# Testing CPT Symmetry with CMB Measurements: Update after WMAP5

###### Abstract

In this Letter we update our previous result on the test of CPT symmetry with Cosmic Microwave Background (CMB) measurements. A CPT violating interaction in the photon sector gives rise to a rotation of the polarization vectors of the propagating CMB photons. Recently the WMAP group used the newly released polarization data of WMAP5 to measure this rotation angle and obtained deg (). However, in their analysis the BOOMERanG 2003 data is not included. Here we revisit this issue by combining the full data of WMAP5 and BOOMERanG 2003 angular power spectra for the measurement of this rotation angle and find that deg at a confidence level.

###### Subject headings:

Cosmology: theory (Cosmology:) cosmic microwave background (Cosmology:) cosmological parameters## 1. Introduction

The fundamental CPT symmetry which has been proved to be exact in the framework of the standard model of particle physics and Einstein gravity could be dynamically violated in the expanding universe. This type of cosmological CPT violation mechanism investigated in the literature Li:2001st (); Li:2002wd (); Li:2004hh () has an interesting feature that the CPT violating effect at present time is small enough to satisfy the current laboratory experimental limits, but large enough in the early universe to account for the generation of the matter anti-matter asymmetry. More importantly, it could be accumulated to be observable in the cosmological experiments Feng:2004mq (); Li:2007 (). With the accumulation of high quality data on the CMB measurements, cosmological observations become a powerful tool to test this fundamental symmetry.

For a phenomenological study in the photon sector the CPT violation can be parameterized in terms of an effective lagrangian Carroll:1989vb (); Carroll:1990zs ():

(1) |

where is a Chern-Simons term, is an external vector and is the dual of the electromagnetic tensor. This Lagrangian is not generally gauge invariant, but the action is gauge independent if . This may be possible if is constant in spacetime or the gradient of a scalar field in the quintessential baryo-/leptogenesis Li:2001st (); Li:2002wd (); quin_baryogenesis () or the gradient of a function of the Ricci scalar in gravitational baryo-/leptogenesis Li:2004hh (); R (). The Chern-Simons term violates Lorentz and CPT symmetries, and also the and symmetries when the background field does not vanish.

For the CMB measurements the Chern-Simons term induces a rotation of the polarization Li:2007 (); Xia:2007qs () with the rotation angle given by

(2) |

where denotes the time component of and is the comoving distance of the CMB photon emitted at the last scattering surface and indicates the present time. In Eq.(2) we have assumed is a constant. For a more general case please see our previous companion paper Xia:2007qs ().

For the standard theory of CMB, the TB and EB cross-correlation power spectra vanish. In the presence of the CPT violating term (Eq.(1)) the polarization vector of each photon is rotated by an angle , and one expects to observe nonzero TB and EB power spectra, even if they are zero at the last scattering surface. Denoting the rotated quantities with a prime, one gets Feng:2004mq (); Lue:1998mq ():

(3) | |||||

while the CMB temperature power spectrum remains unchanged.

In Xia et al. (2007), using the full data of BOOMERanG 2003 and
the WMAP3 angular power spectra we have performed the analysis on
the determination of the rotation angle and find
that deg (). This result
improves the measurement given by our previous paper
Feng:2006dp () and the paper by Cabella et al.
(2007)^{1}^{1}1For the implications of this measurement on the
possible new physics, please also see papers
LiuCPT (); KosteleckyCPT (); GengCPT (); Ni:2007ar (); FinelliCPT ()..
Recently the Wilkinson Microwave Anisotropy Probe (WMAP)
experiment has published the 5-year results for the CMB angular
power spectra which include the TB and EB information
WMAP51 (); WMAP52 (). They use the polarization power spectra of
WMAP5, TE/TB () and EE/BB/EB (),
to determine this rotation angle WMAPCPT (), and find that
deg ().

Besides the WMAP measurement, the BOOMERanG 2003 data also provide the TB and EB polarization power spectra B031 (); B032 (); B033 (), which have been shown to give an interesting constraint on this rotation angle Feng:2006dp (); Xia:2007qs (). Thus it will be interesting and necessary to combine the full data of these two experiments for the analysis, which is the aim of this Letter.

## 2. Method and Results

In our study we make a global analysis on the CMB data with the
public available Markov Chain Monte Carlo package
CosmoMC^{2}^{2}2http://cosmologist.info/. Lewis:2002ah (),
which has been modified to allow the rotation of the power spectra
discussed above, with a new free parameter . We
assume the purely adiabatic initial conditions and impose the
flatness condition motivated by inflation. In our analysis the
most general parameter space is: , where and
are the physical baryon and cold
dark matter densities relative to the critical density,
is the ratio of the sound horizon to the angular
diameter distance at decoupling, is the optical depth to
re-ionization, and characterize the primordial
scalar power spectrum, is the tensor to scalar ratio of the
primordial spectrum. For the pivot of the primordial spectrum we
set Mpc. In our calculation we have assumed
that the cosmic rotation angle is constant at all multipoles and
does not depend on . Furthermore, we think that this rotation
angle is not too large and imposed a conservative flat prior
.

In our calculations we combine the full data of WMAP5 and
BOOMERanG 2003 (B03). We calculate the likelihood of CMB power
spectra using the routine for computing the likelihood supplied by
the WMAP^{3}^{3}3Legacy Archive for Microwave Background Data
Analysis (LAMBDA), http://lambda.gsfc.nasa.gov/. and BOOMERanG
groups. Furthermore, we make use of the Hubble Space Telescope
(HST) measurement of the Hubble parameter h km s Mpc by multiplying a Gaussian likelihood
function Hubble (). We also impose a weak
Gaussian prior on the baryon density
() from the Big Bang
Nucleosynthesis BBN (). Simultaneously we will also use a
cosmic age tophat prior as 10 Gyr 20 Gyr.

Firstly we do a consistency test by comparing two methods used by us and WMAP group. The WMAP group fixed the parameters except for and in their analysis WMAPCPT (). The polarization spectra they considered are TE/TB/EE/BB/EB at and TE/TB at . In our analysis, we vary all of the parameters in the parameter space and use the full WMAP5 data including the CMB TT power spectrum. With the WMAP5 data only we find that our result on is consistent with that given by the WMAP group WMAPCPT (). Therefore, in the study below, we follow our method to do the calculation with the combination of the WMAP5 and B03 data.

In Fig.1 we plot our one dimensional constraints on the rotation angle from the CMB data. The blue dashed line shows our previous result on rotation angle from WMAP3 and B03 data. The red dash-dot line shows the limit on the full data of WMAP5. And the black solid line is our final result from the full data of WMAP5 and B03 data. The best fit value of the rotation angle is deg. Marginalizing over the posterior distributions of other parameters, we find that the mean value of the rotation angle is:

(4) |

This constraint is tighter than all of the previous results on , say, the error bar is decreased by a factor of 2, which is profited from the accurate WMAP5 polarization data. On the other hand, this negative rotation angle is slightly preferred by the TC and GC information of B03. In the B03 data, the TC power at and are both negative, whereas it is positive at . The GC power at , and are all negative. Based on the Eq.(3), we can see that the TC and GC power spectra of B03 really help to obtain this negative rotation angle.

## 3. Summary

In this Letter we have determined the rotation polarization angle
with the combined CMB data from BOOMERanG 2003 and
the newly released WMAP5 data, and obtained
deg (), which shows a mild
detection of a nonzero rotation angle and a weak evidence for
cosmological CPT violation. With the near future CMB measurements
our result on the CPT violation could be confirmed or the CPT
symmetry can be verified with a higher precision. For example,
with the
Planck^{4}^{4}4http://sci.esa.int/science-e/www/area/index.cfm?fareaid=17/.
and the Spider measurements spider () the standard deviation
of the rotation angle will be significantly reduced to
deg Xia:2007gz () and deg
Xia:Spider (), respectively.

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