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(e.g., Cavaliere, Morrison & Wood, 1971 ; Soltan, 1982 ; Chokshi & Turner, 1992 ; Small & Blandford, 1992 ; Marconi et al., 2004 ; Shankar et al., 2004 ; Tamura, Ohta & Ueda, 2006 . Such calculations on the cosmological evolution of massive black holes were usually carried out by adopting two free parameters: the radia... |
[MATH] for AGN. The derived BHMF of AGN relics in this way is required to match that estimated either from velocity dispersion or the luminosity of the spheroid of its host galaxy, which always leads to [MATH] , i.e., almost all AGN are required to be accreting close to the Eddington limit |
(e.g., Yu & Tremaine, 2002 ; Marconi et al., 2004 ; Shankar et al., 2004 ; Tamura, Ohta & Ueda, 2006 In the last decade, several approaches for measuring the masses of the central black holes in AGN have been developed, in which the reverberation mapping may be the most effective one (Peterson, 1993 ; Kaspi et al., 200... |
[MATH] at [MATH] to [MATH] at [MATH] Warner, Hamann & Dietrichet ( 2004 also derived the Eddington ratio distribution for a sample of [MATH] 500 AGN with redshifts [MATH] . As pointed by Kollmeier et al. ( 2006 , both these derived Eddington ratios are heavily weighted towards high-luminosity objects due to the limited... |
In this work, the Eddington ratio distribution at fixed luminosity given by Kollmeier et al. ( 2006 is converted to that at fixed black hole mass by using an AGN LF. We integrate the continuity equation for black hole number density adopting the derived Eddington ratio distributions of AGN to calculate the BHMF of AGN ... |
[MATH] [MATH] , and [MATH] have been adopted in this work. The Eddingtion ratio distribution of active galactic nuclei The Eddington ratio distribution of AGN for given bolometric luminosity can be approximated as a log-normal distribution: |
[EQUATION] where [MATH] [MATH] [MATH] and [MATH] (see Kollmeier et al., 2006 , for the details) . We can derive the BHMF of AGN from the bolometric LF |
[MATH] [EQUATION] where [MATH] [MATH] , and [MATH] is in units of solar mass. Using the bolometric LF [MATH] of AGN, the Eddington ratio distribution for given black hole mass [MATH] can be calculated with |
[EQUATION] where [MATH] , and the BHMF of AGN, [MATH] , is available with Equation ( ). The mean Eddington ratio of AGN with [MATH] at [MATH] is |
[EQUATION] In this work, we adopt the luminosity-dependent density evolution (LDDE) bolometric LF calculated from the rest-frame optical, soft and hard X-ray, and near- and mid-IR bands in the redshift interval |
[MATH] by Hopkins, Richards & Hernquist ( 2007 [EQUATION] [EQUATION] The density function [MATH] is given by [EQUATION] with [EQUATION] |
and [EQUATION] [EQUATION] All the parameters of the LF are as follows: [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] , and |
[MATH] (see Table 4 in Hopkins, Richards & Hernquist, 2007 In Fig. , we plot the Eddington ratio distributions [MATH] for fixed black hole mass derived from that for fixed luminosity given by Kollmeier et al. ( 2006 . We find that the derived Eddington ratio distributions are close to the lognormal distribution, while ... |
[MATH] The evolution of massive black holes The evolution of massive black hole number density is described by (Small & Blandford, 1992 |
[EQUATION] where [MATH] is the mass function of massive black holes including both active and inactive black holes, [MATH] is the mean mass accretion rate for the black holes with |
[MATH] , and [MATH] describes the effect of black hole mergers on the BHMF. The total black hole mass density will not be altered by mergers, if the mass loss caused by the gravitational radiation is neglected. Shankar et al. ( 2007 assessed the importance of black hole mergers on the evolution of the BHMF using a simp... |
[MATH] of equal mass mergers per Hubble time, similar to the models of Richstone et al. ( 1998 . They found that the effect of black hole mergers is to slightly lower the number density of small black holes and increase the number density of massive black holes, if [MATH] is adopted (see Fig. 13 in Shankar et al., 2007... |
(e.g., Yu & Tremaine, 2002 ; Marconi et al., 2004 ; Tamura, Ohta & Ueda, 2006 . The mean mass accretion rate is [EQUATION] where the duty cycle of active black holes is defined as |
[EQUATION] Substituting Eqs. ( 11 ) and ( 12 ) into Eq. 10 ), we can rewrite the black hole evolution equation as [EQUATION] The AGN LF plays an important role in the study of the cosmological evolution of massive black holes. The optical quasar LF was adopted in Yu & Tremaine ( 2002 , however, the optical quasar LF |
(e.g., Boyle et al., 2000 has missed faint AGN (either intrinsic low-luminosity or obscured AGN). The hard X-ray surveys ( [MATH] keV) can trace the whole AGN population, including obscured type II AGN. The hard X-ray LF derived by Ueda et al. ( 2003 was used in some works on the cosmological evolution of massive black... |
(e.g., Marconi et al., 2004 ; Tamura, Ohta & Ueda, 2006 ; Shankar et al., 2007 . The number density of Compton-thick AGN is still quite uncertain, which is not included in the hard X-ray LF. The contribution of Compton-thick AGN to the black hole evolution was taken into account by multiplying a correction factor of 1.... |
The hard X-ray ( [MATH] ) and the mid-IR ( [MATH] bands are optimal for detection of AGN with column densities [MATH] (e.g., Treister & Urry, 2005 . The observations with the International Gamma-Ray Astrophysics Laboratory (INTEGRAL) and the Swift Burst Alert Telescope (BAT) indicate that the fraction of the absorbed A... |
(Markwardt et al., 2005 ; Bassani et al., 2006 (but also see Wang & Jiang, 2006 , which is confirmed by the mid-IR Spitzer observations of 25 luminous and distant quasars |
(Maiolino et al., 2007 Maiolino et al. ( 2007 suggested that the fraction of the obscured AGN to the total can be well fitted with |
[EQUATION] where [EQUATION] Müller & Hasinger ( 2007 found that the fraction of type II AGN detected in the hard X-ray band can be described by this function (Equation |
14 ) quite well (see Fig. 3 in their paper). Hopkins, Richards & Hernquist ( 2007 suggested that the fraction of Compton-thick to the total also decreases with luminosity and it is less than [MATH] per cent based on a variety of very hard X-ray/soft gamma-ray observations on AGN (see their paper for the detailed discus... |
[EQUATION] where [MATH] , as [MATH] is adopted (e.g., Kaspi et al., 2000 . We change the numerator in Eq. 14 ) to [MATH] here, so that [MATH] reduces to [MATH] for low-luminosity AGN, which is the same as that in |
Marconi et al. ( 2004 , while [MATH] for luminous AGN. In most of the previous works, both the radiative efficiency of [MATH] and Eddington ratio [MATH] are free parameters, and the comparisons between the BHMF of AGN relics and the measured local BHMF of galaxies always require: [MATH] and |
[MATH] (e.g., Yu & Tremaine, 2002 ; Marconi et al., 2004 ; Tamura, Ohta & Ueda, 2006 . As we have derived the mean Eddington ratio distributions as functions of black hole mass and redshift in the last section, there is only one free parameter [MATH] in our calculations for the cosmological evolution of massive black h... |
[MATH] . The final results are insensitive to the initial conditions at [MATH] . In all our calculations, [MATH] is adopted, because the Eddington ratio distributions are calculated from a sample of AGN with [MATH] |
(Kollmeier et al., 2006 . The resulted BHMFs of AGN relics at low redshifts are insensitive to the value of [MATH] , because the fraction of local black hole mass accreted at high redshifts can be neglected. We plot our results with different values of [MATH] in the upper panel of Fig. |
, which indicates that the measured local BHMF cannot be fitted with any values of [MATH] . This is due to the mean Eddington ratios derived in this work being [MATH] , which deviates significantly from [MATH] suggested in most of the previous works (e.g., Yu & Tremaine, 2002 ; Marconi et al., 2004 . In our calculation... |
The radiative efficiency of black hole accretion is closely related to the black hole spin. The massive black holes will be spun up through accretion, as the black holes acquire mass and angular momentum simultaneously through accretion. The spins of the massive black holes may also be affected by mergers of black hole... |
(e.g., Volonteri, Sikora & Lasota, 2007 ; King, Pringle & Hofmann, 2008 . This is also supported by the observations that single accretion events last [MATH] years in Seyfert galaxies and their total active lifetime is [MATH] |
years (Kharb et al., 2006 ; Ho, Filippenko & Sargent, 1997 ; Volonteri, Sikora & Lasota, 2007 Volonteri, Sikora & Lasota ( 2007 studied on how the accretion from a warped disc influences the evolution of black hole spins and concluded that within the cosmological framework, one indeed expects most supermassive black ho... |
(Volonteri, Sikora & Lasota, 2007 ; King, Pringle & Hofmann, 2008 . Motivated by their results on the cosmological spin evolution of black holes, we tentatively adopt a [MATH] -dependent radiative efficiency, in which [MATH] |
remains constant for [MATH] and increases as a power-law with black hole mass for [MATH] [EQUATION] in our calculations. We find that the measured local BHMF can be roughly reproduced by the BHMF of AGN relics provided [MATH] and [MATH] are adopted (see the lower panel in Fig. |
). Tamura, Ohta & Ueda ( 2006 tried to derive the BHMFs with redashifts up to [MATH] from the spheroid LF of early-type galaxies using the correlation between the spheroid luminosity and black hole mass (see their paper for the details), which provide further constraints on the model calculations for the cosmological e... |
Discussion As in the most previous works, we implicitly assume that the black hole growth is dominated by mass accretion in bright AGN, while some inactive black holes may still be accreting gases, though their mass accretion rates are very low. If the duration of the accretion in these objects is as long as the Hubble... |
Cao ( 2005 suggested that the accretion of such low-luminosity objects can be constrained by the hard X-ray background, though the emission from most of these individuals cannot be detected by any facilities now. It was found that less than [MATH] per cent of the local black hole mass density was accreted during the AD... |
(see Cao, 2007 , for the details) Hopkins, Narayan & Hernquist ( 2006 considered the distribution of local supermassive black hole Eddington ratios and accretion rates, accounting for the dependence of radiative efficiency and bolometric corrections on the accretion rate. They also found that black hole mass growth was... |
The main difference of this work from the previous works is that the Eddington ratio distributions are derived from an AGN sample with measured Eddington ratios (Kollmeier et al., 2006 . The Eddington ratio distributions for fixed black hole mass derived in our work approximate to the lognormal distribution (see Fig. )... |
Watarai et al. ( 2000 ’s calculations on the slim discs showed that the radiative efficiency will not deviate significantly from that for standard thin discs if [MATH] , which implies that the present adopted radiative efficiency independent of Eddington ratio [MATH] is indeed a good assumption. |
There is only one free parameter [MATH] in our calculations for the cosmological evolution of massive black holes. We find that the resulted BHMF of AGN relics is unable to reproduce the measured local BHMF for any value of [MATH] adopted, provided the radiative efficiency [MATH] is independent of black hole mass, as t... |
(e.g., Yu & Tremaine, 2002 ; Marconi et al., 2004 ; Tamura, Ohta & Ueda, 2006 . The mean Eddington ratios adopted in our calculations are derived from an AGN sample, which are in the range of [MATH] . Thus, it is not surprising that the local BHMFs cannot be reproduced by our calculations with any constant radiative ef... |
). We also use the hard X-ray LF derived from an AGN sample at high redshifts by Silverman et al. ( 2008 in stead of the bolometric LF of Hopkins, Richards & Hernquist ( 2007 in the calculations. It is found that the main results of this work change very little and the main conclusion is not altered. |
Volonteri, Sikora & Lasota ( 2007 studied on how the accretion from a warped disc influences the evolution of black hole spins and concluded that within the cosmological framework, one indeed expects most supermassive black holes in elliptical galaxies to have on average higher spin than black holes in spiral galaxies,... |
[MATH] and increases with black hole mass for [MATH] . This [MATH] -dependent radiative efficiency is qualitatively consistent with the results of Volonteri, Sikora & Lasota ( 2007 . It is found that the measured BHMFs can be fairly well reproduced by our model calculations with this [MATH] -dependent radiative efficie... |
[MATH] for [MATH] . This provides evidence for most massive black holes being spinning very rapidly. It is interesting to find that [MATH] is also required by the fitting of the residual hard X-ray background with the emission from the ADAFs in the low-luminosity objects |
(Cao, 2007 Our calculations can be improved if the mean spin parameter [MATH] as a function of black hole mass is available from the work within the cosmological framework, which is beyond the scope of present work. |
In our present calculations of the black hole evolution, the black hole mergers have been neglected. Shankar et al. ( 2007 assessed the importance of the black hole mergers on the evolution of the BHMF. They found that the impact of black hole mergers on the cosmological evolution of BHMF may probably be small compared... |
Acknowledgments We thank the anonymous referee for the helpful comments/suggestions, T. G. Wang and Y. F. Yuan for discussion, and N. Tamura for providing us the data of the spheroid-BHMFs. This work is supported by the NSFC (grant 10773020), and the CAS (grant KJCX2-YW-T03). |
# Source: arxiv 0808.0888 # Title: Self-organized periodicity of protein clusters in growing bacteria # Sections: all # Downloaded: 2026-03-03T05:15:34.004491+00:00 |
Current Address:]Department of Physics and Optical Science, University of North Carolina at Charlotte, Charlotte, North Carolina 28223 |
Self-organized periodicity of protein clusters in growing bacteria Abstract Chemotaxis receptors in E. coli form clusters at the cell poles and also laterally along the cell body, and this clustering plays an important role in signal transduction. Recently, experiments using fluorescence imaging have shown that, during... |
pacs: 87.16.A-, 05.50.+q, 87.15.Vv Spatial organization of proteins is important in many cellular processes including growth, division, movement, and establishment of polarity Kroos03 . Over the past few years, advances in imaging techniques such as fluorescence microscopy have led to an increased appreciation of the s... |
ThiemPositioningofchemosensoryEJ07 Sourjik08 demonstrated that clusters of chemotaxis receptors are approximately periodically positioned along the cell wall, independent of any known positioning mechanism such as the Min system MinOscillations . Other examples of periodically positioned protein clusters have emerged a... |
Lindner08 . The question arises – could such periodic positioning arise spontaneously or does it require the existence of an unknown positioning system? |
Here we demonstrate, within the context of a minimal lattice model, that protein clustering and periodic positioning of clusters can emerge spontaneously in growing cells. Lattice models have been used before to study clustering of membrane proteins with short-range interactions Goldman04 . In our model, existing clust... |
In our model, the cell membrane is represented by a square lattice, whose [MATH] -axis coincides with the long axis of the cell (see Fig. ). We employ periodic boundary conditions in the [MATH] direction to account for the cylindrical shape of bacteria like E. coli . The protomers (independently diffusing protein units... |
for each particle with any neighbors, which accounts for the loss of internal entropy when a particle associates with a cluster or a second protomer. The total energy of the system (in units of [MATH] ) is |
[EQUATION] where [MATH] is the total number of particles with one or more neighbors ( i.e. , the number of particles that are in clusters of size two or greater). Experiments indicate that the lateral receptor clusters are relatively immobile while individual membrane proteins are typically free to diffuse ThiemPositio... |
We use a Metropolis Monte Carlo algorithm to simulate the system. A randomly selected particle is moved to one of its unoccupied neighboring sites with an acceptance probability [MATH] where [MATH] is the energy change due to the proposed displacement of the particle. One Monte Carlo time step corresponds to one attemp... |
In the absence of the conformational energy cost ( [MATH] ), the thermodynamic system described by our energy function can be mapped to a two-dimensional Ising model, for which the critical interaction strength is known to be |
[MATH] Onsager44 . When the interaction strength is low, [MATH] , the system has one stable homogeneous phase, while for [MATH] , the system can phase separate into regions of high and low density. The conformational energy cost increases [MATH] |
To account for growth of the bacterial cell, we allow the lattice to expand in the [MATH] direction according to [MATH] , where [MATH] is the initial length of the bacterium and [MATH] is the growth rate. The expansion of the lattice is implemented by random insertion of empty columns at a rate [MATH] with equal probab... |
[MATH] so that, on average, [MATH] remains fixed at [MATH] For our simulations, the cell circumference was fixed at [MATH] , with [MATH] and interaction strength [MATH] , considerably greater than the critical strength [MATH] The system was initialized with a cluster at each end of the cell to mimic the existing cluste... |
The emergence of a self-organized periodicity of clusters can be understood by noting the positions of new clusters when they first appear. At the start of the simulation the clusters at each end of the cell act as sinks for newly inserted particles. As the cell grows and these two clusters move apart, a new cluster fo... |
To quantify the separation between clusters, we obtained the distribution of the separations between neighboring clusters for systems grown to [MATH] |
(see Fig. ). Due to stochastic fluctuations, a cluster is not thermodynamically stable until it reaches a critical size. We found that clusters of size [MATH] were stable and did not disappear; we thus used size 50 as a criterion to identify a cluster. The separation between neighboring clusters is defined to be the di... |
[MATH] ) clusters. For comparison, the distribution of inter-cluster separation would be an exponential if the cluster centers were positioned randomly (dotted line in Fig. ). |
To investigate the mechanism responsible for cluster positioning, we studied how the position of a newly formed cluster depends on the positions of existing clusters. For simplicity, we used periodic boundary conditions in the [MATH] direction and initialized simulations with [MATH] , that is, with only one column of s... |
[MATH] , yielding [MATH] As the system grows, the deposited particles aggregate to form first one cluster and, later, a second stable cluster. We recorded the position of the second cluster (once it had reached a size [MATH] ) with respect to the first. The distribution of the separations between the new cluster and th... |
To understand why the second cluster formed near midcell, we investigated the density profile of particles in the dilute region between two stable clusters during cell growth. Simulations were performed, as for Fig. 4(a), using periodic boundary conditions in [MATH] and starting from a single column. In Fig. 4(b), we p... |
The observed quadratic particle density profile can be understood as follows. The average local particle density [MATH] in regions that do not contain clusters satisfies the diffusion equation [MATH] where [MATH] is the particle diffusion coefficient and [MATH] is the particle insertion rate. Consider a region flanked ... |
[EQUATION] The peak of [MATH] is located at [MATH] , precisely at the midpoint between two cluster edges. The maximum particle density is [MATH] . Notice that it scales quadratically with cluster spacing. |
Since the particle density peaks at the midpoint between two neighboring clusters, this is the most likely place for a new cluster to nucleate. Of course, nucleation of a new cluster is a stochastic event; nevertheless, the probability of nucleating a new cluster is a highly nonlinear function of the local density, so ... |
The existence of a relatively sharp density threshold for cluster nucleation predicts scaling relations for the mean cluster separation. For steady-state growth of a long cell, on average a new cluster must appear between neighboring old clusters every time the cell doubles ( e.g. , see Fig. ). If there is a sharp dens... |
footnote2 , which are verified in Figs. 4(c) and 4(d). We have demonstrated that an interplay of protein creation or insertion in the membrane, protomer diffusion, aggregation, and cell growth can lead to periodically spaced protein clusters. In addition to the larger stable clusters, our stochastic nucleation mechanis... |
For chemoreceptors in E. coli , the observed inter-cluster spacing is of the order of [MATH] m. Assuming a membrane diffusion constant of [MATH] /sec Diffusion_constant and doubling time of around 60 minutes, we estimate [MATH] , implying that the vast majority of the chemoreceptors are bound to clusters rather than ex... |
Finally, we consider the biological significance of the periodicity of protein clusters. In the case of chemoreceptors, periodic cluster spacing ensures that each daughter cell receives at least one cluster following cell division or fragmentation of a filamentous cell ThiemPositioningofchemosensoryEJ07 This “equiparti... |
We thank K.C. Huang, Y. Meir, and V. Sourjik for helpful discussions and suggestions. NSW and RM acknowledge support from NIH grant R01 GM073186. |
# Source: arxiv 0808.1285 # Title: Adiabatic fluctuations from cosmic strings in a contracting universe # Sections: all # Downloaded: 2026-03-02T07:59:06.869891+00:00 |
Adiabatic fluctuations from cosmic strings in a contracting universe Abstract We show that adiabatic, super-Hubble, and almost scale invariant density fluctuations are produced by cosmic strings in a contracting universe. An essential point is that isocurvature perturbations produced by topological defects such as cosm... |
pacs: 98.80.Cq Introduction The cosmic microwave background (CMB) anisotropies observed by the Wilkinson Microwave Anisotropy Probe (WMAP) have revealed that the primordial density fluctuations are almost adiabatic and scale invariant |
. Inflationary cosmology is currently the favored scenario to explain such primordial fluctuations. According to it, density perturbations are produced as quantum vacuum fluctuations on sub-Hubble scales and then stretched to super-Hubble scales during the phase of accelerated expansion of space. However, inflationary ... |
), and thus it is important to study scenarios alternative to inflation. In the 1980s, topological defect models such as those based on cosmic strings were investigated intensely as a possible alternative to generate primordial density fluctuations |
. However, the fluctuations induced by defects in an expanding universe are isocurvature and, even if they might mimic the inflationary predictions for the temperature-temperature (TT) correlation of the CMB |
, observations of the anti-correlation of the temperature and the E-mode polarization (TE), precisely measured by WMAP, confirmed that such fluctuations could not be the dominant source of CMB anisotropies |
. Thus, causal scaling seed models are ruled out as a main component of primordial density fluctuations. On the other hand, in recent years other types of scenarios alternative to inflation motivated by developments in string theory have been proposed. Examples are the Pre-Big-Bang model (PBB) |
and the Cyclic/Ekpyrotic scenario The common feature of these models is that the universe begins in a contracting phase before emerging into the expanding phase of Standard Big Bang cosmology after a cosmological bounce. In the contracting phase, comoving scales exit the Hubble radius unless the contraction is too rapi... |
. There are models which yield an almost scale invariant spectrum (see e.g. ) after the bounce. In this paper, we suggest the possibility that primordial density fluctuations are produced by causal seeds such as cosmic strings in the contracting phase, and show that they could generate adiabatic, almost scale invariant... |
where quantum fluctuations source the primordial density fluctuations, here cosmic strings seed the perturbations. Of course, in the cyclic scenario, topological defects may be dangerous because they may dominate the energy density of the universe, as pointed out by Avelino et al. in Refs. |
. However, Avelino et al. also give a solution to that problem: they point out that a relatively long period of cosmic acceleration at low energies (late period of one cycle) can dilute topological defects in order that they do not overdominate the universe. A second possibility is to consider the birth of the universe... |
Density fluctuations produced by causal seed models naturally become super-Hubble in the contracting phase. More specifically, the key point is that the defect-seeded perturbations which are initially isocurvature in nature seed a growing adiabatic mode. At the time when the symmetry (whose breaking yields the topologi... |
Adiabatic fluctuations from cosmic strings in a contracting universe The evolution of cosmic strings in a contracting universe was investigated in Refs. |
. As in these references, we will assume that the distribution of strings on super-Hubble scale is like a random walk. We make the simplest assumption that the universe is initially matter and then radiation dominated in the contracting phase. In this context, it was shown that cosmic strings obey the scaling solution ... |
On super-Hubble scales, the dynamics of the defect network sets up isocurvature fluctuations which in turn act as a seed for growing curvature perturbations. As the universe contracts, the temperature of radiation increases, and eventually leads to symmetry restoration and the disappearance of the topological defects. ... |
On super-Hubble scales, the equation for the evolution of the curvature perturbation on uniform total density hypersurfaces, [MATH] , is given by |
[EQUATION] where [MATH] is the Hubble parameter, [MATH] and [MATH] are the total energy density and pressure, respectively. A dot represents a derivative with respect to the cosmic time. The non-adiabatic pressure perturbation [MATH] is defined as [MATH] with [MATH] being the total density fluctuation and [MATH] being ... |
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