# Gravitational waves generated by second order effects during inflation

###### Abstract

The generation of gravitational waves during inflation due to the non-linear coupling of scalar and tensor modes is discussed. Two methods describing gravitational wave perturbations are used and compared: a covariant and local approach, as well as a metric-based analysis based on the Bardeen formalism. An application to slow-roll inflation is also described.

## I Introduction

The generation of gravitational waves (GW) is a general prediction of an early inflationary phase infgw . Their amplitude is related to the energy scale of inflation and they are potentially detectable via observations of B-mode polarization in the cosmic microwave background (CMB) if the energy scale of inflation is larger than GeV MKam ; USeljak ; MKes ; LKnox ; USeljakH . Such a detection would be of primary importance to test inflationary models.

Among the generic predictions of one-field inflation inflgen are the existence of (adiabatic) scalar and tensor perturbations of quantum origin with an almost scale invariant power spectrum and Gaussian statistics. Even if non-linear effects in the evolution of perturbations are expected, a simple calculation fu02 , confirmed by more detailed analysis maldacena , shows that it is not possible to produce large non-Gaussianity within single field inflation as long as the slow-roll conditions are preserved throughout the inflationary stage. Deviations from Gaussianity can be larger in, e.g., multi-field inflation scenarios fu02 ; bu and are thus expected to give details on the inflationary era.

As far as scalar modes are concerned, the deviation from Gaussianity has been parameterized by a (scale-dependent) parameter, . Various constraints have been set on this parameter, mainly from CMB analysis komatsu (see Ref. NGreview for a review on both theoretical and observational issues). Deviation from Gaussianity in the CMB can arise from primordial non-Gaussianity, i.e. generated during inflation, post-inflation dynamics or radiation transfer transfert . It is important to understand them all in order to track down the origin of non-Gaussianity, if detected.

Among the other signatures of non-linear dynamics is the fact that the Scalar-Vector and Tensor (SVT) modes of the perturbations are no longer decoupled. This implies in particular that scalar modes can generate gravity waves. Also, vector modes, that are usually washed out by the evolution, can be generated. In particular, second-order scalar perturbations in the post-inflation era will also contribute to B-mode polarization SMoll or to multipole coupling in the CMB prunet , and it is thus important to understand this coupling in detail.

In this article, we focus on the gravitational waves generated from scalar modes via second order dynamics. Second-order perturbation theory has been investigated in various works Tomita ; Mena ; Matarr ; 2ndorder ; clarkson1 ; langlois ; ananda ; nakamura ; MAB and a fully gauge-invariant approach to the problem was recently given in Ref. nakamura . Second-order perturbations during inflation have also been considered in Refs. Acq ; maldacena , providing the prediction of the bispectrum of perturbations from inflation.

Two main formalisms have been developed to study perturbations, and hence second order effects: the covariant formalism EB in which exact gauge-invariant variables describing the physics of interest are first identified and exact equations describing their time and space evolution are then derived and approximated with respect to the symmetry of the background to obtain results at the desired order, and the coordinate based approach of Bardeen Bardeen in which gauge-invariants are identified by combining the metric and matter perturbations and then equations are found for them at the appropriate order of the calculation. In this article we carry out a detailed comparison of the two approaches up to second order, highlighting the advantages and disadvantages of each method, thus extending earlier work on the linear theory BDE . Our paper also extends the work of Ref. langlois , in which the relation between the two formalisms on super-Hubble scales is investigated. In particular, we show that the degree of success of one formalism over the other depends on the problem being addressed. This is the first time a complete and transparent matching of tensor perturbations in the two formalisms at first and second order is presented. We also show, using an analytical argument, that the power-spectrum of gravitational waves from second-order effects is much smaller than the first order on super-Hubble scales. This is in contrast to the fact that during the radiation era the generation of GW from primordial density fluctuations can be large enough to be detected in principle, though this requires the inflationary background of GW to be sufficiently small ananda .

This paper is organized as follows. We begin by reviewing scalar field dynamics in Section II within the 1 + 3 covariant approach. In Section III, we formulate the problem within the covariant approach followed by a reformulation in the coordinate approach in Section IV. A detailed comparison of the two formalisms is then presented in Section V. In Section VI, we study gravitational waves that are generated during the slow-roll period of inflation. In particular, we introduce a generalization of the parameter to take into account gravity waves and we compute the three point correlator involving one graviton and two scalars. Among all three point functions involving scalar and tensor modes, this correlator and the one involving three scalars are the dominant maldacena . Finally, we conclude in Section VII.

## Ii Scalar field dynamics

Let us consider a minimally coupled scalar field with Lagrangian density^{1}^{1}1We use
conventions of Ref. bi:wald . Units in which are used throughout this article, Latin indices run from 0 to 3, whereas Latin
indices run from to . The symbol represents the usual covariant
derivative and corresponds to partial differentiation. Finally the Hilbert-Einstein
action in presence of matter is defined by

(2) |

where is a general (effective) potential expressing the self interaction of the scalar field. The equation of motion for the field following from is the Klein - Gordon equation

(3) |

where the prime indicates a derivative with respect to . The energy - momentum tensor of is of the form

(4) |

provided , equation (3) follows from the conservation equation

(5) |

We shall now assume that in the open region of spacetime that we consider, the momentum density is timelike:

(6) |

This requirement implies two features: first, is not constant in , and so specifies well-defined surfaces in spacetime. When this is not true (i.e., is constant in ), then by (4),

(7) |

in , [the last being necessarily true due to the conservation law (5))] and we have an effective cosmological constant in rather than a dynamical scalar field.

### ii.1 Kinematical quantities

Our aim is to give a formal description of the scalar field in terms of fluid quantities;
therefore, we assign a 4-velocity vector to the scalar field itself. This will allow us to
define the dot derivative, i.e. the proper time derivative along the flow lines:
. Now, given the
assumption (6), we can choose the 4-velocity field as the unique timelike vector
with unit magnitude () parallel to the normals of the hypersurfaces
bi:madsen ^{2}^{2}2In the case of more than one scalar field, this choice can still be made
for each scalar field 4-velocity, but not for the 4-velocity of the total fluid. A number of frame choices exist for the 4-velocity of the total fluid, the most common being the energy frame, where the total energy flux vanishes (see DBE for a detailed description of this case.
,

(8) |

where we have defined the field to denote the
magnitude of the momentum density (simply momentum from now on). The choice
(8) defines as the unique timelike eigenvector of the
energy - momentum tensor (4).^{3}^{3}3The quantity
will be positive or negative depending on the initial
conditions and the potential ; in general could oscillate
and change sign even in an expanding phase, and the determination of
by (8) will be ill - defined on those surfaces where
(including the surfaces of maximum
expansion in an oscillating Universe). This will not cause us a
problem however, as we assume the solution is differentiable and
(6) holds almost everywhere, so determination of
almost everywhere by this equation will extend (by continuity) to
determination of everywhere in .

The kinematical quantities associated with the “flow vector” can be obtained by a standard method bi:ellis1 ; lang-vern . We can define a projection tensor into the tangent 3-spaces orthogonal to the flow vector:

(9) |

with this we decompose the tensor as

(10) |

where is the spatially totally projected covariant derivative operator orthogonal to (e.g., ; see the Appendix of Ref. bi:BDE for details), is the acceleration (), and the shear ( Then the expansion, shear and acceleration are given in terms of the scalar field by

(11) | |||||

(12) | |||||

(13) |

where the last equality in Eq. (11) follows on using the Klein - Gordon equation (3). We can see from Eq. (13) that is an acceleration potential for the fluid flow bi:ellis2 . Note also that the vorticity vanishes:

(14) |

an obvious result with the choice (8), so that is the covariant derivative operator in the 3-spaces orthogonal to , i.e. in the surfaces . As usual, it is useful to introduce a scale factor (which has dimensions of length) along each flow - line by

(15) |

where is the usual Hubble parameter if the Universe is homogeneous and isotropic. Finally, it is important to stress that

(16) |

which follows from our choice of via equation (8), a result that will be important for the choice of gauge invariant (GI) variables and for the perturbations equations.

### ii.2 Fluid description of a scalar field

It follows from our choice of the four velocity (8) that we can represent a minimally coupled scalar field as a perfect fluid; the energy - momentum tensor (4) takes the usual form for perfect fluids

(17) |

where the energy density and pressure of the scalar field “fluid” are given by

(18) | |||||

(19) |

If the scalar field is not minimally coupled this simple representation is no longer valid, but it is still possible to have an imperfect fluid form for the energy - momentum tensor bi:madsen .

Using the perfect fluid energy - momentum tensor (17) in (5) one obtains the energy and momentum conservation equations

(20) | |||||

(21) |

If we now substitute and from Eqs. (18) and (19) into Eq. (20) we obtain the 1+3 form of the Klein - Gordon equation (3):

(22) |

an exact ordinary differential equation for in any space - time with the choice (8) for the four - velocity. With the same substitution, Eq. (21) becomes an identity for the acceleration potential . It is convenient to relate and by the index defined by

(23) |

This index would be constant in the case of a simple one-component fluid, but in general will vary with time in the case of a scalar field:

(24) |

Finally, it is standard to define a speed of sound as

(25) |

### ii.3 Background equations

The previous equations assume nothing on the symmetry of the spacetime. We now specify it further and assume that it is close to a flat Friedmann-Lemaître spacetime (FL), which we consider as our background spacetime. The homogeneity and isotropy assumptions imply that

(26) |

where is any scalar quantity; in particular

(27) |

The background (zero - order) equations are given by bi:scalar :

(28) | |||

(29) | |||

(30) |

where all variables are a function of cosmic time only.

## Iii gravitational waves from density perturbations: covariant formalism

### iii.1 First order equations

The study of linear perturbations of a FL background are relatively straightforward. Let us
begin by defining the *first-order gauge-invariant* (FOGI) variables
corresponding respectively to the spatial fluctuations in the energy density,
expansion rate and spatial curvature:

(31) |

The quantities are FOGI because they vanish exactly in the background FL spacetime bi:SW ; bi:BS . It turns out that a more suitable quantity for describing density fluctuations is the co-moving gradient of the energy density:

(32) |

where the ratio allows one to evaluate the magnitude of the energy density perturbation relative to its background value and the scale factor guarantees that it is dimensionless and co-moving.

These quantities exactly characterize the inhomogeneity of any fluid; however we specifically want to characterize the inhomogeneity of the scalar field: this cannot be done using the spatial gradient because it identically vanishes in any space-time by virtue of our choice of 4-velocity field . It follows that in our approach the inhomogeneities in the matter field are completely incorporated in the spatial variation of the momentum density: , so it makes sense to define the dimensionless gradient

(33) |

which is related to by

(34) |

where we have used Eq. (18) and is given by Eq. (23); comparing
Eq. (33) and Eq. (13) we see that is proportional to the
acceleration: it is a gauge-invariant
measure of the spatial variation of proper time along the flow lines of
between two surfaces const. (see Ref. bi:ellis1 ).
The set of linearized equations satisfied by the FOGI variables consists of the
*evolution equations*

(35) | |||

(36) | |||

(37) | |||

(38) | |||

(39) |

and the *constraints*

(40) | |||||

(41) | |||||

(42) | |||||

(43) | |||||

(44) | |||||

(45) |

The operator is defined by is the completely antisymmetric tensor with respect to the spatial section defined by , being the volume antisymmetric tensor such that . The divergence of a rank tensor is a rank tensor defined by . where

Because the background is homogeneous and isotropic, each FOGI
vector may be uniquely split into a *curl-free* and
*divergence-free* part, usually referred to as scalar and
vector parts respectively, which we write as

(46) |

where and . Similarly, any tensor may be invariantly split into scalar, vector and tensor parts:

(47) |

where ,

Let us now concentrate on scalar perturbations at linear order. It is clear from the above discussion that pure scalar modes are characterized by the vanishing of the magnetic part of the Weyl tensor: , so the above set of equations reduce to a set of two coupled differential equations for and :

(48) | |||

(49) |

and a set of coupled evolution and constraint equations that determine the other variables

(50) | |||

(51) | |||

(52) | |||

(53) | |||

(54) |

### iii.2 Gravitational waves from density perturbations

The preceding discussion deals with first-order variables and their behavior at linear order. It is important to keep in mind that we were able to set only because pure scalar perturbations in the absence of vorticity implies that at first order. The vanishing of the magnetic part then follows from equation (43). However, at second order . We denote the non-vanishing contribution at second order by clarkson1

The new variable is *second-order and gauge-invariant* (SOGI), as it vanishes at all lower orders bi:SW . It should be
noted that the new variable is just the magnetic part of the Weyl tensor subject to the conditions
mentioned above i.e.

(55) |

We are interested in the properties inherited by the new variable from the magnetic part of the Weyl tensor. In particular, it can be shown that the new variable is transverse and traceless at this order and is thus a description of gravitational waves. It should be stressed that in full generality, there are tensorial modes even at first order. By assuming that there are none, we explore a particular subset in the space of solutions. From the ”iterative resolution” point of view, this means that we constrain the equations in order to focus on second order GWs sourced by terms quadratic in scalar perturbations. In doing so, we artificially switch off GW perturbations at first order.

### iii.3 Propagation equation

The propagation of the new second-order variable now needs to be investigated using a covariant set of equations that are linearized to second-order about FL. We make use of Eqs. (20), (21) and the following evolution equations which are up to second order in magnitude;

(56) | |||||

(57) |

together with the constraint

(58) |

Unlike at first-order, where the splitting of tensors into their scalar, vector and tensor parts is possible, at second order this can only be achieved for SOGI variables.

We may isolate the tensorial part of the equations by decoupling : since it is divergence free it is already a pure tensor mode, whereas is not. The wave equation for the gravitational wave contribution can be found by first taking the time derivative of (57) and making appropriate substitutions using the evolution equations and keeping terms up to second order. The wave equation for then reads:

(59) |

where the source is given by the cross-product of the electric-Weyl curvature and its divergence (or acceleration):

(60) | |||||

To obtain this, we have used the fact that with a flat background space-time

and used the commutation relation

We have also used Eqs. (24) and (25) to eliminate from the source term. It can also be shown that is transverse, illustrating that Eq. (59) represents the gravitational wave contribution at second order. Note that this is a local description of gravitational waves, in contrast to the non-local extraction of tensor modes by projection in Fourier space. Since contains exactly the correct number of degrees of freedom possible in GW, any other variable we may choose to describe GW must be related by quadrature, making this a suitable master variable. The situation is analogous to the description of electromagnetic waves: Should we use the vector potential, the electric field, or the magnetic field for their description? Mathematically it doesn’t matter of course – each variable obeys a wave equation and the others are related by quadrature. Physically, however, it’s the electric and magnetic fields which drive charged particles through the Lorentz force equation – the electromagnetic analogue of the geodesic deviation equation.

In order to express the gravitational wave equation in Fourier space, we define our normalised tensor harmonics as

(61) |

where is the polarization tensor, which satisfies the (background) tensor Helmholtz equation: . As is required to satisfy in the background, it can thus be identified with a 3-vector and will subsequently written in bold when necessary. We denote harmonics of the opposite polarization with an overbar. Amplitudes of may be extracted via

(62) |

with an analogous formula for the opposite parity. This implies that our original variable may be reconstructed from

(63) |

The same relations hold for any transverse tensor. Hence, our wave equation in Fourier space is

(64) |

with an identical equation for the opposite polarization. We have converted to conformal time , where a prime denotes derivatives with respect to , and we have defined the conformal Hubble parameter as . The source term is composed of a cross-product of the electric part of the Weyl tensor and its divergence. At first-order, the electric Weyl tensor is a pure scalar mode, and can therefore be expanded in terms of scalar harmonics. To define these, let , be a solution to the Helmholtz equation: Beginning with this basis, it is possible to derive vectorial and (PSTF) tensorial harmonics by taking successive spatial derivatives as follows:

(65) | |||||

This symmetric tensor has the additional property Using this representation we can express our source in Eq. (64) in terms of a convolution in Fourier space, by expanding the electric Weyl tensor as

(67) |

Then, the right hand side of Eq. (60) expressed in conformal time, accompanied by appropriate Fourier decomposition of each term and making use of the normalization condition for orthonormal basis, yields:

(68) |

where

(69) |

with a similar expression for the other polarization.

In principle we can now solve for the gravitational wave contribution , and calculate the power spectrum of gravitational waves today. For this however, we need initial conditions for the electric Weyl tensor (or, alternatively ).

## Iv gravitational waves from density perturbations: coordinate based approach

In this formalism, we consider perturbations around a FL universe with Euclidean spatial sections and expand the metric as

(70) |

where is the conformal time and the scale factor. We perform a scalar-vector-tensor decomposition as

(71) |

and

(72) |

where , are transverse (), and is traceless and transverse (). Latin indices are lowered by use of the spatial metric, e.g. . We fix the gauge and work in the Newtonian gauge defined by so that and are the two Bardeen potentials. As in the previous sections, we assume that the matter content is a scalar field that can be split into background and perturbation contributions: . The gauge invariant scalar field perturbation can be defined by

(73) |

where . We denote the field perturbation in Newtonian gauge by so that . Introducing

(74) |

the equation of state (23) takes the form . We thus have two expansions: one concerning the perturbation of the metric and the other in the slow-roll parameter .

### iv.1 Scalar modes

Focusing on scalar modes at first order in the perturbation, it is convenient to introduce

(75) |

and

(76) |

in terms of which the action (1) takes the form

(77) |

when expanded to second order in the perturbations. It is the action of a canonical scalar field with effective square mass . is the canonical variable that must be quantized bmf . It is decomposed as follows

(78) |

Here is solution of the Klein-Gordon equation

(79) |

and the annihilation and creation operators satisfy the commutation relation, . We define the free vacuum state by the requirement for all .

From the Einstein equation, one can get the expression for the Bardeen potential (recalling that )

(80) |

and for the curvature perturbation in comoving gauge

(81) |

Once the initial conditions are set, solving Eq. (79) will give the evolution of during inflation, from which and can be deduced, using the previous expressions.

Defining the power spectrum as

(82) |

one easily finds that

(83) |

Note also that and are related by the simple relation

(84) |

so that

(85) |

### iv.2 Gravitational waves at linear order

At first order, the tensor modes are gauge invariant and their propagation equation is given by

(86) |

since a minimally coupled scalar field has no anisotropic stress. Defining the reduced variable

(87) |

the action (1) takes the form

(88) |

when expanded tosecond order. Developing , and similarly , in Fourier space:

(89) |

where is the polarization tensor, the action (88) takes the form of the action for two canonical scalar fields with effective square mass

(90) |

If one considers the basis of the 2 dimensional space orthogonal to then .

are the two degrees of freedom that must be quantized bmf and we expand them as

(91) | |||||

is solution of the Klein-Gordon equation

(92) |

where we have dropped the polarization subscript. The annihilation and creation operators satisfy the commutation relations, and . We define the free vacuum state by the requirement for all and .

Defining the power spectrum as

(93) |

one easily finds that

(94) |

where the two polarizations have the same contribution.

### iv.3 Gravitational waves from density perturbations

At second order, we split the tensor perturbation as . The evolution equations of is similar to Eq. (86), but inherits a source term quadratic in the first order perturbation variables and from the transverse tracefree (TT) part of the stress-energy tensor

(95) |

It follows that the propagation equation is

(96) |

where is a TT tensor that is quadratic in the first order perturbation variables.

Working in Fourier space, the TT part of any tensor can easily be extracted by means of the projection operator

(97) |

where (note that is not analytic in and is a non-local operator) from which we get

(98) | |||||

The source term is now obtained as the TT-projection of the second order Einstein tensor quadratic in the first order variables and of the stress-energy tensor

(99) |

The three terms respectively indicate terms involving products of first order scalar quantities, first order scalar and tensor quantities and first order tensor quantities. The explicit form of the first term is

(100) |

The first term was considered in Ref. gwpot and the second term was shown to be the dominant contribution for the production of gravitational waves during preheating gwphi . In Fourier space, it is given by

(101) | |||||

can be decomposed as in Eq. (89), using the same definition (87) at any order. The two polarizations evolve according to

(102) |

Since the polarization tensor is a TT tensor, it is obvious that , so that

(103) |

From the equation (102), we deduce that the source term derives from an interaction Lagrangian of the form

(104) |

It describes a two-scalars graviton interaction. In full generality the interaction term would also include, at lowest order, cubic terms of three scalars, two gravitons-scalar and three gravitons. They respectively correspond to second order scalar-scalar modes generated from gravitational waves and second order tensor modes. As emphasized previously, we do not consider these interactions here.

## V Comparison of the two formalisms

Before going further it is instructive to compare the two formalisms and understand how they relate to each other. Note that we go beyond Ref. bi:BDE , where a comparison of the variables was made at linear order. Here we investigate how the equations map to each other and extend the discussion to second order for the tensor sector. At the background level the scale factors and expansion rates introduced in each formalism agree, which explains why we made use of the same notation.

The perturbations of the metric around FL space-time has been split into a first-order and a second-order part according to

(105) |

We make a similar decomposition for the quantities used in the covariant formalism. As long as we are interested in the gravitational wave sector, we only need to consider the four-velocity of the perfect fluid describing the matter content of the universe which we decompose as

(106) |

Its spatial components are decomposed as

(107) |

being the vector degree of freedom and the scalar degree of freedom. As has only three independent degrees of freedom since satisfies , its temporal component is linked to other perturbation variables. We assume that the fluid has no vorticity (), as it is the case for the scalar fluid we have in mind and consequently we will also drop the vectorial perturbations ().

### v.1 Matching at linear order

At first order, the spatial components of the shear, acceleration and expansion are respectively given by

(108) |

(109) |

(110) |

The electric and magnetic part of the Weyl tensor take the form

(111) |