Potential of Radio Telescopes as High-Frequency Gravitational Wave Detectors

Potential of Radio Telescopes as High-Frequency Gravitational Wave Detectors

Valerie Domcke (Deutsches Elektronen-Synchrotron DESY, Hamburg; Theoretical Physics Department, CERN, Geneva; Institute of Physics, Laboratory for Particle Physics and Cosmology, EPFL, Lausanne) and Camilo Garcia-Cely (Deutsches Elektronen-Synchrotron DESY, Hamburg).

Physical Review Letters 126, 021104 (2021). Received 11 June 2020; revised 6 August 2020; accepted 7 December 2020; published 14 January 2021. DOI 10.1103/PhysRevLett.126.021104.

Abstract

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, a characteristic strain below 10 to the minus 21 (10 to the minus 12) at 78 MHz, for the strongest (weakest) cosmic magnetic fields allowed by current astrophysical and cosmological constraints. Similarly, the results of ARCADE 2 imply a characteristic strain below 10 to the minus 24 (10 to the minus 14) 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.

Introduction

Gravitational waves (GWs) produced in the early Universe can traverse cosmic distances without experiencing any interactions, making them a unique probe of very high energy physics. Since the comoving Hubble horizon grows with time, GWs produced at energies around the scale of grand unification have frequencies in the MHz and GHz regime today, far beyond the reach of the laser interferometers LIGO, VIRGO, or KAGRA. Some existing laboratory bounds exist at these frequencies.

Here we focus on searching for high-frequency GWs exploiting the (inverse) Gertsenshtein effect, which describes the conversion of GWs into photons in the presence of a magnetic field. As an immediate consequence of general relativity and classical electromagnetism, this is a purely Standard Model process. Involving gravity, the conversion probability is extremely small which may, however, be compensated by considering a "detector" of cosmological size. In fact, magnetic fields with cosmological correlation lengths might well permeate our Universe, with certain astrophysical observations strongly suggesting a lower limit of order 10 to the minus 16 gauss, and the CMB setting an upper bound in the picogauss to nanogauss range.

The pioneering study proposed the inverse Gertsenshtein effect in cosmic magnetic fields to search for GWs but neglected the plasma mass of photons, as later pointed out. The idea was revisited in a study suggesting an observable effect; however, as noted elsewhere, decoherence effects were not correctly accounted for. More recently, the production of GWs from CMB photons has been studied. In this Letter, we focus on CMB distortions arising from the Gertsenshtein effect during the dark ages, that is, the period between recombination and reionization. Because of the small fraction of free electrons in this period, the effective plasma mass of the photons is suppressed, increasing the conversion probability between GWs and photons. Taking into account inhomogeneities in the thermal plasma and in the cosmic magnetic fields, we demonstrate that existing measurements of the Rayleigh-Jeans tail of the CMB spectrum, performed, for example, by ARCADE 2 and by EDGES, can be translated into constraints on GWs in the MHz-GHz regime. These are competitive with, or even exceed, current laboratory constraints, depending on the assumptions on the cosmic magnetic fields.

The Gertsenshtein effect

Calculating the conversion rate for this oscillation process requires solving Maxwell's equations for the vector potential, describing the electromagnetic radiation, together with the linearized Einstein equations for the metric written as the flat metric plus a perturbation, in which the perturbation describes the GWs. In this work we adopt the mostly-minus signature and work with natural Heaviside-Lorentz units, except in this section, where we keep fundamental constants explicitly to emphasize that the Gertsenshtein effect is a classical phenomenon.

Let us ignore the Universe expansion first and consider a GW propagating in the third direction inside a fixed box of size given by a path length, containing a uniform transverse magnetic field and a non-negligible uniform density of free electrons. Without loss of generality, we assume that the magnetic field points in the first direction. In this coordinate system we introduce the cross and plus polarisations of the wave and the corresponding components of the vector potential. The equations can be elegantly cast as a pair of coupled wave equations (Equation 1): the wave operator acting on the vector potential, with an added term equal to the squared plasma frequency over the squared speed of light, is sourced by minus the magnetic field times the gradient of the metric perturbation along the propagation direction; and the wave operator acting on the metric perturbation is sourced by the squared gravitational coupling times the magnetic field times the gradient of the vector potential.

Here the polarisation index runs over the plus and cross states, the path coordinate is the third component, the wave operator is the usual d'Alembertian, and the gravitational coupling is the square root of sixteen pi times Newton's constant, divided by the speed of light squared. We include the plasma frequency, the square root of the electron charge squared times the electron number density divided by the electron mass, which acts as an effective mass term and gives electromagnetic waves of a given frequency a refractive index equal to the square root of one minus the squared ratio of plasma frequency to wave frequency when the magnetic field is switched off. These equations also apply for arbitrary uniform fields with the magnetic field interpreted as the corresponding transverse component.

Assuming a plane wave travelling in the positive direction with frequency at or above the plasma frequency, the exact solution can be written as a two-component state — the refractive-index-weighted vector potential and the metric perturbation divided by the gravitational coupling — evolving as a plane wave in time multiplied by the exponential of a Hermitian mixing matrix times the path length (Equations 2 and 3). The diagonal entries of that matrix carry the wave frequency corrected by the magnetic term, and the off-diagonal entries carry the mixing, proportional to the gravitational coupling times the magnetic field.

It is convenient to introduce this state because its magnitude is conserved, which follows from the unitarity of the evolution matrix. In particular, the state entering the box is a pure GW state, and consequently after leaving the box it carries both components. Since the two probabilities sum to one, the squared off-diagonal entry can be interpreted as the probability of GW conversion after traversing a distance. Simple algebra shows that this probability is the squared mixing entry times the squared oscillation length times the squared sine of the ratio of path length to oscillation length (Equation 4), where the inverse oscillation length is the square root of the squared wave frequency times the squared difference of the refractive index from one, divided by the speed of light squared, plus half the squared product of the gravitational coupling and the magnetic field. These expressions reduce to the approximated formulae previously found.

Although cosmic magnetic fields are not expected to be perfectly homogeneous, coherent oscillations take place in highly homogeneous patches, for which the oscillation length is far shorter than the patch size, and therefore the conversion probability is half the squared mixing entry times the squared oscillation length on average. Taking into account inhomogeneities in the electron density and in the magnetic field, the coherence of the graviton-photon oscillations is lost on distances larger than the patch size, that is, the smallest distance on which the magnetic field and the electron density are uniform. Denoting the total distance travelled by the GW as D, this corresponds to traversing a number of independent regions equal to D divided by the patch size, with the same conversion probability each. As long as the total stays far below one, this gives a total conversion probability of D times the squared mixing entry times the squared oscillation length, divided by twice the patch size, corresponding to an average conversion rate — that is, probability per unit time — given by the speed of light times the squared mixing entry times the squared oscillation length, divided by twice the patch size (Equation 5).

In the Supplemental Material 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 graviton-photon oscillations.

We now include the effect of the Universe expansion during the dark ages. This is the period between photon decoupling and reionization, from a redshift of about 1100 down to a redshift of about 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. This allows us to adopt the rate formula, after a few modifications.

The conversion probability in an adiabatic expanding Universe is simply the line-of-sight integral of the rate (Equation 6), where we use null geodesics to convert time into redshift, an initial redshift no larger than the decoupling redshift is an initial condition to be specified below, and the Hubble parameter during the matter-dominated dark ages scales as the three-halves power of the temperature ratio. Furthermore, the average magnetic energy density of the Universe, half the squared field, redshifts as the fourth power of one plus redshift. Additionally, such a field is associated with a coherence length that scales inversely with one plus redshift, because it is not expected to be homogeneous everywhere.

Concerning these two quantities we emphasize three important facts: first, a recent CMB analysis gives a present-day field at or below 47 picogauss; second, blazar observations strongly suggest a lower limit on the present-day field, because otherwise their gamma-ray spectra cannot be explained under standard cosmological assumptions; and third, magnetohydrodynamic turbulence damps out large magnetic fields at small distances, imposing an additional theoretical upper limit.

In addition, the electron number density during this epoch is the present baryon number density, 0.251 per cubic metre, times the cube of one plus redshift, times the ionization fraction, which takes values 1, 0.68, 0.0002 and 0.15 at redshifts 0, 10, 20 and 1100 respectively. This gives plasma frequencies today lying in the hertz range, which allows the deviation of the refractive index from one to stay far below one for waves of gigahertz frequency today. Moreover, a present-day field at or below 47 picogauss results in the oscillation length being numerically dominated by the plasma frequency. This gives an oscillation length far below a parsec, itself far below the patch size, as anticipated above. Here, in order to account for electron inhomogeneities we conservatively take the patch size to be the smaller of the magnetic coherence length and the characteristic comoving scale for the onset of structure formation, 95 megaparsecs divided by two pi, corresponding to the perturbation mode entering the horizon at matter-radiation equality.

Putting all this together, we obtain a conversion probability of about 6.3 times 10 to the minus 19, scaled by the squared ratio of the present-day magnetic field to a nanogauss, by the ratio of a megaparsec to the patch size, by the squared ratio of the observed frequency to the CMB temperature scale, by 10 to the minus 6, and by a redshift integral (Equation 7). Here the CMB temperature scale is 2.725 kelvin divided by two pi, or 56.78 gigahertz, and the redshift integral runs over the inverse three-halves power of one plus redshift divided by the squared ionization fraction. The left panel of Figure 2 displays contours of the rescaled conversion probability in the parameter space of cosmic magnetic fields; the largest contribution arises from a redshift of about 10, which explains the weak dependence on the initial redshift.

CMB distortions

The CMB photon distribution retains its equilibrium form during the dark ages, that is, it is given by a blackbody spectrum. Our aim here is to calculate deviations from such a spectrum.

The spectrum of GWs is commonly characterized by the density parameter, which parametrizes the corresponding energy density per logarithmic frequency bin. This quantity can be used to introduce — in an analogous manner to the photon distribution — the distribution function for GWs. More precisely, in terms of it, the energy density is the logarithmic frequency integral of the fourth power of frequency times the distribution, divided by pi squared, which equals the total energy density of the Universe times the logarithmic frequency integral of the GW density parameter (Equation 8).

Both distributions satisfy a Boltzmann equation in which the Liouville operator acting on either distribution equals the average conversion rate times the difference between them. Its solution leads to a fractional distortion of the photon distribution equal to the difference between the initial GW distribution and the equilibrium photon distribution, multiplied by the conversion probability, plus higher-order corrections in that probability (Equation 9).

We solve the Boltzmann equations from an initial temperature — when the photon distribution is a blackbody spectrum — until today. If decoupling is prior to the GW emission, the latter fixes the initial temperature. Otherwise, we set the initial temperature to the decoupling temperature, because the ionization fraction sharply drops after decoupling, rendering any prior contribution negligible. This is illustrated in the inset of Figure 2, left panel, which also shows that the conversion rate is in any case largely insensitive to the precise value of the initial temperature.

Equation (9) can alternatively be derived by considering the density-matrix formalism. In that case, the photon and GW distributions are proportional to the diagonal entries of such a matrix, which evolves by means of the Hamiltonian associated with the mixing matrix. The fact that using both methods we obtain the same result is reassuring and indicates that decoherence effects are properly taken into account. Because of this, as well as the way we treat inhomogeneities, our results differ from those of an earlier study.

Constraints on the stochastic GW background

In this Letter we focus on the Rayleigh-Jeans part of the CMB, that is, frequencies far below the temperature, implying that the equilibrium distribution is approximately the temperature divided by the frequency. In this regime, a subdominant GW contribution to the total radiation energy density is compatible with a GW distribution far exceeding the photon distribution, and can thus produce an enhancement of the low-frequency CMB tail through the first term of the distortion formula. More precisely, the assumption that the GW distribution exceeds the photon one translates into a condition on the ratio of the GW density parameter to the photon density parameter, namely that it exceed fifteen over pi to the fourth times the cubed ratio of frequency to temperature. Even a scale-invariant GW spectrum as small as a density parameter of 10 to the minus 15 satisfies this at a frequency-to-temperature ratio of 10 to the minus 3.

With frequency far below temperature and the GW distribution far above the photon distribution, the distortion formula reads: the fractional distortion of the photon distribution equals pi to the fourth over fifteen, times the cubed ratio of temperature to frequency, times the ratio of the GW density parameter to the photon density parameter, times the conversion probability (Equation 10).

For a given detector sensitivity and a given value of the conversion probability, this relation sets stringent bounds on the GW spectrum, which can be expressed in terms of the characteristic strain: the strain is the square root of three times the squared Hubble constant divided by four pi squared, times the GW density parameter, divided by the squared frequency (Equation 11). This is related to the one-sided power spectral density in the usual way. Figure 2 contrasts the resulting constraints with existing bounds in the literature.

Figure 2, left panel: parameter space for cosmic magnetic fields today, with grey shaded areas showing the exclusions discussed in the text and coloured curves indicating contour lines for the rescaled conversion probability. Right panel: upper bounds on the stochastic GW background derived from ARCADE 2 and EDGES in this work, compared to existing laboratory bounds from a superconducting parametric converter, a waveguide, a 0.75 metre interferometer, a magnon detector, and a magnetic conversion detector. The solid lines indicate the allowed parameter space for cosmic magnetic fields; the dashed lines mark the effective-number-of-species constraint for broad GW spectra and for a peaked spectrum of narrow relative width.

The effective-number-of-species bound

GWs contribute to the energy budget of the Universe in the form of radiation and are as such constrained by the big bang nucleosynthesis and CMB bounds on the effective number of massless degrees of freedom: the GW energy density must stay below seven eighths times the four-thirds power of four elevenths, times the allowed shift in that effective number, times the photon energy density (Equation 12), with the allowed shift at or below 0.1.

For a spectrum which is approximately scale invariant between a minimum and a maximum frequency separated by a factor of order one in the logarithm, this implies that the ratio of the GW density parameter to the photon density parameter obeys the same bound (Equation 13), whereas for a narrow spectrum peaked at a given frequency with a width below it, this bound is relaxed by the ratio of the peak frequency to the width. Note that this bound applies only to GWs present already at CMB decoupling.

Probing the Rayleigh-Jeans tail of the CMB spectrum

Below a frequency-to-temperature ratio of about 10 to the minus 2, galactic foregrounds dominate the radio sky. Here we focus on the results reported by ARCADE 2, which covers the sweet spot of the low-frequency Rayleigh-Jeans spectrum before galactic foregrounds become important, at 3, 8, 10, 30 and 90 gigahertz, and by EDGES, which is a recent measurement of the global 21 cm absorption signal at 78 megahertz.

ARCADE 2 was a balloon experiment equipped with a radio receiver measuring the blackbody temperature of the sky. The cleanest frequency band is around 10 gigahertz, enabling a millikelvin resolution, a fractional temperature sensitivity at or below 4 times 10 to the minus 4 at a frequency-to-temperature ratio of about 0.18. At smaller frequencies, ARCADE 2 observed a significant radio excess beyond the expected galactic foreground whose origin remains an open question. 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 gigahertz frequency bands. In Figure 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, that is sky-averaged, 21 cm absorption signal was reported by the EDGES Collaboration. 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 a fractional distortion at or below one at a frequency-to-temperature ratio of 1.4 times 10 to the minus 3, that is 78 megahertz. The width of the observed absorption feature, 19 megahertz, determines the width of the frequency coverage.

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 effective-number-of-species bound. Some examples are axion inflation, metastable cosmic strings and evaporating light primordial black holes. Further significant contributions may be expected from preheating and first order phase transitions occurring above 10 to the 7 GeV. The sensitivity of radio telescopes can, however, not yet compete with the cosmological effective-number-of-species bound, unless one considers essentially monochromatic signals, which may arise, for example, from large monochromatic scalar perturbations.

Since the dominant contribution to the conversion probability arises around reionization, particularly interesting targets are GW sources active between redshifts of about 10 and 1000, which would not be constrained by the effective-number-of-species 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 to the minus 9 to 10 to the minus 7 solar masses, would result in GHz GW signals today. Superradiant axion clouds around spinning black holes yield an essentially monochromatic GW signal at or below MHz frequencies, 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 telescopes — driven in particular by the advances in 21 cm cosmology — or 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 100 square metres per kelvin in the 0.1–10 GHz range, a few hours of observation will lead to sensitivities in the ballpark of microjansky, which must be compared against CMB fluxes of at least 1000 jansky. SKA measurements are thus very promising although sufficient foreground subtraction will be extremely challenging.

Acknowledgements

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.

(The figures, the Supplemental Material and the numbered reference list are omitted here; the complete text, with its equations set as mathematics, is at the source.)