Frame dragging and Eulerian frames in General Relativity

The physical interpretation of cold dark matter perturbations is clarified by associating Bertschinger's Poisson gauge with a Eulerian/observer's frame of reference. We obtain such an association by using a Lagrangian approach to relativistic cosmological structure formation. Explicitly, we begin with the second-order solution of the Einstein equations in a synchronous/comoving coordinate system---which defines the Lagrangian frame, and transform it to a Poissonian coordinate system. The generating vector of this coordinate/gauge transformation is found to be the relativistic displacement field. The metric perturbations in the Poissonian coordinate system contain known results from standard/Eulerian Newtonian perturbation theory, but contain also purely relativistic corrections. On sub-horizon scales these relativistic corrections are dominated by the Newtonian bulk part. These corrections however set up non-linear constraints for the density and for the velocity which become important on scales close to the horizon. Furthermore, we report the occurence of a transverse component in the displacement field, and find that it induces a non-linear frame dragging as seen in the observer's frame, which is sub-dominant at late-times and sub-horizon scales. Finally, we find two other gauges which can be associated with a Eulerian frame. We argue that the Poisson gauge is to be preferred because it comes with the simplest physical interpretation.Comment: 14 pages, results unchanged, extended discussion, matches published versio

Hermann Hankel's "On the general theory of motion of fluids", an essay including an English translation of the complete Preisschrift from 1861

The present is a companion paper to "A contemporary look at Hermann Hankel's 1861 pioneering work on Lagrangian fluid dynamics" by Frisch, Grimberg and Villone (2017). Here we present the English translation of the 1861 prize manuscript from G\"ottingen University "Zur allgemeinen Theorie der Bewegung der Fl\"ussigkeiten" (On the general theory of the motion of the fluids) of Hermann Hankel (1839-1873), which was originally submitted in Latin and then translated into German by the Author for publication. We also provide the English translation of two important reports on the manuscript, one written by Bernhard Riemann and the other by Wilhelm Eduard Weber, during the assessment process for the prize. Finally we give a short biography of Hermann Hankel with his complete bibliography.Comment: 44 pages, see the companion paper by Frisch, Grimberg and Villone (2017), v2: minor revisions including change of title, accepted for publication in EPJ

Zel'dovich approximation and General Relativity

We show how the Zel'dovich approximation and the second order displacement field of Lagrangian perturbation theory can be obtained from a general relativistic gradient expansion in \Lambda{}CDM cosmology. The displacement field arises as a result of a second order non-local coordinate transformation which brings the synchronous/comoving metric into a Newtonian form. We find that, with a small modification, the Zel'dovich approximation holds even on scales comparable to the horizon. The corresponding density perturbation is not related to the Newtonian potential via the usual Poisson equation but via a modified Helmholtz equation. This is a consequence of causality not present in the Newtonian theory. The second order displacement field receives relativistic corrections that are subdominant on short scales but are comparable to the second order Newtonian result on scales approaching the horizon. The corrections are easy to include when setting up initial conditions in large N-body simulations.Comment: 5 pages, corrected a typo, accepted for publication in MNRAS Letter

Shell-crossing in quasi-one-dimensional flow

Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions. This paper is about quasi-one-dimensional (Q1D) flow that depends on all three coordinates but differs only slightly from a strictly 1D flow, thereby allowing a perturbative treatment of shell-crossing using the Euler--Poisson equations written in Lagrangian coordinates. The signature of shell-crossing is then just the vanishing of the Jacobian of the Lagrangian map, a regular perturbation problem. In essence the problem of the first shell-crossing, which is highly singular in Eulerian coordinates, has been desingularized by switching to Lagrangian coordinates, and can then be handled by perturbation theory. Here, all-order recursion relations are obtained for the time-Taylor coefficients of the displacement field, and it is shown that the Taylor series has an infinite radius of convergence. This allows the determination of the time and location of the first shell-crossing, which is generically shown to be taking place earlier than for the unperturbed 1D flow. The time variable used for these statements is not the cosmic time $t$ but the linear growth time $\tau \sim t^{2/3}$. For simplicity, calculations are restricted to an Einstein--de Sitter universe in the Newtonian approximation, and tailored initial data are used. However it is straightforward to relax these limitations, if needed.Comment: 9 pages; received 2017 May 24, and accepted 2017 June 21 at MNRA

Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering

We investigate the Lagrangian perturbation theory of a homogeneous and isotropic universe in the non-relativistic limit, and derive the solutions up to the fourth order. These solutions are needed for example for the next-to-leading order correction of the (resummed) Lagrangian matter bispectrum, which we study in an accompanying paper. We focus on flat cosmologies with a vanishing cosmological constant, and provide an in-depth description of two complementary approaches used in the current literature. Both approaches are solved with two different sets of initial conditions---both appropriate for modelling the large-scale structure. Afterwards we consider only the fastest growing mode solution, which is not affected by either of these choices of initial conditions. Under the reasonable approximation that the linear density contrast is evaluated at the initial Lagrangian position of the fluid particle, we obtain the nth-order displacement field in the so-called initial position limit: the nth order displacement field consists of 3(n-1) integrals over n linear density contrasts, and obeys self-similarity. Then, we find exact relations between the series in Lagrangian and Eulerian perturbation theory, leading to identical predictions for the density contrast and the peculiar-velocity divergence up to the fourth order.Comment: 31 pages, matches published version in JCAP, added an extra section which discusses and motivates the choice of initial conditions, extended the title for the sake of precisio

Initial conditions for cold dark matter particles and General Relativity

We describe the irrotational dust component of the universe in terms of a relativistic gradient expansion and transform the resulting synchronous metric to a Newtonian coordinate system. The two metrics are connected via a space-like displacement field and a time-like perturbation, providing a relativistic generalization of the transformation from Lagrangian to Eulerian coordinates. The relativistic part of the displacement field generates already at initial time a non-local density perturbation at second order. This is a purely relativistic effect since it originates from space-time mixing. We give two options, the passive and the active approach, on how to include the relativistic corrections for example in N-body simulations. In the passive approach we treat the corrections as a non-Gaussian modification of the initial Gaussian field (primordial non-Gaussianity could be incorporated as well). The induced non-Gaussianity depends on scale and the redshift at which initial conditions are set, with f_NL ~ few for small enough scales and redshifts. In the active approach we show how to use the relativistic trajectory to obtain the initial displacement and velocity of particles for N-body simulations without modifying the initial Gaussian field.Comment: Title adjusted, added a table for clarity, matches published versio

Relativistic Lagrangian displacement field and tensor perturbations

We investigate the purely spatial Lagrangian coordinate transformation from the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for extracting the relativistic displacement field from a given solution in the Lagrangian frame. These techniques are (a) from defining a local set of Eulerian coordinates embedded into the Lagrangian frame; (b) from performing a specific gauge transformation; and (c) from a fully non-perturbative approach based on the ADM split. The latter approach shows that this decomposition is not tied to a specific perturbative formulation for the solution of the Einstein equations. Rather, it can be defined at the level of the non-perturbative coordinate change from the Lagrangian to the Eulerian description. Studying such different techniques is useful because it allows us to compare and develop further the various approximation techniques available in the Lagrangian formulation. We find that one has to solve the gravitational wave equation in the relativistic analysis, otherwise the corresponding Newtonian limit will necessarily contain spurious non-propagating tensor artefacts at second order in the Eulerian frame. We also derive the magnetic part of the Weyl tensor in the Lagrangian frame, and find that it is not only excited by gravitational waves but also by tensor perturbations which are induced through the non-linear frame-dragging. We apply our findings to calculate for the first time the relativistic displacement field, up to second order, for a $\Lambda$CDM Universe in the presence of a local primordial non-Gaussian component. Finally, we also comment on recent claims about whether mass conservation in the Lagrangian frame is violated.Comment: 19 pages, two figures, improved discussion, matches published versio

The recursion relation in Lagrangian perturbation theory

We derive a recursion relation in the framework of Lagrangian perturbation theory, appropriate for studying the inhomogeneities of the large scale structure of the universe. We use the fact that the perturbative expansion of the matter density contrast is in one-to-one correspondence with standard perturbation theory (SPT) at any order. This correspondence has been recently shown to be valid up to fourth order for a non-relativistic, irrotational and dust-like component. Assuming it to be valid at arbitrary (higher) order, we express the Lagrangian displacement field in terms of the perturbative kernels of SPT, which are itself given by their own and well-known recursion relation. We argue that the Lagrangian solution always contains more non-linear information in comparison with the SPT solution, (mainly) if the non-perturbative density contrast is restored after the displacement field is obtained.Comment: 12 pages, clarified the notation in the appendi