I_c(r)={\displaystyle {\int }_0^r}{\mathrm{\Omega }}_c^2(\stackrel{\sim }{r})\mathrm{d}\stackrel{\sim }{r}$, and ${\mathit{k}}_o=k_o{\widehat{\mathit{k}}}_o$, with k_o = 1/λ_o = Gm²/ε_o = const.

To formulate a quantum theory, it is required that its quantum orbits are labeled by positive integers (the principle quantum numbers), which limits the values of the cosine function to be either the maxima (±2nπ) or minima [±(2n + 1)π], where n = 0, 1, 2, 3, …, and the plus and minus signs correspond to the orbits inside and outside the halo, respectively. By choosing n = 0 to label the orbit at the edge of the halo r = R_h, all the remaining quantum orbits inside and outside the halo must be labeled by using ±2nπ because ±(2n + 1)π makes the labeling inconsistent with the selected reference orbit at r = R_h. In the remaining parts of this paper, only the orbits inside the halo are considered.

Thus, the separation constant is given by

$$\begin{array}{c}\hfill {\mu }^2=\frac{1}{r}{I}_c(r)+n\pi {\omega }_o^2|{\widehat{\mathit{k}}}_o·\widehat{\mathit{r}}|\frac{{ħ}^2}{G{m}^{3}r}·\end{array}$$(7)

Finding μ² requires r = R_h, $\widehat{\mathit{r}}={\widehat{\mathit{R}}}_h$, and evaluating I_c(R_h), which gives

$$\begin{array}{c}\hfill I_c(R_h)={\displaystyle {\int }_0^{R_h}}{\mathrm{\Omega }}_c^2(\stackrel{\sim }{r})\mathrm{d}\stackrel{\sim }{r}=C_{\rho }{R}_h{\mathrm{\Omega }}_h^2,\end{array}$$(8)

where C_ρ is a dimensionless constant that depends on the density profile of the halo, and ${\mathrm{\Omega }}_h^2$ is the orbital frequency at the edge of the halo, given as ${\mathrm{\Omega }}_h^2=G{M}_h/R_h^3$. It must be noted that ${\mathrm{\Omega }}_h^2$ corresponds to the orbital velocity $v_h^2$, which is twice lower than the escape velocity $v_{\text{esc}}^2$ at the edge of the halo.

Taking ${\mu }^2=C_{\rho }{\mathrm{\Omega }}_n^2$, where Ω_n represents the quantized orbital frequencies. The quantization condition is

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_n^2={\mathrm{\Omega }}_h^2+n\pi {\kappa }_h{\omega }_o^2,\end{array}$$(9)

where

$$\begin{array}{c}\hfill {\kappa }_h=\frac{{ħ}^2}{G{m}^3{C}_{\rho }{R}_h}\end{array}$$(10)

is a dimensionless constant. Multiplying Eq. (9) by ℏ², the spectrum of quantum energies that corresponds to Eq. (9) is obtained, $E_n^2=E_h^2+n\pi {\kappa }_h{\epsilon }_o^2$.

In the above derivation of ${\mathrm{\Omega }}_n^2$ and $E_n^2$, it is assumed that $|{\widehat{\mathit{k}}}_o·{\widehat{\mathit{R}}}_h|=1$. This is justified by the fact that for spherically symmetric halos, the unit vector ${\widehat{\mathit{R}}}_h$ is not restricted and can always be aligned with the unit vector ${\widehat{\mathit{k}}}_o$. However, for the case of asymmetric DM halos, $|{\widehat{\mathit{k}}}_o·{\widehat{\mathit{R}}}_h|\ne 1$, and its value would account for deviations from perfect spherical symmetry, making the evolution of the wavefunction also dependent on direction.

The location r_n of the quantized orbits Ω_n inside the halo can be calculated by using the conservation of angular momentum (Musielak 2023b), and the result is

$$\begin{array}{c}\hfill r_n=R_h\sqrt{\frac{{\mathrm{\Omega }}_h}{{\mathrm{\Omega }}_n}},\end{array}$$(11)

where ${\mathrm{\Omega }}_n=\sqrt{{\mathrm{\Omega }}_h^2+n\pi {\kappa }_h{\omega }_o^2}$, and for the quantized orbits inside the halo, r_n ≤ R_h.

The developed model demonstrates that the halo has its quantum structure, which resembles atoms with their available energy levels. The model is now applied to a galactic halo with a given NFW density profile (Einasto & Haud 1989; Navarro et al. 1996, 2010; Merritt et al. 2006).

4. Halo with an NFW density profile and its quantum structure

To construct a quantum model of a galactic halo, the density profile for the halo must be specified. Of the different halo models simulated by Navarro et al. (2010), the NFW density profile given by

$$\begin{array}{c}\hfill \rho (r)=\frac{{\rho }_s}{\left(\frac{r}{r_s}\right){(1+\frac{r}{r_s})}^2}\end{array}$$(12)

was considered, with its parameters ρ_s and r_s being specified as the numerical model Aq-A-3 in Table 2 by Navarro et al. (2010). Then, r_s = r₋₂ = 1.1 × 10¹ (kpc h⁻¹) and ρ_s = ρ(r_s) = 4ρ₋₂ = 3.0 × 10⁶ (10¹⁰ h² M_⊙ Mpc⁻³), where h is the dimensionless Hubble parameter. It must be noted that ρ → ∞ when r → 0, but in the same limit, the term ρ_sr² → 0. After the numerical model was selected, the results of Table 1 in Navarro et al. (2010) were used to find the total mass of the halo, which is M_h = M₂₀₀ = 1.3 × 10¹² (M_⊙ h⁻¹), and its radius R_h = r₂₀₀ = 1.8 × 10² (kpc h⁻¹).

The NFW density profile given by Eq. (12) can be used to obtain the resulting mass distribution M(r),

$$\begin{array}{c}\hfill M(r)=4\pi {\rho }_s{r}_s^3[ln(1+\frac{r}{r_s})-\frac{r}{r+r_s}],\end{array}$$(13)

which shows that M(r → 0) = 0. Taking r = R_h, the total mass M_h = M(R_h) was calculated, and its value was consistent with M_h given in Table 1 of Navarro et al. (2010). Similarly, M_s = M(r = r_s) can also be determined, and its value is $M_s=0.76\pi {\rho }_s{r}_s^3$. Thus, the ratio of these two masses of the halo is M_h/M_s ≈ 10, which means that the amount of DM mass inside the radius r_s is ten times lower that the total mass M_h of the halo.

After we obtained M(r), M_h, and M_s, Eq. (3) was used to find the orbital frequencies corresponding to the masses M_h and M_s. Their values are Ω_h ≃ 3.0 × 10⁻¹⁷ (Hz h) and Ω_s ≃ 3.5 × 10⁻¹⁵ (Hz h). The orbital velocities of DM particles moving on the orbits with these frequencies are v_h = 1.8 × 10² (km s⁻¹) and v_s = 1.1 × 10³ (km s⁻¹), respectively.

Construction of the quantum model of the galactic DM halo with an NFW density profile requires evaluation of the integral given by Eq. (8). However, since ρ(r) diverges as r → 0, so does Ω_c(r) and the integral I_c(R_h). By changing the lower limit of the integration in Eq. (8), a core inside the halo was introduced. The existence of such a core has been verified for other density profiles (Musielak 2023b). Since the NFW density profile directly depends on r_s, let r_s be the lower limit of the integral, and thereby the radius of the core. However, if the core is smaller, and its radius r_c < r_s, then r_c would be the lower limit of the integration, with r_c to be determined from the theory.

Let x = r/r_s be a new variable that allows us to write the integral I_c(r_s, R_h) in the following form:

$$\begin{array}{c}\hfill I_c(r_s,R_h)=4\pi G{\rho }_s{r}_s{\displaystyle {\int }_1^{R_h/r_s}}[ln(1+x)-\frac{x}{1+x}]\frac{\mathrm{d}x}{x^3}·\end{array}$$(14)

After using the values of R_h, r_s, and ρ_s, the integration gives I_c(r_s, R_h)≃1.1 × 10⁻¹¹ (m s⁻² h). This result can be compared to that of Eq. (8). With $R_h{\mathrm{\Omega }}_h^2\simeq 6.11\times {10}^{-22}$ (m s⁻² h), the dimensionless constant is C_ρ = 1.8 × 10¹⁰ for a halo with an NFW density profile.

The developed quantum model of a galactic halo with an NFW density profile predicts that the halo has two components, namely, the core of radius r_s and density ρ(r ≤ r_s) = ρ_s= const, and an envelope that extends from r = r_s to r = R_h. The physical properties of the core and envelope are described below.

The core in the above halo model extends from r = 0 to r = r_s, which means that its radius is 11 kpc, and it occupies 6% of the size of the halo. Moreover, the core contains 10% of the total mass of the halo. Since the density is constant inside the core, the orbital frequency Ω_s has the same value at every point in the core, 0 < r ≤ r_s, and as a result, the core has no quantized orbits. Instead, the DM particles are allowed to move randomly and to collide frequently with each other. The core may also contain a large number of free quanta of energy ε_o = ℏω_o, which can be exchanged between the DM particles inside the core. The quanta of energy are called dark gravitons (e.g., Musielak 2022, 2023b). The sea of dark gravitons in the core may contribute to the gravitational wave background (e.g., Romano & Cornish 2017), which may be different than that of the envelope, and which might therefore be detectable (see Sect. 5).

The envelope that surrounds the core extends from the edge of the core r = r_s to the edge of the halo at r = R_h. It therefore occupies 94% of the size of the halo and contains 90% of the total halo mass. The density in the envelope is described by an NFW profile. As a result, the orbital frequencies Ω_n in the envelope range from Ω_h (lowest) to Ω_s (highest), and they are quantized according to the rule given by Eq. (9). The location of the quantized orbits r_n in the envelope can be calculated using Eq. (11). The quantization rule shows that the number of orbits increases rapidly when r → r_s and that density of the orbits in the vicinity of the core is high, so that DM particles on orbits like these may directly interact with the particles and dark gravitons of the core, causing the exchange of some particles.

However, away from the core, DM particles are confined to their quantized orbits because the differences ΔΩ_n = Ω_n + 1 − Ω_n are not exact multiples of ω_o as they also depend on κ_h given by Eq. (14). Since the DM particles may only emit or absorb the characteristic frequency ω_o, which is fixed for DM, there are no quantum jumps of the particles between the quantized orbits (see Sect. 5). Instead, the DM particles are spinless and zero-charged and are confined to their orbits without any other quantum restrictions imposed on the population of these orbits. In other words, the theory presents physical reasons, which are confinements of the DM particles to their quantum orbits, for DM to be collisionless that is typically assumed while constructing models of DM halos, such as the NFW density profile used in this paper.

Clearly, the described quantum structure of a galactic DM halo with an NFW density profile resembles an atom whose size is enormous as it exists on the galactic scale. Similar general conclusions were presented by Musielak (2022, 2023b) and were obtained for a simple linear density profile of DM, which is not realistic for typical galactic DM halos (e.g., Einasto & Haud 1989; Navarro et al. 1996, 2010). In this paper, the same conclusion is drawn from the results obtained with an NFW density profile, which is commonly used in studies of galactic DM halos. This implies that the predicted atomic structure of galactic DM halos is independent from density profiles. The structure ceases to exist for the constant density profile (Musielak 2023b), however. In the following, the developed model with an NFW density profile is used to make theoretical predictions that can be verified by astronomical observations.

5. Characteristic frequency and its detection

According to Eq. (9), the quantized orbital frequency Ω_n depends on the orbital frequency Ω_h at the edge of the halo, the dimensionless parameter κ_h, and the DM characteristic frequency ω_o. For the considered model of a galactic halo with an NFW density profile, Ω_h ≃ 3 × 10⁻¹⁷ (Hz h). The parameter κ_h is a sensitive function of the mass m of DM particles, which is currently unknown. When the values of the universal constants ℏ and G, the value of R_h, and C_ρ = 1.8 × 10¹⁰ are used for a halo with an NFW density profile, then the constant given by Eq. (10) becomes

$$\begin{array}{c}\hfill {\kappa }_h\simeq 1.6\times {10}^{-90}{\mathrm{m}}^{-3},\end{array}$$(15)

and it allows us to write the quantization condition given by Eq. (9) as

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_n^2=1.1\times {10}^{-33}+1.6\times {10}^{-90}n\pi {\mathrm{m}}^{-3}{\omega }_o^2.\end{array}$$(16)

This value shows that the quantization rule for ${\mathrm{\Omega }}_n^2$ may be used to estimate the DM characteristic frequency ω_o when the value of m is specified. In the considered halo model, ${\mathrm{\Omega }}_n^2$ can range from ${\mathrm{\Omega }}_h^2$ to ${\mathrm{\Omega }}_s^2$, which are the lowest and highest frequencies, respectively. Thus, the quantized orbits in the halo envelope and their number depend on the values of ω_o and m. Neither value is known, however. When one of these two quantities is specified, then the other can be estimated (see Table 1).

When Ω_n is known, the location of the quantized orbits in the envelope can be calculated by finding r_n from Eq. (11), whose explict form for a galactic halo with an NFW density profile is given by

$$\begin{array}{c}\hfill r_n\simeq 1.8\times {10}^2\sqrt{\frac{{\mathrm{\Omega }}_h}{{\mathrm{\Omega }}_n}}(\mathrm{kpc}{h}^{-1}).\end{array}$$(17)

By finding r_n, the number of quantized orbits located in the envelope between r = r_s and r = R_h can be calculated. The number of quantized orbits is very sensitive to the mass (m⁻³) of DM particles; in general, the higher the mass, the more quantized the orbits. Our model predicts that most quantized orbits are located near the core and that the separation between these orbits is small and decreases toward the core. Since the DM particles are confined to the quantized orbits, the dense population of the orbits near the core contains a large amount of the total mass of the halo. On the other hand, the outer orbits contain less mass, and their separation increases with the distance from the core. This structure of quantum orbits makes the entire galactic halo a very stable and long-lasting object.

The developed quantum model of galactic DM halos with an NFW density profile predicts that the core has a radius of 11 kpc and that its mass is 1.3 × 10¹¹ M_⊙, with most of the total mass of the halo confined to many quantized orbits located near the core. The size and mass of a typical galaxy located at the center of the halo can be similar to (or larger and higher than) those of the core. Moreover, the surroundings of the core is full of quantum orbits with a high population of DM particles. Thus, the observed rotational curves of different galaxies can be explained by this inner structure of the quantum halo. However, the structure may lead to small wiggles on otherwise flat rotational curves at the edge of the central galaxy. These wiggles might be caused by the quantum shells of DM near the core and away from it.

When we assume that the developed quantum theory correctly describes a galactic halo with an NFW density profile and that quantized orbits exist in the envelope, then the value of ω_o can be estimated for a given m. In the following, these estimates are made by specifying the mass m of DM particles and expressing it in terms of the proton mass m_p = 0.938 GeV = ≃1.7 × 10⁻²⁷ kg. The estimates were made using Eq. (16), and the results are presented in Table 1.

The developed quantum model of DM predicts the quantum structure of galactic halos described in Sect. 4. According to this theory, DM may emit (absorb) radiation of only one fixed frequency ω_o. Now, if DM is described by the theory, then there is a relation between the mass of DM particles and the characteristic frequency of radiation that DM may emit and absorb. The relation is given by Eq. (16), which requires that the term ${10}^{-90}{\text{m}}^{-3}{\omega }_o^2$ is approximately on the same order as 10⁻³³ because ${\mathrm{\Omega }}_n^2$ is restricted to the interval ${\mathrm{\Omega }}_h^2,{\mathrm{\Omega }}_s^2$. If this condition is valid, then the quantum rule given by Eq. (16) can be applied to galactic halos, and it predicts its quantum structure. The estimates were made by using the relation, and the obtained results are presented in Table 1. These preliminary estimates only allow a prediction an order of magnitude.

The results of Table 1 show that the characteristic DM frequency is a very sensitive function of the DM particle mass. The presented theory cannot estimate the mass, and thus it must be determined by experiments. There are no experimental or observational constraints on this value to date. The masses of DM particles we used in the estimates in Table 1 span many orders of magnitude, from approximately the mass of the Higgs boson to the mass of ultralight axions suggested by Rogers et al. (2023). The results given in Table 1 show that higher masses give higher frequencies ω_o and shorter wavelengths Λ_o of radiation that can be emitted and absorbed by DM particles. In general, the resulting frequencies are very low, and they become extremely low for extremely light bosons and axions, but then the masses were only included in the estimates for comparison. An interesting result is that when the mass of DM particles is increased by one order of magnitude, the resulting DM frequency increases by two orders of magnitude. This relation between the mass of DM particles and the characteristic DM frequency is important as it allows us to determine the frequency when the mass is determined experimentally, or to determine the mass when DM radiation is detected.

After predicting the values of the characteristic DM frequency ω_o for different masses of DM particles, the possibilities of a detection remain to be explored. The first question is what type of radiation can be emitted by DM. There is no observational evidence so far for any radiation associated with DM (e.g., Planck Collaboration VI 2020). The quantum theory of DM presented in this paper only accounts for gravitational interactions between the DM particles and the halo, but it does not account for any interaction between the particles. The quanta of energy ε_o = ℏω_o that DM emits and absorbs are called dark gravitons (see Sect. 4) to emphasize that they are quanta of gravitational interactions of DM particles.

The dark gravitons in the halo envelope, where the orbits are quantized, are limited by the fact that the DM particles are confined to their orbits because there are no quantum jumps between the orbits (see Sect. 4). However, dark gravitons may be abundant in the halo core, where orbits are not quantized and the DM particles move randomly and emit and absorb dark gravitons, which carry the quanta of energy ε_o. The abundance of dark gravitons in the core may contribute to the gravitational wave background (GWB; e.g., Romano & Cornish 2017), which was recently discovered by the NANOGrav detector (Agazie et al. 2023; Caprini 2024).

The GWB has two main components: a cosmological and an astrophysical component. This is based on their origin or on the sources that generate it. The two components are characterized by different frequency ranges (in Hz): from 10⁻¹⁶ (or lower) to 1 for the former, and from 10⁻¹⁰ to 10² for the latter (after the NASA Beyond Einstein Program). Comparing these frequencies to ω_o given in Table 1, we observe that for the masses ranging from the electron mass to the mass of the Higgs boson, the characteristic frequency ω_o of dark gravitons may also contribute to the observed GWB. However, the mass range that corresponds to extremely light bosons and axions gives ω_o, whose wavelengths Λ_o exceed the radius of the Universe by many orders of magnitude. They are therefore not observable.

According to the quantum model of galactic DM halos we presented, the contributions to the GWB should mainly come from the halo cores, where dark gravitons are abundant; the contributions from halo envelopes should be significantly smaller. Since the cores are much smaller than the envelopes, the core contributions should be highly localized. In other words, the predicted low-frequency emission should be enhanced at the center of galactic halos, which can also be identified with the centers of their host galaxies. This localized nature of the predicted emission should be distinguishable from the other sources of the GWB that contribute more uniformly to it. Therefore, it is possible that a localized enhancement of the GWB near the centers of galactic DM cores can be detected. Observationally, the NANOGrav detector (Agazie et al. 2023), future missions devoted to studies of the GWB, or more specifically, to the detection of very low frequency gravitational waves, could be used to validate the halo models we presented in this paper.

6. Conclusions

A quantum model of DM was developed based on a new equation that is complementary to the Schrödinger equation. The new quantum theory predicts that DM emits and absorbs radiation of one fixed frequency, whose quanta are called dark gravitons, and that DM is collisionless as a result of quantum effects. Moreover, applications of the model to a galactic DM halo with an NFW density profile showed that a halo is composed of a core and envelope that have different physical properties. In the core, the DM particles move randomly and frequently collide with each other. However, in the envelope the particles are confined to their quantized orbits. The population of orbits in the envelope remains fixed as there are not quantum jumps between them, except in the close vicinity of the core, where the density of orbits is high. The quantum structure of the halo is significantly different from those proposed by previous models of DM (e.g., Sin 1994; Hu et al. 2000; Spivey et al. 2013, 2015; Schive et al. 2014; Zhang et al. 2017).

The new quantum model predicts relations between the quantum structure of the halo and its global physical parameters, such as mass and radius. It requires its density profile to be specified. Moreover, the quantization rule gives a relation between the mass of DM particles and the characteristic frequency of radiation that DM may emit and absorb. Since neither the mass of DM particles nor the frequency of DM radiation are currently known, the model permits us to estimate the frequency that corresponds to a given mass (see Table 1). The estimated range of frequencies corresponds to dark gravitons, whose presence in the core may contribute to the gravitational wave background by making it different from that observed in the envelope. We proposed to consider using the NANOGrav detector, or future instruments designed to observe low frequency gravitational waves, to determine the difference observationally.

The new quantum model is based on analytical solutions to a new governing equation, which require a DM density profile to be specified, for which an NFW density profile was used. In future work, the requirement of specifying a density profile can be removed by simultaneously solving the new governing equation and the Poisson equation. The latter allows us to determine the DM potential by using the square of the wave function as the forcing term (e.g., Sin 1994; Spivey et al. 2013). However, simultaneous solutions like this can only be obtained numerically. They will therefore be presented elsewhere.

Acknowledgments

I appreciate very much valuable comments, stimulating questions and insightful suggestions made by two anonymous referees, which allowed me to significantly improve the original version of this paper. The author also thanks Dora Musielak for comments and suggestions on the earlier version of this manuscript.

References

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