Extra polarisations of relic gravitational waves from a Brans-Dicke theory with axion-gauge dynamics

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Università di Pisa

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Extra polarisations of relic gravitational

waves from a Brans–Dicke theory with

axion–gauge dynamics

Candidate:

Supervisor:

Giulia Pagano

Dr. Giancarlo Cella

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Introduction

In analogy with the electromagnetic interaction, where radiation emission is due to the accelerated motion of charges, accelerated masses may emit gravita-tional radiation, provided that the motion of the system satisfies certain properties. Therefore, the signal reaching our detectors may arise from a variety of sources: in particular, one can distinguish among individual ones and a stochastic back-ground. The former addresses to those systems whose gravitational emission can be individually resolved, at least in principle, such as coalescing binary systems (formed by neutron stars, black holes or both), supernovae explosions or

asym-metric pulsars. It is worth noting that neither these cases have the same difficulty of detection, nor the theoretical predictions are known with the same accuracy, as pointed out in [6]. The latter, instead, refers to all those events which can’t be individually detected, but nevertheless give rise to a signal: many random, independent processes, in fact, can combine and produce a stochastic outcome.

The background signal receives two contributions: a recent one, due to the random sources which are still operating in the sky, and a primordial one, which originated in the early life of the Universe. In this context, in fact, the term “recent” should be intended with respect to the first moments of the evolution of the Universe.

The fist kind of contribution is called a background of astrophysical origin and includes the signal arising from the aforementioned individual sources, in those cases in which it can’t be individually resolved. Its modelling has its own interest, as well as being necessary to single out the “true” contribution of relic gravitational waves.

According to current cosmological models, though, the second type of con-tribute is also present: it is the result of events which are thought to have hap-pened in the early life of the Universe, some examples of which will be given later in this introduction. If this is the case, the understanding of the features of this kind of signal is fundamental for the study of the early age cosmology.

Thus, the present work will focus on the stochastic background of cosmological origin. The main interest in studying such a signal is that, the gravitational interaction being the weakest of the four fundamental forces, its perturbations are expected to decouple from all the other fields early in the life of the Universe (see [26] for a quantitative discussion of the issue in the domain of general relativity). In fact, equilibrium is only maintained as long as the rate Γ at which the interactions occur is greater than the rate of expansion, namely Γ · H−1 1. Here H is the

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Introduction

Hubble parameter, which is an order-of-magnitude estimate of the second scale. Before going further, a clarification is needed: the assumption the last state-ment relies upon is that, since its birth, the Universe undergoes a phase of expan-sion (whether accelerated or decelerated, depending on the kind of fluid which dominated in each epoch), which is a common feature of most of the current cosmological models and is indeed in good agreement with the observed large scale isotropy and homogeneity.

Coming back to the point, since Γ depends on the strength of the force via the cross section, gravitational waves (GWs from now on) are expected to decouple earlier than electromagnetic radiation. The weakest the interaction, in fact, the sooner the above condition will drop. Given that no interaction occurs after decoupling, the signal maintains all the features it had in the primordial epoch of production. Therefore, it provides an insight into energy scales which are not accessible through the study of the cosmic microwave background (CMB).

To get a sense of the energies involved, a rough estimate for the scale of decoupling of GWs is provided by the typical mass related to the gravitational interaction, the Planck mass, whose value in natural units is Mp ∼ 1019GeV. For

the electromagnetic radiation, instead, decoupling takes place with nucleosynthesis, because atoms formation deprives photons of their targets. In that case, the scale to be considered is the minimum energy required for the formation of the hydrogen atom, EH ∼ 10 eV. This means that GWs decoupling takes place at much higher

energies than in the CMB case, or, in other words, at much earlier epochs. Among the mechanisms which generate a stochastic signal of cosmological origin, the one this thesis is going to investigate is the parametric amplification of vacuum fluctuations. The subject can be introduced in the following way: experimental observations of the spatial curvature and the evolution of the horizon (the portion of three-dimensional space which is causally connected with the observer at the present time, which is, that part of the Universe from which one has already received light signals) pose some problems in the standard cosmological model based on general relativity. First, one needs to understand why the spatial sections are almost flat (“flatness problem”), secondly, the present horizon contains several parts which should have been uncorrelated with one another back in time. If this is the case, one should explain how it is possible for the observable Universe to be homogeneous and isotropic and for the CMB to have almost the same temperature at every point in the sky. Different regions, in fact, shouldn’t have been able to tune their properties with one another (“horizon problem”). A possible way out consists in introducing a period of accelerated expansion in the early times which, if long enough, fixes all those difficulties [20]. This is called inflation and is also responsible for the amplification of fluctuations. While quantitative effects will be taken into account extensively in the following chapters, it is important to give a qualitative insight of how such a mechanism works. A simple framework for an understanding of the subject is that of general relativity in an expanding, spatially flat space-time: one starts by expanding the metric tensor around a classical background, the Friedmann-Lemaître-Robertson-Walker

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Introduction

metric in this case, which is the standard procedure for studying GWs. By doing so, the metric reads: gµν = ¯gµν + δgµν, where ¯g is the background and δg its

quantum fluctuation. When introducing the normal modes for the perturbations, in the linearised theory one finds that the normal variables obey the equation of a harmonic oscillator whose frequency is allowed to change with time. Hence, they behave as propagating waves as long as the frequency squared is positive. This is true unless the wavenumber of the mode under consideration is greater than the inverse of the typical spatial scale of the Universe, i.e., of the horizon (namely, k H). When this condition does not hold anymore, the square of the frequency turns negative and the solutions become exponential ones.

This corresponds to the wavelength of the mode exceeding the horizon scale, so that it cannot be regarded as a wave anymore, because no oscillation occurs within the relevant length. From then on, the mode is “freezed”, as it exceeded the scale of causal physics. This means, in particular, that its amplitude is protected from decreasing, which would be the case if the mode were on sub-horizon scales.

This is exactly what happens during inflation: the wavelengths are stretched up to the horizon scale and eventually cross it. When the phase of accelerated ex-pansion ends, they re-enter the horizon and the related amplitudes start decreasing again. If compared to the modes which remained sub-horizon, whose amplitudes decreased during the whole period, this results in an amplification of the first ones. This is, qualitatively, what is called parametric amplification of vacuum fluctuation, the word “vacuum” referring to the fact that the fields are taken to be in their vacuum state. The understanding of this process is fundamental, since an amplification of quantum fluctuations after inflation is always expected.

Of course, this is only one of the mechanisms which can source GWs of cos-mological origin. For the sake of completeness, a few other examples will be listed below, following references [6] and [11].

First of all, one should notice that quantum fluctuations of any field are always present, due to their quantum nature. This means that the previous considerations apply irrespectively of the initial state of the field, in which case the process is dubbed amplification of quantum fluctuations.

Relic gravitational radiation, though, can also arise from the pre-heating pro-cess taking place at the end of inflation. In most cosmological models, in fact, the accelerated expansion is driven by a scalar field called inflaton. At the end of inflation, this field starts oscillating and eventually decays producing radiation of different kinds, including gravitational one. This mechanism is called pre-heating because the produced radiation will only subsequently reach thermal equilibrium.

Other possible sources resulting in a primordial background are cosmic strings and phase transitions. The former are one-dimensional (string-like in fact) topolog-ical defects which may have formed when the Universe experienced the transition from a single unified force to the different components of the interactions. In the resulting network, strings can cross each other and form smaller loops by re-connection. These loops oscillate relativistically and decay emitting gravitational radiation, since they are stable against all other types of decay.

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Introduction

The last example to be discussed are phase transitions: as the Universe cooled, in fact, it may have undergone first-order phase transitions. If this is the case, the transition occurred through “bubbles” of the new phase which expanded in the old one, moving at relativistic speeds. When they collided, they merged sourcing GWs and eventually turning the old phase into the new one. Needless to say, the resulting spectra have specific features depending on the process under consideration.

After these preliminary considerations, the next step is to define the framework in which the cosmological background will be investigated, which is, the underlying theory of gravitation.

The up-to-now most satisfying theory of gravity is Einstein’s general relativity. Its main feature is that of considering the space-time as a dynamical entity whose evolution is determined together with the evolution of the mass and the energy which fill it. This is done by introducing the aforementioned metric tensor, which mathematically describes the properties of the four-dimensional space-time at each point and for each time. Such a formulation represents a marked change in the conception of space and time as absolute entities, as they were regarded in Newton’s gravity, and has been capable of high-precision predictions. In fact, it successfully explained the anomalous precession of the perihelion of Mercury (a rotation of the orbit of the planet around the Sun, whose amount was not satisfactorily predicted in Newton’s theory), but also envisioned interesting new features. Just to name a few examples, one can mention the light deflection due to massive bodies, the gravitational redshift of the wavelengths of photons (caused by a change in the intensity of the gravitational field), which have been confirmed by high-precision experiments, and gravitational waves of course, which have been probed much more recently.

Despite such validations, however, general relativity brings about some issues, as discussed in the works from Gasperini, [19], [20]. First of all, it is a classical theory and as such it can’t provide a quantum unified description of all forces. Indeed, all the attempts of quantisation have up to now proved unsatisfactory. Besides, it leads to a singularity of the equations: the big explosion which is put at t = 0 as the beginning of the Universe, called the Big Bang, could be a hint that the underlying theory is not reliable at extreme regimes, rather than being a prediction. In addition to that, it does not naturally solve the horizon and flatness problems.

Another unexplained observation is that the total energy density of the Uni-verse seems to be smaller than the amount required by a flat space-time. This fact is known as the “missing mass problem” and can be solved by postulating the existence of a non-relativistic kind of matter which interacts only gravitationally, so that it is invisible, from which the name “dark matter”. Furthermore, its distri-bution allows the speed of the stars to stabilise at a constant value as the distance from the centre of the galaxy increases, which is another observed discrepancy with pure general relativity’s predictions.

Last but not least, it has been confirmed that the Universe is now experiencing an accelerated expansion. This can’t be attributed to the vacuum energy density

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Introduction

(represented by the addition of a constant term Λ to the Lagrangian), since a simple estimate shows that Λvac ∼ Mp4, while the experimental value leads to1

Λ ∼ 10−122M_p4 [20]. Given the smallness of the result, one could invoke some symmetry principle in order to set its value to zero, in which case the symmetry should be broken, because Λ differs from zero in fact. In any case, the scale of the symmetry breaking gives an inferred value which disagrees from the observed one for several order of magnitudes.

Another way of addressing the problem is to consider the Λ term as an addi-tional contribution to the cosmic fluid, called “dark energy”, which turns out to have negative pressure. One then is left with the problem of interpreting where such an exotic fluid comes from, and why the related energy density is of the same order of magnitude as the matter energy density at the present time, which is the “coincidence problem”.

All these reasons justify the query for a more general theory of gravitation, which nevertheless has to reproduce the results of general relativity in the low-energy limit. Given the lack of information about the quantum regime which took place at very early epochs, one way of selecting candidates is that of considering the effective actions arising from models which unify all the fundamental forces, such as superstring, supergravity and Grand-Unified Theories ([10], [18]).

In doing so, one always finds two new kinds of contributions to the action: higher order terms in the curvature invariants (in other words, non linear terms in the derivatives of the metric tensor, or terms with higher-than-second-order derivatives of the metric) or scalar fields non-minimally coupled to the metric. The former have been shown to be equivalent to Einstein’s gravity plus a number of scalar fields via a conformal transformation, [10], so that the presence of such fields seems to be a natural requirement to an extended model. The latter, instead, refers to interaction terms which can’t be directly obtained from the Lagrangian of flat space-time, as will be discussed later on. This last kind of theories, which indeed involve scalar fields coupled to the metric tensor in a non trivial way, are called scalar-tensor theories.

Other attempts to extend general relativity have been made by replacing the scalar curvature in the usual action with a function of the same variable, and are therefore called f (R) theories. The trivial case f (R) = R reproduces Einstein’s theory of course. This class of models, though, have been proven to be equivalent to scalar-tensor theories, as shown in [10], so that the last ones seem to be in some sense more general.

Due to these considerations and because the introduction of a scalar field is the simplest modification to the original formalism, the present work will consider the framework of scalar-tensor theories. In particular, the original version proposed by C. Brans and R. H. Dicke will be adopted [8].

The thesis is organised as follows: in chapter 1 results for standard polarisa-tions from general relativity are discussed, the compatibility with observational bounds and the observative perspectives are presented. In chapter 2 the model is

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Introduction

introduced, on the basis of the one developed in [13], with the modification of the gravitational action as previously stressed. The presence of the scalar field, specific to this work, is expected to give rise to an extra polarisation state, due to the additional scalar degree of freedom. The study of its evolution and its interaction with the usual polarisations is indeed the aim of the thesis.

The equations of motion for each field are derived and linearised in the fluctu-ations. Then, the study of the background is performed in order to calibrate the parameters of the model; the equations for the perturbations are then investigated. In chapter 3 the stochastic background is analysed: the study of both usual and emerging polarisations is performed, as well as the quantisation of the fluctua-tions. The general procedures and expressions for the computation of the spectra are presented and applied to some particular cases. In chapter 4 conclusions are offered.

Conventions and notations

In the following chapters the natural units ~ = c = kB = 1 are adopted, where

kB is the Boltzmann constant. Greek indices α, β, ... stand for space-time indices,

0, 1, 2, 3; Latin indices from the middle of the alphabet i, j, k, ... stand for spatial indices 1, 2, 3 and Latin indices from the beginning of the alphabet a, b, c, ... stand for gauge indices. The convention of summation over repeated indices holds, unless otherwise mentioned.

The Minkowskian metric is

ηµν = diag (−1, 1, 1, 1)

and, for a generic metric,

g = det gµν

is the determinant of the covariant components. The Christoffel symbols are

Γα_µν = 1 2g

ασ

[∂µgσν + ∂νgσµ− ∂σgµν]

and the covariant derivative is denoted by ∇µ. The Riemann tensor is defined as

Rα_βµν = ∂µΓαβν+ Γ α

µσΓ σ

βν− {µ ←→ ν}

and the Ricci tensor as

R_µν = Rσ_µσν.

The convention for the antisymmetric symbol µνρσ is 0123 = 1

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Introduction

and Mp indicates the reduced Planck mass,

M_p2 = 1 8πG. The convention for the spatial Fourier transform is

Q(x, t) = Z d3k (2π)3/2e ik·x_Q_˜ k(t). (1)

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Chapter 1 Overview: gravitational waves in

general relativity

Metric theories of gravitation describe the space-time as a dynamic entity through a metric tensor. Therefore, it is meaningful to study the effects of the perturbations of such a quantity. Indeed, the usual approach is to distinguish, in the complete tensor, a background contribution from a perturbation. The latter is assumed to be small, in a sense that will be clarified later in this chapter, and the equations of the theory are expanded as a power series in this variable.

The outcome of this procedure at the first order is a “linearised” theory endowed with a gauge freedom. This is a consequence of the invariance under general coordinate transformations, which is a feature preserved even by alternative metric theories. The freedom is removed by choosing a gauge, i.e., a reference frame: a smart choice provides a wave equation for the perturbations, which are interpreted as gravitational waves propagating throughout the space-time at the speed of light.

A simple case where these points are evident is that of a flat background metric ηµν in the framework of general relativity. This example allows for a preliminary

discussion on sensitive issues like the gauge choice and the detector response to different polarisations, which is particularly important in this thesis. Therefore, a brief review of the results of general relativity will be provided in the next pages, following reference [28].

1.1 Linearisation around flat space-time

The equations of general relativity relate the metric of the space-time gµν to

the energy-momentum tensor Tµν. The latter results from the variation of the

non-gravitational action with respect to the metric and describes the energetic balance of the other fields:

δgSM ≡ −

1 2

Z

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Chapter 1. Overview: gravitational waves in general relativity

The subscript M symbolically stays for matter, though SM could include massless

components like the electromagnetic field. The gravitational action, instead, is

Sg = M2 p 2 Z d4x√−gR, Mp = 1 √ 8πG. (1.2) Varying the whole action with respect to the metric, one obtains the well-known Einstein’s equations 1

Rµν−

1

2gµνR = 8πGTµν (1.3) where Rµν and R are the Ricci tensor and the scalar curvature respectively and

G is the Newtonian gravitational constant.

Linearisation then proceeds introducing a perturbation to the Minkowskian background metric: gµν = ηµν+ hµν, with |hµν| 1. The last condition defines the

smallness of the perturbation, and must be interpreted as a constraint on every component of hµν. Its physical meaning is that one can choose only reference

frames in which this condition holds, thus reducing arbitrary coordinate trans-formations to the ones which do not spoil it. Equation (1.3) is then expanded to the linear order in hµν, the other contributions being negligible if the smallness

condition holds. In doing so, one has that the indices are raised and lowered by the background metric ηµν, thus considering the perturbation as a field “living” in

a flat space-time.

Since the Christoffel symbols are trivial at the 0-th order, the curvature ten-sors are at least first order. This implies that further metrics appearing when a contraction is performed from the Riemann to the Ricci tensor, or from the Ricci tensor to the scalar curvature, reduce to the background metric ηµν. Besides, the

terms containing products of Christoffel symbols are ΓΓ ∼ O(h2_{) and can be}

neglected.

A further simplification comes from introducing the trace-reversed perturba-tion: ¯ hµν = hµν− 1 2ηµνh, h = η µν hµν. (1.4)

With such definition, it is straightforward to verify that ¯h = −h and that the smallness condition is preserved. In terms of this variable the linearised equations assume a simple form:

¯hµν + ηµν∂ρ∂σ¯hρσ− ∂ρ∂ν¯hµρ− ∂ρ∂µ¯hνρ = −16πGTµν, ≡ ηµν∂µ∂ν. (1.5)

The structure of the previous formula suggests that the gauge freedom could be used to select a reference frame where ∂µ¯hµν = 0, in order to get rid of the last

three terms at the left-hand side and turn to a wave equation. To show that this is always possible, suppose to make a transformation of coordinates

xµ−→ x0µ = xµ+ ξµ(x). (1.6)

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1.2. The transverse-traceless gauge

Accordingly, the full metric in the new frame is

g0_µν(x0) = ∂x

ρ

∂x0µ

∂xσ

∂x0νgρσ(x). (1.7)

If |∂µξν| is of the same order of magnitude as the perturbation, the transformation

can be expanded to the linear order in both these quantities. This gives the transformation rule for ¯hµν,

¯

hµν(x) −→ ¯h0µν(x 0

) = ¯hµν− (∂µξν + ∂νξµ− ηµν∂ρξρ). (1.8)

Thus, the requirement that |∂µξν| be of the same order of magnitude as the

perturbation does not spoil the linear approximation, but constraints the co-ordinate transformations to be infinitesimal in this sense. In the new frame, ∂µ_h¯0

µν = ∂µ¯hµν − ξν and the desired constraint can be imposed by choosing

the transformation so that _ξν = ∂µ¯hµν: this is always possible because the

d’Alembertian operator is invertible. Such choice turns equation (1.5) into a wave equation in the variable ¯hµν, where the prime has been dropped for simplicity.

The wave propagates at the speed of light and the energy-momentum tensor plays the role of a source term:

¯hµν = −16πGTµν, ∂µ¯hµν = 0. (1.9)

This gauge choice is known as the De Donder gauge and corresponds to four conditions. Therefore, the ten initial degrees of freedom of the matrix hµν are

reduced to six independent components in ¯hµν.

1.2 The transverse-traceless gauge

Within the De Donder gauge it is possible to perform a further transformation:

xµ−→ x0µ = xµ+ ξµ(x), _ξµ= 0. (1.10) The last condition ensures that the perturbation satisfies both the wave equation and the De Donder constraint (1.9) in the new frame, as it can be seen by using the transformation property of ¯hµν, equation (1.8). Therefore, one can consider

the transformed field ¯h0_µν = ¯hµν − ∂µξν − ∂νξµ+ ηµν∂ρξρ in place of the original

one.

Outside the source, Tµν = 0 and the perturbation satisfies the free-wave

equation:

¯hµν = 0 (1.11)

where the prime has been dropped for simplicity. In this case the form of ¯hµν can

be simplified by using the four arbitrary functions ξµ _{to set further constraints.}

In particular, one can impose the conditions ¯

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The second constraint has been written in terms of hµν, because the traceless

condition ensures that hµν and ¯hµν coincide. From the De Donder gauge with

ν = 0, then, one has that ∂µhµ0 = ∂0h00 = 0. This means that the full temporal

component is constant and contributes to the static field, rather than to the gravitational wave itself, which represents a propagating wave. Therefore, all the temporal components of the pure gravitational wave are trivial and the “true” hµν

satisfies the relations

hµ0 = 0, hii = 0, ∂jhij = 0. (1.13)

The last two equations are the traceless condition and the De Donder gauge, given that the perturbation has only spatial indices.

The solutions of equation (1.11) are plane waves, hence the perturbation is a superposition of components of the form hij(x, t) = Aij(k)eik

µ_x

µ_{, where k}µ_{= (k, k)}

is the wavenumber and k ≡ |k|. Of course, only the real part of the previous expression must be considered.

For a monochromatic wave, in particular, hij coincides with the previous

ex-pansion and the gauge conditions imply that the amplitude must be traceless and transverse to the direction of propagation, Ai

i = kjAij = 0. Thus, the constraints

(1.13) are said to define the transverse-traceless gauge (TT gauge).

It is worth noting that such a choice is possible only in vacuum. If this isn’t the case, the functions ξµν ≡ ∂µξν + ∂νξµ− ηµν∂ρξρ can’t be used to set to zero

any component of the perturbation. In fact, in order for them to cancel a specific component, the corresponding d’Alembertian should be non-vanishing. This is impossible since the requirement _ξµ = 0 also implies that ξµν = 0.

Consider a monochromatic wave propagating along the z axis, then k = kˆz and the transverse condition requires that h13 = h23 = h33 = 0. Recalling that

the perturbation is symmetric and traceless in this gauge, it is easily written as

hij(z, t) =   h+(t − z) h×(t − z) 0 h×(t − z) −h+(t − z) 0 0 0 0  . (1.14)

where h+,× are periodic functions of time. In the general case, Aij(k) is the Fourier

transform of the previous matrix and can be written as Aij(k) = Aij(2πf ˆz) =

P

A

hA(f, ˆz)Aij(ˆz), where 2πf = |k|.

The matrices A

ij(ˆz) are called polarisation tensors and describe the

indepen-dent ways of oscillation that the wave brings with it. They depend on the direction of propagation because they must be transversal with respect to it, while the de-pendence on the frequency is carried by the amplitudes hA(f, ˆz). In this particular

case they are

+_ij(ˆz) =   1 0 0 0 −1 0 0 0 0  , ×_ij(ˆz) =   0 1 0 1 0 0 0 0 0  . (1.15)

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1.2. The transverse-traceless gauge

The corresponding amplitudes, h+ and h×, are called plus and cross polarisations

respectively. They are the two residual degrees of freedom, whose values are given with respect to the chosen axes in the transverse plane. Thus, the two gauge transformations which set up the TT gauge have eliminated eight redundant degrees of freedom from the ten original components of hµν, leaving only the two

physical polarisations h+ and h×.

A generic orientation of the axes in the transverse plane is recovered through the rotation by an arbitrary angle −ψ of the original ones. Accordingly, the polarisations change as

h+ −→ h+cos 2ψ − h×sin 2ψ, h× −→ h+sin 2ψ + h×cos 2ψ. (1.16)

Recalling that the full metric is gµν = ηµν+ hµν, the complete line element, which

gives the infinitesimal distance between two space-time points, is

ds2 = −dt2+ [1 + h+(t − z)]dx2+ [1 − h+(t − z)]dy2+ 2h×(t − z)dxdy + dz2. (1.17)

Thus, each polarisation deforms the proper distance between two events in a characteristic way. This is important when looking for the effects on test masses, because it discriminates among the different degrees of freedom. Besides, it makes it possible to recognise and classify any additional polarisation in alternative theories.

The previous discussion has shown that the choice of a gauge which eliminates the redundant degrees of freedom and the study of the related tensors A_ij(ˆn) is essential for the comprehension of the polarisation content of the theory and of its effects.

In the general case, though, a wave is a superposition of monochromatic com-ponents propagating along different directions. Therefore, its most general form is hij(x, t) = Z _d3_k (2π)3/2Aij(k)e ikµ_x µ_{+ c.c.} = Z ∞ 0 (2π)3/2f2df Z

d2nAˆ ij(2πf ˆn)e−2πif (t−ˆn·x)+ c.c

= Z ∞ −∞ (2π)3/2f2df Z d2n Aˆ ij(2πf ˆn)e−2πif (t−ˆn·x). (1.18)

In the previous formula, d2_{n = d cos θdφ is the solid angle and “c.c.” denotes}_ˆ

the complex conjugate, which is necessary because the expression must be real. For the previous considerations, one can rewrite the amplitudes in terms of the polarisation tensors,

(2π)3/2f2Aij(2πf ˆn) =

X

A=+,×

hA(f, ˆn)Aij(ˆn) (1.19)

and the final form for a generic wave becomes

hij(x, t) = X A=+,× Z ∞ −∞ df Z d2n hˆ A(f, ˆn) Aij(ˆn)e −2πif (t−ˆn·x)_. _(1.20)

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Since the perturbation is real, each Fourier amplitude satisfies the constraint hA(f, ˆn) = h∗A(−f, ˆn).

Besides, since the polarisation tensors are transverse with respect to the di-rection of propagation of the respective component, they can be written as the tensor product of a basis in the transverse plane. Consider two unit vectors, ˆu and ˆv, orthogonal to each other and to ˆn. Then, the tensors can be written as

+_ij(ˆn) = ˆuiuˆj− ˆvivˆj, ×ij(ˆn) = ˆuivˆj + ˆviuˆj. (1.21)

Accordingly, they are subjected to the orthogonality condition

A_ij(ˆn)A0,ij(ˆn) = 2δAA0. (1.22) Choosing ˆu = ˆx and ˆv = ˆy leads to the matrices (1.15) of the previous example. A generic orientation of the basis in the transverse plane is achieved by rotating the axes ˆu and ˆv by an arbitrary angle −ψ. The corresponding transformation of the polarisation tensors, then, is:

+_ij −→ + ijcos 2ψ − × ijsin 2ψ, × ij −→ × ijcos 2ψ + + ijsin 2ψ. (1.23)

1.3 Effects of GWs on test masses

A way to discriminate among the different polarisations of a gravitational wave is to observe the evolution of the proper distances between test particles. Consider first a single mass in absence of non-gravitational forces: its motion is determined by the geodesic equation:

Duµ

Dτ = 0. (1.24)

In the previous formula, x(τ ) is the trajectory as a function of the proper time,

uµ = dx

µ

dτ (1.25)

is the related four-velocity and

Duµ= duµ+ Γµ_αβuαdxβ (1.26) is the covariant differential. To find the evolution of the displacement between different masses, one considers two geodesics with an infinitesimal separation ξµ_{(τ ).}

Combining the geodesic equations of the two trajectories, xµ(τ ) and xµ(τ ) + ξµ(τ ), and expanding to the linear order in the displacement gives the geodesic separation equation: D2_ξµ Dτ2 = −R µ νρσξ ρdxν dτ dxσ dτ (1.27)

where the curvature tensor is computed at the position of the referential body, xµ_.

Thus, the trajectories are subjected to a tidal force whose effects are encoded in the Riemann tensor.

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1.3. Effects of GWs on test masses

The previous equation is covariant and holds in every coordinate system. When performing an experiment, though, one usually considers the freely falling frame associated with one of the masses. In this frame the Christoffel symbols vanish along the trajectory of the particle, where the metric is Minkowskian: it is a standard result of general relativity that these conditions can always be imposed [20]. If the position of the referential mass defines the origin, one has that gµν(0) = ηµν and Γαµν(0) = 0. The geodesic equation, then, implies that a

single particle is in “free fall”, because all the effects of gravity disappear from its equation of motion.

When considering two bodies, the geodesics are taken at equal times, therefore the temporal component of the displacement is trivial: ξ0_{(τ ) = 0. In addition,}

the left-hand side of equation (1.27) turns into an ordinary differential, because Γµ_αβ(0) = 0 at every instant, meaning that temporal derivatives of the connec-tion must vanish at the origin. Moreover, the two masses are non-relativistic, as they are an extremely simplified version of a detector, for which this condition is satisfied. Therefore, the leading term in their four-velocities is given by the tem-poral components: ν, σ = 0. With this at hand, the geodesic separation equation becomes d2_ξi dτ2 = −R i 0j0ξ j dx0 dτ 2 . (1.28)

This result can be further simplified: it turns out that the linearised Riemann tensor is invariant under the gauge transformation (1.8), hence it can be computed in the TT gauge. This is a huge simplification, because the perturbations assume a simple form within this choice. Indeed, one has that

Ri_0j0 ' 1 2 ∂j∂0h T T 0i + ∂i∂0hT T0j − ∂i∂jhT T00 − ∂ 2 0h T T ij = − 1 2 ¨ hT T_ij . (1.29) If the masses are initially at rest, dx0 _{= dτ to the linear order and the evolution}

of the displacement is ¨ ξi ₌ 1 2 ¨ hT T_ij ξj. (1.30) Thus, gravitational waves affect the coordinate separation between two nearby masses in this frame.

The previous results have been found to the linear order in the displacement, thus they require the condition |x_Li2|2

B

1, where |xi_{| ≡ |ξ}i_{| is the position of the}

second particle with respect to the origin, where the other mass is located, and LB

is the typical scale of variation of the metric. This means that this scale, whose estimate is the wavelength of the gravitational wave, must be grater than the linear dimensions of the simplified detector. If this condition holds, the metric is Minkowskian to O|x_Li2|2

B

[28] and the coordinate separation ξi _{appearing in}

(1.30) coincides with the physical distance.

To get the actual time-dependence of the induced deformations, consider a ring of test masses initially at rest in the x−y plane. For each of them, the displacement with respect to the centre can be written as: ξi =x0+ δx(t), y0+ δy(t), where

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Chapter 1. Overview: gravitational waves in general relativity x y (a) - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 x y (b)

Figure 1.1: (a) Deformations induced by a gravitational wave propagating along the z axis on a circular ring of test masses, in the case of plus polarisation. The ring lies in the x − y plane and the direction of propagation is marked by the circled dot at the top of the picture. The full line and the dashed line refer to two different times t1 and t2. The lines of force (b) of the force field induced by the

same polarisation at t1 are also shown for comparison. The arrows indicate the

direction of the force. Both the images are given in arbitrary units.

x0 and y0 mark the position in absence of gravitational waves. A GW propagating

along the z axis has the form (1.14) in the TT gauge and, combined with the previous equation, ensures that the masses don’t accelerate in the z direction. Consequently, the motion takes place in the transverse plane.

Applying equation (1.30) to the plus polarisation, one finds that:

δ ¨x = −h+(t) 2 k 2 (x0+ δx) ' − h+(t) 2 x0k 2 δ ¨y = h+(t) 2 k 2_(y 0+ δy) ' h+(t) 2 y0k 2 (1.31)

The terms hδx and hδy, instead, have been neglected, because they are ∼ O(h2₎

(recall that the induced displacements must vanish in absence of perturbations). The integration of the previous equations immediately gives:

δx(t) = h+(t)

2 x0 δy(t) = − h+(t)

2 y0. (1.32) The additional terms resulting from the integration must vanish, because they would lead either to a non-vanishing displacement when no GWs are present, or

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1.4. Detector response x y (a) - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 x y (b)

Figure 1.2: (a) Deformations induced by a gravitational wave propagating along the z axis on a circular ring of test masses, in the case of cross polarisation. The ring lies in the x − y plane and the direction of propagation is marked by the circled dot at the top of the picture. The full line and the dashed line refer to two different times t1 and t2. The lines of force (b) of the force field induced by the

same polarisation at t1 are also shown for comparison. The arrows indicate the

direction of the force. Both the images are given in arbitrary units.

to a displacement which grows unbounded, which of course is unphysical. With analogous calculations, one gets the separation induced by the cross polarisation:

δx(t) = h×(t)

2 y0 δy(t) = h×(t)

2 x0. (1.33) The resulting deformations, together with the force fields induced over the all space are shown in Figs. 1.1 and 1.2. They are related simply by a rotation of ϕ = π/4 in the transverse plane.

1.4 Detector response

The previous sections have shown that the gravitational signal reaching a detector is described by a tensor. The output of the experimental device, however, is a scalar quantity.

For gravitational waves with long wavelengths with respect to the linear size of the detector, the spatial dependence in the exponential of equation (1.20) can be neglected. Note that this is also the same requirement for the applicability of the geodesic deviation results. Therefore, for a given direction of propagation, the

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spatial components of the perturbation become:

hij(t) = X A A_ij(ˆn) Z ∞ −∞ df hA(f )e−2πif t = X A A_ij(ˆn)hA(t). (1.34)

Thus, disregarding the noise of the instrument, the detector response can be modelled in the same large wavelength approximation as:

h(t) = Dijhij(t), (1.35)

where h(t) is the output signal and Dij is called detector tensor. It describes the coupling of the instrument to the perturbations and depends on the geometry of the apparatus.

It is now convenient to define the antenna pattern functions, FA(ˆn) ≡ DijAij(ˆn),

which models the response of the instrument to the different polarisations. By comparing this definition with equations (1.35) and (1.34), in fact, it follows that: h(t) = h+(t)F+(ˆn) + h×(t)F×(ˆn). (1.36)

In the previous equation, the polarisations are referred to the chosen basis (ˆu, ˆv) in the orthogonal plane with respect to the direction of propagation ˆn. Under a generic rotation of the axes by an angle −ψ on this plane, the polarisation tensors change according to equations (1.23). Since the detector tensor is constant, this implies that the antenna pattern functions transform as:

F+(ˆn) −→ F+(ˆn) cos 2ψ−F×(ˆn) sin 2ψ, F×(ˆn) −→ F×(ˆn) cos 2ψ+F+(ˆn) sin 2ψ.

(1.37) Finally, combining these transformation rules with that ones of the polarisations, equations (1.16), one finds that the output of the detector described in equation (1.36) is independent of ψ, which is, of the chosen basis in the orthogonal plane.

In the search for gravitational waves, a major role is played by ground-based interferometers. These devices are provided with two orthogonal arms of the same length L and measure variations in the optical path of a laser beam travelling back and forth from these arms. In particular, the beam is split at the junction point of the arms by a beam splitter, so that a part travels along one direction and the other one along the orthogonal way. At the end of each arm, a mirror reflects the beam, reversing its direction of propagation. When the two beams reconnect, they are out of phase, so that no light signal is measured by the detector. If a gravitational wave perturbs the length of the arms, though, a phase shift may be introduced and light measured.

Ground-based interferometers have a sensibility range in frequency which goes from a few Hz up to a few kHz. Consequently, the wavelengths of the detectable GWs go from around 3 × 105m to 3 × 108m and are much greater than the three-kilometre-long arms of an interferometer like Virgo. In this case, the phase shift is proportional to the difference between the lengths of the arms and the results of the previous subsection hold. One can choose the beam splitter as reference mass

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1.4. Detector response

which sets the origin of the freely falling frame and the arms as representative of the x and y axes. Then, the coordinates of the mirror at the end of the x arm are ~

ξa = ~La+ δ~La, where ~La = (L, 0, 0) is the position in absence of GWs and δ~La is

the induced variation in each direction. Equation (1.30) says that to the linear order its coordinates change according to

¨ ξ_ax = 1 2 ¨ hT T_xxL, ξ¨_ay = 1 2 ¨ hT T_yxL (1.38) where the contributions ∼ hδL have been neglected at the right-hand sides. Therefore, the mirror experiences two kinds of motions: one along the y axis, driven by ¨hT T

yx, and another one along the x axis, driven by ¨hT Txx. The first one is

transverse with respect to the direction of propagation of the laser beam and does not change its optical path. Thus, only the second one contributes to the phase shift. Similarly, for the second mirror one has that ~ξb = ~Lb+ δ~Lb ≡ (0, L, 0) + δ~Lb

and the position is driven by

¨ ξy_b = 1 2 ¨ hT T_yy L, ξ¨x_b = 1 2 ¨ hT T_xy L. (1.39) Again, only the first one of the previous equations must be considered when computing the optical path.

Integrating twice these relations, one has that the relative phase shift is pro-portional to ξa− ξb L = 1 2 h T T xx − h T T yy . (1.40)

As in the previous example, the additional terms resulting from the integration must vanish if the phase shift is taken to be zero at the initial time.

The previous equation gives, up to a multiplicative factor, the output signal of the interferometer. Therefore, the detector tensor of equation (1.35) turns out to be:

Dij =

1

2(ˆxixˆj− ˆyiyˆj). (1.41) Since the phase shift is proportional to the difference between the lengths of the arms, this detector tensor corresponds to a specific choice of the normalisation constant of the output. In particular, it is the optimal output for a wave with plus polarisation, propagating along the z axis. In fact, in this case the only non-trivial components of hT T_ij are hT T_xx = −hT T_yy = h+ and the instrument detects the whole

amplitude of the wave: O(t) = h+(t). In all the other cases, the detected signal is

smaller than the full amplitude of the polarisation.

With this at hand, one can compute the antenna pattern functions of a ground-based interferometer. Consider a wave propagating along a generic direction ˆn in the reference frame xOy of the interferometer. Then, there exists a frame x0O0y0 whose axis z0coincides with the direction of propagation of the wave. In that frame, the components of the perturbation in the TT gauge are h0T T_xx = −h0T T_yy = h+ and

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respect to the xOy frame, so that the transformation which brings z0 into z is the composition of a rotation by an angle θ around the y axis and a rotation by an angle φ around the z axis, R ≡ RφRθ. The situation is illustrated in Fig. 1.3.

φ θ x z y z0

Figure 1.3: Orientation of the z0 axis with respect to the detector frame. The angles are defined as positive counter-clockwise.

The perturbations in the detector frame then are: hT T _{= Rh}0T T_RT_{. After such}

transformation, one finds that the amplitudes can be written as hT T_xx = h+(cos2θ cos2φ − sin2φ) + h×cos θ sin 2φ

hT T_yy = h+(cos2θ sin2φ − cos2φ) − h×cos θ sin 2φ.

(1.42)

Thus, the relative phase shift between the two arms is proportional to 1 2 h T T xx − h T T yy = 1 2h+ 1 + cos 2_{θ cos 2φ + h} ×cos θ sin 2φ. (1.43)

A comparison with equation (1.36) shows that the antenna pattern functions for a wave propagating along a generic direction are:

F+(ˆn) =

1

2(1 + cos

2_{θ) cos 2φ,} _F

× = cos θ sin 2φ. (1.44)

Given a polarisation state, ground-based interferometers are “blind” to some di-rections, which correspond to the wave deforming both the arms in the same way. In this case, in fact, no phase shift is produced. For example, in the case of the plus polarisation, the output signal O(t) vanishes for φ ∈ {π₄,3π₄ ,5π₄ , 7π₄ }.

The antenna pattern functions can be represented in a polar plot, which makes it explicit the coupling of the device to the different polarisations. Consider h+, for

example: for each direction one can plot a vector whose magnitude is F2

+(ˆn), where

the square is taken to avoid negative quantities. This produces a three-dimensional figure where, by construction, the farther from the origin the surface of the picture in one direction, the highest the value of the antenna pattern function is in that case. Therefore, points which lie in the origin of this plot correspond to blind directions for the given polarisation.

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1.5. Stochastic backgrounds

Fig. 1.4 illustrates such plots for the cases of plus and cross polarisation individually, while Fig. 1.5 shows the case in which both the polarisations are present. y x z (a) y x z (b)

Figure 1.4: Polar plot of the antenna pattern functions. For each direction ˆn, the distance from the origin of the related point of the surfaces is equal (a) to F₊2(ˆn), and (b) to F_×2(ˆn). Therefore, the two pictures represent the coupling of the interferometer to (a) the plus and (b) the cross polarisation as a function of the direction of propagation. A thermal map is used: the warmer the colour, the greatest the distance from the origin and hence the response to the given polarisation. All axes are dimensionless. Interferometer’s arms are aligned with the x and y directions.

1.5 Stochastic backgrounds

The detection of a stochastic background of gravitational waves is constrained by the fact that the expected signal is much lower than the intrinsic noise of the instrument. As a matter of fact, taking into account the noise of the detector, the output of an interferometer is

s(t) = h(t) + n(t) (1.45) where h(t) is the signal produced by the gravitational wave and n(t) is the noise of the detector. The signal has been defined in equation (1.35).

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y x

z

Figure 1.5: Polar plot of the antenna pattern functions for a wave with both the polarisation components. For each direction ˆn, the distance from the origin of the related point of the figure is equal to F2

+(ˆn) + F×2(ˆn). Since the two polarisation

states are independent, this quantity measures the coupling of the instrument to a generic wave, as a function of the direction of propagation. A thermal map is used: the warmer the colour, the greatest the distance from the origin and hence the coupling to the wave. All axes are dimensionless. Interferometer’s arms are aligned with the x and y directions.

As it can be seen looking at Figure 1.6, a stochastic background is an intrin-sically weak signal compared to the sensitivities of earthbound detectors. There are no realistic possibilities of discriminating a stochastic signal from the noise of a single detector.

The approach, then, is to correlate the outputs of at least two detectors, so that, owing to the spatial separation between the instruments their, respective noises become uncorrelated. In this way, the ensemble average of the product of the noises vanishes.

In practice, one considers the quantity

S = Z T /2 −T /2 dt Z T /2 −T /2 dt0s1(t)s2(t0)Q(t − t0), (1.46)

where si is the output of the i-th detector, T is the total observation time and

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Figure 1.6: Typical amplitudes of the stochastic backgrounds from different models are plotted together with the sensitivities of previous generation Virgo and LIGO and upper bounds. See [2] for a full discussion.

(optimal filtering). The signal-to-noise-ration (SNR), is defined as

SN R = hSi2 h(S − hSi)2_i 1/4 . (1.47)

As anticipated, for uncorrelated noises the ensemble average of their Fourier com-ponents is diagonal [26], meaning that

h˜n∗_i(f )˜nj(f0)i =

1

2δijδ(f − f

0

)S_ni(f ). (1.48) This also defines the noise squared spectral density Si

n. A similar definition is

given for the Fourier amplitudes appearing in (1.20):

hh∗_A(f, ˆn)hA0(f0, ˆn0)i =

1

2δAA0δ(f − f

0

)δ(2)(ˆn, ˆn0)Sh(f ). (1.49)

If the background is Gaussian, isotropic and stationary, the spectrum is fully characterised by the spectral energy density Ωgw(f ), in terms of which the power

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spectrum is

Sh(f ) =

2ρc

f3 Ωgw(f )2. (1.50)

For an optimal filtering, then, the previous results and decomposition (1.20) imply that the SNR can be rewritten as [26]

SN R = 2T Z ∞ 0 dfS 2 h(f ) S2 n(f ) Γ2(f ) 1/4 . (1.51)

The function Sn is defined as Sn =pSn1(f )Sn2(f ) for a two-detector correlation,

thus it depends only on the noise of the apparatus. Γ(f ), instead, is the overlap reduction function: its explicit form is

Γ(f ) = 1 4π

X

A

Z

d2ˆnF_A1(ˆn)F_A2(ˆn)e2πif d ˆm·ˆn, d2n = d cos θdφˆ (1.52) where d ˆm = x1− x2 is the vector separation between the detectors. If one

po-larisation only is present, the summation must be removed from the previous definition.

The overlap reduction function can be interpreted as a weighted average over all the possible instruments [29]: the exponential function weights the integrand depending on the delay of propagation between the detectors. Therefore, Γ de-pends on the geometry of the apparatus (relative orientation and position of the detectors), but it is independent of the spectrum of the signal.

On the other hand, the spectral density is determined exclusively by the spec-trum of the background, thus (1.51) can be rewritten as:

SN R = 8ρ2_cT Z ∞ 0 df f6 Ω2_gw(f ) S2 n(f ) Γ2(f ) 1/4 . (1.53)

This result implies that the observations time must be long enough to overcome the lower ratio between the spectrum of the signal and the spectrum of the noise of the apparatus. Therefore, the study of models which could source a sensible stochastic background is of particular interest.

For the Advanced Virgo and Advanced LIGO interferometers, the behaviour of pSn(f ) is represented in Fig.1.7.

A comparison between current and future sensitivities (using new generation detectors), instead, is illustrated in Fig. 1.8.

2_{In the case of scalar polarisation, the density Ω}

gwis related to the physical energy density

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Figure 1.7: The expected evolution of sensitivities for advanced detectors [3]. The distances quoted in the legend refer to a figure of merit which quantifies the sensitivity for a coalescence of binary neutron stars.

Figure 1.8: Sensitivities of past, current and future detectors. The current genera-tion corresponds to Advanced Virgo and Advanced LIGO.[23]

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Chapter 2 The model

In the following chapter, the action of the model is presented and the study of the unperturbed fields (background) is performed. The equations for the pertur-bations are derived through a linearisation procedure and a suitable gauge choice is made. The system is then investigated in order to determine the complete set of equations for the gravitational waves in the generalised theory.

2.1 The action of the model

Despite the achievements of general relativity, many efforts have been put into the development of alternative theories over the years. As pointed out in the introduction, they address to the issues of Einstein’s theory and try to provide possible extensions.

The aim of this thesis is to study the specific features of a modified-gravity theory, combined with a non-gravitational sector which has been shown a promising context for the production of gravitational waves. The action of the model is mainly based on the one presented by Dimastrogiovanni, Fasiello and Fujita in [13]. In their article, they proposed a modified chromo-natural inflation [4] in order to avoid frictions with current observations.

Chromo-natural inflation envisages a pseudo-scalar field φ subjected to an axion-like potential V (φ) ∼ 1 + cos φ_f, f being the decay constant. It is a well-known result that a field slowly rolling down its potential is a possible explanation for the inflationary epoch, the slow motion of the inflaton field triggering both the start and the duration of inflation [20]. As anticipated in the introduction, such stage is necessary to explain the flatness and horizon problems and must be long enough to agree with current observations. For a sub-Plankian decay constant f < Mpl, though, the potential gets steeper and the field rolls faster, not providing

a long-enough accelerated expansion. This can be avoided by embedding an SU (2) gauge field, Aa_µ, which couples to the inflaton through a term φF_µνa F˜aµν, where F is the field strength of the gauge field and ˜F its dual. Such coupling provides a friction term to the equation of motion of φ, which allows it to inflate long enough [4].

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Chapter 2. The model

A remarkable feature of this model is that the tensor fluctuations of the gauge field couple to that ones of the metric tensor and induce an enhancement in one of the helicities of the gravitational waves. This happens because during their evolution, the tensor modes experience an instability which leads to a transient growth in one of their polarisations [15]. As a result, the corresponding helicity of GWs to which they are coupled is enhanced as well, leading to a chiral signal. The chirality property is a specific signature of this model and it is in principle experimentally detectable [12].

Despite such successes, though, the model is unable to satisfy the current bounds on both the tensor-to-scalar ratio r and the scalar spectral index ns

to-gether. These quantities are the ratio between the power spectrum of gravitational waves and that one of curvature fluctuations (taken at the same conventional scale) and the tilt of the scalar power spectrum respectively and are constrained by CMB experiments

The solution proposed in [13] is to decouple the inflaton sector from the other fields, up to gravitational interactions. In their proposal, the authors considered a scalar inflaton φ, whose Lagrangian contains no direct couplings with the other sectors. An axion field χ is introduced, which interacts with the gauge sector through the term χFa

µνF˜aµν. Such field was first introduced by Peccei and Quinn to

explain the CP conservation in strong interactions [35] and is now a candidate for dark matter [36]. Both the inflaton and the axion are subjected to a potential of the type V (∗) ∼ 1+cos _f∗

∗. This set-up relaxes the friction with observational bounds,

while retaining the specific signature of a chiral enhancement of gravitational waves.

The action of the matter fields, then, is:

SM = Sφ+ Sχ+ SA (2.1) where Sφ = Z d4x√−g −(∂φ) 2 2 − V (φ) (2.2) Sχ= Z d4x√−g −(∂χ) 2 2 − U (χ) (2.3) and SA = Z d4x√−g −1 4F a µνF aµν₊ λχ 4fχ F_µνa F˜aµν . (2.4)

V (φ) and U (χ) are the potentials of the inflaton and of the axion respectively, and are of the form previously discussed, while Fa

µν and ˜Faµν are the field strength

tensor and its dual:

F_µνa = ∂µAaν − ∂νAaµ− ˜g abc_Ab µA c ν, F˜ aµν ₌ µνρσ 2√−gF a ρσ. (2.5)

In the previous equations, ˜g is the gauge coupling and abc _{are the structure}

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2.2. Evolution of the background

The original feature of this thesis is to consider a Brans-Dicke gravitational sector in place of the usual Einstein-Hilbert term. This means introducing a real scalar field non-minimally coupled to gravity: such modification was first introduced by Jordan and lately improved by C. Brans and R. H. Dicke [8]. One of their efforts was that of embedding Mach’s principle into the description of gravity. Basically Mach’s principle is the idea that the nature of a local inertial frame is determined by the matter distribution of the Universe. This suggests the possibility that what determines the motion of a system depends on the surrounding context: in other words, even physical “constants” may have a dependence on the space-time point at which the observer is placed, gravitational constant G included. In fact, if Mach’s principle is correct, one can show by a dimensional argument that G is related to the ratio between observable matter and the observable size of the Universe, which by no reason has to be fixed.

This is not the case in general relativity, but such a feature can be implemented by introducing a non-minimally coupled scalar field ϕ. It multiplies the scalar curvature, so that Rϕ replaces the usual1₂M2

pR, resulting in a variable gravitational

“constant”. Actually, only G is supposed to vary, so that all non-gravitational laws of physics are unaffected by the matter distribution of the surrounding Universe. In this way, non-gravitational laws in free-falling frames are independent of the rest of the Universe and the weak equivalence principle is preserved.

Therefore, the complete action considered in this thesis is:

S = Z d4x√−g Rϕ − ω ϕ(∂ϕ) 2 + SM. (2.6)

Since ϕ has the dimensions of a mass squared in natural units, its kinetic term is divided by the field itself so that the coupling constant ω is dimensionless. Actually, Brans and Dicke considered ϕBD≡ 16πϕ in their original formulation, which has

been rescaled in this context to avoid numerical factors in the non-gravitational sector of the action.

As pointed out in the introduction, scalar fields are expected when one attempts to generalise Einstein’s theory. The Brans-Dicke model is one of these possible extensions: besides embedding Mach’s principle, it also preserves the metric nature of the theory. This is important because the description of the space-time in terms of a metric tensor has proven a successful choice in explaining many observations.

The presence of the scalar field, though, brings about an additional degree of freedom, which is expected to become evident in the gravitational content of the theory. In particular, one expects a new polarisation state to appear, in addition to the two orthogonal helicities predicted by general relativity.

2.2 Evolution of the background

As anticipated, in standard theories of gravitation GWs are perturbations of the metric tensor. In this case, though, even the fluctuations of the Brans-Dicke scalar are expected to give rise to an additional polarisation state.

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In order to determine the related equations of motion, one can perform the linearisation procedure presented in section 1.1: since such method consists in linearising around the background (unperturbed) fields, these same quantities appear in the evolution of the perturbations. Therefore, a possible approach consists in determining first the behaviour of the background alone: the results are then inserted into the first-order equations, whose residual unknowns are the fluctuations.

The evolution of the unperturbed fields is obtained through the least action principle. Not to complicate the discussion, however, calculations of the equations are presented in Appendix A, while the computation of the energy-momentum tensor and of the related quantities is wholly performed in Appendix B. In this section, only results are given. Consider first the gravitational field: an arbitrary, infinitesimal variation is performed, which in turn induces a variation on the action:

gµν(x) −→ gµν(x) + δgµν(x) S −→ S + δgS.

(2.7)

The equation of motion for the metric then follows from the requirement that the first order variation of the action vanishes, δgS = 0, for each arbitrary δgµν which

vanishes on the boundary.

Applying this method to the action (2.6), one finds that

Rµν − 1 2gµνR = 1 2ϕTµν − ω ϕ2 1 2gµνg αβ_∂ αϕ∂βϕ − ∂µϕ∂νϕ + 1 ϕ ∇µ∂νϕ − gµνϕ . (2.8)

Note that one recovers general relativity if the scalar field is constant and ϕ = 1₂M2 p.

The same argument leads to the equations for all the other fields. Consider the Brans-Dicke scalar: then

ϕ(x) −→ ϕ(x) + δϕ(x) S −→ S + δϕS (2.9) which leads to 3 + 2ω ϕ ϕ = T 2ϕ, (2.10)

where the energy-momentum tensor Tµν appearing in both the equations is

Tµν =gµν −(∂φ) 2 2 − V (φ) − (∂χ)2 2 − U (χ) − 1 4F a ρσF aρσ + + ∂µφ∂νφ + ∂µχ∂νχ + Fµρa F a νσg ρσ . (2.11)

Now consider an infinitesimal variation of the axion field: χ(x) −→ χ(x) + δχ(x)

S −→ S + δχS.

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By means of the least action principle, one gets

∇µ∂µχ − ∂U (χ) ∂χ + λ 8fχ √ −g µνρσ_Fa µνF a ρσ = 0. (2.13)

The evolution of the inflaton, then, is obtained through the substitutions χ −→ φ

U (χ) −→ V (φ) fχ −→ fφ

λ −→ 0

(2.14)

in the previous expression. Therefore, it is

∇µ∂µφ −

∂V (φ)

∂φ = 0. (2.15)

The last field appearing in the action is the gauge field. Thus, one considers the variation

Aa_µ(x) −→ Aa_µ(x) + δAa_µ(x) S −→ S + δAS.

(2.16)

The least action principle, then, leads to

(DµFµν)a− λχ 2fχ √ −g µνρσ_(D µFρσ)a= λ 2fχ √ −g∂µχ µνρσ_Fa ρσ (2.17) where (DµFµν)a = ∇µFaµν+ ˜gabcAcµFbµν (DµFρσ)a = ∇µFρσa + ˜g abc_Ac µF b ρσ (2.18)

are the covariant derivatives of the SU (2) field strength.

2.2.1 Components

The equations above are still covariant and must be expressed in terms of the components of each field. As pointed out in the introduction, the metric tensor must describe an isotropic, homogeneous and spatially flat expanding Universe. The geometry which sums up all these requirements is the Friedmann-Lemaître-Robertson-Walker Universe, whose line element is

ds2 = −dt2+ a(t)2dxidxjδij, (2.19)

where a(t) is the scale factor giving the expansion. Therefore,

gµν = diag (−1, a2, a2, a2). (2.20)

The isotropy and homogeneity properties also imply that the background fields must depend on time only. Thus, only temporal derivatives survive when comput-ing explicitly the equations of motion.

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For what concerns the gauge field, one can follow the ansatz of reference [13]: Aa₀ = 0, Aa_i = a(t)Q(t)δ_ia (2.21) which has been shown to be the attractor solution in general relativity [30]. The addition of the Brans-Dicke scalar is expected not to spoil this condition, because it interacts with the gauge field merely through the metric.

As shown in Appendix C.1 and C.2, equation (2.8) specialised to µ, ν = 0 and to µ = i, ν = j leads to 6H2ϕ = ˙ φ2 2 + V (φ) + ˙ χ2 2 + U (χ) + 3 2(HQ + ˙Q) 2₊ 3 2g˜ 2_Q4_{+ ω}ϕ˙2 ϕ − 6H ˙ϕ, − 4 ˙H − 6H2 = 1 ϕ " ˙_φ2 2 − V (φ) + ˙ χ2 2 − U (χ) + 1 2(HQ + ˙Q) 2₊ 1 2g˜ 2_Q4 # + ωϕ˙ 2 ϕ2 − 2H ˙ ϕ ϕ+ 1 ϕ(2 ¨ϕ + 6H ˙ϕ) (2.22) where H(t) = ˙a a (2.23)

is known as Hubble parameter. All the other components are trivial identities because of the isotropy of the background.

For what concerns the Brans-Dicke scalar, by inserting the trace of the energy-momentum tensor (see Appendix B for its full calculation) into equation (2.10), one immediately has that

¨

ϕ + 3H ˙ϕ = 1 4ω + 6

h

4V (φ) − ˙φ2+ 4U (χ) − ˙χ2i. (2.24) For the axion and the inflaton fields, instead, Appendix C.3 shows that the evolu-tion is determined by ¨ χ + 3H ˙χ +∂U (χ) ∂χ + 3 λ˜g fχ (HQ + ˙Q)Q2 = 0 ¨ φ + 3H ˙φ + ∂V (φ) ∂φ = 0. (2.25)

Again, the equation for the inflaton was obtained from that one for the axion through the substitutions (2.14).

The component equation for the gauge field, instead, has been computed in Appendix C.4 for the choice ν = i and is

¨

Q + 3H ˙Q + ( ˙H + 2H2)Q + 2˜g2Q3 = ˜g λ fχ

χQ2. (2.26) Selecting ν = 0, one finds a trivial identity which does not add any information to the evolution of the field.

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The previous results are expressed in terms of dimensional quantities, therefore, one can rescale them all in order to obtain relations among dimensionless variables. This allows for the comparison of all the variables involved with the same energy scale.

Since the aim of the model is to study slight modifications to general relativity, which has been proven a valuable theory in many aspects, it seems natural to rescale the fields with the appropriate power of the reduced Planck mass:

1 2M 2 p = 1 16πG. (2.27)

If the Brans-Dicke field is constant and ϕ = 1₂M2

p, in fact, all the previous equations

reproduce the results of Einstein’s theory. Therefore, one may write:

A = _M2 p 2 d ˜ A, (2.28)

where A is a generic quantity and d represent its mass dimensions, so that ˜A is dimensionless.

The exponents for the different quantities are summarised in Table 2.1. After

quantity d ϕ 1 H 1/2 φ 1/2 χ 1/2 Q 1/2 t -1/2 fφ 1/2 fχ 1/2 µφ 1/2 µχ 1/2

Table 2.1: Scaling exponents of all the quantities entering the model.

the scaling, the constant (2.27) factorises from all the equations, so that the system expressed in terms of dimensionless quantities is formally equivalent to that one concerning the original fields. To lighten the notation, however, the superscripts will be dropped from now on and all the quantities appearing in the rest of the thesis must be understood as scaled according to Table 2.1, unless otherwise mentioned.

Extra polarisations of relic gravitational waves from a Brans-Dicke theory with axion-gauge dynamics

Università di Pisa

Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Extra polarisations of relic gravitational

waves from a Brans–Dicke theory with

axion–gauge dynamics

Candidate:

Supervisor:

Giulia Pagano

Dr. Giancarlo Cella

Contents

Introduction

Conventions and notations

Chapter 1

Overview: gravitational waves in

general relativity

1.1

Linearisation around flat space-time

1.2

The transverse-traceless gauge

1.3

Effects of GWs on test masses

1.4

Detector response

1.5

Stochastic backgrounds

Chapter 2

The model

2.1

The action of the model

2.2

Evolution of the background

2.2.1

Components