Cosmic viscosity as a remedy for tension between PLANCK and LSS data
Sampurn Anand, Prakrut Chaubal, Arindam Mazumdar, Subhendra Mohanty
CCosmic viscosity as a remedy fortension between PLANCK and LSSdata
Sampurn Anand, a Prakrut Chaubal, a Arindam Mazumdar, a Subhendra Mohanty a a Physical Research Laboratory, Ahmedabad, 380009, IndiaE-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
Measurements of σ from large scale structure observations show a discordancewith the extrapolated σ from Planck CMB parameters using Λ CDM cosmology. Similardiscordance is found in the value of H and Ω m . In this paper, we show that the presenceof viscosity, shear or bulk or combination of both, can remove the above mentioned conflictssimultaneously. This indicates that the data from Planck CMB observation and different LSSobservations prefer small but non-zero amount of viscosity in cold dark matter fluid. Keywords: self interacting dark matter, shear viscosity, large scale structures, CMB a r X i v : . [ a s t r o - ph . C O ] O c t ontents σ - Ω m tension 84 Resolving H - Ω m tension 95 Planck-LSS combined viscous-cosmological parameters 106 Discussion and Conclusion 11 Over the past few decades, several observations have indicated that our Universe is dominatedby dark components, namely, dark matter (DM) and Dark energy (DE) [1–8]. In the light ofcurrent observations from Cosmic Microwave Background (CMB) and Large Scale Structure(LSS) observations, most favorable theoretical construct to understand the evolution of ourUniverse is provided by the Cold Dark Matter with cosmological constant, Λ , also referredas the standard model of cosmology ( Λ CDM model), which is characterized by six parametersonly. Predictions of Λ CDM cosmology have been seen successfully in the CMB observations.However, LSS observations have shown some conflicts with it. In this paper we will addressthese issues and will ameliorate these conflicts by introducing dissipative effects in the system.The standard six parameters of Λ CDM model are : the ratio of density of cold darkmatter and baryonic matter to the critical density, Ω and Ω b respectively, evaluated today,the acoustic scale Θ MC , the amplitude ( A s ) and the spectral index ( n s ) of the primordialdensity perturbations and the optical depth to the epoch of reionization ( τ reion ). The value ofHubble parameter at current epoch H is related to the acoustic scale, Θ MC . As a consequence,one of them is considered as input model parameter leaving other as a derived quantity. Theseparameters are inferred from two different observations namely CMB and LSS. This type ofindirect determination has mostly provided the value of H lower than the direct measurementfrom type-IA supernova [9]. However, before the release of Planck data [7] the inferred valueof H from CMB was in agreement with that of the LSS observations.The success of Λ CDM model lies in its capability of describing the observed quantities inlarge scales and small scales in a single theoretical framework. The primary CMB anisotropiesprovide an estimate of the amplitude of the matter fluctuations at the last scattering surface.Given a cosmological model, these primary fluctuations can be extrapolated to provide anestimate of matter fluctuations at a later time in the Universe. However, the above describedframework for Λ CDM model predicts the value of σ , r.m.s fluctuation of perturbations at8 h − Mpc scale, which is not agreement with other low-redshift observations of large-scalestructure [10–16]. These disagreements in the value of H and σ inferred from CMB and LSSobservations are typically attributed either to the signals of new physics or to the systematic– 1 –rrors. The recent result from dark energy survey has also shown the similar tension in σ − Ω m plane [8, 17], which indicates some new physics might be responsible for this mismatch.Several attempts have been made in this regard to address this discordance betweenCMB and LSS observations. It has been argued that the interaction between dark matterand dark energy [18–20] as well as dark matter and dark radiation [21–23] can resolve thistension to some extent. However, in most of the cases such models resolve one of the abovementioned tensions but fails to solve the other one. More importantly, interaction betweenthe dark sectors can also modify the scale corresponding to matter radiation equality [20]which might introduce greater problem than σ mismatch. Some other attempts have beenmade by modifying the neutrino sector [24–26]. Addition of massive sterile neutrino in thesystem has been reported to reduce some tension in σ − Ω m plane but not in H − Ω m plane [24, 25]. Recently, it has been claimed that quartessence models, where a single darkcomponent mimics both dark matter and dark energy can reduce the tension in σ − Ω m [27].It has been discussed that the viscosity in CDM has the ability of reducing power on thesmall length scales [28–30]. The effect of bulk has been investigated extensively in [29]. Onthe other hand, the effect of shear has also been investigated to some extent [31]. Attemptsto quantify the dissipative effects in dark matter has been done in ref. [32] from BaryonAcoustic Oscillation (BAO) data. For recent review on this topic, refer to [33]. The bulkviscosity suppresses the growth of structures by imparting a negative pressure against thegravitational collapse while the shear viscosity reduces the amount of velocity perturbationswhich in turn stops the growth. Therefore, on the small length scales, where the homogeneityand isotropy are broken due to velocity gradients, effects of shear viscosity is expected to playcrucial role. Although the physics of these viscosities are different, we will show that thereeffect on large scale structure is more or less similar.This paper is organized as follows: in section 2, we discuss the basic setup of viscouscosmology. This section is divided into two subsections 2.1 and 2.2. In 2.1 we outline thecosmological perturbation theory and derive the perturbation equations in presence of thetwo viscosities. Further, in 2.2 we show the effect of these viscosities on the growth of densityperturbations which in turn effects the matter power spectrum. Having discussed the effectson the matter power spectrum, we move on to perform Markov-Chain-Monte-Carlo (MCMC)analysis with data from different CMB and LSS observations. In section 3, we show thatthe inclusion of viscosities removes the tension between the Planck and LSS data in σ − Ω m plane. Similarly, in section 4 we show the concordance between Planck and LSS data in H − Ω m plane due to viscosities. We also perform a joint analysis of Planck and LSS datawith viscosities to infer the parameters of viscous cosmology in section 5. Finally, we concludeand discuss our results in section 6. Cosmological perturbations have been computed with the assumption of homogeneity andisotropy on large scale in presence of an ideal fluid [34–37]. Although violation of thesesymmetries might be prominent on small length scales, their effect might not be substantialon large scales. The reason behind it is that velocity gradient on smaller scales becomessignificant. Therefore, we go beyond the perfect fluid approximation by considering dissipativeeffects in the system. In this section we will describe the cosmological perturbation theorywith non-ideal fluid in the presence of shear and bulk viscosities.– 2 –he energy momentum tensor for the non-ideal fluid is given as [38] T µνvf = ρ u µ u ν + ( p + p b ) ∆ µν + π µν , (2.1)where ρ is the energy density and p is the pressure in the rest frame of the fluid, p b = − ζ ∇ µ u µ is the bulk viscous pressure with bulk viscosity ζ . π µν is the shear-viscous tensor which takesthe following form π µν = − η σ µν = − η (cid:20)
12 (∆ µα ∇ α u ν + ∆ να ∇ α u µ ) −
13 ∆ µν ( ∇ α u α ) (cid:21) , (2.2)with η being the shear viscosity and ∆ µν = u µ u ν + g µν being the projection operator whichprojects to the subspace orthogonal to the fluid velocity. It is evident that π µµ = 0 = u µ π µν .Conservation of energy momentum, ∇ ν T µν = 0 , leads to the viscous fluid dynamic equa-tions [28] u µ ∇ µ ρ + ( ρ + p ) ∇ µ u µ − ζ ( ∇ µ u µ ) − ησ µν σ µν = 0 , (2.3) ( ρ + p + p b ) u µ ( ∇ µ u α ) + ∆ αµ ∇ µ ( p + p b ) + ∆ αν ∇ µ π µν = 0 . (2.4) Perturbation in the matter field is related to the perturbation in the metric through Einstein’sequation and vice-versa. In general, metric tensor g µν can have scalar, vector and tensorperturbations which are independent of each other in linear order. In this analysis we consideronly scalar perturbations in the conformal-Newtonian gauge given as ds = a ( τ ) (cid:2) − (1 + 2 ψ ( τ, (cid:126)x )) dτ (1 − φ ( τ, (cid:126)x )) dx i dx i (cid:3) , (2.5)where ψ ( τ, (cid:126)x ) and φ ( τ, (cid:126)x ) are space-time dependent functions. We assume a spatially flatuniverse consisting of one species of viscous CDM along with cosmological constant( Λ ). Nor-malization of the flow field u µ as u µ u µ = − allows us to express it in terms of coordinatevelocity v i and the metric perturbations as u µ = 1 a (cid:112) ψ − (1 − φ ) v (1 , v i ) ≈ a (1 − ψ, v i ) + higher order terms . (2.6)We parametrize the density and pressure in terms of isotropic background and spatiallyvarying small perturbations as ρ m ( τ, (cid:126)x ) = ρ m ( τ ) + δρ ( τ, (cid:126)x ) ,p ( τ, (cid:126)x ) = p ( τ ) + δp ( τ, (cid:126)x ) , (2.7)with δρ, δp (cid:28) ρ m . The background field satisfies the following equations: H = (cid:18) ˙ aa (cid:19) = 8 π G ρ m + Λ) a , (2.8) ˙ ρ m + 3 H ( ρ m + p m ) = 0 , (2.9)where H = ˙ a/a is the Hubble parameter and dot denotes the derivative with respect to theconformal time τ . For the analysis below, we consider perturbations, up to linear order, inthe variables δρ , δp , v i , φ and ψ . – 3 –n order to expand the fluid dynamic equations, we use the normalized density contrast δ = δρ/ρ m and the velocity divergence θ = ∇ i v i . Moreover, δp is related to the densityperturbations through w , the equation of state parameter, as w = p m ρ m , c s = δ pδρ , c = ˙ p m ˙ ρ m = w − ˙ w H (1 + w ) , (2.10)where c s is the speed of sound in the medium and c ad is the adiabatic sound speed.Inclusion of these perturbations in eq. (2.3) and eq. (2.4) leads to the dynamical equationswhich govern the evolution of cosmological perturbations. In Fourier space these equationscan be written as [28], ˙ δ = − (1 + w ) (cid:16) θ − φ (cid:17) − H (cid:0) c s − w (cid:1) δ , (2.11) ˙ θ = −H θ + k ψ + (cid:18) c s w (cid:19) k δ + 3 c H θ − k η (1 + w ) a ρ m θ . (2.12)For viscous matter we will evaluate the quantities defined in eq.(2.10). Equation of state, w : For baryonic matter and CDM, pressure p m = 0 . However,in presence of bulk viscosity the effective pressure of CDM is equal to the bulk pressure p b = − ζ ∇ µ u µ = − ζ H /a . Therefore, the equation of state parameter for CDM is w = − ζ H a ρ cdm = − ˜ ζ a Ω cdm ˜ H , (2.13)where ˜ ζ = 8 πG ζ/ H is a dimensionless parameter and ˜ H = H / H . Throughout themanuscript ‘0’ in the superscript or subscript denotes the value of the quantity evaluatedtoday. Sound speed, c s : Assuming a constant bulk viscosity, one can calculate c s = − ζ θa ρ cdm δ = w θ H δ = − (cid:32) ˜ ζ a Ω cdm ˜ H (cid:33) (cid:18) θ H δ (cid:19) . (2.14) Adiabatic sound speed, c : Using eq.(2.13) and performing a little bit of mathematicalmanipulation, we obtain c ad = 2 w (cid:18) − Ω cdm (cid:19) . (2.15)In terms of quantities defined above, the evolution equation for δ of CDM (eq. (2.11)) takesthe following form ˙ δ = − (cid:32) − ˜ ζ a Ω cdm ˜ H (cid:33) ( θ − φ ) + (cid:32) ˜ ζ a Ω cdm ˜ H (cid:33) θ − (cid:32) H ˜ ζ a Ω cdm (cid:33) δ (2.16)and the evolution equation for θ of CDM (eq. (2.1)) becomes ˙ θ = −H θ + k ψ − k a θ H (Ω cdm ˜ H − ˜ ζ a ) (cid:18) ˜ ζ + 4˜ η (cid:19) − H θ (cid:18) − Ω cdm (cid:19) (cid:32) ˜ ζ a Ω cdm ˜ H (cid:33) , (2.17)where ˜ η = 8 π Gη/ H is a dimensionless parameter.– 4 – − a − δ no viscosity ˜ η = 10 − ˜ η = 10 − (a) − a − δ no viscosity ˜ ζ = 10 − ˜ ζ = 10 − (b) − a − δ no viscosity ˜ η = 10 − ˜ ζ = 10 − (c) Figure 1 : Effect of viscosity on the growth of linear over-density ( δ ) as a function of scalefactor has been plotted for different values of : (a) shear (b) bulk viscosities. In figure (c) wehave compared the effect of bulk and shear viscosities. We have set k ∼ . h Mpc − in allthe three plots.The Poisson equation for viscous cosmology remain unchanged and given as ∇ φ = 3 H ˙ φ + a ρ m ( δ + 2 ψ ) . (2.18)The Euler equation in Fourier space gives ˙ φ = 3 a ( p b + ρ m ) k θ − H ψ + ηak (3 ˙ θ − k ψ + 3 H θ ) . (2.19) It is evident from eq.(2.16) and eq.(2.17) that the growth of the overdense region gets affectedby shear as well as bulk viscosity. While bulk viscosity directly slows down the collapse ofthe overdense region, shear viscosity imparts similar effect through the velocity perturbation.Thus, it is important to investigate the effects of these viscosities on the evolution of δ and θ which in turn effects the matter power spectrum [29, 31].We start by considering effect of shear viscosity on the evolution of perturbations. Forthis purpose, we set ˜ ζ = 0 and neglect ˙ φ . Therefore, eq. (2.16) and eq. (2.17) can be rewrittenas ˙ δ = − θ , (2.20) ˙ θ = −H θ + k ψ − η a H Ω cdm k H θ , (2.21)with k ψ = −
32 Ω cdm H δ . (2.22)Note that there are two dissipative terms in the right hand side of eq.(2.21), namely theHubble expansion and the shear viscosity term. If the shear term is greater than the Hubbleterm then the evolution of θ is governed by shear along with the potential ψ . By comparing the– 5 – − − − k [ hMpc − ]10 − − P ( k ) [ h − M p c ] Shear Viscosityno viscosity ˜ η = − ˜ η = − (a) − − − k [ hMpc − ]10 − − P ( k ) [ h − M p c ] Bulk Viscosityno viscosity ˜ ζ = − ˜ ζ = − (b) Figure 2 : The effect of viscosities on the matter-power-spectrum is shown for different valuesof (a) shear (b) bulk viscosities. It is evident that these viscosities play important role onlarge k .first and last term of eq.(2.21), we can do an order of magnitude estimate for ˜ η to influence theevolution of the velocity perturbation which ultimately influences the matter power spectrum.For a = 1 case, ˜ η turns out to be ˜ η = (cid:16) k H (cid:17) − Ω . For k ∼ − the value of shearviscosity is ˜ η ∼ O (10 − ) .We combine the set of equations eq. (2.20) and eq. (2.21) in a single second orderdifferential equation in terms of a as, δ (cid:48)(cid:48) + (cid:34) a + H (cid:48) H + 49 k ˜ η a ˜ H Ω (cid:35) δ (cid:48) −
32 Ω a ˜ H δ = 0 , (2.23)where prime represents derivative with respect to the scale factor a . Solution of this equationgives the growth of linear over density δ with a which has been plotted in Fig. (1a). Forsolving this equation the initial value of δ at a = 10 − has been set to a Ω m and the value of δ (cid:48) is a Ω m . We see in Fig. (1a) that the growth of linear overdensity gets suppressed at latetime.On the other hand, effect of bulk viscosity appears in two different ways. First, it mod-ifies the background evolution of cold dark matter and second, it changes the perturbation-equation for δ as well as θ . Since bulk viscosity of dark matter changes the equation of state(see eq. (2.13)), the evolution ρ cdm with a also gets modified which is depicted through thecontinuity equation (eq. (2.9)) as ρ (cid:48) cdm + 3 ρ cdm a (cid:32) − ˜ ζρ ρ cdm (cid:20) Ω b a + Ω Λ + ρ ρ tot (cid:21) / (cid:33) = 0 . (2.24)We solve this equation numerically and fit the solution (see Fig. (3a)), for numerical work,– 6 – .01 0.05 0.10 0.50 1110100100010 a ρ c d m ρ t o t (a) Figure 3 : Change in the evolution of ρ cdm in the presence of bulk viscosity. The blue-dashedline shows the numerical solution of eq. (2.24) for ˜ ζ ∼ . . The value of ζ has been taken toolarge for demonstration purpose. The green-solid line shows the fitting function given in eq.(2.25) with β = 0 . . The red-solid line represents standard ρ cdm = ρ a .with a function of the following form ρ cdm ( a ) = α ρ a + β ρ a , (2.25)where normalization at a = 1 ensures that α = 1 − β . We have verified that this form fitswell even for large range of ˜ ζ . This form of ρ cdm ( a ) will be used in numerical solution usingCLASS [39, 40] later in the paper. The value of β for ˜ ζ = 10 − turns out to be . × − .To consider the effect of bulk on the perturbation equations, we set ˜ η = 0 in equations(eq. (2.16) and eq. (2.17)) and solve them numerically. The solution is plotted and shownin Fig. (1b). Therefore, we can see that bulk viscosity has similar effect as that of the shearviscosity. However, the effect of shear viscosity is slightly more than that of the bulk viscosityon the growth of delta as shown in Fig. (1c).The suppression of δ at late time due to viscosities shows its effect on matter powerspectrum. We have seen in eq. (2.17) that the bulk and shear viscosities come into theequation multiplied with a k factor. Therefore on small length scales their effects becomeprominent. This is expected as the velocity gradients are more effective on small length scalesresulting in large viscosity and hence suppressing the growth at those scales. Consequently,one would expect that the shear viscosity may influence σ . To get the matter power-spectrumwe have used publicly available CLASS code [39, 40]. We have not used non-linear halo-fitfor evolution of δ at large k , since non-linear evolution of viscous dark-matter is beyond thescope of this paper. We will use this power spectrum to get the value of σ which correspondsto k ∼ . h MPc − , the scale which is expected not to be effected by non-linearities. Thepower-spectrum has been plotted in Fig. (2a) and Fig. (2b) where we can see as expectedthe larger k modes of δ gets more suppressed. In these figures we have extended the linearanalysis beyond k = 1 h MPc − only for the purpose of demonstration.– 7 – .270 0.285 0.300 0.315 0.330 Ω m − ˜ η (a) Ω m σ Planck(shear viscosity)LSS(shear viscosity) LSSPlanck (b)
Figure 4 : (a) The best-fitted range of ˜ η with 1- σ and 2- σ contour for LSS data (PlanckSZ + lensing, BAO-BOSS, SPT and CFHTLens) is shown. The central value of best fit is . × − . (b) We show the Planck (high- (cid:96) + low - (cid:96) ) and LSS fitted region of σ − Ω m which clearly shows the discordance. The discordance ends when the best-fit value of shearviscosity is used. Ω m − ˜ ζ (a) Ω m σ Planck(bulk viscosity)LSS(bulk viscosity) LSSPlanck (b)
Figure 5 : (a) The best-fitted range of ˜ ζ with 1- σ and 2- σ contour for LSS data (PlanckSZ + lensing, BAO-BOSS, SPT and CFHTLens) is shown. The central value of best fit is . × − . (b) We show the Planck (high- (cid:96) + low - (cid:96) ) and LSS fitted region of σ − Ω m which clearly shows the discordance. The discordance ends when the best-fit value of bulkviscosity is used. σ - Ω m tension In this section we will show that there exist some tension between the LSS observations andPlanck CMB observation in σ − Ω m plane. We will also demonstrate that small but non-zeroamount of viscosity in cold-dark matter can remove this tension.– 8 –n order to quantify our analysis we proceed in four steps. First, we find the best-fitvalues of σ and other cosmological parameters without viscosity from Planck high- (cid:96) dataand low- (cid:96) data (hereafter Planck data) using MCMC analysis [41]. Throughout our analyses,we have considered massless neutrinos which changes the values of the parameters slightlycompared to the values obtained in ref. [7]. Since A s and n s have the same origin in the earlyuniverse, we fix the priors on these quantities from the Planck parameter estimation whichgives ln(10 × A s ) = 3 . ± . and n s = 0 . ± . .In the next step, we find the best-fit values of σ and other cosmological parameters with-out viscosity from LSS data which include Planck SZ survey [42], Planck lensing survey [43],Baryon Acoustic Oscillation data from BOSS [44, 45], South Pole Telescope (SPT) [46, 47]and CFHTLens [48, 49] (hereafter LSS data). We keep τ reion fixed at . , since τ reion doesnot have much effect on LSS. These LSS surveys altogether indicate a value of σ to be . +0 . − . at 2- σ level whereas Planck CMB observations predicts it to be . +0 . − . at2- σ level. Therefore, there exits a mismatch between these two observations which is evidentin σ − Ω m plane shown in Fig. (4b) and Fig. (5b).We proceed to the next step which is to obtain the best-fit value for the viscosity param-eters ˜ η and ˜ ζ . In this step we keep A s and n s prior as obtained from the analysis of Planckdata. The best-fit value for ˜ η turns out to be (2 . ± . × − at 1- σ level, as shown inFig. (4a) and the best-fit value of ˜ ζ turns out to be (2 . ± . × − at 1- σ level (Fig.(5a)).Further, we set the values of viscosity parameters ˜ η and ˜ ζ to their best-fit values obtainedin the previous step. With these values we perform MCMC analysis for Planck data to obtainthe statistical estimates of standard cosmological parameters { Ω b , Ω cdm , A s , n s , Θ MC , τ reion } and the derived parameters H and σ . Finally, we perform similar analysis with LSS databy setting ˜ η and ˜ ζ to their best-fit values. For this last step we keep the values of A s and n s obtained from the Planck as their prior.As we have discussed in previous section, the viscosity in the cold dark matter reducesthe power on small scales in matter power spectrum. As a consequence σ shifts downwardin σ − Ω m fitting plane for both the LSS and Planck CMB data. The result with viscosityshows a clear overlap of entire σ region for σ − ω m which had earlier a σ discordance (Fig.(4b) and Fig. (5b)). H - Ω m tension Measurements of the value of Hubble parameter are done in two different ways. One isthe direct measurement from type-IA supernova and another one is the indirect estimationthrough LSS and CMB observations. From LSS observations H is estimated as the requireddamping term in the growth of the over-densities, while from CMB observations H is inferredfrom the scale Θ MC of baryon acoustic oscillation.The tension between the direct and indirect measurements is well known in the liter-ature [50]. However, there was no disagreement between indirect measurements until theadvent of Planck data. The WMAP 7 year result [51] has given such values of H − Ω m whichcan accommodate the LSS results. On the other hand, MCMC analysis done with Planckdata, as described in the previous section shows some tension with LSS result obtained sim-ilarly with the Planck prior set on { A s , n s , τ reion } . Planck analysis gives the value of H tobe . ± . with Ω m = 0 . ± . where as the best-fitted value for H turns out be– 9 – .26 0.28 0.30 0.32 0.34 Ω m H LSS(bulk viscosity)LSS PlanckPlanck(bulk viscosity) (a) Ω m H LSS(shear viscosity)LSS PlanckPlanck(shear viscosity) (b)
Figure 6 : Bestfit parameter range for H − Ω m shows a clear discordance in σ level. Thisdiscordance gets resolved by introduction of (a)bulk viscosity or (b) shear viscosity in theCDM. . ± . and Ω m is . ± . for LSS data at 1- σ level. We see that there is cleardiscordance between these best-fit parameter regions in Fig. (6a) and Fig. (6b).In viscous cosmology both Hubble and viscosities play similar kind of role. Hubble actsas an over-all damping term in eq. (2.17) and ˜ η , ˜ ζ go with scale dependent damping terms.Therefore on small scales viscosity compensates the effect of Hubble reducing the value of H . On the other hand, the value of H estimated from CMB does not get affected muchas it is inferred from the acoustic scale Θ MC . The role of viscosity in changing Ω m is notstraightforward. However, it can be understood from eq. (2.17) and eq. (2.23) that Ω cdm or Ω comes in the denominator in the term with viscous parameters. Therefore, introductionof viscosity drags the value of Ω towards higher values to compensate the effect of viscousterm.The MCMC analysis done in four steps, which had been described in last section, showsthese shifts in H − Ω m plane. This shift ensures that the discordance between LSS and CMBobservations disappear. We have seen that bulk and shear viscosity removes the tension between Planck and LSSobservations. Therefore, we proceed to do a joint MCMC analysis using these two data sets.We keep bulk and shear viscosity parameters, ˜ ζ and ˜ η , varying to find the best-fit value forthem; then in two different analyses, we kept one of the viscosity parameters to be zero toget the best-fit value of other. Since bulk and shear viscosities play almost similar role thebest-fit value of shear(bulk) in the absence of bulk(shear) viscosity is different than that ofthe combined analysis.We do not find any significant change in all the cosmological parameters from the Planck-fitted results. The only significant shift is visible in the derived parameter σ which settlesdown to a lower value than the Planck-fitted value. The value of the newly fitted parameters– 10 –arameter 1- σ value 2- σ value Cosmological parameters Ω b h . ± . . ± . cdm h . ± . . +0 . − . MC . ± . . +0 . − . ln(10 A s ) 3 . ± .
023 3 . +0 . − . n s . ± . . +0 . − . τ reion . ± .
012 0 . +0 . − . Viscosity parameters ˜ η . +0 . − . × − . +1 . − . × − ˜ ζ . +0 . − . × − . +2 . − . × − In absence of bulk viscosity ˜ η . +0 . − . × − . +1 . − . × − In absence of shear viscosity ˜ ζ . +0 . − . × − . +1 . − . × − Derived parameters H (Km/sec/Mpc) . ± .
56 68 . +1 . − . σ . ± .
011 0 . +0 . − . Table 1 : Best-fit values of cosmological parameters along with the viscosity parameters andthe derived parameters for viscous cosmology are shown here. These values are obtained fromPlank-LSS joint analyses in the presence of both bulk and shear viscosities. These valuesremain almost unchanged for the analyses with only one type of viscosity.are shown in table-1. The best fit value of σ obtained from the analysis done with eithertype of viscosity does not change significantly from that of the bulk-shear combined analysis. Through out this paper we have discussed the effect of two different viscosities on large scalestructure formations and CMB. We have found that either of the two viscosities or theircombination affects the growth of linear overdensity which in turn changes the matter powerspectrum at small length scales. Motivated by this, we move on to quantify the amountof viscosity supported by cosmological observations. Therefore, we consider the viscositycoefficients as model parameters and perform MCMC analysis with Planck and LSS data. Inthe analyses with LSS data, the values of amplitude of primordial perturbations and scalarspectral index is set to be equal to the value obtained from Planck CMB analysis. We findthat some amount of viscosity is preferred by LSS observations. Most interestingly this bestfitvalue of viscosity resolves the conflict between Planck CMB and LSS observations, both in σ − Ω m plane and H − Ω m plane, simultaneously. It is interesting to note that the value of H inferred from Planck does not change significantly due to the viscosities, while the sameobtained from LSS changes significantly. This is due to the following reasons: H is obtainedfrom the baryon acoustic oscillation scale and depends on the value of Ω m [52]. The LSSexperiments constrain σ and Ω m jointly [48] which gives a scope to accommodate lower σ by increasing Ω m . However, in the case of Planck data, σ is a derived parameter which comesdown to a lower value, due to inclusion of viscosity, without affecting Ω m . Therefore, in the– 11 –ase of σ , both Planck and LSS fitted values change on inclusion of viscosities, but for the H only LSS value gets affected.We find that the required value of bulk and shear viscosity parameters ( ˜ ζ and ˜ η ) , asobtained from MCMC analyses, are of same order ( O (10 − ) ) and have similar effects. It isalmost impossible to distinguish the effects arising from these two viscosities. The best-fitvalues for commonly used viscosity parameters η and ζ ∼ × Pa.sec .The origin of these viscosities can have different sources. The fundamental viscositygenerated by the self-interaction between the dark matter particles might be one source. An-other source can be the large-scale integrated effect of the small scale non-linear gravitationalphenomenon [28]. One might also attribute this kind of viscosity to the corrections in thegravity sector of the Einstein-Hilbert action [53].The fundamental viscosity generated by the self-interaction between the dark matterparticles can be written in terms of the cross-section to mass ratio, σ/m [54]. However itdepends on the details of the decoupling history of dark matter. So the relation betweenthe viscous coefficients η, ζ and σ/m is model dependent. The observational bound frombullet-cluster [55] on the cross-section to mass ratio σ/m for self-interacting dark matter is σ/m ≤ . / g . This gives the value of mean free path of the dark matter particle greaterthan the horizon size at present day. Therefore a hydrodynamic description might not be validduring late time, but it can work in the past when mean free path was within the horizon.Previous attempts in the literature to remove the tension between LSS and Planck CMBfor σ and H either include sterile neutrinos or exotic interaction in dark sectors. However,these attempts fails to resolve both the discordances simultaneously. We, on the other hand,did not introduce any extra matter component to the Λ CDM cosmology. Moreover, wesolve the two issues of σ and H with introduction of only one parameter, either bulk orshear viscosity. The origin of these dissipative effects requires a thorough investigation of theproperties of dark matter in future. Acknowledgement:
We would like to thank Thejs Brinckmann and Miguel Zumalacàrregui for their help relatedto MontePython.
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