Testing dark energy models in the light of \(\sigma _8\) tension
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Abstract
It has been pointed out that there exists a tension in \(\sigma _8\varOmega _m\) measurement between CMB and LSS observation. In this paper we show that \(\sigma _8\varOmega _m\) 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.
1 Introduction
The \(\varLambda \)CDM 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, \(\sigma _8\), the r.m.s. fluctuation of density perturbations at 8 \(h^{1}\)Mpc scale, inferred from PlanckCMB data and that from LSS observations do not agree. There have been many generalizations of the \(\varLambda \)CDM model to attempt the reconciliation between the two sets of results. For example, it has been shown that self interaction in dark matterdark energy sector [9, 10, 11, 12, 13, 14] and several other scenarios [15, 16, 17, 18, 19, 20, 21] can reconcile the \(\sigma _8\) tension. There is also a tension in the inference of Hubble constant \(H_0\) from CMB observations and that determined from LSS observations [4]. The value of \(H_0\) is also directly measured from SupernovaIa (SNIa) observation (referred as local \(H_0\)) [22] which is also not in agreement with \(H_0\) obtained from CMB observation (referred as local \(H_0\) tension). The \(H_0\) discrepancy between CMB and LSS observations can be resolved by including active massive neutrinos [4]. Whereas addition of a sterile massive neutrino helps resolving the local \(H_0\) tension [23]. Varying the equation of state parameter w and effective number of relativistic degrees of freedom \(N_{eff}\) freely has also been shown to resolve the local \(H_0\) tension [24, 25, 26]. Extended parameter space of \(\varLambda \)CDM model have also been shown to resolve local \(H_0\) tension and \(\sigma _8\) tension [27, 28]. It has been shown recently that both \(\sigma _8\) and \(H_0\) tension between CMB and LSS observations can be resolved simultaneously by invoking a viscous dark matter [29] and effective cosmological viscosity [30]. The bound on neutrino mass is also different in different models of cosmology [31, 32, 33, 34, 35].
The main conceptual problem with \(\varLambda \)CDM model is that there is no explanation of the origin and the unusually small value of the cosmological constant (\(\varLambda \)). One popular class of models which addresses this is the f(R) gravity [36] models, in which the cosmological constant is generated dynamically from the curvature. We consider the Hu–Sawicki f(R) gravity model which also satisfies the constraints from solar system tests [36]. One may also take a phenomenological approach of generalizing 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 phantom crossing. For earlier works on cosmological parameter estimation with DDE models and f(R) gravity models see [28, 37, 38].
In this paper we explore the aspect of structure formation in HS Model and DDE model. In Hu–Sawicki model, we compute the power spectrum and constrain the parameters with PlanckCMB and LSS data. We find that the tension in \(\sigma _8\) between PlanckCMB and LSS observations worsens in the HS model compared to the \(\varLambda \)CDM model. The second model we examine is DDE, nonphantom (equation of state \(w \ge 1\)) model of dark energy. We choose the values of two model parameters in this model such that the nonphantom condition is maintained and obtain \(\sigma _8\) from PlanckCMB and LSS data sets. We find that in the DDE model the \(\sigma _8\) tension is eased as compared to \(\varLambda \)CDM 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 \(\sum m_{\nu } \le 0.157\) in \(\varLambda \)CDM model changes to \(\sum m_{\nu } \le 0.318\) in the HS model and \(\sum m_{\nu } \le 0.116\) in the DDE model from CMB observations. Whereas, \(\sum m_{\nu } = 0.364\pm 0.095\) in \(\varLambda \)CDM model changes to \(\sum m_{\nu } =0.333\pm 0.093 \) in the HS model and \(\sum m_{\nu } =0.275\pm 0.095\) in the DDE model from LSS observations. All these bounds are at 68\(\%\) c.l. All these constraints are obtained considering that all three neutrinos have degenerate mass. 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 [4] 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 phenomenological 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 \(\sigma _8\) has been discussed briefly. We also explain the effect 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.
2 f(R) theory: Hu–Sawicki model
The fact that scalar field couples to the matter fields would result in violations of the Einstein Equivalence Principle [40] and signatures of this coupling would appear in nongravitational experiments based on universality of free fall and local Lorentz symmetry [41] 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 [42] in which \(V(\phi )\) and \(A(\phi )\) are chosen in such forms that \(V_{eff}(\phi )\) has density dependent minimum, i.e., \(V_{eff}(\phi )_{min}=V_{eff}(\phi (\rho ))\). The required screening will be achieved if either the coupling is very small at the minimum of \(V_{eff}(\phi )\) or the mass of the scalar field is extremely large.
2.1 Evolution equations
3 Dynamical dark energy model
4 Massive neutrino in cosmology
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 nonrelativistic neutrinos stream freely on the scales \(k > k_\mathrm{fs}\), where \(k_\mathrm{fs}\) is the neutrino freestreaming scale. On the scales \(k > k_\mathrm{fs}\), the freestreaming 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 \(k < k_\mathrm{fs}\) [69, 70].
4.1 Evolution equations for massive neutrinos
5 Matter power spectrum and \(\sigma _8\)
Parameters with flat prios are listed in this table
Parameters  Planck + BAO  LSS 

\(\varOmega _{c}h^2\)  [0.001, 0.99]  [0.001, 0.99] 
\(\varOmega _{b}h^2\)  [0.005, 0.1]  [0.005, 0.1] 
\(\tau _{reio}\)  [0.01, 0.8]  
100\(\varTheta _\mathrm{MC}\)  [0.5, 10]  [0.5, 10] 
\(\ln (10^{10}A_{s})\)  [2, 4]  [2, 4] 
\(n_{s}\)  [0.8, 1.2]  [0.8, 1.2] 
\(\sum M_{\nu }\)  [0, 5.0]  [0, 5.0] 
\(\log _{10}f_{R_{0}}\)  \([9.0,10]\)  \([9.0,10]\) 
The best fit values with 1\(\sigma \) error for all the parameters with fixed \(\sum m_\nu \), obtained from the MCMC analyses for all the models considered are listed here
Parameter  \(\varLambda \)CDM  DDE  HS  

Planck + BAO  LSS  Planck + BAO  LSS  Planck + BAO  LSS  
\(\varOmega _b h^2 \)  \(0.02227\pm 0.00020 \)  \(0.02274\pm 0.00081 \)  \(0.02236\pm 0.00020\)  \(0.02292\pm 0.00080 \)  \(0.02243\pm 0.00022 \)  \(0.02225\pm 0.00070 \) 
\(\varOmega _c h^2 \)  \(0.1190\pm 0.0013 \)  \(0.1159\pm 0.0016 \)  \(0.1178\pm 0.0013 \)  \(0.1146\pm 0.0016 \)  \(0.1185\pm 0.0013\)  \(0.1153\pm 0.0015\) 
\(100\theta _{MC} \)  \(1.04098\pm 0.00042 \)  \(1.0425\pm 0.0011 \)  \(1.04116\pm 0.00042 \)  \(1.0427\pm 0.0011 \)  \(1.04108\pm 0.00043 \)  \(1.0419\pm 0.0010 \) 
\(\tau _{reio} \)  \(0.081\pm 0.018 \)  0.08  \(0.086\pm 0.018 \)  0.086  \( 0.063\pm 0.020 \)  0.65 
\(\mathrm{{ln}}(10^{10} A_s)\)  \(3.094\pm 0.035 \)  \(3.080\pm 0.012 \)  \(3.101\pm 0.036 \)  \(3.095\pm 0.011 \)  \(3.058\pm 0.041\)  \(3.053\pm 0.011\) 
\(n_s \)  \(0.9673\pm 0.0044 \)  \(0.905\pm 0.019 \)  \(0.9701\pm 0.0045\)  \(0.910\pm 0.019 \)  \(0.9691\pm 0.0047\)  \(0.941\pm 0.011 \) 
\(H_0 \)  \(67.65\pm 0.57 \)  \(69.81^{+0.73}_{0.82}\)  \(66.02\pm 0.52 \)  \(67.98\pm 0.72 \)  \(68.00\pm 0.61\)  \(69.26\pm 0.72 \) 
\(\varOmega _m \)  \(0.3102\pm 0.0077 \)  \(0.2862\pm 0.0071\)  \(0.3230\pm 0.0077\)  \(0.2991\pm 0.0073 \)  \(0.3062\pm 0.0079 \)  \(0.2882\pm 0.0071 \) 
\(\sigma _8 \)  \(0.829\pm 0.015 \)  \(0.7917\pm 0.0074 \)  \(0.808\pm 0.015\)  \(0.7745\pm 0.0074 \)  \(1.10^{+0.12}_{0.030}\)  \(0.7948\pm 0.0068 \) 
\(S_8 \)  \(0.840\pm 0.018\)  \(0.7732^{+0.0139}_{0.0119}\)  \(0.834\pm 0.018\)  \(0.7732^{+0.0141}_{0.0122}\)  \(1.105^{+0.110}_{0.020}\)  \(0.7790^{+0.0137}_{0.0119}\) 

As we discussed in Sect. 4, massive neutrinos stream freely on the scales \(k > k_\mathrm{fs}\) and they can escape out of the high density regions on those scales. The perturbations 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 at length scales even larger than the 8 \(h^{1}\)Mpc which requires a large \(\varOmega _m\) which in turn disfavors the compatibility of \(\sigma _8\varOmega _m\) between the two observations.

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.
The best fit values with 1\(\sigma \) error for all the parameters with varying \(\sum m_\nu \), obtained from the MCMC analyses for all the models considered are listed here
Parameter  \(\varLambda \)CDM  DDE  HS  

Planck + BAO  LSS  Planck + BAO  LSS  Planck + BAO  LSS  
\(\varOmega _b h^2 \)  \(0.02228\pm 0.00020 \)  \(0.02277\pm 0.00080 \)  \(0.02236\pm 0.00020\)  \(0.02283\pm 0.00081\)  \(0.02254\pm 0.00025 \)  \(0.02216^{+0.00064}_{0.00073} \) 
\(\varOmega _c h^2 \)  \(0.1188^{+0.0015}_{0.0014}\)  \(0.1141\pm 0.0018 \)  \(0.1177\pm 0.0014\)  \(0.1134\pm 0.0017 \)  \(0.1172^{+0.0020}_{0.0017}\)  \(0.1139\pm 0.0017 \) 
\(100\theta _{MC} \)  \(1.04097\pm 0.00042 \)  \(1.0430\pm 0.0011\)  \(1.04114\pm 0.00042\)  \(1.0431\pm 0.0011 \)  \(1.04120\pm 0.00045 \)  \(1.0434\pm 0.0010 \) 
\(\tau _{reio} \)  \(0.082^{+0.018}_{0.020} \)  0.082  \(0.086\pm 0.018 \)  0.086  \(0.065\pm 0.021 \)  0.065 
\(\mathrm{{ln}}(10^{10} A_s)\)  \(3.096\pm 0.037 \)  \(3.114\pm 0.015\)  \(3.101\pm 0.036 \)  \(3.117\pm 0.015 \)  \(3.058\pm 0.042 \)  \(3.085\pm 0.014 \) 
\(n_s \)  \(0.9676^{+0.0045}_{0.0050}\)  \(0.911\pm 0.019 \)  \(0.9702\pm 0.0047 \)  \(0.913\pm 0.019 \)  \(0.9723^{+0.0053}_{0.0060}\)  \(0.943\pm 0.011 \) 
\(H_0 \)  \(67.56\pm 0.65 \)  \(67.80\pm 0.99 \)  \(66.05\pm 0.57 \)  \(66.65\pm 0.97 \)  \(67.64\pm 0.74 \)  \(67.45\pm 0.96 \) 
\(\varOmega _m \)  \(0.3112\pm 0.0082 \)  \(0.306\pm 0.010 \)  \(0.3227\pm 0.0079 \)  \(0.314\pm 0.011 \)  \(0.3096\pm 0.0089 \)  \(0.307\pm 0.010 \) 
\(\sigma _8 \)  \(0.826^{+0.022}_{0.017} \)  \(0.735\pm 0.028 \)  \(0.809^{+0.019}_{0.016}\)  \(0.735\pm 0.020 \)  \(1.115^{+0.091}_{0.034}\)  \(0.743\pm 0.020 \) 
\(\varSigma m_\nu \)  \(< 0.157 \)  \(0.364\pm 0.095\)  \(< 0.116 \)  \(0.275\pm 0.095\)  \(< 0.318 \)  \(0.333\pm 0.093 \) 
\(S_8\)  \(0.838\pm 0.022\)  \(0.7431^{+0.0201}_{0.0181}\)  \(0.835\pm 0.020\)  \(0.7522^{+0.0201}_{0.0172}\)  \(1.127^{+0.089}_{0.031}\)  \(0.7522^{+0.0201}_{0.0175}\) 
6 Datasets and analysis
The descrepancy level between the \(\sigma _{8}\) values inferred from Planck + BAO and LSS data for \(\varLambda \)CDM, DDE and HS model are listed here
\(\varLambda \)CDM  DDE  HS  

With fixed \(\sum M_{\nu }\)  \(1.722\sigma \)  \(1.522\sigma \)  \(1.793\sigma \) 
With varying \(\sum M_{\nu }\)  \(1.960\sigma \)  \(1.796\sigma \)  \(2.273\sigma \) 
In our analysis for \(\varLambda \)CDM model we have total six free parameter which are standard cosmological parameters namely, density parameters for cold dark matter(CDM) \(\varOmega _{c}\) and baryonic matter \(\varOmega _{b}\), optical depth to reionization \(\tau _{reio}\), angular acoustic scale \(\varTheta _\mathrm{MC}\), amplitude \(A_{s}\) and tilt \(n_{s}\) of the primordial power spectrum. We fix \(\sum m_{\nu } =0.06\)eV to satisfy the neutrino oscillation experiments results. We also have two derived parameters \(H_{0}\) and \(\sigma _{8}\). 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 \(\varLambda \)CDM model. Since \(\tau _{reio}\) does not affects the LSS observation therefore we also use the best fit value of \(\tau _{reio}=0.08\), obtained from analysis with Planck + BAO data, as fixed prior. We have listed all the parameters with flat in Table 1. These analyses give \(\sigma _{8}=0.829\pm 0.015\) for the Planck + BAO data and \(\sigma _{8}=0.7917\pm 0.0074\) and for LSS data. We plot the parameter space \(\sigma _{8}\varOmega _m\), obtained from two different analysis (Fig. 2). It is clear from the Fig. 2 that there is a mismatch between the values of \(\sigma _{8}\) 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 \(\varLambda \)CDM model, in the analysis of this model with LSS data, we fixed the \(\tau _{reio}=0.065\). The best fit values for \(\sigma _{8}\) in this analysis are \(1.10^{+0.12}_{0.030}\) with Planck + BAO data and \(0.7948\pm 0.0068\) with LSS data. We plot the parameter space \(\sigma _{8}\varOmega _m\), obtained from analysis with two different data sets, see Fig. 2. We found that tension between the values of \(\sigma _{8}\) inferred from Planck + BAO data and that from LSS data is increases.
Next 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. First we do the MCMC analysis for both the data sets keeping \(w_a\) and \(w_0\) as free parameters and get the 2\(\sigma \) allowed ranges which are shown in Fig. 3. Next we put the nonphantom constraints represented by region above blue and dashed blue lines in Fig. 3. We see that the region allowed by both the data sets and also satisfies the non phantom conditions \(w_a+w_0\ge 1 \) and \(w_0 \ge 1\) is very small and close to \(w_{0}=0.9\) and \(w_{a}=0.1\). Therefore we choose these values for our further analysis, and do MCMC analysis scan over the remaining six parameters. We repeat the same procedure as we did for \(\varLambda \)CDM 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 \(\tau _{reio}=0.086\) (This value is obtained in the analysis with Planck + BAO data). We plot the parameter space \(\sigma _{8}\varOmega _m\), obtained from analysis with two different data sets, see Fig. 2. We find that tension between the values of \(\sigma _{8}\) values inferred from Planck + BAO data and that from LSS data is somewhat alleviated in the DDE model. Constraints on \(\sigma _{8}\), \(H_{0}\), and other parameters for each model are listed in Table 2.
Next, we use sum of massive neutrino \(\sum m_{\nu }\) 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 \(\sigma _{8}\varOmega _m\) for each model, see Fig. 4. Constraints on \(\sum M_{\nu }\), \(\sigma _{8}\), \(H_{0}\) and other parameters for each model are listed in Table 3. The triangle plots for all the three models with \(\sum m_\nu =0.06\)eV are shown Figs. 5, 6 and 7. The triangle plots for all the three models with \(\sum m_\nu \) as free parameter are shown Figs. 8, 9 and 10. The corresponding \(1\sigma \) and \(2\sigma \) contours for \(\sum M_{\nu }\) are shown in Fig. 11.
7 Discussion and conclusion
Galaxy surveys and CMB lensing measure the parameter \(\sigma _8 \varOmega _{m}^{\alpha }\), where \(\varOmega _{m}^{\alpha }\) represents a model dependent growth function. In \(\varLambda \)CDM \(\alpha =0.5\) but it could be different for other DMDE models. In CMB measurement of temperature anisotropy spectrum \(C_l\) and BAO determine \(\varOmega _m\). The discrepancy between the CMB and LSS measurement is determined by the model dependent growth function \(\varOmega _{m}^{\alpha }\). 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 \(\sigma _8\varOmega _m\) observations. We find that in the HS model the \(\sigma _8\varOmega _m\) tension worsens compared to the \(\varLambda \)CDM model. On the other hand in the DDE model there is slight improvement in the concordance between the two data sets. The discrepancy levels between values inferred from Planck + BAO and LSS data for \(\varLambda \)CDM, DDE and HS model are listed in Table 4. We also find that adding active massive neutrinos allow us to have larger value of \(\varOmega _m\). H(z) in \(H(z)=H_0 \sqrt{\varOmega _m (1+z)^3 + \varOmega _{\varLambda }}\) is determined by the observation, therefore a larger value of \(\varOmega _m\) brings down the \(H_0\) value to satisfy the observation. Thus, we find that the \(H_0\) tension between CMB and LSS observations is resolved by using active massive neutrinos. However, this increases the mismatch between \(H_0\) values obtained from LSS and SNIa observations. In all three models, the \(n_s\) values obtained in the analysis with LSS data is smaller as compared to \(n_s\) value obtained from Planck + BAO. Which gives rise to another tension between the two data sets. The tilt of the primordial spectrum is calculated at a particular pivot scale(\(k_{*}\)). In our analysis the pivot scale is 0.05 \(\mathrm Mpc^{1}\). The \(n_s\) discrepancy may be due to the fact that Planck data and LSS data have different pivot scale which can be a signature of running tilt of the primordial spectrum. This can be checked in future works. 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 \(\sigma _8\) 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 \(\sigma _8\varOmega _m\) and then help in selecting the correct f(R) and DDE theory.
Notes
Acknowledgements
We thank the anonymous referee for many useful insights and suggestions which helped in improving the overall quality of the work. We also acknowledge the computation facility, 100TFLOP HPC Cluster, Vikram100, at Physical Research Laboratory, Ahmedabad, India.
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