Upper Limits on the Amplitude of Ultra-High-Frequency Gravitational Waves from Graviton to Photon Conversion

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

Upper limits on the amplitude of ultra-high-frequency gravitational waves 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, UK 2Birmingham University, School of Physics and Astronomy, Edgbaston Park Rd, Birmingham B15 2TT, UK 3Department of Physics, Novosibirsk State University, 2 Pirogova Street, Novosibirsk, 630090, Russia

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Abstract In this work, we present the first experimental up- and the abundance of sources facilitated the first detections. per limits on the presence of stochastic gravitational waves Coalescences of compact objects such as black holes and in a frequency band with frequencies above 1 THz. We ex- neutron stars have been detected, and spinning neutron stars, clude gravitational waves in the frequency bands from (2.7 − 14)supernovae× and stochastic signals are likely future sources. 1014 Hz and (5 − 12)×1018 Hz down to a characteristic am- Since in principle, GWs can be emitted at any frequency, min −26 min −28 plitude of hc ≈ 6 × 10 and hc ≈ 5 × 10 at 95% they are expected over many decades of frequency below confidence level, respectively. To obtain these results, we the audio band, but also above it. At lower frequencies, the used data from existing facilities that have been constructed space-based laser interferometer LISA is firmly planned to and operated with the aim of detecting WISPs (weakly in- cover the 0.1 − 10 mHz band [7,8], targeting, for example, teracting slim particles), pointing out that these facilities are black hole and white dwarf binaries. At even lower frequen- also sensitive to gravitational waves by graviton to photon cies in the nHz regime, the pulsar timing technique promises conversion in the presence of a magnetic field. The princi- to facilitate detections of GWs from supermassive black holes ple applies to all experiments of this kind, with prospects of [9–11]. constraining (or detecting), for example, gravitational waves Frequencies above 10 kHz have been much less in the from light primordial black hole evaporation in the early uni- focus of research and instrument development in the past, verse. but given the blooming of the field, it seems appropriate to not lose sight of this domain as technology progresses. One of the main reasons to look for such high frequencies 1 Introduction of GWs is that several mechanisms that generate very high- frequency GWs are expected to have occurred in the early With the first detections of gravitational waves (GWs) by universe just after the big bang. Therefore, the study of such the ground-based laser interferometers LIGO and VIRGO, frequency bands would give us a unique possibility to probe a new tool for astronomy, astrophysics and cosmology has the very early universe. However, the difficulty in probing been firmly established [1,2]. GWs are spacetime perturba- such frequency bands is explained by the fact that laser- arXiv:1908.00232v2 [gr-qc] 11 Jan 2020 tions predicted by the theory of general relativity that propa- interferometric detectors such as LIGO, VIRGO and LISA gate with the speed of light and can be predominantly char- work in the lower frequency part of the spectrum and their acterised by their frequency f and the dimensionless (char- working technology is not necessarily ideal for studying very- acteristic) amplitude h . Based on these two quantities and c high-frequency GWs. The characteristic amplitude of a stochas- the abundance of sources across the full gravitational-wave tic background of GWs h , for several models of GW gener- spectrum, as well as the availability of technology, it be- c ation, decreases as the frequency f increases. Consequently, comes clear that different parts of the gravitational-wave to study GWs with frequencies in the GHz regime or higher spectrum are more accessible than others. requires highly sensitive detectors in terms of the character- Current ground-based detectors are sensitive in the fre- istic amplitude h . quency band from about 10 Hz to 10 kHz [3–6] where the c One possible way to construct detectors for very-high- intersection of efforts in the development of the technology frequency GWs is to make use of the partial conversion of aCorresponding author: [email protected] GWs into electromagnetic waves in a magnetic field. In- 2 deed, as general relativity in conjunction with electrodynam- magnetic field. The generated electromagnetic waves can be ics predicts, the interaction of GWs with electromagnetic processed with standard electromagnetic techniques and can fields, in particular, static magnetic fields, generate propa- be detected, for example, by single-photon counting devices gating electromagnetic radiation at the same frequency as at a variety of wavelengths. Following the classification of the incident GW. In other words, gravitons mix with photons high frequency source in the paper [18], there appear to be in electromagnetic fields. This effect has been studied in the four kinds of potential GW sources in the VHF and UHF literature by several authors in the context of a static labo- bands: ratory magnetic field [12–15] and in astrophysical and cos- 1) Discrete sources [26]: the authors examined the ther- mological situations [16–20]. The effect of graviton-photon mal gravitational radiation from stars, mutual conversion of (also denoted as GRAPH) mixing is the inverse process of gravitons and photons in static fields and focusing the main photon-graviton mixing studied in Refs. [15, 21–24]. attention to the phenomenon of primordial black-hole evap- Based on the graviton-photon (or GRAPH) mixing, in oration, with a backgrounds at the high-frequency region. this work we point out that the existing experiments that are 2) Cosmological sources [27]: another mechanism which conceived for the detection of weakly interacting slim par- generates a very broadband energy density of GWs noise in ticles (WISPs) are also GW detectors in a sense mentioned the form of non-equilibrium of cosmic noise generated as a above: they provide a magnetic field region and detectors for consequence of the super-adiabatic amplification at the very electromagnetic radiation. In this work, we make use of ex- early universe. An upper bound on the energy density in- isting data of three such experiments to set first upper limits dependent of the spectrum of any cosmological GWs back- on ultra-high frequency GWs. As technology may progress ground prediction is given from the nucleosynthesis bound −5 further, future detectors based on the graviton to photon con- of ΩGW ≥ 10 [28]. version effect may be able to reach sensitivities for GW am- 3) Braneworld Kaluza-Klein (KK) mode radiation [29, plitudes near the nucleosynthesis constraint at the very high- 30]: the authors suppose the existence of the fifth dimension frequency regime. in higher-dimensional gravitational models of black holes This paper is organised as follows: In Sec.2, we give derives emission of the GWs. The GWs are generated due an overview of high-frequency GW sources and generating to orbital interactions of massive objects with black holes mechanisms as well as previously existing experimental up- situated on either our local, “visible", brane or the other, per limits. In Sec.3 we discuss the working mechanism of “shadow", brane which is required to stabilise the geome- current WISP detectors and the possibility to use them as try. These KK modes have frequencies which may lie in the GW detectors. In Sec.4, we consider the minimum GW UHF frequency with large amplitudes since the gravity is amplitude that can be detected by current WISP detectors. supposed to be very strong in bulk with a large number of In Sec.5, we discuss the prospects to detected ultra-high- modes. frequency GWs with current and future WISP detectors and 4) Plasma instabilities [30]: the authors have modelled in Sec.6 we conclude. In this work we use the metric with the behaviour of magnetised plasma example supernovae, signature ηµν = diag[1,−1,−1,−1] and work with the ra- active galaxies and the gamma-ray burst. They have devel- tionalised Lorentz-Heaviside natural units (kB = h¯ = c = oped coupled equations linking the high-frequencies elec- 2 ε0 = µ0 = 1) with e = 4πα if not otherwise specified. tromagnetic and gravitational wave modes. Circularly polarised electromagnetic waves travelling parallel to plasma background magnetic field would generate gravitational-wave 2 Overview of high frequency GW sources and with the same frequency of the electromagnetic wave. detection amplitude upper limits Except for the discrete sources and plasma instabilities, the GW radiation is usually emitted isotropically in all direc- The gravitational wave emission spectrum has been fully tions for several generation models, see below. Normally, a classified from 10−15 − 1015 Hz, as for example, more re- GW detector should be oriented toward the source in order cently in [25]. For the frequency region of interest to this to efficiently detect GWs, except for the majority of cosmo- paper, the high-frequency GW bands are given as: logical sources. Indeed, cosmological sources are expected – High-Frequency band (HF), (10 − 100 kHz), to generate a stochastic, isotropic, stationary, and Gaussian – Very High-Frequency band (VHF), (100 kHz − 1 THz), background of GWs that in principle can be searched for – Ultra High-Frequency band (UHF), (above 1 THz). with an arbitrary orientation of the detector. The upper limits on the GW amplitudes that we derive in this paper are lim- A viable detection scheme in the VHF and UHF bands ited to the cosmological sources since the detectors (see next (but in principle at all frequencies), is the graviton to pho- section), except for one experiment which pointed towards ton conversion effect. Based on this effect, it seems feasible the sun, cannot point deliberately to the emitting sources, so to search for GWs converted to electromagnetic waves in a their measurements are most sensitive to sources creating an 3 isotropic background of GWs. We list some possible sources So far, dedicated experiments to detect GWs in the VHF of GWs: region are based on two designs: polarisation measurement on a cavity/waveguide detector and cross-correlation measurement of two laser interferometers. The cavity/waveguide 1) Stochastic background of GWs prototype measured polarisation changes of the electromag- The stochastic background of GWs is assumed to be netic waves, which in principle can rotate under an incoming isotropic [28, 31, 32] and must exist at present as a re- GW, providing an upper limit on the existence of GWs back- sult of an amplification of vacuum fluctuations of grav- ground to a dimensionless amplitude of h ≤ 1.4 × 10−10 at itational field to other mechanisms that can take place c 100 MHz [40]. The two laser interferometer detectors with during or after inflation [33]. Inflationary processes and 0.75 m long arms have used a so-called synchronous recy- the hypothetical cosmic strings are potential candidates cling interferometer and provided an upper limit on the ex- of the GW background with some differences in the pre- istence of the GW background to a dimensionless amplitude dicted intensity and spectral features [28, 31, 32]. These of h ≤ 1.4 × 10−12 at 100 MHz [41, 42]. spectrum would have cutoff frequency approximately in c The most recent upper limit on stochastic VHF GWs has the region ν ∼ 1011 Hz. The prediction for the cutoff c been set from by a graviton-magnon detector which mea- frequency in some cosmic string models gives the cutoff sures resonance fluorescence of magnons [44]. They used shifted to 1013 − 1014 Hz [28, 31, 32]. The metric per- experimental results of the axion dark matter using magne- turbation at the cutoff frequency 1011 Hz corresponds to −32 tised ferromagnetic samples to derive the upper limits on an estimated strain amplitude of hc ∼ 10 . stochastic GWs with characteristic amplitude: hc ≤ 9.1 × 10−17 at 14 GHz and h ≤ 1.1 × 10−15 at 8.2 GHz. 2) GWs from primordial black holes c Another facility, the Fermilab Holometer, has performed In Ref. [34], the authors proposed the existence of Pri- measurement at slightly lower frequencies. The Fermilab mordial Black Hole (PBH) binaries and estimated the ra- Holometer [43] consists of separate, yet identical Michel- diated GW spectrum from the coalescence of such bina- son interferometers, with 39 m long arms. The upper limits ries. In addition, the mechanism of evaporation of small found within 3 , on the amplitude of GWs are, in the range mass black holes gives rise to the production of high σ h < 25 × 10−19 at 1 MHz down to a h < 2.4 × 10−19 at and ultra-high frequency GWs. An estimation of the ef- c c 13 MHz. ficiency of this emission channel which might compen- sate the deficit of high-frequency gravitons in the relic GW background has been thoroughly studied in [26]. A detailed calculation of the energy density of relic GWs 3 WISP search experiments and their relevance to UHF emitted by PBHs has been performed in [35]. The au- GWs thor’s analysed and calculated the energy density of GWs from PBH scattering in the classical and relativistic regimes, PBH binary systems, and PBH evaporation due to the Hawking radiation.

3) GWs from thermal activity of the sun A third class that does generate a stochastic, but not an isotropic background of GWs, which is relevant to this work, is the GW emission from the sun [36]. The high temperature of the sun in the proton-electron plasma pro- duces isotropic gravitational radiation noise due to thermal motion [37–39]. The emission comes to the detector from the direction of the sun, and the observations have the potential to set limits on this process. The frequency 15 of the collisions of νc ∼ 10 Hz determines the gravitational wave frequency, and the highest frequency cor- Fig. 1 Simplified schematic of the experimental setup aiming at the 18 detection of WISPs. In the upper panel left-hand side, the electromag- responds to the thermal limit at ωm = kT/h¯ ∼ 10 Hz. netic waves interacting with the magnetic field produce the hypotheti- Using the plasma parameters in the centre of the Sun cal WISPs, and at the right-hand side electromagnetic waves are pro- the estimation of the “thermal gravitational noise of the duced by the decay of WISPs in the constant magnetic field. Our work Sun” reaching the earth provides a stochastic flux at “op- is illustrated by the lower panel ignoring the transparent left-hand side. −41 On the right hand side, the photons detected could be due to the passage tical frequencies” of the order hc ∼ 10 [39]. of GWs propagating in the constant magnetic field. 4

The experiments ALPS [45], OSQAR [46] and CAST 9 T and 9 m length was used. The solar Axions with [36] have not been designed to detect GWs in the first place. expected energies in the keV range can convert into X- However, in this work their results are used to compute new rays in the constant magnetic field, and an X-ray detector upper limits on GW amplitudes and related parameters. The has been used to performed runs in the time period 2013 experiments performed by ALPS and OSQAR are usually - 2015 [49]. To increase the cross section, both the two called “light shining through the wall experiment” where parallel pipes which pass through the magnet have been the hypothetical WISPs, that are generated within the exper- used which provide an area of 2×14.5 cm2 focused into iment, mediate the “shining through the wall” process, and a Micromegas detector. The CAST detector mounted on decaying successively into photons. In contrast, the CAST the pointing system had a telescope with a focal length experiment searches directly for WISPs emitted from the of 1.5 m installed for the (0.5 − 10) keV energy range. core of the sun. Though all these experiments are not designed to detect gravitational waves, they provide a high sensitivity measurement of single photons generated in their 4 Minimal-detection of GW amplitude constant magnetic field which is the crucial ingredient for the detection of graviton to photon conversion. The main characteristics of the ALPS, OSQAR and CAST !" -:;<=->?@A-B-@BCBA-DDE experiments are: !# -:;<=-FA'>G-B-HIJDDE- $#"II -:;<=-FA'>G-BB-KIC#LEE-DDE 1) ALPS experiment at DESY !$ The ALPS (I) experiment has performed the last data $!% taking run in 2010, and the specific characteristics of the $!& experiment are found in Ref. [47]. A general schematic $!"

of the principle is shown in the upper panel of the Fig. '()*+(,-.//010.*12 $!# 1. The production of WISPs and their re-conversion into $!$ electromagnetic radiation is located in one single HERA "$$ &$$ %$$ $$$ superconducting magnet where an opaque wall in the 3)4.5.*6+7-8*,9 middle separates the two processes. The HERA dipole !" provides a magnetic ﬁeld of 5 T in a length of 8.8 m. The electromagnetic radiation, generated by the decay of the "!# WISPs in the magnetic ﬁeld, passed a lens of 25.4 mm "!$ diameter and focal length 40 mm. The lens focuses the light onto a ≈ 30 µm diameter beam spot on a CCD "!%

camera. "!& '()*+(,-.//010.*12 -->?@A-BCDE-FGH)2-I.+.1+JH 2) OSQAR experiment at CERN "!" & 3 % 4 $ 5 # 6 The OSQAR experiment has performed the last data tak- "! 7)8.9.*:+;-<*,= ing in 2015, and the specific characteristics of the experimental setup are found in [46, 48]. The OSQAR col- Fig. 2 Quantum efficiency as a function of the wavelength. Left-hand laboration has used two LHC superconducting dipole side panel: the quantum efficiency of the detectors using the method “light shining through a wall”; in the right-hand panel: the quantum magnets separated by an optical barrier, (for a concep- efficiency of the Micromegas X-ray detector used in the CAST experi- tual scheme see the upper panel of the Fig.1). The LHC ment. The detector bandwidth and their normalised quantum efficiency dipole magnets provide a constant magnetic field of 9 T, function are used to compute the upper limits un GWs detectors. along a length of 14.3 m. To focus the generated photons of the beam onto the CCD, an optical lens of 25.4 mm diameter and a focal length of 100 mm was used, installed In this section, we show how we compute upper limits in front of the detector similar to the Fig.1. Data acqui- on the GW dimensionless amplitude hc based on the charac- sition has been performed in two runs with two different teristics of the experiments described above that are sensi- CCD’s having different quantum efficiencies. tive to an isotropic background of GWs from cosmological sources and to the thermal activity in the sun. We ignore 3) CAST experiment at CERN the generation of WISPs (Fig.1: lower panel) and focus on The CERN Axion Solar Telescope (CAST) experiment the second half of the magnetic field for the case of ALPS has the aim to detect or set upper limits on the flux of and OSQAR experiments, and, for the CAST experiment we the hypothetical low-mass WISPs produced by the Sun. consider the full magnet region. These experiments measure A refurbished CERN superconducting dipole magnet of a number of photons per unit time with their CCD detectors, 5

2 εγ (ω) Nexp (mHz) A (m ) B (T) L (m) ∆ f (Hz) ALPS I Fig2 0.61 0.5 × 10−3 5 9 9 × 1014 OSQAR I Fig2 1.76 0.5 × 10−3 9 14.3 5 × 1014 OSQAR II Fig2 1.14 0.5 × 10−3 9 14.3 1 × 1015 CAST Fig2 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 and transverse magnetic ﬁeld.

namely Nexp in an energy band ∆ω with efficiency εγ and where we took z = L with L being the spatial extension of in a cross-section A. In what follows we assume that in the the external magnetic field. All the three experiments listed CCD, the background dark current fluctuation is a stochastic in the section above their upper limits are compatible to process with uniform probability distribution and stationary the background fluctuation of the detector which allows us in ω. In this case, the energy flux of photons generated in an to express the relation: N (ω,t) = Nexp/∆ω, where ∆ω = energy bandwidth [ωi,ω f ] is given by: ω f − ωi. Finally, by putting the units in explicitly, we get the following expression for the minimum detectable GW Z ω f 1 N (ω,t) ω amplitude: ΦCCD z,ω ;t = dω (1) γ f A(z) ε (ω) ωi γ s 4N where N (ω,t) is the number of photons per unit of time min exp −16 hc (0,ω) ' 2 2 ' 1.6 × 10 × and energy. Now, we have to compare the measured en- AB L εγ (ω) ∆ω s ergy flux of photons with the intrinsic energy flux of pho- 2 1 T 1 m Nexp 1 m 1 Hz 1 tons generated in the graviton to photon conversion in pres- × (4) B L 1 Hz A ∆ f ε ence of an isotropic background of GWs. The analytical γ treatment for an isotropic background of GWs converted where ω = 2π f with f being the frequency. In order min into electromagnetic waves, is described in detail in Ap- to compute the minimum detectable GW amplitude, hc , pendix2,3. In Appendix3, different useful quantities are we have extracted the following quantities from the ALPS, calculated, for stochastic GWs propagating in a transverse OSQAR and CAST experiments: and constant magnetic field. Since all the experiments oper- – Nexp the total detected number of photons per second in ated under vacuum condition and the propagation distance the bandwidth ∆ω, z is small with respect to the oscillation length of the parti- – A cross-section of the detector, cles, we can safely take ∆x,y z 1. The variable ∆x,y defined – B magnetic field amplitude, in the Eq. A.7 is a function of the magnetic field, the mag- – L distance extension of the magnetic field, nitude of the photon and graviton wave-vectors and Newto- – εγ (ω) quantum efficiency of the detector, nian constant. Then the energy flux of photons generated in – ∆ f operation bandwidth of the CCD. the magnetic field of length z given by expression (B.16), in These quantities permit to compute the equivalent mini- the same energy bandwidth [ωi,ω f ] becomes: min mum amplitude hc of a stochastic GW background which would generate photons through graviton to photon conver- Z ω " 2 2 # graph x 2 f sin (∆xz) sin (∆yz) sion, equivalent number of background photons that the CCD Φγ z,ω f ;t = Mgγ 2 + 2 × ωi ∆x ∆y has read. The data accounted for the photon detection in the h2 (0,ω) ω constant magnetic field for the three experiments ALPS, OS- × c dω 2 κ2 QAR and CAST, exclude the detection of physical signals Z ω 2 2 2 with fluxes bigger or comparable to the background count f B z hc (0,ω) ω ' (2) of the CCD detectors at 95% confidence level which allows ω 4 i putting upper limits on the minimal detectable GW ampli- Comparing the energy fluxes in expressions (1) and (2), min tude hc at the same confidence level. and requiring that for detection, the energy flux in (2) must The extracted quantities used to compute the upper lim- be bigger or equal to the energy flux in (1), we get min its on hc are summarised in Table1. These experiments at- tempting to detect WISPs have used subsequently improved

2 4N (ω,t) CCDs during different run phases which is taken into ac- hc (0,ω) ≥ 2 2 , (3) B L A(L) εγ (ω) count in the analysis. The quantum efﬁciencies in Table1 are 6 represented graphically in Fig.2 as a function of the wavelength. We have taken into account that Nexp is normalised to the quantum efﬁciency the working frequency of the WISPs experiments, and the range of the expected photons is im- posed by the sensitive frequency range of the CCD. The cross-section reported in Table1 has been computed, for the ALPS and OSQAR experiments, considering the area enclosed by the diameter of the lens [45, 46]. Instead, the CAST experiment uses the whole cross-section of the two beam pipes.

Fig. 4 In the upper panel conceptual scheme of the experimental setup ALPS IIc and a possible follow-up named JURA where we note the addition of the FP cavity in the right-hand side. Our prediction for the min sensitivity of the minimal amplitude of hc used the right-hand side process, where the photons generated via graviton to photon conversion are resonantly enhanced in the FP cavity. In the lower panel, the FP resonator concept is described where Egraph is the electric field generated from the graviton to photon conversion in the cavity, Ecirc is circulating electric field accumulated inside the resonator after transmission losses on both mirrors, Etrans is transmitted electric field through the mirrors and L the length of the cavity.

min Fig. 3 Plots of the minimum detectable GW amplitude hc as a function of the frequency f , deduced from the measured data of the denoted experiments. graviton-photon mixing process, in two frequency regions infrared and X-rays:

Using the data of the Table1 and expression (4) it is possible to produce an upper limit plot of the GW ampli- 5.1 Infrared tude, see Fig.3, due to the conversion of GWs into photons. The region above each curve is the excluded region. To our One of the most important changes that ALPS IIc, with re- knowledge, these are the first experimental upper limits in spect to the ALPS I and OSQAR, is the use of a Fabry-Perot these frequency regions. (FP) cavity to enhance the decay processes of WISPs into photons, see Fig.4. The FP cavity will allow just a range of 5 Prospects on detecting Ultra-High Frequency GWs electromagnetic waves to be built up resonantly, within the from primordial black holes cavity bandwidth: ∆ωc = ∆ωFSR/F where F = π/(1 − R) is the cavity finesse, ∆ωFSR = π/L is the cavity free spectral Graviton to photon conversion maybe a useful path towards range, and R is the reflectance of the mirrors. The FP cavity the detection of UHF GWs. The actual technology has made enhances the decay rate of WISPs to photons [54]. This is further progress in the detection of single photons and new an essential aspect because it will also account for the tran- facilities are intended for WISP search, using higher values sition of gravitons into photons [55]. Stochastic broadband of B and L in order to achieve higher sensitivities. One fa- GWs converted into electromagnetic radiation would excite cility which plans to do so is the ALPS IIc proposal which several resonances of the cavity at frequencies ωc ±n∆ωFSR, consists of two 120 m long strings of 12 HERA magnets where ∆ωc is the cavity frequency bandwidth, and n is an in- each, with a magnetic field of 5.3 T. The scheme of gen- teger number with its range depending on the coating of the eration and conversion of the WISPs is still the same, ex- mirrors. To calculate the response of the FP resonator, we pected an optical resonator is added to the reconversion re- use of the circulating field approach [56, 57], as displayed gion. A follow-up of the CAST telescope is the proposed in the lower panel of Fig.4. We assume a steady state ap- International Axion Observatory (IAXO). Tab.2 represent proximation to derive the circulating electric field Ecirc in- the detector parameters of ALPS IIc [50], a possible follow- side the cavity and the mirrors have the same reflectance R up named JURA [51], and of the IAXO proposal [53]. and transmittance T. Defining the phase shift after one round Since the working frequency of the detectors is differ- trip 2φ (ω) = 2ωL, the accumulated electric field Ecirc after ent we compute the sensitivity to detected GWs, with the a large number of reflections (which can be assumed infinite 7

2 εγ Ndark (Hz) A (m ) 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 the graviton to photon conversion in their constant and transverse magnetic field: εγ is the efficiency photodetector at 1064 nm, Ndark correspond to the 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 the magnetic field length and F is the finesse of the cavity. in the calculations below) of the electric field Egraph gener- flux generated in the graviton to photon conversion without ated in the GRAPH mixing is: the cavity

circ Ecirc (z,t) = Egraph (z,t) dΦγ (L,ω;t) 1 x,y x,y Γ (φ) = = . circ graph 2 2 2 dΦγ (L,ω;t) (1 − R) + 4R sin [φ (ω,L)] × 1 + Re−i2φ(ω,L) + Re−i2φ(ω,L) + ··· (8) ∞ n graph −i2φ(ω,L) For a given length L and for frequencies matching the = Ex,y (z,t) ∑ Re n=0 cavity resonance, or φ (ω,L) = nπ where n is a positive 1 integer, the internal gain enhancement factor is maximum: = Egraph (z,t) . (5) x,y −i2φ(ω,L) 2 1 − Re Γcirc (nπ) = (F /π) . In the same way, we derive the transmitted gain on both sides of the cavity, which expression is The circulating flux, in a time t > τ where τ = F L/π is the given by: circ charge time of the cavity, at a distance z = L is Φγ (L,t) ≡ circ 2 circ 2 graph h|E (z,t)| i + h|E (z,t)| i. By expanding Ex,y (z,t) x y trans as a Fourier integral and doing the same steps to derive the dΦγ (L,ω;t) 1 − R Γ (φ) = = . trans graph 2 2 flux as shown in Appendix (Appendix A) from Eq. B.9 to dΦγ (L,ω;t) (1 − R) + 4R sin [φ (ω,L)] Eq. B.16, then the circulating flux simplifies to the follow- (9) ing expression: Unlike before, the transmitted flux from the cavity will exhibit transmitted light peaks which gain factor, for fre- Z +∞ 1 Φcirc (L,t) ' quencies matching the cavity resonance condition, reduces γ 2 2 0 (1 − R) + 4R sin [φ (ω,L)] to: Γtrans (nπ) = (F /π). Now we can write explicitly the B2 L2 h2 (0,ω) ω equation of the flux produced from graviton to photon con- × c dω. (6) 4 version transmitted from the cavity in a energy bandwidth [ωi,ω f ]: where we have consider that the propagation distance z = L is small with respect to the oscillation length of the particles, Z ω f trans graph and we can safely take ∆x,yz 1. Taking the differential of Φγ L,ω f ;t = Γtrans (φ) dΦγ (L,ω;t) (10) ω the circulating flux in (6) for a given energy interval [ωi,ω], i Z ω 2 2 2 we find the the following relation f BxL hc (0,ω) = Γtrans (φ) ω dω. ωi 4 A photo-detector placed at the transmission line of the dΦ graph (L,ω;t) dΦ circ (L,ω;t) = γ . (7) cavity will measures an energy flux, within its bandwidth γ 2 2 (1 − R) + 4R sin [φ (ω,L)] ∆ω defined in Eq.1. According to the previous discussion, a cavity of length L will transmit light peaks for frequen- graph ∗ where dΦγ (L,ω;t) is the differential of Eq. B.16 in a cies ω = nπ/L, and such frequency should be in the inter- given energy interval [ωi,ω], which correspond to the flux val bandwidth ∆ω. Reminding that the flux in expressions of photons generated through the graviton to photon conver- (10) is calculated for a stochastic process and taking into ac- sion without the cavity. Now we can derive the gain factor, count the bandwidth of the photodetectors of ALPS IIc and namely Γcirc (φ), as the differential of the circulating energy JURA such condition is satisfied. Now, considering the en- flux in the resonator relative to the differential of the energy ergy flux of a photodetector limited by the dark count rate 8

Ndark, where N (ω,t) = Ndark/∆ω, and comparing with the energy ﬂuxes in expressions (10), and solving for hc (0,ω) in SI units, at the maximum transmission ω∗ = nπ/L, becomes:

1 T1 m hmin (0,ω∗) ' 2.8 × 10−16 × (11) c B L s 1 N 1 m2 1 Hz 1 × dark . F 1 Hz A ∆ f εγ

From the above equation, we can observe that to compute the sensitivity in amplitude hc (0,ω), with respect to the case without cavity Eq.4, in addition, we need to know the ﬁ- min Fig. 5 Graphical representation of the amplitude hc as a function −3 4 nesse factor F . The minimum detectable GW amplitude, of frequency for the: PBH evaporation of masses mBH (10 , 10 g, considering the photodetector background rate in one sec- 108 g), the upper limits of graviton-photon conversion data in Fig.3, ond, limited by the dark counting rate, a frequency band- the estimated amplitude sensitivity for the ALPS IIc and JURA (red f ≈ × 14 and grey lines) at the infrared region using their detection scheme. width ∆ 4 10 Hz [52], and the relevant character- The dotted blue line is the estimated sensitivity for the solar telescope istic of Tab.5, the sensitivity for the minimal amplitude is: JAXO successors of CAST experiment. The two dashed lines represent ALPS IIc −30 JURA −32 hc ≈ 2.8×10 and hc ≈ 2×10 , which is two the nucleosynthesis amplitude upper limit and the predicted amplitude orders of magnitude better than in the case without cavity. from the thermal GW emitted from the sun. Here for simplicity we have assumed a value of the PBH density parameter at their production −7 times equal to Ωp ' 10 . Both amplitude, hc, and frequency axes are in Log10 units. 5.2 X-rays

The core element of IAXO will be a superconducting toroidal graviton-photon mixing in cosmic magnetic ﬁelds [19]. It magnet, and the detector will use a large magnetic ﬁeld dis- is especially the evaporation of GWs due to Hawking radi- tributed over a large volume to convert solar axions into de- ation which generates a substantial amount of GWs in the tectable X-ray photons. The central component of IAXO is frequency regime compatible with the ALPS IIc and JURA a superconducting toroidal magnet of 25 m length and 5.2 m working frequency. The spectral density parameter of GWs diameter. Each toroid is assembled from eight coils, gen- at present is given in Ref. [35] and it reads erating 2.5 T in eight bores of 600 mm diameter. The X-ray detector would be an enhanced Micromegas design to match − 2 2 4 the softer 1 10 keV spectrum. The X-rays are then focused 2 −57 Neff 1g f h0Ωgw ( f ;t0) = 1.36 × 10 at a focal plane in each of the optics read by pixelised planes 100 mBH 1 Hz with a dark current background level of N = 104 [53]. For √ dark Z zmax 1 + z × dz the process of graviton to photon conversion, the sensitivity 2π f (1+z) (12) IAXO −29 0 T on the minimal amplitude of hc ≈ 1 × 10 . e 0 − 1

where T0 is the PBH temperature redshifted to the present time, m is the PBH mass, N is number of particle species 5.3 Ultra-High Frequency GWs from Primordial Black BH eff with masses smaller than the BH temperature T , and z Holes BH max is the maximum redshift. The PBH temperature at present In order to describe the potential of ALPS IIc and JURA on and the maximum redshift are given by [35] probing the GW background at very high frequencies an ex- plicit example of a GW source can be considered. One of 13 T0 = 1.43 × 10 the most promising sources of VHF and UHF GWs in the s 100 100 m frequency range of interest regarding ALPS IIc is the evap- × BH (Hz) oration of very light PBHs that would have been formed just Neff gS (T (τBH)) 1g after the big bang. As shown in detail in Ref. [35], these 2/3 4/3 32170 mBH 1/3 black holes would emit GWs by different mechanism as zmax = Ω − 1, (13) N m P scattering, binary black hole, and evaporation by hawking eff Pl radiation which in principle could contribute to the spec- where ΩP is the density parameter of PBH at their pro- trum of cosmic electromagnetic X-ray background due to duction time and mPl is the Planck mass and gS (T (τBH)) 9 is the number of species contributing to the entropy of the 6 Conclusions primeval plasma at temperature T (τBH) at the evaporating life-time τBH [58, 59] . The number of PBHs that take part A broad spectrum of the emission of GWs is predicted to in this process is included in the density parameter ΩP, see exist in the universe, and some sources could generate GWs Ref. [35] for details. with frequencies higher than THz. The predicted conversion Now in order to extract the characteristic amplitude due of gravitons into photons, due to the propagation in a static to the stochastic background of GWs due to PBH evapora- magnetic field, is not out of reach for current technologies. tion we need an expression which connects hc to the density The technique of various detectors having the aim of count- 2 ing single photons, at a narrow wavelength in a static mag- parameter h0Ωgw. By using the definition of the density parameter in (B.17) and the expression for the energy density netic field has been developed as detectors for the measure- of GWs in (B.18), we get ment of WISPs, a dark matter candidate, decaying to photons in the transverse magnetic field. Though the WISPs detection setups were not particularly designed to detect GW q 1Hz −18 2 conversion, the generation of electromagnetic radiation as hc (0, f ) ' 1.3 × 10 h0Ωgw ( f ;t0) . (14) f GWs propagate in a static magnetic field provides the possibility of using the published data, currently for the ALPS, Now by using expression (12) into expression (14), we get OSQAR, CAST collaborations, to set the first upper limits the following expression for the characteristic amplitude of on the amplitude of isotropic Ultra-High-Frequency GWs. GWs due to PBH evaporation We exclude the detection of GWs down to an amplitude min −26 14 min hc ≈ 6 × 10 at (2.7 − 14) × 10 Hz and hc ≈ 5 × 10−28 at (5 − 12) × 1018 Hz at 95% confidence level. Many −47 Neff f 1 g theoretical potential ultra-high-frequency GW sources could hc (0, f ) = 4.8 × 10 100 1 Hz mBH be searched for using such similar experimental setups. The v √ next generation experiments, such as the ALPS IIc and JURA uZ zm 1 + z × u z. facilities, or similar experiments using high static magnetic t 2π f (1+z) d (15) 0 e T0 − 1 fields, are potential detectors for the graviton to photon conversion as well. The predicted ALPS IIc data taking or even- tually JURA will be able to produce more stringent upper In order to have an overview of the upper limit derived limits on the amplitude of the stochastic wave background and the perspective to detect UHF GWs, in Fig.5 is shown: of GWs generated from PBH evaporation models. the upper limits derived in the previous section, the estimated minimum detectable amplitude for the ALPS IIc and JURA considering the photo detector dark count rate, the maximum GWs amplitude generated in the production cav- 7 Acknowledgement −7 ity, the estimated GWs amplitude for Neff = 100, Ωp = 10 and BH masses (10−3, 104, 108) g, the prediction of the GWs We are grateful to Prof. Bernard Schutz for helpful com- ments on the manuscript, and we recognise the support from from the sun and the nucleosynthesis upper limit, Ωgw ≈ 10−5 [28]. The sensitivity to GW detection for ALPS IIc and the Leverhulme Emeritus Fellowship EM 2017-100. JURA 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.√ In Appendix A: Propagation of GWs in a constant such a situation, the signal-to-noise ratio improves as T. magnetic field An alternative method, without the signal modulation, is to correlate the data stream from two different photodetectors. Here we review the graviton-photon mixing in a static ex- The electromagnetic wave is generated inside the FP cavity, ternal magnetic field, which can result in the conversion of and the transmission is on both mirrors of the cavity which gravitational waves into photons, the process which we de- are placed on the sides of the magnet. Having two photode- scribe in the main paper operating in the detectors designed tectors mounted on both sides of the magnet and correlating to detect WISPs. In this section we closely follow Ref. [20]. their signal in time would allow lowering the statistical noise To start with it is necessary first to write the total Lagrangian of the detector. So, the time integration would let to a further density L of the graviton-photon system. In our case, it is min gain in the sensitivity amplitude hc . Ultimately the ALPS given by the sum of the following terms IIc would be able to explore new amplitude regions of GWs which target source could be the predicted GWs generated from the evaporation of PBHs. L = Lgr + Lem, (A.1) 10 where Lgr and Lem are respectively the Lagrangian den- where the elements of the mass mixing matrix M are sities of the gravitational and electromagnetic fields. These given by: terms are respectively given by x ¯ Mgγ = κ kBx/(ω + k), y ¯ 1 Mgγ = κ kBy/(ω + k), Lgr = R, (A.2) κ2 Mx = −Πxx/(ω + k), 1 1 Z L = − F F µν − d4x0A (x)Π µν x,x0A x0, My = −Πyy/(ω + k), em 4 µν 2 µ ν MCF = −Πxy/(ω + k). where R is the Ricci scalar, g is the metric determinant, Fµν Here MCF is a term which includes a combination of the 2 is the total electromagnetic field tensor, κ ≡ 16πGN with Cotton-Mouton and Faraday effects in a plasma and which µν GN being the Newtonian constant and Π is the photon depends on the magnetic field direction with respect to the polarisation tensor in a magnetised medium. photon propagation. Here ω is the total energy of the fields, The equations of motion from (A.1) and (A.2) for the namely ω = ωgr = ωγ . In this work all the particles partic- propagating electromagnetic and GW fields components, Aµ ipating in the mixing are assumed to be relativistic, namely and hi j, in an external magnetic field are given by [20] ω + k ' 2k with k being the magnitude of the photon and graviton wave-vectors. The system of differential equations (A.4) does not have ∇2A0 = 0, closed solutions in the case when the mixing occurs in ar- Z bitrary matter that evolves in space and time, namely in the i 4 0 i j 0 0 i µ A + d x Π x,x A j x + ∂ ∂µ A = case when the system of differential equations is with variable coefficients such as in cosmological scenarios. How- = κ ∂ [hµβ F¯ i − hiβ F¯ µ ], µ β β ever, in the case of mixing in a laboratory magnetic field, hi j = −κ (BiB¯ j + B¯iB j + B¯iB¯ j), (A.3) the system (A.4) can be simplified by considering a specific propagation of GWs with respect to the magnetic field direc- where Aµ = (φ,A) is the incident electromagnetic vector- tion and by considering the propagation in the magnetic field potential with magnetic field components Bi and B¯i are the without gas or a plasma present (see below). For example, components of the external magnetic field vector B¯ . In ob- first one can choose the magnetic field to be transverse to the taining the system (A.3) we made use of the TT-gauge con- photon direction of propagation such as B¯ = (B¯x,0,0) where y i j we have M = 0 and M = 0. Second, if there is a gas or a ditions for the GWs tensor h0µ = 0,hi = 0 and ∂ hi j = 0. gγ CF As shown in details in Ref. [20], the system (A.3) can be plasma in addition to the external magnetic field, usually we linearized by making use of the slowly varying envelope ap- have that Mx 6= My which essentially means that the trans- proximation (SVEA) which is a WKB-like approximation verse photon states have different indexes of refraction. In for a slowly varying external magnetic field of spacetime co- the case when one is able to achieve almost a pure vacuum ordinates. In this approximation, and for propagation along in the laboratory, the contribution of the gas or plasma to the observer’sz ˆ axis in a given cartesian coordinate system, the index of refraction can be safely neglected while there equations (A.3) can be written as [20] is also still present a contribution to the index of refraction due to the vacuum polarisation in the magnetic field. How- ever, the latter contribution is completely negligible because the magnitude of the laboratory magnetic field is usually few (ω + i∂z)Ψ (z,ω,zˆ)I + M (z,ω)Ψ (z,ω,zˆ) = 0, (A.4) Teslas and consequently is too small to have any appreciable effect on Mx and My. As discussed above, let us consider first the case when where in (A.4) I is the unit matrix, Ψ (z,ω,zˆ) = (h ,h ,A ,A )T × + x y the external magnetic field is completely transverse with re- is a four component field with h being the usual GW ×,+ spect to the photon direction of propagation where My = 0 cross and plus polarisation states and A are the usual prop- gγ x,y and M = 0. The fact that M = 0 is because B¯ = 0,B¯ = 0 agating transverse photon states. In (A.4) M (z,ω) is the CF CF y z and consequently the term corresponding to the Faraday ef- mass mixing matrix which is given by fect is absent since this effect occurs only when the magnetic field has a longitudinal component along the electromagnetic wave direction of propagation. In addition, in MCF  x y  0 0 −iMgγ iMgγ term it is also zero the term corresponding to the Cotton- y x  0 0 iMgγ iMgγ  Mouton effect in plasma because by convention we have M (z,ω) =  x y , (A.5) ¯  iMgγ −iMgγ Mx MCF  chosen that By = 0. After these considerations several terms y x ∗ −iMgγ −iMgγ MCF My in the mixing matrix M (z,ω) are zero and in the case of the 11 medium in the laboratory being homogeneous in space (in- to simply replace h×,+ (0,t) = h˜×,+ (0,t)/κ, where h˜×,+ are cluding the magnetic field), then the mass mixing matrix M now dimensionless amplitudes. does not depend on the coordinate z. In this case the commu- Consider the case when GWs enter a region of constant tator [M (z,ω),M (z0,ω)] = 0 and the solution of the system external magnetic field in the laboratory and that initially (A.4) is given by taking the exponential of M (z,ω). Conse- there are no electromagnetic waves. quently, we obtain the following solutions for the fields The assumption of no initial electromagnetic waves means that Ax (0,ω,zˆ) = Ay (0,ω,zˆ)=0 in the solutions (A.6). There- Mx sin(∆xz) i(ω+ Mx )z fore, the expressions for the electromagnetic field compo- h× (z,ω,zˆ) = cos(∆xz) − i e 2 h× (0,ω,zˆ) 2∆x nents, in the graviton to photon mixing for a transverse prop- x M sin(∆xz) gγ i(ω+Mx/2)z agation with respect to magnetic field, are given by + e Ax (0,ω,zˆ), ∆x M z My y sin(∆y ) i ω+ 2 z h+ (z,ω,zˆ) = cos(∆yz) − i e h+ (0,ω,zˆ) x 2∆y M sin(∆xz) gγ i(ω+Mx/2)z x Ax (z,ω,zˆ) = − e h˜× (0,ω,zˆ), M sin(∆yz) gγ i(ω+My/2)z κ∆x − e Ay (0,ω,zˆ), ∆y x Mgγ sin(∆yz) x i(ω+My/2)z ˜ M sin( z) M Ay (z,ω,zˆ) = e h+ (0,ω,zˆ) (B.8) gγ ∆x i(ω+ x )z Ax (z,ω,zˆ) = − e 2 h× (0,ω,zˆ) κ∆y ∆x Mx sin(∆xz) i(ω+Mx/2)z + cos(∆xz) + i e Ax (0,ω,zˆ), The expressions for the electromagnetic field compo- 2∆x x nents in (B.8), even though very important, are not much M sin(∆yz) My gγ i ω+ 2 z Ay (z,ω,zˆ) = e h (0,ω,zˆ) useful for practical purposes since we usually detect electro- ∆ + y magnetic radiations through their transported energy to the M z My y sin(∆y ) i ω+ 2 z + cos(∆yz) + i e Ay (0,ω,zˆ),(A.6) detector. For this reason is better to work with the Stokes 2∆y parameter Iγ (z,t) ≡ Φγ (z,t) of the electromagnetic radiation generated in the graviton to photon mixing and which where h×,+ (0,ω,zˆ) and Ax,y (0,ω,zˆ) are respectively the quantifies the energy flux (or energy density) of the elec- GW and electromagnetic wave initial amplitudes at the ori- tromagnetic radiation. The Stokes Φγ parameter, at a given gin of the coordinate system z = 0, namely the amplitudes point in space z, is defined as before entering the region where the magnetic field is located. In addition, we have defined q 2 2 2 x 2 Φγ (z,t) ≡ h|Ex (z,t)| i + h|Ey (z,t)| i, (B.9) Mx,y + 4 Mgγ ∆ ≡ . (A.7) x,y 2 where Ex and Ey are the components of the electric field of electromagnetic radiation and the symbol h(·)i denotes Appendix B: Electromagnetic energy fluxes generated temporal average of quantities over many oscillation peri- with laboratory graviton-photon mixing ods of electromagnetic radiation. The components of the electric field E are related to that of the vector-potential 0 In the previous appendix we found the solutions of the lin- A through the relation Ex,y (z,t) = −∇A (z,t) − ∂t Ax,y (z,t). earised equations of motion (A.4) for the GW fields h×,+ For a globally neutral medium (if there is one except the 0 and for the electromagnetic wave fields Ax,y. In this sec- magnetic field) in the laboratory we can choose A = 0 from tion we use these solutions to find the energy flux of the the first equation in (A.3) and after we simply get Ex,y (z,t) = electromagnetic radiation generated in the laboratory for the −∂t Ax,y (z,t). graviton-photon mixing (in equations abbreviated as graph). In order to calculate the energy density of the electro- Before proceeding further is important to stress that in (A.6), magnetic radiation and related quantities in the graviton to the GW amplitudes h×,+ are not dimensionless, as they are photon mixing, we need first to make some assumptions on commonly defined in some textbooks, but have energy di- the GW signal which interacts with the magnetic field in the mension units. This is due to the fact that in Ref. [20] the laboratory. In this work we concentrate on our study on a metric tensor is expanded as gµν = ηµν + κhµν where the stochastic background of GWs with astrophysical or cosmo- GW tensor hµν has the physical dimensions of an energy. logical origin. It is rather natural to assume that the stochas- But in many cases one also writes gµν = ηµν + hµν where tic background of GWs is isotropic, unpolarized and station- in this case hµν is a dimensionless quantity. Since the latter ary [28, 32]. In order to make more clear what these assump- case is quite common in the theory of GWs and because we tions mean, we write the GW amplitude tensor h˜i j (z,t) at want to conform to the literature, in expression (A.6) one has z = 0 as a Fourier integral 12

Z +∞ Z +∞ d Z x 2 dω ˜ ω 2 ˜ λ −iωt Φ (z,t) = M × (B.15) hi j (0,t) = d nˆ hλ (0,ω,nˆ)ei j (nˆ)e , γ gγ ∑ 2 0 2π λ=×,+ −∞ 2π S " 2 2 # 2 (i, j = x,y,z), (B.10) sin (∆xz) sin (∆yz) ω H (0,ω) × 2 + 2 2 , ∆x ∆y κ wheren ˆ is a unit vector on the two sphere S2 which denotes an arbitrary direction of propagation of the GW, d2nˆ = where in obtaining the expression (B.16) we used also d (cosθ)dφ, λ is the polarisation index of the GW with the expression (B.11). In addition, we may note that both ∆x λ and ∆y implicitly depend on ω through Mx and My and thus usual cross and plus polarisation states and ei j (nˆ) is the GW λ i j explain the reason why ∆x,y do appear under the integral sign polarisation tensor which has the property ei j (nˆ)eλ0 (nˆ) = x in (B.16). On the other hand, Mgγ does not depend on ω 2δ 0 . The assumptions that the stochastic background is λλ since we are considering relativistic particles with ω ' k. isotropic, unpolarised and stationary means that the ensem- Now in order to relate the total energy density of the ble average of the Fourier amplitudes satisfies formed electromagnetic radiation in the graviton-photon mixing, we may note from expression (B.16) that the energy 2 0 ∗ 0 0 0 δ (nˆ,nˆ ) density contained in a logarithmic energy interval, is given hh˜ (0,ω,nˆ)h˜ 0 0,ω ,nˆ i = 2πδ ω − ω δ 0 λ λ 4π λλ by H (0,ω) × , (B.11) graph " 2 2 # 3 2 dΦγ (z,ω;t) 2 sin (∆xz) sin (∆yz) ω H (0,ω) = Mx + d (logω) gγ ∆ 2 ∆ 2 2πκ2 where H (0,ω) is defined as the spectral density of the x y x 2 " 2 2 # 2 2 stochastic background of GW and it has the physical dimen- Mgγ sin (∆xz) sin (∆yz) ω h (0,ω) = + c . −1 2 0 0 0 2 2 2 sions of Hz and δ (nˆ,nˆ ) = δ (φ − φ )δ (cosθ − cosθ ) 2 ∆x ∆y κ is the covariant Delta function on the two sphere. One can (B.16) check by using (B.10) and (B.11), that the ensemble average graph i j The expression dΦγ (z,ω;t)/d (logω) in (B.16) is an hh˜i j (0,t)h˜ (0,t)i, is given by expression which tells us how much of the total energy density is contained in a logarithmic energy interval. The ex- Z +∞ i j dω pression (B.16) can be written also in an equivalent form in hh˜i j (0,t)h˜ (0,t)i = 2 H (0,ω) (B.12) −∞ 2π terms of the density parameter of photons Ωγ and the density Z +∞ d (logω) parameter of GWs, Ω . The density parameter of a specie = 4 ω H (0,ω) gw 0 2π i of particles at the energy ω is defined as Z +∞ 2 ≡ 2 d (logω)hc (0,ω), (B.13) 0 1 dρi (z,ω;t) Ωi (z,ω;t) ≡ , (B.17) where the last equality in (B.12) defines the characteristic ρc d (logω) amplitude, h (dimensionless), of a stochastic background c where ρc is the critical energy density to close the universe, of GWs. In obtaining (B.12) we used the fact that for an un- 2 2 ρc = 6H0 /κ , where H0 is the Hubble parameter H0 = 100h0 polarised stochastic background, we have that on average, (km/s/Mpc) with h0 being a dimensionless parameter which ˜ 2 ˜ 2 h|h× (0,ω)| i = h|h+ (0,ω)| i 6= 0 while the ensemble aver- is determined experimentally. In addition since the energy age of the mixed terms vanish identically. We may observe density (or energy flux Φ) of GWs is given by that by comparing the two last equalities in (B.12) we get 2 hc (0,ω) = 2ωH (0,ω)/(2π). i j hh˙i j (0,t)h˙ (0,t)i With the expressions (B.10)-(B.12) in hand we are at the ρ (0,t) = gw 2κ2 position to calculate Φγ and relate it with hc or H. Let us at Z +∞ 2 2 this point write the components of the vector-potential for ω hc (ω) = d (logω) 2 , (B.18) nˆ = zˆ as Fourier integrals 0 κ where we used (B.10)-(B.11), we have from (B.16), (B.17) Z +∞ Z and (B.18) that dω 2 −iωt Ax,y (z,t) = d zAˆ x,y (z,ω,zˆ)e . (B.14) −∞ 2π x 2 " 2 2 # graph Mgγ sin (∆xz) sin (∆yz) The by using expression (B.14) in E (z,t) = −∂ A (z,t) h2Ω (z,ω;t) = + × x,y t x,y 0 γ 2 ∆ 2 ∆ 2 and then putting all together in the expression of the Stokes x y 2 parameter (B.9), we get × h0Ωgw (0,ω;t). (B.19) 13

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