Potential of Radio Telescopes as High-Frequency Gravitational Wave Detectors

In the presence of magnetic fields, gravitational waves are converted into photons and vice versa. We demonstrate that this conversion leads to a distortion of the cosmic microwave background (CMB), which can serve as a detector for MHz to GHz gravitational wave sources active before reionization. The measurements of the radio telescope EDGES can be cast as a bound on the gravitational wave amplitude, h c < 10 − 21 ð 10 − 12 Þ at 78 MHz, for the strongest (weakest) cosmic magnetic fields allowed by current astrophysical and cosmological constraints. Similarly, the results of ARCADE 2 imply h c < 10 − 24 ð 10 − 14 Þ at 3 – 30 GHz. For the strongest magnetic fields, these constraints exceed current laboratory constraints by about 7 orders of magnitude. Future advances in 21 cm astronomy may conceivably push these bounds below the sensitivity of cosmological constraints on the total energy density of gravitational waves.

size Δl that contains a uniform transverse magnetic field B and a non-negligible uniform density of free electrons, n e . Without loss of generality, we assume that the magnetic field points in theê 1 direction. See Fig. 1. In this coordinate system we introduce h × ¼ h 12 ¼ h 21 and A × ¼ A 1 as well as h þ ¼ −h 22 ¼ h 11 and A þ ¼ −A 2 . This is because the aforementioned equations can be elegantly cast as [12,15] [30] where λ ∈ fþ; ×g, l is the third component, We include the plasma frequency ω pl ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 n e =m e p , which acts as an effective mass term and gives electromagnetic waves of frequency ω a refractive index μ ¼ Equation (1) also applies for arbitrary uniform fields with B interpreted as the corresponding transverse component. See the Supplemental Material [31] for more details. Assuming a plane wave traveling in the positive direction with ω ≥ ω pl , the exact solution of Eqs. (1) (see also Ref. [16]) can be written as with K being the Hermitian matrix It is convenient to introduce ψ because its magnitude jψðt; lÞj 2 is conserved. This easily follows from the unitarity of the matrix UðlÞ ¼ e iKl . In particular, ψð0; 0Þ ¼ ð0; h λ;0 =κÞ for a pure GW state entering the box, and, consequently, ψðt; ΔlÞ ¼ e −iωt ½U 12 ðΔlÞ; U 22 ðΔlÞh λ;0 =κ after leaving the box. Since jU 12 ðΔlÞj 2 þ jU 22 ðΔlÞj 2 ¼ 1, the quantity PðΔlÞ ≡ jU 12 ðΔlÞj 2 can be interpreted as the probability of GW conversion after traversing a distance Δl. Simple algebra shows PðΔlÞ ¼ jK 12 j 2 l 2 osc sin 2 ðΔl=l osc Þ; ð4Þ These expressions reduce to the approximated formulae previously found (see, e.g., Refs. [12,33]).
Although cosmic magnetic fields are not expected to be perfectly homogeneous, coherent oscillations take place in highly homogeneous patches, for which l osc ≪ Δl and therefore PðΔlÞ ¼ jK 12 j 2 l 2 osc =2 on average. Taking into account inhomogeneities in n e [34] and B, the coherence of the g ↔ γ oscillations is lost on distances larger than Δl, that is, the smallest distance on which B and n e are uniform. Denoting the total distance traveled by the GW as D, this corresponds to traversing N ¼ D=Δl independent regions with a conversion probability PðΔlÞ each. As long as NPðΔlÞ ≪ 1, this gives a total conversion probability of PðDÞ ¼ DjK 12 j 2 l 2 osc =ð2ΔlÞ [25,26], corresponding to an average conversion rate (i.e., probability per time) [31] given by In the Supplemental Material [31] we demonstrate that this simple estimate correctly captures the essential features of a more involved computation based on the expected power spectrum of the magnetic field. Note that any additional inhomogeneities would further enhance the conversion rate by limiting the coherence of the g ↔ γ oscillations. We now include the effect of the Universe expansion during the dark ages. This is the period between photon decoupling and reionization, z dec ≃ 1100 ≳ z ≳ z rei ≃ 10, beginning with the formation of the CMB and ending when the first stars were formed. During this time, the refractive index of MHz-GHz CMB photons is determined by the tiny electron density, with the contributions of neutral hydrogen, helium, and birefringence being subdominant [39][40][41]. This allows us to adopt Eq (5), after a few modifications. The conversion probability in an adiabatic expanding Universe is simply the line-of-sight integral of the rate P ≡ Z l:o:s: where we use null geodesics Also, z ini ≤ z dec is an initial condition to be specified below and H ¼ H dec ðT=T dec Þ 3=2 is the Hubble parameter during the dark ages, which are matter dominated. Furthermore, the average magnetic energy density of the Universe [42]. Additionally, such a field is associated with a coherence length, λ B ¼ λ 0 B =ð1 þ zÞ, because it is not expected to be homogeneous everywhere. Concerning these two quantities we emphasize three important facts here and refer the reader to Ref. [23] for a more comprehensive discussion: (i) a recent CMB analysis gives B 0 ≲ 47 pG [20], (ii) blazar 021104-2 observations strongly suggest a lower limit on B 0 [43] because otherwise their gamma-ray spectra cannot be explained under standard cosmological assumptions [17][18][19]44], and (iii) magnetohydrodynamic turbulence damps out large magnetic fields at small distances, imposing an additional (theoretical) upper limit [23]. Figure 2 shows these constraints.
In addition, the electron number density during this epoch is n e ðzÞ ¼ n b0 ð1 þ zÞ 3 X e ðzÞ, where n b0 ¼ 0.251 m −3 is the baryon number density today [45] and X e ðzÞ is the ionization fraction, taking values 1,0.68,0.0002, and 0.15 at z ¼ 0, 10, 20, and 1100, respectively [46]. This gives plasma frequencies today, ω pl;0 , lying in the Hz range, which allows us to take 1 − μðzÞ ¼ ð1 þ zÞX e ðzÞω 2 pl;0 = ð2ω 2 0 Þ ≪ 1, for waves of frequency ω ¼ ω 0 ð1 þ zÞ with ω 0 ∼ GHz. Moreover, B 0 ≲ 47 pG results in the oscillation length being numerically dominated by the plasma frequency so that l −1 osc ¼ ð1 þ zÞ 2 X e ðzÞω 2 pl;0 =ð4ω 0 cÞ. This gives l osc ≪ 1 pc ≪ Δl, as anticipated above. Here, in order to account for electron inhomogeneities we conservatively take Δl ¼ Δl 0 =ð1 þ zÞ to be given by Mpc is the characteristic comoving scale for the onset of structure formation (corresponding to the perturbation mode entering the horizon at matter-radiation equality). Putting all this together, we obtain The left panel of Fig. 2 displays contours of ðT 0 =ω 0 Þ 2 P in the parameter space of cosmic magnetic fields. The inset shows I 0 ðz ini Þ, explaining the weak redshift dependence of Iðz ini Þ, with the largest contribution arising from z ∼ 10.
CMB distortions.-The CMB photon distribution, f γ ðω; TÞ, retains its equilibrium form during the dark ages, i.e., is given by a blackbody spectrum, f eq ¼ 1=ðe ω=T − 1Þ with ω=T ¼ ω 0 =T 0 . Our aim here is to calculate deviations from such a spectrum, The spectrum of GWs is commonly characterized by Ω GW , which parametrizes the corresponding energy density per logarithmic frequency bin. This quantity can be used to introduce-in an analogous manner to f γ -the distribution function for GWs, f g . More precisely, in terms of it, the energy density is given by with ρ c ðTÞ denoting the Universe total energy density.  6) and (7). Right: Upper bounds on the stochastic GW background derived from ARCADE2 and EDGES (this work), compared to existing laboratory bounds from (a) superconducting parametric converter [3], (b) waveguide [4], (c) 0.75 m interferometer [5], (d) magnon detector [6], and (e) magnetic conversion detector [7]. The solid lines indicate the allowed parameter space for cosmic magnetic fields, as given in the left panel. The dashed lines mark the N eff constraint for broad GW spectra and for a peaked spectrum with Δω=ω ¼ 10 −3 . For reference, the dotted lines indicate ρ g ¼ ρ c .
Both distributions satisfy the Boltzmann equation Lf γ=g ¼ AEhΓ g↔γ iðf g − f γ Þ, whereL ≡ ∂ t − Hω∂ ω ¼ −HðT∂ T þ ω∂ ω Þ is the corresponding Lioville operator. Its solution leads to δf γ ðω 0 ; T 0 Þ ¼ ½f g ðω ini ; T ini Þ − f eq P þ OðP 2 Þ; ð9Þ with P defined as in Eq. (6). We solve the Boltzmann equations from an initial temperature T ¼ T ini -when the photon distribution is a blackbody spectrum, i.e., f γ ðω; T ini Þ ¼ f eq ðω=T ini Þ-until today. If decoupling is prior to the GW emission, the latter fixes T ini . Otherwise, we set T ini ¼ T dec because the ionization fraction sharply drops after z ∼ z dec rendering any prior contribution negligible. This is illustrated in the inset of Fig. 2 (left panel), which also shows that the conversion rate is anyways largely insensitive to the precise value of T ini .
Equation (9) can alternatively be derived by considering the density-matrix formalism. In that case, f γ and f g are proportional to the diagonal entries of such a matrix, which evolves by means of the Hamiltonian associated with Eq. (3). See the Supplemental Material [31] for details. The fact that using both methods we obtain the same result-i.e., Eq. (9)-is reassuring and indicates that decoherence effects are properly taken into account [15]. Because of this as well as the way we treat inhomogeneities, our results differ from those of Ref. [26].
With ω ≪ T and f g ≫ f γ , Eq. (9) reads For a given detector sensitivity δf γ =f γ and a given value of the conversion probability P, relation (10) sets stringent bounds on the GW spectrum, which can be expressed in terms of the characteristic strain by means of [49] This is related to the one-sided power spectral density S h as h c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi fS h ðfÞ p . Figure 2 contrasts the resulting constraints with existing bounds in the literature. N eff bound.-GWs contribute to the energy budget of the Universe in the form of radiation and are as such constrained by the BBN and CMB bounds on the effective number of massless degrees of freedom N eff [50], with ΔN eff ≲ 0.1 [45,51]. For a spectrum Ω GW which is approximately scale invariant between f min and f max with lnðf max =f min Þ ∼ Oð1Þ, this implies whereas for a narrow spectrum peaked atω with width Δω ≲ω this bound is relaxed by a factor ðω=ΔωÞ. Note that this bound applies only to GWs present already at CMB decoupling.
Probing the Rayleigh-Jeans tail of the CMB spectrum.-Below ω 0 =T 0 ∼ 10 −2 , galactic foregrounds dominate the radio sky. Here we focus on the results reported by ARCADE2 [28] which covers the sweet spot of the lowfrequency Rayleigh-Jeans spectrum before galactic foregrounds become important [f ¼ ω 0 =ð2πÞ ¼ 3, 8, 10, 30, and 90 GHz] and by EDGES [29], which is a recent measurement of the global 21 cm absorption signal at 78 MHz.
ARCADE2 was a balloon experiment equipped with a radio receiver measuring the blackbody temperature of sky [28]. The cleanest frequency band is around 10 GHz enabling a mK resolution, δf γ =f γ ¼ δT=T CMB ≲ 4 × 10 −4 at ω 0 =T 0 ≃ 0.18. At smaller frequencies, ARCADE2 observed a significant radio excess beyond the expected galactic foreground whose origin remains an open question (see, e.g., Refs. [52,53]). Assuming that this excess is entirely astrophysical, we can impose an upper bound on an additional contribution from a stochastic GW background using the 3, 8, 10, and 30 GHz frequency bands. In Fig. 2, these frequencies are marked by crosses, the solid lines connecting them serve only to guide the eye.
Recently, the first observation of the global (i.e., skyaveraged) 21 cm absorption signal was reported by the EDGES Collaboration [29]. The absorption feature was found to be roughly twice as strong as previously expected, which if true, would indicate that either the primordial gas was significantly colder or the radiation background was significantly hotter than expected. Conservatively, we may assume that the deviation from the expected value is due to foreground contamination, and place a bound on any stochastic GW background by using δf γ =f γ ≲ 1 at ω=T ¼ Discussion.-Cosmological sources of GWs typically produce stochastic GW backgrounds with a frequency roughly related to the comoving Hubble horizon at the time of production. Processes in the very early Universe at energy scales far beyond the reach of colliders thus generically produce GWs in the MHz and GHz regime. Despite the large amount of redshift, these violent processes can produce sizable GW signals, saturating the N eff bound (12). Some examples are axion inflation [54,55], metastable cosmic strings [57] and evaporating light primordial black holes [58,59]. Further significant contributions may be expected from preheating [60][61][62][63][64] and first order phase transitions occurring above 10 7 GeV [65][66][67][68][69]. The sensitivity of radio telescopes can, however, not yet compete with the cosmological N eff bound, unless one considers essentially monochromatic signals (which may arise, e.g., from large monochromatic scalar perturbations [70,71]).
Since the dominant contribution to P arises around reionization, particularly interesting targets are GW sources active around 10 ≲ z ≲ 10 3 , which would not be constrained by the N eff bound. During the dark ages, there is no generic reason to expect GW production in the GHz regime but there are models which predict such a signal for suitable parameter choices. For example, mergers of light primordial black holes in this epoch (with masses of about 10 −9::−7 M ⊙ ) would result in GHz GW signals today [49], see Refs. [72,73] for a discussion of possible rates. Superradiant axion clouds around spinning black holes yield an essentially monochromatic GW signal with f ≲ MHz [74][75][76], with higher frequency possible when considering primordial black holes with masses below the Chandrasekhar limit.
We emphasize that the use of radio telescopes allows us to search for GWs in a wide frequency regime. While the absence of any excess radiation can already constrain some models under the assumption of strong cosmic magnetic fields, the potential of this method will truly unfold with further improvements in the sensitivity of radio telescopesdriven in particular by the advances in 21 cm cosmologyor in the case of a positive detection of excess radiation.
An example of future advances in radio astronomy is the case of the Square Kilometer Array (SKA). Assuming an effective area per antenna temperature of at least 10 2 m 2 =K [77,78] in the 0.1-10 GHz range, a few hours of observation will lead to sensitivities in the ballpark of μJy, which must be compared against CMB fluxes of at least 10 3 Jy. SKA measurements are thus very promising although sufficient foreground subtraction will be extremely challenging.
It is a pleasure to thank Nancy Aggarwal, Sebastien Clesse, Mike Cruise, Hartmut Grote, and Francesco Muia for insightful discussions on the Gertsenshtein effect and high-frequency GW sources at the ICTP workshop "Challenges and opportunities of high-frequency gravitational wave detection." Likewise, we would also like to thank Torsten Bringmann, Damian Ejlli, Kohei Kamada, and Kai Schmidt-Hoberg. This work was partially funded by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy-EXC 2121 "Quantum Universe"-390833306. C. G. C. is supported by the Alexander von Humboldt Foundation.