Constraints on Neutrino Mass and Light Degrees of Freedom in Extended Cosmological Parameter Spaces

Constraints on Neutrino Mass and Light Degrees of Freedom
in Extended Cosmological Parameter Spaces

Shahab Joudaki Center for Cosmology, Dept. of Physics & Astronomy, University of California, Irvine, CA 92697
August 24, 2019
Abstract

From a combination of probes including the cosmic microwave background (WMAP7+SPT), Hubble constant (HST), baryon acoustic oscillations (SDSS+2dFGRS), and supernova distances (Union2), we have explored the extent to which the constraints on the effective number of neutrinos and sum of neutrino masses are affected by our ignorance of other cosmological parameters, including the curvature of the universe, running of the spectral index, primordial helium abundance, evolving late-time dark energy, and early dark energy. In a combined analysis of the effective number of neutrinos and sum of neutrino masses, we find mild (2.2) preference for additional light degrees of freedom. However, the effective number of neutrinos is consistent with the canonical expectation of 3 massive neutrinos and no extra relativistic species to within when allowing for evolving dark energy and relaxing the strong inflation prior on the curvature and running. The agreement improves with the possibility of an early dark energy component, itself constrained to be less than 5% of the critical density (95% CL) in our expanded parameter space. In extensions of the standard cosmological model, the derived amplitude of linear matter fluctuations is found to closely agree with low-redshift cluster abundance measurements. The sum of neutrino masses is robust to assumptions of the effective number of neutrinos, late-time dark energy, curvature, and running at the level of 1.2 eV (95% CL). The upper bound degrades to 2.0 eV (95% CL) when further including the early dark energy density and primordial helium abundance as additional free parameters. Even in extended cosmological parameter spaces, Planck alone could determine the possible existence of extra relativistic species at confidence and constrain the sum of neutrino masses to 0.2 eV (68% CL).


I Introduction

Observations of the cosmic microwave background (CMB) Spergel:2003cb ; Spergel:2006hy ; Komatsu ; Komatsu:2010fb , large-scale structure Tegmark:2003uf ; Tegmark:2006az ; Cole:2005sx , and type Ia supernovae (SNe) RiessSNe ; Perlmutter have established a flat CDM model, with nearly scale-invariant, adiabatic, Gaussian primordial fluctuations as providing a consistent description of the global properties of our universe. At the same time, we do not yet understand the microscopic identities of the dark energy (), cold dark matter (CDM), and inflaton (primordial fluctuations) that enter our standard cosmological model.

Parameter Symbol Prior
Baryon density
Dark matter density
Angular size of sound horizon
Optical depth to reionization
Scalar spectral index
Amplitude of scalar spectrum
Effective number of neutrinos
Sum of neutrino masses
Constant dark energy EOS
Running of the spectral index
Curvature of the universe
Primordial helium abundance
Present dark energy EOS
Derivative of dark energy EOS
Early dark energy density
Table 1: We impose uniform priors on the above cosmological parameters. In addition, we always consider the Poisson point source power , the clustered power , and the SZ power as nuisance parameters constrained by the CMB data Keisler:2011aw . Moreover, we always derive , the amplitude of linear matter fluctuations on scales of at . Beyond a constant dark energy equation of state (EOS), we also consider a time-varying expansion , and an early dark energy model described in Sec. III.4. In this table, the first 6 parameters are defined as “vanilla” parameters.
CDM CDM CDM CDM CDM CDM
++ ++ +++
Primary
Extended
Derived
  • Mean of the posterior distribution of cosmological parameters along with the symmetric 68% confidence interval about the mean. We report the 95% upper limit on the sum of neutrino masses . The primordial helium mass fraction is enforced consistent with standard BBN unless we allow it to vary as a free parameter.

Table 2: Constraints on Cosmological Parameters using SPT+WMAP++BAO.

The neutrino sector is another area that the standard model is yet unable to fully describe, with open questions related to the effective number of neutrinos and their masses . The effective number of neutrinos is sensitive to both the number of neutrinos along with additional particle species that were relativistic at the photon decoupling epoch (e.g. Komatsu ; Mangano:2005cc ). A joint analysis of CMB data from WMAP7 with baryon acoustic oscillation (BAO) distances from SDSS+2dF and Hubble constant from HST reveals a weak preference for extra relativistic species (Komatsu:2010fb . When further combined with small-scale CMB data from ACT or SPT, this preference mildly increases and reaches the level ( with addition of ACT dunkleyact and with addition of SPT Keisler:2011aw ; further see Hamann:2010bk ; Archidiacono:2011gq ; Riess:2011yx ; Smith:2011es ; Hou:2011ec ; Hamann:2007pi ; Hamann:2011hu ; GonzalezMorales:2011ty ; Calabrese:2011hg ; Smith:2011ab ; Hamann:2011ge ; Giusarma:2011ex ; Giusarma:2011zq ; Fischler:2010xz ; deHolanda:2010am ; Nakayama:2010vs ; Joudaki:2012uk ).

A primary objective of this manuscript is to clarify how robust these recent indications of additional light degrees of freedom are to assumptions of the underlying cosmology, in particular to alternative models of the dark energy, curvature of the universe, running of the spectral index, primordial helium abundance, and to the sum of neutrino masses, which we know is nonzero from neutrino oscillation experiments hirata92 ; davis68 ; Eguchi:2002dm ; Ahn:2002up .

Constraining with cosmology is mainly achieved through tight CMB measurements of the redshift at matter-radiation equality , the baryon density , the angular size of the sound horizon , and the angular scale of photon diffusion  Hou:2011ec . Keeping and fixed as increases can be achieved by increasing the dark matter density (assuming massless neutrinos), which manifests in a large correlation with (shown in Fig. 2). Meanwhile, an increase in and both yield an enhanced Silk damping effect Hu:1996vq ; Hu:1995fqa ; Bashinsky:2003tk ; Hou:2011ec , and by fixing it can be shown that  Hou:2011ec , where is proportional to . As a consequence, the suppression of the CMB damping tail can be picked out as a signature of extra relativistic species when is known, while the constraints on are relaxed when allowing for as a free parameter.

An increase in further shifts the acoustic peak locations Bashinsky:2003tk , but this has been shown to be a small effect Hou:2011ec . Instead, the constraint on can be improved by the inclusion of low-redshift distances and a prior on the Hubble constant, , as these are useful in constraining and by extension . However, when allowing for evolving dark energy, the ability to improve constraints on from observations of the expansion history becomes diminished, as illustrated by the error ellipses for in Fig. 2. Therefore, the inclusion of SN data becomes critical to a precise determination of the effective number of neutrinos.

CDM CDM CDM CDM CDM CDM
++ ++ ++ ++ +++
+ + +
Primary
Extended
Derived
  • Same as Table 2 but with the addition of . Due to the large correlation between and at our pivot scale , we quote values for at a less correlated scale . For the “CDM” case where is closest to to the boundary at 3, we also considered a run where we impose a hard prior of . Here, we find , where the two sets of upper and lower boundaries denote 68% and 95% CLs, respectively. The changes to the sum of neutrino masses and other parameters that weakly correlate with are small (). While all within 1, the largest changes are seen in (compared to ), (compared to ), (compared to ), (compared to ), and (compared to ). This particular configuration of parameter space and datasets shows the largest extent to which parameters may change with an prior as compared to our other runs. The changes to the parameters are more modest when including SNe because of the preference for larger values of , as seen in Table 4.

Table 3: Constraints on Cosmological Parameters using SPT+WMAP++BAO.

The dark energy equation of state (EOS) is moreover anti-correlated with the sum of neutrino masses Hannestad:2005gj ; Komatsu ; Ichikawa:2004zi ; Komatsu:2010fb . In the CMB temperature power spectrum, the sum of neutrino masses shifts the first peak position to lower multipoles by changing the fraction of matter to radiation at decoupling, which can be compensated by a reduction in the Hubble constant (similar to the case for positive universal curvature) Hannestad:2003xv ; Ichikawa:2004zi ; Komatsu . BAO distances and an prior can therefore be used to reduce correlations between the sum of neutrino masses and the dark energy EOS, but also the curvature density.

The strongest limits on the sum of neutrino masses from the CMB combined with probes of the expansion history and matter power spectrum place it at sub-eV level Hannestad:2003xv ; Seljak:2006bg ; Goobar:2006xz ; Ichiki:2008ye ; Tereno:2008mm ; Komatsu ; Komatsu:2010fb ; Reid:2009xm ; Thomas:2009ae ; dePutter:2012sh ; Spergel:2003cb ; Spergel:2006hy ; Benson:2011ut ; RiemerSorensen:2011fe ; Reid:2009nq ; Tegmark:2003ud ; Crotty:2004gm ; Barger:2003vs ; Allen:2003pta ; Elgaroy:2004rc ; Hannestad:2006mi ; Xia:2012na . We take the conservative approach in only combining CMB data with low-redshift measurements of the expansion history. While SN observations play an important role in constraining the dark energy EOS and thereby reduce the correlation between and , these observations are not powerful in constraining the curvature of the universe and therefore less helpful in reducing the correlation between and (e.g. RiessSNe ; Perlmutter ; Komatsu ).

Beyond the vanilla parameters and the three additional parameters , we relax the commonly employed strong inflation prior on the universal curvature and running of the spectral index . Given that most popular models of inflation predict  Kosowsky:1995aa ; Baumann:2008aq and (e.g. liddlelyth ; weinberg ; Baumann:2008aq ), at the level of precision of present CMB data it is generally justified to fix these two parameters to their fiducial values of zero. However, given the mild preference for  Komatsu:2010fb ; dunkleyact ; Keisler:2011aw , we allow for the possible existence of inflationary models with large curvature or running. In particular, may be generated in models of open inflation in the context of string cosmology Freivogel:2005vv ; Baumann:2008aq , while a large negative running may be produced by multiple fields, temporary breakdown of slow-roll, or several distinct stages of inflation Easther:2006tv ; Burgess:2005sb ; Joy:2007na ; Baumann:2008aq .

Among alternatives to the cosmological constant with , the most popular are scalar field models with potentials tailored to give rise to late-time acceleration and current equation of state for the dark energy, , close to -1 Ford:1987de ; Ratra:1987rm ; Wetterich88 ; Peebles:1987ek ; Zlatev ; Ferreira ; Caldwell:1997ii ; Chiba . Like a cosmological constant, these models are fine-tuned to have dark energy dominate today. However, the requirement currently, does not imply that dark energy was negligible at earlier times, specifically redshift , where we have no direct constraints. Given the degeneracy between dark energy and the sum of neutrino masses, we further consider a model that describes dark energy as non-negligible in the early universe in Sec. III.4.

We describe our analysis method in Section II. In Section III, we provide constraints on a CDM model with three massive neutrinos and additional light degrees of freedom, then follow up with successive additions of a constant dark energy equation of state, universal curvature, running of the spectral index, and primordial helium abundance (all parameters defined in Table 1). We also explore the constraints for a time-varying dark energy equation of state, including an early dark energy model. Lastly, we compare the constraints from present data to the constraints expected from the Planck experiment. Section IV concludes with a discussion of our findings.

CDM CDM CDM CDM
++ ++ +++
+ +
Primary
Extended
Derived
  • Same as Table 3 but with the addition of supernova distance measurements from the Union2 compilation. For the case “CDM+,” we also considered a run where we impose a hard prior of . Here, we find , where the two sets of upper and lower boundaries denote 68% and 95% CLs, respectively. The largest changes this prior induces in other parameters are in (compared to ), (compared to ), and (compared to ). All other parameters are modestly affected by our choice of prior ().

Table 4: Constraints on Cosmological Parameters using SPT+WMAP++BAO+SNe.

Ii Methodology

We employed a modified version of CosmoMC Lewis:2002ah ; cosmomclink in performing Markov Chain Monte Carlo (MCMC) analyses of extended parameter spaces with CMB data from WMAP7 Komatsu:2010fb and SPT Keisler:2011aw , BAO distance measurements from SDSS+2dFGRS Percival:2009xn , the Hubble constant from HST Riess:2011yx , and SN distances from the SCP Union2 compilation Amanullah:2010vv . In determining the convergence of our chains, we used the Gelman and Rubin statistic gelmanrubin , where is defined as the variance of chain means divided by the mean of chain variances. To stop the runs, we generally required the conservative limit , and checked that further exploration of the tails does not change our results.

The CMB temperature and E-mode polarization power spectra were obtained from a modified version of the Boltzmann code CAMB LCL ; camblink . We approximated the effect of a dark energy component with a time-varying EOS by incorporating the PPF module by Fang, Hu, & Lewis (2008) Fang into CosmoMC. Given that the small scale CMB measurements of SPT come with much smaller error bars than ACT dunkleyact ; Keisler:2011aw , the further inclusion of the ACT dataset would not lead to significant improvements in our constraints, as we explicitly checked.

All parameters are defined in Table 1. In our analyses, we always include the “vanilla” parameters, given by the full set . Our constraints correspond to the mean of the posterior distribution of cosmological parameters along with the symmetric 68% confidence interval about the mean. We impose uniform priors on the cosmological parameters, and let the prior ranges to be significantly larger than the posterior, such that the parameter estimates are unaffected by the priors. When including the sum of neutrino masses and early dark energy density, we report the 95% upper limit on these parameters.

When allowing for nonzero neutrino rest mass, we distribute the sum of neutrino masses () equally among 3 active neutrinos. We treat additional contributions to as massless, such that , where denotes the massless degrees of freedom. In principle, these additional states could be massive, for example see recent treatments in Refs. Hamann:2010bk ; Hamann:2011ge ; Joudaki:2012uk .

Since we impose , the number of relativistic species is always positive at early times. At late times, our prior on implies that the number of relativistic species can be negative when the three active neutrinos are massive (). However, the total radiation energy density is always positive (at late times when the three active neutrinos are massive). We choose this particular prior on in order for the data itself to rule out a given part of parameter space. In Figs 2-4, we find that the marginalized contours on close before the lower end of our prior, such that the data itself is constraining the radiation content from below. For completeness, we also considered several conventional runs with the prior , such that , and we find no qualitative changes in our results. For complete details, see the captions of Tables 346.

As part of our analysis of extended parameter spaces, we consider cases with the primordial fraction of baryonic mass in helium as an unknown parameter to be determined by the data. However, when we do not allow to vary freely, it is determined in a BBN-consistent manner within CAMB via the PArthENoPE code Pisanti:2007hk , which enforces

(1)

Here encapsulates deviations from standard BBN Kneller:2004jz ; Simha:2008zj ; Steigman:2007xt , and we let in agreement with the SPT analysis. Aside from the derived limits on , we explicitly checked that our results do not significantly change () when passing to PArthENoPE instead.

Furthermore, in our analysis we either consider “enforcing the strong inflation prior” on the curvature and running, by which , or “relaxing the strong inflation prior” such that are allowed to vary as free parameters to be constrained by the data. We define the running of the spectral index via the dimensionless power spectrum of primordial curvature perturbations:

(2)

where the pivot scale . Due to the large correlation between and at this scale, we always quote our values for at a scale , where the tilt and running are less correlated, such that ) =  Cortes:2007ak . An example of the remaining correlation between the spectral index and its running is shown in Fig. 3.

Figure 1: Joint two-dimensional marginalized constraints on against . The black confidence regions (inner 68%, outer 95%) are for the extended parameter combination “vanilla” using the data from “WMAP7+SPT++BAO+SNe,” while the vertical red lines denote the 68% confidence interval about the mean from the local () galaxy cluster abundance measurement of Vikhlinin et al. (2009) Vikhlinin:2008ym .
Figure 2: Joint two-dimensional marginalized constraints (inner 68% CL, outer 95% CL) for the extended parameter combination “vanilla,” showing against and against . The purple shaded confidence regions are for “WMAP+SPT++BAO” and the regions enclosed by black lines are for “WMAP+SPT++BAO+SNe,” where the BAOs and SNe are from SDSS+2dF and Union2, respectively. For Planck (T, E, ) in dashed white, we have centered the error ellipses on the fiducial values of the parameters that went into computing the Fisher matrix. The exception to this convention is , which we have shifted down to 0 eV from its fiducial value of 0.17 eV for simpler visual comparison with the upper bounds from present data.
Figure 3: Same as Fig. 2, but for against and against .

Iii Results

We now explore the constraints on extended parameter spaces with the CMB (WMAP7+SPT), BAO distances (SDSS+2dFGRS), and an HST prior on the Hubble constant. Beginning with Sec. III.3 we always also consider SN distance measurements from the Union2 compilation. In Sec. III.5, we discuss the expected constraints from Planck planckbb ; plancksite .

iii.1 CDM with Massive Neutrinos

iii.1.1 Enforcing the inflation prior on

In Table 2, we begin by allowing the effective number of neutrinos, sum of neutrino masses, and primordial helium abundance to vary as free parameters, both separately and jointly, in a CDM universe.

First for CDM alone, then with and added separately, we reproduce the results in Ref. Keisler:2011aw . In particular, with in the parameter space given by “vanilla,” we recover the reported deviation from canonical  Keisler:2011aw ; dunkleyact . Given the well known degeneracy between and  Keisler:2011aw ; dunkleyact ; Hou:2011ec (also see discussion in Sec. I), we find when further allowing to vary as a free parameter irrespectively of the BBN expectation. Allowing for three active neutrinos to have mass, and treating additional contributions to as massless, we find an even larger deviation with the standard value as for the parameter combination “vanilla+” (consistent with Keisler:2011aw ).

Here, the upper bound on the sum of neutrino masses is 0.67 eV (95% CL) and we find the spectral index to be consistent with unity within 1 (). The neutrino mass constraint is to be compared with 0.45 eV at 95% CL in “vanilla+” (consistent with Riess:2011yx ). This 0.45 eV constraint is competitive with the robust upper bound of 0.36 eV when including CMASS dePutter:2012sh , the conservative upper bound of 0.34 eV from the MegaZ photometric redshift catalog of luminous red galaxies Thomas:2009ae , and the conservative upper bound of 0.41 eV from the CFHTLS galaxy angular power spectrum Xia:2012na .

When we consider “vanilla+,” the statistical significance of the deviation is reduced from to , and the upper bound on the sum of neutrino masses moderately weakens to 0.73 eV (95% CL). While the primordial helium abundance from the CMB+BAO+ has been found mildly in tension  dunkleyact ; Keisler:2011aw with that from observations of metal-poor extragalactic H II regions Peimbert:2007vm ; Izotov:2007ed ; Izotov:2010ca ; Aver:2010wq ; Aver:2010wd ; Aver:2011bw , we find constraints on consistent to within with these observations. This is mainly due to the strong negative correlation between and (as reported in dunkleyact ; Keisler:2011aw ; Hou:2011ec and detailed in Sec. I). For instance, Aver, Olive, & Skillman (2011) Aver:2011bw determine via an MCMC analysis that accounts for both statistical and systematic uncertainties, which agrees with in “vanilla” and with in “vanilla+.”

Moreover, when and are analyzed in a joint setting, we find that the data is both consistent with higher values of and lower values of , which perfectly agrees with low-redshift measurements of from the abundance of clusters Vikhlinin:2008ym ; Rozo:2009jj ; Vanderlinde:2010eb ; Mantz:2009fw ; Sehgal:2010ca (as also noted in Ref. Keisler:2011aw ). This is because the amount of suppression in matter clustering by the free-streaming of light neutrinos increases with mass Bond:1980ha ; Bond:1983hb ; Ma:1996za ; Spergel:2006hy , which gives a large anti-correlation between and , an example of which can be seen in Fig. 1.

iii.1.2 Relaxing the inflation prior on

Let us now relax the strong inflation prior on the curvature of the universe and running of the spectral index by considering the parameter combination “vanilla+” in Table 3.

Here, becomes increasingly consistent with the canonical value at (down from ), mainly as a result of the positive correlation with , which also brings the tilt down to (from ). Further, we find that the correlation between and degrades the upper bound on the sum of neutrino masses by close to a factor of 2 to (95% CL). As a consequence of the well known anti-correlation with the sum of neutrino masses, which increases when relaxing the strong inflation prior, the amplitude of matter fluctuations is seen to prefer smaller values at (as compared to when and are held fixed).

However, as the strong inflation prior is relaxed, both the running and curvature are consistent with zero to . It is therefore far from certain that shifts in parameters other than that come about from relaxing the strong inflation prior are true manifestations that will hold with improved data.

iii.2 CDM with Massive Neutrinos

iii.2.1 Enforcing the inflation prior on

In the previous section, we considered cases with neutrinos as massless and cases with neutrinos as massive. However, as it is well established that neutrinos are indeed massive hirata92 ; davis68 ; Eguchi:2002dm ; Ahn:2002up , we account for the sum of neutrino masses as a free parameter in all further treatments of neutrinos. In Table 3, we explore possible degeneracies between and a constant EOS of the dark energy ().

Beginning with “vanilla,” we constrain a constant dark energy EOS: (as compared to without SPT). Considering in conjunction with “vanilla,” we find a reduction in (down from ), rendering it consistent with the canonical value to within . This is caused by the correlation discussed in Sec I and shown in Fig. 2. Expectedly, we also find a correlation between the dark energy EOS and the sum of neutrino masses (discussed in Sec. I), the latter of which degrades by close to a factor of 2 to (95% CL).

The joint impact of on the dark energy EOS is to weaken the constraint on it by roughly a factor of 3 to . Moreover, with the introduction of , the amplitude of linear matter fluctuations is mildly shifted to smaller values at (compared to ) because of the anti-correlation between and that mainly enters through the growth function (e.g. see Komatsu ). The spectral index shifts further away from unity to (down from ).

CDM CDM CDM CDM
++ ++ +++
+ +
Primary
Extended
Derived
  • Same as Table 4 but for a time-dependent parameterization of the dark energy equation of state, of the form (as opposed to time-independent ).

Table 5: Constraints on Cosmological Parameters using SPT+WMAP++BAO+SNe.

iii.2.2 Relaxing the inflation prior on

In Table 3, we now consider the parameter combination “vanilla” in conjunction with the running of the spectral index and curvature of the universe. We also consider a case with the primordial helium abundance as a free parameter.

Given that we already identified separate degeneracies between and in past sections, it is not surprising that we obtain to be in even closer agreement with the canonical value for our extended parameter space. This is a result of the even more negative values preferred by and , shown in Figs. 2 and 3. However, the upper bound on is robust to the further expansion of the parameter space, such that this bound holds for all three cases: “vanilla”, “vanilla”, as well as “vanilla”.

In addition, when allowing for as an independent parameter (i.e. considering the case “vanilla”), the upper bound on the sum of neutrino masses is only mildly weakened, while the error bars are large enough that the effective number of neutrinos is consistent with values of both 3 and 4 (to within 68% CL). In all of the above cases we continue to find consistent with that from cluster abundance measurements to within 68% CL, while the spectral index is consistent with unity to within 95% CL.

While our results are based on the construction of 3 massive neutrinos, and massless degrees of freedom (as discussed in Sec. II), we also considered imposing a hard prior of in a new run with “vanilla”, as this is the case where would be the most affected by the prior. Given the weak correlation between and , the upper bound on the sum of neutrino masses doesn’t change, while the data is still consistent with no extra relativistic species as , where the two sets of upper and lower boundaries denote 68% and 95% CLs, respectively. It is clear that our findings are qualitatively unchanged with this alternative choice of prior on the effective number of neutrinos (also see captions of Tables 34, and 6).

eCDM eCDM eCDM eCDM
++ ++ +++
+ +
Primary
Extended
Derived
  • Same as Table 4 but for an early dark energy model with present EOS and density at high redshift (as opposed to time-independent ). We report the 95% upper limit on (and as before). For the “eCDM” case, we also considered a run where we impose a hard prior of . Here, we find , where the two sets of upper and lower boundaries denote 68% and 95% CLs, respectively. The largest changes this prior induces in other parameters are seen in (compared to ), (compared to ), (compared to ), and (as compared to ). All of the other parameters are modestly affected () by our choice of prior on . For the vanilla case, we also considered a run with a hard prior , for which we find at 95% CL (as compared to in Ref. Reichardt:2011fv ).

Table 6: Constraints on Cosmological Parameters using SPT+WMAP++BAO+SNe.

iii.3 CDM with Massive Neutrinos, Running, and Curvature: Including Supernovae

Since much of the work in bringing in agreement with the canonical value is done by the possibility of evolving dark energy, for which the constraints from the CMB, , and BAO measurements that we have considered are relatively weak, we further include SN data from the Union2 compilation in order to more effectively constrain a constant dark energy EOS and parameters with which it strongly correlates.

In Table 4, we find that the addition of SN observations help constrain the dark energy EOS to when analyzed along with the vanilla parameters (35% reduction in uncertainty compared to no SNe). This constraint degrades to when expanding the parameter space to further include , but is still a factor of 4 stronger than the equivalent case where SNe are not included in the analysis. Improving the constraint on is helpful in breaking much of the degeneracy between dark energy and the effective number of neutrinos, resulting in for the case of “vanilla+” (as compared to without SNe, and as compared to with a prior ). However, as before, when relaxing the strong inflation prior on we find the effective number of neutrinos becomes consistent with the canonical value to 68% CL (as ).

Low-redshift SN measurements are useful in reducing the correlation between , which drives the 1.2 eV (95% CL) upper bound on the sum on neutrino masses for the case “vanilla” down to 0.9 eV (Tables 2 and 3). However, since SN observations do not much improve the constraint on the curvature when added to CMB++BAO (shown in Fig. 2), the SNe are unable to lower the upper bound on the sum of neutrino masses from 1.2 eV (95% CL) when relaxing the strong inflation prior (i.e. for the case “vanilla”). The parameter that most strongly increases the upper bound on the sum of neutrino masses when singularly added to “vanilla+” is the curvature, which renders (95% CL) in “vanilla+.”

Expanding the parameter space to allow to vary as an independent parameter (i.e. considering “vanilla”), we find a mild shift in (as compared to ), and a stronger shift in (as compared to at 95% CL). Meanwhile, shows a preference for lower values but is still consistent with measurements of from low-metallicity H II regions Peimbert:2007vm ; Izotov:2007ed ; Izotov:2010ca ; Aver:2010wq ; Aver:2010wd ; Aver:2011bw . For all of the non-minimal cases considered in Table 4, is consistent with unity to at least 95% CL, and mildly prefers values less than 0.8 but is still greatly consistent with cluster abundance measurements as shown in Fig. 1. Moreover, we find that larger values of the dark matter density are preferred, as generally lives around .

For the particular parameter combination that shifts the closest to a value of 3 from above (i.e. “vanilla”), we also considered a run with the prior imposed. Here, we continue to find the effective number of neutrino species to be consistent with the standard value, as , where the two sets of upper and lower boundaries denote 68% and 95% CLs, respectively. The constraints on other parameters such as and change by less than 10% with this alternative choice of prior.

Next, we move on to other parameterizations of the dark energy, such as the popular expansion and an early dark energy model in which the equation of state of the dark energy tracks the equation of state of the dominant component in the universe.

Figure 4: Joint two-dimensional marginalized constraints on the early dark energy density against for the extended parameter combination “vanilla.” The black confidence regions (inner 68%, outer 95%) are for “WMAP+SPT++BAO+SNe,” while the forecasted error ellipses for Planck temperature, E-mode polarization, and lensing potential power spectra (T, E, ) are shown in dashed red. Although the Fisher matrix constraints on the parameters were evaluated at , they have been shifted down to for simpler visual comparison with the upper bounds from present data.

iii.4 Alternative Dark Energy Parameterizations

Given our ignorance of the nature of dark energy, once we move away from a cosmological constant, there is no adequate reason to restrict our analyses to a constant equation of state from the point of view of particle physics, in particular if we wish to describe dark energy as a scalar field or modification of gravity (e.g. Frieman:2008sn ; Zlatev ; Huterer:2000mj ; Caldwell:1997ii ; Linder:2004ng ). Thus, as an extension of the previous section, we now consider models of the dark energy in which the EOS varies with time. While SN measurements proved useful in breaking parameter degeneracies with a constant EOS, we aim to understand how well these degeneracies are broken for less constrained dark energy models.

iii.4.1 Late-Time Dark Energy with Evolving
Equation of State

The first of our alternative parameterizations for the dark energy is given by the two-parameter model Linder:2002et ; Linder:2004ng ; Chevallier:2000qy advocated in the report of the Dark Energy Task Force Albrecht:2006um :

(3)

where is the EOS at present, while conventionally  Linder:2002et ; Linder:2004ng . Eqn. 3 may also be viewed as a first order Taylor expansion of the EOS, where .

As compared to the case with a constant dark energy EOS, we find that the new parameterization for late-time dark energy doesn’t significantly change our constraints on other cosmological parameters. Expectedly, the two most sensitive parameters are and , the constraints on which degrade by less than 15% and 30%, respectively.

In Table 5, we constrain and for the extension “vanilla.” When we instead use SNe from the “Constitution” compilation Hicken:2009dk , we find and , which are consistent with the constraints on these parameters in Ref. Komatsu:2010fb (perfect agreement when excluding SPT). The difference in constraints may be traced to the larger number of SNe in the Union2 compilation (557 SNe) as compared to the Constitution compilation (397 SNe), along with the use of the SALT2 light curve fitter for the Union2 compilation as compared to the SALT fitter for the Constitution compilation. Clearly, precise SN measurements are critical to understanding the true values of these EOS parameters. In further extensions of our parameter space, the constraint on degrades by up to 20%, while the constraint on degrades by up to a factor of 2.

iii.4.2 Early Dark Energy

Late-time dark energy models suffer from the well known coincidence problem. The value of the dark energy density has to be fine-tuned so that it only affects the dynamics of the universe at present. This coincidence problem motivates the exploration of models in which the evolution of the dark energy density is such that it is large enough to affect the universal dynamics even at .

A realization of early dark energy (EDE) is given by the “tracker” parameterization of Doran Robbers (2006) DorRob , where the dark energy tracks the dominant component in the universe. In a sense, it is simpler to parameterize the dark energy density evolution directly, rather than express it in terms of an evolving equation of state. We use a modified form of the original parameterization that tracks the equation of state of the dominant energy DorRob ; Joudaki:2011nw ,

(4)

where