Cosmological inflation remains to be a unique mechanism of generation of plausible initial conditions in the early universe. In particular, it generates the primordial quasiclassical perturbations with power spectrum determined by the fundamental principles of quantum field theory. In this work, we pay attention to the fact that the quasiclassical perturbations permanently generated at early stages of inflation break homogeneity and isotropy of the cosmological background. The evolution of the small-scale quantum vacuum modes on this inhomogeneous background results in statistical anisotropy of the primordial power spectrum, which can manifest itself in the observable large-scale structure and cosmic microwave background. The effect is predicted to have almost scale-invariant form dominated by a quadrupole and may serve as a non-trivial test of the inflationary scenario. Theoretical expectation of the magnitude of this statistical anisotropy depends on the assumptions about the physics in the trans-Planckian region of wavenumbers.
We review an inflationary scenario with the anisotropic expansion rate. An anisotropic inflationary universe can be realized by a vector field coupled with an inflaton, which can be regarded as a counter example to the cosmic no-hair conjecture. We show generality of anisotropic inflation and derive a universal property. We formulate cosmological perturbation theory in anisotropic inflation. Using the formalism, we show anisotropic inflation gives rise to the statistical anisotropy in primordial fluctuations. We also explain a method to test anisotropic inflation using the cosmic microwave bac…
We study inflation with multiple vector fields. In the presence of non-trivial couplings between the inflaton and the vector fields, it turns out that no-hair conjecture does not hold and vector-hair appears. In the case of uniform couplings, nevertheless, we find that the universe approaches an isotropic final state after transient anisotropic inflationary phases. For general couplings, we numerically show attractors are anisotropic inflation. Even in these cases, it turns out that the inflation always tends to minimize the anisotropy in the expansion of the universe.
We perform a detailed analysis on a primordial gravitational-wave background amplified during a Kasner-like pre-inflationary phase allowing for general triaxial anisotropies. It is found that the predicted angular distribution map of gravitational-wave intensity on large scales exhibits topologically distinctive patterns according to the degree of the pre-inflationary anisotropy, thereby serving as a potential probe for the pre-inflationary early universe with future all-sky observations of gravitational waves. We also derive an observational limit on the amplitude of such anisotropic gravitat…
An error was made when calculating the amplitude of the perturbed scalar mode after Hubble-radius crossing. When corrected, it eliminates the leading term in the statistical anisotropy of the power spectrum, reducing it from the previously claimed upper estimate of $\lesssim 10^{-2}$ to the level of at most $10^{-6}$, which is beyond the possibility of detection. The general conclusion is that a single-field inflationary scenario cannot produce statistical anisotropy of appreciable magnitude.
Anisotropic inflation is an interesting model with an U(1) gauge field and it predicts the statistical anisotropy of the curvature perturbation characterized by a parameter $g_*$. However, we find that the background gauge field does not follow the classical attractor solution due to the stochastic effect. We develop the stochastic formalism of a vector field and solve Langevin and Fokker-Planck equations. It is shown that this model is excluded by the CMB constraint $g_*\le 10^{-2}$ with a high probability about $99.999\%$.
Solid inflation is an effective field theory of inflation in which isotropy and homogeneity are accomplished via a specific combination of anisotropic sources (three scalar fields that individually break isotropy). This results in specific observational signatures that are not found in standard models of inflation: a non-trivial angular dependence for the squeezed bispectrum, and a possibly long period of anisotropic inflation (to drive inflation, the "solid" must be very insensitive to any deformation, and thus background anisotropies are very slowly erased). In this paper we compute the expe…
In the model of solid / elastic inflation, inflation is driven by a source that has the field theoretical description of a solid. To allow for prolonged slow roll inflation, the solid needs to be extremely insensitive to the spatial expansion. We point out that, because of this property, the solid is also rather inefficient in erasing anisotropic deformations of the geometry. This allows for a prolonged inflationary anisotropic solution, providing the first example with standard gravity and scalar fields only which evades the conditions of the so called cosmic no-hair conjecture. We compute th…
We consider the non-minimal model of gravity in $Y(R) F^2$-form. We investigate a particular case of the model, for which the higher order derivatives are eliminated but the scalar curvature $R$ is kept to be dynamical via the constraint $Y_RF_{mn}F^{mn} =-\frac{2}{κ^2}$. The effective fluid obtained can be represented by interacting electromagnetic field and vacuum depending on $Y(R)$, namely, the energy density of the vacuum tracks $R$ while energy density of the conventional electromagnetic field is dynamically scaled with the factor $\frac{Y(R)}{2}$. We give exact solutions for anisotropic…
We present a review of vector field models of inflation and, in particular, of the statistical anisotropy and non-Gaussianity predictions of models with SU(2) vector multiplets. Non-Abelian gauge groups introduce a richer amount of predictions compared to the Abelian ones, mostly because of the presence of vector fields self-interactions. Primordial vector fields can violate isotropy leaving their imprint in the comoving curvature fluctuations zeta at late times. We provide the analytic expressions of the correlation functions of zeta up to fourth order and an analysis of their amplitudes and …
We derive the primordial power spectrum of density fluctuations in the framework of quantum cosmology. For this purpose we perform a Born-Oppenheimer approximation to the Wheeler-DeWitt equation for an inflationary universe with a scalar field. In this way we first recover the scale-invariant power spectrum that is found as an approximation in the simplest inflationary models. We then obtain quantum gravitational corrections to this spectrum and discuss whether they lead to measurable signatures in the CMB anisotropy spectrum. The non-observation so far of such corrections translates into an u…
Single field inflationary models that seek to greatly enhance small scale power in order to form primordial black holes predict both a squeezed bispectrum that is enhanced by this small scale power and a potentially detectable enhancement of CMB spectral distortions. Despite this combination, spectral distortion anisotropy on CMB scales remains small since the squeezed bispectrum represents an unobservable modulation of the scale rather than local amplitude for the short wavelength acoustic power that dissipates and forms the $μ$ spectral distortion. The leading order amplitude effect comes fr…
We outline the modifications in the numerical Boltzmann code Cosmic Linear Anisotropy Solving System (CLASS) in order to include extra inflationary fields. The functioning of the code is first described, how and where modifications are meant to be done are later explained. In the present study, we focus on the modifications needed for the implementation of a two-field inflationary model, with canonical kinetic terms and a polynomial potential with no cross terms, presenting preliminary results for the effect of the second field on the spectra. The adaptability of the code is exploited, making …
Provided the quantum fluctuations are amplified in the presence of a classical gauge field configuration the resulting curvature perturbations exhibit a mild statistical anisotropy which should be sufficiently weak not to conflict with current observational data. The curvature power spectra induced by weakly anisotropic initial states are computed here for the first time when the electric and the magnetic gauge couplings evolve at different rates as it happens, for instance, in the relativistic theory of van der Waals interactions. After recovering the results valid for coincident gauge coupli…
We study the statistical descriptors for some cosmological inflationary models that allow us to get large levels of non-gaussianity and violations of statistical isotropy. Basically, we study two different class of models: a model that include only scalar field perturbations, specifically a subclass of small-field slow-roll models of inflation with canonical kinetic terms, and models that admit both vector and scalar field perturbations. We study the former to show that it is possible to attain very high, including observable, values for the levels of non-gaussianity f_{NL} and \tao_{NL} in th…
We study an anisotropic inflation model with a gauge kinetic function for a non-abelian gauge field. We find that, in contrast to abelian models, the anisotropy can be either a prolate or an oblate type, which could lead to a different prediction from abelian models for the statistical anisotropy in the power spectrum of cosmological fluctuations. During a reheating phase, we find chaotic behaviour of the non-abelian gauge field which is caused by the nonlinear self-coupling of the gauge field. We compute a Lyapunov exponent of the chaos which turns out to be uncorrelated with the anisotropy.
Inflationary models can correlate small-scale density perturbations with the long-wavelength gravitational waves (GW) in the form of the Tensor-Scalar-Scalar (TSS) bispectrum. This correlation affects the mass-distribution in the Universe and leads to the off-diagonal correlations of the density field modes in the form of the quadrupole anisotropy. Interestingly, this effect survives even after the tensor mode decays when it re-enters the horizon, known as the fossil effect. As a result, the off-diagonal correlation function between different Fourier modes of the density fluctuations can be th…
We show that perturbations generated during the anisotropic pre-inflationary stage of cosmic evolution may affect cosmological observations today for a certain range of parameters. Due to the anisotropic nature of the universe during such early times, it might explain some of the observed signals of large scale anisotropy. In particular we argue that the alignment of CMB quadrupole and octopole may be explained by the Sachs-Wolfe effect due to the large scale anisotropic modes from very early times of cosmological evolution. We also comment on how the observed dipole modulation of CMB power ma…
We examine cosmological perturbations in a dynamical theory of inflation in which an Abelian gauge field couples directly to the inflaton, breaking conformal invariance. When the coupling between the gauge field and the inflaton takes a specific form, inflation becomes anisotropic and anisotropy can persist throughout inflation, avoiding Wald's no-hair theorem. After discussing scenarios in which anisotropy can persist during inflation, we calculate the dominant effects of a small persistent anisotropy on the primordial gravitational wave and curvature perturbation power spectra using the "in-…
We study the most general contributions due to scalar field perturbations, vector field perturbations, and anisotropic expansion to the generation of statistical anisotropy in the primordial curvature perturbation ζ. Such a study is done using the δN formalism where only linear terms are considered. Here, we consider two specific cases that lead to determine the power spectrum P_ζ(k) of the primordial curvature perturbation. In the first one, we consider the possibility that the n-point correlators of the field perturbations in real space are invariant under rotations in space (statistical iso…
In this paper we operate under the assumption that no tensors from inflation will be measured in the future by the dedicated experiments and argue that, while for single-field slow-roll models of inflation the running of the spectral index will be hard to be detected, in multi-field models the running can be large due to its strong correlation with non-Gaussianity. A detection of the running might therefore be related to the presence of more than one active scalar degree of freedom during inflation.
We study an inflationary scenario with a vector field coupled with an inflaton field and show that the inflationary universe is endowed with anisotropy for a wide range of coupling functions. This anisotropic inflation is a tracking solution where the energy density of the vector field follows that of the inflaton field irrespective of initial conditions. We find a universal relation between the anisotropy and a slow-roll parameter of inflation. Our finding has observational implications and gives a counter example to the cosmic no-hair conjecture.
We study the statistical nature of primordial fluctuations from an anisotropic inflation which is realized by a vector field coupled to an inflaton. We find a suitable gauge, which we call the canonical gauge, for anisotropic inflation by generalizing the flat slicing gauge in conventional isotropic inflation. Using the canonical gauge, we reveal the structure of the couplings between curvature perturbations, vector waves, and gravitational waves. We identify two sources of anisotropy, i.e. the anisotropy due to the anisotropic expansion of the universe and that due to the anisotropic coupling…
We display some simple cosmological solutions of gravity theories with quadratic Ricci curvature terms added to the Einstein-Hilbert lagrangian which exhibit anisotropic inflation. The Hubble expansion rates are constant and unequal in three orthogonal directions. We describe the evolution of the simplest of these homogeneous and anisotropic cosmological models from its natural initial state and evaluate the deviations they will create from statistical isotropy in the fluctuations produced during a period of anisotropic inflation. The anisotropic inflation is not a late-time attractor in these…
We consider cosmological inflationary models in which vector fields play some role in the generation of the primordial curvature perturbation $ζ$. Such models are interesting because the involved vector fields naturally seed statistical anisotropy in the primordial fluctuations which could eventually leave a measurable imprint on the cosmic microwave background fluctuations. In this article, we estimate the scale and shape dependent effects on the non-Gaussianity (NG) parameters due to the scale dependent statistical anisotropy in the distribution of the fluctuations. For concreteness, we use …
The gravitational baryogensis may not generate a sufficient baryon asymmetry in the standard thermal history of the Universe when we take into account the gravitino problem. Hence it has been suggested that anisotropy of the Universe can enhance the generation of the baryon asymmetry through the increase of the time change of the Ricci scalar curvature. We study the gravitational baryogenesis in the presence of anisotropy, which is produced at the end of an anisotropic inflation. Although we confirm that the generated baryon asymmetry is enhanced compared with the original isotropic cosmologic…
The phenomenon of stochastic eternal inflation is studied for a chaotic inflation potential in a Bianchi Type I spacetime background. After deriving the appropriate stochastic Klein-Gordon equation, we give details on the conditions for eternal inflation. It is shown that for eternal inflation to occur, the amount of anisotropy must be small. In fact, it is shown that eternal inflation will only take place if the shear anisotropy variables take on values within a small region of the interior of the Kasner circle. We then calculate the probability of eternal inflation occurring based on techniq…
The presence of a light vector field coupled to a scalar field during inflation makes a distinct prediction: the observed correlation functions of the cosmic microwave background (CMB) become statistically anisotropic. We study the implications of the current bound on statistical anisotropy derived from the Planck 2013 CMB temperature data for such a model. The previous calculations based on the attractor solution indicate that the magnitude of anisotropy in the power spectrum is proportional to $N^2$, where $N$ is the number of $e$-folds of inflation counted from the end of inflation. In this…
We investigate the possibility of embedding the vector curvaton paradigm in D-brane models of inflation in Type IIB string theory in a simple toy model. The vector curvaton is identified with the U(1) gauge field that lives on the world volume of a D3-brane, which may be stationary or undergoing general motion in the internal space. The dilaton is considered as a spectator field which modulates the evolution of the vector field. In this set up, the vector curvaton is able to generate measurable statistical anisotropy in the spectrum and bispectrum of the curvature perturbation assuming that th…
We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat $Λ$CDM model. Unlike earlier analyses of non-flat models, which assumed an inconsistent power-law power spectrum of energy density inhomogeneities, we find that the Planck 2015 data alone, and also in conjunction with baryon acoustic oscillation measurements, are reasonably well fit by a closed $Λ$CDM model in which spatial curvature contributes a few percent of the current cosmological …
We perform Markov chain Monte Carlo analyses to put constraints on the non-flat $ϕ$CDM inflation model using Planck 2015 cosmic microwave background (CMB) anisotropy data and baryon acoustic oscillation distance measurements. The $ϕ$CDM model is a consistent dynamical dark energy model in which the currently accelerating cosmological expansion is powered by a scalar field $ϕ$ slowly rolling down an inverse power-law potential energy density. We also use a physically consistent power spectrum for energy density inhomogeneities in this non-flat model. We find that, like the closed-$Λ$CDM and clo…
Recent BICEP2 detection of low-multipole B-mode polarization anisotropy in the cosmic microwave background radiation supports the inflationary universe scenario and suggests a large inflaton field range. The latter feature can be achieved with axion fields in the framework of string theory. We present such a helical model which naturally becomes a model with a single cosine potential, and which in turn reduces to the (quadratic) chaotic inflation model in the super-Planckian limit. The slightly smaller tensor/scalar ratio $r$ of models of this type provides a signature of the periodic nature o…
We propose a model of cosmological evolution of the early and late Universe which is consistent with observational data and naturally explains the origin of inflation and dark energy. We show that the de Sitter accelerated expansion of the FLRW space with no matter fields (hereinafter, empty space) is its natural state, and the model does not require either a scalar field or cosmological constant or any other hypotheses. This is due to the fact that the de Sitter state is an exact solution of the rigorous mathematically consistent equations of one-loop quantum gravity for the empty FLRW space …
In this work we study the imprints of a primordial cosmic string on inflationary power spectrum. Cosmic string induces two distinct contributions on curvature perturbations power spectrum. The first type of correction respects the translation invariance while violating isotropy. This generates quadrupolar statistical anisotropy in CMB maps which is constrained by the Planck data. The second contribution breaks both homogeneity and isotropy, generating a dipolar power asymmetry in variance of temperature fluctuations with its amplitude falling on small scales. We show that the strongest constra…
We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat XCDM model. Here dark energy is parameterized using a fluid with a negative equation of state parameter but with the speed of fluid acoustic inhomogeneities set to the speed of light. We use this simple parameterization of dynamical dark energy, that is relatively straightforward to use in a computation, in a first attempt to gain some insight into how dark energy dynamics and non-zero s…
Searches for inflationary gravitational wave signals in the CMB B-mode polarisation are expected to reach unprecedented power over the next decade. A major difficulty in these ongoing searches is that galactic foregrounds such as dust can easily mimic inflationary signals. Though typically foregrounds are separated from primordial signals using the foregrounds' different frequency dependence, in this paper we investigate instead the extent to which the galactic dust B-modes' statistical anisotropy can be used to distinguish them from inflationary B-modes, building on the work of Kamionkowski a…
A suitable coupling of the inflaton phi to a vector kinetic term F^2 gives frozen and scale invariant vector perturbations. We compute the cosmological perturbations zeta that result from such coupling by taking into account the classical vector field that unavoidably gets generated at large scales during inflation. This generically results in a too anisotropic power spectrum of zeta. Specifically, the anisotropy exceeds the 1% level (10% level) if inflation lasted ~5 e-folds (~50 e-folds) more than the minimal amount required to produce the CMB modes. This conclusion applies, among others, to…
Both the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck observations reported the hemispherical asymmetry of the cosmic microwave background (CMB) temperature fluctuation. The hemispherical asymmetry might be stemmed from the primordial statistical anisotropy during the inflationary era of the universe. In this paper, we study possible implications of the primordial power spectra with dipolar anisotropy on the CMB temperature fluctuation and polarizations. We explicitly show that the statistical dipolar anisotropy may induce the off-diagonal (\(\ell'\neq\ell\)) $TT$, $EE$, $BB$, and $T…
Measurements of the cosmic microwave background (CMB) spectral $y$-distortion anisotropy offer a test for the statistical isotropy of the primordial density perturbations on $0.01\lesssim k{\rm Mpc}\lesssim 1$. We compute the 1-point ensemble averages of the $y$-distortion anisotropies which vanish for the statistically isotropic perturbations. For the quadrupole statistical anisotropy, we find $4π\langle y_{2m}\rangle=-6.8A_2\times 10^{-9}Y_{2m}(\mathbf d )$ with the quadruple Legendre coefficient of the anisotropic powerspectrum $A_2$ and the $\ell=2$ spherical harmonics $Y_{2m}(\mathbf d )$…
Inflation as the leading paradigm depicting the very early universe physics could leave imprints on the cosmic microwave background (CMB) radiation. Using currently available CMB observations, we give the tightest constraints on inflation so far. We discuss the theoretical implications of our results including the energy scale of inflation, inflaton field excursion, Hubble expansion rate during inflation, equation of state of inflation, primordial tensor non-Gaussianity, primordial tensor power spectrum, B-mode anisotropy and inflationary gravitational wave background.
We examine the effect of the thermal vacuum on the power spectrum of inflation by using the thermal field dynamics. We find that the thermal effect influences the CMB anisotropy at large length scale. After removing the divergence by using the holographic cutoff, we observe that the thermal vacuum explains well the observational CMB result at low multipoles. This shows that the temperature dependent factor should be considered in the study of power spectrum in inflation, especially at large length scale.
The Wilkinson Microwave Anisotropy Probe (WMAP) mapped the distribution of temperature and polarization over the entire sky in five microwave frequency bands. These full-sky maps were used to obtain measurements of temperature and polarization anisotropy of the cosmic microwave background with the unprecedented accuracy and precision. The analysis of two-point correlation functions of temperature and polarization data gives determinations of the fundamental cosmological parameters such as the age and composition of the universe, as well as the key parameters describing the physics of inflation…
In solid inflation the single field non-Gaussianity consistency condition is violated. As a result, the long tensor perturbation induces observable clustering fossils in the form of quardupole anisotropy in large scale structure power spectrum. In this work we revisit the bispectrum analysis for the scalar-scalar-scalar and tensor-scalar-scalar for the general parameter space of solid. We consider the parameter space of the model in which the level of non-Gaussianity generated is consistent with Planck constraints. Specializing to this allowed range of model parameter, we calculate the quadrup…
We study an inflationary scenario with a two-form field to which an inflaton couples non-trivially. First, we show that anisotropic inflation can be realized as an attractor solution and that the two-form hair remains during inflation. A statistical anisotropy can be developed because of a cumulative anisotropic interaction induced by the background two-form field. The power spectrum of curvature perturbations has a prolate-type anisotropy, in contrast to the vector models having an oblate-type anisotropy. We also evaluate the bispectrum and trispectrum of curvature perturbations by employing …
We study curvature perturbations in the anisotropic inflationary model with a complex scalar field charged under a $U(1)$ gauge field in Bianchi I universe. Due to Abelian Higgs mechanism, the gauge field receives an additional longitudinal mode. We verify that the dominant contributions into statistical anisotropies come from matter fields perturbations and one can neglect the contributions from the metric perturbations. It is shown that the contribution of longitudinal mode into the statistical anisotropy power spectrum, though exponentially small, has an opposite sign compared to the corres…
Solid inflation can support a long period of anisotropic inflation. We calculate the statistical anisotropies in the scalar and tensor power spectra and their cross-correlation in anisotropic solid inflation. The tensor-scalar cross-correlation can either be positive or negative, which impacts the statistical anisotropies of the TT and TB spectra in CMB map more significantly compared with the tensor self-correlation. The tensor power spectrum contains potentially comparable contributions from quadrupole and octopole angular patterns, which is different from the power spectra of scalar, the cr…
This thesis is centered on three main subjects within the theory of inflation and cosmological perturbations: loop corrections to the power spectrum of curvature fluctuations generated during inflation; evolution of cosmological fluctuations in anisotropic pre-inflationary cosmologies; statistical anisotropy and non-Gaussianity predictions of models of inflation populated with vector fields. Currently, what makes even more interesting the study of 2-nd and higher order corrections to cosmological correlation functions as well as the computation of higher-than-two order correlators, is the almo…
We quantitatively study how the primordial density fluctuations are imprinted on the anisotropy of the phase transition gravitational wave (PTGW). Generated long before recombination and free from Silk damping, the anisotropic PTGW might reveal the density perturbation seeded from inflation or alternatives. We find new behaviors of the PTGW anisotropy power spectrum. The PTGW anisotropy is stronger than the anisotropy of the cosmic microwave background temperature at all scales, and the high-$\ell$ multiples are enhanced about 1 order due to the early integrated Sachs-Wolfe effect. Furthermore…
When a light scalar field is present during inflation, its value will vary on superhorizon scales, modulating the preheating process at the end of inflation. Consequently, the amplitude of the gravitational wave (GW) background produced during preheating will also be modulated. The observed energy density of this background will therefore appear anisotropic at different angles in the sky. We provide a master formula for the angular power spectrum C_l of the anisotropies in the GW background from preheating, valid for any scenario where the anisotropies are due to the superhorizon modulation of…
It is shown that, in supergravity models of inflation where the gauge kinetic function of a gauge field is modulated by the inflaton, we can obtain a new inflationary attractor solution, in which the roll-over of the inflaton suffers additional impedance due to the vector field backreaction. As a result, directions of the scalar potential which, due to strong Kaehler corrections, become too steep and curved to normally support slow-roll inflation can now naturally do so. This solves the infamous eta-problem of inflation in supergravity and also keeps the spectral index of the curvature perturb…
We propose a single field inflationary model by generalizing the inverse power law potential from the intermediate model. We study the implication of our model on the primordial anisotropy of cosmological microwave background radiation. Specifically, we apply the slow-roll approximation to calculate the scalar spectral tilt $n_s$ and the tensor-to-scalar ratio $r$. The results are compared with the recent data measured by the Planck satellite. We found that by choosing proper values for the parameters, our model can well describe the Planck data.
Primordial inflation may represent the most powerful collider to test high-energy physics models. In this paper we study the impact on the inflationary power spectrum of the comoving curvature perturbation in the specific model where massive higher spin fields are rendered effectively massless during a de Sitter epoch through suitable couplings to the inflaton field. In particular, we show that such fields with spin $s$ induce a distinctive statistical anisotropic signal on the power spectrum, in such a way that not only the usual $g_{2M}$-statistical anisotropy coefficients, but also higher-o…
Several early Universe scenarios predict a direction-dependent spectrum of primordial curvature perturbations. This translates into the violation of the statistical isotropy of cosmic microwave background radiation. Previous searches for statistical anisotropy mainly focussed on a quadrupolar direction-dependence characterised by a single multipole vector and an overall amplitude $g_*$. Generically, however, the quadrupole has a more complicated geometry described by two multipole vectors and $g_*$. This is the subject of the present work. In particular, we limit the amplitude $g_*$ for differ…
In this paper, we consider a possibility that the temperature anisotropy of cosmic microwave background (CMB) is dominantly generated by the primordial fluctuations of QCD-axion like particles under a circumstance that inflaton's perturbation is too small to explain the CMB anisotropy. Since the axion potential is generated by acquiring its energy from radiation, the primordial fluctuations of the axion field generated in the inflation era are correlated with the CMB anisotropies. Consequently, the observations stringently constrain a model of the axion and the early universe scenario. The fol…
