[Paper Review] Constraints on the extensions to the base $Λ$CDM model from BICEP2, Planck and WMAP

Cheng Cheng, Qing-Guo Huang+1 more|arXiv (Cornell University)|Apr 14, 2014

Cosmology and Gravitation TheoriesPhysics and Astronomy3 citations

TL;DR

This paper investigates tensions between BICEP2, Planck (P13), and WMAP CMB data regarding primordial gravitational waves, finding that extending the base ΛCDM model with a running spectral index and running of running significantly reduces discrepancies. It shows that a positive running of running is preferred at 1.7σ, suggesting more complex early-Universe physics beyond simple inflation models.

ABSTRACT

Recently Background Imaging of Cosmic Extragalactic Polarization (B2) discovered the relic gravitational waves at $7.0σ$ confidence level. However, the other cosmic microwave background (CMB) data, for example Planck data released in 2013 (P13), prefer a much smaller amplitude of the primordial gravitational waves spectrum if a power-law spectrum of adiabatic scalar perturbations is assumed in the six-parameter $Λ$CDM cosmology. In this paper, we explore whether the $w$CDM model and the running spectral index can relax the tension between B2 and other CMB data. In particular, we find that a positive running of running of spectral index is preferred at $1.7σ$ level from the combination of B2, P13 and WMAP Polarization data.

Motivation & Objective

• Address the tension between BICEP2's high tensor-to-scalar ratio (r ≈ 0.2) and Planck/WMAP's lower constraints (r < 0.11) in the base ΛCDM model.
• Investigate whether extensions to ΛCDM—specifically the $w$CDM model and models with running spectral index—can reconcile the conflicting CMB data.
• Assess whether higher-order terms in the scalar perturbation spectrum, such as running of running, are required by the combined B2+P13+WP dataset.
• Determine if the observed discrepancy arises from limitations in the standard ΛCDM+tensor model or from systematic issues in data or assumptions.

Proposed method

• Use CosmoMC to perform Bayesian parameter estimation on combined BICEP2, Planck (P13), and WMAP polarization (WP) datasets.
• Fix background parameters to Planck best-fit values and vary only scalar amplitude, spectral index, and tensor-to-scalar ratio using low-multipole TT and TE data.
• Extend the ΛCDM model to include a constant dark energy equation-of-state parameter $w$ and to include running of spectral index ($dn_s/d\ln k$) and running of running ($d^2n_s/d\ln k^2$).
• Apply the consistency relation $n_t = -r/8$ to link tensor spectral index to tensor-to-scalar ratio.
• Compare model fits using $\Delta\chi^2$ to assess statistical preference for higher-order extensions.
• Use Markov Chain Monte Carlo sampling to derive posterior constraints and credible intervals on cosmological parameters.

Experimental results

Research questions

• RQ1 Can the $w$CDM model resolve the tension between BICEP2 and Planck/WMAP on the primordial gravitational wave amplitude?
• RQ2 Does including a running spectral index ($dn_s/d\ln k$) reduce the discrepancy between BICEP2 and other CMB datasets?
• RQ3 Is there statistical evidence for a non-zero running of running ($d^2n_s/d\ln k^2$) in the scalar power spectrum when combining B2, P13, and WP data?
• RQ4 Do the combined datasets favor a model beyond the standard six-parameter ΛCDM+tensor cosmology?
• RQ5 How do higher-order terms in the scalar perturbation spectrum affect constraints on $r_{0.002}$ and $n_s$?

Key findings

• The $w$CDM model does not significantly alleviate the tension between BICEP2 and Planck/WMAP data on the tensor-to-scalar ratio.
• A model including running of spectral index and running of running ($\Lambda\text{CDM}+n_{\rm run}+n_{\rm run\,run}+r$) provides a better fit, with $\Delta\chi^2 = -3.12$, indicating a preference at more than 1σ level.
• The running of spectral index is preferred at 2.2σ significance, with best-fit $dn_s/d\ln k = -0.108$ and 68% limits $-0.157$ to $-0.059$.
• A positive running of running is preferred at 1.7σ, with best-fit $d^2n_s/d\ln k^2 = 0.033$ and 68% limits $0.015$ to $0.051$.
• The scalar spectral index $n_s$ exceeds 1 at 2.2σ significance, with best-fit $n_s = 1.1344$ and 68% limits $1.0732$ to $1.1956$.
• The tensor-to-scalar ratio at $k_p = 0.002$ Mpc⁻¹ is constrained to $r_{0.002} = 0.24_{-0.07}^{+0.05}$, consistent with BICEP2's $r = 0.20_{-0.05}^{+0.07}$.