David Langlois

Filippo Vernizzi

A geometrical approach to nonlinear perturbations in relativistic cosmology

We give a pedagogical review of a covariant and fully non-perturbative approach to study nonlinear perturbations in cosmology. In the first part, devoted to cosmological fluids, we define a nonlinear extension of the uniform-density curvature perturbation and derive its evolution equation. In the second part, we focus our attention on multiple scalar fields and present a nonlinear description in terms of adiabatic and entropy perturbations. In both cases, we show how the formalism presented here enables one to easily obtain equations up to second, third and higher orders.

I. INTRODUCTION

The relativistic theory of cosmological perturbations (see e.g. [1][2][3][4][START_REF] Langlois | Inflation, quantum fluctuations and cosmological perturbations[END_REF][6][START_REF] Langlois | Lectures on inflation and cosmological perturbations[END_REF]) is an indispensable tool to interpret cosmological data such as the Cosmic Microwave Background (CMB) anisotropies, and thus to connect the scenarios of the early universe, such as inflation, to cosmological observations. Because the temperature anisotropies of the CMB are so small (δT /T ∼ 10 -5 ), considering only linear perturbations is an excellent first approximation and this is why most of the efforts devoted to the study of cosmological perturbations have dealt with the linear theory.

However, with the rapidly increasing precision of cosmological data, we have now reached a stage where nonlinear features of primordial perturbations could be observationally accessible. In the last few years, this has motivated an intensive development of the theory of cosmological perturbations beyond linear order. Two main strategies have been considered. The first one is based on extending the traditional coordinate-based approach by adding secondorder perturbations in the metric and in the matter fields and by computing the equations of motion for the metric and matter perturbations by brute force [6,[START_REF] Tomita | [END_REF][9][10][11][12]. The second strategy is based on a fully nonlinear treatment, but restricted to super-Hubble scales and working directly in a coordinate system [13][14][15][16][17][18][19]. This strategy is related to the so called separate universe picture that represents our universe, on scales larger than the Hubble radius, as juxtaposed Friedmann-Lemaitre-Robertson-Walker (FLRW) universes with slightly different scale factors [20,21].

The approach that we present here, based on [22][23][24][25], relies on a geometrical perspective. From a computational point of view, it can be seen as a "middle way" between the two main strategies discussed above. Inspired by the so-called covariant formalism [26,[START_REF] Ellis | Relativistic Cosmology[END_REF] for cosmological perturbations developed in [START_REF] Ellis | [END_REF][29][30][31][32][33] (see also the detailed review [34]), the fundamental idea is to use a nonlinear and covariant approach as long as possible and try to construct covariant quantities that mimic the traditional quantities that have been useful in the theory of linear perturbations (see also [35] for an alternative application of the covariant formalism to nonlinear perturbations). Since these quantities are tensors, they are intuitively much easier to understand from a geometrical point of view. Once these covariant objects have been defined and their evolution equation obtained in a fully nonlinear and covariant form, it is then possible to choose a coordinate system and to expand the quantities and equations to the desired order, in order to make concrete quantitative calculations.

In this review we discuss two illuminating examples, where this approach is particularly efficient in comparison with traditional methods. The first is the nonlinear extension of the familiar linear curvature perturbation on uniformdensity hypersurfaces, usually denoted ζ, introduced in [36]. This quantity, defined for a cosmological perfect fluid, plays a crucial role in the linear theory, because it is conserved on large scales for adiabatic perturbations [37,38], and therefore enables one to very easily relate cosmological perturbations in the very early universe and in the late universe, as long as the scales remain super-Hubble. The second example is useful in the context of multi-field inflation. In the linear theory, it is useful to decompose the perturbations of the scalar fields into (instantaneous) adiabatic and entropic modes, which are simply the projections of the perturbations along directions that are, respectively, tangential and orthogonal to the background trajectory in field space. Once again, it is possible to generalize this decomposition into adiabatic and entropic modes at the nonlinear level. This can be used to compute the evolution of nonlinear perturbations and the resulting non-Gaussianities.

This review is divided into two parts. In the next section we consider the treatment of a cosmological fluid, while the third, and final, section is devoted to the case of several scalar fields.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 4 May 2010

II. COSMOLOGICAL FLUIDS

A. Covariant approach

In this section we consider a single perfect fluid, characterized by a comoving four-velocity u a (u a u a = -1), a proper energy density ρ and a pressure P . This fluid can describe any of the standard species, such as relativistic matter or cold dark matter, as long as a fluid description is relevant and provided that the fluid does not interact with the other fluids that might be present. As will be shown in the last subsection, our approach can also be extended to dissipative and interacting fluids and could thus be applied to models of this type that describe dark energy or dark matter.

The energy-momentum tensor associated to the perfect fluid is given by

T a b = (ρ + P ) u a u b + P g a b . (1)
To fully characterize the fluid, one needs an equation of state relating P to ρ and, possibly, to other physical quantities if the fluid is not barotropic.

The spatial projection tensor orthogonal to the fluid velocity u a is defined by

h ab = g ab + u a u b , (h a b h b c = h a c , h b a u b = 0). ( 2
)
It is also useful to introduce the familiar decomposition

∇ b u a = σ ab + ω ab + 1 3 Θh ab -a a u b , (3)
where one finds on the right-hand side the (symmetric) shear tensor σ ab , the (antisymmetric) vorticity tensor ω ab , the volume expansion Θ ≡ ∇ a u a and the acceleration a a ≡ u c ∇ c u a . The integration of Θ along the fluid world lines with respect to the associated proper time τ ,

α ≡ 1 3 dτ Θ, (4)
can be used to define, for each observer comoving with the fluid, a local scale factor S = e α . It follows that Θ = 3 α, where the dot of a scalar quantity denotes its derivative along u a , i.e. α ≡ u a ∇ a α. This quantity α, which can be seen as a (covariantly defined) local number of e-folds, plays a crucial rôle in the definition of our covariant and nonlinear extension of the familiar linear curvature perturbation on uniform-density hypersurfaces, usually denoted ζ, introduced in [36].

Our starting point is the conservation of the energy-momentum tensor,

∇ a T a b = 0, (5)
which yields, after substituting (1) and projecting along u b , the continuity equation

ρ + 3 α(ρ + P ) = 0. (6)
The spacetime gradient of this expression can be written as

(∂ a ρ) . + 3 (∂ a α) . (ρ + P ) + 3 α ∂ a (ρ + P ) = 0, (7)
where the dot acting on a spacetime gradient, which is a covector, is defined as the Lie derivative (see e.g. [39]) with respect to u a . For any covector X a , this means

Ẋa ≡ L u X a ≡ u b ∂ b X a + X b ∂ a u b . (8)
More generally, a dot acting on any tensor will denote the Lie derivative of this tensor with respect to u a . This definition is compatible with our notation for the scalars since the Lie derivative coincides with the derivative along u a for scalars (and we have used the property ∂ a ( ρ) = (∂ a ρ)

. to obtain [START_REF] Langlois | Lectures on inflation and cosmological perturbations[END_REF]).

After some simple manipulations, one finds that Eq. ( 7) is equivalent to [22,23]

ζa = - Θ 3(ρ + p) ∂ a p - ṗ ρ ∂ a ρ , (9)
where the left-hand side consists of the time derivative, i.e. the Lie derivative with respect to u a , of the covector

ζ a ≡ ∂ a α - α ρ ∂ a ρ . (10)
This covector can also be re-expressed, using (6), in the form

ζ a = ∂ a α + ∂ a ρ 3(ρ + P ) (11)
and, if w ≡ P/ρ is constant, this is a total gradient since

ζ a = ∂ a α + 1 3(1 + w) ln ρ (w = P/ρ = const). ( 12
)
On the right-hand side, the quantity

Γ a ≡ ∂ a P - Ṗ ρ ∂ a ρ (13)
is a nonlinear generalization of the non-adiabatic pressure. It vanishes for purely adiabatic perturbations, for instance when the pressure P is solely a function of the density ρ.

Eq. ( 9) has a form very similar to the conservation equation for ζ of the linear theory, which will be rederived in the next section. At the non-linear level, our covector ζ a is closely related to the non-linear perturbation introduced in [19],

ζ = δN + 1 3 ρ ρ dρ ρ + P , (14)
where ρ(t) is the homogeneous background density, while ρ(t, x) is the local energy density and N = dt Θ/3 is the number of e-folds, defined with respect to the time coordinate t. A similar non-linear quantity has been introduced earlier in [17]. The advantage of the definition (10) is that one does not need to introduce a coordinate system or to restrict oneself to super-Hubble scales, and that its evolution equation ( 9) is exact, fully non-perturbative and valid at all scales. Moreover, it is worth stressing that this equation is a direct consequence of the conservation of the energy-momentum tensor and is independent of the underlying theory of gravitation.

B. Link with the coordinate approach

We now relate the covariant approach with the more familiar coordinate based formalism [1][2][3][4][START_REF] Langlois | Inflation, quantum fluctuations and cosmological perturbations[END_REF][6][START_REF] Langlois | Lectures on inflation and cosmological perturbations[END_REF]. We first examine the linear perturbations and consider later the perturbations at second and third orders.

Linear theory

A cosmological spacetime closed to FLRW geometry can be described by the perturbed metric

ds 2 = -(1 + 2A)dt 2 + 2aB i dx i dt + a 2 (γ ij + H ij ) dx i dx j , (15)
where a = e ᾱ is the background scale factor. As usual, one can decompose the linear perturbations into so-called scalar, vector and tensor modes,

B i = ∇ i B + B V i , H ij = -2ψγ ij + 2 ∇ i ∇ j E + 2 ∇ (i E V j) + 2E T ij , (16)
where B V i and E V i are transverse, i.e., ∇ i B V i = 0 = ∇ i E V i , and E T ij is transverse and traceless, i.e., ∇ i E T ij = 0 and γ ij E T ij = 0. Here ∇ i denotes the three-dimensional covariant derivative with respect to the homogeneous spatial metric γ ij (which is also used to lower or raise the spatial indices).

The corresponding matter content is a perfect fluid with perturbed energy density and pressure, ρ(t, x i ) = ρ(t) + δρ(t, x i ), P (t, x i ) = P (t) + δP (t, x i ), and four-velocity

u µ = ūµ + δu µ , δu µ = {-A, v i /a}, v k = ∇ k v + v V k , (17)
where v V i is transverse, ∇ i v V i = 0. It is easy to check that the covector ζ a automatically vanishes in the background. At linear order, its spatial components are simply [22,23]

ζ i = ∂ i ζ, ζ ≡ δα - H ρ δρ, (18)
where H denotes the Hubble rate, H = a /a = ᾱ and we recall that a prime denotes the partial derivative with respect to the cosmic time t. The quantity ζ defined above differs from the familiar definition [36]

ζ B ≡ -ψ -H δρ ρ , (19)
because δα differs from -ψ. They are related by [23] δα

= -ψ + 1 3 ∇ 2 E + 1 3 dt ∇ 2 v a , (20)
and they coincide when the gradient terms are negligible.

The components of the non-adiabatic term Γ a = ∂ a P -( Ṗ / ρ)∂ a ρ can be deduced directly from the components of ζ a by substituting P to α. One finds

Γ i = ∂ i Γ, Γ ≡ δP - P ρ δρ. (21)
Putting all together Eq. ( 9) gives, at first order,

ζ = ζ B + 1 3 ∇ 2 (E + v/a) = - H ρ + P δP - P ρ δρ . (22)
For adiabatic perturbations, the right-hand side vanishes and the quantity ζ is conserved at all scales because it comes directly from the covector ζ a . By contrast, ζ B is conserved only when the gradient terms are negligible.

Second-order perturbations and beyond

The covariant approach turns out very useful to go beyond linear order in perturbation theory. Any function X can be decomposed in the form X(t, x i ) = X(t) + δX (1) (t, x i ) + δX (2) (t, x i ) + δX (3)

(t, x i ) + . . . , (23)
where a quantity with a superscript (n) is the n-th order contribution to X in the perturbation theory expansion [9] (note that here we do not follow the convention of including a numerical factor 1/n! in front of the n-th contribution). Expanding ζ i = ∂ i α -( α/ ρ)∂ i ρ at second order one finds that, in contrast with the first order expression, ζ (2) cannot be written as the gradient of a scalar perturbation. Indeed, after some manipulations one finds, at second order [22,23],

ζ (2) i = ∂ i ζ (2) + δρ ρ ∂ i ζ (1) , (24)
with

ζ (2) = δα (2) - H ρ δρ (2) - 1 ρ δα δρ + H ρ 2 δρδρ + 1 2ρ H ρ δρ 2 . ( 25
)
On large scales, using

α ≈ ln a -ψ -ψ 2 , (26)
one can show that ζ (2) is directly related to the conserved second-order quantity defined by Malik and Wands [40]:

ζ (2) ≈ ζ (2) MW -(ζ (1)
MW ) 2 (see also the discussion in [41]). We use the symbol ≈ to denote an equality only valid on large scales. Now that we have identified a second-order expression for ζ, we can expand Eq. ( 9) at second order. On making use of the second line of Eq. ( 8) to reexpress the Lie derivative along u a in terms of the coordinate time derivative, and retaining only terms at second order, we have

ζ(2) i = ζ (2) i -Aζ (1) i + v j a ∂ j ζ (1) i + ζ (1) j ∂ i v j a . (27)
Finally, on making use of Eqs. ( 18), (24), and (25), and that Θ = 3(1 -A)α up to first order, we can explicitly write the conservation equation ( 9) up to second order and on all scales:

ζ (2) = - H ρ + P Γ (2) - 1 ρ + P Γ (1) ζ (1) - v j a ∂ j ζ (1) . ( 28
)
The definition of Γ (2) can be read from the second-order expression of ζ (2) by substituting P to α. For adiabatic perturbations, we find that at second order the scalar variable ζ is conserved only on large scales, when the last term on the right-hand side of Eq. ( 28) can be neglected.

One can extend straightforwardly the procedure presented here to higher orders in the perturbation expansion [42,43]. Expanding Eq. ( 10) at third order, after some manipulations one can write

ζ (3) i = ∂ i ζ (3) + δρ ρ ∂ i ζ (2) + δρ (2) ρ ∂ i ζ (1) - ρ 2ρ 3 δρ 2 ∂ i ζ (1) + δρ 2 2ρ 2 ∂ i ζ (1) , (29)
with

ζ (3) =δα (3) - H ρ δρ (3) - 1 ρ δα - H ρ δρ δρ (2) - δρ δρ ρ + ρ δρ 2 2ρ 2 - δρ ρ δα (2) - H ρ δρ (2)
+ δρ 2 2ρ 2 δα - H ρ δρ + 1 ρ H ρ δρ (2) δρ - 1 ρ 2 H ρ δρ 2 δρ - 1 6ρ 1 ρ H ρ δρ 3 . (30)
Its evolution equation up to third order can be found by expanding Eq. ( 9). On large scales it reads [43] ζ (3) ≈ -H ρ + P Γ (3) -

1 ρ + P (Γ (1) ζ (2) + Γ (2) ζ (1) ) - 1 3H(ρ + P ) Γ(ζ ) 2 , (31)
where Γ (3) is defined in the same way as ζ (3) in Eqs. ( 29) and ( 30) with the replacements α → P, ζ → Γ.

In [42] one can find a discussion on perturbations beyond the third order, as well as a general expression for a quantity ζ (n) , defined at any order n, which is conserved on large scales for adiabatic perturbations.

The issue of gauge-invariance

One can check that, on large scales, ζ (2) behaves as a gauge-invariant quantity. Given a vector ξ a = n 1 n! ξ a (n) , the gauge transformation it generates is defined by the transformation law of tensors [6] (whose coordinate functions are a particular case),

T → e L ξ T. ( 32
)
Since ζ a vanishes at zeroth order, ζ a is automatically gauge-invariant at first order, according to the first expression above. However, ζ i is not gauge-invariant at second order and the corresponding gauge transformation is given by

ζ (2) i → ζ (2) i + L ξ (1) ζ (1) i ≈ ζ (2) i + ξ 0 (1) ∂ 0 ζ (1) i , (33)
where for the last equality we have neglected the terms of higher order in spatial gradients, concentrating on large scales. By noting that at first order δρ/ρ → δρ/ρ + ξ 0 (1) , it is easy to see using Eq. ( 24) that ∂ i ζ (2) , or equivalently ζ (2) , is gauge-invariant at second order, on large scales. This can also be checked by directly applying the gauge transformation (32) on ζ (2) defined in Eq. ( 25). This analysis can be repeated at third [43] or any higher order n [42] and one can show that the quantity ζ (n) is gauge invariant on large scales.

C. Dissipative and interacting fluids

It is possible to extend our covariant approach to the case of dissipative and interacting fluids, as discussed in detail in [24]. In that case, the energy-momentum tensor is more complicated,

T ab = ρu a u b + P h ab + q a u b + q b u a + π ab , (34)
where the energy flow q a and anisotropic stress π ab satisfy

q a u a = 0, π ab = π ba , π a a = 0, π ab u b = 0. ( 35
)
The continuity equation in this case reads

ρ + Θ(ρ + P ) = D, D ≡ -π ab σ ab + D a q a + 2q a a a , (36)
where D a denotes the spatially projected covariant derivative, orthogonal to the four-velocity u a . For a generic tensor T , the definition is (see e.g. [START_REF] Ellis | [END_REF])

D a T c... b... ≡ h d a h e b . . . h c f . . . ∇ d T f ... e... . (37)
As a consequence of the presence of the dissipative terms on the right-hand side of (36), the evolution equation for ζ a is slightly more involved and now reads

ζa = 3 α2 ρ (Γ a + Σ a ) , (38)
where

Σ a ≡ D a β - β ρ D a ρ , β ≡ - D Θ . (39)
This means that dissipative terms affect the evolution of the covector ζ a in addition to the non-adiabatic term.

III. COSMOLOGICAL SCALAR FIELDS

We now consider the situation where matter is composed of several scalar fields, concentrating on the case [25]. We study N scalar fields minimally coupled to gravity with Lagrangian density

L = - 1 2 ∂ a ϕ I ∂ a ϕ I -V (ϕ K ), ( 40
)
where V is the potential and the summation over the field indices (I, J, . . .) is implicit. For simplicity we assume here canonical kinetic terms but e will discuss below Lagrangians with non-canonical kinetic terms. The energy-momentum tensor derived from the above Lagrangian reads

T ab = ∂ a ϕ I ∂ b ϕ I - 1 2 g ab ∂ c ϕ I ∂ c ϕ I + 2V . (41)
Given an arbitrary unit timelike vector field u a , it is always possible to decompose this energy momentum tensor in the form (34) with, in our case,

ρ= 1 2 φI φI + D a ϕ I D a ϕ I + V, p = 1 2 φI φI - 1 3 D a ϕ I D a ϕ I -V, (42)
q a =-φI D a ϕ I , π ab = D a ϕ I D b ϕ I - 1 3 h ab D c ϕ I D c ϕ I . ( 43
)
The evolution equations for the scalar fields are the N Klein-Gordon equations given by

-∇ a ∇ a ϕ I + ∂V ∂ϕ I = φI + Θ φI + V ,ϕI -D a D a ϕ I -a a D a ϕ I = 0, ( 44
)
where the second expression is obtained by using a decomposition into (covariant) time-like and space-like gradients defined with respect to u a .

A. Adiabatic and entropy covectors

For simplicity we now restrict our analysis to the case of two scalar fields, which we will denote by φ and χ. In the two-field case it is possible to introduce a particular basis in the field space in which various field dependent quantities are decomposed into so-called adiabatic and entropy components. In the linear theory this decomposition was first introduced in [44] for two fields. This decomposition is particularly convenient to follow the time evolution of the curvature perturbation, which is sourced by the entropic perturbations. For the multi-field case it is discussed in [45] in the linear theory and in [46] in the nonlinear context.

In our case, the corresponding basis consists, in the two-dimensional field space, of a unit vector e I σ defined in the direction of the velocity of the two fields, and thus tangent to the trajectory in field space, and of a unit vector e I s defined along the direction orthogonal to it, namely

e I σ ≡ ( φ/ σ, χ/ σ) = (cos θ, sin θ) , e I s ≡ (-χ/ σ, φ/ σ) = (-sin θ, cos θ) , (45)
where we have introduced the formal notation σ ≡ ( φ2 + χ2 ) 1/2 . Note that this notation can be misleading as, in general, in the nonlinear context σ is not the derivative along u a of a scalar field σ.

The angle θ, which characterizes the rotation angle between the initial field basis and the adiabatic/entropic basis, is here an inhomogeneous quantity which depends on time and space. By taking the time derivative of the basis vectors e I σ and e I s , we get

ėI σ = θe I s , ėI s = -θe I σ . ( 46
)
Making use of the basis (45), one can then introduce two linear combinations of the scalar field gradients and thus define two covectors, respectively denoted by σ a and s a , as [25]

σ a ≡e I σ ∂ a ϕ I = cos θ ∂ a φ + sin θ ∂ a χ, (47)
s a ≡e I s ∂ a ϕ I = -sin θ ∂ a φ + cos θ ∂ a χ. (48)
We will call these two covectors the adiabatic and entropy covectors, respectively, by analogy with the similar definitions in the linear context [44]. Whereas the entropy covector s a is orthogonal to the four-velocity u a , i.e., u a s a = 0, this is not the case for σ a which contains a "longitudinal" component: u a σ a = σ. It is also useful to introduce the spatially projected version of ( 47) and ( 48),

σ ⊥ a ≡ e I σ D a ϕ I = σ a + σu a , s ⊥ a ≡ e I s D a ϕ I = s a . (49)
The "adiabatic" combination of the Klein-Gordon equations, i.e., the contraction of ( 44) by e I σ , yields

σ + Θ σ + V ,σ = ∇ a σ ⊥ a -Y (s) , (50)
where we have defined

V ,σ ≡ e I σ V ,ϕI , Y (s) ≡ 1 σ ( ṡa + θσ ⊥ a )s a . ( 51
)
Since e I s φI = θ σ, the "entropic" combination of the Klein-Gordon equations, i.e., the contraction of ( 44) with e I s , gives simply

σ θ + V ,s = ∇ a s a + Y (σ) , (52)
with

V ,s ≡ e I s V ,ϕI , Y (σ) ≡ 1 σ ( ṡa + θσ ⊥ a )σ ⊥ a . ( 53
)
In summary, the Klein-Gordon equations for the fields φ and χ have been replaced by the equations ( 50) and ( 52), whose left-hand side has exactly the same form as the homogeneous equations in a Friedmann universe. However, these equations capture the fully nonlinear dynamics of the scalar fields and their right-hand side contains nonlinear (quadratic) terms represented by Y (s) and Y (σ) , sourcing the adiabatic and entropy equations respectively.

We now consider the evolution equations for the covectors σ a and s a , or rather for their spatial projections. The adiabatic evolution equation is given by

(σ a ) ⊥ + Θ( σa ) ⊥ + σD a Θ + V ,σσ + θ V ,s σ σ ⊥ a -D a (∇ c σ ⊥ c ) = θ - V ,s σ ṡa + θ -V ,σs + Θ θ s a -D a Y (s) , (54)
where we have introduced the notation

V ,σσ ≡ e I σ e J σ V ,ϕI ϕJ , V ,ss ≡ e I s e J s V ,ϕI ϕJ , V ,sσ ≡ e I s e J σ V ,ϕI ϕJ , (55)
for the second derivatives of the potential. Note that the spatial projection and the time derivative do not commute for σ a . The entropic evolution equation reads sa -

1 σ (σ + V ,σ ) ṡa + (V ,ss -θ2 )s a -D a (∇ c s c ) = -2 θ( σa ) ⊥ + θ σ (σ + V ,σ ) -θ -V ,σs σ ⊥ a + D a Y (σ) , (56)
where the covectors ṡa and sa are purely spatial, i.e. ( ṡa ) ⊥ = ṡa and (s a ) ⊥ = sa .

Starting from the fully nonlinear Klein-Gordon equations, we have thus managed to obtain a system of two coupled equations ( 54) and (56), which govern the evolution of our nonlinear adiabatic and entropy components [25]. Remarkably, they are rather simple and they look very similar to the linear equations for the perturbations δσ and δs (see [44]). Furthermore, since these equations are exact, it is straightforward to expand them up to second or higher orders, as we will show explicitly in Sec. III D.

B. Generalized curvature perturbations and large-scale evolution

Let us now turn to the generalized curvature perturbation ζ a , which we introduced in Eq (10) for a cosmological fluid. Although a system of several scalar fields cannot be described as a perfect fluid, its energy-momentum tensor is of the form (34) with

ρ = 1 2 σ2 + Π + V, P = 1 2 σ2 - 1 3 Π -V, (57)
q a =-σσ ⊥ a , π ab = Π ab - 1 3 h ab Π, (58)
where we have defined

Π ab ≡ σ ⊥ a σ ⊥ b + s a s b , Π ≡ σ ⊥ c σ ⊥ c + s c s c . (59)
The quantity ζ a for the two-fluid system therefore satisfies an evolution equation of the form (38).

In the linear theory, an alternative quantity to describe the primordial perturbation is the comoving curvature perturbation. A natural nonlinear extension of this quantity in our formalism is (see also [47] for an alternative definition)

R a ≡ -D a α - α ( φJ φJ ) q a . (60)
As in the linear case, this expression can be directly related to the analog of the Mukhanov-Sasaki variables [48,49], if we generalize the latter by defining

Q I a ≡ D a ϕ I - φI α D a α (61)
for each scalar field. The comoving curvature perturbation R a is then proportional to the adiabatic combination of the Q I a , i.e.

Q a ≡ e σI Q I a = σ ⊥ a - σ α D a α, (62)
since one can write R a = α φI Q I a /( φJ φJ ). Comparing the definitions of ζ a and R a shows that they satisfy the relation

ζ a + R a = - α ρ a , a ≡ D a ρ - ρ σ σ ⊥ a , (63)
where the covector a can be interpreted as a covariant generalization of the comoving energy density perturbation.

Let us now concentrate on the large scale limit by neglecting terms of higher order in spatial gradients. Using the energy constraint u a G ab u b = 8πGρ and the momentum constraint u b G bc h c a = 8πGq a , it can be shown that the comoving energy density perturbation, a , defined in Eq. ( 63), can be neglected on large scales if the shear can also be neglected in this limit. This is the case in an expanding perturbed FLRW universe, where the shear rapidly decreases on large scales.

Then, from their definition (49), σ ⊥ a and s a are first-order quantities with respect to spatial gradients while the scalars Y (s) and Y (σ) are second order, since they are quadratic in σ ⊥ a and s a (or their time derivatives). Hence, the right-hand side of Eq. ( 50) and of Eq. ( 52) can be neglected on large scales. Taking into account these simplifications and neglecting on large scales a and the terms containing Π, one finds that the evolution equations in Eqs. ( 54) and ( 56) simplify to give

(σ a ) ⊥ + Θ( σa ) ⊥ + σD a Θ + (V ,σσ -θ2 )σ ⊥ a ≈ 2( θs a ) . -2 θ V ,σ σ s a , (64)
and

sa + Θ ṡa + (V ,ss + 3 θ2 )s a ≈ 0. ( 65
)
While the entropy mode appears as a source term in the adiabatic equation, the entropy equation involves only the covector s a . On large scales, neglecting a in Eq. ( 63) implies that the uniform density and comoving perturbations ζ a and R a coincide (up to a sign),

ζ a + R a ≈ 0. ( 66
)
Moreover, in the same limit, the source term in the evolution equation for ζ a , Eq. ( 38), is dominated by the gradient of the potential V along the entropic direction, so that ζa ≈ -2 3

Θ 2 ρ V ,s s a . ( 67
)
This equation is sufficient to describe the evolution of the adiabatic mode in the large scale limit, making Eq. ( 64) redundant. The analogs of the spatial components of Eqs. ( 67) and ( 65) have also been derived in [46].

C. Extended formalism for non-standard kinetic terms

In this subsection, we briefly discuss a large class of models described by Lagrangians with non-canonical terms of the form

L = P (X, φ I ) , X ≡ - 1 2 G IJ g ab ∇ a φ I ∇ b φ J , (68)
where G IJ ≡ G IJ (φ K ) is a metric defined in "field space", which can be used to raise and lower the field indices. The cosmological perturbations of these models, at first and second order, have been studied in [50] and [51], respectively. Their covariant nonlinear analysis has been developed in [52], from which we extract the results given here. These models can be seen as multi-field extensions of k-essence or k-inflation, although they are not as general as those studied in [53,54]. Instead of ( 43), the various components of the effective energy-momentum tensor for (68) now read ρ=P ,X φI φI -P ,

p = P + 1 3 P ,X G IJ D a φ I D a φ J , (69)
q a =-P ,X φI D a φ I , π ab = P ,X G IJ D a φ I D b φ J - h ab 3 G IJ D c φ I D c φ J , (70)
where P ,X denotes the derivative of the Lagrangian P with respect to the variable X.

It is then possible to proceed along the same lines discussed above and to introduce, in the two-field case, the adiabatic and entropy covectors σ a and s a , defined respectively along the directions e I σ and e I s , characterized by

e I σ ≡ φI σ σ ≡ G IJ φI φJ , G IJ e I s e J s = 1, G IJ e I s e J σ = 0 . (71)
The corresponding equations can be found in [52]. On large scales, the equations that generalize ( 67) and ( 65) are given by ζa ≈ -

Θ Ξ 3 σ s a , sa + Θ + Ṗ ,X P ,X ṡa + µ 2 s + Ξ 2 c 2 s s a ≈ 0 , (72)
where R I KLJ is the Riemann tensor associated to the metric G IJ . The above equations look very similar to their linearized versions given in [50], but they are here fully nonlinear. For standard kinetic terms P = X -V , and one can check that (72) reduces to ( 67) and ( 65), since c 2 s = 1, Ξ = -2V ,s / σ and µ 2 s = V ,ss -V 2 ,s / σ2 . In the more general case considered here, one sees that new effects arise from the entropic projection of the mixed derivative P ,XI , which appears in both µ 2 s and Ξ.

D. Link with the coordinate approach

Linear theory

We now relate our covariant approach with the more familiar coordinate based formalism, starting with the linear theory. We use the metric and notation of Sec. II B 1 and the scalar fields are decomposed into a background and a perturbed part, ϕ I (t, x i ) = φI (t) + δϕ I (t, x i ). We now need to specify the components of the unit vector u a , which defines the time direction in our covariant approach. In contrast with the perfect fluid case, where u a has a physical meaning, the unit vector u a is here arbitrary. In a given coordinate system it is convenient to choose u µ such that u i = 0 at first order. This implies that the components of u µ and of the "acceleration" vector are given respectively by

u µ = {1 -A, -B i /a}, a µ = {0, ∇ i A/a 2 } . ( 75
)
The background equations of motion can be deduced immediately from Eqs. ( 50) and ( 52) and read

σ + 3H σ + V,σ = 0, σ θ + V,s = 0. ( 76
)
From their definition, one finds that the spatial components of σ a and s a , at linear order, can be expressed as

δσ i = ∂ i δσ, δσ ≡ cos θ δφ + sin θ δχ, ( 77
)
δs i = ∂ i δs, δs ≡ cos θ δχ -sin θ δφ, (78)
which coincide with the notations of [44]. Linearizing the evolution equations for σ a and s a , eqs. ( 54) and ( 56), one easily obtains the linearized equations for δσ and δs. The latter can be written as

δs + 3Hδs + ( V,ss + 3 θ 2 )δs - 1 a 2 ∇ 2 δs = -2 θ σ δ , (79)
where on the right-hand side appears the first-order comoving energy density perturbation δ , defined by

δ i = ∂ i δ , δ ≡ δρ - ρ σ δσ, (80)
which follows from the definition (63) of a . The quantity δ can in general be neglected on large scales, since it is second order in spatial gradients, as can be seen from the relativistic Poisson-like equation.

Let us now turn to the adiabatic equation. Since δσ is not gauge-invariant, in contrast with δs, it is more useful to consider the gauge invariant Sasaki-Mukhanov variable Q SM , defined as [48,49]

Q SM ≡ δσ + σ H ψ. (81)
Note that the above traditional definition coincides with the scalar quantity Q that can be extracted from our definition of Q a given earlier in Eq. ( 62), via Q i = ∂ i Q, only in the large scale limit, because ψ and -δα coincide only in this limit. The evolution equation of Q SM reads [44,55]

Q SM + 3HQ SM + V,σσ -θ 2 -2 H H V,σ σ + H H - σ σ - ∇ 2 a 2 Q SM = 2( θ δs) -2 θ V,σ σ + H H δs. ( 82
)
On large scales the adiabatic evolution is simpler and governed by the first integral

Q SM + H H - σ σ Q SM -2 θ δs ≈ 0 , (83)
which simply expresses that δ , written in terms of Q SM and δs, is negligible on large scales. One can easily check that the large-scale limit of ( 82) follows from the first integral (83).

Let us now consider the evolution equation for ζ a . The spatial components of ζ a , at linear order, are given by eq. ( 18) as for the fluid case:

ζ i ≡ ∂ i ζ. Similarly, one can define R by R i = ∂ i R.
Note that R coincides with the familiar comoving curvature perturbation only in the large scale limit, because of the difference between α and ψ, as already discussed for ζ. Eq. ( 66) implies a simple relation between ζ and R,

ζ + R = - ᾱ ρ δ , (84)
which shows that ζ and -R coincide on large scales.

The evolution equation for ζ on large scales follows from eq. ( 67) which, after linearization, yields

ζ ≈ - 2H σ θ δs. (85)
One thus recovers the familiar linear result [44] that the entropy perturbation is sourcing the evolution of the uniform density curvature perturbation on large scales.

Second-order perturbations

As for the perfect fluid case in section II B 2, we now expand the equations governing σ a , s a and ζ a at second order in the perturbations. For simplicity, we will restrict ourselves to large scales and we will thus start from the equations expanded in spatial gradients discussed in Sec. III B.

The second-order evolution equations for the perturbations of a single scalar field, in a coordinate based approach, has been considered in several references (see for instance [11,12,56,57]). The multi-field case has been first studied by Malik in [58,59] and, using the separate universe approach, in [60,61]. Here, we obtain directly the second-order evolution equations in terms of the adiabatic and entropic perturbations.

By expanding Eq. ( 47) and Eq. ( 48) up to second order, one can write [25] δσ

i =∂ i δσ (2) + θ σ δσ∂ i δs - 1 σ V i , (2)
i =∂ i δs (2)

+ δσ σ ∂ i δs , (87)
with

δσ (2) ≡ φ σ δφ (2) + χ σ δχ (2) + 1 2σ δsδs , (88)
δs (2) ≡- χ σ δφ (2) + φ σ δχ (2) - δσ σ δs + θ 2 δσ , (89)
and where we have defined the spatial vector

V i ≡ 1 2 (δs∂ i δs -δs ∂ i δs) . (90)
The definition of δs (2) is chosen such that it is gauge invariant on large scales. Since the adiabatic component σ a does not vanish at zeroth order, δσ (2) is not a gauge invariant variable and can be chosen for convenience. Its definition here is such that the momentum perturbation almost vanishes (up to V i ) when δσ (2) = 0. The presence of the spatial vector V i on the right-hand side of Eq. ( 86) is due to the fact that σ a defined in Eq. ( 47) is not hypersurface orthogonal. Indeed, one can check using its definition that σ [a ∇ b σ c] = σ-1 σ [a ṡb s c] . As the momentum δq i is proportional to δσ ⊥ i , Eq. ( 58), this implies that if V i does not vanish one cannot define at second-order a comoving gauge, i.e., such that δq (1) i = 0 and δq (2) i = 0, in contrast with the linear theory or the single-field case [46]. However, the evolution equation for V i on large scales reads

V i + 3HV i ≈ 0, (91)
which implies that in an expanding universe V i decays as a -3 and rapidly become negligible even if it is nonzero initially. Consequently, in the following we will ignore V i on large scales. Since δσ (2) is not gauge-invariant, it is useful to consider the Sasaki-Mukhanov variable at second order Q

SM , which on large scales coincides with the scalar quantity Q (2) which can be extracted by expanding at second order Q a defined in Eq. ( 62),

Q (2) i = ∂ i Q (2) + δα H ∂ i Q + θ σ Q∂ i δs - 1 σ V i . (92)
The Sasaki-Mukhanov Q

SM can be found by replacing Eq. ( 26) in the definition of Q (2) , which yields

Q (2) SM ≡ δσ (2) + σ H (ψ (2) + ψ 2 ) + ψ H Q (1)
SM -

1 2 σ H ψ -θ δs . (93)
Restricted to a single scalar field, this definition coincides with the one given in [58]. By using the energy and momentum constraint equations at second order it is possible to derive a first integral for Q

SM [25], extending at second order Eq. ( 83).

Here we will simply derive the large-scale evolution equation for ζ at second order, which is enough to describe the evolution of the adiabatic mode. Expanding up to second order Eq. (67) yields

ζ (2) ≈ - H σ 2 2 θ σ δs (2) -V,ss + 4 θ 2 δs 2 + V,σ σ δsδs . (94)
It is also useful to express our results in terms of R a . The spatial components of R a can be decomposed as

R (2) i = ∂ i R (2) + δσ σ ∂ i R (1) - H σ 2 V i , (95)
with

R (2) ≡ -δα (2) + H σ δσ (2) + δσ σ -R (1) + 1 2 H σ δσ + θ H σ δs . (96)
The last term in Eq. ( 95) comes from the fact that, in contrast to ζ a , R a is defined in terms of the spatial momentum which cannot be expressed in general as a pure gradient. However, in an expanding universe this term can be neglected and R (2) coincides with the second-order comoving curvature perturbation defined in [57,62].

It is easy to derive a first-order (in time) evolution equation for R (2) by noting that ζ (2) and R (2) are related on large scales. Indeed, by using Eqs. ( 24) and (95), neglecting δ (1) and V i , and making use of ζ (1) + R (1) ≈ 0, one can show that the spatial component of Eq. (66) yields ζ (2) + R (2) ≈ 0 [57]. From this relation and the second-order evolution equation of ζ, Eq. (94), one can find a large-scale evolution equation for R at second order,

R (2) ≈ H σ 2 2 θ σ δs (2) -V,ss + 4 θ 2 δs 2 + V,σ σ δsδs . (97)
The second-order perturbation R (2) can be related on large scales to Q

SM by combining Eqs. ( 93) and (96) and using Eq. (26). One obtains

R (2) ≈ H σ Q (2) SM - 1 σ Q SM -θ δs Q SM - 1 2H H σ Q 2 SM , (98)
which can be used to show that Eq. ( 97) can be rewritten as a first integral for Q

SM . Let us now discuss the second order evolution of the entropy perturbation δs. On large scales this is obtained by simply expanding the spatial components of Eq. ( 65) up to second order, which gives δs (2) + 3Hδs (2)

+ V,ss + 3 θ 2 δs (2) ≈ - θ σ δs 2 - 2 σ θ + θ V,σ σ - 3 2 H θ δsδs - 1 2 V,sss -5 θ σ V,ss -9 θ 3 σ δs 2 .
(99)

As in the linear theory, the entropy perturbation evolves independently of the adiabatic component on large scales.

Third-order perturbations

It is straightforward to extend the same procedure at third order, as discussed in detail in [43]. The evolution equation for ζ (3) on large scales can be obtained by expanding up to third order Eq. ( 67), which yields an equation of the form

ζ (3) ≈ S (3)
ζ [δs, δs , δs (2) , δs (2) ],

(100

)
where the source term on the right-hand side contains terms cubic in first order perturbations (δs or δs ) and terms that are products of second order perturbations (δs (2) or δs (2) ) with first order perturbations (δs or δs ). The explicit definition of δs (3) used in this expansion is

δs (3) ≡ ēsI δφ I(3) - δσ (2) σ δs + θ δσ - δσ σ δs (2) - δσ 2 2σ 2 (δs - σ σ δs + θ 2 δs) - δσ 3 6σ θ σ - θ 2σ 2 δsδs δσ, (101)
which can be determined by an appropriate expansion of s a at third order, similar to the second order expansion (87).

To complete this equation, one needs to derive the second order (in time) equation of motion at third order on large scales. This is done by expanding Eq. ( 65) up to third order, which yields an equation of the form δs (3) + 3Hδs (3) + ( V,ss + 3 θ 2 )δs (3) ≈ S (3) s [δs, δs , δs (2) , δs (2) ].

The explicit expressions of the source terms S ζ and S s , which have been derived explicitly in [43], are given in the Appendix.

E. An application: super-Hubble generation of non-Gaussian perturbations

An important application of the above formalism is the computation of the non-Gaussianities of the primordial curvature perturbation. The computation of ζ and δs, by integration of their respective evolution equations up to the required order, can be seen as an alternative to the δN formalism, described in the article by Wands in this issue [63]. It is especially useful when the background evolution is complicated so that one cannot express analytically the number of e-folds as a function of the initial scalar field amplitudes.

Formally, one can write the solutions of the evolution equations as

ζ = ζ * + T (1)
ζ δs * + T

ζ δs 2 * + T

(3)

ζ δs 3 * + . . . , (103)
δs = T (1) s δs * + T (2) s δs 2 * + T (3) s δs 3 * + . . . , (104)
where the T Neglecting slow-roll corrections, the field perturbations Q SM * and δs * at Hubble crossing can be treated as Gaussian, with respective power spectra P Q * (k) and P δs * (k) (see Appendix of [64] for an explicit calculation of their 3-point functions). Note that there exist models with non-standard kinetic terms where the perturbations just after horizon crossing are non-Gaussian (see [START_REF] Langlois | Lectures on inflation and cosmological perturbations[END_REF] and the article of K. Koyama in this issue [65] for a general discussion on multi-field models of this type) and where the entropy perturbations can play a crucial rôle [66]. However, we will not consider these models here.

The perturbations Q SM * and δs * are also independent random fields, at least at leading order in slow-roll [67,68]. Moreover, although the relation between ζ and Q SM is in principle nonlinear (as illustrated for example by Eq. ( 98) for R which coincides with ζ on large scales), at horizon crossing the linear relation dominates in slow-roll models so that ζ * -(H/σ )Q SM * [64].

ζ k ζ k = (2π) 3 δ( k + k )P ζ (k), P ζ (k) ≡ (2 * M 2 P ) -1 P Q * (k) + (T (1) ζ ) 2 P δs * (k), (105)
ζ k1 ζ k2 ζ k3 2(T (1)
ζ ) 2 T

(2)

ζ [P δs * (k 1 )P δs * (k 2 ) + 2 cyclic] , (106)
ζ k1 ζ k2 ζ k3 ζ k4 4(T (1) ζ ) 2 (T (2)
ζ ) 2 [P δs * (k 1 )P δs * (k 2 )P δs * (k 13 ) + 11 perms] +6(T

(1) ζ ) 3 T (3) ζ [P δs * (k 1 )P δs * (k 2 )P δs * (k 3 ) + 3 cyclic] , (107)
where we have introduced the slow-roll parameter ≡ 4πGσ 2 /H 2 and M P ≡ (8πG) -1/2 . If the power spectrum is dominated by its entropy contribution, i.e. (T

ζ ) 2 P δs * (2 * M 2 P ) -1 P Q * , then one can easily relate the non-Gaussianities generated by the isocurvature field to the local nonlinear parameters f NL , τ NL and g NL , defined by [60,62,69,70]

ζ k1 ζ k2 ζ k3 = 6 5 f NL [P ζ (k 1 )P ζ (k 2 ) + 2 cyclic] , (108)
From Eqs. (106) and (107) one finds

f NL = 5 3 T (2) ζ (T (1) ζ ) 2 , τ NL = 4 (T (2) ζ ) 2 (T (1) ζ ) 4 , g NL = 25 9
T (3) ζ (T (1) ζ ) 3
.

These expressions can be applied to compute the non-Gaussianities generated in multifield models. For instance, in the case of two fields, for separable potentials -potentials that can be written as the sum or product of two functions dependent only on one of the fields, the transfer functions can be computed analytically either from the evolution equations for adiabatic and isocurvature perturbations derived in this section or by using the δN -formalism [61,71,72]. It has been shown that the non-Gaussianity generated after inflation are slow-roll suppressed in these cases (see also the review by Byrnes and Choi [73]). The non-linear evolution of the isocurvature field during inflation can generate a large non-vanishing 3 and 4-point functions [74,75], as reviewed by Bernardeau in this issue [76]. From Eq. (110), the conditions to obtain large nonlinear parameters are T 110), together with the evolution equations for adiabatic and isocurvature perturbations derived in this section, have also been applied to compute the non-Gaussianities generated during the new Ekpyrotic scenario [43,[77][78][79] with results that agree with those obtained using the δN -formalism [43,80].

A similar approach (although keeping the original scalar fields instead of a decomposition into adiabatic and entropic modes) has been used to compute the evolution of ζ up to second order, and thus obtain the non-Gaussianities, in the context of hybrid inflation [START_REF] Enqvist | Non-Gaussian perturbations in hybrid inflation[END_REF][82][83][84]. In these works the curvature perturbation at second order was expressed as a time integral of an expression quadractic in the linear perturbations of the tachyonic field. The same procedure can also been used to study the preheating phase at the end of hybrid inflation [83,84].

Isocurvature perturbations

It is possible, although not necessary, that the entropy perturbations generated during inflation survive after inflation, in the form of traditional entropy or isocurvature perturbations, i.e. fluctuations of the relative number of particles between two species (for instance, the number of photons per cold dark matter particle). In this case, the accessible cosmological data would result from a combination of curvature and isocurvature perturbations, which could be correlated [85]. So far, there has been no detection of an isocurvature perturbation in the CMB data and the constraints on the power spectra are already tight [86]. One could also envisage that an isocurvature component might be detectable through its own higher-order correlations or its higher-order correlations with the adiabatic component, as discussed in [64,[87][88][89][90]. Such a possibility can arise for instance in the mixed curvaton and inflaton scenario introduced in [91].

In the radiation dominated era the adiabatic perturbation coincides with the perturbation in the radiation fluid which, using Eq. ( 12) with w = 1/3, can be written as

ζ r a = ∂ a ζ, ζ ≡ δα + 1 4 ln ρ r ρr . ( 111
)
The isocurvature perturbation between the cold dark matter and radiation, say, can be defined as

Formally, one can write the relation between the entropy perturbation during inflation and the final isocurvature perturbation in the form

S = T (1)
S δs * + T

S δs 2 * + T

(3)

S δs 3 * + . . . , (113)
where the transfer functions T

S are given for instance in [64].

The observational upper limit on the fraction of isocurvature fluctuations allowed by data (see [86]) implies that the linear transfer coefficient T (1) S must be very small. Interestingly, however, T

(2) S and T

(3) S could still be large and the non-adiabatic perturbation detectable through its effect on higher-order correlation functions. Indeed, neglecting T (116) It would thus be interesting to obtain some observational constraints on the isocurvature non-Gaussianities from the future data, such as those collected by the Planck satellite.

APPENDIX: SOURCE TERMS AT THIRD ORDER

The source terms that appear on the right hand side of (100) and (102) are respectively given by [43]

S (3)

ζ [δs, δs , δs (2) , δs (2) ] ≡ -H σ 2 2 θ σ δs (3) -2( V,ss + 4 θ 2 )δsδs (117) and S (3) s [δs, δs , δs (2) , δs (2) ] = -2 θ σ δs δs (2) -2 σ θ + θ V,σ σ -3 2 H θ (δsδs (2) ) -V,sss -10 θ V,ss σ -18 (118)

  functions, for ζ and δs, respectively, at the n-th order and ζ * , δs * are their initial conditions at Hubble exit. In particular, we are interested in the values of ζ and δs after inflation, at the onset of the radiation dominated era.

1 .

 1 Curvature perturbations Using Eqs. (103) and (104) one finds that the final 2-point, 3-point and 4-point functions of ζ, which are in principle observable, are given respectively by

ζ

  k1 ζ k2 ζ k3 ζ k4 = τ NL [P ζ (k 1 )P ζ (k 2 )P ζ (k 13 ) + 11 perms] + 54 25 g NL [P ζ (k 1 )P ζ (k 2 )P ζ (k 3 ) + 3 cyclic] .

S a = 3

 3 (ζ c a -ζ r a ) = ∂ a S,

  details of the model. Analytically solvable examples of T

( 1 )SS ) 2 PSS

 12 S , one finds that the non-vanishing 3 and 4-point correlation functions are given byζ k1 ζ k2 S k3 2P δs * (k 1 )P δs * (k 2 ),(114)ζ k1 ζ k2 S k3 S k4 4δs * (k 1 )P δs * (k 2 ) [P δs * (k 13 ) + P δs * (k 23 )] , (115) ζ k1 ζ k2 ζ k3 S k4 4{P δs * (k 1 )P δs * (k 2 ) [P δs * (k 13 ) + P δs * (k 23 )] + 2 cyclic(1, 2, P δs * (k 1 )P δs * (k 2 )P δs * (k 3 ).

4 σ 2

 42 δs 3 .


! " # $ % &' ( ( ") ) ) ') * + , " - . / ) $ " # % ! " " )0 1 $ 1 " )! )0 ,$ )0 # % ! " ( 2 1 --1 " )! 3 ! " 4 4 ! " # $ % " # # $ # % & !)! !) / * 3( ) 5 (6 3( ) 2 / " ' ! " #$ %#$ & ! '$$ " ( ! )* *+) ,-+ . . . ! ) / 0 / 0 & & & 1 " ! + ! ( )* ' !

+

The world-class Imiter mine is the 10 th largest worldwide and the largest African deposit with ~210 metric tons of silver produced in 2009. Since 1960, the total production is estimated at ~4,300 t of Ag. The deposit's total reserves are approximately 8,000 t of Ag and a 30 year mine life is estimated. Ore grades range between 500 and 700 g/t. Mercury is exploited as a byproduct of silver, but no production data are available regarding this metal. The deposit, located at the northern edge of the West African craton (Fig. 1A), belongs to the Moroccan Pan-African chain (Neoproterozoic orogeny: 750 to 540 Ma). The Imiter deposit is assumed to be a late Neoproterozoic epithermal vein deposit, formed during regional-scale strike slip faulting and is associated with felsic volcanism dated at 550 ± 3 Ma by ion-probe U/Pb on zircons (Cheilletz et al. 2002;Tuduri et al. 2006).

To date, the supergene enrichment of the silver-rich

Imiter deposit has been poorly documented, even though it is being exploited. Few details exist in mining reports but one can note interesting grades related to secondary enrichments that can reach 400 to 500 g/t with locally more than 300 kg/t. In this study, the processes of supergene alteration of the hypogene ore were evaluated, with an emphasis on the deeper part of the underground mine. Eventually, such supergene conditions have been modelled using rare earth elements (REE) as proxies of low temperature processes.

, $ ) ) #

The primary hypogene ore is mainly related to the E-W 10-km long Imiter fault system that separates Cryogenian black shales to the south from Ediacaran plutonic, volcanic and volcaniclastic rocks to the north (Fig. 1B).

. / #

The supergene enrichment zone depth reaches 50 to 150 m below the surface and is vertically zoned. This supergene part has been intensively mined since the discovery of the deposit and is still exploited. The Imiter deposit is herein considered as a pyrite-poor deposit type (Sillitoe 2009) at least in the first 200 m of exploitation below the surface.

The main supergene alteration displays a generalized downward progression with ( ) massive Ag 0 veins (up to 300 kg/t, from surface to 50-75 m depth), Ag-rich galena and Ag-rich sphalerite and related Mn oxides, and ( ) powdery black and rusty materials in the deeper part, from 50-75 m and up to 150 m below the surface (400-500 g/t). Such material was studied and determined by metallographic observations, SEM-EDS and XRD analyses. The alteration of the hypogene mineralization consists of cerussite, mimetite, acanthite, cinnabar, perroudite/imiterite and various Ag-Hg arsenates, coronadite (PbMn 8 O 16 ), synchisite (Ca(Ce,La)(CO 3 ) 2 F), hematite and goethite that can be attributed to two main alteration stages: ( ) the former is characterized by the crystallization of supergene sulfides associated with carbonates and arsenates; ( ) the latter is mainly oxidic (Fig. 2). According to XRD data, quartz, illite and kaolinite are predominant in the oxidized altered rocks. Dolomite occurs only in the least oxidized varieties.

) #

The first alteration type (sulfidic) is defined by a redistribution and local enrichment by accumulation of secondary silver phases. Arsenopyrite, primary acanthite and Ag-Hg alloys are not preserved. Thus Ag, As, Hg and Pb are solubilized and further incorporated into new minerals depending on geochemical conditions (e.g., pe, pH). In such cases, galena may be replaced by acanthite and cerussite, or only by cerussite. According to SEM and XRD analysis, imiterite and perroudite (Hg 5 Ag 4 S 5 (Cl,I,Br) 4 ) are also present. Generally cerussite is well developed until formation fluids favor mimetite precipitation. At this moment, cerussite and mimetite may display complex intergrowths. Then mimetite occurs as the common crystal and free growth textures of euhedral prismatic to acicular crystals may be observed. Scarce inclusions of acanthite and cinnabar have also been observed within mimetite (Fig. 2).

The second alteration type occurs as overgrowths on the first alteration type of oxidized minerals such as iron oxyhydroxides, rare earth element fluorocarbonates and Pb-Mn oxides. It generally consists of syncrystallization of a goethite-hematite mixture with synchisite overgrown by fibrous botryoidal aggregate textures of complex manganates such as coronadite (Fig. 3). Mimetite may also occur in the earlier oxidic processes associated with iron oxyhydroxides.

-

.' SEM backscattered electron image showing syncrystallization of iron oxyhydroxide (feox), with synchisite (sch) overgrown by botryoidal coronadite (cor).

2

Both alteration processes described above are mainly oxidizing phenomena (i.e., increasing (O 2 ) and decreasing (S)) that are superimposed on pre-existing primary mineralization. The oxidation of sulfide releases dissolved Ag, Hg, S and Pb into formation fluids. Subsequently, the dissolved species undergo oxygenation and/or complexation by other ligands, which then concentrate them to form new insoluble phases. The secondary concentration observed in the first alteration type documented in this study, results in a zonation from sulfide to sulfohalide, probably in response to an increasing oxidation state (Gammons and Yu 1997). Thus, the replacement of galena by acanthite suggests neutral to alkaline pH values, a relatively low oxidation state and high total sulfide (hatched area, Fig. 4). Moreover, release of S, Ag, Pb, As and Hg during sulfide alteration (i.e., acanthite and primary minerals), probably influenced by halide-rich solutions, provides the necessary ingredients for imiterite/perroudite crystallization.

-0' pe-pH stability diagram for Ag, Hg, Pb minerals and dissolved species of Ag-Hg-Pb-Cl-S-C at 25°C. Arrow shows pe-pH trend for supergene alteration stages.

3

The weathering process has resulted in the mobilization and redistribution of the REE affected by the nature of secondary mineral formation (e.g., Mn oxides, fluorocarbonates). The presence of aqueous ligands, notably fluoride, carbonate or hydroxide in water strongly affects REE solubility. Indeed, formation of secondary REE-minerals (i.e., synchisite) can give some clues for deciphering the probable ligands present in the supergene ore-forming fluid. In order to establish the supergene conditions, REE speciation in two multiligand solutions (reduced seawater and F-rich groundwater) was calculated. Results show that ( ) in the reduced seawater at near-neutral pH, lanthanides exist as free ionic, carbonate and chloride species; ( ) in Fenriched groundwater, lanthanides occur as free ionic, carbonate and fluoride complexes. While there is no indication for the presence of alkaline fluids, there is evidence for the presence of F-bearing fluids. Indeed, the precipitation of F-REE-bearing synchisite suggests the presence of F and CO 2 in the aqueous fluid.

Because the Ag enrichment zone occurs in response to chloride-rich, neutral to alkaline pH and reduced formation fluid circulation, two alternative ore-forming models are proposed below: (i) deep Phanerozoic basinal brines channelled within Precambrian basement fractures influence primary ore destabilization during the long and complex history of the south-Moroccan area; (ii) downward infiltration of surficial water such as modified rainwater are progressively reduced in response to fluidrock interaction (F-rich groundwater resulting from interactions with volcano-sedimentary host-rocks). Organic matter encountered within black shales hosting the mineralization (Pašava 1994) would participate in redox processes. The paragenetic evolution from mimetite, Ln-fluoro-carbonate and iron oxyhydroxide to coronadite (Fig. 3), observed in the second alteration type, implies a drastic increase of the oxidation states with probable neutral to alkaline pH preservation conditions and reinforces a continuum model of supergene weathering (arrow on Fig. 4).

In conclusion, supergene processes provide Ag-rich sulfide concentration in response to saline groundwater circulations. This suggests that supergene enrichment processes of Ag deposits in the Imiter mine are dynamic in nature and do not exclusively require the presence of meteoric water. As compared to other ore deposits (e.

  11th Biennal meeting SGA 2011, Sep 2011, Antofagasta, Chile. pp.982. hal-00601984

  map of the West African Craton, B) Simplified geological map of the Imiter silver mine. The Imiter fault consists of the association between N090°E and N060-070°E lineament arrays and defines a succession of apparent left-lateral pull-apart textures at district scale (Fig. 1B). According to Tuduri et al. (2006) a simplified two stage hypogene model is herein proposed. ( ) Stage one would correspond to the silver-rich low sulfidation epithermal deposit model. Quartz is the primary gangue, associated with minor dolomite, chlorite, sericite and scarce adularia. Metallic ore (Fig. 2) is mainly composed of Ag-Hg alloys, arsenopyrite, acanthite-argentite, Ag sulfosalts and imiterite (Ag 2 HgS 2 ). Along N060-070°E structures, large open voids are observed, as a result of important dissolution of carbonates during hydrothermal alteration. ( ) Stage two corresponds to a reactivation of the Imiter fault system. As a consequence, this stage reopened and reused stage 1 structures along the shear-zone. Structures correspond to veins filled with massive pink dolomite (Σ Fe+Mg ≈0.1 atoms per formula unit). Scarce quartz and a sulfide-rich paragenesis characterize this stage. Sulfides are mainly Ag-rich galena and sphalerite, pyrite, chalcopyrite, with minor amounts of argentite and arsenopyrite (Fig. 2). -,' Paragenetic diagram of Imiter hypogene vein filling and supergene alteration minerals, earliest formed minerals to the left.

  g., Cu, Reich et al. 2008; Ag, Sillitoe 2009), development of such mineralization suggests prevalence of arid to semiarid conditions during the supergene enrichment. Processes controlling the supergene products of the giant Imiter silver deposit belong to the pyrite-poor deposit type and are controlled by neutral pH solutions (Sillitoe 2009). Cheilletz A, Levresse G, Gasquet D, Azizi-Samir M, Zyadi R, Archibald DA, Farrar E (2002) The giant Imiter silver deposit: Neoproterozoic epithermal mineralization in the Anti-Atlas, Morocco. Miner Deposita 37:772-781 Gammons CH, Yu Y (1997) The stability of aqueous silver bromide and iodide complexes at 25-300°C: Experiments, theory and geologic applications. Chem Geol 137:155-173 Pašava J (1994) Geochemistry and the role of anoxic sediments in the origin of the Imiter silver deposit in Morocco. Vĕstnik Českého geologického ústavu 69, pp 1-12 Reich M, Palacios C, Parada MA, Fehn U, Cameron EM, Leybourne MI, Zuniga A (2008) Atacamite formation by deep saline waters in copper deposits from the Atacama Desert, Chile: evidence from fluid inclusions, groundwater geochemistry, TEM, and 36 Cl data. Miner Deposita 43:663-675 Sillitoe RH (2009) Supergene silver enrichment reassessed. In: Titley SR (ed) Supergene environments, processes and products, Soc Econ Geol Spec Publ 14:15-32 Tuduri J, Chauvet A, Ennaciri A, Barbanson L (2006) Modèle de formation du gisement d'argent d'Imiter (Anti-Atlas oriental, Maroc). Nouveaux apports de l'analyse structurale et minéralogique. Comptes Rendus Geoscience 338:253-261


Enrique Henestroza Anguiano

Pascal Denis
email: pascal.denis@inria.fr

Fredist

Enrique Henestroza
email: henestro@inria.fr

Anguiano Pascal

Denis Alpage

FreDist: Automatic construction of distributional thesauri for French

Keywords: thésaurus distributionnel, similarité sémantique, méthodes non supervisées, lexique distributional thesaurus, semantic similarity, unsupervised methods, lexicon

come

Introduction

We present FreDist, software that implements methods for the automatic construction of distributional thesauri. Distributional lexical resources are appealing because they can be constructed automatically from raw text corpora, and are useful for alleviating data sparseness in many NLP applications (e.g. parsing and coreference resolution). Moreover, we believe that open software like FreDist can be useful to the NLP community by providing an easy way to generate distributional thesauri from any text corpus using adjustable settings.

We base our work on that of [START_REF] Lin | Automatic retrieval and clustering of similar words[END_REF], which uses word context relations to calculate lexical distributional similarity, and the subsequent work of [START_REF] Curran | From distributional to semantic similarity[END_REF], which distinguishes between weight and measure functions and evaluates different functions on a semantic similarity task for English. We build on their work by considering the joint use of different types of context relations, and evaluating distributional similarity metrics for French.

Current lexical resources for French that have been semi-automatically created include the work of [START_REF] Sagot | The lefff, a freely available, accurate and large-coverage lexicon for french[END_REF] on the Lefff , a large-coverage morphosyntactic lexicon, and [START_REF] Sagot | Building a free french wordnet from multilingual resources[END_REF] on the WOLF, a semantic resource based on the Princeton WordNet. Our work differs by providing a fully automatic approach to the creation of a lexical resource. Previous work on distributional methods for French includes that of [START_REF] Bourigault | Upery : un outil d'analyse distributionnelle étendue pour la construction d'ontologies à partir de corpus[END_REF] on UPERY, a distributional analysis module that calculates proximities between words and their contexts, and the work of [START_REF] Ferret | Discovering word senses from a network of lexical cooccurrences[END_REF], which uses distributional similarity to build word senses from a network of lexical cooccurrences. Our work differs by focusing on the construction and evaluation of distributional thesauri, combining different types of context relations, and making FreDist and our best distributional thesauri freely-available. 1 véhicule voiture-0.546, camion-0.401, engin-0.301, camionnette-0.291, bus-0.276, fourgon-0.271, avion-0.269, appareil-0.254, tracteur-0.249, moto-0.248, fourgonnette-0.242, train-0.234, automobile-0.234, bateau-0.227, scooter-0.225, matériel-0.225, berline-0.224 ... tragique dramatique-0.331, cruel-0.269, douloureux-0.260, terrible-0.237, triste-0.230, sanglant-0.188, traumatisant-0.179, fatal-0.174, funeste-0.173, regrettable-0.173, effroyable-0.171, fâcheux-0.162, spectaculaire-0.160, violent-0.159, étrange-0.159, drôle-0.158, inquiétant-0.155 ... Section 2 explains distributional lexical methods, Section 3 describes FreDist and the construction of our distributional thesauri, Section 4 evaluates different distributional similarity metrics, and we conclude in Section 5.

Distributional methods

The distributional hypothesis states that words occurring in the same contexts tend to have similar meanings, as posited by [START_REF] Harris | Distributional structure[END_REF]. We focus on methods that generate distributional thesauri from a large collection of lexical terms and contexts. An entry in a distributional thesaurus contains, for each lexical term, a list of neighboring terms ordered by similarity. Example entries are shown in Figure 1.

Context relations

Basing our terminology on the work of [START_REF] Lin | Automatic retrieval and clustering of similar words[END_REF]) and of [START_REF] Curran | From distributional to semantic similarity[END_REF], we define a context relation as the tuple (w, r, w ′ ), where w is a primary lexical term (we use lemmas) that occurs in a particular context ; in our work, contexts consist of a relation r and a secondary lexical term w ′ . Commonly-used contexts for w include syntactic dependencies, fixed-size windows (such as bigrams), and bag-of-words representations of documents.

The choice of context dictates the semantic relationship obtained between primary lexical terms : [START_REF] Agirre | A study on similarity and relatedness using distributional and wordnet-based approaches[END_REF] find that syntactic dependencies and fixed-size windows best capture semantic similarity (synonymy and hypernymy/hyponymy), while bag-of-words approaches capture broader semantic relatedness (particularly shared topic). We use syntactic dependencies and linear bigrams, since we are interested in semantic similarity. If a relation r is bigram, it can take values of either -1 or +1, indicating that w ′ appears either before or after w. If r is syntactic, it can take values of either gov or dep, indicating that w either governs w ′ or is a dependent of w ′ : e.g. the context relation (aboyer, gov, chien). After context relations are extracted from a corpus, each primary lexical term w can be represented as a frequency vector v w ∈ R d , where d is the number of unique contexts appearing in the corpus, and v w i = f req(w, r, w ′ ), where i corresponds to the context c i = (r, w ′ ).

Term similarity

Term similarity metrics are used to calculate similarities between pairs of primary lexical terms w 1 and w 2 using frequency vectors v w1 and v w2 . [START_REF] Curran | From distributional to semantic similarity[END_REF] breaks term similarity metrics down into two components : a weight function transforms the raw frequency of each context relation by determining the informativeness of the context, while a measure function subsequently calculates the similarity between two weighted frequency vectors.

We experiment with the following weight functions for a context relation (w, r, w ′ ) : relative frequency (RELFREQ), which normalizes the frequency of (w, r, w ′ ) with respect to the frequency of its primary lexical term w ; t-test (TTEST), where the null hypothesis states that the primary lexical term w and the context (r, w ′ ) are independent, and the test compares their product distribution to their observed joint distribution ; and the pointwise mutual information function (PMI), which calculates the mutual information between w and (r, w ′ ).

We experiment with the following measure functions for a pair of primary lexical terms w 1 and w 2 and their respective weighted vectors v w1 and v w2 : cosine similarity (COSINE), which measures the cosine of the angle between v w1 and v w2 ; the Jaccard measure (JACCARD), which compares the number of common contexts to the number of unique contexts between w 1 and w 2 ; and the similarity measure that [START_REF] Lin | Automatic retrieval and clustering of similar words[END_REF]) uses (LIN), which is an information theoretic measure to determine the similarity between w 1 and w 2 . The formulas for the weight and measure functions used in our experiments are listed in Tables 1 and2. The symbol * as an argument to a function is shorthand for taking the sum of the function over all possible values for that argument ; p denotes the probability of a context relation, where p(w, r, w ′ ) is estimated as f (w, r, w ′ )/f ( * , * , * ) ; and wgt denotes the application of some weight function to a context relation count.

f (w,r,w ′ ) f (w, * , * ) TTEST p(w,r,w ′ )-p( * ,r,w ′ )p(w, * , * ) √ p(w,r,w ′ )/f ( * , * , * ) PMI log f (w,r,w ′ ) f ( * ,r,w ′ )f (w, * , * )

Resource construction

In order to generate distributional thesauri for French, we introduce the FreDist package for the Python programming language. FreDist provides functionality for each step of distributional thesauri construction : (1) Extraction of context relations from a text corpus in CONLL 2 format ; (2) Weighting of context relations according to a specified weight function ;

(3) Generation of a similarity matrix for primary lexical terms according to a specified measure function ; (4) Construction of a distributional thesaurus from a similarity matrix. FreDist is highly flexible, with parameters including : context relation type(s), weight function, measure function, term frequency thresholding, part-of-speech (POS) restrictions, filtering of numerical terms, etc.

We now discuss the steps we took to build our distributional thesauri for French. We first describe the preprocessing used to tag and parse our chosen raw corpus. We then describe the settings we used for FreDist to extract context relations and perform similarity calculation.

Corpus preprocessing

We chose to use the freely-available L'Est Républicain corpus, which contains 125 million words of French journalistic text. The corpus was first preprocessed using simple tokenization and sentence segmentation tools. POS tagging was then conducted using MElt 3 , a freely-available POS tagger for French. Subsequently, we performed lemmatization and morphological analysis using the previously mentioned Lefff 4 : the Lefff was queried with a word+POS pair to obtain a corresponding lemma (in case of ambiguity, the first lemma was chosen) as well as a set of morphological features. The corpus was then parsed with the MaltParser 5 , a fast and highly accurate system for data-driven dependency parsing.

Both MElt and MaltParser were trained on the standard training section of the French Dependency Treebank (FTB) as described in [START_REF] Candito | Statistical french dependency parsing : Treebank conversion and first results[END_REF], which is based on the original French Treebank with constituent structure (Abeillé & [START_REF] Barrier | Enriching a french treebank[END_REF]. The FTB contains a total of 12,531 sentences from the Le Monde newspaper, and the annotation scheme contains 28 unique tags in its POS tagset. On the FTB development set, MElt obtains 97.7% tagging accuracy and MaltParser obtains 89.3% unlabeled attachment accuracy.

Context relation extraction

Once the corpus had been automatically annotated with lemmas, POS tags, and syntactic dependency relations between words, we extracted context relations. Although the base lexical term used in distributional lexical methods is often the inflected form or the lemma, we chose to use a base lexical term consisting of lemma+POS to distinguish between homonyms (as in dîner+noun vs. dîner+verb) within contexts.

We extracted context relations exclusively for primary lexical terms w that had a POS tag of adjective, adverb, common noun, or verb, and that appeared at least 100 times in the corpus. A frequency threshold of 100 is often used in the literature, and is applied because distributional similarity methods are known to suffer degraded performance for terms that appear infrequently in a corpus [START_REF] Gorman | Scaling distributional similarity to large corpora[END_REF]. Secondary lexical terms w ′ were also subject to the POS tag restriction, and contexts (r, w ′ ) were subject to the frequency thresholding. Additionally, lexical terms containing numbers were replaced with a num token. Primary lexical terms above the frequency threshold totaled 4,126 adjectives, 802 adverbs, 10,997 common nouns, and 3,562 verbs.

Bigram context relations were generated in a straightforward manner. For each token of a primary lexical term w in the corpus, we placed it in the relation -1 with the preceding token's lexical term (or a generic term beg if w was the first token in a sentence) and in the relation +1 with the subsequent token's lexical term (or a generic term end if w was the last token in a sentence). Unique bigram contexts above the frequency threshold totaled 6,680 for adjectives, 3,935 for adverbs, 6,134 for common nouns, and 6,436 for verbs.

Syntactic context relations were extracted in accordance with the FTB annotation style. For each occurrence of a primary lexical term w in the corpus, we placed it in a relation r with its governing lexical term (or a generic term root if w rooted a dependency tree), where r is the dependent relation dep. Then for each dependent of w, we placed w in a relation r with that dependent, where r is the governor relation gov. 6 Some dependencies were collapsed : for prepositional/coordinating phrases, we chose to fold the preposition/conjunction into r in order to include both the head of the prepositional/coordinated phrase and the head of the modified/preceding phrase. Some dependencies were ignored : none involving a punctuation mark was included, due to the underspecification of punctuation attachment in the FTB. Unique syntactic contexts above the frequency threshold totaled 6,389 for adjectives, 3,141 for adverbs, 33,881 for common nouns, and 25,624 for verbs.

Note the particularly large number of syntactic contexts extracted for common nouns and verbs. This is perhaps indicative of participation in rich and varied long-range dependencies for common nouns and verbs, as opposed to more local dependencies for adjectives and adverbs that can be largely accounted for with bigram contexts.

Similarity calculation

For each of the four relevant POS categories, we applied weight and measure functions to its collection of context relations and pairs of primary lexical terms. The resulting thesauri provided the basis for our evaluation. Since we wanted to test each combination of weight and measure functions, as well as three settings for context relations (bigram, syntactic, bigram+syntactic), we generated in total 27 test thesauri for each relevant POS category.

Evaluation

In order to evaluate our distributional thesauri, we used two wordnets for French as references : the French Euro WordNet (FREWN) 7 , which is manually validated, and the WOLF 8 , which is not manually validated. While the FREWN covers only verbs and common nouns, the WOLF covers all four relevant POS categories. We thus evaluated verbs and common nouns using FREWN, and adjectives and adverbs using WOLF.

During evaluation, we considered only those primary lexical terms appearing in both the distributional thesauri and the wordnet reference. This reduction was carried out in order to prevent unnecessary penalization of the thesauri due to potential incompleteness of the wordnet references. This gave us 3,018 common nouns and 1,426 verbs for the FREWN evaluation, and 374 adjectives and 195 adverbs for the WOLF evaluation.

6. Dependency labels were not used in our evaluation for computational efficiency reasons, but the option is available in FreDist. For our evaluation metric, we chose to use average inverse rank (INVR), a standard information retrieval metric.

For each term w, we considered all terms appearing in a synset with w in the wordnet reference to be relevant, while other terms were considered irrelevant. In the distributional thesaurus to be evaluated, the entry for w was treated as a ranked list of query results (neighbor terms ranked by descending similarity). The INVR metric returns the sum, over relevant neighboring terms, of the inverse of that term's rank in the list. The average INVR is taken over all terms to be evaluated, providing an evaluation metric for the quality of a distributional thesaurus. One downside to this evaluation approach is that we ended up evaluating our distributional thesaurus using strict synonymy, which ignores pairs of words that may be otherwise semantically similar. Due to the scarcity of appropriate and/or manually validated resources for evaluating distributional thesauri, we believe that a synset-based evaluation is nonetheless useful and allows for a comparison of the relative quality of different thesauri settings.

The top five scoring distributional thesauri for each wordnet+POS pairing are shown in Table 3. Our primary finding is that the combination of bigram+syntactic context relations, PMI weight function, and COSINE measure function consistently produces the best thesauri across all four POS categories (though the results were not tested for significance). The finding that PMI and COSINE outperform other combinations is consistent with a recent comparison of distributional similarity metrics for English [START_REF] Ferret | Similarité sémantique et extraction de synonymes à partir de corpus[END_REF], although in that work a TOEFL test was used for evaluation. A possible explanation for PMI weight function performing best is that RELFREQ does not weight contexts on the basis of informativeness, and TTEST has the downside of erroneously assuming normal distributions for the probabilities of particular context relations appearing in a corpus [START_REF] Church | Introduction to the special issue on computational linguistics using large corpora[END_REF]. COSINE simply appears to work better empirically than the other measure functions. The finding that bigram and syntactic contexts are most effective when used together is remarkable : to our knowledge, the combination of different types of contexts has not been explored before. A possible explanation is that bigram and syntactic contexts provide different views of the distribution of lexical terms, resulting in better informed similarity estimates.

Finally, we also looked at the running time of FreDist. We used the best setting (bigram+syntactic contexts, PMI weight function, and COSINE measure function), starting from a parsed corpus and ending with distributional thesauri for the four relevant POS categories. When the input contained 1/2 of the L'Est Républicain corpus (62.5 million words) the cpu time was just under 8 hours. For the full corpus (125 million words) the cpu time was just over 18 hours. Trials were run on a Linux machine with a 2.4GHz processor and 8GB of memory.

Conclusion

We have presented FreDist, a freely-available software package that implements lexical distributional methods for the automatic construction of distributional thesauri from text corpora. FreDist is highly customizable and can be used with any type of text corpus in a number of different languages. Its running time is currently acceptable for medium-sized corpora of up to a few hundred million words (with the particular settings we used in our evaluation), but we hope to optimize the code in future versions in order to handle larger corpora of 1 billion words or more. In addition to the software, we are also releasing distributional thesauri for French that were created using the best settings from our evaluation.

A second goal of this work was to evaluate distributional methods on French data, and we have obtained results similar to those of past work on English : our finding that the PMI weight function and COSINE measure function work best for French mirrors the results of [START_REF] Ferret | Similarité sémantique et extraction de synonymes à partir de corpus[END_REF] for English. We also experimented with a novel approach that involved joint consideration of bigram and syntactic context relations, and found that it works better than either type of context relation on its own. Given this interesting result, an avenue for future work might be to include additional context relation types, such as document or paragraph co-occurrence, and determine if they can help further improve the quality of our distributional thesauri.

FIGURE 1 -

 1 FIGURE 1 -Distributional thesaurus entries for the noun véhicule and the adjective tragique.

  COSINE wgt(w1, * r, * w ′ )×wgt(w2, * r, * w ′ ) √ wgt(w1, * , * ) 2 × wgt(w2, * , * ) 2 JACCARD min(wgt(w1, * r, * w ′ ),wgt(w2, * r, * w ′ )) max(wgt(w1, * r, * w ′ ),wgt(w2, * r, * w ′ )) LIN wgt(w1, * r, * w ′ )+wgt(w2, * r, * w ′ ) wgt(w1, * , * )+ wgt(w2, * , * )

TABLE 1 -

 1 Weight functions

TABLE 2 -

 2 Measure functions

TABLE 3 -

 3 Average INVR evaluation scores for the top 10 distributional thesauri (out of 27) by POS category. Each setting name is a combination of the context relation type (bigram, syntactic, or bigram+syntactic), weight function (RELFREQ, TTEST, or PMI), and measure function (COSINE, JACCARD, or LIN).

http://alpage.inria.fr/~henestro/fredist.html

Acknowledgements

We would like to thank Marie Candito for helpful discussions. This work was partially funded by the ANR project Sequoia ANR-08-EMER-013.


Annie Simon
email: simon@icmcb-bordeaux.cnrs.fr

Jean Ravez

NEW LEAD-FREE PEROVSKITE RELAXOR CERAMICS DERIVED FROM BaTiO 3 AND CONTAINING SCANDIUM

Keywords: lead-free, perovskite, relaxor, BaTiO 3, scandium

New lead-free compositions of ABO 3 perovskite type were synthesized in the BaTi 1-x (Nb 1/2 Sc 1/2 ) x O 3 (0 ≤ x ≤ 1) system, with regards to the comparable electronic configurations of the various 6coordinated cations. Special solid state reaction was used. The effect of the replacement of titanium by scandium and niobium in the B-site on structural and dielectric properties was investigated. For this, we have used X-ray diffraction and dielectric characterizations. Dielectric measurements performed on ceramics with 0 ≤ x ≤ 0.25 composition revealed that the small change in composition from BaTiO 3 has a strong influence on the dielectric properties: fast decreasing of the ferroelectric transition temperature for low values of x and evolution from classical ferroelectric for 0 ≤ x ≤ 0.075 to relaxor behavior for 0.075 < x ≤ 0.25.

Introduction

In AMO 3 -type perovskites, relaxor behavior mainly occurs in lead-based compositions (PMN, PSN, PIN, PLZT, etc…) with more than one type of ion occupying the equivalent 6-coordination number (C.N.) crystallographic site. 1 Lead-free compositions could be of great interest for environment-friendly applications (dielectric for capacitors, actuators, etc…). Studies have therefore been performed in detail on ceramics with BaTiO 3 -derived compositions. Thanks to frequency dielectric studies, these materials have been shown to be classical or relaxor ferroelectrics depending on both the composition and the type of ionic substitution.

The aim of the present work was to prepare and to characterize some new lead-free ceramics with perovskite compositions derived from BaTiO 3 and with scandium content. In spite of the relative proximity of the electronic configuration of Ti 4+ (3d 2 4s 2 ) and Sc 3+ (3d 1 4s 2 ) neither dielectric properties nor crystallographic study was previously performed. However such element was already used in the case of Pb(Sc 1/2 Nb 1/2 )O 3 which is a well-known lead content relaxor. [START_REF] Smolenski | [END_REF]25

Preparation

The various compositions were prepared by solid state reactions from pecursors with mixing, grinding, calcination and sintering to obtain ceramics the most compact as possible. In a first time, such a process was conceivable using here the various departure used compositions BaCO 3 (Cerac 99.9%), TiO 2 (Cerac 99.9%), Nb 2 O 5 (Cerac 99.99%) and Sc 2 O 3 (Fluka 99.9%, containing less than 0.01% Yb 2 O 3 and less than 0.02% Y 2 O 3 ) by solid state synthesis using the following chemical reaction:

BaCO 3 + (1-x)TiO 2 + (0.5x/2)Nb 2 O 5 + (0.5x/2)Sc 2 O 3  BaTi 1-x (Nb 1/2 Sc 1/2 ) x O 3 + CO 2 .
In spite of a systematic study with different parameters of preparation (grinding type, reaction temperature and times, annealing or quenching), it was not possible to obtain single phase ceramics; the BaTiO 3 composition was always present with the solid solution. This result was well evidenced by dielectric measurements where the peak of relative real permittivity ' r was always present at T C (BaTiO 3 ) = 400K. Solid state process reactions remaining, we were successful to obtain a monophased solid solution carrying out the previous mixing and intimate grinding with three steps: firstly Nb 2 O 5 + Sc 2 O 3 , then we add TiO 2 to the previous mixing and in the end we add BaCO 3 to the transition oxides. Thermal treatments were then carried out: a calcination for 15h at 1250°C which ensure the chemical reaction followed by 4h sintering at 1450 or 1500°C depending on the composition. Both heat treatments were carried out under a dry oxygen atmosphere. This process was necessary due to our relative low value of temperature heat treatments compared to the refractory effect of Sc 2 O 3 (T melting (Sc 2 O 3 ) ~ 2430°C). 26

X-ray diffraction study

Powder X-ray diffraction measurements were performed on a Philips diffractometer using Cu-K radiation at room temperature. For compositions 0 ≤ x ≤ 1, a pure perovskite X-ray diffraction appeared in each case. For instance, the figure 1 gives the X-ray pattern for the composition corresponding to x = 0.25 .The diameter shrinkages  [ init .- fin / init. ] of the ceramics were in the range 14-16%. The densities of the sintered samples were 95-97 % of the theoretical values. No sintering additives were here used.

Dielectric measurements

Dielectric measurements were performed on ceramic discs after deposition of gold electrodes on the circular faces by dc sputtering. The real and imaginary parts of the permittivity ' r and " r were measured under helium as a function of both temperature (85-500K) and frequency (10 2 -2.10 5 Hz) using a Wayne-Kerr component analyzer.

For compositions with 0 ≤ x ≤ 0.075, the ceramics showed conventional ferroelectric properties with a sharp maximum of ' r at the Curie temperature T C and a small anomaly at T 1 corresponding to the orthorhombictetragonal transition in the ferroelectric region. There was a small dielectric frequency dispersion, but the value of T C was not dependent on frequency. As an example we reported on figure 2 the result obtained for ceramics corresponding to x = 0 (BaTiO 3 ) and x = 0.025. The values of the Curie temperature vary slowly from 400 to 320K as x increased. The temperature dependence of 1/' r at 1 kHz was also in good agreement with a Curie-Weiss law in the paraelectric region (T > T C ). For example, figure 3 shows the temperature dependence of 1/' r for a ceramic with composition x = 0.05. Here the transition was of first order (T C -T 0  20K, where T 0 is the Curie-Weiss temperature), as it was also for pure BaTiO 3 . 27 For compositions 0.075 < x ≤ 0.25 only one broad peak of ' r was observed; in fact, the limit value 0.25 corresponds only to that where dielectric anomalies were detectable by us with regards to our measurement limitation temperature (85K). For example, figure 4 shows the temperature and frequency dependences of ' r for x = 0.15. In addition, the temperature T m of ' r maximum was shifted to higher values at greater frequencies. A frequency dispersion took place for T ≤ T m , the value of ' r decreasing when the frequency increased. The temperature and frequency variations of " r were also specific (Fig. 5): the temperature T m of " r maximum was shifted to higher values at greater frequencies, but contrary to the evolution of ' r , the evolution of the frequency dispersion was here characterized by an increase of " r when frequency increased. In addition there was a deviation from the Curie-Weiss law. The value of the Curie-Weiss temperature T 0 was greater than that of T m (T 0 > T m ) and that of T dev. -T m was about of 104K for the composition corresponding to x = 0.15 (Fig. 6).

A dielectric dispersion was evidenced leading to a Vogel-Fulcher behavior. It has been found that in many relaxors the relationship between the frequency f of the ac measurement field and the temperature T m can be described using the Vogel-Fulcher relationship: log f = log f 0 -E a /(k B (T m -T VF )) where f 0 is the high temperature extrapolation of the attempt frequency, E a the activation energy, k B the Boltzman constant and T VF the static freezing temperature. [28][29][30] This is known to be one of the characteristics of relaxor systems. For example, T VG = 88K for x = 0.15. Figure 7 illustrates the good fit of the data log f vs T m with a Vogel-Fulcher law. The fitting parameters of Vogel-Fulcher relation were:

E a = 0.255 eV, f o = 2.7.10 22 Hz and T f = 87.7 K All these dielectric characteristics are typical of a relaxor behavior for 0.075 < x ≤ 0.25. 1 As x increased, the value of T dev. -T m increased, implying that the relaxor effect also increased (Table 1). This is known to be one of the characteristics of relaxor systems.

Figure 8 shows the composition variation of T C (or T m ) at 1 kHz for the solid solution BaTi 1-x (Nb 1/2 Sc 1/2 ) x O 3 . The value of T C or T m decreased from T C (BaTiO 3 ) = 400K as x increased. We also obtained an increase in T m when x increased. These results are in good agreement with those previously obtained. [START_REF] Simon | Lead free relaxors[END_REF]

Conclusion

Ceramics samples of niobium and scandium doped barium titanate with general composition BaTi 1-x (Nb 1/2 Sc 1/2 ) x O 3 have been elaborated by a special solid state reaction method. It was shown by X-ray diffraction at room temperature that the structure of the compounds is of perovskite type for 0 ≤ x ≤ 1. Dielectric measurements show that the permittivity peak is sharp, without frequency dependence for 0 ≤ x ≤ 0.075: it corresponds to a typical ferroelectric-paraelectric phase transition. For compositions in the region 0.075 < x ≤ 0.25, the ' r peak becomes diffuse and characterizes a relaxor behaviour. The degree of relaxor character represented by T m increases when the ratio of Nb and Sc increases up to T m = 20K for x = 0.25. Up to this last value no permittivity anomaly was detected due only to our low temperature dielectric measurement limitations.

25 Fig. 1 .Fig. 2 .

 2512 Figure

Fig. 3 . 15 Fig. 4 . 15 Fig. 5 .

 3154155 Fig. 3. Temperature dependence of 1/ e' r for a ceramic with composition corresponding to x = 0.05.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Temperature dependence of 1/e' r for a ceramic with composition corresponding to x = 0.15.