# A note of clarification: BICEP2 and Planck are not in tension

###### Abstract

The apparent discrepancy between the value of the tensor-to-scalar ratio reported by the BICEP2 collaboration, at 68% CL, and the Planck upper limit, at 95% CL, has attracted a great deal of attention. In this short note, we show that this discrepancy is mainly due to an ‘apples to oranges’ comparison. The result reported by BICEP2 was measured at a pivot scale Mpc, assuming , whereas the Planck limit was provided at Mpc, assuming the slow-roll consistency relation . One should obviously compare the BICEP2 and Planck results under the same circumstances. By imposing , the Planck constraint at Mpc becomes at CL, which can be compared directly with the BICEP2 result. Once a plausible dust contribution to the BICEP2 signal is taken into account (DDM2 model), is reduced to and the discrepancy becomes of order only.

## I The pivot scale confusion

A generic prediction of the inflationary paradigm is that the primordial scalar and tensor power spectra from inflation are nearly scale invariant. The deviation from scale invariance is then quantified by specifying a spectral index and possibly a running at a pivot scale . The primordial power spectra are then given by

(1) | ||||

(2) |

where and . The primordial tensor amplitude is, by convention, always substituted for the ratio

(3) |

evaluated at a given scale , which is often, but not always, taken to be
. The analysis of the temperature anisotropies by the Planck
collaboration Ade et al. (2013) was done at a pivot scale , but their reported constraint at Confidence Level (CL)
was given at . The BICEP2
collaboration Ade et al. (2014), on the other hand, reported the amplitude at 68% CL, evaluated at the Planck pivot scale^{1}^{1}1Note that
this is not explicitly stated in Ade et al. (2014), at least not in the current
arXiv version at the time of writing this note, but it has been confirmed to us
by the BICEP2 collaboration. . To avoid
any confusion, it is then convenient to denote the tensor-to-scalar ratios
evaluated at and
by and respectively. That BICEP2 indeed has
as best fit becomes evident in Fig. 1, where the
low- B-mode angular power spectrum are plotted for (red
lines) and (blue lines), considering for both cases , and assuming . The curve is
clearly not a good fit to the data points given by BICEP2.

## Ii The likelihood confusion

The confusion related to the scale is enhanced by the following
circumstances: i) the BICEP2 collaboration used a different likelihood in their own
analyses than the publicly released Python likelihood code, and ii) the
best-fit of the public code is actually , and *not*
as found in the BICEP2 analysis. This difference in best-fit
is due to two separate facts: a) different methods are being used for
computing the likelihood, the public one using the Hamimeche & Lewis code,
whereas the private one uses the formula introduced in Barkats et al. (2013),
paragraph 9.3.1, and b) the public code uses information from all nine
bandpower bins, whereas the internal one makes use of only the five first ones.

It should be noted that the difference between the best-fit values of the two likelihoods is well below . So this is not alarming in any way, but it leads nonetheless to an overestimation of the tension with Planck when using the public code.

In conclusion, the only data product matching exactly the BICEP2 internal analysis is the tabulated likelihood, obtained for a fixed cosmology with different values of , represented in green in the top panel of Fig. 2. This corresponds to the advertised value of . Reference Ade et al. (2014) also discusses several dust models, retaining DDM2 (Data Driven Model 2) as the most plausible one. After removing the DDM2 contamination, the BICEP2 collaboration obtains (68% CL). In the lower panel of Fig. 2 we present an approximative posterior after dust removal.

## Iii Comparison with Planck

Using Eqs. (1) and (2), one can convert from BICEP2 to for comparison with Planck, or vice versa. For instance, the curve in Fig. 1, which is the best fit to the data (for ), is equivalent to . However, this would still be an ‘apples to oranges’ comparison, since the Planck analysis used a tensor spectral index inferred from the single-field slow-roll consistency condition , while BICEP2 used . This means that the underlying tensor primordial spectra was not of the same form, so it is in principle meaningless to compare the two parameters: If one experiment fits to the data while the other fits , we certainly should not compare and .

We derived the posterior probability for assuming a flat
model and the Planck+WP dataset. In any Bayesian
parameter extraction, the posterior depends on the choice of prior. Here, we
choose to restrict ourselves to physical models by imposing a prior (a different choice is advocated in the recent analysis of
Smith et al. (2014)). After running the class and Monte Python
codes, we obtained at 95% CL, which is not in significant
tension with the BICEP2 result, as shown in Fig. 2. Even before
subtracting the dust model, the two posteriors overlap at the 9% CL
(corresponding to 1.7). After removing dust contamination (under the
DDM2 assumption), the compatibility increases^{2}^{2}2Here the compatibility
is quantified by searching for the confidence level of each likelihood above
which there is an overlap. Another statistical test of the compatibility
between two such likelihoods is presented in Smith et al. (2014). to the
level of 17%, corresponding to a 1.3 overlap.

With such an overlap between the two likelihoods, we can conclude (even without calculating Bayesian evidence ratios) that there is no compelling reason at the moment to invoke extra ingredients in the cosmological model, in order to alleviate a would-be tension between the Planck 2013 and BICEP2 measurements. In particular, there is no convincing case for introducing a non-zero scalar running of the order of , which would be incompatible with the simplest and most elegant slow-roll inflationary paradigm.

## Acknowledgements

We would like to thank Clem Pryke and Stefan Fliescher from the BICEP2 collaboration for their prompt and very helpful answers to all our questions.

## References

- Ade et al. (2013) P. Ade et al. (Planck Collaboration) (2013), eprint 1303.5076.
- Ade et al. (2014) P. Ade et al. (BICEP2 Collaboration) (2014), eprint 1403.3985.
- Barkats et al. (2013) D. Barkats et al. (BICEP1 Collaboration) (2013), eprint 1310.1422.
- Smith et al. (2014) K. M. Smith, C. Dvorkin, L. Boyle, N. Turok, M. Halpern, et al. (2014), eprint 1404.0373.