Effective Description of Dark Matter As a Viscous Fluid

Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 Effective Description of Dark Matter as a Viscous Fluid 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 1.5 million galaxies)...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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/ρ0 and its spectrum

⟨δ(k)δ(q)⟩ ≡ δD(k + q)P(k).

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid 31 timation method in its entirety, but it should be equally valid. 7.3. Comparison to other results Figure 35 compares our results from Table 3 (modeling approach) with other measurements from galaxy surveys, but must be interpreted with care. The UZC points may contain excess large-scale power due to selection function effects (Padmanabhan et al. 2000; THX02), and the angular SDSS points measured from the early data release sample are difficult to interpret because of their extremely broad window functions. Only the SDSS, APM and angular SDSS points can be interpreted as measuring the large- scale matter power spectrum with constant bias, since the others have not been corrected for the red-tilting effect of luminosity-dependent bias. The Percival et al. (2001) 2dFGRS analysis unfortunately cannot be directly plotted in the figure because of its complicated window functions. Figure 36 is the same as Figure 35, but restricted to a comparison of decorrelated power spectra, those for SDSS, 2dFGRS and PSCz. Because the power spectra are decorrelated, it is fair to do “chi-by-eye” when examining this Figure. The similarity in the bumps and wiggles between

Fig. 36.— Same as Figure 35, but restricted to a comparison of decorrelated power spectra, those for SDSS, 2dFGRS and PSCz. Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions The similarity in the bumps and wiggles between the three power spectra is intriguing.

Fig. 35.— Comparison with other galaxy power spectrum measurements. Numerous caveats must be borne in mind when interpreting this figure. Our SDSS power spectrum measurements are those from Figure 22, corrected for the red-tilting effect of luminositydependent bias. The purely angular analyses of the APM survey (Efstathiou &Moody2001)andtheSDSS(thepointsarefromTegmarket al. Fig. 37.— ∗ Comparison of our results with other P (k)constraints. 2002 for galaxies in the magnitude range 21 reionization optical depth τ), so requiring consistency with from the analysis of HTX02, and like the PSCz points (HTP00) and SDSS constrains these cosmological parameters without ass. umptions...... the UZC points (THX02) have not been corrected for this effect, about the primordial power spectrum. This figure is for. the. ca. se. of. a...... whereas the Percival et al. 2dFGRS analysis should be unafflicted “vanilla” flat scalar scale-invariant model with Ωm =0.28, h =0.72 N. Tetradis University of Athens by such red-tilting. The influential PD94 points (Table 1 fromPea- and Ωb/Ωm =0.16, τ =0.17 (Spergel et al. 2003; Verde et al. 2003, cock & Dodds 1994), summarizing the state-of-the-artEffective a Description decade ago, of Dark MatterTegmark as a Viscouset al. Fluid2003b), assuming b∗ =0.92 for the SDSS galaxies. are shown assuming IRAS bias of unity and the then fashionable density parameter Ωm =1. Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Figure: Matter power spectrum in the range of baryon acoustic oscillations.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 = atoday/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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 ≃ 0.05 − 0.2 h/Mpc): Mildly nonlinear regime of perturbation theory. Higher-order corrections dominate for k ≃ 0.3 − 0.5 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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Lessons from the functional renormalization group

Coarse-graining: Integrate out the modes with k > km and replace them with effective couplings in the low-k theory.

Wetterich equation for the coarse-grained effective action Γk [ϕ]: [ ] ( ) ∂Γ [ϕ] 1 −1 ∂Rˆ k = Tr Γ(2)[ϕ] + Rˆ k . ∂t 2 k k ∂ ln k

For a standard kinetic term and potential Uk [ϕ], with a sharp cutoff, the ﬁrst step of an iterative solution gives ∫ 1 Λ d d q ( ) (ϕ) = (ϕ) + 2 + ′′(ϕ) . Ukm V d ln q V 2 km (2π) The low-energy theory contains new couplings, not present in the tree-level action. It comes with a UV cutoff km.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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.

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Scales

kΛ ∼ 1 − 3 h/Mpc (length ∼ 3 − 10 Mpc): The ﬂuid description becomes feasible.

Scales k > kΛ correspond to virialized structures, which are essentially decoupled.

km ∼ 0.5 − 1 h/Mpc (length ∼ 10 − 20 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 > km. The form of the power spectrum ∼ k −3 implies that the effective viscosity is dominated by k ≃ km.

km 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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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) = f0(p)[1 + δf (x, p, pˆ, τ)] can be expanded in Legendre polynomials: ∑∞ ˆ − n [n] ˆ · ˆ δf (k, p, p, τ) = ( i) (2n + 1)δf (k, p, τ)Pn(k p). n=0 The Vlasov equation leads to: [n] [ ] dδ n + 1 n − f = kv δ[n+1] − δ[n 1] , n ≥ 2, dτ p 2n + 1 f 2n + 1 f

with vp = p/am the particle velocity. [n] ∼ H The time τ available for the higher δf to grow is 1/ . A ﬂuid description is possible for kvp/H ∼< 1. Estimate the particle velocity from the ﬂuid velocity v at small

length scales...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

At the comoving scale k, the linear evolution indicates that (θ/H)2 ∼ k 3PL(k),

with θ = ⃗k⃗v and PL(k) the linear power spectrum. −3 The linear power spectrum scales roughly as k above ∼ km. k 3PL(k) is rougly constant, with a value of order 1 today. Its time 2 dependence is given by DL , with DL the linear growth factor. If the maximal particle velocity is identiﬁed with the ﬂuid velocity at the scale km, we have H vp ∼ DL. km [n] The dimensionless factor characterizing the growth of higher δf is kvp/H ∼ DLk/km. Scales with k ≫ km/DL require the use of the whole Boltzmann hierarchy.

In practice, the validity of the ﬂuid description. extends...... beyond...... km...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

The shear viscosity is estimated as η/(ρ + p) ∼ lfreevp, with lfree the mean free path.

For the effective viscosity we can estimate lfree ∼ vp/H, with H = H/a. In this way we obtain

η H2 H eff H ∼ ∼ 2 νeff = lfreevpH 2 DL . (ρ + p)a km There is also a nonzero speed of sound. The linearized treatment of the Boltzmann hierarchy gives

3 ν H = c2 eff 5 s on the growing mode.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Plan of the talk

Basic formalism Determination of the effective viscosity Calculation of the spectrum Conclusions

S. Floerchinger, N. T., U. Wiedemann arXiv:1411.3280[gr-qc], Phys. Rev. Lett. 114: 9, 091301 (2015) D. Blas, S. Floerchinger, M. Garny, N. T., U. Wiedemann arXiv:1507.06665[astro-ph.CO], JCAP 1511, 049 (2015) S. Floerchinger, M. Garny, N. T., U. Wiedemann arXiv:1607.03453[astro-ph.CO], JCAP 1701 no.01, 048 (2017)

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Covariant hydrodynamic description of a viscous ﬂuid Work within the ﬁrst-order formalism. Energy-momentum tensor:

µν µ ν µν µν T = ρu u + (p + πb)∆ + π .

ρ: energy density p: pressure in the ﬂuid rest frame πb: bulk viscosity µν µν µ π : shear viscosity, satisfying: uµπ = π µ = 0 ∆µν projector orthogonal to the ﬂuid velocity: ∆µν = gµν + uµuν New elements: ρ Bulk viscosity: πb = −ζ∇ρu Shear viscosity tensor: ( ) 1 1 πµν = −2ησµν = −2η (∆µρ∇ uν + ∆νρ∇ uµ) − ∆µν (∇ uρ) . 2 ρ ρ 3 ρ ...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Dynamical equations

Einstein equations: Gµν = 8πGN Tµν , µν Conservation of the energy momentum tensor (∇ν T = 0):

µ µ µ 2 µν u ∇µρ + (ρ + p)∇µu − ζ (∇µu ) − 2ησ σµν = 0 µ∇ ρ ρµ∇ ρ ∇ µν (ρ + p + πb)u µu + ∆ µ(p + πb) + ∆ ν µπ = 0.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Subhorizon scales Ansatz for the metric: [ ] ds2 = a2(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 uµ = dx µ/ −ds2 can be expressed through the coordinate velocity v i = dx i /dτ and the potentials Φ and Ψ: 1 uµ = √ (1, ⃗v). a 1 + 2Ψ − (1 − 2Φ)⃗v 2 Neglect vorticity and consider the density δ = δρ and velocity ρ0 ⃗ θ = ∇⃗v ﬁelds. Combine them in a doublet φ1,2 = (δ, −θ/H), where H = a˙ /a is the Hubble parameter. The spectrum is

⟨φa(k, τ)φb(q, τ)⟩ ≡ δD(k + q)Pab(k, τ).

P(k, τ) ≡ P11(k, τ)...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Convergence of perturbation theory (no viscosity)

103 linear ideal fluid 102 D 3 L 1 h

10 H L

Mpc SPT 1 L H 0 @ 10 L 0 = z

, -1

k 10 H P SPT 2 L 10-2

10-3 10-2 10-1 100 101 k h Mpc

Figure: Linear power spectrum and the one- and two-loop corrections in standard perturbation theory (SPT) at z @= 0. D ...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Higher-order (loop) corrections dominate for k ∼> 0.3 − 0.5 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 ∫ ∞ ∫ ∞ 2 4π L 4π 2 L σd (η) = dq P (q, η) = DL (η) dq P (q, 0), 3 0 3 0

with η = ln a = − ln(1 + z) and DL(η) the linear growth factor: δ(k, η) = DL(η)δ(k, 0) on the growing mode. For the spectrum, the complete expression is (Blas, Garny, Konstandin 2013) ( ) 61 25 − 1 loop − 105 21 2 2 L Pab (k, η) = 25 9 k σd P (k, η). 21 5

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Effective description (in terms of pressure and viscosity) D. Blas, S. Floerchinger, M. Garny, N. T., U. Wiedemann arXiv:1507.06665[astro-ph.CO], 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 ∼> km and replace them with effective couplings (viscosity, pressure), determined in terms of

∫ ∞ ∫ ∞ 2 4π L 4π 2 L σdk (η) = dq P (q, η) = DL (η) dq P (q, 0). 3 km 3 km

For a spectrum ∼ 1/k 3, the integral is dominated by the region near km. The deep UV does not contribute signiﬁcantly.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Parameters of an effective viscous theory

ρ(x, τ) = ρ0(τ) + δρ(x, τ) p(x, τ) = 0 + δp(x, τ) δρ δ = θ = ∇⃗ ⃗v ρ0 a˙ 1 H = H = H a a δ H2 ηH H2 2 p H 3 cs (τ) = = αs(τ) 2 ν(τ) = = αν (τ) 2 . δρ km ρ0a 4 km

with αs, αν = O(1) today.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 H2/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 2 2 2 We assume that cs , νH are of order H /km. Keeping the dominant terms, we obtain

δ˙ + ∇⃗ ⃗v + (⃗v∇⃗ )δ + δ∇⃗ ⃗v = 0 ⃗˙ + H⃗ + (⃗∇⃗ )⃗ + ∇⃗ Φ + 2( − δ)∇⃗ δ v v v v ( cs 1 ) 1 −ν(1 − δ) ∇2⃗v + ∇⃗ (∇⃗ ⃗v) = 0. 3 3 ∇2Φ = Ω H2δ. 2 m

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Use Fourier-transformed quantities to obtain ∫ ˙ 3 3 δk + θk + d p d q δ(k − p − q) α1(p, q) δp θq = 0 ( ) ( ) ˙ H 4 2 3 H2 − 2 2 θk + + νk θk + Ωm cs k δk ∫ 3 2 + d 3p d 3q δ(k − p − q) ( )

β1(p, q) δp δq + β2(p, q) θp θq + β3(p, q) δp θq = 0, with (p + q)q α (p, q) = 1 q2 2 β1(p, q) = cs (p + q)q (p + q)2p · q β (p, q) = 2 2p2q2 4 β3(p, q) = − ν(p + q)q...... 3 ...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Deﬁne the doublet     δk(τ) φ1(k, η)     =   , θ (τ) φ (k, η) − k 2 H

where η = ln a(τ). The evolution equations take the form ∫ 3 3 ∂ηϕa(k) = −Ωab(k, η)φb(k)+ d p d q δ(k−p−q)γabc(p, q, η) φb(p) φc(q),

where ( ) 0 −1 ′ Ω(k, η) = − 3 k 2 H k 2 . Ωm + αs 2 1 + H + αν 2 2 km km

and a prime denotes a derivative with respect to η. The nonzero elements of γabc are expressed in terms of α1(p, q), β1(p, q), β2(p, q), β3(p, q)...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

The evolution of the spectrum

Deﬁne the spectra, bispectra and trispectra as

⟨φa(k, η)φb(q, η)⟩ ≡δD(k + q)Pab(k, η)

⟨φa(k, η)φb(q, η)φc(p, η)⟩ ≡δD(k + q + p)Babc(k, q, p, η)

⟨φa(k, η)φb(q, η)φc(p, η)φd (r, η)⟩ ≡δD(k + q)δD(p + r)Pab(k, η)Pcd (p, η)

+δD(k + p)δD(q + r)Pac(k, η)Pbd (q, η)

+δD(k + r)δD(q + p)Pad (k, η)Pbc(q, η)

+δD(k + p + q + r)Qabcd (k, p, q, r, η).

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 ∂ P (k, η) = −Ω P (k, η) − Ω P (k, η) η ab ∫ ac cb bc ac [ 3 + d q γacd (k, −q, q − k)Bbcd (k, −q, q − k) ] +γbcd (k, −q, q − k)Bacd (k, −q, q − k) ,

∂ηBabc(k, −q, q − k) = −Ωad Bdbc(k, −q, q − k) − Ωbd Badc(k, −q, q − k) −Ω B (k, −q, q − k) cd∫ abd [ 3 +2 d q γade(k, −q, q − k)Pdb(q, η)Pec(k − q, η)

+γ (−q, q − k, k)P (k − q, η)P (k, η) bde dc ea ] +γcde(q − k, k, −q)Pda(k, η)Peb(q, η) .

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Alternative approach

Expand the ﬁelds in powers of the initial perturbations at η = η0, ∑ ∫ ∑ 3 3 3 (3) ϕa(k, η) = d q1 ··· d qn (2π) δ (k − qi ) n i × ··· Fn,a(q1,..., qn, η)δq1 (η0) δqn (η0) . From the equation of motion we can get evolution equations for the kernels Fn,a

(∂ηδab + Ωab(k, η))Fn,b(q1,..., qn, η) = ∑n γabc(q1 + ··· + qm, qm+1 + ··· + qn) m=1

×Fm,b(q1,..., qm, η)Fn−m,c(qm+1,..., qn, 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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Matching the perfect-fluid and viscous theories For the matching one could use the propagator ⟨ ⟩ δϕ ( , η˜) ′ (3) − ′ a k Gab(k, η,˜ η˜ )δ (k k ) = ′ ′ , δϕb(k , η˜ ) Define appropriate fields for the background to be effectively Einstein-de Sitter to a very good approximation: DL(˜η) = exp(˜η). The one-loop propagator of the perfect-fluid theory is ( ) ′ 61 61 ′ − ′ − 2 η˜−η˜ 2 350 525 Gab(k, η,˜ η˜ ) = gab(˜η η˜ ) k e σd (˜η) 27 9 , 50 25 where the linear propagor for the growing mode is ( ) η˜−η˜′ ′ e 3 2 g (˜η − η˜ ) = . ab 5 3 2 The contribution from k > k can be isolated by taking m ∫ π ∞ 2 −→ 2 ≡ 4 L σd (˜η) σdk (˜η) exp(2η˜) dq P (q, 0) . 3 ...... km...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Compute the propagator of the viscous theory at linear order, for kinematic viscosity and sound velocity of the form η H2 δ H2 H H 2η˜ 2 p 3 2η˜ ν = = βν e 2 , cs = = βs e 2 . ρ0 a km δρ 4 km The propagator contains a contribution ( ) 2 3η˜ − k e 3 2 δgab(k, η˜) = 2 (βν + βs) km 45 9 6 in addition to the perfect-ﬂuid linear contribution (for η˜ ≫ η˜′). ∼ 2 2 ∼ 2 H Identify the linear contribution k cs and k ν with the ∼ 2 2 one-loop correction k σdk of the perfect-ﬂuid propagator. This can be achieved with 1% accuracy, and gives 27 β + β = k 2 σ2 (0). s ν 10 m dk Solve for the nonlinear spectrum in the effective viscous theory

with an UV cutoff km in the momentum integrations...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Particular features

There are no free parameters in this approach.

The results are independent of the ratio βν /βs to a good approximation. They depend mainly on the value of βν + βs. 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.

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P∆∆ k,z=0 , km =0.6h Mpc ç

ç æ ç ç æ ç lin viscous æ 1.6 H L ç æ ç æ 1 L viscous ç L ç ç æ

z ç

, æ 2 L viscous ç k æ

H H L 1.4 æ æ ç æ

ideal H L , æ Αs ΑΝ=1 ç lin ç ç ç æ P H L ç 1.2 æ L æ z ç æ æ , æ

k ç H ç ç ç æ

P ç æ æ æ æ æç æç ç æ 1.0 æç æç æç æç æç æ

0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 k h Mpc

Figure: Power spectrum obtained in the viscous theory for redshift z = 0, @ D normalized to Plin,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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆∆ k,z=0.375 , km =0.6h Mpc

lin viscous 1.6 H L æ ç ç ç 1 L viscous ç ç æ

L ç æ

z æ

, æ 2 L viscous ç k æ H 1.4 H L ç æ ç ç ç æ

ideal H L ç , Α Α = æ s Ν 1 æ æ lin ç æ

P H L 1.2 ç æ L ç ç ç

z ç æ

, æ æ æ æ

k ç

H æ ç ç ç ç

P æ æç æ æ æ 1.0 æç æç æç æç æç æç æç æç

0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 k h Mpc

Figure: One-loop spectrum in the effective theory at z = 0.375. @ D

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆∆ k,z=0.833 , km =0.6h Mpc

lin viscous 1.6 H L ç 1 L viscous L z , æ 2 L viscous k

H H L 1.4 çæ æç ç æç ç æ ideal H L , Α Α = ç æ s Ν 1 æ lin ç ç ç ç æ P H L ç æ æ 1.2 æ L ç æ

z æ

, ç ç ç ç æ

k ç æ æ æ H ç æ ç æ P ç æç æç æ æç æ 1.0 æç æç æç æç æç æç æç æç

0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 k h Mpc

Figure: One-loop spectrum in the effective theory at z = 0.833. @ D

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆∆ k,z=1.75 , km =0.6h Mpc

lin viscous 1.6 H L ç 1 L viscous L z , æ 2 L viscous k

H 1.4 H L

ideal H L , Αs ΑΝ=1 lin

P H L

æ 1.2 ç æç æç ç L æç æ

z æç

, æç ç æç æç æç k ç æ H ç æ æç æç æç æç æ P æç æç 1.0 æç æç æç æç æç æç æç æç æç æç æç æç

0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 k h Mpc

Figure: One-loop spectrum in the effective theory at z = 1.75. @ D

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆∆ k,z=0 çó í çóí æà ó óí í ç ò ç à óí æ á ç á áà ìá à æ ì ò ò ì 2 L viscous á æ óí à ì 1.6 H L ç æ ò á à ì á ç í ó 1 L viscous ó æ ò çí à ì

L ó ò óí çóí çí á æ z ç à ì , à á á á km=0.4 h Mpc çóí ò k á æ

H H L à ì 1.4 à æ ò óí à æ ì æ = çá æ ò km 0.6 h Mpc à ìò ideal H L ì , æ ò çáóí ìò lin ì ó à km=0.8 h Mpc çáí çáóí çáóí æ P çáóí à ìò 1.2 æ L à à à æ ìò z ò = çáóí à ìæ ìæ ì , km 1.0 h Mpc ò ò ò ìæò k çóí à

H á çáóí çáóí çáóí ìæ à ò P çáóí à à æ æà ìæò ìæò ìæà ìò ìæçàáòóí ìæçàáòóí çáóí çáóí óí çáóí ìò ò 1.0 ìæàò ìæàò ìæçàáò ìæçàáòóí ìæçàáòóí ìæàò Αs ΑΝ=1 lin viscous 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 H L k h Mpc

Figure: One-loop spectrum in the effective theory at z = 0 for various values @ D of km.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆∆ k,z=0.375

à æ ì ò 2 L viscous 1.6 H L óí ó çæ á ç í ó óí çí à 1 L viscous çóí ç æà ò ó ì ìá L í à ç á ìæá áò z á æà ò , à ì km=0.4 h Mpc çóí à ò k á ìæ H 1.4 H L à ò çóí æ á ì æ k =0.6 h Mpc çóí çóí çóí à ò m á á á ìæ ideal H L çóí à ò , á à à ìæ æ ìæò ò lin ì çóí à ìò km=0.8 h Mpc á ìæò P à 1.2 çáóí ìæ

L ò çáóí çáóí çáóí à z ò = çáóí à à à ìæò , km 1.0 h Mpc æ æ æ æà ìò ìò ìò k çáóí ìò H çáóí ìæàò çáóí çáóí çáóí à P çáóí ìæàò ìæà à ìæò çáóí ìæàò ò ìæò 1.0 ìæçàáòóí ìæçàáòóí ìæçàáòóí ìæçàáòóí ìæçàáòóí ìæçàáòóí ìæçàáòóí ìæàò Αs ΑΝ=1 lin viscous 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 H L k h Mpc

Figure: One-loop spectrum in the effective theory at z = 0.375 for various @ D values of km.

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P∆∆ k,z=0 PN-body, viscous theory

1.2 à æ ì ò 2 L viscous á ç í ó 1 L viscous ó à H L í km=0.4 h Mpc ç óí ó ó ó í ó í í ç ç í ç L æ ç k =0.6 h Mpc ç á z m ó á á , H L H çí Lá á

k á

H 1.1 ì = óí á km 0.8 h Mpc ó ç à à çóí çóí çí á çóí á á á body à ò á à æ - k =1.0 h Mpc m çáóí æ

N æ çáóí à ì ì P çóí çáóí à æ ó á à à ì ò L çáí ìæçàáòóí çáóí à æ ò z çáóí à æ ò , ó à æ æ ì çáí çáóí à à à à æ ì ò k çáóí ìæà à à à à à æ æ ì H ò æ ì ì 1.0 ìæçàáòóí çáóí ìæà ìæò ìæ ìæ æ ì æ æ æ æ ì ò ò ìæçàáòóí ìæàò ò ò ò ìò ò ìò ì ì ì ì ì ò ò ò P ò ò ò ò ò

lin viscous

0.9 0.00 0.05 0.10 0.15 0.20 0.25 H L k h Mpc

Figure: Comparison of results for the power spectrum obtained within the viscous theory normalized to the N-body@ resultD at z = 0. The grey band corresponds to an estimate for the error of the N-body result.

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P∆∆ k,z=0 PN-body, SPT with cutoff

1.2 2 L SPT 1 L SPT H L L=0.4 h Mpc

L L=0.6 h Mpc z , H L H L k H 1.1 L=0.8 h Mpc body

- L=1.0 h Mpc

N P

L®¥ L z , k H 1.0 P

lin SPT 0.9 0.00 0.05 0.10 0.15 0.20 0.25 kH h MpcL

Figure: One- and two-loop results in standard perturbation theory (shown as dashed and dotted lines, respectively),@ computed D with various values of an ad-hoc cutoff Λ (coloured lines), as well as in the limit Λ → ∞ (black lines).

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PΘΘ k,z=0 P∆∆, viscous theory

lin viscous ìçæáàóòí ìçæáàóòí 1.0 ìçæáàóòí ìçæáàò óí ìæàò çáóí ìæàò H L çáóí ìæò áà çóí ìæàò çáóí ìæò áà çóí ìæàò H L

L á çóí ìæàò z 0.8 çáóí ìæò , à çáí ì k ó æáàò H çóí ìçæàáòí ∆∆ ó çáí ìæàòó çáí P ó çá ìæàò óí á L æà ç ìò óí çá z ìæà í çá á , ò ó óí çí á 0.6 æà ó çí k ìò ó çáí H ó á á ç í ó æà çí çá ç 1 L viscous ì ó óí áí ΘΘ ò à ó ìæ P à æ ì ò 2 L viscous ò à ìæ à ò ìæ à à km=0.4 h Mpc ò æ à ìò æ ì à 0.4 æ km=0.6 h Mpc ò æ H L ì ì k =0.8 h Mpc N-body ò m H L J12 ò km=1.0 h Mpc - N bodyHAA14 0.05 0.10 0.15 0.20 0.25 k h Mpc

Figure: The velocity-velocity spectrum obtained within the viscous theory, compared to results from N-body simulations@ D at z = 0. The pink band corresponds to a 10% error for the N-body results.

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PΘΘ k,z=0 P∆∆, SPT with cutoff

1.0 lin SPT H L L

z 0.8 H L , k H ∆∆ P L z , 0.6 1 L SPT k H 2 L SPT ΘΘ

P L=0.4 h Mpc L=0.6 h Mpc L=0.8 hHMpcL 0.4 H L L=1.0 h Mpc N-body J12 L®¥ - N bodyHAA14 0.05 0.10 0.15 0.20 0.25 k h Mpc

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆Θ k,z=0 P∆∆, viscous theory

ìçæáàóòí lin viscous 1.0 ìçæáàóòí ìçæáàóòí ìçæáàóòí ìçæáàóòí ìçæáàóòí H L ìçæáàóòí ìçæáàóòí ìçæáàóòí 0.9 ìçæàáóòí ìæçàáóòí L ìæçàáóòí H L z ìæçàáóí , ò çá óí çá k ìæàò óí çá H óí à çáí ìæò ó çá ∆∆ 0.8 í æà ó çáí çá ìò à ó óí çáóí P ìæò çáóí

ç á ç í ó æà áóí ç L 1 L viscous ìò áóí ç æà áóí çáóí z ìò , à æ ì ò æà k 0.7 2 L viscous ìò à H ìæ à ∆Θ à ò æ à k =0.4 h Mpc ìò æ à P m ì ò æ à H L ìò æ à æ = ì 0.6 km 0.6 h Mpc ò æ ì H L ò ì km=0.8 h Mpc N-body J12 0.5 ò km=1.0 h Mpc N-body HAA14 0.05 0.10 0.15 0.20 0.25 k h Mpc

Figure: The density-velocity spectrum obtained within the viscous theory, compared to results from N-body simulations@ D at z = 0. The pink band corresponds to a 10% error for the N-body results.

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

P∆Θ k,z=0 P∆∆, SPT with cutoff

1.0 H L lin SPT 0.9 L z ,

k H L H 1 L SPT ∆∆ 0.8 P

L 2 L SPT z ,

k 0.7 L=0.4 h Mpc H

∆Θ L=0.6 hHMpcL P H L 0.6 L=0.8 h Mpc L=1.0 h Mpc N-body J12 0.5 L®¥ N-body HAA14 0.05 0.10 0.15 0.20 0.25 k h Mpc

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

We guessed H 3 H2 2 H2 < ν(τ) = αν (τ) 2 , cs (τ) = αs(τ) 2 , with αν , αs ∼ 1. 4 km km At one loop we found H 3 2η H2 2 2η H2 27 2 2 ν = βν e 2 , cs = βs e 2 , with βs + βν = kmσ (0). 4 km km 10 dk We generalize the framework by considering H 3 κ(k)ηH2 2 κ(k)ηH2 ν = 4 λν (k) e , cs = λs(k) e . The scale dependence of the parameters can be determined through renormalization-group methods. This requires a functional representation of the problem.

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Functional representation S. Floerchinger, M. Garny, N. T., U. Wiedemann arXiv:1607.03453[astro-ph.CO], JCAP 1701 no.01, 048 (2017) We follow and extend Matarrese, Pietroni 2007. We are interested in solving − ∂ηϕa(k) = Ω∫ab(k, η)ϕb(k) 3 3 (3) + d p d q δ (k − p − q)γabc(p, q, η) ϕb(p) ϕc(q)

with stochastic initial conditions determined by the primordial 0 power spectrum Pab(k). This can be achieved by computing the generating functional ∫ { 1 Z[J, K ; P0] = DϕDχ exp − χ (0)P0 χ (0) 2 a ab b ∫ }

+ i dη [χa(δab∂η + Ωab)ϕb − γabcχaϕbϕc + Jaϕa + Kbχb] ,

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

One can now deﬁne the generating functional of connected Green’s functions

W [J, K ; P0] = −i log Z[J, K ; P0].

The full power spectrum Pab and the propagator Gab can be obtained through second functional derivatives of W ,

2 δ W ′ ′ = iδ(k − k ) Pab(k, η, η ) , δJ (−k, η) δJ (k′, η′) a b J, K =0 2 δ W ′ R ′ = −δ(k − k )G (k, η, η ) , δJ (−k, η) δK (k′, η′) ab a b J, K =0 2 δ W − ′ ′ = 0 . δKa( k, η) δKb(k , η ) J, K =0

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

The effective action is the Legendre transform ∫ 0 3 0 Γ[ϕ, χ; P ] = dηd k {Jaϕa + Kbχb} − W [J, K ; P ],

where ϕa(k, η) = δW /δJa(k, η), χb(k, η) = δW /δKb(k, η). The inverse retarded propagator satisﬁes ∫ ′ R ′ R ′ ′′ − ′′ dη Dab(k, η, η )Gbc(k, η , η ) = δac δ(η η ) .

It can be computed from the effective action

2 δ Γ = 0 , δϕ (−k, η) δϕ (k′, η′) a b J, K =0 2 δ Γ ′ R ′ = −δ(k − k )D (k, η, η ) , δχ (−k, η) δϕ (k′, η′) ab a b J, K =0 2 δ Γ − − ′ ′ − ′ ′ = iδ(k k )Hab(k, η, η ) δχa( k, η) δχb(k , η ) J, K =0 ...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

“Renormalized” field equations δ Γ[ϕ, χ] = Ja(x, η), δϕa(x, η) δ Γ[ϕ, χ] = Ka(x, η), δχa(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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Renormalization-group improvement

Modify the initial power spectrum so that it includes only modes with wavevectors |q| larger than the coarse-graining scale k:

0 0 | | − Pk (q) = P (q) Θ( q k).

The coarse-grained effective action satisﬁes the Wetterich equation {( ) } 1 ( ) −1 ∂ Γ [ϕ, χ] = Tr Γ(2)[ϕ, χ] − i P0 − P0 ∂ P0 . k k 2 k k k k

...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Use an ansatz of the form [ ∫ ∫ ( ) 3 ˆ Γk [ϕ, χ] = dη d q χa(−q, η) δab∂η + Ωab(q, η) ϕb(q, η) ∫ 3 3 3 (3) − d k d p d q δ (k − p − q)γabc(k, p, q)χa(−k, η)ϕb(p, η)ϕc(q, η) ∫ ] i − d 3q χ (q, η)H (q, η, η′)χ (q, η′) + ... , 2 a ab,k b

where ( ) 0 −1 Ω(ˆ , η) = ′ . q − 3 κ(k)η 2 H κ(k)η 2 2 Ωm + λs(k) e q 1 + H + λν (k) e q Derive differential equations for the k-dependence of λ (k), λ (k), κ(k). ν s ...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

Comments A prescription is needed in order to project the general form of the inverse retarded propagator R ′ ′ − ′ − ′ R ′ Dab(q, η, η ) = δabδ (η η ) + Ωab(q, η)δ(η η ) + Σab(q, η, η ) to the form ( ) R − ˆ − ′ Dab(q, η, ∆η) = δab∂η Ωab(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 31 78 λ (k) = σ2 , λ (k) = σ2 , κ(k) = 2. s 70 dk ν 35 dk This validates our intuitive matching through the propagator.

At the next level, the k-dependence of λν (k), λs(k), κ(k) can be derived...... N. Tetradis University of Athens Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

20 2.2 80 D D

2 15

2 2.1 L L h h 60

10 Κ 2.0 Mpc Mpc 40 H H @ @ s Ν Λ Λ 20 5 1.9

0 0 1.8 10-3 10-2 10-1 100 10-3 10-2 10-1 100 10-3 10-2 10-1 100 k h Mpc k h Mpc k h Mpc

@ D @ D @ D Figure: RG evolution of λν (k), λs(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.

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103

D 2 2 10 L h 101 Mpc H

@ 0

s 10 Λ + Ν -1 Λ 10

10-2 10-3 10-2 10-1 100 k h Mpc

Figure: RG evolution of the sum λ (k) + λ (k). The various lines correspond ν @ sD 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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

3.0

2.5

Κ 2.0

1.5

1.0 10-3 10-2 10-1 100 k h Mpc

Figure: RG evolution of the power law index κ(k) characterizing the @ D 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 Effective Description of Dark Matter as a Viscous Fluid Motivation Framework Perturbation theory Effective viscosity Results FRG improvement Conclusions

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 0.4 − 1 h/ Mpc.

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