Issues in Non-Linear Cosmological Dynamics

Transcription

1 Issues in Non-Linear Cosmological Dynamics Marco Bruni Institute of Cosmology and Gravitation University of Portsmouth NLCP Workshop - Kyoto - 22/05/09

2 Outline a couple of reminders on Newtonian cosmology how the universe got its skewness shaking fists generates gravitational waves (so, be careful and gentle) post-newtonian cosmology? more issues (open ended)...

3 a brief reminder on Newtonian cosmology

4 linear Newtonian Perturbations: take 2 Some results of linear theory can be turned into an ansatz for the mildly non-linear regime; let s first look at these results. Peculiar velocity V:, Consider an EdS background, and re-scale variables using a(t) as time variable, so that

5 take 2 continue.. From these exact equation for a p=0 fluid in a EdS background, if we linearize and neglect the decaying mode it can be seen that the growing mode δ + a solution corresponds to: This further implies that the fluid is irrotational and in free fall motion (in the re-scaled variables):

6 Remarks on exact Newtonian theory Assume irrotational motion for the non-linear fluid and define the re-scaled kinematical variables:, Then the exact system for these is: Ẽ ij the tidal field has no evolution eq.: effect of action-atthe-distance in Newtonian gravity (Poisson is elliptic).

7 non-linear perturbations: Zel dovich approximation Using the shear and expansion equations, extrapolate to the nonlinear regime the results of linear theory: With this Zel dovich ansatz, the brackets in the equations vanish focus on collapse and use as time ; then we end up with a planar autonomous dynamical system for the two dimensionless rescaled shear vars.:

8 non-linear perturbations: Zel dovich approximation The shear variables related to the dimension-less eigenvalues are These determine the evolution of the fluid element through where the l i are the 3 scale factors along the shear eigen-directions. Then the definitions follows: pancake collapse: filament collapse: spherical collapse: Σ 1 =Σ 2 = 1/3, Σ 3 =2/3 Σ 1 =Σ 2 =1/6, Σ 3 = 1/3 Σ 1 =Σ 2 =Σ 3 =0

9 Pancakes as attractors from stability analysis Plotting the evolution and finding fixed points and eigenvalues of the linearized system we get:

10 How the Universe got its Skewness* (*) work with Irene Milillo and Kazuya Koyama (in preparation)

11 How the Universe got its Skewness a classical result (Peebles 1980) of non-linear perturbation theory in the matter dominated era (aka CDM) is: < δ 3 >= 34 7 ξ(0)2 Q1: how and when this value is reached, starting from before equidensity? Q2: can we do better than II-order in the dynamics of CDM?

12 CDM in a mixed background Continuity equation Euler equation Poisson equation Lagrangian approach: convective time derivative:

13 Meszaros Approximation Meszaros-Peebles approximation (see e.g. Peebles 1980, Dodelson 2003): the radiation background is smooth (negligible radiation density perturbation) This is a good approximation only for small scales, i.e. fluctuations before matter radiation equality are already well inside the horizon. The growth of perturbation is initially inhibited by the fast expansion in the radiation era. Lower limit on scales given by free-streaming length of CDM: negligible with respect to the scales of interest. Approximation valid below critical density ratio, deep inside radiation era (see e.g. Weinberg 2008): Translates into lower time limit.

14 Meszaros Approximation As we want to study the transition era between radiation and matter, it is convenient to rescale the time variabile (y > y crit ) and the velocity perturbation: Taking the divergence of the the first and the time derivative of the second equation and linearizing we obtain the Meszaros equation

15 First order: Meszaros equation Solution: growing and decaying mode well known stagnation effect: the density contrast is approximately constant before the matter-radiation equality: y<<1 Translates in the power spectrum in the bending for k modes that cross the horizon before equidensity.

16 Velocity field: expansion, shear and vorticity We can write the non-perturbative system in terms of velocity components defining expansion,shear, vorticity. The complete non-perturbative system is now.

17 First order Writing perturbatively every variable as with a small parameter, we can separate the linearized equations. Growing modes: From now on we only consider irrotational (scalar) modes setting vorticity to zero Important outcome: growing modes of velocity variables are constant, as for pure CDM Growing modes: the square brackets vanish, generalising matter era known result.

18 Q1: Second order Analytic solution: f,a,b and c are functions of σ (1) 2 and θ (1) 2, with a,b,c > 0; C 1 and C 2 are the two integration constants to be found through initial conditions. From the first order result Which relation for expansion and shear? No relation between their local values, but from their definitions we can infer a simple proportionality low for their variances (see e.g.peebles 1980):

19 Skewness we can now compute the skewness exactly as in Peebles book to obtain in the limit of large y this gives < δ 3 >= 34 7 ξ(0)2 a similar calculation can be done for the velocity perturbation related work: cf. arxiv

20 y=3.3 (last scattering in a vanilla ΛCDM) S=4.198 cf. S =34/7=4.857

21 Q2: Generalizing the Zeldovich approximation We can now generalize the Zeldovich approximation back into the radiation era and derive the dynamic behaviour of perturbations using a non linear approximate system. Zeldovich approximation : just CDM is present. From the first order growing mode, constant first order velocity makes the square brackets vanish. Same result for growing mode in Meszaros approximation: Our generalized Zeldovich ansatz:

22 Using it in the non-perturbative system The tidal force dependence disappears and we remain with a local and closed system, formally the same as that of pure CDM. The shear and tide matrices now commute, so the system can be written in the shear-tide eigenframe (with eigenvalues σ 1, σ 2, σ 3 = σ 1 σ 2 ). Solutions for shear and expansion are formally identical to those in the matter era: θ (Zel ) = 3 λ i, σ (Zel ) i = (1+ λ i (y y I ) i=1 δ (Zel ) = (δ I +1) 3 i=1 (1+ λ i (y y I )) 1 1 λ i 1+ λ i (y y I ) θ 3,

23 Conclusions the Mes zaros approx. is a useful tool to study the evolution of CDM perturbations at small scales, even in the mildly non-linear regime A1: the skewness of CDM perturbation evolves rapidly from 0 to LS, close to the 34/7 4.9 asymptotic value A2: in the Mes zaros approx., the Zel dovich approx. is valid before equidensity, with: linear Meszaros recovered at first order in the λi standard, CDM era, Zel dovich Next questions: relation with NG calculation of Pitrou et al. CDM collapse is unstoppable: what s the fate of rare large perturbations? Can we form BH s?

24 GW produced by density perturbations (a classically generated GW background)

25 A classically generated GW background as we all have learned as students, moving/evolving matter generates gravitational waves Problem 18.1 A Massachusetts motorist shakes his fist angrily at another motorist. What fraction of his expended energy goes into gravitational radiation? Problem book in General Relativity, Lightman et al (Princeton UP 1975)

26 A classically generated GW background matter perturbations exist (CMB, galaxies) matter motion/evolution generates GW effect is quadratic in the first-order scalar perturbations (density perturbations) GW detectors can place limits on cosmological density perturbations at scales vastly different from those probed by CMB and galaxy surveys

27 A classically generated GW background two approaches for two problems: 1. small scales, λ << H -1, during reheating at the end of inflation: large non-linear density perturbations can be treated in a Newtonian approx., and an effective Π is derived 2. full II-order relativistic perturbation theory is required to compute the effect of small density perturbations at all scales in both cases, the wave eq. has a source term: h k +2 a a h k + k 2 h k = S k

28 1): preheating/reheating requires detailed numerical simulations various attempts: Thelebnikov & Tkachev 1997, Garcia-Bellido et al , Easter et al. 2006, Dufaux et al. 2007, Nakayama et al. arxiv: physics partly unknown/speculative, results are thus model dependent

29 2): II-order relativistic perturbation theory II-order brings in mode coupling: tensor (TT) modes can be generated by scalar (density) perturbations (Tomita 1967, Matarrese et al , Bruni et al 1997, Matarrese Mollerach & Bruni 1998) understanding of gauge issues at II-order is required (cf. Bruni et al 1997, Nakamura , Malik & Wands 2008) extraction of the TT part of the source term non trivial (Ananda, PhD thesis, Ananda et al 2006)

30 2): II-order perturbation Recent work with emphasis on different aspects: Formalism (Bruni et al 1997, Matarrese Mollerach & Bruni 1998, Noh & Hwang 2004, Nakamura ) Effects on the CMB (Mollerach et al 2004) Evolution of the power spectrum: radiation dominated era (Ananda et al 2006) radiation to matter transition (Baumann et al 2007) early matter dominated era at the end of inflation (Assadullahi & Wands 2009)

31 radiation era consider scalar (matter) perturbations evolving from λ>>h -1 to λ<<h -1 (Ananda et al 2006) for each k, the source term Sk for the GW equation is non zero when λ>>h -1, then the density perturbation and Sk rapidly decay when λ<<h -1, and we are left with free decaying waves

32 matter era In general the source Sk is complicated function of the first order scalar metric perturbations for a pressurless fluid, there is a single scalar metric perturbation variable, a gravitational potential which is constant in time at all scales: Sk becomes constant hk evolves to a constant value: TT but not wave!

33 from radiation to matter Baumann et al 2007: vanilla ΛCDM The spectrum of GWs evolves differently for different k modes: it is red-shifted for modes that have entered the horizon well before equidensity, and enhanced for modes that enter the horizon in the matter era

34 pre-radiation matter era: entering the horizon during reheating

35 pre-radiation matter era at the end of inflation, oscillations of a scalar field around the minimum of the potential produce an effective matter era, p=0, ρ a -3 study the effect of density perturbation coming into the horizon during this period (Assadullahi & Wands, 2009) these produce a constant source term Sk, generating constant TT modes that become initial conditions for GW in the following rad. era crucial enhancement factor appears in the spectrum, with respect to the naive expectation

36 pre-radiation matter era Assadullahi & Wands, 2009

37 pre-radiation matter era schematic Baumann et al 2007: vanilla ΛCDM Schematic of the spectrum that could result from an early matter era, starting at kdom and ending at kdec, extrapolating the linear results into the non-linear regime (courtesy of Assadullahi & Wands, 2009)

38 Some Conclusions mode coupling generates TT-GWs from scalar density perturbations these GW are the leading order in this classical generation mechanism useful to place limits, with GW detectors, on the spectrum of density perturbations at scales vastly different from those probed by CMB and galaxies on small CMB scales, scalar induced tensor modes could even dominate over the primordial background during an early matter dominated era at the end of inflation GW can be generated in the frequency range of GW detectors, possibly at a level interesting for advanced ground based detectors, but more work is needed to refine predictions and study the non-linear regime statistic χ 2, maybe interesting/useful for detection?