Evolution of the angular momentum of protogalaxies from tidal torques: Zel'dovich approximation

1996MNRAS.282..436C Mon. Not. R. Astron. Soc. 282, 436-454 (1996) Evolution of the angular momentum of protogalaxies from tidal torques: Zel'dovich approximation Paolo Catelan and Tom Theuns Department of Physics, Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH Accepted 1996 April 12. Received 1996 April 4; in original form 1995 September 14 The growth of the angular momentum L of protogalaxies induced by tidal torques is reconsidered. We adopt White's formalism and study the evolution of L in Lagrangian coordinates; the motion of the fluid elements is described by the Zel'dovich approximation. We obtain a general expression for the ensemble expectation value of the square of L in terms of the first and second invariant of the inertia tensor of the Lagrangian volume r enclosing the collapsing mass of the proto-object. We then specialize the formalism to the particular case in which r is centred on a peak of the smoothed Gaussian density field and approximated by an isodensity ellipsoid. The result is the appropriate analytical estimate for the rms angular momentum of peaks to be compared against simulations that make use of the Hoffman-Ribak algorithm to set up a constrained density field that contains a peak with given shape. Extending the work of Heavens & Peacock, we calculate the joint probability distribution function for several spin parameters and peak mass M using the distribution of peak shapes, for different initial power spectra. The probability distribution for the rms final angular momentum <Li)l!2 on the scales corresponding to common bright galaxies, M ~ 1011 M o ' is centred on a value of ~ 1067 kg m2 s - \ for any cosmologically relevant power spectrum, in line with previous theoretical and observational estimates for L f • Other astrophysical consequences are discussed. In particular, we find that typical values O?)l!2 ~ 0.1 of the dimensionless spin parameter for peaks smoothed on galactic scales and of height v", 1, usually associated with late-type galaxies, may be recovered in the framework of the Gaussian peak formalism. This partially relaxes the importance attributed to dissipative processes in generating such high values of centrifugal support fot spiral galaxies. In addition, the values of the specific angular momentum versus mass - as deduced from observations of rotational velocities and photometric radii of spiral galaxies - are well fitted by our theoretical isoprobability contours. In contrast, the observed lower values for the specific angular momentum for ellipticals of the same mass cannot be accounted for within our linear-regime investigation, highlighting the importance of strongly non-linear phenomena to explain the spin of such objects. Key words: galaxies: formation - large-scale structure of Universe. 1 INTRODUCTION It has been argued that tidal coupling between the inhomo- geneities in the primordial matter distribution, in the context of a gravitational hierarchical scenario, may explain the acquisition of the jlngular momentum by a protogalaxy. This idea, originally due to Hoyle (1949) and applied by Sciama (1955), has been first thoroughly examined by Peebles (1969), who demonstrated that the tidal spin growth of the matter contained in a spherical (Eulerian) volume is proportional to t 513 in an Einstein-de Sitter universe (t is the standard cosmic time). Specifically, Peebles's analysis is based on a second-order perturbative description, since a spherical volume does not gain angular momentum from © 1996 RAS © Royal Astronomical Society • Provided by the NASA Astrophysics Data System ABSTRACT 1996MNRAS.282..436C Linear evolution of the tidal angular momentum 2 ANGULAR MOMENTUM We carry out our analysis of the evolution of the angular momentum in three steps. First, we review White's method for obtaining an expression for L involving the shape of the proto-object and the distribution of the surrounding matter. Next, we simplify the expression for L by performing the ensemble average in Section 2.2. Finally, we specialize the formalism to the case where the object is centred on a peak of the (Gaussian) underlying density distribution. 2.1 Dynamical description Let us assume that at the epoch of structure formation the matter may be described on the relevant scales as a Newtonian collisionless cold fluid (dust) embedded in an expanding Friedmann universe with arbitrary density parameter Q (but, for simplicity, vanishing cosmological constant). We indicate the comoving Eulerian spatial coordinate by x. The physical distance is r=a(t)x, with a(t) the expansion scale factor. In the Einstein-de Sitter model, a (t) IX t 2/3 • The angular momentum L of the matter contained at time t in a volume Vof the Eulerian x-space is L(t)=f drprxv=Pba4f dx(l+15)xxu. a 3V (1) V Here, p=Pb(l + b) denotes the matter density field, Ph the background mean density, 15 the density fluctuation field; v=dr/dt indicates the velocity field, and u=a dx/dt is the so-called 'peculiar' velocity (see, e.g., Peebles 1980). The origin of the Cartesian coordinate system is assumed to coincide with the centre of mass. Since we are interested in the intrinsic angular motion, we disregard the centre-ofmass motion. It is important to note that the previous integral over the Eulerian volume V may equally be written as an integral over the corresponding Lagrangian volume r: L(t)=IJoa 2 dS f r dq(q+S)x-, dt (2) where Pba3 = poa~ == IJo in the matter-dominated era. Equation (2) can be obtained from equation (1) using the mapping x(q, t)=q+S(q, t), (3) from Lagrangian coordinates q to comoving coordinates x, where S is the displacement vector. The determinant J of the Jacobian of the mapping q--+x(q, t), dxJ-I=dq, is related to the density fluctuation through the continuity equation, 1 + 15 [x(q, t), t] =J(q, t)-I. (4) Before the occurrence of shell crossing (caustic formation process) J is non-vanishing (see, e.g., Shandarin & Zel'dovich 1989). The expression for L(t) in equation (2) allows us to apply the Lagrangian theory directly: we stress that it is an exact relation. Consequently, when applying perturbation theory to equation (3), perturbative corrections to S (Bouchet et a1. © 1996 RAS, MNRAS 282, 436-454 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System tidal torques in linear approximation, as pointed out by Zel'dovich and reported by Doroshkevich (1970). An important result of Doroshkevich's paper is that a generic non-spherical volume enclosing the protogalaxy acquires tidal angular momentum proportionally to the cosmic time t during the linear regime. This theoretical prediction has been confirmed by the N-body simulations of White (1984). In addition, White showed that the second-order growth described by Peebles is due to convective motion of matter across the surface of the initial volume r containing the proto-object. An important point of Peebles's and White's theoreti (...truncated)