Massive Primordial Black Holes in Hybrid Inflation

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介绍黑洞的引言英文作文

介绍黑洞的引言英文作文Introduction to Black Holes。

Black holes, one of the most fascinating objects in the universe, have been a subject of scientific research for decades. These mysterious objects are formed when a massive star collapses under its own gravity, creating a region in space where the gravitational pull is so strong that nothing, not even light, can escape. The concept of black holes was first proposed by the physicist John Michell in 1783 and was later refined by Albert Einstein's theory of general relativity in 1915.Black holes are invisible to the naked eye, as they do not emit any light or radiation. However, their presence can be detected by observing the effects of their immense gravitational pull on nearby stars and gas. The area surrounding a black hole is known as the event horizon, which marks the point of no return. Anything that crosses the event horizon is pulled into the black hole and is lostOne of the most intriguing aspects of black holes is their ability to distort time and space. The intensegravity near a black hole causes time to slow down, and space to be warped and stretched. This phenomenon, known as gravitational time dilation, has been confirmed by observations of stars orbiting black holes.Black holes come in different sizes, ranging from a few times the mass of the sun to billions of times the mass of the sun. Supermassive black holes, found at the centers of galaxies, are thought to play a crucial role in the evolution of galaxies, as they can influence the motion of stars and gas.Despite their mysterious nature, black holes have become a topic of intense research in astrophysics and cosmology. Scientists are using a variety of techniques, such as gravitational wave detection and observations of the effects of black holes on nearby objects, to study these enigmatic objects and unlock the secrets of theIn conclusion, black holes are one of the most intriguing and mysterious objects in the universe. They are formed by the collapse of massive stars and have a gravitational pull so strong that nothing can escape. Their ability to distort time and space and influence the motion of nearby objects makes them a subject of intense research in astrophysics and cosmology.。

Supermassive Black Holes Their Formation, and Their Prospects as Probes of Relativistic Gra

arXiv:astro-ph/9912346v1 16 Dec 1999
Supermassive Black Holes: Their Formation, and Their Prospects as Probes of Relativistic Gravity
Martin J. Rees
Institute of Astronomy Madingley Road, Cambridge, CB3 0HA, UK
The situation was transformed by remarkable observations of stars in the near infrared band, where obscuration by intervening material is less of an obstacle. These are presented by Ekhart and by Ghez at this meeting. The speeds scale as r−1/2 with distance from the centre, consistent with a hole of mass 2.5 × 106 M⊙.
Abstract. The existence of supermassive collapsed objects in the cores of most galaxies poses still-unanswered questions. First, how did they form, and how does their mass depend on the properties of the host galaxy? Second, can observations probe the metric in the strong-ﬁeld domain, testing whether it indeed agrees with the Kerr geometry predicted by general relativity (and, if so, what the spin is)?

黑洞的由来的英语作文

黑洞的由来的英语作文The Origin of Black Holes: A Journey into Cosmic Mysteries。

Introduction。

Black holes, enigmatic entities lurking in the depthsof space, have captivated the imagination of scientists and laypersons alike. Their origins, shrouded in cosmic mystery, have been the subject of intense study and speculation. In this essay, we embark on a journey to unravel the secretsof black holes, exploring their formation, properties, and significance in the universe.Formation of Black Holes。

The genesis of black holes begins with the demise of massive stars. When a massive star exhausts its nuclear fuel, it undergoes a cataclysmic event known as a supernova explosion. During this explosive phase, the outer layers ofthe star are ejected into space, while its core undergoes gravitational collapse. If the core's mass exceeds acritical threshold, it collapses into a singularity—a point of infinite density—giving birth to a black hole.The process of black hole formation can also occur through the gravitational collapse of dense stellar remnants, such as neutron stars, or through the merger of two compact objects, such as neutron stars or black holes. These pathways lead to the creation of different types of black holes, ranging from stellar-mass black holes to supermassive black holes found at the centers of galaxies.Properties of Black Holes。

关于黑洞的演化

Title: Evolution of Black HolesAbstract: Black holes are fascinating objects in the universe that have a significant impact on the surrounding matter. Understanding their evolution is crucial to comprehend the dynamics of the universe. This paper discusses the formation, growth, and evolution of black holes.Introduction: Black holes are immensely dense and compact objects that result from the collapse of massive stars. The gravitational pull of a black hole is so intense that not even light can escape it. Hence, studying the behavior of black holes can provide valuable insights into the workings of the universe. In this paper, we delve into the evolution of black holes, starting from their formation to their current state.Formation of Black Holes: The formation of black holes occurs when a massive star runs out of fuel and its core collapses under its own gravity. The core collapses to a point of zero volume and infinite density, forming a singularity. The point of no return, also known as the event horizon, forms around the singularity, marking the boundary beyond which nothing can escape.Growth of Black Holes: Black holes continue to grow over time by accreting matter from the surrounding environment. The matter in the vicinity of a black hole can be pulled towards it due to its strong gravitational pull. As matter falls towards the black hole, it heats up and releases energy in the form of radiation. This process is known as accretion.Evolution of Black Holes: As black holes grow larger, they can merge with other black holes to form ones even bigger. When two black holes merge, the resultant black hole's size is proportional to the sum of the masses of the merging black holes. The final size of the black hole's event horizon increases, allowing it to capture even more matter.Conclusion: In conclusion, understanding the evolution of black holes is crucial in understanding the mechanics of the universe. Black holes form when massive stars run out of fuel and collapse under their own gravity. They continue to grow by accreting matter from their surroundings and can merge to form even larger ones. Therefore, further research is necessary to explore the different stages in the evolution of black holes.。

Supermassive Black Holes and Galaxy Formation

a r X i v :a s t r o -p h /0109325v 1 19 S e p 2001Supermassive Black Holes and Galaxy Formation1JOSEPH SILKDepartment of Physics,Nuclear and Astrophysics Laboratory,Keble Road,Oxford OX13RH,United KingdomABSTRACTThe formation of supermassive black holes (SMBH)is intimately related to galaxy formation,although precisely how remains a mystery.I speculate that formation of,and feedback from,SMBH may alleviate problems that have arisen in our understanding of the cores of dark halos of galaxies.1.INTRODUCTIONGalaxy formation theory is not in a very satis-factory state.This stems ultimately from our lack of any fundamental understanding of star formation. There is no robust theory for the detailed proper-ties of galaxies.In contrast,the hierarchical forma-tion of large-scale structure in a cold dark matter-dominated Friedmann-Lemaˆitre universe has success-fully confronted essentially all observations,ranging from deep surveys of the galaxy distribution,the for-mation of galaxy clusters,and the temperatureﬂuc-tuations in the cosmic microwave background.2.The apparent demise of CDMThe advent of high resolution N-body simulations has revealed challenges for galaxy formation.Dark halos contain considerable sub-structure,amounting to of order10%of the halo mass,and continuing down to unresolved scales of106M⊙or smaller.This has led to two problems.One is that the predicted number of dwarf gallaxy satellites exceeds that observed around the Milky Way by an order of magnitude.Even very low surface brightness dwarfs are easily observed in our local environment,and eﬃcient mass loss prior to star formation has been invoked to resolve this dis-crepancy(Gnedin2001).However,since stars will form before the gas is en-tirely stripped,the dwarfs should be visible.Star for-mation is a local process that occurs in localized in-homogeneities,and the local free-fall time is short be-cause the sound crossing time is short,whereas strip-ping is a global process that requires star formation toﬁrst occur,even if externally imposed,e.g.by an ionizing UV radiationﬁeld.Another manifestation of excessive substructure in CDM halos is the failure of the CDM model to ac-count for the velocity dispersion distribution of low ionization damped Lyman alpha systems at z>1.5 (Prochaska and Wolfe2001).The kinematics reveal too few clouds with velocity widths of100km/s or more compared to the number of low velocity width systems.It is unlikely that feedback plays much of a role in these systems,selected by absorption in low ionization,neutral gas,and with velocity widths that are unlikely to be aﬀected by photoionization.A second problem is that while the initial angular momentum of a typical protogalaxy when the halo ﬁrst collapsed can reproduce the observed size dis-tribution of disks if angular momentum is approxi-mately conserved during baryon infall and disk ac-cretion,the numerical simulations show that most of the angular momentum is actually lost to the dark halo.The clumpiness induces strong angular momen-tum transfer via tidal torquing and dynamical friction from the dissipating baryons to the energy-conserving dark matter.The resulting disks have too little spe-ciﬁc angular momentum by an order of magnitude (e.g.,Navarro and Steinmetz2000a;Eke,Navarro and Steinmetz2001).It seems likely that stellar feedback can in principle provide suﬃcient input of momen-tum to the infalling baryons via supernova explosions and remnant formation to avoid much of the angular momentum loss,at the price however of a signiﬁcant delay in disk formation.Giant disks,in particular, would form relatively late,in possible conﬂict both with the observational evidence at z∼1(Faber et al.2001)and with ages in the outer parts of nearby disks(Ferguson and Johnson2001).A more serious concern for those theoretically inclined is that there is as yet no robust prescription for the required feed-back,although progress is being made(Couchman and Thacker2001).Another result from the high resolution simulations of galaxy halos is that the dark halo has a central concentration and a central cusp.The halo proﬁle is described by a universal proﬁleρ=Aical studies of nearby barred galaxies conclude that self-gravity of the bars,which consist primarily of stars,must dominate the inner gravitational poten-tial(Sellwood and Kosowsky2000).At most,a10% contribution by spherically or axially-symmetrically distributed dark matter can be allowed within the bar region to avoid conﬂict with the observed kine-matics of certain barred galaxies.Both the bars and the matter within the inner galaxy out to a bar ra-dius are required to be baryon-dominated by the need for gravitational instability to explain the existence of bars.In contrast,a recent study of a non-barred spi-ralﬁnds that detailed modelling of the rotation curve with a maximal disk requires an axially symmetric dark matter contribution of about30%within an op-tical radius(Kranz,Slyz and Rix2001).This is not inconsistent with bar formation by secular instability of a cold disk,but raises the question of why there is an apparently large spread in dark matter content among disk galaxies.The pioneering study by Navarro,Frenk and White (1997)found a cusp proﬁle ofγ≈1,but this has been superceded by higher resolution simulations which al-most invariablyﬁnd thatγ≈1.5for galaxy mass halos(Moore et al.1999;Jing and Suto2000;Klypin et al.2001).There is another observational hurdle to overcome. Low surface brightness spirals are everywhere dark matter-dominated,and so provide outstanding labo-ratories for dark matter studies via rotation curves. High resolution Hαrotation curves reveal a wide ar-ray of central proﬁles.Most systems have soft cores without any indication of a central cusp,but some do show indications ofγbeing as large as unity.None however approach the predicted value of1.5,and the best studied examples are in clear conﬂict with sucha steep cusp(van den Bosch and Swaters2001).3.Resurrection of CDMCold dark matter is seriously challenged.Whether it is actually dead is quite another matter.Never-theless,to confront this possibility,there have been numerous attempts to resurrect CDM.These come under two distinct guises:tinkering with the particle physics or elaborating on the astrophysics.Particle physics variations include the introduction of self-interacting dark matter.Self-interactions,via elastic scattering,allow the dark matter to develop a smooth core.If the scattering mean free path is ad-justed to be of the order of the core size,one can avoid developing an excessive central concentration of dark matter.This comes at a price,however:for example, the dark halo is found to be spherical,in apparent contrast to the elongated shapes inferred via gravi-tational lensing of dominant massive cluster galaxy halos(Miralda-Escude2000).Warm dark matter has provided a possible resolu-tion of the cold dark matter“crisis”.A sterile neu-trino with a mass of around1keV can constitute the dark matter,since its sterility enables the neu-trino abundance to be suppressed relative to the ordi-nary neutrinos,and thereby avoid the hot dark matter mass bound for overclosing the universe.At1keV, the free-streaming length is of order the comoving scale of a dwarf galaxy.Substructure is suppressed. However high resolution simulationsﬁnd that warm dark matter still produces a central cusp,although withγ≈1.(Bode et al.2001;Knebe et al.2001). Possibly more serious is another consequence of the suppression of small-scale structure,which results in late formation of dwarf galaxies.If nearby dwarfs contain genuinely old stellar populations,this would be diﬃcult to understand in the context of warm dark matter.Perhaps more serious maybe the need for a substantial spatial density of ultraluminous quasars at z>6,each requiring a central SMBH of mass somewhat in excess of109M⊙(Fan et al.2001).The diminished power of WDM makes it diﬃcult to recon-cile observations of this type with WDM.The abun-dance of massive(∼1012M⊙)halos is unaﬀected,but these form late:it is necessary to have small-scale power at high redshift to form the dense cores within which the SMBH formed.More complex versions of particle dark matter have been proposed.These include self-interacting warm dark matter,a model which inspires little conﬁdence given the diﬃculties encountered by warm and inter-acting dark matter,and more exotic variants such as shadow dark matter.Modiﬁed particle dark matter can no doubt be developed to explain all of the re-quired dark matter properties.However the seductive simplicity of the SUSY LSP as an attractive candidate for CDM is lost.An alternative approach is via the astrophysics of galaxy formation.Can the dark matter proﬁle be modiﬁed by astrophysical processes?The an-swer is perhaps.Consider the following sequence of events.Supermassive black holes form at the cen-tres of dark halos,possibly contemporaneously with, and certainly coupled to,the formation of the stellar spheroid.This sequence of events is strongly moti-vated by the observed correlation between supermas-sive black hole mass and spheroid mass(Ferrarese et al.2000;Gebhardt et al.2000)as well as by the super-solar abundances found in quasar broad emis-sion line regions(Hamann et al.2001).Now consider a merger between a dwarf spheroid, containing a massive black hole,and a much larger galaxy.The smaller massive black hole spirals in un-der dynamical friction and eventually merges with the SMBH of the dominant galaxy.The details of the merger are not clear,but it seems likely that the smaller SMBH decays into tighter and tighter or-bits as stars are ejected into the regime where grav-itational radiation eventually takes over.Gas is es-sential for the initial formation of the SMBH,but dark matter cores and stars are crucial for the SMBH to undergo merger-induced growth.The process is well matched to galaxy formation.The dwarf galaxy stars are stripped and help feed and regulate spheroid growth.However there is feedback on the dark matter.The SMBH merger results in heating of the dark mat-ter cusp.The region that undergoes heating can be quite extensive as a transient rapidly rotating gaseous bar forms during the merger,and is slowed by dy-namical friction.The dark matter is heated and ac-quires angular momentum.The result is that the concentration and cusp are likely to be modiﬁed. Simulations suggest that the inner proﬁleﬂattens to γ≈0.5(Nakano and Makino1999;Merritt,Cruz and Milosavljevic2001)This is considerablyﬂatter than the initial cusp,and is insensitive to the initial density proﬁle.The substructure is unaﬀfected.How-ever the angular momentum acquired by the halo will help in reinjecting angular momentum into infalling gas clouds that form the disk over a time-scale of a gigayear or longer.A more dramatic interaction of the black hole with the dark matter may be imagined.Suppose there is an early accretion phase onto the SMBH,perhaps driven by transient bar formation.The activated SMBH will produce a vigorous outﬂow.The inner region is baryon-dominated.If enough baryonic mass loss occurs,the inner dark matter proﬁle will be less con-centrated and more uniform(Binney,Gerhard and Silk2001).The modiﬁcations occur within the re-gion where the baryon content changes from being dominant to being sub-dominant,i.e.of order half of the baryonic mass must be driven out,possibly just into the halo.LSB dwarfs may be extreme examples where such mass loss has occurred.It is precisely these objects that provide possible evidence for’discrepant’CDM proﬁles.Even in the absence of strong outﬂows,im-portant dynamical heating can occur via dynamical friction on the rotating bar.Hence LSB galaxies, galaxies which had primeval bars(and so may be vulnerable to formation of a second bar if there is a suitable supply of gas),and galaxies with soft stellar cores,where SMBH mergers may have initiated stel-lar ejection,are the prime examples where the initial CDM proﬁle is likely to have been modiﬁed.Evidently,there is noﬁrm prediction about dark matter proﬁles,whether in LSB galaxies or in lumi-nous galaxies.I turn now to observations of dark matter,and discuss whether one can indeed observe the inner proﬁle of dark halos.4.”Observing”dark matterThe favoured candidate for CDM is the lightest stable SUSY relic particle.This must be neutral (to avoid already having been detected)and its mass is constrained by accelerator searches and theoreti-cal considerations of thermal freeze-out to lie in the range50GeV to a few TeV.The relic density is de-termined when annihilations and pair production go out of thermal equilibrium in the early universe at T∼m x/20k,and one infers thatΩx∝σ−1ann,where σann is the annihilation cross-section extrapolated to the low temperature limit.For typical weak interac-tion values ofσann,oneﬁnds thatΩx∼0.3is re-quired to account for the dark matter content of the universe.Via studying a grid of SUSY models,one can infer a range of particle masses from the anni-hilation cross-section.Were it not for the accelerator bounds on the sparticle masses,the uncertainty in m x would span some5orders of magnitude.The annihilation cross-section and particle mass is constrained.So also is the elastic scattering cross-section once the annihilation cross-section is speciﬁed. This means that one can now consider possible detec-tion schemes.The obvious one is direct detection by elastic e of annual modulation of the in-cidentﬂux on a terrestrial detector has led to a tenta-tive detection(DAMA)that requires an implausibly large cross-section given the suite of minimal SUSYmodels and is marginally inconsistent with another experiment(CDMS).Annihilations result in hadronic jets that decay into gamma rays,high energy electron-positron pairs,proton-antiproton pairs and neutrinos, all of which are potentially detectable as galactic halo signals.The EGRET gamma ray detector on CGRO has reported a diﬀuse high latitude gamma rayﬂux that can only be accounted for by annihilations if the halo is clumpy by a factor<n2>/<n>2∼100 (Calcaneo-Roldan and Moore2000).In fact,the ob-served signal has a spectral signature that does not resemble that expected for annihilations.Presumably it is due to unresolved distant sources.Another possible signature of CDM is associated with a feature in the cosmic ray positron spectrum at very high energy.The HEAT experiment re-ported a feature near100GeV that,if real,cannot be explained by secondary interactions between cos-mic rays and the interstellar medium.Annihilation products of a100GEV neutralino would provide an excellentﬁt to the data,except that one requires the annihilation cross-section to be boosted by a factor ∼100relative to typical models.Clumpiness in the dark halo again provides a possible explanation in terms of a WIMP signal(Baltz and Edsjo1999).Perhaps the most exciting prospect comes from the Galactic Centre,where annihilations may already have been seen.The supermassive black hole at the Galactic Centre of2.6×106M⊙most likely formed by baryonic dissipation within the already existing dark halo.If the growth process is approximately adia-batic,the neutralinos form a central cusp with slope ρ∝r−γ′,whereγ′=9−2γP.Bode,J.Ostriker and N.Turok,Astrophys.J.556 (2001)93.F.van den Bosch and R.Swaters,MNRAS325(2001)1017C.Calcaneo-Roldan and B.Moore,PRD,62(2000)123005.H.Couchman and R.Thacker,Astrophys.J.555(2001)L17.V.Eke,J.Navarro and M.Steinmetz,Astrophys.J.554(2001)114.S.Faber et al.,in Galaxy Disks and Disk Galaxies, ASP Confrence Series Vol.230,ed.J.Funes andE.Corsini(2001),p.517.X.Fan et al.,Astr.J.(2001),in press.A.Ferguson and R.Johnson,Astrophys.J.Letters(2001),in press.L.Ferrarese and D.Merritt,Astrophys.J.539(2001) 9.K.Gebhardt et al.Astrophys.J.539(2001)13.N.Gnedin,Astrophys.J.542(2000)535.F.Hamann et al.,Astrophys.J.(2001),in press.Y.Jing and Y.Suto,astro-ph/0001288(2000),4th RESCUE Symposium on Birth and Evolution of the Universe,in press.A.Klypin et al.,Astrophys.J.554(2001)903A.Knebe et al.,MNRAS in press(2001).T.Kranz, A.Slyz and H.-W.Rix,Astrophys.J.(2001)in press.D.Merritt, F.Cruz and osavljevic,astro-ph/0008497(2001),Dynamics of Star Clusters and the Milky Way,ASP Conference Series,eds.S.Deiters et al.,in press.J.Miralda-Escude,astro-ph/0002050(2000).T.Nakano and J.Makino,Astrophys.J.510(1999) 155.J.Navarro,C.Frenk and S.White,Astrophys.J.490 (1997)493.J.Navarro and M.Steinmetz,Astrophys.J.538 (2000a)477.J.Navarro and M.Steinmetz,Astrophys.J.528 (2000a)607.J.Prochaska and A.Wolfe,astro-ph/0108154,Astro-phys.J.,submitted(2000).。

Ejection of Supermassive Black Holes from Galaxy Cores

a r X i v :0708.0771v 2 [a s t r o -p h ] 31 J a n 2008D RAFT VERSION F EBRUARY 1,2008Preprint typeset using L A T E X style emulateapj v.10/09/06EJECTION OF SUPERMASSIVE BLACK HOLES FROM GALAXY CORESA LESSIA G UALANDRIS AND D AVID M ERRITTCenter for Computational Relativity and Gravitation,Rochester Institute of Technology,78Lomb Memorial Drive,Rochester,NY 14623Draft version February 1,2008ABSTRACTRecent numerical relativity simulations have shown that the emission of gravitational waves during the merger of two supermassive black holes (SMBHs)delivers a kick to the ﬁnal hole,with a magnitude as large as 4000kms −1.We study the motion of SMBHs ejected from galaxy cores by such kicks and the effects on the stellar distribution using high-accuracy direct N -body simulations.Following the kick,the motion of the SMBH exhibits three distinct phases.(1)The SMBH oscillates with decreasing amplitude,losing energy via dynamical friction each time it passes through the core.Chandrasekhar’s theory accurately reproduces the motion of the SMBH in this regime if 2 ln Λ 3and if the changing core density is taken into account.(2)When the amplitude of the motion has fallen to roughly the core radius,the SMBH and core begin to exhibit oscillations about their common center of mass.These oscillations decay with a time constant that is at least 10times longer than would be predicted by naive application of the dynamical friction formula.During this phase,the SMBH is typically displaced from the peak of stellar density by roughly the core radius.(3)Eventually,the SMBH reaches thermal equilibrium with the stars.We use straightforward scaling arguments to estimate the time for the SMBH’s oscillations to damp to the Brownian level in real galaxies and infer times as long as ∼1Gyr in the brightest galaxies.The longevity of the oscillations makes this mechanism competitive with others that have been proposed to explain double or offset nuclei.Ejection of SMBHs also results in a lowered density of stars near the galaxy center;mass deﬁcits as large as ﬁve times the SMBH mass are produced for kick velocities near the escape velocity.We compare the N -body density proﬁles with luminosity proﬁles of early-type galaxies in Virgo and show that even the largest observed cores can be reproduced by the kicks,without the need to postulate “hypermassive”binary SMBHs.Implications for displaced AGNs and helical radio structures are discussed.Subject headings:galaxies:nuclei -stellar dynamics1.INTRODUCTIONThe recent breakthroughs in numerical relativity (Pretorius 2005;Campanelli et al.2006;Baker et al.2006a)have al-lowed a number of groups to evolve binary black holes (BHs)to full coalescence.The ﬁnal inspiral is driven by emission of gravitational waves,and in typical (asymmetric)inspirals,a net impulse is imparted to the system due to anisotropic emis-sion of the waves (Bekenstein 1973;Fitchett &Detweiler 1984;Favata et al.2004).Early arguments that the magni-tude of the recoil velocity would be modest for non-spinning BHs (Redmount &Rees 1989)were conﬁrmed by the simula-tions,which found V kick 200kms −1in the absence of spins (Baker et al.2006b;González et al.2007b;Herrmann et al.2007).The situation changed dramatically following the ﬁrst simulations of “generic”binaries,i.e.,binaries in which the individual BHs were spinning and in which the spins were allowed to have arbitrary orientations (Campanelli et al.2007b).Kicks as large as ∼2000kms −1have now been conﬁrmed (Campanelli et al.2007a;González et al.2007a;Tichy &Marronetti 2007),and simple scaling arguments sug-gest that the maximum kick velocity would probably increase to ∼4000kms −1in the case of maximally-spinning holes (Campanelli et al.2007a).The most propitious conﬁguration for the kicks consists of an equal-mass binary in which the in-dividual spin vectors are oppositely aligned and oriented par-allel to the orbital plane.The kick amplitude also depends sensitively on the angle between the BH spin vectors and their linear momenta shortly before the plunge (Campanelli et al.2007c).Electronic address:alessiag,merritt@Galaxy escape velocities are 3000kms −1(Merritt et al.2004),which means that gravitational wave recoil can in prin-ciple displace coalescing supermassive black holes (SMBHs)arbitrarily far from galaxy centers,or even eject them com-pletely.The actual distribution of kick velocities is very uncertain,since it depends on the unknown distribution of binary mass ratios and spins,but most kicks are probably 103kms −1.A SMBH that is kicked with less than escape velocity will travel some maximum distance from the galaxy center after which its orbit decays due to dynamical friction;most of the energy loss takes place during passages through the galaxy center.Removal of the SMBH from the core has the effect of transferring kinetic energy to the stars and low-ering the core density (Redmount &Rees 1989;Merritt et al.2004;Boylan-Kolchin et al.2004).This implies a more grad-ual return of the SMBH to a zero-velocity state than in a galaxy with ﬁxed density.In fact,however,the SMBH is not expected to ever reach a state of zero kinetic energy.When its energy falls to a value12m ⋆v 2⋆(1)with respect to the galaxy central potential,where m ⋆andv ⋆are a typical stellar mass and velocity respectively,ran-dom gravitational perturbations from stars act to accelerate the SMBH as often as they decelerate it.This is the regime of gravitational Brownian motion (Young 1977;Bahcall &Wolf 1976;Merritt et al.2007).A natural deﬁnition of the “re-turn time”of a kicked SMBH is the time required for dy-namical friction to reduce the SMBH’s mean kinetic energy to the Brownian value.Applying standard expressions for2Gualandris and Merrittthe dynamical friction force leads one to the conclusion that this would occur in a relatively short time,of order a few or-bital periods,after dynamical friction has returned the kicked SMBH to the core.The N -body simulations presented here were designed to test these expectations by evaluating the return times of kicked SMBHs and by quantifying the induced changes in galaxy structure.These processes can not be studied accurately using classical dynamical friction theory since the SMBH substan-tially modiﬁes the core as it recoils and falls back.Approx-imate N -body schemes,e.g.tree or grid codes,are also not well suited to the problem since they can not robustly follow both the early (collisionless)and late (collisional)evolution of the rge particle numbers are required in order to cleanly separate the collisional and collisionless regimes.These various requirements can currently be met only with parallel,direct N -body codes running on special-purpose supercomputers.Our simulations use the φGRAPE integrator (Harfst et al.2007)as implemented on gravitySimulator ,a 32-node supercomputer em-ploying GRAPE-6A accelerator boards (Fukushige et al.2005).Our ﬁndings are surprising in one important respect.After returning to the core,the kicked SMBH exhibits long-lived oscillations with amplitude comparable to the core radius 1.These oscillations eventually decay but with a time constant that is at least an order of magnitude longer than would be predicted by a straightforward application of the dynamical friction equation.We demonstrate that the existence,ampli-tude and damping time of these oscillations are independent of the number N of “star”particles used in the simulations,for N up to 2×106.The oscillations are similar to those ﬁrst re-ported by ler and collaborators (Miller &Smith 1992;Miller 1996)in their pioneering N -body studies of the central regions of galaxies.A number of other authors have reported low effective values of the dynamical friction force as it acts on massive objects that inspiral into constant-density cores (Bontekoe 1988;Bertin et al.2003;Read et al.2006)or on rotating bars (Weinberg &Katz 2002;Valenzuela &Klypin 2003).Our use of a high-accuracy,direct-summation N -body code combined with large particle numbers greatly reduces the possibility that our results are an artifact of the potential calculation scheme,an issue that has plagued the interpreta-tion of similar results in the past (Zaritsky &White 1988).§2describes the initial models and the N -body algorithm.Evolution of the SMBH’s orbit is described in detail in §3,and the induced changes in galaxy structure are described in §4,where the N -body models are compared to luminosity proﬁles of core galaxies.§5presents estimates of the SMBH return times in real galaxies,and §6discusses some of the observable consequences of the kicks.2.INITIAL MODELS AND NUMERICAL METHODSThe light proﬁles of elliptical galaxies and the bulges of spi-ral galaxies are generally well described in terms of the Sér-sic model (Sérsic 1963;Sersic 1968),which is a generaliza-tion of the de Vaucouleurs (1948,1959)law.The most lumi-nous elliptical galaxies depart systematically from the Sérsiclaw near the center,where they show evidence for partially depleted stellar cores (Faber et al.1997;Milosavljevi´c et al.2002;Graham 2004;Ferrarese et al.2006).Formation of1A movie showing the oscillations is available at/Research/Publications.php?paper=0708.0771.TABLE 1P ARAMETERS OF THE INITIAL MODELS .A1 4.0 2.00.0140.551.0×10−3A2 4.0 2.00.00950.551.0×10−3B4.02.00.0270.553.0×10−3rα γ/αr α+r αb /R αe−p /αe −b [(r α+r αb )/R αe ]1/n α(2)withρ′=ρb 2(p −γ)/αrbEjection of Supermassive Black Holes from Galaxy Cores3 speed V esc:V kick=(0.1,0.2,..,1.1)×V esc;the latter was com-puted numerically from the initial N-body models.In orderto guarantee energy conservation,we used a time-step accu-racy parameterη=0.01.This ensures a relative energy errorsmaller than one part in106.An accuracy parameter twice asbig would approximately halve the integration time but wouldresult in a relative energy error of10−4,which we do not con-sider acceptable for this study.A softening lengthǫ=10−4was assigned to both the stars and the BH.Such a small soft-ening length has been shown not to affect even the Brownianmotion of a massive particle in models like ours(Merritt et al.2007).Throughout the paper we adopt units according to which thegravitational constant G,the effective radius R e in equation(2),and the total galaxy mass M gal are unity.The models canbe scaled to physical units as follows:[T]= GM gal1011M⊙ −1/2 R eR e 1/2(6)=378kms−1 M gal3kpc −1/2.(7)3.THE BLACK HOLE MOTION3.1.General RemarksFigure1(which can be compared with Fig.1of Madau&Quataert2004shows BH trajectories in models A1, A2and B for V kick/V esc=(0.4,0.7,0.9,1.1).For V kick≥V esc the black hole escapes the galaxy on an unbound orbit.The maximum displacement of the BH(r max)is shown in Figure2. The data from the simulations(points)are compared to theo-retical(dotted lines)and numerical(dashed lines)estimates of r max in the absence of dynamical friction.The theoretical and numerical estimates are obtained from the initial N-body data by assuming conservation of total energy for the BH:r max is the distance at which the gravitational potential of the system equals the initial total energy of the BH.For the theoretical so-lution we use the expression of the potential in a core-Sérsic model(see equations7through13of Terzi´c&Graham2005) while for the numerical solution we compute the potential at different radii from the N-body data.The two estimates are for practical purposes indistinguishable.Dynamical friction affects the maximum displacement of the BH only for moderately large kicks,where the data points appear systematically lower than the theoretical curves.Val-ues of r max larger than the expected turning points in theﬁrst orbit are due to the rapidly-expanding core.During the initial outward journey,dynamical friction does not strongly inﬂuence the motion of the BH,and the maxi-mum displacement is similar to that of an energy-conserving orbit.We note that a kick velocity larger than about0.3V esc is necessary to bring the BH beyond the core.Due to the combined effect of the kick and dynamical friction,the BH displays a damped oscillatory motion.The number of radial oscillations increases with V kick;for V kick=0.9V esc the BH ex-periences∼5full radial oscillations before returning to the core.Almost all of the energy loss to dynamical frictiontakesF IG. 1.—BH trajectories in models A1,A2and B,for V kick/V esc=0.4 (blue/lower),0.7(green),0.9(red)and1.1(black).place during the short intervals that the BH passes through the core.This is shown in Figure3which plots the evolution of the BH speciﬁc energy E in Model A1with V kick=0.9V esc, whereE≡V24Gualandris andMerrittF IG.2.—Maximum displacement of the BH from the galaxy center.Thedata points show the results from the simulations while the lines are estimatesin the absence of dynamical friction.The dashed lines represent numericalestimates from the computation of the potential of the N-body system at timet=0while the dotted lines represent theoretical estimates from the analyticexpression of the potential in a core-Sérsicmodel.F IG.3.—Upper panel:Speciﬁc energy of the BH particle versus time inModel A1with V kick=0.9V esc.Almost all of the energy loss occurs duringpassages through the core.Lower panel:Mean density in a sphere of radius0.05centered on the point of maximum density in the core of the galaxy(excluding the BH).and the summation is over the“star”particles3.The energylost during the initial emergence from the core appears to beless than during subsequent passages,suggesting that dynam-3Unless otherwise noted,upper-case variables X and V refer to the BHparticle while lower-case symbols are reserved for the starparticles.F IG.4.—Stellar mass bound to the BH in the initial models.ical friction requires aﬁnite time to“turn on”after the kick.During theﬁrst few oscillations,the BH’s motion remains es-sentially rectilinear,but eventually the Y-and Z-componentsof the motion become important due to non-sphericities in thegalaxy potential and also to perturbations from stars.At latetimes,the BH’s motion is essentially random,similar to thatof a Brownian particle in aﬂuid.Figure3also shows themean density in a sphere ofﬁxed radius whose center is lo-cated at the estimated density peak(computed via the algo-rithm described in§3.2).The core density decreases rapidlyfollowing the initial ejection,then more gradually as the BHreturns again and again to the core,losing energy to the starseach time.Figure4shows the mass in stars bound to the BH at t=0.The bound mass was computed by counting all the stars,within the inﬂuence radius r h,which formed a bound two-body system with the BH particle.The inﬂuence radius wasdeﬁned as the radius containing a mass in stars equal to twiceM BH.The bound mass decreases steeply with V kick,as notedin earlier studies(Merritt et al.2004;Boylan-Kolchin et al.2004),and is ignorable for V kick 0.6V esc.In all cases where the kick velocity was large enough to re-move the BH completely from the core(i.e.V kick 0.3V esc),we observed three distinct regimes of the motion.In PhaseI,the BH’s motion is well predicted by Chandrasekhar’s dy-namical friction theory,after taking into account the chang-ing size of the galaxy core where most of the friction occurs.This is the phase illustrated in Figure1;in Figure3,Phase Iextends until t≈20.Phase II begins roughly when the ampli-tude of the BH’s motion had decayed to the size of the core.In this phase,the energy of the BH’s orbit continues to decaybut with a much longer time constant than predicted by Chan-drasekhar’s formula.The BH and the core oscillate about theircommon center of mass in this regime.In Phase III,the BH’senergy has dropped to the thermal level.Phase II is generallylonger than Phase I,and this would presumably be even moretrue in real galaxies since the amplitude of thermal oscilla-tions is much lower than in our simulations implying a longerEjection of Supermassive Black Holes from Galaxy Cores5time to reach the Brownian regime.We discuss these three regimes in detail below.3.2.Phase IThe extent of Phase I is clearly indicated in the plots of BH energy vs.time(e.g.Fig.3):a distinct“knee”appears in the E(t)curves marking the end of this phase.Values of T I,the elapsed time from the kick until the end of Phase I,are given in Table2.We compared the evolution of the BH’s motion in Phase I with the predictions of Chandrasekhar’s dynamical friction theory(Chandrasekhar1943).Such comparisons are prob-lematic since much of the energy exchange between BH and stars occurs during passages through the galaxy’s core,and the core density changes signiﬁcantly with time due to the BH’s motion.We dealt with this problem by breaking the BH’s motion into segments,each containing one passage through the center,and assuming that the galaxy’s density re-mained constant during each segment.Chandrasekhar(1943)derived his expression for the dy-namical friction acceleration F df assuming an inﬁnite,homo-geneous and unchanging background of perturbers(stars).In the limit that the mass of the heavy object greatly exceeds the masses of the stars,the acceleration is predicted to beF df≈−2πG2ρM BH ln(1+Λ2)V−2N(<V,r),(9) whereρ(r)is the mass density of stars at the BH’s position, (1/2)ln(1+Λ2)is the Coulomb logarithm,V is the BH’s in-stantaneous velocity,and N(<V,r)is the fraction of stars at r that are moving(in the frame of the galaxy)with velocities less than V.Some care must be taken in the deﬁnition of the Coulomb logarithm.One commonly writesln 1+Λ2 ≈2lnΛ≈2ln(p max/p min)(10) where p min and p max are the minimum and maximum effective impact parameters of the stars that contribute to the frictional force,and p max≫p min.However,p min depends on theﬁeld-star velocity(White1949;Merritt2001)and p max is likewise ill-deﬁned since a realistic stellar system is inhomogeneous and has no outer boundary.Numerous N-body simulations have been carried out to evaluate Chandrasekhar’s formula in the case of a mas-sive particle inspiraling toward the center of a galaxy (White1983;Bontekoe&van Albada1987;Bontekoe1988; Weinberg1989;Cora et al.1997;Bertin et al.2003).Early work was typically based on approximate N-body schemes and the results were often discrepant from study to study (Zaritsky&White1988).These differences appear to have been resolved in the last few years through the use of direct-summation codes(Spinnato et al.2003;Merritt2006),which consistentlyﬁnd4 lnΛ 6for inspiral of massive point particles,on circular or near-circular orbits,into the centers of galaxies with steeply-rising density proﬁles.Fewer exper-iments have been done with highly eccentric orbits,although Just&Peñarrubia(2005),using an approximate method,ﬁnd 2 lnΛ 3for orbits with moderate eccentricities.In general,we expect the effective value of lnΛto be smaller for radial orbits than for circular motion.The dy-namical friction force arises from a polarization of the stellar density which produces an over-dense region,or wake,be-hind the massive object(Mulder1983).Aﬁnite time,of order a galaxy crossing time,is presumably required for this wake to be set up.In the case of a gradually-decaying circular or-bit,the galaxy is able to reach a quasi-steady state after a few orbits of the massive object.In our case,the position and velocity of the BH are changing dramatically over one cross-ing time,so that the wake never has a chance to establish its steady-state amplitude;indeed just after apocenter passages, the over-dense region can be seen to lie in front of the BH.In order to determine the effective value of lnΛin the N-body integrations,we computed BH trajectories using Chan-drasekhar’s formula(equation9)with various values of the ln 1+Λ2 term(henceforth written simply as2lnΛ)and compared them with the N-body trajectories.The following procedure was followed.1.The density center of the galaxy moves slightly with respect to the origin of the coordinates due to transfer of mo-mentum from the kicked BH to the galaxy.In order to accu-rately determine the distance of the BH from the galaxy center as a function of time,we recorded full snapshots of the par-ticle positions at frequent intervals,then used the Casertano-Hut(1985)algorithm toﬁnd the density center of the stars in each snapshot.A smoothing spline wasﬁt through the measured positions to give a continuous estimate of the center displacement as a function of time,and this displacement was subtracted from the BH positions.(The instantaneous velocity of the density center was ignored,which is a good approxima-tion at least until the end of Phase I.)The resulting correction was at most∼0.02;at late times the displacement reached a constant value since the center-of-mass velocity of the system was zero by construction.2.In order to apply Chandrasekhar’s formula we needed to specify the galaxy model.The galaxy’s mass distribu-tion changes with time due to the BH’s motion;most of this change takes place in the core just after the BH passes through.We thereforeﬁxed all the parameters in equation(2) except for the core radius r b.We determined the effective value of r b at the discrete times when the BH passed through the galaxy center by assuming aﬂat core(γ=0)andﬁnding the value of r b such that the mass contained within r b accord-ing to equation(2),withα=2,was the same as the mass in the N-body model in a sphere of radius r b centered on the BH. This procedure was always found to yield a unique r b and ac-curately recovered the known value of r b in the initial models.3.BH trajectories were then computed in a piecewise fashion using Chandrasekhar’s formula,starting from one ex-tremum in the BH displacement and continuing until the next extremum,using the value of r b corresponding to the central passage lying between the two extrema.This was repeated for several values of lnΛ.We used equation(5)of Szell et al. (2005)to compute N(<V,r)in equation(9)from the assumed ρ(r).Figure5shows the results for Model A1with V kick/V esc= 0.7and Model B with V kick/V esc=0.8.During each inward leg of the trajectory,the dynamical friction force hardly af-fects the motion;only when passing through the dense center is the motion signiﬁcantly non-ballistic.(This could be seen already in Figures2and3.)The best-ﬁt value of lnΛwas found to lie in the range2 lnΛ 3,and for such values, Chandrasekhar’s formula did a good job of reproducing the motion.We found no evidence of a systematic change in the effective value of lnΛfrom one time interval to the next.3.3.Phase IIIThe BH trajectories in Figure5are displayed until the am-plitude of the oscillations has decayed down to roughly the6Gualandris and MerrittF IG.5.—Comparison between BH trajectories computed via the N-bodyintegrations(open circles)and via Chandrasekhar’s formula(9)(lines).TheN-body models were A1(M BH=0.001)with V kick=0.7V esc(a)and B(M BH=0.003)with V kick=0.8V esc(b).Theoretical trajectories were computed in apiecewise manner,starting from extrema in the BH’s trajectory(vertical solidlines)and continuing until the next extremum;the core radius r b of the galaxymodel was adjusted as described in the text to give the same core density asin the N-body model at the time when the BH passed through the center.Horizontal dashed lines show the adopted values of r b.Line colors/stylescorrespond to different values of lnΛ:1(blue/solid),2(magenta/dashed),3(red/dash-dotted),4(black/dotted).core radius.As discussed above,the BH’s motion is well pre-dicted by Chandrasekhar’s dynamical friction formula in thisregime.Shortly after returning to the core,however,the BH’smotion was found to depart strikingly from the predictions of Chandrasekhar’s formula.A detailed discussion of the mo-tion in“Phase II”is presented below.Before doing so,weconsider the motion of the BH at still later times,“Phase III,”when it has reached thermal equilibrium with the stars.Figure6shows the squared velocity of the BH,V2=V2x+V2y+V2z,over the full integration interval,for kick veloci-ties V kick≥0.3V esc in Model B.For V kick 0.4V esc the BHmoves substantially beyond the core during itsﬁrst oscillation(Fig.2).At late times,the motion of the BH in each of these integrations appears to be stochastic(i.e.non-quasi-periodic)but with roughly constant amplitude.The dashed(blue)lines in thisﬁgure show V2 ,the meansquare velocity of the BH averaged over Phase III.(The pre-cise deﬁnition of the start of Phase III is given below.)Alsoshown(dotted red lines)are estimates of the expected valueof V2 for the BH once it reaches statistical equilibrium withthe stars.The latter velocity,V2Brown,was computed usingm⋆V2Brown=3Ejection of Supermassive Black Holes from Galaxy Cores7F IG .7.—RMS amplitude of the BH oscillations in the Brownian regime,Phase III,for models A1(black/circles),A2(blue/squares)and B (red/triangles).virial theorem,V 2 ≈4(M gal /m ⋆)/N ,i.e.∼50for M gal =109M ⊙and∼500for M gal =1011M ⊙.The time required for a BH to reach these lower kinetic energies would also presumably be longer than in our simulations,as discussed in more detail below.3.4.Phase IIAs noted above,the motion of the BH after returning to the core,and before reaching the Brownian regime,is not well described by Chandrasekhar’s formula.Here we consider the motion in this regime (“Phase II”).Figure 6reveals the following qualitative features.1.The motion in Phase II is essentially oscillatory,with a period similar to that at the end of Phase I,i.e.roughly equal to the period of oscillation of a test particle moving in the stellar core.2.There is evidence of additional frequencies affecting the BH’s motion.For instance,the amplitude of the oscillations sometimes appears to increase temporarily over several peri-ods in a manner suggestive of beats.3.Averaged over many periods,the mean amplitude of the oscillations decays,but with a time constant that is much longer than observed toward the end of Phase I.4.Near the end of Phase II,the motion becomes increas-ingly stochastic,presumably due to perturbations from indi-vidual stars.Eventually the BH rms velocity falls to the Brow-nian (thermal)level marking the start of Phase III.5.Phase II always begins roughly when the stellar mass in-terior to the BH’s orbit is equal to M BH .When V kick 0.3V esc ,the BH never escapes the core,and its motion appears to tran-sition directly from Phase I to Phase III.Based on Figure 6,the elapsed time in Phase II can be sub-stantially longer than the time spent in Phase I.Understanding the character of the motion in this regime is therefore crucial for predicting the expected displacement of a supermassive BH in a real galaxy following a kick.We begin by considering a simple model for damped oscil-lations of a massive particle in a constant-density core.While this model will fail to quantitatively reproduce the motion in Phase II,it provides a useful framework for discussing what is observed in the simulations.In the absence of dynamical friction,and neglecting the in-ﬂuence of the massive particle’s presence on core structure,the motion of the massive particle is simple harmonic oscilla-tion with frequency ωc =8πσ3c6G 3/2ρ1/2c M BH ln Λ(15)=√9F 3M c9G ρc r 2c .(17)F ≈2for our models.Thus2ωc T df ≈4M c。

神秘的银河中心

MCG-6-30-15
II : The Theory of AGN

What powers AGN? Define efficiency of an AGN
E Mc

2
Think about efficiency of different processes
Chemical burning, 10-9 Nuclear fusion, 0.007 Accretion onto a black hole, 0.1 Matter/anti-matter annihilation, =1
Now believe that AGN are indeed accreting supermassive black holes Matter can’t fall straight in… forms an accretion disk. Accretion disks power jets.
There’s something strange at the center of our galaxy…

Modern large telescopes can track individual stars at Galactic Center

Need infra-red (to penetrate dust?) Need very good resolution.
Faster than a speeding light ray?

Superluminal motion

Jets have “blobs” Some blobs appear to move faster than light! Motion faster than light is not allowed by the Special Theory of Relativity

黑洞是怎样形成的英语作文

黑洞是怎样形成的英语作文英文回答：Black holes are formed when massive stars reach the end of their lives and collapse under their own gravity. The core of the star collapses to form a singularity, an infinitely dense point of zero volume and infinite gravity. The gravitational pull of the singularity is so strong that nothing, not even light, can escape it. The region of space around the singularity where nothing can escape is called the event horizon.The formation of a black hole can be divided into three main stages:1. Core Collapse: As a massive star burns through its nuclear fuel, it gradually loses mass. When the star has lost enough mass, the gravitational forces acting on its core become too strong for the star's internal pressure to withstand. The core collapses under its own gravity,forming a singularity.2. Formation of the Event Horizon: The singularity has an infinite gravitational pull, but its physical size is zero. As the core collapses, the gravitational field around it becomes so strong that nothing can escape from it, not even light. This region of space around the singularity is called the event horizon.3. Black Hole Formation: Once the event horizon has formed, the star has effectively become a black hole. Nothing can escape from the black hole's gravitational pull, so it appears to be black to an outside observer.中文回答：黑洞是由质量巨大的恒星在生命的尽头时因自身引力而坍缩而形成的。

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Evidently its numerical value is rather small unless M >> MP . It is worth to note that the hybrid model [4] was invented just to avoid the problem with very small coupling constants. Meantime Eq. (4) indicates unambiguously that λ << 1 at reasonable values of parameters and hence this problem remains in the hybrid inﬂation model. If average ﬁeld value approaches too close to critical line χ = 0, the ﬂuctuations of the ﬁeld in some space domains could cross this line. In future, these domains will be ﬁlled by vacuum, say, χ− surrounded by a sea of another vacuum χ+ . The two vacua are
3
One of the condition of slow motion for the ﬁeld χ is
′′ Vχχ η≡ << 1. 3H 2
Estimation for this value can be easily performed for σ ≈ σin, χ = 0 and H ≈ 8π/3κ M/MP , MP is Plank mass. The result is

Serious problem arises when we take into account ﬁeld ﬂuctuations during inﬂation. Indeed, the classical motion along the line χ = 0 describes an average motion. The fact is that the space is constantly divided into increasing number of causally disconnected domains. Each of them is characterized by a ﬁeld value slightly diﬀerent from neighbor ones. Evidently, vast majority of them contain ﬁeld χ = 0. Thus, at the end of inﬂation, i.e. when the ﬁeld σ reaches the critical value σc huge amount (about 1078) domains has been produced. Half of them, those with χ < 0,
′′ 1 2 2 2 2 Vχχ 2 λ σin − κ M ≃ η≡ 3H 2H 2 NU − 1 3H 2
2m2
4 2 m 2NU MP ≈ , (8π )2 κ 2 M 6
what leads to inequality
4 2 m 6NU MP << 1. η= 2 2 (8π ) κ M 6
It is well known that energy density ﬂuctuations at the early Universe give rise primordial black holes (BH) formation [1]. Widespread opinion is that these BHs are small enough with masses somewhere in the range MBH ∼ 10−5 − 1020gram depending on speciﬁc model. What could be said about BHs with masses in the range 1020 − 1040gram? There exist a few number of inﬂationary models that predict BH production at some conditions [2], [3]. Here I discuss necessary conditions for BH formation in the framework of hybrid inﬂation [4]. It will be shown that it is hard to avoid BH production at early stage of the Universe evolution described by this model. New ﬂuctuating mechanism of BH formation elaborated in [3] proved to be signiﬁcant one if a potential of the inﬂaton ﬁeld possesses at least two minima as it is in the case of hybrid inﬂation. Potential of hybrid inﬂation has the form V (χ, σ ) = κ 2 M 2 − χ2 /4
4
separated by a closed wall as it was discussed above. A number of such a walls depends in initial conditions at the moment of our Universe creation. Let us estimate energy and size of the closed walls. To proceed, suppose that the ﬁeld in the volume in question crossed critical line at e-folds number N before the end of inﬂation. Its size is about Hubble radius, H −1 and it will be increased in eN times during inﬂation. Surface energy density of the domain wall after inﬂation is √ 8 2 ǫ= κM 3. (5) 3 Thus the energy Ewall contained in the wall after inﬂation is at least 2 √ MP −1 N 2 e2N . (6) Ewall ≈ 4πǫ H e =4 2 κM Numerical value N varies in the interval (0 < N < NU ≈ 60) . Gravitational radius of the wall could be easily calculated √ 8 2 2N 2 rg = 2Ewall /MP ≈ e , κM √ what is much larger than the wall width d = 2 2/(κ M ) for any e-fold N . It means that this wall will collapse into BH with mass MBH ≈ Ewall [3]. Let us estimate masses of such a BHs for ordinary values of the parameters κ = 10−2 and M = 1016GeV. If N = 40 we obtain the mass of BH MBH ≈ 3 · 1059GeV ∼ 100 Solar mass. The same estimation for a mass of smallest BHs which are created at the e-fold number N = 1 before the inﬂation is ﬁnished gives MBH,small ≈ 106MP . Thus, hybrid inﬂation leads to BH production in the wide range 1025 ÷ 1059 GeV. A number of the massive Bhs depends on how
(2)
Combining it with the formula for temperature ﬂuctuations [6] , [7] 16π 1/2 λκ 2M 5 δT ∼ ≈ 10−5, (3) 3 2 45 MP m T we obtain an estimation for the parameter λ (8π )2 45 δT M λ≈ η . 6NU 16π T MP (4)
1
When the ﬁeld σ passes this value, the motion along the line χ = 0, σ < σc becomes unstable and the ﬁeld χ quickly moves to one of the minima χ± = ±2M, σ = 0. The ﬁeld motion is ruled by classical equations 1 σ ¨ + 3H σ ˙ + λ2 χ2 σ + m2 σ = 0, 2 1 χ ¨ + 3H χ ˙ − κ 2χ M 2 − χ2 /4 + λ2 χσ 2 = 0, 2 were H is Hubble parameter at this period. Slow rolling, which is necessary condition for inﬂation, means slow variation of the ﬁeld σ along the valley χ = 0 what takes place while σ > σc . This condition looks like m << H for simple estimation. Neglecting second derivative as usual, solution of Eq. (1) acquires the form m2 σ (t) = σin exp − t , 3H