The influence of scalar fields in protogalactic interactions
M.A. Rodriguez-Meza, J. Klapp, J.L. Cervantes-Cota, H. Dehnen
aa r X i v : . [ a s t r o - ph . C O ] J u l The influence of scalar fieldsin protogalactic interactions
M. A. Rodr´ıguez-Meza ∗ , Jaime Klapp † and Jorge L. Cervantes-Cota ‡ Departamento de F´ısica,Instituto Nacional de Investigaciones Nucleares (ININ)Apartado postal 18–1027, M´exico D.F. 11801, M´exicoandHeinz DehnenFakult¨at f¨ur PhysikUniversit¨at Konstanz, Postfach 5560 M 677D-78434 Konstanz, Germany
Abstract
We present simulations within the framework of scalar-tensor the-ories, in the Newtonian limit, to investigate the influence of massivescalar fields on the dynamics of the collision of two equal sphericalclouds. We employ a SPH code modified to include the scalar field tosimulate two initially non-rotating protogalaxies that approach eachother, and as a result of the tidal interaction, intrinsic angular mo-mentum is generated. We have obtained sufficient large values of
J/M to suggest that intrinsic angular momentum can be the result of tidalinteractions. ∗ On leave from Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla. † e-mail:[email protected] ‡ e-mail:[email protected] Introduction
In recent years there has been achieved important progress in understandingthe dynamics that led to the formation of galaxies. Two and three dimen-sional N-body simulations of galaxies and protogalaxies have been computedusing up to a few millions of particles, giving a more realistic view of howgalaxies, quasars, and black holes could have formed (Barnes & Hernquist1992, Barnes 1998).The Universe’s composition at the time galaxies formed could be, theoret-ically, very varied, including baryonic visible and dark matter, non baryonicdark matter, neutrinos, and many cosmological relics stemming from sym-metry breaking processes predicted by high energy physics (Kolb & Turner1990). All these particles, if present, should have played a role in the structureformation. Then, galaxies are expected to possess dark matter componentsand, in accordance with the rotational curves of stars and gas around thecentres of spirals, they are in the form of halos (Ostriker & Peebles 1973)and must contribute to at least 3 to 10 times the mass of the visible matter(Kolb & Turner 1990).Whatever the Universe composition was, protogalaxies were originateddue to a spectrum of scale-invariant perturbations (Harrison 1970; Zel’dovich1972) that was present within the cosmological background at the beginningof structure formation; the inflationary cosmology is the most convincingscenario that explains its origin (Mukhanov et al 1992). Protogalactic struc-tures began to acquire some momenta, e.g. tidal torques (Fall & Efstathiou1980), because of local gravitational instabilities to provoke plenty of colli-sions, mergings, fly-bys, etc, between these original, cosmic structures. Asa result of their evolution, galaxies, as we presently know them, must haveformed. The dynamics of protogalaxies has been studied intensively, for areview see Barnes & Hernquist (1992). There has been much interest to un-derstand how galaxies acquired their present features, especially how theirinternal angular momentum (spin) has been gotten,
J/M ∼ O (10 ) cm / s.A very important issue about it is how the transfer of angular momentumbetween protogalaxies took place to give rise to the observed elliptical andspiral galaxies with their known mass and rotational properties. As an ini-tial condition can be thought that protogalaxies were gravitationally isolated.However, there are some indications that the orbital angular momentum inspiral galaxies in pairs is few times larger than their spins, so pairs seem to benot dynamically isolated (Oosterloo 1993). Part of this angular momentum2ould had its origin in the cosmic expansion (Caimmi (1989,1990), Andriani& Caimmi (1994)), where it has been computed the torques at the beginningof strong decoupling from the Hubble flow of spherical-symmetric densityperturbations. Moreover, observations of spin angular momentum of variousthousands of disc galaxies are compatible with the mechanism of generationof spin via tidal torques (Sugai & Iye 1995). Other theoretical and numericalstudies of evolution of angular momentum of protogalaxies from tidal torquesare in line with observations (Chernin 1993, Catelan & Theuns 1996).In the present work we investigate how the transfer of orbital to spinangular momentum is achieved when two equal, spherical clouds pass by, andin some cases when they collide; this type of interactions are to be expected inthe tidal torques scheme. Studies of interacting spherical systems very relatedto ours have been done using the Newtonian theory of gravity, including anumber of topics: mergings (White 1978, 1979), mixing processes (White1980), simulation of sinking satellites (White 1983a), and mass and energylost in tidal interactions (Aguilar & White 1985), among others. However,we made our calculations within the framework of scalar-tensor theories, inthe Newtonian limit, to investigate the influence of masive scalar fields onthe dynamics.This paper is organized as follows: In section 2 we present the Newtonianapproximation of a typical scalar field theory. In section 3, we present ourmodels of protogalaxies, the initial conditions, and the results. The conclu-sions are in section 4. We consider a typical scalar field theory given by the following Lagrangian L = √− g π " − φR + ω ( φ ) φ ( ∂φ ) − V ( φ ) + L M ( g µν ) (1)from which we get the gravity equations, R µν − g µν R = 8 πφ T µν + V φ g µν + ωφ ∂ µ φ∂ ν φ − ωφ ( ∂φ ) g µν + φ ; µν φ − g µν φ φ (2)3nd the scalar field equation φ + φV ′ − V ω = 13 + 2 ω h πT − ω ′ ( ∂φ ) i (3)We expect to have nowadays small deviations of the scalar fields aroundthe background defined here as h φ i = 1. If we define ¯ φ = φ − R = 12 ∇ h = 4 πρ − ∇ ¯ φ (4) ∇ ¯ φ − m ¯ φ = − παρ (5)where we have done φV ′ − V ω = m ¯ φ − m k ¯ φ + . . . and α = 1 / (3 + 2 ω ).The solution of these equations is¯ φ = 2 αu λ h = − u − αu λ (6)where u = X a m a | r − r a | u λ = X a m a | r − r a | exp [ −| r − r a | /λ ] (7) λ = 1 /m , where m is the mass given through the potential. This mass canhave a variety of values depending on the particular particle physics model.The potential u is the Newtonian part and u λ is the dark matter contributionwhich is of Yukawa type. The total force on a particle of mass m i is X F = − ∇ h = m i a . (8)4 Protogalactic Cloud Models and Results
Original protogalaxies could have very irregular forms, but we use as a firstapproximation spherical clouds for simplicity, and because this form seems toresemble the global shape of both visible and dark matter of many galaxies,i.e., taking into account their spherical halos (Ostriker & Peebles 1973, White1983b). The initial clouds are in polytropic equilibrium with small internalvelocities, compared to what they would need to be in dynamical, gravita-tional equilibrium. This feature avoids a large initial spin, in accordancewith the fact that there were no primordial rotational motions in the uni-verse (Ozernoy & Chernin 1968; Parijskij 1973; Boynton & Partridge 1973;Peebles & Silk 1990). Then, the clouds are sent to approach each other, andonly after their gravitational interaction takes place, spin will be gained.For the simulations, each protogalaxy is constructed using the Plummermodel given by the potential-density pair (Aarseth et al P ( r ) = − GM √ r + b , ρ P ( r ) = (cid:18) M πb (cid:19) r b ! − / (9)where G is the gravitational constant, M is the total mass and b is a param-eter which determines the dimensions of the cloud. Particle velocities arechosen everywhere isotropic which gives a system initially in steady state.The total energy of the cloud is E = − (3 π/ GM b − . We are using unitsin which G = M = − E = 1.We take two identical 3-D clouds consisting of N = 2 particles. Theinitial separation between the clouds is 10, and the velocity of the center ofmasses are V = (0 . , . ,
0) and V = ( − . , . ,
0) in our units. The initialvelocities are given so that the kinetic energy is a fraction of the potentialenergy and we consider a range that is consistent with the observed velocitiesof galaxies in clusters that goes from 50 km/s up to about 1000 km/s. Forthe present investigation we consider that the protogalaxies moves initiallyin the plane ( x, y ), and the angle of both protogalaxies is the same. Moregeneral initial conditions will be considered in a future communication.A three-dimensional hydrodynamic code based on the TREESPH algo-rithm formulated by Hernquist & Katz (1989) was used for the computationsof this paper. The code combines the method of SPH, developed by Lucy(1977) and Gingold & Monaghan (1977), with the hierarchical tree algorithmof Hernquist (1987) for the calculation of the gravitational acceleration forces.5
20 0 20 40 60 80 100 120 140 160 1800.00.20.40.60.81.0 w ith o u t s c a la r fie ldJ /M -20 0 20 40 60 80 100 120 140 160 1800.00.10.20.30.40.50.60.70.80.9 l = b /1 6 -20 0 20 40 60 80 100 120 140 160 1800.00.20.40.60.8 l = b /3 2 -20 0 20 40 60 80 100 120 140 160 1800.00.20.40.60.81.0 l = b /6 4 tim e Figure 1: Evolution of the angular momentum without scalar field and withscalar field for different values of λ i isdetermined by solving Euler’s equation d r i dt ≡ v i (10) d v i dt = − ρ ∇ p i − ∇ ( h ) i + A visc,i , (11)where p i and A visc,i denote, respectively, the gas pressure and the artificialviscous acceleration associated with particle i . This quantities are introducedbecause we are considering that the protogalaxies are gaseous. The codewas modified (Rodr´ıguez-Meza 2000) to include the effect of the scalar field,through the term h given by Eq. (6).The simulation of the interaction of two protogalactic models starts withthe clouds separated by a distance of 10 and on the x -axis. The selectedseparation is large enough to ensure that tidal effects are important butsmall enough that the calculation is possible in a reasonable computing time.Each particle in the initial steady state clouds is given an additional velocity( V or V , corresponding to cloud 1 or 2) so that its magnitude is muchbigger than the internal velocities they have at the equilibrium describedabove. In this way, initial clouds are almost spinless, and the given velocities V , V imply kinetic energies associated with each cloud. The evolution ofthe intrinsic angular momentum ( J/M ) with respect to the center of massof each cloud is shown in Fig. 1. Continuous lines indicate cloud 1 anddashed lines cloud 2. The first plot is without scalar field, the other plotsconsider values of λ of b/ b/
32, and b/
64, respectively. We observe thatthe intrinsic angular momenta start from their initial values to a constantmean value approximately of 0.75 which in the cgs units is of the order of10 cm /s. This transient stage is slower without scalar field than the oneswhich consider scalar field. The faster transient occurs when λ is bigger. InFig. 2 we show plots of phase space v r versus r of the whole system and forthe same cases as Fig. 1. The scalar field extends the phase space in the v r direction. 7 w ith o u t s c a la r fie ldV r l = b /1 6 l = b /3 2 l = b /6 4 r Figure 2: Phase space v r versus r of the combined system after the transientstage without scalar field and with scalar field for different values of λ Conclusions
We have made other computations varying the kinetic energy from 4 to 1/16times the potential energy. For large values of the kinetic energy the deflec-tion is small, but for small values there is a considerable deflection, and insome cases we got almost a head-on collision. This is consistent with theknown fact that the merging probability in an encounter of two clouds is en-hanced significantly when the encounter takes place at relatively low speed(see for instance Makino & Hut 1997). We found that only close encountersand mergings permit the original spinless clouds to gain rotational velocitiesas is observed in typical galaxies nowadays. Similar studies have being done(Namboodiri & Kochhar 1991) considering point mass perturbers. In ourapproach the perturber is itself a protogalaxy and therefore the dynamics ismore complicate, especially in close encounters. We have also found that thetransient time to spin up the clouds depends on the scalar field. The tran-sient stage is faster than the one without scalar field. When the scalar fieldis included faster transients occur for bigger values of λ . The phase space v r versus r of the combined system is also more extended in the v r directionwith scalar field than the one without scalar field. Acknowledgments
The computations of this paper were performed using the Silicon Graph-ics Oring2000 computer of the Instituto de F´ısica, Benem´erita UniversidadAut´onoma de Puebla, M´exico. This work was supported in part by theDAAD and CONACyT grant numbers 33278-E and 33290-E.