Effective Description of Dark Matter as a Viscous Fluid

Transcription

1 Effective Description of Dark Matter as a Viscous Fluid Nikolaos Tetradis University of Athens Work with: D Blas, S Floerchinger, M Garny, U Wiedemann N Tetradis University of Athens

2 Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions Distribution of dark and baryonic matter in the Universe Figure: 2MASS Galaxy Catalog (more than 15 million galaxies) N Tetradis University of Athens

3 Inhomogeneities Inhomogeneities are treated as perturbations on top of an expanding homogeneous background Under gravitational attraction, the matter overdensities grow and produce the observed large-scale structure The distribution of matter at various redshifts reflects the detailed structure of the cosmological model Define the density field δ = δρ/ρ 0 and its spectrum δ(k)δ(q) δ D (k + q)p(k) N Tetradis University of Athens

4 Figure: Matter power spectrum (Tegmark et al 2003) N Tetradis University of Athens

5 Figure: Matter power spectrum in the range of baryon acoustic oscillations N Tetradis University of Athens

6 A scale from the early Universe The characteristic scale of the baryon acoustic oscillations is approximately 150 Mpc (490 million light-years) today It corresponds to the wavelength of sound waves (the sound horizon) in the baryon-photon plasma at the time of recombination ( z = a today /a 1 = 1100) It is also imprinted on the spectrum of the photons of the cosmic microwave background Comparing the measured with the theoretically calculated spectra constrains the cosmological model The aim is to achieve a 1% precision both for the measured and calculated spectra Galaxy surveys: Euclid, DES, LSST, SDSS N Tetradis University of Athens

7 Problems with standard perturbation theory In the linearized hydrodynamic equations each mode evolves independently Higher-order corrections take into account mode-mode coupling Calculation of the matter spectrum beyond the linear level (Crocce, Scoccimarro 2005) Baryon acoustic oscillations (k h/mpc): Mildly nonlinear regime of perturbation theory Higher-order corrections dominate for k h/mpc The theory becomes strongly coupled for k > 1 h/mpc The deep UV region is out of the reach of perturbation theory Way out: Introduce an effective low-energy description in terms of an imperfect fluid (Baumann, Nicolis, Senatore, Zaldarriaga 2010, Carrasco, Hertzberg,Senatore 2012, Pajer, Zaldarriaga 2013, Carrasco, Foreman, Green, Senatore 2014) N Tetradis University of Athens

8 Lessons from the functional renormalization group Coarse-graining: Integrate out the modes with k > k m and replace them with effective couplings in the low-k theory Wetterich equation for the coarse-grained effective action Γ k [ϕ]: [ ] Γ k [ϕ] = 1 ( t 2 Tr Γ (2) k [ϕ] + ˆR ) 1 ˆRk k ln k For a standard kinetic term and potential U k [ϕ], with a sharp cutoff, the first step of an iterative solution gives U km (ϕ) = V (ϕ) Λ k m d d q (2π) d ln ( q 2 + V (ϕ) ) The low-energy theory contains new couplings, not present in the tree-level action It comes with a UV cutoff k m N Tetradis University of Athens

9 Why is this intuition relevant for the problem of classical cosmological perturbations? The primordial Universe is a stochastic medium The fluctuating fields (density, velocity) at early times are Gaussian random variables with an almost scale-invariant spectrum The generation of this spectrum is usually attributed to inflation The coarse graining can be implemented formally on the initial condition for the spectrum at recombination N Tetradis University of Athens

10 The question Question: Is dark matter at large scales (in the BAO range) best described as a perfect fluid? I shall argue that there is a better description in the context of the effective theory Going beyond the perfect-fluid approximation, the description must include effective (shear and bulk) viscosity and nonzero speed of sound Formulate the perturbative approach for viscous dark matter N Tetradis University of Athens

11 Scales k Λ 1 3 h/mpc (length 3 10 Mpc): The fluid description becomes feasible Scales k > k Λ correspond to virialized structures, which are essentially decoupled k m 05 1 h/mpc (length Mpc): The fluid parameters have a simple form The description includes effective viscosity and speed of sound, arising through coarse-graining The viscosity results from the integration of the modes k > k m The form of the power spectrum k 3 implies that the effective viscosity is dominated by k k m k m acts as an UV cutoff for perturbative corrections in the large-scale theory Good convergence, in contrast to standard perturbation theory N Tetradis University of Athens

12 Dark matter can be treated as a fluid because of its small velocity and the finite age of the Universe Dark matter particles drift over a finite distance, much smaller than the Hubble radius The phase space density f (x, p, τ) = f 0 (p)[1 + δ f (x, p, ˆp, τ)] can be expanded in Legendre polynomials: δ f (k, p, ˆp, τ) = ( i) n (2n + 1)δ [n] (k, p, τ)p n (ˆk ˆp) n=0 The Vlasov equation leads to: dδ [n] f dτ [ n + 1 = kv p 2n + 1 δ[n+1] f n ] 2n + 1 δ[n 1] f, n 2, with v p = p/am the particle velocity The time τ available for the higher δ [n] f to grow is 1/H A fluid description is possible for kv p /H < 1 Estimate the particle velocity from the fluid velocity v at small length scales N Tetradis University of Athens f

13 At the comoving scale k, the linear evolution indicates that (θ/h) 2 k 3 P L (k), with θ = k v and P L (k) the linear power spectrum The linear power spectrum scales roughly as k 3 above k m k 3 P L (k) is rougly constant, with a value of order 1 today Its time dependence is given by D 2 L, with D L the linear growth factor If the maximal particle velocity is identified with the fluid velocity at the scale k m, we have v p H k m D L The dimensionless factor characterizing the growth of higher δ [n] f is kv p /H D L k/k m Scales with k k m /D L require the use of the whole Boltzmann hierarchy In practice, the validity of the fluid description extends beyond k m N Tetradis University of Athens

14 The shear viscosity is estimated as η/(ρ + p) l free v p, with l free the mean free path For the effective viscosity we can estimate l free v p /H, with H = H/a In this way we obtain ν eff H = η eff (ρ + p)a H l freev p H H2 k 2 m D 2 L There is also a nonzero speed of sound The linearized treatment of the Boltzmann hierarchy gives on the growing mode ν eff H = 3 5 c2 s N Tetradis University of Athens

15 Plan of the talk Basic formalism Determination of the effective viscosity Calculation of the spectrum Conclusions S Floerchinger, N T, U Wiedemann arxiv: [gr-qc], Phys Rev Lett 114: 9, (2015) D Blas, S Floerchinger, M Garny, N T, U Wiedemann arxiv: [astro-phco], JCAP 1511, 049 (2015) S Floerchinger, M Garny, N T, U Wiedemann arxiv: [astro-phco], JCAP 1701 no01, 048 (2017) N Tetradis University of Athens

16 Covariant hydrodynamic description of a viscous fluid Work within the first-order formalism Energy-momentum tensor: T µν = ρu µ u ν + (p + π b ) µν + π µν ρ: energy density p: pressure in the fluid rest frame π b : bulk viscosity π µν : shear viscosity, satisfying: u µ π µν = π µ µ = 0 µν projector orthogonal to the fluid velocity: µν = g µν + u µ u ν New elements: Bulk viscosity: π b = ζ ρ u ρ Shear viscosity tensor: ( 1 π µν = 2ησ µν = 2η 2 ( µρ ρ u ν + νρ ρ u µ ) 1 ) 3 µν ( ρ u ρ ) N Tetradis University of Athens

17 Dynamical equations Einstein equations: G µν = 8πG N T µν, Conservation of the energy momentum tensor ( ν T µν = 0): u µ µ ρ + (ρ + p) µ u µ ζ ( µ u µ ) 2 2ησ µν σ µν = 0 (ρ + p + π b )u µ µ u ρ + ρµ µ (p + π b ) + ρ ν µ π µν = 0 N Tetradis University of Athens

18 Subhorizon scales Ansatz for the metric: ds 2 = a 2 (τ) [ (1 + 2Ψ(τ, x)) dτ 2 + (1 2Φ(τ, x)) dx dx ] The potentials Φ and Ψ are weak Their difference is governed by the shear viscosity We can take Φ Ψ 1 The four-velocity u µ = dx µ / ds 2 can be expressed through the coordinate velocity v i = dx i /dτ and the potentials Φ and Ψ: u µ 1 = a (1, v) 1 + 2Ψ (1 2Φ) v 2 Neglect vorticity and consider the density δ = δρ ρ 0 and velocity θ = v fields Combine them in a doublet φ 1,2 = (δ, θ/h), where H = ȧ/a is the Hubble parameter The spectrum is P(k, τ) P 11 (k, τ) φ a (k, τ)φ b (q, τ) δ D (k + q)p ab (k, τ) N Tetradis University of Athens

19 Convergence of perturbation theory (no viscosity) 10 3 Pk,z 0 Mpch SPT 1 L linear ideal fluid SPT 2 L k hmpc Figure: Linear power spectrum and the one- and two-loop corrections in standard perturbation theory (SPT) at z = 0 N Tetradis University of Athens

20 Higher-order (loop) corrections dominate for k > h/mpc The theory becomes strongly coupled for k 1 h/mpc The deep UV region is out of the reach of perturbation theory Higher-order corrections are increasingly more UV sensitive For small k, the one-loop depends on the dimensionful scale σ 2 d (η) = 4π 3 dq P L (q, η) = 4π 3 D2 L (η) dq P L (q, 0), 0 0 with η = ln a = ln(1 + z) and D L (η) the linear growth factor: δ(k, η) = D L (η)δ(k, 0) on the growing mode For the spectrum, the complete expression is (Blas, Garny, Konstandin 2013) ) P 1 loop ab (k, η) = k 2 σd 2 PL (k, η) ( N Tetradis University of Athens

21 Effective description (in terms of pressure and viscosity) D Blas, S Floerchinger, M Garny, N T, U Wiedemann arxiv: [astro-phco], JCAP 1511, 049 (2015) Introduce an effective low-energy description in terms of an imperfect fluid (Baumann, Nicolis, Senatore, Zaldarriaga 2010, Carrasco, Hertzberg,Senatore 2012, Pajer, Zaldarriaga 2013, Carrasco, Foreman, Green, Senatore 2014) Integrate out the modes with k > k m and replace them with effective couplings (viscosity, pressure), determined in terms of σ 2 dk (η) = 4π 3 k m dq P L (q, η) = 4π 3 D2 L (η) k m dq P L (q, 0) For a spectrum 1/k 3, the integral is dominated by the region near k m The deep UV does not contribute significantly N Tetradis University of Athens

22 Parameters of an effective viscous theory ρ(x, τ) = ρ 0 (τ) + δρ(x, τ) p(x, τ) = 0 + δp(x, τ) δ = δρ ρ 0 θ = v H = ȧ a H = 1 a H cs 2 (τ) = δp δρ = α s(τ) H2 km 2 ν(τ)h = ηh ρ 0 a = 3 4 α ν(τ) H2 km 2 with α s, α ν = O(1) today N Tetradis University of Athens

23 We rely on a hierarchy supported by linear perturbation theory for subhorizon perturbations 1 We treat δ and θ/h as quantities of order 1, v as a quantity of order H/k and Ψ, Φ as quantities of order H 2 /k 2 2 We assume that a time derivative is equivalent to a factor of H, while a spatial derivative to a factor of k 3 We assume that c 2 s, νh are of order H 2 /k 2 m Keeping the dominant terms, we obtain δ + v + ( v )δ + δ v = 0 v + H v + ( v ) v + Φ + cs 2 (1 δ) δ ( ν(1 δ) 2 v + 1 ) ( 3 v) = 0 2 Φ = 3 2 Ω mh 2 δ N Tetradis University of Athens

24 Use Fourier-transformed quantities to obtain δ k + θ k + d 3 p d 3 q δ(k p q) α 1 (p, q) δ p θ q = 0 θ k + (H + 43 ) ( ) 3 νk 2 θ k + 2 Ω mh 2 cs 2 k 2 δ k + d 3 p d 3 q δ(k p q) ) (β 1 (p, q) δ p δ q + β 2 (p, q) θ p θ q + β 3 (p, q) δ p θ q = 0, with α 1 (p, q) = (p + q)q q 2 β 1 (p, q) = c 2 s (p + q)q β 2 (p, q) = (p + q)2 p q 2p 2 q 2 β 3 (p, q) = 4 3 ν(p + q)q N Tetradis University of Athens

25 Define the doublet φ 1 (k, η) φ 2 (k, η) = δ k (τ) θ k(τ) H, where η = ln a(τ) The evolution equations take the form η ϕ a (k) = Ω ab (k, η)φ b (k)+ d 3 p d 3 q δ(k p q)γ abc (p, q, η) φ b (p) φ c (q), where Ω(k, η) = ( 3 2 Ω m + α s k 2 k 2 m H H + α ν k 2 k 2 m ) and a prime denotes a derivative with respect to η The nonzero elements of γ abc are expressed in terms of α 1 (p, q), β 1 (p, q), β 2 (p, q), β 3 (p, q) N Tetradis University of Athens

26 The evolution of the spectrum Define the spectra, bispectra and trispectra as φ a (k, η)φ b (q, η) δ D (k + q)p ab (k, η) φ a (k, η)φ b (q, η)φ c (p, η) δ D (k + q + p)b abc (k, q, p, η) φ a (k, η)φ b (q, η)φ c (p, η)φ d (r, η) δ D (k + q)δ D (p + r)p ab (k, η)p cd (p, η) +δ D (k + p)δ D (q + r)p ac (k, η)p bd (q, η) +δ D (k + r)δ D (q + p)p ad (k, η)p bc (q, η) +δ D (k + p + q + r)q abcd (k, p, q, r, η) N Tetradis University of Athens

27 Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy Essential (rather crude) approximation: Neglect the effect of the trispectrum on the evolution of the bispectrum (Pietroni 2008) In this way we obtain η P ab (k, η) = Ω ac P cb (k, η) Ω bc P ac (k, η) + d 3 q [ γ acd (k, q, q k)b bcd (k, q, q k) +γ bcd (k, q, q k)b acd (k, q, q k) ], η B abc (k, q, q k) = Ω ad B dbc (k, q, q k) Ω bd B adc (k, q, q k) Ω cd B abd (k, q, q k) +2 d 3 q [ γ ade (k, q, q k)p db (q, η)p ec (k q, η) +γ bde ( q, q k, k)p dc (k q, η)p ea (k, η) +γ cde (q k, k, q)p da (k, η)p eb (q, η) ] N Tetradis University of Athens

28 Alternative approach Expand the fields in powers of the initial perturbations at η = η 0, ϕ a (k, η) = d 3 q 1 d 3 q n (2π) 3 δ (3) (k q i ) n i F n,a (q 1,, q n, η)δ q1 (η 0 ) δ qn (η 0 ) From the equation of motion we can get evolution equations for the kernels F n,a ( η δ ab + Ω ab (k, η))f n,b (q 1,, q n, η) = n γ abc (q q m, q m q n ) m=1 F m,b (q 1,, q m, η)f n m,c (q m+1,, q n, η) When neglecting the pressure and viscosity terms, the solution is known analytically for an Einstein-de Sitter Universe For non-zero pressure and viscosity, the time-dependence does not factorize We solve the differential equations numerically N Tetradis University of Athens

29 Matching the perfect-fluid and viscous theories For the matching one could use the propagator G ab (k, η, η )δ (3) (k k δϕa (k, η) ) = δϕ b (k, η, ) Define appropriate fields for the background to be effectively Einstein-de Sitter to a very good approximation: D L ( η) = exp( η) The one-loop propagator of the perfect-fluid theory is ( 61 ) G ab (k, η, η ) = g ab ( η η ) k 2 e η η σd 2 61 ( η) , where the linear propagor for the growing mode is g ab ( η η ) = e ( ) η η The contribution from k > k m can be isolated by taking σ 2 d ( η) σ2 dk ( η) 4π 3 exp(2 η) dq P L (q, 0) k m N Tetradis University of Athens 9 25

30 Compute the propagator of the viscous theory at linear order, for kinematic viscosity and sound velocity of the form νh = η ρ 0 a H = β 2 η H2 ν e km 2, cs 2 = δp δρ = 3 4 β 2 η H2 s e km 2 The propagator contains a contribution δg ab (k, η) = k 2 e3 η (β ν + β s ) with an UV cutoff k m in the momentum integrations k 2 m 45 ( ) in addition to the perfect-fluid linear contribution (for η η ) Identify the linear contribution k 2 cs 2 and k 2 νh with the one-loop correction k 2 σdk 2 of the perfect-fluid propagator This can be achieved with 1% accuracy, and gives β s + β ν = k 2 mσ 2 dk (0) Solve for the nonlinear spectrum in the effective viscous theory N Tetradis University of Athens

31 Particular features There are no free parameters in this approach The results are independent of the ratio β ν /β s to a good approximation They depend mainly on the value of β ν + β s They are also insensitive to the (mode-mode) couplings proportional to the viscosity or the shound velocity The (mode-mode) couplings of the perfect-fluid theory are the dominant ones N Tetradis University of Athens

32 P k,z 0, k m 06hMpc Pk,zP lin,ideal k,z lin viscous 1 L viscous 2 L viscous Α s Α Ν k hmpc Figure: Power spectrum obtained in the viscous theory for redshift z = 0, normalized to P lin,ideal The open (filled) circles show the one-loop (two-loop) result in the viscous theory The solid blue line is the linear spectrum in the viscous theory, and the red points show results of the Horizon N-body simulation N Tetradis University of Athens

33 P k,z 0375, k m 06hMpc Pk,zP lin,ideal k,z lin viscous 1 L viscous 2 L viscous Α s Α Ν k hmpc Figure: One-loop spectrum in the effective theory at z = 0375 N Tetradis University of Athens

34 P k,z 0833, k m 06hMpc Pk,zP lin,ideal k,z lin viscous 1 L viscous 2 L viscous Α s Α Ν k hmpc Figure: One-loop spectrum in the effective theory at z = 0833 N Tetradis University of Athens

35 P k,z 175, k m 06hMpc Pk,zP lin,ideal k,z lin viscous 1 L viscous 2 L viscous Α s Α Ν k hmpc Figure: One-loop spectrum in the effective theory at z = 175 N Tetradis University of Athens

36 P k,z 0 Pk,zP lin,ideal k,z L viscous 1 L viscous k m 04 hmpc k m 06 hmpc k m 08 hmpc k m 10 hmpc 08 Α s Α Ν k hmpc lin viscous Figure: One-loop spectrum in the effective theory at z = 0 for various values of k m N Tetradis University of Athens

37 P k,z 0375 Pk,zP lin,ideal k,z L viscous 1 L viscous k m 04 hmpc k m 06 hmpc k m 08 hmpc k m 10 hmpc 08 Α s Α Ν k hmpc lin viscous Figure: One-loop spectrum in the effective theory at z = 0375 for various values of k m N Tetradis University of Athens

38 P k,z 0P Nbody, viscous theory Pk,zP Nbody k,z L viscous k m 04 hmpc 1 L viscous k m 06 hmpc k m 08 hmpc k m 10 hmpc lin viscous k hmpc Figure: Comparison of results for the power spectrum obtained within the viscous theory normalized to the N-body result at z = 0 The grey band corresponds to an estimate for the error of the N-body result N Tetradis University of Athens

39 P k,z 0P N body, SPT with cutoff Pk,zP Nbody k,z L SPT 04 hmpc 06 hmpc 08 hmpc 10 hmpc 1 L SPT lin SPT k hmpc Figure: One- and two-loop results in standard perturbation theory (shown as dashed and dotted lines, respectively), computed with various values of an ad-hoc cutoff Λ (coloured lines), as well as in the limit Λ (black lines) N Tetradis University of Athens

40 P ΘΘ k,z 0P, viscous theory 10 lin viscous P ΘΘ k,zp k,z L viscous 2 L viscous k m 04 hmpc k m 06 hmpc k m 08 hmpc k m 10 hmpc k hmpc Nbody J12 Nbody HAA14 Figure: The velocity-velocity spectrum obtained within the viscous theory, compared to results from N-body simulations at z = 0 The pink band corresponds to a 10% error for the N-body results N Tetradis University of Athens

41 P k,z 0P, SPT with cutoff 10 lin SPT P ΘΘ k,zp k,z L SPT 2 L SPT 04 hmpc 06 hmpc 08 hmpc 10 hmpc k hmpc Nbody J12 Nbody HAA14 Figure: One- and two-loop results in standard perturbation theory (shown as dashed and dotted lines, respectively), computed with various values of an ad-hoc cutoff Λ (coloured lines), as well as in the limit Λ (black lines) N Tetradis University of Athens

42 P Θ k,z 0P, viscous theory 10 lin viscous P Θ k,zp k,z L viscous 2 L viscous k m 04 hmpc k m 06 hmpc k m 08 hmpc k m 10 hmpc Nbody J12 Nbody HAA k hmpc Figure: The density-velocity spectrum obtained within the viscous theory, compared to results from N-body simulations at z = 0 The pink band corresponds to a 10% error for the N-body results N Tetradis University of Athens

43 P Θ k,z 0P, SPT with cutoff P Θ k,zp k,z L SPT 2 L SPT 04 hmpc 06 hmpc 08 hmpc 10 hmpc lin SPT Nbody J12 Nbody HAA k hmpc Figure: One- and two-loop results in standard perturbation theory (shown as dashed and dotted lines, respectively), computed with various values of an ad-hoc cutoff Λ (coloured lines), as well as in the limit Λ (black lines) N Tetradis University of Athens

44 We guessed ν(τ)h = 3 4 α ν(τ) H2, c 2 km 2 s (τ) = α s (τ) H2, with α km 2 ν, α s < 1 At one loop we found νh = 3 4 β 2η H2 ν e, c 2 2η H2 km 2 s = β s e, with β km 2 s + β ν = k mσ 2 dk 2 (0) We generalize the framework by considering νh = 3 4 λ ν(k) e κ(k)η H 2, cs 2 = λ s (k) e κ(k)η H 2 The scale dependence of the parameters can be determined through renormalization-group methods This requires a functional representation of the problem N Tetradis University of Athens

45 Functional representation S Floerchinger, M Garny, N T, U Wiedemann arxiv: [astro-phco], JCAP 1701 no01, 048 (2017) We follow and extend Matarrese, Pietroni 2007 We are interested in solving η ϕ a (k) = Ω ab (k, η)ϕ b (k) + d 3 p d 3 q δ (3) (k p q)γ abc (p, q, η) ϕ b (p) ϕ c (q) with stochastic initial conditions determined by the primordial power spectrum Pab 0 (k) This can be achieved by computing the generating functional { Z [J, K ; P 0 ] = DϕDχ exp 1 2 χ a(0)pab 0 χ b(0) } + i dη [χ a (δ ab η + Ω ab )ϕ b γ abc χ a ϕ b ϕ c + J a ϕ a + K b χ b ] N Tetradis University of Athens

46 One can now define the generating functional of connected Green s functions W [J, K ; P 0 ] = i log Z [J, K ; P 0 ] The full power spectrum P ab and the propagator G ab can be obtained through second functional derivatives of W, δ 2 W δj a ( k, η) δj b (k, η ) = iδ(k k ) P ab (k, η, η ), J, K =0 δ 2 W δj a ( k, η) δk b (k, η ) = δ(k k )Gab R (k, η, η ), J, K =0 δ 2 W δk a ( k, η) δk b (k, η ) = 0 J, K =0 N Tetradis University of Athens

47 The effective action is the Legendre transform Γ[ϕ, χ; P 0 ] = dηd 3 k {J a ϕ a + K b χ b } W [J, K ; P 0 ], where ϕ a (k, η) = δw /δj a (k, η), χ b (k, η) = δw /δk b (k, η) The inverse retarded propagator satisfies dη Dab R (k, η, η )Gbc R (k, η, η ) = δ ac δ(η η ) It can be computed from the effective action δ 2 Γ δϕ a ( k, η) δϕ b (k, η ) = 0, J, K =0 δ 2 Γ δχ a ( k, η) δϕ b (k, η ) = δ(k k )Dab R (k, η, η ), J, K =0 δ 2 Γ δχ a ( k, η) δχ b (k, η ) = iδ(k k )H ab (k, η, η ) J, K =0 N Tetradis University of Athens

48 Renormalized field equations δ δϕ a (x, η) Γ[ϕ, χ] = J a(x, η), δ δχ a (x, η) Γ[ϕ, χ] = K a(x, η), For vanishing source fields J = K = 0, we have χ = 0 and the first equation is trivially satisfied N Tetradis University of Athens

49 Renormalization-group improvement Modify the initial power spectrum so that it includes only modes with wavevectors q larger than the coarse-graining scale k: P 0 k (q) = P0 (q) Θ( q k) The coarse-grained effective action satisfies the Wetterich equation k Γ k [ϕ, χ] = 1 { ( 2 Tr Γ (2) k [ϕ, χ] i ( } Pk 0 P0)) 1 k Pk 0 N Tetradis University of Athens

50 Use an ansatz of the form Γ k [ϕ, χ] = ( ) dη[ d 3 q χ a ( q, η) δ ab η + ˆΩ ab (q, η) ϕ b (q, η) d 3 k d 3 p d 3 q δ (3) (k p q)γ abc (k, p, q)χ a ( k, η)ϕ b (p, η)ϕ i ] d 3 q χ a (q, η)h ab,k (q, η, η )χ b (q, η ) +, 2 where ( ˆΩ(q, η) = Ω m + λ s (k) e κ(k)η q H H + λ ν(k) e κ(k)η q 2 Derive differential equations for the k-dependence of λ ν (k), λ s (k), κ(k) N Tetradis University of Athens )

51 Comments A prescription is needed in order to project the general form of the inverse retarded propagator D R ab (q, η, η ) = δ ab δ (η η ) + Ω ab (q, η)δ(η η ) + Σ R ab (q, η, η ) to the form ) Dab (δ R (q, η, η) = ab η ˆΩ ab (q, η) δ(η η ) The projection is performed through a Laplace transform At the first order of an iterative solution of the exact RG equation, one finds λ s (k) = σ2 dk, λ ν(k) = σ2 dk, κ(k) = 2 This validates our intuitive matching through the propagator At the next level, the k-dependence of λ ν (k), λ s (k), κ(k) can be derived N Tetradis University of Athens

52 Λ Ν Mpch k hmpc Λ s Mpch k hmpc Κ k hmpc Figure: RG evolution of λ ν (k), λ s (k) and κ(k) We have initialized the flow at k = Λ = 1 h/ Mpc with the one-loop values The solid lines correspond to the solution of the full flow equations, while the dashed lines correspond to the solution of the one-loop approximation N Tetradis University of Athens

53 10 3 Λ Ν Λ s Mpch k hmpc Figure: RG evolution of the sum λ ν(k) + λ s(k) The various lines correspond to the RG evolution obtained when imposing initial values at Λ = 1 h/mpc (light blue) or Λ = 3 h/mpc (dark blue), respectively The dashed line shows the perturbative one-loop estimate for comparison N Tetradis University of Athens

54 30 25 Κ k hmpc Figure: RG evolution of the power law index κ(k) characterizing the time-dependence of the effective sound velocity and viscosity The various blue lines show the RG evolution when initializing the RG flow at Λ = 1 h/mpc (light blue) or Λ = 3 h/mpc (dark blue), respectively N Tetradis University of Athens

55 Conclusions The nature of dark matter is still unknown It is reasonable to consider possibilities beyond an ideal, pressureless fluid Standard perturbation theory cannot describe reliably the short-distance cosmological perturbations It is possible to integrate out the short-distance modes in order to obtain an effective description of the long-distance modes One must allow for nonzero speed of sound and viscosity, whose form and time-dependence can be computed through the FRG The nonlinear spectrum computed through the effective theory is in good agreement with results from N-body simulations Perturbation theory seems to converge quickly for the effective theory if the UV cutoff is taken in the region 04 1 h/ Mpc N Tetradis University of Athens