For whom the black hole tolls: from ringdown to tests of general relativity

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Dipartimento di Fisica

Tesi di Laurea Magistrale in Fisica

For whom the black hole tolls:

from ringdown to tests of general relativity

Candidato Relatore

Gregorio Carullo

Dr. Walter Del Pozzo

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Capisci adesso, Bulkington,

un bagliore tu devi aver veduto di una verita’ che mortalmente ci inquieta.

Essa ci insegna che ogni pensare serio e profondo, altro non e’ che l’intrepido tentativo dell’anima, di conservare l’aperta indipendenza del proprio mare, mentre i venti piu’ selvaggi del cielo e della terra, cospirano, per risbatterla indietro,

sulle coste traditrici e servili.

Soltanto nell’assenza di terra risiede la verita’ piu’ alta, senza rive, senza limiti, senza Dio.

E per questo, meglio e’ morire, in quell’immane infinito, che ingloriosamente farsi gettare dal vento a terra, anche se quello, sarebbe l’unico sistema per salvarsi. Sara’ vana tutta questa agonia?

Oh, terrore.

E allora coraggio, coraggio Bulkington, aggrappati al timone semidio,

il tuo trionfo balzera’ verso il cielo

su dalla schiuma del tuo morire, d’oceano. Herman Melville

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Abstract

The several observations of gravitational waves, generated by binary black holes coalescences, as well as by binary neutron stars mergers, officially ushered the observational era of observational gravitational waves physics. Access to the most violent stages of the coalescence provides un-precedented insights in the general relativistic strong-field dynamics.

The information contained inside such signals has already been used by the LIGO and Virgo collaborations to estimate the parameters of such binaries and to produce some of the most stringent tests of general relativity [11].

However, current methods are not suited to fully extract information about the black hole born at the end of the coalescence process.

In fact, the remnant black hole mass and spin are inferred from measurements of the initial stages of the coalescence in conjunction with predictions from numerical relativity simulations.

In this thesis we show that, using recent models of the latest stage of the coalescence known as the “ringdown”, the final parameters of the coalescence can be directly extracted from the signal. Combining numerical simulations and parameter estimation methods, we provide a procedure to measure the parameters of the remnant black hole. For the first time, we demonstrate that ground-based gravitational waves observatories will allow the direct measurement of the mass and the spin of the remnant black hole, exclusively from the analysis of the ringdown part of the signal.

We are also able to determine the “effective start time” of the ringdown, as well as the mass ratio of the initial black holes, thanks to the imprint it leaves on the ringdown regime.

Independent methods from numerical relativity confirm our results (see [18]).

The start of the ringdown indicates the transition from the non-linear to the linear regime of the theory of gravity, where perturbation methods can be applied. Its knowledge allows to test for the presence of violations of general relativity that can be searched for only in the quasi-linear regime. We quantify the constraints on parametric violations of general relativity with single ringdown observations and show their improvements as the number of observed events increases.

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Introduction and motivations

Einstein’s theory of General Relativity (GR) stands as one of the pillars of modern physical understanding of the Universe.

Its applicability ranges from the Solar System, to cosmological scales. No deviations from the theory predictions have yet been observed. On the contrary, exactly after 100 years of its pub-lication, one of its most astonishing predictions has been directly1 confirmed: the existence of gravitational waves.

The signal which passed through the Earth on September 14, 2015 at 09:50:45 UTC through the twin LIGO detectors ([10]) proved that gravitational waves are observable with current earth-based detectors. It also proved for the first time the existence of black holes binary and the fact that these objects merge within Hubble time.

The importance of this measurement to investigate the nature of black holes, cannot be under-stated.

Given that black holes are purely made of space-time and interact only through their gravita-tional force, the gravitagravita-tional channel is the privileged one to provide observation of strong-field effects predicted by general relativity, which may arise in such extremely violent collisions. Historically, collisions at high energy of bodies which seemed to have a fundamental structure, is the way in which physicists have uncovered the limits of existing "fundamental" theories. These experimental probes have traced the path that led from Chemistry to Atomic, then to Nuclear and finally to Particle physics.

Each of these theories apply to a completely different energy regime and each of them recovers the previous theory in the appropriate limit.

This picture, formalized in the contest of Wilson’s Renormalization group [51], leads the modern understanding of fundamental and effective field theory: whatever "fundamental theory" could lie beyond the known fundamental forces, the accessible observables can be computed through a lower-energy effective field theory that emerges from the fundamental "mother" theory.

Whether this picture will hold or not in the context of GR, the pursuit of an answer is in itself valuable, since it requires an unprecedented deep understanding of the general relativistic two-body problem, whose solution is notoriously challenging.

Eventual deviations from GR predictions are expected to be found especially during the merger phase of the two black holes and in the subsequent ringdown phase, in which the single black hole just formed radiates away all its anisotropies, until the stable Kerr solution is reached. In fact, gravitational waves emitted during these phases can probe scales close to the black holes event horizon. During these last phases, these black holes reach relativistic velocities (the black holes coalescing during the GW150914 event reached a relative velocity v ∼ 0.6c) , whose peak gravitational wave luminosity was L_GW ∼ 1056_{erg/s, where a large fraction of the energy is}

1

The "indirect" existence of gravitational waves was proved by the observation of the Hulse–Taylor binary [54].

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emitted in less than a hundred of a second during the merger phase.

General relativity has already been extensively tested through gravitational waves observations [11], passing all such tests with flying colours.

Nevertheless, among all these tests, the one which searched for a ringdown phase inside the signals, although rigorously implemented, only partially exploited the current knowledge of the ringdown phase of the coalescence.

With this work we aim to bridge the gap between frontier-theoretical knowledge and experimen-tal exploitation of this knowledge, to uncover new physics that could lie in the increasingly new amount of data coming from GW observations, specifically during the ringdown phase.

Throughout all theoretical computations, we will use geometric units c = G = 1. We will restore SI units in cases of numerical interest.

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Chapter 1

Gravitational waves primer

We will follow the conventions and derivations presented in [24,40,49].

General Relativity (GR) is a theory of space, time and gravity relying on a unified vision of space and time, modelized as a unique geometrical entity. This geometrical entity is identified with a real differentiable manifold M equipped with a Lorentzian metric gµν, a symmetric tensor of

rank two that has signature (-, +, +, +). This unique entity is called space-time. GR is based on the equivalence principle:

Any local physical experiment not involving gravity will have the same results if performed in a freely falling inertial frame as if it were performed in the flat space-time of special relativity. GR is also based on the assumption that all the observable quantities, can be computed starting from the metric g_µν. Thus, it is called a metric theory of gravity.

Notable functions of the metric, which are needed in order to investigate the properties of curved manifolds are: • Christoffel symbols: Γα_µν = 1 2g αβ_(∂ νgβµ+ ∂µgβν− ∂βgµν) (1.1)

• Riemannian curvature tensor:

R_βµνα := ∂µΓαβν− ∂νΓαβµ+ ΓασµΓσβν− ΓασνΓσβµ (1.2)

• Ricci tensor:

Rαβ := Rµ_αµβ (1.3)

• Ricci scalar:

R := gµνRµν (1.4)

• Covariant derivative for a vector:

DαVβ := ∂αVβ− ΓµαβVµ:= Vβ;α (1.5)

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Generalizing Newton’s law of gravity in a form which is valid for all the observers (i.e. "covari-ant"), one arrives to the Einstein’s equations:

Rµν−

1

2gµνR = 8πG

c4 Tµν (1.6)

The metric is found as a solution of these ten non-linear partial differential equations, given the appropriate boundary conditions.

The tensor Tµν(defined as the flux of the α component of momentum across a surface of constant xβ), encodes all the information on matter and energy density in a covariant way. Its single components reduce to the density and pressure functions in the appropriate physical scenarios. The classical equations of motion for a test mass in a space-time described by the metric gµν are

then: d2xµ dτ2 + Γ µ νρ dxν dτ dxρ dτ = 0 (1.7)

where τ is the proper time, the time measured by a clock carried along the trajectory of such a test mass.

In the absence of gravitational interaction, space-time is flat and the metric tensor gµν reduces

to the Minkowskian metric:

ηµν =     −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1    

The connection vanishes:

Γα_µν = 0 (1.8)

and the space-time manifold becomes R4, impling:

Rα_βµν = 0 (1.9)

It can be proven that also the converse is true, a null Riemann tensor implies that space is flat. A notable set of exact solution of Einstein is:

• Schwarzschild metric: this metric describes spacetimes characterized by spherical sym-metry. It can be used to describe a non-spinning black hole (BH);

• Kerr metric: this metric describes spacetimes characterized by axial symmetry. It can be used to describe a spinning BH;

• Kerr-New metric: this metric describes spacetimes characterized by axial symmetry, when an electric charge is present. This metric can be used to describe a spinning charged BH;

• Friedman-Lemaitre-Robertson-Walker metric: this metric describes spacetimes char-acterized by homogeneity and isotropy.

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1.1 Linearized Einstein equations

Being interested to observe gravitational effects in regions of the space-time in which extremely dense matter is absent, which means that curvature is small, we can decompose the metric as:

gµν = ˚gµν+ hµν (1.10)

where ˚gµν is a fixed background metric and we defined hµν as a small perturbation to the

back-ground metric.

We require that |hµν| 1. To require that the components of a tensor are small does not make

sense in a general case, because one can always find a coordinate transformation in which they are big. In this context, we have explicitly broken the covariance of Einstein’s equation and specialized to a specific coordinate system in which the Eq.1.10 is valid.

In other words, we suppose that there exists a reference frame, admitting a set of coordinates, in which the metric has the components specified above (which is actually the case for realistic physical scenarios).

For the present discussion, we specialize to the simplest case, in which ˚gµν = ηµν. In Chapter 2

we will show how the same derivation can be repeated in the presence of a curved background. We proceed to linearize Einstein’s equations, keeping only terms linear in the perturbation h_µν. Christoffel’s symbols are:

Γλ_µν = 1 2η

λρ_(∂

µhρν+ ∂νhµρ− ∂ρhµν) (1.11)

The Ricci tensor is then:

Rµν =

1

2(∂λ∂νh

λ

µ+ ∂λ∂µhλν − ∂µ∂νh − hµν) (1.12)

where _{= η}λρ∂λ∂ρ is the D’Alambertian in flat spacetime and h denotes the trace of hµν:

h := hµµ.

Consequently, the Ricci scalar is:

R = ∂λ∂µhλµ− h (1.13)

We can now compute the right-hand side of the Einstein’s equations: Rµν− 1 2ηµνR = 1 2(∂λ∂νh λ µ− ηµν∂µ∂νhµν+ ηµνh − hµν) (1.14)

To solve this equations, the first step is to define a new perturbation, the trace-reversed pertur-bation:

¯

hµν = hµν− ηµνh (1.15)

With this transformation, we do not lose any information, since: ¯

h = h − 1

24h = −h (1.16)

Moreover one can prove that we can always find a coordinate transformation such that:

∂µ¯hµν = 0 (1.17)

this is called the harmonic gauge condition.

Applying these two transformations to Einstein’s equations we find: Rµν−

1

2ηµνR = − 1

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Summarizing, the linearized Einstein’s equations in presence of matter reduce to a wave equation for the perturbation, with a source term given by the energy-momentum tensor:

¯hλν = −

16πG

c4 Tλν (1.19)

To find the general solution of these equation given an arbitrary T_µν, one can apply standard techniques of PDE, which amounts to find the "inverse" of the D’Alambertian operator, i.e. to find the solution of the equation:

G(x; x0) = δ(4)(x − x0) =⇒ G(x; x0) = −δ(t

0_{− [t − |x − x}0_|])

4π|x − x0_| (1.20)

where G is called the Green function.

Then, the general solution is found integrating the Green function over the support of Tµν:

¯ hµν(t, x) = 4 Z d3x0 Tµν(t − |x − x 0_{|, x}0₎ |x − x0_| (1.21)

1.2 Gauge degrees of freedom

We must now verify that we actually impose the the Lorentz gauge condition.

For our procedure to be valid, the gauge condition must hold in every reference frame. Under a coordinate transformation such as:

x0µ= xµ+ ξµ(xβ) (1.22)

with |ξµ| 1 and |ξ,νµ| 1 using the tensorial law of transformation of gµν we see that the

perturbation transforms like: ¯

h0µν = ¯hµν− ∂µξν − ∂νξµ+ ηµν∂γξγ (1.23)

If we require the gauge condition to be valid also for ¯h0 _{we obtain:}

ξν = ∂µ¯hµν (1.24)

with appropriate boundary conditions (in this case the asymptotic flatness of space) this equation always admits a solution, so it is a valid gauge condition. This does not fix all the gauge freedom of our coordinate choice, since every transformation of the kind:

ξ0_ν = ξν + αν (1.25)

with αν solution of:

αν = 0 (1.26)

is still a valid gauge transformation. This means that within the Lorentz gauge hµν still possesses

non-physical degrees of freedom, which just reflects the arbitrary coordinate choice that we can still make.

To remove these spurious degrees of freedom we impose two further conditions: ¯

h = 0 ¯h0,µ= 0 (1.27)

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1.3 Gravitational waves in vacuum

TT).

The fact that this is a valid gauge condition can be proved through a Fourier decomposition and subsequently applying the transformation law of ¯hµν exactly as we have already done.

Without entering in the details of this last computation, one passage must be noted.

We use Eq. 1.23to remove four spurious degrees of freedom of hµν, by subtracting the function

ξµν := ∂µξν+ ∂νξµ− ηµν∂γξγ

which respects:

ξµν = 0

This is true only in vacuum, since in the presence of matter (inside the source): hµν 6= 0

So, we cannot eliminate four degrees of freedom of hµν using a function which respectsξµν = 0.

The TT gauge in the presence of matter, cannot be imposed.

Thus, to remove the spurious degrees of freedom in the general case solution, first we must use all the allowed gauge conditions to simplify the equations inside the source. After solving these equations, if we are interested in the solution outside the source, one may always project the final result onto the TT-frame, where only the physical degrees of freedom remain. An example of this procedure will be given below.

Recalling the definition of ¯hµν, within the TT gauge, we find: ¯hµν = hµν.

In conclusion, the previously found wave equation becomes now valid directly for the metric perturbation.

1.3 Gravitational waves in vacuum

In vacuum we are left with the following equation:

hµν = 0 (1.28)

together with the gauge conditions already specified.

A natural way to write a complete solution of this equation is to write a plane-wave decomposi-tion, using the property of completeness of complex exponentials in, e.g., L2:

hµν = Cµν(k)eikµx

µ

(1.29) Using the TT gauge conditions, we find:

C_µµ= 0 C0,µ= 0 kµCµν = 0 (1.30)

with kµ = (ω, 0, 0, k) for our choice of axis orientations.

We find that the most general expression for our polarization tensor, given the physical restric-tions on which the 2-rank symmetric tensor hµν is subjected, is:

Cµν =     0 0 0 0 0 h+ h× 0 0 h× −h+ 0 0 0 0 0    

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The wave associated with space-time perturbations of the metric, propagates through vacuum space with speed c, frequency ω = |k| and two possible independent polarizations, the"plus" and "cross" polarizations .

The eventual particle associated with this perturbation of the space time would be a massless spin 2 particle with 2 different values of elicity.

Since in our gauge the temporal part of the metric perturbation is 0, we can introduce the notation for the spatial part:

Cij(k) = h+(k)+ij(k) + h×(k) × ij(k) (1.31) where: +_ij =   1 0 0 0 −1 0 0 0 0   ×_ij =   0 1 0 1 0 0 0 0 0   and i,j = 1,2,3.

Within the space of all the possible 2-rank symmetric tensors that could describe the propagation of gravitational waves, only a subspace of it is physically allowed and a basis of this subspace is given by +_ij and ×_ij (0-padded into 4D).

Before having a general description of gravitational waves in vacuum, we must write down a form independent on the direction of propagation we chose (the z-direction).

To write the solutions in a generic reference frame, we can express the polarization tensors as: += (êx⊗ êx− êy⊗ êy) × = (êx⊗ êy+ êy⊗ êx) (1.32)

We now take a cartesian system of axes in which θ and φ are the spherical angles, ˆn = ˆer is a

generic versor on the direction of propagation of the wave and ˆeθ, ˆeφare the orthonormal versors

defined in the plane orthogonal to the direction of propagation. We can then define:

ˆ

u = cos(φ)êθ+ sin(φ)êφˆv = −sin(φ)êθ+ cos(φ)êφ (1.33)

to obtain finally a direction generic definition for our polarization tensors:

+= (û ⊗ û − ˆv ⊗ ˆv) × = (û ⊗ ˆv + ˆv ⊗ û) (1.34)

1.4 Gravitational waves in the presence of matter

We can now proceed to a perturbative expansion to get an explicit solution in the presence of matter.

We want to observe gravitational waves produced at a very large distance from our detectors, from a source which has a finite extension, much smaller than this distance.

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1.4 Gravitational waves in the presence of matter

If L is the typical scale of the system (e.g.. the radius of the black hole) and λ is the wavelength of the emitted gravitational wave, then:

L λGW =⇒ LωGW 2πc (1.35)

Since:

ωGW ∼ ωorb (1.36)

the previous equation implies:

v c (1.37)

i.e. we are performing a low-velocity expansion. In terms of the equation 1.21, this means:

|x0| |x| (1.38)

We are then led to perform a multipolar expansion of the function _|x−x1 0_|.

Defining |x| := r: 1 |x − x0_|= 1 r + 1 r2(ˆx · x 0_{) + ...} _(1.39) By substitution: ¯ hµν(t, x) = 4 r Z d3x0 Tµν(t − r, x0) (1.40)

Coherently with our low-velocity expansion, we can now write an adiabatic solution. The idea is that, although the system is radiating gravitational waves and losing energy and angular momen-tum, in the initial orbits the energy radiated away is negligible, so we can apply the conservation of energy and fcompute the emission of gravitational waves.

We will then use the waves emitted to compute the amount of energy and angular momentum lost by the system. Subsequently, we will use the amount of energy and angular momentum loss, to set new initial conditions, finding the corrections to the 0-order solution. The process goes on iteratively, until we will break our assumptions.

In the context of a binary coalescence, this procedure is valid only in the initial phase of the co-alescence when the motion is actually adiabatic. It will eventually break up when the separation between the orbiting bodies will become small and the large curvature (or eventual tidal forces) will break up these assumptions.

Energy conservation is expressed by the equation:

∂µTµν = 0 (1.41)

which is not generally true in GR, since it is not a covariant equation.

Nevertheless, we can use this equation in the linearized case, since all contributions coming from the curvature of space-time are proportional to Γ · T Since T ∼ h and Γ ∼ h, all such contributions would be quadratic in h and thus negligible.

Using the previously specified gauge conditions and the energy conservation: Tkl(t, x) = ∂tt(

Z

d3x0T00(t − r, x0)x0_kx0_l) (1.42) where k, l = 1, 2, 3.

Defining the quadrupole moment of the system as: qij(t − r) =

Z

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we arrive to the quadrupolar formula: ¯ hij =

2

r∂ttqij(t − r) (1.44)

This field is still not a traceless solution, we still need to project it onto the transverse-traceless solution subspace.

The projection can be performed by starting from the transverse projector of a vector:

P = I − ˆn ⊗ ˆn or Pij = δij − ninj = Pji (1.45)

If the direction of propagation is ˆn, projecting a longitudinal vector (parallel to ˆn):

Pijaj = ai− ajnjni = kni− knjnjni = 0 (1.46)

while for a transverse vector (orthogonal to ˆn):

Pijbj = bi− bjnjni= bi− 0ni= bi (1.47)

For a matrix we get:

P_ijklT T(ˆn) = PikPjl−

1

2PijPkl (1.48)

which eliminates the trace since:

P_iiklT T(ˆn) = PikPil−

1

2PiiPkl= PikPil− PikPil = 0 (1.49) Applying this projector to the previously found solution, we get:

hT T_ij (x, t) = 2 rP

T T

ijkl(ˆn)∂ttqij(t − r) (1.50)

1.5 Radiation from the coalescence of a compact binary in

circu-lar orbit

The discussion up to this point is completely general, not being tied to any specific source. The source emitting gravitational waves could have been a single rotating star, a binary of stars orbiting each other in a circular or elliptical orbit etc. The object has only to possess a time-varying quadrupole moment.

In the remainder of the section, we will be focusing on a specific case: a compact binary coales-cence (CBC).

The objects composing the binary can in principle be: two black holes (BBH), two neutron stars (BNS), a neutron star and a black hole (NS-BH), a binary of white dwarfs etc. One may also consider more exotic objects as wormholes, in which case most of these considerations still apply, with notable distinctions especially in the last part of the emission (see Ref [22]).

The shape, range of frequency, phase and amplitude of the signal will of vary within all these different scenarios.

We will restrict to BBH system, thus ignoring any effect related to matter deformability as tidal forces or other effects that may arise from exotic compact objects.

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1.5 Radiation from the coalescence of a compact binary in circular orbit • Inspiral: This is the initial phase of the coalescence. The radiation emission slowly shrinks

the orbit and the separation between the two BHs decreases.

The frequency of the gravitational wave goes up as the system shrinks its orbit and the or-bital frequency increases. This phase can be modeled in a Post-Newtonian approximation, adding corrective terms coming from a perturbative expansion in powers of v_c.

• Plunge and merger: As the BHs approach one another, the radial angular momentum will give an increasingly large contribution to the dynamics of the system, until its value becomes bigger than the azimuthal component of the angular momentum. At this point, the system cannot sustain a stable orbit and the BHs plunge one into another. During this process a common horizon appears and a single final BH is formed as result of the coalescence. The peak of the waveform belongs to this phase.

The final BH is highly distorted and quickly radiates away all its anisotropies to reach a stable Kerr state.

The merger phase is the most difficult phase of the coalescence to model. The non-linearity of the Einstein equations during this regime makes a theoretical modelization extremely difficult. An effective modelization can be achieved by adding a set of free parameters in the inspiral expansion and fit them to reproduce a set of simulated waveforms, obtained through numerical solution of the full Einstein equations. Further details on this approach will be given in Chapter 3.

• Ringdown: The ringdown is the final transient phase when the final BH is formed and ends when the final BH has relaxed to a stable state and gravitational radiation is no longer emitted.

It can be partly theoretically modeling using perturbation theory and a detailed derivation will be given in Chapter 2.

A theoretical model which aims to describe the full coalescence process, has to cope with a variety of physical effects: distortion of the space-time between the two objects from mass ratio effects, interaction of the intrinsic angular momenta ("spin") of the BHs (spin-spin interactions), spin-orbit interactions, precession effects, scattering of GW onto the curved background (tail effects), backreaction due to the emission of energy and angular momentum etc.

A full-treatment of all such effects can be found in Ref [19].

Two classes of theoretical models that have been developed during the last decades, are:

• EOB models: the effective-one-body (EOB) approach is a technique first introduced by Damour and Buonanno in Ref [20]. This approach recasts the general relativistic two-body problem into a problem where a point particle orbits a single Kerr BH. The BH metric effectively acquires a distortion proportional to the mass ratio and spins of the initial binary. The inspiral phase is modeled using perturbative expansions (post-Newtonian) techniques combined with series-resummation techniques. The merger phase is described using a set of coefficients, fit to Numerical Relativity solutions, which constitutes the corrections needed to characterize the plunge phase of the coalescence. The ringdown predictions from perturbation theory of GR are attached through another set of coefficients, fitted to ensure the continuity of the waveform. This model has proven to be in striking agreement with numerical solutions and has been extensively employed to perform parameter estimation of GW signals.

• IMR Phenomenological models: the Inspiral-Merger-Ringdown (IMR) phenomenolog-ical approach [33] starts from standard post-Newtonian techniques, in which higher terms of the expansion, still not known analytically, are added using free parameters. These free

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parameters are then fit to numerical relativity simulations in order to faithfully represent the binary evolution up to the merger. This "gluing" procedure requires an alignment at a specific frequency between the post-Newtonian model and numerical data. Then, the final phase of the coalescence is parametrized in the frequency domain by a Lorentzian (corresponding to the frequency representation of one single mode of the ringdown phase), in which a set of arbitrary coefficients is added.

This set of coefficients is then calibrated to numerical data to attach this last part of the waveform to the Inspiral-Merger representation.

1.5.1 Leading order GW from CBC

We will now derive the lowest order approximation to the gravitational wave signal emitted by two nonspinning point masses. We will then compare the result of this simple computation with the complete result coming from an IMR model.

Consider two point objects of masses m1 and m2 orbiting around their center of mass with

constant orbital frequency ω, whose reference frame is represented in Figure 1.1.

Figure 1.1: Source frame, from [13].

If we define M = m₁ + m2, R as the distance coordinate from the center, setting the orbital

phase to 0 at the origin of the time axis, we get: x1 = m2 MRcos(ωt) x2= − m1 MRcos(ωt) (1.51) y1= m2 M Rsin(ωt) y2 = − m1 MRsin(ωt) (1.52) z1 = 0 z2 = 0 (1.53)

The quadrupole moment is:

qij = m1xi1x j

1+ m2xi2x j

2 (1.54)

Defining the reduced mass as µ = m1m2

m1+m2, using Kepler’s third law:

ω2= (m1+ m2)

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1.5 Radiation from the coalescence of a compact binary in circular orbit

and inserting this expression inside the quadrupolar formula we obtain:

hT T_ij (x, t) = M 5 34ω23 r P T T ijkl(ˆn)   −cos(2ωt) −sin(2ωt) 0 −sin(2ωt) cos(2ωt) 0 0 0 0   lk

which is the solution for the radiation observed from an arbitrary inclination angle, without energy loss.

The frequency at which gravitational waves are emitted in this case is exactly equal to double the orbital frequency.

As the system evolves, gravitational waves carry away energy and angular momentum from the system, this changes the energy momentum tensor of the binary system, which will in turn modify the emission.

The gravitational waves have an energy tensor of their own, which can be computed applying Noether’s theorem, obtaining:

tµν = 1 32πh∂

µ_hαβ_∂ν_h

αβi (1.56)

Without entering into the details of the computation, there is a crucial detail that needs to be mentioned in order to understand the motivation of the averaging procedure.

The energy tensor represents the local energy density of a field. For the gravitational field, this concept is intrinsically ill-defined as a consequence of the equivalence principle, since the gravitational field can always be locally set to 0. Thus an average process (in this case over several wavelengths of GW) is required.

The same issue appears in the computation of the electromagnetic energy tensor, where the problem is solved requiring Noether’s currents (not only conserved charges, which are the only true physical observables) to be gauge invariant or with a similar averaging procedure.

The power lost by the system is: dE dt = c3_r2 32πG Z dΩh ˙hT T_ij ˙hT T_ij i = 32 5 (Mω) 10 3 (1.57)

To first approximation, we can compute this effect by using the Newtonian expression for the orbital energy of the system:

Eorbit= − m1m2 r + m1m2 2r = − m1m2 2r (1.58)

Combining these equations we obtain the differential equation:

˙ ω = 96

5 M

5/3_ω11/3 _(1.59)

for the orbital frequency, solved by:

ω(t) = (−256 5 M 5 3(t − t_c))− 3 8 (1.60)

where tcis the time of coalescence, defined as the time at which the frequency formally diverges,

signaling that our assumptions break up, the signal is no longer inspiralling in circular orbits, but the two masses are plunging one onto another.

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1.5.2 Polarizations of the signal

The polarizations of a GW signal can be defined as: • Linear polarization:

A gravitational wave is linearly polarized, if it can be decomposed as:

h(x, t) = sin(ωt)(c1(x)e++ c2(x)e×) (1.61)

with c₁, c2 time independent factors.

• Circular polarization:

A gravitational wave is circularly polarized, if it can be decomposed as:

h(x, t) = A(x)(cos(ωt)e++ sin(ωt)e×) (1.62)

with A a time independent factor. • Elliptical polarization:

A gravitational wave is elliptically polarized if it can be decomposed as:

h(x, t) = A(x)cos(ωt)e++ B(x)sin(ωt)e× (1.63)

with A 6= B time independent factors.

Decomposing the gravitational wave radiation as:

  −cos(2ωt) −sin(2ωt) 0 −sin(2ωt) cos(2ωt) 0 0 0 0  = −cos(2ωt)[êx⊗ êx− êy⊗ êy] − sin(2ωt)[êx⊗ êy+ êy⊗ êx]

and recalling that:

P_ijklT T(ˆex⊗ ˆey) = PijxyT T (1.64)

we can study the observed polarization for different relative angles between the direction of propagation ˆn and a xyz reference frame on the Earth.

The inclination angle is defined as the relative angle between the z-axis and ˆn. It is equivalent to polar angle of spherical coordinates.

• Face-on binary: ˆn = ˆz (zero inclination). The configuration is face-off, if ˆn = −ˆz. We obtain: P_ijklT T(ˆn)   −cos(2ωt) −sin(2ωt) 0 −sin(2ωt) cos(2ωt) 0 0 0 0   lk = (−1)(cos(2ωt)ˆe++ sin(2ωt) ˆe×)

The radiation emitted when the binary is face-on/off, is a composition of harmonic motions with a gravitational wave frequency equal to the double of the orbital frequency of the binary. The radiation is circularly polarized.

• Edge-on binary:: ˆn = ˆx (π/2 inclination).

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1.6 Detection of gravitational waves P_ijklT T(ˆn)   −cos(2ωt) −sin(2ωt) 0 −sin(2ωt) cos(2ωt) 0 0 0 0   lk = 1₂cos(2ωt)ˆe+

The radiation emitted in the situation when the binary is edge-on, is an harmonic motions with a gravitational wave frequency equal to double the orbital frequency of the binary. The radiation is linearly polarized onto the ˆe+ polarization axis.

An example of a waveform generated with this simple model, is shown in Figure 1.2. Note the characteristic chirp behaviour.

In the upper plots of Figure 1.3we show instead the strain and the reconstructed waveform for GW150914. The waveform was generated by a full inspiral-merger-ringdown numerical relativity simulation, thus representing a complete prediction of general relativity of the full dynamics of the two body problem. The simple waveform mode that we derived, is able to only reproduce the initial phase of the coalescence. The amplitude rapidly increases in the last few cycles, breaking the assumptions underlying our derivation. Note the fast transient phase just after the peak, corresponding to the ringdown regime, during which the final BH relaxes towards a stable state. The bottom panel represents the evolution of the frequency with time, showing the characteristic "chirp" behaviour typical of compact binary coalescence signals.

Figure 1.2: Plus polarization computed with the simplified model derived in this section.

1.6 Detection of gravitational waves

1.6.1 Response of test masses

How would a test mass on Earth respond to the passing perturbation of space-time?

To answer, one may use the geodesic equation, which, evaluated at the origin of the proper time axis, yields: d2xi dτ2|τ =0= −[Γ i 00( dx0 dτ ) 2_] τ =0 (1.65)

where we already used that our object was at rest, so dx_dτi = 0. Evaluating the right-hand side in the TT gauge we find:

Γi₀₀= 1

2(2∂0h0i− ∂ih00) (1.66) which using our gauge conditions vanishes.

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Figure 1.3: Top panels:Strain and correspondent numerical relativity waveform correspondent to the event GW150914, inside the two LIGO detectors.

Bottom panels: residues after waveform subtraction (consistent with noise) and time-frequency plot of GW150914. The color-scale quantifies the power relesed into the detector. From [10]

of the gravitational wave.

What happened here is a simple manifestation of the equivalence principle and of the invariance under diffeomorphisms of general relativity.

Every gravitational field can be set to zero at exactly one point in space by an appropriate choice of coordinates.

The TT gauge is then a choice of coordinates whose spatial motion is synchronized with the perturbation of spacetime induced by a gravitational wave.

What cannot be set to zero is instead a tidal effect between two points. It will be convenient to study this effect in the laboratory frame, defined as a freely falling frame with a flat metric (on sufficiently small scales) before the GW passage.

In such frame, the separation between two spatially close geodesics (here close means that the separation distance, indicated as ξ = (x, y, z), is much smaller than the wavelength of the gravitational perturbation), obeys the equation:

¨ ξi₌ 1 2 ¨ hT T ij ξj (1.67)

In the proper detector frame, when the separation of the test masses is much smaller than the wavelength of the gravitational wave, such waves can be described as Newtonian forces.

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1.6 Detection of gravitational waves

Assuming a plus-polarized GW propagating along the z-direction and a ring of test masses placed on the (x,y) plane at z=0, we obtain:

δx(t) = h+

2 x0sin(ωt) δy(t) = − h+

2 y0sin(ωt) (1.68) while if the wave is ×-polarized:

δx(t) = h×

2 y0sin(ωt) δy(t) = h×

2 x0sin(ωt) (1.69) These solutions can be visualized in figure 1.4. The lines of force given by Eq. 1.67, are shown

Figure 1.4: Effect of a passing gravitational wave on a ring of test masses, fromLearner visuals

in Figure 1.5.

Figure 1.5: Force lines of the gravitational radiation, from Centrella et al.

1.6.2 Gravitational wave detectors

A simple strategy to detect a gravitational wave is to use two or more small masses connected, for example, by a spring. This system, resonating at its proper frequency, would respond to gravitational waves of specific frequencies. This idea was implemented in a much more sophis-ticated version, through the resonant bar detectors. These "Weber" bars (by the named of the physicist who pioneered this kind of experiment) were aluminium cylinders, 2 meters in length and 1 meter in diameter, isolated in vacuum chambers with a resonance frequency of 1660 Hz (which is compatible with what expected by supernovae bursts of gravitational waves).

No confident detection was ever reached with such an experimental apparatus.

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the end of two L-shapes interferometer’s arms and use the fact that the proper distance experi-enced by particles travelling on null geodesics (photons that constitute a laser beam in this case) would change when a gravitational wave passes. Injecting a laser beam between the two mirrors and measuring the dephasing induced by the change in distance on the laser beams travelling in the two opposite arms, would allow to measure the perturbations of spacetime passing through the interferometer. The effect of the gravitational waves would also be amplified by the fact that the response pattern to GW is opposite for the two arms, i.e. when one arm shrinks, the other is extended. Detecting a deformation of spacetime induced by a gravitational wave requires ex-traordinary precision and sophisticated isolation techniques. In fact, the typical change in length of an interferometer’s arms is 10−18 m, while already daily seismic vibrations of the Earth are of the order of 10−6 m, 12 orders of magnitude bigger. Moreover, an interferometer experiences several noise sources different than pure seismic noise:

• Mechanical vibrations: these vibrations are induced, for example, by the coupling of Earth vibrations or by the coupling of the surrounding environment on the suspensions of the mirrors. Also, the optical apparatus needed to inject the laser, can experience this kind of vibrations. The most sophisticated isolation system, the "Superattenuator", increasing the sensitivity at very low frequency by the means of multi-stage pendula was initially built in the Virgo detector[5].

• Thermal noise: the finite temperature of the mirrors, induces small vibrations that can mimic the passage of a gravitational wave. Moreover, the heating experienced by the mirror, due to the high-power laser shining on the mirror, can also increase the temperature of the system. Gravitational waves interferometers are sensitive to frequencies far from the resonance frequencies of the apparatus, minimizing the effect of thermal noise. A cryogenically cooled gravitational wave detector (KAGRA), exploring the improvement in sensitivity due to the reduction of thermal noise, is currently under construction in Japan. • Radiation pressure: the pressure due to the laser photons hitting the mirrors, can perturbate the system and induce couplings with other noise sources, limiting the sensitivity at low-frequencies. Although the decreasing of laser power would decrease this noise source, it would increase at the same time the shot noise.

• Shot noise: this noise source is due to the Poisson statistics of the photons hitting the mirrors, inducing random fluctuations of the number of photons hitting on the mirrors surfaces, which limit the high-frequency sensitivity of an interferometer. The increase in laser power would increase the number of photons hitting the mirrors, decreasing the statistical fluctuations induced by this noise source.

To limit all the disturbances induced by these noise sources, a network of instruments has been build all over the globe. These new-generation interferometers employ resonant Fabry-Perot cavities on each arm, Superattenuators on each of the interferometer component, signal recycling cavities to increase laser power, multiple reflections between the mirrors to increase the effective path travelled by light, ultra-vacuum systems to prevent dust particles hitting the mirrors or the suspensions. A simplified scheme of the Virgo interferometer is shown in Figure 1.6.

The noise sources mentioned above only includes the stationary and known noise sources. Non-stationary sources, usually referred to as non-Gaussianities, coming from environmental distur-bances, electromagnetic coupling of the electronics etc., can affect the detector status. Monitoring environmental disturbances and exclude these noise "glitches" (i.e. transients), is a key aspect of gravitational waves detection.

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1.6 Detection of gravitational waves

Figure 1.6: Simplified scheme of the Virgo interferometer, situated in Cascina, near Pisa (Italy). After the laser injection, only the gaussian mode of the laser passes through the input mode cleaner (left of the figure). The beam is separated by the beam splitter (BC) and enters the Fabry-Perot cavities, reflecting several times through the arms (IM-EM). The two beams are then recomposed, going through the output mode cleaner (OMC) and the arms are locked in such a way that if no gravitational wave passes, no signal is present on the output port (bottom of the figure). From [5]

(accelerometers, magnetometers, etc.) are used to correlate the output of the interferometer with noise induced by the surrounding environment.

1.6.3 A network of detectors

A global network of interferometers is required to improve both the detection false alarm rate and the localization of astrophysical sources in the sky. Independent observations of an astrophysical signal are required in order to confidently claim a detection, since the detectors continuously experience noise transient which can mimic the passage of gravitational waves. Sometimes these transients can also mimic the time-frequency evolution of a gravitational wave and not all the transients have a known origin, making difficult to classify the transient as a signal or just noise, with a single detector.

Most of these issues can be overcome by employing multiple detectors thousands of km far from each other, whose environmental noise is completely uncorrelated. Throughout the years, great effort was spent by GW physicists to create detection pipelines able to combine information from multiple detectors, thus greatly increasing the statistical confidence of detections. For an introduction see Ref. [45].

The network of first generation detectors consisted of the British-German GEO600 detector, located near Hannover, the TAMA300 detector in Japan, the Virgo detector near Pisa in Italy and the two LIGO detectors located in Hanford (Washington) and Livingston (Louisiana). The

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initial LIGO-Virgo detectors performed six consecutive science runs.

Starting from 2011 the LIGO and Virgo detectors underwent a series of upgrades, among which the inclusion of more powerful lasers, enhanced electronics and optical system, aiming to improve their sensitivity, in distance, of a factor of 10, the equivalent of a factor 1000 in volume.

The Advanced LIGO detectors entered in science mode during September 2015. Their first ob-servational run included the first observation of gravitational waves coming from a binary black hole merger [10], together with a second detection [7].

During the second science run (O2) of LIGO, the Advanced Virgo detector entered in science mode for the first time during August 2017.

The network of three interferometers was indispensable to enhance the capability of the instru-ments to locate the source of gravitational waves. While with only two instruinstru-ments the location of an astrophysical source is an almost hopeless effort, the localization region covering hundreds of square degrees in the sky, with three detectors triangulation techniques can be applied and the localization greatly improved.

An example of such an improvement is shown in Figure 1.7. The advanced detector network

Figure 1.7: Improvement of source sky localization with the addition of the Virgo interferometer, during the event GW170814. The yellow contour shows the rapid sky localization using only the LIGO detectors. The green one shows the rapid sky localization when Virgo is taken into account. The purple contour shows the sky localization coming from a full Bayesian analysis. From [8].

achieved its first joint detection on the 14th of August. Having a three detector network not only enables a far better localization of the source, but also some new science results not possible with only two detectors, such as the measurement of the GW polarizations [8]. These measurements can be used to put stringent constraints on alternative theories of gravity.

Only three days later, an extremely loud signal passed through the Earth and was soon identi-fied by the three gravitational waves interferometers, while the gamma-ray-burst electromagnetic counterpart triggered 2 seconds later the FERMI GBM telescope. The signal, coming from a binary neutron star merger, was the first event not generated by black holes and the loudest gravitational wave event ever observed so far [9]. The follow-up campaign performed

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through-1.6 Detection of gravitational waves

out the whole electromagnetic spectrum, enabling an extremely detailed study of the emission evolution, identified as a kilonova event [2]. This event also proved how gravitational waves can provide insights into the nuclear equation of state, from which the deformability properties of the objects arise. These deformability properties are imprinted in the phase of gravitational waves, enabling fundamental-physics measurements through gravitational waves astronomy [9]. The constraints placed on the deformation parameters of the star from a preliminary analysis, are shown in Figure 1.8. Another extraordinary science result coming from this event, was the

Figure 1.8: 2-D posterior distribution of the measured deformability parameters of the stars. The grey contours mark the predictions of some known equation of states. This measurement already excludes some known equations of states for GW170817. From [9].

independent measurement of the Hubble constant [1], see Figure 1.9. This result was possible since the redshift of the signal was very well determined by optical observations. Combining this measurement with the luminosity distance inferred by gravitational waves, it is possible to measure the expansion rate of the universe, without any reference to "cosmic distance ladders", needed by electromagnetic observations.

Two other gravitational waves observatories are currently under constructions in Japan (KA-GRA) and in India (LIGO India), who are expected to join the network in the next years, enabling new and more precise measurement on gravitational polarizations, other than local-ization improvements, allowing to test the fundamental nature of gravity with ever-increasing precision.

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Figure 1.9: 2-D posterior distribution of the measured Hubble constant and system inclination. The correlation is due to the high correlation between distance and amplitude. From [1].

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Chapter 2

Linear perturbations of black holes:

ringdown regime

2.1 Introduction

Normal modes solutions are ubiquitous in physical theories. They arise, for example, in classical mechanics as solutions of the equations of motion in stationary systems, when the Newtonian potential is close to a minimum. Normal modes are also of paramount importance in quantum mechanics when characterizing the time-evolution of stationary states and the quantum field theory formalism is based on a normal mode expansion.

The complete dynamical solutions of such systems are built as superpositions of these normal modes solutions, who constitute a "basis" in the functional space of the solutions. Thus, com-puting the normal modes allows (in linearized systems) to find the most general solution, whose specific form will be fixed by the initial conditions:

A(x, t) =

∞

X

n=0

aneiωntAn(x) (2.1)

where A_n(x) are the normal modes of the system, ωn is the frequency characterizing each mode

(real) and an are the amplitudes of each mode. A common feature of all these solutions is that

the frequencies emitted are characterized by intrinsic properties of the system (examples are the elastic constant of a spring and the mass in a classical system or the mass and spin of a particle in a quantum system), while the amplitude of the perturbation depends on the initial conditions. Thus, given different initial conditions, the different modes will be more or less excited and the magnitude of the a_n will then fix the normal mode content of a specific solution.

In this simple argument, we heavily relied on the assumption of stationarity. In non-stationary systems, some features of the normal modes solutions can be recovered, but in general the picture will be different.

In matter systems such as stars, the presence of these oscillations is understood as a collective motion of the matter fields present inside the star. However, in general relativity, spacetime itself is a dynamical quantity. Consequently, although black holes do not posses matter fields that can propagate such oscillations, we expect to be able to apply a normal modes analysis also to space-time oscillations around black holes, within the appropriate regime.

Differently from other physical systems, black holes have two specific properties which will affect their normal mode content. First, their event horizon, which prevents radiation from escaping

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once it has fallen inside the horizon surface. Second, they are purely made of gravitational en-ergy.

A consequence of the presence of the event horizon is that black holes are intrinsically a dissi-pative system. In classical systems, when for example considering friction, we are used to first solve an idealized version of a realistic system where no friction is taken into account. After the frictionless system has been solved, the friction is then "turned on". The contribution of friction to oscillatory systems will generally be a damping of the initial oscillations, as the system loses its energy in favor of the surrounding environment. The resulting spectrum is a "quasi" normal-mode (QNM) spectrum in which the normal modes frequency spectrum is modified by an exponential term which damps the amplitudes of the oscillations.

Due to the horizon, such an approach is not possible when dealing with a black hole, since a removal of the dissipative term (the horizon itself), would drastically change the nature of the investigated system.

Instead, a consequence of their pure gravitational nature is their normal modes content will be affected by long-range configurations of the space-time surrounding the black hole, since gravita-tional force cannot be screened. This situation is different from the oscillatory modes of matter fields in a star, since the matter field has compact supported in a limited region of the space-time. The fluctuations we are going to study are properties of the space-time itself and as such a similar formalism can be applied also to the stars, giving rise to purely relativistic spectra unpresent in Newtonian theory.

We will show what is the normal mode content of a black hole and the differences, compared to a linearized system whose stationary state can be completely characterized by its normal mode expansion. Throughout the chapter, we will closely follow the derivations present in [44,

15, 37]. We will present the derivation and detailed solution of the perturbation equations for Schwarzschild spacetimes, since the derivation for Kerr spacetimes follows similar lines, but is complicated by technical details. We will then discuss the main results for Kerr solutions, how these solutions reflect in the gravitational waveforms observed at large distances from the source, together with the phenomenology of their spectra.

2.2 Perturbation theory: the Regge-Wheeler-Zerilli equations

In this section, we will outline a procedure through which the evolution of linear perturbations of a Schwarzschild black hole can be reduced to a single wave equation.

The metric of the perturbed space-time can be decomposed as: ¯

gµν = ˚gµν+ hµν (2.2)

where ˚gµν is the unperturbed Schwarzschild metric and hµν is the perturbation metric (small

compared to the unperturbed metric in the same sense outlined in Chapter 1). We want to linearize Einstein equations in the hµν metric. The Christoffel symbols are:

Γk_µν = 1 2¯g kα_(∂ ν¯gαµ+ ∂µg¯αν − ∂α¯gµν) = ˚Γkµν+ δΓkµν (2.3) where: δΓk_µν = 1 2˚g kα_(D µhαν+ Dνhαµ− Dαhµν) (2.4)

and the covariant derivative is computed from the unperturbed metric. In vacuum Einstein equations reduce to:

¯

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2.2 Perturbation theory: the Regge-Wheeler-Zerilli equations Since: ˚ Rµν = 0 (2.6) and δRµν = DνδΓαµα− DαδΓαµν (2.7)

we obtain the equation:

DνδΓαµα= DαδΓαµν (2.8)

To solve these equations we would like to separate the angular variables from the radial one. This can be done by developing the metric in tensorial spherical harmonics. Under the rotation group the metric component transform as:

hµν =   S S Vi S S Vj Vi Vj Tkl  

where the scalar part is associated with the familiar "scalar" spherical harmonics Y_lm(θ, φ) which satisfy the equation:

sin2(θ) ∂ 2 ∂θ2 + ∂2 ∂φ2 + cosθsinθ ∂ ∂θ + sin 2_{(θ)l(l + 1)} Ylm(θ, φ) = 0 (2.9)

Instead, vector and tensor harmonics can be built from the scalar harmonics through: (V_lmx)a= ∂aYlm(θ, φ) (V_lmy )a= γbcac∂bYlm(θ, φ) (T_lmx )ab = DaDbYlm(θ, φ) (T_lmy )ab = γabYlm(θ, φ) (T_lmz )ab = 1 2[ c aDbDcYlm(θ, φ) + cbDaDcYlm(θ, φ)]

a,b,c = 2,3; γ = diag(1, sin2(θ)) is the metric on the 2-sphere of radius 1 and

= sinθ0 −1 1 0

is the anti-symmetric tensor in D. The covariant derivatives are to be computed with the 2-sphere metric.

These tensors can be divided in two different families, which give rise to two different kinds of perturbations, in terms of their parity.

We call polar contributions terms acquiring a factor of (-1)lunder a parity transformation, while we call axial contributions those terms acquiring a factor of (-1)l+1.

According to this classification, Vy and Tz are axial, while the rest of the tensors are polar. Since the unperturbed metric is parity-invariant, the perturbation equations will not mix different parity contributions and can be studied separately.

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2.2.1 The Regge-Wheeler gauge

As in the case outlined in chapter one, after perturbing the metric we are left with a gauge freedom, that we can use to simplify our equations.

We require this gauge fixing to preserve the angular-radial variables separation and the distinction between polar and axial contributions, thus we will fix the gauge by performing the infinitesimal transformation:

x0µ= xµ+ ηµ (2.10)

where η is a polar or axial vector.

2.2.1.1 Axial perturbations

In the axial case, a general perturbation takes the form:

hµν=     

0 0 −h0(r, t)_sinθ1 ∂Y_∂φlm −h0(r, t)sinθ∂Y_∂θlm

0 0 −h1(r, t)_sinθ1 ∂Y_∂φlm −h1(r, t)sinθ∂Y_∂θlm

∗ ∗ 1 2h2(r, t) 1 sinθXlm − 1 2h2(r, t)sinθWlm ∗ ∗ ∗ −1₂h2(r, t)sinθXlm     

No scalar component is present since the scalar spherical harmonics acquire a factor of (-1)l under a parity transformation.

The functions X and W are defined as: Xlm= 2 ∂ ∂θ ∂ ∂φ − cotθ ∂ ∂φ Ylm (2.11) Wlm= ∂2 ∂θ2 − cotθ ∂ ∂θ − 1 sin2_θ ∂2 ∂φ2 Ylm (2.12)

In this case, the gauge vector takes the form:

ηµ= Λ(r, t) · 0, 0, − 1 sinθ ∂ ∂φYlm, sinθ ∂ ∂θYlm (2.13)

As shown in Chapter 1, we can compute the change of the independent components of the metric in the case of axial perturbations:

δh0 = ∂ ∂tΛ(r, t) δh1 = ∂ ∂tΛ(r, t) − 2Λ(r, t) r δh2 = −2Λ(r, t)

and we fix the gauge by setting to zero the last component (Regge-Wheeler gauge):

Λ = 1

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2.2 Perturbation theory: the Regge-Wheeler-Zerilli equations

Inserting the axial contribution inside Einstein’s equations we obtain: 1 1 −2M_r ∂h0 ∂t − ∂ ∂r 1 −2M r h1 = 0 1 1 −2M_r ∂2_h 1 ∂t2 − ∂2h0 ∂t∂r + 2 r ∂h0 ∂t + 1 r2(l(l + 1) − 2)h1 = 0 1 2 1 −2M r ∂2_h 0 ∂r2 − ∂2h1 ∂t∂r − 2 r ∂h0 ∂t + 1 r2 r ∂ ∂r 1 −2M r −1 2l(l + 1) h0 = 0

These are three differential equations for two independent components. It turns out that the third equation can be obtained from the first two, after a time derivative. This last equation is needed to fix the arbitrarity arising from adding a function of r to the set of solutions of the first two equations (since the solutions of the first two equations are determined only up to a time derivative). Defining: ZA:= 1 r(1 − 2M r )h1 (2.15)

and changing the radial coordinate to the "tortoise" coordinate: x = r + 2M ln r

2M − 1

(2.16) we obtain the wave equation:

∂2 ∂t2 − ∂2 ∂x2 + VA(x) ZA(x, t) = 0 (2.17) where: VA(x) = 1 − 2M r(x) l(l + 1) r(x)2 − 6M r(x)3 (2.18) is the Regge-Wheeler potential describing the axial contribution ZA, a gauge-invariant quantity.

2.2.1.2 Polar perturbations

With similar computations it can be proved (see [44]) that the polar contributions give again rise to a wave equation:

∂2 ∂t2 − ∂2 ∂x2 + VP(x) ZP(x, t) = 0 (2.19)

where the potential now is: VP(x) = (1 − 2M r(x)) 72M3 r5_λ(x)2 − 12M3 r3_λ(x)2(l − 1)(l + 2)(1 − 3M r + (l − 1)(l + 2)l(l + 1) r2_λ(x) ) (2.20) with λ(x) = l(l +1)−2+_r(x)6M. Although the two potential look rather different in their functional form, their values are actually very close, as shown by Figure 2.1.

Moreover, they are characterized by the "isospectrality" property both in the Schwarzschild and Kerr cases, namely they give rise to the same QNM spectrum, see [15].

Note that the rotational invariance of the initial problem translates into the potential through the degeneracy in the m index. Since the m index does not appear inside the potential, the quasi-normal-mode (QNM) spectrum will be degenerate with respect to m.

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Figure 2.1: Potential for axial and polar perturbations for l=2 and l=3. From Nollert [44]

2.3 Solutions of the Regge-Wheeler-Zerilli equations:

quasi-normal-modes

A standard method to study the possible solutions of a wave equation is to assume a harmonic perturbation of the system and study the subsequent evolution. If such a perturbation grows indefinitely in time, then the system is not stable, if it decays in time, the system is stable. Indeed, we are trying to perform a normal-mode analysis. We will see that this approach is very useful to exploit the physical intuition we have on the problem and search for specific solutions. Nevertheless, a normal-mode analysis cannot completely solve the problem at hand, due to the intrinsic dissipative nature of the system.

We will show which problems arise in trying to apply this method and how to tackle them, re-covering a set of solutions with specific frequencies and damping times, the quasi-normal-modes. Assuming a harmonic time-perturbation:

Qω(x, t) = eiωtφ(x) (2.21)

and inserting this ansatz into the time-dependent equation, we obtain:

ψ00(x) + (ω2− V (x))ψ = 0 (2.22) Since we can obtain a general solution by the Fourier composition of all the single harmonic contribution, we will attempt to solve this equation and then recover the complete solution by superposition.

The second order differential Eq. (2.22) admits an infinite set of solutions. Imposing boundary conditions, this infinite set will be reduced to only two independent solutions. What are appro-priate boundary conditions to impose? Since the potential is always positive, this system does not admit bound states.

A reasonably physically motivated choice is to impose boundary conditions such that the system does not receive any external disturbance (i.e. no incoming waves from spatial infinity) and such that no waves would propagate from inside the horizon.

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2.3 Solutions of the Regge-Wheeler-Zerilli equations: quasi-normal-modes

Accordingly, we require that:

ψ(x, ω) ∼ eiωx x → −∞ and e−iωx x → +∞ (2.23) since in tortoise coordinates, the horizon is placed at spatial −∞.

Note that this implementation of boundary conditions, which allows no external disturbance from the outside to interfere, is a simplified one. In the perturbed system, we do not have a Schwarzschild black hole. Thus, the horizon will not be the one of the unperturbed black hole, which is the one we used to place the first condition. The "true" horizon will be different than the spherical unperturbed horizon and it will change with time. This boundary condition is then a simplification of the "true" boundary condition "no outcoming waves from the horizon". Also imposing the second condition is less trivial than it seems, since in a curved background as the one we are considering, the radiation can be scattered. Part of it will be reflected and will give rise to a contribution to the solution.

The effect of considering a more robust boundary condition at infinity than the one just presented, has already been studied and will be presented in the remainder of the section (this complication will correspond to the non-analytical solution, in the language of the next sections). Instead, to the best of our knowledge, effects connected to a generalized version of the boundary condition at the horizon has not been already taken into account. In order to investigate new effects that may arise from a generalized boundary condition at the horizon, a detailed study should be performed. Perturbations of the type presented here has been shown by Vishveshwara [53] to have a nega-tive imaginary part (corresponding to a neganega-tive damping time and as such to an exponential decrease) and subsequently by Wald [36] to be bounded.

The solutions identified by the boundary conditions (2.23) turns out to not be enough, in order to single out a specific solution of the equation. Equation (2.22) and the boundary conditions (2.23) admits multiple solutions. Moreover, they do not allow to represent perturbations arising from finite initial data. This is because each of the two solutions diverges at one of the extremes of the x-space. As such, a finite initial perturbation cannot be represented at all points by con-tributions of diverging modes.

To find the correct boundary conditions, we perform a Laplace transform. We define the Laplace transform of a solution in time domain as:

ˆ

f (x, s) = Z ∞

0

e−stψ(x, t)dt (2.24)

which is analytic for Re(s)>0. This function satisfies: ˆ

f00(x, s) − (s2+ V (x)) ˆf (x, s) = I(x, s) (2.25) where the source term I(x, s) is given by:

I(x, s) = −sψ|t=0−

∂ψ

∂t|t=0 (2.26)

Given any two independent solutions of the homogeneous equation f+ and f−, the general

solu-tion is then constructed as: ˆ

f (x, s) = Z +∞

−∞

G(x, x0, s)I(x0, s)ds0 (2.27) where G is the Green function of the problem:

G(s, x, x0) = 1

For whom the black hole tolls: from ringdown to tests of general relativity

Dipartimento di Fisica

Tesi di Laurea Magistrale in Fisica