stochastic background of gravitational waves from amplification of vacuum fluctuations in a bimetric theory of gravity

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Dipartimento di Fisica E. Fermi

Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Stochastic background of gravitational waves

from amplification of vacuum fluctuations

in a bimetric theory of gravity

Candidato :

Relatore :

Chiara Animali

Dr. Giancarlo Cella

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Notations and Terminology

Constant and units

We adopt the convention of natural units, in which the speed of light, the Boltzmann constant and the Planck constant are set equal to one, c = ~ = kB≡1.

The Newton constant G acquires dimension of a squared mass and is linked to the reduced Planck mass MP l and to the reduced Planck lenght λP l by the relation

( 8 π G )−1_{= M}2

P l= λ

−2

P l

where is adopted the normalization with the factor 8 π commonly used in particle physics and cosmology.

Mass, energies and temperature will be commonly expressed in eV and its multiples, therefore in natural units

MP l= ( 8 π G )−1/2 '2.4 × 1018_GeV,

where 1GeV = 109_eV.

Indices, metric signature, four-vectors

Greek indices such as α, β, ... or µ, ν, ... run over the four spacetime coordinate labels 0, 1, 2, 3 or t, x, y, z.

Latin indices i, j, ... run over three spatial coordinate labels, usually 1, 2, 3 or x, y, z. Cartesian three-vectors are indicated by boldface type.

Our metric signature is (−, +, +, +), so that ds2 _{= −dt}2_{+ dx}2 _{for Minkoswi space, which is the}

signature commonly used in General Relativity.

An over-dot denotes derivative with respect to cosmic time, whereas a prime denotes derivative with respect to conformal time.

Partial derivative is denoted by ∂µψ or ψ,µ where convenient. Covariant derivative is denoted by ∇µψ or ψ;µ where convenient. Fourier Transform

Our conventions on the n-dimensional Fourier transform are

F(x) = Z _dn_k (2 π)nFe(k) e i k x_, e F(k) = Z dnx F(x) e−i k x.

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1

Introduction

Einstein theory of General Relativity (GR), together with Quantum Mechanics, is one of the cornerstones of modern physics. The theory, that was introduced by Einstein more than a century ago, allows to give a physical description of the current Universe, and explains the forces that stars, galaxies, galaxies clusters exert upon one another at the cosmic distance scales corresponding to their huge spatial separations.

The theory is built on the astonishing idea of describing gravity in terms of a geometrical tensor field, extending and generalizing the fundamental principles underlying special relativity: the physical laws, which in special relativity are the same in all inertial frames, become the same in all reference frames, disregarding the coordinates transformations between them. This feature encodes the so called principle of general covariance, according to which the physical laws have to be invariant under any coordinate transformation, and not only under Lorentz transformations between inertial observers. The geometrical consequence of this feature is a whole revolution of the concept of spacetime: from the rigid structure of the Minkowski spacetime of special relativity, one moves to a deformable and curved spacetime of Riemannian type described by a general metric.

It is really the connection between spacetime curvature and gravitational interaction the most fundamental innovation of the theory: the motion of bodies in a curved spacetime deviates from straight paths to follow curved trajectories, as if there is a force to which they are subject. Hence the gravitational interaction can be viewed as embedded into the spacetime geometry, and the Einstein equations, which represent the field equations of GR, are the expression of the link between spacetime curvature and gravitational properties of material bodies.

So far GR has been spectacularly successful, describing accurately the dynamics of astronom-ical objects over a vast range of sizes, and has passed all precision tests with extreme precision. However, most of these tests, with few possible exceptions, are probes of weak field gravity, i.e., they probe gravity at intermediate lengths ( 1 µm ≤ l ≤ 1AU ∼ 1011_{m ) and therefore at}

inter-mediate energy scales (see [1] for a review of the state of the art on experimental tests of GR). Experimentally, however, we do not know how gravity behaves at very short or long distances, and on such scales it could look completely differently from what stated by GR. On the other side there is growing theoretical evidence that modifications of the theory at both small and large distances are inevitable.

Indeed from the theoretical point of view, GR is a purely classical theory and does not know anything about quantum mechanics: this constitutes a limitation of the theory when trying

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to extend it to the description of regimes of extremely high energies, as the ones coming into play in the primordial Universe when approaching the Planckian era, when quantum effects can become relevant. So far, a consistent theory of gravitation even in a quantum fashion is not available, indeed GR can not be embedded into the framework of standard quantum field theory, as power counting arguments indicate that what is obtained is not renormalizable in the standard quantum field theory sense. Strong field modifications may provide a solution to this problem: the theory becomes renormalizable if we add quadratic curvature terms, i.e., high-energy corrections [2], however these theories manifest other problems as violation of unitarity and causality, at least if they are formulated in the usual way. Furthermore quantum theories of gravity, as loop quantum gravity and string theory, make specific and potentially testable predictions of how GR must be modified at high energies.

Moving to the field of cosmology, the standard picture for our present cosmological model, the ΛCDM model, relies on GR as the theory of gravity, and although successfully describes the expansion history of the Universe, still faces several problems. Recent high precision observations revealed that the expansion of our Universe is accelerating [3]. GR also predicts the expansion of the Universe as a whole, but the theory has fallen in explaining that the expansion takes place at an accelerating rate. According to GR, indeed, the sum of all known radiation, visible and dark matter should slow down the rate of expansion over time. To account for acceleration, physicist have been forced to introduce a cosmological constant, or vacuum energy, which counteracts gravity by exerting a constant negative effective pressure [4]. This interpretation however poses serious issues, commonly included in the so called "cosmological constant problem" [5]. In addition, in this model, about 95% of the energy content in our Universe is made of unknown constituents, commonly referred to as dark matter and dark energy (if not fully accounted for by the cosmological constant term Λ.)

Hence, it seems reasonable to consider modification of GR to cosmological scales, introducing low energy (infrared) corrections. All these arguments suggest that GR should be modified at both low and high energies, and such modifications should be consistent with GR in the intermediate energy regime, where the theory is well tested.

Theoretically, there are countless nonequivalent ways to modify GR [6], and also the experi-mental search for physics beyond GR is becoming an active and well motivated area of research. A very useful tool to classify all the possible modifications and extensions of the Einstein theory of gravity is given by the Lovelock theorem [7, 8], which naturally conducts to the Einstein equations, and consequently to the Einstein-Hilbert action for gravity, asserting that the only divergence-free symmetric rank-2 tensor built from the metric and its derivatives up to second order, and preserving diffeomorphism invariance, is the Einstein tensor plus the cosmological term. This theorem can be circumvented in different ways by violating its several underlying assumptions, providing a useful tool to classify modified theories of gravity (Figure 1.1).

In this thesis we take into consideration the class of modified gravity models in which extra fields are added, in particular we consider the additional field to be a tensor spin-2 field which has its own dynamics, so that the theory can be included in the class of bimetric models of gravity.

Commonly, when talking about bimetric gravity, one has in mind the Hassan-Rosen bigravity model [10], which describes in a fully covariant treatment the theory of two interacting spin-2 fields, of which one is massless and one is massive, which is ghost free and consistently stable. In this theory both the spin-2 fields are given its own dynamics, and the role of real physical metric is only selected by the choice of matter couplings. The importance of introducing the second metric lies in the fact that it becomes possible to construct a covariant nonlinear interaction term for a spin-2 field. Moreover, the interaction term, whose form is uniquely determined by a construction of polynomials of the two metrics, supplies the mass term for the graviton,

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Figure 1.1. Classification of possible modified models of gravity guided by the Lovelock theorem. Each

yellow box represents a class of modified theories of gravity which arise from the violation of one of the theorem’s assumptions. The branches represent the possible subclasses. (Figure taken from [9])

endowing the graviton with a non-zero mass in a not puzzling way, avoiding the propagation of a ghost mode [11] that for a long time had been the biggest obstacle to find a massive extension of gravity, starting from the early attempts by Pauli and Fierz [12].

Nevertheless, we focus on a bimetric model, recently introduced by Hossenfelder [13], that moves away from this standard picture of bimetric gravity well known in literature. In particular, it fulfills specific symmetry requirements: direct coupling terms between the two metrics are absent and both gravitons remain massless, making the theory symmetric under the exchange of the two metrics. An additional sector of matter, that could be a copy of the usual Standard Model, is embedded into the model, as a dark component whose dynamics is determined by the additional metric: the specific geometrical construction of the model and the double matter content make possible to switch the sign of the relation between inertial and gravitational mass of bodies, opening the possibility to have an antigravitating behavior of matter, therefore to have and additional source term to the Einstein equations with properties unlike those of standard matter, and that acts as an effective stress-energy tensor with a minus sign.

This model seems to have a good potential to understand some of the issues of the stan-dard cosmological model, for instance the problems related to the unknown constituents of the Universe and the incomplete understanding of its gravitational dynamics, opening also the pos-sibility to a different evolution history of the Universe, depending on the mixture of matter and antigravitating matter which could govern the evolution of the Universe during time, however it has not been studied in a thorough way. Therefore, after having analyzed and set the con-struction of the symmetric bimetric model and its cosmological background solutions, we start investigating its cosmology studying the perturbations around such background solutions at the linear order.

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titative cosmology, since it is the framework which provides the link between models of the very early Universe and the wealth of high-precision cosmological data on the large-scale-structure of the Universe (mainly the Cosmic Microwave Background (CMB) is the observational window which has yielded the most information in recent years). In particular, we focus on the tensor part of the perturbations, studying the gravitational waves (GWs) of cosmological origin arising in the model. Such gravitational waves are amplified by the phase of inflationary expansion characterizing the evolution of the primordial Universe during the first 10−36_{s, and give rise to}

the primordial stochastic background of gravitational waves (SGWB) which is actually expected to permeate the whole spacetime [14].

It is well known that, as in the context of electromagnetism, where, according to Maxwell equations, the oscillations of electromagnetic and magnetic fields propagate from one point to another at the speed of light, in the context of GR, according to Einstein equations, oscillations in the geometry can propagate from one point to another at a speed which, -in vacuum- coincides with the speed of light. As electromagnetic waves are produced by accelerated motion of electrical charges, GWs are produced by the accelerated motion of masses, which generate perturbations of the local geometry that propagate and are transmitted to the whole surrounding spacetime as a wave, bringing information about how the gravitational field (i.e., the curvature of the spacetime) varies with time.

Processes characterizing the primordial Universe are associated with a copious production of GWs which have filled the Universe and should still be present as relics of the primordial cosmological epochs. In this case the emission of GWs is not linked to the motion of accelerated masses, instead it is the whole spacetime which accelerates and produces GWs, according to a mechanism called parametric amplification of vacuum fluctuations. To understand the origin of this phenomenon is necessary to turn to the quantum mechanical description of fields in a curved spacetime.

Indeed, if we consider the spacetime geometry at the classical and macroscopic level, it is fully determined by the mass and the energy distribution of the gravitational sources. However, at the microscopic level, spacetime can be viewed as a "sea" which shows a huge number of ripples, due to continuous fluctuations in the geometry, according to quantum mechanics, for which all types of fields (including the gravitational field, hence the geometry), can fluctuate. These fluctuations can be considered as small virtual GWs which are not freely propagating, but which are continuously emitted and reabsorbed by the spacetime itself: in a quantum language these tiny perturbations of the geometry can be considered as virtual gravitons that are produced and suddenly destroyed, and which are always produced in pairs to avoid conservation laws violations.

The crucial point of the amplification mechanism is that, if the geometry is static and without horizons, for instance in the case of the flat and empty Minkowski space, the pairs are formed and destroyed, leading to a null net result, i.e., in average the number of gravitons is still zero. In some situations, if the geometry expands rapidly, as happens during inflation, is possible for two gravitons to be dragged away from one another so rapidly that they are no longer able to come back together and annihilate, leading to a net production of gravitons, and therefore of GWs, directly form the spacetime itself. This is in strong analogy with the mechanism leading to the production of black hole radiation due to creation of virtual pairs of particles in the region near the event horizon of a black hole [15]; the difference is that, in the cosmological case, the virtual pair is separated not by the black hole horizon, but by the Hubble horizon associated to the phase of accelerated expansion. Indeed, during the inflationary expansion, the physical wavelengths of these fluctuations increase and are stretched until they exit the Hubble radius RH ≡1/H(t), which sets the dimension of the causally connected region, which instead remains quasi-constant in time. On such scales the amplitude of the fluctuations remains frozen, therefore, at the end

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of inflation, when they re-enter the horizon, it results in an effective amplification of the wave with respect to the fluctuations that always remained sub-horizon.

The importance of studying such a production mechanism is that the number of particles (that in our case are gravitons but could be photons, dilatons etc.) produced as a function of their energy, named spectral distribution, give a sort of snapshot of the primordial Universe taken at different times and energies, therefore the combination of the various information encoded in these spectra make possible to reconstruct the history of our Universe.

Particularly, the stochastic background of GWs is potentially very interesting because due to the weakness of gravity, GWs are decoupled from the rest of matter and radiation components in the Universe upon production. Therefore they can propagate freely toward us maintaining their original spectrum, carrying pristine information about the state of the Universe at epochs and energies unreachable by other experiments and giving us a snapshot of the Universe about 50 orders of magnitude younger that the Universe we see in the CMB (which gives us the earliest electromagnetic view of the Universe at the time of recombination, about 4 × 105 _{years after the}

Big Bang).

CMB Gravitational Waves

Photons SGWB

Figure 1.2. Key events in the history of the Universe and their associated time and energy scales, with

the investigation tool and probes available in each regime. It is evident the difference between the energy regime of the CMB photons and the GWs of the stochastic background, coming from energies exceeding the energy currently reached by particle accelerators. [Credits to: Particle Data Group, LBNL, 2015]

In the context of GR, the commonly used model of single-field slow-roll inflation, in which the epoch of accelerated expansion is supplied by the potential energy of a scalar field which slowly evolves toward the minimum of its potential, predicts a quasi-scale invariant spectrum for

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tensor perturbations, with a very small red tilt, close to the Harrison-Zeldovich scale invariant power spectrum predicted by the exact de Sitter solution, which results suppressed with respect to the scalar one and whose deviations depend on the specific inflationary dynamics [16].

These features, at the present time, set the SGWB power spectrum far below the sensitivity of present ground-based and proposed future space-based interferometers, as LISA [17]. Given the great potential of the SGWB in exploring the very early Universe and in shedding light on the specific mechanism of inflation, is thus appealing to consider modifications of the underling theory of gravitation, and/or going beyond the single-field slow-roll inflationary scenario: this is well motivated from the theoretical point of view, since our knowledge about gravity and the expansion history of the Universe is incomplete.

On the other hand, this is also phenomenologically interesting since different predictions on the power spectrum of the inflationary GWs could place it in the frequency range accessible to LISA and to next-to-next generation of space-based observatories [18], and the features of the sourced GWs could reflect properties of the underlying theory of gravity different from those of GR.

The thesis is organized as follows.

In Chapter 2 we introduce how GWs emerge from GR, and what are their properties, using the most straightforward approach of linearized theory of gravity, then using standard tools of GR such as the geodesic equation and the geodesic deviation we show how these waves interact with a detector. We then introduce the SGWB, focusing on the cosmological one, we present several cosmological sources that can source such a primordial background and we discuss some aspects related to its detection, showing the currents upper bounds on the amplitude of several SGWB from different sources, and the potential of current and planned ground-based and space-based detectors in a future detection.

In Chapter 3 we introduce the modified model of gravity under our investigation: after a concise discussion of some aspects related to the introduction of the second metric and of the additional copy of the SM sector, we present in detail the construction of the theory with all the necessary stuff to write the action of the model. The Einstein equations are derived and the FRW cosmological solutions for the two metrics are presented, together with a discussion about their modification with respect to the standard case of GR with the consequent implications about the evolution of the Universe.

In Chapter 4 we focus on the theory of cosmological perturbations: after a review in the context of GR, we apply the theory of cosmological perturbations to the model, in order to determine the evolution equations for tensor perturbations of both metrics. These describe the propagation of GWs of both types travelling across cosmological distances, and the only difference from the case of GR is the presence of an additional anisotropic stress contribution due to the additional matter source. Since we are interested in the amplification mechanism of such perturbations due to a period of inflation, at first we introduce the mechanism of amplification of vacuum fluctuations appealing to the quantization of fields in a curved background in a very general setting, then we show a basic example of computation in the case of GR for the simplest inflationary model described by a de Sitter scenario. Nevertheless, a more realistic model of inflation is described by the slow-roll inflationary scenario. After a briefly introduction of it for the case of GR, we find a generalization of slow-roll inflation for the bimetric model, we discuss its validity conditions and we define the modified slow-roll parameters. Is then presented, at first-order in the slow-roll expansion, the spectrum of the GW background of inflationary origin. In Chapter 5 the conclusion and some discussions are offered, focusing also on potential future investigations of the model.

In Appendix A we propose a compendium about the subject of differential geometry, focusing on differentiable manifolds and on Riemannian geometry, whereas Appendix from B to F are

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2

Gravitational waves in

General Relativity

2.1 Linearized theory of gravity

A natural way to introduce GWs in GR is that of linearized gravity, and consists of expanding the Einstein equations around the flat Minkowski metric ηµν, considering a small perturbation hµν over the fixed background

gµν(x) = ηµν+ hµν(x) , |hµν(x)| 1 , (2.1)

then expanding the equations of motion to linear order in hµν. The resulting theory is called linearized theory.

GR is invariant under a huge symmetry group, the group of all possible coordinate transfor-mations

xµ→ x0µ(x) , (2.2)

where x0µ _{is an arbitrary smooth function of x}µ_{. More precisely, we require x}0µ_{(x) to be}

in-vertible, differentiable, and with differentiable inverse, i.e. x0µ _{is an arbitrary diffeomorphism.}

Under (2.2), the metric transforms as gµν(x) → g0 µν(x 0_{) =} ∂xρ ∂x0µ ∂xσ ∂x0ν gρσ(x) . (2.3)

We can call this symmetry as the gauge symmetry of GR. The condition |hµν| 1 implies that only weak gravitational fields are allowed and that only a restricted set of coordinate systems where (2.1) holds are admitted.

Choosing a reference frame breaks the invariance of GR under coordinate transformations, however, after choosing a frame where (2.1) holds, a residual gauge symmetry remains. If we consider the transformation of coordinates

xµ→ x0µ= xµ_{+ ξ}µ_{(x) ,} _(2.4)

where the derivatives |∂µξν| are at most of the same order of smallness as |hµν|, using the transformation law of the metric (2.3), we find that the transformation of hµν, to lowest order, is

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2.1 Linearized theory of gravity 9

If |∂µξν|are at most of the same order of smallness as |hµν|, the condition |hµν| 1 is preserved, hence these slowly varying diffeomorphism are a symmetry of the linearized theory. 1

To linear order in the perturbation hµν, the Christoffel symbols are Γρ µν = 1 2gρσ ( ∂νgσµ+ ∂µgσν− ∂σgµν) = 1 2 ∂νhρ_µ+ ∂µhρ_ν− ∂ρhµν . (2.6) Using this expression we can write, always to linear order, the Riemann tensor, the Ricci tensor and the Ricci scalar as

Rρ_µνσ = ∂νΓρ_µσ− ∂σΓρ_µν = 1 2 ( ∂µ∂νhρσ + ∂σ∂ρhνµ− ∂ν∂ρhµσ− ∂σ∂µhρν) , (2.7) Rµν= Rρµρν = 1 2 ∂ν∂ρhρµ+ ∂µ∂ρhνρ− ∂µ∂νh − hµν , (2.8) R= Rµ_µ= ( ∂ρ∂σhρσ− h ) , (2.9) where h ≡ hρ

ρ is the trace of the metric perturbation, ≡ ∂ρ∂ρ and indices are raised and lowered by the Minkowski metric. From these expressions we can construct the Einstein tensor, again to first order in the metric perturbation, as

Gµν = Rµν−1₂ηµνR = 1₂ ∂ν∂ρhρ_µ+ ∂µ∂ρhνρ− ∂µ∂νh − hµν− ηµν∂ρ∂σhρσ+ ηµνh = 1₂ ∂ρ∂νhρµ+ ∂ρ∂µhνρ− hµν− ηµν∂ρ∂σh ρ σ , (2.10)

where in the last line, for convenience, we have introduced a new metric perturbation

hµν ≡ hµν−1₂ηµνh . (2.11)

As the trace of hµνhas opposite sign to that of hµν,2_hµν _{is referred to as the trace-reversed metric}

perturbation (writing Gµν in terms of hµν has the advantage that the trace terms disappear). The expression of Gµν in eq. (2.10) can be further simplified exploiting the invariance of the linearized theory under the slowly varying infinitesimal coordinate transformations in eq. (2.4), indeed the trace-reversed perturbation transforms as

h0µν(x0) = hµν(x) + ξµν(x) , ξµν(x) ≡ ηµν∂ρξρ− ∂µξν− ∂νξµ. (2.12) In light of the expression of Gµν in terms of hρσ it seems convenient to make a coordinate transformation such that the metric perturbation verifies

∂µhµν(x) = 0 . (2.13)

This gauge choice is known as the Lorentz gauge (also called the Hilbert gauge, or the harmonic gauge, or the De Donder gauge) and it is always possible. To prove this, let us start with an

1_{Notice that under a Lorentz transformation x}0

µ= Λµνxν, g0µν(x 0 ) = Λ ρ µ Λνσgρσ(x), preservation of eq. (2.1) requires ΛµρΛνσhρσ(x)

1, so that it remains true that h

0

µν(x

0

) 1. Rotations do not spoil the condition |hµν| 1, but boost could, therefore they must be restricted to those that do not spoil such condition. As hµν(x)

is invariant under constant displacements x0µ → xµ

+ aµ, linearized gravity is also invariant under Poincaré transformations.

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arbitrary perturbation hµν for which ∂µ_h

µν 6= 0. The Lorentz gauge condition, using eq. (2.12), transform as

∂0µh0_µν(x0) = ∂µhµν(x) − ξν, (2.14)

so that we can always require that ∂0µ _h0_µν(x0_{) = 0, as long as}

ξν = fν(x) , fν(x) ≡ ∂µhµν(x) . (2.15)

Since the d’Alambertian operator is invertible, is always possible to find solutions to the above equation.

Restricting the coordinate systems to those that verify the Lorentz-gauge condition leads to a very simple expression for the Einstein tensor

G(L)_µν = −1

2 hµν, (2.16)

where (L) _{denotes the Lorentz gauge. In this gauge the linearized Einstein equation reduces to}

hµν = −16 π G Tµν, (2.17)

which has the form of a wave equation in presence of a source.

We can note that the Lorentz gauge condition in eq. (2.13) gives four conditions that re-duce the ten independent components of the symmetric 4 × 4 matrix hµν to six independent components.3

2.1.1 The transverse-traceless gauge

Equation (2.17) is the basic result for computing the generation of GWs within linearized theory. We now restrict to vacuum asymptotically flat spacetimes, therefore we assume Tµν(x) = 0 at every spacetime point and hµν →0 as |x| → 0, so that equation (2.17) becomes

hµν = 0 . (2.18)

We can further observe that the Lorentz condition (2.13) does not fix the gauge completely, indeed under an infinitesimal coordinate transformation the trace-reversed metric perturbation transforms as in equation (2.14). Therefore the condition ∂µ_h

µν = 0 is not spoiled by a further coordinate transformation xµ_{→ x}µ_{+ ξ}µ _{with ξµ}_{= 0. If ξµ}_{= 0 then also ξµν} _{= 0 (where} ξµν is defined in eq. (2.12)), since the flat space d’Alambertian commutes with the partial derivatives. Hence we can consider a further transformation ξµ appropriately chosen to verify ξµ = ξµν = 0, and use it to impose four conditions over hµν, in order to eliminate four degrees of freedom.

Therefore, actually, only two independent degrees of freedom are present, which are the truly physical propagating components of the metric perturbation. In particular it is common to choose ξ0 _{such that the trace h = 0, which implies the equality hµν} _{= hµν} _{and the three}

functions ξi_{(x) such that h}0i_{= 0.}

The Lorentz condition (2.13) with µ = 0 reads

∂0h00+ ∂ih0i= 0 , (2.19)

and, since we have fixed h0i = 0, simplifies to ∂0h00 = 0, i.e h00 becomes constant in time. A

time-independent term h00 corresponds to the static part of the gravitational interaction, i.e. 3_{We observe that equations (2.13) and (2.17) together imply for consistency ∂}ν_T

µν = 0, which is the

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to the Newtonian potential of the source which generates the GW. As GWs are only concerned with the time-dependent part of the gravitational interaction, we may set h00= 0.

Therefore we have fixed all the four components h0µ = 0 and we have left only the spatial

components hij for which the Lorentz gauge condition and the vanishing trace condition read respectively as ∂i_h

ij = 0 and hii= 0. In summary, we have specialized the gauge to

h0µ = 0, hii = 0, ∂ihij = 0, (2.20)

that defines the transverse-traceless gauge or TT gauge. In this gauge, the counting of the residual degrees of freedom becomes clearer: having eliminated all temporal components hµ0, we are left with six degrees of freedom in the spatial components hij. From these, three are further eliminated from the transversality conditions ∂ihij = 0 and one from the traceless condition hi_i = 0. Hence, once the TT gauge is adopted, the gauge freedom is saturated and two surviving propagating degrees of freedom remain.

Equation (2.18) has plane wave solutions, hT T

ij (x) = eij(k) eikx,with kµ= (ω, k) and ω = |k|. In these definition eij(k) is called the polarization tensor of the wave.

As an example, we can consider a single plane wave with wave-vector k propagating in direction ˆn = k/|k|. As we can see from equation (2.20), the non-zero components of hT T

ij are in the plane transverse to ˆn, indeed on a plane wave the transversality condition ∂i_h

ij = 0 becomes nihij = 0. Without loss of generality, we can fix ˆn along the z axis, then imposing hij to be symmetric and traceless we have

hT T_ij (t, z) =    h+ h× 0 h× −h+ 0 0 0 0    ij cos ( ω ( t − z ) ) , (2.21)

where h+ and h× are called the amplitudes of the plus and cross polarizations of the wave.

In terms of the interval ds2 _{we can write}

ds2= −dt2+ dz2+ [1 + h+ cos ( ω ( t − z ) ) ] dx2+

+ [ 1 − h+ cos ( ω ( t − z ) ) ] dy2+ 2 h× cos ( ω ( t − z ) ) dx dy.

(2.22) It is now useful to define the plane wave expansion in the TT gauge, where hT T

ij can be expanded as hT T_ij (x) = Z _d3_k (2π)3 A_ij(k) ei k x+ A∗_ij(k) e−i k x. (2.23) Since kµ _{is related to ω by k}µ _{= (ω, k), with |k| = ω = 2πf and k/|k| =}

b

n, we can write

d3_k_{= |k|}2_{d|k| d}_{Ω = (2 π)}3_f2_{df d}_{Ω, with f > 0. Denoting d}2 b

n = d cos θdφ the integration over

the solid angle, the above expansion reads hT T_ij (x) = Z ∞ 0 df f2 Z d2ˆn A_ij(f,n_b) e−2 π i f ( t−ˆn·x )+ c.c. . (2.24) We can observe that the contribution written in parentheses and its complex conjugate refer to a wave traveling in the +ˆn direction, since the only dependence is on the combination (t − ˆn · x), and that only frequencies f > 0 enter in the expansion. The TT gauge condition (2.20) gives Ai_{i(k) = 0 and k}i_A

ij(k) = 0.

If we consider a GW emitted by a single astrophysical source,4 _{the direction of propagation}

of the wave ˆn0 is well defined and we can express Aij(k) as

A_ij(k) = Aij(f) δ(2)(ˆn − ˆn₀) , (2.25)

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Of course, in a superposition of waves with different propagation directions, hij does not reduce to a 2×2

matrix (for instance h12 gets contributions from the waves with k3 _{6= 0, h13} _{gets contributions from the waves} with k26= 0 and so on): this is important when one considers the study of stochastic backgrounds of GWs.

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so that the expansion (2.24) reads hT T_ij (t, x) = Z ∞ 0 df ˜hij(f, x) e−2 π i f t+ ˜h∗ij(f, x) e2 π i f t , (2.26) with ˜hij(f, x) = f2 Z d2ˆn Aij(f, ˆn) e2 π i f ˆn·x = f2A_ij(f) e2 π i f ˆn0·x, (2.27) where, because of the transversality condition, the only non-zero components are in the plane transverse to the propagation direction ˆn0.

For ground based detectors the linear dimension is much smaller than the reduced wavelength ¯λ = λ/(2π) of the GWs to which they are sensitive, hence choosing the origin of the coordinate system on the detector, we have exp{2π i f ˆn · x} = exp{i ˆn · x/¯λ} ' 1 all over the detector. Therefore we can neglect all the x-dependency and simply write

hT T_ij (t) = Z ∞ 0 df ˜hij(f) e−2 π i f t+ ˜h∗ij(f) e2 π i f t , (2.28)

with ˜hij(f) = ˜hij(f, x = 0). From equation (2.21) it follows that ˜hij(f) =    ˜h+(f) ˜h×(f) 0 ˜h×(f) −˜h+(f) 0 0 0 0    ij . (2.29)

Notice that the + and × polarizations are defined with respect to a given choice of axes in the transverse plane. If we rotate by an angle ψ the system of axes used for their definition, h+

and h× transform as

h+→ h+ cos 2ψ − h× sin 2ψ, h×→ h+ sin 2ψ + h× cos 2ψ . (2.30)

Moreover, until now we have only considered positive frequencies, however we can extend the definition of ˜hij(f, x) to negative frequencies defining

˜hij(−f, x) = ˜h∗ij(f, x) , (2.31)

so that equation (2.28) becomes

hT T_ij (t) =

Z ∞

−∞ df ˜hij(f) e −2 π i f t

. (2.32)

We can further define another useful form for the plane wave expansion introducing the polarization tensors eA

ij(ˆn), with A = +, × as

e_ij+(ˆn) = ûiûj −ˆviˆvj, e_ij×(ˆn) = ûiˆvj + ˆviûj, (2.33) where û, ˆv are unit vectors orthogonal to ˆn and to each other. Their normalization is eA

ij(ˆn) eA

0_{, ij}

(ˆn) = 2 δA A0_{. Choosing ˆn = ˆz, ˆu = ˆx and ˆv = ˆy}

e_ij+=    1 0 0 0 −1 0 0 0 0    ij , e_ij×=    0 1 0 1 0 0 0 0 0    ij . (2.34)

We can define the amplitude in a generic frame as f2Aij(f, ˆn) =

X

A = +, ×

˜hA(f, ˆn) eA

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2.2 Interaction of GWs with test masses 13

so that equation (2.24) becomes hT T_ij (t, x) = X A = +, × Z ∞ −∞ df Z d2ˆn ˜hA(f,nb) e A ij(ˆn) e−2 π i f ( t−ˆn·x ), (2.36) where ˜hA(−f, ˆn) = ˜h∗ A(f, ˆn).

2.2 Interaction of GWs with test masses

Until now we have introduced the description of GWs, now we want to discuss the interaction of GWs with a detector, which we idealize for the moment as a set of test masses. We start briefly recalling some basic notions of GR, for more details and proofs refer to [19, 20].

In a curved background described by the metric gµν, and in absence of external non-gravitational forces, the equation of motion of a test mass is 5

d2xµ dτ2 + Γ µ_νρ(x)dxν dτ dxρ dτ = 0 , (2.37)

which is called the geodesic equation. We can consider two nearby geodesics, parameterized by xµ(τ) and xµ(τ)+ξµ(τ) respectively: the first satisfies the geodesic equation (2.37), whereas the second the geodesic equation

d2(xµ+ ξµ) dτ2 + Γ µ_{νρ(x + ξ)}d(xν+ ξν) dτ d(xρ+ ξρ) dτ = 0 . (2.38)

If |ξ| is much smaller than the typical scale of variation of the gravitational field, taking the difference between eqs. (2.37) and (2.38) and expanding to first order in ξ we obtain

d2ξµ dτ2 + 2 Γ µ νρ(x) dxν dτ dξρ dτ + ξ σ_∂σ_Γµ νρ(x) dxν dτ dxρ dτ = 0 . (2.39)

This equation can be rewritten in a more elegant way by using the covariant derivative along the curve xµ_(τ) D2ξµ Dτ2 = −R µ νρσξρ dxν dτ dxσ dτ = −R µ νρσξρuνuσ, (2.40)

where uν _{≡ dx}ν_/dτ _{is the four-velocity. This equation is called the equation of geodesic deviation} and shows that two nearby geodesics experience a tidal gravitational force, determined by the Riemann tensor.

2.2.1 The TT frame

In subsection 2.1.1 we have introduced the TT gauge, in which GWs have an especially simple form. We can denote the corresponding frame as the TT frame, and we can consider which is the physical meaning to be in this reference frame by looking at the geodesic equation. Therefore we consider the geodesic equation for a test mass initially at rest: from eq. (2.37) we obtain

d2xi dτ2 |τ = 0 = − Γi_νρ(x)dxν dτ dxρ dτ τ = 0 = −  Γi₀₀ dx0 dτ !2  τ = 0 , (2.41)

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where we used that dxi_/dτ _{= 0 at τ = 0. Using the expression of the Christoffel symbols in the} linearized theory given in eq. (2.6) one obtains

Γi

00=

1

2 (2 ∂0h0i− ∂ih00) , (2.42)

but this quantity in the TT gauge vanishes, because both h00 and h0i are set to zero by the

gauge condition. This means that also d2_xi_/dτ2 _{vanishes, and consequently dx}i_/dτ _{is zero at all} times.

We have obtained that in the TT frame, if a mass is at rest at the initial time, it remains at rest even during and after the passage of the GW, i.e. the coordinates of the TT frame stretch themselves in response to the passage of the wave, in order to make possible that the position of free test masses initially at rest does not change. 6

A physical implementation of the TT gauge can be achieved using the free test masses themselves to define the coordinates: one mass can be used to define the origin, then the other masses to define points with determined coordinates. During the passage of the GW, these masses still mark the origin and the points they defined before the passage of the wave, thus also their coordinate separation must remain constant. We can check explicitly, using the equation of geodesic deviation in the TT frame, that the separation ξi_{between the coordinates of two test} masses initially at rest does not change. We can use the spatial component µ = i of eq. (2.39) that gives d2ξi dτ2|τ = 0 = − 2 Γi 0ρ dξρ dτ + ξ σ_∂ σΓi00 τ = 0 . (2.43)

Now, in the TT gauge Γi

00 vanishes, hence the last term is equal to zero, whereas in the first

term the ρ index has to be a spatial one, therefore from (2.6) we can obtain Γi

0j = (1/2) ∂0hij: this implies that equation (2.43) becomes

d2ξi dτ2|τ = 0 = − ˙hij dξi dτ ! τ = 0 , (2.44)

and therefore, if we have dξi_/dτ _{= 0 at τ = 0, also d}2_ξi_/dτ2 _{= 0, and the separation ξ}i _remains constant at all times.

Obviously the example of the TT gauge does not means that the passage of GWs has no physical effect, but only expresses that in GR physical effects are not manifest in what happens to the coordinates, since the theory is invariant under coordinate transformations (we have exactly used this freedom to choose a frame in which the coordinates do not change).Physical effects can be found in considering proper distances. As instance we can consider two events at (t, x1,0, 0)

and at (t, x2,0, 0) respectively: the proper distance s between them is from eq. (2.22)

s= ( x2− x1) ( 1 + h+cos(ωt) )1/2' L

1 +1₂h+ cos(ωt)

, (2.45)

where L = x2 − x1 and we have only retained the term linear in h+. Therefore, even if the

coordinate distance L remains constant in time, the proper distance s changes periodically in time because of the passage of the GW.

6

This is true at linear order, if we also include terms O(h2), then Γi00 no longer vanishes. Nevertheless on Earth one typically expects GWs with at most h = O(10−21), hence going beyond the linear order is here not interesting [19].

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2.2.2 The proper detector frame

The TT frame is not the frame commonly used by experimentalists to describe the experimental apparatus, indeed, in a laboratory, positions are not marked by freely falling particles as in the TT frame, rather, after choosing an origin, one ideally uses a "rigid ruler" to define the coordinates. Hence in this frame a test mass which is free to move will be displaced by the passage of GWs. The proper detector frame puts in evidence the changes in the proper distance: it can be appropriate to give a description of an interferometer using it, when the interferometer’s size is much smaller than the typical wavelength of the GW as is the case for earth–bound detectors in their sensitivity band. In this case one can approximate the entire detector to be in a local Lorentz frame ( freely falling frame) even in the presence of GWs, thus, if one restricts to a sufficiently small region of space can choose coordinates (t, x) so that the metric in nearly flat ds2' −dt2_{+ δij}_dxi_dxj_{+ O} xixj L2 B ! , (2.46)

where LB denotes the typical variation scale of the metric.

We consider two test masses in free fall separated by ξi _{and we want to know the influence} of GWs on these two test masses. Hence we consider the equation for geodesic deviation given by eq. (2.39) and we rewrite it as

d2_ξµ dτ2 + 2 Γ µ νρ dxν dτ dxρ dτ + ξ σ_Γµ νρ,σ dxν dτ dxρ dτ = 0 . (2.47)

We assume that the two test masses are moving non-relativistically so that we can neglect dxi_/dτ _{compared to dx}0_/dτ_{, moreover the term proportional to Γ}µ

νρ is negligible in a freely falling frame, consequently eq. (2.47) reduces to

d2_ξi dτ2 + ξ σ_Γi 00,σ dx0 dτ !2 = 0 . (2.48) Moreover placing ξσ_Γi

00,σ ≈ ξjΓi00,j and since in a freely falling frame we have Ri0j0= Γi00,j−

Γi

0j,0= Γi00,j we can further simplify to

d2ξi dτ2 + R i 0j0ξj dx0 dτ !2 = 0 . (2.49)

If we limit to linear order in h we can write t = τ, so that dx0_{/dτ ≈}_{1, and this allows to write}

¨ξi _{= −R}i

0j0ξj, (2.50)

where the dots denote the derivatives with respect to the coordinate time t. Finally, we have to compute the Riemann tensor Ri

0j0: since in the linearized theory the Riemann tensor is

invariant, we can compute it in an arbitrary frame. For simplicity the best choice is the TT frame, where it has the simple form

Ri_0j0= Ri0j0= −1

2¨hT Tij . (2.51)

Inserting eq. (2.51) in eq. (2.50), the geodesic deviation equation in the proper detector frame takes the form

¨ξi ₌ 1

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This equation simply states that in the proper detector frame the effect of GWs on a point particle of mass m can be described in terms of a Newtonian force, allowing us to analyze the response of the detector to GWs in a purely Newtonian language.

At this point it seems necessary to make a clarification: the geodesic deviation equation is only valid if ξ is small compared to the typical variation scale of the metric. For a GW, this length-scale is the reduced wavelength ¯λ, therefore if a detector has a characteristic linear size L, the condition

L ¯λ (2.53)

has to be satisfied. For instance, for LIGO or Virgo ( ¯λ ≈ 105_{m and L ≈ 10}3_{m ), the geodesic}

deviation equation is valid, whereas for LISA ( ¯λ ≈ 1010_{m and L ≈ 10}9_{m ) is no longer valid and}

a full GR treatment is needed to study the influence of GWs on masses [19].

2.2.3 Ring of test masses

At this point we can use eq. (2.52) to study the effect of GWs on test masses and we use the proper detector frame, hence we consider a ring of test masses initially at rest and we fix the origin in the center of the ring: in this way ξi _{describes the distance of a test mass with respect} to the fixed origin. Eq. (2.52) describes the evolution of these positions due to the passage of a GW. For simplicity we consider a GW propagating along the z direction and a ring of test masses located in the (x, y) plane, centered at z = 0. Since hT T

ij is transverse to the propagation direction, the GW will only have influence in the (x, y) plane and the displacements will be confined on it.

We first consider the + polarization (choosing the origin of time such that hT T

ij = 0 at t = 0) given by hT T_ij = h+ sin ωt    1 0 0 0 −1 0 0 0 0    . (2.54)

We denote the location of a test mass as ξa(t) = ( x0+ δx(t), y0+ δy(t) ), with (x0, y0)

unper-turbed positions and δx(t), δy(t) the displacements. Eq. (2.52) gives

δ¨x = −h+

2 ( x0+ δx ) ω2 sin ωt , (2.55a)

δ¨y = +h+

2 ( y0+ δy ) ω2 sin ωt . (2.55b)

Neglecting the terms δx, δy which are O(h) with respect to the constant parts x0, y0, the above

equations can be easily integrated, obtaining δx(t) = h+

2 x0 sin ωt , (2.56a)

δy(t) = h+

2 y0 sin ωt . (2.56b)

Similarly for the cross polarization, we get δx(t) = h×

2 y0 sin ωt , (2.57a)

δy(t) = h×

2 x0 sin ωt . (2.57b)

These deformations can be visualized by considering a ring of test masses located in the (x, y) plane as shown in Figure 2.1

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2.3 Detector response to GWs 17

Figure 2.1. The deformation of a ring of test masses caused by the + and × polarizations of a GW for

various phases of the wave. Figure taken from [21].

2.3 Detector response to GWs

We briefly review what happens in a detector, see [19, 22] for a full treatment of the argument The output of any GW detector is a combination of the true GW signal and of noise: to understand how signal and noise combine, it is useful to think of a GW detector as a linear system. The input and the output of the detector are scalar quantities, whereas the GW is described by a tensor, hence commonly the input has the form

h(t) = Dijhij(t), (2.58)

where Dij _{is a constant tensor called detector tensor, which depends on the detector geometry,} and the output, in absence of noise, is a linear function (in frequency space) of the input h(t)

e

hout(f) = T (f)h(f) ,e (2.59)

where T (f) is the transfer function of the system.

However the output of any real detector also contains a noise, hence the truly output is in general of the form

sout(t) = hout(t) + nout(t) . (2.60)

More precisely, since a detector can be modeled as a linear system with many stages, the full transfer function is the product of the separate transfer functions of each stage, and also the noise can be generated at each of these stages, propagating to the output with the corresponding transfer function depending on the point of the linear system at which it first appeared [16]. For simplicity, and because it is the quantity that we directly compare with h(t) (the effect of GWs), we can define a fictitious function

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that, if pumped in at the detector input reproduce the output noise nout(t) really observed. Thus we can consider the output to be composed of a noise n(t) and a GW signal h(t) as

s(t) = h(t) + n(t) , (2.62)

and the detection problem deal with the distinction of h(t) from n(t). The advantage of referring everything to the input is that n(t) quantifies the minimum value of h(t) that can be detected and h(t), apart from the factor Dij _{which is always O(1), actually depends on the GW only.}

Therefore the detection noise is assumed to be n(t), and if it is stationary the different Fourier components are uncorrelated, conducting to the result7

h˜n∗(f) ˜n(f0) i = δ(f − f0)1

2Sn(f) . (2.63)

This expression defines the function Sn(f), known as noise spectral density which has dimensions Hz−1 _{and characterizes the noise in a detector. Equivalently, the noise can also be characterized}

by p

Sn(f), which is called spectral strain sensitivity and has dimensions Hz−1/2_.

2.3.1 Patter functions and angular sensitivity

Recalling eq. (2.36) a GW propagating in direction ˆn can be written as hij(t, x) = X A = +, × e_ijA(ˆn) Z ∞ −∞ df ˜hA(f) e−2 π i f ( t−ˆn·x ), (2.64) with eA

ij(ˆn) the polarization tensors given in eq. (2.33). We can set x = 0 as the location of the detector, and, assuming that the reduced GW wavelength is much larger than the size of the detector, we have 2 π f ˆn · x = ˆn · x/¯λ 1 over the whole detector, hence we can neglect the spatial dependence of hij(t, x). Therefore we can write

hij(t) = X A = +, × e_ijA(ˆn) Z ∞ −∞ df ˜hA(f) e−2 π i f t = X A = +, × e_ijA(ˆn) hA(t) . (2.65) Combining this result with eq. (2.58) we can express the contribution of GWs to the scalar output as

h(t) = X

A = +, ×

Dije_ijA(ˆn) hA(t) . (2.66)

We define now other useful objects called detector pattern functions FA(ˆn) as

FA(ˆn) = Dije_ijA(ˆn) , (2.67)

which depend on the direction ˆn = (θ, φ) of propagation of the wave, hence equation (2.66) can be written as

h(t) = h+(f) F+(θ, φ) + h×(t) F×(θ, φ) . (2.68)

To define the above equations we have chosen a system of axes (ˆu, ˆv) in the plane orthogonal to ˆn, with respect to which we have defined the polarizations h+ and h×. If we change the system

performing a rotation by an angle ψ the axes are rotated to

ˆu0 _{= ˆu cos ψ − ˆv sin ψ ,} _(2.69a)

7

Here the ensemble average has to be intended as a time average: a time-scale T is implicit in this procedure. This also implies that for f = f0 the following equation does not diverge, since, if we restrict to a time interval −T /2 < t < T /2 we have δ(f = 0) = T [16].

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ˆv0 _{= ˆu sin ψ + ˆv cos ψ .} _(2.69b)

In the new frame, the polarization tensors are given by (e+ ij) 0_{(ˆn) = ˆu}0 iˆu 0 j−ˆv 0 iˆv 0 j = eij+(ˆn) cos 2ψ − e × ij(ˆn) sin 2ψ , (2.70a) (e× ij) 0₍ b n) =ub 0 ivb 0 j+bv 0 iub 0 j = e+ij(nb) sin 2ψ + e × ij(nb) cos 2ψ . (2.70b)

The pattern functions FA depend on the polarization tensors through eq. (2.67), but since the detector tensor is a fixed quantity, in the new frame we find

F₊0 (ˆn) = F+(ˆn) cos 2ψ − F×(ˆn) sin 2ψ , (2.71a)

F×0 (ˆn) = F+(ˆn) sin 2ψ − F×(ˆn) cos 2ψ . (2.71b)

On the other hand, from eqs. (2.30) we see that the amplitudes of the plus and cross polarizations are related in the two frames by

h0₊= h+ cos 2ψ − h×sin 2ψ , (2.72a)

h0×= h+ sin 2ψ + h×cos 2ψ . (2.72b)

Combining the transformations of the patter functions in eqs. (2.71) with the transformations of the amplitudes in eqs. (2.72), we see that h(t) in eq. (2.68) is independent of ψ.

The detector pattern functions encode the response of an interferometer to GWs with arbi-trary direction and polarization. We have seen that in the proper detector frame the equation of geodesic deviation is given by eq. (2.52); for a ground based interferometer we are interested in the displacements of the mirrors, in particular for the mirror located at ξj _{= (L, 0, 0) we are} interested in its displacement along the x direction

¨ξx= 1₂¨hxxL , (2.73)

whereas for the mirror located at ξj _{= (0, L, 0) we are interested in its displacement along the} y direction

¨ξy = 1

2¨hyyL . (2.74)

These equations govern the change in the length of the x-arm and the y-arm of the Michelson interferometer respectively. The relative phase shift between the x and y arms is determined by (1/2)(¨hxx− ¨hyy), hence integrating twice we obtain that the shift is proportional to (1/2)(hxx− hyy).8 Since this quantity is proportional to the output signal of the interferometer, the detector tensor for an interferometer with arms along the ˆx and ˆy directions turns out to be

Dij = 1₂

ˆxiˆxj −ˆy_iˆy_j. (2.75)

This allows us to compute the detector pattern functions for a ground based interferometer, whose geometry is given in Figure 2.2. The two arms of the interferometer are located along the x and y axes, whereas the propagation direction of the GW coming from an arbitrary source coincides with the z0 _{axis of a second reference frame, which defines polar angles θ and φ with}

8_{When the wave comes from the z direction, we have h}

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Figure 2.2. Basic sketch of the geometry used in the computation of the pattern functions. The arms

of the interferometer are highlighted by the red axes in the (x, y, z) frame, the propagation direction of the gravitational wave is denoted by the blue axis in the (x0_{, y}0_{, z}0_{) frame. Are also displayed the polar}

angles defined by the GW direction propagation in the interferometer’s system.

respect to the (x, y, z) frame. The polarizations h+ and h× are defined with respect to the

(x0_{, y}0_{) axes, hence in the (x}0_{, y}0_{, z}0_{) frame we have}

h0_ij =    h+ h× 0 h× −h+ 0 0 0 0    ij , (2.76)

therefore, to bring this definition to the (x, y, z) frame we have to perform a rotation by an angle θ around the y axis and a rotation by an angle φ around the z axis, i.e.,

hij = RikRjlh0kl, (2.77)

where RikRjl denotes the combination of the two rotations given by

   cos φ sin φ 0 −sin φ cos φ 0 0 0 1       cos θ 0 sin θ 0 1 0 −sin θ 0 cos θ    . (2.78) Hence we obtain hxx = h+

cos2_θ_cos2_{φ −}_sin2_θ

+ 2h×cos θ sin φ cos φ , (2.79a)

hyy= h+

cos2_θ _sin2_{φ −}_cos2_θ

+ 2h×cos θ sin φ cos φ , (2.79b)

and therefore 1

2 ( hxx− hyy) = 1₂h+

1 + cos2_θ

cos 2φ + h× cos θ sin 2φ . (2.80)

This allows us, comparing with eq. (2.68), to define the detector pattern function as F+(θ, φ) = 1

2

1 + cos2_θ_{cos 2φ ,} _F

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2.4 Stochastic Gravitational Waves Background 21

The form of the pattern functions reveals that they are functions of the position of the source in the sky, and also that there are blind directions for GW interferometers due to their forms: as instance for a GW with plus polarization the direction identified by φ = π/4 is blind, since F+ = 0. This is due to the fact that the displacements produced in the x and y directions are

equal, hence they cancel reciprocally in the definition of the shift. However, apart from some blind directions, they allows the GWs detector to have a large sky coverage of almost 4π.

Eqs. (2.81) are represented in Figure 2.3 by a spherical polar plot in which the radial coor-dinate corresponds to the sensitivity given by the magnitude |FA|. Therefore points which are located in the origin of these plots correspond to blind directions for the given polarization. In the figure the interferometric detector has to be imagined as placed with its vertex at the center of each plot and arms along the x and y axes represented by black lines in the figure.

Figure 2.3. Polar plot of the antenna pattern functions of ground based interferometers for the two

tensor plus (a) and cross (b) modes allowed in GR. A thermal map is used to show the sensitivity to the given polarization: the warmer is the color, the highest is the response of the detector to that polarization. On the contrary, as the distance from the origin decreases, the color becomes colder, as a sign of the decreased response, up to the origin which is the location of blind directions.

2.4 Stochastic Gravitational Waves Background

A possible target of GW experiments is given by stochastic backgrounds of GWs, which could be both of astrophysical or cosmological origin.

The emission of GWs from a large number of unresolved astrophysical sources can create a stochastic background of GWs (see [23] for a review on astrophysically produced stochastic backgrounds), which can give important information on the state of the Universe at redshift z ∼ 2-5, and which can provide a probe of star formation rates, supernova rates, mass distribution of black holes births, angular momentum distributions and black hole growth mechanisms. Among the possible astrophysical sources we mention the supernovae collapses to black holes, the GWs produced from hydrodynamic waves in rotating neutron stars, the GWs emitted by mass mul-tipoles of rotating neutron stars and the GWs emitted by unresolved galactic and extragalactic binaries.

However, in this thesis, we focus on the stochastic gravitational wave background (SGWB) of cosmological origin [24, 25, 22].

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2.4 Stochastic Gravitational Waves Background 22

2.4.1 SGWB of cosmological origin

GW sourced form the early Universe typically lead to stochastic background of GWs today. The SGWB of cosmological origin can be considered as the gravitational analogous of the 2.7 K microwave photon background (CMB). Apart from its obvious intrinsic interest, it would carry extraordinary information on the state of the very early Universe and on physics at correspond-ingly high energies.

CMB, which is a thermal radiation, gives us currently our earliest view ot the state of the young Universe when it became electromagnetically thin, i.e. when it was about 4 × 105 _years

old, which is the time of recombination.

GWs couple much more weakly to matter, unless they interact with gravitational fields where the ratio GM/rc2 _∼_{1, i.e. with black holes. Obviously this can happen in rare, isolated}

locations, but the only time it could have happened to all the radiation in the Universe is the Planck time.

To be more quantitative, we know that particles which decoupled from the primordial plasma at time t ∼ tdec when the Universe had a temperature Tdec give a "snapshot" of the state of the Universe at that epoch, whereas all information when the particles still were in thermal equilib-rium has been obliterated by the successive interactions. The condition for thermal equilibequilib-rium is that the rate Γ of the processes that maintain equilibrium is larger than the rate of expansion of the Universe, as measured by the Hubble parameter H. The rate is given by Γ = n σ|v|, with n number density of the particles in exam (which is n ∼ T3 for massless or light particles in equilibrium at temperature T), |v| ∼ 1 is the typical velocity and σ ∼ G2_T2 _{is the cross-section}

of the process. Comparing the rate of interaction of GWs with the Hubble rate one gets [22] Γ(T ) H(T ) ∼ G2T5 T2_/M P l = T MP l 3 , (2.82)

where we assumed H(T ) ∼ T2_{/MP l} _{in the radiation dominated era. This estimate shows that}

the GW interaction rate is smaller than the Hubble rate essentially at any temperature in the Universe T < MP l for which our knowledge about gravitation holds, hence GWs are decoupled below the Planck scale MP l∼1019 _{GeV, already 10}−44_{sec after the Big-Bang.}

Thus, due to the weakness of gravity, GWs are decoupled from the rest of matter and radiation components in the Universe upon production and they have not interacted with matter in any significant way since being generated: in other words, GWs produced in the very early Universe can propagate freely until us maintaining their original spectrum, typical frequency and intensity unalterated, hence carrying pristine informations about the state of the Universe at epochs and energies unreachable by any other experiment. Therefore, if GWs from the cosmological stochastic background would be detected, they will show us a view of the Universe about 50 orders of magnitude younger than the Universe we see in the CMB. Nevertheless the same reason why GWs are potentially so interesting, that is their very small cross section, is at the basis of the difficulty of their detection.

2.4.1.1 Possible sources of cosmological SGWB

The main cosmological mechanisms that can give rise to a SGWB today and that have been proposed during years are:

• Inflation

The inflationary period, defined as an early phase of accelerated expansion, provides a natural explanation for the physical origin of the primordial density perturbations, required to start