An Effective Field Theory for Large Scale Structures based on 1301.7182 with M. Zaldarriaga, 1307.3220 with L. Mercolli, 1406.4135 with T. Baldauf, L. Mercolli & M. Mirbabayi Enrico Pajer Utrecht University
Outline Summary Motivations Effective Field Theory for Large Scale Structures Results: spectrum, bispectrum Outlook
Summary A program of applying Effective Field Theory (EFT) methods to Large Scale Structures (LSS) Today: the distribution of Dark Matter (EFT of LSS) Towards observations: Bias, Redshift Space Distortion and non-gaussianity Why bother? Fun field theory
The EFT Program Inflationζ(x) Matterδ m (x) non-gaussianity Quintessence Mod. GR Tracers δ m (x) Redshiftδ m (z) non CDM
The EFT of LSS EFT: LSS as a generalized fluid Analytical predictions for the distribution of matter on scales larger than a Mpc Consistently account for all non-linearities and gives physically meaningful predictions EFT Accurate theoretical predictions No speculations, this is the actual system
Why bother? We should spend at least as much time understanding the standard model (LCDM) as its modifications Analytical understanding can lead to new ways to extract information from the data, for example the reconstruction of the BAO Discover new relations among observables Simulations make assumptions as well and contain several unphysical parameters whose effect on the predictions might be hard to estimate
Fun Field Theory EFT of LSS features a range of interesting phenomena UV-divergences to be renormalized by counterterms IR-divergences and IR-safe observables Cutoff regularization or dimensional regularization to keep the symmetries explicit Running coupling constants and RG-flow Renormalization of contact operators
Motivations Define Large Scale Structures (LSS) LSS teach us about: Dark Matter, Dark Energy, primordial perturbations, modifications of GR,... Analytical understanding of LSS is a milestone of our cosmological model
History of the universe Size ä>0 ä<0 ä>0 Inflation? radiation matter dark E <<sec 60 kyear 10 Gyear 14 Gyear Time
The hard problem Size CMB physics is linear, i.e. the physics of waves Inflation? radiation matter dark E <<sec 60 kyear 10 Gyear 14 Gyear Time LSS physics involves large nonlinearities Now we have to deal with the hard problem!
Structure formation
LSS and Dark Energy Dark Energy can be probed studying the expansion history of the universe The Baryon Acoustic Oscillations (BAO) provide a standard ruler of 150 Mpc The BAO peak has a width of O(10) Mpc which gets broaden by non-linear effects
LSS and primordial perturbations LSS give us a very clean probe of initial conditions LSS are compatible with 10-5 perturbations with a scale-invariant initial power spectrum 3D information: many more modes (lower cosmic variance) in LSS than in the CMB which is 2D
Large Scale Structures The distribution of matter in the universe is very inhomogeneous, with very dense clumps of matter (e.g. galaxies) separated by big voids On scales much larger than the average galaxy-galaxy distance, i.e. O(1) Mpc, the density of clumps (e.g. galaxies) is very homogeneous Large Scale Structures (LSS) have a small density contrast (x) = (x) 1
Standard Perturbation Theory A Boltzmann equation for collisionless Dark Matter particles: the Vlasov equation On large scales (before shell crossing) one can truncate the hierarchy and get the fluid equations (Bernardeau et al 01) Problem 1: there is no clear expansion parameter Problem 2: missing deviations from a perfect pressureless fluid Problem 3: predictions are UV-divergent and hence unphysical
Vlasov Equation Since there is 6 times more DM than baryons, we focus on a system of collisionless DM particles interacting only gravitationally The corresponding Boltzmann equation @f @t + known as the Vlasov Equation, describes the evolution of the phase-space density f(~p, ~x) = X i pi @f @f ma 2 @x i m@ i @p i =0 3 (~x ~x i ) 3 (~p ma~v) The Poisson s equation determines ϕ
Fluid equations Let us define density and velocity ma 3 R d 3 pf(x, p) v i R d 3 pp i f(x, p) Taking the first two moment of the Vlasov eqution leads the continuity and the Euler equations @ + @ i (1 + ) v i l =0 @ v i + Hv i + @ i + v k @ k v i =0 Can we solve it perturbatively?
Problem 1 Because of shell-crossing the density diverges a short scales No clear expansion parameter for perturbation theory Even when applying to large scales, this makes it hard to estimate the theoretical errors in the computation
Problem 2 The fluid equations are those of a perfect pressureless fluid Since the short scales cannot be model correctly, there is no way to exclude non-linear exchanges of energy and/or momentum with the large scales, leading to dissipation More generally, there is NO symmetry forbidding a pressure term or any higher derivative corrections, e.g. viscosity. Why would they not be there?
Problem 3 Perturbation theory as (dubious) expansion in δ n R GF (k, k 0 ) m (k 0 ) n m (k k 0 ) Corrections to correlators, e.g. to the power spectrum P(k), are organized in loops. E.g. linear and 1-loop: P =P lin + P 22 + P 13 +... h i =h 1 1 i + h 2 2 i +2h 1 3 i +... and similarly for v and higher n-point functions
Problem 3 Loop corrections indeed have loop integrals Z P 22 (k!1) ' k 4 dq q 2 P in(q) 2 Z P 13 (k!1) ' k 2 P in (k) dqp in (q) For generic initial conditions these are UV-divergent, and hence unphysical P in = Ak n
Effective Field Theory of Large Scale Structures Consistently integrate out short-scales [Baumann et al 10] Problems 1: smoothed density and velocity are a good expansion parameters Problem 2: effective corrections to a perfect pressureless fluid arise (EFT philosophy) Problem 3: effective corrections are exactly the needed counterterms to renormalize the theory
Smoothing We smooth all fields on a certain scale Λ<k NL R! [ ] = dx 0 W (x x 0 ) (x 0 ) Short modes can combine to create long-wavelength perturbations We can expand short modes in the background of long modes (f g) l = f l g l +(f s g s ) l + 1 2 rf l rg l +... We get long, stochastic and higher derivative terms
Short scales We do not know how to describe the short scales, but we can parameterize our ignorance @hf s g s i (f s g s ) l = hf s g s i 0 + l +(f s g s ) hf s g s i +... @ l There are numerical and stochastic unknown coefficients These coefficients can be determined by simulations or by fitting the observations (Carrasco et al 12, Hertzberg 12) As always in EFT, the theory becomes predictive once we have more observables than parameters
Effective corrections Smoothing the Vlasov equation leads to @ + @ i (1 + ) v i l =0, @ v i l + Hv i l + @ i + v k l @ k v i l = c 2 s@ i +(c 2 sv + c 2 bv) @2 v i l H + c2 sv H @i @ j v j l J i... A pressure, viscosity and a stochastic terms, plus (infinitely many) higher derivatives These are all the terms allowed by the symmetries of the problem, as in the EFT philosophy
Problems 1 & 2 Every field is now smoothed on a scale Λ<k NL therefore δ, v << 1 providing good expansion parameters The short scales are now consistently accounted for, through the effective terms Collisionless dark matter on large scales shows indeed deviations from a perfect pressureless fluid, that vanish as k goes to 0 What about perturbation theory?
Perturbation theory For simplicity let us focus just on δ ' c 2 sk 2 J + R F (k, q) (k q) (q) F is the usual interaction kernel in SPT, while J and c s are the new effective terms The terms on the rhs are treated perturbatively J = R GJ c s = R Gc 2 sk 2 1 New corrections to the correlators, e.g. power spectrum h 1 cs i P cs h J J i P J
Stochastic term x 0 Mass and momentum conservation restrict the effects of short scales on large scales Z [Peebles] (k) = d 3 x (x) e ikx ' Z x0 0 d 3 x (x) 1+ikx kx 2 / (kx 0 ) 2 )hjji k 4 + O(k 5 )
Regularization The smoothing has regularized the theory. For P = k n Z P 22 (k!1) ' k 4 dq q 2 P in(q) 2 k 4 2n 1 Z P 13 (k!1) ' k 2 P in (k) dqp in (q) k 2 P in (k) n 1 But now we have extra (conter)terms P J = hjji( ) k 4 f( ) P c 2 s = c 2 s( )k 2 P in (k) Precisely the right k-dependence to cancel the UVdivergences
Cancellation of UV-divergences Although we show it just at one loop the cancellation of divergences takes place at all loops This is ensured by the EFT construction: if all terms compatible with the symmetries are included, there is always a term with the same structure as the UV-divergences The cancellation ensures that the result is independent of the cutoff Λ, and hence physically meaningful (unlike for SPT)
EdS & Self-similarity During 3300<z<1 our universe was matter dominated (Einstein de Sitter universe, Ω m =1). I assume EdS in this presentation because it s more transparent. I use LCDM for the actual calculation EdS with no-scale initial power spectrum P = A a 2 k n enjoys self-similarity: space (or momentum) dependence fixes time dependence univocally Only one scale in the problem, e.g. the non-linear scale k 3+n NL 2 2 Aa 2 / 4 2 lin = Everything must be function of k/k NL n+3 k k NL
Power spectrum Because of self-similarity, knowing the k- dependence of every term fixed the form of all correlators. E.g. the power spectrum is 2 = k + k NL k k NL 3+n + (n) 2(3+n) apple 5+n k + k NL (n)+ (n)ln k k NL (n) k k NL +... 7
Relative importance of corrections Relative importance of terms as k->0 depends on n For our universe n = 1.5 hence c s is more important than 2- loops J is less important than 3- loops d D 2 dlnk 10 1-loop 2 c s J 8 6 4 2-3 -2-1 0 1 2 3 4 n Only one fitting parameter at leading order
LCDM matter spectrum [Carrasco et al. 13] [Senatore & Zaldarriaga 14]
LCDM matter spectrum [Baldauf, Mercolli, Mirbabayi, EP 14]
LCDM matter bispectrum The bispectrum requires 3 new counterterms @ j ij c 2 s@ i + e 1 @ i 2 + e 2 @ i (,jk,jk )+e 3,ij (@ i ) The structure of UV-divergences relate them all to cs, which is fixed by the power spectrum The bispectrum hence has no fitting parameters! [Baldauf, Mercolli, Mirbabayi, EP 14]
Drawing the bispectrum Tree level Counter-terms 1 Loop
SPT matter bispectrum Tree level and 1-loop matter bispectrum from Standard Perturbation Theory
EFT corrections c s e 1 e 2 e 3
EFT matter bispectrum Good fit to scales twice as short (no params!) [Baldauf, Mercolli, Mirbabayi, EP 14]
Conclusions SPT is unsatisfactory for at least three reasons 1.there is no clear expansion parameter 2.deviation from perfect pressureless fluid are missing 3.predictions are UV-divergent and hence unphysical The EFT approach is to consistently integrate out the short scales. This addresses all the above problems smoothed fields are small everywhere pressure, dissipation and stochastic noise appear couterterms cancel UV-divergences a make the theory predictive
Conclusions The LCDM matter spectrum has one fitting parameter at leading order The 1 and 2-loop spectrum agrees with sims much better than SPT The bispectrum has no fitting parameters (at leading order) and LCDM matter bispectrum agrees with sims to scales twice as short (2^3=8 times more modes)
Outlook Study the EFT of bias in redshift space Understand the fitting coefficient from some rough model of the short scales Extensions of the standard model: non-gaussianity
Apparent violation of self-similarity When UV-divergences are present, in cutoff regularization terms appear of the form + These violate self-similarity (Frieman & Scoccimarro 96) But also the counterterms violate self-similarity in such a way that the final result, after the cancellation, is self-similar ## k NL # k k NL Dimensional regularization instead preserves selfsimilarity in all steps of the computation
Comparison with simulations β
Dim reg computation
Dimensional regularization β andγare n-dependent fitting parameters, that can be determined comparing with observations or simulations (e.g. Carrasco et al 12, Hertzberg 12) α and α-tilde are n-dependent numbers predicted by perturbation theory. They are most easily computed in dimensional regularization (dim reg) Dim reg preserved the scaling symmetry (unlike the cutoff regularization) of EdS, hence no violation of self-similarity appears anywhere in the computation