Eur. Phys. J. C manuscript No.(will be inserted by the editor)

Upper limits on the amplitude of ultra-high-frequency gravitationalwaves from graviton to photon conversion

A. Ejllia,1, D. Ejlli3, A. M. Cruise2, G. Pisano1, H. Grote1

1Cardiff University, School of Physics and Astronomy, The Parade, Cardiff, CF24 3AA, UK2Birmingham University, School of Physics and Astronomy, Edgbaston Park Rd, Birmingham B15 2TT, UK3Department of Physics, Novosibirsk State University, 2 Pirogova Street, Novosibirsk, 630090, Russia

the date of receipt and acceptance should be inserted later

Abstract In this work, we present the first experimental up-per limits on the presence of stochastic gravitational wavesin a frequency band with frequencies above 1 THz. We ex-clude gravitational waves in the frequency bands from (2.7−14)×1014 Hz and (5−12)×1018 Hz down to a characteristic am-plitude of hmin

c ≈ 6× 10−26 and hminc ≈ 5× 10−28 at 95%

confidence level, respectively. To obtain these results, weused data from existing facilities that have been constructedand operated with the aim of detecting WISPs (weakly in-teracting slim particles), pointing out that these facilities arealso sensitive to gravitational waves by graviton to photonconversion in the presence of a magnetic field. The princi-ple applies to all experiments of this kind, with prospects ofconstraining (or detecting), for example, gravitational wavesfrom light primordial black hole evaporation in the early uni-verse.

1 Introduction

With the first detections of gravitational waves (GWs) bythe ground-based laser interferometers LIGO and VIRGO,a new tool for astronomy, astrophysics and cosmology hasbeen firmly established [1, 2]. GWs are spacetime perturba-tions predicted by the theory of general relativity that propa-gate with the speed of light and can be predominantly char-acterised by their frequency f and the dimensionless (char-acteristic) amplitude hc. Based on these two quantities andthe abundance of sources across the full gravitational-wavespectrum, as well as the availability of technology, it be-comes clear that different parts of the gravitational-wavespectrum are more accessible than others.

Current ground-based detectors are sensitive in the fre-quency band from about 10 Hz to 10 kHz [3–6] where theintersection of efforts in the development of the technology

aCorresponding author: EjlliA@cardiff.ac.uk

and the abundance of sources facilitated the first detections.Coalescences of compact objects such as black holes andneutron stars have been detected, and spinning neutron stars,supernovae and stochastic signals are likely future sources.Since in principle, GWs can be emitted at any frequency,they are expected over many decades of frequency belowthe audio band, but also above it. At lower frequencies, thespace-based laser interferometer LISA is firmly planned tocover the 0.1−10 mHz band [7, 8], targeting, for example,black hole and white dwarf binaries. At even lower frequen-cies in the nHz regime, the pulsar timing technique promisesto facilitate detections of GWs from supermassive black holes[9–11].

Frequencies above 10 kHz have been much less in thefocus of research and instrument development in the past,but given the blooming of the field, it seems appropriateto not lose sight of this domain as technology progresses.One of the main reasons to look for such high frequenciesof GWs is that several mechanisms that generate very high-frequency GWs are expected to have occurred in the earlyuniverse just after the big bang. Therefore, the study of suchfrequency bands would give us a unique possibility to probethe very early universe. However, the difficulty in probingsuch frequency bands is explained by the fact that laser-interferometric detectors such as LIGO, VIRGO and LISAwork in the lower frequency part of the spectrum and theirworking technology is not necessarily ideal for studying very-high-frequency GWs. The characteristic amplitude of a stochas-tic background of GWs hc, for several models of GW gener-ation, decreases as the frequency f increases. Consequently,to study GWs with frequencies in the GHz regime or higherrequires highly sensitive detectors in terms of the character-istic amplitude hc.

One possible way to construct detectors for very-high-frequency GWs is to make use of the partial conversion ofGWs into electromagnetic waves in a magnetic field. In-

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deed, as general relativity in conjunction with electrodynam-ics predicts, the interaction of GWs with electromagneticfields, in particular, static magnetic fields, generate propa-gating electromagnetic radiation at the same frequency asthe incident GW. In other words, gravitons mix with photonsin electromagnetic fields. This effect has been studied in theliterature by several authors in the context of a static labo-ratory magnetic field [12–15] and in astrophysical and cos-mological situations [16–20]. The effect of graviton-photon(also denoted as GRAPH) mixing is the inverse process ofphoton-graviton mixing studied in Refs. [15, 21–24].

Based on the graviton-photon (or GRAPH) mixing, inthis work we point out that the existing experiments that areconceived for the detection of weakly interacting slim par-ticles (WISPs) are also GW detectors in a sense mentionedabove: they provide a magnetic field region and detectors forelectromagnetic radiation. In this work, we make use of ex-isting data of three such experiments to set first upper limitson ultra-high frequency GWs. As technology may progressfurther, future detectors based on the graviton to photon con-version effect may be able to reach sensitivities for GW am-plitudes near the nucleosynthesis constraint at the very high-frequency regime.

This paper is organised as follows: In Sec. 2, we givean overview of high-frequency GW sources and generatingmechanisms as well as previously existing experimental up-per limits. In Sec. 3 we discuss the working mechanism ofcurrent WISP detectors and the possibility to use them asGW detectors. In Sec. 4, we consider the minimum GWamplitude that can be detected by current WISP detectors.In Sec. 5, we discuss the prospects to detected ultra-high-frequency GWs with current and future WISP detectors andin Sec. 6 we conclude. In this work we use the metric withsignature ηµν = diag[1,−1,−1,−1] and work with the ra-tionalised Lorentz-Heaviside natural units (kB = h = c =

ε0 = µ0 = 1) with e2 = 4πα if not otherwise specified.

2 Overview of high frequency GW sources anddetection amplitude upper limits

The gravitational wave emission spectrum has been fullyclassified from

(10−15−1015

)Hz, as for example, more re-

cently in [25]. For the frequency region of interest to thispaper, the high-frequency GW bands are given as:

– High-Frequency band (HF), (10−100 kHz),– Very High-Frequency band (VHF), (100 kHz −1 THz),– Ultra High-Frequency band (UHF), (above 1 THz).

A viable detection scheme in the VHF and UHF bands(but in principle at all frequencies), is the graviton to pho-ton conversion effect. Based on this effect, it seems feasibleto search for GWs converted to electromagnetic waves in a

magnetic field. The generated electromagnetic waves can beprocessed with standard electromagnetic techniques and canbe detected, for example, by single-photon counting devicesat a variety of wavelengths. Following the classification ofhigh frequency source in the paper [18], there appear to befour kinds of potential GW sources in the VHF and UHFbands:

1) Discrete sources [26]: the authors examined the ther-mal gravitational radiation from stars, mutual conversion ofgravitons and photons in static fields and focusing the mainattention to the phenomenon of primordial black-hole evap-oration, with a backgrounds at the high-frequency region.

2) Cosmological sources [27]: another mechanism whichgenerates a very broadband energy density of GWs noise inthe form of non-equilibrium of cosmic noise generated as aconsequence of the super-adiabatic amplification at the veryearly universe. An upper bound on the energy density in-dependent of the spectrum of any cosmological GWs back-ground prediction is given from the nucleosynthesis boundof ΩGW ≥ 10−5 [28].

3) Braneworld Kaluza-Klein (KK) mode radiation [29,30]: the authors suppose the existence of the fifth dimensionin higher-dimensional gravitational models of black holesderives emission of the GWs. The GWs are generated dueto orbital interactions of massive objects with black holessituated on either our local, “visible", brane or the other,“shadow", brane which is required to stabilise the geome-try. These KK modes have frequencies which may lie in theUHF frequency with large amplitudes since the gravity issupposed to be very strong in bulk with a large number ofmodes.

4) Plasma instabilities [30]: the authors have modelledthe behaviour of magnetised plasma example supernovae,active galaxies and the gamma-ray burst. They have devel-oped coupled equations linking the high-frequencies elec-tromagnetic and gravitational wave modes. Circularly po-larised electromagnetic waves travelling parallel to plasmabackground magnetic field would generate gravitational-wavewith the same frequency of the electromagnetic wave.

Except for the discrete sources and plasma instabilities,the GW radiation is usually emitted isotropically in all direc-tions for several generation models, see below. Normally, aGW detector should be oriented toward the source in orderto efficiently detect GWs, except for the majority of cosmo-logical sources. Indeed, cosmological sources are expectedto generate a stochastic, isotropic, stationary, and Gaussianbackground of GWs that in principle can be searched forwith an arbitrary orientation of the detector. The upper limitson the GW amplitudes that we derive in this paper are lim-ited to the cosmological sources since the detectors (see nextsection), except for one experiment which pointed towardsthe sun, cannot point deliberately to the emitting sources, sotheir measurements are most sensitive to sources creating an

3

isotropic background of GWs. We list some possible sourcesof GWs:

1) Stochastic background of GWsThe stochastic background of GWs is assumed to beisotropic [28, 31, 32] and must exist at present as a re-sult of an amplification of vacuum fluctuations of grav-itational field to other mechanisms that can take placeduring or after inflation [33]. Inflationary processes andthe hypothetical cosmic strings are potential candidatesof the GW background with some differences in the pre-dicted intensity and spectral features [28, 31, 32]. Thesespectrum would have cutoff frequency approximately inthe region νc ∼ 1011 Hz. The prediction for the cutofffrequency in some cosmic string models gives the cutoffshifted to 1013− 1014 Hz [28, 31, 32]. The metric per-turbation at the cutoff frequency 1011 Hz corresponds toan estimated strain amplitude of hc ∼ 10−32.

2) GWs from primordial black holesIn Ref. [34], the authors proposed the existence of Pri-mordial Black Hole (PBH) binaries and estimated the ra-diated GW spectrum from the coalescence of such bina-ries. In addition, the mechanism of evaporation of smallmass black holes gives rise to the production of highand ultra-high frequency GWs. An estimation of the ef-ficiency of this emission channel which might compen-sate the deficit of high-frequency gravitons in the relicGW background has been thoroughly studied in [26]. Adetailed calculation of the energy density of relic GWsemitted by PBHs has been performed in [35]. The au-thor’s analysed and calculated the energy density of GWsfrom PBH scattering in the classical and relativistic regimes,PBH binary systems, and PBH evaporation due to theHawking radiation.

3) GWs from thermal activity of the sunA third class that does generate a stochastic, but not anisotropic background of GWs, which is relevant to thiswork, is the GW emission from the sun [36]. The hightemperature of the sun in the proton-electron plasma pro-duces isotropic gravitational radiation noise due to ther-mal motion [37–39]. The emission comes to the detectorfrom the direction of the sun, and the observations havethe potential to set limits on this process. The frequencyof the collisions of νc ∼ 1015 Hz determines the gravi-tational wave frequency, and the highest frequency cor-responds to the thermal limit at ωm = kT/h ∼ 1018 Hz.Using the plasma parameters in the centre of the Sunthe estimation of the “thermal gravitational noise of theSun” reaching the earth provides a stochastic flux at “op-tical frequencies” of the order hc ∼ 10−41 [39].

So far, dedicated experiments to detect GWs in the VHFregion are based on two designs: polarisation measurementon a cavity/waveguide detector and cross-correlation mea-surement of two laser interferometers. The cavity/waveguideprototype measured polarisation changes of the electromag-netic waves, which in principle can rotate under an incomingGW, providing an upper limit on the existence of GWs back-ground to a dimensionless amplitude of hc ≤ 1.4×10−10 at100 MHz [40]. The two laser interferometer detectors with0.75 m long arms have used a so-called synchronous recy-cling interferometer and provided an upper limit on the ex-istence of the GW background to a dimensionless amplitudeof hc ≤ 1.4×10−12 at 100 MHz [41, 42].

The most recent upper limit on stochastic VHF GWs hasbeen set from by a graviton-magnon detector which mea-sures resonance fluorescence of magnons [44]. They usedexperimental results of the axion dark matter using magne-tised ferromagnetic samples to derive the upper limits onstochastic GWs with characteristic amplitude: hc ≤ 9.1×10−17 at 14 GHz and hc ≤ 1.1×10−15 at 8.2 GHz.

Another facility, the Fermilab Holometer, has performedmeasurement at slightly lower frequencies. The FermilabHolometer [43] consists of separate, yet identical Michel-son interferometers, with 39 m long arms. The upper limitsfound within 3σ , on the amplitude of GWs are, in the rangehc < 25× 10−19 at 1 MHz down to a hc < 2.4× 10−19 at13 MHz.

3 WISP search experiments and their relevance to UHFGWs

Fig. 1 Simplified schematic of the experimental setup aiming at thedetection of WISPs. In the upper panel left-hand side, the electromag-netic waves interacting with the magnetic field produce the hypotheti-cal WISPs, and at the right-hand side electromagnetic waves are pro-duced by the decay of WISPs in the constant magnetic field. Our workis illustrated by the lower panel ignoring the transparent left-hand side.On the right hand side, the photons detected could be due to the passageof GWs propagating in the constant magnetic field.

4

The experiments ALPS [45], OSQAR [46] and CAST[36] have not been designed to detect GWs in the first place.However, in this work their results are used to compute newupper limits on GW amplitudes and related parameters. Theexperiments performed by ALPS and OSQAR are usuallycalled “light shining through the wall experiment” wherethe hypothetical WISPs, that are generated within the exper-iment, mediate the “shining through the wall” process, anddecaying successively into photons. In contrast, the CASTexperiment searches directly for WISPs emitted from thecore of the sun. Though all these experiments are not de-signed to detect gravitational waves, they provide a highsensitivity measurement of single photons generated in theirconstant magnetic field which is the crucial ingredient forthe detection of graviton to photon conversion.

The main characteristics of the ALPS, OSQAR and CASTexperiments are:

1) ALPS experiment at DESYThe ALPS (I) experiment has performed the last datataking run in 2010, and the specific characteristics of theexperiment are found in Ref. [47]. A general schematicof the principle is shown in the upper panel of the Fig.1. The production of WISPs and their re-conversion intoelectromagnetic radiation is located in one single HERAsuperconducting magnet where an opaque wall in themiddle separates the two processes. The HERA dipoleprovides a magnetic field of 5 T in a length of 8.8 m. Theelectromagnetic radiation, generated by the decay of theWISPs in the magnetic field, passed a lens of 25.4 mmdiameter and focal length 40 mm. The lens focuses thelight onto a ≈ 30 µm diameter beam spot on a CCDcamera.

2) OSQAR experiment at CERNThe OSQAR experiment has performed the last data tak-ing in 2015, and the specific characteristics of the exper-imental setup are found in [46, 48]. The OSQAR col-laboration has used two LHC superconducting dipolemagnets separated by an optical barrier, (for a concep-tual scheme see the upper panel of the Fig. 1). The LHCdipole magnets provide a constant magnetic field of 9 T,along a length of 14.3 m. To focus the generated photonsof the beam onto the CCD, an optical lens of 25.4 mm di-ameter and a focal length of 100 mm was used, installedin front of the detector similar to the Fig. 1. Data acqui-sition has been performed in two runs with two differentCCD’s having different quantum efficiencies.

3) CAST experiment at CERNThe CERN Axion Solar Telescope (CAST) experimenthas the aim to detect or set upper limits on the flux ofthe hypothetical low-mass WISPs produced by the Sun.A refurbished CERN superconducting dipole magnet of

9 T and 9 m length was used. The solar Axions withexpected energies in the keV range can convert into X-rays in the constant magnetic field, and an X-ray detectorhas been used to performed runs in the time period 2013- 2015 [49]. To increase the cross section, both the twoparallel pipes which pass through the magnet have beenused which provide an area of 2×14.5 cm2 focused intoa Micromegas detector. The CAST detector mounted onthe pointing system had a telescope with a focal lengthof 1.5 m installed for the (0.5−10) keV energy range.

4 Minimal-detection of GW amplitude

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Fig. 2 Quantum efficiency as a function of the wavelength. Left-handside panel: the quantum efficiency of the detectors using the method“light shining through a wall”; in the right-hand panel: the quantumefficiency of the Micromegas X-ray detector used in the CAST experi-ment. The detector bandwidth and their normalised quantum efficiencyfunction are used to compute the upper limits un GWs detectors.

In this section, we show how we compute upper limitson the GW dimensionless amplitude hc based on the charac-teristics of the experiments described above that are sensi-tive to an isotropic background of GWs from cosmologicalsources and to the thermal activity in the sun. We ignorethe generation of WISPs (Fig. 1: lower panel) and focus onthe second half of the magnetic field for the case of ALPSand OSQAR experiments, and, for the CAST experiment weconsider the full magnet region. These experiments measurea number of photons per unit time with their CCD detectors,

5

εγ (ω) Nexp (mHz) A (m2) B (T) L (m) ∆ f (Hz)

ALPS I Fig 2 0.61 0.5×10−3 5 9 9×1014

OSQAR I Fig 2 1.76 0.5×10−3 9 14.3 5×1014

OSQAR II Fig 2 1.14 0.5×10−3 9 14.3 1×1015

CAST Fig 2 0.15 2.9×10−3 9 9.26 1×1018

Table 1 Relevant characteristics of the experimental setups, as operated for the detection of WISPs, and used for GW upper limits in this work.The reported quantities are used to estimate the minimum detectable GW amplitude through the graviton to photon conversion in a constant andtransverse magnetic field.

namely Nexp in an energy band ∆ω with efficiency εγ andin a cross-section A. In what follows we assume that in theCCD, the background dark current fluctuation is a stochasticprocess with uniform probability distribution and stationaryin ω . In this case, the energy flux of photons generated in anenergy bandwidth [ωi,ω f ] is given by:

ΦCCDγ

(z,ω f ; t

)=∫

ω f

ωi

1A(z)

N (ω, t) ω

εγ (ω)dω (1)

where N (ω, t) is the number of photons per unit of timeand energy. Now, we have to compare the measured en-ergy flux of photons with the intrinsic energy flux of pho-tons generated in the graviton to photon conversion in pres-ence of an isotropic background of GWs. The analyticaltreatment for an isotropic background of GWs convertedinto electromagnetic waves, is described in detail in Ap-pendix 2, 3. In Appendix 3, different useful quantities arecalculated, for stochastic GWs propagating in a transverseand constant magnetic field. Since all the experiments oper-ated under vacuum condition and the propagation distancez is small with respect to the oscillation length of the parti-cles, we can safely take ∆x,y z 1. The variable ∆x,y definedin the Eq. A.7 is a function of the magnetic field, the mag-nitude of the photon and graviton wave-vectors and Newto-nian constant. Then the energy flux of photons generated inthe magnetic field of length z given by expression (B.16), inthe same energy bandwidth [ωi,ω f ] becomes:

Φgraphγ

(z,ω f ; t

)=(Mx

gγ

)2∫

ω f

ωi

[sin2 (∆xz)

∆ 2x

+sin2 (∆yz)

∆ 2y

]×

× h2c (0,ω) ω

2 κ2 dω

'∫

ω f

ωi

B2 z2 h2c (0,ω) ω

4(2)

Comparing the energy fluxes in expressions (1) and (2),and requiring that for detection, the energy flux in (2) mustbe bigger or equal to the energy flux in (1), we get

h2c (0,ω)≥ 4N (ω, t)

B2 L2 A(L) εγ (ω), (3)

where we took z = L with L being the spatial extension ofthe external magnetic field. All the three experiments listedin the section above their upper limits are compatible tothe background fluctuation of the detector which allows usto express the relation: N (ω, t) = Nexp/∆ω , where ∆ω =

ω f −ωi. Finally, by putting the units in explicitly, we getthe following expression for the minimum detectable GWamplitude:

hminc (0,ω)'

√4Nexp

AB2 L2 εγ (ω) ∆ω' 1.6×10−16×

×(

1 TB

)(1 mL

)√(Nexp

1 Hz

)(1 m2

A

)(1 Hz∆ f

)(1εγ

)(4)

where ω = 2π f with f being the frequency. In orderto compute the minimum detectable GW amplitude, hmin

c ,we have extracted the following quantities from the ALPS,OSQAR and CAST experiments:

– Nexp the total detected number of photons per second inthe bandwidth ∆ω ,

– A cross-section of the detector,– B magnetic field amplitude,– L distance extension of the magnetic field,– εγ(ω) quantum efficiency of the detector,– ∆ f operation bandwidth of the CCD.

These quantities permit to compute the equivalent mini-mum amplitude hmin

c of a stochastic GW background whichwould generate photons through graviton to photon conver-sion, equivalent number of background photons that the CCDhas read. The data accounted for the photon detection in theconstant magnetic field for the three experiments ALPS, OS-QAR and CAST, exclude the detection of physical signalswith fluxes bigger or comparable to the background countof the CCD detectors at 95% confidence level which allowsputting upper limits on the minimal detectable GW ampli-tude hmin

c at the same confidence level.The extracted quantities used to compute the upper lim-

its on hminc are summarised in Table 1. These experiments at-

tempting to detect WISPs have used subsequently improvedCCDs during different run phases which is taken into ac-count in the analysis. The quantum efficiencies in Table 1 are

6

represented graphically in Fig. 2 as a function of the wave-length. We have taken into account that Nexp is normalised tothe quantum efficiency the working frequency of the WISPsexperiments, and the range of the expected photons is im-posed by the sensitive frequency range of the CCD.

The cross-section reported in Table 1 has been computed,for the ALPS and OSQAR experiments, considering the areaenclosed by the diameter of the lens [45, 46]. Instead, theCAST experiment uses the whole cross-section of the twobeam pipes.

Fig. 3 Plots of the minimum detectable GW amplitude hminc as a func-

tion of the frequency f , deduced from the measured data of the denotedexperiments.

Using the data of the Table 1 and expression (4) it ispossible to produce an upper limit plot of the GW ampli-tude, see Fig. 3, due to the conversion of GWs into photons.The region above each curve is the excluded region. To ourknowledge, these are the first experimental upper limits inthese frequency regions.

5 Prospects on detecting Ultra-High Frequency GWsfrom primordial black holes

Graviton to photon conversion maybe a useful path towardsthe detection of UHF GWs. The actual technology has madefurther progress in the detection of single photons and newfacilities are intended for WISP search, using higher valuesof B and L in order to achieve higher sensitivities. One fa-cility which plans to do so is the ALPS IIc proposal whichconsists of two 120 m long strings of 12 HERA magnetseach, with a magnetic field of 5.3 T. The scheme of gen-eration and conversion of the WISPs is still the same, ex-pected an optical resonator is added to the reconversion re-gion. A follow-up of the CAST telescope is the proposedInternational Axion Observatory (IAXO). Tab. 2 representthe detector parameters of ALPS IIc [50], a possible follow-up named JURA [51], and of the IAXO proposal [53].

Since the working frequency of the detectors is differ-ent we compute the sensitivity to detected GWs, with the

Fig. 4 In the upper panel conceptual scheme of the experimental setupALPS IIc and a possible follow-up named JURA where we note theaddition of the FP cavity in the right-hand side. Our prediction for thesensitivity of the minimal amplitude of hmin

c used the right-hand sideprocess, where the photons generated via graviton to photon conversionare resonantly enhanced in the FP cavity. In the lower panel, the FP res-onator concept is described where Egraph is the electric field generatedfrom the graviton to photon conversion in the cavity, Ecirc is circulatingelectric field accumulated inside the resonator after transmission losseson both mirrors, Etrans is transmitted electric field through the mirrorsand L the length of the cavity.

graviton-photon mixing process, in two frequency regionsinfrared and X-rays:

5.1 Infrared

One of the most important changes that ALPS IIc, with re-spect to the ALPS I and OSQAR, is the use of a Fabry-Perot(FP) cavity to enhance the decay processes of WISPs intophotons, see Fig. 4. The FP cavity will allow just a range ofelectromagnetic waves to be built up resonantly, within thecavity bandwidth: ∆ωc = ∆ωFSR/F where F = π/(1−R)is the cavity finesse, ∆ωFSR = π/L is the cavity free spectralrange, and R is the reflectance of the mirrors. The FP cavityenhances the decay rate of WISPs to photons [54]. This isan essential aspect because it will also account for the tran-sition of gravitons into photons [55]. Stochastic broadbandGWs converted into electromagnetic radiation would exciteseveral resonances of the cavity at frequencies ωc±n∆ωFSR,where ∆ωc is the cavity frequency bandwidth, and n is an in-teger number with its range depending on the coating of themirrors. To calculate the response of the FP resonator, weuse of the circulating field approach [56, 57], as displayedin the lower panel of Fig. 4. We assume a steady state ap-proximation to derive the circulating electric field Ecirc in-side the cavity and the mirrors have the same reflectance Rand transmittance T . Defining the phase shift after one roundtrip 2φ (ω) = 2ωL, the accumulated electric field Ecirc aftera large number of reflections (which can be assumed infinite

7

εγ Ndark (Hz) A (m2) B (T) L (m) F

ALPS IIc 0.75 ≈ 10−6 ≈ 2×10−3 5.3 120 40000

JURA 1 ≈ 10−6 ≈ 8×10−3 13 960 100000

IAXO 1 ≈ 10−4 ≈ 21 2.5 25 -

Table 2 Parameters of ALPs IIc, JURA and IAXO proposals used to estimate the predicted minimum detectable GW amplitude through thegraviton to photon conversion in their constant and transverse magnetic field: εγ is the efficiency photodetector at 1064 nm, Ndark correspond tothe number of photons per unit of time limited by the dark count sensitivity, A is the cross-section, B (T) is the magnetic field magnitude, L is themagnetic field length and F is the finesse of the cavity.

in the calculations below) of the electric field Egraph gener-ated in the GRAPH mixing is:

Ecircx,y (z, t) = Egraph

x,y (z, t)

×(

1+Re−i2φ(ω,L)+(

Re−i2φ(ω,L))2

+ · · ·)

= Egraphx,y (z, t)

∞

∑n=0

(Re−i2φ(ω,L)

)n

= Egraphx,y (z, t)

11−Re−i2φ(ω,L)

. (5)

The circulating flux, in a time t > τ where τ = FL/π is thecharge time of the cavity, at a distance z = L is Φ circ

γ (L, t)≡〈|Ecirc

x (z, t) |2〉+ 〈|Ecircy (z, t) |2〉. By expanding Egraph

x,y (z, t)as a Fourier integral and doing the same steps to derive theflux as shown in Appendix (Appendix A) from Eq. B.9 toEq. B.16, then the circulating flux simplifies to the follow-ing expression:

Φcircγ (L, t) '

∫ +∞

0

1

(1−R)2 +4R sin2 [φ (ω,L)]

× B2 L2 h2c (0,ω) ω

4dω. (6)

where we have consider that the propagation distance z = Lis small with respect to the oscillation length of the particles,and we can safely take ∆x,yz 1. Taking the differential ofthe circulating flux in (6) for a given energy interval [ωi,ω],we find the the following relation

dΦcircγ (L,ω; t) =

dΦgraphγ (L,ω; t)

(1−R)2 +4R sin2 [φ (ω,L)]. (7)

where dΦgraphγ (L,ω; t) is the differential of Eq. B.16 in a

given energy interval [ωi,ω], which correspond to the fluxof photons generated through the graviton to photon conver-sion without the cavity. Now we can derive the gain factor,namely Γcirc (φ), as the differential of the circulating energyflux in the resonator relative to the differential of the energy

flux generated in the graviton to photon conversion withoutthe cavity

Γcirc (φ)=dΦ circ

γ (L,ω; t)

dΦgraphγ (L,ω; t)

=1

(1−R)2 +4R sin2 [φ (ω,L)].

(8)

For a given length L and for frequencies matching thecavity resonance, or φ (ω,L) = nπ where n is a positiveinteger, the internal gain enhancement factor is maximum:Γcirc (nπ) = (F/π)2. In the same way, we derive the trans-mitted gain on both sides of the cavity, which expression isgiven by:

Γtrans (φ)=dΦ trans

γ (L,ω; t)

dΦgraphγ (L,ω; t)

=1−R

(1−R)2 +4R sin2 [φ (ω,L)].

(9)

Unlike before, the transmitted flux from the cavity willexhibit transmitted light peaks which gain factor, for fre-quencies matching the cavity resonance condition, reducesto: Γtrans (nπ) = (F/π). Now we can write explicitly theequation of the flux produced from graviton to photon con-version transmitted from the cavity in a energy bandwidth[ωi,ω f ]:

Φtransγ

(L,ω f ; t

)=∫

ω f

ωi

Γtrans (φ) dΦgraphγ (L,ω; t) (10)

=∫

ω f

ωi

B2xL2hc (0,ω)2

4Γtrans (φ) ω dω.

A photo-detector placed at the transmission line of thecavity will measures an energy flux, within its bandwidth∆ω defined in Eq. 1. According to the previous discussion,a cavity of length L will transmit light peaks for frequen-cies ω∗ = nπ/L, and such frequency should be in the inter-val bandwidth ∆ω . Reminding that the flux in expressions(10) is calculated for a stochastic process and taking into ac-count the bandwidth of the photodetectors of ALPS IIc andJURA such condition is satisfied. Now, considering the en-ergy flux of a photodetector limited by the dark count rate

8

Ndark, where N (ω, t) = Ndark/∆ω , and comparing with theenergy fluxes in expressions (10), and solving for hc (0,ω)

in SI units, at the maximum transmission ω∗ = nπ/L, be-comes:

hminc (0,ω∗) ' 2.8×10−16

(1 TB

)(1 mL

)× (11)

×

√(1F

)(Ndark

1 Hz

)(1 m2

A

)(1 Hz∆ f

)(1εγ

).

From the above equation, we can observe that to computethe sensitivity in amplitude hc (0,ω), with respect to the casewithout cavity Eq. 4, in addition, we need to know the fi-nesse factor F . The minimum detectable GW amplitude,considering the photodetector background rate in one sec-ond, limited by the dark counting rate, a frequency band-width ∆ f ≈ 4× 1014 Hz [52], and the relevant character-istic of Tab. 5, the sensitivity for the minimal amplitude is:hALPS IIc

c ≈ 2.8×10−30 and hJURAc ≈ 2×10−32, which is two

orders of magnitude better than in the case without cavity.

5.2 X-rays

The core element of IAXO will be a superconducting toroidalmagnet, and the detector will use a large magnetic field dis-tributed over a large volume to convert solar axions into de-tectable X-ray photons. The central component of IAXO isa superconducting toroidal magnet of 25 m length and 5.2 mdiameter. Each toroid is assembled from eight coils, gen-erating 2.5 T in eight bores of 600 mm diameter. The X-raydetector would be an enhanced Micromegas design to matchthe softer 1−10 keV spectrum. The X-rays are then focusedat a focal plane in each of the optics read by pixelised planeswith a dark current background level of Ndark = 104 [53]. Forthe process of graviton to photon conversion, the sensitivityon the minimal amplitude of hIAXO

c ≈ 1×10−29.

5.3 Ultra-High Frequency GWs from Primordial BlackHoles

In order to describe the potential of ALPS IIc and JURA onprobing the GW background at very high frequencies an ex-plicit example of a GW source can be considered. One ofthe most promising sources of VHF and UHF GWs in thefrequency range of interest regarding ALPS IIc is the evap-oration of very light PBHs that would have been formed justafter the big bang. As shown in detail in Ref. [35], theseblack holes would emit GWs by different mechanism asscattering, binary black hole, and evaporation by hawkingradiation which in principle could contribute to the spec-trum of cosmic electromagnetic X-ray background due to

Fig. 5 Graphical representation of the amplitude hminc as a function

of frequency for the: PBH evaporation of masses mBH (10−3, 104 g,108 g), the upper limits of graviton-photon conversion data in Fig. 3,the estimated amplitude sensitivity for the ALPS IIc and JURA (redand grey lines) at the infrared region using their detection scheme.The dotted blue line is the estimated sensitivity for the solar telescopeJAXO successors of CAST experiment. The two dashed lines representthe nucleosynthesis amplitude upper limit and the predicted amplitudefrom the thermal GW emitted from the sun. Here for simplicity wehave assumed a value of the PBH density parameter at their productiontimes equal to Ωp ' 10−7. Both amplitude, hc, and frequency axes arein Log10 units.

graviton-photon mixing in cosmic magnetic fields [19]. Itis especially the evaporation of GWs due to Hawking radi-ation which generates a substantial amount of GWs in thefrequency regime compatible with the ALPS IIc and JURAworking frequency. The spectral density parameter of GWsat present is given in Ref. [35] and it reads

h20Ωgw ( f ; t0) = 1.36×10−57

(Neff

100

)2( 1gmBH

)2( f1 Hz

)4

×∫ zmax

0

√1+ z

e(

2π f (1+z)T0

)−1

dz (12)

where T0 is the PBH temperature redshifted to the presenttime, mBH is the PBH mass, Neff is number of particle specieswith masses smaller than the BH temperature TBH, and zmaxis the maximum redshift. The PBH temperature at presentand the maximum redshift are given by [35]

T0 = 1.43×1013

×

√(100Neff

)(100

gS (T (τBH))

)(mBH

1g

)(Hz)

zmax =

(32170

Neff

)2/3(mBH

mPl

)4/3

Ω1/3P −1, (13)

where ΩP is the density parameter of PBH at their pro-duction time and mPl is the Planck mass and gS (T (τBH))

9

is the number of species contributing to the entropy of theprimeval plasma at temperature T (τBH) at the evaporatinglife-time τBH [58, 59] . The number of PBHs that take partin this process is included in the density parameter ΩP, seeRef. [35] for details.

Now in order to extract the characteristic amplitude dueto the stochastic background of GWs due to PBH evapora-tion we need an expression which connects hc to the densityparameter h2

0Ωgw. By using the definition of the density pa-rameter in (B.17) and the expression for the energy densityof GWs in (B.18), we get

hc (0, f )' 1.3×10−18√

h20Ωgw ( f ; t0)

(1Hz

f

). (14)

Now by using expression (12) into expression (14), we getthe following expression for the characteristic amplitude ofGWs due to PBH evaporation

hc (0, f ) = 4.8×10−47(

Neff

100

)(f

1 Hz

)(1 g

mBH

)

×

√√√√∫ zm

0

√1+ z

e(

2π f (1+z)T0

)−1

dz. (15)

In order to have an overview of the upper limit derivedand the perspective to detect UHF GWs, in Fig. 5 is shown:the upper limits derived in the previous section, the esti-mated minimum detectable amplitude for the ALPS IIc andJURA considering the photo detector dark count rate, themaximum GWs amplitude generated in the production cav-ity, the estimated GWs amplitude for Neff = 100, Ωp = 10−7

and BH masses (10−3, 104, 108) g, the prediction of the GWsfrom the sun and the nucleosynthesis upper limit, Ωgw ≈10−5 [28]. The sensitivity to GW detection for ALPS IIc andJURA could reach better results for longer integration time,for example, T = 106 − 107 s. A straightforward method,to integrate in time, is to modulate the field amplitude. Insuch a situation, the signal-to-noise ratio improves as

√T .

An alternative method, without the signal modulation, is tocorrelate the data stream from two different photodetectors.The electromagnetic wave is generated inside the FP cavity,and the transmission is on both mirrors of the cavity whichare placed on the sides of the magnet. Having two photode-tectors mounted on both sides of the magnet and correlatingtheir signal in time would allow lowering the statistical noiseof the detector. So, the time integration would let to a furthergain in the sensitivity amplitude hmin

c . Ultimately the ALPSIIc would be able to explore new amplitude regions of GWswhich target source could be the predicted GWs generatedfrom the evaporation of PBHs.

6 Conclusions

A broad spectrum of the emission of GWs is predicted toexist in the universe, and some sources could generate GWswith frequencies higher than THz. The predicted conversionof gravitons into photons, due to the propagation in a staticmagnetic field, is not out of reach for current technologies.The technique of various detectors having the aim of count-ing single photons, at a narrow wavelength in a static mag-netic field has been developed as detectors for the measure-ment of WISPs, a dark matter candidate, decaying to pho-tons in the transverse magnetic field. Though the WISPs de-tection setups were not particularly designed to detect GWconversion, the generation of electromagnetic radiation asGWs propagate in a static magnetic field provides the pos-sibility of using the published data, currently for the ALPS,OSQAR, CAST collaborations, to set the first upper limitson the amplitude of isotropic Ultra-High-Frequency GWs.We exclude the detection of GWs down to an amplitudehmin

c ≈ 6× 10−26 at (2.7− 14)× 1014 Hz and hminc ≈ 5×

10−28 at (5−12)×1018 Hz at 95% confidence level. Manytheoretical potential ultra-high-frequency GW sources couldbe searched for using such similar experimental setups. Thenext generation experiments, such as the ALPS IIc and JURAfacilities, or similar experiments using high static magneticfields, are potential detectors for the graviton to photon con-version as well. The predicted ALPS IIc data taking or even-tually JURA will be able to produce more stringent upperlimits on the amplitude of the stochastic wave backgroundof GWs generated from PBH evaporation models.

7 Acknowledgement

We are grateful to Prof. Bernard Schutz for helpful com-ments on the manuscript, and we recognise the support fromthe Leverhulme Emeritus Fellowship EM 2017-100.

Appendix A: Propagation of GWs in a constantmagnetic field

Here we review the graviton-photon mixing in a static ex-ternal magnetic field, which can result in the conversion ofgravitational waves into photons, the process which we de-scribe in the main paper operating in the detectors designedto detect WISPs. In this section we closely follow Ref. [20].To start with it is necessary first to write the total Lagrangiandensity L of the graviton-photon system. In our case, it isgiven by the sum of the following terms

L = Lgr +Lem, (A.1)

10

where Lgr and Lem are respectively the Lagrangian den-sities of the gravitational and electromagnetic fields. Theseterms are respectively given by

Lgr =1

κ2 R, (A.2)

Lem = −14

Fµν Fµν − 12

∫d4x′Aµ (x)Π

µν(x,x′)

Aν

(x′),

where R is the Ricci scalar, g is the metric determinant, Fµν

is the total electromagnetic field tensor, κ2 ≡ 16πGN withGN being the Newtonian constant and Π µν is the photonpolarisation tensor in a magnetised medium.

The equations of motion from (A.1) and (A.2) for thepropagating electromagnetic and GW fields components, Aµ

and hi j, in an external magnetic field are given by [20]

∇2A0 = 0,

Ai +∫

d4x′Π i j (x,x′)A j(x′)+∂

i∂µ Aµ =

= κ ∂µ [hµβ F iβ−hiβ Fµ

β],

hi j = −κ (BiB j + BiB j + BiB j) , (A.3)

where Aµ =(φ ,A) is the incident electromagnetic vector-potential with magnetic field components Bi and Bi are thecomponents of the external magnetic field vector B. In ob-taining the system (A.3) we made use of the TT-gauge con-ditions for the GWs tensor h0µ = 0,hi

i = 0 and ∂ jhi j = 0.As shown in details in Ref. [20], the system (A.3) can belinearized by making use of the slowly varying envelope ap-proximation (SVEA) which is a WKB-like approximationfor a slowly varying external magnetic field of spacetime co-ordinates. In this approximation, and for propagation alongthe observer’s z axis in a given cartesian coordinate system,equations (A.3) can be written as [20]

(ω + i∂z)Ψ (z,ω, z) I +M (z,ω)Ψ (z,ω, z) = 0, (A.4)

where in (A.4) I is the unit matrix, Ψ (z,ω, z) = (h×,h+,Ax,Ay)T

is a four component field with h×,+ being the usual GWcross and plus polarisation states and Ax,y are the usual prop-agating transverse photon states. In (A.4) M (z,ω) is themass mixing matrix which is given by

M (z,ω) =

0 0 −iMx

gγ iMygγ

0 0 iMygγ iMx

gγ

iMxgγ −iMy

gγ Mx MCF

−iMygγ −iMx

gγ M∗CF My

, (A.5)

where the elements of the mass mixing matrix M aregiven by:

Mxgγ = κ kBx/(ω + k) ,

Mygγ = κ kBy/(ω + k) ,

Mx = −Πxx/(ω + k) ,

My = −Πyy/(ω + k) ,

MCF = −Πxy/(ω + k) .

Here MCF is a term which includes a combination of theCotton-Mouton and Faraday effects in a plasma and whichdepends on the magnetic field direction with respect to thephoton propagation. Here ω is the total energy of the fields,namely ω = ωgr = ωγ . In this work all the particles partic-ipating in the mixing are assumed to be relativistic, namelyω + k ' 2k with k being the magnitude of the photon andgraviton wave-vectors.

The system of differential equations (A.4) does not haveclosed solutions in the case when the mixing occurs in ar-bitrary matter that evolves in space and time, namely in thecase when the system of differential equations is with vari-able coefficients such as in cosmological scenarios. How-ever, in the case of mixing in a laboratory magnetic field,the system (A.4) can be simplified by considering a specificpropagation of GWs with respect to the magnetic field direc-tion and by considering the propagation in the magnetic fieldwithout gas or a plasma present (see below). For example,first one can choose the magnetic field to be transverse to thephoton direction of propagation such as B = (Bx,0,0) wherewe have My

gγ = 0 and MCF = 0. Second, if there is a gas or aplasma in addition to the external magnetic field, usually wehave that Mx 6= My which essentially means that the trans-verse photon states have different indexes of refraction. Inthe case when one is able to achieve almost a pure vacuumin the laboratory, the contribution of the gas or plasma tothe index of refraction can be safely neglected while thereis also still present a contribution to the index of refractiondue to the vacuum polarisation in the magnetic field. How-ever, the latter contribution is completely negligible becausethe magnitude of the laboratory magnetic field is usually fewTeslas and consequently is too small to have any appreciableeffect on Mx and My.

As discussed above, let us consider first the case whenthe external magnetic field is completely transverse with re-spect to the photon direction of propagation where My

gγ = 0and MCF = 0. The fact that MCF = 0 is because By = 0, Bz = 0and consequently the term corresponding to the Faraday ef-fect is absent since this effect occurs only when the mag-netic field has a longitudinal component along the electro-magnetic wave direction of propagation. In addition, in MCFterm it is also zero the term corresponding to the Cotton-Mouton effect in plasma because by convention we havechosen that By = 0. After these considerations several termsin the mixing matrix M (z,ω) are zero and in the case of the

11

medium in the laboratory being homogeneous in space (in-cluding the magnetic field), then the mass mixing matrix Mdoes not depend on the coordinate z. In this case the commu-tator [M (z,ω) ,M (z′,ω)] = 0 and the solution of the system(A.4) is given by taking the exponential of M (z,ω). Conse-quently, we obtain the following solutions for the fields

h× (z,ω, z) =[

cos(∆xz)− iMx sin(∆xz)

2∆x

]ei(ω+ Mx

2 )z h× (0,ω, z)

+Mx

gγ sin(∆xz)∆x

ei(ω+Mx/2)zAx (0,ω, z) ,

h+ (z,ω, z) =[

cos(∆yz)− iMy sin(∆yz)

2∆y

]ei(

ω+My2

)z h+ (0,ω, z)

−Mx

gγ sin(∆yz)∆y

ei(ω+My/2)zAy (0,ω, z) ,

Ax (z,ω, z) = −Mx

gγ sin(∆xz)∆x

ei(ω+ Mx2 )z h× (0,ω, z)

+

[cos(∆xz)+ i

Mx sin(∆xz)2∆x

]ei(ω+Mx/2)z Ax (0,ω, z) ,

Ay (z,ω, z) =Mx

gγ sin(∆yz)∆y

ei(

ω+My2

)z h+ (0,ω, z)

+

[cos(∆yz)+ i

My sin(∆yz)2∆y

]ei(

ω+My2

)z Ay (0,ω, z) ,(A.6)

where h×,+ (0,ω, z) and Ax,y (0,ω, z) are respectively theGW and electromagnetic wave initial amplitudes at the ori-gin of the coordinate system z = 0, namely the amplitudesbefore entering the region where the magnetic field is lo-cated. In addition, we have defined

∆x,y ≡

√M2

x,y +4(Mx

gγ

)2

2. (A.7)

Appendix B: Electromagnetic energy fluxes generatedwith laboratory graviton-photon mixing

In the previous appendix we found the solutions of the lin-earised equations of motion (A.4) for the GW fields h×,+and for the electromagnetic wave fields Ax,y. In this sec-tion we use these solutions to find the energy flux of theelectromagnetic radiation generated in the laboratory for thegraviton-photon mixing (in equations abbreviated as graph).Before proceeding further is important to stress that in (A.6),the GW amplitudes h×,+ are not dimensionless, as they arecommonly defined in some textbooks, but have energy di-mension units. This is due to the fact that in Ref. [20] themetric tensor is expanded as gµν = ηµν + κhµν where theGW tensor hµν has the physical dimensions of an energy.But in many cases one also writes gµν = ηµν + hµν wherein this case hµν is a dimensionless quantity. Since the lattercase is quite common in the theory of GWs and because wewant to conform to the literature, in expression (A.6) one has

to simply replace h×,+ (0, t) = h×,+ (0, t)/κ , where h×,+ arenow dimensionless amplitudes.

Consider the case when GWs enter a region of constantexternal magnetic field in the laboratory and that initiallythere are no electromagnetic waves.

The assumption of no initial electromagnetic waves meansthat Ax (0,ω, z) = Ay (0,ω, z)=0 in the solutions (A.6). There-fore, the expressions for the electromagnetic field compo-nents, in the graviton to photon mixing for a transverse prop-agation with respect to magnetic field, are given by

Ax (z,ω, z) = −Mx

gγ sin(∆xz)κ∆x

ei(ω+Mx/2)z h× (0,ω, z) ,

Ay (z,ω, z) =Mx

gγ sin(∆yz)κ∆y

ei(ω+My/2)z h+ (0,ω, z) (B.8)

The expressions for the electromagnetic field compo-nents in (B.8), even though very important, are not muchuseful for practical purposes since we usually detect electro-magnetic radiations through their transported energy to thedetector. For this reason is better to work with the Stokesparameter Iγ (z, t) ≡ Φγ (z, t) of the electromagnetic radia-tion generated in the graviton to photon mixing and whichquantifies the energy flux (or energy density) of the elec-tromagnetic radiation. The Stokes Φγ parameter, at a givenpoint in space z, is defined as

Φγ (z, t)≡ 〈|Ex (z, t) |2〉+ 〈|Ey (z, t) |2〉, (B.9)

where Ex and Ey are the components of the electric fieldof electromagnetic radiation and the symbol 〈(·)〉 denotestemporal average of quantities over many oscillation peri-ods of electromagnetic radiation. The components of theelectric field E are related to that of the vector-potentialA through the relation Ex,y (z, t) =−∇A0 (z, t)−∂tAx,y (z, t).For a globally neutral medium (if there is one except themagnetic field) in the laboratory we can choose A0 = 0 fromthe first equation in (A.3) and after we simply get Ex,y (z, t)=−∂tAx,y (z, t).

In order to calculate the energy density of the electro-magnetic radiation and related quantities in the graviton tophoton mixing, we need first to make some assumptions onthe GW signal which interacts with the magnetic field in thelaboratory. In this work we concentrate on our study on astochastic background of GWs with astrophysical or cosmo-logical origin. It is rather natural to assume that the stochas-tic background of GWs is isotropic, unpolarized and station-ary [28, 32]. In order to make more clear what these assump-tions mean, we write the GW amplitude tensor hi j (z, t) atz = 0 as a Fourier integral

12

hi j (0, t) = ∑λ=×,+

∫ +∞

−∞

dω

2π

∫S2

d2n hλ (0,ω, n)eλi j (n)e−iωt ,

(i, j = x,y,z) , (B.10)

where n is a unit vector on the two sphere S2 which de-notes an arbitrary direction of propagation of the GW, d2n =

d (cosθ)dφ , λ is the polarisation index of the GW with theusual cross and plus polarisation states and eλ

i j (n) is the GW

polarisation tensor which has the property eλi j (n)ei j

λ ′ (n) =2δλλ ′ . The assumptions that the stochastic background isisotropic, unpolarised and stationary means that the ensem-ble average of the Fourier amplitudes satisfies

〈hλ (0,ω, n) h∗λ ′(0,ω ′, n′

)〉 = 2πδ

(ω−ω

′) δ 2 (n, n′)4π

δλλ ′

× H (0,ω)

2, (B.11)

where H (0,ω) is defined as the spectral density of thestochastic background of GW and it has the physical dimen-sions of Hz−1 and δ 2 (n, n′) = δ (φ −φ ′)δ (cosθ − cosθ ′)is the covariant Delta function on the two sphere. One cancheck by using (B.10) and (B.11), that the ensemble average〈hi j (0, t) hi j (0, t)〉, is given by

〈hi j (0, t) hi j (0, t)〉 = 2∫ +∞

−∞

dω

2πH (0,ω) (B.12)

= 4∫ +∞

0

d (logω)

2πω H (0,ω)

≡ 2∫ +∞

0d (logω)h2

c (0,ω) , (B.13)

where the last equality in (B.12) defines the characteristicamplitude, hc (dimensionless), of a stochastic backgroundof GWs. In obtaining (B.12) we used the fact that for an un-polarised stochastic background, we have that on average,〈|h× (0,ω) |2〉= 〈|h+ (0,ω) |2〉 6= 0 while the ensemble aver-age of the mixed terms vanish identically. We may observethat by comparing the two last equalities in (B.12) we geth2

c (0,ω) = 2ωH (0,ω)/(2π).With the expressions (B.10)-(B.12) in hand we are at the

position to calculate Φγ and relate it with hc or H. Let us atthis point write the components of the vector-potential forn = z as Fourier integrals

Ax,y (z, t) =∫ +∞

−∞

dω

2π

∫d2zAx,y (z,ω, z)e−iωt . (B.14)

The by using expression (B.14) in Ex,y (z, t) = −∂tAx,y (z, t)and then putting all together in the expression of the Stokesparameter (B.9), we get

Φγ (z, t) =(Mx

gγ

)2∫ +∞

0

dω

2π× (B.15)

×

[sin2 (∆xz)

∆ 2x

+sin2 (∆yz)

∆ 2y

]ω2H (0,ω)

κ2 ,

where in obtaining the expression (B.16) we used alsoexpression (B.11). In addition, we may note that both ∆xand ∆y implicitly depend on ω through Mx and My and thusexplain the reason why ∆x,y do appear under the integral signin (B.16). On the other hand, Mx

gγ does not depend on ω

since we are considering relativistic particles with ω ' k.Now in order to relate the total energy density of the

formed electromagnetic radiation in the graviton-photon mix-ing, we may note from expression (B.16) that the energydensity contained in a logarithmic energy interval, is givenby

dΦgraphγ (z,ω; t)d (logω)

=(Mx

gγ

)2

[sin2 (∆xz)

∆ 2x

+sin2 (∆yz)

∆ 2y

]ω3H (0,ω)

2πκ2

=

(Mx

gγ

)2

2

[sin2 (∆xz)

∆ 2x

+sin2 (∆yz)

∆ 2y

]ω2h2

c (0,ω)

κ2 .

(B.16)

The expression dΦgraphγ (z,ω; t)/d (logω) in (B.16) is an

expression which tells us how much of the total energy den-sity is contained in a logarithmic energy interval. The ex-pression (B.16) can be written also in an equivalent form interms of the density parameter of photons Ωγ and the densityparameter of GWs, Ωgw. The density parameter of a speciei of particles at the energy ω is defined as

Ωi (z,ω; t)≡ 1ρc

dρi (z,ω; t)d (logω)

, (B.17)

where ρc is the critical energy density to close the universe,ρc = 6H2

0/κ2, where H0 is the Hubble parameter H0 = 100h0(km/s/Mpc) with h0 being a dimensionless parameter whichis determined experimentally. In addition since the energydensity (or energy flux Φ) of GWs is given by

ρgw (0, t) =〈hi j (0, t) hi j (0, t)〉

2κ2

=∫ +∞

0d (logω)

ω2h2c (ω)

κ2 , (B.18)

where we used (B.10)-(B.11), we have from (B.16), (B.17)and (B.18) that

h20Ω

graphγ (z,ω; t) =

(Mx

gγ

)2

2

[sin2 (∆xz)

∆ 2x

+sin2 (∆yz)

∆ 2y

]×

× h20Ωgw (0,ω; t) . (B.19)

13

The expressions (B.16) and (B.19) essentially give a com-plete description of how the graviton to photon mixing prop-agates in space in a transverse and constant magnetic fieldand uniform medium. Both (B.16) and (B.19) are equallyimportant and can be used in different contexts in order tocompare with experimental data. It is very important to an-alyze these expressions in some limiting cases. Considerthe case when in the laboratory there is a medium (gas orplasma) in addition to the magnetic field and when Mx =

My. The last condition essentially means that both propagat-ing transverse states of the electromagnetic radiation havethe same index of refraction. When Mx = My, we have that∆x = ∆y and consequently we get for (B.19) that

h20Ω

graphγ (z,ω) =

(Mx

gγ

)2[

sin2 (∆xz)∆ 2

x

]h2

0Ωgw (0,ω) .

(B.20)

Another important situation is when in the laboratorythere is not a medium but only a magnetic field in vacuum.In this case we have that ∆x = ∆y = Mx

gγ and the graviton-photon mixing is maximal or resonant

h20Ω

graphγ (z,ω) = [sin2 (Mx

gγ z)]h2

0Ωgw (0,ω) . (B.21)

Expression (B.21) tell us that the graviton-photon mixinghas an oscillatory behaviour with the distance z in the max-imum mixing case. For a long distance of travelling, thereare values of z which make sin2 (Mx

gγ z)= 1 and in that case

we have that all GWs are transformed into electromagneticwaves. However, since Mx

gγ is usually a very small quantity,one needs huge distances of travelling in order to achievethis situation. In many practical cases one has that Mx

gγ z 1

and we can approximate sin2 (Mxgγ z)'(Mx

gγ z)2 in the max-

imum mixing case.

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