Gravitational wave search through electromagnetic telescopes

We study the graviton-photon conversion in the magnetic fields of the Earth, the Milky Way Galaxy, and intergalactic regions. Requiring that the photon flux converted from gravitons does not exceed the observed photon flux with telescopes, we derive upper limits on the stochastic gravitational waves in frequency ranges from $10^{7}$Hz to $10^{35}$Hz. Remarkably, the upper limits on $h^2 \Omega_{{\rm GW}}$ could be less than unity in the frequency range of $10^{18}$-$10^{23}$ Hz in a specific case. The detection of gravitational waves using telescopes would open up a new avenue for high frequency gravitational wave observations.


I. INTRODUCTION
In 2015, LIGO/Virgo collaborations detected gravitational waves from a binary black hole merger for the first time [1].Recently, stochastic gravitational waves around nHz were detected with pulsar timing arrays [2,3].Undoubtedly, it is important to push forward multifrequency gravitational wave observations in order to understand and investigate the evolution of our universe [4].In the low frequency range around 10 −18 − 10 −16 Hz, a promising method for observing gravitational waves is to seek for the B-mode polarization of the cosmic microwave observations [5][6][7].On the other hand, detection of high frequency gravitational waves above kHz is still under development and even new ideas are required [8][9][10][11][12][13][14][15][16][17][18], though such high frequency gravitational waves are theoretically interesting as probes of new physics [8,13,.
One natural approach for detecting high frequency gravitational waves is to employ tabletop experiments.This is because gravitational wave detectors tend to achieve high sensitivity when their size is on the same scale as the wavelength of the gravitational waves.For example, the magnon gravitational wave detector utilizing resonant excitation of magnons by gravitational waves was proposed for detecting gravitational waves around GHz [9][10][11].Under the presence of a background magnetic field, gravitational waves can also be converted into photons [40,41].In this regard, there have been significant proposals for new high frequency gravitational wave detection methods utilizing axion detection experiments [8,[12][13][14][15][16][17].
Another possible way is to utilize astrophysical observations of photons with various frequencies.References [42][43][44][45] put forward the idea that the detection of microwave/X-ray/gamma-ray photons can impose limits on the presence of stochastic gravitational waves at their respective frequencies.The constraints depend on the magnitude of the primordial/Galactic magnetic field. 1  Recently, it was also pointed out that the gravitonphoton conversion happens in the magnetosphere of a planet [50] or a pulsar [51].In this paper, we further investigate the possibility of the gravitational wave detection with various telescopes.We study the gravitonphoton conversion in magnetic fields of the Earth, in the Milky Way Galaxy, and in intergalactic regions and calculate photon flux expected to be observed with telescopes.Comparing it with the observed photon spectra with various astrophysical photon observations, we show that the gravitational wave detection with telescopes is quite promising.Therefore, the detection of gravitational waves using telescopes would open up a new avenue and advance multi-frequency gravitational wave observations in the high frequency range.

II. GRAVITON-PHOTON CONVERSION
We consider the action where M pl represents the reduced Planck mass, R is the Ricci scalar, g is the determinant of the metric g µν .The field strength of electromagnetic fields is defined by where A µ is the vector potential.
We now expand the vector potential and the metric as (3) 1 The cosmic background photon conversion into gravitons has also been studied in Refs.[46][47][48][49].

arXiv:2309.14765v2 [gr-qc] 21 Feb 2024
Here, Āµ consists of background magnetic fields Bi = ϵ ijk ∂ j Āk .η µν stands for the Minkowski metric, and h µν (x) is a traceless-transverse tensor representing canonically normalized gravitational waves.Below we take the gauge A 0 = 0 and only consider two transverse modes.Let us consider gravitons and/or photons propagating along z-direction and the background magnetic field orthogonal to the propagation direction,2 which is taken to be y-direction, B = (0, B, 0).One can also choose the polarization bases for the vector and the tensor as Using the basis (4), the electromagnetic field and the gravitational wave can be expanded as follows: We can now derive coupled equations of motion for the photon and the graviton of each polarization modes from Eqs. ( 1)- (6).We also take into account the plasma effect and the vacuum polarization [53].As for the vacuum polarization effect, we consider corrections to the dispersion relation from both the background magnetic field and the cosmic microwave background (CMB) [54][55][56][57].The derived equations are λ σ takes the value of 2 (7/2) for σ = + (σ = ×), α is the fine structure constant, T represents the temperature of the CMB, n e is the electron number density, and m e is the electron mass.As deriving the equations, we have assumed that the scale of conversion between photons and gravitons is much longer than k −1 and photons are ultrarelativistic, i.e., ω ≃ k.Also, we neglected spatial derivative of B. From Eq. ( 8), we see that while gets significant in higher frequency regime in general.In the next section, we will give upper limits on stochastic gravitational waves converted to photons in various magnetic fields in the universe with the use of telescope observations.

III. UPPER LIMITS ON STOCHASTIC GRAVITATIONAL WAVES
In this section, we consider the graviton-photon conversion in the geomagnetic fields, in the Milky Way Galaxy, and in intergalactic regions.We then calculate the photon flux converted from stochastic gravitational waves and compare it with the observed data of photon flux by various telescope in order to give upper limits on the abundance of stochastic gravitational waves.

A. Graviton to photon conversion in the Earth's magnetic fields
It is known that the geomagnetic field envelops the Earth, extending from the magnetic North Pole to the magnetic South Pole.The magnitude of the geomagnetism on the surface of the Earth is B E = 0.45G on average [58].Since the geomagnetic field has a dipole like structure, the amplitude of the magnetic field scales as where r E = 6367km is the averaged radius of the Earth.
There is no plasma in the troposphere (0 ∼ 10 km) and the stratosphere (10 ∼ 50 km).In the ionosphere (50 ∼ 1000 km), there exists plasma whose density ranges from n e,1 = 10 2 cm −3 to 10 6 cm −3 [59,60].In the inner magnetosphere (1000 ∼ 60000 km), there also exists plasma with the number density n e,2 ∼ 10 − 10 Outside the inner magnetosphere, it is not obvious how the magnetic field and the plasma density distribute due to substantial effects of solar wind.Therefore, we will only consider the graviton-photon conversion within the altitude of 60000km to obtain a conservative result.Photon flux converted from stochastic gravitational waves in the geomagnetic field can be calculated by solving Eq. ( 7) with Eq. ( 9) numerically.We then give upper bounds on the stochastic gravitational waves by comparing the predicted photon flux with the observational results by various telescopes; the CMB [63], the cosmic photon background (CPB) [63], active galactic nuclei [64], the Crab nebula [65], and ultra high energy photons [66].
The result is shown in Fig. 2 where limits on the characteristic amplitude h c defined by and the energy density parameter, which is related to h c and the Hubble constant H 0 (= h × 100 km/s Mpc) as Ω GW = 2π 2 f 2 h 2 c /3H 2 0 , are depicted.Note that the upper bound on Ω GW derived in this paper is much larger than unity in the most cases.Such a large value of Ω GW could still be meaningful if it is interpreted as the local quantity rather than the average over the whole universe.The pink, black, deep green, red, brawn, and navy lines (dashed lines) are obtained from observations of the CMB on the ground [63], the CPB in space [63] (COBE: 900km [67] , AKARI: 750km [68], CIBER: 577km [69], HST: 579km [70], EUVE: 527km [71], MAXI: 400km [72], COMPTEL: 500km [73], FERMI: 535km [74]), active galactic nuclei [64] (FERMI: 535km [74], MAGIC: on the ground), the Crab nebula [65] (LHASSO: on the ground) and ultra high energy photons [66] (Pierre Auger: on the ground), respectively, for n e,1 = 10 2 cm −3 and n e,2 = 10cm −3 (n e,1 = 10 6 cm −3 and n e,2 = 104 cm −3 ).Field of view angle is taken as 0.25 • for FERMI [64], 1.5 • for MAGIC [75], 1 • for LHASSO [65], and (approximately) 360 • for other telescopes.In the lower frequency range where the plasma effect, which prevents the conversion, becomes important, the constraints depend on the plasma density.Indeed, the result depends on the height from the ground of each detector because the graviton-photon conversion efficiently occurs in the troposphere and the stratosphere for gravitational waves in the lower frequency range.This is also the reason why the pink line and dashed line are almost indistinguishable.Furthermore, in the frequency range higher than ≳ 10 13 Hz, the solid and dashed black, deep green, red, brawn, and navy lines are almost degenerates since the plasma effect is not significant in the frequency range.
It should be noted that we focused on photon flux due to graviton-photon conversion from the direction of space even for satellite telescopes.Then, one could think that the conversion rate of gravitational waves to electromagnetic waves is parameterized by altitude alone, approximately.Instead, one can consider photon flux due to graviton-photon conversion from the direction of Earth, which allows us to block background photons, as is done in [50].In such a case, the photon flux can be more anisotropic and sensitive to the detector's field and direction of view.However, in our case, the direction dependence of photon flux is less important and can be neglected.
One might consider that the attenuation of photons by the air should be taken into account for telescopes on the ground when considering the graviton-photon conversion in the geomagnetic field.In Fig. 2, there are three types of experiments conducted on the ground: the CMB experiment represented by the pink line, MAGIC, and LHASSO.For the frequency range of the CMB experiment, the attenuation of photons by the air is small and negligible.For MAGIC and LHASSO, the attenuation effect is utilized for the detection of gamma-rays as air showers.Therefore, the attenuation effect in the air is not relevant to our discussion.

B. Graviton to photon conversion in the Milky
Way Galaxy There exists magnetic fields in our galaxy [76,77].The typical strength of the magnetic field B G ranges from 1µG to 10µG.There are two kind of main magnetic field components; one is large scale magnetic fields, which have large coherence and directional dependence along the spiral of the galaxy and the other is small scale magnetic fields whose coherence length is about 1pc ∼ 100pc [78][79][80].We focus on the latter one, which has isotropic random distribution. 4We then adopt the smoothly connected cellular model for the distribution of the magnetic field, namely there exists homogeneous magnetic fields in a domain whose size is l G and many such cells are contained in the Milky Way Galaxy whose size is ∼ 10kpc.The conversion probability between gravitons and photons in a cell can be calculated by diagonalizing Eq. (7).After passing through N G ∼ 10kpc lG cells, the total conversion probability may be multiplied N G times [81], that is where we defined the oscillation length Note that we replaced B G by its averaged transverse component BG = 2/3B G .When the oscillation length l os is smaller than the coherence length of magnetic fields l G , the sin function in Eq. ( 10) can be replaced by its average, 1/2, approximately.Therefore, the conversion probability can be evaluated as follows: Here is a remark.In Eq. ( 13), we implicitly assumed that the direction of the magnetic field suddenly changes at the boundary of the cells, i.e., the variation of the magnetic field is non-adiabatic.If l os ≫ l G , this assumption is reasonable.In the actual situation, however, the transition between two cells may be smooth and hence it would be regarded as adiabatically changing background if l os ≪ l G [82].In such a case there is no enhancement by a factor N G .Therefore, N G has been replaced by 1 in Eq. ( 12).The oscillation length is plotted in Fig. 1.Note that this calculation method is conservative compared to previous works where the factor N G remains even when l os < l G [43].Using the above method for calculating the conversion probability, one can evaluate the photon flux converted from stochastic gravitational waves in our galaxy with an approximation that the photon flux converted in our galaxy reaches us isotropically.The electron density varies against the direction and distance from the galactic center [83].In the calculation of gravitonphoton conversion probability, we take a conservative value, n e = 7 × 10 −2 cm −3 [83].Comparing the calculated photon flux with the observed flux of the CMB [63], CPB [63], active galactic nuclei [64], the Crab nebula [65], and ultra high energy photons [66], we obtained constraints on stochastic gravitational waves.
The obtained limits are shown in Fig. 3.For higher frequencies than f ∼ 10 33 Hz for l G = 1pc (f ∼ 10 31 Hz for l G = 100pc), one finds that the constraints become stronger for B G = 1µG than the case of B G = 10µG.It is because that ω 4 QED,σ ∝ B 4 G plays an important roll in the denominator of Eq. ( 12) when l os < l G .Thus, the turning points correspond to the frequency when the oscillation length equals the coherent length, i.e., l os = l G , as one can see in Fig. 1.

C. Graviton to photon conversion in intergalactic regions
There also exists magnetic fields whose strength B IG is considered to range from 10 −17 G [64] to 0.1nG [84][85][86][87], whose coherence length of magnetic fields is l IG ∼ 1Mpc−4000Mpc, in intergalactic regions [64].Since there are no known dynamical mechanisms to generate such magnetic fields in the void regions, they are expected to be produced in the early universe.We then consider graviton-photon conversion after the photon decoupling to the present time while taking into account the redshift effect, namely each relevant parameter is promoted to a redshift dependent parameter: and n e → n b0 (1 + z) 3 χ e (z).n b0 = 0.25m −3 is the current baryon number density [88] and χ e (z) is the ionization fraction (we use the model in [89]).Let us again adopt the smoothly connected cellular model as explained in Sec.III B. Repeating the similar discussion to Sec.III B, we obtain the conversion probability per time as for l os (z) > l IG (z).( 14) (15) For l os (z) > l IG (z), the total conversion rate would be zi(ω) where we used the relation dt = −dz H(z)(1+z) and H(z) = H 0 0.69 + 0.31(1 + z) 3 with H 0 = 67km/s Mpc is the Hubble parameter [88].The integration range is taken from z = 0 to z = z i (ω); z i (ω) represents the time when a photon with a frequency ω(z) starts to propagate transparently until now.For example, z i (ω) = 1100 for pho-tons with lower frequencies than ω(z = 1100) < 13.6eV.However, absorption by atoms can be significant for photons with frequencies higher than 13.6eV [90].Moreover, pair creation of an electron and a positron becomes efficient for extremely high energy photons in the presence of CMB or CPB [91].We will set the cutoff of the integration z i (ω) at the point where the optical depth equals (approximately) unity [90,91].The pair creation of an electron and a positron affects not only the absorption but also the dispersion relation of photons.As discussed in [57], the modified ω CMB (z) can be evaluated with the use of the Krasmers-Kronig relation as where the integration denotes the Cauchy principal value.ω 0 = 2m 2 e π 4 T0/30ζ(3) = 1.25 × 10 30 Hz is the threshold energy for the pair creation.Now, as noted in Sec.III B, we must be careful about the use of the cellular model for the intergalactic magnetic fields, as each coherent magnetic field may be more smoothly connected.Firstly, whether l os (z) is larger or smaller than l IG (z) depends on the redshift.During a period when l os (z) > l IG (z), the conversion probability per unit time (15) accumulates.Thus, it should be integrated as shown in Eq. ( 16).On the other hand, in a period when l os (z) < l IG (z), the total conversion probability would be determined solely by l IG P (σ) at the point where l os (z) < l IG (z) breaks.Therefore, when calculating the total conversion probability, we do not integrate P (σ) during a period when l os (z) < l IG (z), instead, we add the conversion probability l IG P (σ) at the time when l os (z) < l IG (z) breaks.We note that this method generally yields conservative results by neglecting the accumulation effect when l os (z) < l IG (z), compared to previous works [42,44].
Using the above method, we calculated photon flux from graviton-photon conversion with Eqs. ( 14)- (16).Comparing the calculated photon flux with the observed flux of the CMB [63], CPB [63], active galactic nuclei [64], and ultra high energy photons [66], we ob-tained constraints on the abundance of stochastic gravitational waves.The obtained limits are shown in Fig. 4. The coherence length of magnetic fields is set as l IG = 1Mpc for upper figures and l IG = 4000Mpc for lower ones.The pink, black, deep green, red, and navy lines (dashed lines) are obtained from observations of the CMB [63], the CPB (COBE, AKARI, CIBER, HST, EUVE, MAXI, COMPTEL, FERMI) [63], active galactic nuclei of FERMI [64] and of MAGIC [64], and ultra high energy photons (Pierre Auger) [66], respectively, for B IG = 0.1nG (B IG = 10 −17 G).We note that the observation of PeV gamma-ray [65], which was used in Sec.III A and Sec.III B, has not been used here, since the attenuation effect of PeV gamma-ray by CMB is significant in the extragalactic scale [91].In the low frequency range around GHz, there are also constraints from CMB observations with EDGES and ARCADE2 [44].The given constraints are stronger than ours for two reasons: they utilized the information of CMB spectral distortion, and we also employed the smoothly connected cellular model.Remarkably, one can see that for l IG = 4000Mpc and B IG = 0.1nG, the upper limits on h 2 Ω GW could be less than unity in the frequency range of 10 18 -10 23 Hz.

IV. CONCLUSION
We investigated the graviton-photon conversion under the magnetic fields of the Earth, the Milky Way Galaxy, and intergalactic regions.To calculate the probability of graviton-photon conversion in the Milky Way Galaxy and in intergalactic regions, we investigated the smoothly connected cellular model that considers the smooth connections among regions containing coherent magnetic fields.Requiring that the calculated photon flux converted from gravitons does not exceed the observed photon spectra with various telescopes allows us to constrain the amount of stochastic gravitational waves.The gravitational wave detection with telescopes enables us to observe high frequency gravitational waves in wide frequency ranges above the radio frequency ∼ 10 7 Hz.Remarkably, it pushes the boundaries of frequency in gravitational wave observations to 10 35 Hz.Moreover, the upper limits on h 2 Ω GW could be less than unity in the frequency range of 10 18 -10 23 Hz in a specific case.
It would be desired to prepare templates of photon spectra converted from gravitational wave spectra predicted by each source of high frequency gravitational waves [8,13, in various background magnetic fields to detect it in data of telescope observations, as partly demonstrated in this paper.Although we considered stochastic gravitational waves as a demonstration in this paper, it is also possible to observe event-like gravitational waves from such as binaries of light primordial black holes and black hole superradiance [8,36].In this case, we have the chance to observe gravitational waves of h 2 Ω GW ≥ 1.For example, if primordial black holes with a mass of ∼ 10 −15 M ⊙ form all of dark matter, we expect a merger event from a distance 10pc per year, which emits a gravitational wave of ∼ 10 19 Hz at the final stage of the coalescence [36].The photon flux from such a single transient event (not from the accumulated stochastic signals of the merger events) through the graviton-photon conversion in our galaxy is calculated as ∼ 0.001 erg/s cm 2 , where we assumed B G = 10µG and l G = 1pc.Apparently, this is detectable with current telescopes.However, we note that the emission of the gravitational wave only lasts for ∼ 10 −20 s [36].This is considerably shorter than the time resolution of current telescopes, which indicates that further efforts/ideas are needed to push forward detecting gravitational waves with telescopes.Nevertheless, we believe that gravitational wave detection with telescopes offers exciting opportunities to probe new physics which predicts high frequency gravitational waves [8,13,.
f/Hz h c h 2 ⌦ GW f/Hz FIG. 3. Limits on stochastic gravitational waves converted to photons in the Milky Way Galaxy with various telescope observations are shown.The coherence length of magnetic fields is set as lG = 1pc for upper figures and lG = 100pc for lower ones.The pink, black, deep green, red, brawn, and navy lines (dashed lines) are obtained from observations of the CMB [63], the CPB (COBE, AKARI, CIBER, HST, EUVE, MAXI, COMPTEL, FERMI) [63], active galactic nuclei of FERMI [64] and of MAGIC [64], the Crab nebula (LHASSO) [65] and ultra high energy photons (Pierre Auger) [66], respectively, for BG = 10µG (BG = 1µG).The light blue line represents limits from pulasr observations [51].The blue (grey) and the orange (lime green) lines respectively represents the constraints with EDGES and ARCADE2 for maximal (minimum) amplitude of cosmological magnetic fields [44].The violet line is the upper limit from 0.75 m interferometer [92].The wine red, green, and magenta lines represents constraints with ALPS, OSQAR, and CAST, respectively [12].Limits on stochastic waves converted to photons in inter galaxies with various telescope observations are shown.The coherence length of magnetic fields is set as lIG = 1Mpc for upper figures and lIG = 4000Mpc for lower ones.The pink, black, deep green, red, and navy lines (dashed lines) are obtained from observations of the CMB [63], the CPB (COBE, AKARI, CIBER, HST, EUVE, MAXI, COMPTEL, FERMI) [63], active galactic nuclei of FERMI [64] and of MAGIC [64], and ultra high energy photons (Pierre Auger) [66], respectively, for BIG = 0.1nG (BIG = 10 −17 G).The light blue line represents limits from pulasr observations [51].The blue (grey) and the orange (lime green) lines respectively represents the constraints with EDGES and ARCADE2 for maximal (minimum) amplitude of cosmological magnetic fields [44].The violet line is the upper limit from 0.75 m interferometer [92].The wine red, green, and magenta lines represents constraints with ALPS, OSQAR, and CAST, respectively [12].
FIG.4.Limits on stochastic waves converted to photons in inter galaxies with various telescope observations are shown.The coherence length of magnetic fields is set as lIG = 1Mpc for upper figures and lIG = 4000Mpc for lower ones.The pink, black, deep green, red, and navy lines (dashed lines) are obtained from observations of the CMB[63], the CPB (COBE, AKARI, CIBER, HST, EUVE, MAXI, COMPTEL, FERMI)[63], active galactic nuclei of FERMI[64] and of MAGIC[64], and ultra high energy photons (Pierre Auger)[66], respectively, for BIG = 0.1nG (BIG = 10 −17 G).The light blue line represents limits from pulasr observations[51].The blue (grey) and the orange (lime green) lines respectively represents the constraints with EDGES and ARCADE2 for maximal (minimum) amplitude of cosmological magnetic fields[44].The violet line is the upper limit from 0.75 m interferometer[92].The wine red, green, and magenta lines represents constraints with ALPS, OSQAR, and CAST, respectively[12].