Weighing neutrinos in the presence of a running primordial spectral index
Abstract
The threeyear WMAP(WMAP3), combined with other cosmological observations from galaxy clustering and Type Ia Supernova (SNIa), prefers a nonvanishing running of the primordial spectral index independent of the low CMB multipoles. Motivated by this feature we study cosmological constraint on the neutrino mass, which severely depends on what prior we adopt for the spectral shape of primordial fluctuations, taking possible running into account. As a result we find a more stringent constraint on the sum of the three neutrino masses, eV (2 ), compared with eV (2 ) for the case where powerlaw prior is adopted to the primordial spectral shape.
PACS number(s): 98.80.Cq
The three year Wilkinson Microwave Anisotropy Probe observations (WMAP3) Spergel:2006hy ; Page:2006hz ; Hinshaw:2006 ; Jarosik:2006 ; WMAP3IE have marked another milestone on the precision cosmology of the Cosmic Microwave Background (CMB) Radiation. The simplest sixparameter powerlaw CDM cosmology is in remarkable agreement with WMAP3 together with the large scale structure(LSS) of galaxy clustering as measured by 2dF Cole:2005sx and SDSS Tegmark:2003uf and with the Type Ia Supernova (SNIa) as measured by the Riess ”gold” sample Riess:2004nr and the first year SNLS snls . This agreement between the above “canonical” cosmological model and observations can be used to test a number of possible new physics, such as the equation of state of dark energy, neutrino masses, time variation of fundamental constants, etc.
Among them, the constraint on the neutrino or hot dark matter mass can be obtained from the freestreaming modification of the transfer function of the matter power spectrum. We should note, however, that if one allowed any shape of primordial spectrum, the freestreaming effect could easily be compensated by some nontrivial shape of the primordial spectrum, so that one cannot obtain sensible limit on the neutrino mass. That is, we can obtain a nontrivial bound on neutrino mass if and only if we adopt some prior on the shape of primordial power spectrum such as a simple powerlaw. From the above argument, we expect that as we allow more degrees of freedom on the primordial spectrum beyond a powerlaw, the constraint on the neutrino mass would be less stringent in general.
As for the shape of the primordial power spectrum, it is noteworthy that a significant deviation has been observed by WMAP3 from the simplest HarrisonZel’dovich spectrum, and that this feature is more eminent with the combination of all the currently available CMB, LSS and SNIa (dubbed the case of ”All” in WMAP3IE ). Moreover, a nontrivial negative running of the scalar spectral index , whose existence was studied even before WMAP epoch Lewis:2002ah ; Feng:2003nt , was favored by the firstyear WMAP papers wmap1a ; wmap1b ; wmap1c . But its preference was somehow diminished as corrections to the likelihood functions were made Slosar:2004fr . However, the new WMAP3 data prefers again a negative running in the ”All” combination WMAP3IE .
If confirmed, a nonvanishing running of would not only constrain inflationary cosmology significantly Feng:2003mk ; Kawasaki:2003zv ; Yamaguchi:2003fp ; Yamaguchi:2004tn ; Chen:2004nx ; Ballesteros:2005eg , but also affect the cosmological constraint on the neutrino mass. In the LSS power spectrum, the effect of massive neutrino may be compensated by a nonvanishing running of primordial spectrum. On the other hand, if negativeness of the running is established, it will lead to even more stringent constraints on neutrino mass compared with fittings in the constant scalar spectral index() cosmology, because both of them lead to a damped power on small scales.
The actual problem, however, cannot be solved by the above simple onetoone correspondence, because the effects of a nonvanishing neutrino mass on CMB is much less dramatic than a nonvanishing running . Alternatively, in the scales probed by CMB, especially near the third peak, there is a large degeneracy between and the matter density Dodelson:1995es ; Hannestad:2006zg ; Lesgourgues:2006nd ; Ichikawa:2004zi ; Fukugita:2006rm .
Hence in the concordance analysis the correlation between running and neutrino mass needs to be addressed in a combined study of CMB, LSS and SNIa, where SNIa helps significantly to determine the matter density. We report the results of such a combined analysis in this paper using Markov Chain Monte Carlo method in constraining the total neutrino mass, eV. Here is the density parameter of the neutrino and is the Hubble constant in unit of 100km/s/Mpc.
To break possible degeneracy among the cosmological parameters, we make a global fit to the forementioned current data with the publicly available Markov Chain Monte Carlo package cosmomc Lewis:2002ah ; IEMCMC . Our most general parameter space is:
(1) 
where and are the physical baryon and cold dark matter densities relative to critical density, characterizes the ratio of the sound horizon and angular diameter distance, is the optical depth and is defined as the amplitude of initial power spectrum. The pivot scale for and is chosen at Mpc.
Assuming a flat Universe and in terms of the Bayesian analysis, we vary the above 8 parameters and fit the theory to the observational data with the MCMC method. For CMB we have only adopted the WMAP3. The bias factors of LSS have been used as nuisance parameters and hence essentially we have used only the shapes of 2dF and SDSS power spectra. As for the SNIa data, while the WMAP team uses both SNLS and Riess sample simultaneously in their “All” dataset, here we adopt only one of them, namely, the Riess ”gold” sample rather than combining with SNLS. This is because these two groups use somewhat different methods in their analysis and it would not be appropriate to put them together simplyfootnt .
Regarding the firstyear WMAP data Bridle et al. Bridle:2003sa found that the claimed preference of a negative was merely due to the lowest WMAP multipoles. In order to probe the sensitivity of the running to the lower multipoles we analyze the running and neutrino properties using the CMB data with and without the contributions of lower multipoles which suffer from large cosmic variance. Specifically, we truncate naturally at given the current likelihood of WMAP3 Spergel:2006hy ; Page:2006hz ; Hinshaw:2006 ; WMAP3IE .
As a result we find a more stringent constraint on the neutrino mass in the presence of running compared with the analysis with constant . We also find that currently the preference of a negative running is fairly independent of the WMAP3 low CMB quadrupoles and hence relatively robust.
TABLE 1. Mean with 1 (2) constrains on the spectral index, the running, and the neutrino mass based on LSS and SNIa with WMAP3(with/without CMB contributions) and with/without introducing a running of the primordial spectral index and with/without massive neutrinos. The last column shows the corresponding reduction of values compared with the power law CDM cosmology.

Normal WMAP3  dropped  


+  +  



set to  set to  

set to  set to  

2  1  

In Table 1 we show the mean 1 (2) constrains on the relevant cosmological parameters combining 2dF, SDSS and Riess ”gold” sample with WMAP3. We have addressed the cases with/without CMB contributions, with/without introducing and with/without massive neutrinos. The last column shows the corresponding reduction of values compared with the power law CDM cosmology. For normal LSS SNIa WMAP in the 7 parameter fittings we get at 2. Our results are considerably less stringent than the ”All” combination by WMAP team WMAP3IE , as we have not included SNLS and other CMB observations. On the other hand the preference of nonvanishing is larger than 2 in both cases. The preference to negative running still remains even if we drop the WMAP3 contributions, when we get at 2. This may imply a negative running is indeed preferred nontrivially in the combination of WMAP+LSS+SNIa, even without the presence of WMAP3 small contributions. This can also be understood through the Akaike information criterion (AIC AIC ), which is defined as defined as
(2) 
where is the maximum likelihood, is the number of parameters of the model and is the number of datapoints. From Table I we can find the reduction of is larger than 2 in the case of running compared with the simple power law CDM cosmology, hence we have smaller values of AIC with running, although currently not strong to decisive levels.
The preference of a negative running can also be obtained in the twodimensional posterior contours of , as depicted in Fig. 1. For the 7parameter case with one additional parameter of , although the contour without WMAP3 low contributions is larger than that in the left panel, a constant lies close to the 2 lines in both cases. The enlarged contour without is easily understood due to the degeneracy.
The correlation between and the shape of the primordial spectrum is obvious, as can be seen from Table 1 and Fig. 1. In the presence of massive neutrinos the error bars on and get increased disregard with or without small CMB contributions. And it is noteworthy in the presence of massive neutrinos, a scaleinvariant primordial spectrum is consistent with the observations at if we drop the small WMAP3 contributions.
We find that, while almost all of the remaining parameters get less stringently constrained, the neutrino mass is an exception: a more tightened bound on is achieved in the presence of a nonzero than the case with constant , as shown in Table 1 and in the one dimensional constraints as in Fig. 2. This has shown the fact that running is indeed strongly correlated with neutrino mass, which is mainly due to the physics on LSS.
In Fig. 3 we display the two dimensional posterior constraints on the sum of neutrino masses versus matter density in the same case as Fig. 2. While the allowed parameter space for is significantly enlarged in the presence of running, neutrino mass is constrained stringently in cases with nonzero . The correlation between and is rather strong and the accumulation of the observational data, such as SNAP, PLANCK and SDSS will help significantly to break the degeneracy, to detect the features of the primordial spectrum as well as the nature of neutrinos.
It has been claimed that a running of the spectral index will be excluded in the presence of SDSS Lyman observations Seljak:2004sj ; Hannestad06x ; Viel:2006yh ; Seljak:2006bg , the systematics of Lyman data are relatively less constrained Spergel:2006hy ; Lesgourgues:2006nd and we leave this in a separate investigation to be reported in Feng:2006ui , and detailed analysis with other additional possible degeneracies in toappear2 .
For many years neutrinos have played a fundamental role in both physics and astrophysics, and provided venues of new physics such as the parity violation and oscillations with tiny masses. Surely neutrinos will continue to play a crucial role for our understanding of the Universe. While the resulting reduction on neutrino mass in this paper is less than 0.2 eV, such an effect would hopefully help to change our understandings on the ultimate detection of neutrino mass with future cosmological surveys and the difference is already larger than the low limit of neutrino mass from oscillation experiments. On the other hand there are also some mild tensions in the determinations of the background cosmological parameters with current CMB, LSS and SNIa. We may still need some better understandings on each dataset before entering the precision cosmology Bridle:2003yz and in cases when all of the observations have similar tendencies in the preference of a negative running one can hopefully get more eminent effects in the probe of neutrino mass.
The distinctive feature probed in the current paper will also open new windows on relevant studies, such as probing neutrino mass with a nonzero running in gravitational lensing surveys, Nbody simulations in the presence of running and massive neutrinos.
Acknowledgments: We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. We have performed our numerical analysis on the Shanghai Supercomputer Center(SSC). We used a modified version of CAMB Lewis:1999bs ; IEcamb which is based on CMBFAST cmbfast ; IEcmbfast . We thank Steen Hannestad, Antony Lewis, Chris Lidman, Hiranya Peiris and Pengjie Zhang for helpful discussions. The work of J.Y. is supported partially by the JSPS GrantinAid for Scientific Research No. 16340076 and B.F. is by the JSPS fellowship program. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 90303004, 10533010 and 19925523 and by Ministry of Science and Technology of China under Grant No. NKBRSF G19990754.
References
 (1) D. N. Spergel et al., “Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology,” arXiv:astroph/0603449.
 (2) L. Page et al., “Three Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis,” arXiv:astroph/0603450.
 (3) G. Hinshaw et al., “ThreeYear Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis,” arXiv:astroph/0603451.
 (4) N. Jarosik et al., “ThreeYear Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Profiles, Data Processing, Radiometer Characterization and Systematic Error Limits,” arXiv:astroph/0603452.
 (5) Available at http://lambda.gsfc.nasa.gov/product/map/current/ .
 (6) S. Cole et al. [The 2dFGRS Collaboration], “The 2dF Galaxy Redshift Survey: Powerspectrum analysis of the final dataset and cosmological implications,” Mon. Not. Roy. Astron. Soc. 362 (2005) 505 [arXiv:astroph/0501174].
 (7) M. Tegmark et al. [SDSS Collaboration], “The 3D power spectrum of galaxies from the SDSS,” Astrophys. J. 606, 702 (2004) [arXiv:astroph/0310725].
 (8) A. G. Riess et al. [Supernova Search Team Collaboration], “Type Ia Supernova Discoveries at z¿1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution,” Astrophys. J. 607, 665 (2004) [arXiv:astroph/0402512].
 (9) P. Astier et al., “The Supernova Legacy Survey: Measurement of , and w from the First Year Data Set,” Astron. Astrophys. 447, 31 (2006) [arXiv:astroph/0510447].
 (10) A. Lewis and S. Bridle, “Cosmological parameters from CMB and other data: a MonteCarlo approach,” Phys. Rev. D 66, 103511 (2002) [arXiv:astroph/0205436].
 (11) B. Feng, X. Gong and X. Wang, “Assessing the Effects of the Uncertainty in Reheating Energy Scale on Primordial Spectrum and CMB,” Mod. Phys. Lett. A 19, 2377 (2004) [arXiv:astroph/0301111].
 (12) C. L. Bennett et al., “First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results,” Astrophys. J. Suppl. 148, 1 (2003) [arXiv:astroph/0302207].
 (13) D. N. Spergel et al. [WMAP Collaboration], “First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters,” Astrophys. J. Suppl. 148, 175 (2003) [arXiv:astroph/0302209].
 (14) H. V. Peiris et al., “First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation,” Astrophys. J. Suppl. 148, 213 (2003) [arXiv:astroph/0302225].
 (15) See e.g. A. Slosar, U. Seljak and A. Makarov, “Exact likelihood evaluations and foreground marginalization in low resolution WMAP data,” Phys. Rev. D 69, 123003 (2004) [arXiv:astroph/0403073].
 (16) B. Feng, M. z. Li, R. J. Zhang and X. m. Zhang, “An inflation model with large variations in spectral index,” Phys. Rev. D 68, 103511 (2003) [arXiv:astroph/0302479].
 (17) M. Kawasaki, M. Yamaguchi, and J. Yokoyama, “Inflation with a running spectral index in supergravity,” Phys. Rev. D 68, 023508 (2003) [arXiv:hepph/0304161].
 (18) M. Yamaguchi and J. Yokoyama, “Chaotic hybrid new inflation in supergravity with a running spectral index,” Phys. Rev. D 68, 123520 (2003) [arXiv:hepph/0307373].
 (19) M. Yamaguchi and J. Yokoyama, “Smooth hybrid inflation in supergravity with a running spectral index and early star formation,” Phys. Rev. D 70, 023513 (2004) [arXiv:hepph/0402282].
 (20) C.Y. Chen, B. Feng, X.L. Wang, and Z.Y. Yang, “Reconstructing large runningindex inflaton potentials”, Class. Quant. Grav. 21, 3223 (2004) arXiv:[astroph/0404419].
 (21) G. Ballesteros, J. A. Casas and J. R. Espinosa, “Running spectral index as a probe of physics at high scales”, JCAP. 0603, 001 (2006) arXiv:[hepph/0601134].
 (22) S. Hannestad, “Primordial neutrinos,” arXiv:hepph/0602058.
 (23) See e.g. J. Lesgourgues and S. Pastor, “Massive neutrinos and cosmology,” arXiv:astroph/0603494.
 (24) S. Dodelson, E. Gates and A. Stebbins, “Cold + Hot Dark Matter and the Cosmic Microwave Background,” Astrophys. J. 467, 10 (1996) [arXiv:astroph/9509147].
 (25) K. Ichikawa, M. Fukugita and M. Kawasaki, “Constraining neutrino masses by CMB experiments alone,” Phys. Rev. D 71, 043001 (2005) [arXiv:astroph/0409768].
 (26) M. Fukugita, K. Ichikawa, M. Kawasaki and O. Lahav, “Limit on the neutrino mass from the WMAP three year data,” arXiv:astroph/0605362.
 (27) Available from http://cosmologist.info.
 (28) Currently the SNLS and Riess ”gold” sample are comparable in the determination of cosmological parameters, as shown in Ref. WMAP3IE , and each dataset has its own nice features. In the present work we use only the Riess sample.
 (29) S. L. Bridle, A. M. Lewis, J. Weller and G. Efstathiou, “Reconstructing the primordial power spectrum,” Mon. Not. Roy. Astron. Soc. 342, L72 (2003) [arXiv:astroph/0302306].
 (30) H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Auto. Control, 19, 716 (1974).
 (31) U. Seljak et al., “SDSS galaxy bias from halo massbias relation and its cosmological implications,” Phys. Rev. D 71, 043511 (2005) [arXiv:astroph/0406594].
 (32) A. Goobar, S. Hannestad, E. Mortsell and H. Tu, “The neutrino mass bound from WMAP3, the baryon acoustic peak, the SNLS supernovae and the Lymanalpha forest,” arXiv:astroph/0602155.
 (33) M. Viel, M. G. Haehnelt and A. Lewis, “The Lymanalpha forest and WMAP year three,” arXiv:astroph/0604310.
 (34) U. Seljak, A. Slosar and P. McDonald, “Cosmological parameters from combining the Lymanalpha forest with CMB, galaxy clustering and SN constraints,” arXiv:astroph/0604335.
 (35) B. Feng, J. Q. Xia and J. Yokoyama, “Scale dependence of the primordial spectrum from combining the threeyear WMAP, Galaxy Clustering, Supernovae, and Lymanalpha forests,” arXiv:astroph/0608365.
 (36) B. Feng et al., at press.
 (37) See e.g. S. L. Bridle, O. Lahav, J. P. Ostriker and P. J. Steinhardt, “Precision Cosmology? Not Just Yet,” Science 299 (2003) 1532 [arXiv:astroph/0303180].
 (38) A. Lewis, A. Challinor and A. Lasenby, “Efficient Computation of CMB anisotropies in closed FRW models,” Astrophys. J. 538, 473 (2000) [arXiv:astroph/9911177].
 (39) Available at http://camb.info .
 (40) U. Seljak and M. Zaldarriaga, “A Line of Sight Approach to Cosmic Microwave Background Anisotropies,” Astrophys. J. 469, 437 (1996) [arXiv:astroph/9603033].
 (41) Available at http://cmbfast.org/ .