Radial Oscillations of Dark Matter Stars Admixed with Dark Energy

Abstract

We imagine spherically symmetric configurations made of both dark matter and dark energy in the halo of spiral galaxies. Adopting a polytropic equation of state for dark matter and the Extended Chaplygin gas equation of state for dark energy, we model the same object with three different dark matter–dark energy compositions. We compute the frequencies and the corresponding eigenfunctions of the ten lowest modes, integrating the equations for the radial perturbations by imposing the appropriate boundary conditions at the center and the surface of the object. Also, a comparison between the different models is made.

1. Introduction

The concordance cosmological model, the $\Lambda$CDM model, describes the formation and structure of the Universe at large scales, and is based on cold dark matter and $\Lambda$, the cosmological constant. The existence of dark matter, still of unknown nature and origin, originates from old studies that showed that the Universe is mostly composed of an invisible material that affects galaxies and galaxy clusters only gravitationally. The pioneer works belong to F. Zwicky [1] in the 30s, and several decades later to V. Rubin, who studied the Andromeda’s rotation curve and discovered that at any distance from the bulge of the galaxy, the rotation speed was constant [2]. These studies showed that approximately 85% of the mass of the Universe corresponds to dark matter [3].

In addition to dark matter, there is another dark component in the Universe, called dark energy. This is also of unknown nature as of today, but has a repulsive rather than attractive character, since the current accelerating expansion of the Universe is produced by it [4,5,6]. Current modern data seem to show that approximately 70% of the Universe is made of dark energy, 25% of dark matter, and 5% of ordinary matter [7].

Compact stars [8], the final products of very massive stars, are extremely dense objects with a mass of the order of one solar mass, and a radius of a few kilometers, such as neutron stars [9] and strange quark stars [10,11,12,13,14,15], although the latter class of compact objects remain hypothetical for the time being. In the same way, there are studies that contemplate the possibility that relativistic exotic stars may exist, composed of dark matter [16,17,18], or dark energy even [19,20,21,22]. In a recent study, we considered configurations composed of both dark matter and dark energy adopting the two-fluid formalism [23]. In that study we gave a relativistic description based on the field equations of General Relativity.

To determine the radius and mass of those stars, we require the EoS of each fluid component, i.e., one for dark matter and another for dark energy, under the condition that there is no direct interaction between the fluids other than the gravitational force. Dark matter may be viewed as a Bose–Einstein condensate [24], and so we adopt the polytropic equation of state for dark matter [25,26]. Regarding dark energy, we consider the Extended Chaplygin gas equation of state [27,28], which was introduced with the aim of unifying dark energy and non-relativistic matter working with a single fluid instead of two.

Regarding plausible scenarios and astrophysical mechanisms that may produce dark stars in the Universe, it has been proposed that the first stars to exist in the Cosmos, contrary to ordinary main sequence stars, were powered by dark matter heating rather than fusion [29,30]. Dark matter particles might have accumulated in the interior of the first stars and been annihilated, producing a heat source that can power such stars. Moreover, if Bose–Einstein condensation took place in the very early Universe, at very high redshifts, as suggested by the analysis performed in [16], at the moment of star formation most of the DM was already in a condensate form. Hence, one may naturally expect the formation of stellar type objects made of pure condensate dark matter stars [16].

Asteroseismology is a widely used technique, since the frequencies of the oscillation modes of pulsating stars are very sensitive to the inner structure and composition of the object. It has been proven to be a powerful tool at our disposal, since it allows us to probe the interior of the Sun, solar-like stars, and other classes of stars [31,32], such as nuclear reactions, their compositions, equations of state, differential rotation rates, and meridional circulation. Currently, asteroseismology has been extended to the study of compact stars [33,34,35,36], like white dwarfs [37] and neutron stars [38]. In addition, there are studies of exotic stars such as strange quark stars [39], dark matter stars [39,40,41], and dark energy stars [21]. For more details, the interested reader may consult for instance [42,43,44,45,46,47,48,49,50].

Regarding excitation and detectability, there are numerous astrophysical mechanisms that may excite oscillation modes of stars, such as tidal effects in binaries, starquakes caused by cracks during supernova explosions, magnetic reconfiguration, or any other form of instability [38,51,52,53,54,55]. The Kepler and CoRoT missions have already measured the oscillation spectra of solar-like, white dwarf, and red giant stars [56,57,58,59,60,61]. Actual wave detectors, like LIGO [38], do not detect the radial oscillations studied here, due to their low sensitivity at the kHz frequency range. However, the third-generation of ground-based gravitational wave detectors are expected to have a much higher sensitivity (by an order of magnitude), such as the Cosmic Explorer [62] and the Einstein Telescope [63]. These detections could give us information related to neutron star masses, frequencies, tidal Love numbers, amplitudes of the modes, damping times, and moments of inertia.

In the present work, we compute the frequencies and corresponding eigenfunctions of the 10 lowest radial oscillation modes of spherical configurations from the dark sector, namely both dark matter and dark energy. To that end, we model an object of a given mass and radius with three different models, which differ in their internal structure and composition.

The present article is structured as follows: after this introductory section, we present the structure equations governing the hydrostatic equilibrium and the equations of state employed here. Radial oscillations are discussed in Section 3, and we present and discuss our main results in Section 4. Finally, we conclude our work in Section 5.

Throughout the manuscript, we adopt the mostly positive metric signature (−, +, +, +) and we work in natural geometrical units, setting G = c = 1 for the Newton’s constant and speed of light in vacuum, respectively. All units used here are measured in GeV: 1 m $=5.068\times {10}^{15}/\mathrm{GeV}$, 1 kg $=5.610\times {10}^{26}\mathrm{GeV}$, 1 s $=1.519\times {10}^{24}/\mathrm{GeV}$, and the Planck mass $M_{pl}=1.22\times {10}^{19}\mathrm{GeV}$.

2. Relativistic Spheres in General Relativity

We assume that the objects are non-rotating and are electrically neutral. Also, we work with relativistic configurations in four dimensions, with a vanishing cosmological constant.

2.1. Structure Equations

The metric tensor in Schwarzschild coordinates (t, r, $\theta$, $\varphi$) for static and spherically symmetric space-times is given by

$$d{s}^2=-e^{\nu }d{t}^2+e^{\lambda }d{r}^2+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2).$$

(1)

To obtain interior solutions describing hydrostatic equilibrium of stars composed of dark matter and dark energy, we need to integrate the Tolman–Oppenheimer–Volkoff (TOV) equations [64,65] for two-fluid formalism [66,67], which are given by

$$\begin{array}{cc}\hfill {\displaystyle m^{\prime }\left(r\right)}& =4\pi r^2\rho \left(r\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\displaystyle {\nu }^{\prime }\left(r\right)}& =2\frac{m\left(r\right)+4\pi r^{3}p\left(r\right)}{r^2\left(1-2m\left(r\right)/r\right)},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle p_M^{\prime }\left(r\right)}& =-[{\rho }_M\left(r\right)+p_M\left(r\right)]\frac{{\nu }^{\prime }\left(r\right)}{2},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle p_E^{\prime }\left(r\right)}& =-[{\rho }_E\left(r\right)+p_E\left(r\right)]\frac{{\nu }^{\prime }\left(r\right)}{2},\hfill \end{array}$$

(5)

which we consider a spherical configuration composed of dark matter and dark energy, whose pressure and energy density are $p_M$ and ${\rho }_M$, for dark matter, and $p_E$ and ${\rho }_E$ for dark energy, respectively. $m\left(r\right)$ is the mass function, and a prime denotes differentiation with respect to the radial coordinate r.

The total pressure p and total density $\rho$ are the sum of the pressures and densities of both fluids, i.e.,

$$p=p_M+p_E,$$

(6)

$$\rho ={\rho }_M+{\rho }_E.$$

(7)

The TOV equations are to be integrated imposing the following initial conditions at the centre of the star

$$m\left(0\right)=0,$$

(8)

$$p_M\left(0\right)=p_{cM},$$

(9)

$$p_E\left(0\right)=p_{cE},$$

(10)

where $p_{cM}$ and $p_{cE}$ are the central pressure for dark matter and dark energy, respectively. Moreover, the the surface conditions of the star that must be satisfied are

$$p\left(R\right)=0,$$

(11)

$$m\left(R\right)=M,$$

(12)

with R being the radius of the star and M its mass.

2.2. Equations of State

To solve the TOV equations, we will require the equations of state. Since each fluid is independent, dark matter and dark energy have their own equation of state.

Dark matter may be interpreted as a Bose–Einstein condensate [24], since it solves the core–cusp problem [68]. Hence, a polytropic EoS of the following form is obtained [16,25,26]

$$p_M\left(r\right)=k{\rho }_M^2\left(r\right),$$

(13)

where k is the constant of proportionality, determined by [16,26,39]

$$k=\frac{2\pi l}{m^3},$$

(14)

with m being the mass of the dark matter particle, and l being the scattering length that determines the elastic cross-section at low energies, $\sigma =4\pi l^2$ [16,26,39]. Each model is characterized by its own numerical value of k, which can be seen in

Table 1. Numerical values of the parameters A, B, k and f for each model. For all three models considered in this work we assume a sphere of mass M = 1.216 $M_{\odot }$ and radius R = 8.145 km.

	A	B	k	f
Model 1	$\sqrt{0.40}$	$2.299\times {10}^{-4}/{\mathrm{km}}^2$	$45.140{\mathrm{GeV}}^{-4}$	0.60
Model 2	$\sqrt{0.55}$	$2.595\times {10}^{-4}/{\mathrm{km}}^2$	$55.567{\mathrm{GeV}}^{-4}$	0.75
Model 3	$\sqrt{0.70}$	$2.848\times {10}^{-4}/{\mathrm{km}}^2$	$66.942{\mathrm{GeV}}^{-4}$	0.90

. Nowadays, the DM self-interacting cross section must satisfy the constraint [69,70,71]

$$1.75\times {10}^{-4}{\mathrm{cm}}^2/\mathrm{g}<\frac{\sigma }{m}<(1-2){\mathrm{cm}}^2/\mathrm{g}.$$

(15)

The numerical values of k displayed in Table 1 may be achieved if, e.g., $l\sim 1fm$ and $m\sim$ 1 GeV, and the bounds on $\sigma$ are satisfied. We comment in passing that regarding the fuzzy DM model based on a free ultra-light scalar field, recent works have put an upper bound on its mass, $m<{10}^{-23}\mathrm{eV}$ [72,73], violating the ${10}^{-22}\mathrm{eV}$ from the linear matter power spectrum.

As far as dark energy is concerned, following [20], we consider the Extended Chaplygin equation of state [27,28].

$${\displaystyle p_E\left(r\right)=-\frac{B^2}{{\rho }_E\left(r\right)}+A^2{\rho }_E\left(r\right),}$$

(16)

where A is dimensionless and B has dimensions of energy density and pressure, both positive constants. Note that the equation of state of Chaplygin gas is given by $p_{Ch}=-B^2/{\rho }_{Ch}$ [74], while the additional barotropic term, $A^2{\rho }_E$, provides us with another viable dark energy model [75]; as it predicts a transition from deceleration to acceleration, it fits supernovae data, while at the same time the equation-of-state parameter w interpolates between 0 and −1, unifying thus non-relativistic matter and DE. What is more, at low red shift, the energy density becomes a constant, mimicking thus a positive cosmological constant, see Equation (11) of [75]. Table 1 shows the numerical values for A and B that we have considered.

It must be taken into account that given the form of the EoS, the pressure and density cannot become zero at the same time. Therefore, a vanishing pressure at the surface of the star implies a non-vanishing value for the energy density, which is computed to be ${\rho }_s=B/A$.

Additionally, it is convenient to introduce a new dimensionless factor related to the central conditions of the stars defined as

$${\displaystyle f=\frac{p_{cM}}{p_{cM}+p_{cE}}.}$$

(17)

The numerical values for A, B, k, and f that we have chosen for three different models are displayed in Table 1 in order to compare stars of the same mass and radius but with different inner composition and obtain realistic solutions. The values are M = 1.216 $M_{\odot }$ for the mass, and R = 8.145 km for the radius. The chosen values are motivated by the fact that there are a few observed objects with known masses and radii around those values, see for instance Table 1 of [76].

In a recent work, we studied some properties of dark stars made of dark matter and dark energy [23], and here we propose to study radial oscillations and frequencies for this class of objects.

3. Radial Oscillations

The equations for radial perturbations of the stars, given below, are satisfied by the radial displacement, $\Delta r$, and the pressure perturbation, $\Delta p$, defined by

$${\displaystyle \xi \equiv \frac{\Delta r}{r},}$$

(18)

$${\displaystyle \eta \equiv \frac{\Delta p}{p},}$$

(19)

which satisfy the following first order differential equations [77,78]:

$$\begin{array}{cc}\hfill {\displaystyle {\xi }^{\prime }\left(r\right)}& =-\left(\frac{3}{r}+\frac{p^{\prime }}{\zeta }\right)\xi -\frac{1}{r\Gamma }\eta ,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\displaystyle {\eta }^{\prime }\left(r\right)}& ={\omega }^2\left[r\left(1+\frac{\rho }{p}\right)e^{\lambda -\nu }\right]\xi -\left[\frac{4p^{\prime }}{p}+8\pi \zeta r{e}^{\lambda }-\frac{r{\left(p^{\prime }\right)}^2}{p\zeta }\right]\xi -\left[\frac{\rho p^{\prime }}{p\zeta }+4\pi \zeta r{e}^{\lambda }\right]\eta ,\hfill \end{array}$$

(21)

where $\zeta$ is defined to be

$${\displaystyle \zeta \equiv p+\rho ,}$$

(22)

$e^{\lambda }$ and $e^{\nu }$ are the metric potentials, which are computed as follows

$${\displaystyle e^{\lambda }\left(r\right)=\frac{1}{1-\frac{2m\left(r\right)}{r}},}$$

(23)

$${\displaystyle \nu \left(r\right)=ln\left(1-\frac{2M}{R}\right)+2{\int }_R^r\frac{m\left(x\right)+4\pi x^{3}p\left(x\right)}{x^2\left(1-2m\left(x\right)/x\right)}dx,}$$

(24)

with M the mass of the star and R its radius, while $\omega =s{\omega }_0$ is the frequency oscillation mode, where s is a dimensionless number, while the constant ${\omega }_0$ is computed by

$${\displaystyle {\omega }_0=\sqrt{\frac{M}{R^3}}.}$$

(25)

Finally

$$\begin{array}{cc}\hfill {\displaystyle \Gamma }& =c_s^2\left(1+\frac{\rho }{p}\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\displaystyle c_s^2}& =\frac{dp}{d\rho },\hfill \end{array}$$

(27)

with $\Gamma$ being the relativistic adiabatic index, and $c_s$ is the the speed of sound.

The unknown frequencies are determined solving the Sturm–Liouville boundary value problem imposing the following boundary conditions [38], both at the center $r=0$ and at the surface of the star $r=R$

$$\begin{array}{cc}\hfill {\displaystyle \frac{\eta }{\xi }{|}_{r=0}}& =-3\Gamma \left(0\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\eta }{\xi }{|}_{r=R}}& =\left[-4+{\left(1-\frac{2M}{R}\right)}^{-1}\left(-\frac{M}{R}-\frac{{\omega }^2{R}^3}{M}\right)\right].\hfill \end{array}$$

(29)

since ${\xi }^{\prime }\left(r\right)$ must be finite as $r\to 0$, and ${\eta }^{\prime }\left(r\right)$ must be finite at the surface as $\rho$, so $p\to 0$. Furthermore, since the system of equations is linear, the scaling factor is free and therefore the oscillating functions may be normalized, setting $\xi \left(0\right)=1$ [78].

Before we proceed, a comment is in order here. When dealing with radial oscillations of objects made of two fluid components, in general there are two approaches [79]. In the first, a consistent general relativistic formalism to deal with the couplings between oscillation amplitudes and Lagrangian perturbations for each fluid was developed in [36,80]. Unfortunately, dealing with a system of highly coupled and non-linear differential equations requires very time-consuming numerical calculations. In the second approach, which is in fact simpler, only the radial oscillation of the whole admixed star, namely treated as a single fluid sphere, is studied without explicitly considering the gravitational coupling between the two fluid components, see, e.g., [39,81]. This is the approach we are taking in the discussion to follow.

4. Results and Discussion

We have studied radial oscillations and frequencies of dark stars made of dark matter and dark energy for their first 10 lowest modes shown in

Table 2. Frequencies (in kHz) of radial oscillation modes for the three models discussed here, see text for more details.

Mode Order n	Model 1	Model 2	Model 3
0	4.56	5.50	6.25
1	12.03	13.81	15.28
2	18.74	21.39	23.61
3	25.30	28.83	31.79
4	31.81	36.21	39.91
5	38.29	43.57	48.01
6	44.76	50.92	56.09
7	51.21	58.25	64.16
8	57.66	65.58	72.23
9	64.11	72.90	80.29

. For this, we have selected three different models, which all have the same radius R = 8.145 km and mass M = 1.216 $M_{\odot }$ but different dark energy mass fractions. The precise numerical values of the parameters A, B, k, and f, are shown in Table 1. The three models considered in this work are as follows

$$\begin{array}{ccc}\hfill \mathrm{Model}1& \to & 49\%\mathrm{DE}\mathrm{mass}\mathrm{fraction}\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill \mathrm{Model}2& \to & 44\%\mathrm{DE}\mathrm{mass}\mathrm{fraction}\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \mathrm{Model}3& \to & 41\%\mathrm{DE}\mathrm{mass}\mathrm{fraction}\hfill \end{array}$$

(32)

In Figure 1, we show the eigenfunctions $\eta \left(r\right)$ and $\xi \left(r\right)$ versus normalized (dimensionless) radial coordinate $x=r/R$ for low, intermediate, and highly excited modes for all three models considered here. We recall that in a Sturm–Liouville boundary value problem, the number of zeros of the eigenfunctions corresponds to the overtone number n. In the graphs, it can be observed that this is indeed the case. For instance, the fundamental mode $n=0$ has no zeros, while the first excited mode $n=1$ and the second excited mode have one and two zeros, respectively. And so on and so forth for higher modes.

In Table 2, where the numerical values of the frequencies are shown, we see that the frequencies are higher for Model 3 and lower for Model 1. In Figure 2, the large frequency separations defined as [38]

$${\displaystyle \Delta {\nu }_n\equiv {\nu }_{n+1}-{\nu }_n,}$$

(33)

are displayed for all three models, where $n=0,1,2,\cdots$ is the overtone number. As usual, for highly excited modes, the large frequency separations tend to a constant which is different from one model to another. They are also larger for Model 3 and smaller for Model 1. It is worth mentioning that Model 3 is characterized by the largest sound speed, see Figure 5 below, while regarding the fundamental mode, the $\xi$ function of Model 3 decreases faster than the corresponding function of the other two models. Those features are in agreement with the results of [38], where radial oscillations of neutron stars were studied, and a similar behavior was observed.

Figure 3 and Figure 4 show the perturbations $\eta$ and $\xi$, respectively, for the three models and for the fundamental mode and the first excited mode. Depending on the model, the values of $\eta \left(0\right)$ start with different values, whereas for $\xi$ they all start with $\xi \left(0\right)=1$. In the three models, the values of $\xi \left(r\right)$ are very similar for each mode. However, in $\eta \left(r\right)$ they differ, except for the zeros, which for each mode the three models are the same, as is $\xi$.

We obtained that the dark energy mass fraction differs in all models, obtaining 49%, 44%, and 41% in Models 1, 2, and 3, respectively. If radial oscillations are discovered in dark stars, it will be possible to use their frequencies to infer the properties of their internal structure, and check if there is evidence for dark matter and dark energy inside the star, and their percentages or mass fraction.

Furthermore, we have checked that the results comply with the criteria for realistic solutions. The Buchdahl limit $C\equiv M/R<4/9$ [82] is satisfied since $C=0.222$. Stability, $\Gamma >4/3$ [83], energy conditions $\rho \ge 0$ and $\rho \ge \left|p\right|$ [21,84,85,86,87], and causality, $0<c_s^2<1$ [21], are fulfilled as well, as can be seen in Figure 5. The values for the speed of sound of Model 3 are the highest and those of Model 1 are the lowest.

5. Conclusions

To summarize our work, in the present article, we studied radial oscillations of spherical, non-rotating configurations made of both dark matter and dark energy. For the latter, we adopted the Extended Chaplygin gas EoS, while for the former, viewed as Bose–Einstein condensate, we assumed a polytropic EoS with index $n=1$. We modeled the same object with mass $M=1.216M_{\odot }$ and radius $R=8.145$ km in three different dark matter–dark energy compositions. We integrated the equations for the perturbations, imposing the appropriate boundary conditions, both at the center and the surface of the objects, and we computed the frequencies as well as the corresponding eigenfunctions for the 10 lowest modes, namely fundamental (f mode) and nine excited modes. The large frequency separations are displayed as well for all three models considered here. A direct comparison between the three models is made showing (i) the two eigenfunctions of the fundamental and first excited mode as well as (ii) the speed of sound of the interior solution. We find that as the objects contain less and less dark energy, the sound speed and the frequencies increase.