Exploring Gravitational-Wave Signatures of Modified Gravity in Black Hole Ringdown

Thesis BETA-BEXCH Bachelor in Physics Exploring Gravitational-Wave Signatures of Modified Gravity in Black Hole Ringdown Author: Unai Razkin Directors: Tanja Hinderer and Pablo Bosch 2023, Unai Razkin Utrecht, 17th June, 2024 1 Contents 1 Introduction and Objectives 6 2 Basics of General Relativity 9 2.1 MetricSpace ......................... 9 2.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . 10 2.3 Tensors ............................ 10 2.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . 11 2.5 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . 12 2.6 RiemannTensor........................ 13 2.7 Ricci Tensor and Scalar . . . . . . . . . . . . . . . . . . . . 14 2.8 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . 14 3 Theoretical Introduction to GWs from perturbed BHs 16 3.1 Gravitational Waves in Flat Spacetime . . . . . . . . . . . 16 3.1.1 Solutions and Polarization . . . . . . . . . . . . . . 17 3.1.2 Gravitational Wave Sources . . . . . . . . . . . . . 18 3.2 Gravitational Waves from Newtonian Binary Systems . . . 19 3.3 Gravitational Waves from Perturbed Black Holes . . . . . 23 3.3.1 Spherical Harmonics Decomposition . . . . . . . . . 24 3.3.2 Equations of Motion of The Perturbations . . . . . 25 3.3.3 Radial Wave Equation . . . . . . . . . . . . . . . . 26 3.3.4 Solutions to the Radial Differential Equation . . . . 28 3.3.5 Isospectrality . . . . . . . . . . . . . . . . . . . . . 28 4 Ringdown waveforms 30 4.1 RingdownModel ....................... 30 4.2 Numerical Relativity data . . . . . . . . . . . . . . . . . . 31 4.3 Comparing Ringdown Model with NR Data . . . . . . . . 35 5 Modified gravity 37 5.1 Lovelock’s theorem . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Alternative theories . . . . . . . . . . . . . . . . . . . . . . 37 5.2.1 Einstein-dilaton-Gauss-Bonnet . . . . . . . . . . . . 38 5.2.2 Dynamical Chern-Simons gravity . . . . . . . . . . 38 5.2.3 Effective-field-theory of gravity . . . . . . . . . . . 39 5.3 Deviations on the Spectra of the Quasinormal Modes . . . 39 2 6 Analysis on the Signatures of Modified Gravitational Theories 41 6.1 Deviation Effects on the Frequencies . . . . . . . . . . . . 41 6.2 Alternative Gravity Theories and NR Data . . . . . . . . . 42 6.3 Effect on Parameter Inference . . . . . . . . . . . . . . . . 43 7 Results and Discussion 47 8 Conclusions and Outlook 48 A Gauge Freedom 50 A.1 The Residual Gauge from Lorenz Gauge . . . . . . . . . . 51 A.2 Gauge Fixing on Black Hole Perturbation Theory . . . . . 52 B Gravitational Wave Polarization 54 B.1 Plus polarisation . . . . . . . . . . . . . . . . . . . . . . . 54 B.2 Cross polarisation . . . . . . . . . . . . . . . . . . . . . . . 55 C Gravitational Wave Detection 56 D Teukolsky Equation 58 3 4 Acknowledgements First, I would like to thank my two supervisors, Tanja and Pablo, for their kindness, patience and especially for the priceless help I have received from them. I also extend my gratitude to the entire research group for introducing me to the fascinating field of gravitational waves and for always being willing to help with a smile. Eskerrik sakonenak familia guztiari txikitatik emandako balditzarik gabeko maitasun guztiagatik. Gehien bat nire gurasoei, zuen alboan oparitu didazuen bizitzagatik. Orainartean irakatsi didazuen guztiagatik eta ikasteko geratzen zaidan guztiagatik. Baita Leireri ere, eskerrik asko, nirekiko daukazun konfiantza eta enpatiagatik, besarkada nekaezinengatik, zure umore sarkastikoagatik eta zoritxarrez inoiz barneratzen ez ditudan ikasgaiengatik ere. Oraindik ez dakit isilik geratzen ”-Zeinek?” bezelako galderetan... Finalmente, doy las gracias a todos mis amigos, a la ”exclusiva” lista de amigos que he hecho aqu´ı en Utrecht (xD), y singularmente, a todos aquellos que han estado siempre a mi lado, a quienes he echado irremediablemente de menos este a˜no. En particular, agradezco de coraz´on a Javi y a Juan por haber hecho este a˜no mucho m´as llevadero y por sacarme una sonrisa d´ıa s´ı y d´ıa tambi´en con vuestro humor incansable. 5 1 Introduction and Objectives General relativity (GR) has an undeniable relevance in modern physics. It was proposed by Einstein in 1916, but over a century later, it continues to be a cornerstone in many fields, such as astrophysics, cosmology, particle physics, string theory, and quantum field theory [1]. This theory stands as one of the most important and influential theories of all times and remains a highly active subject of study. General relativity’s predictions have had a huge influence not only in physics but also on various aspects of modern life. For instance, it is essential in the Global Positioning System (GPS) [2] as we have to consider time corrections for the data we receive from the satellites due to spacetime curvature. The greater the distance between the Earth and the satellite, the more the metric changes, and according to Einstein’s theory, this creates time dilation, causing the satellite’s clock to deviate by approximately 35µs per day, leading to an error of about 10 km in global positioning each day [3]. Moreover, GR has numerous measurable consequences, which, along with strong experimental evidence, have established a firm supporting base for the theory [4]. GR has predicted not only the time-dilatation mentioned above or frequency shifts but also light deflection, the precession of apsides, orbital decay, and gravitational waves, none of which can be explained by Newtonian gravity. This thesis will focus on the last phenomenon: gravitational waves. Gravitational waves (GW) are ripples that propagate through spacetime. They deform spacetime -even the vacuum itselfand carry radiation energy. As Newtonian gravity cannot explain them, they are an interesting feature supporting Einstein’s theory [5]. Moreover, the interest in these waves relies on the fact that they can penetrate regions of space that electromagnetic waves cannot. For instance, they could offer the possibility to observe the birth of the Universe even before the cosmic microwave background [6], which is impossible by traditional means (optical telescopes or radio telescopes) as there was no electromagnetic radiation before the Cosmic Microwave Background. Furthermore, they allow the observation of the merger of black holes, which we will study in this dissertation. The first detection of GW effects was made in 1974 with the Hulse–Taylor pulsar [7], which led to the Nobel Prize in 1993 [8]. Through observations of the binary pulsar over many years, they could measure the orbital decay of it, which accurately agreed with the loss of energy due to gravitational waves. Throughout this dissertation, we discuss binary systems and their energy loss through the emission of gravitational waves (Sec. 3.2). Besides, the first measurement of a GW signal from a merging binary system was possible thanks to the LIGO observatory in 2015 [9] (we explain the methodology used to do so in Appendix C). Again, this led to another Nobel prize in 2017 [10]. In addition to LIGO and Virgo, more observatories have been constructed, such as KARGA [11], and more will be built in the coming years (sensitive to other frequency bands), like a space-based observatory, the Laser Interferometer Space Antenna (LISA) [12], and ground-based detectors, the Einstein Telescope (ET) [13], and Cosmic Explorer (CE) [14]. As we mentioned, in particular, we study GWs from the merger of black holes (BH); thus, just as GW waves, BH is a recurrent topic in this dissertation. Their popularity in 6 science is due to the mysteries they treasure: causality prevents any information about what happens inside the BH from exiting (the boundary beyond which nothing, not light nor information, can escape is the Event Horizon). Additionally, we expect them to have a singularity, a region with infinite curvature where all the mass is contained and where physical laws are still unknown due to the rise of quantum effects [15]. On top of that, the strong curvature creates fascinating phenomena of huge scientific interest, such as gravitational lensing and accretion discs. If that was not enough, they are also part many proposed solutions for explaining dark matter [16]. Finally, the no-hair theorem conjectures that black holes are characterized by only three observable properties: mass, electric charge, and angular momentum. Therefore, the theoretical treatment of BHs is relatively straightforward. Like GWs, BHs rely on a myriad of evidence. Since 1995, astronomers have tracked stars near the centre of our galaxy, and their motions revealed that the central object Sagittarius A* needed to have 4.3×106M⊙[17]. Although not conclusive enough, this was strong evidence supporting the existence of BH. On 10 April 2019, after the Event Horizon Telescope (EHT) observations in 2017, the first direct image of a black hole was published of the supermassive black hole in Messier 87’s galactic centre [18] and later also of Sagittarius A* [19]. Moreover, the aforementioned first GW detection led the LIGOVirgo collaboration to announce the first direct detection of a BH merger and many more since then, which is yet another indication for both BH and GW [9]. All these observational results clearly show that GR has been thoroughly scrutinized through an ample list of tests. Although we have specifically focused on BHs and GWs, all observations have consistently proven GR to be accurate, just as the ones we have presented. Nonetheless, singularities are regarded as a breakdown point of the theory of GR. Likewise, theoretical physicists aim to create a theory that reconciles quantum mechanics and gravity at this point—quantum gravity [15]. Hence, the theory is not the final word and requires ongoing testing. With this objective in mind, in this thesis, we study a method to test GR and explore possible alternative theories. For this purpose, the dissertation focuses on the GWs emitted during the BH Ringdown (RD). The RD is the ”relaxation” process that a perturbed BH -for instance, the remnant one after the collision of binary BHsexperiences as it emits energy by GWs until it reaches the stationary state. The GWs emitted during this process can be described as a sum of quasinormal modes (QNM), as seen in Sec. 3.3.4. QNM are oscillations that, unlike normal modes, decay over time due to energy dissipation, which is why the BH returns to a stable state after all. These modes have different frequencies; for this reason, the field is called Black Hole Spectroscopy, one of the most promising fields for testing general relativity and other theories. The project aims to build a model for the RD waveforms (Sec. 4) that accurately describes them using theory-based frequencies predicted from GR and theories beyond this framework. Nevertheless, regarding alternative theories, the geometry of rotating BHs is not always known, and perturbative or numerical approaches are usually the best we can get. Accordingly, there are generally no separable differential equations for the perturbation of these theories, and in turn, the frequencies are challenging to compute. As a result of these challenges, and in order to compare all theories within a common framework, we 7 need to introduce the ”ParSpec” framework (which is developed in Ref.[20]). The ”ParSpec” framework is based on a dimensionless spin parameter (related to the angular momentum) expansion of the frequencies. Recall that due to the no-hair theorem, the spectrum of RD frequencies uniquely depends on their angular momentum (J) and mass (M). Ref. [21] has computed the coefficients of this spin expansion for linear order (also the deviations from GR), combining the frequencies computed in Refs. [22], [23], [24] and [25] of slowly rotating BHs. Our thesis implements these deviations to quantify the waveform discrepancies caused by alternative theories. On top of that, we study the effect of these theories on parameter inference, namely mass and spin, and briefly discuss the biases of the inference. Lastly, we discuss the ability to constrain fundamental parameters of modified theories. This thesis is structured as follows. First, we present the foundational concepts in Sec. 2. Then, in Sec. 3, we introduce the theory of the physical phenomena of GWs on perturbed BHs, such as the derivation of the GWs from Einstein’s equations and the quasinormal modes in the ringdown of a perturbed BH. Next, some ringdown analyses, including one from Numerical Relativity, are presented in Sec. 4. Subsequently, we introduce the Modified Gravity theories in Sec. 5, followed by an analysis of the signatures that these have on the ringdown in Sec. 6. Finally, we present some results and a brief discussion in Sec. 7. In addition, the appendices contain supplementary material on gauge freedom, GW polarization direction, GW detection, spherical harmonic decomposition, and GW signal modelling. 8 Gνµ ≡Rµν −1 2Rgµν =⇒ ∇νGνµ = 0.(2.29) Altogether, the Einstein field equations become: Rµν −1 2Rgµν =8πG c4Tµν .(2.30) The proportionality constant (8πG c4) is obtained with the Newtonian limit. You are encouraged to see [1] for more details. Lastly, the Einstein-Hilbert action reproduces the Einstein field equations (Eqs. 2.30) via the minimum action principle. The action reads as SH=c4 16πG Z√−gRd4x. (2.31) 15 3 Theoretical Introduction to GWs from perturbed BHs Once we have introduced the mathematical background, we now explore the physics of GWs. The main subject of the thesis are GWs originated from perturbed BHs after a binary merger. Nonetheless, we first consider the basics of GWs in flat spacetime and discuss important features, such as their polarization or the sources of these ripples. 3.1 Gravitational Waves in Flat Spacetime The first step for understanding Gravitational Waves is to consider a flat spacetime (given by the Minkowski metric ηµν from Eq. 2.3) and apply a small perturbation (hµν) to it. gµν =ηµν +hµν,where |hµν| ≪ 1.(3.1) As |hµν| ≪ 1 we only regard linear order in perturbation. Einstein field equations (Eqs. 2.30) reveal the differential equations that these perturbations must obey. To do so, we need to compute the Ricci and Riemann tensor, but in turn, we first need to compute the Christoffel symbols. Using Eq. 2.21, we are left with the following condition: Γλ µν =1 2gλσ (∂µgνσ +∂νgσµ −∂σgµν) = 1 2ηλσ (∂µhνσ +∂νhσµ −∂σhµν).(3.2) From the definition of the curvature 2.23 and using 3.2, we are left with: Rσρµν =1 2ησλ (∂µ∂ρhνλ −∂µ∂λhνρ −∂ν∂ρhµλ +∂ν∂λhµρ).(3.3) The linearized Ricci tensor is (by contracting the Riemann tensor as in Eq. 2.24): Rµν =1 2(∂ρ∂µhνρ +∂ρ∂νhµρ −□hµν −∂µ∂νh),(3.4) where h=hµµ(the trace of hµν) and where □=∂µ∂µ. Likewise, the Ricci scalar is obtained from the contraction of 3.3 with η, since all terms are already linear in perturbation (h), R=ηνµRµν =∂µ∂νhµν −□h. (3.5) Altogether, substituting Eqs. 3.4 and 3.5 in Eq. 2.29 the Einstein tensor is Gµν =1 2[∂ρ∂µhνρ +∂ρ∂νhµρ −□hµν −∂µ∂νh−(∂ρ∂σhρσ −□h)ηµν].(3.6) To simplify this expression, we define the following quantity: ¯ hµν ≡hµν −1 2ηµνh, where h=ηαβhαβ. Subsequently, Einstein field equations (Eq. 2.30) result in □¯ hµν +ηµν∂ρ∂σ¯ hρσ −∂ρ∂ν¯ hµρ −∂ρ∂µ¯ hνρ =−16πG c4Tµν .(3.7) 16 Note, first, that as the equation above is only considering small perturbations around flat space, the gravitational part of Tµν also needs to be very weak/small. Secondly, the equation is linear, and thus, as long as we are just considering first-order perturbations, solutions can be written as superpositions of different solutions. Finally, Eq. 3.7 can be further simplified if we cleverly select a specific coordinate system. This process is called gauge fixing (discussed in Appendix A), and for our purposes, we adopt the Lorenz Gauge (or harmonic gauge). In the Lorenz gauge, we have ∂ρ¯ hαρ = 0, resulting in: □¯ hµν =−16πG c4Tµν .(3.8) 3.1.1 Solutions and Polarization The equation we obtained previously, Eq. 3.8, reveals that small perturbations follow the wave equation, namely, an inhomogeneous wave equation, whose solutions can be expressed as follows: ¯ hµν(t, x) = −16πG c4Zdt′d3x′Tµν (t′,x′)G(t−t′,x−x′),(3.9) where Gis the solution for □G(t, x) = δ4(t, x), which gives: G(t, x) = −1 4π 1 |x|δt−|x| c.(3.10) Regarding the notation in this section, we are using the bold symbols to refer to the spatial part of the coordinates xν; this is, x≡xi, (often also referred to as x ≡x). If we consider the energy-momentum tensor to be 0 (Tµν=0), due to the remoteness of any source, then Eq. 3.8 reduces to the well-known homogeneous wave-equation: □¯ hµν(t, x) = 0 −→ −1 c2 ∂2 ∂t2+∇2¯ hµν(t, x) = 0.(3.11) Even though we have selected the Lorenz gauge condition, some non-physical freedoms should still be restricted (see Appendix A.1 for the residual gauge freedom). In a vacuum, we can impose the following four conditions: ¯ h0i= 0 ¯ hµµ= 0 Lorenz gauge −−−−−−−→ ∂ihij = 0 ¯ hii= 0.(3.12) Under these conditions, we can limit our following procedure to the space coordinates, which is indicated with Latin indexes (i∈1,2,3) as mentioned before. The solution of Eq. 3.11 can be expressed as the superposition of plane waves. If these plane waves are propagating in the z direction, we know that they satisfy hij = aij cos(ω(t−z/x)). We can now impose the before-mentioned conditions Eq. 3.12, which results in azj = 0 = hzj. As Eq 3.12 shows, the matrix is traceless. Therefore, 17 hTT ij =  h+h×0 h×−h+0 0 0 0 =  a+a×0 a×−a+0 0 0 0 cos(ω(t−z/c)).(3.13) These two parameters h+and h×determine the polarization of the gravitational wave in the transverse-traceless gauge (which is what TT stands for). The spacetime metric can be written like: gµν =ηµν +hTT ij ,(3.14) which gives the following line element (recalling Eq. 2.4): ds2=−c2dt2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2.(3.15) To interpret this physically, we first need to imagine a circle of test points when h+= h×= 0. Then, at any time when either h+= 0 or h×= 0, these test points would appear elliptical as illustrated in Fig. 2, where we can see the two polarisations that a gravitational wave can have: Plus (h×= 0) and Cross (h+= 0) polarization, which are explained in detail in Appendix B. x yt (a) Test points under the effect h+polarization gravitational wave x yt (b) Test points under the effect h×polarization gravitational wave Figure 2: Space deformation effects of the two polarization types of gravitational waves illustrated with test points. 3.1.2 Gravitational Wave Sources Radiation of gravitational waves can be analogous to electromagnetic radiation as both emit waves that carry energy at the speed of light c. The inhomogeneous equations for electric and magnetic vector potentials are almost identical to Eq. 3.9. Hence, we can understand sources of GWs in linearized gravity by performing a similar analysis in multipole expansion as for electromagnetic potentials. In the electromagnetic expansion, oscillations of the center of the charge density produce dipole radiation. Notwithstanding, in the case of an isolated source the center of mass of the system cannot accelerate. Consequently, the principal and most crucial difference in the analogy relies on the fact that gravity does not have dipole radiation. Thus the quadruple moment is the responsible for the leading order in this multiple expansion, 18 which is written as (again you are encouraged to see Ref. [1] for a proper derivation of the following equation) ¯ hij(t, r) = −2G c4r¨ Iij(t−r/c),(3.16) where IT ij =ZV′ ρ(r′)3r′ ir′ j−||r′||2δijdV ′.(3.17) We could rewrite this formula as hTT ij (t, r) = −2G c4rΛijkl(n)¨ Mkl,(3.18) where Mij is the mass quadruple moment, Mij ≡Zd3xρxixj, and Λijkl, the traceless transverse projection operator, is defined as Λijkl ≡PikPjl −1 2PijPkl where Pij =δij −ninj, where niis the unit vector of the propagation direction. 3.2 Gravitational Waves from Newtonian Binary Systems A remarkable example of a gravitational wave source is a binary system. In this subsection, we analyze a system composed of two point masses m1, m2, tracing circular orbits in the center of the mass reference frame. We can reduce two-body problems to one-body problems, where the reduced mass µ=m1m2 m1+m2takes the place of the test mass. Be aware that although linearized gravity is not strictly applicable to the binary system, it turns out that rigorous and more sophisticated calculations show that the quadrupole formula (Eq. 3.18) captures the leading-order result even for such a gravitational source. On top of that, for binary systems, we can further simplify the quadrupole formula (Eq. 3.18) to the following expression: h+=1 r G c4¨ M11 −¨ M22, h×=2 r G c4¨ M12,(3.19) where r is the distance to the source and Mij is reduced to Mij =µR2NiNj,(3.20) 19 where R is the relative separation and Niis the unit vector pointing to the reduced mass, whose components are N=  1 0 0 0 cos(ι)−sin(ι) 0 sin(ι) cos(ι)  | {z } Rx(ι)   cos(φ) sin(φ) 0 =  cos(φ) cos(ι) sin(φ) sin(ι) sin(φ) , ι x y z x′ y′ z′ Observer ˆ N Figure 3: Orbital plane inclination and finally, where φrepresents the angle relative to the x-axis within the orbital plane, and ιindicates the inclination of the orbital plane to our line of sight (refer to Figure 3.2). Rx(ι) refers to the matrix for rotations around the x-axis. From the analysis of the equations of motion of the binary system it follows that the radius and frequency of the circular orbits are related by R= (GM)1/3ω−2/3,(3.21) where ω= ˙φ, M is the total mass M=m1+m2. The orbital energy (Eo) is Eo=1 2µR2ω2−GMµ R=−1 2µ(GMω)2/3,(3.22) where in the second equality we used Eq 3.21. The average power radiated in gravitational waves (dEGW dt ) is computed via (See Ref [1] for a closer look at this formula) dEGW dt =r2c3 16πG Z2π 0 dφ Z1 −1 dcos ι˙ h2 ++˙ h2 ×.(3.23) To calculate h+and hxproduced by the source, we first use Eq. 3.20, which results in ¨ M11 =−2µω2R2cos(2φ) ¨ M22 = 2µω2R2cos(2φ) cos2(ι) ¨ M12 = 2µω2R2sin(2φ) cos(ι). Introducing these expressions in Eq. 3.19 we are left with h+=1 r G c4(¨ M11 −¨ M22) = −2µR2ω2G rc4cos(2φ)[1 + cos2(ι)] h×=2 r G c4¨ M12 =−2µR2ω2G rc4sin(2φ) cos(ι) ,(3.24) and finally from Eq. 3.23 we obtain 20 dEGW dt =r2c3 16πG Z2π 0 dφ 56 15 16µ2R4ω6G2 r2c8sin2(2φ) + 2 3 64µ2R4ω6G2 r2c8cos2(2φ) dEGW dt =r2c3 16G56 15 16µ2R4ω6G2 r2c8+8 3 16µ2R4ω6G2 r2c8=32 5 Gµ2R4ω6 c5.(3.25) The only source of energy loss we consider is GW emission (EGW ); other sources of energy loss are neglected. Thus, the energy loss of the orbital decay (Eo) approximately satisfies dEGW dt +dEo dt = 0.(3.26) With this in mind, we can compute the frequency change in time by differentiating Eq. 3.22 with respect to ωand also using Eq. 3.25. ˙ω=dEo/dt (dEo/dω)=−dEGW /dt (dEo/dω)=96 5 3 √G5M5ω11 c5,(3.27) where M=µ3/5M2/5 If we integrate this differential equation from the current time and frequency {t, ω}to the (formal) values at the time tcof coalescence {tc,∞} we are left with τ=−5c5 256(GM)5/3ω8/3ω(τ) = 1 8 8 s125 c3 GM5 τ−3/8,(3.28) where τ=t−tc. We can plot the angular frequency against τto illustrate how it changes over time. Figure 3: Time evolution of the orbital angular frequency in the last 0.2 seconds before coalescence for two binary systems. The blue line system has masses of m1=m2= 4M⊙ while the orange m1=m2= 16M⊙ Frequency magnitudes differ by a factor of 2.78 for these systems, but qualitatively, both figures’ frequency approaches a divergence in t=0, as expected by the formula. In reality, this divergence stops when both objects start merging. Finally, phase as a function of time can be obtained by integrating Eq. 3.28. 21 φ(τ) = −c3τ 5GM5/8 +φc,(3.29) where φcis the integration constant. Substituting Eq 3.29 in 3.24 results in h+=−GM 2rc2 4 r−5GM c3τcos "2c3τ 5GM5/8 + 2φc#(1 + cos2ι) h×=−GM rc2 4 r−5GM c3τsin "2c3τ 5GM5/8 + 2φc#cos(ι) .(3.30) If we plot them, we get: Figure 4: Time evolution of the polarisation amplitude h+in the last 0.2 seconds before coalescence for two different binary systems (both φc=tc= 0 and ι= 0). In the left panel a binary system 40Mpc distance away with masses m1=m2= 4M⊙(plotted until t=- 0.05s). Right panel a binary system 40Mpc distance away with masses m1=m2= 16M⊙ (plotted until t=-0.1s). There is a significant difference in amplitude between the left panel (a) and right panel (b) systems, as the amplitude of the second system is 5.6 times larger. There is a way of obtaining the ratio of amplitudes analytically. Using either 3.19, 3.24 or 3.30, but omitting time dependency and also the orbital plane inclination parameter (ι) the expression of the amplitude reads A+i=−GM 2rc2 4 r−5GM c3τ. The ratio between them is, A+ A′ + =r′ rm1m2 m′ 1m′ 23/4m′ 1+m′ 2 m1+m21/4 −→ A+a A+b≈0.177 or A+b A+a≈5.657 .(3.31) The same derivation can be done for the ratio of frequencies using the formula 3.28. f f′=m′ 1m′ 2 m1m23/8m1+m2 m′ 1+m′ 21/8 −→ fa fb≈2.38.(3.32) 22 Therefore, the first binary (a) would have a frequency that is 2.38 times faster than that of the second binary (b). On top of that, notice that the Eq. 3.32 is not distance-dependent, and hence, if the black hole binary were 5.657 times farther away (r=226.28Mpc), the frequency ratio would not have changed, yet the amplitudes of both systems for that case would be the same. Frequency contains relevant information about the mass of the binary system, which is essential for distinguishing between different systems. Amplitude, however, is not only mass dependent but also depends on extrinsic variables such as distance (r) or the orbital inclination (ι). 3.3 Gravitational Waves from Perturbed Black Holes In the evolution of a binary system, we have three main phases. The inspiral part, which is the phase we have studied in the previous section; the merger, when both BH collide; and the RD, when the merger results in a single perturbed BH, which emits GWs. In this subsection we show how the remnant perturbed BH emits GWs. To do so, we add a perturbation to the BH metric (as illustrated in Fig. 5) and, subsequently, we get the equations of motion of these perturbations. We generalize the procedure conducted in 3.1 based on the lectures from Eric Poisson (Ref. [26]). Figure 5: Illustration of the first order perturbative approach in a Black Hole To begin with, we use a one-parameter (λ) family of metrics gαβ(λ, x). Next, we Taylor expand them under the small λparameter. Even though, for our purposes, we only use the first-order perturbation, it is insightful to see how higher-order perturbations could be added to the metric. gαβ(λ, x) = g(0) αβ +p1 αβ(x)λ+1 2p2 αβ(x)λ2+O(λ3) pn αβ(x) = ∂ngαβ(λ, x) ∂λnλ=0 .(3.33) In the context of this expansion, note that xrefers to all of the spacetime coordinates xν. In the following sections, we work with the Schwarzschild metric for the order 0 metric; this is, 23 g(0) µν =     −1−2GM rc20 0 0 01 (1−2GM rc2)0 0 0 0 r20 0 0 0 r2sin2θ      .(3.34) Nevertheless, note that this metric does not include rotating BH, as we are not considering the Kerr metric, only the Schwarzschild metric. Although spinning BHs are fundamental in our thesis, performing a similar analysis to the one developed in the following chapters is not possible. Kerr metric requires another kind of technique, as the one used by Teukolsky, who obtained the decoupled and separable Kerr-metric equations [27], also known as the Teukolsky equations (see Appendix D). Moreover, in alternative theories, the geometry of rotating BHs is only known perturbatively or numerically, and there is, in general, no analog of the Teukolsky equation [20]. As a result, the following chapters provide a partial description of GW-emission phenomena. They motivate the phenomenon on static BH, a similar enough system to have a broad idea of how this phenomenon should look, as the general features remain the same. 3.3.1 Spherical Harmonics Decomposition The metric introduced in the previous Eq. 3.33 can be expressed differently due to the gauge freedom. This is discussed in Appendix A. In particular, we use the so-called Regge-Wheeler or Zerilli’s gauge ([28], [29]) named after Tullio Regge, John Archibald Wheeler, and Frank J. Zerilli. However, before exploring the gauge fixing, we should first decompose the perturbation in spherical harmonics to take advantage of the spherical symmetry of the background. Likewise, we also take advantage of the spherical coordinates (xν= (T, r, θ, ϕ)). If we constrain our space to the surface of T=constant and r=constant (i.e. S2), or in other words, if we only do rotations of the reference frame around the origin, the ten components of the perturbing tensor (hνµ) transform like 3 scalars (h00, h01, h11), 2 vectors (h02, h03;h12, h13), and a second-order tensor (h22, h23, h32, h33). However, under the appropriate gauge transformations (see Appendix A.2), the 10 variables of the perturbations can be reduced to 6. A scalar function decomposes in spherical harmonics as h(θ, ϕ) = ∞ X l=0 m=l X m=−l hlmYlm(θ, ϕ).(3.35) We can perform a similar analysis for vectors and tensors. The formalism of vector and tensor harmonics is introduced in Appendix A.2 (see [28]). It turns out that sectors with different parity decouple. Parity transformation is the reflection through the origin, which in mathematical lingo is: r −→ −r, which in spherical coordinates means θ−→ θ+πand ϕ−→ π−ϕ, which in turn implies ∂ ∂θ −→ ∂ ∂θ and 24 which for a certain direction in read as (see [37]) hRD lm (t) = N−1 X n=0 Almne−iωlmn(t−tlmn match),(4.3) where n is the QNM overtone number, and N is the number of overtones included in the theoretical model. Additionally, ωlmn =ωR lmn −i/τlmn. Finally, tlmn match is equivalent to the ϕlm. We only uses this notation in the context of this formula for clarity’s sake. Nevertheless, it is important to highlight that we do not consider overtones (n) from now on. Our model is a simplified version of Eq. 4.3 neglecting the overtones n > 0. hRD lm (t) = Alme−iωlm(t−tlm match),(4.4) where Alm =Alm0and ωlm =ωlm0. This is due to the fact that data for the modified theories we consider in Sec. 5 is only available for the first overtone. Although it seems like a clumsy simplification, it does not make a big difference since our model is still significantly accurate compared with NR data. Theoretically, the higher the overtones, the shorter the decay time, and consequently, they fade much faster than n=0, which justifies why we can neglect higher overtones. 4.2 Numerical Relativity data We have previously noted the importance of NR data in checking how accurate our RD model is in comparison. Recall that this data is one of the most accurate methods available and, thus, the best way to check for the sharpness of our method. It also constitutes a method to extract the initial conditions of the RD, such as the amplitude factors or phases resulting from the merger. Consequently, in this subsection, we introduce and analyze this data. NR is a GW signal modeling method that numerically solves Einstein field equations. In particular, we use the data from the SXS collaboration, which uses The Spectral Einstein Code (SpEC) for solving partial differential equations [39]. Unfortunately, there are some parameters related to the binary system of BH that this method finds challenging to cope with, such as big mass ratios (q > 20) or high spins (|χ| ≥ 0.9). Specifically for our analysis we use the simulation SXS:BBH:1580, which we downloaded from Ref. [40]. This simulation has an interesting parameter χ≈0.2 (χ=J/M2), which is not particularly relevant at this point, but it is for the analysis we conduct later in the dissertation. All in all, the time-series waveform on this simulation for an arbitrary direction of the detector is θd= 0.1rad, and ϕd= 0.2rad is: 31 Figure 7: GW polarisation amplitudes measured from a detector at a relative position of θ= 0.1 and ϕ= 0.2, for the SXS:BBH:1580 simulation. As we said, we generally focus on the normal modes; in particular, we want the dominant mode, which, from the differential equation, we expected to be the (2,2) mode. Let us then study all of them and ensure the (2,2) mode is dominant. Figure 8: QNM’s amplitudes for the simulation SXS:BBH:1580. It is apparent that the prevalent mode is the l=m=2 mode, as the next greatest mode, l=2 m=1, is about 10 times smaller in magnitude. We can numerically extract strategic information, such as complex frequencies of each QNM mode. We first focus on the study of the (2,2) mode, which time-series amplitude is illustrated in Fig. 9. 32 Figure 9: The h+polarisation amplitude of the (2,2) mode of the simulation SXS:BBH:1580. Frequency values can be obtained, on the one hand, by differentiating the phase () of the waveform to get the frequency (ω); on the other hand, doing a linear regression of the logarithm of the waveform to get the decay rate of it (τ). Figure 10: Restoring complex frequency values for the (2,2) mode waveform from SXS:BBH:1580 Numerical Relativity simulation. Left panel rh22 waveform and frequency (obtained by numerically differentiating). The red horizontal dashed line indicates the average frequency of the RD. Right panel log(r|h22|/M) waveform (blue) and a linear approximation of it (red). We only consider data on the right side of the red vertical dashed line, where the linear regime starts. Thus, ω22 ≈0.4113 −0.0896i, and then τ22 =−1/Im(ω22)≈11.16. We can do the same analysis for different QNMs as illustrated in Figs. 11 and 12: 33 Figure 11: Restoring wR lm frequency values for some (l,m) QNM waveforms using SXS:BBH:1580 Numerical Relativity simulation rhlm waveforms and frequencies (obtained by numerically differentiating). The red horizontal dashed lines indicate the average frequencies of the RD. Figure 12: Restoring wI lm part of the frequency values for some (l,m) QNM waveforms using SXS:BBH:1580 Numerical Relativity simulation log(r|hlm|/M) waveforms (blue) and linear approximations of them (red). We only regard data on the right-hand side of the red vertical dashed line. These two figures show more significant numerical errors on the data than those from the 34 (2,2) mode. Additionally, note how even in higher-order modes, the red dashed vertical line still points broadly to the beginning of the linear regime. 4.3 Comparing Ringdown Model with NR Data To see the accuracy of our model we now compare the RD model built with the theorybased prediction with NR. At this point, it is essential to recall that these modes are functions of the remnant mass (M) and angular momentum (J). We use the spin parameter χinstead of J, defined as χ≡J/M2. For the case of this simulation, the final spin is χ≈0.2077. By performing a linear interpolation of the data from the BH perturbation calculations extracted from Ref. [32], we get that for the (2,2) mode, the theoretical frequency is ω22 ≈0.4113 −0.0896i, and we can do the same for other modes too. We summarize the theory-based predictions for different QNM modes and the frequencies extracted in Figs. 11 and 12 in the following table. (l,m) mode ωNR ωGR (2,2) 0.4113 −0.0896i0.4119 −0.0901i (2,1) 0.3971 −0.0889i0.3971 −0.0903i (3,3) 0.6581 −0.0939i0.6604 −0.0939i (3,2) 0.6496 −0.0927i0.6444 −0.0940i (4,4) 0.8906 −0.0985i0.8929 −0.0954i (4,3) 0.8847 −0.0981i0.8761 −0.0956i (5,4) 1.0949 −0.1032i1.1013 −0.0963i Table 1: Complex frequencies obtained for (l,m) modes, both from the BH perturbation calculations (labeled as ωGR) and the ones obtained from the NR data (labeled as ωNR albeit it is based on GR too). Let us compare the frequencies obtained in Figs. 11 and 12 with the theory-based predictions. To do so, we compute the relative difference. This is, |ωGR −ωNR|/|ωGR|. Figure 13: Relative difference in percentage of theory-based prediction frequencies compared to the ones inferred from NR data. 35 As we can see in the figure, the theoretical frequencies are close to the ones we obtained from NR data. The greatest differences are in higher modes (4,4), (5,4) and (3,2), which reach 1 % of relative difference. The relative difference gets bigger in higher modes because of their smaller amplitudes relative to the (2,2) mode, which in turn results in more numerical errors as we noticed in Figs. 11 and 12. However, the most significant mode, the (2,2,0), has 0.2 % of relative error. We can also reproduce both waveforms at once, Figure 14: Phase aligned RD waveforms. Blue corresponds to Fig 9 (NR data), while red corresponds to the frequencies obtained theoretically. The red wave equation is phase-aligned in the dashed vertical red line (t/M=19.6); likewise, we set the amplitude to be the same at that point. We stress how accurate the waveform is, albeit only considering the first overtone and neglecting any higher than n > 0. We have conducted the same analysis for higher modes and other simulations, such as the SXS:BBH:0315 (the most similar simulation to the first LIGO detection), with similar results. Figure 15: Phase aligned RD waveforms. Left panel: 3,3 mode of the SXS:BBH:1580 simulation. Right panel: 22 mode of the SXS:BBH:315 simulation. Thus, we have proven that our model with only one overtone is effective for the GW signals, and hence, we can now go beyond the GR regime. 36 5 Modified gravity Before we include variations into our RD model, we will briefly motivate these alternative theories and present a crucial theorem to understand better and classify alternative theories. Einstein’s theory faces two crucial problems at the time. First, there is no quantum description of gravity, and second, the cosmic acceleration constant is inexplicably small [41]. For these reasons, GR is thought to be incomplete, or it is believed to require modifications. One of the most essential theorems for modified gravity is Lovelock’s theorem. 5.1 Lovelock’s theorem According to Lovelock’s theorem [42] in a 4-D spacetime, a rank (2,0) tensor whose components are functions of the metric tensor gνµ and its first and second derivatives (linear in the second) and these are also symmetric and divergence-free, then the only possible option is: Aνµ =aGνµ +bgνµ (5.1) These are Einstein field equations (Eqs. 2.30), so any modified theory of gravity is forced to: •Adding more than second-order derivatives of the metric •Using other fields rather than metric tensor •Using more or fewer than 4 space dimensions •Considering a non-local theory The first option (Adding more than second-order derivatives) might be the most plausible. However, higher derivatives generally lead to problematic instabilities, according to Ostrogradsky’s theorem [43]. Another option is adding higher-order curvature terms in the Einstein-Hilbert action. It turns out that in standard quantum field theory, a significant obstacle on the route to quantum gravity is renormalizing the Einstein-Hilbert (recall 2.30) [44]. It was shown that quadratic curvature terms fix this problem. On top of that, low-energy effective string theory suggests that the Eisntein-Hilbert action could be thought of as the first term in an expansion containing all possible curvature invariants [44]. Therefore, we mainly stick to higher-order curvature theories. 5.2 Alternative theories After the general analysis of modified gravity, which has given us an overall picture, it is time to specify the theories we use. Thus, this subsection briefly introduces the modified gravity theories for which QNMs have been computed. These theories have the same crucial feature: They all affect the GR’s BH QNM spectra. On top of that, they all include 37 higher-curvature corrections to GR. The theories we consider are the same as the ones from Ref. [21]. We include alternative theories introducing variations to the EinsteinHilbert action 2.31 and with small-coupling approximations, to be more exact we impose the fundamental lengthscale, the coupling factor, to be lth ≤GM/c2. Alternative theories could break the isospectral property shown in Sec. 3.3.5. Accordingly, the frequencies for the (2,2,0) mode will vary depending on whether the perturbation is even or odd. How this affects the GW signal will not be studied in this dissertation. Nevertheless, we will follow the practical approach of Ref. [21], which merely chooses the least damped parity. As long as other parity QNM are not excited more, the least damped parity will prevail. Thus, simply choosing this one frequency is a sensible decision. 5.2.1 Einstein-dilaton-Gauss-Bonnet Einstein-dilaton-Gauss-Bonnet (EdGB) theory is described by the action: SEdGB =Zd4x√−g1 16πR−1 2(∂ϕ)2+1 4l2 EdGBf(ϕ)G,(5.2) where g≡det(gνµ), R is the Ricci scalar, ϕis a scalar field, with kinetic term (∂ϕ)2= gνµ∂νϕ∂µϕ, which couples to the Gauss-Bonnet invariant G=RνµρσRνµρσ −4RνµRνµ +R2, by the coupling function f(ϕ) and the coupling factor lEdGB (which has length units). EdGB gravity has BH with the so-called ”secondary hair” scalar depending on the scalar function f(ϕ). The ’secondary’ term refers to its dependency on the black hole’s mass, spin, and other theory parameters, such as lEdGB. Thus, the charge is not an independent quantity. According to Ref. [21], we can associate a monopole scalar charge to this scalar field, just like the electric charge to the electric potential. Similarly, this charge decays inversely proportional to the distance rfrom the black hole (1/r). Namely, as the scalar charge is linked to the curvature scalar, the scalar charge is inversely proportional to the BH mass. This scalar field has observable consequences. On the one hand, BH binaries are a source of scalar-dipole radiation; thus, the orbit should decay faster. Ref. [45] constrained the coupling factor to lEdGB ≤7.1km. On the other hand, the coupling with a scalar field affects the even perturbation, which in turn breaks with the isospectrality of the Swarzchild BH. Using perturbation theory, the QNMs have been computed up to first order in rotation in Ref. [22]. 5.2.2 Dynamical Chern-Simons gravity Dynamical Chern-Simons gravity (dCS) is described by the following action: SdCS =Zd4x√−g1 16πR−1 2(∂Θ)2+ 4l2 dCSΘ∗RR,(5.3) where Θ is a pseudoscalar field which couples to the Pontryagin density ∗RR =∗RνµρσRνµρσ, where ∗Rνµρσ =ϵνµγδRγδρσ/2, and ϵνµγδ is the Levi-Civita tensor and finally, where lEdGB 38 is the coupling factor (which has length units). As before, these BHs have a scalar field. This field decays r−2, to which we can associate a scalar dipole charge (as in the electromagnetic case). However, most importantly, the spectra QNM has some deviations. In this case, odd parity perturbations couple with perturbations of the scalar fields again breaking isospectrality. Ref. [23] and [24] studied the QNM spectra of slowly-rotating BHs in dCS gravity. 5.2.3 Effective-field-theory of gravity The following action describes the Effective Field Theory (EFT) of GR: SEF T =Zd4x√−g R+X n≥2 l2n−2 EF T L(2n)!,(5.4) where L(2n)are higher order curvature corrections and lEFT is the coupling factor (which has length units). We treat higher-order curvature theories separately, and we only consider L(6) and L(8). L(6) =λeRµνρσRρσγδRγδµν +λoRµνρσRρσγδ ˜ Rγδµν L(8) =ϵ1C2+ϵ2˜ C2+ϵ3C˜ C, (5.5) where C=RγδρσRγδρσ ˜ C=RνµγδϵνµρσRρσγδ.(5.6) Finally, λo,e and ϵi, are free parameters, although we set both λo=λe=1 and ϵ1= 1, ϵ2=ϵ3= 0 which leave the coupling factor as the only free parameter. In the case of EFT, the spectra of QNM of rotating BHs were studied to the first spin order corrections in Ref. [25]. 5.3 Deviations on the Spectra of the Quasinormal Modes As previously discussed, alternative theories create modifications in the QNM spectra. This subsection details the framework to account for these deviations in modified theories. We base the model on the ParSpec framework (Ref [20]). According to it, we can express the deviation by performing the following spin expansion: ωlmn =1 Mf Nmax X j=0 χj fω(j) lmn 1 + γδω(j) lmn(5.7) τlmn =Mf Nmax X j=0 χj fτ(j) lmn 1 + γδτ(j) lmn,(5.8) where according to Ref. [21] in geometric units γ=ℓth Ms fp .(5.9) 39 The lth is the ”coupling factor” that depends on the theory, which is a crucial factor as it is the one responsible for coupling the scalar/pseudoscalar fields or the higher order curvature tensor to the original GR formulation, as we have seen in the Eqs. 5.2, 5.3 and 5.4. In other words, this is the parameter we would like to constrain. The closer this parameter is to zero, the less influence this theory has, and from a philosophical point of view, this translates to the validity of the theory (needless to say, this does not prove the theory wrong, but we could argue its validity/reliability). Ref [21] computes the ParSpec framework coefficients of the (l,m,n)=(2,2,0) mode of the least damped parity perturbations based on the previously mentioned theory-by-theory frequencies computed in Refs. [22], [23], [24] and [25]. According to Ref. [21], axial parity perturbations are the least damped for all 4 theories we are studying. We summarize these coefficients in the following table (Tab. 2). GR EdGB(p=4) dCS(p=4) cubic EFT(p=4) quartic EFT(p=6) ω(0) 0.3737 δω(0) 0.0107 3.1964 -0.5813 -0.2114 τ(0) 11.2407 δτ(0) 0.0044 6.3619 -0.2114 -0.6070 ω(1) 0.1258 δω(1) -0.2480 41.199 6.4439 -1.5263 τ(1) 0.2522 δτ(1) -1.1014 794.66 265.12 171.35 Table 2: Deviation parameters for the complex frequencies of the 22 quasinormal mode for different alternative theories (See Ref. [21]). With this data, we can expand alternative theories up to first order in spin. Therefore, we must consider small spins, |χf|<< 1. For the 2,2 mode in GR, this translates to spins between χ∈[0,0.2] (Ref [20]), the region on which relative errors in frequencies are not greater than 1%. Mfωlmn =γhδω(0) lmnω(0) lmn +χfδω(1) lmnω(1) lmni+ Nmax X j=0 χj fω(j) lmn (5.10) τlmn Mf =γhδτ(0) lmnτ(0) lmn +χfδτ(1) lmnτ(1) lmni+ Nmax X j=0 χj fτ(j) lmn.(5.11) We, however, consider the following expansion: Mfωlmn =Mfω(GR) lmn +γhδω(0) lmnω(0) lmn +χfδω(1) lmnω(1) lmni+ Nmax X j=2 χj fδω(j) lmnω(j) lmn (5.12) τlmn Mf =τ(GR) lmn Mf +γhδτ(0) lmnτ(0) lmn +χfδτ(1) lmnτ(1) lmni+ Nmax X j=2 χj fδτ(j) lmnτ(j) lmn.(5.13) The purpose of this variation is to recover the frequencies from GR when we impose l= 0, even for χ > 0.2. This however does not influence the rest of the theories as difference between the expansion we consider and the pure ParSpac framework is: 40 7 Results and Discussion This dissertation is based on the results from Ref [21], which computes the ParSpec framework coefficients of the (l,m,n)=(2,2,0) mode of the least damped parity perturbations combining the theory-by-theory frequencies that were calculated in Refs. [22], [23], [24] and [25]. In Ref [21] they essentially tried to constrain the coupling factor using the two loudest two ringdown signals observed (GW150914 and GW200129). Performing Bayesian inference they place upper bounds on the coupling factors, namely, in dCS ldCD ≤38.7km (%90 of credible level) and lcEFT ≤38.2km (%90 of credible level) and lqEFT ≤51.3km (%90 of credible level). In this thesis, however, we delve into the deviations of the RD waveforms in particular, and we produce some novel results. More precisely, we have quantified the bias on the parameters inferred by GR under the regime of alternative theories or the bias on the inference of parameters from alternative theories. In particular, our results are: First, Table 3, which shows the misinterpreted data we would infer by assuming GR in a universe governed by alternative theories (all alternative theories have assumed that the remanent BH’s Mf= 1 and spin χf= 0.2). Second, Table 4 which contains the inferred mass and spin for GR, EdGB, dCD, cEFT, qEFT theories from the SXS:BBH:1580 NR waveform (according to the catalogue it corresponds to the parameters Mf= 0.98 and χf= 0.2077). Note that these tables show the parameters that minimize the mismatch but are not necessarily the most probable. Nevertheless, we do not expect a big difference between the mismatch plot from Fig. 21 and the probability distribution plot, as the most probable set of parameters should also be close to the ones that match best. The results show that the mass or spin inferred using GR tends to be higher than those inferred using alternative theories. The recovered spin or mass is more significant than the original alternative theory in the first table (table 3). In the second table (table 4), alternative theories predict lower spin or mass than those predicted by GR. This disparity is because the energy of scalar or pseudoscalar fields in alternative theories is interpreted as additional mass or spin in GR. In other words, these scalar or pseudoscalar fields interfere with spacetime curvature, but since in GR there are no coupled fields; the BH’s mass or spin must completely compensate for the modification on spacetime that scalar/pseudoscalar fields introduce. Regarding Table 4, GR has the best mismatch (the smallest one). Nevertheless, qEFT is close to having a better one, and all the other alternative theories are also close to being as good. We should recall Fig. 16 to make sense of this. In this figure, we can broadly see which complex frequency corresponds to a certain spin for GR. With this in mind, we can now try to match the same frequencies on alternative theories by sweeping through the spin and re-scaling with mass. As these theories have to match 2 parameters (ωRand τ) and for doing so, they can sweep through 2 parameters (Mand χ), this resembles a system of equations with two unknowns and two variables. Hence, this system is flexible enough to match the ”right” frequencies and obtain the best mismatch that our model has to offer (as all theories have a similar precision, it is apparent that every theory has reached its limit). In this sense, we could say that GR had the least mismatch by ”pure 47 chance,” as every other theory has the same ability to match the ”right” frequency. We could also try sweeping through the coupling factor (l). Nonetheless, adding a new parameter like that would not ”a priory” improve the mismatch, which can be interpreted with the same intuition as in the above paragraph. If we add a new parameter to the system of equations, we would add a third variable but keep the two unknowns. Accordingly, the system is flexible enough to match the right frequency for each l. Thus, adding l to our analysis would not improve the mismatch but instead lead to a degeneracy of mismatch through l. As a result, the analysis is only worthwhile if we can constrain some of the free parameters for each theory. If we could constrain any of them, we could perform the twounknown two-variable system ”game” described above to match the complex frequencies but now using ”l” instead of either the spin or the mass, and subsequently, we could look for an upper or lower bound for l. 8 Conclusions and Outlook In this thesis, starting from the Theory of General Relativity, we have introduced gravitational wave as a linear perturbation of the flat space. We have also explored the sources of these waves, emphasizing binary systems inspirals as exemplified. Furthermore, we have studied the perturbation of a BH (Schwarzschild) within General Relativity, revealing a relaxation process known as Ringdown - a stage where perturbations eventually dissipate. Secondly, we have analyzed data from Numerical Relativity, notably from the SXS collaboration (Sec. 4.2), some of the most reliable sources concerning binary merger processes. We have built an effective model with one overtone and contrasted NR data to prove our model’s accuracy (Sec. 4.1). Next, we introduced modified theories (Sec. 5) and considered their impact on the complex frequencies of QNMs to include them in our RD model. Additionally, we have computed the time series of the waveforms and plotted them with different coupling factors to see the effect. Thirdly, we have produced RD waveforms based on our alternative-theory-based model and tried to infer the key parameters χand Mfrom them using the GR-based model. Our study has demonstrated significant differences in inferred parameters depending on the underlying gravitational theory, emphasizing the necessity of considering alternative models in GW data analysis to avoid misinterpretation of the experimental data. Furthermore, we have proposed the underlying reason for these discrepancies: As no coupled fields exist in GR, alternative theories need less mass or spin on the BH to create the same curvature effects. Finally, we have used some NR data as ”text” experimental data. We have inferred the data with our GR-based model and ensured the model was good enough to recover the original data used by the NR simulation. Alternatively, we have inferred the data with modified theories and concluded that they would firmly deviate. With this in mind, we have discussed some ways to constrain the coupling parameters, nevertheless we have not obtained any such constrains, as opposed to Ref. [21], as we have seen on the results. In a nutshell, we have comprehended some gravitational wave phenomena, including their sources and characteristics, and focused particularly on BH perturbations and their GW 48 radiation in the RD phase. On top of that, we have understood how alternative theories affect the data analysis and physically interpret the reason for these discrepancies. Ultimately, and as an outlook, we introduce further research paths based on this work, regarding the constraints for the coupling factor: •By exploring other modes, such as the (2,1) mode, and inferring parameters through the same analysis, we can potentially uncover inner inconsistencies of alternative theories with the experimental or NR data. This could lead to constraints on the coupling factors. •We can include overtones, such as the first overtone, and perform a similar analysis to the one above, and if we uncover inconsistencies, we can find constraints on the coupling factor. •We can infer parameters from the inspiral signal and employ it to acquire the final mass and spin of the remnant BH. Under these constraints, we can bound las explained in Sec. 7. However, acquiring the final mass and spin from the initial conditions is quite challenging for GR (we can do it using NR simulations) and even more so for alternative theories. Nonetheless, according to the second law of Thermodynamics S1+S2< Sf, since entropy (S) is proportional to the square of the mass S∝M2, we can constrain the final mass, which in turn can lead to constraints on the coupling factor. Lastly, an alternative path could involve a numerical approach for computing complex frequencies for GR (Teukolsky Equation) or alternative theories. For a theoretical approach, a thrilling path could also be to obtain the GR frequencies with analytical solutions or get to the Teukolsky equation. Additionally, it could be interesting to obtain a perturbative geometry for rotating BH in alternative theories. 49 Appendix A: Gauge Freedom Based on a one-parameter (λ) family of metrics gαβ(λ, x). We will Taylor expand it as: gαβ(λ, x) = g(0) αβ +p1 αβ(x)λ+1 2p2 αβ(x)λ2+··· pn αβ(x) = ∂ngαβ(λ, x) ∂λnλ=0 .(A.1) Note that in this case, xrefers to all coordinates xν. All metrics can be expressed in different coordinate systems. By making infinitesimal rotations or boosts, for example. A general coordinate transformation of this type can be done as follows: xα→x′α(λ) = xα−ξα(x)λ−1 2ξα 2(x)λ2+···.(A.2) Using the tensors transformation rules under the previous transformation, we can get to: gα′β′(λ, x′) = ∂xα ∂x′α′ ∂xβ ∂x′β′gαβ(λ, x) = g(0) α′β′+λ(p1 α′β′(x) + g(0) α′β(∂(β′ξβ))) + O(λ2), gα′β′(λ, x′) = g(0) α′β′+λ(p1 α′β′(x) + ∂(β′ξα′)) + O(λ2). (A.3) Note that we have lowered the index using g(0) instead of g. This is only partially accurate. However, the difference is of second order since the expression was already a perturbation (linear in λ). As a result, g(0) α′β′=g(0) αβ p1 αβ =p1 α′β′(x) + ∂(β′ξα′).(A.4) Therefore, as stated by Equation A.3, by selecting a coordinate system (that follows Equation A.2), we have altered the expression for the perturbation. However, this does not change the underlying physics; the perturbation remains the same but is represented in different coordinates. This is why we call it gauge freedom. We can use these transformations (called gauge fixing) to our advantage by carefully choosing coordinate transformations to simplify equations or expressions. In total, we can impose 6 conditions, one for each element of the tensor ∂(β′ξα′). It is not a coincidence that the degrees of freedom permitted by Lorentz transformations (Section 2) are the freedoms provided by the tensor ∂(β′ξα′)from Equation A.3; ). Boost and rotations present the same structure as equation A.2. We can show this using a perturbative approach. For instance, an example of such Lorentz transformation is a Lorentz boost. 50 Λνµ=    cosh ϕ−sinh ϕ0 0 −sinh ϕcosh ϕ0 0 0 0 1 0 0 0 0 1     ϕ<<1 −−−→       1−ϕ0 0 −ϕ1 0 0 0 0 1 0 0 0 0 1 .       (A.5) Thus, the coordinate transformation is given by ct′=ct −ϕx x′=−ϕct +x y′=y z′=z .(A.6) If we replace ϕwith v, this equation will resemble the usual Galilean transformation. As expected in a perturbative analysis of Lorentzian boosts. To recover the equation A.2, we must identify ϕto be λand ξµto be (x, ct, 0,0). A.1 The Residual Gauge from Lorenz Gauge Throughout this dissertation, we have used the Lorenz Gauge several times. For example, in Section 3.1, we opted for it (Lorenz or harmonic gauge), this is: ∂βhαβ = 0 (Lorenz gauge condition) hαβ ≡p1 αβ −1 2ηαβh h =ηµνp1 µν (A.7) Hence, field equations were □hαβ =−16πTαβ[g] (A.8) However, these conditions still leave room for further gauge conditions. Specifically, as long as ξµobeys the following condition, the Lorenz gauge is satisfied: □ξµ= 0.(A.9) In order to show this, we will use Eq. A.4. hαβ −→ h′αβ ≡p1 αβ(x) + ∂(βξα)−1 2ηαβh′h′=ηµν p1 µν(x) + ∂(νξµ) (A.10) Using ∂βhαβ = 0 and hαβ =ηανηβµhνµ, we get ∂βhαβ = 0 and with this in mind: ∂βh′αβ =∂β(p1 αβ(x) + ∂(βξα)−1 2ηαβh′) =∂β(p1 αβ(x)−1 2ηαβh) + ∂β∂(βξα)−1 2ηαβηνµ∂(νξµ) =  ∂βhαβ +1 2□ξα+   1 2∂β∂αξβ−1 2∂µ∂αξµ ∂βh′αβ =1 2□ξα (A.11) 51 Thus, some residual gauge freedom remains to be specified, as the original requirement (Eq. A.7) is still conserved under all gauge transformations that satisfy □ξα= 0. A.2 Gauge Fixing on Black Hole Perturbation Theory Suppose we decompose the perturbation in spherical harmonics as prescripted in Sec. 3.3.1, without gauge fixing, this is with 10 different perturbation functions, which are decomposed in spherical harmonics. We are used to decomposing scalar functions into spherical harmonics as h(θ, ϕ) = ∞ X l=0 m=l X m=−l hlmYlm(θ, ϕ).(A.12) However, vectors (recall the transformation rules from 2.13)- can also be decomposed in the so-called vector spherical harmonics. In spherical coordinates (r, θ,ϕ) they go as: Ylm =rYlm(θ, ϕ) Ψlm =∇Ylm(θ, ϕ) Φlm =r×∇Ylm(θ, ϕ), (A.13) where ∇=∂f ∂r ˆ r+1 r ∂f ∂θ ˆ θ+1 rsin θ ∂f ∂φ ˆ φ In S2, this is, in r=ct, Ylm is a scalar, while Ψlm and Φlm are vectors, and they can be written as follows in tensor notation: Ψlm A=∂AYlm(θ, ϕ) Φlm B=ϵBA∂AYlm(θ, ϕ),(A.14) where ϵϕϕ=ϵθθ= 0, ϵθϕ=−1/sin(θ) and ϵϕθ=sin(θ). Moreover, the bold notation refers to vectors. Finally, tensors can be decomposed as follows: Ψlm AB =∇A∇BYlm(θ, ϕ) Φlm AB =γABYlm(θ, ϕ) χlm AB =1 2ϵBCΨlm AC +ϵACΨlm CB, (A.15) where again, ϵϕϕ=ϵθθ= 0, ϵθϕ=−1/sin2(θ) and ϵϕθ=sin2(θ). Additionally, γθϕ = γϕθ = 0, γθθ = 1, γϕϕ =sin2(θ), this is, γνµ =gνµ/r2. Most importantly these are the unique vector or tensors we can construct in S2. In that case, our two families of perturbation depending on the parity. On the one hand, the so-called ”odd” parity (or Regge-Wheeler or axial) perturbations (as they transform as pl(−r)=(−1)l+1pl(r)) are decomposed without loss of generality as: 52 p1 odd µν =         0 0 −h01 sin θ ∂ ∂ϕ −h0sin θ∂ ∂θ  0 0 −h11 sin θ ∂ ∂ϕ −h1sin θ∂ ∂θ  sym sym h21 sin θ ∂2 ∂θ∂ϕ −cos θ sin2θ ∂ ∂ϕ sym sym sym 1 2h21 sin θ ∂2 ∂ϕ∂ϕ + cos θ∂ ∂θ −sin θ∂2 ∂θ∂θ −h2sin θ∂2 ∂θ∂ϕ −cos θ∂ ∂ϕ          Ylm, (A.16) where h0=h0(T, r), h1=h1(T, r), h2=h2(T, r) and ”sym” refers to the symmetry property the metric has (pνµ =pµν). On the other hand, the ”even” parity (or Zerilli or polar) perturbations (as they transform as pl(−r) = (−1)lpl(r)) can be decomposed as p1 even µν =         fH0H1¯ h0∂ ∂θ ¯ h0∂ ∂ϕ  H1f−1H2¯ h1∂ ∂θ ¯ h1∂ ∂ϕ  sym sym r2[K+G∂2 ∂θ2] sym sym sym r2G∂2 ∂θ∂ϕ −cos θ∂ sin θ∂ϕ r2sin2θhK+G1 sin2θ ∂2 ∂θ∂ϕ −cos θ sin θ ∂ ∂ϕ i         Ylm, (A.17) where f=1−2GM c2r,¯ h0=¯ h0(T, r), ¯ h1=¯ h1(T, r) and H0=H0(T, r) , H1=H1(T, r), H2=H2(T, r). When we first derived the gravitational wave equation in section 3, we used the Lorenz gauge transformation to simplify the Eq. 3.7. In this section, we will perform a similar but more convoluted gauge transformation to simplify the expressions for the perturbations (see [28]). As we will see in Appendix A, under the following coordinate transformation: xα→x′α(λ) = xα−ξα(x)λ−1 2ξα 2(x)λ2+··· (A.18) the perturbation transforms as p1 αβ =p1 α′β′(xµ) + ∂(β′ξα′)(A.19) For the odd perturbations, we can choose ξµto be ξ0= 0 ξ1= 0 ξ2= ∆(T, r)∂ ∂θYlm(θ, ϕ)ξ3= ∆(T, r)∂ sin2θ∂ϕYlm(θ, ϕ).(A.20) The direct computation will result in p1 µν =    0 0 0 h0(r) 0 0 0 h1(r) 0 0 0 0 h0(r)h1(r) 0 0    ×e−iωT sin θ∂ ∂θPL(cos θ).(A.21) 53 The function h0and h1are treated now as plane waves, Fourier Transformed versions, as we expect them to have a wave-like behaviour, and the treatment of them is easier. For the even perturbations, we can choose ξµto be ξ0=M0(T, r)Ylm(θ, ϕ)ξ1=M1(T, r)Ylm(θ, ϕ) ξ2=M(T, r)∂ ∂θYlm(θ, ϕ)ξ3=M(T, r)∂ sin2θ∂ϕYlm(θ, ϕ),(A.22) which, in turn, will result in p1 µν =    fH0H10 0 H1f−1H20 0 0 0 r2K0 0 0 0 r2Ksin2(θ)    ×e−iωT sin θ∂ ∂θPL(cos θ).(A.23) These are the simplified parity perturbations. Gauge fixing reduces the previous 10 variables to 6 Appendix B: Gravitational Wave Polarization As we have seen in Sec. 3 plane waves propagating in the zdirection (in Lorenz gauge) follow Eq. 3.13 and thus the associated line element is: ds2=−c2dt2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2.(B.1) The parameters h+and h×will determine the polarization of the gravitational wave. To get some intuition, in this appendix, we will first impose h×to be zero and then make h+= 0 instead. B.1 Plus polarisation First, as explained, we will impose h×to be zero. Thus, for a fixed time (dt=0), the line element is ds2= (1 + h+)dx2+ (1 −h+)dy2.(B.2) If we propose the following coordinate change (x′=p1 + h+x y′=p1−h+y, (B.3) they will restore the flat metric; this is, ds2=dx′2+dy′2.(B.4) We can interpret this as follows: The dilation/contraction of the new coordinates match with the spacetime deformation and thus, the flat spacetime will be recovered. This dilation/contraction varies over time, as shown in Equation 3.13. In particular, we can see how the spacetime deformation would affect a ring of test points in space that satisfied the equation x2+y2=R2: 54 x2+y2=R2=⇒ x′ p1 + h+!2 + y′ p1−h+!2 =R2←Ellipse (B.5) This means that if we had a circle of test points when h+= 0, these test points would appear elliptical at any time when h+= 0. The valid equation is the ellipse as opposed to the circle since the ellipse lies in the flat coordinates (the prime coordinates), where the equations of polygons correspond to the ”real” polygons. Again, this is because our notion of distance only holds on flat spaces. x y t Figure 22: Test points under the effect h+polarization gravitational wave propagating along z direction B.2 Cross polarisation Similarly, if we had set h+= 0 instead of h×, we would face the same phenomena but with a 45ºpolarization angle rotation. To show this, we will start with the line element of a GW propagating along z direction in flat spacetime, ds2=−c2dt2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2,(B.6) where according to equation 3.13 h+and h×are h+=a+cos(ω(t−z/c)) h×=a×cos(ω(t−z/c)).(B.7) Performing the following transformation Lorentz transformation (a -45ºrotation in the x-y plane: x′ y′=cos 45 sin 45 −sin 45 cos 45x y=⇒       x′=x+y √2 y′=x−y √2 ⇐⇒        x=x′+y′ √2 y=x′−y′ √2 .(B.8) dx =∂x ∂x′dx′+∂x ∂y′dy′=dx′+dy′ √2 dy =∂x ∂x′dx′+∂x ∂y′dy′=dx′−dy′ √2 =⇒ dx2=dx′2+ 2dx′dy′+dy′2 2 dy2=dx′2−2dx′dy′+dy′2 2 .(B.9) The new line element is similar to the old one, but h+and h×are now swapped. ds2=−c2dt2+ (1 + h×)dx2+ (1 −h×)dy2+ 2h+dxdy +dz2.(B.10) The physical interpretation of htimes is just the same as the one for h+since it is the same spacetime transformation, although it is shifted 45ºrelatively. 55 x yt Figure 23: Test points under the effect h×polarization gravitational wave Appendix C: Gravitational Wave Detection Current GW detectors, such as Ligo and Virgo, are laser interferometers. They operate by splitting a laser beam into two separate beams, which travel along 2 perpendicular long arms as depicted in figure 24. They bounce at the mirrors so that the effective length of the arms is much greater, and then they are recombined in the detector. Whenever a GW crosses the detector, the lengths of these paths change due to GW’s spacetime deformation. Any differences in the light paths result in interference patterns, and by analyzing these patterns, the interferometer can detect minuscule changes in distance, which, in turn, enables us to detect GWs. Laser Beam-splitting mirror Mirrors Detector L L Figure 24: A schematic diagram of a laser interferometer We first need to see the perturbation to see how the lengths are changed when GWs go through the detector. 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