Testing Dark energy models in the light of $\sigma_8$ tension

It has been pointed out that there exists a tension in $\sigma_8-\Omega_m$ measurement between CMB and LSS observation. In this paper we show that $\sigma_8-\Omega_m$ observations can be used to test the dark energy theories. We study two models, (1) Hu-Sawicki(HS) Model of $f(R)$ gravity and (2) Chavallier-Polarski-Linder(CPL) parametrization of dynamical dark energy (DDE), both of which satisfy the constraints from supernovae. We compute $\sigma_8$ consistent with the parameters of these models. We find that the well known tension in $\sigma_8$ between Planck CMB and large scale structure (LSS) observations is (1) exacerbated in the HS model and (2) somewhat alleviated in the DDE model. We illustrate the importance of the $\sigma_8$ measurements for testing modified gravity models. Modified gravity models change the matter power spectrum at cluster scale which also depends upon the neutrino mass. We present the bound on neutrino mass in the HS and DDE model.


Introduction
The ΛCDM model is conventional paradigm which is invoked to explain the observations of CMB temperature anisotropy and matter power spectrum [1]. However it has been pointed out [2][3][4][5][6][7][8] that there is some discordance between CMB and LSS observations. Specifically, σ 8 , the r.m.s. fluctuation of density perturbations at 8 h −1 Mpc scale, inferred from Planck-CMB data and that from LSS observations do not agree. There have been many generalizations of the ΛCDM model to attempt the reconciliation between the two sets of results. For example, it has been shown that self interaction in dark matter-dark energy sector [9][10][11][12][13][14] and several other scenarios [15][16][17][18] can reconcile the σ 8 tension. There is also a tension in the inference of Hubble constant H 0 from CMB observations and that determined from LSS observations [4,19]. The H 0 discrepancy can be resolved by including massive neutrinos [4,19], see fig. 1. In addition, it has been shown recently that both these anomalies can be resolved simultaneously by invoking a viscous dark matter [20] and effective cosmological viscosity [21]. By changing the theories of structure formation the bounds on neutrino masses are also affected.
The main conceptual problem with ΛCDM model is that there is no explanantion to why the cosmological constant is of the same order as the matter density in the present epoch. One popular model which addresses this is the Hu-Sawicki model [22] which relates the cosmological constant to the curvature in an f (R) gravity theory. One may also take a phenomenological approach of generalising the cosmological constant to a dynamical variable and determine from observation how it changes in time. An example of this is the DDE model which avoids the problem of phanton crossing.
In this paper we explore the aspect of structure formation in HS Model and DDE model. The Hu-Sawicki f (R) gravity model provides a good description of dark energy and in addition satisfies the constraints from solar system tests [22]. We compute the power spectrum in this model and constrain the parameters with Planck-CMB and LSS data. We find that the tension in σ 8 between Planck-CMB and LSS observations worsens in the HS model compared to the ΛCDM model. The second model we examine is DDE, non-phantom (  such that the non-phantom condition is maintained and obtain σ 8 from Planck-CMB and LSS data sets. We find that in the DDE model the σ 8 tension is eased as compared to ΛCDM model. Neutrino mass cuts the power at small length scales due to free streaming. The cosmology bound on neutrino mass changes in modified gravity models. We find that the constraint on neutrino mass m ν ≤ 0.157 in ΛCDM model changes to m ν ≤ 0.318 in the HS model and m ν ≤ 0.116 in the DDE model from CMB observations. Whereas, m ν = 0.379 +0.138 in the DDE model from LSS observations. We also check the H 0 inconsistency and find that it is being resolved on inclusion of neutrino mass in both these models, consistent with the earlier findings [19] that neutrino mass resolves the H 0 conflict. The structure of this paper is as follows. In Sect. 2 we briefly discuss the Hu-Sawicki f (R) model and the modification in the evolution equations. In Sect. 3 we describe the phemomenological parametrization of DDE model. We describe the role of massive neutrinos in cosmology and their evolution equations in Sect. 4. In Sect. 5 matter power spectrum and it's relation to σ 8 has been discussed briefly. We also explian the efffect of HS, DDE model parameters and massive neutrinos on the matter power spectrum in this section followed by the description of data sets used and analyses done in Sect. 6. We conclude with discussion in Sect. 7.

f(R) Theory: Hu-Sawicki Model
Scalar-tensor theories are generalized Brans-Dicke [23] theories. The general action for scalartensor theories is where S m (g µν , ψ) is the action for the matter fields, g µν is Jordan frame metric andg µν is Einstein frame metric which are related by the conformal transformation g µν = A 2 (φ)g µν , and φ is the scalar field which couples to Einstein metric as well as to matter fields ψ. The scalar field brings in an additional gravitational interaction between matter fields and the net force on a test particle modifies to and the dynamics is governed by the effective potential where Ψ is Newtonian potential and ρ is density. The fact that scalar field couples to the matter fields would result in violations of the Einstein Equivalence Principle [24] and signatures of this coupling would appear in nongravitational experiments based on universality of free fall and local Lorentz symmetry [25] in the matter sector. These experiments severely constrain the presence of a scalar field and can be satisfied if either the coupling of the scalar field with the matter field is always very small or there is some mechanism to hide this interaction in the dense environments. One such mechanism is called chameleon mechanism [26] in which V (φ) and A(φ) are chosen in such forms that V ef f (φ) has density dependent minimum, i.e., V ef f (φ) min = V ef f (φ(ρ)). The required screening will be achieved if either the coupling is very small at the minimum of V ef f (φ) or the mass of the scalar field is extremely large.
If the scalar field stays at its density dependent minimum, φ(ρ), the theory can be parametrized into two functions, the mass function m(ρ) and the coupling β(ρ) at the minimum of the potential [27,28] where m Pl is the Planck mass and mass of the scalar field m(ρ) and the coupling parameter β(ρ) are respectively given as Using the evolution of the matter density given by ρ(a) = ρ 0 a −3 , m(ρ) and β(ρ) can be represented as functions of scale factor a, i.e, m(a) and β(a).
As discussed in Sect. 1 that dark energy can be explained alternatively by modified gravity. Simplest modified gravity model is the f (R) gravity. In general relativity (GR) Lagrangian deisity is given by Ricci scalar R, whereas it is a non linear function of R in the f (R) gravity. Hence the action for an f (R) theory is given as where f (R) is a non linear function of R. The scalar degree of freedom in the f (R) theories has been utilized as the quintessence field to explain DE. It has been shown [29,30] that f (R) theory is the equivalent to a scalar-tensor theory with an equivalence relation and potential corresponding to extra scalar degree of freedom where f R = ∂f /∂R. There are many form of f (R) proposed which explain the type Ia supernovae observation. In this paper, we consider the Hu-Sawicki model, which explains DE while evading the stringent tests from solar system observations. In HS model the modification in the action is given as where R ≥ R 0 and R 0 is the curvature at present. Here f R 0 and n are the free parameters of the HS model. Using equivalence relation 2.8 and eq. 2.9, we find that The coupling function β(a) is constant for all the f (R) models i.e β(a) = 1 √ 6 , whereas the mass function is a model dependent quantity [27,28,31]. In particular for the HS model, for which form of f (R) is given by eq. 2.10, we have mass function These parameters contains all the imformation of the model, where Ω Λ and Ω m are the matter density fraction for dark energy and matter today. In the next subsection, we will derive the evolution equations in terms of these parameters.

Evolution Equations
In GR the evolution of metric perturbation potentials and density perturbations is given by the following linearized equations, Where denotes the derivative with respect to the conformal time, δ is the comoving density contrast and Φ and Ψ are the space-time dependent perturbations to the FRW metric, In the modified gravity models and other dark energy models these relation can be different.
To incorporate the possible deviations from ΛCDM evolution there are several parametrization [32][33][34][35][36] present in the literature. We use the following parametrization which was introduced in [32] where µ(k, a) and γ(k, a) are two scale and time dependent functions introduced to incorporate any modified theory of gravity. Note the appearance of Ψ instead of Φ in the first equation.
In the quasi-static approximation µ(k, a) and γ(k, a) can be expressed as [27] µ(k, a) = A 2 (φ)(1 + (k, a)), Modification in the evolution of Ψ and Φ in turn modifies the evolution of matter perturbation to as where H = a /a.

Dynamical Dark Energy model
The current measurements of cosmic expansion, indicate that the present Universe is dominated by dark energy (DE). The most common dark energy candidate is cosmological constant Λ representing a constant energy density occupying the space homogeneously. The equation of state parameter for DE in cosmological constant model is However a constant Λ makes the near coincidence of Ω Λ and Ω m in the present epoch hard to explain naturally. This gives way for other models of DE such as quintessence [37][38][39], interacting dark energy [40] and phenomenological parametrization of DE such as DDE [41][42][43][44][45]. In the phenomenological DE models the equation of state parameter is taken to be a variable, dependent on the scale factor (equivalently redshift),i.e., where w n are parameters fixed by observations and x(z) is function of redshift. The most commonly followed w(z) dependence are phantom fields(w(z) < −1) and non phantom field(−1 ≤ w(z) ≤ 1). In this paper we use the Chavallier-Polarski-Linder(CPL) [41,42] parametrization of DDE. The equation of state parameter for DE in CPL parametrization is where w 0 and w a are the CPL parameters. Choosing w 0 = −1 and w a = 0 eq. 3.2 gives back the ΛCDM model. As a result of this parametrization the evolution of DE density fraction is given by the equation where Ω DE,0 is the DE density at present.

Massive neutrino in cosmology
Neutrinos play an important role in the evolution of the Universe. Several neutrino experiments have established that neutrinos are massive. Massive neutrinos can affect the background as well as matter perturbation which in turn can leave its imprint on cosmological observations. In the early universe, neutrinos are relativistic and interact weakly with other particles. As the temperature of the Universe decreases, the weak interaction rate becomes less than the Hubble expansion rate of the Universe and neutrinos decouple from rest of the plasma. Since neutrinos are relativistic, their energy density after decoupling is given [46,47] ρ ν = 7 8 4 11 where ρ γ is the photon energy density. N eff is the effective number of relativistic neutrinos at early times and its value is equal to 3.046 [48]. When the temperature of the Universe goes below the mass of the neutrinos, they turn into non-relativistic particles.The energy density fraction of neutrinos in the present universe depends on the sum of their masses and is given as where m ν is the sum of neutrino masses. Neutrinos in the present Universe contribute a very small fraction of energy density however they can affect the formation of structure at large scales.
After neutrinos decouple, they behave as collisionless fluid with individual particles streaming freely. The free streaming length is equal to the Hubble radius for the relativistic neutrinos, whereas non-relativistic neutrinos stream freely on the scales k > k fs , where k fs is the neutrino free-streaming scale. On the scales k > k fs , the free-streaming of the neutrinos damp the neutrino density fluctuations and suppress the power in the matter power spectrum. On the other hand neutrinos behave like cold dark matter perturbations on the scales k < k fs . [46,47]

Evolution equations for massive neutrinos
Massive neutrinos obey the collisionless Boltzmann equation, therefore we solve the Boltzmann equation for the neutrinos to get their evolution equations. The energy momentum tensor for neutrinos is given as where f (x i , P j , τ ) and P µ are the distribution function and the four momentum of neutrinos respectively. We expand the distribution function around the zeroth-order distribution function f 0 as where χ is the perturbation in the distribution function. Using 4.3 in 4.4 and equating the zeroth order terms, we get the unperturbed energy density and pressure for neutrinos ρ = 4π a −4 q 2 dq f 0 (q),P = 4πa −4 3 q 2 dq q 2 f 0 (q). Similarly, We get the purterbed quantities by equating the first order terms where q i = qn i is the co-moving momentum and = (q, τ ) = q 2 + m 2 ν a 2 . It is clear from eqs. 4.6 and 4.7 that we can not simply integrate out the q dependence as is the function of both τ and q. Hence, we will use the Legendre series expansion of the perturbation χ to get the perturbed evolution equations for the massive neutrino. Legendre series expansion of the perturbation χ is given as χ( k,n, q, τ ) = ∞ l=0 (−i) l (2l + 1) χ l ( k, q, τ )P l (k.n) . (4.8) Using eq. 4.8 in the eqs. 4.6 and 4.7, we get the perturbed evolution equations for the massive neutrino [49] δρ h = 4πa −4 q 2 dq f 0 (q)χ 0 , where the Boltzmann equation governs the evolution of χ l . In the Newtonian gauge Boltzmann equations for χ l are given aṡ We use CAMB [50] to generate the matter power spectrum for DDE model, whereas we use MGCAMB [32,33] to obtain the matter power spectrum for HS model. In order to see the effect of modified gravity models and massive neutrinos we plot matter power spectrum for some bench mark values of m ν , HS model parameters and DDE model parameters. The power spectrums are shown in fig. 2.
• As we discussed in Sect. 4, massive neutrinos stream freely on the scales k > k fs and they can escape out of the high density regions on those scales. The perurbations on length scales smaller than neutrino free streaming length will be washed out and therefore power spectrum gets suppress on these scales. Neutrino mass cuts the power • DDE cuts the power spectrum at all length scales. Since, in the DDE model, dark energy density increases with the redshift, therefore, in the early time when the dark energy density is large, the power cut is more prominent at small scales.
• On the other hand, the power spectrum gets affected in an opposite manner for HS model as the power increases slightly on small length scales.

Datasets and Analysis
As discussed in Sect. 1 there is a discrepancy in the values of H 0 and σ 8 reported by the large scale surveys and Planck CMB observations. In this paper we analyse ΛCDM, HS and DDE model. For analysing these models, we use Planck CMB observations [1] for temperature anisotropy power spectrum over the multipole range ∼ 2 − 2500 and Planck CMB polarization data for low only. We refer to these data sets combined as Planck data. We also use the Baryon acoustic oscillations(BAO) data from 6dF Galaxy Survey [51], BOSS DR11 [52,53] and SDSS DR7 Main Galaxy Sample [54]. In addition we use the cluster count data from Planck SZ survey [55], lensing data from Canada France Hawaii Telescope Lensing Survey (CFHTLens) [56,57] and CMB lensing data from Planck lensing survey [58] and South Pole Telescope (SPT) [59,60]. We also use the data for Redshift space distortions (RSD) from BOSS DR11 RSD measurements [61]. We combine Planck SZ data, CFHTLens data, Planck lensing data, SPT lensing data and RSD data and refer them as LSS data. We perform Markov Chain Monte Carlo(MCMC) analysis for ΛCDM, HS and DDE model with both Planck and LSS data. We use CosmoMC [62] to perform the MCMC analysis for ΛCDM  [32,33] to it for HS model. MGCosmoMC patch includes the µ(k, a) and γ(k, a) parametrization discussed in Sect. 2.
In our analysis for ΛCDM model we have total six free parameter which are standard cosmological parameters namely, density parameters for cold dark matter(CDM) Ω c and baronic matter Ω b , optical depth to reionization τ reio , angular acoustic scale Θ s , amplitude A s and tilt n s of the primordial power spectrum. We fix m ν = 0.06eV to satisfy the neutrino oscillation experiments results. We also have two derived parameters H 0 and σ 8 . First we perform MCMC analysis with Planck+BAO data with these parameters and get constraints for each parameter. Next, we run the MCMC analysis with LSS data for ΛCDM model. In shows that with m ν as free parameter, the H 0 tension is resloved and σ 8 tension is worsened. Whereas in Panel (b) it is shown that H 0 tension is eased and σ 8 tension is alleviated slightly. order to avoid the over fitting of data we use Θ s = 1.0413 ± 0.0063 and n s = 0.9675 ± 0.0075, obtained from analysis with Planck+BAO data, as gaussian prior in our run of MCMC analysis with LSS data. Since τ reio does not affects the LSS observation therefore we also use the best fit value of τ reio = 0.08, obtained from analsis with Planck+BAO data, as fixed prior. These analyses give the H 0 = 67.7 +0.8 −0.9 and σ 8 = 0.829 +0.021 −0.023 for the Planck+BAO data and H 0 = 69.4 +1.0 −0.9 and σ 8 = 0.804 +0.009 −0.009 and for LSS data. We plot the parameter space H 0 − Ω m and σ 8 − Ω m , obtained from two different analysis ( fig. 1). It is clear from the fig.  1 that there is a mismatch between the values of H 0 and σ 8 inferred from Planck+BAO data and that from LSS data.
In our analysis for HS model we have total eight free parameter of which six are standard cosmological parameters, two are HS model parameters namely, f R 0 and n as defined in Sect. 2. Here we fix n = 1 and allowed f R 0 to vary in the range [10 −9 ,10]. We repeat the whole procedure to do the analysis with Planck+BAO and LSS data for HS model and obtain constraints for each parameter. Similar to the analysis for ΛCDM model, in the analysis of this model with LSS data, we fixed the τ reio = 0.078 and use Θ s = 1.0411 ± 0.00064 and n s = 0.9684 ± 0.0067 as gaussian prior for the same reason. The best fit values for H 0 and σ 8 in this analysis are 67.9 +1.0 −0.9 and 1.097 +0.133 −0.077 with Planck+BAO data and 69.3 +0.8 −1.0 and 0.804 +0.006 −0.010 with LSS data respectively. We plot the parameter space H 0 − Ω m and σ 8 − Ω m , obtained from analysis with two different data sets, see fig. 3. We found that tension between the values of σ 8 inferred from Planck+BAO data and that from LSS data is increases, whereas the tension in H 0 value decreses in this model.
Similarly we do the analysis for DDE model. In our analysis for DDE model, in addition to the six standard parameters, we have two model parameters w 0 and w a as defined in Sect. 3 making a total of eight parameters. We fix w 0 and w a to be −0.9 and −0.1 respectively (satisfying w a + w 0 = −1 to keep the field non phanton) and do MCMC analysis scan over the remaining six parameters. We repeat the same procedure as we did for ΛCDM and HS model. First we do analysis with Planck+BAO data and get constraints on all the free parameters. In the analysis with LSS data, we fix τ reio = 0.085 and use Θ s = 1.0411 ± 0.00065 and n s = 0.9601 ± 0.0069 as prior (These values are obtained in the analysis with Planck+BAO data). We plot the parameter space H 0 − Ω m and σ 8 − Ω m , obtained from analysis with two different data sets, see fig. 3. We find that tension between the values of σ 8 Table 1: The best fit values with 1-σ error for m ν , σ 8 and H 0 obtained from the MCMC analyses for all the models considered are listed here.
Next, we use sum of massive neutrino m ν as a free parameter and allow it to vary in the range [0,5]eV in our analysis for all three models. We repeat the whole procedure and obtain constraints for each parameter. We plot the parameter space H 0 − Ω m and σ 8 − Ω m for each model, see fig. 1 and 4. The constraint on M ν in ΛCDM, HS and DDE model is listed in table 1. The corresponding 1σ and 2σ contours are shown in fig. 5a and 5b. We also list the constraints on σ 8 and H 0 in each model with Planck+BAO and LSS data, in table 1. models. In CMB measurement of temperature anisotropy spectrum C l and BAO determine Ω m . The discrepancy between the CMB and LSS measurement is determined by the model dependent growth function Ω α m . The growth function can thus be used for testing theories of gravity and dynamical DE. In the present paper we tested HS and DDE models in the context of σ 8 −Ω m observations. We find that in the HS model the σ 8 −Ω m tension worsens compared to the ΛCDM model. On the other hand in the DDE model there is slight improvement in the concordance between the two data sets. The bounds on neutrino mass become more stringent in the DDE model. In the HS model there is a loosening in the analysis with Planck data and not much effect in the analysis with the LSS data.
In conclusion we see that σ 8 measurement from CMB and LSS experiments can be used as a probe of modified gravity or quintessence models. Future observations of CMB and LSS may shrink the parameter space for σ 8 − Ω m and then help in selecting the correct f (R) and DDE theory.