Viscosity in cosmic fluids

We consider the fluid description of Dark-matter in the presence of dissipation. In the standard scenario, dark-matter is considered to be cold and collisionless, in the effective fluid description the dissipative effects arise when the nonlinearity is present due to gravitational coupling. We show that the bulk and shear viscosities due to both the nonlinearity and the self-interaction are inversely added. Further, we discuss the consequences of our results.


I. INTRODUCTION
In order to study large scales structures in the Universe there are two important length-scales: One is comoving Hubble scale H −1 over which Universe is homogeneous and isotropic and the another is the non-linear scale k −1 NL ∼ a few Mpc, describing the scales at which gravitational collapse takes place. The dynamics of the perturbations can be analyzed in terms of a parameter ǫ k = k NL /k, where k is the inverse length scale. The hierarchy between these two scales is quantified by parameter ǫ k ≫ 1 which is responsible for the success of linear perturbation theory in describing the observed large scale structures (LSS) (for a recent review see [1] and also [2]). The dark energy (cosmological constant Λ) plus cold dark-matter (CDM) model, (i.e. ΛCDM) is highly successful in predicting the large scale structure of the Universe. The model is consistent with observations from the length scales typically of the order of ∼1 Mpc (i.e. intergalactic scale) to the scale of the horizon (∼15000 Mpc) [1]. In this model structure formation in the dark-matter ( DM ) sector occurs more rapidly than the baryonic matter. The structure formation in the dark sector provides the gravitational potential for the baryonic matter and hence gives the information of the distribution of visible matter in the universe. Although this model provides extensive agreement with the large scale structure and Cosmic Microwave Background (CMB) radiation observations, it faces difficulty at small length scale ( 1Mpc). These problems include 'missing satellite problem' [3,4] (prediction of too many dwarf galaxies within the viral radius of the Milky way from the N-body simulations than observed), the 'cusp-core problem' [5] (Observations shows nearly constant dark-matter density in the inner parts of galaxies but simulations shows a steeper density behavior) and the 'too-big-to-fail problem' [6,7] (from simulations it is not possible to explain the dynamics of the massive satellites in the Milky way galaxy). Especially, these problems become more * Electronic address: jeet@prl.res.in † Electronic address: pravin@prl.res.in ‡ Electronic address: arunp77@gmail.com evident in studying the galaxy rotation curve [1,8].
There has been attempts to address some of these issues within ΛCDM and also by modifying the ΛCDM model (see the review [1] and references therein). One of the interesting proposals to resolve the issues related with the small scales is by introducing self-interaction between dark-matter particles. Such models are called self-interacting dark matter (SIDM) models. In these models typical mean free path of dark matter particle is taken to be in the range of 1 kpc to 1 Mpc, proposed as a remedy for tension between observations and numerical simulations at scale of a few Mpc (ǫ k ≪ 1) [9][10][11]. Inclusion of interaction can introduce dissipation in the dark-matter fluid and one can define coefficients of bulk and shear viscosities [12]. This small scale physics can affect the large scale behavior of the Universe-it has been shown that the viscous effect can lead to an accelerated expansion of the Universe [12][13][14][15][16][17]. Further, dissipative dynamics of dark matter can resolves the tension between Planck CMB and LSS observations [18]. In other scenarios, viscous cosmology can also be used for constraint the neutrino mass [19]. It also explains the cosmic chronometer and type Ia supernova data [20,21]. Dissipative effect may arise due to dark matter-baryon interaction also. Recently a systematically inclusion of baryon-DM interaction has been incorporated in the Boltzmann-Fokker-Planck formalism [22,23]. It ought to be noted that the baryon-DM interaction has also been considered in the literature to explain 21-cm line [24][25][26]. The damping of the gravitational waves in the viscous fluid can be used to constraint the mean free path and the DM mass [27,28].
We would like to note that the dissipative effects may arise even for the case of cold-collisionless dark matter (CCDM) in presence of self-gravity. In Ref. [2] effective fluid theory of long-wavelength universe was obtained by integrating out the short-wavelength perturbations. The effective fluid behaves as a viscous medium coupled to gravity. Here the short wavelength contributes to the viscous stress tensor of the DM fluid which depends on the gravitational potential. The effective fluid description of CCDM is based on truncation of the Boltzmann hierarchy [2,29]. This stress tensor can potentially change the bias parameters in the galaxy bispectrum [30]. The perturbations contributing to the background in the viscous effective fluid may affect the baryon acoustic oscilla-tion [2,31].
In the present work we consider the Boltzmann kinetic equation in the relaxation time approximation to obtain the effective fluid description for the dark matter particles. We consider two relaxation times in our scheme: first relaxation time which is inspired by the effective fluid considered in case of CCDM [2] and the second relaxation time is based on the cross-section for SIDM [12]. Thus one can allow for more than one sources of viscosity in dark-matter fluid. In such a situation, different viscosities can combine in a particular way. For example, in quark-gluon plasma shear viscosity η A due to turbulence and kinetic viscosity η c combined to give effective shear viscosity η eff as 1/η eff = 1/η A + 1/η c [32]. Using the relaxation times we show that two different viscosity sources combine in the above way. We believe that this additional contribution to the viscosity can significantly alter the dynamics of the dark-matter fluid and provide useful insight into long wavelength dynamics of dark-matter fluid.
The present work is divided in following sections: Section [II], contains the fluid approximation for the collisionless cold dark-matter in the presence of self-gravity; in section [III], we have calculated the relaxation times for collisionless cold dark-matter and self-interacting dark-matter; Spatial perturbation in the Maxwell-Boltzmann (MB) distribution of darkmatter fluid is discussed in section [IV]; in section [V], we get the shear and bulk viscosity for cosmic fluid. finally we have given results obtained in present work and a brief conclusion in sections [VI]. All Latin indexes in the manuscript represents the spatial indices.

II. FLUID APPROXIMATION FOR CCDM
In this section, we consider an identical, non-relativistic, collisionless cold dark matter particles, coupled gravitationally with each other. Dynamics of phase-space distribution of the particles can be described by Boltzmann Equation [29] ∂ f ∂τ where f ≡ f (x i , τ, q,q i ) is the phase-space distribution and I c represents the collisions between particles. Here variables are: comoving position of the particle x i , comoving-time τ, comoving-momentum q, andq is the unit vector along q. d 3 q I c = 0, leads to the total conservation of phase-space distribution.
In the presence of anisotropies and inhomogeneities, distribution function can be written as where back-ground distribution depends only on time and momentum-amplitude, and Ψ(x, τ, q) is the first order perturbation in phase-space distribution which depends on position, momentum and time. For the length-scale ǫ k ≫ 1, the DM consistent with the ΛCDM is nonrelativistic and noninteractive matter (CCDM), for which zeroth order distribution func-tion can be written as [33] where a(τ) is the scale factor, m is the mass of DM particles and T (τ) is temperature of the CCDM, scales as a −2 (τ). Hence i.e. only depend on particle's comoving momentum ( q ≡ a(τ) p, where |p| ∝ 1/a(τ), is particle's momentum). We have considered the line element in the conformal Newtonian gauge as [34] where φ ≡ φ(x, τ) and ψ ≡ ψ(x, τ) are scalar perturbations and corresponds to the Newtonian potential and the perturbation to the spatial curvature respectively. Since ∂ f /∂q i and dq i /dτ, both are the first-order quantity, we can neglect the last term of L.H.S up-to the first-order contribution in the equation (1). For CCDM, the Boltzmann equation takes the form, where ǫ ≡ ǫ(q, τ) = q 2 + (am) 2 is the comoving energy of a particle [35]. Taking Fourier transformation of the linear perturbation Ψ(x, τ, q) and expanding in the form of Legendre polynomials P l , where ς =k ·q ,k i is the unit vector of k i and Ψ l (k, τ, q) are coefficients of the Legendre polynomials. We get the differential equations for moments (or Boltzmann hierarchy), where v p = q/ǫ is the particle's peculiar velocity. The time evolution of moments can be taken to the order of the Hubble time at long wavelength, Thus, it is clear that higher order moments can be written in terms of second order moment for l > 2. If the factor of Ψ 2 in equation (10) is smaller than unity (i.e. kv p H −1 ≪ 1), then it implies the fluid approximation or truncation of the Boltzmann hierarchy. Taking a bound on the maximum possible particle velocity from the velocity in the non-linear regime [2], therefore, if ǫ k ≫ 1 then kv p H −1 ≪ 1, implies fluid approximation (i.e. l max = 2). Thus, for linear scale ǫ k ≫ 1, the higher moments are suppressed (i.e. Ψ l ≪ Ψ 2 for l > 2). Ψ 1 and Ψ 2 give energy flux and shear stress respectively. This hierarchy depends on non-linear scale and it comes due to the gravitational coupling of fluid.

III. RELAXATION TIME FOR CCDM AND SIDM
In this section we will calculate the mean free-time (relaxation time) for collisionless cold dark-matter and selfinteracting dark-matter. In above section [II] we have obtained the fluid approximation for CCDM. In this section we state the equivalent relaxation time approximation for these.

A. Collisionless cold dark-matter
If τ cb is the mean free path for the CCDM fluid, equation (12) can be written as Multiplying equation (13) by f o (q), where q = v p ǫ v p a m, and taking integral over d 3 q herev p q/(a m) is the mean peculiar velocity, where n(x, τ)q = 1/a 4 d 3 q q f o (q); n(x, τ) = 1/a 3 d 3 q f o (q) is the number density, andv p τ cb is the mean free path. Therefore Here particles are gravitationally bound and during a Hubble time particles move only up-to the nonlinear scale. In the absence of gravitational coupling or non-linear scale ( k NL ), the mean free-path can be infinitely long.

B. Self-interacting dark-matter
In the above subsection we have obtained the relaxation time τ cb for CCDM in presence of nonlinearities. For the case of self-interacting dark-matter [9][10][11], relaxation time can be written as [36,37], where < ·· > represents the ensemble average, n is the number density for the particles, σ is the differential cross-section and v p = |v p | is dark-matter particle's peculiar velocity.

IV. SPATIAL PERTURBATION IN THE MB DISTRIBUTION OF DM
In the present case, the relaxation time comes from two different processes, one from the gravitational coupling of the DM particle's and other one from DM self interaction. The collision term (I c ) in equation (1), can be approximated by "relaxation time approximation". Thus for the present case, the collision term I c becomes [36,[38][39][40][41] where τ −1 eff = τ −1 cb + τ −1 si and f = f (x, τ, q) are the inverse effective relaxation time and phase-space distribution function respectively. At the lowest order approximation, we can assume f o to be the Maxwell-Boltzmann distribution where we have used the metric (4) and q · U (τ, x) = δ i j q i U j (τ, x) ; g represents the degree-of-freedom and U (τ, x) is the comoving fluid velocity. Writing f (x, τ, q) = f o (x, τ, q) + δ f (x, τ, q), where δ f (x, τ, q) is the variation from the MB distribution. The Boltzmann equation in this case, takes the form Assuming δ f ≪ f o , we can neglect δ f on the L.H.S., implies Obtained δ f ≡ δ f (x, τ, q) depends on the effective relaxation time of the fluid. In the above equation, first term is related with the heat conduction [42]. Second term defines the spatial changes in the fluid with velocity i.e. related with spatial-dissipation in the fluid. In the third term, comoving time-derivative of momentum q can be written in terms of the comoving time-derivative and spatial-derivative of the scalar perturbations φ and ψ respectively (dq/dτ = qφ − ǫq i ∂ i ψ). This term signifies the effect of the over/under-dense regimes on the fluctuations in the phase-space distribution of the DM particles. Viscosity in the fluid is defined by the spatial derivative of the fluid velocity and in the distribution function ( f o (x, τ, q)), only fluid velocity depends on spatial component. Accordingly we evaluate only the second term of the equation (21), where δ fs ≡ δ fs (x, τ, q) is the spatial perturbation in the phasespace distribution , θ = ∂ i U i is the velocity divergence and it's related with the bulk-viscosity. The quantity, in the curly bracket, is known as the shear tensor W il [42,43].

V. VISCOSITY IN THE DARK-MATTER FLUID
The stress-energy tensor for imperfect fluid can be written as, where ρ is the energy density, p is the pressure density, U is the fluid velocity, ζ is the bulk viscosity, Σ i j is the viscous stress-tensor, γ i j = g i j + U i U j and g i j is metric. Here, We are interested in the bulk-viscosity and shear-viscosity as the dissipation in DM fluid. The viscous stress-tensor defined as where η is shear viscosity. Thus the dissipative stress-energy tensor The stress-energy tensor can be described in the terms of the distribution function [29] T i j + ∆T i j = 1 a 4 q 2 dqdΩ q i q j ǫ(q, τ) here we are interested only in spatial dissipation, therefore we have taken only δ fs , and T i j is the background energymomentum tensor. Substituting equation (22) into equation (26) and comparing with equation (25), we get the expression for the effective bulk viscosity as and for the effective shear viscosity as [44,45], For the cold (non-relativistic) DM, the comoving energy (ǫ) can be approximated as ǫ −2 ≃ (am) −2 − q 2 /(a 2 m 2 ) 2 . Hence (29) and, (30) where n(x, τ) = a −3 d 3 q f o , is the number density of the DM and ρ = m n(x, τ) is the energy density of the DM. We can write equation (29) and (30) as and Where ζ SIDM and ζ CCDM are bulk-viscosities due to selfinteracting DM and gravitational coupling of DM respectively, and defined as similarly η SIDM and η CCDM are shear-viscosities due to SIDM and gravitational coupling of DM respectively, and defined as here S ≡ S (x, τ), We get the effective bulk-viscosity (31) and shear-viscosity (32) due to two different relaxation time because of two different processes, as in the reference [32], and it's inversely additive.

VI. RESULT AND DISCUSSION
In the present work we have considered the possibility where the viscosity coefficients of a dark-matter fluid can arise due to two different processes. For this purpose we have used Boltzmann equation with the effective relaxation time (18) which contains contributions from the nonlinear scale and the self-interaction between the dark-matter particles. Here we note that the relaxation times for the different processes in the Boltzmann equation are inversely additive. This leads to the expressions of the effective bulk (31) and shear (32) viscosities. In terms of relaxation time one can write the effective (shear or bulk) viscosity η eff or ζ eff ∝ ( τ cb τ si )/( τ cb + τ si ). Thus the smaller relaxation time is dominates in determining the viscous contribution. For example, when relaxation time due to the self-interaction is larger than relaxation time due to the nonlinearities, the effective viscosity is dominated by the smaller time scale i.e. nonlinearities. Consider, for example, the relaxation time for self-interacting dark-matter for a typical cluster, with size 2-10 Mpc, average velocity ∼ 1000 km/s, < σ > /m ∼ 0.1 cm 2 /g and energy density ρ ≡ mn ∼ 5 × 10 −1 M ⊙ pc −3 [12], is ∼ 3 × 10 15 sec, and relaxation time due to nonlinearities is ∼ (0.6−3.1)×10 17 sec for the redshift z = 0. Therefore at this scale self-interaction in darkmatter will determine the viscosity of cosmic fluid. However, for the galactic scale, e.g. for a typical dwarf galaxy with size ∼1 Kpc, average velocity ∼ 5 × 10 6 cm/s, < σ > /m ∼ 2 cm 2 /g and energy density ρ ≡ mn ∼ 1 × 10 −1 M ⊙ pc −3 [12], is ∼ 10 16 sec, and relaxation time due to nonlinearities is ∼ 10 14 sec for the redshift z = 0. Thus, we have shown that if the self-interaction in dark-matter are present, they can manifest themselves at the larger scales, where as the effective fluid description may remain valid on the smaller scales. We believe that our results can provide an useful insight in understanding how different dissipative effects can combine at the different length scales in doing numerical simulations.