Bounds on Sum of Neutrino Masses in a 12 Parameter Extended Scenario with Non-Phantom Dynamical Dark Energy ()
We performed a Bayesian analysis and obtained constraints on a 12 parameter extended cosmological scenario including non-phantom dynamical dark energy (NPDDE hereafter) with CPL parametrization ( approach with at all times). Along with NPDDE we also include the six CDM parameters, number of relativistic neutrino species () and sum over active neutrino masses (), tensor-to-scalar ratio and running of the spectral index. We constrain the parameter space using different combinations of these latest datasets: Cosmological Microwave Background (CMB) Data from Planck 2015; Baryon Acoustic Oscillation (BAO) Measurements from SDSS BOSS DR12, MGS, and 6dFS; Supernovae Type Ia Luminosity Distance Measurements from the latest Pantheon Sample; CMB B-mode polarization data from BICEP/Keck collaboration (BK14); weak lensing measurements from CFHTLenS; Planck lensing data; and a prior on Hubble constant ( km/sec/Mpc) from local measurements (R16). We find that it is possible to constrain cosmological parameters effectively even in such an extended scenario. Especially, we found strong bounds on the sum of the active neutrino masses. Our strongest bound of 0.123 eV on comes from Planck+BK14+BAO. We also find that inclusion of the R16 prior leads to the standard value of being discarded at more than 68% C.L., implying a small preference for dark radiation.
a,b,1]Shouvik Roy Choudhury,11footnotetext: Corresponding author. c]Abhishek Naskar
Bounds on Sum of Neutrino Masses in a 12 Parameter Extended Scenario with Non-Phantom Dynamical Dark Energy ()
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi, Allahabad 211019, India
Homi Bhabha National Institute
Training School Complex, Anushaktinagar, Mumbai - 400094, India
Indian Statistical Institute, Kolkata
203 BT Road, Kolkata-700108, India
Recent observations suggest the universe can be modelled according to the six parameter CDM model, where structure formation is explained by cold dark matter physics (CDM) and recent acceleration of the universe is explained by vacuum energy which is the candidate for dark energy. There are however possible extensions to the standard CDM. Cosmic neutrino background and Inflationary Gravitational waves (IGWs/tensors) are theoretically well motivated. Among them, cosmic neutrino background () is indirectly confirmed by the CMB measurements of the Planck satellite  where the current preferred value of the effective number of extra radiation species at recombination, (68%, Planck TT+lowP+BAO) in a minimal , is very far away away from the value of . The theoretically predicted value of  considering three active neutrinos as the only relativistic species apart from photons during recombination, is completely compatible with this bound, implying consistency with CDM. In standard model of particle physics, neutrinos are massless. But terrestrial neutrino oscillation experiments have strongly confirmed that neutrinos have small masses. While strongest upper bounds on the sum of masses of the three active neutrino mass eigenstates) come from cosmology, it is still unable to provide any lower bound, indicating that the standard model assumption of is consistent with current data. For instance, Planck collaboration  provided a bound of eV in the minimal . Again, while is indirectly detected, existence of IGWs is still to be confirmed. The main probe for IGWs is the CMB B-mode polarization, and the corresponding important parameter is the tensor-to-scalar ratio (). The currently available observations can only put an upper bound on the tensor to scalar ratio: (95% C.L.; at a pivot scale of ) , implying that is consistent with current data.
While CDM has its success there are also parameter tensions between CMB and non-CMB data within the CDM model. One of the most important limitations of CDM is that high redshift (CMB) and low redshift (local universe) measurements gives different values of Hubble constant. The Planck 2016 intermediate results  provide km/sec/Mpc (68% C.L.) in CDM (with fixed at 0.06 eV) and recent direct measurement gives km/sec/Mpc (68% C.L., hereafter R16) . There is a inconsistency between these datasets. Recent strong lensing observations from the H0LiCOW program  provides km/sec/Mpc (68% C.L.) and partially confirms the tension. CMB data also has tensions in the measurements of and with x-ray galaxy cluster measurements  or cosmic shear surveys like CFHTLenS  and KiDS-450 . For instance, the KiDS-450 survey measures a combined quantity which has a 2.3 tension with Planck, which prefers a much higher value of .
Apart from inconsistencies among high and low redshift datasets, there are several internal inconsistencies in the Planck data itself. Parameter estimations in CDM differ when considering small scale () and high or intermediate scale () temperature data separately . This is especially true for the measured value of which is much lower when obtained from the data than when obtained from the data. Another puzzling inconsistency in CDM with Planck data is that the latest measurement of lensing parameter by Planck 2016 re-analysis (95% C.L.) in a model  is 2 level higher than CDM prediction of .
A possible explanation for these tensions is the systematics of the observations. But it is also possible that we need physics beyond CDM and standard model of particle physics. These inconsistencies in model and different datasets have motivated several studies of cosmological scenarios in extended parameter spaces [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Recent studies have also analyzed models with as large as twelve parameters [11, 12, 13]. The motivation behind studying such a large parameter space is that currently seems to be an over-simplification. Indeed, there is no reason to fix to 0.06 eV, since it is only approximately the minimum sum of masses required for normal hierarchy of neutrinos and this mass might not be an accurate one. Massive neutrinos produce distinct effects on CMB and large scale structure data and this has been widely studied [39, 40, 41, 42, 43, 44]. Again, the discrepancy with Planck and R16 might be explained by a dark radiation species contributing to  and varying is useful if we take the conflict between Planck and R16 at face value. Similarly, existence of tensor perturbations are theoretically well motivated and there seem no reason to not to include them in a analysis.
Apart from massive neutrinos and tensors, another extension to CDM which has recently received a lot of attention is dynamical dark energy, where the dark energy equation of state (EoS) is not fixed at , but is varied with time. In this analysis we have followed this approach with non-phantom dark energy only ( where is the cosmological redshift). We specifically consider a non-phantom scenario since in a universe with phantom dark energy (), the dark energy density reaches infinity in a finite time leading to dissociation of all bound states, i.e., the so called Big Rip, and seems unphysical in that sense [45, 46]. While the data currently favours a phantom dark energy  Dark energy models with a single scalar field are not able to go across the line (i.e., the phantom barrier) and more general models that allow it demand extra degrees of freedom to supply stability gravitationally . Phantom dark energy accommodating field theories are usually plagued with one or more of the following problems like Lorentz violation, unstable vacuum, superluminal modes, ghosts, non-locality, or instability to quantum corrections. On the other hand, however, it is possible to make theories free of such abnormalities by using effects like photon-axion conversion or modified gravity which leads to an apparent (see  for a brief review). Nonetheless, single scalar field theories like quintessence [49, 50, 51] are non-phantom in nature and in this work we limit ourselves to such theories.
In this work we have first considered a 12 parameter extended scenario with 6 usual parameters, two dynamical dark energy parameters ( approach, CPL parametrization) with , two neutrino parameters ( and ), and two inflationary parameters ( and the running of the spectral index, ). We performed a Bayesian analysis to constrain parameters using different combinations of latest available datasets: (1) Cosmological Microwave Background temperature and polarization data from Planck 2015; (2)the latest data released from the BICEP/Keck Collaboration for the BB mode of the CMB spectrum (BK14); (3) Baryon Acoustic Oscillation Measurements from SDSS III BOSS DR12, MGS and 6dFGS; (4) Supernovae Type Ia Luminosity Distance Measurements from the newly released Pantheon Sample, (5) Weak lensing measurements from CFHTLenS; (6) Planck 2015 lensing data; and (7) the R16 Gaussian prior ( km/sec/Mpc) on Hubble constant. Next we turned off the tensor perturbations (i.e., removed ) and constrained this 11 parameter scenario with the same datasets except BK14. Finally we add a new parameter and again constrain this 12 parameter expended space with the mentioned datasets. We emphasize here that this is the first time someone has evaluated the non-phantom dark energy scenario in a 12 parameter extended space. Our main focus in this paper is on sum of neutrino masses, however we provide the constraints on all the varying parameters.
This paper is arranged as follows: in section 2 we describe the cosmological models used in this paper and the prior ranges of parameters used, along with a brief description of the CPL parametrization. In section 3 we briefly describe the datasets used in this work. In section 4 we present our analysis results. We provide a discussion and summary in section 5. The main results are in tables 2, 4, and 5.
In this work we have considered 3 different cosmological scenarios to obtain bounds on the cosmological parameters. Below we list the vector of parameters to vary in each of these cosmological scenarios.
For NPDDE11+ model with 12 parameters:
For NPDDE11 model with 11 parameters:
For NPDDE11+ model with 12 parameters :
In this analysis, the first model, NPDDE11+, comprises of six additional parameters on top of CDM model. The six parameters of CDM are: present day cold dark matter energy density , present day baryon energy density , reionization optical depth , spectral tilt and amplitude of primordial scalar power spectrum and (evaluated at pivot scale ) and is the ratio between the sound horizon and the angular diameter distance at decoupling.. For our analysis we are adding the following parameters: two dark energy parameters and , effective number of relativistic species at recombination , total neutrino mass , the tensor-to-scalar ratio (evaluated at pivot scale ) and the running of spectral index of primordial power spectrum (). In this model, the gravitational lensing amplitude of the CMB angular spectra is fixed at the CDM predicted value of unity.
We also consider two other scenarios. In the NPDDE11 model, we do not run the tensor perturbations and constrain the parameter space considering scalar only perturbations. In the NPDDE11+ model we also allow the parameter to vary. This is since the cause of the -anomaly is unknown and therefore it is important to look into the effect of varying on the constraints of rest of the parameter space.
CPL Parametrization: For dark energy dynamics we use the famous Chevallier-Polarski-Linder (CPL) parametrization [52, 53] which uses a varying equation of state in terms of the redshift and two parameters and :
This uses the Taylor expansion of the equation of state in powers of the scale factor and takes only the first two terms. Here is the dark energy EoS at present day (), whereas is the dark energy EoS in the beginning of the universe; and is a monotonic function between these two times. Therefore, to constrain only the NPDDE region of the parameter space i.e. it is enough to apply these hard priors:
For the cosmological parameters mentioned in eqs. 2.1–2.3, we have assumed flat priors which are listed in table 1, along with hard priors given in eq. 2.5. We obtain the constraints using the Markov Chain Monte Carlo (MCMC) sampler CosmoMC  which uses CAMB  as the Boltzmann code and the Gelman and Rubin statistics  to estimate the convergence of chains.
Below, we provide a description of the datasets used in our analyses. We have used different combinations of these datasets.
Cosmic Microwave Background: Planck 2015:
We have used measurements of the CMB temperature, polarization, and temperature-polarization cross-correlation spectra from the Planck 2015 data release . We use a combination of the high- (30 2508) and low- (2 29) TT likelihood. Along with that, we also include the high- (30 1996) EE and TE likelihood and the low- (2 29) polarization likelihood. We refer to this whole dataset as Planck.
Baryon Acoustic Oscillations (BAO) Measurements:
We use measurements of the BAO signal obtained from different galaxy surveys in this work. We include the SDSS-III BOSS DR12 Consensus sample ( which includes LOWZ and CMASS galaxy samples at 0.38, 0.51 and 0.61). Along with it, we also include the DR7 MGS at , and the 6dFGS survey at . We denote this full combination as BAO. Here is the effective redshift of a survey.
Luminosity Distance Measurements from Type Ia Supernovae (SNe Ia):
We also use Supernovae Type-Ia (SNe Ia) luminosity distance measurements from the Pantheon Sample . It comprises of data from 279 Pan-STARRS1 (PS1) Medium Deep Survey SNe Ia () and distance estimates of SNe Ia from SDSS, SNLS, various low-z and HST samples. This combined sample comprises of data from a total of 1048 SNe Ia with a redshift range of and is the largest one till date. We refer to this data as PAN from now on. This dataset supersedes the Joint Light-curve Analysis (JLA) sample which comprises of information from 740 spectroscopically confirmed SNe Ia .
BB Mode Spectrum of CMB:
We use the latest data available from BICEP/Keck collaboration for the B mode polarization of CMB, which includes all data (range: ) taken up to and including 2014 . This dataset is denoted as BK14.
Hubble Parameter Measurements:
We use a Gaussian prior of km/sec/Mpc on . This result is a recent 2.4% determination of the local value of the Hubble parameter by  which combines the anchor NGC 4258, Milky Way and LMC Cepheids. We denote this prior as R16.
Planck Lensing Measurements:
We also use the lensing potential measurements via reconstruction through the four point functions of Planck 2015 measurements of CMB . We simply refer to this data as lensing.
Weak Lensing Measurements from CFHTLenS:
We have split the results in the three smaller sections for the three different models we have studied. The description of models and datasets are given at section 2 and section 3 respectively. We have presented the results in the following order: first the NPDDE11+ model, then the NPDDE11 model and lastly the NPDDE11+ model. All the marginalized limits quoted in the text or tables are at 68% C.L. whereas upper limits are quoted at 95% C.L.
4.1 Npdde11+ model
Bounds on the NPDDE11+ model parameters are presented in table 2 while the bounds on the model parameters are presented in table 3. We do not include the bounds from CMB only data as the bounds are not strong enough in the NPDDE11+ model model, a finding that corroborates with a recent study  which had varied the dark energy EoS in both phantom and non-phantom regions. However adding either BAO or R16 with CMB data seems to provide strong bounds on cosmological parameters. Comparing with the bounds on the parameters in the model however we can see that the 68% C.L. spreads of the relevant parameters have increased to different degrees for different parameters. This is an expected phenomenon given the number of parameters has been doubled. Overall the six parameters have been estimated in the NPDDE11+ model with reasonable spreads, showing that it is possible to constrain cosmology effectively in a large parameter space with current datasets.
We also find tight bounds on in this model. The 1-D posteriors for and are given in figure 1. Our most aggressive bound in this paper is found in this model with Planck+BAO dataset: 0.123 eV which is very close to the minimum mass of 0.1 eV required for inverted hierarchy of neutrinos (for normal hierarchy, the minimum required is around 0.06 eV) . Without the BAO data, only Planck and BK14 together provide a bound of eV whereas only using Planck in the same model gives us a bound of eV which is incidentally very close to the bound of eV reported by Planck collaboration  using the same data in the minimal model. Recent studies [33, 38] in smaller parameter spaces have shown that the models comprising of NPDDE provide stronger bounds on than , because of a degeneracy present between the dark energy EoS and  which leads to the phantom region of the dark energy parameter space preferring larger masses and the non-phantom region preferring smaller masses. However, cosmological datasets usually prefer the phantom region more when the dark energy EoS is allowed to vary both in the phantom and non-phantom regions, which usually leads to weaker bounds on . This work shows that even as a 12 parameter model, the NPDDE11+ is very efficient in constraining . Contrary to what happens in lower dimensional parameter spaces, the R16 prior does not lead to stronger bounds on , as the correlation between and is very small in this model. However we found a strong positive correlation still present with , which leads to a large increase in the value of with the use of R16 prior (the correlations can be visualized in figure 2). Indeed, while Planck+BK14+BAO prefers a km/sec/Mpc and , the inclusion of the R16 prior to this data combination leads to higher values of km/sec/Mpc and both. This has an important consequence that the standard value of is excluded at 68% C.L., favouring a dark radiation component, albeit it being allowed in 95% C.L. This is a general feature in all the dataset combinations that include the R16 prior in this model, signifying the large tension present between Planck and R16. The R16 prior also prefers higher values of . This model does not help the conflict between Planck and CFHTLenS regarding the value of . Visual depiction of this can be found in figure 3 in the plane. Inclusion of the lensing and WL data lead to worsening of the mass bounds whereas bounds on are almost unaffected. These datasets however lower the preferred values.
The SNe Ia luminosity distance measurements provide information about evolution of luminosity distance as a function of redshift ( for the Pantheon sample). This can be used to measure the evolution of the scale factor  and is helpful in constraining the dark energy EoS. We found that addition of the PAN data did help in constraining the dark energy parameters more tightly. For Planck+BK14+BAO, we have a bound of (95% C.L.), which shrinks to (95% C.L.) with the addition of PAN. On the other hand, Planck+BK14+BAO produces a bound of (68% C.L.), whereas Planck+BK14+BAO+PAN leads to (68% C.L.). We see that the 68% deviations of have shrunk. This has also been depicted in figure 4. The R16 prior also has similar but less strong effect. With Planck+BK14+BAO+R16 we have 0.908 and . In all cases we found that the cosmology is compatible with a cosmological constant (i.e., , ).
As far as values of the tensor-to-scalar ratio is concerned, we find that if we run the chains without the BK14 data, we get a bound of 0.155 with Planck+BAO, which is higher than the bound of 0.12 set by Planck collaboration . However, inclusion of the BK14 data leads to a bound of 0.075, which is close to the limit set by the BICEP/Keck collaboration . The value of remains almost unchanged across all the datasets as long as the BK14 data is included.
Dynamical dark energy models which encompass both the phantom and non-phantom region, usually find better fit than . Here we compare the NPDDE11+ model with . We define, , when defined for the same dataset. For Planck data only, . For Planck+BAO, we find, . For Planck+R16, we get, . This shows that while Planck data itself does not provide a better fit, combining with low redshift data like BAO and R16 leads to improvements in fit compared to . Note that we do not include the BK14 data in this computation.
4.2 NPDDE11 model
In this section we consider the NPDDE11 model where we turn off the tensor perturbations and also do not include the BK14 data. This does not affect the bounds much as can be seen from table 4 and comparing with table 2, which verifies the stability of the results in a smaller parameter space.
The 1-D posteriors for and for selected datasets are given in figure 5. We again find strong bounds on the sum of neutrino masses. We notice that the removal of BK14 data has a small effect on which persists over different datasets. For instance, in NPDDE11+, for Planck+BAO, we find a eV, which is reduced to eV when we add the BK14 data. In the NPDDE11, this bound is eV with Planck+BAO, which is our best bound in this model. The strengthening of the bound from NPDDE11+ to NPDDE11 with Planck+BAO might simply be due to reduction in the parameter space volume. On the other hand it seems BK14 prefers a lower . However even then the changes are small. BK14 data also seems to prefer slightly larger values of , thereby increasing the tension with CFHTLenS.
Also, the inclusion of R16 prior again seems to discard the standard value of at 68% C.L. but again, not at 95% C.L., and also it doesn’t lead to stronger , as before in the NPDDE+ model, due to a large positive correlation between and but a only small correlation between and . This can be visualized in figure 6. The PAN dataset provides stricter bounds on and , as before. We depict that in figure 7.
Here we compare the NPDDE11 model with . As before, we define, , when defined for the same dataset. For Planck data only, . For Planck+BAO, we find, . For Planck+R16, we get, . Here again, with only Planck the fit is worse than . With Planck+BAO, the fit is almost the same, whereas, with Planck+R16 a better fit is found.
4.3 Npdde11+ model
We present the limits on the cosmological parameters in table 5. A number of important changes happen with the introduction of the new varying parameter . Considering that our main goal in this paper is to constrain neutrino masses, we see a substantial relaxation in the bounds on . In previous cases we had fixed 1. However now that is varied we find that the data prefers a large and discards the value of at more than 95% C.L. (except in case of inclusion of Planck lensing data, which prefers a much lower , implying a tension between Planck and lensing). The increasing of the lensing amplitude has the same effect as the decreasing of . So to compensate for the increase in , the neutrino masses are also increased. The 1-D plots for and for selected datasets are given in figure 8. In this model, the Planck data is almost insensitive to neutrino masses eV. Our tightest bound of eV again comes with Planck+BAO data. This bound, while weaker than the previous cases, is still close to the eV bound provided by Planck collaboration . The preferred values are also higher in NPDDE11+ compared to the previous cases. The addition of the R16 data leads to even higher which leads to the value being disallowed even at 95% C.L. with Planck+R16, for which the 68% and 95% limits are and respectively. This signifies the presence of tension between Planck and R16 in this model, as it was in previous models.
Another important change is the change in bounds on the optical depth to reionization, . With Planck+BAO, the NPDDE11 model preferred a value of , whereas this model prefers , which is actually closer to the bound of given by Planck 2016 intermediate results . Again, while the NPDDE11+ and NPDDE11 models failed to reconcile Planck with weak lensing measurements like CFHTLenS, the NPDDE11+ model prefers lower values of and the agreement with CFHTLenS is considerable. This can be visualized in figure 9. The bounds on the dynamical dark energy parameters are weaker than in the other two models. The cosmological constant is however compatible with the data even in this model.
Here we compare the NPDDE11+ model with . As before, we define, , when defined for the same dataset. For Planck data only, . For Planck+BAO, we find, . For Planck+R16, we get, . Unlike the previous two models, here we get a better fit even with only Planck data. And then there is a substantial improvement in fit with the R16 data.
In this work we have studied three different extended cosmological scenarios with non-phantom dynamical dark energy (NPDDE) with a focus on constraining sum of neutrino masses. We have presented bounds on all the varying parameters in these extended scenarios and described the main effects we observed. In the first model, NPDDE11+, we consider 12 parameters: the 6 parameters, two dynamical dark energy parameters with CPL parametrization ( and ) with hard priors to satisfy the non-phantom requirement, number of effective relativistic neutrino species at recombination ( and sum of neutrino masses (), and the running of the inflation spectral index () and the tensor-to-scalar ratio (). We used different combinations of recent datasets including Planck 2015 temperature and polarization data, CMB B-mode spectrum data from BICEP/Keck collaboration (BK14), BAO SDSS III BOSS DR12, MGS and 6dFS data, SNe Ia Pantheon sample (PAN), the R16 prior (), the data from Planck lensing and CFHTLenS weak lensing survey (WL). We found that CMB only data is not very effective in constraining the cosmological parameters. However use of BAO or R16 improved the constraints considerably with providing slightly better fits than . The 1 deviations for the parameters were however increased in this model compared to due to the doubling of number of parameters. Our best bound on neutrino masses in this model came from Planck+BK14+BAO: eV which is a strong bound close to the minimum mass of 0.1 eV required for inverted hierarchy of neutrino masses. We also found that inclusion of the R16 prior lead to a preference for dark radiation at 68% C.L. but not at 95%, while without the R16 prior the data is completely consistent with the standard value of . This model did not improve the tension present in the plane between Planck and CFHTLenS. All combinations of data are also compatible with a cosmological constant (). The Pantheon sample improved the bounds on the dark energy parameters.
We tested the stability of these results in a lower parameter space (model:NPDDE11) where we turned off the tensor perturbations and also did not use the BK14 data. We found that the general conclusions made for NPDDE11+ were also true in this model. The tightest bound of 0.126 eV also came from Planck+BAO. This model however provided worse fits than the NPDDE11+ model.
Finally we studied the NPDDE11+ model where we also varied the lensing amplitude. We found that the fits were considerably better than with a for Planck+R16. We found that except when Planck lensing data is included, the value predicted by was rejected at more than 95% C.L. by the datasets. Due to this, the bounds also worsened with our best result in this model: eV coming from Planck+BAO again. This result is, however, still close to the eV bound by Planck collaboration , showing that the NPDDE sector is effective in constraining neutrino masses. The R16 prior also preferred a dark radiation component but this time also at 95% C.L. level, as this model also prefers higher values of . On the other hand, we found that this model helps relieve the tension between Planck and CFHTLenS considerably.
We would like to add a final remark that we have obtained the bounds while taking the datasets at face value. However unresolved systematics present in the dataset could have affected our results and conclusions. For instance the tension between Planck and R16 prior can be due to a dark radiation species, but can also be due to systematics present in both the datasets. Thus there is still a lot to learn about robustness of datasets and also about dynamics of dark energy.
SRC thanks the cluster computing facility at HRI (http://cluster.hri.res.in) and the Department of Atomic Energy (DAE) Neutrino Project of HRI. AN thanks ISI Kolkata for financial support through Senior Research Fellowship. The authors also thank Archita Bhattacharyya for valuable help. This project has received funding from the European Union’s Horizon 2020 research and innovation programme InvisiblesPlus RISE under the Marie Sklodowska-Curie grant agreement No 690575. This project has received funding from the European Union’s Horizon 2020 research and innovation programme Elusives ITN under the Marie Sklodowska-Curie grant agreement No 674896.
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