Cosmic microwave background polarization in non-commutative space-time

In the standard model of cosmology (SMC) the B-mode polarization of the CMB can be explained by the gravitational effects in the inflation epoch. However, this is not the only way to explain the B-mode polarization for the CMB. It can be shown that the Compton scattering in the presence of a background, besides generating a circularly polarized microwave, can lead to a B-mode polarization for the CMB. Here we consider the non-commutative (NC) space-time as a background to explore the CMB polarization at the last scattering surface. We obtain the B-mode spectrum of the CMB radiation by scalar perturbation of metric via a correction on the Compton scattering in NC-space-time in terms of the circular polarization power spectrum and the non-commutative energy scale. It can be shown that even for the NC scale as large as 20TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$20\,\mathrm{TeV}$$\end{document} the NC-effects on the CMB polarization and the r parameter are significant. We show that the V-mode power spectrum can be obtained in terms of linearly polarized power spectrum in the range of micro- to nano-kelvin squared for the NC scale of about 1–20 TeV, respectively.


Introduction
The polarization anisotropy and temperature inhomogeneities of the cosmic microwave background radiation (CMBR) can provide a way for exploring the physics of the early universe.The light polarization can be parameterized in terms of the Stokes parameters (Q, U and V ).A nonzero values for Q and/or U show linearly polarized radiations while a circular polarized radiation has a non vanishing value for the Stokes parameter V [1].An anisotropic Thomson scattering due to the temperature inhomogeneity around the recombination phase leads to the linear polarization about 10% of CMBR [2,3].Meanwhile, according to the standard model of cosmology there is no physical mechanism to generate a circular polarized radiation at the last scattering surface or V = 0.However, a linearly polarized radiation through its propagation in a magnetic field can be partially circular polarized, a property known as the Faraday effect.The Stokes parameter V in this mechanism evolves as where ∆φ F C is the Faraday conversion phase shift [4].Linear polarization of the CMB radiation from the last scattering can be converted to the circular polarization due to effects of background fields, particle scattering and temperature fluctuations.The conversion probability of the CMB linear polarization to the circular polarization has been discussed in many papers [5][6][7][8][9].Furthermore, since Q and U are frame-dependent, by decomposing the linear polarization into the E and B components one can extract more information from the polarization pattern on the sky.The Thomson scattering at the last scattering surface only produces the E mode which can be converted to the B mode through the vector and tensor perturbations.Meanwhile, the gravitational waves, if exist, due to the tensor mode perturbation arising from the inflation epoch generate the B-mode polarization for the CMB radiation.In early 2014, the BICEP2 team announced a nonzero measurement on the B-mode polarization for the CMB radiation as an evidence for the primordial gravitational wave [10].This result is not consistent with the Planck limit, r < 0.11(98%CL).However, at this time there is no conclusive evidence of primordial gravitational waves from a joint analysis of data provided Planck and BICEP2 experiments.The recent Bicep/Keck Array observation reported upper bounds on the tensor-to-scalar ratio, r 0.05 < 0.09 and r 0.05 < 0.07 at (95%) C.L. by using B-modes alone and combining the B-mode results with Planck temperature analysis, respectively [11].In fact, to distinguish the tensor and scalar components, a tensor-to-scalar ratio can be calculated by measuring the polarization angles on the sky, which Plank has been reported this ratio to be about r ∼ 0.12.Therefore, to find out the contribution of the gravitational wave on the B-mode one should consider all contribution from the other sources.Although, in the standard model of cosmology the B-mode dose not receive contribution from the scalar mode one can consider the B-mode as a result of Faraday rotation of the E-mode polarization [12,13].Furthermore, the Compton scattering in presence of a background can potentially lead to a B-mode polarization for the CMB even for the scalar perturbation.The contribution to the observed B-mode spectrum from the interaction between CMB photon and the Cosmic Neutrino Background(CNB) in the scalar perturbation background has been considered in [14].
Here we would like to explore the effects of non-commutative background on the B-mode polarization.In ref. [15] the energy scale of the non-commutativity of space-time has been constrained by using CMB data from PLANCK.They find that PLANCK data put the lower bound on the non-commutativity energy scale to about 20 TeV, which is about a factor of 2 larger than the previous bound that was obtained using data from WMAP, ACBAR and CBI.In this paper we study the possibility of generating circular polarization of CMB radiation by considering Compton scattering on Non-Commutative background.In Sec II we review the Stokes parameters and Boltzmann equation formalism.In Sec.III we give a brief introduction on Non-Commutative standard model.In Sec.IV the time evolution of Stokes parameters by using the scalar mode perturbation of metric and the generation of circular polarization on NonCommutative space is computed.Then We calculate circular, E-and B-modes spectrum of CMB.By comparing our results with experimental data the lower limit of non-commutative energy scale is obtained.

Stokes parameters and Boltzmann equation
For a monochromatic electromagnetic wave propagating in the ẑ direction, the electric field components can be given as where a x and a y are the amplitudes and θ x and θ y are the phase angles.The electromagnetic field can be parameterized in terms of the Stokes parameters which is the total intensity and for the linear polarization.Q and U are defined as the difference in brightness between the two linear polarization at 90 o and 45 o , respectively, and the circular polarization is One can see that under a right handed rotation of the coordinate axes perpendicular to the direction n on the sky, Q and U with a rotation's angle ψ transform to and the Stokes parameters I and V remain unchanged.The density matrix in terms of the Stokes parameters is defined as Meanwhile, a system of photons can be described by the density operator [2] ρ = 1 tr(ρ) where ρ ij (k) is the general density-matrix which is related to the photon number operator D 0 ij (k) ≡ a † i (k)a j (k).The canonical commutation relations of the creation and annihilation operators for photons and electrons are defined as where s and r show the photon polarization and electron spin while the bold and plain momenta represent the three dimensional vectors and the four-momentum vectors, respectively.For the expectation value of the number operator one has and in the Heisenberg picture, the time evolution of the operator D 0 ij (k) can be obtained as where H is the total Hamiltonian.Therefore, the time evolution of the densitymatrix can be written as follows where H 0 I (t) is the interacting Hamiltonian at the lowest order.The first term on the right-hand side of ( 12) is a forward scattering term that is called damping term, while the second term is the higher order collision term.To calculate the right hand side of (12) one needs the operator expectation values as where for the two point functions using the the commutation relations ( 9) one has where n f (q) represents the number density of fermions (electrons and protons) with momentum q per unit volume.Meanwhile, the distribution of fermions in the x space are defined as 3 Noncommutative standard model The noncommutative space-time is one of the consequence of the string theory.Indeed, as is predicted in [16] and shown by Seiberg and Witten [17], endpoints on D-branes in a constant B-field background live on a noncommutative spacetime.In the canonical form one has where θ µν is a real, constant and antisymmetric matrix; , and Λ N C is the noncommutative scale.The noncommutative parameter, θ µν , can be divided into two parts: the time-space components (θ 01 , θ 02 , θ 03 ) which denote the electric-like part and the magnetic-like part which contains the space-space components (θ 23 , θ 31 , θ 12 ).To construct noncommutative filed theories, one should replace the ordinary products between fields in the corresponding commutative versions with the Moyal-⋆ product which is defined as follows By applying this correspondence, there would be two approaches to construct the noncommutative standard model.The first one is based on the Moyal-⋆ product and the Seiberg-Witten maps in which the gauge group is Y , the number of particles and gauge fields are the same as the ordinary standard model [18].Furthermore, matter fields, gauge fields and gauge parameters should be expanded via the Seiberg-Witten map as a power series of θ [18] in terms of the commutative fields as Where the hats show the noncommutative fields which reduce to their counterparts in the ordinary space in the limit θ → 0.
In the second approach the noncommutative fields and the ordinary fields are the same while the gauge group is Y by an appropriate symmetry breaking [19].In the both versions, besides corrections on the usual standard model interactions, many new interactions would appear.For instance, in the QED part of the NCSM through the first approach there is a correction on the photon-fermion vertex f f γ that can be derived up to the first order of θ as [20] Meanwhile, in the noncommutative QED there are vertices which have not any counterpart in the ordinary QED.For example, two photons can directly couple to two fermions in NC space as follows where photons momenta are taken to be incoming and θ µνρ is a totally antisymmetric quantity which is defined as 4 Compton scattering in NC space-time The photons in the cosmic Microwave background can be scattered from all charged particles with the scattering rate proportional to the inverse mass squared.Therefore, in the ordinary space as a good approximation the Compton scattering on the electrons is usually considered.There are 5 diagrams in the NCSM to describe the Compton scattering on noncommutative space-time.By replacing the ordinary couplings with the NC vertices, four diagrams can be obtained which is shown in Fig 1 .The fifth diagram in which two fermions directly couple to two photons is given in (21).Therefore, the amplitude of the Compton scattering in the NCQED can be given as where and p • θ • q ≡ p µ θ µν q ν .Now the leading-order interacting Hamiltonian can be obtained as follows where dq ≡ d 3 q (2π) 3 m f q 0 and dp ≡ d 3 p (2π) 3 2p 0 , and the similar expressions for dp ′ and dq ′ , respectively.

Density matrix elements and CMB Polarization on NC space-time
In the preceding section we found the amplitude for the Compton scattering in the NCQED.Now substituting the NC interacting Hamiltonian (25) in (12) leads to Therefore, the time evaluation of the density matrix after a little algebra can be obtained as where and It should be noted that A and B are related to the NC contribution on the time evaluation of the density matrix from the vertices given in ( 20) and ( 21), respectively.As we show below the first contribution depends on the time-like component of the NC parameter θ 0i while the second contribution depends on the space-like component of the NC parameter θ ij .To this end we individually evaluate the contribution of each part on the time evolution of the stokes parameters as follows

Part A:
For the A term of (27) which is due to the diagrams given in Fig.
(1) the time evolution of the density matrix is By using Eq.( 16) and assuming and where θ ij ; i, j ∈ {1, 2, 3} are the space-like components of the NC-parameter, one can rewrite (30) as follows where the polarization four-vectors ε µi (k) with i, j, s and s ′ run over 1, 2, represent two transverse polarization of photon, nf represents the number density of Fermions and v f is the Fermion bulk velocity which is a small quantity.Note that the first term in the third line of (33) vanishes due to the anti-symmetric property of θ and the second one depends on v 2 f which is negligible in comparison with the first term.Therefore, the time derivative of the components of the density matrix can be cast into where m e is the mass of electron, σ T is the Thomson cross section, α = e 2 /4π and Q 2 f = 1.Using the density operator matrix elements, time variation of the stokes parameters, linear polarization intensities Q and U and the difference between the left and right handed polarization V in the NC space can be obtained as follows where These equations show that the contribution of the A term on the time evolution of the stokes parameters depends on the mass and bulk velocity of Fermion and the time-like components of the NC-parameter θ 0i and as is already claimed there is not any contribution from the space-space part of the NC-parameter.
In contrast with the usual Compton scattering which has a larger cross section for the particles with lower masses the evolution of the stokes parameters in the NC space are directly proportional to the Fermion masses which leads to the larger values for the scattering from Fermions with larger masses, see (38-40).In fact, since the average number of electrons ne approximately equals to the average number of protons np due to electric neutrality in cosmology, in the NC space-time the contribution of photon-proton forward scattering is larger than photon-electron scattering on the evolution of the Stokes parameters by a factor m p /m e .Nevertheless, to have any significant effects from the A term on the CMB polarization, the factor 3 8 Λ 2 should be comparable to one.It should be noted that m f k 0 is much larger than unity which can compensate the smallness of

Part B:
For the B term of ( 27) which is coming from the direct vertex (21) we have which after some calculations, lead to where and Here the time evolution of the stokes parameters depend on the space-space part of the NC-parameter as well.However, in order to have any significant effect on the CMB polarization, the value of σ| N Cdv should be comparable to σ T .For the photon-proton scattering one has which is too small to be considered.Therefore, we can neglect the B term with respect to the A term to evaluating the CMB polarization in the next sections.

Time-evolution of polarized CMB photons
In this section we expand the primordial scalar perturbations (S) in the Fourier modes which is characterized by a wave-number K.For a given Fourier mode K, one can select a coordinate system where K ẑ and (ê 1 , ê2 ) = (ê θ , êφ ).We consider the electron and baryon bulk velocity directions as v e = v b K and the photon polarization vectors are taken to be ε1x = cos θ cos ϕ, ε1y = cos θ sin ϕ, ε1z = − sin θ, ε2x = − sin ϕ, ε2y = cos ϕ, ε2z = 0. (47) Meanwhile, temperature anisotropy (I) and polarization (Q,U) of the CMB radiation can be expanded in an appropriate spin-weighted basis as follows [21] ∆ (S) ∆ (S) For each plane wave, the scattering can be described as the transport through a plane parallel medium [22,23], which leads to the Boltzmann equations as (S) here C I eγ , C ± eγ and C V eγ indicate the contributions from the usual photon-electron Compton scattering to the time evaluation of I, ∆ ±(S) P and V parameters, respectively, their expressions can be found for example in [2,21,24].In (51-53) µ = n • K = cos θ, θ is the angle between the direction of the CMB photon n = k/|k| and the wave-vectors K and where C and D are given in (41) and a(η) is the normalized scaling factor.The values of ∆ and ∆ (S) in which x = K(τ 0 − τ ), C is defined in (41) and ∆ (S) where Π = ∆ (S) The differential optical depth τeγ (τ ) and total optical depth τ eγ (τ ) due to the Thomson scattering at time τ are defined as As is shown in (51), the temperature anisotropy ∆ (S) I doesn't have any source due to the forward Compton scattering in the NC space-time therefore, we only focus on the other equations to explore the NC effects.Meanwhile, (55) and (56) indicate that the effect of non-commutativity on the linear and circular polarization can be valuable for a significant value of κNC τeγ which is defined as follows which leads to larger values for Protons than the electrons.

CMB power spectrum in NC space-time
In the preceding section we found that the Compton scattering in the NC space changes the Boltzmann equations for the time evolution of the polarized CMB photons.Here, we are ready to find the power spectra of I, B, E and V in the NC background.To this end, we consider the power spectrum as where which for the circularly polarized part of the CMB photons by using (56) in the power spectrum C V l , one has where and P v (K) is the velocity power spectrum which can be expressed in terms of the primordial scalar spectrum P (S) φ as [25] P Now ( 67) and ( 70) can provide an estimate on C V l in terms of the linearly polarized power spectrum C P l as follows for a conservative value of the NC scale of the order of Λ ∼ 10T eV .Meanwhile, for the more or less accepted value Λ ∼ 1T eV [27], the circular polarization power spectrum C V l can be obtained in a range as follows which is in the range of achievable experimental values.
In addition to C V l the Compton scattering in the NC space, in contrast with the ordinary space, can also generate the B-mode polarization.To explore such a property we give the CMB polarization in terms of the divergence-free part (B-mode ∆ where ð and ð are spin raising and lowering operators respectively [24].As Eqs.( 56), ( 77) and ( 63) show the B-mode power spectrum C Bl due to the forward electron and proton Compton scattering in the NC space-time depends on the circular polarization power spectrum which can be estimated as where S indicates the scalar mode of the matter perturbation.Furthermore, the B-mode power spectrum depends on the scale of NC parameter Λ, through κ which can have a significant effect on the value of the r-parameter even for Λ ∼ 10T eV .In fact, by using Eqs.( 71

= 3 4 m e + m p T 0 1 α m 2 eΛ 2 ≃ 1 ( 2 eΛ 2 ≃
10T eV /Λ) 2 , 10 −3 (10T eV /Λ) 2 , (73) in which k 0 = T 0 and k 0 = T lss are the energies of the CMB photons at the present time and the last scattering epoch, respectively.Using the experimental value for the linearly polarized power spectrum of the CMB photons which is of the order of 0.1µK 2 for l < 250 [26], one finds from (71)-(73) an estimation on the range of C V l as 0.1nK 2 ≤ C V l ≤ 0.1µK 2 ,