Measuring growth index in a universe with massive neutrinos: A revisit of the general relativity test with the latest observations
Abstract
We make a consistency test for the general relativity (GR) through measuring the growth index in a universe with massive (sterile/active) neutrinos. We employ the redshift space distortion measurements to do the analysis. To constrain other cosmological parameters, we also use other cosmological measurements, including the Planck 2015 cosmic microwave background temperature and polarization data, the baryon acoustic oscillation data, the type Ia supernova JLA data, the weak lensing galaxy shear data, and the Planck 2015 lensing data. In a universe with massive sterile neutrinos, we obtain , with the tension with the GR prediction at the 1.48 level, showing that the consideration of sterile neutrinos still cannot make the true measurement of be well consistent with the GR prediction. In a universe with massive active neutrinos, we obtain for the normal hierarchy case, for the degenerate hierarchy case, and for the inverted hierarchy case, with the tensions with GR all at beyond the 2 level. We find that the consideration of massive active neutrinos (no matter what mass hierarchy is considered) almost does not influence the measurement of the growth index .
I Introduction
The current astronomical observations have indicated that the universe is undergoing an accelerated expansion [1, 2, 3, 4, 5]. To explain this accelerated expansion, in the context of general relativity (GR), the socalled dark energy (DE), an unknown component with negative pressure, is proposed [6, 7, 8, 9, 10]. On the other hand, the modification of gravity (MG) can also account for the accelerated expansion by mimicking the behavior of DE within GR for the whole expansion history at the background level [11, 12, 13]. Both of them can in principle describe the same expansion, but they are different in nature. To distinguish between MG and GR, the precise largescale structure (LSS) measurements are required because they have different histories of growth of structure.
A way to describe the growth of scalar (density) perturbations in nonrelativistic matter component (cold dark matter and baryons) is provided by the parametrization , proposed in Ref. [14], where is the growth rate for linear perturbations, is the fractional matter density, and is called the growth index. Both of the growth index and the evolution of matter density depend on the specific model (for details see the latest review [15]). For dark energy models with slowly varying equation of state, within GR, an approximation of is derived. For example, based on the CDM model, is given [16]. However, for MG models, different theoretical values of are derived; e.g., for the DvaliGabadadzePorrati (DGP) model, is obtained [17, 18, 19].
The growth index in a cosmological model can be constrained by using the redshift space distortion (RSD) observation. RSD is a significant probe for the growth of structure, which provides an important way of measuring the growth rate at various redshifts. In practice, RSD measures the product of and , namely, , where is the rootmeansquare mass fluctuation in a sphere of radius at the redshift . However, using RSD to constrain the growth index (based on the CDM model), it is found that there is a deviation of the value from the GR’s theoretical prediction of at the 2–3 confidence level (see, e.g., Ref. [20]). Recently, GilMarin et al. [21] used the latest BOSS CMASS and LOWZ DR12 measurements combined with the Planck 2015 temperature and polarization spectra to constrain , and they obtained , which reveals a beyond tension with the GR prediction. The similar situation can be found in the previous studies [22, 23].
It can be noticed that the measurements of have been always higher than the GR prediction. That is to say, the actual observed growth of structure is faster than that predicted by GR. One way to reconcile them is to consider massive (active or sterile) neutrinos in the cosmological model, since the freestreaming property of neutrinos could help suppress the growth of structure on small scales. In 2014, two of the authors of the present paper (JingFei Zhang and Xin Zhang) and another collaborator (YunHe Li) [24] considered this scheme, and they found that if massive sterile neutrinos are involved in the cosmological model, then the constraint value of and the theoretical prediction of will become well consistent.
In Ref. [24], Zhang, Li, and Zhang used the RSD data in combination with other observations (at that time) including the cosmic microwave background (CMB) anisotropy data from the Planck 2013 temperature spectrum [25] and the WMAP 9yr polarization data [26], baryon acoustic oscillations (BAO) measurements from the 6dFGS [27], SDSS DR7 [28], WiggleZ [29], and BOSS DR11 [30] surveys, the Hubble constant measurement with the value of [31], the Planck SunyaevZeldovich cluster counts data [32], and the cosmic shear data from CFHTLenS survey [33], to constrain , and they obtained , well consistent with the prediction value of GR of . See Refs. [34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 43, 44, 50] for other previous works discussing the issue of using massive sterile neutrinos to relieve tensions among cosmological observations.
However, it should be pointed out that it is the time to revisit this issue with the latest cosmological observations. In the past years, numerous more accurate data were released, which would update the previous results derived in Ref. [24] and even would change the conclusive statements.
In this paper, we will revisit the study of the constraints on the growth index, based on the CDM cosmology with massive (sterile/active) neutrinos, using the latest cosmological measurements, including the Planck 2015 temperature and polarization power spectra and the latest RSD data. Moreover, since the lensing observations including the weak lensing and the CMB lensing can capture the effects of massive neutrinos on the matter power spectra, they can provide useful constraint on the neutrino mass. In addition, the growth index is related to not only the structure’s growth, but also the expansion of the universe, and thus the independent geometric observations such as BAO and type of Ia supernova (SN) are also needed. We will use these latest observations to study the measurement of the growth index .
Ii Method and data
In this paper, we place constraints on the growth index in the CDM cosmology with massive (sterile/active) neutrinos with the latest observations. Within GR, as long as the equationofstate parameter of DE is slowly varying, the theoretical predictions of for DE models are almost the same, i.e., . Thus, in this paper, we only consider the CDM cosmology.
For the base CDM model, there are six base parameters, which are the baryon density , the cold dark matter density , the ratio of the angular diameter distance to the sound horizon at last scattering , the reionization optical depth , and the amplitude and the tilt of the primordial scalar fluctuations.
To constrain the growth index , we use the parametrization to describe the density perturbations in the CDM cosmology, and thus we introduce an extra parameter into the model. We use the RSD measurements of to set constraints on . We follow the procedure of Sec. 9.1 in Ref. [23] to include as an additional parameter. Here, it is helpful to briefly describe how the parameter product is derived in the theoretical calculations by the following two steps: (i) Since in this description the value of depends on , we have to recalculate this value by using the parametrization of . We first calculate the growth factor, , where is the scale factor at the effective redshift . Then, we derive by the extrapolation from the matter dominated epoch to the effective redshift, , where is calculated at which is in the deep matterdominated regime, where . (ii) We calculate the growth rate by using the parametrization . Thus, now, we can obtain the parameter product in the numerical calculations.
If we further consider massive neutrinos in cosmology, we need to add the total neutrino mass for the case of active neutrino and the effective mass and the effective number of relativistic species for the case of sterile neutrino.
We make our analysis by employing several important cosmological probes. We use the Planck 2015 full temperature and polarization power spectra at . We refer to this dataset as “Planck TT,TE,EE” (note that we do not use “+lowP”, as the Planck collaboration used, for simplicity). In addition to the CMB dataset described above, we consider the combination with the following cosmological measurements:

The SN data: For the type Ia supernova observation, we adopt the “JLA” sample, compiled from the SNLS, SDSS, and the samples of several lowredshift SN data [53].

The WL and CMB lensing data: We use the cosmic shear measurement of weak lensing from the CFHTLenS survey, and we apply the “conservative” cuts for the shear data according to the recipe of Ref. [59]. We denote the dataset of shear measurement (weak lensing) as “WL” in this paper. We also use the CMB lensing power spectrum from the Planck 2015 lensing measurement [60], which is denoted as “lensing” in this paper.
The analysis is done with the latest version of the publicly available MarkovChain Monte Carlo package CosmoMC [61], with a convergence diagnostic based on the Gelman and Rubin statistics.
In this paper, tensions between different observations for some cosmological parameters will occasionally be mentioned, so it is helpful to clearly describe how to estimate the degree of tension between two observations for some parameter in this place. Assume that, for a parameter , we have its 68% confidence level ranges from an observation (O1) and from another observation (O2). The statement that “the tension between O1 and O2 is at the level” means that we have for the case , and vice versa. In this work, we estimate the degree of tension between different observations by this simple way.
Iii Results
iii.1 Sterile neutrinos and growth index
We constrain the growth index in the CDM cosmology with sterile neutrinos by using the data combination of Planck TT,TE,EE+BAO+SN+RSD+WL+lensing. We show the constraint results in Fig. 1 and Table 1. Here, with the purpose of visually showing the effect of sterile neutrinos on the constraints of the growth index, we also perform an analysis for the CDM+ model (without sterile neutrinos), to make a comparison, and the detailed results are presented in Fig. 1 and Table 1.
Model  CDM+  CDM+++ 

[eV]  
Figure 1 displays the onedimensional posterior distribution for in the top panel and the twodimensional, joint, marginalized constraints (68% and 95% confidence level) in the plane in the bottom panel. First, let us have a look at the constraint results of (see Table 1). We obtain for the CDM+ model and for the CDM+++ model, which indicates that the tensions of with the GR prediction are at the level and the level for the two cases, respectively. We find that, in the CDM+++ model, the tension is milder than that of the model without sterile neutrinos. In the top panel of Fig. 1, the onedimensional posterior distributions of show that the consideration of sterile neutrinos indeed leads the fit value of to be more consistent with the GR theoretical value of , presented by the grey dotted line. According to the fitting results, we can clearly see that once a light sterile neutrino species is considered in the universe, a smaller value of is indeed derived, but it is not enough to lead the true measurement of to be well consistent with the GR prediction.
The tension with GR might be related to the matter density in the CDM model fitting to the CMB data [21]. The change in also depends on a change in determination of using the CMB and other data. We show the contours in the plane in the bottom panel of Fig. 1. We find that the correlation between and is weak for the both models. We also see that once sterile neutrinos are considered in the model, the value of is increased and the value of is somewhat lowered, although the whole range of the contour is amplified due to the addition of extra two parameters.
We also compare our results with those of the previous study [24]. In Ref. [24], it is found that when a sterile neutrino species is considered in the model, the tension with GR will be at the less than 1 level. But, in this study, we find that even though sterile neutrinos are considered, the tension with GR will still be at the more than 1 level (but less than 2 level). In Ref. [24], for the constraint on the parameters of sterile neutrino, the authors obtain and eV, indicating the preference for at the 1.4 level and for nonzero mass of sterile neutrino at the 3.4 level. But, in the present study, we only obtain upper limits on both and , namely, and eV. Recently, the neutrino oscillation experiments by Daya Bay and MINOS collaborations [62], as well as the cosmic ray experiment by the IceCube collaboration [63], found no evidence for a massive sterile neutrino species, in good consistency with our present result.
We also wish to simply address the issue of other tensions. Including sterile neutrinos in a universe can enhance the fit value of (due to the positive correlation between and existing in the CMB fit), and hence reconcile the tension between Planck and the direct measurement of the Hubble constant ( [64]) [65, 66, 67, 68, 69, 47, 48]. We present the constraint values of and tensions with the direct measurement of the Hubble constant for the CDM+ model and the CDM+++ model in Table 2. From the table, we can see that for the CDM+ model and for the CDM+++ model, indicating that the tensions with the local determination of the Hubble constant are at the level and the level, respectively. This means that the consideration of sterile neutrinos offers only a marginal improvement for the issue of tension.
Model  CDM+  CDM+++ 

tension 
Next, we examine the tension between Planck and the recent observations of largescale structure, mainly in terms of constraining the parameter . In the last year, the tomographic weak gravitational lensing analysis of 450 from the Kilo Degree Surveys (KiDS450) gave their latest cosmological parameter constraints, in which they obtained assuming a flat CDM model [70]. More recently, the cosmological result from a combined analysis of galaxy clustering and weak gravitational lensing, using 1321 of imaging data from the first year of the Dark Energy Survey (DES Y1) was presented in Ref. [71]. They obtained for the CDM cosmology. In the present work, we also calculated the values for the two models. In Table. 3, we show our fit results. We obtain for the CDM+ model and for the CDM+++ model. According to our fit results, in the CDM+ model, the tension with KiDS450 is at the level and the tension with DES Y1 is at the level; in the CDM+++ model, the tension is somewhat relieved, i.e., the tension with KiDS450 is relieved to be at the level and the tension with DES Y1 is relieved to be at the level (see also Table. 3).
Model  CDM+  CDM+++ 

tension (KiDS450)  
tension (DES Y1) 
iii.2 Active neutrinos and growth index
Model  CDM+  CDM++ (NH)  CDM++(DH)  CDM++ (IH) 

[eV]    
13765.68  13764.51  13764.65  13765.55 
In this subsection, we constrain the growth index in a universe with massive active neutrinos. In this case, is fixed at 3.046 and is freely varied within a prior range.
In this investigation, we also consider the mass splitting of neutrinos. The solar and reactor experiments measured eV, and the atmospheric and accelerator beam experiments gave eV, indicating that there are two possible mass orders, i.e., the normal hierarchy (NH) with and the inverted hierarchy (IH) with . According to these two measured values, for the NH case, the neutrino mass spectrum can be written as , in terms of a free parameter ; and for the IH case, the neutrino mass spectrum is expressed as , in terms of . We also consider the degenerate hierarchy (DH) case, i.e., , where is a free parameter. Note that the input lower bound of is eV, 0.10 eV, and for NH, IH, and DH, respectively. See Refs. [72, 73, 74, 75, 76, 77] for details about the consideration of the mass hierarchies of neutrinos.
We use the same data combination, i.e., Planck TT,TE,EE+BAO+SN+WL+RSD+lensing, to do the analysis. The detailed fit results are shown in Table 4 and Fig. 2. In Table 4, for a direct comparison, we duplicate the results of the CDM+ model here (see also Table 1). We obtain for the CDM++ (NH) model, for the CDM++ (DH) model, and for the CDM++ (IH) model. In the CDM+ model, we have . So, we find that the consideration of neutrino mass (no matter what mass hierarchy is considered) does not influence the measurement of the growth index .
Figure 2 shows the onedimensional posterior distributions for (the top panel) and the twodimensional marginalized contours in the plane (the bottom panel) for the CDM+ and CDM++ models. From the top panel of Fig. 2, we can clearly see that the posterior distribution curves are almost in coincidence. The tensions with the standard value of GR prediction are still at the , , and level, for the NH, DH, and IH cases, respectively. In the bottom panel of Fig. 2, we can see that, after the consideration of neutrino mass, the range of is enlarged (and the value of is also enhanced), and the range of is nearly unchanged.
Next, we report the constraint results of the active neutrino mass in a universe with the density perturbation parametrized by the growth index . Using the Planck TT,TE,EE+BAO+SN+RSD+WL+lensing data combination, we obtain eV for the CDM++ (NH) model, eV for the CDM++ (DH) model, and eV for the CDM++ (IH) model. These constraints are not the most stringent, compared to other recent upper limits of the neutrino mass such as reported in Refs. [34, 78, 72, 73, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. This is because in this work we consider the neutrino mass in a cosmological model with extra parameter (and also we use different observational data). In this work, we obtain the almost same values of for the three mass hierarchy cases, indicating that the current cosmological observations still cannot diagnose the mass hierarchy of neutrinos in a universe with the density perturbation parametrized by the growth index .
Iv Conclusion
We measure the growth index in a universe with massive neutrinos, through which we make a consistency test for GR. We employ the RSD measurements (eleven data points) to do the analysis. In order to constrain other cosmological parameters, we combine with the Planck 2015 CMB temperature and polarization data, the BAO data, the SN JLA data, the WL galaxy shear data, and the Planck 2015 CMB lensing data. We consider the both cases of sterile neutrino and active neutrino.
In the standard cosmology (the CDM+ model), we have , with the tension with the standard value of GR prediction at the 2.30 level. When massive sterile neutrinos are considered, i.e., in the CDM++ model, we obtain , with the tension relieved to be at the 1.48 level. This result shows that the consideration of massive sterile neutrinos although can lead to a smaller value of , is not capable of making the true measurement of be well consistent with the GR prediction. In this case, we have and eV. We also discuss the issue of other tensions of and , and make comparison with the latest direct measurement of the Hubble constant and the weak lensing measurements of KiDS450 and DES Y1.
We also consider the case with massive active neutrinos, and we obtain for the CDM++ (NH) model, for the CDM++ (DH) model, and for the CDM++ (IH) model. We find that the consideration of massive active neutrinos (no matter what mass hierarchy is considered) almost does not influence the measurement of the growth index . For the three cases, the tensions with the standard value of GR prediction are still at the , , and level, respectively. For the neutrino mass, we have eV for the CDM++ (NH) model, eV for the CDM++ (DH) model, and eV for the CDM++ (IH) model. We find that, in a universe with the density perturbation parametrized by the growth index , the current cosmological observations still cannot diagnose the mass hierarchy of neutrinos.
Acknowledgements.
This work was supported by the National Natural Science Foundation of China (Grants No. 11522540 and No. 11690021), the TopNotch Young Talents Program of China, and the Provincial Department of Education of Liaoning (Grant No. L2012087).References
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