Dark Matter (DM), almost five times more abundant than ordinary matter, is theorized as one of the basic constituents of the Universe. Many experiments are running worldwide to detect these DM particles but till now the interaction of these particles with the ordinary matter has proven to be so feeble that they have escaped direct detection [1]. In the cosmic structure formation, the lightest objects would have formed first, i.e. the structure formation is a bottom-up scenario. It is of interest to note that the earliest objects to form could perhaps have been primordial planets dominantly composed of DM. The formation of such objects and their presence in large numbers in our galaxy could significantly reduce the number of free DM particles moving around in the Universe. The typical mass of such objects, made up mostly of DM particles of mass $m_D$, is given by [2,3]. where $M_{Pl}$ is the Planck mass given by $M_{Pl}={\left(\raisebox{1ex}{$\hslash c$}\!\left/ \!\raisebox{-1ex}{$G$}\right.\right)}^{1/2}\approx 2\times {10}^{5}g$. If we consider the mass of DM particles to be 60𝐺𝑒𝑉, favored from the detection of excess of gamma rays from the galactic centre, attributed to the decay of 60𝐺𝑒𝑉 DM particles [4], the mass of the DM object works out to be ${10}^{29}g$ which is the mass of Neptune. The radius of the DM object is given by [5] where 𝐺 is the gravitational constant and $m_D\sim 60GeV$ is the DM particle mass [6].

As the density of these objects fall of as $M^2$, the objects formed at later epochs would have a lower mass. If we consider the object density at a value 100 times the ambient density, say at $z=10$, we get a lower mass limit of the object as ≈ ${10}^{14}g$ (typical asteroid mass). So the mass range of these DM objects will be from ${10}^{14}g$ (asteroid mass) extending to the mass of Neptune. These objects could have formed in the early epochs of the Universe (when local DM density was much higher) and be in existence even now.

Existence of (baryonic) primordial planets have been considered earlier by many authors [7,8]. In the recent paper [9] we had discussed possibility of DM at high redshifts forming primordial planets composed entirely of DM to be one of the reasons for not detecting DM as the flux of ambient DM particles would be consequently reduced. The evolution of these DM primordial planets is discussed in detail in [10]. In [11], we had proposed that the hypothesised Planet 9, in our solar system ([12,13]), could indeed be such a DM planet, with a mass about that of Neptune. This might explain why it has not been visibly detected so far.

Here we discuss the possibility of baryons getting mixed with the DM particles in forming these primordial objects. During the phase of formation of these primordial objects, as the primeval ambient cloud collapses we consider the presence of baryonic matter in addition to the DM particles.

If $M_D$ is the total mass of DM particles and $M_B$ is the total mass of baryonic particles in forming the primordial planet, then the total mass ($M_T$) of the planet is given by

The gravitational binding energy density ($P_{planet}$) of such a planet of radius 𝑅 admixed with both baryons and DM is given by

As the DM particles gets admixed with baryonic particles in forming these structures, the DM particles will exert pressure and the DM degeneracy pressure is given by [14] where 𝑅 is the radius of planet and $m_D$ is the mass of DM particle forming the planet. On the other hand, baryonic particles when they get admixed, in addition to degeneracy pressure like DM particles, they will also exert thermal and radiation pressure. The degeneracy pressure exerted by the baryons is given by [15] where $m_p$ is proton mass. It is assumed that the gravitational self energy $~\frac{{GM}_T^2}{R}$ goes into heating only the baryonic particles of mass $M_B$ as it is assumed that the DM does not interact with radiation. Thus $M_B{R}_{g}T=\frac{{GM}_T^2}{R}$. This leads to equation (7) and equation (8). The thermal pressure and the radiation pressure exerted by the baryonic particles is given by where $R_g$ is the gas constant, and 𝑇 is the temperature of baryonic matter given by where $k_B$ is Boltzmann constant. The radiation pressure is given by where 𝑎 is Stefan’s constant.

For forming the planet admixed with DM and baryonic particles, the gravitational binding energy density of the planet must be in balance with the radiation pressure, thermal pressure and degeneracy pressure of baryonic and DM particles. Thus, where $M_T$ is the total mass of the planet, $M_B$ is the total mass of baryonic particles, $M_D$ is the total mass of DM particles, 𝑅 is the radius of the planet, $m_d$is the mass of DM particle, $m_p$ is mass of proton, $a$ is Stefan’s constant and $k_B$ is Boltzmann constant.

If we assume a fraction $f$ of baryonic particles getting mixed with $(1-f)$ fraction of DM particles, then $M_B$ and $M_D$ in equation (1) can be replaced by $f{M}_T$ and ${(1-f)M}_T$ respectively. Thus equation (10) becomes,

The mass of the planet formed with these particles [3] will be given by where $m_{eff}$ is the effective mass of the constituent particles forming the planet given by

When the effective mass is considered for the planetary formation, the mass of the object will increase (can be more than Jupiter mass) beyond the maximum limit (${10}^{29}g$) proposed for DM planets [11]. This happens because the effective mass $m_{eff}$ is reduced, since $m_D\gg m_B$.

Consider such a planet with 50% of DM (assuming $m_D$ of 60GeV) and 50% of baryonic matter, the mass of the planet is $~2M_J$, where $M_J$ is the mass of Jupiter. For the planet of this mass, the baryonic radiation pressure will be very small compared to the degeneracy and thermal pressures. Hence the radiation pressure can be neglected from equation (11). Thus

Thus, the radius of the object from the above equation becomes

For the planet with mass $~2M_J$, as discussed in the above case, the radius works out to be $9.8\times {10}^5$cm.

Table 1 shows the mass and radius of the primordial planets with an admixture of DM and baryonic particles of different fractions and for 60GeV DM particles. Figure 1 shows that the mass of DM planets with an admixture of baryonic particles increases their mass limit beyond the Neptune mass. For planets made entirely of DM, the maximum mass limit was Neptune mass and it can go down to asteroid mass. But with an admixture of DM and baryonic particles, the mass of the planet increases with the increase in the fraction of baryonic matter. When the baryonic fraction to that of DM particle increases, the planetary mass can increase and go beyond the Jupiter mass (up to about fifty Jupiter mass). It is also found that some objects can have sub stellar masses (above 50 Jupiter mass), like that of brown dwarfs. The radius of these planets (admixed with DM and baryonic matter) also increases with an increase in the fraction of baryonic matter. It is found from Figure 2 that if the DM particles involved in the formation of these planets are heavier, the size of the planets increases compared to the planets being formed by lighter DM particles.

In the previous section, we had discussed the possibility of formation of DM planets with an admixture of baryonic and DM particles. There could be one such object within half a light year for the density of DM $\approx 0.1\mathit{G}\mathit{e}\mathit{V}/\mathit{c}\mathit{c}$ around the solar neighbourhood [11]. These objects can rotate about their axis, thus emitting gravitational waves.

Consider such an object with $50\%$ DM and $50\%$ baryonic matter made up of DM particles of mass ${\mathit{m}}_{\mathit{D}}=60\mathit{G}\mathit{e}\mathit{V}$. The mass and radius of the object from Table 1 is ${\mathit{M}}_{\mathit{o}\mathit{b}\mathit{j}}=2{\mathit{M}}_{\mathit{J}}$ and ${\mathit{R}}_{\mathit{o}\mathit{b}\mathit{j}}=9.8\times {10}^5\mathit{c}\mathit{m}$, where ${\mathit{M}}_{\mathit{J}}$ is Jupiter mass. The gravitational wave energy emitted per unit time by the rotating individual object is given by where $\mathit{I}$ is the moment of inertia of object given by $\mathit{I}=\frac{2}{5}{\mathit{M}}_{\mathit{o}\mathit{b}\mathit{j}}{\mathit{R}}_{\mathit{o}\mathit{b}\mathit{j}}^2$, ${\mathit{\omega }}_{\mathit{o}\mathit{b}\mathit{j}}$ is the frequency of rotating object and $\mathit{ϵ}$ is the ellipticity of object. Then equation (16) becomes

The rotational frequency of DM object spinning close to breakup is given by where $\mathit{G}$ is the gravitational constant. Thus, the maximum rotational frequency of the DM object is given by

Considering $\mathit{ϵ}=0.1$ for the object of mass $2{\mathit{M}}_{\mathit{J}}$ and ${\mathit{R}}_{\mathit{o}\mathit{b}\mathit{j}}=9.8\times {10}^5\mathit{c}\mathit{m}$, energy emitted per second works out to be $7.2\times {10}^{39}\mathit{e}\mathit{r}\mathit{g}/\mathit{s}$ with the frequency of $518\mathit{H}\mathit{z}$ which is well within the frequency of LIGO [16]. The typical period of rotation of object is $\frac{2\mathit{\pi }}{\mathit{\omega }}=0.012\mathit{s}$. As the object rotates, it can break up emitting gravitational waves. The binding energy of the object will be emitted as gravitational waves. The binding energy of the object is given by

For the above object, the binding energy works out to be $9.82\times {10}^{47}\mathit{e}\mathit{r}\mathit{g}\mathit{s}$. The strain $\mathit{h}$ (in the gravitational wave detector) is calculated using the formula where $\mathit{E}$ is the binding energy of the object, $\mathit{c}$ is speed of light and $\mathit{r}$ is the distance of object from Earth. If we consider this object at a distance of $10\mathit{A}.\mathit{U}$. from Earth, then the corresponding strain is ${10}^{-14}$.

These primordial planetary objects can form binary systems. Considering a binary system with each component of mass $2M_J$ and size of $9.8\times {10}^5$𝑐𝑚. and separation about ten times their size, the orbital period 𝑃 is given by where $R$ is the orbital radius and $M_T$ is the total mass of the system. The orbital period works out to be 𝑃 = 0.27𝑠 and the corresponding frequency is 𝜔 = 23𝐻𝑧. The binary system will be emitting gravitational wave energy as it revolves and the energy emitted per unit time is given by where 𝜖 is the eccentricity of orbit and 𝜇 is the reduced mass of system given by

$M_1$ and $M_2$ are the masses of individual objects in the binary system. During their orbit around one another they lose energy and the orbital radius keeps decreasing until it becomes $2R_{obj}$. The final merger period and merger frequency of the binary system will be $P=0.0243s$ and $\omega =260Hz$. This frequency is also within the existing range of LIGO. The binding energy of the binary system will be emitted as gravitational waves and is given by where $R=2R_{obj}$

Figure 3 shows the orbital frequency of binary system of primordial objects admixed with baryonic matter versus fraction of baryonic matter. From the figure we can conclude that most of the frequency emitted by these binary systems falls under the LIGO range of frequency. Table 2 shows the gravitational wave energy emitted by the binary system for different fractions of baryons admixed with DM particles of mass ranging from 20 to 100 GeV in forming the planet. It is found that for greater mass DM particles, the energy emitted as gravitational waves decreases. As the fraction of baryons increases, the energy emitted by the binary system will increase. If we consider these binary systems to be situated at distance 𝑟 distance from Earth, then the strain, ℎ on earth due to the gravitational radiation emission from them is given by equation (21). If this binary system is assumed to be at distances $1kpc$ and $10kpc$ from Earth, then the strain due to the gravitational wave is $2\times {10}^{-23}$ and $2\times {10}^{-24}$. The corresponding flux on Earth at these distances will be $8\times {10}^{-9}ergs/m^{2}s$ and 8 × ${10}^{-11}ergs/m^{2}s$.

If a primordial degenerate object (as discussed in the above sections) of mass $M_T$ and radius $R$ approaches a BH of mass $M_{BH}$ , the tidal force is given by where d is the separation between the BH and the primordial object. The self-gravitational force of the object is given by

For the object to break up, the tidal force must be greater than the self-gravitational force of the object, i.e.,

Considering the distance between the BH and object to be around 10 times Schwarzchild radius ($d\approx 10\left(\frac{2G{M}_{BH}}{c^2}\right)$), the minimum mass of BH required for tidal break up of the object is given by

Figure 1 shows the mass of BH required for the tidal break-up of the primordial planet when it comes near the BH. The mass of the BH required for tidal break up increases with increase in mass of DM particles as well as with the fraction of baryons in the primordial planet. As these objects orbit the BH, they lose energy according to equation (23). Table 3 shows the gravitational wave energy emitted per second by the DM object consisting of different mass DM particles with different fractions of baryons in forming the primordial object. It is found that energy decreases with increase in the mass of DM particles.

When they lose energy, the orbital radius keeps decreasing until the radius becomes equal to Schwarzchild radius (𝑅𝑠𝑐ℎ). At the swarschild radius the frequency is given by [17] where ${\mathit{M}}_{\mathit{T}}$ is the total mass of the system and the orbital radius $\mathit{R}={\mathit{R}}_{\mathit{s}\mathit{c}\mathit{h}}$. The orbital binding energy will be emitted as gravitational waves at this frequency. The time of merger of the primordial object with the BH is given by [18] where 𝑐 is speed of light, 𝑟_𝑖 is the initial orbital radius, 𝑀 is the total mass of system involving BH and object, 𝜇 is reduced mass of system and G is gravitational constant. For an object with 50% DM and 50% baryonic matter and made of 60GeV DM particle the merger time is ${10}^7\mathit{s}$.

Figure 5 shows the relation between the merger time and fraction of baryons in forming the primordial planet. It is found that the merger time increases with increase in DM particle mass. Also, the merger time increases with fraction of baryons, reaches a maximum for planets made of 60% baryonic matter.

At the center of our galaxy is a BH of mass 4 million solar masses (${\mathit{M}}_{\mathit{B}\mathit{H}})$. It appears reasonable to assume that stars near the Galactic center (several stellar clusters are known to exist near galactic center [19,20] have planets and other small orbiting bodies, such as asteroids and comets. When the parent star approaches the central black hole, tidal interaction may either strip these bodies off their parent stars or cause them to become more tightly bound. If we consider the primordial object orbiting around this BH, at $\mathit{R}=10{\mathit{R}}_{\mathit{s}\mathit{c}\mathit{h}}$. where R is the orbital radius, then according to Kepler’s law, the orbital frequency is given by

The system will lose energy as the object orbits around the central BH and the orbital radius keeps decreasing. The energy emitted per second is given by equation (23). For the primordial planet of mass $2{\mathit{M}}_{\mathit{J}}$ and ${\mathit{R}}_{\mathbf{o}\mathbf{b}\mathbf{j}}=9.8\times {10}^5\mathit{c}\mathit{m}$, energy emitted per unit time works out to be $1.6\times {10}^{33}\mathit{e}\mathit{r}\mathit{g}/\mathit{s}$. At orbital radius equal to Schwazchild radius of central BH, the binding energy will be emitted as gravitational waves. The final orbital frequency and Energy emitted as gravitational waves for the above object works out to be $0.0178\mathit{H}\mathit{z}$ and $1.7\times {10}^{51}\mathit{e}\mathit{r}\mathit{g}\mathit{s}$. The flux per unit time falling on Earth from the gravitational waves is given by where $\dot{E}$ is the energy emitted per unit time from the system and 𝑟 is the distance of Earth from center of galaxy equal to $8kpc$. The merger time of the object with the BH is given by equation (31) and for the above object $t\approx {10}^{19}s$.

Table 4 shows the variation of merger time with different fractions of baryons mixed with DM and with different mass of DM particles ranging from $20GeV$ to $100GeV$.It is found that the merger time of the planet with the BH decreases as fraction of baryons increases in the primordial planet and also increases with increase in mass of DM particles in forming the planet. Considering the primordial planet at center of galaxy $(\approx 8kpc$ from Earth) the flux per unit time on Earth is very low, of the order of ${10}^{-12}ergs/m^{2}s$.

If the objects were made of pure DM, the binding energy of the merging objects would be released as gravitational waves. In the case of primordial planet admixed with DM and baryons, the binding energy due to baryons will be emitted as electromagnetic waves (like merging neutron stars). Thus, during the merger, there will be emission of gravitational waves and electromagnetic waves by the baryonic particles inside the object. For objects with equal proportion of DM and baryonic matter, the energy released as EM waves will be of the order of ${10}^{47}\mathit{e}\mathit{r}\mathit{g}\mathit{s}$ and it will be emitted in the frequency range of gamma radiation. These gamma rays emitted would be in short duration bursts with the period same as final orbital period before merger (around 5 milli-second).

Here we discussed the possibility of admixture of baryons to the DM primordial planets with the DM particles varying in mass from 20GeV to 100 GeV. We have considered different fractions of admixture to form the planet. The mass of the primordial planet made completely of DM, ranges from asteroid mass to Neptune mass. Whereas, the mass of primordial planets (admixed with DM and baryonic matter) is found to increase with the fraction of baryonic matter in the planets and the mass of these objects can go well beyond the mass of Jupiter (around 40 times Jupiter mass) and can also approach sub stellar mass (Brown dwarf mass). So far, thousands of exoplanets have been discovered by the Kepler mission and more will be found by NASA’s Transiting Exoplanet Survey Satellite (TESS) mission, which is observing the entire sky to locate planets orbiting the nearest and brightest stars. Many exoplanets (Exo Jupiters) discovered so far fall in this mass range and we are not very sure whether these exoplanets are entirely made of baryons. Some of the exoplanets with mass several times Jupiter mass could be possible signatures of the presence of primordial planets with an admixture of baryonic and DM particles. It is also found that some of these planets could reach even sub stellar mass $\left({10}^{32}\mathit{g}\right)$ like that of brown dwarf [21]. Also, even if a small fraction of DM particles is trapped in these objects, the flux of ambient DM particles would be reduced significantly. This could be one of the many reasons for not detecting the DM particles in various experiments like XENON1T experiment etc. as suggested earlier. If two such primordial planets merge, they will release a lot of energy. The energy released and the time scale of merger of these objects is found to increase with the mass of primordial objects. The frequency of gravitational waves emitted during the merger is found to match with the frequency range of LIGO. The objects near the galactic center could consists of such primordial objects, planets, comets etc. Here we have also discussed the possibility of tidal break up of these primordial objects in the presence of a BH. The mass of BH required for tidal break up is calculated and it is found that the mass of BH required for tidal break up increases with the DM particle mass and also with the increase in fraction of baryons in these objects. The energy released in the form of gravitational waves as well as the frequency of emission is tabulated and again it is found that the frequency falls in the sensitivity range of LIGO.