#  Complete Gravitational-Wave Spectrum of the Sun

Camilo García-Cely  Instituto de Física Corpuscular (IFIC), Universitat de
València-CSIC, Parc Científic UV, C/ Catedrático José Beltrán 2, E-46980
Paterna, Spain  Andreas Ringwald  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

The high-temperature plasma in the solar interior generates stochastic
gravitational waves (GWs). Due to its significance as the primary source of
high-frequency GWs in the solar system, we reexamine this phenomenon
highlighting some physical processes, including the contribution of
macroscopic hydrodynamic fluctuations. Our analysis builds upon several
studies of axion emission from the Sun, particularly in relation to the
treatment of plasma effects. Similar to many well-motivated Early Universe
signals, we find that the resulting GW spectrum is several orders of magnitude
below the current sensitivities of axion helioscopes such as (Baby)IAXO.

††preprint: DESY-24-111

Introduction. While current observational efforts [1, 2, 3, 4, 5, 6] to detect
gravitational waves (GWs) have predominantly focused on frequencies below a
few kHz, a growing community is seriously considering higher frequencies with
the hope of detecting Early Universe signals [7]. This prompts the question of
identifying GW backgrounds associated with the Standard Model (SM) at this
frequency range. The dominant source of such GWs is the Sun.

A seminal work by Weinberg [8] estimated a total power of approximately
$6\times 10^{7}\,\mathrm{W}$. Although this figure turns out to be an
excellent approximation (we find $1.3\times 10^{8}\,\mathrm{W}$), his spectrum
was calculated only for soft gravitons from proton and electron collisions and
neglected radial dependences –particularly of plasma effects [9, 10]– as well
as additional contributions such as that of photoproduction [11, 12].
Employing a state-of-the-art solar model, we revisit the computation of the
solar GW spectrum. For this we rely on several studies of axion emission from
the Sun and show that they can be adapted to solar GWs, particularly those
related to the treatment of plasma effects. Additionally, we point out the
contribution from hydrodynamical fluctuations, previously overlooked, but
essential for completing the spectrum in its low-frequency range. Our
prediction of the complete solar GW power spectrum is summarized in Fig. 1.

We organize the discussion as follows. We start by reviewing properties of the
solar plasma that will be relevant throughout, then we compile all the
contributions to the GW spectrum of the Sun and clarify what has been omitted
in the literature. Subsequently, we discuss implications for axion
helioscopes. In the Appendix we provide the details of our calculations.
Throughout we adopt Heaviside units, take $\hbar=c=1$, and utilize the
Minkowski metric $\eta_{\mu\nu}=\text{diag}(+---)$.

Figure 1:  GW power per unit logarithmic frequency, originating from the solar
thermal plasma as a function of the frequency, $\omega=2\pi f$.

The solar plasma. We model the Sun as a non-relativistic plasma composed of
electrons ($e$) and nucleons ($Z$) following a Maxwell-Boltzmann distribution.
We take the corresponding temperature and number density distributions from
the B16-GS98 Solar model [13], and estimate the electron density assuming full
ionization. Elements heavier than Helium are negligible and are excluded from
the analysis.

GWs originate through microscopic and macroscopic mechanisms, respectively
corresponding to particle collisions emitting gravitons and GW emission
sourced by hydrodynamic fluctuations. The former corresponds to frequencies
much larger than that of collisions in the solar plasma, so that there is
sufficient time for them not to interfere with each other [14]. Below all
collision frequencies of the plasma, fluctuations are large compared with the
characteristic microscopic dimensions, allowing a hydrodynamical approach to
calculate GW emission [15, 16].

The estimation of the collision frequencies for each species,
$\omega^{(i)}_{c}$, is therefore of utmost importance for this work. By
following the approach detailed next, we obtain the results presented in Fig.
2. We first calculate the viscosity cross section, $\sigma_{V}$. For instance,
for $eZ$,

$\sigma_{V}^{(eZ)}=\int\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}^{(eZ)}\sin^{2}\theta\mathrm{d}\Omega\simeq\frac{2\pi\alpha^{2}Z^{2}}{E_{i}^{2}}L_{\text{Coulomb}}\,,$
(1)

and similarly for $ee$. The collision frequencies are then estimated from the
corresponding scattering rate as
$\omega^{(e)}_{c}=n_{e}\sigma_{V}^{(ee)}v+\sum_{Z}n_{Z}\sigma_{V}^{(eZ)}v$.

Screening gives rise, in the non-relativistic limit, to an effective Yukawa
potential with a range given by the Debye-Hückel screening scale,
$\kappa=\left(4\pi\alpha(n_{e}+\sum_{Z}Zn_{Z})/T\right)^{1/2}$ [9], see Fig.
2. For instance, for $eZ$ collisions,
$V_{\text{Coulomb}}=-Z\alpha/r\to-Z\alpha e^{-\kappa r}/r$. For non-
relativistic collisions, the scattering rate associated with these potentials
may be non-perturbative. Nevertheless, in the Sun the scattering largely
occurs either in the Born regime, where perturbation theory is applicable, or
in the semi-classical regime, where classical physics can be employed. In
these cases, the viscosity cross section is given by the last equality in Eq.
(1), where the Coulomb logarithm, $L_{\text{Coulomb}}$, equals $\log
2p_{i}/\kappa$ and $\log E_{i}/(Z^{2}\alpha\kappa)$, in the Born and semi-
classical approximations, respectively [17, 18].

We note that the collision frequencies should be regarded as transitions
scales, particularly because Eq. (1) is valid only up to logarithmic accuracy
[19] and different prescriptions exist for the velocity averaging [17]. Here,
we follow [14] and take $v$ equal to its thermal average. For further details,
see the Appendix.

Figure 2:  Screening scale (dotted), temperature (solid), plasma frequency
(dashed), and collision frequencies (dot-dashed) as a function of distance
from the center for the adopted solar model [13]. See text for details.

GWs from hydrodynamical fluctuations. We closely follow Ref. [15] to calculate
this contribution. In essence, GWs in this limit are sourced by tensor
fluctuations of the energy-momentum tensor, whose Fourier transform is largely
independent of the frequency and wave number. Being purely hydrodynamical, the
tensor fluctuations are simply proportional to the shear viscosity, $\eta$, of
the plasma. Plugging them in Einstein’s Equations, one obtains

$\frac{\mathrm{d}P}{\mathrm{d}\omega}\Bigg{|}_{\mathrm{Hydrodynamics}}=\frac{16G\omega^{2}}{\pi}\int_{\text{Sun}}\\!\\!\\!\\!\mathrm{d}^{3}\mathbf{r}\,\eta
T\,.$ (2)

While this emission has not been previously investigated for the Sun or
stellar plasmas, it is known that it contributes to the GW emission from the
Early Universe plasma, that is, to the Cosmic Gravitational Microwave
Background (CGMB) [15, 20, 21]. The only conceptual difference here is that
viscosity must be computed differently because the solar plasma is non-
relativistic.

Due to the significant mass ratio between protons and electrons, the primary
source of shear viscosity may be attributed to momentum transfer involving
protons. For this we note that proton-proton collisions in the Sun are semi-
classical. Moreover, the proton component of the solar plasma is not strongly
correlated, meaning that the average Coulomb potential energy is much smaller
than the thermal energy (which follows from $\alpha\,n_{p}^{1/3}/T\sim
0.03-0.07\ll 1$). This justifies to employ the numerical fits to the state-of-
the-art simulations reported by Ref. [22] –more precisely, its Eq. (31)– for
the calculation of the viscosity. In practice, this is just a slight
modification of the Landau-Spitzer theory, which predicts $\eta\sim
m_{p}v_{p}/\sigma^{(pp)}_{V}$ [19]. Let us remark that Eq. (2) loses its
validity when the sound waves associated with the hydrodynamical fluctuations
are damped by viscosity [23], which happens for
$\omega\gtrsim\omega^{(p)}_{c}$ in the one-component approximation.

The solar GW power spectrum from hydrodynamic fluctuations is given by the
dashed-dotted line in Fig. 1, where we conservatively put a cut-off for
frequencies above $0.05\,\omega_{c}^{(p)}\big{|}_{\rm Center}$ in light of
damping.

Collision | $\frac{\mathrm{d}\Gamma}{\mathrm{d}\omega\mathrm{d}V}$ |
---|---|---
Photoproduction | $n_{\gamma}n_{Z}GZ^{2}\alpha\pi\,\delta(\omega-E_{i})\frac{p_{i}}{\omega}\int\mathrm{d}\\!\cos\theta\left(\cot^{2}\\!\frac{\theta}{2}[1+\cos^{2}\theta]|F(\theta)|^{2}+{\cal O}(\omega_{\rm Pl}^{2}/\omega^{2})\right)$ | $|F(\theta)|^{2}=\frac{\left(2\omega\sin\\!\frac{\theta}{2}\right)^{2}}{\kappa^{2}+\left(2\omega\sin\\!\frac{\theta}{2}\right)^{2}}$
$\gamma\,Z/e\to Z/e\,h$
Bremsstrahlung | $\frac{32n_{e}n_{Z}GZ^{2}\alpha^{2}p_{i}}{15\omega}\left(\frac{1}{m_{e}}+\frac{1}{m_{Z}}\right)\left(3(1+\xi^{2}){L}+10\xi+{\cal O}(\xi_{s}^{2})\right)$ |  $\xi=p_{f}/p_{i}$, $\xi_{s}=\kappa/p_{i}$
$eZ\to eZ\,h$ | $\omega=E_{i}(1-\xi^{2})$
Bremsstrahlung | $\frac{16n_{e}^{2}G\alpha^{2}p_{i}}{15\omega m_{e}}\left(\left(6(1+\xi^{2})-\frac{3(1-\xi^{2})^{4}+7(1-\xi^{4})^{2}}{2(1+\xi^{2})^{3}}\right){L}+20\xi-\frac{6\xi(1+\xi^{4})}{(1+\xi^{2})^{2}}+{\cal O}(\xi_{s}^{2})\right)$ | $L=\log\sqrt{\frac{(1+\xi)^{2}+\xi_{s}^{2}}{(1-\xi)^{2}+\xi_{s}^{2}}}$
$ee\rightarrow ee\ h$
Table 1: Emission rates of gravitons, $h$, from the indicated process in the
non-relativistic limit. $p_{i}$ ($p_{f}$) denotes the initial (final) momentum
of the colliding particles in the center-of-mass frame, and $E_{i}$ is total
kinetic energy.

GWs from particle collisions. This contribution to the solar GW spectrum is
obtained by thermally averaging individual graviton emission rates,

$\frac{\mathrm{d}P}{\mathrm{d}\omega}\Bigg{|}_{\mathrm{Collisions}}=\int_{\text{Sun}}\\!\\!\\!\\!\mathrm{d}^{3}\mathbf{r}\,\sum_{i}\omega\left\langle\frac{\mathrm{d}\Gamma^{(i)}(\mathbf{r})}{\mathrm{d}\omega\mathrm{d}V}\right\rangle\,.$
(3)

The rates entering here are calculated by squaring the relevant amplitudes
using CalcHEP [24] and FeynRules [25], and taking the non-relativistic limit
for electrons and nucleons. Such rates, at leading order in the screening
scale, are summarized in Table 1, while full expressions are reported in the
Appendix. We assume spherical symmetry for the integration in the Sun.

_Bremsstrahlung._ We account for plasma effects following the bremsstrahlung
treatment of Ref. [10]. This entails modifying the propagator of the photon
exchanged by the charged particles with an effective mass equal to $\kappa$,
as suggested by the aforementioned Yukawa potential. In addition to
corrections of order $\kappa^{2}/p_{i}^{2}$, this introduces an infrared cut-
off. This is essential because, in the absence of screening, the rates exhibit
a logarithmic divergence for soft gravitons.

Notably, our explicit computations reveal that the bremsstrahlung rates depend
on a common regulating logarithm, which for soft gravitons approaches the one
entering the viscosity cross section of Eq. (1). This is not a coincidence,
soft theorems [8] arising from basic principles of quantum field theory
dictate that
$\omega\mathrm{d}\Gamma^{(eZ)}/\mathrm{d}\omega\mathrm{d}V|_{\mathrm{soft}}=(32G/5\pi)E_{i}^{2}\sin^{2}\theta\,n_{e}n_{Z}\sigma^{(eZ)}_{V}v$
for $\omega\ll E_{i}\sim T$, and a similar expression for $ee$ with a factor
$1/2$ 111Although Refs. [8, 14] seemingly omitted it, this symmetrization
factor is explicitly included in the $pp$ chain [61] and in axion
bremsstrahlung from $ee$ [10]. . Applying this to the center of the Sun is how
Weinberg [8, 14] originally found a flat power spectrum,
$\mathrm{d}P/\mathrm{d}\omega$, with only a minor logarithmic dependence.
Interestingly, despite its quantum-mechanical origin, this formula coincides
[14, 27, 28] with the classical emission rate of GWs associated with the
quadrupole of the $eZ/ee$ hyperbolic orbit for sufficiently soft radiation
222Note that this is in agreement with the arguments presented in [62]
concerning the treatment of a classical GW as a collection of individual
gravitons (see also [63])..

Weinberg’s calculation motivated studies of bremsstrahlung beyond the soft
limit. Reference [30, 31] reported rates associated with $eZ$ –but not $ee$–
using similar methods to ours but assuming an infinitely heavy nucleon and no
screening. Likewise, bremsstrahlung rates neglecting screening were calculated
assuming that the emission comes from one electron moving in the Coulomb
potential of a nucleon [32, 33] or another electron [27]. Our results agree
with these studies in the appropriate limit 333A recent attempt to interpolate
between the external field approximation and the semi-classical approach has
been given in Ref. [28]. Our results with no screening do not match theirs as
they neglect electrons’ indistinguishability..

It is worth mentioning that Ref. [27] calculated $ee$ bremsstrahlung rates in
the non-relativistic limit through the following clever insight: _photon_
bremsstrahlung from $ee$ results from their quadrupole, as their dipole
vanishes by their indistinguishability, which allows adapting the $ee$
electromagnetic emission to gravitational radiation, with the adjustment being
a mere global factor. Our rates, obtained through a different approach,
support this conclusion.

Finally, let us remark that our procedure relies on the validity of the Born
approximation, which for $ee$ and $eZ$ holds throughout the Sun, except in the
outermost regions, approximately the last 15% of the solar radius. Since
bremsstrahlung is dominated by processes at the center, neglecting non-
perturbative corrections to the graviton emission is justified. The solar GW
power spectrum from bremsstrahlung is given by the solid line in Fig. 1, where
we put a cut-off for frequencies below $\omega_{c}^{(e)}\big{|}_{\rm Center}$.
We remark that for this we use the full rates reported in the appendix, where
further details are provided.

_Photoproduction._ This is analogue to the Primakoff and Compton production of
axions. Our results in Table 1 with $\kappa=\omega_{\rm Pl}=0$ coincide with
the differential rates for graviton emission reported in Ref. [11].
Nevertheless, from this it is impossible to make sense of the integrated rate,
as it diverges due to the collinear divergence appearing at $\theta=0$. To
tackle this problem, we again employ insights from axion physics [10].
Concretely, we note that charged particles in the solar plasma mutually
interact through their Coulomb fields, leading to a pair correlation function
[35, 10, 36]. This gives rise to an effective form factor –$F(\theta)$, in
Table 1– that depends on the screening scale $\kappa$ and regularizes the
total rate.

If one neglects such correlations and instead consider photons scattering off
individual screened charges, one finds the rate given in Table 1, but with
$|F(\theta)|\to|F(\theta)|^{2}$ [37]. In practice, this corresponds to
modifying the photon propagator with an effective mass, a picture which is
appropriate if the scattering process is so slow that the charged particles
can move around and rearrange themselves [38]. This issue has been revisited
recently for the Primakoff process of axions, accounting for the velocity in
the form factor [39]. Despite the differences between axions and gravitons,
from this study corrections to the choice given in Table 1 are expected to be
negligible.

Note also that the dispersion relation of the initial-state photon is
influenced by the surrounding plasma, which imparts an effective mass to
transverse photons equal to the plasma frequency,
$\omega_{\mathrm{Pl}}=\left(4\pi n_{e}\alpha/m_{e}\right)^{1/2}$. However,
since the solar temperature far exceeds it (see Fig. 2), the GW spectrum
remains largely insensitive to the photon’s effective mass, a fact we have
verified explicitly. The main consequence is the phase-space factor
$p_{i}/\omega$, that provides a lower cut-off to the photoproduction
contribution at each point of the Sun. We note that this is completely
analogous to the Primakoff effect for axions, which we discuss in the appendix
for completeness. It is noteworthy that Ref. [12] calculated photoproduction
rates using semi-classical methods and including the effect of the plasma
frequency, but did not account for screening, which is the dominant effect.

The resulting solar GW power spectrum from photoproduction is given by the
dashed line in Fig. 1.

Transition regime and other processes. As it shifts from macroscopic to
microscopic fluctuations, $\mathrm{d}P/\mathrm{d}\omega$ transitions from
$\propto\omega^{2}$ to $\propto\omega^{0}$ within the range
$\omega^{(p)}_{c}\lesssim\omega\lesssim\omega^{(e)}_{c}$. Similar to the case
of the CGMB, a thorough examination of this regime necessitates a
hydrodynamical simulation. Nevertheless, we postulate an interpolating factor
$\omega^{2}/(\omega^{2}+\omega_{0}^{2})$. Eqs. (2)-(3) and the Landau-Spitzer
viscosity for protons give
$\omega_{0}\approx(\omega^{(p)}_{c}\omega^{(e)}_{c})^{1/2}\propto 1/\eta$.
Notably, the interpolating factor with the predicted scale $\omega_{0}\propto
1/\eta$ is also found in analytical studies of the energy-momentum tensor for
relativistic plasmas (in the so-called shear channel) [15]. It is also worth
mentioning that power shifts from $\mathrm{d}P/\mathrm{d}\omega\propto\eta\sim
1/\sigma_{V}$ to soft bremsstrahlung, where
$\mathrm{d}P/\mathrm{d}\omega\propto\sigma_{V}$. Interestingly, the same
behavior is found for the CGMB because $\sigma_{V}\sim\alpha^{2}/T^{2}$. Our
interpolation of the solar GW power spectrum between the hydrodynamic and
bremsstrahlung regime is given by the orange solid line in Fig. 1.

We might consider other emission processes known to exist for axions,
including small features associated with atomic emission lines [40, 41] as
well as photoproduction exploiting the solar magnetic field at frequencies
$\omega_{\mathrm{Pl}}$ [42, 43, 44, 39], i.e. from longitudinal photons.
However, we emphasize that gravitons exhibit a unique coupling to photons and
fermions indicating fixed proportions for the different contributions of the
solar spectrum. In contrast, axions possess model-dependent couplings to
photons and fermions, allowing scenarios where bremsstrahlung is subdominant
compared to photoproduction processes induced by the solar magnetic field. In
the light of this, the latter hardly contribute to the GW spectrum around
$\omega_{\mathrm{Pl}}\sim 0.3\,\mathrm{keV}$ or lower where bremsstrahlung
dominates by more than five orders of magnitude.

As explained above, similar to axions, the emission rates of gravitons from
atomic levels can be derived by multiplying ordinary atomic transition rates
by a global factor, see e.g. Refs. [45, 46]. Being quadrupole transitions,
these are largely suppressed in comparison to those induced by dipole terms,
to which axions couple [40, 41], and hence we neglect them.

Figure 3:  Characteristic GW strain versus frequency. Exclusion bound and
corresponding projected sensitivities of current (CAST) and future (BabyIAXO,
IAXO) axion helioscopes, together with the corresponding predictions from the
solar GW spectrum in Fig. 1, an estimate of the stochastic background from the
main-sequence stars in the Milky Way, the CGMB spectrum, and the bound from
the total radiation in the Universe.

Discussion and outlook. Our interpolation of the solar GW power spectrum
including all processes is presented in Fig. 1. In terms of power, the
relative contribution of each process is as follows: 25% for photoproduction,
33% for $ee$ bremsstrahlung, 24% for $ep$ bremsstrahlung, 17% for $eHe$
bremsstrahlung, with the rest attributed to the hydrodynamical contribution.

Axion helioscopes [47] can be used as detectors of gravitational radiation
exploiting the inverse Gertsenshtein effect [48, 49], by which GWs convert to
electromagnetic waves in the presence of an external magnetic field. Reference
[50] presented an exclusion bound from CAST [51], sensitivity prospects for
IAXO [52], and compared them against Weinberg’s prediction of the solar GW
spectrum. In Fig. 3 we present their findings in terms of the GW strain,
$h_{c}\equiv 2(G\,\mathrm{d}P/\mathrm{d}\omega)^{1/2}/d_{\rm
Sun}\omega^{1/2}$, together with the complete spectrum as derived from Fig. 1.
For comparison, we also display the CGMB, whose overall normalization depends
upon the (unknown) maximum temperature of the primordial plasma [21]. In
addition, we show the cosmic dark radiation bound, arising from the fact that
GWs contribute to the energy budget of the Universe in the form of radiation
and are thus limited by Big Bang Nucleosynthesis (BBN) and Cosmic Microwave
Background (CMB) observations [7]. In addition, we derive also prospects for
BabyIAXO following Refs. [21, 52]. For further comparison, we include an
estimate of the stochastic background associated with main-sequence stars in
the galaxy. Noting that these stars have similar masses to the Sun [53], we
approximate their GW spectrum using the one of the Sun, leading to
$h_{c\,\text{stars}}=\sqrt{\int dV\,n_{\text{stars}}\,d^{2}_{\rm
Sun}/d^{2}}\,h_{c\,\text{Sun}}$. A similar estimation has been performed in
Ref. [54] for stellar neutrinos. Adapting their results for the
${}^{7}\mathrm{Be}$ line, we find $h_{c\,\text{stars}}\sim
10^{-4}h_{c\,\text{Sun}}$. We note that this figure should be regarded as an
order-of-magnitude estimate.

Our calculation of solar GWs reveals a clear similarity between gravitons and
axions. Expanding on this idea, we could consider the emission of massive
spin-2 particles from the Sun, which could also constitute the dark matter of
the Universe, see e.g. [55, 56, 57]. Extending the methods discussed here to
such particles, we find that the resulting spectrum is nearly identical to
that of the solar GWs, differing only by an overall factor associated with the
coupling of spin-2 particles to SM fields. In a separate publication [58], we
will discuss this synergy and present observational prospects at helioscopes.

Our study represents a step towards establishing a multi-frequency GW
background from SM processes, similar to that of neutrinos [59]. In fact, our
approach can be extended to other stellar plasmas 444See [27] for a discussion
on white dwarfs and neutron stars and to the early stages of the Universe
after nucleosynthesis, where plasmas are also non-relativistic. Figure 3
indicates that current experiments are still many orders of magnitude away
from detecting signals from these backgrounds. Whether we can aspire to probe
such strain sensitivities in the future remains an open question.

Acknowledgements. We thank Diego Blas, Andrea Caputo, Valerie Domcke, Javier
Galán, Maurizio Giannotti, Igor Irastorza, Mikko Laine, Jamie McDonald, Maxim
Pospelov and Georg Raffelt for discussions. CGC is supported by a Ramón y
Cajal contract with Ref. RYC2020-029248-I, the Spanish National Grant
PID2022-137268NA-C55 and Generalitat Valenciana through the grant
CIPROM/22/69. A.R. acknowledges support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC
2121 Quantum Universe - 390833306 and under - 491245950. This article is based
upon work from COST Action COSMIC WISPers CA21106, supported by COST (European
Cooperation in Science and Technology).

## References

  * [1] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 no. 16, (2017) 161101, arXiv:1710.05832 [gr-qc].
  * [2] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116 no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc].
  * [3] EPTA, InPTA: Collaboration, J. Antoniadis et al., “The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals,” Astron. Astrophys. 678 (2023) A50, arXiv:2306.16214 [astro-ph.HE].
  * [4] D. J. Reardon et al., “Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett. 951 no. 1, (2023) L6, arXiv:2306.16215 [astro-ph.HE].
  * [5] NANOGrav Collaboration, G. Agazie et al., “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,” Astrophys. J. Lett. 951 no. 1, (2023) L8, arXiv:2306.16213 [astro-ph.HE].
  * [6] H. Xu et al., “Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,” Res. Astron. Astrophys. 23 no. 7, (2023) 075024, arXiv:2306.16216 [astro-ph.HE].
  * [7] N. Aggarwal et al., “Challenges and opportunities of gravitational-wave searches at MHz to GHz frequencies,” Living Rev. Rel. 24 no. 1, (2021) 4, arXiv:2011.12414 [gr-qc].
  * [8] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140 (1965) B516–B524.
  * [9] E. E. Salpeter, “Electrons Screening and Thermonuclear Reactions,” Australian Journal of Physics 7 (Sept., 1954) 373.
  * [10] G. G. Raffelt, “Astrophysical Axion Bounds Diminished by Screening Effects,” Phys. Rev. D 33 (1986) 897.
  * [11] N. A. Voronov, “Gravitational Compton effect and photoproduction of gravitons by electrons,” Sov. Phys. JETP 37 (1973) 953–958.
  * [12] D. V. Galtsov, “Photocoulomb gravitons and gravitational emittance of the sun,” Zh. Eksp. Teor. Fiz. 67 (1974) 425–427.
  * [13] N. Vinyoles, A. M. Serenelli, F. L. Villante, S. Basu, J. Bergström, M. C. Gonzalez-Garcia, M. Maltoni, C. Peña Garay, and N. Song, “A new Generation of Standard Solar Models,” Astrophys. J. 835 no. 2, (2017) 202, arXiv:1611.09867 [astro-ph.SR].
  * [14] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity.  No. Sec. 10.4. John Wiley and Sons, New York, 1972.
  * [15] J. Ghiglieri and M. Laine, “Gravitational wave background from Standard Model physics: Qualitative features,” JCAP 07 (2015) 022, arXiv:1504.02569 [hep-ph].
  * [16] E. Lifshitz and L. Pitaevskii, Statistical Physics, Part 2.  No. §88-§89. Butterworth-Heinemann, Oxford, 1980.
  * [17] S. Weinberg, “Soft Bremsstrahlung,” Phys. Rev. D 99 no. 7, (2019) 076018, arXiv:1903.11168 [astro-ph.GA].
  * [18] S. A. Khrapak, A. V. Ivlev, G. E. Morfill, and S. K. Zhdanov, “Scattering in the Attractive Yukawa Potential in the Limit of Strong Interaction,” Phys. Rev. Lett. 90 no. 22, (2003) 225002.
  * [19] L. Pitaevskii and E. Lifshitz, Physical Kinetics: Volume 10.  No. $\S$41\. Elsevier Science, 2012.
  * [20] J. Ghiglieri, G. Jackson, M. Laine, and Y. Zhu, “Gravitational wave background from Standard Model physics: Complete leading order,” JHEP 07 (2020) 092, arXiv:2004.11392 [hep-ph].
  * [21] A. Ringwald, J. Schütte-Engel, and C. Tamarit, “Gravitational Waves as a Big Bang Thermometer,” JCAP 03 (2021) 054, arXiv:2011.04731 [hep-ph].
  * [22] J. Daligault, K. O. Rasmussen, and S. D. Baalrud, “Determination of the shear viscosity of the one-component plasma,” Phys. Rev. E 90 (Sep, 2014) 033105. https://link.aps.org/doi/10.1103/PhysRevE.90.033105.
  * [23] P. Klose, M. Laine, and S. Procacci, “Gravitational wave background from non-Abelian reheating after axion-like inflation,” JCAP 05 (2022) 021, arXiv:2201.02317 [hep-ph].
  * [24] A. Belyaev, N. D. Christensen, and A. Pukhov, “CalcHEP 3.4 for collider physics within and beyond the Standard Model,” Comput. Phys. Commun. 184 (2013) 1729–1769, arXiv:1207.6082 [hep-ph].
  * [25] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, “FeynRules 2.0 - A complete toolbox for tree-level phenomenology,” Comput. Phys. Commun. 185 (2014) 2250–2300, arXiv:1310.1921 [hep-ph].
  * [26] Although Refs. [8, 14] seemingly omitted it, this symmetrization factor is explicitly included in the $pp$ chain [61] and in axion bremsstrahlung from $ee$ [10].
  * [27] R. J. Gould, “The graviton luminosity of the sun and other stars,” Astrophys. J.  288 (Jan., 1985) 789–794.
  * [28] A. M. Steane, “Gravitational bremsstrahlung in plasmas and clusters,” Phys. Rev. D 109 no. 6, (2024) 063032, arXiv:2309.06972 [astro-ph.HE].
  * [29] Note that this is in agreement with the arguments presented in [62] concerning the treatment of a classical GW as a collection of individual gravitons (see also [63]).
  * [30] B. M. Barker, S. N. Gupta, and J. Kaskas, “Graviton bremsstrahlung and infrared divergence,” Phys. Rev. 182 (1969) 1391–1396.
  * [31] G. Papini and S. R. Valluri, “Gravitons in Minkowski Space-Time. Interactions and Results of Astrophysical Interest,” Phys. Rept. 33 (1977) 51–125.
  * [32] D. V. Gal’Tsov and Y. V. Grats, “Gravitational radiation from Coulomb collisions,” Soviet Physics Journal 17 no. 12, (Dec., 1974) 1713–1717.
  * [33] R. J. Gould, “Quadrupole Bremsstrahlung in the Scattering of Identical Charged Bosons and Fermions,” Phys. Rev. A 23 (1981) 2851–2857.
  * [34] A recent attempt to interpolate between the external field approximation and the semi-classical approach has been given in Ref. [28]. Our results with no screening do not match theirs as they neglect electrons’ indistinguishability.
  * [35] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1.  No. §79. Butterworth-Heinemann, Oxford, 1980.
  * [36] G. G. Raffelt, “Plasmon Decay Into Low Mass Bosons in Stars,” Phys. Rev. D 37 (1988) 1356.
  * [37] W. K. De Logi and A. R. Mickelson, “Electrogravitational Conversion Cross-Sections in Static Electromagnetic Fields,” Phys. Rev. D 16 (1977) 2915–2927.
  * [38] G. G. Raffelt, Stars as laboratories for fundamental physics: The astrophysics of neutrinos, axions, and other weakly interacting particles.  5, 1996.
  * [39] S. Hoof, J. Jaeckel, and L. J. Thormaehlen, “Quantifying uncertainties in the solar axion flux and their impact on determining axion model parameters,” JCAP 09 (2021) 006, arXiv:2101.08789 [hep-ph].
  * [40] J. Redondo, “Solar axion flux from the axion-electron coupling,” JCAP 12 (2013) 008, arXiv:1310.0823 [hep-ph].
  * [41] M. Pospelov, A. Ritz, and M. B. Voloshin, “Bosonic super-WIMPs as keV-scale dark matter,” Phys. Rev. D 78 (2008) 115012, arXiv:0807.3279 [hep-ph].
  * [42] A. Caputo, A. J. Millar, and E. Vitagliano, “Revisiting longitudinal plasmon-axion conversion in external magnetic fields,” Phys. Rev. D 101 no. 12, (2020) 123004, arXiv:2005.00078 [hep-ph].
  * [43] C. A. J. O’Hare, A. Caputo, A. J. Millar, and E. Vitagliano, “Axion helioscopes as solar magnetometers,” Phys. Rev. D 102 no. 4, (2020) 043019, arXiv:2006.10415 [astro-ph.CO].
  * [44] E. Guarini, P. Carenza, J. Galan, M. Giannotti, and A. Mirizzi, “Production of axionlike particles from photon conversions in large-scale solar magnetic fields,” Phys. Rev. D 102 no. 12, (2020) 123024, arXiv:2010.06601 [hep-ph].
  * [45] S. Boughn and T. Rothman, “Aspects of graviton detection: Graviton emission and absorption by atomic hydrogen,” Class. Quant. Grav. 23 (2006) 5839–5852, arXiv:gr-qc/0605052.
  * [46] T. Rothman and S. Boughn, “Can gravitons be detected?,” Found. Phys. 36 (2006) 1801–1825, arXiv:gr-qc/0601043.
  * [47] P. Sikivie, “Experimental Tests of the Invisible Axion,” Phys. Rev. Lett. 51 (1983) 1415–1417. [Erratum: Phys.Rev.Lett. 52, 695 (1984)].
  * [48] M. E. Gertsenshtein, “ Wave Resonance of Light and Gravitational Waves ,” Sov. Phys. JETP 14 (1962) 84.
  * [49] D. Boccaletti, V. De Sabbata, P. Fortini, and C. Gualdi, “Conversion of photons into gravitons and vice versa in a static electromagnetic field,” Il Nuovo Cimento B (1965-1970) 70 no. 2, (Dec, 1970) 129–146. https://doi.org/10.1007/BF02710177.
  * [50] A. Ejlli, D. Ejlli, A. M. Cruise, G. Pisano, and H. Grote, “Upper limits on the amplitude of ultra-high-frequency gravitational waves from graviton to photon conversion,” Eur. Phys. J. C 79 no. 12, (2019) 1032, arXiv:1908.00232 [gr-qc].
  * [51] CAST Collaboration, V. Anastassopoulos et al., “New CAST Limit on the Axion-Photon Interaction,” Nature Phys. 13 (2017) 584–590, arXiv:1705.02290 [hep-ex].
  * [52] IAXO Collaboration, E. Armengaud et al., “Physics potential of the International Axion Observatory (IAXO),” JCAP 06 (2019) 047, arXiv:1904.09155 [hep-ph].
  * [53] J. Bovy, “Stellar inventory of the solar neighbourhood using gaia dr1,” Monthly Notices of the Royal Astronomical Society 470 no. 2, (May, 2017) 1360–1387. http://dx.doi.org/10.1093/mnras/stx1277.
  * [54] E. Brocato, V. Castellani, S. Degl’Innocenti, G. Fiorentini, and G. Raimondo, “Stars as galactic neutrino sources,” Astron. Astrophys. 333 (1998) 910, arXiv:astro-ph/9711269.
  * [55] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermäe, and M. von Strauss, “Bigravitational origin of dark matter,” Phys. Rev. D 94 no. 8, (2016) 084055, arXiv:1604.08564 [hep-ph].
  * [56] E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermäe, and M. von Strauss, “Heavy spin-2 Dark Matter,” JCAP 09 (2016) 016, arXiv:1607.03497 [hep-th].
  * [57] X. Chu and C. Garcia-Cely, “Self-interacting Spin-2 Dark Matter,” Phys. Rev. D 96 no. 10, (2017) 103519, arXiv:1708.06764 [hep-ph].
  * [58] J. Galán, C. García-Cely, and A. Ringwald In preparation (2024) .
  * [59] E. Vitagliano, I. Tamborra, and G. Raffelt, “Grand Unified Neutrino Spectrum at Earth: Sources and Spectral Components,” Rev. Mod. Phys. 92 (2020) 45006, arXiv:1910.11878 [astro-ph.HE].
  * [60] See [27] for a discussion on white dwarfs and neutron stars.
  * [61] H. A. Bethe and C. L. Critchfield, “The formation of deuterium by proton combination,” Phys. Rev. 54 (1938) 248–254.
  * [62] D. Carney, V. Domcke, and N. L. Rodd, “Graviton detection and the quantization of gravity,” Phys. Rev. D 109 no. 4, (2024) 044009, arXiv:2308.12988 [hep-th].
  * [63] F. Dyson, “Is a graviton detectable?,” Int. J. Mod. Phys. A 28 (2013) 1330041.
  * [64] R. Flauger and S. Weinberg, “Absorption of Gravitational Waves from Distant Sources,” Phys. Rev. D 99 no. 12, (2019) 123030, arXiv:1906.04853 [hep-th].
  * [65] Although this agrees with [64], the viscosity cross section reported by Weinberg in his textbook [14] is $1/2$ of this.
  * [66] S. Y. Choi, J. S. Shim, and H. S. Song, “Factorization and polarization in linearized gravity,” Phys. Rev. D 51 (1995) 2751–2769, arXiv:hep-th/9411092.
  * [67] J. F. Donoghue, “General relativity as an effective field theory: The leading quantum corrections,” Phys. Rev. D 50 (1994) 3874–3888, arXiv:gr-qc/9405057.
  * [68] H. H. Patel, “Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals,” Comput. Phys. Commun. 218 (2017) 66–70, arXiv:1612.00009 [hep-ph].
  * [69] J. Pradler and L. Semmelrock, “Accurate Gaunt Factors for Nonrelativistic Quadrupole Bremsstrahlung,” Astrophys. J. 916 no. 2, (2021) 105, arXiv:2103.03248 [astro-ph.CO].

##  Appendix A Viscosity cross sections in the Born and semi-classical regimes

Figure S1:  Scattering regimes of the indicated solar collisions due to the
screened Coulomb potential. Perturbation theory is applicable in the Born
regime, while classical physics is valid in the semi-classical regime.
Different points correspond to various distances from the center of the Sun.
See the text for details.

Viscosity cross sections can be used to estimate collision frequencies, and as
discussed in the main text, Weinberg’s soft theorem relates them to graviton
bremsstrahlung. Hence, they are of utmost importance for this work. They are
defined by

$\sigma_{V}^{(ab)}=\int\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}^{(ab)}\sin^{2}\theta\,\mathrm{d}\Omega\,.$
(S1)

For a Coulomb potential, $V(r)=-Z_{a}Z_{b}\alpha/r$, between two particles of
charge $Z_{a}$ and $Z_{b}$ (in units of $e$), this cross section
logarithmically diverges, because the corresponding
$\mathrm{d}\sigma^{(ab)}/{\mathrm{d}\Omega}$ scales as $1/\sin^{4}(\theta/2)$.
This is not really a problem because the Coulomb potential is screened. In the
non-relativistic limit, this is described by an effective Yukawa potential
with a range given by the Debye-Hückel screening scale,

$\displaystyle V(r)=-\frac{Z_{a}Z_{b}\alpha}{r}e^{-\kappa r}\,,$ and
$\displaystyle\kappa^{2}=\frac{4\pi\alpha}{T}\left(n_{e}+\sum_{Z}Zn_{Z}\right)\,.$
(S2)

Determining $\mathrm{d}\sigma^{(ab)}/{\mathrm{d}\Omega}$ for non-relativistic
particles interacting by means of such a potential is non-trivial due to non-
perturbative effects. Nevertheless, as we show in the sequel, for the relevant
collisions in the Sun the scattering largely occurs either in the Born regime,
where perturbation theory is applicable, or in the weakly coupled semi-
classical regime, where classical physics can be employed. In both scenarios,
one obtains

$\sigma_{V}^{(ab)}\simeq\frac{2\pi\alpha^{2}Z_{a}^{2}Z_{b}^{2}}{E_{i}^{2}}L_{\text{Coulomb}}\,,$
(S3)

where $L_{\text{Coulomb}}$ is the Coulomb logarithm. We emphasize this is
valid to logarithmic accuracy, meaning that this expressions neglects
quantities smaller than $L_{\text{Coulomb}}$. For a comprehensive discussion
see e.g. Ref. [19]. The Born regime applies for large velocities, namely
$Z\alpha\ll v$, or for sufficiently small couplings, concretely
$Z\alpha\ll\kappa/\mu$ [19], where $\mu$ is the reduced mass. Ordinary quantum
perturbation theory employing the potential in Eq. (S2) gives 555 Although
this agrees with [64], the viscosity cross section reported by Weinberg in his
textbook [14] is $1/2$ of this.

$L_{\text{Coulomb}}=\log\left(\frac{2\mu v}{\kappa}\right)\,.$ (S4)

The semi-classical regime takes place when the de Broglie wavelength is much
smaller than the range of the potential, that is, $\kappa\ll\mu v$ [19]. In
this case there is a critical parameter, $\beta=|Z_{a}||Z_{b}|\kappa\alpha/\mu
v^{2}$, determining different sub-regimes [18]. For $\beta\ll 1$, of interest
in this work, the plasma is weakly-coupled and the Coulomb logarithm is given
by [19, 18]

$L_{\text{Coulomb}}=\log\left(\frac{\mu
v^{2}}{|Z_{a}||Z_{b}|\alpha\kappa}\right)\,.$ (S5)

The range of applicability of either regime along the position in the Sun is
sketched in Fig. S1. The green portion of the lines corresponds to the 15%
outermost part, while the blue part corresponds to the innermost 85%. For
simplicity we only show $ee$, $ep$, $eHe$ and $pp$ collisions. We also show
the line $\beta=1$, above which the plasma is weakly coupled and Eq. (S5) can
be used. Note that pp collisions are semi-classical, as mentioned in the main
text.

##  Appendix B Graviton emission rates and Feynman rules

The emission rate per unit volume of one graviton in the collision of two
particles is given by

$\frac{d\Gamma}{\mathrm{d}\omega\mathrm{d}V}\left(1+2\rightarrow
h_{\lambda}\cdots\right)=\int dn_{1}dn_{2}|{\cal M}(\lambda)|^{2}{\rm
d(PS)}(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-p-\sum_{k}p_{k}\right)\,,$ (S6)

where $p=(\omega,\bf{p})$ and $\lambda$ are respectively the graviton momentum
and helicity, while ${\rm d(PS)}$ is the phase space associated with the
final-state particles,

${\rm d(PS)}=\frac{\omega^{2}{\rm
d}\Omega_{\bf{p}}}{(2\pi)^{3}2\omega}\prod_{k}\frac{{\rm
d}^{3}p_{k}}{(2\pi)^{3}2E_{k}}\,.$ (S7)

In addition, ${\cal M}(\lambda)$ is the graviton emission amplitude, which can
be cast as

${\cal M}(\lambda)={\cal M}_{\mu\nu}\epsilon^{\mu\nu}_{(\lambda)}(p)\,,$ (S8)

where polarizations tensors are given by

$\displaystyle\epsilon^{\mu\nu}_{\pm
2}(p)=\epsilon_{\pm}^{\mu}(p)\epsilon_{\pm}^{\nu}(p)\,,$ with
$\displaystyle\epsilon_{\pm}^{\mu}(p)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
\mp\cos\theta\cos\phi+i\sin\phi\\\ \mp\cos\theta\sin\phi-i\cos\phi\\\
\pm\sin\theta\end{array}\right)\,,$ and $\displaystyle
p^{\mu}=\left(\begin{array}[]{c}\omega\\\ |\bf{p}|\sin\theta\cos\phi\\\
|\bf{p}|\sin\theta\sin\phi\\\ |\bf{p}|\cos\theta\end{array}\right)\,.$ (S17)

When perturbation theory applies, ${\cal M}_{\mu\nu}$ in Eq. (S6) can be
calculated from tree-level diagrams, with the vertices obtained from

$\mathcal{L}=\frac{1}{2}\partial_{\rho}h_{\mu\nu}\partial^{\rho}h^{\mu\nu}-\frac{1}{2}\partial_{\rho}h\partial^{\rho}h+\partial_{\rho}h\partial_{\nu}h^{\rho\nu}-\partial_{\rho}h_{\mu\nu}\partial^{\nu}h^{\mu\rho}+\frac{1}{2}\kappa_{G}\,h_{\mu\nu}T^{\mu\nu}\,,$
(S18)

where $T^{\mu\nu}$ is the stress-energy-momentum tensor of the Standard-Model
particles, and $g_{\mu\nu}=\eta_{\mu\nu}+\kappa_{G}h_{\mu\nu}$ with
$\kappa_{G}=\sqrt{32\pi G}$. The corresponding Feynman rules of relevance for
this work are
$C^{\rm(fermion)}_{\mu\nu}(p_{1},p_{2})=\frac{i}{8}\kappa_{G}\left[2\eta_{\mu\nu}(\not{p}_{1}+\not{p}_{2}-2m_{f})-(p_{1}+p_{2})_{\mu}\gamma_{\nu}-\gamma_{\mu}(p_{1}+p_{2})_{\nu}\right]\,,$
$C^{\rm(photon)}_{\alpha\beta\mu\nu}(p_{1},p_{2})=-\frac{i}{2}\kappa_{G}\left[\eta_{\mu\nu}\eta_{\alpha\beta}\,p_{1}\cdot
p_{2}-\eta_{\mu\nu}p_{1\beta}p_{2\alpha}+\eta_{\nu\alpha}p_{1\beta}p_{2\mu}-\eta_{\alpha\beta}p_{1\nu}p_{2\mu}+\eta_{\mu\beta}p_{1\nu}p_{2\alpha}\right.$
$\qquad\qquad\qquad\left.-\eta_{\nu\alpha}\eta_{\mu\beta}p_{1}\cdot
p_{2}+\eta_{\nu\beta}p_{1\mu}p_{2\alpha}-\eta_{\alpha\beta}p_{1\mu}p_{2\nu}+\eta_{\mu\alpha}p_{1\beta}p_{2\nu}-\eta_{\nu\beta}\eta_{\mu\alpha}p_{1}\cdot
p_{2}\right]\,,$
$C^{\rm(quartic)}_{\alpha\mu\nu}=-\frac{i}{4}eZ\kappa_{G}\left[2\eta_{\mu\nu}\gamma_{\alpha}-\eta_{\alpha\mu}\gamma_{\nu}-\eta_{\alpha\nu}\gamma_{\mu}\right]\,.$

The fermion line is either $e$ or $Z$. These rules agree with existing
literature (see e.g. [66, 67]), and were derived implementing Eq. (S18) in
FeynRules [25]. This process also gives the input needed by CalcHEP [24] for
cross section calculations. Using Package-X [68], we manipulate the symbolic
output thus obtained to compute ${\cal M}^{\mu\nu}$. For all processes, we
verify that

$p^{\mu}{\cal M}_{\mu\nu}=0\,,$ (S19)

as follows from energy-momentum conservation.

Validity of the Born approximation.  The procedure just described relies on
the validity of the Born approximation, which holds true for photoproduction,
$\gamma a\to ah$, but becomes problematic for bremsstrahlung processes, $ab\to
ab\,h$. (Here $a$ or $b$ stands for an electron or a nucleon, as above). Let
us discuss bremsstrahlung first. This issue is closely connected to the
perturbativity in the $ab$ collisions, discussed above. This can be understood
as follows. The Born approximation assumes that particles in the initial and
final states are approximately described by plane waves. While this assumption
holds true for gravitons, it does not necessarily apply to electrons or
nucleons, as they are subject to mutual interaction. The extent to which a
plane wave is a good approximation depends on whether this mutual interaction
is perturbative. If it is, the wave function of electrons or nucleons can be
expanded perturbatively, resulting in a plane wave with a small additional
contribution, whose impact on the scattering rate appears beyond the Born
approximation.

The previous argument makes it clear that whenever perturbativity is a problem
for graviton bremsstrahlung, it is also a problem for the bremsstrahlung of
axion or photons. Interestingly, Weinberg has recently revisited the latter
case, providing approximation formulas for photon bremsstrahlung valid beyond
the Born approximation [17] (see also Ref. [69]). Such formulas arise from the
non-perturbative wave functions and provide substantial corrections when
$v/Z\alpha$ is small. In light of Fig. S1, this shows that for the main
processes contributing to graviton bremsstrahlung in the Sun –namely, $ee$,
$ep$ and $eHe$– the Born approximation works well except in the last 15% of
the solar radius. There, the temperature is comparatively lower resulting in
$v/Z\alpha<1$, invalidating the plane-wave assumption. Since bremsstrahlung is
dominated by processes at the center, neglecting non-perturbative corrections
to the graviton emission is justified. Likewise, this argument applies to
axion bremsstrahlung, where accounting for non-perturbative wave functions
gives corrections below 10% (see e.g. Ref. [39]).

Finally, the aforementioned argument justifies the use of the Born
approximation for photoproduction because the charged particles in the initial
state, although non-relativistic, do not experience a distortion of their wave
function, as in the case of bremsstrahlung.

###  B.1 Photoproduction

Figure S2: Feynman diagrams for photoproduction

Having justified the Born approximation, let us note that diagrammatically
such collisions correspond to Feynman diagrams where the photon emits $h_{\pm
2}$ and then collides with a charged particle (Primakoff-like scattering), or
to one photon impinging on a charged particle that subsequently emits $h_{\pm
2}$ (Compton-like scattering), or to a diagram involving a contact interaction
with all particles. Using Eq. (S7), the amplitude can be cast as

${\cal
M}_{\alpha\beta}=\overline{u}_{3}\Gamma_{\alpha\beta\mu}u_{2}\,\epsilon^{\mu}(p_{1})\,,$
(S20)

where the subscripts $2$ and $3$ refer to the fermion in the initial and final
state, respectively, while $1$ refers to the incoming photon. Following the
procedure outlined above, we find

$\displaystyle\Gamma_{\alpha\beta\mu}=$ $\displaystyle-$
$\displaystyle\frac{Ze}{(p_{2}-p_{3})^{2}}C_{\mu\lambda\alpha\beta}^{\rm(photon)}(p_{1},p_{3}-p_{2})\gamma^{\lambda}$
$\displaystyle+$
$\displaystyle\frac{Ze}{(p_{1}+p_{2})^{2}-m_{f}^{2}}C^{\rm(fermion)}_{\alpha\beta}(p_{1}+p_{2},p_{3})\left(\not{p_{1}}+\not{p_{2}}+m_{f}\right)\gamma_{\mu}$
$\displaystyle+$
$\displaystyle\frac{Ze}{(p_{1}-p_{3})^{2}-m_{f}^{2}}\gamma_{\mu}\left(-\not{p_{1}}+\not{p_{3}}+m_{f}\right)C^{\rm(fermion)}_{\alpha\beta}(p_{2},-p_{1}+p_{3})$
$\displaystyle+$ $\displaystyle C^{\rm(quartic)}_{\mu\alpha\beta}\,.$

In addition to making sure that this satifies Eq. (S19), we also verify gauge
invariance by means of the Ward–Takahashi relation

$\displaystyle
p_{1}^{\mu}\,\left(\overline{u}_{3}\Gamma_{\alpha\beta\mu}u_{2}\,\epsilon^{\alpha\beta}_{(\lambda)}(p)\right)=0\,.$
(S22)

Plugging Eq. (B.1) into Eq. (S6), we obtain the photoproduction rates. Taking
the non-relativistic limit, $m_{f}\gg\omega$, we obtain the differential
emission rate

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}\omega\mathrm{d}V\mathrm{d}\Omega}\left(\gamma
Z\to Z\,h\right)=n_{\gamma}n_{Z}\delta(\omega-
p_{i})\,Z^{2}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\bigg{|}_{e}\,,$ with
$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\bigg{|}_{e}=\frac{1}{2}\,\alpha
G\,\cot^{2}\\!\frac{\theta}{2}\left(1+\cos^{2}\theta\right)\,.$ (S23)

This cross section was reported in Ref. [37, 11]. As usual, here $\theta$ is
the scattering angle in the photoproduction process, i.e. the angle between
the graviton and the incoming particles. So far, we have assumed that the
photon is massless. However, the surrounding plasma gives photons in the
external line an effective mass given by the plasma frequency, $\omega_{\rm
pl}$. A similar modification is required in the photon propagator and
vertices. Nevertheless, the solar temperature far exceeds $\omega_{\rm pl}$
(see Fig. 2), allowing us to calculate photoproduction rates by expanding on
$\omega_{\rm pl}$. In practice, this approach is effectively equivalent to
treating the external photon as massless while incorporating the plasma
frequency into the phase space, as reported in the Table 1 with $\kappa=0$.
The resulting cross section agrees with semi-classical calculations [12]
accounting for a non-trivial dispersion (without screening).

As is clear from Eq. (S23), the total rate diverges due to the pole at
$\theta=0$. The origin of the divergence is the photon propagator in the
amplitude of the Primakoff-like diagram, whose denominator
$-(p_{2}-p_{3})^{2}$ equals in the non-relativistic limit the square of the
three-momentum transfer, $|{\bf q}|=2\omega\sin(\theta/2)$. To gain further
intuition on this, let us discuss the case of axions, where a similar
divergence occurs for the Primakoff effect. As shown in [10], the divergence
is effectively regularized by accounting for correlations among plasma
particles. This is closely related to screening, and despite the differences
between axions and gravitons, the same remains true for the latter in non-
relativistic limit, $m_{f}\gg\omega$. We repeat the argument here for
completeness.

In the non-relativistic limit, $|{\bf q}|/m_{f}\ll 1$, and therefore the
charged particles off which the photon scatters may be regarded as at rest
during the collision. As a result, the photon scatters off a charge
distribution, $\rho({\bf r})$, with a cross section

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\bigg{|}_{e}\left|F(\mathbf{q})\right|^{2}\,,$
where $\displaystyle F({\bf q})=\frac{1}{e}\int\mathrm{d}{\bf r}\,e^{i{\bf
q}\cdot{\bf r}}\rho({\bf r})\,.$ (S24)

The cross section with subscript $e$ refers to electrons as introduced in Eq.
(S23). For $N$ point charges $Z_{i}e$ located at $\mathbf{r}_{i}$, a explicit
computation shows that $F({\bf q})=\sum_{i}^{N}Z_{i}e^{i{\bf q}\cdot{\bf
r}_{i}}$ as well as
$\left|F(\mathbf{q})\right|^{2}=\sum_{i}^{N}Z_{i}^{2}+\sum_{\begin{subarray}{c}i\neq
j\end{subarray}}^{N}Z_{i}Z_{j}\cos\left(\mathbf{q}\cdot\mathbf{r}_{ij}\right)$
where $\mathbf{r}_{ij}\equiv\mathbf{r}_{i}-\mathbf{r}_{j}$. This must be
averaged accounting for the statistical ensemble describing the plasma.
Clearly, the first term is associated with individual charges, while the
second one describes interference effects. The latter averages to zero for
very diluted plasmas ($\kappa\to 0$) because their particles are uncorrelated.
In that case, the total rate in Eq. (S24) simply consists of adding the
contribution of individual charges. Nevertheless, in real plasmas particles
mutually interact through their Coulomb fields and their motion is slightly
correlated. Concretely, the potential, $\phi_{i}$, around a given particle is
sourced by a charge cloud with density

$\displaystyle\rho_{i}(r)=Z_{i}e\left(\delta^{3}(\mathbf{r})-\frac{\kappa^{2}}{4\pi}\frac{e^{-\kappa
r}}{r}\right)=-\nabla^{2}\phi_{i}\,.$ (S25)

That is, the Coulomb field of any charge is screened over distances larger
than about $\kappa^{-1}$. To determine the aforementioned average, and hence
the effect of correlations, we note that the potential energy associated with
Eq. (S25) is given by $U=\frac{1}{2}e\sum^{N}_{i\neq j}Z_{i}\phi_{i}$.
Employing the Boltzmann factor associated with this potential energy,
$e^{-U/T}$, at leading order in $\alpha$ one finds that thermal average is

$\displaystyle\sum^{N}_{j\neq
i}Z_{j}\langle\cos\left(\mathbf{q}\cdot\mathbf{r}_{ij}\right)\rangle=-\frac{Z_{i}\kappa^{2}}{\kappa^{2}+|{\bf
q}|^{2}}\,,$ which leads to
$\displaystyle\left\langle\left|F(\mathbf{q})\right|^{2}\right\rangle=\sum_{i=1}^{N}Z_{i}^{2}\left(1-\frac{\kappa^{2}}{\kappa^{2}+|\mathbf{q}|^{2}}\right)\,.$
(S26)

Hence, Eq. (S24) implies that [10]

$\displaystyle\left\langle\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\right\rangle=\sum_{i=1}^{N}Z_{i}^{2}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\bigg{|}_{e}\frac{|\mathbf{q}|^{2}}{\kappa^{2}+|\mathbf{q}|^{2}}\,.$
(S27)

In summary, one must add individual cross sections introducing a global factor
determined by the screening scale. This clearly regularizes the divergence at
$|{\bf q}|=0$. Following this procedure, we obtain the rates reported in Table
1. A few comments are in order.

While the Compton process for axions arises from a renormalizable coupling,
$g_{aee}$, leading to a cross section independent of the momentum transfer in
the non-relativistic limit, the Primakoff process results from a non-
renormalizable coupling, $g_{a\gamma\gamma}$, and leads to a differential
cross section scaling as the fourth power of the momentum transfer [10]. Thus,
only the latter exhibits a divergence, raising the question of whether the
Compton-like and Primakoff-like diagrams for gravitons must be separated
somehow. Nonetheless, unlike photoproduction of axions, where such a
separation is feasible, for gravitons that cannot be done in a meaningful
manner. First, because they depend on the same coupling, namely, the
gravitational coupling in Eq. (S18). Moreover, although the identity in Eq.
(S22) is separately satisfied by the Compton-like diagrams of Fig. S2, that is
not the case of the identity associated with energy-momentum conservation, Eq.
(S19). In the light of this, we apply Eq. (S27) to photoproduction of
gravitons as a whole.

One might ignore the correlations given by Eq. (S26) and assume that the
incoming photon scatters off a single charge distribution given by Eq. (S25).
This is the regularization advocated by Ref. [37]. The resulting cross section
is that in Eq. (S27) with
$|\mathbf{q}|^{2}/(\kappa^{2}+|\mathbf{q}|^{2})\to|\mathbf{q}|^{4}/(\kappa^{2}+|\mathbf{q}|^{2})^{2}$.
As explained in the main text, this issue has been revisited recently for the
Primakoff process [39], accounting for the velocity in the form factor.
Corrections to Eq. (S27) have been deemed negligible. Coupled with this, let
us remark that, as opposed to bremsstrahlung or two-two scattering, the effect
of screening here cannot be understood as an effective mass for the photon for
several reasons. First, the effective mass of transverse photons in the
external line is given by the plasma frequency, $\omega_{\rm pl}$, and not by
the Debye-Hückel screening scale, $\kappa$, which in the Sun satisfies
$\omega_{\rm pl}\ll\kappa$. See Fig. 2. In fact, even if one adopts an ad-hoc
prescription in which the virtual photon in Fig. S2 has a mass $\kappa$, while
the incoming external photon has a mass $\omega_{\rm pl}$, one quickly finds
inconsistencies. For instance, this approach spoils Eq. (S19).

###  B.2 Bremsstrahlung

Figure S3:  Feynman diagrams for bremsstrahlung in $eZ$ collisions.

Figure S4:  _Left:_ Contribution of photoproduction processes to the solar GW
spectrum. _Right:_ Contribution of bremsstrahlung processes to the solar GW
spectrum. Solid lines are the full results determined by Eqs. (B.2) and (S29).
The dashed lines are the leading results in the screening scale, reported in
Table 1.

In the Born approximation, there are seven Feynman diagrams associated with
bremsstrahlung in $eZ$ collisions, see Fig. S3. For $ee$ collisions,
additional diagrams are obtained by symmetrizing the final state of those
shown in Fig. S3. Note that bremsstrahlung from two-nucleon states are
negligible. To calculate the corresponding $M^{\mu\nu}$, we follow the
procedure outlined above. Concretely, we manipulate the symbolic output
provided by CalcHEP [24] using Package-X [68]. Crucially, we verify that Eq.
(S19) is satisfied. Subsequently, we use this together with the appropriate
polarization tensors to compute ${\cal M}(\lambda)$. To account for screening,
we modify the photon propagators with an effective mass equal to the screening
scale. As argued above, that is the correct prescription for slow particles
such as the non-relativistic plasma components (in contrast to the external
photon in the photoproduction process described above). From this, we
calculate the differential cross sections in energy. To the best of our
knowledge, the expression for these have never been reported in the literature
in a general form. This is not surprising as the aforementioned procedure is a
daunting task, even in the the non-relativistic limit. After algebraic
manipulation, in such a limit we find

$\displaystyle\frac{{\mathrm{d}}\sigma\\!v}{{\mathrm{d}}\omega}\Bigg{|}_{eZ\to
eZ\,h}$ $\displaystyle=$
$\displaystyle\frac{32Z^{2}\alpha^{2}Gp_{i}}{15\omega}\left(\frac{1}{m_{e}}+\frac{1}{m_{Z}}\right)\left[3(1+\xi^{2}+\xi_{s}^{2})L\right.$
$\displaystyle\left.+10\xi+\frac{1}{3}\xi_{s}^{2}\left(\frac{(1-\xi)^{2}\left(18(1+\xi)^{4}+29(1+\xi)^{2}\xi_{s}^{2}+12\xi_{s}^{4}\right)}{\left((1+\xi)^{2}+\xi_{s}^{2}\right)^{3}}-(\xi\to-\xi)\right)\right]\,,$

with notation $L$, $\xi$ and $\xi_{s}$ introduced in Table 1. In this
calculation we do not take the limit $m_{Z}\gg m_{e}$, we simply assume that
both nucleons and electrons are non-relativistic. Likewise

$\displaystyle\frac{{\mathrm{d}}\sigma\\!v}{{\mathrm{d}}\omega}\Bigg{|}_{ee\to
ee\,h}=\frac{32\alpha^{2}Gp_{i}}{15\omega
m_{e}}\left\\{20\xi-\frac{6\xi(1+\xi^{4})}{(1+\xi^{2})^{2}}+\left[6(1+\xi^{2})-\frac{3(1-\xi^{2})^{4}+7(1-\xi^{4})^{2}}{2(1+\xi^{2})^{3}}\right.\right.$
(S29)
$\displaystyle\left.\left.\frac{1}{2}\xi_{s}^{2}\left(\frac{6(\xi^{4}+1)(1-\xi^{2})^{2}}{(\xi^{2}+1)^{2}(\xi^{2}+\xi_{s}^{2}+1)^{2}}+\frac{2(\xi^{4}-4\xi^{2}+1)(1-\xi^{2})^{2}}{(\xi^{2}+1)(\xi^{2}+\xi_{s}^{2}+1)^{3}}+\frac{13\xi^{8}+22\xi^{4}+13}{(\xi^{2}+1)^{3}(\xi^{2}+\xi_{s}^{2}+1)}+15\right)\right]L\right.$
$\displaystyle+\left.\xi_{s}^{2}\left[2(1-\xi^{2})^{2}\left(-\frac{(1-\xi)^{4}-80\xi^{2}}{16\,\xi^{2}(\xi+1)^{2}((\xi+1)^{2}+\xi_{s}^{2})}-\frac{3(\xi^{4}+1)\xi_{s}^{2}+4(\xi^{6}+1)}{8\xi(\xi^{2}+1)^{2}(\xi^{2}+\xi_{s}^{2}+1)^{2}}+\frac{6(\xi+1)^{2}+5\xi_{s}^{2}}{3((\xi+1)^{2}+\xi_{s}^{2})^{3}}\right)\right.\right.$
$\displaystyle\left.\left.-(\xi\to-\xi)\right]\right\\}\,.$

For $\xi_{s}=0$, these cross sections agree fully with those reported by Gould
in Refs. [33, 27]. The total emission rate is given by

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}\omega\mathrm{d}V}\Bigg{|}_{\rm
Bremsstrahlung}=\frac{1}{2}n_{e}^{2}\frac{{\mathrm{d}}\sigma\\!v}{{\mathrm{d}}\omega}\Bigg{|}_{ee\to
ee\,h}+\sum_{Z}n_{e}n_{Z}\frac{{\mathrm{d}}\sigma\\!v}{{\mathrm{d}}\omega}\Bigg{|}_{eZ\to
eZ\,h}\,.$ (S30)

The partial contributions of each channel are shown in Fig. S4. To leading
order in the screening scale the resulting expressions are reported in Table
1. While these approximations are quite accurate for $eZ$ bremsstrahlung, they
are less precise for $ee$ bremsstrahlung, as shown in Fig. S4. This
underscores the critical importance of incorporating screening effects in the
bremsstrahlung rates.

Soft emission. Let us consider the initial state $eZ$. A soft graviton leads
to $\xi\to 1$. Taking $\xi_{s}\to 0$, the effect of screening appears only in
the logarithm with $L\to L_{\rm Coulomb}$, and the corresponding emission rate
can be cast as

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}\omega\mathrm{d}V}\Bigg{|}_{eZ\to
eZ\,h}=\frac{64\alpha^{2}Z^{2}Gv}{5}L_{\rm
Coulomb}n_{e}n_{Z}=\frac{8G}{5\pi}n_{e}n_{Z}\,\mu^{2}v^{5}\sigma_{V}^{(eZ)}\,.$
(S31)

In the last equation, we use the viscosity cross section obtained in Eq. (S3)
with $E_{i}=\mu v^{2}/2$. Note that
$v=p_{i}\left(\frac{1}{m_{e}}+\frac{1}{m_{Z}}\right)$. Similarly, for $ee$ Eq.
(S29) reduces to

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}\omega\mathrm{d}V}\Bigg{|}_{ee\to
ee\,h}=\frac{32\alpha^{2}Z^{2}Gv}{5}L_{\rm
Coulomb}n_{e}^{2}=\frac{4G}{5\pi}n_{e}^{2}\,\mu^{2}v^{5}\sigma_{V}^{(ee)}\,.$
(S32)

Here $v=2p_{i}/m_{e}$. As explained in the text, this was first proven by
Weinberg [8], who used it to estimate the total power emitted by the Sun.