Distinctive electromagnetic signals caused by gravitational waves (of sub-solar mass primordial black hole binary mergers) interacting with galactic magnetic fields
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Distinctive electromagnetic signals caused by gravitational waves (of sub-solarmass primordial black hole binary mergers) interacting with galactic magneticfields
Hao Wen ∗ Physics Department, Chongqing University, Chongqing 401331, China. (Dated: July 10, 2019)As a candidate of dark matter, and related to many fundamental physics issues, the primordialblack hole (PBH) is a crucial topic. However, so far the existence of PBHs is still not confirmed, andcurrently running gravitational wave (GW) detectors are still not able to distinguish them from thenormal astrophysical black holes. In this article, we propose that the GWs (of PBH binary mergers)could interact with the very widespread background galactic magnetic fields in the Milky way, toproduce the perturbed electromagnetic waves (EMWs) with unique characteristics of frequencies,waveforms, spectra and polarizations. In order to be distinguished from astrophysical black holes,only the PBHs with masses less than the solarmass are considered here, and their binary mergers willradiate GWs in frequencies much higher above the plasma frequency of interstellar medium (ISM),so corresponding perturbed EMWs (in the same frequencies to such GWs) can propagate throughthe ISM until the Earth. Our estimations show that, for the sub-solar mass PBH binary mergerswithin the Milky way (disk or halo), the strengths of the perturbed EMWs turn into constant levelsaround ∼ − Tesla (for magnetic components) and ∼ − W att · m − (for energy flux densities)at the Earth, generally for all cases of different PBH masses (and not dependent on the distanceof sources), and the same mass ratio of the PBH binary gives the same strength (at the Earth) ofperturbed EMWs despite different PBH masses (GW frequencies) or binary distances. Differently,for the sub-solar mass PBH binary mergers outside the Milky way, the perturbed EMWs at Earthhave lower strengths (and depend on the distance of sources), but for some part of distance range,they would also be detectable. If such EM signals and special EM counterpart of GWs from PBHscould be detected by space- or land-based EMWs detectors, it may provide direct evidence of thePBHs. PACS numbers:
Keywords: primordial black holes, gravitational waves, galactic magnetic fields, electromagnetic responseto gravitational waves
I. INTRODUCTION
The PBHs (produced in the early Universe bycollapse of large energy density fluctuations) areconsidered as a promising candidate of the darkmatter and also related to many crucial cosmolog-ical issues and fundamental physical problems, sothey have been massively studied[1–20]. However,the existence of PBH is still not confirmed. Inrecent years, the LIGO scientific collaboration andVirgo collaboration have reported lots of gravita-tional wave (GW) events from black hole mergers[21–27] (with frequencies around 30Hz to 450Hzand dimensionless amplitudes ∼ − to ∼ − near the Earth). These black hole binaries could beformed by astrophysical black holes or primordialblack holes, but so far, they cannot be distinguished ∗ [email protected] from these different origins.Here, we particularly focus on the PBHs withmasses less than the solar mass, due to such smallblack holes can only be the PBHs but not astro-physical black holes. The binary mergers of thesesub-solar mass PBHs will produced GWs in higherfrequencies, e.g., the frequency at innermost stablecircular orbit ( f ISCO ) from ∼ ∼ − M ⊙ PBHs) to ∼ Hz (by ∼ − M ⊙ PBHs). If thePBHs could be one component of the dark matter,they should distribute in the vast range of Universe,within or outside the Milky way. On the other hand,according to contemporary observations, there arevery widespread background galactic magnetic fields(GMFs, strength around ∼ − to 10 − Tesla)[28]within the Milky way. Therefore, according to theelectrodynamics in curved spacetime, in the frameof EM response to GWs, we propose that suchhigh-frequency GWs of sub-solar mass PBH binarymergers will interact with the GMFs in the Milky arth galacticmagneticfield ( )
GMF perturbed as GW
EMWs signals and counterpart
GMF
GMF
GMF from primordialblack hole binary merger
GWs
PBH binarymerger m u l t i - m e ss enge r G W s + pe r t u r bed E M W s PBH binarymerger
PBH binarymerger
GMF
M i l k y w a y kpc
ExtragalacticPBHbinarymergerExtragalacticPBHbinarymerger
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FIG. 1:
An intuitive view (not in scale):
The GWs from sub-solar mass primordial black hole (PBH) binary mergers(within or outside the Milky way) can interact with the galactic magnetic fields (GMFs, typically ∼ − Tesla, widespread inMilky way), and then lead to perturbed EMWs [in the same frequencies to the GWs, e.g., f ISCO from ∼ ∼ − M ⊙ PBHs) to ∼ Hz (by ∼ − M ⊙ PBHs); these frequencies are all far above the plasma frequency of interstellar medium(ISM) and thus the perturbed EMWs can propagate through the ISM]. The perturbed EMWs propagate until the Earthalmost synchronously with these GWs, and they could be a new type of signals and special EM counterparts of GWs from thePBH binary mergers. Such EM signals would have characteristic frequencies, waveforms (related to waveforms of the GWs,after modifications by the ISM), spectra and polarizations. These distinctive perturbed EMWs cannot be caused by GWsfrom normal astrophysical black holes, and thus if they could be captured by high sensitive land- or space-based EMWdetectors, it may provide direct evidence of the PBHs. way, to produce the perturbed EM waves (EMWs,in the same frequencies to such GWs), as a newtype of signals and special EM counterparts of theGWs from PBHs (see Fig. I).The effect of EM response to GWs had been longstudied[29–46], and previous works[30, 35–37, 45] in-dicated that the strengths of perturbed EMWs de-pends on both strengths and spatial scales (accumu-lation distance) of the background magnetic fields.Thus, the GMFs will provide a huge accumulationdistance to compensate the weakness of their verylow strengths, and then would lead to considerableEM signals. Besides, for the effect of EM response to GWs, the strengths of the perturbed EMWs areproportional to both the amplitude and the frequen-cies of the GWs. Therefore, although the amplitudeof GWs from sub-solar mass PBH binary mergersare much lower than that from astrophysical blackhole binary mergers, their very high frequencies ef-fectively compensates the weakness of their low am-plitudes.Also, the frequencies of such perturbed EMWsare all far above the plasma frequency of the in-terstellar medium (ISM). Thus, they can propagatethrough the ISM together with the GWs until theEarth, and they would have waveforms similar to(or related to, after some modifications by the ISM)2he waveforms of the GWs of PBH binary mergers.The perturbed EMWs could also have particular po-larizations which may reflect some new informationabout tensorial and possible nontensorial polariza-tions of the GWs from the PBH binary mergers. Ifthe perturbed EMWs are captured by current land-based or space-based high sensitive EMW detectors,such characteristics would be helpful to extract theseEM signals from noise by methods similar to the wayfor searching GW signals from data of LIGO, Virgo,KAGRA, etc, of using the matched filtering based onwaveform templates. Such perturbed EMWs withunique properties cannot be caused by GWs fromastrophysical black holes, so they would be directsignals of the PBHs.In Sect.II, we estimate strengths at the Earthof the perturbed EMWs caused by GWs (fromsub-solar mass PBH binary mergers) interactingwith the GMFs, and in Sect.III, we give a shortconclusion and discussion. R acc r dr A (r) =amplitude of GWat position of r r , start point ofaccumulation o Earth r galacticmagnetic fields B galactic ( ) r = galacticmagnetic fieldsat position of r binaryprimordialblack holemergerm1 m2 ( ) G s from PBH W binary merger galacticmagnetic fields FIG. 2:
Calculation for accumulated perturbedEMWs caused by GWs (of sub-solar mass PBHbinary mergers) interacting with GMFs.
II. ESTIMATION OF STRENGTHS OFPERTURBED EMWS CAUSED BY GWS (OFPRIMORDIAL BLACK HOLE BINARYMERGERS) INTERACTING WITHGALACTIC MAGNETIC FIELDS
Here we estimate the strengths (around the Earth)of the perturbed EMWs caused by interaction be-tween the GWs of sub-solar PBH binary mergersand the galactic magnetic fields in the Milky way.At first, we calculate the cases that the PBH bi-nary mergers are within the Milky way (disk orhalo), and later we will extend the calculation to the cases for PBH binary mergers outside the Milkyway.If we consider the GWs of binary PBH mergerscontain not only the tensor polarizations but alsopossible nontensorial polarizations, they can be gen-erally express as: h µν = A + + A b A × A x A × − A + + A b A y A x A y √ A l e i ( k g · r − ωt ) , (1)the +& × , x & y , b & l respectively represent the cross-&plus- (tensor mode), x -& y - (vector mode), b -& l -(scalar mode) polarizations. Interaction of theseGWs of PBH binary mergers with the GMFs in theMilky way, will generate the perturbed EMWs, andsuch effect can be calculated by the electrodynamicsequations in curved spacetime:1 √− g ∂∂x ν [ √− gg µα g νβ ( F (0) αβ + ˜ F (1) αβ )] = µ J µ , ∇ µ F να + ∇ ν F αµ + ∇ α F µν = 0 , ∇ α F µν = F µν,α − Γ σµα F σν − Γ σνα F µσ , (2)Due to previous works[30, 36, 37, 40, 41, 45], the E(electric) and B (magnetic) components of perturbedEMWs caused by planar GWs in an interaction dis-tance (accumulation distance) of ∆ L can be given:˜ E (1) = A ˆ B (0) galactic k g c ∆ L exp[ i ( k g · r − ωt )] , ˜ B (1) = A ˆ B (0) galactic k g ∆ L exp[ i ( k g · r − ωt )] , (3)here, “ A ” is the GW amplitude of tensorial modes( A + , A × ), or of nontensorial modes [here, onlyfor ( A x , A y ), but not for ( A b , A l ), the reasonis explained below]. The ˆ B (0) galactic can be trans-verse components of the galactic magnetic fields[perpendicular to direction of GW propagation,interacting with tensorial polarizations of the GWsof PBH binary mergers], or can be longitudinalcomponents of the galactic magnetic fields [alongthe direction of GW propagation, interacting withvector polarizations of the GWs of PBH binarymergers[45]].Importantly, the tensorial GWs can interact withthe transverse magnetic fields but cannot with thelongitudinal magnetic fields, and contrarily, thenontensorial GWs can interact with the longitudinalmagnetic fields but cannot with the transverse mag-netic fields. Thus, in this article, we only considerthe vector modes of ( A x , A y ) for the nontensorial3 a) (b)(c) (d) FIG. 3: Examples of strengths of magnetic components and energy flux densities of the accumulatedperturbed EMWs caused by GWs (from sub-solar PBH binary mergers) interacting with galactic magneticfields, for different cases of PBH masses from ∼ − M ⊙ to ∼ − M ⊙ , corresponding to the f ISCO from ∼ Hz to ∼ − Tesla(for magnetic components) and 10 − W att · m − (for energy flux densities) until the Earth generally for allcases of different PBH masses and frequencies of GWs. The R acc is from the start point of accumulation r until the Earth.GWs, because the the longitudinal magnetic fieldscan only interact with ( A x , A y ) GWs and cannotinteract with A b or A l GWs[45].To estimate the strengths, we can calculate bythe scheme explained in Fig. 2, where the binaryevolution is already very close to the merger time( t = 0, defined as the time when the amplitude ofGW reaches the maximum), e.g., only several pe-riods before the separation reaching the innermoststable circular orbit (ISCO). As shown in Fig. 2, wecan integrate the contributions [given by Eq. (3), re-place the ∆ L by dr ] of generation of the perturbedEMWs of every small accumulation distance “ dr ”(in such very small distance the spherical GWs canbe treated as planar waves), from the r (start pointof the accumulation, e.g., set as 10 times of ISCO)until end point of accumulation distance R acc (this “ R acc ” is from the start point r to the position ofEarth, or the source-Earth distance R pbh , and r = 0means the position of PBH binary). Every part ofcontribution of the perturbed EMWs in the “ dr ” willdecay from the position r until the end point of ac-cumulation R acc , so there will be a term of “ r/R acc ”in the formula, see below. Therefore, we work outthe accumulated perturbed EMWs in the form:˜ B (1) prtbd = Z R acc r A ( r ) ˆ B (0) galactic ( r ) ωc rR acc dr, (4)where, the subscript “ prtbd ” means “perturbedEMWs”. The ω is the angular frequency. The term A ( r ) represents the amplitude of GWs at the posi-4 a) (b)(c) (d) FIG. 4: Examples of strengths of magnetic components and energy flux densities of the accumulatedperturbed EMWs caused by GWs (from sub-solar PBH binary mergers) interacting with galactic magneticfields, for more cases of various PBH masses. It is found the accumulated strengths also generally turn intolevels around 10 − Tesla (for magnetic components) and 10 − W att · m − (for energy flux densities) untilthe Earth, consistent to the Fig. 3.tion of r , and this can be expressed by[47, 48]: A ( r ) = 4 . × − ( 10 kpcr )( M c . M ⊙ ) / ( f − Hz ) / , (5)where, M = m m µ = m ∗ m /M , M c = µ / M / . For aconservative estimation, we can take the frequency f in the above A ( r ) as the f ISCO (the GW frequencyat the ISCO), and f ISCO = (6 / πM ) − [49].Actually, during the accumulation process, thegalactic magnetic fields ˆ B (0) galactic ( r ) vary along theline of sight, or, it should be the function of “r”,but as the first step of a estimation for the order ofmagnitude, we can treat the ˆ B (0) galactic generally in level of 10 − Tesla[28].The estimated levels (magnetic components andenergy flux densities) of the perturbed EMWs areshown in Fig. 3 for very large range of massesof the PBHs, from ∼ − M ⊙ to ∼ − M ⊙ ,corresponding to the f ISCO from ∼ ∼ Hz. The results are interesting that, theaccumulated strengths increase and asymptoticallyturn into constant levels around ∼ − Tesla (formagnetic components) and ∼ − W att · m − (forenergy flux densities) until the Earth, generally forall cases of different PBH masses (and thus differentGW frequencies) or different distance of sources.This phenomenal is on account of two reasons: (1)the composite effect of both the accumulation ofperturbed EMWs caused by GWs and the decaying5ABLE I: Strengths of perturbed EMWs caused by GWs (of sub-solar mass PBH binary mergers in Milkyway) interacting with galactic magnetic fields (set as 10 − Tesla here). The “B of perturbed EMWs”represents the magnetic component of the perturbed EMWs, and “EFD” means energy flux density. It isshown that the same mass ratio of the PBH binary gives the same strength (at the Earth) of perturbedEMWs despite different PBH masses (GW frequencies) or binary distances.PBH binary f ISCO A ( r ) at Earth of GWs from PBH B of perturbed EFD of perturbedmasses of GWs binaries with various distance EMWs at EMWs at( M ⊙ ) (Hz) 10 kpc 50 kpc 100 kpc Earth (Tesla) Earth ( W/m )10 − , 10 − . × . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − − , 2 ∗ − . × . × − . × − . × − . × − . × − − , 3 ∗ − . × . × − . × − . × − . × − . × − − , 4 ∗ − . × . × − . × − . × − . × − . × − − , 4 ∗ − . × . × − . × − . × − . × − . × − − , 3 ∗ − . × . × − . × − . × − . × − . × − − , 2 ∗ − . × . × − . × − . × − . × − . × − TABLE II: Similar to the Table I, but for sub-solar PBH binary mergers outside the Milky way. SuchGWs propagate and decay from the far away source until the Milky way, and then interact with the GMFs(set as 10 − Tesla) in the Milky way (the accumulation distance of GMF scale taken account to theinteraction, is set to only 10 kpc for conservative estimation) to produce the perturbed EMWs. The “B ofperturbed EMWs” represents the magnetic component of the perturbed EMWs, and “EFD” means energyflux density.PBH binary f ISCO A ( r ) at Earth of GWs from PBH EFD of perturbed EMWs at Earth formasses of GWs binaries with various distance binaries with various distance ( W/m )( M ⊙ ) (Hz) 3 Mpc 1 Gpc 15 Gpc 3 Mpc 1 Gpc 15 Gpc10 − , 10 − . × . × − . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − . × − − , 10 − . × . × − . × − . × − . × − . × − . × − − , 2 ∗ − . × . × − . × − . × − . × − . × − . × − − , 3 ∗ − . × . × − . × − . × − . × − . × − . × − − , 4 ∗ − . × . × − . × − . × − . × − . × − . × − − , 4 ∗ − . × . × − . × − . × − . × − . × − . × − − , 3 ∗ − . × . × − . × − . × − . × − . × − . × − − , 2 ∗ − . × . × − . × − . × − . × − . × − . × − of these GWs in a spherical ratio during theirpropagations from the source until the Earth. (2)binary mergers with lower PBH masses produceGWs with lower amplitudes, but with higherfrequencies; whereas, in the frame of EM responseto GWs, the strengths of the perturbed EMWsare proportional to both the amplitude and the frequencies of the GWs, i.e., higher GW frequencieslead to higher strengths of the perturbed EMWs,and thus the effect of lower amplitudes of GWs isoffset. Fig. 4 similarly shows strengths but in anarrower frequency range, for more cases of variousPBH masses (equal or unequal), and these resultsare consistent to the Fig. 3. The Table I gives6 a)(b) FIG. 5:
Similar to Figs. 3 and 4, but for sub-solarPBH binary mergers outside the Milky way. Here,all cases have the same distance of binary mergersof 3 Mpc from the Earth. some concrete results for different parameters, andit is shown that the same mass ratio of the PBHbinary gives the same strength (at the Earth) ofperturbed EMWs despite different PBH masses(GW frequencies) or binary distances. For suchstrengths, these EM signals from sub-solar massPBHs would already be detectable by currentspace-based or land-based EMWs detectors.The above calculation can also be extended forthe sub-solar mass PBH binary mergers outside theMilky way. Therefore, differently to Figs. 3, 4 andTable I, the Fig. 5 and Table II present the extra-galactic cases. Such GWs propagate and decay fromthe far away source until the Milky way, and theninteract with the GMFs (set as 10 − Tesla), andthe accumulation distance in the Milky way is setto only 10 kpc for conservative estimation. For suchcases, the interaction range (accumulation distance)does not cover the whole range of propagation ofthe GWs from source to Earth, but only cover avery small part of the full distance (in the last stage of the propagation to the Earth), i.e., before theGWs arrive the GMF of Milky way, we considerthat there is not interaction between GWs andbackground magnetic fields (therefore no generationand no accumulation of the perturbed EMWs), soin such cases, the accumulation behavior is differentto that for the binary mergers inside the Milkyway (where the accumulation covers the wholepropagation), and thus, corresponding perturbedEMWs will be dependent on the distance of GWsources, as shown in the Table II. It is clear that forthe cases of binary mergers outside the Milky way,the perturbed EMWs will be generally weaker, andthey would be detectable only for some part of thedistance range.
III. CONCLUSION AND DISCUSSION
We estimate the strengths of perturbed EMWscaused by the GWs (of sub-solar mass primordialblack hole binary mergers within or outside theMilky way) interacting with the widespread galacticmagnetic fields (in the Milky way disk or halo).Frequencies ( f ISCO ) of such GWs are from ∼ ∼ − M ⊙ PBHs) to ∼ Hz (by ∼ − M ⊙ PBHs), and the strength of GMF is considered as ∼ − Tesla.For cases of PBH binary mergers withinMilky way, the estimated strengths of the per-turbed EMWs turn into constant levels around ∼ − Tesla (for magnetic components) and ∼ − W att · m − (for energy flux densities) untilthe Earth, generally for all cases of different PBHmasses (and not dependent to source distance,see Figs. 3, 4, Table I). It is also found that thesame mass ratio of the PBH binary gives the samestrength (at the Earth) of perturbed EMWs despitedifferent PBH masses (GW frequencies) or binarydistances (Table I).This interesting phenomenal is on account oftwo reasons: (1) the composite effect of bothaccumulation of perturbed EMWs and decaying ofGWs during their propagations. (2) the strengthsof perturbed EMWs are proportional to bothamplitude and frequencies of the GWs, so thehigher GW frequencies lead to higher strengths ofthe perturbed EMWs to compensate the effect oflower amplitudes of the GWs.For cases of sub-solar mass PBH binary mergersoutside the Milky way, differently, the strengthsof perturbed EMWs are dependent on the source7istance, and they are generally weaker than thecase of binary mergers within the Milky way (Fig.5 and Table II), and would be detectable only forsome part of the distance range.With above estimated strengths (especially forthe cases within the Milky way), these EM signalsand counterpart of GWs from sub-solar mass PBHbinary mergers might be detectable by currentspace-based or land-based EMWs detectors forvarious bands, even if considering the depressionof the strengths of perturbed EMWs at the Earthcaused by the dispersion. Further, although theywill be recorded with massive noise, they couldbe distinguished by methods similar to the wayfor searching GW signals from data of LIGO,Virgo, KAGRA, etc, of using the matched filter-ing based on waveform templates. However, tobuild such template bank of the waveforms ofthe perturbed EMWs, we need to consider theimpact of some other factors, e.g., the interstellarmedium (ISM) along the propagation path of theperturbed EMWs, the variation of the GMFs andthe inhomogeneous distribution of electron densityin the ISM. The impact of ISM brings dispersion,refraction and scattering, etc. The dispersion causesthe frequency-dependent dispersive delay (thus thededispersion will be required). Refraction of signalEMWs caused by inhomogeneities in the ISM andturbulence of electron density lead to interstellarscintillation[50], and the presence of GMFs in theISM also results in the Faraday rotation. Therefore,the above effects should be included to build proper template bank of the modified waveforms(or spectra) of the perturbed EMWs, with moreaccurate numerical calculation based on detailedastrophysical observation data. This would be quitecomplicated treatments, and such issues will beinvestigated in our subsequent works.Moreover, the GWs from sub-solar mass PBHbinary mergers would contain tensorial polarizationsand possible nontensorial polarizations, and thisleads to that the perturbed EMWs caused by suchGW polarizations will be different to those causedby only the tensorial GW polarizations. On theother hand, the effect of Faraday rotations and thedetailed distribution of directions and strengths ofthe GMFs, will also influence the polarizations ofthe perturbed EMWs[45] along the propagationpath from the source to the Earth. 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