Lagrangian Space Research Articles

Evidence for quadratic tidal tensor bias from the halo bispectrum

The relation between the clustering properties of luminous matter in the form of galaxies and the underlying dark matter distribution is of fundamental importance for the interpretation of ongoing and upcoming galaxy surveys. The so called local bias model, where galaxy density is a function of local matter density, is frequently discussed as a means to infer the matter power spectrum or correlation function from the measured galaxy correlation. However, gravitational evolution generates a term quadratic in the tidal tensor and thus non-local in the density field, even if this term is absent in the initial conditions (Lagrangian space). Because the term is quadratic, it contributes as a loop correction to the power spectrum, so the standard linear bias picture still applies on large scales, however, it contributes at leading order to the bispectrum for which it is significant on all scales. Such a term could also be present in Lagrangian space if halo formation were influenced by the tidal field. We measure the corresponding coupling strengths from the matter-matter-halo bispectrum in numerical simulations and find a non-vanishing coefficient for the tidal tensor term. We find no scale dependence of the bias parameters up to k=0.1 h/Mpc and that the tidal effect is increasing with halo mass. While the Lagrangian bias picture is a better description of our results than the Eulerian bias picture, our results suggest that there might be a tidal tensor bias already in the initial conditions. We also find that the coefficients of the quadratic density term deviate quite strongly from the theoretical predictions based on the spherical collapse model and a universal mass function. Both quadratic density and tidal tensor bias terms must be included in the modeling of galaxy clustering of current and future surveys if one wants to achieve the high precision cosmology promise of these datasets.

An Investigation of the Short-Term Predictability of Precipitation Using High-Resolution Composite Radar Observations

AbstractThe short-term predictability of precipitation patterns observed by meteorological radar is an important concept as it establishes a means to characterize precipitation and provides an upper limit on the extent of useful nowcasting. Predictability also varies on the basis of spatial and temporal scales of the observed meteorological phenomena. This paper describes an investigation of the short-term predictability of precipitation patterns containing microalpha (0.2–2 km) to mesobeta (20–200 km) scales using high-resolution (0.5 km–1 min–1 dBZ) composite radar reflectivity data, extending the analysis presented in previous work to smaller space and time scales. An experimental approach is used in which continuous and categorical lifetimes of radar reflectivity fields in Eulerian and Lagrangian space are used to quantify short-term predictability. The space–time scale dependency of short-term predictability is analyzed, and a practical upper limit on the extent of Lagrangian persistence-based nowcasting is estimated. Connections to the predictability of larger scales are made within the context of previous work. The results show that short-term predictability estimates in terms of lifetime are approximately 14–15 and 20–21 min in Eulerian and Lagrangian space, respectively, and suggest that a linear relationship exists between predictability and space–time structure from microalpha to macrobeta (2000–10 000 km) scales.

Gravity and large-scale nonlocal bias

The relationship between galaxy and matter overdensities, bias, is most often assumed to be local. This is however unstable under time evolution, we provide proofs under several sets of assumptions. In the simplest model galaxies are created locally and linearly biased at a single time, and subsequently move with the matter (no velocity bias) conserving their comoving number density (no merging). We show that, after this formation time, the bias becomes unavoidably non-local and non-linear at large scales. We identify the non-local gravitationally induced fields in which the galaxy overdensity can be expanded, showing that they can be constructed out of the invariants of the deformation tensor (Galileons). In addition, we show that this result persists if we include an arbitrary evolution of the comoving number density of tracers. We then include velocity bias, and show that new contributions appear, a dipole field being the signature at second order. We test these predictions by studying the dependence of halo overdensities in cells of fixed matter density: measurements in simulations show that departures from the mean bias relation are strongly correlated with the non-local gravitationally induced fields identified by our formalism. The effects on non-local bias seen in the simulations are most important for the most biased halos, as expected from our predictions. The non-locality seen in the simulations is not fully captured by assuming local bias in Lagrangian space. Accounting for these effects when modeling galaxy bias is essential for correctly describing the dependence on triangle shape of the galaxy bispectrum, and hence constraining cosmological parameters and primordial non-Gaussianity. We show that using our formalism we remove an important systematic in the determination of bias parameters from the galaxy bispectrum, particularly for luminous galaxies. (abridged)

Finite volume element method with Lagrangian cubic functions

This paper establishes a new finite volume element scheme for Poisson equation on triangular meshes. The trial function space is taken as Lagrangian cubic finite element space on triangular partition, and the test function space is defined as piecewise constant space on dual partition. Under some weak condition about the triangular meshes, the authors prove that the stiffness matrix is uniformly positive definite and convergence rate to be O(h 3) in H 1-norm. Some numerical experiments confirm the theoretical considerations.

Variability of the Norwegian Atlantic Current and associated eddy field from surface drifters

The Norwegian Atlantic Current (NwAC) and its eddy field are examined using data from surface drifters. The data set used spans nearly 20 years, from June 1991 to December 2009. The results are largely consistent with previous estimates, which were based on data from the first decade only. With our new data set, statistical analysis of the mean fields can be calculated with larger confidence. The two branches of the NwAC, one over the continental slope and a second further offshore, are clearly captured. The Norwegian Coastal Current is also resolved. In addition, we observe a semipermanent anticylonic eddy in the Lofoten Basin, a feature seen previously in hydrography and in models. The eddy kinetic energy (EKE) is intensified along the path of the NwAC, with the largest values occurring in the Lofoten Basin. The strongest currents, exceeding 100 cm s−1, occur west of Lofoten. Lateral diffusivities were computed in five domains and ranged from 1–5 × 107 cm2 s−1. The Lagrangian integral time and space scales are 1–2 days and 7–23 km, respectively. The data set allows studies of seasonal and interannual variations as well. The strongest seasonal signal is in the NwAC itself, as the mean flow strengthens by approximately 20% in winter. The EKE and diffusivities on the other hand do not exhibit consistent seasonality in the sampled regions. There are no consistent indications of changes in either the mean or fluctuating surface velocities between the 1990s and 2000s.

Numerical simulation of the transient temperature distribution inside moving particles

AbstractThis work is devoted to the two‐dimensional (2D) numerical simulation of heat and fluid flow by granular mixing in a horizontally rotating kiln. The heat and fluid flow in the gas phase are solved directly using a fixed Eulerian grid. At the same time, the particle dynamics and their collisions are solved on a Lagrangian grid. The no‐slip boundary condition on the particle surface is implemented using the fictitious boundary method. The heat transfer inside the particles is calculated utilising two models: the first is the direct solution of the energy conservation equation in Lagrangian and Eulerian space and the second is the so‐called linear model, which assumes a homogeneous distribution of the temperature inside each particle. Numerical simulations showed that if the thermal diffusivity of the gas phase significantly exceeds the same parameter of the particles, the linear model over‐predicts the heating rate of the particles. The analysis of the time‐averaged flow field inside the kiln showed that in the gas phase a double vortex structure is formed which increases the convective heat transfer in the upper part of the particulate bed. The influence of the particle size, the angular velocity of the drum and the fluid on the heating rates of particles is studied and discussed.

Direct numerical simulation of particulate flows with heat transfer in a rotating cylindrical cavity

The purpose of this work was the direct numerical simulation of heat and fluid flow by granular mixing in a horizontal rotating kiln. To model particle behaviour and the heat and fluid flow in the drum, we solve the mass conservation, momentum and energy conservation equations directly on a fixed Eulerian grid for the whole domain including particles. At the same time the particle dynamics and their collisions are solved on a Lagrangian grid for each particle. To calculate the heat transfer inside the particles we use two models: the first is the direct solution of the energy conservation equation in the Lagrangian and Eulerian space, and the second is our so-called linear model that assumes homogeneous distribution of the temperature inside each particle. Numerical simulations showed that, if the thermal diffusivity of the gas phase significantly exceeds the same parameter of the particles, the linear model overpredicts the heating rate of the particles. The influence of the particle size and the angular velocity of the drum on the heating rates of particles is studied and discussed.

An unfitted interface penalty method for the numerical approximation of contrast problems

We aim to approximate contrast problems by means of a numerical scheme which does not require that the computational mesh conforms with the discontinuity between coefficients. We focus on the approximation of diffusion–reaction equations in the framework of finite elements. In order to improve the unsatisfactory behavior of Lagrangian elements for this particular problem, we resort to an enriched approximation space, which involves elements cut by the interface. Firstly, we analyze the H 1 -stability of the finite element space with respect to the position of the interface. This analysis, applied to the conditioning of the discrete system of equations, shows that the scheme may be ill posed for some configurations of the interface. Secondly, we propose a stabilization strategy, based on a scaling technique, which restores the standard properties of a Lagrangian finite element space and results to be very easily implemented. We also address the behavior of the scheme with respect to large contrast problems ending up with a choice of Nitscheʼs penalty terms such that the extended finite element scheme with penalty is robust for the worst case among small sub-elements and large contrast problems. The theoretical results are finally illustrated by means of numerical experiments.

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.

Mathematical model of Stewart platform motion

A mathematical model of the mechanism of movement of an undeformed platform with six degrees of freedom is proposed. The equations of spatial motion of particles of the platform are derived using the principle of superposition of motions in Lagrangian space. Their derivation with respect to time allows any kinematic and associated energy characteristics of the movement to be determined. The conditions of energy balance are converted to a system of equations for determining the forces in kinematic pairs in all versions of the platform motion. The model can be used to improve the mechanisms of technological machines, including the positioning of their working groups.

Merging and fragmentation in the Burgers dynamics

We explore the noiseless Burgers dynamics in the inviscid limit, the so-called "adhesion model" in cosmology, in a regime where (almost) all the fluid particles are embedded within pointlike massive halos. Following previous works, we focus our investigations on a "geometrical" model, where the matter evolution within the shock manifold is defined from a geometrical construction. This hypothesis is at variance with the assumption that the usual continuity equation holds but, in the inviscid limit, both models agree in the regular regions. Taking advantage of the formulation of the dynamics of this "geometrical model" in terms of Legendre transforms and convex hulls, we study the evolution with time of the distribution of matter and the associated partitions of the Lagrangian and Eulerian spaces. We describe how the halo mass distribution derives from a triangulation in Lagrangian space, while the dual Voronoi-like tessellation in Eulerian space gives the boundaries of empty regions with shock nodes at their vertices. We then emphasize that this dynamics actually leads to halo fragmentations for space dimensions greater or equal to 2 (for the inviscid limit studied in this paper). This is most easily seen from the properties of the Lagrangian-space triangulation and we illustrate this process in the two-dimensional (2D) case. In particular, we explain how pointlike halos only merge through three-body collisions while two-body collisions always give rise to two new massive shock nodes (in 2D). This generalizes to higher dimensions and we briefly illustrate the three-dimensional case. This leads to a specific picture for the continuous formation of massive halos through successive halo fragmentations and mergings.

LAGRANGIAN STATISTICS OF DARK HALOS IN A ΛCDM COSMOLOGY

New statistical properties of dark matter halos in Lagrangian space are presented. Tracing back the dark matter particles constituting bound halos resolved in a series of N-body simulations, we measure quantitatively the correlations of the proto-halo's inertia tensors with the local tidal tensors and investigate how the correlation strength depends on the proto-halo's sphericity, local density and filtering scale. It is shown that the majority of the proto-halos exhibit strong correlations between the two tensors provided that the tidal field is smoothed on the proto-halo's mass scale. The correlation strength is found to increase as the proto-halo's sphericity increases, as the proto-halo's mass increases, and as the local density becomes close to the critical value, delta_{ec}. It is also found that those peculiar proto-halos which exhibit exceptionally weak correlations between the two tensors tend to acquire higher specific angular momentum in Eulerian space, which is consistent with the linear tidal torque theory. In the light of our results, it is intriguing to speculate a hypothesis that the low surface brightness galaxies observed at present epoch correspond to the peculiar proto-halos with extreme low-sphericity whose inertia tensors are weakly correlated with the local tidal tensors.

Calibrating the baryon oscillation ruler for matter and halos

We characterize the nonlinear evolution of the baryon acoustic feature as traced by the dark matter and halos, using a combination of perturbation theory and $N$-body simulations. We confirm that the acoustic peak traced by the dark matter is both broadened and shifted as structure forms, and that this shift is well described by second-order perturbation theory. These shifts persist for dark matter halos, and are a simple function of halo bias, with the shift (mostly) increasing with increasing bias. Extending our perturbation theory results to halos with simple two parameter bias models (both in Lagrangian and Eulerian space) quantitatively explains the observed shifts. In particular, we demonstrate that there are additional terms that contribute to the shift that are absent for the matter. At $z=0$ for currently favored cosmologies, the matter shows shifts of $\ensuremath{\sim}0.5%$, $b=1$ halos shift the acoustic scale by $\ensuremath{\sim}0.2%$, while $b=2$ halos shift it by $\ensuremath{\sim}0.5%$; these shifts decrease by the square of the growth factor $D(z)$ at higher redshifts. These results are easily generalized to galaxies within the halo model, where we show that simple galaxy models show marginally larger shifts than the correspondingly biased halos, due to the contribution of satellites in high mass halos. While our focus here is on real space, our results make specific predictions for redshift space. For currently favored cosmological models, we find that the shifts for halos at $z=0$ increase by $\ensuremath{\sim}0.3%$; at high $z$, they increase by $\ensuremath{\sim}0.5%{D}^{2}$. Our results demonstrate that these theoretical systematics are smaller than the statistical precision of upcoming surveys, even if one ignored the corrections discussed here. Simple modeling, along the lines discussed here, has the potential to reduce these systematics to below the levels of cosmic variance limited surveys.

Unitarity in technicolor

We investigate the longitudinal $WW$ scattering in models of dynamical electroweak symmetry breaking featuring a spin one axial and vector state and a composite Higgs. We also investigate the effects of a composite spin two state which has the same properties of a massive graviton. Any model of dynamical electroweak symmetry breaking will feature, depending on the dynamics, some or all these basic resonances as part of the low energy spectrum. We suggest how to take limits in the effective Lagrangian parameter space to reproduce the dynamics of different types of underlying gauge theories, from the traditional Technicolor models to the newest ones featuring nearly conformal dynamics. We study the direct effects of a light composite Higgs and the indirect ones stemming from the presence of a light axial resonance on the longitudinal $WW$ scattering.

The origin of `Great Walls'

A new semi-analytical model that explains the formation and sizes of the `great walls' - the largest structures observed in the universe is suggested. Although the basis of the model is the Zel'dovich approximation it has been used in a new way very different from the previous studies. Instead of traditional approach that evaluates the nonlinear density field it has been utilized for identification of the regions in Lagrangian space that after the mapping to real or redshift space (depending on the kind of structure is studied) end up in the regions where shell-crossing occurs. The set of these regions in Lagrangian space form the progenitor of the structure and afterthe mapping it determines the pattern of the structure in real or redshift space. The particle trajectories have crossed in such regions and the mapping is no longer unique there. The progenitor after mapping makes only one stream in the multi-stream flow regions therefore it does not comprise all the mass. Nevertheless, it approximately retains the shape of the structure. The progenitor of the structure in real space is determined by the linear density field along with two non-Gaussian fields derived from the initial potential. Its shape in Eulerian space is alsoaffected by the displacement field. The progenitor of the structure in redshift space also depends on these fields but in addition it is strongly affected by two anisotropic fields that determine the pattern of great walls as well as their huge sizes. All the fields used in the mappingsare derived from the linear potential smoothed at the current scale of nonlinearity which isRnl = 2.7 h−1 Mpc for the adopted parameters of the ΛCDM universe normalized to σ8 = 0.8. The model predicts the existence of walls with sizes significantly greater than 500 h−1 Mpcthat may be found in sufficiently large redshift surveys.

Erratum: Nonlinear perturbation theory with halo bias and redshift-space distortions via the Lagrangian picture [Phys. Rev. D78, 083519 (2008)

The nonlinear perturbation theory of gravitational instability is extended to include effects of both biasing and redshift-space distortions, which are inevitable in predicting observable quantities in galaxy surveys. The precise determination of scales of baryon acoustic oscillations is crucial to investigate the nature of dark energy by galaxy surveys. We find that a local Lagrangian bias and redshift-space distortions are naturally incorporated in our formalism of perturbation theory with a resummation technique via the Lagrangian picture. Our formalism is applicable to any biasing scheme which is local in Lagrangian space, including the halo bias as a special case. Weakly nonlinear effects on halo clustering in redshift space are analytically given.We assume only a fundamental idea of the halo model: haloes form according to the extended Press-Schechter theory, and the spatial distributions are locally biased in Lagrangian space. There is no need for assuming the spherical collapse model to follow the dynamical evolution, which is additionally assumed in standard halo prescriptions. One-loop corrections to the power spectrum and correlation function of haloes in redshift space are explicitly derived and presented. Instead of relying on expensive numerical simulations, our approach provides an analytic way of investigating the weakly nonlinear effects, simultaneously including the nonlinear biasing and nonlinear redshift-space distortions. Nonlinearity introduces a weak scale dependence in the halo bias. The scale dependence is a smooth function in Fourier space, and the bias does not critically change the feature of baryon acoustic oscillations in the power spectrum. The same feature in the correlation function is less affected by nonlinear effects of biasing.

Nonlinear perturbation theory with halo bias and redshift-space distortions via the Lagrangian picture

The nonlinear perturbation theory of gravitational instability is extended to include effects of both biasing and redshift-space distortions, which are inevitable in predicting observable quantities in galaxy surveys. Weakly nonlinear effects in galaxy clustering on large scales recently attracted great interest, since the precise determination of scales of baryon acoustic oscillations is crucial to investigate the nature of dark energy by galaxy surveys. We find that a local Lagrangian bias and redshift-space distortions are naturally incorporated in our formalism of perturbation theory with a resummation technique via the Lagrangian picture. Our formalism is applicable to any biasing scheme which is local in Lagrangian space, including the halo bias as a special case. Weakly nonlinear effects on halo clustering in redshift space are analytically given. We assume only a fundamental idea of the halo model: haloes form according to the extended Press-Schechter theory, and the spatial distributions are locally biased in Lagrangian space. There is no need for assuming the spherical collapse model to follow the dynamical evolution, which is additionally assumed in standard halo prescriptions. One-loop corrections to the power spectrum and correlation function of haloes in redshift space are explicitly derived and presented. Instead of relying on expensive numerical simulations, our approach provides an analytic way of investigating the weakly nonlinear effects, simultaneously including the nonlinear biasing and nonlinear redshift-space distortions. Nonlinearity introduces a weak scale dependence in the halo bias. The scale dependence is a smooth function in Fourier space, and the bias does not critically change the feature of baryon acoustic oscillations in the power spectrum. The same feature in the correlation function is less affected by nonlinear effects of biasing.

Propagators in Lagrangian space

It has been found recently that propagators, e.g. the cross correlation spectra of the cosmic fields with the initial density field, decay exponentially at large k in an Eulerian description of the dynamics. We explore here similar quantities defined for a Lagrangian space description. We find that propagators in Lagrangian space do not exhibit the same properties: they are found not to be monotonic functions of time, and to track back the linear growth rate at late time (but with a renormalized amplitude). These results have been obtained with a novel method which we describe alongside. It allows the formal resummation of the same set of diagrams as those that led to the known results in Eulerian space. We provide a tentative explanation for the marked differences seen between the Eulerian and the Lagrangian cases, and we point out the role played by the vorticity degrees of freedom that are specific to the Lagrangian formalism. This provides us with new insights into the late-time behavior of the propagators.

Canonical Lagrangian dynamics and general relativity

Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a ‘Lagrangian one-form’ to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein–Cartan gravity, to which we apply it explicitly. We find that one can compute the constraint algebra and demonstrate the closure of the constraints without gauge fixing the Lorentz group or introducing primary constraints on the phase space variables. Ultimately our aim is towards a more covariant approach for canonical quantum gravity, and we discuss a possible route to quantization. Applying these techniques and using methods from geometric quantization, we find a new representation of the pre-quantum operator corresponding to the Hamiltonian constraint, which in contrast to the standard representation, is a kinematical operator that simply generates timelike diffeomorphisms on functionals of the Lagrangian phase space. This opens the possibility for a kinematical spacetime–diffeomorphism invariant Hilbert space.

Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space

Let Gr L ⊂ Gr ( n , V ) be the space of all Lagrangian subspaces of C 2 n equipped with the standard symplectic form. Let X ˜ be a universal cover of a compact connected Riemann surface X . We consider all immersions f : X ˜ ⟶ Gr L satisfying the following two conditions: (1) the map f is equivariant with respect to some homomorphism into Sp ( 2 n , C ) of the Galois group of the covering X ˜ ⟶ X , and (2) the symmetric bilinear form on the pullback, to X ˜ , of the tautological vector bundle over Gr L is fiberwise nondegenerate. Two such maps are called equivalent if they differ by the action of some fixed element of Sp ( 2 n , C ) . We prove that the equivalence classes of all such maps are bijectively parametrized by pairs of the form ( P , ( F , ∇ ) ) , where P is a projective structure on X and ( F , ∇ ) is an equivalence class of flat O ( n , C ) -connection on X . Two flat O ( n , C ) -bundles are equivalent if the corresponding flat PO ( n , C ) -bundles are isomorphic.