Noncommutative Inflation Research Articles

Anisotropic constant-roll inflation with noncommutative model and swampland conjectures

In this paper, we study a constant-roll inflationary model in the presence of a noncommutative parameter with a homogeneous scalar field minimally coupled to gravity. The specific noncommutative inflation conditions proposed new consequences. On the other hand, we use anisotropic conditions and find new anisotropic constant-roll solutions with respect to noncommutative parameter. Also, we will plot some figures with respect to the specific values of the corresponding parameter and the swampland criteria which is raised from the exact potential obtained from the constant-roll condition. Finally, different of figures lead us to analyze the corresponding results and also show the effect of above mentioned parameter on the inflationary model.

Stochastic gravitational-wave background in quantum gravity

Among all cosmological quantum-gravity or quantum-gravity-inspired scenarios, only very few predict a blue-tilted primordial tensor spectrum. We explore five of them and check whether they can generate a stochastic gravitational-wave background detectable by present and future interferometers: non-local quantum gravity, string-gas cosmology, new ekpyrotic scenario, Brandenberger-Ho non-commutative inflation and multi-fractional spacetimes. We show that non-local quantum gravity is unobservable, while all the other models can reach the strain sensitivity of DECIGO but not that of LIGO-Virgo-KAGRA, LISA or Einstein Telescope. Other quantum-gravity models with red-tilted spectra (most loop quantum cosmologies) or with exceptionally tiny quantum corrections (Wheeler-DeWitt quantum cosmology) are found to be non-detectable.

Strong Planck constraints on braneworld and non-commutative inflation

We place observational likelihood constraints on braneworld and non-commutative inflation for a number of inflaton potentials, using Planck, WMAP polarization and BAO data. Both braneworld and non-commutative scenarios of the kind considered here are limited by the most recent data even more severely than standard general-relativity models. At more than 95 % confidence level, the monomial potential V(ϕ)∝ϕp is ruled out for p ⩾ 2 in the Randall-Sundrum (RS) braneworld cosmology and, for p > 0, also in the high-curvature limit of the Gauss-Bonnet (GB) braneworld and in the infrared limit of non-commutative inflation, due to a large scalar spectral index. Some parameter values for natural inflation, small-varying inflaton models and Starobinsky inflation are allowed in all scenarios, although some tuning is required for natural inflation in a non-commutative spacetime.

Conceptual problem for noncommutative inflation and the new approach for nonrelativistic inflationary equation of state

In a previous paper, we connected the phenomenological non-commutative inflation of Alexander, Brandenberger and Magueijo (2003) and Koh S and Brandenberger (2007) with the formal representation theory of groups and algebras and analyzed minimal conditions that the deformed dispersion relation should satisfy in order to lead to a successful inflation. In that paper, we showed that elementary tools of algebra allow a group like procedure in which even Hopf algebras (roughly the symmetries of non-commutative spaces) could lead to the equation of state of inflationary radiation. In this paper, we show that there exists a conceptual problem with the kind of representation that leads to the fundamental equations of the model. The problem comes from an incompatibility between one of the minimal conditions for successful inflation (the momentum of individual photons is bounded from above) and the group structure of the representation which leads to the fundamental inflationary equations of state. We show that such a group structure, although mathematically allowed, would lead to problems with the overall consistency of physics, like in scattering theory, for example. Therefore, it follows that the procedure to obtain those equations should be modified according to one of two possible proposals that we consider here. One of them relates to the general theory of Hopf algebras while the other is based on a representation theorem of Von Neumann algebras, a proposal already suggested by us to take into account interactions in the inflationary equation of state. This reopens the problem of finding inflationary deformed dispersion relations and all developments which followed the first paper of Non-commutative Inflation.

Explosive particle production in non-commutative inflation

We consider a model of inflation which has recently been proposed in the literature and where inflation is induced by corrections to the energy density coming from the non-commutativity of spacetime. We show that the very rapid inflationary expansion typical of this model is responsible for a burst of particle production which ends inflation and leads to a radiation-dominated phase. We analytically estimate the energy density of these particles and we confront the results with more precise numerical calculations. We estimate the number of inflationary e-folds before the back-reaction of the radiation energy density overcomes the non-commutative effects and terminate inflation naturally.

Generalized non-commutative inflation

Non-commutative geometry indicates a deformation of the energy–momentum dispersion relation for massless particles. This distorted energy–momentum relation can affect the radiation-dominated phase of the universe at sufficiently high temperature. This prompted the idea of non-commutative inflation by Alexander et al (2003 Phys. Rev. D 67 081301) and Koh and Brandenberger (2007 JCAP06(2007)021 and JCAP11(2007)013). These authors studied a one-parameter family of a non-relativistic dispersion relation that leads to inflation: the α family of curves f(E) = 1 + (λE)α. We show here how the conceptually different structure of symmetries of non-commutative spaces can lead, in a mathematically consistent way, to the fundamental equations of non-commutative inflation driven by radiation. We describe how this structure can be considered independently of (but including) the idea of non-commutative spaces as a starting point of the general inflationary deformation of . We analyze the conditions on the dispersion relation that leads to inflation as a set of inequalities which plays the same role as the slow-roll conditions on the potential of a scalar field. We study conditions for a possible numerical approach to obtain a general one-parameter family of dispersion relations that lead to successful inflation.

A new approach to non-commutative inflation

We propose an inflationary scenario inspired by a recent formulation, in terms of coherent states, of non-commutative quantum field theory. We consider the semiclassical Einstein equations, and exploit the ultraviolet finiteness of the non-commutative propagator to construct the expectation value of the energy–momentum tensor associated with a generic scalar field. It turns out that the latter is always finite and dominated by an effective cosmological constant. By combining this general feature with the intrinsic fuzziness of spacetime, we show that non-commutativity governs the energy density of the early Universe in such a way that the strong energy condition is violated. Thus, there might be a bounce and a subsequent inflationary phase, which does not need any ad hoc scalar field.

CMB statistics in noncommutative inflation

Noncommutative geometry can provide effective description of physics at very short distances taking into account generic effects of quantum gravity. Inflation amplifies tiny quantum fluctuations in the early universe to macroscopic scales and may thus imprint high energy physics signatures in the cosmological perturbations that could be detected in the CMB. It is shown here that this can give rise to parity-violating modulations of the primordial spectrum and odd non-Gaussian signatures. The breaking of rotational invariance of the CMB provides constraints on the scale of noncommutativity that are competitive with the existing noncosmological bounds, and could explain the curious hemispherical asymmetry that has been claimed to be observed in the sky. This introduces also non-Gaussianity with peculiar shape- and scale-dependence, which in principle allows an independent cross-check of the presence of noncommutativity at inflation.

TRANS-PLANCKIAN PHYSICS AND NON-COMMUTATIVE INFLATION

Non-commutativity of spacetime at the Planck scale may deform the usual dispersion relations. And these deformed dispersion relations could lead to the accelerating phase without a scalar field. In this paper, we have calculated the spectral index and the running of spectral index in a non-commutative inflation model. Non-commutative inflation with thermal radiation gives a scale invariant spectrum in the limit w → −1 and negative running spectral index which are consistent with the WMAP 3-year results.

NONCOMMUTATIVE INFLATION AND NON-GAUSSIANITY

We investigate the physical implications of the noncommutative geometry in inflationary scenario. Firstly we study the noncommutative inflation with a time-dependent noncommutativity between space and time and find that the scalar perturbation power spectrum depends sensitively on the time varying of the spacetime noncommutativity. Secondly, we discuss the spatially noncommutative effect on the non-Gaussianities of primordial cosmological perturbation. In the lowest order of noncommutativity and slow-roll parameter, we find that in the models with small speed of sound the noncommutative modifications could be observable if assuming a relatively low string scale. It turns out that the corrections are sensitive to the speed of sound and the choice of string length scale, and the shapes of the corrected non-Gaussianities are distinct from that of ordinary ones.

Evolution of gravitational perturbations in non-commutative inflation

We consider the non-commutative inflation model in which it is the unconventionaldispersion relation for regular radiation which drives the accelerated expansion of space. Inthis model, we study the evolution of linear cosmological perturbations through thetransition between the phase of accelerated expansion and the regular radiation-dominatedphase of standard cosmology, a transition which is analogous to the reheating period inscalar field-driven models of inflation. If matter consists of only a single non-commutativeradiation fluid, then the curvature perturbations are constant on super-Hubble scales. Onthe other hand, if we include additional matter fields which oscillate during the transitionperiod, e.g. scalar moduli fields, then there can be parametric amplification of the amplitudeof the curvature perturbations. We demonstrate this explicitly by numerically solvingthe full system of perturbation equations in the case where matter consists ofboth the non-commutative radiation field and a light scalar field which undergoesoscillations. Our model is an example where the parametric resonance of the curvaturefluctuations is driven by the oscillations not of the inflaton field, but of the entropymode.

Generalized space–time noncommutative inflation

We study the noncommutative inflation with a time-dependent noncommutativity betweenspace and time. From the numerical analysis of power law inflation, there are clues that theCMB spectrum indicates a nonconstant noncommutative inflation. Then we extend ourtreatment to the inflation models with more general noncommutativity and find that thescalar perturbation power spectrum depends sensitively on the time varying of thespacetime noncommutativity. This stringy effect may be probed in the future cosmologicalobservations.

Slow-roll reconstruction for running spectral index

We reconstruct from the Wilkinson microwave anisotropy probe three-year data the slow-roll parameters in the standard slow-roll inflation model and the noncommutative inflation model. We also investigate the evolution of these slow-roll parameters. Requiring that slow-roll inflation lasts more than 20 e-folds after cosmic microwave background scales leave the horizon, we find that the potential at the last stage of inflation takes the form $V(\ensuremath{\phi})={V}_{0}(1+\frac{{\ensuremath{\eta}}_{c}}{2}\frac{(\ensuremath{\phi}\ensuremath{-}{\ensuremath{\phi}}_{c}{)}^{2}}{{M}_{p}^{2}})$, where ${\ensuremath{\eta}}_{c}$ is a constant. A natural mechanism to end inflation at $\ensuremath{\phi}={\ensuremath{\phi}}_{c}$ is the hybrid-type inflation.

Cosmological perturbations in non-commutative inflation

We compute the spectrum of cosmological perturbations in a scenario in which inflationis driven by radiation in a non-commutative space–time. In this scenario, thenon-commutativity of space and time leads to a modified dispersion relation forradiation with two branches, which allows for inflation. The initial conditionsfor the cosmological fluctuations are thermal. This is to be contrasted with thesituation in models of inflation in which the accelerated expansion of space is drivenby the potential energy of a scalar field, and in which the fluctuations are ofquantum vacuum type. We find that, in the limit that the expansion of space isalmost exponential, the spectrum of fluctuations is scale invariant with a slightred tilt. The magnitude of the tilt is different from what is obtained in a usualinflationary model with the same expansion rate during the period of inflation. Theamplitude also differs, and can easily be adjusted to agree with observations.

A note on non-commutative brane inflation

In this paper, we investigate the non-commutative KKLMMT(Kachru–Kallosh–Linde–Maldacena–McAllister–Trivedi) -brane inflation scenario in detail. Incorporation of the brane inflationscenario and the non-commutative inflation scenario can nicely explainthe large negative running of the spectral index as indicated by WMAPthree-year data and can significantly release the fine-tuning for the parameterβ. Using the WMAP three-year results (blue-tilted spectral index with large negativerunning), we explore the parameter space and give the constraints and predictionsfor the inflationary parameters and cosmological observables in this scenario.We show that this scenario predicts a quite large tensor/scalar ratio and, whatis more, a too large cosmic string tension (assuming that the string couplinggs is in its likely range from 0.1 to 1) to be compatible with the present observational bound.A more detailed analysis reveals that this model has some inconsistencies according to thefit to WMAP three-year results.

Running of the running of the spectral index and WMAP three-year data

The three-year WMAP (Wilkinson microwave anisotropy probe) data imply not only anegative running of the spectral index with large absolute value, but also a largepositive running of the running of the spectral index with the order of magnitude10−2. We calculate the running of the running in the usual inflation modeland the non-commutative inflation model. A large tensor–scalar ratior≥1.23 is needed in order to fit the WMAP data in the non-commutative inflation model, whichroughly saturates the observational upper bound on it.

Gauss–Bonnet braneworld and WMAP three year results

We compare predictions for the spectral index and tensor-scalar ratio in models of patch inflation with the WMAP three year data. There are three cases of these models of inflation, which arise in the Gauss-Bonnet braneworld scenario: Gauss-Bonnet (GB), Randall-Sundrum (RS), and 4D general relativity (GR). We consider the large-field potential $V \propto \phi^p$ in both commutative and noncommutative spacetimes, and find that in the cases of the GB and GR patch cosmologies, the quadratic potential is observationally favored, while the quartic potential is ruled out in most patches. Strong noncommutative inflation is excluded in all cases because it leads to a blue-tilted scalar spectral index.

Running spectral index in noncommutative inflation and WMAP three year results

A model independent analysis shows that the running of the spectral index of the three year WMAP results can be nicely realized in noncommutative inflation. We also re-examine some concrete noncommutative inflation models. We find that a large tensor–scalar ratio is required, corresponding to a low number of e-folds before the end of inflation in some simple models.

Noncommutative chaotic inflation and WMAP three year results

Abstract 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 space–time uncertainty principle and can be used to place constraints on the parameters of the noncommutative inflation model. In this Letter 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 p 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.

RUNNING OF THE SPECTRAL INDEX IN NONCOMMUTATIVE INFLATION

We study the cosmological implications of the space–space noncommutative inflation and present formulae for the spectral index and its running. Our results show that deviations from the spectral index and its running depend on the space–space noncommutativity length scale. We conclude that in the slow-roll regime of a typical inflationary scenario, space–space noncommutativity has negligible effects on both the spectral index and its running. Two classes of examples have been studied and comparisons made with the standard slow-roll formulae. The results show that the correction terms to the commutative case are positive for the spectral index and negative for the running of the spectral index. Nevertheless the spectral index in noncommutative spaces is less than one owing to the very small values of the correction terms.