Inflation: 1980–201X

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This is an introductory review of theoretical developments and observational investigation of inflationary cosmology with a particular emphasis on its role in the theoretical physics of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index E80


Introduction
The classical Big Bang cosmology [1] was the first successful theory of the Universe based on physical science that succeeded in interpreting the three fundamental cosmological observations of the time in a unified manner, namely, the homogeneous cosmic expansion originally discovered by Hubble [2] and Lemaitre [3], the cosmic microwave background radiation (CMB) discovered by Penzias and Wilson [4], and the abundance of light elements [5,6]. On the other hand, it also suffered from fundamental difficulties such as the horizon problem, the flatness problem [7], and the initial singularity problem. Since they are related with the initial condition of the Universe and there were no reliable theories to describe the very early Universe, these problems had not been seriously studied for a long time, even after the classical Big Bang cosmology was established by the discovery of CMB.
In the late 1970s and early 1980s, however, the situation drastically changed in association with the development of the grand unified theories (GUTs) of elementary interactions [8] which claim that the three fundamental interactions are unified at an extremely high energy scale around M GUT = 10 15-16 GeV. Since this energy scale is higher than the scale experimentally accessible through particle accelerators by more than ten digits, there emerged a trend to use the high temperature and density regime in the early Universe as an arena for their verification. The most representative positive outcome was the study of baryogenesis initiated by Yoshimura [9]. We also note that since the GUT scale was just a step off from the Planckian scale, M Pl = ( c 5 /G) 1/2 = 1.2 × 10 19 GeV, where quantum gravitational effects become important, cosmology based on GUTs stimulated serious study of the birth of the Universe.
At the same time, however, GUTs also brought about a serious difficulty for the classical Big Bang cosmology, namely, overproduction of magnetic monopoles [10,11] at the GUT phase transition [12] which would over-close the Universe [13] with the density parameter monopole ∼ 10 15 ! Fortunately, however, the resolution of this serious problem was already prepared in the GUT itself, as observed independently by Sato [14][15][16] and Guth [17]. Inflationary cosmology thus came into being.
These pioneers observed that in the symmetric state of the Higgs field, which is realized thermally shortly after the Big Bang, the field has a large potential energy density of the order of M 4 GUT , and that if this state is metastable with a long enough lifetime, the Universe would be dominated by the false vacuum energy density which would not decrease as long as the Higgs field remains in that state no matter how much the Universe expands. As a result, an exponential cosmic expansion is realized which was referred to as cosmic inflation. Since both the particle horizon and the curvature radius are exponentially stretched during inflation, it can in principle provide solutions to the horizon and the flatness problems.
In practice, however, this first model, now referred to as old inflation, was not successful because inflation could not be terminated to realize a hot Friedmann Universe over the comoving scale corresponding to the current Hubble horizon. On the other hand, this model led a conceptual revolution in cosmology that our Universe is not unique but there can be multiple universes that are causally disconnected from each other [18,19].
The idea of solving various cosmological problems with an accelerated expansion was so attractive that a number of attempts to realize inflation followed by a transition to a radiation-dominated regime were made immediately after the original proposal. Now the standard paradigm of inflation is that the accelerated expansion is realized when a scalar field, dubbed the inflaton, slowly rolls over its potential down to a global minimum in a time scale longer than the Hubble time [20][21][22]. The remarkable feature of the slow-roll inflation is that it can not only explain the global properties of the observed Universe, but also provide seeds of density and curvature fluctuations that evolve into large-scale structures [23][24][25][26]. The temperature (and polarization) anisotropies generated in the CMB at the same time provide indirect cosmological tests of the inflation paradigm (see, e.g. [27][28][29] among others).
In fact, the slow-roll inflation driven by a potential is not the only inflation scenario prevailing today, and there are two alternatives. One is models realizing inflation without introducing any inflaton field but modifying gravity from Einstein's general relativity. Starobinsky was the first to show that quasiexponential expansion was realized by incorporating higher-order curvature terms in the action based on quantum corrections [30]. Nowadays its simpler version, including a square scalar curvature term besides the Einstein action, is referred to as the Starobinsky model. Since Starobinsky's original paper [30], which was written in an attempt to avoid the initial singularity, was earlier than the old inflation models of Sato [14] and Guth [17], papers by these three authors are now regarded as the original references of the inflationary cosmology [31].
The other alternative is models that make use of a scalar field with a higher-order kinetic function. If we can realize a state with a constant canonical kinetic function, its energy density can have the same equation of state as the cosmological constant to drive exponential inflation. Such models are called k-inflation models [32].
All these models can be regarded as a subclass of the generalized G-inflation model [33,34] which is the most general theory of single-field inflation with its field equations given by second-order differential equations, as with most of the other theories of fundamental physics.

Resolution of fundamental problems
Although the original inflation models of Sato and Guth realized exactly exponential cosmic expansion, it is not a necessary condition to resolve horizon and flatness problems. All we need is a 2/26 Downloaded from https://academic.oup.com/ptep/article-abstract/2014/6/06B103/1562252 by guest on 30 July 2018 sufficiently long period of accelerated expansion, when the particle horizon increases more rapidly than t and the Hubble horizon, so the horizon problem can be solved. From the Einstein equations for the cosmic scale factor, a(t), we see that such an expansion is possible if cosmic energy density, ρ, and pressure, P, satisfy ρ + 3P < 0. Then the energy density decreases less rapidly than a −2 (t) so that the curvature term in the Friedmann equation (1), K /a 2 (t), becomes unimportant. As a result, the flatness problem is also solved simultaneously. In this review we take c = = k B = 1 unless these quantities are explicitly shown.
Of course, such a stage being dominated by a new energy component should not last forever but the Universe should be converted to a state dominated by radiation to realize the initial state of the hot Big Bang cosmology, since we do not wish to demolish its success. The accelerated or inflationary expansion must be followed by the creation of radiation and entropy. In fact it is this epoch rather than the inflationary expansion itself through which the monopole and other unwanted relic problems such as domain walls, gravitinos, etc. are solved, since the current state of the Universe is characterized by the CMB temperature T = 2.73 K and the corresponding entropy density, and we measure the abundance of these relics in units of the entropy density.
Let us quantify how much inflationary expansion is required to solve the horizon and flatness problems in terms of entropy consideration. First, the entropy contained in the current Hubble radius H −1 0 = 4.2 × 10 3 Mpc is given by that of the CMB photon and neutrino background with the effective temperature T ν = 1.95 K as S 0 = 2.6 × 10 88 . We consider the condition that the comoving Hubble volume at the beginning of inflation, 4π 3 H −3 inf at t = t i , contains more entropy than S 0 after the reheating, so that the currently observable region is well inside the initial Hubble radius at the onset of inflation.
For simplicity, let us assume that during inflation the energy density takes a constant value ρ inf with the corresponding Hubble parameter H inf , and that inflation continues from t = t i to t f to stretch the scale factor by a f /a i ≡ e N , and that after inflation the Universe is dominated by a component with equation-of-state P = wρ until radiation domination with the reheating temperature T R is achieved. At this time the initial Hubble radius is stretched to which contains the entropy where g * represents the number of relativistic degrees of freedom at reheating. Requiring this to be larger than S 0 we find ln g * 106.75 + 1 + 3w 6(1 + w) ln r 0.01 where r is the tensor-to-scalar ratio to be defined later in Eq. (99). As mentioned above, this is the condition that the initial Hubble radius at the onset of inflation is stretched to be larger than the current Hubble radius. However, while density perturbation on this scale is observed to be 10 −5 , we expect that the initial amplitude of fluctuation on this scale can be close to unity just to initiate inflation [35]. Thus it is not sufficient to have N = N min but we need more inflation by a factor of (10 −5 ) − 1 2 ∼ 500 to suppress the amplitude of fluctuations to an acceptable level [36]. Thus the minimal condition to solve the horizon problem is actually N > N min + ln 500 = N min + 6.2.
Taking this extra number of e-folds into account we find for w = 0, and for w = 1.
Next let us consider the evolution of the total density parameter tot using We find Thus, if the horizon problem is solved by inflation, the flatness problem is also solved automatically and such inflation models predict that the total density parameter today is equal to unity with four to five digits' accuracy.

Three mechanisms
As mentioned in the introduction one can classify inflation realization mechanisms into three broad categories.
The most standard category is the slow-roll inflation models. Consider a scalar field φ, the inflaton, with a canonical kinetic term X ≡ − 1 2 g μν ∂ μ φ∂ ν φ and a potential V [φ]. Taking the variation of the action with respect to the metric tensor, one finds the energy-momentum tensor For homogeneous field configuration, the energy density, ρ, and the pressure, P, are given by respectively. Hence if the potential is so flat that V [φ] >φ 2 is satisfied, the equation-of-state parameter w ≡ P/ρ is smaller than −1/3 and an accelerated expansion is realized. Next, in order to see the impact of a non-canonical kinetic term, let us rewrite the Lagrangian in more general form as L = K (X, φ). Then calculating the energy-momentum tensor in the same way as (11), we find in this case. Thus an accelerated expansion is possible, even without any potential, if X K X < −K is satisfied. This is what is called k-inflation [32]. In particular, in the case when K X = ∂ K /∂ X = 0 holds we find de Sitter expansion. But of course, the solution must be found by solving the field equation with the Einstein equations. One can find a simple model of k-inflation by expanding the Lagrangian as a power series of X , like Let us consider a regime where A(φ) can be approximated by unity, taking, say, and assuming φ φ f initially with λφ f /M G being a positive constant well above unity. Then we find a solution X ∼ = M 4 =const., and de Sitter inflation with 18 GeV is the reduced Planck scale.
The type of k-inflation in (15) is terminated when A(φ) flips its sign as it crosses φ f . Then X starts to decrease rapidly and only the first term becomes relevant. Now the Universe is dominated by the kinetic energy of a free scalar field whose energy density dissipates in proportion to a −6 (t) with the equation-of-state parameter w = 1. This abrupt change from de Sitter to a power-law a(t) ∝ t 1/3 induces gravitational particle production to reheat the Universe in this model. We note that despite the fact that the lowest-order kinetic term has the wrong sign during inflation, the theory is shown to be perturbatively stable throughout the cosmic evolution [34,37].
So far we have implicitly assumed the gravity sector is expressed by the Einstein-Hilbert action and considered a single scalar field as a matter ingredient. We may realize inflation even without any scalar field matter by modifying gravity as where f (R) is a function of scalar curvature R. The trace of the field equation is given by One can therefore find a de Sitter solution, A pure R 2 model cannot terminate inflation nor has a proper Einstein limit, so we adopt This theory should not be regarded as a mere perturbative extension of the Einstein gravity because it contains an additional scalar degree of freedom called the scalaron.

Inflation scenario
Let us describe cosmic evolution in a slow-roll inflation model as a prototype. Once inflationary expansion sets in, the Universe rapidly becomes homogeneous, isotropic, and spatially flat practically fairly soon, and the energy density other than the inflaton is soon diluted away. Hence we may consider a spatially flat FLRW Universe from the beginning, except when we discuss the feasibility of inflation in generic inhomogeneous and anisotropic spacetime. Thus the scalar field equation and the Einstein equation readφ respectively.
To realize successful inflation the potential V [φ] must dominate ρ φ for a sufficiently long time, that is, the scalar field must remain practically constant in the cosmic expansion time scale. For this purposeφ must be negligibly small in the equation of motion. Then the field equations read which are called slow-roll equations of motion. For these approximate equations to hold, we require the potential to satisfy Here V and η V are called the (potential) slow-roll parameters. Conversely, if these slow-roll conditions (24) are satisfied, inflationary expansion sets in soon. Inflation or accelerated expansion is terminated whenφ 2 becomes larger than V [φ] or when |Ḣ | becomes larger than H 2 . Under the slow-roll approximation, (22) and (23), the former occurs at V = 3/2 and the latter at V = 1. The discrepancy is due to the invalidity of the slow-roll approximation then. Using the slow-roll approximation, the number of e-folds of inflation after φ has crossed a value φ N is given by where φ f is the value at the end of inflation. Figure 1 shows the typical shape of the inflaton's potential. Near the local maximum at φ = 0 the potential is so flat that inflation may be possible. Models of such a class are called small-field models. On the other hand, if the potential increases at most with a power-law, one can see that the slow-roll conditions (24) are satisfied for super-Planckian field values, so that inflation is realized there. Such a model is called a large-field model.
In both cases the slow-roll conditions will no longer hold once the field approaches the global minimum at φ = v, and it starts to oscillate around it. Thus the potential energy density is transferred to field oscillation energy, which will eventually decay to particles interacting with φ. The Universe will then be heated up to a radiation-dominated regime. This is the physical origin of the Big Bang in the modern context.  In fact, Starobinsky's R 2 inflation can be regarded as a variant of a large-field model because the modified gravity action can be converted to the Einstein gravity plus a scalar field with a potential. Specifically, in terms of conformal transformation, we find the action is equivalent with Thus the system is equivalent to scalar field matter in the Einstein gravity. For φ M G = κ −1 , the potential has a plateau with a height 3M 2 G M 2 /4 where inflation can occur. Inflation is followed by scalar field oscillation around the origin and reheating proceeds through gravitational decay of the inflaton φ.

Slow-roll inflation models
Let us focus on specific models of slow-roll inflation with a canonical kinetic term.

Large-field model
This model was originally proposed by Linde under the name chaotic inflation [22], since it makes use of the chaotic initial condition of the Universe at the Planckian time t Pl = ( G/c 5 ) 1/2 = 5.4 × 10 −44 s, when we expect large quantum fluctuations were dominant. Let us consider the following simple Lagrangian as an example: By virtue of the uncertainty principle, at the Planckian time we expect both gradient energy and potential energy densities were fluctuating with the Planckian magnitude in each coherent domain. Taking m M Pl we find φ can take a large value up to φ ∼ M 2 Pl /m M Pl . If the magnitude of φ is saturated, from the gradient energy constraint we find the coherent length of the field, L, Pl . In this case, φ is homogeneous over the Compton wavelength which is ∼ 10 5 times the horizon scale at the Planckian regime for m ∼ 10 13 GeV. To initiate inflation we do not need such an extremely homogeneous configuration, but simply the inhomogeneity over the initial horizon scale should remain smaller than unity [35].
Then, over the horizon scale we may regard the field as homogeneous and apply the homogeneous field equations which were derived in the previous section. Solving the field equation (20) and the Einstein equation (21) we find Inflationary expansion is terminated around φ M Pl / √ 4π when V becomes unity and |φ/φ| becomes as large as H . Therefore, in order to solve the horizon problem one only needs the initial The most stringent constraint on this model comes from the amplitude of the density and curvature perturbation generated during inflation, which sets the mass m ∼ = 10 13 GeV and constrains the selfcoupling of the form λφ 4 /4 as λ 10 −12 [38] (see Sect. 10.1).

Small-field model
Soon after the old inflation model was proposed and its problems were elucidated, Linde proposed a new inflation model based on the Coleman-Weinberg type potential of the GUT Higgs field, in which inflation occurs as the scalar field induces a slow-roll over phase transition toward its zero-temperature minimum after thermal correction of the potential has disappeared due to cosmic expansion [20,21]. This was the first slow-roll inflation model and the first small-field inflation model using a concave potential, which is sometimes called hill-top inflation [39]. Unfortunately, however, the original new inflation predicted too short inflation and too large density fluctuations [24] so that it could not serve as the right model of inflation. Furthermore, as realized by Linde himself, the Universe would have been far from a thermal state at the GUT era, hence one cannot expect thermal symmetry restoration to set the appropriate initial condition for new inflation. These considerations motivated chaotic inflation.
The required initial condition for small-field models can be achieved, however, without resorting to thermal symmetry restoration [40,41]. Consider a simple Lagrangian which has a local maximum at φ = 0. Since φ is a real scalar field this model admits a domain wall solution connecting two vacuum states φ = ±v. Neglecting gravitational effects for the moment we 8 that describes a domain wall on the x y-plane. Its thickness d 0 can be estimated by equating (∇φ On the other hand, the Hubble horizon scale corresponding to the vacuum energy density V c is given by which is smaller than the thickness of the wall if v M Pl . In this case one can find a region with a large potential energy density V ∼ V c whose dimension is larger than the Hubble scale near the center of the domain wall. Such a field configuration provides a sufficient condition to initiate inflation there without being affected by the configuration outside the Hubble horizon.
In this model, an initially random field configuration in the global space naturally creates domain walls inside which inflation sets in near the central core. In this sense this model provides a natural mechanism of small-field inflation on condition that the Universe continues to expand until the energy density of the domain wall is locally dominant. We can also show that inflation ends after a finite time except for the locus with φ = 0, and the reheating process proceeds just as in the chaotic inflation model. Since the origin of inflation is provided by a topological defect, this model is called topological inflation.

Hybrid inflation
This is a model to induce inflation by a false vacuum energy density through a non-thermal symmetry restoration by virtue of an extra scalar field [42]. The simplest model of hybrid inflation consists of a real scalar field (φ) and a complex scalar field (χ ) with a Lagrangian From we find that χ = 0 is a potential minimum if the inequality g 2 φ 2 > λv 2 is satisfied initially. Then the potential reads If the first term dominates the energy density of the Universe, inflation takes place. It is terminated at φ < √ λv g when χ induces a phase transition, which typically occurs within one Hubble time unless fine-tuning of parameters is applied. This model is attractive from the particle-physics viewpoint because adequate inflation is possible even if φ is much smaller than the Planck scale [43], and neither λ nor g requires fine-tuning. Instead the initial field configurations of the two scalar fields do need tuning to realize inflation [44,45] In this model, since χ is a complex scalar field, a network of strings will be formed after the phase transition. Were χ a real scalar field, a network of domain walls would have been formed and over-dominated the energy density. Other mechanisms of formation of topological defects have been proposed in [46][47][48][49][50] near or at the end of inflation. Hence there may well remain some relic defects even if their energy scale is higher than the maximum temperature after inflation so that symmetry restoration by thermal effects is impossible.

Reheating
The entropy production process required after inflation to realize an appropriate initial state of the hot Big Bang cosmology is called reheating, although in modern inflationary universe models it is at this time that the Universe was dominated by radiation for the first time.
As is clear from the discussion in the previous section, the Universe is dominated by coherent scalar field oscillation after potential-driven slow-roll inflation. Let us take the large-field massive scalar model as an example. As the inflaton turns to satisfy |φ| M Pl / √ 4π the scalar field starts to oscillate around the origin with a period 2π/m. Such field oscillation is equivalent to a condensation of the homogeneous zero-mode of the scalar field which decreases its amplitude through cosmic expansion, to decay into radiation finally. In the initial oscillatory regime when its amplitude is large, some non-perturbative particle production also takes place efficiently to some extent, which is called preheating [51][52][53]. But the final reheating stage is always dominated by the perturbative decay of the scalar field, which is understood by incorporating a decay term φφ in its equation of motion [54][55][56]. Here φ is the decay rate of a φ particle which is much smaller than m since the inflaton must be weakly coupled with other fields to suppress quantum fluctuations to an acceptable level, as discussed in the following sections.
Multiplying (20) withφ after introducing the abovementioned dissipation term, we find d dt Since φ is rapidly oscillating in the cosmic expansion time scale, we can replaceφ 2 on the right-hand side by an average over the oscillation period,φ 2 , which is identical to the total energy density of φ, ρ φ , thanks to the virial theorem [57]. Thus we find a Boltzmann equation which is associated with that for radiation energy density ρ r : These two equations can be solved to yield Thus the Universe turns to be dominated by radiation at t −1 φ , when the radiation temperature T R is given by where g * is the effective number of relativistic degrees of freedom.

Generation of quantum fluctuations that eventually behave classically
Thanks to the smallness of the slow-roll parameters, the quantum-field-theoretic properties of the inflaton during inflation are quite similar to those of a massless minimally coupled scalar field ϕ(x, t) in de Sitter space: As is well known, such a scalar field shows anomalous growth of the square vacuum expectation value as [58][59][60] ϕ(x, t) 2 = H 2π 2 Ht.
Decomposing the scalar field as we find the momentum conjugate to ϕ(x, t) reads π(x, t) = a 3 (t)φ(x, t), hence the canonical Thusâ † k andâ k satisfy the usual commutation relation and act as creation and annihilation operators of the k-mode, respectively.
The mode function ϕ k (t) satisfies as derived from the Klein-Gordon equation in de Sitter space, which is solved as under the condition (49). Here η is the conformal time defined by Note that since (50) is a second-order differential equation there are two independent solutions proportional to H Note also that −kη = k Ha(t) is nothing but the ratio of physical wavenumber to the Hubble parameter. In the super-horizon limit k a(t)H , we find so that we find ϕ * k (t) = −ϕ k (t) and can regard thatφ holds when the wavelength corresponding to k is much longer than the Hubble radius. Then the momentum conjugate tô , which means that bothφ k andπ k have the same operator dependence in the super-horizon regime.
Thus in the super-horizon regime where the decaying mode is negligible, the quantum operatorŝ ϕ k andπ k commute with each other, so that quantum fluctuations behave as classical statistical fluctuations [61]. Furthermore, the absolute square of mode function behaves as that is, it takes a constant value proportional to k −3 . Multiplying the phase space density over a logarithmic frequency interval, 4π k 3 (2π) 3 d ln k, we find the dispersion takes a constant over each logarithmic frequency interval.
One can immediately see that (47) can be reproduced from this mode function by summing up super-horizon fluctuations only as where the infrared cutoff is taken as the mode that left the Hubble radius at the beginning of inflation (set to t = 0) [62]. The behavior that the square expectation value increases in proportion to time is the same as that of Brownian motion with a step ±H/(2π) at each time interval H −1 . Thus one can describe the quantum fluctuations generated during inflation as follows: Finally, as a preparation for the calculations of curvature and tensor perturbations, we write down the action of a massive scalar field ϕ using conformal time with the line element ds 2 = a 2 (η)(−dη 2 + dx 2 ): which is equivalent to the action of a scalar field with a time-dependent mass term in Minkowski space.
In de Sitter space with a(η) = −1/(H η) the mode function is determined by where we have taken m = 0. Its solution is given by in agreement with (51) with the normalization condition χ χ * − χχ * = i, equivalent to (49).

Cosmological perturbation
In order to discuss quantum mechanical generation of cosmological perturbation, we incorporate metric perturbation to the spatially flat homogeneous and isotropic background spacetime, as where all new variables are functions of time and position [63][64][65]. As is well known, a spatial vector B j can be decomposed to a rotation-free component and a divergence-free component as Similarly, the spatial tensor H T i j can be decomposed as H T T i j is the transverse-traceless tensor perturbation corresponding to gravitational waves. Thus perturbation variables are classified to scalar, vector, and tensor variables according to the spatial transformation law. Variables of different type are not mixed in linear perturbation theory and we can treat them independently. Since vector perturbation has only a decaying mode under normal circumstances in the linear perturbation theory [63,64], we do not consider it hereafter and focus on scalar and tensor perturbations. First let us consider scalar perturbation, which is related to density and curvature fluctuations, keeping only A, B, H L , and H T . In fact, not all of them are geometrical quantities, but include gauge modes. They come from the arbitrariness of the choice of the background spacetime with respect to which we define the perturbation variables. One should remember that the real physical entity is an inhomogeneous and anisotropic spacetime, and that it may be regarded as homogeneous and isotropic only after taking some average whose definition is not unique in general. 13 To see the gauge dependence, let us consider two background spacetimes with coordinates x μ and x μ , and assume that these two coordinates are related by a coordinate transformation where δx μ are small quantities whose relative magnitudes are of the order of the perturbation variables. Here T and L are functions of space and time coordinates. By this coordinate transformation, we find that the scalar perturbation variables at the same coordinate values in two different backgrounds are related as respectively. Here T and L represent gauge degrees of freedom on scalar perturbation. In order to avoid their appearance, one may either find combinations of perturbation variables that remain unchanged after gauge transformation, or fix the gauge completely. For example, if we set H T = 0, L is fixed. If we further set B = 0, T is also fixed and there remains no gauge mode. In this case the perturbed metric is given by where we have rewritten A = and H L = to stress their gauge invariance. This is called the longitudinal gauge. In order to define the gauge, one may combine transformation properties of matter variables such as a scalar field, which transforms as that is, This may be used to fix T .

Generation of curvature fluctuations in inflationary cosmology
Here we consider the generation of curvature perturbations in single-field inflation models that cover not only potential-driven slow-roll models but also k-inflation. For the most general single-field inflation model with second-order field equations, see [34]. We start with an action In the spatially flat homogeneous and isotropic spacetime (60), the background equations read In order to derive second-order action with respect to the curvature perturbation in the comoving gauge, R, which is conserved outside the Hubble horizon if there is only adiabatic fluctuations, it is convenient to use (3 + 1) formalism: in which constraint equations are easily obtained for perturbation variables [66]. Since we are concerned with scalar-type fluctuations we put N = 1 + α and N i = ∂ i ψ, where α and ψ are also perturbation variables of the same order of R. Correspondence with the general perturbed metric, (61) through (63), is given as with all the other vector and tensor modes set to zero. From (65) we immediately find that the gauge L is fixed in the above expression, and equation (68) tells us that T can also be fixed by taking δφ = 0. Then, indeed, R coincides with the comoving curvature perturbation, and there is no more gauge degree of freedom.
Writing the action (69) in this gauge, we find where terms replaced by · · · are all total derivative terms, so they do not affect the dynamics of the system. Here R (3) is scalar curvature of 3-space and | denotes covariant derivative with respect to h i j . Differentiating the action (74) with respect to N we find the Hamiltonian constraint which yields The momentum constraint obtained by differentiation with respect to N i reads which gives α =Ṙ/H up to a term that does not depend on spatial coordinates. Inserting these relations in the action we obtain an action for R only as Introducing new variables z ≡ a(2 ) 1 2 /c s , v ≡ M G zR, and using the conformal time, we find  (57), is an action of a free scalar field with a time-dependent mass term with the important difference that the "sound velocity" c s may not be identical to unity. Still, one can quantize v in the same way as a free scalar field if the time variation of c s is negligible. q is a small number consisting of slow-roll parameters that can be read off from (81).
Using the de Sitter scale factor a = −1/(H η) one can obtain the mode function in the same way as (59), which approaches a constant in the long wave limit. Thus the power spectrum of curvature perturbations multiplied by the phase space density reads Here each quantity has weak time dependence and it should be evaluated at the time k-mode left the sound horizon when |kc s η| = 1 was first satisfied. Then the spectral index of curvature perturbation is defined and given by In the case when the scalar field has a canonical kinetic term, as in the standard theory of particle physics, we find c s = 1 and H =φ 2 /(2M 2 G H 2 ). Thus we find Here δϕ = H/(2π) is the amplitude of scalar field fluctuation, as discussed in Sect. 6. This equality is intuitively understandable since it represents the relative fluctuation of local scale factor where N ≡ ln a. Taking H L = 0 and identifying δφ with δϕ through an appropriate gauge transformation by T to the flat gauge, we find the above expression is indeed equivalent with the comoving curvature perturbation. This δ N method [67,68] is often used in the literature. For slow-roll inflation models with a canonical kinetic term, from (22) and (23) we find H = V and η H = −2η V + 4 V , so the spectral index is given by Furthermore, the scale dependence of the spectral index which is called the running spectral index is given by These are fundamental observables of CMB anisotropy.

Tensor perturbation
Next we consider tensor perturbations generated during inflation as quantum gravitational waves [69][70][71]. First we incorporate a small metric perturbation h μν in Minkowski space as g μν = η μν + h μν . The indices of h μν are raised by the Minkowski tensor. We again need to calculate the action up to second order in the perturbed variables, so we should use g μν = η μν − h μν + h μα h ν α . Taking the longitudinal and traceless gauge, we find h 00 = h 0i = 0, h α α = h i i = 0, h , j i j = 0, with which the Ricci scalar is given by Here Greek indices run from 0 to 3 while Latin indices from 1 to 3. Of course, what we need is a perturbed action in cosmological background spacetime with the metric which is obtained from (61) through (63) by identifying H T T i j = h i j and putting all other perturbation variables to zero.
Taking = a(η) we find the Ricci scalar in the metric (90) is given bỹ Since we wish to calculate properties of quantum gravitational waves generated during inflation, let us assume inflation is driven by a constant vacuum energy density equivalent to a cosmological constant and calculate the second-order action, to yield where we have used a 4 = 2aa − a 2 . This action has the same form as a free scalar field again, so we can quantize it in the same way as before. Indeed, taking new variables as z T ≡ a/2, u i j ≡ M G z T h i j , we find which is indeed of the same form as (57). For exact de Sitter space with H = 0, we have a = −1/(H η), whereas in the case H = 0, one can express the scale factor around an arbitrary time η * using the Hubble parameter at that time H * as Therefore we find the normalized mode function Thus the power spectrum of long-wave tensor perturbation is given by with the spectral index Finally, the ratio of this power spectrum to the curvature perturbation, is called the tensor-to-scalar ratio. It is one of the most important quantities subject to ongoing and future observational tests of inflation. See Ref. [34] for the relation between r and n t in more generic theories.

Inflationary cosmology and observations
Nowadays much effort is being made to compare various predictions of inflationary cosmology to observations of CMB anisotropy and large-scale structures. Since the first-year WMAP result was disclosed [28,72,73], the cosmological parameters of a homogeneous and isotropic universe, as well as properties of fluctuations, have been determined quite accurately with some error bars. Indeed, until the WMAP papers were published, although we had the concordance cosmology, we did not have any sensible error bars because the concordance values of the cosmological parameters had been obtained by combining a number of different observational results and we did not have a means to calculate error bars by making a sensible weighted average. Contemporary precision cosmology makes use of the Markov Chain Monte Carlo (MCMC) method to constrain values of cosmological parameters as well as the amplitude and spectrum of primordial fluctuations using the temperature and E-mode polarization data of the CMB [74]. Note, however, that the final values of these parameters, as well as the magnitude of errors obtained by these procedures, change considerably depending on which observational data sets one uses, as well as what model parameters one wishes to fit.
Current cosmological data as a whole show good agreement with a spatially flat universe with nonvanishing cosmological constant ( ) and cold dark matter (CDM) based on the simple single-field inflation paradigm that predicts almost scale-invariant adiabatic fluctuations. Thus it makes sense to estimate various parameters of the CDM model using the MCMC technique.
Below we quote results from the seven-and nine-year WMAP reports [75,76], as well as the Planck 2013 results [77], to get an idea of what constraints will survive in future independent of the subtleties of each experiment. The abbreviations in parentheses after each quoted number refer to the dataset used, whose meaning should be self-explanatory.
First, as the most important test of the global spacetime, the spatial curvature is constrained as which is consistent with (9). Hence, from now on we fix K 0 = 0, since hte inflationary cosmology we discuss here predicts so with much higher accuracy than (100), as discussed in Sect. 2.    These constraints are often graphically displayed in an (n s , r ) plane [78] with one-and two-sigma contours and compared with various inflation models. We do not do the same here for two reasons. First, constraints vary appreciably depending on what dataset one uses and what underlying model one adopts, as seen above. Second, occupying the center of the likelihood contour at the current level of observations does not necessarily mean that such a model is the right one, and we should be open minded for further development of observations. Instead, below we calculate these observables in each class of models discussed in Sect. 3.1.

Large-field models
Consider a massive scalar model as an example of large-field models. The number of e-folds of inflation after φ crossed φ N is given from (32) as Hence the slow-roll parameters read which gives the amplitude of curvature perturbation, (83), Setting N = 55 on the pivot scale tentatively, we find m = 1.6 × 10 13 GeV and the quartic coupling is constrained as λ 8 × 10 −13 in the case that such a term coexists as V [φ] = 1 2 m 2 φ 2 + λ 4 φ 4 and the first term dominates the observable regime of inflation, namely, the last ∼ 60 e-folds.
The spectral index and its running are given by in agreement with observations. The amplitude of the tensor perturbation is large enough to be observable soon, r = 16 V = 0.15. Let us also consider R 2 inflation in the Einstein frame as a kind of large-field model. From (24), (25), and (28) we find which yield

Non-canonical models and multi-field models
So far we have focused on the amplitude and spectral index of primordial curvature and tensor perturbations. Indeed, in single-field slow-roll inflation there is only one fluctuating degree, so only adiabatic fluctuations are generated with no isocurvature counterparts. Furthermore, the curvature perturbation generated in slow-roll inflation behaves as a practically free massless scalar field during inflation, and so its vacuum fluctuation is Gaussian distributed. This is why the power spectrum fully quantifies the properties of fluctuations.
Much work has been done in recent years to quantify the skewness or the bispectrum of the temperature anisotropy, and a number of non-standard models were proposed that predicted observationally falsifiable magnitudes of the deviation from Gaussian using the expected Planck data [89]. As a result, these models were indeed basically falsified by Planck, which confirmed Gaussian distribution with a relatively high accuracy [90]. So we do not go into further detail.
Another possible signature of multi-field models would be the presence of isocurvature modes. But, again, CMB observations do not need any isocurvature components, at least on relevant scales [76].

Conclusion
The fact that we live in a large and old Universe with a large amount of entropy can be regarded as a primary consequence of inflation in the early Universe. Any attempt to avoid inflation by providing alternative explanations to the other predictions of inflationary cosmology, namely, generation of adiabatic density perturbation with nearly scale-invariant spectrum as well as tensor perturbation should also have a sound explanation for the aforementioned global properties of the spacetime.
Some people, however, try to refute this viewpoint by claiming that the global properties of the Universe, namely, homogeneity, isotropy, and longevity, had been known long before the first inflation 22 [14,17,30] were proposed so that they cannot be regarded as the predictions of inflationary cosmology. This is true indeed. One should recognize, however, that theories in physics have double roles, namely, one to give a new prediction which may be justified or falsified by experiments and observations, and the other to provide sensible explanations to known mysteries of the nature. Inflationary cosmology certainly provides a simple mechanism to explain fundamental properties of the global structure of space, playing the latter role of theoretical physics. Thus the abovementioned refutation is a long way off the mark.
Furthermore, the simplest class of slow-roll inflation models predicted nearly scale-invariant, adiabatic, and Gaussian curvature and density fluctuations, all of which are now being proven to be true by dedicated CMB experiments. In particular, the observed angular correlation between temperature anisotropy and E-mode polarization clearly indicates that the observed fluctuations were generated as super-Hubble fluctuations, as predicted by inflation [74].
We are tempted to interpret the fact that Planck did not confirm any sizable non-Gaussianity [90] despite the recent efforts of many theorists to make models of non-Gaussian fluctuations as indicating that nature preferred the simplest inflation models. This, on the other hand, highlights the difficulty in singling out the right model responsible for our Universe from the small numbers of observational clues available. For example, among the models occupying the central regions of the likelihood contours in the (n s , r ) plane at the moment, it is very difficult to distinguish between the R 2 model [30,91,92] and a Higgs inflation model with a large and negative non-minimal coupling [93][94][95] from the observations of perturbation variables alone. We should also consider the reheating processes of these models for more detailed comparison, which may be observationally probed by future space-based laser interferometers such as DECIGO [96,97].
Our final goal is to identify the inflaton in a particle physics context. As argued above, the more observations agree with the standard prediction, the harder it is to single out the inflaton from purely observational grounds, so particle-physics considerations are also very important. Currently we know only one scalar field in nature experimentally. In this sense, the case where the standard Higgs field is responsible for inflation is interesting and deserves serious consideration. Since its self coupling is too large to produce an appropriate amplitude of curvature perturbation, several remedies have been proposed besides the non-minimally coupled model mentioned above [93][94][95], namely new Higgs inflation [98], Higgs G-inflation [99], and running kinetic inflation [100], all of which can be treated on an equal footing as a variant of generalized G-inflation [101]. Since the predictions of each model are different, if particle physics identity is specified, observations may show the form of these additional interactions.
Further open questions are how inflation started at the birth of the universe, and how it ended, that is, how the Universe was reheated. Observationally, as mentioned above, future space-based laser interferometers such as DECIGO [96,97] can provide us with useful information on the reheating history after inflation [102][103][104]. Hence we hope these satellites will be put into action as soon as possible.

Note Added
After the original version of this manuscript was submitted on February 14, 2014, on March 17, 2014 the BICEP2 collaboration announced detection of B-mode polarization of the CMB [105] and claimed that it originates from tensor perturbations generated during inflation corresponding to r 0.2 (or r 0.16 after subtracting foregrounds based on the best foreground model), disfavoring synchrotron or dust at 2.3 σ and 2.2 σ , respectively. While their discovery would have profound implications [106], further confirmation is certainly desired to determine the actual value of r through multi-frequency observations in a wider area of the sky. Hence I decided not to change Sect. 10 of the manuscript and left "X" in the title as it is. We hope the forthcoming Planck data will achieve the above desire, so that we should be able to put X = 4 to refer to the year of the discovery of the primordial tensor mode.