High frequency background gravitational waves from spontaneous emission of gravitons by hydrogen and helium

A direct consequence of quantization of gravity would be the existence of gravitons. Therefore, spontaneous transition of an atom from an excited state to a lower-lying energy state accompanied with the emission of a graviton is expected. In this paper, we take the gravitons emitted by hydrogen and helium in the Universe after recombination as a possible source of high frequency background gravitational waves, and calculate the energy density spectrum. Explicit calculations show that the most prominent contribution comes from the $3d-1s$ transition of singly ionized helium $\mathrm{He}^{+}$, which gives a peak in frequency at $\sim10^{13}$ Hz. Although the corresponding energy density is too small to be detected even with state-of-the-art technology today, we believe that the spontaneous emission of $\mathrm{He}^{+}$ is a natural source of high frequency gravitational waves, since it is a direct consequence if we accept that the basic quantum principles we are already familiar with apply as well to a quantum theory of gravity.

In the present paper, we are concerned with another possible source of high frequency gravitational waves, i.e. the spontaneous emission of gravitons by hydrogen and helium after recombination. Recombination is a stage at which the free electrons became bound into hydrogen and helium atoms, ending the scattering of photons. The atoms get excited due to the background thermal radiation, and spontaneous emission occurs. The emitted quanta can be photons, and can also be gravitons if gravity can be quantized. Similar to the electromagnetic spectrum emitted by atoms, the gravitational emission spectrum is expected to be a unique set of discrete spectral lines. However, due to the expansion of the Universe, the emitted gravitons are redshifted, so the spectrum expected to be observed today should be continuous instead of discrete.
In this paper, firstly, we will derive the graviton emission rate for hydrogen atoms.
Let us note that this topic has been studied by several authors [10,[29][30][31]. In Weinberg's book [10], the result is obtained in a semi-classical approach by taking the quantummechanical quadrupole transition matrix elements into the classical formula for the gravitational quadrupole radiation. Following the same approach, Kiefer derived a result which is 4 orders of magnitude larger than Weinberg's result, and pointed out a numerical error in Weinberg's calculation [29]. In Ref. [30], the result is obtained based on the standard perturbation theory with the interaction Hamiltonian H I = − 1 2 h µν T µν . For a hydrogen atom, this Hamiltonian describes the influence of the fluctuating gravitational fields on the elec-tromagnetic interaction between the electron and the nucleus. Note that this Hamiltonian is not gauge invariant. Of course, a gauge dependent Hamiltonian does not necessarily mean that computed physical observables depend on a particular gauge, but much care should be taken when a gauge-dependent quantity is involved in a calculation to ensure that the outcome is not gauge dependent. In Ref. [30], it has been shown that to ensure the gauge invariance of the result, one must consider how the electromagnetic stress-energy is changed in the presence of the gravitational field if one chooses to work in a frame which is not local inertial. Therefore, it is desirable to explore if we can address this issue with a gauge invariant Hamiltonian. This is what we are going to do in the present paper. We assume that the hydrogen atom is subjected to a bath of fluctuating quantum gravitational fields in vacuum which must exist if we accept that the basic quantum principles we are already familiar with apply as well to a quantum theory of gravity since they are necessitated by the uncertainty principle. Under the influence of quantum fluctuations of spacetime, an instantaneous quadrupole moment will be induced in the hydrogen atom. We describe the interaction between the instantaneous quadrupole moment of a hydrogen atom and the fluctuating gravitational fields with the quadrupolar interaction Hamiltonian [32], which does not involve the electromagnetic degrees of freedom and is gauge invariant. So, here not only the issue is dealt with a gauge invariant Hamiltonian, but also the underlying physical picture is different from that in Ref. [30]. We will work out the gravitational polarizability of the hydrogen atom and the emission rate based on the formalism first proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji (DDC) [33,34], which has also recently been applied to study the spontaneous excitation of an accelerated atom coupled with quantum fluctuations of spacetime [35]. In the DDC approach, we can separately calculate the contributions of vacuum fluctuations and radiation reaction to the emission rate. Secondly, we will calculate the density spectrum of gravitational waves we observe today from the spontaneous emission of hydrogen atoms in the Universe. We also consider the contribution to the background gravitational waves from helium since it is the second most abundant element in the Universe. Natural units = c = 16πG = 1 will be used in this paper.

II. GRAVITON EMISSION RATE FOR MULTILEVEL ATOMS
We plan to study the spontaneous emission of gravitons for a multilevel atom in interaction with a bath of fluctuating quantum gravitational fields in vacuum, and consider it as a source of high frequency gravitational waves. We assume that the atoms comove with the expansion of the Universe, so the proper time of the atoms τ coincides with the cosmic time. Here, for simplicity, we neglect the effect of the cosmic expansion when calculating the spontaneous emission rate, but take it into account later when calculating the density spectrum of gravitational waves. Therefore, we assume that the spacetime metric g µν can be expanded as the metric of the Minkowski spacetime η µν and a linearized perturbation h µν . The Hamiltonian of a multilevel atom can be written as where τ is the proper time, σ nn (τ ) = |n n|, and |n denotes the eigenstate of the atom with energy ω n . The Hamiltonian of the quantum gravitational field takes the form in which k denotes the wave vector, a † k and a k are the creation and annihilation operators respectively. The quadrupolar interaction between the atom and the quantum gravitational fields can be expressed as where is the gravitational quadrupole moment operator of the atom with ρ M (x) describing the mass distribution, and E ij = C i0j0 , with C i0j0 being the Weyl tensor, which can be regarded as the trace-free part of the Riemann tensor R i0j0 . It has been shown in Ref. [36] that the Riemann tensor is gauge invariant in linearized theory of gravity, and so is the Weyl tensor.
Therefore, the quadrupolar interaction Hamiltonian (3) we use here is gauge invariant, which is different from the gauge dependent one used in Ref. [30]. The derivation of the interaction Hamiltonian can be found, e.g., in Ref. [32].
The Heisenberg equations of motion for dynamical variables of the atom and the gravitational field can be derived from the total Hamiltonian H = H A (τ ) + H F (τ ) + H I (τ ).
Following the DDC formalism [33,34], the equation of motion for the atomic energy H A can be separated into two parts, i.e. the vacuum fluctuations (VF) and the radiation reaction (RR) with the symmetric ordering of variables between the atom and field. Assume that initially the field is in the vacuum state |0 , and the atom is in state |b . The expectation of the rate of change of the atomic energy in state |b can then be expressed as where | =|b, 0 . Here, C F ijkl and χ F ijkl are the symmetric correlation function and linear susceptibility of the gravitational field, which are defined as Similarly, (C A ijkl ) b and (χ A ijkl ) b are the symmetric correlation function and the linear susceptibility of the atom, We assume that the atom is static and located at the origin, so its trajectory can be written as where τ is the proper time. In the following, we work in the transverse-traceless (TT) gauge, so there are only spatial components h ij in the gravitational perturbations. In the quantum linearized theory of gravity, the quantized spacetime perturbation h ij can be written as [37] where H.c. represents the Hermitian conjugate, λ labels the polarization states, e µν (k, λ) is the polarization tensor, and ω = |k| = (k 2 where a dot denotes derivative with respect to t. The two-point function of E ij in the vacuum state can then be obtained as The summation of the polarization tensors in the transverse traceless gauge takes the following form [37], wherek i = k i /k. According to Eqs. (7) and (8), the field statistical functions C F ijkl and χ F ijkl can be calculated as where The nonzero components of C F ijkl satisfy the following relations, which are the same for χ F ijkl . Inserting a complete set of states into Eqs. (9) and (10), it can be shown that the explicit forms of the statistical functions of the atom (C A ijkl ) b (τ, τ ) and respectively, where ω bd = ω b − ω d . With a substitution u = τ − τ , and an extension of the range of integration to infinity 1 , the contributions of vacuum fluctuations and radiation reaction to the average rate of change of the atomic energy can be calculated from Eqs. (5) and (6) as, and 1 Here it is assumed that the time τ is much larger than the correlation time of the bath of fluctuating gravitational fields τ c , so the contribution to the integration comes mainly from the interval [0, τ c ], and it is safe to extend the integration to infinity [33,34]. where are the Fourier transforms of C F ijkl and χ F ijkl . Note that the quadrupole-dependent terms in Eqs. (21) and (22) are equal. This leads to the cancellation of the contributions from vacuum fluctuations and radiation reaction for an inertial atom when ω bd < 0, i.e. transitions to higher-lying levels are not allowed for inertial atoms in vacuum, as expected. When ω bd > 0, the total average rate of change of the atomic energy is Here we have returned to the SI units, and have defined the gravitational polarizability as

III. THE DENSITY SPECTRUM OF GRAVITATIONAL WAVES
First, we investigate the density spectrum of background gravitational waves emitted by hydrogen atoms. We assume that the hydrogen atoms are in thermal equilibrium with the background radiation and satisfy the Boltzmann distribution, so the population decreases significantly as the principal quantum number n increases. On the other hand, gravitons are expected to be spin-2 particles, so the change of the orbital angular momentum quantum number ∆l should be 2 after transition. Therefore, the most prominent process would be the transition from 3d to 1s.
The wavefunctions of hydrogen atoms in 3d and 1s states are where m e is the mass of an electron, and a is the Bohr radius. Therefore, which agrees with the previous results derived from a semi-classical approach [29], and the standard perturbation theory [30]. At first glance, it may seem puzzling that the rate of change of the atomic energy d dτ H A (τ ) is proportional a 4 while the polarizability α ijkl is proportional to a 2 . Recall that the Bohr radius a = 4π 0 2 mee 2 , and the transition frequency ω mn = − mee 4 2(4π 0 ) 2 3 1 m 2 − 1 n 2 , so the Bohr radius a, the transition frequency ω bd , and the mass of electrons m e are not independent physical quantities. We write the rate of change of the atomic energy and the polarizability in the way above, since we are trying to avoid the Coulomb constant 1 4π 0 and the charge of electron e in an issue with gravitation as the main concern.
At a redshift z ∼ 1100, free electrons and protons became bound to form hydrogen atoms, and the Universe became transparent, which is known as recombination [38]. In this paper, we consider gravitational waves produced by the spontaneous emission of gravitons of hydrogen atoms in the Universe after recombination. Due to the expansion of the Universe, the gravitons emitted by hydrogen atoms are redshifted, and the redshifts are different for gravitons emitted at different time. Therefore, the spectrum expected to be observed today should be continuous instead of a series of discrete frequencies.
In the following, we calculate the current energy density ρ normalized with respect to the critical energy density ρ c = The gravitational energy density emitted by hydrogen atoms from the time τ to τ + dτ can be expressed as where N (z) is the number density of hydrogen atoms in the 3d state. Here ρ represents the energy density at the current epoch z = 0, which scales as 1 (1+z) 4 due to the volume dilution as well as the redshift caused by the cosmic expansion. The minus sign indicates that a decrease in the atomic energy means an increase in the gravitational wave energy. For simplicity, we assume that all atoms (ordinary matter) today are hydrogen atoms, and the number is conserved during the expansion of the Universe. Therefore, the number density of hydrogen atoms in the 3d state at redshift z can be estimated as where m H is the mass of a hydrogen atom, P B ≈ 5% is the current percentage of the baryonic matter, and P 3d (z) is the percentage of hydrogen atoms in the 3d state. Here, we neglect the atoms with principal quantum number n ≥ 5 since the percentage is extremely small.
The percentage of atoms in the 3d state can then be expressed as Here where B n = 13.6/n 2 eV is the binding energy, k B is the Boltzmann constant, n 1s is the population in the 1s state, and Taking Eqs. (32)-(37) into Eq. (31), we have Following the same procedures, we have considered all possible transitions for hydrogen atoms with the principal quantum number up to n = 4. The results are shown in Fig. 1.
As expected, the dominant contribution comes from the 3d − 1s transition at the redshift z ∼ 1100, which corresponds to the peak in frequency at ω = 1.67 × 10 13 Hz in Fig. 1 (left), and the relative energy density is ∼ 10 −54 . As ω increases, the energy density drops significantly. Physically, this means that the population of hydrogen atoms in the 3d state significantly decreases as the Universe cools down. There is also a small peak in frequency at ω = 1.76 × 10 13 Hz in Fig. 1 (left), due to the 4d − 1s transition, and the energy density is one order of magnitude smaller than the 3d − 1s one, since the population of hydrogen atoms in the 4d state is smaller than that in the 3d state. In Fig. 1  Apart from hydrogen, helium is the second most abundant element in the Universe, constituting ∼ 24% of the baryonic matter. The binding energy of helium is larger than that of hydrogen, so the recombination comes earlier. The recombination takes place in two steps. The recombination of singly ionized helium He + takes place around redshift z ≈ 6000, and the recombination of neutral helium takes place around redshift z ≈ 2000 [39]. Here, we consider only the contribution from He + ions, since they recombine earlier and therefore the population of higher-lying excited states is larger, and they have a longer time to emit.
Since a singly ionized helium He + is a hydrogen-like atom, following the same calculations which have been done in the case of hydrogen atoms, we find a spectrum whose shape is similar to that of the hydrogen atoms but the signal is much stronger. Here, the dominant contribution comes from the 3d − 1s transition of He + at the redshift z ∼ 6000, which gives a peak in frequency at ω = 1.22 × 10 13 Hz, and the relative energy density is ∼ 10 −48 , which is 6 orders of magnitude larger than that from hydrogen atoms. This significant difference mainly comes from a much larger population of higher-lying excited states since the recombination of He + is much earlier and therefore the Universe is much hotter.
In summary, we take the gravitons emitted by hydrogen and helium in the Universe after recombination as a possible source of high frequency gravitational waves. In order to calculate the energy density spectrum, we first obtain the transition rate for multilevel atoms in interaction with a bath of fluctuating quantum gravitational fields using the DDC formalism in the framework of the quantum linearized theory of gravity. Then we derive the energy density spectrum expected to be observed today. Explicit calculations show that the most prominent contribution comes from the 3d − 1s transition singly ionized helium He + , which gives a peak frequency at ω ∼ 10 13 Hz. Since the population in excited states decreases significantly as the temperature of the Universe cools down, the energy density quickly decreases as the frequency we observe today increases. Although far from the precision of measurement today, we believe that the spontaneous emission of He + is a natural source of high frequency gravitational waves, since it is a direct consequence if we accept that the basic quantum principles we are already familiar with apply as well to a quantum theory of gravity and no hypothetical theories are involved.