Non-commutative Power-law Inflation: Mode Equation and Spectra Index

Following an elegant approach that merge the effects of the stringy spacetime uncertainty relation into primordial perturbations suggested by Brandenberger and Ho, we show the mode equation up to the first order of non-commutative parameter. A new approximation is provided to calculate the mode functions analytically in the non-commutative power-law inflation models. It turns out that non-commutativity of spacetime can provide small corrections to the power spectrum of primordial fluctuations as the first-year results of WMAP indicate. Moreover, using the WMAP data, we obtain the value of expansion parameter, non-commutative parameter and find the approximation is viable. In addition, we determined the string scale ls ≃ 2.0× 10−29cm. e-mail address: [email protected] The cosmological parameters and the properties of inflationary models are tightly constraint by the recent result from Wilkinson Microwave Anisotropy Probe (WMAP)[1], Sloan Digital Sky Survey (SDSS) and Two degree Field (2dF) galaxy clustering analyses [2], and from the latest SNIa data [3]. The standard inflationary ΛCDM model provides a good fit to the observed cosmic microwave background (CMB) anisotropies. The first-year results of WMAP also bring us something intriguing. Some analyses [4, 5, 6, 7] show that the new data of CMB suggest an anomalously low quadrupole and octupole and a larger running of the spectral index of the power spectrum than that predicted by standard single scalar field inflation models satisfying the slow-roll conditions. On the other hand, it is well known that during the period of inflation, the classical gravitational theory, general relativity, might break down due to the very high energies at that time and the correction from string theory may take effect. In the nonperturbative string/M theory, any physical process at the very short distance takes an uncertainty relation, called stringy spacetime uncertainty relation (SSUR), ∆tp∆xp ≥ l s , (1) where tp and xp are the physical time and space, ls is the string length scale. It is suggested that the SSUR is a universal property for strings as well as D-branes [8]. Unfortunately, we now have no ideas to derive cosmology directly from string/M theory. Brandenberger and Ho [9] have proposed a variation of spacetime non-commutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. If the inflation is affected by physics at a scale close to string scale, one expects that spacetime uncertainty must leave vestiges in the CMB power spectrum [10, 11, 12]. It is found that, in the non-commutative inflation context, IR modes are created on scales larger than the Hubble radius and thus are not as squeezed as they would be in the commutative case. Cai [13] show that the choice of initial vacuum has a significant effect on the power spectrum of density fluctuation in a non-commutative spacetime. Following Ref. [9], the scalar fluctuations of tachyon inflation was discussed in non-commutative spacetime [14]. While the standard model is observationally well justified, successful non-commutative models predict that there should be observable deviations from it. Undoubtedly, we should expect that the effect of the non-commutativity of the spacetime may only provide a small correction to the prediction of the standard model. The primary observational test of inflation is observation of CMB. Temperature fluctuations in the CMB is related to perturbations in the metric at the surface at last scattering. During the inflationary epoch, metric perturbations are created by field fluctuation, and quantum fluctuations on small scales are rapidly redshifted to scales much larger than the Hubble radius. The metric perturbations can be decomposed according to their spin with respect to a local rotation of the spatial coordinates on hypersurfaces of constant time. This leads to two types: scalar perturbations which couple to the stress-energy of matter in the universe and form ”seeds” for structure formation, and tensor perturbations which do not couple to matter.