# Inflation Dynamics and Reheating

###### Abstract

We review the theory of inflation with single and multiple fields paying particular attention to the dynamics of adiabatic and entropy/isocurvature perturbations which provide the primary means of testing inflationary models. We review the theory and phenomenology of reheating and preheating after inflation providing a unified discussion of both the gravitational and nongravitational features of multi-field inflation. In addition we cover inflation in theories with extra dimensions and models such as the curvaton scenario and modulated reheating which provide alternative ways of generating large-scale density perturbations. Finally we discuss the interesting observational implications that can result from adiabatic-isocurvature correlations and non-Gaussianity.

###### pacs:

98.80.Cq astro-ph/0507632## I Introduction

Inflation was introduced Sta80 ; Guth81 ; Sato81p1 ; Sato81p2 ; Kazanas80 as a way of addressing pressing problems that were eating away at the foundations of the otherwise rather successful big-bang model. It is a very flexible paradigm, based squarely in semi-classical physics, and has provided a sturdy foundation linking the classical cosmos and the quantum gravity world Lindebook ; LLbook ; Kolbbook .

Inflation can be viewed in many different ways. One approach is to argue that inflationary models (of which there are hundreds) provide a convenient method of parameterising the early universe but that, because they are fundamentally semi-classical, are unlikely to be a true description of the physics underlying the very early universe. The other, probably more common, approach is to argue that an inflationary phase did indeed occur at some stage in the early universe and that the source of inflation is a scalar field whose identity may be found by considering one of the extensions of the standard model based on grand unified theories (GUT’s), supergravity or string theory. In the latter view, we can use inflation as a way both to understand features of quantum gravity/string theory and of particle physics beyond the standard model.

Taking this latter view, it is extremely natural to consider inflation with many fields. As a simple example, consider a grand unified theory (GUT) based on the group . Such a GUT has no pretensions to be a theory of everything and yet it already implies the existence of large numbers (of order ) scalar Higgs fields. Similarly supersymmetry requires the existence of large numbers of superpartners LRreview and string theory rather naturally has dynamical moduli fields corresponding to the geometrical characteristics of compactified dimensions LWCreview . If scalar fields are natural sources of inflation, then modern particle physics is the perfect supplier.

The inflationary paradigm not only provides an excellent way to solve flatness and horizon problems but also generates density perturbations as seeds for large-scale structure in the universe Mukhanov81 ; Guth82 ; Hawking82 ; Sta82 . Quantum fluctuations of the field responsible for inflation – called the inflaton – are stretched on large scales by the accelerated expansion. In the simplest version of the single-field scenario the fluctuations are “frozen” after the scale of perturbations leaves the Hubble radius during inflation. Long after the inflation ends, the perturbations cross inside the Hubble radius again. Thus inflation provides a causal mechanism for the origin of large-scale structure in the universe. An important prediction of inflation is that density perturbations generally exhibit nearly scale-invariant spectra. This prediction can be directly tested by the measurement of the temperature anisotropies in Cosmic Microwave Background (CMB). Remarkably the anisotropies observed by the Cosmic Background Explorer (COBE) in 1992 showed nearly scale-invariant spectra. Fortunately, all existing and constantly accumulating data including WMAP WMAP ; Peiris03 , SDSS Tegmark03 ; Tegmarkdata and 2dF 2dF have continued to confirm the main predictions of the inflationary paradigm within observational errors. We live in a golden age for cosmology in which the physics of the early universe can be probed from high-precision observations.

Recent progress in constructing particle-physics models of inflation has shown us that a key question is “how many light fields exist during inflation?” Here “light” is measured relative to the Hubble constant (which has dimensions of mass). If there is only one light field (typically required to get slow-roll inflation in the first place), then inflation is effectively single-field dominated and the cosmological consequences are rather well-understood. In the case of multiple light fields, the situation is significantly more complicated since the fields may interact and between each light field there will typically be a dynamically important entropy/isocurvature perturbation (we will use these two terms interchangeably).

Further, as the fields evolve, their effective mass can change, renormalised by the expectation values of other fields. Since the spectrum of fluctuations associated with any given perturbation mode depend on its effective mass, there is a rich phenomenology of possible effects associated with time-dependent effective masses.

In this review we will lay out the foundations of inflation and cosmological perturbation theory appropriate for application to cases involving many relevant fields. Our main aim is to provide the reader with a unified framework and set of tools to begin practical application in inflationary cosmology. This review is complementary to the many excellent review on related topics, given in Refs. Robert85 ; CKLLreview ; Kolbbook ; Liddle93 ; LLbook ; LLKCreview ; Lindebook ; LRreview ; Riotto02 ; NP91 ; Gio05 .

Our paper is organised as follows. In Sec. II we introduce the inflationary paradigm as a way to solve several cosmological problems associated with standard big bang cosmology. Inflationary models are classified into four different types. In Sec. III we review cosmological perturbation theory using the gauge-invariant formalism. Sec. IV is devoted to the spectra of scalar and tensor perturbations generated in single-field inflation. In Sec. V we present observational constraints on single-field inflation from CMB and galaxy redshift surveys. In Sec. VI we review density perturbations generated in higher-dimensional models including braneworld, pre-big-bang and ekpyrotic/cyclic cosmologies.

In Sec. VII the definition of adiabatic and entropy perturbations is given with the field space rotation and we show how the correlation between adiabatic and entropy perturbations emerges in the context of two-field inflation. In Sec. VIII we present general features in the CMB from correlations. In Sec. IX we explain the elementary theory of reheating after inflation. Sec. X is devoted to preheating in which particles coupled to the inflaton are resonantly amplified by parametric resonance. In Sec. XI we discuss the evolution of metric perturbations during preheating and possible consequences from it. We review the curvaton scenario in Sec. XII and the modulated reheating scenario in Sec. XIII to generate large-scale density perturbations as alternative models of inflation. Summary and future outlook are given in the final section.

## Ii Dynamics of inflation

### ii.1 The standard big-bang cosmology

The standard big-bang cosmology is based upon the cosmological principle LLbook , which requires that the universe is homogeneous and isotropic on averaging over large volumes. Then the metric takes the Friedmann-Robertson-Walker (FRW) form:

(1) | |||||

Here is the scale factor with being the cosmic time. The constant is the spatial curvature, where positive, zero, and negative values correspond to closed, flat, and hyperbolic spatial sections respectively.

The evolution of the universe is dependent on the material within it with a key role played by the equation of state relating the energy density and the pressure . For example we have

(2) | |||||

(3) |

The dynamical evolution of the universe is known once we solve the Einstein equations of General Relativity:

(4) |

where , , , and are the Ricci tensor, Ricci scalar, energy-momentum tensor and gravitational constant, respectively. The Planck energy, GeV, is related to through the relation . Here and are the Planck’s constant and the speed of light, respectively. Hereafter we use the units . is the cosmological constant originally introduced by Einstein to make the universe static. In what follows we set the cosmological constant to zero () unless otherwise stated, preferring to include any non-zero vacuum energy density in the total energy-momentum tensor.

From the Einstein equations (4) for the background FRW metric (1), we obtain the field equations:

(5) | |||

(6) |

where a dot denotes the derivative with respect to , and is the Hubble expansion rate. Eqs. (5) and (6) are the Friedmann and Raychaudhuri equations, respectively. Combining these relations implies energy conservation,

(7) |

which is known as the continuity or fluid equation.

The Friedmann equation (5) can be rewritten as

(8) |

where

(9) |

Here the density parameter is the ratio of the energy density to the critical density. When the spatial geometry is flat (; ), the solutions for Eqs. (5) and (7) are

(10) | |||

(11) |

In these simple cases, the universe exhibits a decelerated expansion () as confirmed by Eq. (6).

### ii.2 Problems of the standard big-bang cosmology

#### ii.2.1 Flatness problem

In the standard big-bang theory with , the term in Eq. (8) always decreases. This means that tends to evolve away from unity with the expansion of the universe. However, since present observations suggest that is within a few percent of unity today WMAP , is forced to be much closer to unity in the past. For example, we require at the epoch of nucleosynthesis and at the Planck epoch LLbook . This appears to be an extreme fine-tuning of initial conditions. Unless initial conditions are chosen very accurately, the universe either collapses too soon, or expands too quickly before the structure can be formed. This is the so-called flatness problem.

#### ii.2.2 Horizon problem

Consider a comoving wavelength, , and corresponding physical wavelength, , which at some time is inside the Hubble radius, (i.e., ). The standard big-bang decelerating cosmology is characterized by the cosmic evolution of with . In this case the physical wavelength grows as , whereas the Hubble radius evolves as . Therefore the physical wavelength becomes much smaller than the Hubble radius at late times. Conversely any finite comoving scale becomes much larger than the Hubble scale at early times. This means that a causally connected region can only be a small fraction of the Hubble radius.

To be more precise, let us first define the particle horizon which is the distance travelled by light since the beginning of the universe, at time ,

(12) |

Here corresponds to the comoving particle horizon. Setting , we find in the matter-dominant era and in an early hot big bang. We observe photons in the cosmic microwave background (CMB) which are last-scattered at the time of decoupling. The particle horizon at decoupling, , corresponds to the causally connected region at that time. The ratio of the comoving particle horizon at decoupling, , to the particle horizon today, , can be estimated to be

(13) |

This implies that the causally connected regions at last scattering are much smaller than the horizon size today. In fact causally connected regions on the surface of last scattering corresponds to an angle of order .

This appears to be at odds with observations of the cosmic microwave background (CMB) which has the same temperature to high precision in all directions on the CMB sky. Yet there is no way to establish thermal equilibrium if these points were never been in causal contact before last-scattering. This is the so-called horizon problem.

#### ii.2.3 The origin of large-scale structure in the universe

Experiments which observe temperature anisotropies in the CMB find that the amplitude of the anisotropies is small and their power spectrum is close to scale-invariant on large scales WMAP . These fluctuations are distributed on such a large scale that it is impossible to generate them via causal processes in a FRW metric in the time between the big bang and the time of the last scattering. Hence, standard big-bang models can neither explain the FRW metric nor explain the deviations from FRW if a FRW background is assumed.

#### ii.2.4 Relic density problem

The standard paradigm of modern particle physics is that physical laws were simpler in the early universe before gauge symmetries were broken. The breaking of such symmetries leads to the production of many unwanted relics such as monopoles, cosmic strings, and other topological defects Lindebook . The existence of a finite horizon size leads to a maximum causal correlation length during any symmetry breaking transition and hence gives a lower bound on the density of defects. In particular, any grand unified theory based on a simple Lie group that includes the of electromagnetism must produce monopoles. String theories also predict supersymmetric particles such as gravitinos, Kaluza-Klein particles, and weakly coupled moduli fields.

If these massive particles exist in the early stages of the universe then their energy densities decrease as a matter component () once the temperature drops below their rest mass. Since the radiation energy density decreases , these massive relics if they are stable (or sufficiently long-lived) could become the dominant matter in the early universe depending on their number density and therefore contradict a variety of observations such as those of the light element abundances. This problem is known as the relic density problem.

### ii.3 Idea of inflationary cosmology

The problems in the standard big bang cosmology lie in the fact that the universe always exhibits decelerated expansion. Let us assume instead the existence of a stage in the early universe with an accelerated expansion of the universe, i.e.,

(14) |

From the relation (6) this gives the condition

(15) |

which corresponds to violating the strong energy condition. The condition (14) essentially means that increases during inflation and hence that the comoving Hubble radius, , decreases in the inflationary phase. This property is the key point to solve the cosmological puzzles in the standard big-bang cosmology.

#### ii.3.1 Flatness problem

Since the term in Eq. (8) increases during inflation, is rapidly driven towards unity. After the inflationary period ends, the evolution of the universe is followed by the conventional big-bang phase and begins to increase again. But as long as the inflationary expansion lasts sufficiently long and drives very close to one, will remain close to unity even in the present epoch.

#### ii.3.2 Horizon problem

Since the scale factor evolves approximately as with during inflation, the physical wavelength, , grows faster than the Hubble radius, . Therefore physical wavelengths are pushed outside the Hubble radius during inflation which means that causally connected regions can be much larger than the Hubble radius, thus potentially solving the horizon problem. Formally the particle horizon, defined in Eq. (12) diverges as in an inflationary universe.

Of course the Hubble radius begins to grow faster than the physical wavelength after inflation ends, during the subsequent radiation and matter dominant eras. In order to solve the horizon problem, it is required that the following condition is satisfied for the comoving particle horizon:

(16) |

This implies that the comoving distance that photons can travel before decoupling needs to be much larger than that after the decoupling. A detailed calculation shows this is achieved when the universe expands at least about times during inflation, or e-folds of expansion Lindebook ; LLbook ; Riotto02 .

#### ii.3.3 The origin of the large-scale structure

The fact that the Hubble rate, , is almost constant during inflation means that it is possible to generate a nearly scale-invariant density perturbation on large scales. Since the scales of perturbations are well within the Hubble radius in the early stage of inflation, causal physics works to generate small quantum fluctuations. On very small scales we can neglect the cosmological expansion and perturbations can be treated as fluctuations in flat spacetime. But after a scale is pushed outside the Hubble radius (i.e., the first Hubble radius crossing) during inflation, we can no longer neglect the Hubble expansion.

Fluctuations in a light field become over-damped on long-wavelengths, leading to a squeezed state in phase-space, so that the perturbations can effectively be described as classical on these large scales. When the inflationary period ends, the evolution of the universe follows the standard big-bang cosmology, and the comoving Hubble radius begins to increase until the scales of perturbations cross inside the Hubble radius again (the second Hubble radius crossing). The small perturbations imprinted during inflation have amplitudes determined by the Hubble rate which is approximately constant and hence leads to an almost scale-invariant spectrum with constant amplitude on different scales. In this way the inflationary paradigm naturally provides a causal mechanism to generate the seeds of density perturbations observed in the CMB anisotropies.

#### ii.3.4 Relic density problem

During the inflationary phase (), the energy density of the universe decreases very slowly. For example, when the universe evolves as with , we have and . Meanwhile the energy density of massive particles decreases much faster (), and these particles are red-shifted away during inflation, thereby solving the monopole problem, as long as the symmetry breaking transition that produces the monopoles occurs at least 20 or so e-foldings before the end of inflation.

We also have to worry about the possibility of producing these unwanted particles after inflation. In the process of reheating followed by inflation, the energy of the universe can be transferred to radiation or other light particles. At this stage unwanted particles must not be overproduced in order not to violate the success of the standard cosmology such as nucleosynthesis. Generally if the reheating temperature at the end of inflation is sufficiently low, the thermal production of unwanted relics such as gravitinos can be avoided KM95 ; moroi95 .

### ii.4 Inflationary dynamics

Scalar fields are fundamental ingredients in modern theories of particle physics. We will consider a homogeneous single scalar field, , called the inflaton, whose potential energy can lead to the accelerated expansion of the universe. Neglecting spatial gradients, the energy density and the pressure of the inflaton are given by

(17) |

where is the potential energy of the inflaton. Substituting Eq. (17) into Eqs. (5) and (7), we obtain

(18) | |||

(19) |

where . The curvature term, , is dropped in Eq. (18) since it adds nothing concrete to our discussion.

The condition for inflation (15) requires or classically that the potential energy of the inflaton dominates over the kinetic energy. Hence one requires a sufficiently flat potential for the inflaton in order to lead to sufficient inflation. Imposing the slow-roll conditions: and , Eqs. (18) and (19) are approximately given as

(20) | |||

(21) |

One can define the so-called slow-roll parameters

(22) |

We can easily verify that the above slow-roll approximations are valid when and for a prolonged period of time.

The inflationary phase ends when and grow to of order unity, though this does not, of itself, imply reheating of the universe. A useful quantity to describe the amount of inflation is the number of e-foldings, defined by

(23) |

where the subscript denotes evaluation of the quantity at the end of inflation.

In order to solve the flatness problem, is required to be right after the end of inflation. Meanwhile the ratio between the initial and final phase of slow-roll inflation is given by

(24) |

where we used the fact that is nearly constant during slow-roll inflation. Assuming that is of order unity, the number of e-foldings is required to be to solve the flatness problem. This statement is a statement about the measure on the space of initial conditions and is therefore properly in the domain of quantum gravity. It is clear that for any fixed number of e-foldings one can choose an infinite number of such that the flatness problem is not solved. Nevertheless, inflation certainly mitigates the problem. We require a similar number of e-foldings in order to solve the horizon problem and hence is taken as a standard target minimum number of e-foldings for any new model of inflation.

### ii.5 Models of inflation

So far we have not discussed the form of the inflaton potential, . The original “old inflation” scenario Guth81 ; Sato81p1 ; Sato81p2 assumed the inflaton was trapped in a metastable false vacuum and had to exit to the true vacuum via a first-order transition. As Guth pointed out Guth81 this could occur neither gracefully nor completely, problems avoided in the “new inflation” model where inflation ends via a second-order phase transition after a phase of slow roll. We now have many varieties of inflationary models : , new, chaotic, extended, power-law, hybrid, natural, supernatural, extra-natural, eternal, D-term, F-term, brane, oscillating, trace-anomaly driven, k, ghost, tachyon,…, etc…

The different kinds of single-field inflationary models can be roughly classified in the following way Kolb99 . The first class (type I) consists of the “large field” models, in which the initial value of the inflaton is large and it slow rolls down toward the potential minimum at smaller . Chaotic inflation Linde83 is one of the representative models of this class. The second class (type II) consists of the “small field” models, in which the inflaton field is small initially and slowly evolves toward the potential minimum at larger . New inflation Linde82 ; Albre82 and natural inflation Freese90 are the examples of this type. In the first class one usually has , whereas it can change the sign in the second class. The third class (type III) consists of the hybrid inflation models Linde94 , in which inflation typically ends by a phase transition triggered by the presence of a second scalar field. The fourth class (type IV) consists of the double inflation models in which there exist two dynamical scalar fields leading to the two stage of inflation. A simple example is two light massive scalar fields given in Ref. Polarski92 .

We note that several models of inflation
can not be classified in the above four classes.
For example, some models do not have a potential minimum
such as quintessential inflation PV99 and
tachyon inflation FT02 ; Fe02 ; Paddy02 ; PST05 ; Sami02 ; SCQ02 ; TW05 .
Typically these scenarios suffer from a reheating
problem Kofman02 ,
since gravitational particle production is not efficient
compared to the standard non-gravitational particle production
by an oscillating inflaton field.
There exist other models of inflation in which an accelerated
expansion is realised without using the potential of the inflaton.
For example, k-inflation Picon99 and ghost
inflation Arkani03
belong to this class. In this case inflation occurs
in the presence of higher-order kinematic terms of
a scalar field.
Inflation can also be realised when the higher-order curvature
terms are present Sta80 ; MO04 ; CTS05 ; NOZ00 ; NO00 ; NO03 ; HHR01 ; ellis99 ; BB89 ,
even without an inflaton
potential^{1}^{1}1We note that in the simple inflation
model Sta80 the system can be reduced to
a minimally coupled scalar
field with a large-field potential by making a conformal
transformation Maeda89 .
However this transformation is not generally easy in the presence of
more complicated higher-order curvature terms..
Apart from these models, let us
briefly review each class of inflationary models.

#### ii.5.1 Large-field models

The large-field models are typically characterized by the monomial potential

(25) |

The quadratic and quartic potentials in chaotic inflation correspond to and , with inflation occurring for Planckian scale values of (see Fig. 1). Such models lend themselves naturally to randomly distributed initial conditions with regions of spacetime that initially have and are homogeneous on the Hubble scale undergoing inflation and therefore potentially giving rise to our observable universe Lindebook .

It is easy to get analytic forms of solutions under the slow-roll approximation: and . For example in the case of the quadratic potential ( and ) we get the following relation by Eqs. (20) and (21):

(26) | |||

(27) |

where is an integration constant corresponding to the initial value of the inflaton. The relation (27) implies that the universe expands exponentially during the initial stage of inflation. The expansion rate slows down with the increase of the second term in the square bracket of Eq. (27). We require the condition, , in order to have the number of e-foldings which is larger than .

#### ii.5.2 Small-field models

The small-field models are characterized by the following potential around :

(28) |

which may arise in spontaneous symmetry breaking. The potential (28) corresponds to a Taylor expansion about the origin, but realistic small-field models also have a potential minimum at some to connect to the reheating stage.

For example let us consider natural inflation model in which a Pseudo Nambu-Goldstone boson (PNGB) plays the role of inflaton. The PNGB potential is expressed as Freese90

(29) |

where two mass scales and characterize the height and width of the potential, respectively (see Fig. 2). The typical mass scales for successful inflation are of order GeV and GeV. The potential (29) has a minimum at .

One typical property in the type II model is that the second derivative of the inflaton potential can change sign. In natural inflation is negative when inflaton evolves in the region of . This leads to the enhancement of inflaton fluctuations by spinodal (tachyonic) instability Cormier99 ; Cormier00 ; Tsujispi ; Felder01p1 ; Felder01p2 . When the particle creation by spinodal instability is neglected, the number of e-foldings is expressed by

(30) |

In order to achieve a sufficient number of e-foldings (), the initial value of the inflaton is required to be for the mass scale .

#### ii.5.3 Hybrid inflation

Hybrid inflation models involve more than one scalar field. This scenario is particularly motivated from the viewpoint of particle physics Cope94 ; LRreview ; Linde97b . Inflation continues from an initial large value of the inflaton which decreases until it reaches a bifurcation point, after which the field becomes unstable and undergoes a “waterfall” transition towards a global minimum (see Fig. 3). During the initial inflationary phase the potential of the hybrid inflation is effectively described by a single field:

(31) |

Consider the Linde hybrid inflation model with potential Linde94

(32) |

When is large the field rolls down toward the potential minimum at . Then we have

(33) |

The mass-squared of becomes negative for signifying a tachyonic instability. Then the field begins to roll down to one of the true minima at and (and thereby creates domain walls). In this original version of the hybrid inflation Linde94 inflation soon comes to an end after the symmetry breaking () due to the rapid rolling of the field . In this case the number of e-foldings can be approximately estimated by using the potential (33):

(34) |

where is the initial value of inflaton.

#### ii.5.4 Double inflation

Double inflation can occur even for the potential (32) depending on the model parameters. When the condition, , is satisfied, the mass of the field is “light” relative to the Hubble rate around , thereby leading to a second stage of inflation for Randall96 ; Garcia96hybrid ; Tsuji03 . This corresponds to a genuine multi-field inflationary model, with more than one light field, of the type that we will examine in later sections. More generally multi-field inflation may be naturally realised near points of enhanced symmetry in moduli space Kadota03 ; Kadota03v2 . In any model where more than one scalar field is light during inflation then there is no longer a unique attractor trajectory in phase-space and such models can support isocurvature as well as adiabatic perturbations about a particular background solution.

An alternative form of double inflation is also realised in the following simple model:

(35) |

as studied by Polarski and Starobinsky Polarski92 , and later by Langlois Langlois99 who realised that the adiabatic and isocurvature perturbations surviving at the end of inflation will in general be correlated.

## Iii Cosmological perturbations

Having undertaken a rapid tour of standard inflationary theory and models we move to discussion of perturbations. The description of the Universe as a perfectly homogeneous and isotropic FRW model is obviously an idealisation. In practice we are interested in deviations from homogeneity and isotropy that enable us to characterise different models. We will deal with small perturbations, but we will assume that the distribution of perturbations is statistically homogeneous and isotropic, which is an alternative statement of the Copernican principle.

In particular we have so far considered only the dynamics of homogeneous scalar fields driving inflation. But to investigate inflation models in more detail, and to test theoretical predictions against cosmological observations we need to consider inhomogeneous perturbations. In this section we will define the variables and notation used in subsequent sections to describe cosmological perturbations generated by inflation.

We will consider only small perturbations about the homogeneous fields

(36) |

and only keep terms to first-order in .

### iii.1 Metric perturbations

For an inhomogeneous matter distribution the Einstein equations imply that we must also consider inhomogeneous metric perturbations about the spatially flat FRW metric. The perturbed FRW spacetime is described by the line element

where denotes the spatial partial derivative . We will use lower case latin indices to run over the 3 spatial coordinates. Our metric perturbations follow the notation of Ref. Mukhanov90 , apart from our use of rather than as the perturbation in the lapse function.

The metric perturbations have been split into scalar, vector and tensor parts according to their transformation properties on the spatial hypersurfaces. The Einstein equations for the scalar, vector and tensor parts then decouple to linear order. We do not consider second-order cosmological perturbations in this review Acqu03 .

#### iii.1.1 Scalar perturbations

The four scalar metric perturbations , , and are constructed from 3-scalars, their derivatives, and the background spatial metric. The intrinsic Ricci scalar curvature of constant time hypersurfaces is given by

(38) |

where is the spatial Laplacian, and hence we refer to as the curvature perturbation. We can Fourier decompose an arbitrary scalar perturbation with respect to the complete set of eigenvectors of the spatial Laplacian, , with comoving wavenumber indexing the corresponding eigenvalues.

Under a scalar coordinate/gauge transformation

(39) | |||||

(40) |

determines the time slicing and the spatial threading. The scalar metric perturbations then transform as

(41) | |||||

(42) | |||||

(43) | |||||

(44) |

Although and separately are spatially gauge-dependent, the combination is independent of spatial gauge and describes the scalar potential for the anisotropic shear of worldlines orthogonal to constant time hypersurfaces Kodama84 .

We can construct a variety of gauge-invariant combinations of the scalar metric perturbations. The longitudinal gauge corresponds to a specific gauge-transformation to a (zero-shear) frame such that , leaving the gauge-invariant variables

(45) | |||||

(46) |

Matter perturbations are also gauge-dependent. Scalar field, density and pressure perturbations all obey the simple transformation rule

(47) |

The adiabatic pressure perturbation is defined to be

(48) |

and hence the non-adiabatic part of the actual pressure perturbation, or entropy perturbation, is a gauge-invariant perturbation

(49) |

The scalar part of the 3-momentum is given by and this momentum potential transforms as

(50) |

Thus we can obtain the gauge-invariant comoving density perturbation Bardeen80

(51) |

We can construct two further commonly used gauge-invariant combinations in terms of matter and metric perturbations. The curvature perturbation on uniform-density hypersurfaces is given by

(52) |

first defined by Bardeen, Steinhardt and Turner Bardeen83 (see also Refs. Bardeen88 ; Martin98 ; Wands00 ). The comoving curvature perturbation (strictly speaking the curvature perturbation on hypersurfaces orthogonal to comoving worldlines)

(53) |

This has been used by Lukash Lukash80 , Lyth Lyth85 and many others, including Mukhanov, Feldman and Brandenberger in their review Mukhanov90 . (Note that in their review the comoving curvature perturbation is denoted by “” in Ref. Mukhanov90 and defined in terms of the metric perturbations in the longitudinal gauge, but it is equivalent to our definition of in a spatially flat background with vanishing anisotropic stress.) The difference between the two curvature perturbations and is proportional to the comoving density perturbation:

(54) |

For single field inflation we have and hence

(55) |

In slow-roll single-field inflation we have and hence and these two commonly used curvature perturbations, and , coincide.

Finally we note that another variable commonly used to describe scalar perturbations during inflation is the field perturbation in the spatially flat gauge (where ). This has the gauge-invariant definition Mukhanov85 ; Sasaki86 :

(56) |

In single field inflation this is simply a rescaling of the comoving curvature perturbation in (53). We see that what appears as a field perturbation in one gauge is a metric perturbation in another gauge and vice versa.

#### iii.1.2 Vector perturbations

The vector perturbations and can be distinguished from scalar perturbations as they are solenoidal (divergence-free), i.e., .

Under a vector coordinate/gauge transformation

(57) |

the vector metric perturbations transform as

(58) | |||||

(59) |

and hence is the gauge-invariant vector shear perturbation.

#### iii.1.3 Tensor modes

The tensor perturbations are transverse and trace-free . They are automatically independent of coordinate gauge transformations.

These are referred to as gravitational waves as they are the free part of the gravitational field and evolve independently of linear matter perturbations.

We will decompose arbitrary tensor perturbations into eigenmodes of the spatial Laplacian, , with comoving wavenumber , and scalar amplitude :

(60) |

with two possible polarisation states, and .

### iii.2 Field equations

#### iii.2.1 Scalar perturbations

By considering the perturbed Einstein equations , we find that the metric perturbations are related to matter perturbations via the energy and momentum constraints Mukhanov90

(61) | |||

(62) |

These can be combined to give the gauge-invariant generalisation of the Poisson equation

(63) |

relating the longitudinal gauge metric perturbation (46) to the comoving density perturbation (51).

The Einstein equations also yield two evolution equations for the scalar metric perturbations

(64) | |||

(65) |

where the scalar part of the anisotropic stress is given by . Equation (III.2.1) can be written in terms of the longitudinal gauge metric perturbations, and defined in Eqs. (45) and (46), as the constraint

(66) |

and hence we have in the absence of anisotropic stresses.

Energy-momentum conservation gives evolution equations for the perturbed energy and momentum

(67) | |||

(68) |

Re-writing the energy conservation equation (III.2.1) in terms of the curvature perturbation on uniform-density hypersurfaces, in (52), we obtain the important result

(69) |

where is the non-adiabatic pressure perturbation, defined in (49), and is the scalar shear along comoving worldlines Lyth03p1 , which can be given relative to the Hubble rate as

Thus is constant for adiabatic perturbations on super-Hubble scales (), so long as remains finite, in which case the shear of comoving worldlines can be neglected.

If we consider scalar fields with Lagrangian density

(71) |

and minimal coupling to gravity, then the total energy, pressure and momentum perturbations are given by

(72) | |||||

(73) | |||||

(74) |

where . These then give the gauge-invariant comoving density perturbation

(75) |

The comoving density is sometimes used to represent the total matter perturbation but for a single scalar field it is proportional to the non-adiabatic pressure (49):

(76) |

From the Einstein constraint equation (63) this will vanish on large scales () if remains finite, and hence single scalar field perturbations become adiabatic in this large-scale limit.

The anisotropic stress, , vanishes to linear order for any number of scalar fields minimally coupled to gravity.

The first-order scalar field perturbations obey the wave equation

(77) |

#### iii.2.2 Vector perturbations

The divergence-free part of the 3-momentum obeys the momentum conservation equation

(78) |

where the vector part of the anisotropic stress is given by . The gauge-invariant vector metric perturbation is then directly related to the divergence-free part of the momentum via the constraint equation

(79) |

Thus the Einstein equations constrain the gauge-invariant vector metric perturbation to vanish in the presence of only scalar fields, for which the divergence-free momentum necessarily vanishes.

#### iii.2.3 Tensor perturbations

There is no constraint equation for the tensor perturbations as these are the free gravitational degrees of freedom (gravitational waves). The spatial part of the Einstein equations yields a wave equation for the amplitude, defined in Eq. (60), of the tensor metric perturbations:

(80) |

This is the same as the wave equation for a massless scalar field (III.2.1) in an unperturbed FRW metric.

### iii.3 Primordial power spectra

Around the epoch of primordial nucleosynthesis the universe is constrained to be dominated by radiation composed of photons and 3 species of relativistic neutrinos. In addition there are non-relativistic baryons, tightly coupled to the photons by Thomson scattering, and cold dark matter which has decoupled. There is probably also some form of vacuum energy, or dark energy, which eventually comes to dominate the density of the universe at the present day. All of these different components may have different density perturbations, . These are usefully characterised by the gauge-invariant curvature perturbations for each component:

(81) |

These individual remain constant on large scales Wands00 as a consequence of local energy-conservation for photons, neutrinos, baryons and cold dark matter, each of which has a well-defined equation of state and hence . Even when energy is not separately conserved for each individual component it may still be possible to define a conserved perturbation on large scales with respect to some other locally conserved quantity, such as the baryon number so long as the net baryon number is conserved Lyth03p1 . Perfect fluid models of non-interacting dark energy will also have constant on large scales, but scalar field models of dark energy do not in general have a well-defined equation of state and hence is not necessarily constant on large scales Doran ; Malquarti ; Malik04 .

The total curvature perturbation , defined in Eq. (52), is simply given by the weighted sum of the individual curvature perturbations

(82) |

This is often referred to as the adiabatic density perturbation, while the difference determines the isocurvature density perturbations

(83) |

By convention the isocurvature perturbations are defined with respect to the photons, hence these are also referred to as entropy perturbations. The factor of 3 arises so that coincides with the perturbation in the local baryon-photon ratio:

(84) |

The relative isocurvature perturbation, , remains constant on large scales as a consequence of the conservation of the individual . The total curvature perturbation only remains constant on large scales as the universe evolves from radiation to matter domination for adiabatic perturbations with , in agreement with Eq. (69).

The primordial power spectrum of density perturbations in the radiation-dominated era, after inflation but well before matter-domination, is commonly given in terms of either , or the comoving curvature perturbation, in Eq. (53). Combining Eqs. (63) and (54) we have

(85) |

and hence and coincide on large scales.

The power on a given scale is given by the -space weighted contribution of modes with given wavenumber. Thus the power spectrum of scalar curvature perturbations, , is commonly given as

(86) |

This coincides with the definition of used in the review by Lidsey et al LLKCreview and in the Liddle and Lyth book LLbook , and is denoted by the WMAP team Peiris03 . An alternative notation widely used for the scalar power spectrum is the fractional density perturbation when adiabatic density perturbations re-enter the Hubble scale during the matter dominated era LLKCreview ; LLbook

(87) |

An isocurvature power spectrum is naturally defined as

(88) |

The cross-correlation between adiabatic and isocurvature perturbations can be given in terms of a correlation angle :

(89) |

The tensor power spectrum is denoted by

(90) |

where the additional factor of 2 comes from adding the 2 independent polarisations of the graviton. Again there is an alternative notation also widely used LLKCreview ; LLbook

(91) |

The scale dependence of the scalar power spectrum is given by the logarithmic derivative of the power spectrum

(92) |

which is evaluated at Hubble-radius crossing, . We note that for a scale-invariant spectrum by convention. Most authors refer to this as denoting the scalar spectrum. We use to distinguish this from the isocurvature spectrum:

(93) |

where for a scale-invariant spectrum. Similarly for a scale-invariant tensor spectrum.

The best way to distinguish multi-field models for the origin of structure from other inflationary models be the statistical properties of the primordial density perturbation. Inflationary models start with small-scale vacuum fluctuations of an effectively free scalar field, described by a Gaussian random field, with vanishing three-point correlation function. Simple deviations from Gaussianity in multi-field scenarios are conventionally parameterised by a dimensionless parameter , where Komatsu01 ; Komatsu03 ; BKMR04 ; Bartolo02 ; Bernar02

(94) |

and is the potential in the longitudinal gauge, defined in Eq. (45), on large scales in the matter-dominated era and is a strictly Gaussian distribution arising from the first-order field perturbations. For adiabatic perturbations on large scales in the matter dominated era we have and hence this corresponds to

(95) |

This describes a “local” non-Gaussianity where the local curvature perturbation, , is due to the local value of the first-order field perturbation and the square of that perturbation. For example, as we shall see, this naturally occurs in curvaton models and where the local curvaton density is proportional to the square local value of the curvaton field.

### iii.4 formalism

A powerful technique to calculate the resulting curvature perturbation in a variety of inflation models, including multi-field models, is to note that the curvature perturbation defined in Eq. (52) can be interpreted as a perturbation in the local expansion Sasaki95

(96) |

where is the perturbed expansion to uniform-density hypersurfaces with respect to spatially flat hypersurfaces:

(97) |

and must be evaluated on spatially flat () hypersurfaces.

An important simplification arises on large scales where anisotropy and spatial gradients can be neglected, and the local density, expansion, etc, obeys the same evolution equations as the a homogeneous FRW universe Sasaki95 ; Sasaki98 ; Wands00 ; Lyth03p1 ; Rigopoulos03 ; Lyth05 . Thus we can use the homogeneous FRW solutions to describe the local evolution, which has become known as the “separate universe” approach Sasaki95 ; Sasaki98 ; Wands00 ; Rigopoulos03 . In particular we can evaluate the perturbed expansion in different parts of the universe resulting from different initial values for the fields during inflation using homogeneous background solutions Sasaki95 . The integrated expansion from some initial field values up to a late-time fixed density, say at the epoch of primordial nucleosynthesis, is some function . The resulting primordial curvature perturbation on the uniform-density hypersurface is then

(98) |

where and is the field perturbation on some initial spatially-flat hypersurfaces during inflation. In particular the power spectrum for the primordial density perturbation in a multi-field inflation can be written in terms of the field perturbations after Hubble-exit as

(99) |

This approach is readily extended to estimate the effect of non-linear field perturbations on the metric perturbations Sasaki98 ; Lyth03p1 ; Lyth05 . The curvature perturbation due to field fluctuations up to second order is Rodriguez05 ; Seery05b

(100) |

We expect the field perturbations at Hubble-exit to be close to Gaussian for weakly coupled scalar fields during inflation Maldacena02 ; Seery05a ; RS05 ; Seery05b . Hence if the contribution of only one field dominates the perturbed expansion, this gives a non-Gaussian contribution to the curvature perturbation of the “local” form (95), where Rodriguez05

(101) |

## Iv The spectra of perturbations in single-field inflation

In this section we shall consider the spectra of scalar and tensor perturbations generated in single-field inflation.

The perturbed scalar field equation of motion (III.2.1) for a single scalar field can be most simply written in the spatially flat gauge (where ). Using the Einstein constraint equations to eliminate the remaining metric perturbations one obtains the wave equation

(102) |

where a gauge-invariant definition of is given in (56).

Introducing new variables, and , Eq. (102) reduces to Sasaki86 ; Mukhanov88

(103) |

where a prime denotes a derivative with respect to conformal time . The effective mass term, , can be written as Stewart93 ; LLKCreview ; Hwang96

(104) |

where

(105) |

These definitions of the slow-roll parameters coincide at leading order in a slow-roll expansion Liddleetal94 with our earlier definitions in Eq. (22) in terms of the first, second and third derivatives of the scalar field potential.

Neglecting the time-dependence of and
during slow-roll inflation^{2}^{2}2Stewart Stewart01 has
developed a generalised slow-roll approximation to calculate the
spectrum of perturbations that drops the
requirement that the slow-roll parameters are slowly varying.,
and other terms of second and higher order
in the slow-roll expansion, gives

(106) |

and

(107) |

The general solution to Eq. (103) is then expressed as a linear combination of Hankel functions

The power spectrum for the scalar field perturbations is given by

(109) |

Imposing the usual Minkowski vacuum state,

(110) |

in the asymptotic past () corresponds to the choice and in Eq. (LABEL:han). The power spectrum on small scales () is thus

(111) |

and on the large scales () we have

(112) |

where we have made use of the relation for and . In particular for a massless field in de Sitter ( and hence ) we recover the well-known result

(113) |

One should be wary of using the exact solution (LABEL:han) at late times as this is really only valid for the case of constant slow-roll parameters. At early times (on sub-Hubble scales) this does not matter as the precise form of in Eq. (103) is unimportant for . Thus Eq. (112) should be valid some time after Hubble-exit,