Cosmic Web: Nonlinear Formalisms

1. Cosmic Web:Nonlinear Formalisms

2. Nonlinear Descriptions Given that it is not feasible to fully solve the equations of structure formation, a variety of strategies have been formulated. These may be classified (roughly) in various classes: • Fully analytical models of limited validity, usually representing a (crucial) useful reference : - spherical model - the Zel’dovich approximation - Press-Schechter formalism • Approximate analytical descriptions, to be solved numerically: - (homogeneous) ellipsoidal model - Least Action Principle (mixed boundary) formalism • Numerical computer simulations/experiments - N-body (dark matter) simulations - gravity+gasdynamics simulations (e.g. SPH) • Phenomological Models - e.g. Voronoi model of the cosmic web

3. Cosmic Spheres

4. Spherical Model The spherical model (Gunn & Gott 1972) describes the evolution of a spherical mass distribution. It forms THE reference point for all further evaluations of structure formation. ● Because of Birkhoff’s theorem we may see the evolution of each individual mass shell as due only to the integrated mass distribution within its radius. ● As long as two mass shells are not crossing – e.g. due to the faster infall of an outer shell into an overdensity -- the motion of a shell – with radius r -- is simply that of an individual spherical shell attracted by a point mass M(r), with M(r) the integrated mass within radius r. ● Perhaps not surprisingly, the equations of motion for the mass shells are the same as that of Friedmann-Robertson-Walker universes for an equivalent density parameter (r). ● These equations of motion for each mass shell can be solved analytically for any decently behaving mass profile (i.e. the mass profile should be sufficiently centrally concentrated to prevent shell crossing). ● The spherical model is equally valid for overdensities as well as for underdensities.

5. Spherical Model Contraction/Expansion of a shell with initial (Lagrangian) radius ri is described by a scale factor R(t,ri), such that the radius r(t,ri) at time t is given by:

6. Spherical Model The motion is fully determined by the average mass density ∆(r,t) within a radius r, and by the the peculiar velocity vpec,i of the shell. For this we usually take the peculiar velocity predicted by linear theory for the growing mode. It is convenient to describe the density perturbation with respect to a EdS Universe, in terms of ∆i and the velocity perturbation with respect to the Hubble expansion in terms of parameter αi .

7. Spherical Model The solutions for the scale factor of overdense/underdense shells can be written in the same parameterized form, by means of shell angle Θ, as we know from the solutions for FRW universes, with time dependence specified by

8. Spherical Model The corresponding peculiar velocity of the shell can be inferred from with

9. Spherical Model

10. Spherical Model

11. Spherical Model Having determined the evolution of the radius and velocity of each spherical shell of the density perturbation, we may then proceed to derive the corresponding evolution of the density profile of the shell. Here we limit ourselves to the integrated density profile ∆(r,t), whose solution can be specified in terms of a density function f(θ),

12. Spherical Model whose solution can be specified in terms of a density function f(θ), At maximum expansion of an overdense shell, Θ=π, defining the turnaround radius of the matter concentration, we thus find that the integrated overdensity of the shell is ~ 5.6

13. Spherical Model In the “imaginary” situation in which the overdensity would have continued to evolve linearly, it would have reached an overdensity dictated by the linear growth factor D(t) for the corresponding background Universe. For the situation of an Einstein-de Sitter Universe, with a mass overdensity reaches its turnaround at a linear overdensity The consequences of this finding are truely wonderful: the cosmologist may resort to the primordial density field, search for the peaks in this Gaussian field, and assuming they are spherical (which they are not at all), and identify the ones that reach turnaround at some redshift z. Even more useful is the equivalent case for final collapse.

14. Spherical Model Collapse, ie. ∆=∞, happens when the density fluctuation would have reached a linear overdensity of The fact that this is a universal value, valid for any (spherical) density peak, makes it into one of the most crucial numbers in the theory of structure formation. We may thus find the collapse redshifts zcoll for any primordial density peak,

15. Lagrangian Views

16. Lagrangian Perturbation Theory Nonlinear Descriptions In the ``Lagrangian” view of a physical system, we follow along with a fluid element on its path through space. As it moves along with the migration flows through the Universe, a fluid element ● either contracts or expands ● gets distorted and changes shape ● rotates/changes orientation The Lagrangian coordinate of the element is q, its initial position. We restrict ourselves to pure laminar flow, fluid elements should not cross ! q x(q,t)

17. Lagrangian Perturbation Theory Nonlinear Descriptions The transformation from ``Eulerian’’ to ``Lagrangian’’ space is given by q Note that N-body simulations, as well as SPH hydrodynamical simulations, are intrinsically Lagrangian descriptions for the evolution of a system. x(q,t)

18. Lagrangian Perturbation Theory Nonlinear Descriptions The evolution of a fluid element on its path through space can be specified by its velocity gradient, In which θ: velocity divergence, specifying contraction/expansion σ: velocity shear, specifying deformation ω: vorticity, specifying rotation of element

19. Lagrangian Perturbation Theory Nonlinear Descriptions The evolution of a fluid element on its path through space can be specified by its velocity gradient,

20. Lagrangian Perturbation Theory The Lagrangian equations of motion Naturally includes the Lagrangian form of the continuity equation, the Euler equation and the Poisson equation. Note that the Lagrangian Euler equation is exactly the equation of motion, F=ma, which we know from Newtonian mechanics. In comoving space, however, it includes the extra “Hubble drag term” (due to the expanding Universe). It is easy to observe that therefore in the absence of a gravitational field perturbation an existing peculiar velocity will die out.

21. Lagrangian Perturbation Theory:Raychaudhury Equation In addition to these equations of motion for the fluid element, one also needs equations specifying the evolution of its “internal” properties. In other words, one needs to describe the expansion/contraction, the shearing and the vorticity of the fluid element while it moves through the matter distribution. The contraction/expansion of the fluid element is specified by the Raychaudhury equation, which is a rather complex assembly of terms including the density perturbation δ, a quadratic shear term σijσij and a quadratic vorticity term ωijωij, Crucial Observation: the presence of quadratic shear and vorticity terms in the Raychaudhury equation explicitly reveals the influence of shear and vorticity on the (density) evolution of a fluid element. The + of the shear term shows that shear tends to accelerate collapse, while the – of the vorticity term implies a slow-down of collapse. Also note that these terms are quadratic, meaning that their influence is a nonlinear effect.

22. Lagrangian Perturbation Theory:Vorticity Equation The equations for the evolution of the vorticity of a fluid element, the ``Lagrangian Vorticity equation’’, is obtained from the antisymmetric part of the Euler equation. Note that in the absence of any primordial vorticity , the fluid element will remain irrotational. This will remain so as long as the fluid remains laminar, rotation can only be obtained after this breaks down.

23. Lagrangian Perturbation Theory:Shear Equation The equations for the evolution of the shear of a fluid element, the ``Lagrangian Shear equation’’, is obtained from the symmetric part of the Euler equation. This shows that the source term for shear is to be found in the gravitational Tidal Field exerted on the fluid element,

24. Lagrangian Perturbation Theory:Irrotational Flow In summary, the full system of Lagrangian perturbation equations for irrotational flow. Mind you, potential flow as that induced by gravity is irrotational of character (ie. if initially there is no vorticity).

25. Lagrangian Perturbation Theory:Spherical Model Combining the continuity equation with Euler and Poisson equation we obtain the second order evolution equation for the density of a fluid element. When we assume the shear and vorticity to be zero, we get the Lagrangian equation for the spherical model. Notice that in the case of irrotational flow, and in the presence of shear, the density will evolve faster than in the pure spherical model. In other words, Shear accelerates the collapse of a fluid element Spherical Collapse is slowest

26. Zel’dovich Formalism

27. Zel’dovich Formalism Having transformed our view in that of a Lagrangian, we are set to see whether we can proceed by looking at the first linear stages of Lagrangian evolution. This is a SEMINAL step taken by Yakov Zel’dovich, the great Soviet cosmologist. By means of a Lagrangian perturbation analysis Zel’dovich (1970) proved that to first order – typifying early evolutionary phases – the reaction of cosmic patches of matter to the corresponding peculiar gravity field would be surprisingly simple, expressing itself in a plain ballistic linear displacement set solely by the initial (Lagrangian) force field. This framed the well-known Zel’dovich approximation.

28. Zel’dovich Formalism

29. Zel’dovich Formalism The central role of the Zel’dovich formalism in structure formation studies stems from its ability to take any arbitrary initial random density field, not constrained by any specific restriction in terms of morphological symmetry, and mould it through a simple and direct operation into a reasonable approximation for the matter distribution at later nonlinear epochs. The Zel’dovich approximation: ● a rough qualitative outline of the nonlinear matter distribution solely on the basis of the given initial density field. ● represents a surprisingly accurate description up to considerably advanced evolutionary stages - up to the point where matter flows start to cross - even though formally a mere first order perturbative term ● Is of major importance in translating a field of primordial density perturbations characterized by a given power spectrum into the initial conditions for a N-body simulation, specifying initial displacement and velocities of the simulation particles.

30. Zel’dovich Formalism In essence, the Zel’dovich approximation is the solution of the Lagrangian equations for small density perturbations, δ2<<1 . The solution is based upon the first-order truncation of the Lagrangian perturbation series of the trajectories of mass elements When considering successive terms of the displacement from the initial Lagrangian position q,

31. Zel’dovich Formalism The truncation at the x(1) term then involves a simple linear prescription for the displacement of a particle from its initial (Lagrangian) comoving position q to an Eulerian comoving position x, solely determined by the ``displacement potential’’ Ψ(q), In this mapping, the time dependent function D(t) is the growth rate of linear density perturbations, and the time-independent spatial function (q), the displacement potential, is related to the linearly extrapolated gravitational potential φ,

32. Zel’dovich Formalism:Streaming & Caustics Illustration of the formation of caustics due to streaming paths of light through deforming medium

33. Zel’dovich Formalism:Velocity Field The corresponding velocity of a fluid element with Lagrangian coordinate q can be readily inferred from the Zel’dovich transformation, which leads to the relation in which D(t) is the linear growth factor, H(t) the Hubble parameter, and f()~0.6 the linear velocity factor.

34. Zel’dovich Formalism:Potentials The relation between displacement potential and gravitational potential can be immediately understood on the basis of the expression for the velocity, We know that the velocity in the linear regime is linearly proportional to the peculiar gravity g, from which we can establish the direct relationship between φ(x) and Ψ(q),

35. Zel’dovich Formalism:Physical Nature • The Zel’dovich approximation is a first order tracing of the trajectories of matter elements. As such it involves some intrinsic physical approximations. • The continuity equation is always satisfied, implicitly based upon mass conservation. • In general, it does not satisfy the Euler and Poisson equation !!!! • Only in the case of pure one-dimensional perturbations does the Zel’dovich approximation • represent a full solution to all three dynamic equations (approximation worst for spherical config.). • Involves implicitly a linear relation between the deformation tensor ψij, and hence the velocity shear • tensor σij , and the gravitational tidal tensor Tij, This relation between deformation and tidal tensor is the true central core and essential physical significance of the Zel’dovich formalism.

36. Zel’dovich Formalism:Density Evolution Perhaps the one outstanding virtue of the Zel’dovich approximation is its ability to assess the evolution of a density field from the primordial density field itself, through the corresponding linearly extrapolated (primordial) gravitational potential. While in essence a local approximation, the Lagrangian description provides the starting point for a far-reaching analysis of the implied density field development. The density evolution of a fluid element can be simply evaluated by comparing its Lagrangian volume (the volume it had at its initial location q) and its Eulerian volume (the volume it has when it has reached location x), from which the expression for the density perturbation δ(x,t) of the fluid element can be inferred in terms of the Jacobian of the displacement x(q,t)

37. Zel’dovich Formalism:Density Evolution Solving the Jacobian for the first-order displacement approximation leads to a highly insightful equation for the development of structure in the Universe, where the vertical bars denote the Jacobian determinant, and the eigenvalues of the Zel’dovich deformation tensor ψmn

38. Zel’dovich Formalism:Density Evolution The density evolution within the Zel’dovich formalism reveals the fact that gravitational collapse is not proceeding in an isotropic fashion, The generic features of anisotropic collapse within generic media of cold matter distributions can be inferred by assessing the relation between a fluid element’s density evolution and the eigenvalues of its deformation tensor, and hence the tidal force tensor. We can identify 4 configurations: • 1) λ1 > 0, λ2 > 0, λ3 > 0 full 3-D collapse into clump • λ1 > 0, λ2 > 0, λ3 < 0 collapse into elongated filament • λ1 > 0, λ2 < 0, λ3 < 0 collapse into flattened pancake • λ1 < 0, λ2 < 0, λ3 < 0 no collapse, expanding void

39. Zel’dovich Formalism:Density Evolution Density Profile through pancake, at moment of formation and shortly thereafter (multistream)

40. Zel’dovich Formalism Illustration: Caustic formation of weblike matter distribution according to the Zel’dovich formalism. From: Buchert 1989

41. Zel’dovich Formalism:Density Evolution Three instrumental observations concerning the collapse process can be readily made: ●Ultimate Fate Having one or more positive eigenvalues λi(i=1,2,3) is sufficient to guarantee collapse. Unless each individual λm< 0 (i=1,2,3), the fluid element, even if initially underdense, will undergo collapse along at least one of its axes at some stage during its evolution.

42. Zel’dovich Formalism:Density Evolution Three instrumental observations concerning the collapse process can be readily made: ●Collapse Configurations For fluid elements with at least one positive deformation eigenvalue, the ultimate collapse geometry of a fluid element can be one of 3 distinct configurations. Assuming λ3 > λ2 > λ1 , 1) A flattened pancake shape will be attained if only λ3 > 0 and λ2 < 0 While collapse has occurred along the corresponding axis, the element will expand along the 2 others. 2) If λ3 > λ2 > 0 and λ1 < 0 the mass element will evolve into an elongated spindle. 3) A state of full collapse along all three axes will only be reached if all λi > 0 (i=1,2,3)

43. Zel’dovich Formalism:Density Evolution Three instrumental observations concerning the collapse process can be readily made: ●Collapse Time Sequence The collapse will be anisotropic and, dependent on the sign of the deformation eigenvalues, proceeding along a sequence of one, two or three distinct stages. 1) Firstly, collapse will happen along the direction of the largest positive eigenvalue, at a1c = 1/λ1 It occurs upon the fluid element entering a flattened pancake. 2) Subsequently, contraction along a second direction, leading to collapse along second axis at a2c = 1/λ2 This corresponds to the presence in a filament. 3) Ultimately, if also the fluid element will undergo full collapse upon a singular point at a3c = 1/λ3 The physical nature of the emerging object will depend on the scale of the corresponding deformation field, a galaxy halo will emerge through full collapse at a scale < 1 h-1 Mpc, a galaxy cluster from a similar collapse on a scale ~ 4h-1 Mpc

44. Zel’dovich Formalism Illustration: a typical time sequence of the Zel’dovich scheme. Clearly visible is the gradual morphological progression along “pancake” and “filamentary” stages. Comparsion with results of full-scale N-body simulations shows that in particular at early structure formation epochs the predicted Zel’dovich configurations are acccurately rendering the full nonlinear matter distributions.

45. Zel’dovich Formalism Note that ● a comparison of the last 3 frames with the very first stage indicates that the pattern of the resulting large scale matter distribution is already, be it faintly, imprinted in the primordial matter configuration. ● The essence of the Zel’dovich approximation is in the deformation tensor ψmn, directly related to the tidal field tensor Tmn . It indicates the existence of the profound relation between the anisotropic force field as shaping agent and the resulting anisotropic matter concentrations, and the cosmic web of which these are the elementary constituents. ● The 1-1 Lagrangian-Eulerian mapping of the Zel’dovich approximation is only possible as long as a fluid element has not yet crossed the path of another fluid element.

46. Zel’dovich Formalism Limitations: The Zel’dovich approximation startsto fail after the occurrence of orbit crossing. This is clearly observed through the unrealistic “diffusion” of clumps, filaments and walls in the 4th frame. In reality, following orbit crossing we will start to see thorough gravitationally induced orbit mixing and energy exchange proceeding towards a “quasi-equilibrium” state of complete virialization. The Zel’dovich approximation is fundamentally inadequate to describe these advanced nonlinear stages.

47. Zel’dovich Formalism:Probabilities Overall morphology and spatial pattern of a cosmic density field at its “quasilinear” development stage, ▬ the prominence of flattened structures, denser elongated filaments and dense compact clumps as well as their interconnectedness sensitively dependent on the statistical distribution of the values of the eigenvalues λi. Starting from the observation that the deformation tensor ψmn has a Gaussian distribution, Doroshkevich (1970) computed the probability for the eigenvalues λ1 ,λ2 and λ3 , This yields a probability for all eigenvalues to be negative 8% one or more eigenvalues positive 92% A distinct wall- and filament- dominated weblike configuration during the moderate quasi-linear evolution phase is a natural outcome.

48. Zel’dovich Formalism:Addendum: derivation Evaluation of the Jacobian in terms of a perturbative series of the displacement field x(q,t) yields the expression:

49. Zel’dovich Formalism:Addendum: derivation Evaluation of the Jacobian in terms of a perturbative series of the displacement field x(q,t) yields the expression: So that to first order approximation the density evolution is given by:

50. Zel’dovich Formalism:Addendum: derivation Having found a relation between the first order density perturbation δ(1)and the first order displacement term, we can use the Poisson equation and the Euler equation to infer an equation for x(1) (q,t) This equation is exactly similar to that of the linear density evolution in Eulerian linear theory, so that we immediately know the solution for the first order (Lagrangian) displacement,