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One parameter / encodes (he unknown inclination of the filament relative to the plane of the sky. | One parameter $i$ encodes the unknown inclination of the filament relative to the plane of the sky. |
Finally. 2 additional dimensional parameters are required to convert the surface density X caleulated from the models to a predicted 350jn flix density / that can be compared with the observations. | Finally, 2 additional dimensional parameters are required to convert the surface density $\Sigma$ calculated from the models to a predicted $850\micron$ flux density $f$ that can be compared with the observations. |
These are the flux conversion factor FCF. defined by equation 4. below. and a DC flux level fpe: that accounts for the unknown zero point of the 350jmi map. | These are the flux conversion factor FCF, defined by equation \ref{eq:FCF} below, and a DC flux level $f_{DC}$ that accounts for the unknown zero point of the $850\micron$ map. |
The latter parameter is necessary because the JCMT measures only a differential signal between chop positions and is insensitive therefore to the absolute zero point of the flux in the map. | The latter parameter is necessary because the JCMT measures only a differential signal between chop positions and is insensitive therefore to the absolute zero point of the flux in the map. |
The total nmunber of parameters is 6 lor the Ostriker model and 3 for the magnetic models. but all 5 of the dimensional parameters are constrained by independent observations or reasonable assumptions. | The total number of parameters is 6 for the Ostriker model and 8 for the magnetic models, but all 5 of the dimensional parameters are constrained by independent observations or reasonable assumptions. |
Η one is concerned with the density structure only. and not with the magnetic field geometry. then the degenerate parameters 2 and 9of the GS model can be replaced by a single independent parameter & (see discussion following equation 11)). thereby decreasing the parameter space of the GS model to 7 dimensions. | If one is concerned with the density structure only, and not with the magnetic field geometry, then the degenerate parameters $\beta$ and $\theta$of the GS model can be replaced by a single independent parameter $\kappa$ (see discussion following equation \ref{eq:Bstod}) ), thereby decreasing the parameter space of the GS model to 7 dimensions. |
We determined the velocity dispersion within the filament from CO (2.1) and CO maps of the southern half of the GI1.11-0.12. including all of region 1. | We determined the velocity dispersion within the filament from $^{18}$ O (2–1) and $^{13}$ CO (2--1) maps of the southern half of the G11.11-0.12, including all of region 1. |
These observations were obtained with RxA3 on the «ΙΟΝΤ during several observing sessions in 2000 July. | These observations were obtained with RxA3 on the JCMT during several observing sessions in 2000 July. |
Typical line widths were around 0.9 + for CPO a 1.2 ! for CO. | Typical line widths were around 0.9 $^{-1}$ for $^{18}$ O a 1.2 $^{-1}$ for $^{13}$ CO. |
Since the CUO data were often quite noisy. we loosened the constraint on the L-dimensional velocity dispersion of the gas to be : OSkms <12kms 1 | Since the$^{18}$ O data were often quite noisy, we loosened the constraint on the 1-dimensional velocity dispersion of the gas to be : 0.8 1.2 . |
Since the CUO data were often quite noisy. we loosened the constraint on the L-dimensional velocity dispersion of the gas to be : OSkms <12kms 1, | Since the$^{18}$ O data were often quite noisy, we loosened the constraint on the 1-dimensional velocity dispersion of the gas to be : 0.8 1.2 . |
Future studies should also address other open issues. like the presence of a single optical filament observed in FilD and the effects of the pre-shock symmetry of the cloud on the morphology of the optical emission and the on the surface brightness in X-rays. | Future studies should also address other open issues, like the presence of a single optical filament observed in FilD and the effects of the pre-shock symmetry of the cloud on the morphology of the optical emission and the on the surface brightness in X-rays. |
Moreover. we also aim at describing the shock-cloud interaction in RegNE. which is an other bright ray knot. hotter than FilD. discussed in Paper I. The evolution of the shock-cloud interaction has been followed for ~6000 vr. for all the model setups presented in this paper. | Moreover, we also aim at describing the shock-cloud interaction in RegNE, which is an other bright X-ray knot, hotter than FilD, discussed in Paper I. The evolution of the shock-cloud interaction has been followed for $\sim 6000$ yr, for all the model setups presented in this paper. |
We have developed a procedure for the selection of the evolutionary stage of the interaction which best reproduce the observed data. | We have developed a procedure for the selection of the evolutionary stage of the interaction which best reproduce the observed data. |
This procedure is based on the definition of three quantities: Figure Al shows the temporal evolution (¢=0 corresponds to the impact of the shock front with the cloud) of A (upper panel). MPE (central panel). and R (lower panel). for setup Sphl (black diamond) and setup EII2 (red cross). | This procedure is based on the definition of three quantities: Figure \ref{fig:timesel} shows the temporal evolution $t=0$ corresponds to the impact of the shock front with the cloud) of $A$ (upper panel), $MPE$ (central panel), and $R$ (lower panel), for setup Sph1 (black diamond) and setup Ell2 (red cross). |
The corresponding observed values are indicated by the horizontal blue lines. while the vertical green lines indicate the formation of thermal instabilities: no optical emission is therefore present on the left of the green line. | The corresponding observed values are indicated by the horizontal blue lines, while the vertical green lines indicate the formation of thermal instabilities: no optical emission is therefore present on the left of the green line. |
The graphs in Fig. | The graphs in Fig. |
Al indicate that. for both setups. we can identify a temporal range. around 3500+500 yr where the synthesized values globally best approach the observed ones. | \ref{fig:timesel} indicate that, for both setups, we can identify a temporal range, around $3500\pm500$ yr where the synthesized values globally best approach the observed ones. |
Below ¢=3000 yr we do not have optical emission. while above 4000 vr the synthesized values have largest deviations from the observed ones. | Below $t=3000$ yr we do not have optical emission, while above 4000 yr the synthesized values have largest deviations from the observed ones. |
For all the three quantities. the smallest deviations are at ~3500 yr. therefore in this paper we report the results at ¢=3400 yr. for setup Sphl. and ¢=3550. for setup EII2. | For all the three quantities, the smallest deviations are at $\sim 3500$ yr, therefore in this paper we report the results at $t=3400$ yr, for setup Sph1, and $t=3550$, for setup Ell2. |
Figure Al also indicates that. for -3500 yr. setup Ell2 well reproduces the observed values of A and MPE. while in setup Sphl we have a large area of the X-ray emitting knot and a mean photon energy in the bright regions that is larger than the observed one. | Figure \ref{fig:timesel} also indicates that, for $t\sim3500$ yr, setup Ell2 well reproduces the observed values of $A$ and $MPE$, while in setup Sph1 we have a large area of the X-ray emitting knot and a mean photon energy in the bright regions that is larger than the observed one. |
This is in agreement with the results discussed in Sect. 3.. | This is in agreement with the results discussed in Sect. \ref{Results}, |
where we show that in setup Sphl the bright regions do not correspond to the minima in the mean photon energy map. | where we show that in setup Sph1 the bright regions do not correspond to the minima in the mean photon energy map. |
As for setup EII2. the high values of the count-rate 1n the soft region are due to the brightest part in the northern edge of the X-ray knot (see Fig. 5)). | As for setup Ell2, the high values of the count-rate in the soft region are due to the brightest part in the northern edge of the X-ray knot (see Fig. \ref{fig:mappeXAB}) ), |
where we have the minimum in the mean photon energy map (Fig. 7.Fig. | where we have the minimum in the mean photon energy map (Fig. \ref{fig:avgEAB}, |
The well-known mass discrepancies in galaxies (Dosunia 1978: Brocils 1992: Verleijen 1997) aud iu clusters of ealaxies (Zwicky 1937: Carlbere. Yee Ellingson 1997: 'Tvsou. IKochauski DellAntonio 1998) are usually taken to imply the existence of à large fraction of invisible. or “dark.” matter (DAT) in the universe. | The well-known mass discrepancies in galaxies (Bosma 1978; Broeils 1992; Verheijen 1997) and in clusters of galaxies (Zwicky 1937; Carlberg, Yee Ellingson 1997; Tyson, Kochanski Dell'Antonio 1998) are usually taken to imply the existence of a large fraction of invisible, or “dark,” matter (DM) in the universe. |
A popular candidate is Cold Dark Matter. for which the DM particles. whatever thev are. are alinost at rest with respect to the ITubble flow in the carly universe. | A popular candidate is Cold Dark Matter, for which the DM particles, whatever they are, are almost at rest with respect to the Hubble flow in the early universe. |
CDM is often imagined to be a heavy. uon-barvouic relic particle from the carly universe which has essentially ouly eravitational interactions with itself and with normal. or barvonic. matter. | CDM is often imagined to be a heavy, non-baryonic relic particle from the early universe which has essentially only gravitational interactions with itself and with normal, or baryonic, matter. |
The CDM model has been studied intensively for tweuty vears and now has many wellavorked out predictions for the formation of structure in the universe Bertschinger 1998). | The CDM model has been studied intensively for twenty years and now has many well-worked out predictions for the formation of structure in the universe Bertschinger 1998). |
The broad-brush impression one now has is that the curreutly favored ACDAL model boasts a considerable degree of success in predicting larec-scale structure PPearce 11999: Dahicall 11999). | The broad-brush impression one now has is that the currently favored $\Lambda$ CDM model boasts a considerable degree of success in predicting large-scale structure Pearce 1999; Bahcall 1999). |
But it has become apparent iu recent vears that the predictions of almost auv favor of CDM are seriously at variance with the observed properties of ealaxies because the ceutral deusities of collapsed objects and fragments are predicted to be too hieh refsec:cusps) }. | But it has become apparent in recent years that the predictions of almost any flavor of CDM are seriously at variance with the observed properties of galaxies because the central densities of collapsed objects and fragments are predicted to be too high \\ref{sec:cusps}) ). |
A ununuber of authors have therefore beeuu to explore variations of the CDM model: most favor a modification to the properties of the DAL particles rather than the alternative a chanee to the law of eravitv. | A number of authors have therefore begun to explore variations of the CDM model; most favor a modification to the properties of the DM particles rather than the alternative – a change to the law of gravity. |
The simplest is the warm DM matter model CColombi. Dodelson Widrow 1996: Somer-Larsen Doleov 1999: Wogan 1999) in which streaming of tle DM particles in the carly universe suppresses sinall-scale power in the fluctuation spectrum. | The simplest is the warm DM matter model Colombi, Dodelson Widrow 1996; Sommer-Larsen Dolgov 1999; Hogan 1999) in which streaming of the DM particles in the early universe suppresses small-scale power in the fluctuation spectrum. |
In addition. WDAL particles iu halos of greater volume density mst have larger velocity spreads. because of Liouville's theorem. thereby prechiding y.rong density eradieuts. | In addition, WDM particles in halos of greater volume density must have larger velocity spreads, because of Liouville's theorem, thereby precluding strong density gradients. |
Suuulators of the WDAL model suppress small-scale power in the fluctuation spectrum but generally ignore the initial finite velocity spread. which is dificult to include without wrecking the quiet start. | Simulators of the WDM model suppress small-scale power in the fluctuation spectrum but generally ignore the initial finite velocity spread, which is difficult to include without wrecking the quiet start. |
Since the DM in these simulations still has infinite plase space deusity. the resulting halo profiles have deusitv cusps resenibliug those which form iu CDM AIMoore 11999: 1n 22000). | Since the DM in these simulations still has infinite phase space density, the resulting halo profiles have density cusps resembling those which form in CDM Moore 1999; n 2000). |
It the DM particle decoupled from thermal equilibrium at some carly epoch. then its phase space deusitv today can be predicted. | If the DM particle decoupled from thermal equilibrium at some early epoch, then its phase space density today can be predicted. |
Feriuouic DAL would have a finite initial masini f=fu Which leads the ποιον constraint ou neutrino masses (Tremaine Comm 1979). | Fermionic DM would have a finite initial maximum $f = \fm$, which leads the well-known constraint on neutrino masses (Tremaine Gunn 1979). |
Hogan (1999) suggests that halos today may reflect. f. the average over the initial velocity distribution. | Hogan (1999) suggests that halos today may reflect, $\fbar$, the average over the initial velocity distribution. |
But fias is uubounded for bosous. making a single value of f£ rather poorly defined (Madsen. 2000). | But $\fm$ is unbounded for bosons, making a single value of $\fbar$ rather poorly defined (Madsen 2000). |
Thus pliase space density constraints are much weaker for bosonic WDAL careful simulations are needed to predict the structure of halos iu this model. but the πια] fraction of hieh f material males wuld cusps seem likely. | Thus phase space density constraints are much weaker for bosonic WDM; careful simulations are needed to predict the structure of halos in this model, but the small fraction of high $f$ material makes mild cusps seem likely. |
The fne-erained phase space deusitv. fay. of a truly collisionless fluid is strictly conserved. but the maxima coarse-erain deusity. fous can decrease dunus violeut evolution. such as a collapseor merecr. | The fine-grained phase space density, $\ffg$, of a truly collisionless fluid is strictly conserved, but the maximum coarse-grain density, $\fcg$, can decrease during violent evolution, such as a collapseor merger. |
Simulations. which by their nature can track oulv. Az, have fouud. however. that the maximum fen in even quite violent collapses (λίαν van Albada 1981) or mergers (Farouki. Shapiro Duncau 1983: Barnes 1992). | Simulations, which by their nature can track only $\fcg$, have found, however, that the maximum $\fcg$ in even quite violent collapses (May van Albada 1984) or mergers (Farouki, Shapiro Duncan 1983; Barnes 1992). |
These tests were with svstems that have finite fuas at the outset. whereas fj; can. and ecuerally does. decrease during the collapse or merger of svstems with uubounded fiuax. | These tests were with systems that have finite $\fm$ at the outset, whereas $\fcg$ can, and generally does, decrease during the collapse or merger of systems with unbounded $\fm$ . |
Heruquist. Sperecl Ilexl (1993) iuerged. svstenis | Hernquist, Spergel Heyl (1993) merged systems |
Lindblad resonances (Goodman&Ralikov2001). | Lindblad resonances \citep{GOR01}. |
. Instead. thev are staucling waves trapped lh a CANAtv bounded by (he resonance radius and the inner edge of the disk. with viscous dissipation as their principal source of damping. | Instead, they are standing waves trapped in a cavity bounded by the resonance radius and the inner edge of the disk, with viscous dissipation as their principal source of damping. |
Consequently. (he standard Lindblad resonance Lorcque ormula cannot be applied to determine the torque al an apsidal resonance in a protopl:metary disk. | Consequently, the standard Lindblad resonance torque formula cannot be applied to determine the torque at an apsidal resonance in a protoplanetary disk. |
Fortunately. simple physical arguments enable us to estimate the factor by whic1 Lhe apsidal toreue is reduced below the estimate provided by the standard formula. | Fortunately, simple physical arguments enable us to estimate the factor by which the apsidal torque is reduced below the estimate provided by the standard formula. |
Suppose that the apsidal cavity contains Vo wavelength. | Suppose that the apsidal cavity contains $N$ wavelength. |
Comparing the viscous dissipation (iniescae. (v7)1. with that for propagation across a wavelength. 27/(hey)=27Qy/(κος)LD we deduce that the apsidal torque is a factor 27Na smaller (han (he standard torque formula iniplies. | Comparing the viscous dissipation timescale, $(\nu k^2)^{-1}$, with that for propagation across a wavelength, $2\pi/(kv_g)=2\pi\Omega_d/(kc_s)^2$, we deduce that the apsidal torque is a factor $2\pi N\alpha$ smaller than the standard torque formula implies. |
Thus for plausible parameters. V<1 anda ~107.ni it is even smaller than the lorque :wo other first order Lindblad resonances. | Thus for plausible parameters, $N\lesssim 1$ and $\alpha\sim 10^{-3}$, it is even smaller than the torque at other first order Lindblad resonances. |
We have deliberately glossed over several items that merit attention. | We have deliberately glossed over several items that merit attention. |
These are briefly covered below. | These are briefly covered below. |
First order Lindblad resonances are of two kinds. external ones which excite the planet's eccentricitv. and co-orbital ones which damp it. | First order Lindblad resonances are of two kinds, external ones which excite the planet's eccentricity, and co-orbital ones which damp it. |
In a disk of uniform surface density. damping | In a disk of uniform surface density, damping |
indicated by the arrows in Figure {13}. | indicated by the arrows in Figure \ref{lightcurve_times}. |
The increase of the outflow speeds compared to the inflow speeds — especially for the plasma moving within the legs of the loops in the 193 and 211 A filters — suggests the possibility of acceleration during the reconnection process. | The increase of the outflow speeds compared to the inflow speeds – especially for the plasma moving within the legs of the loops in the 193 and 211 $\mbox{\AA}$ filters -- suggests the possibility of acceleration during the reconnection process. |
The localization of the initial fast inflows, along with the short inflow-outflow delay time on order of at most a few minutes, favors fast, Petschek reconnection. | The localization of the initial fast inflows, along with the short inflow-outflow delay time on order of at most a few minutes, favors fast, Petschek reconnection. |
An in depth study of the magnetic fluxes and reconnection rates would be required for confirmation of such speculation. | An in depth study of the magnetic fluxes and reconnection rates would be required for confirmation of such speculation. |
refeq:c,, BS10 found the slope of the mgu—Mhaio relation to be weakly dependent on Τα. | \\ref{eq:c}, BS10 found the slope of the $\mbh-\mhalo$ relation to be weakly dependent on $\reject$. |
At z=0 it varies from no=1.50 for z=0.1 to no=1.61 for x=1.0. | At $z=0$ it varies from $n_0=1.50$ for $x=0.1$ to $n_0=1.61$ for $x=1.0$. |
In order to exactly match the slope of 1.55 that is both observed (Bandaraetal.2009 find 1.55+0.31) and predicted by the simulations (BS10 find 1.55+0.05 for the same simulation as is analyzed here?)), we wouldneed to use z=0.22. | In order to exactly match the slope of 1.55 that is both observed \citealt{band09} find $1.55 \pm 0.31$ ) and predicted by the simulations (BS10 find $1.55 \pm 0.05$ for the same simulation as is analyzed ), we wouldneed to use $x=0.22$. |
By using an NFW density profile and Eq. (5)), | By using an NFW density profile and Eq. \ref{eq:c}) ), |
we have implicitly assumed that the dark matter profile is well described by the results obtained from simulations that include only dark matter. | we have implicitly assumed that the dark matter profile is well described by the results obtained from simulations that include only dark matter. |
Duffyetal.(2010) have recently shown that, on the scales of interest here, the back-reaction of the baryons onto the dark matter is in fact very small if feedback from AGN is included, as required to reproduce the observed stellar and gas properties of groups of galaxies (McCarthyetal.2010;Puchwein2008;2010;Duffyetal. 2010). | \citet{duff10} have recently shown that, on the scales of interest here, the back-reaction of the baryons onto the dark matter is in fact very small if feedback from AGN is included, as required to reproduce the observed stellar and gas properties of groups of galaxies \citep{mcca10,puch08,fabj10,duff10}. |
. If, as argued in BS10, the BH mass is controlled by the DM halo binding energy, then we expect the mau—Mhalo relation to evolve because the halo binding energy depends not only on halo mass, but also on the virial radius and concentration, both of which vary with redshift for a fixed halo mass. | If, as argued in BS10, the BH mass is controlled by the DM halo binding energy, then we expect the $\mbh-\mhalo$ relation to evolve because the halo binding energy depends not only on halo mass, but also on the virial radius and concentration, both of which vary with redshift for a fixed halo mass. |
If the c—mhalo relation did not evolve, then we would expect MBH(Mnalo)ος(14-z) (Eq. 3)). | If the $c-\mhalo$ relation did not evolve, then we would expect $\mbh(\mhalo)\propto (1+z)$ (Eq. \ref{eq:be}) ). |
However, because halo concentration decreases with redshift (Eq. 5)) | However, because halo concentration decreases with redshift (Eq. \ref{eq:c}) ) |
we expect the actual evolution of the mpg—Mhalo relation to be weaker, o,< 1. | we expect the actual evolution of the $\mbh-\mhalo$ relation to be weaker, $\alphas<1$ . |
The resulting relation between BH mass and DM halo binding energy predicts that, at a given mhalo, ΠΙΒΗ increases with redshift and that by z=2 BHs are between 1.5 (for roj/rnaio= 0.1) and 2.6 (for ra/Thalo= 1.0) times more massive than at redshift zero. | The resulting relation between BH mass and DM halo binding energy predicts that, at a given $\mhalo$, $\mbh$ increases with redshift and that by $z=2$ BHs are between 1.5 (for $\reject/\rhalo=0.1$ ) and 2.6 (for $\reject/\rhalo=1.0$ ) times more massive than at redshift zero. |
For our fiducial radius of self-regulation of x=0.22, BHs are 2.1 times more massive, in excellent agreement with the simulation prediction of as=0.65+0.06 (Table 1)). | For our fiducial radius of self-regulation of $x=0.22$, BHs are 2.1 times more massive, in excellent agreement with the simulation prediction of $\alphas=\asmh$ (Table \ref{tab:deltas}) ). |
'The evolution predicted by Eqs. 3 | The evolution predicted by Eqs. \ref{eq:be} |
refeq:c is shown in Fig. 2.. | \\ref{eq:c} is shown in Fig. \ref{fig:mh}. |
The grey shaded region outlines the analytic prediction for the evolution in BH mass over the range Tej/Thalo=0.1—1.0 and the solid black line shows the prediction for rej/Tnalo=0.22 (the value that reproduces the slope of the redshift zero ΠΙΒΗ—Mnalo relation). | The grey shaded region outlines the analytic prediction for the evolution in BH mass over the range $\reject/\rhalo=0.1-1.0$ and the solid black line shows the prediction for $\reject/\rhalo=0.22$ (the value that reproduces the slope of the redshift zero $\mbh-\mhalo$ relation). |
The red, dotted curve shows the simulation prediction for the evolution of the mpH—mna relation, including all BHs with mgu>10?mae«a. | The red, dotted curve shows the simulation prediction for the evolution of the $\mbh-\mhalo$ relation, including all BHs with $\mbh>10^2\,\mseed$. |
At all redshifts the normalisation of the simulated ΊΠΒΗ—Mnaio relation is compatible with that predicted by the analytic model. | At all redshifts the normalisation of the simulated $\mbh-\mhalo$ relation is compatible with that predicted by the analytic model. |
For comparison, the dashed, black line shows the predicted evolution of the ΠΙΒΗ—mao relation if c(mnalo) were independent of redshift. | For comparison, the dashed, black line shows the predicted evolution of the $\mbh-\mhalo$ relation if $c(\mhalo)$ were independent of redshift. |
The analytic model can only reproduce the simulation result if the evolution of the concentration-mass relation is taken into account. | The analytic model can only reproduce the simulation result if the evolution of the concentration-mass relation is taken into account. |
The evolution of the πιΒΗ—mna relation thus provides additional evidence for the idea that the masses of BHs are determined by the binding energies of the haloes in which they reside. | The evolution of the $\mbh-\mhalo$ relation thus provides additional evidence for the idea that the masses of BHs are determined by the binding energies of the haloes in which they reside. |
Considering now only the stellar masses for which the evolution has been measured observationally (m.10!! Mo) and the redshift range for which all of the stellar and BH properties of the galaxies are converged numerically (z« 1), we ask if we can explain how the relations between BH mass and galaxy stellar properties evolve. | Considering now only the stellar masses for which the evolution has been measured observationally $\ms\sim10^{11}\,\msun$ ) and the redshift range for which all of the stellar and BH properties of the galaxies are converged numerically $z<1$ ), we ask if we can explain how the relations between BH mass and galaxy stellar properties evolve. |
BS10 showed that the BH mass is determined by the binding energy of the DM halo, which explains why the mgpu—Uhaio relation does not evolve. | BS10 showed that the BH mass is determined by the binding energy of the DM halo, which explains why the $\mbh-\uhalo$ relation does not evolve. |
We find that the mpy—U. relation also does not evolve, implying that, over the range of redshifts and masses investigated here, U.οςUnaio. | We find that the $\mbh-\us$ relation also does not evolve, implying that, over the range of redshifts and masses investigated here, $\us\propto\uhalo$. |
The top panel of Fig. | The top panel of Fig. |
3 confirms that this is indeed the case in our simulation. | \ref{fig:tracks} confirms that this is indeed the case in our simulation. |
For the binding energy of the galaxy to track that of the halo, we require the two to grow through the same mechanism. | For the binding energy of the galaxy to track that of the halo, we require the two to grow through the same mechanism. |
This condition is met if the galaxies grow primarily through dry mergers. | This condition is met if the galaxies grow primarily through dry mergers. |
In the absence of significant in-situ star formation, both the stellar component, which is predominantly spheroidal for massive galaxies, and the DM halo are collisionless systems and are therefore expected to evolve in a similar manner. | In the absence of significant in-situ star formation, both the stellar component, which is predominantly spheroidal for massive galaxies, and the DM halo are collisionless systems and are therefore expected to evolve in a similar manner. |
'The bottom panel of Fig. | The bottom panel of Fig. |
3 shows that at z«1 thespecific star formation rates (SSFR= m/m.) of galaxies with ms~1011Mo are significantly lower than the inverse of the Hubble time, 1/tg, implying that the galaxies are indeed not growing significantly via in-situ star-formation, in agreement with various observations | \ref{fig:tracks} shows that at $z\ll 1$ thespecific star formation rates $\equiv\dot{m}_{\rm s}/\ms$ ) of galaxies with $\ms \sim 10^{11}\,\msun$ are significantly lower than the inverse of the Hubble time, $1/t_{\rm H}$ , implying that the galaxies are indeed not growing significantly via in-situ star-formation, in agreement with various observations |
where we used | where the later equation is the $V$ -gauge. |
spherical svimmetrv and RM)" J pq—qu" For our choi | The equations of motion for $M_{ij}$ and $\Phi^i_{\phantom{i}\alpha}$ are trivially satisfied by the above assumption, where for the latter we assume a vanishing $SU(2)$ connection. |
ce of charges. ancl thecomplex-valued aoe Forthe U(1) and SU(2) | We therefore remain with the constraint: When using the hypermultiplet, we assume covariantly constant hypermultiplet scalars: The equation of motion for the hyperscalars then gives \ref{DR}) ). |
connections we assume The equationof motionfor theSU(2) vanishing of (he appe | Solving the other equations of motion we will find that for all our cases: For our ansatz to constitute a solution, it must satisfy the equations of motion for the, the moduli $Y^I$ , the auxiliary field $T_{01}^-$, the $U(1)$ connection $A_a$, and either $V_a$ for the nonlinear multiplet or the auxiliary field $D$ for the hypermultiplet. |
arSU(2) Lagrangianalso 17 (2)) always | In the cases that we solved, we observed that when using the hypermultiplet, the equation of motion for $D$ has an overall factor of $(k^3-k_0)^2$ , after substituting the ansatz. |
[13).. quadratic implies When using | For $k^3=k_0$, one has to take this limit only after solving the equations of motion, in order not to lose a constraint. |
(he nonlinear multiplet.the auxilia | From the Einstein-Hilbert term in the Lagrangian \ref{lag}) ), one sees that “Newton's constant” is given by the unscaled Kähhler potential: Usually one fixes $G_N=1$ as the dilatational $D$ -gauge choice. |
ry | This, however, is too restrictive and does not always allow a solution. |
field Dmax be d | Therefore $G_N$ is a function of the radial coordinate, resembling the case of dilaton gravity. |
etermined by(he Mb,=0 | The metric in the Einstein frame is given by TheADM mass (in Planck units) for a non-normalized metric is given by |
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