Noncommutative chaotic inflation and WMAP three year results

Noncommutative inflation is based upon the consideration of some effects of the space-time uncertainty principle motivated by ideas from string/M theory. The CMB anisotropies may carry a signature of this very early Universe correction from the spacetime uncertainty principle and can be used to place constraints on the parameters of the noncommutative inflation model. In this paper we analyze the noncommutative chaotic inflation model by means of the WMAP three year results. We show that the noncommutative chaotic inflation model can produce a large negative running of spectral index within a reasonable range of e-folding number N , provided that the value of p in the potential V (φ) ∝ φ is enhanced roughly to p ∼ 12− 18. Applying a likelihood analysis to the WMAP results, we find the best-fit values of parameters for the noncommutative chaotic inflation model. In addition, this model predicts a rather big gravitational wave which may be tested by the upcoming experiments. Einstein’s general relativity will break down at very high energies in the early Universe when quantum effects are expected to be important. If inflation [1] happens at the very early time and if the period of inflation lasts sufficiently long, then the effects of quantum gravity should in principle leave an imprint on the primordial spectrum of perturbations, since the wavelengths of perturbations emerged from short distances in the early stages of inflation are stretched to the cosmic scales observable today by the rapid expansion during inflation. Though we lack a complete theory of quantum gravity presently, we can still look for the hint of these quantum gravity imprints through cosmological experiments such as WMAP and SDSS. Since string/M theory is a promising framework for quantum gravity, it is of interest to explore specific stringy corrections to the spectrum of fluctuations. Many authors have considered and discussed this problem. For the first discussion on the trace of the trans-planckian effect in the primordial spectrum of perturbations see [2]; for a review containing a comprehensive list of references see [3]. In particular, it should be pointed out that the possible effects of quantum gravity in the spectra of fluctuations in inflationary cosmology may be induced Noncommutative chaotic inflation and WMAP three year results 2 by the consequences of a basic stringy effect, namely the space-time uncertainty relation [4] ∆t∆xphys > L 2 s, (1) where t, xphys are the physical space-time coordinates and Ls is the string scale. It was shown in [5] that this space-time uncertainty principle may yield inflation from pure radiation. A more modest approach was pioneered in [6], where the consequences were studied by imposing Eq.(1) into the action for cosmological perturbations on an inflationary background. On a general ground, if inflation is indeed affected by physics at a scale close to string scale or a related scale, it can be expected that space-time uncertainty must leave traces in the CMB power spectrum [6, 7, 8]. The primordial power spectrum based upon the noncommutative inflation model [6] has been given in [9, 10, 11, 12], and the explicit calculation indeed shows that the effects can be observed. The standard concordance ΛCDM model, which can arise from an inflationary background cosmology in which the quasi-exponential expansion of space is driven by a scalar field, still provides a fairly good fit to the recent three-year WMAP [13] and earlier observations. Assuming that the primordial fluctuations are adiabatic with a power law spectrum, the WMAP three-year data require a spectral index that is significantly less than the Harrison-Zel’dovich-Peebles scale-invariant spectrum (ns = 1, r = 0). This suggests, for power law inflationary models, a detectable level of gravity waves. However, many inflationary models can only predict a much smaller gravity wave amplitude. Moreover, the WMAP three-year data can place significant constraints on inflationary models. For example, the chaotic inflationary model with the inflaton potential, V (φ) ∝ φ, has been tested in light of the current data, with the result that the data prefer the mφ model over the λφ model, assuming a power-law primordial power spectrum. When allowing for a running spectral index, the data will favor a large negative running index and a large tensor amplitude (characterized by r, the ratio of the tensor to scalar power spectrum). However, it is rather hard to produce a large absolute value of the running spectral index within the framework of the usual slowroll inflationary models [14]. Hence, the confirmation of this suggestive trend is very important for our understanding of the early Universe physics, especially of the quantum gravity relevant physics. It has been mentioned that the CMB anisotropies may carry a signature of the very early Universe corrections from the space-time uncertainty principle motivated by string theory, and can be used to place constraints on the parameters appearing in the noncommutative inflation model. A recent paper [15] shows that the noncommutative inflation model can nicely produce a large running spectral index. In [15], some concrete noncommutative inflation models such as the chaotic inflation and power-law inflation have also been re-examined. It is well known that the usual chaotic inflation model can only predict the negligible amount of running index, dns/d ln k ≈ −10 . Though the noncommutative chaotic inflation model can realize a considerable running index, for the mφ model and the λφ model, the authors of [15] find that a low number of e-folds, Noncommutative chaotic inflation and WMAP three year results 3 say N ∼ 10, is required. Needless to say, this is fairly unnatural since such low values of N may not be sufficient to resolve the naturalness problems such as the flatness and the horizon puzzles. A reasonable range of N has been derived in [16], that is 47 < N < 61 (or N = 54± 7). In this paper, we will show that the noncommutative chaotic inflation model can provide a large negative running spectral index within the reasonable range of N , provided that the value of p in the potential V (φ) ∝ φ is enhanced, say p ∼ 15. In addition, we perform a likelihood analysis to find the best-fit values of the parameters of the noncommutative chaotic inflation model to the WMAP results of the spectral index and its running. Thus we are able to constrain the noncommutative model so as to place limits on the space-time noncommutativity in the string scale. The space-time noncommutative effects can be encoded in a new product, star product, replacing the usual algebra product. The evolution of a homogeneous and isotropic background will not change and the standard cosmological equations based upon the Friedmann-Robertson-Walker (FRW) metric remain the same: φ̈+ 3Hφ̇+ V (φ) = 0, (2)