**A new test of the Einstein equivalence principle**

The recent claim from the antartic experiment BICEP2 to have found in the B-mode polarization of the cosmic microwave background (CMB) the footprint of the gravitational waves produced by the inflation of the Universeprovides the first direct proof that inflation actually occurred. The inflation is a quasi-exponential expansion, which should have happened soon after the Big Bang and is necessary to explain the present uniformity and lack of curvature of the universe. It would not have directly measurable effects other than a background of gravitational waves. However, for the parameter characterising the inflationary gravitational waves, i.e. the tensor-scalar ratio, BICEP2 has obtained a value r=0.2 which is nearly twice the upper limit r≤0.11 set about a year ago from a first analysis of the data gathered by Planck, a satellite which was lauched by ESA in 2009 to study the CMB and operated until 2013. The full results of Planck should definitely clarify the situation, but they will not be available before the end of this year. In the mean time the attempts to explain this discrepancy are blooming.

Sperello di Serego Alighieriof the Arcetri Astrophysical Observatory, in collaboration with two chinese cosmologists, Wei-Tou Ni e Wei-Ping Pan, in the paper “New Constraints on Cosmic Polarization Rotation from B-mode Polarization in Cosmic Microwave Background” has proposed two improvements in the analysis of the BICEP2: 1) consider the possible effects of the cosmic polarization rotation (CPR) and 2) take into account also the B-mode polarization data from other experiments, in particular from SPTpol.

The CPR, i.e. a rotation of the plane of polarization for a radiation traveling over cosmological distances, is theoretically foreseen in case some fundamental physical principles were violated. The most famous is the Einstein equivalence principle (EEP), on which all metric theories of gravitation are based, including general relativity. The principle of equivalence between a gravitational field and a uniformly accelerated frame comes in three forms: 1) the week one, also called after Galileo, which states the equivalence for free falling bodies, implies the equivalence between inertial and gravitational masses, and was tested for the first time by Galileo with the famous experiment of bodies falling from the leaning tower in Pisa; 2) the Einstein one (EEP), which extends the equivalence to all experiments not including gravitational forces; 3) the strong one, which includes also gravitational experiments. The weak equivalence has been tested to a very high accuracy (10^{-13}) with torsion balance experiments. The EEP, instead, is tested by gravitational redshift experiments to an accuracy of only about 10^{-4}. In 1960 Schiff has conjectured that any consistent Lorents-invariant theory of gravity which obeys the weak principle would necessarily obey the EEP. If this were true, the EEP would be tested with the same accuracy as the weak principle, thereby greatly increasing the experimental confidence in general relativity. However Wei-Tou Ni in 1977 has found a unique counter-example to Schiff’s conjecture: a pseudoscalar field which couple to electromagnetism leading to a violation of the EEP, while obeying the weak principle. This pseudoscalar field, if it existed, would produce a rotation of the polarization, i.e. the CPR. Astronomers have searched for the CPR for more than 20 years, without finding it, and have set an upper limit of a couple of degrees to any rotation (as discussed by Sperello di Serego Alighieri in "Cosmological Birefringence: an astrophysical test of fundamental physics". However this unsuccessful search does not imply that there is no CPR, just that it is less than a few degrees. The point of Sperello di Serego Alighieri et al. is that the CPR, if it existed, would couple to B-mode polarization,producing a non-negligible signal even for a rotation of just a few degrees: therefore it must be taken into account while interpreting BICEP2 data.

Moreover, the CMB B-mode polarization has been detected also by SPTpol from the South Pole, before BICEP2 although at larger multipoles, where the effect of weak gravitation lensing dominates. Therefore, for an improved interpretation of BICEP2 data, we should also take into account the SPTpol data.

The CMB B-mode polarization power spectrum. The black and blue dots are the BICEP2 and SPTpol data, respectively. The red, brown, and yellow dashed lines show the components due to inflationary gravitational waves, to weak gravitational lensing, and to the CPR coupling, respectively. The latter one derives from the E-mode polarizarization shown by the yellow continous line. The green line is the sum of the 3 components, while the green region shows the error. The α of the CPR is 0.0237 radians (1.36°) in this figure, corresponding to a 0% coupling of CPR with B-mode polarization. |

Sperello di Serego Alighieri, Wei-Tou Ni e Wei-Ping Pan have considered separately the B-mode polarization components due to inflationary garvitational waves, to weak gravitationallensing, and to CPR coupling. The latter one is significant mainly at large multipoles (i.e. at small angular scales), covered by the SPTpol data. The inflationary component is less affected by CPR coupling, but still at a significant level: in fact the tensor-scalar ratio changes from 0.20 to 0.18 when coupling is taken into account. The CPR angle α depends on the coupling percentage, which is unknown, and varies between 0.92° and 1.97° for a coupling percentage between 0% and 100%.

These results lead to another interesting consideration: the data at multipoles larger than 250, mailnly from SPTpol, are in fact very close to the weak lensing component, not leaving much room for the CPR. Therefore from this data it is possible to put an upper limit to the CPR angle, for example at the maximum value α = 1.97°, corresponding to a coupling percentage of 16.7%. This upper limit is similar to previous limits. The present CMB polarization experiments also have the problem taht the polarization angle is calibrated with an accuracy of only 1° or 2°. This results in an error on α at the same level of the current upper limit. However, if the calibration of the polarization angle and the accuracy of the B-mode polarization power specrum data will improve, this new method of studying the CPR promises interesting results, leading to an important confirmation fo the Einsten equivalence principle, or to its astonishing denial.

Edited by S. di Serego Alighieri and A. Gallazzi