Thesis BETA-BEXCH
Bachelo in Physics
Explo ing G a i a ional-Wa e Signa u es o
Modi ied G a i y in Black Hole Ringdown
Au ho :
Unai Razkin
Di ec o s:
Tanja Hinde e and Pablo Bosch
2023, Unai Razkin
U ech , 17 h June, 2024
1
Con en s
1 In oduc ion and Objec i es 6
2 Basics o Gene al Rela i i y 9
2.1 Me icSpace ......................... 9
2.2 Lo en z T ans o ma ions . . . . . . . . . . . . . . . . . . . 10
2.3 Tenso s ............................ 10
2.4 Co a ian De i a i e . . . . . . . . . . . . . . . . . . . . . 11
2.5 Ch is o el Symbols . . . . . . . . . . . . . . . . . . . . . . 12
2.6 RiemannTenso ........................ 13
2.7 Ricci Tenso and Scala . . . . . . . . . . . . . . . . . . . . 14
2.8 Eins ein’s Field Equa ions . . . . . . . . . . . . . . . . . . 14
3 Theo e ical In oduc ion o GWs om pe u bed BHs 16
3.1 G a i a ional Wa es in Fla Space ime . . . . . . . . . . . 16
3.1.1 Solu ions and Pola iza ion . . . . . . . . . . . . . . 17
3.1.2 G a i a ional Wa e Sou ces . . . . . . . . . . . . . 18
3.2 G a i a ional Wa es om New onian Bina y Sys ems . . . 19
3.3 G a i a ional Wa es om Pe u bed Black Holes . . . . . 23
3.3.1 Sphe ical Ha monics Decomposi ion . . . . . . . . . 24
3.3.2 Equa ions o Mo ion o The Pe u ba ions . . . . . 25
3.3.3 Radial Wa e Equa ion . . . . . . . . . . . . . . . . 26
3.3.4 Solu ions o he Radial Di e en ial Equa ion . . . . 28
3.3.5 Isospec ali y . . . . . . . . . . . . . . . . . . . . . 28
4 Ringdown wa e o ms 30
4.1 RingdownModel ....................... 30
4.2 Nume ical Rela i i y da a . . . . . . . . . . . . . . . . . . 31
4.3 Compa ing Ringdown Model wi h NR Da a . . . . . . . . 35
5 Modi ied g a i y 37
5.1 Lo elock’s heo em . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Al e na i e heo ies . . . . . . . . . . . . . . . . . . . . . . 37
5.2.1 Eins ein-dila on-Gauss-Bonne . . . . . . . . . . . . 38
5.2.2 Dynamical Che n-Simons g a i y . . . . . . . . . . 38
5.2.3 E ec i e- ield- heo y o g a i y . . . . . . . . . . . 39
5.3 De ia ions on he Spec a o he Quasino mal Modes . . . 39
2
6 Analysis on he Signa u es o Modi ied G a i a ional The-
o ies 41
6.1 De ia ion E ec s on he F equencies . . . . . . . . . . . . 41
6.2 Al e na i e G a i y Theo ies and NR Da a . . . . . . . . . 42
6.3 E ec on Pa ame e In e ence . . . . . . . . . . . . . . . . 43
7 Resul s and Discussion 47
8 Conclusions and Ou look 48
A Gauge F eedom 50
A.1 The Residual Gauge om Lo enz Gauge . . . . . . . . . . 51
A.2 Gauge Fixing on Black Hole Pe u ba ion Theo y . . . . . 52
B G a i a ional Wa e Pola iza ion 54
B.1 Plus pola isa ion . . . . . . . . . . . . . . . . . . . . . . . 54
B.2 C oss pola isa ion . . . . . . . . . . . . . . . . . . . . . . . 55
C G a i a ional Wa e De ec ion 56
D Teukolsky Equa ion 58
3
4
Acknowledgemen s
Fi s , I would like o hank my wo supe iso s, Tanja and Pablo, o hei kindness,
pa ience and especially o he p iceless help I ha e ecei ed om hem. I also ex end
my g a i ude o he en i e esea ch g oup o in oducing me o he ascina ing ield o
g a i a ional wa es and o always being willing o help wi h a smile.
Eske ik sakonenak amilia guz ia i xiki a ik emandako baldi za ik gabeko mai asun guz-
iaga ik. Gehien ba ni e gu asoei, zuen alboan opa i u didazuen bizi zaga ik. O aina ean
i aka si didazuen guz iaga ik e a ikas eko ge a zen zaidan guz iaga ik. Bai a Lei e i e e,
eske ik asko, ni ekiko daukazun kon ian za e a enpa iaga ik, besa kada nekaezinenga ik,
zu e umo e sa kas ikoaga ik e a zo i xa ez inoiz ba ne a zen ez di udan ikasgaienga ik
e e. O aindik ez daki isilik ge a zen ”-Zeinek?” bezelako galde e an...
Finalmen e, doy las g acias a odos mis amigos, a la ”exclusi a” lis a de amigos que he
hecho aqu´ı en U ech (xD), y singula men e, a odos aquellos que han es ado siemp e a mi
lado, a quienes he echado i emediablemen e de menos es e a˜no. En pa icula , ag adezco
de co az´on a Ja i y a Juan po habe hecho es e a˜no mucho m´as lle ade o y po saca me
una son isa d´ıa s´ı y d´ıa ambi´en con ues o humo incansable.
5
1 In oduc ion and Objec i es
Gene al ela i i y (GR) has an undeniable ele ance in mode n physics. I was p oposed
by Eins ein in 1916, bu o e a cen u y la e , i con inues o be a co ne s one in many
ields, such as as ophysics, cosmology, pa icle physics, s ing heo y, and quan um ield
heo y [1]. This heo y s ands as one o he mos impo an and in luen ial heo ies o all
imes and emains a highly ac i e subjec o s udy.
Gene al ela i i y’s p edic ions ha e had a huge in luence no only in physics bu also
on a ious aspec s o mode n li e. Fo ins ance, i is essen ial in he Global Posi ioning
Sys em (GPS) [2] as we ha e o conside ime co ec ions o he da a we ecei e om
he sa elli es due o space ime cu a u e. The g ea e he dis ance be ween he Ea h
and he sa elli e, he mo e he me ic changes, and acco ding o Eins ein’s heo y, his
c ea es ime dila ion, causing he sa elli e’s clock o de ia e by app oxima ely 35µs pe
day, leading o an e o o abou 10 km in global posi ioning each day [3].
Mo eo e , GR has nume ous measu able consequences, which, along wi h s ong expe -
imen al e idence, ha e es ablished a i m suppo ing base o he heo y [4]. GR has
p edic ed no only he ime-dila a ion men ioned abo e o equency shi s bu also ligh
de lec ion, he p ecession o apsides, o bi al decay, and g a i a ional wa es, none o which
can be explained by New onian g a i y. This hesis will ocus on he las phenomenon:
g a i a ional wa es.
G a i a ional wa es (GW) a e ipples ha p opaga e h ough space ime. They de o m
space ime -e en he acuum i sel - and ca y adia ion ene gy. As New onian g a i y
canno explain hem, hey a e an in e es ing ea u e suppo ing Eins ein’s heo y [5].
Mo eo e , he in e es in hese wa es elies on he ac ha hey can pene a e egions o
space ha elec omagne ic wa es canno . Fo ins ance, hey could o e he possibili y o
obse e he bi h o he Uni e se e en be o e he cosmic mic owa e backg ound [6], which
is impossible by adi ional means (op ical elescopes o adio elescopes) as he e was no
elec omagne ic adia ion be o e he Cosmic Mic owa e Backg ound. Fu he mo e, hey
allow he obse a ion o he me ge o black holes, which we will s udy in his disse a ion.
The i s de ec ion o GW e ec s was made in 1974 wi h he Hulse–Taylo pulsa [7],
which led o he Nobel P ize in 1993 [8]. Th ough obse a ions o he bina y pulsa o e
many yea s, hey could measu e he o bi al decay o i , which accu a ely ag eed wi h
he loss o ene gy due o g a i a ional wa es. Th oughou his disse a ion, we discuss
bina y sys ems and hei ene gy loss h ough he emission o g a i a ional wa es (Sec.
3.2). Besides, he i s measu emen o a GW signal om a me ging bina y sys em was
possible hanks o he LIGO obse a o y in 2015 [9] (we explain he me hodology used
o do so in Appendix C). Again, his led o ano he Nobel p ize in 2017 [10]. In ad-
di ion o LIGO and Vi go, mo e obse a o ies ha e been cons uc ed, such as KARGA
[11], and mo e will be buil in he coming yea s (sensi i e o o he equency bands),
like a space-based obse a o y, he Lase In e e ome e Space An enna (LISA) [12], and
g ound-based de ec o s, he Eins ein Telescope (ET) [13], and Cosmic Explo e (CE) [14].
As we men ioned, in pa icula , we s udy GWs om he me ge o black holes (BH);
hus, jus as GW wa es, BH is a ecu en opic in his disse a ion. Thei popula i y in
6
science is due o he mys e ies hey easu e: causali y p e en s any in o ma ion abou
wha happens inside he BH om exi ing ( he bounda y beyond which no hing, no ligh
no in o ma ion, can escape is he E en Ho izon). Addi ionally, we expec hem o ha e
a singula i y, a egion wi h in ini e cu a u e whe e all he mass is con ained and whe e
physical laws a e s ill unknown due o he ise o quan um e ec s [15]. On op o ha ,
he s ong cu a u e c ea es ascina ing phenomena o huge scien i ic in e es , such as
g a i a ional lensing and acc e ion discs. I ha was no enough, hey a e also pa many
p oposed solu ions o explaining da k ma e [16]. Finally, he no-hai heo em conjec-
u es ha black holes a e cha ac e ized by only h ee obse able p ope ies: mass, elec ic
cha ge, and angula momen um. The e o e, he heo e ical ea men o BHs is ela i ely
s aigh o wa d.
Like GWs, BHs ely on a my iad o e idence. Since 1995, as onome s ha e acked
s a s nea he cen e o ou galaxy, and hei mo ions e ealed ha he cen al objec
Sagi a ius A* needed o ha e 4.3×106M⊙[17]. Al hough no conclusi e enough, his
was s ong e idence suppo ing he exis ence o BH. On 10 Ap il 2019, a e he E en
Ho izon Telescope (EHT) obse a ions in 2017, he i s di ec image o a black hole was
published o he supe massi e black hole in Messie 87’s galac ic cen e [18] and la e also
o Sagi a ius A* [19]. Mo eo e , he a o emen ioned i s GW de ec ion led he LIGO-
Vi go collabo a ion o announce he i s di ec de ec ion o a BH me ge and many mo e
since hen, which is ye ano he indica ion o bo h BH and GW [9].
All hese obse a ional esul s clea ly show ha GR has been ho oughly sc u inized
h ough an ample lis o es s. Al hough we ha e speci ically ocused on BHs and GWs,
all obse a ions ha e consis en ly p o en GR o be accu a e, jus as he ones we ha e
p esen ed. None heless, singula i ies a e ega ded as a b eakdown poin o he heo y
o GR. Likewise, heo e ical physicis s aim o c ea e a heo y ha econciles quan um
mechanics and g a i y a his poin —quan um g a i y [15]. Hence, he heo y is no he
inal wo d and equi es ongoing es ing. Wi h his objec i e in mind, in his hesis, we
s udy a me hod o es GR and explo e possible al e na i e heo ies.
Fo his pu pose, he disse a ion ocuses on he GWs emi ed du ing he BH Ringdown
(RD). The RD is he ” elaxa ion” p ocess ha a pe u bed BH - o ins ance, he em-
nan one a e he collision o bina y BHs- expe iences as i emi s ene gy by GWs un il
i eaches he s a iona y s a e. The GWs emi ed du ing his p ocess can be desc ibed
as a sum o quasino mal modes (QNM), as seen in Sec. 3.3.4. QNM a e oscilla ions ha ,
unlike no mal modes, decay o e ime due o ene gy dissipa ion, which is why he BH
e u ns o a s able s a e a e all. These modes ha e di e en equencies; o his eason,
he ield is called Black Hole Spec oscopy, one o he mos p omising ields o es ing
gene al ela i i y and o he heo ies.
The p ojec aims o build a model o he RD wa e o ms (Sec. 4) ha accu a ely de-
sc ibes hem using heo y-based equencies p edic ed om GR and heo ies beyond his
amewo k. Ne e heless, ega ding al e na i e heo ies, he geome y o o a ing BHs is
no always known, and pe u ba i e o nume ical app oaches a e usually he bes we can
ge . Acco dingly, he e a e gene ally no sepa able di e en ial equa ions o he pe u ba-
ion o hese heo ies, and in u n, he equencies a e challenging o compu e. As a esul
o hese challenges, and in o de o compa e all heo ies wi hin a common amewo k, we
7
need o in oduce he ”Pa Spec” amewo k (which is de eloped in Re .[20]).
The ”Pa Spec” amewo k is based on a dimensionless spin pa ame e ( ela ed o he an-
gula momen um) expansion o he equencies. Recall ha due o he no-hai heo em,
he spec um o RD equencies uniquely depends on hei angula momen um (J) and
mass (M). Re . [21] has compu ed he coe icien s o his spin expansion o linea o de
(also he de ia ions om GR), combining he equencies compu ed in Re s. [22], [23],
[24] and [25] o slowly o a ing BHs. Ou hesis implemen s hese de ia ions o quan i y
he wa e o m disc epancies caused by al e na i e heo ies. On op o ha , we s udy he
e ec o hese heo ies on pa ame e in e ence, namely mass and spin, and b ie ly discuss
he biases o he in e ence. Las ly, we discuss he abili y o cons ain undamen al pa-
ame e s o modi ied heo ies.
This hesis is s uc u ed as ollows. Fi s , we p esen he ounda ional concep s in Sec. 2.
Then, in Sec. 3, we in oduce he heo y o he physical phenomena o GWs on pe u bed
BHs, such as he de i a ion o he GWs om Eins ein’s equa ions and he quasino mal
modes in he ingdown o a pe u bed BH. Nex , some ingdown analyses, including one
om Nume ical Rela i i y, a e p esen ed in Sec. 4. Subsequen ly, we in oduce he Mod-
i ied G a i y heo ies in Sec. 5, ollowed by an analysis o he signa u es ha hese ha e
on he ingdown in Sec. 6. Finally, we p esen some esul s and a b ie discussion in Sec.
7. In addi ion, he appendices con ain supplemen a y ma e ial on gauge eedom, GW
pola iza ion di ec ion, GW de ec ion, sphe ical ha monic decomposi ion, and GW signal
modelling.
8
Gνµ ≡Rµν −1
2Rgµν =⇒ ∇νGνµ = 0.(2.29)
Al oge he , he Eins ein ield equa ions become:
Rµν −1
2Rgµν =8πG
c4Tµν .(2.30)
The p opo ionali y cons an (8πG
c4) is ob ained wi h he New onian limi . You a e en-
cou aged o see [1] o mo e de ails.
Las ly, he Eins ein-Hilbe ac ion ep oduces he Eins ein ield equa ions (Eqs. 2.30) ia
he minimum ac ion p inciple. The ac ion eads as
SH=c4
16πG Z√−gRd4x. (2.31)
15
3 Theo e ical In oduc ion o GWs om pe u bed
BHs
Once we ha e in oduced he ma hema ical backg ound, we now explo e he physics o
GWs. The main subjec o he hesis a e GWs o igina ed om pe u bed BHs a e a
bina y me ge . None heless, we i s conside he basics o GWs in la space ime and
discuss impo an ea u es, such as hei pola iza ion o he sou ces o hese ipples.
3.1 G a i a ional Wa es in Fla Space ime
The i s s ep o unde s anding G a i a ional Wa es is o conside a la space ime (gi en
by he Minkowski me ic ηµν om Eq. 2.3) and apply a small pe u ba ion (hµν) o i .
gµν =ηµν +hµν,whe e |hµν| ≪ 1.(3.1)
As |hµν| ≪ 1 we only ega d linea o de in pe u ba ion. Eins ein ield equa ions (Eqs.
2.30) e eal he di e en ial equa ions ha hese pe u ba ions mus obey. To do so, we
need o compu e he Ricci and Riemann enso , bu in u n, we i s need o compu e he
Ch is o el symbols. Using Eq. 2.21, we a e le wi h he ollowing condi ion:
Γλ
µν =1
2gλσ (∂µgνσ +∂νgσµ −∂σgµν) = 1
2ηλσ (∂µhνσ +∂νhσµ −∂σhµν).(3.2)
F om he de ini ion o he cu a u e 2.23 and using 3.2, we a e le wi h:
Rσρµν =1
2ησλ (∂µ∂ρhνλ −∂µ∂λhνρ −∂ν∂ρhµλ +∂ν∂λhµρ).(3.3)
The linea ized Ricci enso is (by con ac ing he Riemann enso as in Eq. 2.24):
Rµν =1
2(∂ρ∂µhνρ +∂ρ∂νhµρ −□hµν −∂µ∂νh),(3.4)
whe e h=hµµ( he ace o hµν) and whe e □=∂µ∂µ.
Likewise, he Ricci scala is ob ained om he con ac ion o 3.3 wi h η, since all e ms
a e al eady linea in pe u ba ion (h),
R=ηνµRµν =∂µ∂νhµν −□h. (3.5)
Al oge he , subs i u ing Eqs. 3.4 and 3.5 in Eq. 2.29 he Eins ein enso is
Gµν =1
2[∂ρ∂µhνρ +∂ρ∂νhµρ −□hµν −∂µ∂νh−(∂ρ∂σhρσ −□h)ηµν].(3.6)
To simpli y his exp ession, we de ine he ollowing quan i y:
¯
hµν ≡hµν −1
2ηµνh, whe e h=ηαβhαβ.
Subsequen ly, Eins ein ield equa ions (Eq. 2.30) esul in
□¯
hµν +ηµν∂ρ∂σ¯
hρσ −∂ρ∂ν¯
hµρ −∂ρ∂µ¯
hνρ =−16πG
c4Tµν .(3.7)
16
No e, i s , ha as he equa ion abo e is only conside ing small pe u ba ions a ound
la space, he g a i a ional pa o Tµν also needs o be e y weak/small. Secondly, he
equa ion is linea , and hus, as long as we a e jus conside ing i s -o de pe u ba ions,
solu ions can be w i en as supe posi ions o di e en solu ions.
Finally, Eq. 3.7 can be u he simpli ied i we cle e ly selec a speci ic coo dina e sys em.
This p ocess is called gauge ixing (discussed in Appendix A), and o ou pu poses, we
adop he Lo enz Gauge (o ha monic gauge). In he Lo enz gauge, we ha e ∂ρ¯
hαρ = 0,
esul ing in:
□¯
hµν =−16πG
c4Tµν .(3.8)
3.1.1 Solu ions and Pola iza ion
The equa ion we ob ained p e iously, Eq. 3.8, e eals ha small pe u ba ions ollow
he wa e equa ion, namely, an inhomogeneous wa e equa ion, whose solu ions can be
exp essed as ollows:
¯
hµν( , x) = −16πG
c4Zd ′d3x′Tµν ( ′,x′)G( − ′,x−x′),(3.9)
whe e Gis he solu ion o □G( , x) = δ4( , x), which gi es:
G( , x) = −1
4π
1
|x|δ −|x|
c.(3.10)
Rega ding he no a ion in his sec ion, we a e using he bold symbols o e e o he
spa ial pa o he coo dina es xν; his is, x≡xi, (o en also e e ed o as x ≡x).
I we conside he ene gy-momen um enso o be 0 (Tµν=0), due o he emo eness o
any sou ce, hen Eq. 3.8 educes o he well-known homogeneous wa e-equa ion:
□¯
hµν( , x) = 0 −→ −1
c2
∂2
∂ 2+∇2¯
hµν( , x) = 0.(3.11)
E en hough we ha e selec ed he Lo enz gauge condi ion, some non-physical eedoms
should s ill be es ic ed (see Appendix A.1 o he esidual gauge eedom). In a acuum,
we can impose he ollowing ou condi ions:
¯
h0i= 0 ¯
hµµ= 0 Lo enz gauge
−−−−−−−→ ∂ihij = 0 ¯
hii= 0.(3.12)
Unde hese condi ions, we can limi ou ollowing p ocedu e o he space coo dina es,
which is indica ed wi h La in indexes (i∈1,2,3) as men ioned be o e.
The solu ion o Eq. 3.11 can be exp essed as he supe posi ion o plane wa es. I
hese plane wa es a e p opaga ing in he z di ec ion, we know ha hey sa is y hij =
aij cos(ω( −z/x)). We can now impose he be o e-men ioned condi ions Eq. 3.12, which
esul s in azj = 0 = hzj. As Eq 3.12 shows, he ma ix is aceless. The e o e,
17
hTT
ij =
h+h×0
h×−h+0
0 0 0
=
a+a×0
a×−a+0
0 0 0
cos(ω( −z/c)).(3.13)
These wo pa ame e s h+and h×de e mine he pola iza ion o he g a i a ional wa e in
he ans e se- aceless gauge (which is wha TT s ands o ). The space ime me ic can
be w i en like:
gµν =ηµν +hTT
ij ,(3.14)
which gi es he ollowing line elemen ( ecalling Eq. 2.4):
ds2=−c2d 2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2.(3.15)
To in e p e his physically, we i s need o imagine a ci cle o es poin s when h+=
h×= 0. Then, a any ime when ei he h+= 0 o h×= 0, hese es poin s would
appea ellip ical as illus a ed in Fig. 2, whe e we can see he wo pola isa ions ha a
g a i a ional wa e can ha e: Plus (h×= 0) and C oss (h+= 0) pola iza ion, which a e
explained in de ail in Appendix B.
x
y
(a) Tes poin s unde he e ec h+pola iza ion g a i a ional wa e
x
y
(b) Tes poin s unde he e ec h×pola iza ion g a i a ional wa e
Figu e 2: Space de o ma ion e ec s o he wo pola iza ion ypes o g a i a ional wa es
illus a ed wi h es poin s.
3.1.2 G a i a ional Wa e Sou ces
Radia ion o g a i a ional wa es can be analogous o elec omagne ic adia ion as bo h
emi wa es ha ca y ene gy a he speed o ligh c. The inhomogeneous equa ions o
elec ic and magne ic ec o po en ials a e almos iden ical o Eq. 3.9. Hence, we can
unde s and sou ces o GWs in linea ized g a i y by pe o ming a simila analysis in mul-
ipole expansion as o elec omagne ic po en ials.
In he elec omagne ic expansion, oscilla ions o he cen e o he cha ge densi y p oduce
dipole adia ion. No wi hs anding, in he case o an isola ed sou ce he cen e o mass
o he sys em canno accele a e. Consequen ly, he p incipal and mos c ucial di e ence
in he analogy elies on he ac ha g a i y does no ha e dipole adia ion. Thus he
quad uple momen is he esponsible o he leading o de in his mul iple expansion,
18
which is w i en as (again you a e encou aged o see Re . [1] o a p ope de i a ion o
he ollowing equa ion)
¯
hij( , ) = −2G
c4 ¨
Iij( − /c),(3.16)
whe e
IT
ij =ZV′
ρ( ′)3 ′
i ′
j−|| ′||2δijdV ′.(3.17)
We could ew i e his o mula as
hTT
ij ( , ) = −2G
c4 Λijkl(n)¨
Mkl,(3.18)
whe e Mij is he mass quad uple momen ,
Mij ≡Zd3xρxixj,
and Λijkl, he aceless ans e se p ojec ion ope a o , is de ined as
Λijkl ≡PikPjl −1
2PijPkl whe e Pij =δij −ninj,
whe e niis he uni ec o o he p opaga ion di ec ion.
3.2 G a i a ional Wa es om New onian Bina y Sys ems
A ema kable example o a g a i a ional wa e sou ce is a bina y sys em. In his subsec-
ion, we analyze a sys em composed o wo poin masses m1, m2, acing ci cula o bi s
in he cen e o he mass e e ence ame. We can educe wo-body p oblems o one-body
p oblems, whe e he educed mass µ=m1m2
m1+m2 akes he place o he es mass.
Be awa e ha al hough linea ized g a i y is no s ic ly applicable o he bina y sys em,
i u ns ou ha igo ous and mo e sophis ica ed calcula ions show ha he quad upole
o mula (Eq. 3.18) cap u es he leading-o de esul e en o such a g a i a ional sou ce.
On op o ha , o bina y sys ems, we can u he simpli y he quad upole o mula (Eq.
3.18) o he ollowing exp ession:
h+=1
G
c4¨
M11 −¨
M22, h×=2
G
c4¨
M12,(3.19)
whe e is he dis ance o he sou ce and Mij is educed o
Mij =µR2NiNj,(3.20)
19
whe e R is he ela i e sepa a ion and Niis he uni
ec o poin ing o he educed mass, whose componen s
a e
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′
Obse e
ˆ
N
Figu e 3: O bi al plane
inclina ion
and inally, whe e φ ep esen s he angle ela i e o he x-axis wi hin he o bi al plane,
and ιindica es he inclina ion o he o bi al plane o ou line o sigh ( e e o Figu e 3.2).
Rx(ι) e e s o he ma ix o o a ions a ound he x-axis.
F om he analysis o he equa ions o mo ion o he bina y sys em i ollows ha he
adius and equency o he ci cula o bi s a e ela ed by
R= (GM)1/3ω−2/3,(3.21)
whe e ω= ˙φ, M is he o al mass M=m1+m2. The o bi al ene gy (Eo) is
Eo=1
2µR2ω2−GMµ
R=−1
2µ(GMω)2/3,(3.22)
whe e in he second equali y we used Eq 3.21.
The a e age powe adia ed in g a i a ional wa es (dEGW
d ) is compu ed ia (See Re [1]
o a close look a his o mula)
dEGW
d = 2c3
16πG Z2π
0
dφ Z1
−1
dcos ι˙
h2
++˙
h2
×.(3.23)
To calcula e h+and hxp oduced by he sou ce, we i s use Eq. 3.20, which esul s in
¨
M11 =−2µω2R2cos(2φ)
¨
M22 = 2µω2R2cos(2φ) cos2(ι)
¨
M12 = 2µω2R2sin(2φ) cos(ι).
In oducing hese exp essions in Eq. 3.19 we a e le wi h
h+=1
G
c4(¨
M11 −¨
M22) = −2µR2ω2G
c4cos(2φ)[1 + cos2(ι)]
h×=2
G
c4¨
M12 =−2µR2ω2G
c4sin(2φ) cos(ι)
,(3.24)
and inally om Eq. 3.23 we ob ain
20
dEGW
d = 2c3
16πG Z2π
0
dφ 56
15
16µ2R4ω6G2
2c8sin2(2φ) + 2
3
64µ2R4ω6G2
2c8cos2(2φ)
dEGW
d = 2c3
16G56
15
16µ2R4ω6G2
2c8+8
3
16µ2R4ω6G2
2c8=32
5
Gµ2R4ω6
c5.(3.25)
The only sou ce o ene gy loss we conside is GW emission (EGW ); o he sou ces o ene gy
loss a e neglec ed. Thus, he ene gy loss o he o bi al decay (Eo) app oxima ely sa is ies
dEGW
d +dEo
d = 0.(3.26)
Wi h his in mind, we can compu e he equency change in ime by di e en ia ing Eq.
3.22 wi h espec o ωand also using Eq. 3.25.
˙ω=dEo/d
(dEo/dω)=−dEGW /d
(dEo/dω)=96
5
3
√G5M5ω11
c5,(3.27)
whe e M=µ3/5M2/5
I we in eg a e his di e en ial equa ion om he cu en ime and equency { , ω} o he
( o mal) alues a he ime co coalescence { c,∞} we a e le wi h
τ=−5c5
256(GM)5/3ω8/3ω(τ) = 1
8
8
s125 c3
GM5
τ−3/8,(3.28)
whe e τ= − c.
We can plo he angula equency agains τ o illus a e how i changes o e ime.
Figu e 3: Time e olu ion o he o bi al angula equency in he las 0.2 seconds be o e
coalescence o wo bina y sys ems. The blue line sys em has masses o m1=m2= 4M⊙
while he o ange m1=m2= 16M⊙
F equency magni udes di e by a ac o o 2.78 o hese sys ems, bu quali a i ely, bo h
igu es’ equency app oaches a di e gence in =0, as expec ed by he o mula. In eali y,
his di e gence s ops when bo h objec s s a me ging.
Finally, phase as a unc ion o ime can be ob ained by in eg a ing Eq. 3.28.
21
φ(τ) = −c3τ
5GM5/8
+φc,(3.29)
whe e φcis he in eg a ion cons an .
Subs i u ing Eq 3.29 in 3.24 esul s in
h+=−GM
2 c2
4
−5GM
c3τcos "2c3τ
5GM5/8
+ 2φc#(1 + cos2ι)
h×=−GM
c2
4
−5GM
c3τsin "2c3τ
5GM5/8
+ 2φc#cos(ι)
.(3.30)
I we plo hem, we ge :
Figu e 4: Time e olu ion o he pola isa ion ampli ude h+in he las 0.2 seconds be o e
coalescence o wo di e en bina y sys ems (bo h φc= c= 0 and ι= 0). In he le panel
a bina y sys em 40Mpc dis ance away wi h masses m1=m2= 4M⊙(plo ed un il =-
0.05s). Righ panel a bina y sys em 40Mpc dis ance away wi h masses m1=m2= 16M⊙
(plo ed un il =-0.1s).
The e is a signi ican di e ence in ampli ude be ween he le panel (a) and igh panel
(b) sys ems, as he ampli ude o he second sys em is 5.6 imes la ge . The e is a way o
ob aining he a io o ampli udes analy ically. Using ei he 3.19, 3.24 o 3.30, bu omi ing
ime dependency and also he o bi al plane inclina ion pa ame e (ι) he exp ession o
he ampli ude eads
A+i=−GM
2 c2
4
−5GM
c3τ.
The a io be ween hem is,
A+
A′
+
= ′
m1m2
m′
1m′
23/4m′
1+m′
2
m1+m21/4
−→ A+a
A+b≈0.177 o A+b
A+a≈5.657 .(3.31)
The same de i a ion can be done o he a io o equencies using he o mula 3.28.
′=m′
1m′
2
m1m23/8m1+m2
m′
1+m′
21/8
−→ a
b≈2.38.(3.32)
22
The e o e, he i s bina y (a) would ha e a equency ha is 2.38 imes as e han ha o
he second bina y (b). On op o ha , no ice ha he Eq. 3.32 is no dis ance-dependen ,
and hence, i he black hole bina y we e 5.657 imes a he away ( =226.28Mpc), he
equency a io would no ha e changed, ye he ampli udes o bo h sys ems o ha case
would be he same.
F equency con ains ele an in o ma ion abou he mass o he bina y sys em, which is
essen ial o dis inguishing be ween di e en sys ems. Ampli ude, howe e , is no only
mass dependen bu also depends on ex insic a iables such as dis ance ( ) o he o bi al
inclina ion (ι).
3.3 G a i a ional Wa es om Pe u bed Black Holes
In he e olu ion o a bina y sys em, we ha e h ee main phases. The inspi al pa , which
is he phase we ha e s udied in he p e ious sec ion; he me ge , when bo h BH collide;
and he RD, when he me ge esul s in a single pe u bed BH, which emi s GWs.
In his subsec ion we show how he emnan pe u bed BH emi s GWs. To do so, we add
a pe u ba ion o he BH me ic (as illus a ed in Fig. 5) and, subsequen ly, we ge he
equa ions o mo ion o hese pe u ba ions. We gene alize he p ocedu e conduc ed in 3.1
based on he lec u es om E ic Poisson (Re . [26]).
Figu e 5: Illus a ion o he i s o de pe u ba i e app oach in a Black Hole
To begin wi h, we use a one-pa ame e (λ) amily o me ics gαβ(λ, x). Nex , we Taylo
expand hem unde he small λpa ame e . E en hough, o ou pu poses, we only use
he i s -o de pe u ba ion, i is insigh ul o see how highe -o de pe u ba ions could
be added o he me ic.
gαβ(λ, x) = g(0)
αβ +p1
αβ(x)λ+1
2p2
αβ(x)λ2+O(λ3)
pn
αβ(x) = ∂ngαβ(λ, x)
∂λnλ=0
.(3.33)
In he con ex o his expansion, no e ha x e e s o all o he space ime coo dina es xν.
In he ollowing sec ions, we wo k wi h he Schwa zschild me ic o he o de 0 me ic;
his is,
23
g(0)
µν =
−1−2GM
c20 0 0
01
(1−2GM
c2)0 0
0 0 20
0 0 0 2sin2θ
.(3.34)
Ne e heless, no e ha his me ic does no include o a ing BH, as we a e no conside ing
he Ke me ic, only he Schwa zschild me ic. Al hough spinning BHs a e undamen al
in ou hesis, pe o ming a simila analysis o he one de eloped in he ollowing chap-
e s is no possible. Ke me ic equi es ano he kind o echnique, as he one used by
Teukolsky, who ob ained he decoupled and sepa able Ke -me ic equa ions [27], also
known as he Teukolsky equa ions (see Appendix D). Mo eo e , in al e na i e heo ies,
he geome y o o a ing BHs is only known pe u ba i ely o nume ically, and he e is,
in gene al, no analog o he Teukolsky equa ion [20].
As a esul , he ollowing chap e s p o ide a pa ial desc ip ion o GW-emission phenom-
ena. They mo i a e he phenomenon on s a ic BH, a simila enough sys em o ha e a
b oad idea o how his phenomenon should look, as he gene al ea u es emain he same.
3.3.1 Sphe ical Ha monics Decomposi ion
The me ic in oduced in he p e ious Eq. 3.33 can be exp essed di e en ly due o he
gauge eedom. This is discussed in Appendix A. In pa icula , we use he so-called
Regge-Wheele o Ze illi’s gauge ([28], [29]) named a e Tullio Regge, John A chibald
Wheele , and F ank J. Ze illi. Howe e , be o e explo ing he gauge ixing, we should i s
decompose he pe u ba ion in sphe ical ha monics o ake ad an age o he sphe ical
symme y o he backg ound. Likewise, we also ake ad an age o he sphe ical coo di-
na es (xν= (T, , θ, ϕ)).
I we cons ain ou space o he su ace o T=cons an and =cons an (i.e. S2), o in
o he wo ds, i we only do o a ions o he e e ence ame a ound he o igin, he en
componen s o he pe u bing enso (hνµ) ans o m like 3 scala s (h00, h01, h11), 2 ec-
o s (h02, h03;h12, h13), and a second-o de enso (h22, h23, h32, h33). Howe e , unde he
app op ia e gauge ans o ma ions (see Appendix A.2), he 10 a iables o he pe u ba-
ions can be educed o 6.
A scala unc ion decomposes in sphe ical ha monics as
h(θ, ϕ) =
∞
X
l=0
m=l
X
m=−l
hlmYlm(θ, ϕ).(3.35)
We can pe o m a simila analysis o ec o s and enso s. The o malism o ec o and
enso ha monics is in oduced in Appendix A.2 (see [28]).
I u ns ou ha sec o s wi h di e en pa i y decouple. Pa i y ans o ma ion is he e-
lec ion h ough he o igin, which in ma hema ical lingo is: −→ − , which in sphe ical
coo dina es means θ−→ θ+πand ϕ−→ π−ϕ, which in u n implies ∂
∂θ −→ ∂
∂θ and
24
which o a ce ain di ec ion in ead as (see [37])
hRD
lm ( ) =
N−1
X
n=0
Almne−iωlmn( − lmn
ma ch),(4.3)
whe e n is he QNM o e one numbe , and N is he numbe o o e ones included in he
heo e ical model. Addi ionally, ωlmn =ωR
lmn −i/τlmn. Finally, lmn
ma ch is equi alen o he
ϕlm. We only uses his no a ion in he con ex o his o mula o cla i y’s sake.
Ne e heless, i is impo an o highligh ha we do no conside o e ones (n) om now
on. Ou model is a simpli ied e sion o Eq. 4.3 neglec ing he o e ones n > 0.
hRD
lm ( ) = Alme−iωlm( − lm
ma ch),(4.4)
whe e Alm =Alm0and ωlm =ωlm0.
This is due o he ac ha da a o he modi ied heo ies we conside in Sec. 5 is only
a ailable o he i s o e one. Al hough i seems like a clumsy simpli ica ion, i does
no make a big di e ence since ou model is s ill signi ican ly accu a e compa ed wi h
NR da a. Theo e ically, he highe he o e ones, he sho e he decay ime, and con-
sequen ly, hey ade much as e han n=0, which jus i ies why we can neglec highe
o e ones.
4.2 Nume ical Rela i i y da a
We ha e p e iously no ed he impo ance o NR da a in checking how accu a e ou RD
model is in compa ison. Recall ha his da a is one o he mos accu a e me hods a ail-
able and, hus, he bes way o check o he sha pness o ou me hod. I also cons i u es
a me hod o ex ac he ini ial condi ions o he RD, such as he ampli ude ac o s o
phases esul ing om he me ge . Consequen ly, in his subsec ion, we in oduce and
analyze his da a.
NR is a GW signal modeling me hod ha nume ically sol es Eins ein ield equa ions. In
pa icula , we use he da a om he SXS collabo a ion, which uses The Spec al Eins ein
Code (SpEC) o sol ing pa ial di e en ial equa ions [39]. Un o una ely, he e a e some
pa ame e s ela ed o he bina y sys em o BH ha his me hod inds challenging o cope
wi h, such as big mass a ios (q > 20) o high spins (|χ| ≥ 0.9).
Speci ically o ou analysis we use he simula ion SXS:BBH:1580, which we downloaded
om Re . [40]. This simula ion has an in e es ing pa ame e χ≈0.2 (χ=J/M2), which
is no pa icula ly ele an a his poin , bu i is o he analysis we conduc la e in
he disse a ion. All in all, he ime-se ies wa e o m on his simula ion o an a bi a y
di ec ion o he de ec o is θd= 0.1 ad, and ϕd= 0.2 ad is:
31
Figu e 7: GW pola isa ion ampli udes measu ed om a de ec o a a ela i e posi ion o
θ= 0.1 and ϕ= 0.2, o he SXS:BBH:1580 simula ion.
As we said, we gene ally ocus on he no mal modes; in pa icula , we wan he dominan
mode, which, om he di e en ial equa ion, we expec ed o be he (2,2) mode. Le us
hen s udy all o hem and ensu e he (2,2) mode is dominan .
Figu e 8: QNM’s ampli udes o he simula ion SXS:BBH:1580.
I is appa en ha he p e alen mode is he l=m=2 mode, as he nex g ea es mode,
l=2 m=1, is abou 10 imes smalle in magni ude.
We can nume ically ex ac s a egic in o ma ion, such as complex equencies o each
QNM mode. We i s ocus on he s udy o he (2,2) mode, which ime-se ies ampli ude
is illus a ed in Fig. 9.
32
Figu e 9: The h+pola isa ion ampli ude o he (2,2) mode o he simula ion
SXS:BBH:1580.
F equency alues can be ob ained, on he one hand, by di e en ia ing he phase () o he
wa e o m o ge he equency (ω); on he o he hand, doing a linea eg ession o he
loga i hm o he wa e o m o ge he decay a e o i (τ).
Figu e 10: Res o ing complex equency alues o he (2,2) mode wa e o m om
SXS:BBH:1580 Nume ical Rela i i y simula ion. Le panel h22 wa e o m and equency
(ob ained by nume ically di e en ia ing). The ed ho izon al dashed line indica es he
a e age equency o he RD. Righ panel log( |h22|/M) wa e o m (blue) and a linea
app oxima ion o i ( ed). We only conside da a on he igh side o he ed e ical
dashed line, whe e he linea egime s a s.
Thus, ω22 ≈0.4113 −0.0896i, and hen τ22 =−1/Im(ω22)≈11.16.
We can do he same analysis o di e en QNMs as illus a ed in Figs. 11 and 12:
33
Figu e 11: Res o ing wR
lm equency alues o some (l,m) QNM wa e o ms using
SXS:BBH:1580 Nume ical Rela i i y simula ion hlm wa e o ms and equencies (ob ained
by nume ically di e en ia ing). The ed ho izon al dashed lines indica e he a e age e-
quencies o he RD.
Figu e 12: Res o ing wI
lm pa o he equency alues o some (l,m) QNM wa e o ms
using SXS:BBH:1580 Nume ical Rela i i y simula ion log( |hlm|/M) wa e o ms (blue)
and linea app oxima ions o hem ( ed). We only ega d da a on he igh -hand side o
he ed e ical dashed line.
These wo igu es show mo e signi ican nume ical e o s on he da a han hose om he
34
(2,2) mode. Addi ionally, no e how e en in highe -o de modes, he ed dashed e ical
line s ill poin s b oadly o he beginning o he linea egime.
4.3 Compa ing Ringdown Model wi h NR Da a
To see he accu acy o ou model we now compa e he RD model buil wi h he heo y-
based p edic ion wi h NR. A his poin , i is essen ial o ecall ha hese modes a e
unc ions o he emnan mass (M) and angula momen um (J). We use he spin pa am-
e e χins ead o J, de ined as χ≡J/M2.
Fo he case o his simula ion, he inal spin is χ≈0.2077. By pe o ming a linea
in e pola ion o he da a om he BH pe u ba ion calcula ions ex ac ed om Re . [32],
we ge ha o he (2,2) mode, he heo e ical equency is ω22 ≈0.4113 −0.0896i, and
we can do he same o o he modes oo.
We summa ize he heo y-based p edic ions o di e en QNM modes and he equencies
ex ac ed in Figs. 11 and 12 in he ollowing able.
(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 equencies ob ained o (l,m) modes, bo h om he BH pe u ba ion
calcula ions (labeled as ωGR) and he ones ob ained om he NR da a (labeled as ωNR
albei i is based on GR oo).
Le us compa e he equencies ob ained in Figs. 11 and 12 wi h he heo y-based p e-
dic ions. To do so, we compu e he ela i e di e ence. This is, |ωGR −ωNR|/|ωGR|.
Figu e 13: Rela i e di e ence in pe cen age o heo y-based p edic ion equencies com-
pa ed o he ones in e ed om NR da a.
35
As we can see in he igu e, he heo e ical equencies a e close o he ones we ob ained
om NR da a. The g ea es di e ences a e in highe modes (4,4), (5,4) and (3,2), which
each 1 % o ela i e di e ence. The ela i e di e ence ge s bigge in highe modes be-
cause o hei smalle ampli udes ela i e o he (2,2) mode, which in u n esul s in mo e
nume ical e o s as we no iced in Figs. 11 and 12. Howe e , he mos signi ican mode,
he (2,2,0), has 0.2 % o ela i e e o .
We can also ep oduce bo h wa e o ms a once,
Figu e 14: Phase aligned RD wa e o ms. Blue co esponds o Fig 9 (NR da a), while ed
co esponds o he equencies ob ained heo e ically.
The ed wa e equa ion is phase-aligned in he dashed e ical ed line ( /M=19.6); like-
wise, we se he ampli ude o be he same a ha poin . We s ess how accu a e he
wa e o m is, albei only conside ing he i s o e one and neglec ing any highe han
n > 0.
We ha e conduc ed he same analysis o highe modes and o he simula ions, such as he
SXS:BBH:0315 ( he mos simila simula ion o he i s LIGO de ec ion), wi h simila
esul s.
Figu e 15: Phase aligned RD wa e o ms. Le panel: 3,3 mode o he SXS:BBH:1580
simula ion. Righ panel: 22 mode o he SXS:BBH:315 simula ion.
Thus, we ha e p o en ha ou model wi h only one o e one is e ec i e o he GW
signals, and hence, we can now go beyond he GR egime.
36
5 Modi ied g a i y
Be o e we include a ia ions in o ou RD model, we will b ie ly mo i a e hese al e na-
i e heo ies and p esen a c ucial heo em o unde s and be e and classi y al e na i e
heo ies.
Eins ein’s heo y aces wo c ucial p oblems a he ime. Fi s , he e is no quan um
desc ip ion o g a i y, and second, he cosmic accele a ion cons an is inexplicably small
[41]. Fo hese easons, GR is hough o be incomple e, o i is belie ed o equi e
modi ica ions. One o he mos essen ial heo ems o modi ied g a i y is Lo elock’s
heo em.
5.1 Lo elock’s heo em
Acco ding o Lo elock’s heo em [42] in a 4-D space ime, a ank (2,0) enso whose com-
ponen s a e unc ions o he me ic enso gνµ and i s i s and second de i a i es (linea
in he second) and hese a e also symme ic and di e gence- ee, hen he only possible
op ion is:
Aνµ =aGνµ +bgνµ (5.1)
These a e Eins ein ield equa ions (Eqs. 2.30), so any modi ied heo y o g a i y is o ced
o:
•Adding mo e han second-o de de i a i es o he me ic
•Using o he ields a he han me ic enso
•Using mo e o ewe han 4 space dimensions
•Conside ing a non-local heo y
The i s op ion (Adding mo e han second-o de de i a i es) migh be he mos plausi-
ble. Howe e , highe de i a i es gene ally lead o p oblema ic ins abili ies, acco ding o
Os og adsky’s heo em [43].
Ano he op ion is adding highe -o de cu a u e e ms in he Eins ein-Hilbe ac ion. I
u ns ou ha in s anda d quan um ield heo y, a signi ican obs acle on he ou e o
quan um g a i y is eno malizing he Eins ein-Hilbe ( ecall 2.30) [44]. I was shown ha
quad a ic cu a u e e ms ix his p oblem. On op o ha , low-ene gy e ec i e s ing
heo y sugges s ha he Eisn ein-Hilbe ac ion could be hough o as he i s e m in
an expansion con aining all possible cu a u e in a ian s [44]. The e o e, we mainly s ick
o highe -o de cu a u e heo ies.
5.2 Al e na i e heo ies
A e he gene al analysis o modi ied g a i y, which has gi en us an o e all pic u e, i is
ime o speci y he heo ies we use. Thus, his subsec ion b ie ly in oduces he modi ied
g a i y heo ies o which QNMs ha e been compu ed. These heo ies ha e he same c u-
cial ea u e: They all a ec he GR’s BH QNM spec a. On op o ha , hey all include
37
highe -cu a u e co ec ions o GR. The heo ies we conside a e he same as he ones
om Re . [21]. We include al e na i e heo ies in oducing a ia ions o he Eins ein-
Hilbe ac ion 2.31 and wi h small-coupling app oxima ions, o be mo e exac we impose
he undamen al leng hscale, he coupling ac o , o be l h ≤GM/c2.
Al e na i e heo ies could b eak he isospec al p ope y shown in Sec. 3.3.5. Acco dingly,
he equencies o he (2,2,0) mode will a y depending on whe he he pe u ba ion is
e en o odd. How his a ec s he GW signal will no be s udied in his disse a ion.
Ne e heless, we will ollow he p ac ical app oach o Re . [21], which me ely chooses he
leas damped pa i y. As long as o he pa i y QNM a e no exci ed mo e, he leas damped
pa i y will p e ail. Thus, simply choosing his one equency is a sensible decision.
5.2.1 Eins ein-dila on-Gauss-Bonne
Eins ein-dila on-Gauss-Bonne (EdGB) heo y is desc ibed by he ac ion:
SEdGB =Zd4x√−g1
16πR−1
2(∂ϕ)2+1
4l2
EdGB (ϕ)G,(5.2)
whe e g≡de (gνµ), R is he Ricci scala , ϕis a scala ield, wi h kine ic e m (∂ϕ)2=
gνµ∂νϕ∂µϕ, which couples o he Gauss-Bonne in a ian G=RνµρσRνµρσ −4RνµRνµ +R2,
by he coupling unc ion (ϕ) and he coupling ac o lEdGB (which has leng h uni s).
EdGB g a i y has BH wi h he so-called ”seconda y hai ” scala depending on he scala
unc ion (ϕ). The ’seconda y’ e m e e s o i s dependency on he black hole’s mass,
spin, and o he heo y pa ame e s, such as lEdGB. Thus, he cha ge is no an independen
quan i y. Acco ding o Re . [21], we can associa e a monopole scala cha ge o his scala
ield, jus like he elec ic cha ge o he elec ic po en ial. Simila ly, his cha ge decays
in e sely p opo ional o he dis ance om he black hole (1/ ). Namely, as he scala
cha ge is linked o he cu a u e scala , he scala cha ge is in e sely p opo ional o he
BH mass.
This scala ield has obse able consequences. On he one hand, BH bina ies a e a sou ce
o scala -dipole adia ion; hus, he o bi should decay as e . Re . [45] cons ained he
coupling ac o o lEdGB ≤7.1km. On he o he hand, he coupling wi h a scala ield a -
ec s he e en pe u ba ion, which in u n b eaks wi h he isospec ali y o he Swa zchild
BH.
Using pe u ba ion heo y, he QNMs ha e been compu ed up o i s o de in o a ion
in Re . [22].
5.2.2 Dynamical Che n-Simons g a i y
Dynamical Che n-Simons g a i y (dCS) is desc ibed by he ollowing ac ion:
SdCS =Zd4x√−g1
16πR−1
2(∂Θ)2+ 4l2
dCSΘ∗RR,(5.3)
whe e Θ is a pseudoscala ield which couples o he Pon yagin densi y ∗RR =∗RνµρσRνµρσ,
whe e ∗Rνµρσ =ϵνµγδRγδρσ/2, and ϵνµγδ is he Le i-Ci i a enso and inally, whe e lEdGB
38
is he coupling ac o (which has leng h uni s).
As be o e, hese BHs ha e a scala ield. This ield decays −2, o which we can associa e
a scala dipole cha ge (as in he elec omagne ic case). Howe e , mos impo an ly, he
spec a QNM has some de ia ions. In his case, odd pa i y pe u ba ions couple wi h
pe u ba ions o he scala ields again b eaking isospec ali y. Re . [23] and [24] s udied
he QNM spec a o slowly- o a ing BHs in dCS g a i y.
5.2.3 E ec i e- ield- heo y o g a i y
The ollowing ac ion desc ibes he E ec i e Field Theo y (EFT) o GR:
SEF T =Zd4x√−g R+X
n≥2
l2n−2
EF T L(2n)!,(5.4)
whe e L(2n)a e highe o de cu a u e co ec ions and lEFT is he coupling ac o (which
has leng h uni s). We ea highe -o de cu a u e heo ies sepa a ely, and we only con-
side L(6) and L(8).
L(6) =λeRµνρσRρσγδRγδµν +λoRµνρσRρσγδ ˜
Rγδµν
L(8) =ϵ1C2+ϵ2˜
C2+ϵ3C˜
C, (5.5)
whe e
C=RγδρσRγδρσ
˜
C=RνµγδϵνµρσRρσγδ.(5.6)
Finally, λo,e and ϵi, a e ee pa ame e s, al hough we se bo h λo=λe=1 and ϵ1= 1,
ϵ2=ϵ3= 0 which lea e he coupling ac o as he only ee pa ame e .
In he case o EFT, he spec a o QNM o o a ing BHs we e s udied o he i s spin
o de co ec ions in Re . [25].
5.3 De ia ions on he Spec a o he Quasino mal Modes
As p e iously discussed, al e na i e heo ies c ea e modi ica ions in he QNM spec a.
This subsec ion de ails he amewo k o accoun o hese de ia ions in modi ied heo ies.
We base he model on he Pa Spec amewo k (Re [20]). Acco ding o i , we can exp ess
he de ia ion by pe o ming he ollowing spin expansion:
ωlmn =1
M
Nmax
X
j=0
χj
ω(j)
lmn 1 + γδω(j)
lmn(5.7)
τlmn =M
Nmax
X
j=0
χj
τ(j)
lmn 1 + γδτ(j)
lmn,(5.8)
whe e acco ding o Re . [21] in geome ic uni s
γ=ℓ h
Ms
p
.(5.9)
39
The l h is he ”coupling ac o ” ha depends on he heo y, which is a c ucial ac o as
i is he one esponsible o coupling he scala /pseudoscala ields o he highe o de
cu a u e enso o he o iginal GR o mula ion, as we ha e seen in he Eqs. 5.2, 5.3 and
5.4. In o he wo ds, his is he pa ame e we would like o cons ain. The close his
pa ame e is o ze o, he less in luence his heo y has, and om a philosophical poin o
iew, his ansla es o he alidi y o he heo y (needless o say, his does no p o e he
heo y w ong, bu we could a gue i s alidi y/ eliabili y).
Re [21] compu es he Pa Spec amewo k coe icien s o he (l,m,n)=(2,2,0) mode o he
leas damped pa i y pe u ba ions based on he p e iously men ioned heo y-by- heo y
equencies compu ed in Re s. [22], [23], [24] and [25]. Acco ding o Re . [21], axial pa i y
pe u ba ions a e he leas damped o all 4 heo ies we a e s udying. We summa ize
hese coe icien s in he ollowing able (Tab. 2).
GR EdGB(p=4) dCS(p=4) cubic EFT(p=4) qua ic 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: De ia ion pa ame e s o he complex equencies o he 22 quasino mal mode
o di e en al e na i e heo ies (See Re . [21]).
Wi h his da a, we can expand al e na i e heo ies up o i s o de in spin. The e o e,
we mus conside small spins, |χ |<< 1. Fo he 2,2 mode in GR, his ansla es o spins
be ween χ∈[0,0.2] (Re [20]), he egion on which ela i e e o s in equencies a e no
g ea e han 1%.
M ωlmn =γhδω(0)
lmnω(0)
lmn +χ δω(1)
lmnω(1)
lmni+
Nmax
X
j=0
χj
ω(j)
lmn (5.10)
τlmn
M
=γhδτ(0)
lmnτ(0)
lmn +χ δτ(1)
lmnτ(1)
lmni+
Nmax
X
j=0
χj
τ(j)
lmn.(5.11)
We, howe e , conside he ollowing expansion:
M ωlmn =M ω(GR)
lmn +γhδω(0)
lmnω(0)
lmn +χ δω(1)
lmnω(1)
lmni+
Nmax
X
j=2
χj
δω(j)
lmnω(j)
lmn (5.12)
τlmn
M
=τ(GR)
lmn
M
+γhδτ(0)
lmnτ(0)
lmn +χ δτ(1)
lmnτ(1)
lmni+
Nmax
X
j=2
χj
δτ(j)
lmnτ(j)
lmn.(5.13)
The pu pose o his a ia ion is o eco e he equencies om GR when we impose l= 0,
e en o χ > 0.2. This howe e does no in luence he es o he heo ies as di e ence
be ween he expansion we conside and he pu e Pa Spac amewo k is:
40
7 Resul s and Discussion
This disse a ion is based on he esul s om Re [21], which compu es he Pa Spec ame-
wo k coe icien s o he (l,m,n)=(2,2,0) mode o he leas damped pa i y pe u ba ions
combining he heo y-by- heo y equencies ha we e calcula ed in Re s. [22], [23], [24]
and [25]. In Re [21] hey essen ially ied o cons ain he coupling ac o using he wo
loudes wo ingdown signals obse ed (GW150914 and GW200129). Pe o ming Bayesian
in e ence hey place uppe bounds on he coupling ac o s, namely, in dCS ldCD ≤38.7km
(%90 o c edible le el) and lcEFT ≤38.2km (%90 o c edible le el) and lqEFT ≤51.3km
(%90 o c edible le el). In his hesis, howe e , we del e in o he de ia ions o he RD
wa e o ms in pa icula , and we p oduce some no el esul s. Mo e p ecisely, we ha e
quan i ied he bias on he pa ame e s in e ed by GR unde he egime o al e na i e he-
o ies o he bias on he in e ence o pa ame e s om al e na i e heo ies. In pa icula ,
ou esul s a e:
Fi s , Table 3, which shows he misin e p e ed da a we would in e by assuming GR in a
uni e se go e ned by al e na i e heo ies (all al e na i e heo ies ha e assumed ha he
emanen BH’s M = 1 and spin χ = 0.2). Second, Table 4 which con ains he in e ed
mass and spin o GR, EdGB, dCD, cEFT, qEFT heo ies om he SXS:BBH:1580 NR
wa e o m (acco ding o he ca alogue i co esponds o he pa ame e s M = 0.98 and
χ = 0.2077).
No e ha hese ables show he pa ame e s ha minimize he misma ch bu a e no nec-
essa ily he mos p obable. Ne e heless, we do no expec a big di e ence be ween he
misma ch plo om Fig. 21 and he p obabili y dis ibu ion plo , as he mos p obable
se o pa ame e s should also be close o he ones ha ma ch bes .
The esul s show ha he mass o spin in e ed using GR ends o be highe han hose
in e ed using al e na i e heo ies. The eco e ed spin o mass is mo e signi ican han
he o iginal al e na i e heo y in he i s able ( able 3). In he second able ( able
4), al e na i e heo ies p edic lowe spin o mass han hose p edic ed by GR. This
dispa i y is because he ene gy o scala o pseudoscala ields in al e na i e heo ies is
in e p e ed as addi ional mass o spin in GR. In o he wo ds, hese scala o pseudoscala
ields in e e e wi h space ime cu a u e, bu since in GR he e a e no coupled ields; he
BH’s mass o spin mus comple ely compensa e o he modi ica ion on space ime ha
scala /pseudoscala ields in oduce.
Rega ding Table 4, GR has he bes misma ch ( he smalles one). Ne e heless, qEFT is
close o ha ing a be e one, and all he o he al e na i e heo ies a e also close o being
as good. We should ecall Fig. 16 o make sense o his. In his igu e, we can b oadly
see which complex equency co esponds o a ce ain spin o GR. Wi h his in mind, we
can now y o ma ch he same equencies on al e na i e heo ies by sweeping h ough
he spin and e-scaling wi h mass. As hese heo ies ha e o ma ch 2 pa ame e s (ωRand
τ) and o doing so, hey can sweep h ough 2 pa ame e s (Mand χ), his esembles a
sys em o equa ions wi h wo unknowns and wo a iables. Hence, his sys em is lexible
enough o ma ch he ” igh ” equencies and ob ain he bes misma ch ha ou model
has o o e (as all heo ies ha e a simila p ecision, i is appa en ha e e y heo y has
eached i s limi ). In his sense, we could say ha GR had he leas misma ch by ”pu e
47
chance,” as e e y o he heo y has he same abili y o ma ch he ” igh ” equency.
We could also y sweeping h ough he coupling ac o (l). None heless, adding a new
pa ame e like ha would no ”a p io y” imp o e he misma ch, which can be in e p e ed
wi h he same in ui ion as in he abo e pa ag aph. I we add a new pa ame e o he sys-
em o equa ions, we would add a hi d a iable bu keep he wo unknowns. Acco dingly,
he sys em is lexible enough o ma ch he igh equency o each l. Thus, adding l o ou
analysis would no imp o e he misma ch bu ins ead lead o a degene acy o misma ch
h ough l. As a esul , he analysis is only wo hwhile i we can cons ain some o he ee
pa ame e s o each heo y. I we could cons ain any o hem, we could pe o m he wo-
unknown wo- a iable sys em ”game” desc ibed abo e o ma ch he complex equencies
bu now using ”l” ins ead o ei he he spin o he mass, and subsequen ly, we could look
o an uppe o lowe bound o l.
8 Conclusions and Ou look
In his hesis, s a ing om he Theo y o Gene al Rela i i y, we ha e in oduced g a i a-
ional wa e as a linea pe u ba ion o he la space. We ha e also explo ed he sou ces o
hese wa es, emphasizing bina y sys ems inspi als as exempli ied. Fu he mo e, we ha e
s udied he pe u ba ion o a BH (Schwa zschild) wi hin Gene al Rela i i y, e ealing a
elaxa ion p ocess known as Ringdown - a s age whe e pe u ba ions e en ually dissipa e.
Secondly, we ha e analyzed da a om Nume ical Rela i i y, no ably om he SXS collab-
o a ion (Sec. 4.2), some o he mos eliable sou ces conce ning bina y me ge p ocesses.
We ha e buil an e ec i e model wi h one o e one and con as ed NR da a o p o e ou
model’s accu acy (Sec. 4.1). Nex , we in oduced modi ied heo ies (Sec. 5) and consid-
e ed hei impac on he complex equencies o QNMs o include hem in ou RD model.
Addi ionally, we ha e compu ed he ime se ies o he wa e o ms and plo ed hem wi h
di e en coupling ac o s o see he e ec .
Thi dly, we ha e p oduced RD wa e o ms based on ou al e na i e- heo y-based model
and ied o in e he key pa ame e s χand M om hem using he GR-based model.
Ou s udy has demons a ed signi ican di e ences in in e ed pa ame e s depending on
he unde lying g a i a ional heo y, emphasizing he necessi y o conside ing al e na i e
models in GW da a analysis o a oid misin e p e a ion o he expe imen al da a. Fu -
he mo e, we ha e p oposed he unde lying eason o hese disc epancies: As no coupled
ields exis in GR, al e na i e heo ies need less mass o spin on he BH o c ea e he
same cu a u e e ec s.
Finally, we ha e used some NR da a as ” ex ” expe imen al da a. We ha e in e ed he
da a wi h ou GR-based model and ensu ed he model was good enough o eco e he
o iginal da a used by he NR simula ion. Al e na i ely, we ha e in e ed he da a wi h
modi ied heo ies and concluded ha hey would i mly de ia e. Wi h his in mind, we
ha e discussed some ways o cons ain he coupling pa ame e s, ne e heless we ha e no
ob ained any such cons ains, as opposed o Re . [21], as we ha e seen on he esul s.
In a nu shell, we ha e comp ehended some g a i a ional wa e phenomena, including hei
sou ces and cha ac e is ics, and ocused pa icula ly on BH pe u ba ions and hei GW
48
adia ion in he RD phase. On op o ha , we ha e unde s ood how al e na i e heo ies
a ec he da a analysis and physically in e p e he eason o hese disc epancies.
Ul ima ely, and as an ou look, we in oduce u he esea ch pa hs based on his wo k,
ega ding he cons ain s o he coupling ac o :
•By explo ing o he modes, such as he (2,1) mode, and in e ing pa ame e s h ough
he same analysis, we can po en ially unco e inne inconsis encies o al e na i e
heo ies wi h he expe imen al o NR da a. This could lead o cons ain s on he
coupling ac o s.
•We can include o e ones, such as he i s o e one, and pe o m a simila analysis
o he one abo e, and i we unco e inconsis encies, we can ind cons ain s on he
coupling ac o .
•We can in e pa ame e s om he inspi al signal and employ i o acqui e he inal
mass and spin o he emnan BH. Unde hese cons ain s, we can bound las
explained in Sec. 7. Howe e , acqui ing he inal mass and spin om he ini ial
condi ions is qui e challenging o GR (we can do i using NR simula ions) and
e en mo e so o al e na i e heo ies. None heless, acco ding o he second law o
The modynamics S1+S2< S , since en opy (S) is p opo ional o he squa e o he
mass S∝M2, we can cons ain he inal mass, which in u n can lead o cons ain s
on he coupling ac o .
Las ly, an al e na i e pa h could in ol e a nume ical app oach o compu ing complex e-
quencies o GR (Teukolsky Equa ion) o al e na i e heo ies. Fo a heo e ical app oach,
a h illing pa h could also be o ob ain he GR equencies wi h analy ical solu ions o ge
o he Teukolsky equa ion. Addi ionally, i could be in e es ing o ob ain a pe u ba i e
geome y o o a ing BH in al e na i e heo ies.
49
Appendix A: Gauge F eedom
Based on a one-pa ame e (λ) amily o me ics gαβ(λ, x). We will Taylo expand i as:
gαβ(λ, x) = g(0)
αβ +p1
αβ(x)λ+1
2p2
αβ(x)λ2+···
pn
αβ(x) = ∂ngαβ(λ, x)
∂λnλ=0
.(A.1)
No e ha in his case, x e e s o all coo dina es xν.
All me ics can be exp essed in di e en coo dina e sys ems. By making in ini esimal
o a ions o boos s, o example. A gene al coo dina e ans o ma ion o his ype can be
done as ollows:
xα→x′α(λ) = xα−ξα(x)λ−1
2ξα
2(x)λ2+···.(A.2)
Using he enso s ans o ma ion ules unde he p e ious ans o ma ion, we can ge o:
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)
No e ha we ha e lowe ed he index using g(0) ins ead o g. This is only pa ially accu a e.
Howe e , he di e ence is o second o de since he exp ession was al eady a pe u ba ion
(linea in λ).
As a esul ,
g(0)
α′β′=g(0)
αβ p1
αβ =p1
α′β′(x) + ∂(β′ξα′).(A.4)
The e o e, as s a ed by Equa ion A.3, by selec ing a coo dina e sys em ( ha ollows Equa-
ion A.2), we ha e al e ed he exp ession o he pe u ba ion. Howe e , his does no
change he unde lying physics; he pe u ba ion emains he same bu is ep esen ed in
di e en coo dina es. This is why we call i gauge eedom.
We can use hese ans o ma ions (called gauge ixing) o ou ad an age by ca e ully
choosing coo dina e ans o ma ions o simpli y equa ions o exp essions. In o al, we can
impose 6 condi ions, one o each elemen o he enso ∂(β′ξα′). I is no a coincidence
ha he deg ees o eedom pe mi ed by Lo en z ans o ma ions (Sec ion 2) a e he
eedoms p o ided by he enso ∂(β′ξα′) om Equa ion A.3; ).
Boos and o a ions p esen he same s uc u e as equa ion A.2. We can show his using
a pe u ba i e app oach. Fo ins ance, an example o such Lo en z ans o ma ion is a
Lo en z boos .
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, he coo dina e ans o ma ion is gi en by
c ′=c −ϕx
x′=−ϕc +x
y′=y
z′=z
.(A.6)
I we eplace ϕwi h , his equa ion will esemble he usual Galilean ans o ma ion. As
expec ed in a pe u ba i e analysis o Lo en zian boos s.
To eco e he equa ion A.2, we mus iden i y ϕ o be λand ξµ o be (x, c , 0,0).
A.1 The Residual Gauge om Lo enz Gauge
Th oughou his disse a ion, we ha e used he Lo enz Gauge se e al imes. Fo example,
in Sec ion 3.1, we op ed o i (Lo enz o ha monic gauge), his is:
∂βhαβ = 0 (Lo enz gauge condi ion)
hαβ ≡p1
αβ −1
2ηαβh h =ηµνp1
µν
(A.7)
Hence, ield equa ions we e
□hαβ =−16πTαβ[g] (A.8)
Howe e , hese condi ions s ill lea e oom o u he gauge condi ions. Speci ically, as
long as ξµobeys he ollowing condi ion, he Lo enz gauge is sa is ied:
□ξµ= 0.(A.9)
In o de o show his, 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 ge ∂βhαβ = 0 and wi h his 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 esidual gauge eedom emains o be speci ied, as he o iginal equi emen
(Eq. A.7) is s ill conse ed unde all gauge ans o ma ions ha sa is y □ξα= 0.
A.2 Gauge Fixing on Black Hole Pe u ba ion Theo y
Suppose we decompose he pe u ba ion in sphe ical ha monics as p esc ip ed in Sec.
3.3.1, wi hou gauge ixing, his is wi h 10 di e en pe u ba ion unc ions, which a e
decomposed in sphe ical ha monics. We a e used o decomposing scala unc ions in o
sphe ical ha monics as
h(θ, ϕ) =
∞
X
l=0
m=l
X
m=−l
hlmYlm(θ, ϕ).(A.12)
Howe e , ec o s ( ecall he ans o ma ion ules om 2.13)- can also be decomposed in
he so-called ec o sphe ical ha monics. In sphe ical coo dina es ( , θ,ϕ) hey go as:
Ylm = Ylm(θ, ϕ)
Ψlm =∇Ylm(θ, ϕ)
Φlm = ×∇Ylm(θ, ϕ),
(A.13)
whe e ∇=∂
∂ ˆ
+1
∂
∂θ ˆ
θ+1
sin θ
∂
∂φ ˆ
φ
In S2, his is, in =c , Ylm is a scala , while Ψlm and Φlm a e ec o s, and hey can be
w i en as ollows in enso no a ion:
Ψlm
A=∂AYlm(θ, ϕ)
Φlm
B=ϵBA∂AYlm(θ, ϕ),(A.14)
whe e ϵϕϕ=ϵθθ= 0, ϵθϕ=−1/sin(θ) and ϵϕθ=sin(θ). Mo eo e , he bold no a ion
e e s o ec o s.
Finally, enso s can be decomposed as ollows:
Ψlm
AB =∇A∇BYlm(θ, ϕ)
Φlm
AB =γABYlm(θ, ϕ)
χlm
AB =1
2ϵBCΨlm
AC +ϵACΨlm
CB,
(A.15)
whe e again, ϵϕϕ=ϵθθ= 0, ϵθϕ=−1/sin2(θ) and ϵϕθ=sin2(θ). Addi ionally, γθϕ =
γϕθ = 0, γθθ = 1, γϕϕ =sin2(θ), his is, γνµ =gνµ/ 2.
Mos impo an ly hese a e he unique ec o o enso s we can cons uc in S2.
In ha case, ou wo amilies o pe u ba ion depending on he pa i y. On he one hand,
he so-called ”odd” pa i y (o Regge-Wheele o axial) pe u ba ions (as hey ans o m
as pl(− )=(−1)l+1pl( )) a e decomposed wi hou loss o gene ali y as:
52
p1 odd
µν =
0 0 −h01
sin θ
∂
∂ϕ −h0sin θ∂
∂θ
0 0 −h11
sin θ
∂
∂ϕ −h1sin θ∂
∂θ
sym sym h21
sin θ
∂2
∂θ∂ϕ −cos θ
sin2θ
∂
∂ϕ sym
sym sym 1
2h21
sin θ
∂2
∂ϕ∂ϕ + cos θ∂
∂θ −sin θ∂2
∂θ∂θ −h2sin θ∂2
∂θ∂ϕ −cos θ∂
∂ϕ
Ylm,
(A.16)
whe e h0=h0(T, ), h1=h1(T, ), h2=h2(T, ) and ”sym” e e s o he symme y
p ope y he me ic has (pνµ =pµν).
On he o he hand, he ”e en” pa i y (o Ze illi o pola ) pe u ba ions (as hey ans o m
as pl(− ) = (−1)lpl( )) can be decomposed as
p1 e en
µν =
H0H1¯
h0∂
∂θ ¯
h0∂
∂ϕ
H1 −1H2¯
h1∂
∂θ ¯
h1∂
∂ϕ
sym sym 2[K+G∂2
∂θ2] sym
sym sym 2G∂2
∂θ∂ϕ −cos θ∂
sin θ∂ϕ 2sin2θhK+G1
sin2θ
∂2
∂θ∂ϕ −cos θ
sin θ
∂
∂ϕ i
Ylm,
(A.17)
whe e =1−2GM
c2 ,¯
h0=¯
h0(T, ), ¯
h1=¯
h1(T, ) and H0=H0(T, ) , H1=H1(T, ),
H2=H2(T, ).
When we i s de i ed he g a i a ional wa e equa ion in sec ion 3, we used he Lo enz
gauge ans o ma ion o simpli y he Eq. 3.7. In his sec ion, we will pe o m a simila
bu mo e con olu ed gauge ans o ma ion o simpli y he exp essions o he pe u ba-
ions (see [28]).
As we will see in Appendix A, unde he ollowing coo dina e ans o ma ion:
xα→x′α(λ) = xα−ξα(x)λ−1
2ξα
2(x)λ2+··· (A.18)
he pe u ba ion ans o ms as
p1
αβ =p1
α′β′(xµ) + ∂(β′ξα′)(A.19)
Fo he odd pe u ba ions, we can choose ξµ o be
ξ0= 0 ξ1= 0
ξ2= ∆(T, )∂
∂θYlm(θ, ϕ)ξ3= ∆(T, )∂
sin2θ∂ϕYlm(θ, ϕ).(A.20)
The di ec compu a ion will esul in
p1
µν =
0 0 0 h0( )
0 0 0 h1( )
0 0 0 0
h0( )h1( ) 0 0
×e−iωT sin θ∂
∂θPL(cos θ).(A.21)
53
The unc ion h0and h1a e ea ed now as plane wa es, Fou ie T ans o med e sions, as
we expec hem o ha e a wa e-like beha iou , and he ea men o hem is easie .
Fo he e en pe u ba ions, we can choose ξµ o be
ξ0=M0(T, )Ylm(θ, ϕ)ξ1=M1(T, )Ylm(θ, ϕ)
ξ2=M(T, )∂
∂θYlm(θ, ϕ)ξ3=M(T, )∂
sin2θ∂ϕYlm(θ, ϕ),(A.22)
which, in u n, will esul in
p1
µν =
H0H10 0
H1 −1H20 0
0 0 2K0
0 0 0 2Ksin2(θ)
×e−iωT sin θ∂
∂θPL(cos θ).(A.23)
These a e he simpli ied pa i y pe u ba ions. Gauge ixing educes he p e ious 10 a i-
ables o 6
Appendix B: G a i a ional Wa e Pola iza ion
As we ha e seen in Sec. 3 plane wa es p opaga ing in he zdi ec ion (in Lo enz gauge)
ollow Eq. 3.13 and hus he associa ed line elemen is:
ds2=−c2d 2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2.(B.1)
The pa ame e s h+and h×will de e mine he pola iza ion o he g a i a ional wa e. To
ge some in ui ion, in his appendix, we will i s impose h× o be ze o and hen make
h+= 0 ins ead.
B.1 Plus pola isa ion
Fi s , as explained, we will impose h× o be ze o. Thus, o a ixed ime (d =0), he line
elemen is
ds2= (1 + h+)dx2+ (1 −h+)dy2.(B.2)
I we p opose he ollowing coo dina e change
(x′=p1 + h+x
y′=p1−h+y, (B.3)
hey will es o e he la me ic; his is,
ds2=dx′2+dy′2.(B.4)
We can in e p e his as ollows: The dila ion/con ac ion o he new coo dina es ma ch
wi h he space ime de o ma ion and hus, he la space ime will be eco e ed. This
dila ion/con ac ion a ies o e ime, as shown in Equa ion 3.13.
In pa icula , we can see how he space ime de o ma ion would a ec a ing o es poin s
in space ha sa is ied he equa ion x2+y2=R2:
54
x2+y2=R2=⇒ x′
p1 + h+!2
+ y′
p1−h+!2
=R2←Ellipse (B.5)
This means ha i we had a ci cle o es poin s when h+= 0, hese es poin s would
appea ellip ical a any ime when h+= 0. The alid equa ion is he ellipse as opposed
o he ci cle since he ellipse lies in he la coo dina es ( he p ime coo dina es), whe e
he equa ions o polygons co espond o he ” eal” polygons. Again, his is because ou
no ion o dis ance only holds on la spaces.
x
y
Figu e 22: Tes poin s unde he e ec h+pola iza ion g a i a ional wa e p opaga ing
along z di ec ion
B.2 C oss pola isa ion
Simila ly, i we had se h+= 0 ins ead o h×, we would ace he same phenomena bu
wi h a 45ºpola iza ion angle o a ion. To show his, we will s a wi h he line elemen
o a GW p opaga ing along z di ec ion in la space ime,
ds2=−c2d 2+ (1 + h+)dx2+ (1 −h+)dy2+ 2h×dxdy +dz2,(B.6)
whe e acco ding o equa ion 3.13 h+and h×a e
h+=a+cos(ω( −z/c))
h×=a×cos(ω( −z/c)).(B.7)
Pe o ming he ollowing ans o ma ion Lo en z ans o ma ion (a -45º o a ion in he
x-y plane:
x′
y′=cos 45 sin 45
−sin 45 cos 45x
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 elemen is simila o he old one, bu h+and h×a e now swapped.
ds2=−c2d 2+ (1 + h×)dx2+ (1 −h×)dy2+ 2h+dxdy +dz2.(B.10)
The physical in e p e a ion o h imes is jus he same as he one o h+since i is he
same space ime ans o ma ion, al hough i is shi ed 45º ela i ely.
55
x
y
Figu e 23: Tes poin s unde he e ec h×pola iza ion g a i a ional wa e
Appendix C: G a i a ional Wa e De ec ion
Cu en GW de ec o s, such as Ligo and Vi go, a e lase in e e ome e s. They ope a e
by spli ing a lase beam in o wo sepa a e beams, which a el along 2 pe pendicula long
a ms as depic ed in igu e 24. They bounce a he mi o s so ha he e ec i e leng h
o he a ms is much g ea e , and hen hey a e ecombined in he de ec o . Whene e
a GW c osses he de ec o , he leng hs o hese pa hs change due o GW’s space ime
de o ma ion. Any di e ences in he ligh pa hs esul in in e e ence pa e ns, and by
analyzing hese pa e ns, he in e e ome e can de ec minuscule changes in dis ance,
which, in u n, enables us o de ec GWs.
Lase
Beam-spli ing
mi o
Mi o s
De ec o
L
L
Figu e 24: A schema ic diag am o a lase in e e ome e
We i s need o see he pe u ba ion o see how he leng hs a e changed when GWs go
h ough he de ec o . Fo doing so, we will ecall he pe u ba ion ma ix o a z-di ec ed
GW 3.13:
hTT
ij =
h+h×0
h×−h+0
0 0 0
.(C.1)
This pe u ba ion is obse ed om a coo dina e sys em adap ed o he GW so ha i is
p opaga ing in i s z-di ec ion. Typically, he de ec o e e ence sys em will no ma ch he
GW e e ence sys em. Thus, we will gene ally need some o a ions o ela e one sys em
o ano he , as shown in igu e 25.
56
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