The Layze -I ine equa ion in
heo ies wi h non-minimal
coupling be ween ma e and
cu a u e
Cl´audio Filipe Viei a Gomes
A Thesis p esen ed o he deg ee o
Mas e in Physics
Cen o de F´ısica do Po o
Depa amen o de F´ısica e As onomia
Uni e sidade do Po o
Po ugal
July, 2014
Ju y:
P o . Miguel Sousa da Cos a
P o . O eu Be olami Ne o (ad iso )
P o . Jo ge Tiago Almeida P´a amos
Acknowledgemen s
I would like o hank:
•my mo he o he suppo o all hese yea s, and specially o his cu en
one;
•my iends o keeping eminding me o ha e some o -wo k li e;
•my ad iso , P o . O eu Be olami, o his guidance and help ul suppo du ing
my wo k, wi hou which he la e migh no ha e gone ha a ;
• o P o . Jo ge P´a amos o discussion a he ini ial phases o ou s udy o
non-minimally coupled heo ies be ween ma e and cu a u e.
iii
Resumo
Nes e abalho, ´e de i ada a equa¸c˜ao de Layze -I ine no con ex o de eo ias
al e na i as da g a i a¸c˜ao que en ol em um acoplamen o n˜ao m´ınimo en e ma ´e ia
e geome ia. Como aplica¸c˜ao, ´e analisado o caso do enxame de gal´axias Abell 586,
po se no o iamen e es e icamen e sim´e ico e li e de in e a¸c˜oes, eco endo a
alguns pe is de densidade.
Es e abalho baseia-se no abalho desen ol ido na Re . [1].
Pala as-cha e: Rela i idade Ge al, eo ias al e na i as da g a i a¸c˜ao, cos-
mologia, enxame A586.

Abs ac
In his wo k, he Layze -I ine equa ion is de i ed in he con ex o al e na i e
g a i a ional heo ies wi h non-minimal coupling be ween ma e and geome y. As
an applica ion, he case o he sphe ically symme ic clus e Abell 586 is analysed,
assuming some ma e densi y p o iles.
This wo k is based on wo k de eloped in Re . [1].
Keywo ds: Gene al Rela i i y, al e na i e heo ies o g a i y, cosmology, A586
clus e .
ii
Lis o Figu es
5.1 Func ion 2( ) o he clus e A586, in e ms o he discussed densi y
p o iles. .................................. 17
ix
Chap e 2
Non-minimal cu a u e-ma e
coupling
In GR, he ac ion unc ional is exp essed as
S=Z[κR+Lm]√−gd4x(2.1)
whe e Ris he scala cu a u e, Lmis he ma e Lag angian densi y, gis he me ic
de e minan and κ=c4/16πG,Gbeing he New on’s g a i a ional cons an .
In he so-called (R) heo ies [4–6], he scala cu a u e e m in he p e ious
ac ion is eplaced by an a bi a y unc ion o i :
S=Z1
2 (R) + Lm√−gd4x(2.2)
Mo e gene ally, one can hink in a non-minimal coupling be ween cu a u e and
ma e [7]:
S=Z1
2 1(R) + (1 + 2(R)) Lm√−gd4x, (2.3)
whe e 1(R) and 2(R) a e a bi a y unc ions o he scala cu a u e, R. One
no es ha , by se ing 1(R)=2κRand 2(R) = 0, Gene al Rela i i y is eco e ed.
Va ying he ac ion wi h espec o he me ic yields he ield equa ions [7]:
FRµ
ν−1
2δµ
ν 1−(gµσ∇σ∇ν−δµ
ν)F= (1 + 2)Tµ
ν,(2.4)
3

Chap e 2. Non-minimal cu a u e-ma e coupling 4
wi h Fi≡d i/dR(i= 1,2), F≡F1+ 2F2Lm, and Tµν is he ene gy-momen um
enso o ma e de ined as
Tµν =−2
√−g
δ(√−gLm)
δ(gµν)(2.5)
The Bianchi iden i ies o he Eins ein enso , ∇µGµ
ν= 0, and he iden i y
(2∇ν−∇ν2)Fi=Rµν∇µFi(2.6)
imply o he exp ession o Eq.(2.4):
∇µTµ
ν= (Lmδµ
ν−Tµ
ν)∇µln (1 + 2).(2.7)
This is one o he undamen al ea u es o he model (2.3) - he non-conse a ion
o he ene gy-momen um enso . This p ope y induces an ex a o ce ac ing on a
es pa icle, which is o hogonal o he luid ou - eloci y and can be exp essed o
a pe ec luid as:
µ=1
ρ+pF2
1 + 2
(Lm+p)∇νR+∇νphµν,(2.8)
whe e hµν =gµν +uµuνis he p ojec ion ope a o .
Chap e 3
Pe u bed F iedmann-Lemaˆı e-
Robe son-Walke
model
We now aim o achie e he p ime objec i e o his wo k, which is o de i e he
Layze -I ine equa ion o he non-minimal coupling model desc ibed by he ac ion
Eq. (2.3). To do so, we ollow closely he de i a ion pe o med in Re s. [17,19,23,26].
Fi s , we shall conside ha he Uni e se is well desc ibed by a pe ec luid,
whose ene gy-momen um enso eads Tµν = (ρ+p)uµuν+pgµν , whe e uµ=
(1, ui) is he ou - eloci y unde he condi ion uµuµ=−1 . We also admi an
homogeneous and iso opic space ime desc ibed by he Robe son-Walke me ic,γij,
whose pe u ba ions a e gi en by he line elemen
ds2=−(1 + 2Φ) d 2+a2( ) (1 −2Ψ) γijdxidxj.(3.1)
F om now on, we conside he choice o he Lag angian densi y as Lm=−ρ,
which is he mos sui able o desc ibing bound sys ems as discussed in Re . [29].
The o he possible choice, Lm=p, is no e y use ul, since we assume a p essu eless
Uni e se.1De ining he po en ial eloci y in e ms o he componen s o he 4-
1In he con ex o heo ies wi h non-minimal coupling be ween ma e and cu a u e, one b eaks
he degene acy on he Lag angian densi y choice ha exis ed in Gene al Rela i i y [29]. We can
5
Chap e 3. Pe u bed F iedmann-Lemaˆı e-Robe son-Walke model 6
eloci y as ui=−∂i and compu ing he i s o de pe u ba ion in he componen s
δTi
0o he s ess enso o a ma e domina ed epoch, ρ≈ρm, we ge [30]:
˙ +˙
Φc = Φ + δΦc,(3.3)
whe e Φc= ln (1 + 2). This exp ession can be ew i en in e ms o he ou - eloci y
as
˙ui=−∇ (Φ + δΦc− ˙
Φc).(3.4)
We shall make he assump ion ha he low eloci y associa ed o he expansion
a e o he Uni e se is much smalle han he ypical peculia eloci ies o cosmic
s uc u es. Then ui≈a˙xi≡ m i . Unde his condi ion, Eq. (3.4) can be exp essed
in a mo e con enien o m
∂
∂ (a m) = −a∇ Φ + δΦc−˙
Φc .(3.5)
The e olu ion o ma e densi y pe u ba ions is gi en in he Fou ie space by [30]
˙
δρm+ 3Hδρm= 3 ˙
Ψρm−k2
a2
aρm,(3.6)
whe e H= ˙a/a is he expansion a e. In he con igu a ion space, using he no a ion
σm≡δρm, only conside ing peculia eloci ies and in he subho izon app oxima ion
(k/a > H) we can w i e
˙σm+ 3Hσm=−1
a∇x·(ρm−→
m).(3.7)
easily see ha he non-conse a ion o he ene gy-momen um enso , Eq. (2.7), s ongly depends
on he Lag angian densi y. Fo ins ance, o a p essu eless Uni e se, p≃0, such ha ∇νp≃0,
he ex a o ce, Eq. (2.8], anishes o Lm=p, whils o he o he possible, Lm=−ρ, gi es
µ=−∇ν(1 + 2(R)) hµν ,(3.2)
which is, in gene al, di e en om ze o.
Chap e 3. Pe u bed F iedmann-Lemaˆı e-Robe son-Walke model 7
Finally, om he ime componen o he non-conse a ion o he ene gy-momen um
enso , o a p essu eless (w= 0) Uni e se wi h Lag angian densi y L=−ρm, hen2
˙ρm+ 3Hρm= 0.(3.9)
2The gene alisa ion o he p e ious esul is as ollows [31]
˙ρm+ 3H(1 + w)ρm=F2
1 + 2
(α−1) ρm˙
R,(3.8)
whe e α=




1,L=−ρm
−w, L=p
so ha he Lag angian densi y has he o m L=−αρm(see Re . [29]
o a ho ough discussion) and w=p/ρmis he equa ion o s a e pa ame e .
Chap e 4
The Layze -I ine equa ion
We a e now able o de i e he Layze -I ine equa ion. We s a by con ac ing Eq.
(3.5) wi h a−→
mρmd3 , o =ax, and hen in eg a ing o e he olume, we ge :
Zρma−→
m
∂
∂ (a−→
m)d3 =−Za2−→
mρm∇ Φ + δΦc−˙
Φc d3 . (4.1)
Using Eq. (3.9), he le hand side o Eq. (4.1) can be exp essed as ∂
∂ (a2K),
whe e K≡1/2Rρm 2
md3 is he kine ic ene gy associa ed wi h he peculia eloci y.
In i s u n, he igh hand side can be e alua ed pe o ming an in eg a ion by
pa s:
−Za2−→
mρm∇ Φ + δΦc−˙
Φc d3 =
=−Z∇ a2−→
mρmΦ + δΦc−˙
Φc d3 +ZΦ + δΦc−˙
Φc ∇ ·a2−→
mρmd3
=−ZΦ + δΦc−˙
Φc a2( ˙σm+ 3Hσm)d3 .
(4.2)
In he las equali y, he i s in eg al co esponds o a o al de i a i e, which
he e o e anishes. Mo eo e , we ha e eso ed o Eq. (3.7).
Collec ing he esul s, we ge
∂K
∂ + 2HK =−Z(Φ + δΦc−˙
Φc )∂
∂ σmd3 .(4.3)
We will equi e ha each po en ial sa is ies Poisson’s equa ion. We shall de ine
he au oco ela ion unc ion (−→
) o he ma e densi y pe u ba ion ield, σm, as
8

Chap e 4. The Layze -I ine equa ion 9
in Re . [17] as
Dσm(−→
, )σm(−→
0, )E=σ2
m |−→
−−→
0|.(4.4)
F om which we can de ine some as ophysical and cosmological scales. We should
also no e ha hσm(−→
, )i= 0. Addi ionally, we use ha
∂
∂
1
| − 0|=−H
| − 0|.(4.5)
Since we equi e ha he po en ials sa is y he Poisson’s equa ion, hen any o
hem can be exp essed in e ms o he ma e densi y pe u ba ion as
ϕ=−GZσm( 0, )
| − 0|d3 0.(4.6)
Bea ing his in mind, he igh hand side o Eq. (4.3) can be exp essed as
−Zϕ∂
∂ (σmd3 ) = GZ∂
∂ (σmd3 )Zσ0
m
| − 0|d3 0
=GZ∂
∂ (σ0
md3 0)Zσm
| − 0|d3 ,
(4.7)
whe e σm≡σm(−→
, ) and σ0
m≡σm(−→
0, ). Now, ecalling he esul (4.5), he
exp ession (4.7) can be w i en as
GZ∂
∂ (σ0
md3 0)Zσm
| − 0|d3 =−(˙
Uϕ+HUϕ),(4.8)
whe e
Uϕ≡ −G
2Z Z σmσ0
m
| − 0|d3 d3 0=1
2Zϕ σmd3 . (4.9)
No e ha he non-minimal coupling e ec s on he g a i a ional coupling in he
case o clus e s a e negligible, so ha he e ec i e g a i a ional cons an , as de ined
in Re . [30], obeys Ge ≈G. Now we can w i e he Layze -I ine equa ion in he
o m
∂
∂ (K+UΦ+UδΦc−˙
Φc ) + H(2K+UΦ+UδΦc−˙
Φc ) = 0,(4.10)
which can be ea anged in o a mo e con enien o m:
Chap e 4. The Layze -I ine equa ion 10
∂
∂ (K+U+UNMC ) + H(2K+U+UNMC)=0,(4.11)
wi h U≡UΦand
UNMC ≡UδΦc−˙
Φc =1
2ZδΦc−˙
Φc σmd3 . (4.12)
We see ha he non-minimal coupling be ween ma e and geome y induces an
ex a e m in he s anda d gene alised cosmic i ial heo em, which can accoun
o he ”da k componen s” e ec s on se e al sys ems. Fo a elaxed as ophysical
sys em which no longe e ol es in ime, we ge a gene alised e sion o he i ial
heo em o hese g a i a ional heo ies:
2K+U+UNMC = 0.(4.13)
F om his equa ion we can p oceed o analyse clus e s o galaxies and impose
some cons ain s on he non-minimal model. Clea ly, any de ia ion om he usual
i ial a io K/U =−1/2 can be exp essed in e ms o he quo ien :
UNMC
U=−2K
U−1.(4.14)
Chap e 5
The Abell 586 clus e
We conside now he well known elaxed clus e Abell 586, ollowing up he p oce-
du e de eloped in Re s. [19, 22]. We assume he ob ious cases o he op-ha and
iso he mal sphe es densi y p o iles. In o de o es he sensi i i y o he esul s,
we adop en a i ely he Na a o-F enk-Whi e (NFW) densi y p o ile [32], e en
hough his is known o be a p o ile ob ained om N-body simula ions o galax-
ies wi hin he Cosmological S anda d Model, ΛCDM (which assumes ha da k
ene gy is pa ame ised by a cosmological cons an , Λ, da k ma e is aken o be
non- ela i is ic). The NFW model is, he e o e, somewha inacu a e o clus e s.
As we shall see, esul s o UNMC a e dependen on he densi y p o ile choice, e en
hough no s ongly so. I is ele an o bea in mind ha he conside ed densi y is
exclusi ely ba yonic.
5.1 Top-ha densi y p o ile
In his case, one assumes ha he kine ic and po en ial ene gy densi ies a e well
desc ibed by [19]
ρK≃9
8π
M
R3σ2
,(5.1)
ρW≃ − 3
8π
G M2
hRiR3,(5.2)
11
5.2. Na a o-F enk-Whi e densi y p o ile 12
whe e Me R a e he o al ba yonic mass and adius o Abell 586, which include
galaxies and in a-clus e gas, σ is he eloci y dispe sion and hRiis he mean
in e galac ic adius. Since he case we a e s udying has sphe ical symme y, he
o al olume is simply V= 4πR3/3, and he a io be ween o al peculia kine ic
and po en ial ene gies is he same as he a io o he ene gy densi ies, hus:
K
U≡ρK
ρW
=−3σ2
hRi
G M .(5.3)
5.2 Na a o-F enk-Whi e densi y p o ile
The Na a o-F enk-Whi e model is e y use ul in ealis ic N-body simula ions wi hin
he ΛCDM pa adigm. I is cha ac e ised by he ene gy densi y [32]:
ρ( ) = ρ0
01 +
02,(5.4)
whe e is he dis ance om he cen e, ρ0and 0a e he densi y and shape pa am-
e e s, espec i ely. As desc ibed in Re . [22], he o al mass and mean adius can be
calcula ed by in eg a ing Eq. (5.4) o e he olume:
M= 4πZR
0
ρ( ) 2d = 4π 3
0ρ0ln 1 + R
0−R
R+ 0,(5.5)
hRi= 0hR
0−2 ln 1 + R
0+R
R+ 0i
hln 1 + R
0−R
R+ 0i.(5.6)
We poin ou ha 0can be nume ically calcula ed om he mean adius, hRi.
Thus, he densi y pa ame e , ρ0, is immedia ly sol ed nume ically. F om hese
quan i ies we can now es ima e he kine ic and po en ial ene gy densi ies assuming
a cons an a e age eloci y, σ [22] :
ρK=9
8π
M
R3σ2
,(5.7)
Chap e 6. Conclusions 19
di e en densi y p o iles ( op-ha , Na a o-F enk-Whi e, and iso he mal sphe es),
he a io be ween he non-minimal coupling po en ial ene gy and he ba yonic ene gy
po en ial, UNMC/U, is o he o de o ∼7. One can also conclude om he alues
in Table 5.2 ha he eloci y dispe sion alue om X- ay luminosi y is no e y
eliable, as discussed be o e in Re . [22].
Since he new po en ial ene gy e m can be exp essed in e ms o he non-minimal
coupling and he eloci y po en ial, he la e was es ima ed assuming ha i is a lin-
ea unc ion o he dis ance om he clus e ’s cen e. This assump ion is consis en
wi h he ac ha he A586 clus e has al eady i ialised and eached hyd os a ic
equilib ium, since i has no unde gone any me ging p ocess wi hin he las Gy s.
Finally, i was also analysed he 2(R) unc ion o e he dis ance om he clus-
e ’s cen e o he di e en densi y p o iles used in his wo k. F om he summa ised
plo , one concludes ha o singula densi y p o iles a = 0, as he NFW and he
iso he mal sphe es p o iles, he coupling unc ion is na u ally s onge . Whils o
he op-ha , one ind a cons an unc ion o e he dis ance.

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