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Gravitation & Cosmology, Vol. 4 (1998), No. , pp. 1–10 (cid:13)c 1998 RussianGravitational Society CAN QUASARS BE EXPLAINED BY COSMOLOGICAL WAVEGUIDE EFFECTS? S. Capozziello1 and G. Iovane2 Dipartimento di Scienze Fisiche ”E.R. Caianiello”, Universita` di Salerno, 9 9 INFN Sezione di Napoli, Gruppo collegato di Salerno, 9 Via S. Allende, I-84081 Baronissi (SA) Italy. 1 n a A sort of gravitational waveguide effect in cosmology could explain, in principle, the huge luminosities coming J from quasars using the cosmological large scale structures as selfoc–type or planar waveguides. Furthermore, other 8 anomalous phenomena connected with quasars, as the existence of “brothers” or “twins” objects having different brilliancy but similar spectra and redshifts, placed on the sky with large angular distance, could be explained by 1 this effect. Wedescribe thegravitational waveguide theory and then we discuss possible realizations in cosmology. v 2 8 1. Introduction lens–observer. This study is simple if we suppose that 0 these are three points on a plane as well as if we con- 1 Recently, gravitational lensing has become, in actual siderthinlensapproximation: suchhypothesesarerea- 0 fact, a new field of astronomy and astrophysics to in- sonable because of the large distances considered. A 9 9 vestigate the Galaxy, the large scale structure of the theoretical model can be workedout by giving a spec- / universe and to test cosmological models [1]. Acting ified form to the lens density, i.e. fixing its structure. h on all scales, it provides a great amount of applica- Fromthedensityfunction,usingtheequationsderived p - tions like a more accurate determination of the cos- fromthe geometry,wecanhavepredictionsfortheob- o mological parameters as H0, Ω, and Λ [2],[3], the served deviation of the source light and magnitude of r t possibility of describing the potential of lensing galax- every image. s a ies and galaxy clusters from the observation of multi- It is well known that the gravitationallensing may : ply imaged quasars, arcs and arclets [4],[5]. However, be explained using the action of a gravitational field v i the leading role of gravitational lensing is its contri- on the light rays. In this case, the action of media X bution in searching for dark matter. In fact a way with corresponding refraction index is, for weak field r to detect compact objects with masses in the range approximation, completely determined by the Newto- a 10−5M⊙ 100M⊙,intheGalaxyorinnearbygalaxies, niangravitationalpotential whichdeflects andfocuses ÷ is based upon an application of lensing, the so called the light rays. microlensing, which effect is to produce characteristic In optics, however, there exist other types of de- light variations of distant compact sources. The fea- vices, like optical fibers and waveguides which use turesofsuchacurvegivethephysicalpropertiesofthe the same deflection phenomena. The analogy with unseen objects (the Massive Astrophysical Compact the action of a gravitational field onto light rays may Halo Objects, i.e. the MACHOs) which seem greatly be extended to incorporate these other structures on to contribute to the mass of our Galaxy [6]. the light. In other words, it is possible to suppose Microlensing has also cosmological applications. the existence of a sort of gravitational waveguide ef- Particularlypromisingarethe multiplymacro–imaged fect [16],[17], [18]. furthermore, structures like cosmic quasars whose lensing galaxy should have a large op- strings,texture anddomainwalls,whichareproduced tical depth for lensing effects [7],[8] [11] (at least 20 at phase transiton in inflationary models, can evolve objects are identified; see, for example, [12],[13], [14]). into today observed filaments, clusters and groups of Theabovekindsofanalysisarepossibleifwehavea galaxies and behave in a variety of ways with respect modelexplainingthewayofformingimagessuchasthe to the propagation of light. In fact, the lensing by above–mentioned arcs, rings or simply double images cosmic string was suggestedas explaination of the ob- and predicting the effects of the deflector [9],[10]. servation [19] of twins objects with very large angular From a theoretical point of view, lensing must be distance between the partners [21],[22]. treated studying the geometry of the system source– The aim ofthis work is to discuss the properties of 1e-mail: [email protected] possible waveguidesin the universe and to suggestthe 2e-mail: [email protected] explaination of some phenomena, like quasar huge lu- 2 S.Capozziello and G.Iovane minosities and large angular distances between twins, (it is worthwhile to note that such a situation can be asaby–productoftheirexistence. Forexample,afila- easily reproduced in cosmology [14]). mentofgalaxiescanbe consideredasortofwaveguide Forweakgravitationalfields,consideredalsotode- preserving total luminosity of a source, if we have, lo- scribe usual gravitational lensing effects, the metric cally, an effective gravitational potential of the form tensor components are expressedin terms of the New- Φ(r) r2, while the planar structures generated by ton gravitationalpotential Φ as [12],[14],[29] ∼ the motion of cosmic strings (the so called ”wakes”) Φ(r) canyieldcosmologicalstructureswherethetotalfluxof g 1+2 ; (2.5) 00 ≃ c2 lightis preservedandthe brightnessofobjects athigh redshift, whose radiation passes through such struc- Φ(r) g δ 1 2 ; (2.6) tures, appears higher to a far observer. ik ≃− ik − c2 Sec. 2 is devoted to the discussion of the gravi- (cid:18) (cid:19) tational potential intended as the refraction index of where we are assuming the weak field Φ/c2 1 and ≪ geometricaloptics. In Sec. 3, we derivethe Helmholtz the slow motion approximation v c. Then, due | | ≪ scalar equation, starting from the Maxwell equation to relations (2.3), (2.5) and (2.6), the refraction index for the electromagnetic field, which is the dynamical n(r) in (2.4) can be expressed in terms of the gravita- equation for the optical waveguides. tional potential Φ(r) produced by some matter distri- In Sec. 4, we costructthe optical waveguidemodel bution. Such a weak field situation is realized for cos- using the paraxial approximation (the so called Fock– mologicalstructureswhichgiverisetothegravitational Leontovich approximation [23]). We propose a model lensingeffectsconnectedtoseveralobservablephenom- in which we consider,instead of a simple gravitational ena (multiple images, magnification, image distorsion, lensing effect, the effect of a sort of system of lenses arcsandarclets)[14]. Here,weareinterestedtoaspe- which, combined in files or in planes, results as a cificapplicationwhichcouldberealizedbysomekinds waveguide. In Sec. 5, we discuss the eventual cos- of gravitational systems as cosmological string–like or mological realization of such structures and the con- planar–like distributions of matter. nection with observations, in particular with quasars, giving some simulations. Conclusions are drawn in 3. The Helmholtz Equation Sec.5. Inthis section,wederivethe Helmholtz equationfrom 2. The propagation of light in a weak the Maxwellequationsin media reducingthe equation gravitational field with interaction of different light polarizations to a scalar equation. Let us write the Maxwell equations The behaviour of the electromagnetic field without forthe electromagneticfieldinmedia without sources, sources in the presence of a gravitational field is de- i.e. J=0,ρ=0, have the form scribed by the Maxwell equations [30],[31] 1∂B(r,t) rot E(r,t) = , (3.1) ∂Fαβ ∂Fβγ ∂Fγα −c ∂t + + =0; (2.1) ∂xγ ∂xα ∂xβ 1∂D(r,t) rot H(r,t) = , (3.2) 1 ∂ c ∂t √ g∂xβ √−gFαβ =0, (2.2) div B(r,t) = 0, (3.3) − (cid:0) (cid:1) div D(r,t) = 0, (3.4) where Fαβ is the electromagnetic field tensor and √ g is the determinant of the four–dimensional met- where the media contributionis taken into accountby − ric tensor. For a static gravitational field, these equa- the relations tions can be reduced to the usual Maxwell equations D (r)=ε(ω,r)E (r), (3.5) describing the electromagnetic field in media where ω ω the dielectric and magnetic tensor permeabilities are B (r)=µ(ω,r)H (r), (3.6) ω ω connected with the metric tensor g by the equation µν [29] where the subscript ω means the Fourier amplitudes ε =µ = g−1/2[detg ]−1/2g ; (2.3) of the fields, i.e. ik ik − 00 ik ik i,k =1,2,3. D(r,t)= Dω(r)e−iωtdω, (3.7) Z If one has an isotropic model, the metric tensor is di- andanalogouslyfor E,B,H. Then,takingtheFourier agonal and the refraction index may be introduced by transforms with respect to the time variable, we get mimicking the gravitational field iω n(r)=(εµ)1/2, (2.4) rotEω(r) = Bω(r), (3.8) c Can quasars be explained by cosmological waveguide effects? 3 iω rotH (r) = D (r), (3.9) Comparing Eqs.(3.21),(3.22) and (3.23), we con- ω ω − c clude that for divB (r) = 0, (3.10) ω δε divD (r) = 0. (3.11) 1, (3.24) ω ε ≪ Using the relations (3.5), (4.9) we get we can neglect the term depending on the light polar- izationinteractionwithrespecttotheothertwoterms. iω rotE (r) = µ(ω,r)H (r), (3.12) In this approximation we get the scalar Helmholtz ω ω c equation for all the decoupled components of the elec- iω rotH (r) = ε(ω,r)E (r), (3.13) tric vector field, i.e. ω ω − c µdivH (r) = 0, (3.14) ω2 ω E (r)+ n2(ω,r)E (r)=0. (3.25) div[ε(ω,r)E (r)] = 0. (3.15) △ ω c2 ω ω IfonehasasolutionoftheHelmholtzequation E(0)(r), Letusconsiderthemagneticpermeability µ tobecon- ω either exact or approximate one, the influence of the stant. Being the operator equality light polarization interaction may be taken into ac- rotrot=graddiv , (3.16) counttheBornmethodofiteration. Infact,theGreen −△ function given by Eq.(3.25) G(r,r′,ω) or by an ap- we get from Eq.(3.12) proximation of this equation, satisfies the equation graddivEω(r)−△Eω(r)= + ω2n2(ω,r) G(r,r′,ω)=δ(r r′). (3.26) △ c2 − ω2 (cid:20) (cid:21) = µ(ω,r)ε(ω,r)E (r). (3.17) c2 ω Thenthesolutionoftheequationwiththepolarization term has the form Introducing the refractive index E (r)=E(0)(r)+ n2(ω,r)=µ(ω,r)ε(ω,r), (3.18) ω ω E(0)(r) ε(ω,r) we can rewrite Eq.(3.17) as + G(r,r′,ω) ω ∇ dr. (3.27) ∇" ε(ω,r) # Z ω2 E (r)+ n2(ω,r)E (r)= △ ω c2 ω Eq.(3.26) has the form equivalent to the equation for the GreenfunctionoftheSchrödingerequationforthe E (r) ε(ω,r) = ω ∇ . (3.19) energy constant E equal to zero. In fact, if we write −∇ ε(ω,r) down the Hamiltonian operator (cid:20) (cid:21) Here we have used the relation ˆ = 1 +U(r), (3.28) E (r) ε(ω,r) H −2△ divE (r)= ω ∇ . (3.20) ω − ε(ω,r) with h¯ =m=1,andthe equationforthe Greenfunc- tion of the Schrödinger equation G (r,r′,E) which is s One can neglect the term in the right–hand side of the matrix element of the operator (ˆ E)−1 in the Eq.(3.19)ifitismuchlessthanbothtermsinthe left– coordinate representation H− handsideofthesamerelation. Infact,since = fordistancesofanorderofthelightwaveleng△th λ∇, t·h∇e Gs(r,r′,E)= r(ˆ E)−1 r′ , (3.29) h | H− | i bothtermsintheleft–handsideofEq.(3.19)(indepen- which comes from the equation dently of the light polarization) are of the order (ˆ E)G (r,r′,E)=δ(r r′). (3.30) E (r) λ−2E (r), (3.21) H− s − ω ω |△ |∼ The comparisonof this equation in explicit form ω2 c2n2(ω,r)Eω(r)∼λ−2Eω(r). (3.22) 1 +U(r) E G (r,r′,E)=δ(r r′) (3.31) s −2△ − − The term depending on the light polarization interac- (cid:26) (cid:27) tion for the same distances is of the order withEq.(3.26)showsthatthey areidenticalfor E =0 with the replacements E (r) ε(ω,r) δε ω ∇ λ−2 E (r), (3.23) ∇(cid:20) ε(ω,r) (cid:21)∼ ε ω U(r)= 2ω2n2(ω,r), (3.32) − c2 where δǫ is the change of the dielectric permeability for distances of the order of wavelength λ. 2G(r,r′)=G (r,r′,0). (3.33) s − 4 S.Capozziello and G.Iovane Thus,wehaveshownthatifoneknowstheGreenfunc- isalongthe z axis. WerewriteEq.(4.1)neglectingsec- tion G (r,r′,E) of the Schrödinger equation for the ond order derivative in longitudinal coordinate z and s unit mass particle moving in a potential like that in obtain a Schrödinger–like equation for Ψ: Eq.(3.32), the Green function of the Helmholtz equa- ∂Ψ tion (3.26) is given by the equality iλ = (4.3) ∂ξ 1 G(r,r′)= Gs(r,r′,E =0). (3.34) λ2 ∂2Ψ ∂2Ψ 1 −2 = + + n2(z) n2(x,y,z) Ψ, − 2 ∂x2 ∂y2 2 0 − Since the Green function for the Schrödinger equation (cid:18) (cid:19) (cid:2) (cid:3) arestudiedformanypotentials,theresultsobtainedin where λ is the electromagnetic radiation wavelength quantummechanicscanbeappliedforourpurposesto and we adopt the new variable study polarization and waveguiding effects since they z dz′ are formally identical. ξ = , (4.4) n (z′) 0 Z 4. The gravitational waveguide model normalized with respect to the refraction index [16] (for our application, n (z) 1 so that ξ coincides 0 ≃ essentially with z). Following the above procedure for deriving the scalar Helmholtz equationfor the components of the electro- At this point, it is worthwhile to note that if one magnteticfieldfromthefirstorderMaxwellequations, hasthe distributionofthe matter inthe formofcylin- we get (for some arbitrary monochromatic component der with a constant (dust) density ρ0 , the gravita- of the electric field) tionalpotentialinsidehasaparabolicprofileproviding waveguide effect for electromagnetic radiation analo- ∂2E ∂2E ∂2E gous to sel–foc optical waveguidesrealized in fiber op- + + +k2n2(r)E =0, (4.1) ∂z2 ∂x2 ∂y2 tics. In this case, Schrödinger–like equation is that of two–dimensional quantum harmonic oscillator for where k is the wave number. This procedure works which the mode solutions exist in the form of Gauss– if, as we have seen, the relative change of diffraction Hermite polynomials (see, for example, [24]). In the index on distances of the light wavelength is small. case of inhomogeneous longitudinal dust distribution The coordinate z, in Eq.(4.1), is considered as the in the cylinder (that is ρ(z)), the Schrödinger-like longitudinaloneanditcanmeasurethespacedistance equationdescribesthe modeloftwo-dimensionalpara- along the gravitational field structure produced by a metric oscillator for which the mode solutions, in the mass distribution with an optical axis. Such a coordi- formofmodifiedGaussianandGauss–Hermitepolyno- nate may also correspondto a distance along the light mials,existwithparametersdeterminedbythedensity path inside a planar gravitational field structure pro- dependence on longitudinal coordinate. duced by a planar matter–energy distribution in some As a side remark, it is interesting to stress that, regions of the universe. In other words, if one has a consideringagainEq.(4.3),theterminsquarebrackets matterdistributionwithsomeaxislikeacylinderwith in the rhs plays the role of the potential in a usual dust or like a planar slab with dust, it is possible to Schrödinger equation; the role of Planck constant is consider the electromagnetic field radiation propagat- now assumed by λ. Since the refraction index can be ing paraxially. The parabolic approximation [23] is expressed in terms of the Newtonian potential when used for describing light propagation in media and in we consider the propagationof light in a gravitational devices as optical fibers [15]. Below, we will discuss field, we can write the potential in (4.3) as the possibility to use this approximation for describ- 2 ing electromagnetic radiation propagating in a weak U(r)= [Φ(x,y,z) Φ(0,0,z)]. (4.5) c2 − gravitationalfield. Let us consider, the scalar equation (4.1) and the Thewaveguideeffectdependsspecificallyontheshape electric field E of the form of potential (4.5): for example, the radiation from a remotesourcedoesnotattenuateif U r2;thissitua- z ∼ E =n−1/2Ψexp ik n (z′)dz′ ; (4.2) tionisrealizedsupposinga”filamentary”ora”planar” 0 0 massdistribution with constantdensity ρ. Due to the (cid:18) Z (cid:19) Poissonequation,the potentialinside the filamentis a n0 n(0,0,z), quadratic function of the transverse coordinates, that ≡ is of r = x2+y2 in the case of the filament and of where Ψ(x,y,z) is a slowly varying spatial amplitude r = x in the case of the planar structure (obviously p along the z axis, and exp(iknz) is a rapidly oscillat- thelightpropagatesinthe”remaining”coordinates: z ing phase factor. Its clear that the beam propagation for the filament, z,y for the plane). In other words, Can quasars be explained by cosmological waveguide effects? 5 if the radiation, travelling from some source, under- (x2+y2)[(ωlk)2 ωk(i+kl)cotωl] exp − goesawaveguideeffect,itdoesnotattenuatelike 1/R2 × − 2(1 ikl ikωl2cotωl) (cid:18) − − (cid:19) as usual, but it is, in some sense conserved; this fact means that the source brightness will turn out to be a2ωk(i+kl)cotωL exp muchstrongerthanthebrightnessofanalogousobjects × 2(1 ikl ikωl2cotωl) (cid:18) − − (cid:19) located at the same distance (i.e. at the same redshift 2xaωk(1+kl) Z)andtheapparentenergyreleasedbythesourcewill exp . be anomalously large. × −2sinωL(1 ikl ikωl2cotωL) (cid:18) − − (cid:19) To fix the ideas, let us estimate how the electric The parameter l drops out of the denominator of the field (4.2) propagates into an ideal filament whose in- pre–exponentialfactorifthelength L satisfiesthecon- ternal potential is dition 1 4πGρ U(r)= ω2r2, ω2 = (4.6) tanωL= ωl. (4.14) 2 c2 − where ρ is constantand G is the Newton constant. A Eq.(4.13) is interesting in two limits. If ωl 1, we ≪ spherical wave from a source, have 1 l+iλ E =(1/R)exp(ikR), (4.7) Ψ = exp (x+a)2+y2 , (4.15) f iλ − 2λ2l (cid:26) (cid:27) can be represented in the paraxial approximationas (cid:2) (cid:3) whichmeans thatthe radiationemergingfromapoint 1 ikr2 r2 with coordinate (a,0,0) is focused near a point with E(z,r)= exp ikz+ , (4.8) z 2z − 2z2 coordinates ( a,0,l + L) (that is the radius has to (cid:18) (cid:19) − be of the order of the wavelength). This means that, where we are using the expansion when the beamfrom anextended sourceis focused in- R= z2+r2 1/2 z 1+ r2 , r z. (4.9) sidethewaveguideinsuchawaythat,atadistance L, ≈ 2z2 ≪ Eq.(4.14) is satisfied, an inverted image of the source (cid:18) (cid:19) (cid:0) (cid:1) is formed, having the very same geometrical dimen- It is realistic to assume n 1 so that, from (4.4), 0 sionsofthe source. Thewaveguide”draws”the source ≃ ξ = z. Assume now that the starting point of the closer to the observer since, if the true distance of the filament of length L is at a distance l from a source observer from the source is R, its image brightness shifted by a distance a from the filament axis in the willcorrespondtothatofasimilarsourceatthecloser x direction. The amplitude Ψ ofthe field E, entering distance the wave guide is R =R l L. (4.16) 1 ikl 1 eff − − Ψ = exp − (x a)2+y2 , (4.10) in l 2l2 − If we do not have ωl 1, we get (neglecting the term (cid:20) (cid:21) ≪ (cid:0) (cid:1) iλ/l compared with unity) and so in (4.7), we have R= l2+y2+(x a)2 1/2. We can calculate the amplitude of the fi−eld at the 1+(ωl)2 (cid:0) (cid:1) Ψf = (4.17) exit of the filament by the equation iλ × p Ψ (x,y,l+L)= (4.11) 2 f 1+(ωl)2 a exp y2+ x+ , × − 2λ2  1+(ωl)2!  = dx dy G(x,y,l+L,x ,y ,l)Ψ (x ,y ,l),   1 1 1 1 in 1 1 Z  p  from which, in general, the size of the image is de- where G is the Green function of Eq.(4.3). For the creased by a factor 1+(ωl)1/2. The amplitude in- potential (4.6), G has the form creases by the same factor, so that the brightness is p ω (R/R ) times larger. G(x,y,l+L,x ,y ,l)= (4.12) eff 1 1 2πiλsinωλ× In the opposite limit ωl 1, we have tanωL ≫ → , so that L π/ω, that is the shortest focal length iω[cosωL(x2+y2+x2+y2) 2(xx +yy )] ∞ ≃ exp 1 1 − 1 1 , of the waveguide is 2πiλsinωλ (cid:18) (cid:19) πc2 whichisthepropagatoroftheharmonicoscillator. The L = , (4.18) foc integral (4.11) is Gaussian and can be exactly evalu- s4Gρ ated which is the length of focusing of the initial beam of ωl light trapped by the gravitational waveguide. All this Ψ = (4.13) f ωl2cosωL+(l+iλ)sinωl × argumentsapplyifthewaveguidehas(atleastroughly) 6 S.Capozziello and G.Iovane acylindricalgeometry. Thetheoryofplanarwaveguide magnitude magnification of brilliancy. The waveguide is similarbutwe haveto consideronly x astransverse effect may explain the anomalous high luminosity ob- dimensionandnotalso y. Thecosmologicalfeasibility served in quasars. In fact, quasars are objects at very ofawaveguidedepends onthe geometricaldimensions high redshift which appear almost as point sources ofthestructures,ontheconnecteddensitiesandonthe but haveluminosity that areabout one hundredtimes limitsofapplicabilityoftheaboveidealizedscheme. In than that of a giant elliptical galaxy (quasars have thenextsection,weshalldiscussthesefeaturesandthe luminosity which range between 1038 1041 W). For − possiblecandidateswhichcouldgiverisetoobservable example, PKS 2000-330 has one of the largest known effects. redshifts (Z = 3.78) with a luminosity of 1040 W. Such a redshift corresponds to a distance of 3700 Mpc, if it is assumed that its origin is due to the 5. Cosmic structures as waveguides expansion of the universe and the Hubble constant is and quasars assumed H = 75km s−1 Mpc. This means that the light left the quasar when the size of the universe was The gravitationalwaveguideeffect has the same phys- one–fifth of its present age where no ordinary galaxies ical reasonthat has the gravitationallens effect which (includedthesupergiantradio–galaxies)areobserved. is the electromagnetic wave deflection by the gravi- The quasars, often, have both emission and absorp- tational field (equivalent to the deflection of light by tion lines in their spectra. The emission spectrum is refractive media). However, there are essential differ- thought to be produced in the quasar itself; the ab- ences producing specific predictions for observing the sorptionspectrum, ingas cloudsthat haveeither been waveguide effect. The gravitational lenses are usu- ejected from the quasar or just happen to lie along allyconsideredascompactobjectswithstrongenough the same line of sight. The brightness of quasars may gravitational potential. The light rays deflected by vary rapidly, within a few days or less. Thus, the gravitational lenses move outside of the matter which emittingregioncanbenolargerthanafewlight–days, forms the gravitational lens itself. The gravitational i.e. about one hundred astronomic units. This fact waveguide as well as optical waveguide is noncompact excludes that quasars could be galaxies (also if most long structure which may contain small matter den- astronomers think that quasars are extremely active sity and the deflection of light by each element of the galactic nuclei). structure is very small. Due to very large scale sizes Themainquestionishowtoconnectthissmallsize of the structure (we give an extimation below), the with the so high redshift and luminosity. For exam- electromagnetic radiation deflection by the gravita- ple, H.C. Arp discoveredsmallsystems of quasarsand tional waveguide occurs and, in principle, it may be galaxies where some of the components have widely observed. We will mention, for example, a possibil- discrepant redshifts [32]. For this reason, quasar high ity of brilliancy magnification of the long distanced redshift could be produced by some unknown process objects (like quasars) with large red shift as conse- andnotbeingsimplyofcosmologicalorigin. Thisclaim quenceofthewaveguidingstructureexistencebetween is very controversial. However there is a fairly widely the object and the observer. This effect exists also accepted preliminary model which, in principle, could foragravitationallenslocatedbetweentheobjectand unify all the forms of activities in galaxies (Seyfert, theobserver,butthelonggravitationalwaveguidemay radio, Markarian galaxies and BL Lac objects). Ac- givehugemagnification,sincetheradiationpropagates cording to this model, most galaxies contain a com- alongthewaveguidewithfunctionaldependenceofthe pact central nucleus with mass 107 109 M⊙ and di- ÷ intensity on the distance which does not decrease as ameter < 1 pc. For some reason, the nucleus may, 1/R2 , characteristic for free propagation. The sometimes,releaseanamountofenergyexceedingthe ∼ gravitational lens, being a compact object, collects power of all the rest of the galaxy. If there is only lit- much less light by deflecting the rays to the observer tle gas near the nucleus, this leads to a sort of double than the gravitational waveguide structure transport- radiosource. Ifthe nucleuscontainsmuchgas,the en- ing to the direction of observer all trapped energy (of ergy is directly released as radiation and one obtains course, one needs to take into account losses for scat- a Seyfert galaxy or, if the luminosity is even larger, a tering and absorbtion). From that point of view, it is quasar. In fact, the brightest type 1 Seyfert galaxies possible that enormous amount of radiation emitted and faintest quasars are not essentially different in lu- by quasarsis only seemingly existing. The object may minosity( 1038 W)alsoifthequestionofredshifthas ∼ radiate a resonable amount of energy but the waveg- to be explained (in fact quasar are, apparently, much uide structure transmits the energy in high portion to moredistant). Finally,ifthereisnogasatallnearsuch the observer. Similar ideas, related to gravitational an active nucleus, one gets BL Lac objects. These ob- lensing, were discussed in [40] but, since above men- jects are similar to quasar but show no emission lines. tioned reasons, the singular lens or even few aligned Howeverthemechanismtoreleasesuchalargeamount strong lenses cannot produce effect of many orders of ofenergyfromactivenucleiorquasarsisstillunknown. Can quasars be explained by cosmological waveguide effects? 7 Somepeoplesupposethatitisconnectedtothereleas- ing of gravitational energy due to the interactions of internal components of quasars. This mechanism is extremely more efficient than the releasing of energy during the ordinary nuclear reactions. The necessary gravitational energy could be produced, for example, as consequence of the falling of gas in a very deep po- tentialwellasthatconnectedwithaverymassiveblack hole. Only with this assumption, it is possible to justify a huge luminosity, a cosmological redshift and a small size for the quasars1. Analternativeexplainationcouldcomefromwaveg- uiding effects. As we have discussed, if light travels within a filamentary or a planar structure, whose Newtonian gravitational potential is quadratic in the transverse coordinates, the radiation is not attenu- ated, moreover the source brightness is stronger than the brightness of an analogous object located at the same distance (that is at the same redshift). In other words,ifthelightofaquasarundergoesawaveguiding effect, due to some structure along the path between it and us, the apparent energy released by the source Figure 1: The number of quasars against their local will be anomalously large, as the object were at a separation. The excess could be attributed to some distance (4.16). Furthermore, if the approximation lensing effect. It is clear the deviation with respect to ωl 1 does not hold, the dimensions of resulting image≪would be decreased by a factor 1+(ωl)2 while the expectedPoissoniandistribution(CRONAproject the brightness would be (R/R )2, larger, then ex- internal communication). eff p plaining how it is possible to obtain so large emission by such (apparently) small objects. In conclusion, the existence of a waveguidingeffect may prevents to take into consideration exotic mechanism in order to produce huge amounts of energy (as the existence of a massive black hole inside a galactic core) and it may justify why it is possible to observe so distant objects of small geometrical size. Another effect concerning the quasars may be directly connected with multiple images in lensing. The waveguideeffectdoesnotdisappearifthe axisof“filament” or if the guide plane is bent smoothly in space. As in the case of gravitational lenses, we can observe “twin” images if part of the radiation comes to the observer directly from the source, and another part is captured by the bent waveguide. The “virtual” image can then turn out to be brighter than the “real” Figure 2: The random distribution of quasars in red- one (in this case we may deal with “brothers” rather shift Z on a sphere of radius 3000 Mpc. We take ∼ than “twins” since parameters like, spectra, emission into accountanattenuation effect by which we cannot periods and chemical compositions are similar but the see quasars with a distance 1500 Mpc if the lumi- brightnesses are different). Furthermore, such a bend- nosity goes as r−2. After th≥is attenuation, from 105 ing in waveguide could explain large angular separa- initial objects, we observe only 980 of them. tions among the images of the same object which cannot be explained by the current lens models (pointlike lens, thin lens and so on). Thisfeaturecouldaffectsalsothestatisticsexplain- ing why the luminosity distribution of quasars in the 1However, the question is extremely controversial and some skyisnotaPoissondistribution,asexpectedconsider- people do not believe to the presence of the black hole. An ing the number of quasars and their relative distances intriguingalternativeisthatproposedbyViollier[33]whosup- [9]. What is observed is a double peak and the excess poses that a heavy neutrino matter condensation could repro- ducetheverymassivecoreofquasars. could be attributed to some lensing effect (see Fig.1). 8 S.Capozziello and G.Iovane criticaldensityoftheuniverse ρ 10−29 g/cm3 (that c ∼ isthevalueforwhichthedensityparameteris Ω=1), we obtain L 5 104 Mpc which is an order of foc ∼ × magnitude larger than the observable universe and it is completely unrealistic. Onthe contrary,considering a typical galactic density as ρ 10−24 g/cm3, we ob- ∼ tain L 100 Mpc, which is a typical size of large foc ∼ scale structure (e.g. the Great Wall has such dimensions and also a filament of galaxies can have such a length [35]). However, an important issue has to be taken into consideration: the absorption and the scattering of light by the matter inside the filament or the planar structure increase with density and, at certain criti- cal value, the waveguide effect can be invalidated [16]. Figure 3: Superposing a random distribution of 200 For the smaller frequency of broadcast range (due to waveguides(withsizes 100 Mpcoflengthand 100 the strong dependence of the absorbtion cross section ∼ ∼ kpc of thickness) we get that 137 of the above quasars on the electromagnetic wavelength) σ σ (ω/ω )4, T 0 (14%)results“twinsed”. Thissimplesimulationshows whereThomsoncross-section σ =6 10∼−25 cm2 and T that it is quite easy to get the double peak in the dis- the characteristic atomic frequency is· ω 1016 s−1, 0 ∼ tribution of quasars by using waveguides. the ratio ω/ω 1, and the absorbtion is small. 0 ≪ It means that the absorbtion length L = m /ρσ, a p (wherethemassofproton m isapproximatelyequal p Using a simple simulation, which we are going to ex- to the hydrogen atom mass) is larger than the focus- plain, it is possible to implement the effect by using ing length L < L for the electomagnetic waves a foc a distribution of waveguides. Let us consider a uni- ofbroadcastrange. Thus,themagnificationofelectro- form and isotropic distribution of quasars ( 105) in ∼ magneticwavesmaybenotmaskedbyessentialenergy a 3000Mpc sphere,whosepeak is atredshift Z 2.5. ∼ lossesduetolightabsorbtionandscatteringprocesses. We can take into consideration also some attenuation However, no restrictions exist practically if the radio effect which selects only quasars over some brilliancy band and a thickness of the structure r >1014cm are treshold (Fig. 2). Superposing a random distribution considered. of filamentary waveguides ( 200) of length 100 ∼ ∼ In sucha case,the relative density change between Mpc which give rise to “twins effect” as soon as they the background and the structure density is valid till interact with quasars (i.e. as soon as they catch the δρ/ρ 1 . Thismeansthatwehavetostayinalinear radiation emitted by quasars), the double peak distri- ≤ perturbation density regime. butionisreproducedexplainingtheexcess2(seeFig.3). By such hypotheses, practically all the observed Nowtheissueis: aretherecosmicstructureswhich largescalestructureslikefilaments,walls,bubblesand can furnish workable models for waveguides? Have clusters of galaxiescanresultas candidates for waveg- they to be “permanent” structures or may the waveg- uidingeffectiftherestrictionsondensity,potentialand uide effect be accidental (for example an alignment of wavebandarerespected(inopticalband,suchphenom- galaxiesofsimilardensityandstructure,duetocosmic ena are possible but the density has to be chosenwith shear and inhomogeneity, may be available as waveg- some care). uidejustforalimitedintervaloftime[34])? Ingeneral, However, it is well-known that lensing effects re- bothpointsofviewmaybereasonableandherewewill lated to large scale structures, corresponding to den- outlinebothofthem. Furthermorewehavetoconsider sity contrasts equal or smaller than 1, do not give the problem of the abundance of such structures: are strong lensing effects (in these situations, we are deal- they common and everywhere in the universe or are ing with ”weak lensing” phenomena). In our case, they peculiar and located in particular regions (and the effect is different since the light is ”trapped” and eras)? ”guided”insidethestructures. Differently,inordinary We have to do a first remark on the densities of lensing light travels outside the structures. In addi- waveguide structures which allow observable effects tion, taking into account fractal models for large scale [16]. ConsideringEq.(4.18)andintroducinginto itthe structures,one could have stronggravitationallensing 2Byasimilarsimulation,itcouldbepossiblealsotoexplain effects thanks to the self-similarity properties of the the same existence of quasars implementing the above mecha- models (for an exhaustive discussion about the topic, nism. Givenadistributionofgalaxiesorprotogalaxies(quasars seeforexample[20]). Inanycase,thewaveguideeffect are very old objects if their redshift has a cosmological ori- couldbe alsointerestingin a fractalmodel context: in gin) which cannot be revealed, the presence of a distribution ofwaveguides betweenthemandusallowstheirdetection. fact there is no contradiction between them. Can quasars be explained by cosmological waveguide effects? 9 Also primordial structures (produced in inflation- tation to the Earth) could produce the effect of the ary phase transition and surviving later), like cosmic disappearence (or the appearence)of the observedob- string, could furnish waveguides. In fact, in weak en- ject. Forthis analysis,itis crucialtoconsiderlongpe- ergy limit approximation, such objects are internally riod astronomical observations and deep pencil beam described by the Poisson equation 2Φ = ρ and ex- surveys of galaxies and quasars. 0 ∇ ternally by 2Φ = 0 and, furthermore, they act as ∇ gravitational lenses after the formation of the quasar 6. Conclusions [21],[22]. It is easy to recover an internal potential of the form Φ r2 and, considering the dynamical ∼ Inthis paper, we havediscussedthe possible existence evolution after the decoupling, lengths in the required of gravitational waveguide effects in the universe and ranges for waveguide (e.g. 100 Mpc). The main ∼ constructed a radiation propagation model to realize problem is due to the fact that also after the evolu- them. As in the case of gravitational lensing, several tiontomacroscopicsizes,stringsremain“wires”with- phenomena and cosmic structures could confirm their outbecomingcylinders,thatistheirthicknessremains existence, starting from primordial object like cosmic wellbelow r 1014 cm,the minimalvaluerequiredto ∼ strings to temporary alignement of evolved late–type get observable effects. However, we have not consid- galaxies. Furthermore, due to the wavelength considered the scaling solutions (see for example [36]) from ered, they could give observable effects in optic, radio which such wires could evolve in cylindrical structures or microwave bands or, alternatively, considering the (with transverse sizes non trivial with respect to the propagation of other weak interacting particles as the background). neutrinos. The experimental feasibility for the detec- Other two interesting features are connected with tion could have serious troubles due to the need of cosmic strings: the first is that their motion with re- long period observations or due to the discrimination spectto the backgroundproduceswakesandfilaments among data coming from objects which have under- which, later, are able to evolve in large scale struc- gone waveguide effects and objects which not. tures systems of galaxies [37]. For example, at decou- In any case, if such a hypothesis will be confirmed pling (Z 1000), a string can produce a wake, which ∼ insomeoftheabovequotedsenses,weshallneedapro- consists in a planar structure, with side ∆r 1 Mpc andconstantsurfacedensity σ0 ∼3×1011M⊙∼Mpc−2. ftouurendanrdevmisaiotnteorfdoisutrricbountcioenp.tions of large scale struc- Such a feature is interesting for large scale structure Finally we want to stress that our treatment does formationandcanyieldaplanarwaveguidewithtoday notconcernonly electromagneticradiation: actuallya observable effects. The secondfact is that inflationary waveguide effect could be observed also for streams of phase transition can produce a large amount of cos- neutrinos [41], gravitational waves [42], or other par- mic strings which, evolving, can give rise to a string ticles which gravitationally interact with the filament network pervading all the universe [1],[21]. 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