EXACT AND EARTH SCIENCES CENTER
DEPARTMENT OF THEORETICAL AND EXPERIMENTAL PHYSICS GRADUATE PROGRAM ON PHYSICS
Discriminating Cosmological Expansion
Models With Third Generation Gravitational
Wave Detectors
Josiel Mendonça Soares de Souza
Natal-RN February 2019
Discriminating Cosmological Expansion Models With
Third Generation Gravitational Wave Detectors
Master’s Dissertation presented to the Post-graduate Program in Physics of the Federal University of Rio Grande do Norte as a re-quirement to obtain a Master’s Degree in Physics.
Advisor:
Prof. Dr. Riccardo Sturani
Universidade Federal do Rio Grande do Norte - UFRN Departamento de Física Teórica e Experimental - DFTE
Natal-RN February 2019
Souza, Josiel Mendonça Soares de.
Discriminating Cosmological Expansion Models With Third Generation Gravitational Wave Detectors / Josiel Mendonça Soares de Souza. - 2019.
97 f.: il.
Dissertação (mestrado) - Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Programa de Pós-Graduação em Física, Natal, RN, 2019.
Orientador: Prof. Dr. Riccardo Sturani.
1. Gravitational waves Dissertação. 2. Standard sirens -Dissertação. 3. Bayesian model selection - -Dissertação. 4. Einstein telescope - Dissertação. I. Sturani, Riccardo. II. Título.
RN/UF/BCZM CDU 520.2
Catalogação de Publicação na Fonte. UFRN - Biblioteca Central Zila Mamede
First of all, I thanks my Lord Jesus, by his life breath in me and by the strength that he gave me to cross more one stage of my life in the conclusion of my master course. Without him I could do nothing. I thanks, also to my wife by her charisma and love, joint with her help in staying my side and ever support me. I thanks my advisor by his effort in help lead me to more high level in my scientific career. Without hard work, there is no progress. I thanks my friends by help me in my academic activities. God bless you all. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
earth?" Job 38:33 - Holy Bible
At the dawn of Gravitational Wave Astronomy, new avenues have been opened to test and constrain cosmological models. By simulating standard sirens detected by the third generation ground-based gravitational wave detector, Einstein Telescope, we show how we one can discriminate among different cosmological models. In this work, we use 10-year projected data of Einstein Telescope to compare the standard cosmological model (the Lambda CDM) with models using different phenomenological parametrizations of the equation of state for the dark energy.
Keywords: Gravitational Waves, Standard Sirens, Bayesian Model Selection, Ein-stein Telescope.
No alvorecer da Astronomia das Ondas Gravitacionais, novos caminhos têm sido aberto para testar e restringir modelos cosmológicos. Simulando sirenes padrões de-tectadas pelo detector terrestre de terceira geração de ondas gravitacionais, o Einstein Telescope, mostramos como podemos discriminar diferentes modelos cosmológicos. Neste trabalho, usamos dados projetados de 10 anos de medições pelo Einstein Telescope para comparar o modelo cosmológico padrão (o Lambda CDM) com modelos utilizando difer-entes parametrizações fenomenológicas da equação de estado para a energia escura.
Palavras-chave: Ondas Gravitacionais, Sirenes Padrões, Seleção Bayesiana de Mod-elos, Telescópio Einstein.
1 (Left) Signal GW150914 of a gravitational wave emitted by a binary black hole system detected by the LIGO and Virgo. From: Phys. Rev. Lett. 116, 241103 (2016). (Right) Artistic representation of a inpiralling BBS emitting gravitational waves as ripples propagating through the spacetime. From LIGO website. . . 5
1.1 Artistic representation of the spacetime deformation due the presence of a static spherical symmetric body. . . 10
1.2 Light cones inside (shaded region) and outside the black hole horizon. . . . 11
1.3 Artistic representation of a dark matter halo around a spiral galaxy. Figure from author. . . 13
1.4 Photons arrival time interval affected by the cosmological expansion. . . . 14
1.5 (Top) Modulus distances µ of supernovas with respect to their redshifts from [15]. (Bottom) Luminosity distance of supernovas in respect to their redshifts also from [15]. . . 17
1.6 Parameters evaluated for the expansion Eq.(1.12). These estimations was made using the Nested Sampling algorithm [17], through the python im-plementation Nestle [18]. Graphs output by Corner [19]. . . 18
1.7 Evolution of the density parameters for the cosmological contents with respect to the redshift. Looking for high redshifts we are looking to the past. 21
stage of accelerated expansion. . . 21
2.1 Sketch of the system treated in the current section. . . 30
2.2 Binary System . . . 32
2.3 Sloped Binary System Configuration . . . 33
2.4 Behaviour of the gravitational waves produced by a binary system merging. 39 3.1 (Left) LIGO in Hanford-Washington, (Right) LIGO in Livingstone-Louisiana and (Below) Virgo in Italy. From: http://www.ligo.caltech.eduandhttp:// www.virgo-gw.eu/. . . 43
3.2 Rotated system in the source position. Arms of the detector making a angle of 90o between them (the case of the LIGO detector). . . . 45
3.3 Einstein Telescope design project, from http://www.et-gw.eu/index.php/ etimages. . . 47
3.4 Rotated system in the source position. Arms of the detector making a angle of 60o between then (the case of the Einstein Telescope detector). . . 48
3.5 (Left) The fractional error estimation Eq. (Eq: 3.12) in the luminosity distance from standard sirens by the Einstein Telescope. (Right) The error estimation using the ΛCDM model as the fiducial model to compute dL(z) (cosmological parameters from [23]). . . 50
3.6 Logarithmic best-fit with two free parameters. . . 52
3.7 Logarithmic best-fit with three free parameters. . . 53
3.8 Venn diagram for the system in question, the whole set is denoted by S, the events spaces A and B are displayed. . . 54
3.9 Venn diagram for the system in question. Each squared and triangular block represents one set Bi. The shadded part represents the set A. . . 54
3.10 Use of the Bayes Parameter Estimation for the cosmological parameters H0 and ΩM from the supernovae data set. . . 58
4.1 (Left) First point from the standard siren with signal GW170817 ralating its luminosity distance dL= 40+8−14M pc and its respective redshift z = 0.008 [37]. (Right) Illustrative representation of the event GW170817, from http://www.ligo.caltech.edu. . . 64
differents values for the delay τ . . . 67
4.3 Relative error on the measurement of the luminosity distance by Einstein Telescope, following [39]. . . 68
4.4 Uncertainties for luminosity distance measurements with the full error and with a 80% error reduction. ∆dL(z)/dL(z) in the left side, and ∆dL(z)
(Mpc units) in the right side. . . 68
4.5 Data simulations for 1000 standard sirens, with and without delay, and us-ing two percentage in the uncertainty in the luminosity distance measure-ment by the Einstein Telescope. The top part indicates a error percentage of 100% and the bottom part with a error percentage of 20%; these ones is represented by the red curves (dL ± σET). The left side indicates the
simulations without delay, and the right side shows the simulations with a delay τ = 10 Gyr. The blue curves in the center gives the theoretical prediction of the luminosity distance in the ΛCDM model. . . 69
4.6 Logarithm of the /bayes factor for comparison between ΛCDM and ω0CDM
using ΛCDM model as the fiducial one with standard uncertainty in the luminosity distance (top line) and uncertainty reduced to 20% (bottom line), with merger rate equal to star formation rate (left column) and a distribution with a τ = 10Gyr delay between formation and merger (right column). . . 71
4.7 Same as in fig. 4.6 for ΛCDM versus w0waCDM with standard dL
uncer-tainty on the top line and reduced unceruncer-tainty in the bottom line. Left column refers to equal merger and star formation rate, right column to a distribution delay with τ = 10Gyr delay. . . 72
4.8 Average of the 10 curves related to the 10 different shuffles on the catalogs from the figures fig. 4.6 and fig. 4.7. . . 73
4.9 Same as in fig.4.6, Logarithm of the /bayes factor for comparison between ΛCDM and ω0CDM for simulated data following non-local massive gravity
cosmology [51] with standard dL uncertainty on the top line and reduced
uncertainty in the bottom line. Left column refers to equal merger and star formation rate, right column to a distribution delay with τ = 10Gyr delay. 74
tainty on the top line and reduced uncertainty in the bottom line. Left column refers to equal merger and star formation rate, right column to a distribution delay with τ = 10Gyr delay. . . 75
4.11 Average of the 10 curves related to the 10 different shuffles on the catalogs from the figures fif. 4.9 and fig. 4.10. . . 76
Aknowledgements Abstract
Resumo Figures List
Introduction 4
1 Foundations of General Relativity and Cosmology 6
1.1 The Tensorial Tools of the General Relativity . . . 6
1.1.1 Least Path Between Two Points - Geodesics . . . 6
1.1.2 Covariant Derivatives . . . 8
1.2 Gravity as a Spacetime Curvature Manifestation . . . 9
1.2.1 The Einstein Field Equations . . . 9
1.2.2 The Schwarzschild Solution and Black Holes . . . 10
1.3 Modern Cosmology . . . 11
1.3.1 An Universe in Expansion . . . 12
1.3.2 Dark Matter . . . 13
1.3.3 Accelerated Expansion and the Dark Energy . . . 16
1.3.4 Alternative Parametrization to Dark Energy . . . 22
2 Gravitational Wave Theory 23 2.1 Linearized Theory of Gravity . . . 23
2.1.1 Vacuum Einstein’s equations . . . 23
2.1.2 Linearized Theory . . . 24
2.2 Energy From the GWs . . . 26
2.3 Multipole Expansion . . . 28
2.4 Waveforms of Gravitational Waves from Binary Systems . . . 32
2.5 General Binary System . . . 33
2.6 Gravitational Waves Evolutions From Binary Systems . . . 35
3 Data Analysis and Bayesian Inference 41
3.1 Gravitational Waves Detectors . . . 41
3.1.1 The Gravitational Waves Detectors LIGO and Virgo . . . 43
3.1.2 Third Generetion Detector: Einstein Telescope . . . 47
3.2 Least Square Method . . . 50
3.3 Conditional Probabilities and Bayes Theorem . . . 53
3.4 Bayesian Inference . . . 55
3.4.1 Etimating the ΛCDM Parameters . . . 56
3.5 Bayesian Model Selection . . . 58
4 Discriminating Cosmological Expansion Models 62 4.1 Model Selection From Events Like GW170817 . . . 63
4.2 Data Simulations . . . 65
4.2.1 Binary Merger Rate and Source Distributions . . . 66
4.2.2 Measurements Uncertainties . . . 67
4.2.3 Simulation And Events Catalogues Construction . . . 69
4.3 Comparing Cosmological Models Against Data . . . 70
4.3.1 Comparisons Between Nested Cosmological Models . . . 70
5 Conclusions and Perspectives 77
Bibliography 79
Apendix A - Transverse Traceless (TT) Gauge 84
After the General Relativity Theory publication by Albert Einstein in 1915 [1] the gravity natural understanding was totally changed. Now we can study the gravity as a spacetime curvature manifestation. Years later, with the intense study about the possible predictions from the General Relativity (GR) one interesting issue was raised. The GR predicted the existence of spacetime oscillations that can propagate through the Universe due to some kind of cosmological or astrophysical events. These oscillations were called Gravitational Waves (GW). Many projects were proposed with the goal of detecting these kind of waves. With this goal the Laser Interferometer Gravitational-Waves Observatory (LIGO) in Hanford, Washington and Livingstone, Louisiana as well as the Virgo in the Cascina, Italy, were built with the hope to detect the gravitational waves.
With the first detection of a gravitational wave in 2015 emitted by a binary black hole system (BBS) [2], a new era for the astronomy was opened in which we can now use this new type of signal to study the universe evolution due its energetic contents. One of the last signals obtained by LIGO an Virgo, in 2017 and named GW170817 [3], was iden-tified as coming from a distinct event from the other ones already ideniden-tified as binaries black holes systems merging. The former was identified as a binary neutron star system (BNS), in which, after its merging, emitted also, joint with the gravitational waves, a strong electromagnetic beam in a form of short gamma-ray burst. With this event it was possible to directly measure its distance from us only using its gravitational signal. Until 2015 the only method to make measurements of the distance versus cosmological redshift relationship via astronomical observation was by mean of electromagnetic radi-ation measurements coming from physically well known astrophysical events which were called standard candles. Since then, with the rising of the gravitational waves astronomy, the events in which we can directly infer its distance only using gravitational radiation
are now called standard sirens (in analogy with the standard candles). In the present thesis we will consider the forecasts for future detections of gravitational waves measured by third generation detectors as the Einstein Telescope (ET) [4] and the Cosmic Explore (CE) [5] by using them to make analysis of cosmological models to discriminate among these models.
Figure 1: (Left) Signal GW150914 of a gravitational wave emitted by a binary black hole system detected by the LIGO and Virgo. From: Phys. Rev. Lett. 116, 241103 (2016). (Right) Artistic representation of a inpiralling BBS emitting gravitational waves as ripples propagating through the spacetime. FromLIGO website.
This dissertation is divided as follows. We will begin to introduce the basics of the general relativity and the cosmology presenting the tools that will be used along this work. After we will show the linearised theory of gravity in which gravitational waves arise. With the foundations of the linearised theory of gravity in hand we pass to study its applications focusing on the emission of GW by binary systems. The following will be dedicated to statistical methods used to treat the main problem addressed in present dissertation. Finally, in the last chapter, we are going to show that the estimated precision in the luminosity distances measurements by the Einstein Telescope in not good enough to rank cosmological models, suggesting that a few-fold increase in sensitivity is necessary for this purpose. Alternatively the combination of several detectors can improve the precision in distance determination.
Foundations of General Relativity and
Cosmology
As mentioned in the Introduction, after the publication of the famous Einstein Fields Equations in 1915 [1], gravity becomes related to the manifestation of the curvature of the spacetime. In this chapter we are going to see the basic concepts of this gravity theory and its application in the universe dynamics that gives rises to the modern cosmology.
The Tensorial Tools of the General Relativity
To study mathematically the description of the curvature of the four dimensional space (three spacial dimensions plus one temporal) it is necessary to introduce some basics aspects about the spacetime as well as its tensorial formalism.
Least Path Between Two Points - Geodesics
When dealing with four dimensional spaces the infinitesimal distance between two points in this space (line element dS) can be mathematically represented, in the flat Minkowski space1, as dS2 = −dt2+ dx2+ dy2+ dz2 = 3 X µ,ν=0 ηµνdxµdxν ,
1Along this dissertation we are going to use units system in which c = 1.
where the ηµν represents the Minkowski metric and is given in the matrix form as ηµν = −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 .
In the general case, when the metric is different from the one with a flat geometry, we have the following representation of the line element (using the Einstein convention2)
dS2 = gµν(x)dxµdxν .
Using this line element we can compute the least path between two points through the minimization of the spacetime interval ∆S. To do this we have to make use of the variational calculus that gives us the Euler-Lagrange Equations. First, let us make this minimization in terms of the proper time τ looking for the lagrangian that describes the dynamics of goes from one point A to the point B with the minimum time interval ∆τ .
dS = r gµν dxµ dτ dxν dτ dτ , ∆S = SB− SA = Z τB τA r gµν dxµ dτ dxν dτ dτ .
Hence we can see, from the argument of the last integral, the lagrangian that will be used in the Euler-Lagrange Equations to minimize the spacetime interval
L(xα, ˙xα) = r gµν dxµ dτ dxν dτ ,
where the dot denotes the derivatives in relation to τ . Now, evaluating the Euler-Lagrange Equations we have dL dxα + d dτ dL d ˙xα = 0 ⇒ d 2xα dτ2 + 1 2g αω dgωβ dxγ + dgωγ dxβ − dgβγ dxω dxβ dτ dxγ dτ = 0 . From this equation we can compute the shape of the minimized trajectory between
two events in the spacetime. Let us now introduce the definition of the Christoffel Symbol (Γαβγ) to recast the last equation in a more elegant form as follows
Γαβγ = 1 2g αω dgωβ dxγ + dgωγ dxβ − dgβγ dxω . Hence we have the called geodesic equation.
d2xα dτ2 + Γ α βγ dxβ dτ dxγ dτ = 0 . (1.1) Geodesic Equation
It is straightforward to notice from this equation that if the spacetime is flat (the Minkowski spacetime) the Christoffel symbol will be zero and, therefore, the trajectory of travel between two points A and B with the least time interval will be a straight line. For arbitrary metric this trajectory can be curve.
Covariant Derivatives
As we are treating curved spacetime, to maintain the invariance in the physics laws through general coordinates transformations (in a mathematical language, these laws have to maintain their tensorial Covariant formulation), we have to substitute the conventional derivatives by covariant derivatives. These covariant derivatives are defined as follows 3 [6]
∇µVα≡ ∂µVα+ ΓαµνV ν ,
∇µVα ≡ ∂µVα− ΓνµαVν .
This can be derived by performing a general coordinate transformation and imposing that these quantities transform as vectors. For conventional derivatives we have
∂µ0V0α= ∂x ν ∂x0µ ∂x0α ∂xβ ∂νV β+ ∂x ρ ∂x0µ ∂2x0α ∂xρ∂xσV σ , ∂µ0Vα0 = ∂x ν ∂x0µ ∂xβ ∂x0α∂νV β + ∂2x 0σ ∂xµ∂xαVσ .
3From here we are going to use a compact form in to represent partial derivatives as ∂
As can be seen, the second term in the right hand side of each of these equations pre-vents the conventional derivatives from mantaining a covariant character through general coordinates transformations. Already in the case of the covariant derivatives, one can shows that its covariant character is preserved as we can see below
(∇µVα)0 = ∂xν ∂x0µ ∂x0α ∂xβ∇νV β , (∇µVα)0 = ∂xν ∂x0µ ∂xβ ∂x0α∇νVβ .
Gravity as a Spacetime Curvature Manifestation
Here we are going to show how the presence of matter and energy can curves the spacetime within the theory of general relativity.
The Einstein Field Equations
The Einstein’s Field Equations describes the influence of the matter and energy in the spacetime geometry. This equation formulated by Einstein and published in 1915 [1] can be deduced by the Einstein-Hilbert Action given by
SEH =
1 16πG
Z
d4x√−gR + SM ,
where SM is some given action related to the matter-energy. The quantity R is called
Ricci Scalar and is given by the contraction of the Ricci Tensor that is of the following form Rµν = ∂µΓανα− ∂αΓαµν+ Γ α αΓ µν− Γ α νΓ µα .
G is the Newton’s gravitational constant and g is the determinant of the metric gµν.
After we make the minimization of the gravity fundamental action (SEH) we obtain
the famous Einstein’s Field Equations
Rµν −
1
2gµνR = 8πGTµν . (1.2) Einstein’s Field Equations
in the system, was obtained from the matter action SM as Tµν = −√2 g δSM δgµν .
The Schwarzschild Solution and Black Holes
The spherically symmetric solution of the Einstein’s equations in the vacuum (Tµν = 0) obtainded by Karl Schwarzschild in 1916 [7] that receives his name, the Schwarzschild Solution, describes the geometry of the spacetime around static spherical symmetric bodies as stars and black holes. This solution is given by the metric below
dS2 = − 1 −2GM r dt2+ 1 − 2GM r −1 dr2+ r2dΩ2 . (1.3) Schwarzschild Solution
where M is the mass of the spherical body and dΩ2 represents the angular part of the line element
dΩ2 = dθ2+ sin2θdφ2 .
Figure 1.1: Artistic representation of the spacetime deformation due the presence of a static spherical symmetric body.
Studying the radial trajectories of light rays in this geometry (dS2 = 0) we can see
that for initial points at r coordinate less than 2GM the light rays can not escape beyond r = 2GM . Rays within this region will continue to move inward until the singularity r = 0. This limit is called Schwarzschild Radius (Rs = 2GM ). Spherical object of this
type are called Black Holes with Events Horizon equal to Rs. These objects has a gravity
so strong that not even light can escape out.
Figure 1.2: Light cones inside (shaded region) and outside the black hole horizon.
Modern Cosmology
Modern cosmology started with general relativity foundation in 1915. With this new tool in hand it was possible to study the universe evolution under the influence of gravity. In 1929 [8], Edwin P. Hubble, studying the relation between the galaxies distances from us and their redshifts from direct observations, observed that this relationship presents a linear behavior that was afterwards called Hubble-Lemaître Law [9] interpreting the redshifts due the Doppler Effect4 and thus in relation with recession velocity [11]. This
4Recalling that from the Doppler effect, for non-relativistic sources receding from us with a velocity
v the wavelength λobs measured in our frame will be related to the wavelength λemis measured in the
source frame by λobs = (1 + v/c)λemis. Hence, by the definition of the redshift z = λobs/λemis− 1 we
law is given by the linear relation v = H0x, where v is the galaxy recession velocity, H0
is the Hubble-Lemaître Constant, and x is the galaxy distance from us.
This equation shows that, apparently, all the galaxies are moving away from us stating that we are in a privileged position in the universe. However, the real meaning that can be extracted from this law is that the observers in others galaxies will see the same thing in their frames. Hence, the Hubble’s law informs us that our universe is in a state of expansion, contradicting the old cosmological view in which the universe was immutable and eternal.
An Universe in Expansion
Knowing now that the universe is in expansion we would like to study its gravitational dynamics due the cosmic content. First we have to use one metric that describe the universe in expansion or contraction through a scale factor a(t)5 at large scale as per the Cosmological Principle6. This metric called Friedmann-Lemaître-Robertson-Walker (FLRW) metric is expressed below
ds2 = −dt2+ a2(t) dr2 1 − Kr2 + r 2dθ2 + r2sin2θdφ2 . (1.4) Friedmann-Lemaître-Robertson-Walker Solution
where K is the curvature constant that is related with the global universe geometry, for K = 1 the universe is closed, K = −1 the universe is open, and K = 0 the universe is flat (that is our case when we compare the standard model with the cosmological observations). Pluggin this metric into Einstein’s equations, the 00 equation dictates the scale factor time evolution. The combination of the 00 Einstein’s equation with the FLRW metric gives the equation called Friedmann’s equation
H2 = 8πG 3 ρ −
K a2 ,
where H ≡ ˙a(t)/a(t) is called Hubble-Lemaître Factor.
in way that the velocity is related with the redshift by v(z) = c[(1 + z)2− 1]/[(1 + z)2+ 1].
5This scale factor can be understood in the relation r(t) = a(t)x, where r(t) represents the physical
distances that evolves due the cosmological expansion, and x called comoving distance is the coordinate distance which is constant along the evolution.
6This principle states that at scales larger than approximatively 100 M pc the universe can be treated
Dark Matter
Studies shows an intriguing issue as the whole matter content present in the galaxies that can be detect by electromagnetic radiation can not explain its rotational dynamics. It will be necessary much more matter in the galaxies to maintain the rotational behavior showed by the data set. To try to explain this strange behavior, in 1937 [12] it was proposed the existence of an exotic kind of matter called Dark Matter that interact only via gravitational forces and it could be present in all galaxies in a halo shape. Depending on the velocities of this kind of matter it can be classified in three ways, Cold Dark Matter (CDM) for velocities much less than the light velocity, Hot Dark Matter for relativistics velocities, and Warm Dark Matter for intermediate velocities. The cosmology standard model explained the observed universe matter content as mostly made of cold dark matter.
Figure 1.3: Artistic representation of a dark matter halo around a spiral galaxy. Figure from author.
Distance Measures at Cosmological Scales
There are many ways to make distances measures in the universe as the angular distances, comoving distances and the most commom is the luminosity distance (dL).
This is the one that we pretend to use along this dissertation. In astrophysics context we can measure the distance of some star from us if, in principle, we know its intrinsic luminosity L (energy emitted per unit of time through electromagnetic radiation). This
is made from the definition of the Flux that is the quantity we can directly measure at the detector [10]
F = L
4πr2 . (1.5)
The distance r is the quantity that we are interesting to measure. However, when we are making these measurements in cosmological scales, the universe expansion has to be taking in account. This consideration can be easily included in the definition of the flux knowing that the received radiation energy7will be affected by the cosmological expansion through dEreceived = (1 + z)−1dEemmited. In other hand, as the luminosity L is given by
the rate of energy received per unit of detector time, this temporal parameter will be also affected by the expansion as dtreceived = (1 + z)dtemmited. Hence, the luminosity in the
flux equation Eq. (1.5) will be changed by the factor (1 + z)−2 and, therefore, we have
Fd=
L
4π(1 + z)2r2 , (1.6)
where the subscript d denotes the flux in the detector frame. t
r ∆tA
∆tB
rA rB
Figure 1.4: Photons arrival time interval affected by the cosmological expansion.
The quantity r in this equation is going to be related to the comoving radius mentioned before that can be computed by equating to zero the spacetime interval (that characterizes the photons travel) and the angular part of the FLRW metric [11].
7This is the photons energy multiplied by the number of the same ones received in the detector, that
dS2 = −dt2+ a2(t) dr2 1 − Kr2 + r 2 (dθ2+ sin2θdφ2) = 0 , Z t 0 dt a(t) = Z r 0 dr √ 1 − Kr2 = 1 √ Ksinh −1 (√Kr) . Therefore the comoving radius rc will be
r = √1 Ksinh √ K Z t 0 dt a(t) . (1.7)
Rewriting the integral on the hyperbolic function argument in terms of redshifts (here we are choosing a0 = 1) rather than time we have
r = √1 Ksinh √ K Z z 0 dz H(z) (1.8) So, it will be the quantity (1 + z)r that is going to be called as luminosity distance
dL(z) ≡ 1 + z √ K sinh √ K Z z 0 dz H(z) . (1.9) Luminosity Distance
As we do not know about the exact analytic form of the Hubble-Lemaître parameter to evaluate the integral in Eq.(1.8), let us perform a Taylor expansion of H−1(z) around z = 0. 1 H(z) = 1 H0 1 − (1 + q0)z + 1 2(3 + q 2 0 + 4q0− j0+ 2)z2+ ... , (1.10) where the quantities q0 and j0, called deceleration and jerk parameters [11] respectively,
are defined as q0 ≡ − 1 H2 0 ¨a a t=0 , j0 ≡ 1 H3 0 ... a a t=0 . Integrating the equation Eq.(1.10) we have
Z z 0 dz H(z) = 1 H0 z − (1 + q0)z2+ 1 6(3 + q 2 0+ 4q0− j0+ 2)z3+ ... . (1.11)
dL(z) = 1 H0 z −1 2(1 − q0)z 2− 1 6 1 − q0− 3q20+ j0+ K H2 0 z3+ ... . (1.12) Luminosity Distance in Power Series
Accelerated Expansion and the Dark Energy
Beside galaxy recession showing expansion, observations from supernovas type Ia8 show us that this expansion is accelerating. Let us see how we can get this conclusion. First, using data catalogue of supernovas type Ia from [15], we will convert the measure-ments of their modulus distance µ 9 to their respective luminosity distance to get the relationship between dL and z. For this we use [10]
µ = 5(log10 dL− 1) ⇒ dL= 10
µ 5+1 .
The relashionship between the modulus distance of the supernovas type Ia and their respective redshifts as well as the relationship between luminosity distance and red-shifts, using the data set from Pan-STARRS Supernova (PS1 SN) [15] accescible inhttps://archive.stsci.edu/prepds/ps1cosmo/is showed in Fig. 1.5.
8Supernovas type Ia are a kind of supernova in which in their expectral emission lines don’t present
hydrogen lines and have an absolute magnitude well known [10].
9The modulus distance µ is related to the difference between the stellar astrophysical quantities
Figure 1.5: (Top) Modulus distances µ of supernovas with respect to their redshifts from [15]. (Bottom) Luminosity distance of supernovas in respect to their redshifts also from [15].
Now, we are going to compare these data set with the Eq.(1.12) (using methods that will be explaned in the Chapter 3) in a flat universe10 to evaluate the parameters H
0, q0
and j0. The result is showed bellow.
10In principle, analyzing these set of data in a range of redshift up to z = 1.4, we can not constrain the
value of the curvature parameter K, however, analyzing the Cosmic Microwave Background Radiation (CMB) this parameter could be constrained tho a value aprroximately null. For this fact, we are going to study the universe as being flat.
Figure 1.6: Parameters evaluated for the expansion Eq.(1.12). These estimations was made us-ing the Nested Samplus-ing algorithm [17], through the python implementation Nestle [18]. Graphs output by Corner [19].
As the decceleration parameter q0 evaluated is negative (and, therefore, the universe
beeing in a accelerated expansion stage) this implies that the universe apparently contains another exotic content that promotes a repulsive gravity, in contrast with the others contents that ever promotes an attraction. This new component named Dark Energy. In the context of the general relativity, in principle, the dark energy was attributed to the Cosmological Constant Λ inserted by Einstein in his field equations [22] that posteriorly was reject by him as he considered dishonest the introdcution of a cosmological constant to make the cosmological solution static. The inclusion of Λ in the Einstein’s equations
can be seen below
Rµν−
1
2gµνR + gµνΛ = 8πGTµν .
Making use of the FLRW metric in the Einstein’s equation gives us the Friedmann’s equation below H2 = 8πG 3 ρ + Λ 3 − K a2 . (1.13)
To evaluate the universe evolution, let us express this equation in terms of its energy content. Let us consider that the universe, at large scales, can be treated as filled by a perfect fluid in such way that the energy-momentum tensor can be written as
Tµν = (ρ + P )υµυν + gµνP ,
where ρ is the energy density, υµ = (1, 0, 0, 0) is the velocity in the fluid rest frame and P is the pressure. Hence, let us taking into account that the universe conserves its energetic content in such way that the covariant divergence of the energy-momentum tensor is zero
∇νTµν = 0 .
The temporal part of the continuity equation gives Eq. (1.14)
˙
ρ = −3H(ρ + P ) . (1.14)
Friedmann Continuity Equation
Using the equation of state (EoS) P = ωρ for a given parameter ω that depends on the type of energy species in question, we have the equation for the density evolution:
ρ(a) = ρ0
a a0
−3(1+ω) , where ρ0 = ρ(a0). In terms of redshift we have,
Let us now define a new quantity called Density Parameter Ω as
Ω ≡ ρ ρc
,
where ρc is the critical density for which the universe would be flat (K = 0) and is given
by
ρc=
3H2 8πG .
Turning back to the Friedmann’s equation Eq. (1.13) and put in evidence the coefficient of the energy term we can define the new energy density for Λ (related to the dark energy) and ρK (related to the curvature parameter).
H2 = 8πG 3 ρ + Λ 8πG − 3K 8πGa2 = 8πG 3 (ρ + ρΛ+ ρKa −2 ) . Substituting for ρc one obtains
H2 = H
2 0
ρc0
(ρ + ρΛ+ ρKa−2) .
The density ρ in this equation will be the sum of the matter (barionic ρb and cold dark
matter ρdm) and the radiation with their respective parameters to the EoS11. So,
express-ing the Friedmann’s equation in terms of the density parameters Ω we have
H2 H2 0 = ΩM0(1 + z) 3 + Ωrad0(1 + z) 4 + ΩK(1 + z)2+ ΩΛ . (1.15) Friedmann’s Equation
where ΩM0 is the density parameter of matter today and Ωrad0 is the density parameter of
radiation also today. Knowing the values for the density parameters today we can make use of the Friedmann’s equation to evaluate the evolution of the individual parameters by the fact that the summation of all of they has to be one (P
iΩi = 1) [?]. The result
of this one can be seen in the figure below for the value parameters ΩM0 = 0.3103,
Ωrad0 = 9.20962 · 10
−5
, ΩK = 0, and ΩΛ= 0.6897 [23].
11As showed in [?] the parameter to the matter will be ω
Figure 1.7: Evolution of the density parameters for the cosmological contents with respect to the redshift. Looking for high redshifts we are looking to the past.
Solving numerically the equation Eq.(1.15) we have the following evolution of the scale factor:
Figure 1.8: The dynamics of the scale factor a(t) showing the universe in a current stage of accelerated expansion.
Hence, relating the dark energy in a form of a cosmological constant Λ in the Einstein’s equations with its respective density parameter being the bigger one in the universe and with the largest fraction of matter in a form of cold dark matter (CDM) we pass now to
call our standard cosmological model as ΛCDM Model.
The luminosity distance measurements in this model will be given by
dL(z) ≡ 1 + z H0 Z z 0 dz pΩM0(1 + z) 3+ Ω rad0(1 + z) 4 + Ω K(1 + z)2+ ΩΛ . (1.16)
Luminosity Distance in the ΛCDM
Alternative Parametrization to Dark Energy
An alternative to the cosmological constant was proposed by Chevallier, Polarski and Linder [24, 25] (CPL) in which its equation of state parameter ω is given by
ω(a) = ω0+ (1 − a)ωa ,
and in terms of the redshifts z we have
ω(z) = ω0+
z
1 + zωa . (1.17)
CPL Parametrization
The new cosmological model that makes use of this dark energy EoS parametrization will be called ω0ωaCDM Model. Treating only with the first parameter ω0 different from
zero we are going to call it as ω0CDM Model with a constant parameter ω0 but that can
be different from −1 contrary to the case the dark energy is described by a cosmological constant Λ.
Using the energy conservation equation in the Friedmann’s Universe we have a new dark energy density evolution equation in terms of the redshifts.
˙ ρ = −3˙a a[1 + ω(a)]ρ ⇒ ρDE(z) = ρ (0) DE (1 + z)3(1+ω0+ω1)exp −3ω1z 1 + z .
Dividing this equation by the current critical density we have
ρDE(z) ρc0 = ΩDE (1 + z)3(1+ω0+ω1)exp −3ω1z 1 + z .
Gravitational Wave Theory
The General Relativity Theory (GR) with its foundations could make forecasts about the spacetime ripples propagating away due some given cosmic events. As in the electro-dynamics theory, where the motion of charged particles produces electromagnetic waves, in the GR we have something similar in which the motion of bodies can produces oscilla-tions in the spacetime called Gravitational Waves (GW). In this chapter we are going to show the foundations of GW study in the linearised theory of gravity context in which we are dealing with small perturbations in the flat Minkowski spacetime following [26]. More-over we are going to study these small perturbations in a expanding universe background. So, let us begin to evaluate the Einstein’s equations introducing small perturbations in the Minkowski metric in a empty space.
Linearized Theory of Gravity
Vacuum Einstein’s equations
As we want to study, in principle, the propagation of the GW in the vacuum, we have to equate the energy-momentum tensor to zero in the Einstein’s equations and so, evaluate them adding a small tensorial perturbation hµν (with |hµν| 1) to the Minkowski metric
ηµν.
gµν = ηµν + hµν , (2.1)
where ηµν is the Minkowski metric and hµν is the small tensorial perturbation with |hµν|
1 [26].
The Einstein’s equations, in the vacuum, can be simplified such that the Ricci tensor is zero.
Rµν = 0 . (2.2)
Einstein’s equations in the Vacuum
The resulting Einstein’s equation using the perturbed Minkowski metric will be
hµν− ∂α(∂µhαν + ∂νhαµ) + ∂µ∂νh = 0 .
Linearized Theory
The linearized theory of gravity is characterized by small tensorial perturbations in a specific background metric gµν(0). In the following process, we are interested with
pertur-bations in the flat Minkowskian spacetime as in Eq. (2.1).
So, let us evaluate the Einstein’s equations (Gµν = 8πGTµν) with this perturbed
metric, and noting that our result in a specific gauge will gives rise to a wave equation for the hµν, that will describes the gravitational waves.
1First, we are going to compute the Ricci tensor using the metric mentioned above.
Rµν = ∂α∂(µhαν)− 1 2hµν− 1 2∂µ∂νh , where ∂(µhαν) = 12(∂µhαν+ ∂νhαµ).
Its contracted form, the Ricci scalar, will be
R = ∂α∂βhαβ− h .
Taking all together to compute the Einstein tensor Gµν we have
2Gµν = 2∂α∂(µhαν)− hµν− ∂µ∂νh − ηµν∂α∂βhαβ+ ηµνh .
1It is important to underline that in the linearized theory the indices are rised and lowered with the
So, the Einstein’s equations in this formalism will be given by
hµν+ ηµν∂α∂βhαβ+ ∂µ∂νh − ∂α(∂µhαν + ∂νhαµ) − ηµνh = −16πGTµν(xα) . (2.3)
From here we are going to make the following coordinates transformation
x0µ = xµ+ ξµ(x) , (2.4)
with the condition in which |∂µξν| 1. The resulting transformation of the tensor
perturbation and its inverse will be given by
h0µν = hµν− (∂µξν + ∂νξµ), hµν = h0µν+ (∂µξν + ∂νξµ) .
Substituting it in the equation Eq. (2.3) we can see that the left hand side of equation Eq. (2.3) maintain invariant itself2
h0µν + ηµν∂α∂βh0αβ + ∂µ∂νh0 − ∂α(∂µh0αν + ∂νh0αµ) − ηµνh0 = −16πG[Tµν0 (x0λ) + ∂νξαTµα0 (x 0λ ) + ∂µξαTαν0 (x 0λ )] . (2.5) As the terms ∂νξαTµα0 (x
0λ) are of the order of (h
µν)2, we can neglect them such that
we get
h0µν+ ηµν∂α∂βh0αβ + ∂µ∂νh0− ∂α(∂µh0αν + ∂νhαµ0 ) − ηµνh0 = −16πGTµν0 (x 0λ
), (2.6)
that is of the same form of Eq. (2.3).
Let us now define a new tensor perturbation ¯hµν in terms of the old ones as follow
¯
hµν = hµν −
1
2ηµνh . With this definition its inverse relation is given by
2By the fact that |∂
µξν| 1 and ∂α= ∂α0 + ∂αξβ∂β0 we have ∂αh0µν = ∂0αh0µν+ ∂β0ξβ∂αh0µν ≈ ∂α0h0µν,
hµν = ¯hµν −
1 2ηµν
¯ h . Substituting this one in the equation Eq. (2.3) we have
¯hµν+ ηµν∂α∂β¯hαβ − ∂α(∂µ¯hαν + ∂ν¯hαµ) = −16πGTµν(xα) .
Again let us perform the coordinates transformation Eq. (2.4) to get
¯h0µν+ ηµν∂α∂β¯h0αβ− ∂ α (∂µ¯h0αν + ∂ν¯h0αµ) = −16πGT 0 µν(x 0λ ) . (2.7)
Because the invariance of the Einstein’s equation through coordinates transformation Eq. (2.4) we can choose a specific gauge in which the the perturbation is transverse (divergenceless). This gauge is called De Donder or Lorentz Gauge. So, evaluating the previous equation with this gauge we can cancel all the terms that contain divergences, reducing the same in a wave equation.
¯h0µν = −16πGT 0 µν(x
0α
) . (2.8)
Gravitational Waves Equation
Energy From the GWs
To define a energy-momentum tensor for the GW let us go back to the Einstein’s equations looking at their second order expansion term. From the Einstein’s equations
Gµν = Rµν−
1 2gµνg
αβR
αβ = 8πGTµν ,
we can split the Ricc tensor in its first and second order in the perturbation (hµν)
Rµν = R(1)µν + R (2) µν .
The first order of the Ricc tensor we already had computed as
R(1)µν = ∂α∂(µhαν)−
1
2hµν− 1
2∂µ∂νh .
R(2)µν = 1 2h ρσ ∂µ∂νhρσ+ 1 4∂µhρσ∂νh ρσ + ∂σhρν∂[σhρ]µ− hρσ∂ρ∂(µhν)σ +1 2∂σ(h σρ∂ ρhµν) − 1 4∂ ρh∂ ρhµν − ∂σhσρ− 1 2∂ ρh ∂(µhν)ρ .
where ∂[σhρ]µ ≡ 12(∂σhρµ − ∂ρhσµ). With this expansions in hand we can also split the
Ricc scalar in its first and second perturbative order
R = ηµνR(1)µν + ηµνR(2)µν − hµν
R(1)µν .
Hence, splitting the Einstein’s equations in its first perturbative order already com-puted previously, we have
G(1)µν = R(1)µν − 1 2ηµνη αβR(1) αβ = 8πGT (1) µν .
Thus, the treatment of the energy from the gravitational waves will come from the second perturbative order of the these equations
G(2)µν = R(2)µν −1 2ηµν(η αβ R(2)αβ − hαβRαβ(1)) −1 2hµνη αβ R(1)αβ = 8πGTµν(2) .
When we are studying the gravitational waves in the vacuum, the TT gauge (Trans-verse Traceless gauge treated in the Apendix A) is the most indicated to be used in this problem. This gauge can greatly simplify our dynamical equation where the first pertur-bative order in the Ricci tensor vanishes (R(1)µν = 0). So, the Einstein’s equations in this
context will be G(2)µν = R(2)µν − 1 2ηµνη αβR(2) αβ = 8πGT (2) µν .
As we are dealing with waves, to compute the gravitational wave energy it is need to compute the average of the second perturbative order energy-momentum tensor Tµν(2). The
average of the second perturbative order in the Ricci tensor, in addition to the Lorentz condition, will cancel all the terms of the type hhρσ∂
ρ∂σhµνi = −h∂ρhρσ∂σhµνi = 0 (see
[28]). So, in the TT gauge wherehT Tij = 0, the average in the second perturbative order Ricci tensor will be reduced to
hR(2)µνi = −1 4h∂µh
T T
ij ∂νhijT Ti .
Hence, the energy momentum tensor for the GW will be defined from the Einstein’s equation as follow tµν ≡ hTµν(2)i = − 1 8πGhR (2) µνi T T gauge .
Therefore, we have found the energy momentum tensor for GW in the TT gauge as
tµν =
1
32πGh∂µh
T T
ij ∂νhijT Ti . (2.9)
Energy-Momentum Tensor for GW in the TT Gauge
From this one, we can obtain the energy density of the GW taking the 00 component of this tensor. ρ = t00 = 1 16πGh ˙h 2 ++ ˙h 2 ×i . (2.10)
Multipole Expansion
From this point we are interested in the solution of the GW equations for binaries star systems as sources. Let us start with the inhomogeneous wave equation
hµν(xα) = −16πGTµν(xα) .
To solve this partial differential equation in the presence of a source characterized by the energy momentum tensor, we are going to use the method of the Green’s function. This is made by the relation [29]
G(x, t; x0, t0) = δ(3)(x − x0)δ(t − t0) .
Computing the Green’s function that solve this equation above, one can shows that the solution of the GW equation can be computed from the following equation
hµν(xα) = −16πG
Z
The retarded Green’s function related to the d’Alembertian operator is given by 3 G(x, t; x0, t0) = − Θ(t − t 0) 4π|x − x0|δ(|x − x 0| − (t − t0 )) ,
where Θ(t − t0) is the Heaviside step function. Thus, we can substitute it in the equation Eq. (2.11) to obtain hµν(xα) = 4G Z Tµν(x0, t − |x − x0|) |x − x0| d 3 x0 .
The transverse-traceless part of the perturbation hµν can be obtained by apply the
projector tensor defined below (see [26])
Λij,kl ≡ PikPjl−
1
2PijPkl , (2.12)
where Pij ≡ δij − ninj is the vectorial projector tensor that makes the rule of to project
some vector in a given direction n = (n1, n2, n3).
Considering the directional vector n along the z-axis of some coordinates system (n = (0, 0, 1)), we can see the effect of to apply the tensor Λij,kl in an arbitrary matrix (3 × 3)
Aij. Aij = A11 A12 A13 A21 A22 A23 A31 A32 A33 Λij,klAkl= 1 2(A11− A22) A12 0 A21 −12(A11− A22) 0 0 0 0 .
Now, we want to compute the transverse-traceless part of the perturbation tensor on the z-axis hT Tij = Λij,klhkl = 4GΛij,kl Z Tkl(x0, t − |x − x0|) |x − x0| d 3x0 . (2.13)
As we are dealing with sources that are far away from us, we can perform a Taylor ex-pansion in the previous equation to simplify them and so, proceed with the computations.
So, we have the following expansions |x − x0| ≈ r 1 + 1 2 r0 r r0 r − 2cosψ + ... ≈ r − x0 · n , (2.14) 1 |x − x0| ≈ 1 r 1 − 1 2 r0 r r0 r − 2cosψ + ... ≈ 1 r + x0· n r2 ≈ 1 r , (2.15)
where r = |x| is the position where we want to compute the wave forms, r0 = |x0| is the position inside the source, and ψ is the angle between them. Here, n is the unit vector along the x direction.
Substituting Eq. (2.14) and Eq. (2.15) in Eq. (2.13) we have
hT Tij = 4G r Λij,kl
Z
Tkl(x0, tr+ x0· n)d3x0 , (2.16)
where tr≡ t − r is the retarded time.
x’
x
ψ
Figure 2.1: Sketch of the system treated in the current section.
Expanding now Eq. (2.16) around the retarded time tr we have
hT Tij = 4G r Λij,kl Z Tkl(x0, tr)d3x0+ nq Z x0qT˙kl(x0, tr)d3x0+ ... . (2.17)
In this equation we are going to make use of the two defined quantities below
Sij ≡ Z Tijd3x , (2.18) Mij ≡ Z T00xixjd3x , (2.19)
where this last one is called Quadrupole Moment4. These two quantities can be related as Sij = 1 2 ¨ Mij . (2.20)
And, therefore, we have the quadrupole formula from the equation Eq. (2.17) at the leader order hT Tij (t, x) Quadrupole = 2G r Λij,kl ¨ Mkl(tr, x) . (2.21) Quadrupole Formula
From this last equation, we have the expressions for the two gravitational waves po-larizations far away the source.
h+ = G r( ¨M11− ¨M22) t=tr , (2.22) h×= 2G r ¨ M12(tr) . (2.23)
Hence, for gravitational waves emitted by binaries systems propagating in the direction n = (0, 0, 1), the perturbation tensor in the TT gauge in the matrix form will be given by hT Tij = h+ h× 0 h× −h+ 0 0 0 0 .
The line element adding this perturbation to the Minkowski metric (gµν = ηµν+ hT Tµν)
will be
dS2 = −dt2 + (1 + h+)dx2+ (1 − h+)dy2+ 2h×dxdy + dz2 . (2.24)
Waveforms of Gravitational Waves from Binary Systems
In this section we are dealing with deduction of the GW waveforms from binary systems in circular orbits. Using the formalism of two body problem, in which the relative distances are given by R ≡ |r1− r2| =px20+ y20 + z02, we have
x0(t) = R sin(ωst), y0(t) = R cos(wst), z0 = 0 ,
where the two bodies are rotating around its center of mass with angular frequency ωs.
The quadrupole moment for this system is given by
Mij = M XCMi XCMj + µxi0xj0 ,
where M and µ are the total and reduced mass respectively. XCMi = (XCM, YCM, ZCM)
indicates the center of mass position of the system, and xi
0 = (x0, y0, z0) is the relative
distance between the masses.
In this sense, let the origin of the coordinates system be in the binary center of mass shuch that the + and × perturbation components are
h+(t, r) = 4Gµ r ω 2 sR 2cos(2ω stret) , h×(t, r) = 4Gµ r ω 2 sR 2sin(2ω stret) . y x ~ r1 ~ r2 φ φ m1 m2 x y z ~n m1 m2
General Binary System
When the angular momentum of the system is in a arbitrary direction n, for the spherical angles ι and φ, a proper coordinate transformation have to be used to provide us the waveform for the GW from this source. Basically what we have to do is to perform a coordinate transformation through rotation matrices applied in the second time derivative of the quadrupole moment tensor.
hT Tij (t, r) = 2G r Λij,kl
¨
Mkl(tret) .
In this case, the components ¨Mi3 = ¨M3i is non null. The system treated here is
schematically showed in fig. 2.3.
x y z m1 m2 n ι φ x y z x’ y’ n ι φ
Figure 2.3: Sloped Binary System Configuration
The useful rotation matrices for this problem are
Rx0(ι) = 1 0 0 0 cosι sinι 0 −sinι cosι Rz0(φ) = cosφ sinφ 0 −sinφ cosφ 0 0 0 1 .
From these matrices we are going to represent the components of ¨M (second time derivative of the quadrupole moment tensor) in the coordinates (x, y, z) in terms of the same tensor components in the rotated system with z0 along the direction of n. We are
going to use the results of the previous section to these last components. Let us define a suitable rotation operator (given by the product between the previous matrices) to proceed with the computations.
R ≡ Rz0(φ)Rx0(ι) =
cosφ cosι sinφ sinι sinφ −sinφ cosι cosφ sinι cosφ
0 −sinι cosι . With the RTR = RRT = I.
The transformation law for tensors using the given matrix above is such that ¨Mij =
RikRjlM¨kl0 . In the matrix notation we have ¨M = R ¨M 0RT.
The only components we have to compute are the ¨M11, ¨M22, ¨M12 and ¨M21 (knowing
that ¨M110 = − ¨M220 and ¨M120 = ¨M210 ). So we have, ¨
M11= (cos2− sin2φcosι) ¨M110 + cosι sin(2φ) ¨M 0 12
¨
M22= (sin2− cos2φcosι) ¨M110 − cosι sin(2φ) ¨M120
¨
M12= −12(1 + cos2ι)sin(2φ) ¨M110 + cosι cos(2φ) ¨M 0 12
¨
M21= −12(1 + cos2ι)sin(2φ) ¨M110 + cosι cos(2φ) ¨M120
.
Now we are going to use the equations Eq. (2.22) and Eq. (2.23) to compute the components of the strains h+ and h×. The results are showed below.
h+(t) = 4Gµ r ω 2 sR 2 1 + cos2ι 2
cos(2ωstret)cos(2φ) + cosι sin(2ωstret)sin(2φ)
, h×(t) = 4Gµ r ω 2 sR 2
cosι sin(2ωstret)cos(2φ) −
1 + cos2ι
2
cos(2ωstret)sin(2φ)
.
Let us now define a new parameter called chirp mass
Mc≡ µ3/5M2/5 =
(m1m2)3/5
(m1+ m2)1/5
.
Also we are going to use the Newtonian approximation to the equivalence between the gravitational and centripetal forces to found a relation between the angular frequencies ωs, the constant G and the orbit radius R.
GµM R2 = µ(ωsR)2 R ⇒ ω 2 s = GM R3 .
Using the definition of the chirp mass and defining ωgw = 2ωs ⇒ fgw = 2fs, follows
that the two perturbing components will be given by
h+(t, r, ι, φ) = A
1 + cos2ι
2
cos(2πfgwtret)cos(2φ) + cosι sin(2πfgwtret)sin(2φ)
,
h×(t, r, ι, φ) = A
cosι sin(2πfgwtret)cos(2φ) −
1 + cos2ι
2
cos(2πfgwtret)sin(2φ)
,
with A ≡ 4r(GMc)5/3(πfgw)2/3.
For sources in which we can neglect variations of their orbital angular momentum, the angles ι and φ consequently will be constants and we can do one more suitable coordinates transformation such that φ = 0. Thus, we have the perturbatives components below
h+(t, r) = 4 r(GMc) 5/3(πf gw)2/3 1 + cos2ι 2 cos(2πfgwtret) , h×(t, r) = 4 r(GMc) 5/3(πf
gw)2/3cosι sin(2πfgwtret) . (2.25)
Gravitational Waves Evolutions From Binary Systems
Through the emission of gravitational waves, a binary system loses energy and, conse-quently, by the system orbital energy decreasing, its orbit radius decreases too. With this, by the conservation of the angular momentum, its angular frequency increases together with the amplitude of the gravitational waves (which is dependent of these frequencies).
Let us compare the variation of the Newtonian energy of circular orbits with the dissipated energy by the GW emission.
Newtonian Orbit Energy
For classical binaries systems that are maintained connected only by Newtonian grav-itational forces, one can shows that the system angular frequencies ωs can be related to
the Newton’s gravitational constant G, the orbit radius R and its total mass M as
ωs2 = GM R3 ⇒ 1 R = ωs2/3 (GM )1/3 .
The total energy of the system in terms of the quantities µ, M, G, R, and ωs can be
computed as follows E = 1 2µ(ωsR) 2− GM µ R ⇒ E = − GM µ 2R = − (GM )2/3µ 2 ω 2/3 s .
Using the definition of the chirp mass Mc≡ µ3/5M2/5 ⇒ M 5/3 c = µM2/3 we have E = −(GMc) 5/3 2G ω 2/3 s .
With this, the variation in the orbit energy will be given by
dE dt = − (GMc)5/3 3G ω −1/3 s ω˙s .
Defining the angular gravitational wave frequency as ωgw ≡ 2ωs and knowing that
ω = 2πf we have dE dt = − π2/3(GMc)5/3 3G f −1/3 gw f˙gw . (2.26)
Dissipated Energy by the Gravitational Waves Emission
We already saw the gravitational waves energy can be obtained from 00 component of the energy momentum tensor tµν. As the dissipated energy by GW emission is given by
the integral of the time variation of t00 over the whole space, we can equate it with Eq. (2.26) to get an expression which describes the evolution of gravitational radiation. So,
the dissipated energy by GW is5 dE dt = Z ∂0t00d3x = − Z ∂it0id3x .
To solve this integral, after use the divergence theorem, we must make use of a valid property of the tensor tµν for our problem, in which t00= t0r as one can shows. Thus, we
have6 dE dt = −r 2 Z t0rdΩ = −r2 Z t00dΩ = − r 2 16πG Z h ˙h2 ++ ˙h2×idΩ . (2.27)
Computing the time derivatives of Eq. (2.25) for the perturbations components we have
˙h+ = −2ωshc
1 + cos2ι
2
sin(2ωstr) , ˙h× = 2ωshccosι cos(2ωstr) ,
where the coefficient hc of theses ones can be identified as
hc =
4
r(GMc)
5/3
(πfgw)2/3 .
With this, we are able to compute the average of the sum between the square of both. So, h ˙h2 ++ ˙h 2 ×i = 2h2cω 2 sg(ι) , (2.28)
where the angular dependence of this equation is given by
g(ι) ≡ 1 + cos
2ι
2
2
+ cos2ι .
Integrating this function over the solid angle element dΩ we have Z g(ι)dΩ = 2π Z π 0 g(ι)sinι dι = 16π 5 . (2.29)
Therefore, including the Eq. (2.28) in Eq. (2.27) with the obtained result Eq. (2.29),
5Here we are using the Lorentz gauge in which ∂
0t00= −∂iti0. 6dΩ ≡ sinι dι dφ denotes the solid angle element.
we get the dissipated energy by the GW emission dE dt = − 2r2h2 cωs2 5G = − 32(GMc)10/3 5G ω 10/3 s .
And in terms of the its wave frequency fgw we get
dE dt = − 32(πGMc)10/3 5G f 10/3 gw . (2.30)
Binary System Merging
The next step is to equates the equations Eq. (2.26) and Eq. (2.30) to obtain one relation between the chirp mass and the GW frequencies. The result of this one follows below −32(πGMc) 10/3 5G f 10/3 gw = − π2/3(GMc)5/3 3G f −1/3 gw f˙gw , Mc= 1 G " 5 96π8/3 ˙ fgw fgw11/3 #3/5 . (2.31)
Thus, if we can measure the GW frequency and its first time derivative we can compute the value of the chirp mass.
By the end, let us turn back to the waveform of the GW and make use of the previous relation between chirp mass and the GW frequencies to see how their involve in time. The result is ˙ fgw fgw11/3 = 96π 8/3 5 (GMc) 5/3 .
Solving this differential equation we have
fgw(t) = 3 8 5 96π8/3(GM c)5/3 3/8 (t0− t)−3/8 ,
where t0 is the coalescence time of the binary system. Using this frequency evolution
we can include it in the expressions of the waveforms Eq. (2.25). Thus, we can see this behaviour for the h+ in fig. 2.4
Figure 2.4: Behaviour of the gravitational waves produced by a binary system merging.
Remembering that the component h+ is given by:
h+(t, r, ι) = 4 r(GMc) 5/3(πf gw(t))2/3 1 + cos2ι 2 cos 2π Z tret 0 fgw(t)dt .
Gravitational Waves from Cosmological Distances
From the amplitude of the gravitational waves computed in the previous section let us compute its form from cosmological distances. First we already know that the GW amplitudes are of the following form
hA =
4
r(GMc)
5/3
(πfgw)2/3γ(ι) ,
where γ(ι) is the angular dependence of the amplitude.
For cosmological distances we have the frequencies of the GW will be affected by the cosmic expansion of the universe. So, what we have in the previous equation will be the frequency of the waves in the source frame. Let us exchange this one for the observed frequency using the definition of the redshift z.
f(emis) = (1 + z)f(obs) .
respectively.
The distance r in the amplitude of these waves will be substituted for the distance measured in the current value of the scale factor (r → a(t0)r). We also know that this
distance can be computed using the Friedmann’s equation as follows
ds2 = −dt2+ a2(t)dχ2 = 0 ⇒ ∆χ = Z t0 0 dt a(t) = Z 1 0 da a2H = 1 H0 Z z 0 dz E(z) , ∆χ = r = 1 H0 Z z 0 dz E(z) , where E(z) =pΩM(1 + z)3+ ΩR(1 + z)4+ ΩDE .
From the definition of the luminosity distance we have,
dL ≡ (1 + z)∆χ = 1 + z H0 Z z 0 dz E(z) .
Let us now perform more one substitution using a new mass parameter, the shifted chirp mass Mc≡ (1 + z)Mc. Hence we are going to obtain the following equation for the
GW amplitude in cosmological distances
hA =
4 dL(z)
(McG)5/3(πfgw(obs))
2/3γ(ι) . (2.32)
This is a very interesting result, showing that, if we know the orientation of the binary system, the amplitude of its GW signal, as well as its shifted chirp mass (that can be deduced from the wave phasing), we found a direct way to compute the luminosity distance of this cosmological sources. For this reason, these kind of sources passed to be call standard sirens in analogy to the standard candles7.
7Standard candles are kind of sources whose their intrinsic luminosity are well known, in way that
measuring its apparent luminosity (knowing the quadratic decay of the radiation energy with the distance) we can infer their respective distances as already was mentioned inchapter 1.
Data Analysis and Bayesian Inference
In this chapter we are going to show how we can make data analysis starting from the type of signals that we can get from GW detectors, which depends on the detector geometry and the sky position of the source. After this we will introduce some aspects of the statistical analysis focusing on the χ2-test (also used to make data best-fit) and on
the Bayesian analysis.
Gravitational Waves Detectors
The output of a GW detector is, actually, a scalar quantity rather than a tensorial one [26]. What the detector could detect is a linear combination of the GW polarizations through a new tensorial quantity called detector tensor Dij (that depends on the detector
geometry). This can be seen in the following relation
h(t) = Dijhij(t) , (3.1)
where we can express the perturbations hij in terms of its polarized components
hij(t) =
X
A=+,×
eAij(ˆn)hA(t) , (3.2)
where eA
ij(ˆn) is the polarization tensor given by
e+ij(ˆn) = ˆUiUˆj − ˆViVˆj , e×ij(ˆn) = ˆUiVˆj + ˆViUˆj . (3.3)
Polarization Tensors
The unit vectors ˆU and ˆV in the expressions above defines the perpendicular plane to the GW propagation direction.
In the matrix form the polarization tensor, for a GW propagating along the z-direction, can be seen as follows
e+ij = 1 0 0 0 −1 0 0 0 0 , e×ij = 0 1 0 1 0 0 0 0 0 .
Substituting Eq. (3.2) in Eq. (3.1) we have
h(t) = X
A=+,×
[DijeAij(ˆn)]hA(t) .
Let us define the new quantity FA(ˆn) called antenna pattern function as follows
FA(ˆn) = DijeAij(ˆn) . (3.4)
Antenna Pattern Function
Hence, with ˆn pointing in the direction (θ, φ) we have
h(t) = F+(θ, φ)h+(t) + F×(θ, φ)h×(t) .
The kind of detectors that we are interested are the Michelson-Morley interferometers. The principal gravitational waves interferometer detectors around the world are listed below [30]
• LIGO in Livigstone, Lousiana, US and Hanford, Washington, US1; • Virgo in Cascina, Italy2;
• KAGRA in Gifu, Japan3; • GEO600 in Sarstedt, Germany4;
1From http://www.ligo.caltech.edu.
2From http://public.virgo-gw.eu/language/en/.
3From https://gwcenter.icrr.u-tokyo.ac.jp/en/. This detector is currently in its final tests stages to
be enter in operation soon.
The Gravitational Waves Detectors LIGO and Virgo
The couple of detectors called Laser Interferometer of Gravitational Waves Observa-tory (acronym LIGO) located in Hanford, Washington, US and Livingstone, Louisiana, US were funded by the US National Science Foundation5. These observatories consists in interferometers whose arms have a length of 4 km.
Other GW detector called Virgo, located in Cascina, a small town near Italy’s capital in the site of he European Gravitational Observatory (EGO), has been founded by the French Centre National de la Recherche Scientifique (CNRS) and the Italian Istituto Nazionale di Fisica Nucleare (INFN). This observatory, similar to LIGO, consists in a interferometer with perpendicular arms whose lengths reach 3 km each6.
Figure 3.1: (Left) LIGO in Hanford-Washington, (Right) LIGO in Livingstone-Louisiana and (Below) Virgo in Italy. From: http://www.ligo.caltech.eduand http:// www.virgo-gw.eu/.
Having completed its construction in 1999, the LIGOs started their first operation in 2002 until 2010, not achieving any detection [31]. Five years later, with intense work to improve the detector up to ten times at design sensitivity, the LIGOs made their
5Informations extracted from Direct Observation of Gravitational Waves - Educator’s Guide retrieved
fromhttps://www.ligo.caltech.edu/page/educational-resources
first detection in 14 September, 2015 (signal called GW150914) from a gravitational wave emitted by a by two coalescing binary black hole whose masses were estimated as 36+5−4M
and 29+4−4M [31,2]. Up to the end of 2017 the LIGOs joint with Virgo already have made
a catalog with eleven detections listed below [32].
Signal System’s Type m1(M ) m2 (M ) z dL (M pc)
GW150914 BHS 36+5−4 29+4−4 0.09+0.03−0.04 410+160−180 GW151012 BHS 23.3+14.0−5.5 13.6+4.1−4.8 0.21+0.09−0.09 1060+540−480 GW151226 BHS 14.2+8.3−3.7 7.5+2.3−2.3 0.09+0.03−0.04 440+180−190 GW170104 BHS 31.2+8.4−6.0 19.4+5.3−5.9 0.18+0.08−0.07 880+450−390 GW170608 BHS 12+7−2 7+2−2 0.07+0.03−0.03 340+140−140 GW170729 BHS 50.6+16.6−10.2 34.3+9.1−10.1 0.48+0.19−0.20 2750+1350−1320 GW170809 BHS 32.2+8.3−6.0 23.8+5.2−5.1 0.20+0.05−0.07 990+320−380 GW170814 BHS 30.5+5.7−3.0 25.3+2.8−4.2 0.11+0.03−0.04 540+130−210 GW170817 BNS ∈ (1.36, 1.60) ∈ (1.17, 1.36) 0.008+0.002−0.003 40+8−14 GW170818 BHS 35.5+7.5−4.7 26.8+4.3−5.2 0.20+0.07−0.07 1020+430−360 GW170823 BHS 39.6+10.0−6.6 29.4+6.3−7.1 0.34+0.13−0.14 1850+840−840
Table 3.1: List of the signals measured by the detectors LIGO and Virgo with a total of eleven detections. m1 and m2 are the masses of the binary components. Informations from:
[2,33,34,35,36,37,32]. Legend: BHS (Binary Black Hole System); BNS (Binary Neutron Star System).
Let us now pass to the technical part of the type of signal output h(t) (without treating noise) that can be analysed from type interferometers detectors as the LIGO and Virgo. First we are going to introduce the form of the detector tensor for interferometers as follows Dij ≡ 1 2(uiuj − vivj) = 1 2 1 0 0 0 −1 0 0 0 0 ,
where the unit vectors u and v are pointing in the directions of the arms, in our case we are going to put our frame with the axis x an y in the directions of the arms. This form of the tensor is justified by the fact that what the interferometer are going to identify, with the interferences of the beams in the arms, will be the difference in the arms lengths (this
is due the effect of the gravitational wave passage through the detector that will change the lengths of its arms).
When the event whose signal was detected in a angular position (θ, φ), we are dealing with the system showed in the following figures.
~ u ~ v z x y ρ φ z y x z’ X’ Y’ y’ x’ θ θ x’ y’ X’ Y’ ψ ψ z’ X’ ρ z θ θ y x ρ Y’ φ φ
Figure 3.2: Rotated system in the source position. Arms of the detector making a angle of 90o between them (the case of the LIGO detector).
From the system above, the polarization tensors is defined being its more simple form in the primed system (x’, y’, z’). These polarization tensors is given by the following matrices e0ij+ = 1 0 0 0 −1 0 0 0 0 , e0ij×= 0 1 0 1 0 0 0 0 0 .
So, as we are measuring signals in the detector frame (unprimed frame given by the co-ordinates x, y and z), we need to put the polarization tensors above in the same coordinate system. This can be made through the rotation matrices below
Rz(φ) = cosφ −sinφ 0 sinφ cosφ 0 0 0 1 , RY0(θ) = cosθ 0 sinθ 0 1 0 −sinθ 0 cosθ , Rz0(ψ) = cosψ sinψ 0 −sinψ cosψ 0 0 0 1 .
