The Formation of Vortical Motion in Cosmic Large Scale Structure Ruth Durrer Universit´ e de Gen` eve D´ epartment de Physique Th´ eorique and Center for Astroparticle Physics Gravity and Cosmology Workshop, Kyoto 2018 Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 1 / 20
Outline Introduction 1 Perturbative Results 2 Vorticity from N-body simulations 3 Observation of vorticity 4 Conclusions 5 Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 2 / 20
Introduction: rotational motion of galaxies Most galaxies in the Universe rotate. The rotation axes of neighboring galaxies are correlated. New observations find alignments of jets in radio galaxies at z = 1 out to (10-20) Mpc (A. Taylor & P. Jagannathan (2016)). NGC 5457 Can these vortical motions be explained within standard ΛCDM? Can we learn something about cosmology by observing them? Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 3 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Helmholtz’s (third) theorem implies that no vorticity is generated in a perfect fluid hence within the perfect fluid approximation the velocity remains a gradient field. This theorem is valid also in General Relativity. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Helmholtz’s (third) theorem implies that no vorticity is generated in a perfect fluid hence within the perfect fluid approximation the velocity remains a gradient field. This theorem is valid also in General Relativity. Therefore, within Lagrangian or Eulerian, Relativistic or Newtonian perturbation theory, no vorticity is generated at any order. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Helmholtz’s (third) theorem implies that no vorticity is generated in a perfect fluid hence within the perfect fluid approximation the velocity remains a gradient field. This theorem is valid also in General Relativity. Therefore, within Lagrangian or Eulerian, Relativistic or Newtonian perturbation theory, no vorticity is generated at any order. This is not true for the momentum (sometimes called ’mass weighted velocity’) which acquires a rotational component at second order in perturbation theory. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Helmholtz’s (third) theorem implies that no vorticity is generated in a perfect fluid hence within the perfect fluid approximation the velocity remains a gradient field. This theorem is valid also in General Relativity. Therefore, within Lagrangian or Eulerian, Relativistic or Newtonian perturbation theory, no vorticity is generated at any order. This is not true for the momentum (sometimes called ’mass weighted velocity’) which acquires a rotational component at second order in perturbation theory. But CDM is not a (perfect) fluid. It is a collection of free streaming particles which can be accurately described with the Vlasov equation. This distinction is important since a fluid assigns to a given point in space a fixed value of the velocity where as the distribution in phase space allows the full velocity space in each volume element. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Introduction: vorticity in cosmology At first order in perturbation theory the velocity field of dark matter is a gradient field with vanishing vorticity. Helmholtz’s (third) theorem implies that no vorticity is generated in a perfect fluid hence within the perfect fluid approximation the velocity remains a gradient field. This theorem is valid also in General Relativity. Therefore, within Lagrangian or Eulerian, Relativistic or Newtonian perturbation theory, no vorticity is generated at any order. This is not true for the momentum (sometimes called ’mass weighted velocity’) which acquires a rotational component at second order in perturbation theory. But CDM is not a (perfect) fluid. It is a collection of free streaming particles which can be accurately described with the Vlasov equation. This distinction is important since a fluid assigns to a given point in space a fixed value of the velocity where as the distribution in phase space allows the full velocity space in each volume element. In a fluid description shell (orbit)-crossing is a singular process while in phase space it is regular. N-body simulations can accommodate shell crossing without problem, they are actually nothing else than a poor woman’s Vlasov equation solver. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 4 / 20
Perturbative Results: Vlasov eq. One might think that a perturbative approach to the Vlasov equation could be successful but... v v ... the flow of CMD is very cold. Contrary to the case of hot dark matter, a perturbative treatment using the Vlasov equation is not adequate for CDM. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 5 / 20
Perturbative Results: velocity dispersion But one can go to higher moments of the Vlasov equation, beyond the 0th and first moments which yield the continuity and Euler equations for perfect fluids. ∂ t δ + ∇ ((1 + δ ) v ) = 0 , v j + H v j + ∂ j Φ + 1 � ∂ t + v i ∂ i � ρ∂ i ( ρσ ij ) = 0 , ( ∂ t + v k ∂ k ) σ ij + 2 H σ ij + σ ik ∂ k v j + σ jk ∂ k v i = 0 σ ijk = 0 The curl of the Euler eqn. then gives, ω = ∇ ∧ v , � � ∂ ω 1 dt + H ω − ∇ ∧ [ v ∧ ω ] = −∇ ∧ ρ ∇ ( ρσ ) . To lowest order in perturbation theory, the velocity dispersion take the form σ ij = σ 0 3 a − 2 δ ij . Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 6 / 20
Perturbative Results: vorticity power spectrum We have solved the vorticity equation to lowest non-vanishing order (Cusin, Tansella & RD, 2017). � ω (2) ( k , t ) ω (2) ∗ ( k ′ , t ) � = (2 π ) 3 � δ ij − ˆ k i ˆ � δ ( k − k ′ ) P ω ( k , t ) . k j i j � 2 σ 2 � d 3 w � P ω ( k ) = 1 0 D + ( t ) w · ( k − w ) 2 k · w − k 2 � 2 P δ ( w ) P δ ( | k − w | ) | w ∧ k | 2 � 9 H 2 (2 π ) 3 w 2 | k − w | 2 0 Ω m k → 0 k 4 D + ( t ) P ω ( k , t ) → Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 7 / 20
Perturbative Results: vorticity power spectrum The rotational velocity spectrum, P R = k − 2 P ω compared to the gradient velocity spectrum P G = k − 2 P θ , θ = ∇ · v . 1 x 10 4 1 x 10 3 P(k) [h -1 Mpc] 3 100 σ 0 ~ 10 -2 10 1 σ 0 ~ 10 -4 0.1 0.01 1 x 10 -3 σ 0 ~ 10 -6 1 x 10 -4 1 x 10 -5 0.001 0.01 0.1 1 10 k [h/Mpc] σ 0 ~ 10 -2 1 x 10 k [h/Mpc] P(k) [h -1 Mpc] 3 4 1 x 10 3 1 x 10 -4 100 10 0.001 0.01 10 1 0.1 1 x 10 -5 1 x 10 -3 0.01 0.1 1 Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 8 / 20
Vorticity from N-body simulations: Pueblas et al. From Pueblas & Scoccimarro ’09 Using a Delauny tessellation for the velocity field. The vorticity and divergence spectra, P ω and P θ . They find a slope P ω ∝ k 2 . 5 and time dependence P ω ∝ D 7 + . The results shown are from a L = 256Mpc simulations with N = 512 3 particles using Gadget-2 with softening length 0.04. Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 9 / 20
Vorticity from N-body simulations: Zhu etal. From Zhu, Yu &Pen ’17 The gradient and rotational velocity spectra, k 3 P G ( k ) and k 3 P R ( k ). They find a slope P R ∝ k 0 . It is not clear whether these are spectra are Fourier transforms from Eulerian or Lagrangian coordinates. The results shown are from a L = 600Mpc simulations with N = 1024 3 particles on a 512 3 grid using CUBEP 3 M using multi-grid techniques to compute the displacement field and velocity (in Lagrangian coordinates). Ruth Durrer (Universit´ e de Gen` eve, DPT & CAP) Cosmological Vorticiy Kyoto, Feb 27, 2018 10 / 20
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