Distinguishing between inflationary models from CMB
In this paper, inflationary cosmology is reviewed, paying particular attention to its observational signatures associated with large-scale density perturbations generated from quantum fluctuations. In the most general scalar-tensor theories with second-order equations of motion, we derive the scalar spectral index , the tensor-to-scalar ratio , and the nonlinear estimator of primordial non-Gaussianities to confront models with observations of Cosmic Microwave Background (CMB) temperature anisotropies. Our analysis includes models such as potential-driven slow-roll inflation, k-inflation, Starobinsky inflation, and Higgs inflation with non-minimal/derivative/Galileon couplings. We constrain a host of inflationary models by using the Planck data combined with other measurements to find models most favored observationally in the current literature. We also study anisotropic inflation based on a scalar coupling with a vector (or, two-form) field and discuss its observational signatures appearing in the two-point and three-point correlation functions of scalar and tensor perturbations.
The inflationary paradigm was first proposed in the early 1980s to address the horizon, flatness, and monopole problems that plagued Big Bang cosmology [1, 2]. Moreover, inflation provides a causal mechanism for the generation of large-scale density perturbations from the quantum fluctuation of a scalar field (inflaton). In its simplest form the resulting power spectra of scalar and tensor perturbations are nearly scale-invariant and Gaussian . This prediction showed good agreement with the temperature anisotropies of CMB measured by the Cosmic Background Explorer (COBE)  and the Wilkinson Microwave Anisotropy Probe (WMAP) . In March 2013 the Planck team  released more accurate CMB data up to the multipoles . With these new data it is now possible to discriminate between a host of inflationary models.
The first model of inflation, proposed by Starobinsky , is based on a conformal anomaly in quantum gravity. The Lagrangian density , where is a Ricci scalar and is a mass scale of the order of GeV, can lead to a sufficient amount of inflation with a successful reheating . Moreover, the Starobinsky model is favored from the 1-st year Planck observations . The “old inflation” , which is based on the theory of supercooling during the cosmological phase transition, turned out to be unviable, because the Universe becomes inhomogeneous as a result of the bubble collision after inflation. The revised version dubbed “new inflation” [8, 9], where the second-order transition to true vacuum is responsible for cosmic acceleration, is plagued by a fine-tuning problem for spending enough time in false vacuum. However, these pioneering ideas opened up a new paradigm for the construction of workable inflationary models based on theories beyond the Standard Model of particle physics (see e.g., Refs. [10, 11, 12, 13]).
Most of the inflationary models, including chaotic
inflation , are based on a slow-rolling scalar field
with a sufficiently flat potential.
One can discriminate between a host of inflaton potentials
by comparing theoretical predictions of the scalar spectral
index and the tensor-to-scalar ratio
with the CMB temperature anisotropies
(see, e.g., [15, 16, 17, 18]).
The Planck data, combined with the WMAP large-angle polarization (WP)
measurement, placed the bounds (68 % CL)
and (95 % CL) for the pivot wavenumber
Based on the paper , we review the observational bounds
on potential-driven slow-roll inflation constrained from the joint data analysis of
Planck , WP , Baryon Acoustic
Oscillations (BAO) , and high- 
Besides slow-roll inflation, there is another class of models, called k-inflation models , in which the non-linear field kinetic energy plays a crucial role in driving cosmic acceleration. Since the scalar propagation speed in k-inflation is generally different from the speed of light , this can give rise to large non-Gaussianities of primordial perturbations for the equilateral shape in the regime [26, 27]. Using the recent Planck bound on the equilateral non-linear parameter (68 % CL) , it is possible to put tight constraints on most of the k-inflationary models.
There are also other single-field inflationary scenarios constructed in the framework of extended theories of gravity, such as non-minimally coupled models [29, 30], Brans-Dicke theories , Galileons [32, 33, 34, 35], field derivative couplings to gravity [36, 37], and running kinetic couplings [38, 39]. All of these models are covered in Horndeski’s most general scalar-tensor theories with second-order equations of motion [40, 41]. For single-field inflation based on the Horndeski theory, the two-point and three-point correlation functions of scalar and tensor perturbations have been computed in Refs. [42, 43, 44, 45, 46] (see also Refs. ). We shall first review these results and then apply them to concrete models of inflation.
The WMAP5 data indicated that there is an anomaly associated with the broken rotational invariance of the CMB power spectrum . This statistical anisotropy is difficult to address in the context of single-field slow-roll inflation. The power spectrum of curvature perturbations with broken statistical isotropy involves an anisotropy parameter . This parameter was constrained as (68 % CL) from the WMAP5 data by including multipoles up to . With the WMAP9 data, the bound (68 % CL) was derived in Ref. . Recently, Kim and Komatsu obtained the bound (68 % CL) from the Planck data by taking into account the beam correction and the Galactic foreground correction . This result is consistent with the isotropic power spectrum, but there is still a possibility that anisotropy of the order remains.
For the models in which the inflaton field has a coupling to a vector kinetic term , an anisotropic hair can survive during inflation for a suitable choice of coupling [52, 53]. In this case, it is possible to explain the anisotropic power spectrum compatible with the broken rotational invariance of the CMB perturbations [54, 55]. The same property also holds for the two-form field models in which the inflaton couples to the kinetic term , but the types of anisotropies are different from each other. Moreover, these two anisotropic inflationary models can give rise to a detectable level of primordial non-Gaussianities [57, 58, 56, 59]. We shall review the general properties of anisotropic inflation and discuss their observational signatures.
This review is organized as follows. In Sec. 2 we derive the two-point and three-point correlation functions of curvature perturbations and the resulting CMB observables in the Horndeski theory. In Sec. 3 we study observational constraints on potential-driven slow-roll inflation in the light of the Planck data. In Sec. 4 we distinguish between a host of single-field inflationary models that belong to the framework of the Horndeski theory. In Sec. 5 we discuss the current status of anisotropic inflation paying particular attention to their observational signatures. Sec. 6 is devoted to the conclusion.
2 Inflationary power spectra and non-Gaussianities in the most general scalar-tensor theories
where is a determinant of the metric tensor , is the reduced Planck mass, is the Ricci scalar, and
Here, a semicolon represents a covariant derivative, and () are functions in terms of and , and is the Einstein tensor ( is the Ricci tensor). For the partial derivatives with respect to and , we use the notation and .
On the flat Friedmann-Lemaître-Robertson-Walker (FLRW) background described by the line element , the Friedmann equation and the scalar-field equation of motion are given, respectively, by [42, 43, 44]
where is the Hubble parameter (a dot represents a derivative with respect to ), , and
where the slow-variation parameters on the r.h.s. are
defined by ,
, , ,
, and .
The number of e-foldings is defined as , where and are the scale factors at time during inflation and at the end of inflation respectively. On using the relation , it can also be expressed as
For the computations of the two-point and three-point correlation functions of scalar and tensor perturbations, we use the following perturbed ADM metric  on the flat FLRW background
where describe scalar metric perturbations, and is the tensor perturbation. The choice of the ADM metric is particularly convenient for the calculation of non-Gaussianities [62, 63]. Note that, at linear order in perturbations, the coefficient in front of in Eq. (10) reduces to . We introduce the gauge-invariant curvature perturbation 
where is the perturbation in the field . We choose unitary gauge to fix the time component of a gauge-transformation vector . The scalar perturbation , which appears in the metric (10) in the form , is gauged away, so that the spatial component of is fixed (see Refs. [65, 66] for details of the cosmological perturbation theory).
Expanding the action (1) up to second order in perturbations, we can derive the equations of motion for linear perturbations. Variations of the second-order action with respect to and lead to the Hamiltonian and momentum constraints, respectively, by which and can be related to the curvature perturbation . Then, the resulting second-order action of scalar perturbations reads [42, 43, 44, 20]
At leading order in slow-variation parameters we have
As we will see later, the tensor ghost is absent for . As long as and , we can avoid the ghost and Laplacian instabilities of scalar perturbations.
We write the curvature perturbation in terms of Fourier components, as
where is the conformal time, is a comoving wavenumber, and and are the annihilation and creation operators, respectively, satisfying the commutation relations
Since in the de Sitter background, the asymptotic past and future correspond to and , respectively. Introducing a field with the kinetic term in the second-order action (12) can be rewritten as , where a prime represents a derivative with respect to . Hence is a canonical field that should be quantized. In Fourier space the field obeys the differential equation
In the de Sitter background with a slow variation of the quantity , we have . In the asymptotic past () we choose the Bunch-Davies vacuum characterized by the mode function . Then the solution of Eq. (19) reads
The two-point correlation function, some time after the Hubble radius crossing, is given by the vacuum expectation value at . We define the scalar power spectrum , as
On using the solution (20), the resulting power spectrum of is
which is evaluated at (because is nearly frozen for ). The scalar spectral index reads
The running spectral index is defined by , which is of the order of from Eq. (23).
We can decompose the transverse and traceless tensor perturbations into two independent polarization modes, as , where (where ) satisfy the relations , , and . The second-order action for reads [42, 43, 44]
Following a similar procedure to that for scalar perturbations, we obtain the tensor power spectrum
At the epoch when both and become nearly constant during inflation, the tensor-to-scalar ratio can be evaluated as
Then the tensor spectral index is given by
The non-linear estimator, , is commonly used to characterize the level of non-Gaussianities [67, 68, 62]. In Refs. [43, 44] the leading-order bispectrum was derived on the de Sitter background. Reference  evaluated the three-point correlation function by taking into account all possible slow-variation corrections to the leading-order term (along the lines of Ref. ). Under the slow-variation approximation where each term appearing on the r.h.s. of Eq. (8) is much smaller than 1, the non-linear estimator in the squeezed limit (, ) reads 
which is consistent with the result of Refs. [62, 70]. Since , the Planck bound (68 % CL)  is satisfied for all the slow-variation single-field models based on the Horndeski theory. There are some non-slow roll models in which the non-Gaussianity consistency relation (33) is violated , but we do not study such specific cases.
In the limit of the equilateral triangle (), the leading-order non-linear parameter is given by 
For the models in which is much smaller than 1, the nonlinear estimator can be much larger than 1. The Planck team derived the bound (68 % CL)  by using three optimal bispectrum estimators. The primordial non-Gaussianities provide additional constraints on the models to those derived from and .
3 Planck constraints on potential-driven slow-roll inflation
Let us first study observational constraints on standard slow-roll inflation characterized by the functions
where is the field value at the end of inflation known by the condition .
The slow-roll parameter is equivalent to . Under the slow-roll approximation it follows that and , where
For a given inflaton potential these observables can be expressed in terms of . The field value corresponding to is known by Eq. (37).
For observational constraints on inflationary models based on the Planck data, we expand the scalar and tensor power spectra around a pivot wavenumber , as
where . Since the likelihood results are insensitive to the choice of , we fix as in Ref. . Since the runnings and are of the order of under the slow-roll approximation, we also set these values to 0. Using the consistency relation , the three inflationary observables , , and are varied in the likelihood analysis. We also assume the flat CDM model with relativistic degrees of freedom  and employ the standard Big Bang nucleosynthesis consistency relation . In addition to the Planck data , we also use the data of WP , BAO , and high- .
In Fig. 1 we plot the 68 % and 95 % CL boundaries in the plane constrained by the joint data analyses of Planck, WP, BAO, high- data (thick solid curves) and Planck, WP, BAO data (thick dotted curves). With the high- data, the scalar spectral index shifts toward smaller values and the tensor-to-scalar ratio gets slightly smaller. In what follows we place observational constraints on a number of representative inflaton potentials. For observational bounds on other potentials, we refer the reader to Ref. . The Planck constraints on braneworld inflation  and non-commutative inflation  have been studied in Ref.  for several inflaton potentials discussed below, but we do not have any significant observational evidence that they are particularly favored over standard slow-roll inflation.
3.1 Chaotic inflation
Chaotic inflation is characterized by the potential 
where and are positive constants. In this case the slow-roll parameters (38) reduce to and . From Eq. (37) we obtain the relation , where we used . The scalar spectral index and the tensor-to-scalar ratio read
As we see in Fig. 1, the quartic potential () is far outside the 95 % CL contour. For the quadratic potential () we have and for , which is marginally inside the 95 % CL boundary constrained by the Planck+WP+BAO+high- data. For the quadratic potential is outside the 95 % CL region.
The potentials with the powers and appear in the axion monodromy scenario [77, 78]. For , the linear potential is within the 95 % CL region constrained by the Planck+WP+BAO data, but it is outside the 95 % CL boundary by adding the high- data. For the linear potential enters the joint 95 % CL region constrained by the Planck+WP+BAO+high- data due to the decrease of . For the potential with is outside the joint 95 % CL boundary derived by the Planck+WP+BAO+high- data, but for the model marginally lies within the 95 % CL contour.
3.2 Natural inflation
We proceed to natural inflation given by the potential 
where and are constants. From Eq. (37) the number of e-foldings can be estimated as , where . This is inverted to give
The slow-roll parameter is related to via
For given values of and , we can evaluate and by using Eq. (45). In the limit that , inflation occurs in the regime where is close to the potential minimum (). Since and in this limit, we obtain and , which correspond to the values of chaotic inflation with .
In Fig. 1, we plot the theoretical values of and for as a function of . For decreasing , both and get smaller. From the joint analysis of the Planck+WP+BAO+high- data, the symmetry-breaking scale is constrained as
whereas at 95 % CL.
3.3 Hybrid inflation
Hybrid inflation involves two scalar fields: the inflaton and another symmetry-breaking field . During inflation the field is close to 0, in which regime the potential is approximately given by
where is a constant, and depends on . Inflation ends due to a waterfall transition driven by the growth of . Linde’s original hybrid model  corresponds to . Provided that the ratio is much smaller than 1, it follows that
Hence the scalar power spectrum is blue-tiled (). Under the condition , the tensor-to-scalar ratio is bounded as (which is shown as a solid curve in Fig. 1 in the regime ). The hybrid model with is far outside the 95 % CL region.
There is also a supersymmetric GUT model characterized by the potential with . In the regime where is much larger than the field value at the bifurcation point, the observables are given by and , where we have used . Since the second derivative is negative, the spectrum is red-tilted. In Fig. 1 the theoretical curves are plotted for and . The model is outside the 95 % CL region constrained by the Planck+WP+BAO+high- data due to the large scalar spectral index.
3.4 Very small-field inflation
The tensor-to-scalar ratio is related to the variation of the field during inflation. In fact, we obtain the relation from Eqs. (37)-(39). Provided that is nearly constant during inflation, the field variation is approximately given by [82, 83]
The models with are dubbed small-field inflation, in which case is smaller than for . In Ref.  the criterion was used to separate large-field and small-field models. Here we employ a more precise criterion according to Eq. (50).
Small-field inflation can be realized by the potential
For the function the number of e-foldings is given by , in which case and are
For and , it follows that and . The model is inside the 68 % CL boundary constrained by the Planck+WP+BAO+high-