# 1.3 – Gravitational Radiation

If a Holeum in an excited state $n^{^{\prime}}$ goes to a lower energy state $n$, it emits a gravitational radiation of frequency given by

 $\nu=\nu_{g}\left( n^{-2}-n^{^{\prime-2}}\right)$ (eq. 1.55)

where

 $\nu_{g}=\frac{mc^{2}\alpha_{g}^{2}}{4h}$ (eq. 1.56)

which is the gravitational Rydberg constant. Note that the orbital angular momentum must change by a multiple of 2 because of the spin 2 of the graviton. Now $n=n_{r}+l+1$ where $n_{r}$ is the radial quantum number and $l$ is the orbital angular momentum quantum number. Therefore, $n^{^{\prime}}-n$ must be an even integer. Note that we can rewrite equation (1.56) using equation (1.26) as

 $\nu_{g}=\nu_{0}\left(\frac{m}{m_{P}}\right)^{5}$ (eq. 1.57)

where

 $\nu_{0}=\frac{m_{P}c^{2}}{4h}=7.416346\times10^{41}$ Hz (eq. 1.58)

and

 $r_{n}=n^{2}\left(\frac{\pi^{2}}{8}\right)\left(\frac{R_{P}}{R}\right)^{3}R_{P}=R_{g}n^{2}$ (eq. 1.59)

where $R_{P}$ is the Schwarzschild radius of a black hole of mass $m_{P}$. It is given by

 $R_{P}=\frac{2m_{P}G}{c^{2}}=1.609502\times10^{-35}$ m (eq. 1.60)

and

 $R_{g}=\left(\frac{\pi^{2}}{8}\right)\left(\frac{R_{P}}{R}\right)^{3}R_{P}$ (eq. 1.61)

Using equations (1.57) – (1.60), we present the values of several quantities of interest in Table 1.2. We note that presently gravitational wave detectors are being built around frequencies of about $10^{2}$$10^{3}$ Hz. From Table 1.2 we see that Holeums having sizes in the atomic and the nuclear range would emit gravitational radiation covering this range. This can be tested once sufficiently powerful gravitational wave detectors become operational. The corresponding masses of the primordial black holes are in the range of $10^{11}$ GeV/c$^{2}$$10^{12}$ GeV/c$^{2}$.

From equation (12.59) we get

 $\frac{dr_{n}}{dn}=2nR_{g}$ (eq. 1.62)

This quantity, $R_{g}$ given by equation (1.61), is presented in Table 1.2. From the latter we see that $R_{g}$ is extremely small for massive Holeums. This makes the derivative in equation (1.62) negligibly small implying great stability of their orbits. As seen from equations (1.57) and (1.58), the transitions among the discrete levels $n<10$, say, of the massive Holeums are very costly in terms of energy. And, as discussed in [1], the levels with $n\gg1$ have large orbital angular momenta. Hence the transitions among them are very costly in terms of angular momentum. The overall result is that massive Holeums have much more stable orbits than lighter ones. Hence lighter Holeums are more likely to radiate gravitational waves than heavier ones. From Table 1.2 we see that lighter Holeums emit frequencies in the detectable range.

Table 1.2: We present here a table containing data about the physical properties of Holeums having constituent Primordial Black Holes of masses ranging from $1.0\times10^{3}$ GeV/c$^{2}$ to $1.075156\times10^{19}$ GeV/c$^{2}$. The following quantities are displayed in the table: $m$: the mass of the constituent Primordial Black Holes of the Holeum (GeV/c$^{2}$), $E_{n}$: the ionization energy of the Holeum (GeV), $R_{g}$: the ground state radius of the Holeum (m), $m_{H}$: the mass of the Holeum (GeV/c$^{2}$), $\nu_{g}$: the gravitational Rydberg constant (Hz).

 $m$ $(GeV/c^{2})$ $\left\vert E_{n}\right\vert$ $(GeV)$ $R_{g}$ $(m)$ $m_{H}$ $(GeV/c^{2})$ $\nu_{g}$ $(Hz)$ $1.0\times10^{3}$ $1.124437\times 10^{-62}$ $5.869605\times10^{13}$ $2.00\times10^{03}$ $2.716031\times10^{-39}$ $2.5\times10^{3}$ $1.098083\times 10^{-60}$ $3.756547\times10^{12}$ $5.00\times10^{03}$ $2.652374\times10^{-37}$ $5.0\times10^{3}$ $3.513865\times 10^{-59}$ $4.695684\times10^{11}$ $1.00\times10^{04}$ $8.487597\times10^{-36}$ $7.5\times10^{3}$ $2.668341\times 10^{-58}$ $1.391314\times10^{11}$ $1.50\times10^{04}$ $6.445269\times10^{-35}$ $1.0\times10^{4}$ $1.124437\times 10^{-57}$ $5.869605\times10^{10}$ $2.00\times10^{04}$ $2.716031\times10^{-34}$ $2.5\times10^{4}$ $1.098083\times 10^{-55}$ $3.756547\times10^{09}$ $5.00\times10^{04}$ $2.652374\times10^{-32}$ $5.0\times10^{4}$ $3.513865\times 10^{-54}$ $4.695684\times10^{08}$ $1.00\times10^{05}$ $8.487597\times10^{-31}$ $7.5\times10^{4}$ $2.668341\times 10^{-53}$ $1.391314\times10^{08}$ $1.50\times10^{05}$ $6.445269\times10^{-30}$ $1.0\times10^{5}$ $1.124437\times 10^{-52}$ $5.869605\times10^{07}$ $2.00\times10^{05}$ $2.716031\times10^{-29}$ $2.5\times10^{5}$ $1.098083\times 10^{-50}$ $3.756547\times10^{06}$ $5.00\times10^{05}$ $2.652374\times10^{-27}$ $5.0\times10^{5}$ $3.513865\times 10^{-49}$ $4.695684\times10^{05}$ $1.00\times10^{06}$ $8.487597\times10^{-26}$ $7.5\times10^{5}$ $2.668341\times 10^{-48}$ $1.391314\times10^{05}$ $1.50\times10^{06}$ $6.445269\times10^{-25}$ $1.0\times10^{6}$ $1.124437\times 10^{-47}$ $5.869605\times10^{04}$ $2.00\times10^{06}$ $2.716031\times10^{-24}$ $2.5\times10^{6}$ $1.098083\times 10^{-45}$ $3.756547\times10^{03}$ $5.00\times10^{06}$ $2.652374\times10^{-22}$ $5.0\times10^{6}$ $3.513865\times 10^{-44}$ $4.695684\times10^{02}$ $1.00\times10^{07}$ $8.487597\times10^{-21}$ $7.5\times10^{6}$ $2.668341\times 10^{-43}$ $1.391314\times10^{02}$ $1.50\times10^{07}$ $6.445269\times10^{-20}$ $1.0\times10^{7}$ $1.124437\times 10^{-42}$ $5.869605\times10^{01}$ $2.00\times10^{07}$ $2.716031\times10^{-19}$ $2.5\times10^{7}$ $1.098083\times 10^{-40}$ $3.756547\times10^{00}$ $5.00\times10^{07}$ $2.652374\times10^{-17}$ $5.0\times10^{7}$ $3.513865\times 10^{-39}$ $4.695684\times10^{-01}$ $1.00\times10^{08}$ $8.487597\times10^{-16}$ $7.5\times10^{7}$ $2.668341\times 10^{-38}$ $1.391314\times10^{-01}$ $1.50\times10^{08}$ $6.445269\times10^{-15}$ $1.0\times10^{8}$ $1.124437\times 10^{-37}$ $5.869605\times10^{-02}$ $2.00\times10^{08}$ $2.716031\times10^{-14}$ $2.5\times10^{8}$ $1.098083\times10^{-35}$ $3.756547\times10^{-03}$ $5.00\times10^{08}$ $2.652374\times10^{-12}$ $5.0\times10^{8}$ $3.513865\times 10^{-34}$ $4.695684\times10^{-04}$ $1.00\times10^{09}$ $8.487597\times10^{-11}$ $7.5\times10^{8}$ $2.668341\times 10^{-33}$ $1.391314\times10^{-04}$ $1.50\times10^{09}$ $6.445269\times10^{-10}$ $1.0\times10^{9}$ $1.124437\times 10^{-32}$ $5.869605\times10^{-05}$ $2.00\times10^{09}$ $2.716031\times10^{-09}$ $2.5\times10^{9}$ $1.098083\times 10^{-30}$ $3.756547\times10^{-06}$ $5.00\times10^{09}$ $2.652374\times10^{-07}$ $5.0\times10^{9}$ $3.513865\times 10^{-29}$ $4.695684\times10^{-07}$ $1.00\times10^{10}$ $8.487597\times10^{-06}$ $7.5\times10^{9}$ $2.668341\times 10^{-28}$ $1.391314\times10^{-07}$ $1.50\times10^{10}$ $6.445269\times10^{-05}$ $1.0\times10^{10}$ $1.124437\times10^{-27}$ $5.869605\times10^{-08}$ $2.00\times10^{10}$ $2.716031\times10^{-04}$ $2.5\times10^{10}$ $1.098083\times10^{-25}$ $3.756547\times10^{-09}$ $5.00\times10^{10}$ $2.652374\times10^{-02}$ $5.0\times10^{10}$ $3.513865\times10^{-24}$ $4.695684\times10^{-10}$ $1.00\times10^{11}$ $8.487597\times10^{-01}$ $7.5\times10^{10}$ $2.668341\times10^{-23}$ $1.391314\times10^{-10}$ $1.50\times10^{11}$ $6.445269\times10^{00}$ $1.0\times10^{11}$ $1.124437\times10^{-22}$ $5.869605\times10^{-11}$ $2.00\times10^{11}$ $2.716031\times10^{01}$ $2.5\times10^{11}$ $1.098083\times10^{-20}$ $3.756547\times10^{-12}$ $5.00\times10^{11}$ $2.652374\times10^{03}$ $5.0\times10^{11}$ $3.513865\times10^{-19}$ $4.695684\times10^{-13}$ $1.00\times10^{12}$ $8.487597\times10^{04}$ $7.5\times10^{11}$ $2.668341\times10^{-18}$ $1.391314\times10^{-13}$ $1.50\times10^{12}$ $6.445269\times10^{05}$ $1.0\times10^{12}$ $1.124437\times10^{-17}$ $5.869605\times10^{-14}$ $2.00\times10^{12}$ $2.716031\times10^{06}$ $2.5\times10^{12}$ $1.098083\times10^{-15}$ $3.756547\times10^{-15}$ $5.00\times10^{12}$ $2.652374\times10^{08}$ $5.0\times10^{12}$ $3.513865\times10^{-14}$ $4.695684\times10^{-16}$ $1.00\times10^{13}$ $8.487597\times10^{09}$ $7.5\times10^{12}$ $2.668341\times10^{-13}$ $1.391314\times10^{-16}$ $1.50\times10^{13}$ $6.445269\times10^{10}$ $1.0\times10^{13}$ $1.124437\times10^{-12}$ $5.869605\times10^{-17}$ $2.00\times10^{13}$ $2.716031\times10^{11}$ $2.5\times10^{13}$ $1.098083\times10^{-10}$ $3.756547\times10^{-18}$ $5.00\times10^{13}$ $2.652374\times10^{13}$ $5.0\times10^{13}$ $3.513865\times10^{-09}$ $4.695684\times10^{-19}$ $1.00\times10^{14}$ $8.487597\times10^{14}$ $7.5\times10^{13}$ $2.668341\times10^{-08}$ $1.391314\times10^{-19}$ $1.50\times10^{14}$ $6.445269\times10^{15}$ $1.0\times10^{14}$ $1.124437\times10^{-07}$ $5.869605\times10^{-20}$ $2.00\times10^{14}$ $2.716031\times10^{16}$ $2.5\times10^{14}$ $1.098083\times10^{-05}$ $3.756547\times10^{-21}$ $5.00\times10^{14}$ $2.652374\times10^{18}$ $5.0\times10^{14}$ $3.513865\times10^{-04}$ $4.695684\times10^{-22}$ $1.00\times10^{15}$ $8.487597\times10^{19}$ $7.5\times10^{14}$ $2.668341\times10^{-03}$ $1.391314\times10^{-22}$ $1.50\times10^{15}$ $6.445269\times10^{20}$ $1.0\times10^{15}$ $1.124437\times10^{-02}$ $5.869605\times10^{-23}$ $2.00\times10^{15}$ $2.716031\times10^{21}$ $2.5\times10^{15}$ $1.098083\times10^{00}$ $3.756547\times10^{-24}$ $5.00\times10^{15}$ $2.652374\times10^{23}$ $5.0\times10^{15}$ $3.513865\times10^{01}$ $4.695684\times10^{-25}$ $1.00\times10^{16}$ $8.487597\times10^{24}$ $7.5\times10^{15}$ $2.668341\times10^{02}$ $1.391314\times10^{-25}$ $1.50\times10^{16}$ $6.445269\times10^{25}$ $1.0\times10^{16}$ $1.124437\times10^{03}$ $5.869605\times10^{-26}$ $2.00\times10^{16}$ $2.716031\times10^{26}$ $2.5\times10^{16}$ $1.098083\times10^{05}$ $3.756547\times10^{-27}$ $5.00\times10^{16}$ $2.652374\times10^{28}$ $5.0\times10^{16}$ $3.513865\times10^{06}$ $4.695684\times10^{-28}$ $1.00\times10^{17}$ $8.487597\times10^{29}$ $7.5\times10^{16}$ $2.668341\times10^{07}$ $1.391314\times10^{-28}$ $1.50\times10^{17}$ $6.445269\times10^{30}$ $1.0\times10^{17}$ $1.124437\times10^{08}$ $5.869605\times10^{-29}$ $2.00\times10^{17}$ $2.716031\times10^{31}$ $2.5\times10^{17}$ $1.098083\times10^{10}$ $3.756547\times10^{-30}$ $5.00\times10^{17}$ $2.652374\times10^{33}$ $5.0\times10^{17}$ $3.513865\times10^{11}$ $4.695684\times10^{-31}$ $1.000000\times10^{18}$ $2.13\times10^{48}$ $7.5\times10^{17}$ $2.668341\times10^{12}$ $1.391314\times10^{-31}$ $1.500003\times10^{18}$ $1.08\times10^{49}$ $1.0\times10^{18}$ $1.124437\times10^{13}$ $5.869605\times10^{-32}$ $2.000011\times10^{18}$ $3.41\times10^{49}$ $2.5\times10^{18}$ $1.098083\times10^{15}$ $3.756547\times10^{-33}$ $5.001098\times10^{18}$ $1.33\times10^{51}$ $5.0\times10^{18}$ $3.513865\times10^{16}$ $4.695684\times10^{-34}$ $1.003514\times10^{19}$ $2.14\times10^{52}$ $7.5\times10^{18}$ $2.668341\times10^{17}$ $1.391314\times10^{-34}$ $1.526683\times10^{19}$ $1.10\times10^{53}$ $1.0\times10^{19}$ $1.124437\times10^{18}$ $5.869605\times10^{-35}$ $2.112444\times10^{19}$ $3.60\times10^{53}$ $1.075156\times10^{19}$ $1.615446\times10^{18}$ $4.722744\times10^{-35}$ $2.311857\times10^{19}$ $4.90\times10^{53}$

### References

[1] L. K. Chavda and Abhijit L. Chavda, Dark matter and stable bound states of primordial black holes, arXiv:gr-qc/0308054 (2002).

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