Beyond six parameters: extending Cdm
Abstract
Cosmological constraints are usually derived under the assumption of a parameters CDM theoretical framework or simple oneparameter extensions. In this paper we present, for the first time, cosmological constraints in a significantly extended scenario, varying up to cosmological parameters simultaneously, including the sum of neutrino masses, the neutrino effective number, the dark energy equation of state, the gravitational waves background and the running of the spectral index of primordial perturbations. Using the latest Planck 2015 data release (with polarization) we found no significant indication for extensions to the standard CDM scenario, with the notable exception of the angular power spectrum lensing amplitude, that is larger than the expected value at more than two standard deviations even when combining the Planck data with BAO and supernovae type Ia external datasets. In our extended cosmological framework, we find that a combined Planck+BAO analysis constrains the value of the r.m.s. density fluctuation parameter to at c.l., helping to relieve the possible tensions with the CFHTlenS cosmic shear survey. We also find a lower value for the reionization optical depth at % c.l. respect to the one derived under the assumption of CDM. The scalar spectral index is now compatible with a HarrisonZeldovich spectrum to within standard deviations. Combining the Planck dataset with the HST prior on the Hubble constant provides a value for the equation of state at more than two standard deviations while the neutrino effective number is fully compatible with the expectations of the standard three neutrino framework.
pacs:
98.80.k 95.85.Sz, 98.70.Vc, 98.80.CqI Introduction
In the past twenty years, measurements of the Cosmic Microwave Background (CMB, hereafter) anisotropy angular power spectrum have witnessed one of the most impressive technological advances in experimental physics. Following the first detection of CMB temperature anisotropies at large angular scales by the COBE satellite in COBE (), the angular power spectrum has been measured with increasing precision by balloonborne experiments such as BOOMERanG boom (), MAXIMA maxima () and by groundbased experiments as DASI dasi (), showing the unambiguous presence of a ”first peak” and subsequent oscillations in the angular power spectrum at intermediate angular scales (). The spectacular measurements obtained by the WMAP satellite mission WMAP () have not only confirmed the presence of these acoustic oscillations but also provided the first precise measurement of the cross temperaturepolarization angular spectrum and the first constraints on the epoch of reionization. The very smallscale part of the angular temperature power spectrum, and especially the damping tail, has been accurately determined by experiments as ACT ACT () and SPT spt (). This impressive progress in the measurement of the CMB anisotropies temperature angular spectrum has culminated with the cosmicvariance limited measurements of the Planck experiment that has now also provided exquisite results on the polarization and cross temperaturepolarization spectra.
Despite this impressive progress on the experimental side, it is interesting to note that the constraints on cosmological parameters are still presented (as in the latest Planck 2015 data release, planckparams2015 ()) under the assumption of a simple CDM model, based on the variation of just cosmological parameters. While this model still provides a good fit to the data, it is the same model used, for example, in the analysis of the BOOMERanG data (see boom ()), i.e. more than fifteen years ago. While this ”minimal” approach is justified by the good fit to the data that the CDM provides we believe that it does not do adequate justice to the high quality of the most recent datasets. In light of the new precise data, some of the assumptions or simplifications made in the parameters approach are indeed not anymore fully justified. For example, fixing the total neutrino mass to zero or to some small value is completely arbitrary since we know that neutrinos must have masses and that current cosmological datasets are sensitive to variations in the absolute neutrino mass scale of order meV. At the same time, considering that a cosmological constant offers difficulties in any theoretical interpretation, it seems reasonable to incorporate in the analysis a possible dynamical dark energy component. This is certainly plausible (and even preferred if one wants to address the ”Why Now ?” problem), and indeed fixing the dark energy equation of state to is not favoured by any theoretical argument. Most inflationary models predict a sizable contribution of gravitational waves. Given the progress made in the search for Bmode polarization, especially by the recent combined BICEP2+Planck analysis BKP (), it is an opportune moment to allow any such contribution to be directly constrained by the data, without assuming a null contribution as in the 6parameter model. A similar argument can be made for the running of the scalar spectral index . Moreover, the neutrino effective number, could be easily different from the standard expected value of . Even assuming the standard three neutrino framework, nonstandard decoupling, inflationary reheating, dark matter decay and many other physical processes could alter its value. Finally, the Planck 2015 release still hints for an anomalous value for the lensing amplitude cala (). While this parameter is purely phenomenological, one should clearly consider it and check if the cosmology obtained is consistent with other datasets.
Clearly, one should be careful in neglecting these extensions following a motivation based simply on an Occam razor argument. Indeed having more parameters does not necessary means having a more complicated theory. For example, forcing the dark energy component to a cosmological constant means accepting an extreme fine tuning and a physical theory probably vastly more ”complicated” as those arising from a multiverse scenario.
The goal of this Letter is to constrain cosmological parameters in this extended parameter space.
Ii Method
As discussed in the introduction, besides the six parameters of the ”standard” CDM model, i.e. the Hubble constant , the baryon and CDM energy densities , the primordial amplitude and spectral index of scalar perturbations and (at pivot scale ), and the reionization optical depth , we also consider variations in additional parameters: the total mass for the standard neutrinos, , the dark energy equation of state assumed constant with redshift, the tensor/scalar ratio of amplitude at pivot scale , the running of the scalar spectral index , at pivot scale , the amplitude of the lensing signal in the CMB angular spectra, as defined in cala (), the effective number of relativistic neutrinos, . In what follows, we refer to this model as CDM (extended Cold Dark Matter).
We let all these parameters vary freely simultaneously in a range of external, conservative, priors listed in Table 1.
Parameter  Prior 

(eV)  
[3.5,0.5]  
[0.5,0.5]  
[0,0.5]  
[0.05,10]  
[0,10] 
Model  

Dataset  [km/s/Mpc]  
CDM  
Planck TT+LowP              
CDM  
Planck              
CDM  
Planck+ BAO              
CDM  
Planck TT+LowP  
CDM  
Planck  
CDM  
Planck+BAO  
CDM  
Planck+lensing  
CDM  
Planck+HST  
CDM  
Planck+JLA  
CDM  
Planck+WL  
CDM  
Planck+BAORSD  
CDM  
Planck+BKP  

We produce constraints on these cosmological parameters by making use of several, recent, datasets. Firstly, we use the full Planck 2015 release on temperature and polarization CMB angular power spectra. Since unresolved systematics may be present in the Planck 2015 polarization data at small angular scales (see discussion in plancklike2015 ()) we denote as Planck TT+lowP the Planck temperature data plus the low angular scale polarization, while with Planck the dataset that also includes the smallscale polarization data measured by Planck HFI.
We also include information on CMB lensing from Planck trispectrum detection (see plancklens2015 ()). We refer to this dataset as lensing. We add baryonic acoustic oscillation data from 6dFGS beutler2011 (), SDSSMGS ross2014 (), BOSSLOWZ anderson2014 () and CMASSDR11 anderson2014 () surveys as in planckparams2015 (). We refer to this dataset as BAO. We impose a constraint on the Hubble constant from the Hubble Space Telescope HST () dataset. Recently this constraint has been criticized in GHST () where a more conservative value was suggested, a choice adopted in the recent Planck analysis planckparams2015 (). Since the HST prior has been shown to be in tension with the Planck constraints on the Hubble constant obtained under CDM, we choose to use the less conservative HST () determination in order to investigate its compatibility in our extended scenario. We refer to this dataset as HST. We use luminosity distances of supernovae type IA from the ”Joint Lightcurve Analysis” derived from the SNLS and SDSS catalogs JLA (). We refer to this dataset as JLA. We add weak lensing galaxy data from the CFHTlenS WL () survey with the priors and conservative cuts to the data as described in planckparams2015 (). We refer to this dataset as WL. We consider redshift space distortions from RSD () with the prescription give in planckparams2015 (). We refer to this dataset as RSD. Finally, we include upper limits on CMB polarization modes as recently placed by a common analysis of Planck, BICEP2 and Keck Array data BKP (). We refer to this dataset as BKP.
We use the publicly available Monte Carlo Markov Chain package cosmomc Lewis:2002ah () with a convergence diagnostic based on the Gelman and Rubin statistic. We use the July 2015 version which includes support for the Planck data release 2015 Likelihood plancklike2015 () (see http://cosmologist.info/cosmomc/) and implements an efficient sampling using the fast/slow parameter decorrelations Lewis:2013hha (). While in this paper we will focus the attention on cosmological parameters, we also vary foreground parameters using the same technique and parametrization described in plancklike2015 () and planckparams2015 ().
Iii Results
The results of our analysis are reported in Table 2 where we also include, for comparison, the constraints obtained assuming the standard, parameters in CDM. The significant increase in the number of parameters produces, as expected, a relaxation in the constraints on the CDM parameters. Considering the allowable volume of the sixdimensional CDM parameter space to be proportional to the square root of the determinant of the parameter covariance, we find that moving from CDM to CDM expands this volume by a factor in case of the Planck dataset. The parameters that are mostly affected are the Hubble constant and the r.m.s. amplitude of density fluctuations that are now practically undetermined from Planck measuments alone and have significantly larger errors with respect to CDM even when external datasets such as BAO are included. The main reason for this relaxation is the inclusion in the analysis of the dark energy equation of state , that introduces a geometrical degeneracy with the matter density and the Hubble constant. Moreover, marginalizing over the lensing amplitude removes the lensing information in the CMB spectra that could potentially break this geometrical degeneracy. In this respect, it is interesting to note that a Planck+HST analysis provides a value for the equation of state less than the cosmological constant value at more than c.l.. As mentioned, the HST prior we adopt is not conservative. This result shows that in case the HST measurement will be confirmed by future data, the only parameter between the extra we consider that can be varied to acccomodate this tension is the dark energy equation of state . Increasing the neutrino number does not seem to offer a viable solution as was on the contrary the case with the previous Planck 2013 release.
Constraints on the baryon and cold dark matter densities, the scalar spectral index and the optical depth are also much weaker, mainly due to degeneracies between these parameters and and . Apart from the increase in the errors, it is interesting that parameters as and the optical depth are shifted toward lower values respect to CDM. These shifts are clear in Figures 1 and 2 where we plot the and c.l. contour plots in the vs and vs planes, respectively. This is mainly due to the anomalous value of and persists when external datasets as BAO, JLA, WL and RSD are included. Looking at the results in Table 2, the value of is always different from the standard value at more than c.l. when the Planck CMB dataset is combined with external datasets, with the only notable exception of the lensing information from the Planck trispectrum that pushes the value of back to agreement with unity. The nature of the Planck anomaly could be different from the lensing determination but since it also persists in our extended CDM scenario, this clearly deserves further investigation. Moreover, is the only parameter that hints at a tension with standard CDM. Again, by looking at Table 2, apart from , we see no evidence for ”new physics”: we just have (weaker) upper limits on the neutrino mass, the running of the spectral index is compatible with zero, the dark energy equation of state is compatible with (expect when we use the HST prior), and the neutrino effective number is remarkably close to the standard value . It is impressive that even in a parameter space, the neutrino effective number is still constrained with exquisite precision. This is mainly due to the inclusion of the Planck HFI small angular scale polarization data in the analysis. Removing this dataset, but keeping the low angular scale LFI polarization, we get a much weaker constraint from Planck+BAO of at c.l.. The Planck+BAO constraint on neutrino mass of eV at c.l. is significantly weaker with respect to the constraint eV at c.l.. obtained with the same dataset but assuming CDM. The constraint on the tensor/scalar amplitude is about a factor two larger than in CDM. However, when the BKP dataset is included, the c.l. upper limit of is recovered. This clearly shows how a measurement of primordial B modes is crucial to constrain the tensor amplitude in a modelindependent way. The inclusion of the BKP dataset affects only the constraint on the tensor amplltude and leaves the other constraints virtually unchanged.
Iv Conclusions
In this Letter we have presented, for the first time, constraints on cosmological parameters in the framework of an ”extended” cold dark matter model (CDM) that is based on parameters instead of the usual assumed in the CDM model. In this extension some of the parameters usually well constrained under CDM such as the Hubble constant and the amplitude of matter density fluctuations are now unconstrained by CMB observations. Combining the CMB data with several other datasets reveals no statistically significant evidence for any tensions. More specifically, we have found no evidence for ”new physics” beyond the standard CDM model, i.e. there is no island of parameters in our extended theoretical framework that could be preferred to the standard CDM territory. However CDM prefers a lower value of relative to that obtained for 6parameter CDM but still requires a slightly anomalous value of . The lower value of in CDM brings the Planck data in more agreement with the results of the CFHTlenS survey mcrann (). This result motivates further studies that could explain the physical nature of the anomaly.
The tension between the Planck and HST values of the Hubble parameter, in the CDM scenario is solved by a value of the dark energy equation of state while the neutrino effective number remains compatible with the standard value of .
Of course, the number of parameters can be further extended by considering, for example, nonzero curvature, isocurvature primordial perturbations, features in the primordial spectrum, a varying (with redshift) dark energy equation of state, nonstandard Big Bang Nucleosynthesis and a change in the primordial helium abundance , and so on. Further extensions however may be premature because of degeneracies. For example, most effects of varying curvature are degenerate with a variation in . CDM isocurvature modes have a spectrum similar to a GW background and the parameter could account for undetected features in the angular spectrum. Moreover, a change in could account for a change in .
We find it impressive that despite the increase in the number of the parameters, some of the constraints on key parameters are relaxed but not significantly altered. The cold dark matter ansatz remains robust, the baryon density is compatible with BBN predictions, and the neutrino effective number is compatible with standard expectations. The excellent quality of the new data motivates our exploration beyond the overexploited territory of CDM towards new and uncharted frontiers.
JS and EdV acknowledge support by ERC project 267117 (DARK) hosted by UPMC, and JS for support at JHU by National Science Foundation grant OIA1124403 and by the Templeton Foundation. EdV has been supported in part by the Institute Lagrange de Paris. AM acknowledge support by the research grant Theoretical Astroparticle Physics number 2012CPPYP7 under the program PRIN 2012 funded by MIUR and by TASP, iniziativa specifica INFN.
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