Gravitational collapse in f(R) theories

0.32 MB·English

by  J. A. R. Cembranos

Gravitational collapse in f(R) theories
J.A.R.Cembranos(a)∗, A.de la Cruz-Dombriz(b,c)† and B.Montes Nu´n˜ez(d)‡
(a)Departamento de F´ısica Te´orica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain
(b) Astrophysics, Cosmology and Gravity Centre (ACGC),
University of Cape Town, Rondebosch, 7701, South Africa
(c) Department of Mathematics and Applied Mathematics,
University of Cape Town, 7701 Rondebosch, Cape Town, South Africa and
(d) Centro de Investigaciones Energ´eticas, Medioambientales y Tecnolo´gicas (CIEMAT), 28040 Madrid, Spain
(Dated: January 13, 2012)
Westudythegravitationalcollapseinmodifiedgravitationaltheories. Inparticular,weanalyzea
general f(R)model with uniformly collapsing cloud of self-gravitating dustparticles. Thisanalysis
shares analogies with the formation of large-scale structures in the early Universe and with the
formation of stars in a molecular cloud experiencing gravitational collapse. In the same way, this
2
investigationcanbeusedasafirstapproximationtothemodification thatstellarobjectscansuffer
1
in these modified theories of gravity. We study concrete examples, and find that the analysis of
0
gravitational collapse is an important tool to constrain models that present late-time cosmological
2
acceleration.

n
a PACSnumbers: 98.80.-k,04.50.+h
J

2
1 INTRODUCTION successful theories of the twentieth century, it does not
give a satisfactory explanation to some of the latest cos-

] mologicalandastrophysicalobservationswithusualmat-
c In the general study of astrophysical weak gravita-
q tersources. Inthefirstplace,adarkenergycontribution
tional fields, relativistic effects tend to be ignored. How-
- needs to be considered to provide cosmologicalaccelera-
r ever, there are clear examples of stellar objects in which
g tion whereas the baryonic matter content has to be sup-
these effects may have important consequences, such as
[ plemented by a dark matter (DM) component to give
neutron stars, white dwarfs, supermassive stars or black
a satisfactory description of large scale structures, rota-
2 holes. Indeed, it becomes necessary to consider observa-
v tional speeds of galaxies, orbital velocities of galaxies in
tionally consistent gravitational theories to study these
9 clusters, gravitational lensing of background objects by
objects. General Relativity (GR) has been the most
8 galaxy clusters, such as the Bullet Cluster, and the tem-
2 widely used theory but other gravitational theories may
perature distribution of hot gas in galaxies and clusters
1 bestudiedforabetterunderstandingofthefeaturesand
of galaxies. All these evidences have revealed the inter-
. properties of such objects and to compare their predic-
1
esttostudyalternativecosmologicaltheories. Thisextra
0 tions with experimental results.
DM component is required to account for about 20% of
2
The gravitational collapse for a spherically symmet- the energy content of our Universe. Although there are
1
: ric stellar object has been extensively studied in the GR many possible originsfor this component[2], DM is usu-
v framework (see [1] and references therein). By assum- ally assumed to be in the form of thermal relics that
i
X ing the metric of the space-time to be spherically sym- naturally freeze-out with the right abundance in many
r metric and that the collapsing fluid is pressureless, the extensions of the standard model of particles [3]. Fu-
a foundmetricinteriortotheobjectturnstobeRobertson- ture experiments will be able to discriminate among the
Walker type with a parameterplaying the role of spatial large number of candidates and models, such as direct
curvature and proportional to initial density. The time andindirectdetectiondesignedexplicitlyfortheirsearch
lapse and the size of the object are given by a cycloid [4], oreven athigh energycolliders,where they could be
parametricequationwithanangleparameterψ. Further produced [5].
results are that the time when the object gets zero size
A larger number of possibilities can be found in the
is finite and inversely proportional to the square root of
literature to generating the present accelerated expan-
the initialdensity. Finally,the redshift seenby anexter-
sion of the Universe [6]. One of these methods consists
nalobserverisneverthelessinfinitewhentimeapproaches
of modifying Einstein’s gravity itself [7, 8] without in-
the collapse time.
vokingthe presenceofanyexoticdarkenergyamongthe
In spite of the fact that GR has been one of the most
cosmological components. In this context, functions of
the scalar curvature when included in the gravitational
action give rise to the so-called f(R) theories of modi-
fied gravity [9]. They amount to modifying the l.h.s. of
∗E-mail: [email protected]
†E-mail: [email protected] thecorrespondingequationsofmotionandprovideageo-
‡E-mail: [email protected] metricalorigintotheacceleratedcosmologicalexpansion.
2
Although such theories are able to describe the acceler- Reference [22] includes a complete study of neutron
atedexpansiononcosmologicalscalescorrectly[10],they stars in f(R) theories. The most relevant result in this
typically give rise to strong effects on smaller scales. In investigation suggests that f(R) theory allows stars in
any case, viable models can be constructed to be com- equilibrium with arbitrary baryon number, no matter
patible with local gravity tests and other cosmological howlargethey are. Very recentlyauthors in[23] studied
constraints [11]. collapse of charged black holes by using the double-null
Thestudyofalternativetheoriesofgravitationrequires formalism.
establishing methods able to confirmordiscardtheir va- Charged black holes in f(R) gravity can have a new
liditybystudyingthecosmologicalevolution,thegrowing type of singularity due to higher curvature corrections,
of cosmological perturbation and, at astrophysical level, the so-called f(R)-induced singularity, although it is
the existence of objects predicted by GR such as black highly model-dependent.
holes [12] or dust clouds forming compact structures. It
Thepresentworkhasbeenarrangedasfollows: insec-
is well-knownthat f(R) gravitytheories may mimic any
tionII,f(R)modifiedgravitytheorieswillbeintroduced.
cosmological evolution by choosing adequate f(R) mod-
Gravitationalcollapse in f(R) theories will be presented
els,inparticularthatofΛCDM[10]. Thisistheso-called
in section III. After performing some calculations, the
degeneracy problem that some modified gravity theories
evolution equation for the object scale factor will be ob-
present: accordingly, the exclusive use of observations
tained. This equation will be used throughout the fol-
such as high-redshift Hubble diagrams from SNIa [13],
lowing sections. Section IV is then dedicated to achieve
baryonacousticoscillations[14]orCMBshiftfactor[15],
solutionsforthemodifiedequationsinthreequalitatively
basedon differentdistance measurements whichare sen-
different f(R) models, which try to illustrate the broad
sitive only to the expansion history, cannot settle the
phenomenology of the subject. This is therefore the aim
question of the nature of dark energy [16] since identi-
of this section: to study gravitational collapse by calcu-
cal results may be explained by several theories. Nev-
lating the evolution of the object scale factor in particu-
ertheless, it has been proved that f(R) theories - even
larf(R)models. Finally,theconclusionsbaseduponthe
mimickingthestandardcosmologicalexpansion-provide
presented results will be analyzed in detail in section V.
different results from ΛCDM if the scalar cosmological
perturbations are studied [17]. Consequently, the power
spectra would be distinguishable from that predicted by
ΛCDM [18]. II. f(R) THEORIES OF GRAVITY
It is therefore of particular interest, to establish the
predictions off(R) theories concerningthe gravitational
With the aim of proposing and alternative theory to
collapse, and in particular collapse times, for different
GR, a possible modification consists of adding a func-
astrophysical objects. Collapse properties may be either
tionofthescalarcurvature,f(R),totheEinstein-Hilbert
exclusive for Einstein’s gravityor intrinsic to anycovari-
(EH) Lagrangian. Therefore the gravitationalaction be-
ant gravitation theory. On the other hand, obtained re-
comes 1
sults may be shed some light about the models viability
andbeusefultodiscardmodelsindisagreementwithex-
1
pected physical results. S = d4x g (R+f(R)). (1)
G
16πG | |
In [19] the authors studied gravitational collapse of a Z
p
spherically symmetric perfect fluid in f(R) gravity. By
By performing variations with respect to the metric, the
proceeding in a similar way to [1], the object mass was
modified Einstein equations turn out to be
deducedfromthejunctionconditionsforinteriorandex-
teriormetrictensors. Finally,theyconcludedthatf(R )
0
1
(constantscalarcurvatureterm)slowsdownthe collapse (1+f )R (R+f(R))g + f = 8πGT ,
R µν µν µν R µν
− 2 D −
of matter and plays the role of a cosmological constant.
(2)
Authors in [20] paid attention to the curvature singular-
where T is the energy-momentumtensor ofthe matter
µν
ity appearing in the star collapse process in f(R) the- content,f df(R)/dRand g (cid:3)with
R µν µ ν µν
ories. This singularity was claimed to be generated in (cid:3) α≡and is the usuaDl cov≡ari∇ant∇de−rivative.
viable f(R) gravity and can be avoided by adding a Rα ≡ ∇α∇ ∇
These equations may be written `a la Einstein by iso-
term. Theyalsostudiedexponentialgravityandthetime
lating on the l.h.s. the Einstein tensor and the f(R)
scale ofthe singularityappearancein thatmodel. Itwas
shownthatincaseofstarcollapse,thistimescaleismuch
shorter than the age of the universe. Analogous studies
were carriedon by [21] claiming that in this class of the-
1 Inthepresentworkweemploythenaturalunitssysteminwhich
ories, explosive phenomena in a finite time may appear ~=c=1. NotealsothatourdefinitionfortheRiemanntensor
in systems with time dependent increasing mass density. isRσµνκ=∂κΓσµν−∂νΓσµκ+ΓσκλΓλµν−ΓσνλΓλµκ.
3
contribution on the r.h.s. as follows Componentstt,rrandθθforthemodifiedtensorialequa-
tions may be written respectively in terms of the func-
1 1
R Rg = 8πGT tions A(t) and h(r) as follows
µν µν µν
− 2 (1+f ) −
R (cid:20)
1 A¨ 1 A˙ 1
− DµνfR+ 2(f(R)−RfR)gµν(cid:21) (3) 3A = (1+fR)"−8πGρ+3Af˙R+ 2(R+f(R))#,
We can also find the expression for the scalar curvature (11)
by contracting (2) with gµν which gives:
(1−fR)R+2f(R)+3(cid:3)fR =8πGT. (4) AA¨+2A˙2+ rhh′2 = (1+A2f ) f¨R +2AA˙f˙R
R "
Note that, unlike GR where R and T are related alge-
1
braically, for a general f(R) those two quantities are + (R+f(R)) , (12)
dynamically related. In the homogeneous and isotropic 2
(cid:21)
case, the scalar curvature in f(R) theories becomes
1 1 h′
8πGT 2f(R) 3f¨ AA¨+2A˙2+ +
R= − − R (5) r2 − hr2 2rh2
(1 f )
− R A2 A˙ 1
= f¨ +2 f˙ + (R+f(R)) .
where dot means the derivative with respect to cosmic (1+f ) R A R 2
R " #
time.
(13)
Letus pointouttwoimportantaspectsofequations(12)
III. GRAVITATIONAL COLLAPSE IN f(R)
and(13): firstly,termsonthe r.h.sofbothequationsare
equal. Secondly, the term on the l.h.s. exclusively de-
Inthecaseofourinvestigation,weintroducethespher-
pends on r, whereas the term on the r.h.s. only depends
ically symmetric metric
on t in both equations, so that they must be constants2.
ds2 =dt2 U(r,t)dr2 V(r,t)(dθ2+sin2θdφ2) (6) Therefore we may equal l.h.s. of both equations to pro-
− − vide
Ifthecollapsingobjectisapproximatedtobepressureless 1 h′ 1 1 1 h′
p 0, the components of the energy-momentum tensor = + 2k, (14)
≃ rh2 r2 − hr2 2rh2 ≡
can be expressed as follows
where we have equaled both equations (multiplied by a
Tµν =ρuµuν ; Ttt =ρ; Tii =0 if i=r,θ, φ. (7) factor A2) to a constant 2k. The resulting solution is
−
h(r) = (1 kr2)−1. Once we have calculated h(r), the
We may further simplify the collapse model by consider- −
resulting metric can be expressed as follows:
ing ρ independent from the position. Therefore, we can
search-asis actually the usualapproachinthe GR case- dr2
ds2 =dt2 A2(t) +r2(dθ2+sin2θdφ2) ,
a separable solution for this metric as follows − 1 kr2
(cid:20) − (cid:21)
(15)
U(r,t)=A2(t)h(r), V(r,t)=A2(t)r2, (8)
1 2 which is formally the same as the one obtained in the
GRcase[1]. Expression(14)for k maybe substituted in
where a previous reparametrization of the radial coor-
either expression (12) or (13) yielding:
dinate is required. When f(R) modified tensorial equa-
tions are studied in the homogeneous and isotropic case 2
A¨ A˙ 2k 1 A˙
- in which f(R) does not depend on the position-, the 2 = f¨ 2 f˙
trace component provides − A − A! − A2 (1+fR)"− R− A R
1
A˙ A˙ g′ A˙ A˙ (R+f(R)) . (16)
2 1 =0 2 = 1. (9) − 2
A2 − A1! g ⇒ A2 A1 (cid:21)
Takinginto accountρ(t)=ρ(t=0)/A(t)3 (givenby the
From(9),wededucethatA1 andA2 areproportional,in energy motion equation for dust matter) and the results
otherwords,A (t)=C(r)A (t). So,ifwechooseA (t)=
1 2 1
A (t) A(t), the dependence in the radial coordinate is
2
≡
reabsorbed by h(r). Hence:
2 ThisfactisalsosatisfiedintheGRcaseandallowsthesimplifi-
U(r,t)=A2(t)h(r), V(r,t)=A2(t)r2. (10) cationofthecalculus.
4
in (16), equation (11) becomes: to unity. In order to simplify the notation we define
R(t=0) R and ρ(t=0) ρ . Therefore, evaluation
0 0
A˙2 = k+ 1 4πGρ(0)A−1+ 1A2f¨ of (17) at≡t=0 allows to rec≡ast k as follows
R
− (1+f ) 3 2
R (cid:20)
+ 12AA˙f˙R+ A62 (R+f(R)) . (17) k = (1+f1R(R0))(cid:20)4π3Gρ0+ 12f¨R(R0)+ 61(R0+f(R0))(cid:21)
(cid:21)
(18)
Furthermore, provided that the fluid is assumed to be
at rest for t = 0, initial conditions A˙(t = 0) = 0 and Oncekhasbeenexpressedintermsofdifferentquantities
A(t = 0) = 1 hold. This last condition means that the initial values, equations (5) and (18) may be inserted in
scale factor of the object at initial time is normalized (17) to provide
1 1 8πG
A˙2 = 8πGρ (2 f (R )) f(R )(1+f (R )) 3f¨ (R )f (R ) + ρ A−1
−6(1 f2(R )) 0 − R 0 − 0 R 0 − R 0 R 0 (1 f2) 3 0
− R 0 h i − R
1
8πGρ A−1f +3A2f¨ f 3AA˙f˙ (1 f )+A2f(R)(1+f ) . (19)
−6(1 f2) 0 R R R− R − R R
− R h i
The previous expression will be solved perturbatively to Inorderto study the modificationto the gravitational
firstorderinperturbationsfordifferentf(R)models. Let collapse in f(R) theories, we will expand A around A
G
us remind at this stage that the zeroth order solution of and f(R) around the scalar curvature in GR (R=R ):
G
GR is given by the parametricequations of a cycloid [1]:
A=A +g(ψ), (22)
G
ψ+sinψ 1
t= , A = (1+cosψ). (20)
G
2√k 2
f(R) f(R )+f′(R )(R R ). (23)
Expression(20) clearly implies that a sphere with initial ≃ G G − G
density ρ and negligible pressure will collapse from rest
0 The presence of a function f(R) in the gravitationalLa-
toastateofinfiniteproperenergydensityinafinitetime
grangian will represent a correction of first order with
thatwewilldenoteT . Thistimeisobtainedforthefirst
G respect to the usual EH Lagrangian. Hence, g(ψ) as de-
value of ψ such as A =0, i.e. for ψ =π. It means
G finedin(22)willbealsofirstorderatleast. Bysubstitut-
π+sinπ π π 3 1/2 ingtheseriesexpansions(22)and(23)inexpression(19)
T = = = . (21) until first order in ε, we find that equation (19) becomes
G
(cid:18) 2√k (cid:19) 2√k 2 (cid:18)8πGρ0(cid:19)
ψ 1 ψ 1 ψ 1
tg g′ = cos−2 g+ cos2 (f(R )+3kf (R )) f (R )
G0 R G0 R G
2 −2 2 12k 2 − 4
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 ψ ψ 1 ψ
+ sin cos3 f˙ (R ) cos6 f(R ), (24)
R G G
4√k 2 2 − 12k 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
where we have cancelled out the GR exact solution and IV. f(R) THEORY RESULTS
onlykeptfirstorderperturbedterms. Equation(24)will
provide g(ψ) evolution for different f(R) models to be
considered in the next section.
Inthissectionweshallconsiderthreeillustrativef(R)
models and study the gravitational collapse process for
collapsing dust. The models under consideration are
5
Model 1: f(R)=εR2 After some algebra, for this model, equation (24) can
be written as follows:
This function has been proposed both as a viable in-
flation candidate [24] and as a dark matter model [25].
9 ψ
In this last reference, the ε parameter definition reads g′(ψ)+g(ψ)csc(ψ)+ kε sin(ψ)+2tan
8 2
(cid:18) (cid:18) (cid:19)
1 ψ
ε= , (25) 4tan4 csc(ψ) =0. (26)
6m2 − 2
0 (cid:18) (cid:19) (cid:19)
and the minimum value allowed for m is computed as
0
m = 2.7 10−12 GeV at 95 % confidence level, i.e. The homogeneous equation associated to (26) presents
0
ε 2.3 ×1022GeV−2. On the other hand, ε > 0 is the solution g (ψ) cot(ψ/2), which diverges at
hom
≤ × ∝
needed to ensure the stability of the model, since in the ψ = 0. Therefore, its contribution will be ignored in
opposite case, a tachyon is present in the theory. These the upcoming analysis. The analytical full solution of
constraints are in agreement with [26]. (26) becomes
ψ 9 ψ 64 ψ ψ ψ
g(ψ)=c cot kεcot 16(ψ+sin(ψ))+ tan 4 sec2 sec2 2 ,
1
2 − 128 2 − 5 2 − 2 2 −
(cid:18) (cid:19) (cid:18) (cid:19)(cid:26) (cid:18) (cid:19)(cid:20) (cid:18) (cid:19)(cid:18) (cid:18) (cid:19) (cid:19)(cid:21)(cid:27)
(27)
This analytical solution given by (27) can be compared this region, A can be approximated by:
G
with the GR one by plotting them together as shown in
Figure 1. As we see in this Figure, in the first stage of 1 (ψ π)2
A = (1+cosψ) − . (31)
G
thecollapse,thecorrectionisnegative,whatimpliesthat 2 ≃ 4
wehavealargercontraction. Onthecontrary,veryclose
We can use these approximations in the limit ψ π to
to ψ =π, where the solution can be approximated as →
determine the intersection between the particular solu-
tion of (26) and the GR solution: A . This calculation
72kε G
g(ψ) . (28) will help establishing the validity regime of the pertur-
≃ 5(ψ π)4
− bative approach. Therefore, by imposing g(ψ) = A ,
G
| | | |
thesignofthemodificationchangesandthetotalcollapse with g(ψ) given by equation (28), we obtain:
is avoided. Exactly at this moment, the perturbation
1/6
leaves the linear regime and a more complete analysis 288kε
ψ =π | | . (32)
is required. It is interesting to estimate when the linear − 5
(cid:18) (cid:19)
approachfailsandanimportantmodificationisexpected.
At this point, it is necessary to clarify the physical
By usingthe collapsetime parametrization(20)inthe
value for k in order to discuss if the departure from lin-
ψ π limit, one gets
→ earity is important. k is the initial condition given by
ψ+sin(ψ) π+1/6(∆ψ)3 equation (18) that depends on the matter density, the
t = , (29)
2√k ≃ 2√k initial curvature and the particular f(R) model. In our
analysis we are interested in studying the modification
or written in terms of the relative variation: to the gravitational collapse in GR and for this reason
we will assume the same value of k than as givenin GR.
∆t (∆ψ)3
. (30) This implies that the entire modification has a dynam-
t ≃ 6π
GR ical origin and it does not come from a change in the
We are interested in estimating the region of the pa- initial conditions. Therefore, we will assume that k only
rameter space of the model, where the modified col- depends on the matter density:
lapse, and the result for ψ (ψ value for the collapse)
C 8πG
is significantly different from the one predicted by GR k = ρ . (33)
0
3
(ψ π). With this purpose, we can estimate the
C,GR
≡
valuesforwhichA isofthesameorderofitscorrection. For the most physically interesting values of k and ε,
G
As one cansee inFigure 1, this deviationis moreimpor- forwhichwehavestudiedgravitationalcollapseofadust
tantcloseto the finalstageofthe collapse,forψ π. In matter cloud, the value of ψ is quite close to ψ =π and
∼
6
followingsections. InFigure2,wehaverepresentedsome
1.2 ` relevant density values as well as the non linear regime
g@ΨD
region. The loss of the linear regime takes place at high
1.0 AG densities since the correction is directly proportional to
ρ.
`
g@ΨD+AG
0.8
0.6
0.4 Ε= 2.3´1022GeV-2
k= 8.38´10-76GeV2
0.2
0.0
-0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Ψ
Figure 1: Comparisonbetween the solutiongiven by (27)setting
c1=0andtheGRcaseforthemodel1: f(R)=εR2. Theplotted
kvalueisfixedbyEq. (33),wherethedensityisρSF≃1.5×10−38
GeV4 ≃ 3.5×10−18kg/m3, i.e., the matter density in the early
Universe at redshift z ≃ 1100 marking the decoupling of matter
andradiationand thebeginning ofstructure formation(SF). The
modification is extraordinarily small and has been increased 52 Figure2: Validityoftheperturbativeregimeformodel1,showing
ordersofmagnitudetomakeitobservable: gˆ(ψ)=1052g(ψ).
different relevant regions: In blue we show the region where our
linear approach loses its validity. The excluded region is depicted
inyellow and determined by the condition ε≤2.3×1022GeV−2.
Finally, the density marking the beginning of structure formation
therefore the asymptotic approachto obtain (32) is fully (SF)andthedarkenergy(DE)density(ρDE≃2.8×10−47 GeV4)
havealsobeenplottedforreference.
justified. The results are summarized in Figure 2, were
thenon-linearregimeisshownfordifferentvaluesofǫand
initial densities. For example, it is interesting to check
the behaviorfor the matterdensity inthe earlyUniverse Model 2: f(R)=εR−1
atredshiftz 1100,whichmarksthedecouplingofmat-
≃
ter and radiation and the beginning of structure forma-
We will continue our analysis with the f(R) model
tion (SF): ρ 1.5 10−38 GeV4 3.5 10−18kg/m3.
SF ≃ × ≃ × proposed in reference [27] as a dark energy candidate.
In this particular case, the calculated root presents a
Thispossibilityiscurrentlyexcluded,butthismodelisa
slight difference with respect to the solution for GR. In
simpleexamplethathelptounderstandthegravitational
particular, ψ differs from ψ π in the ninth sig-
C C,GR ≡ collapse modifications in models that provide late-time
nificant figure.
acceleration.
Although this modification is not detectable for this
For this model, equation (24) becomes:
first model as we deduce from the last considerations,
it is interesting to stress that the relative modification ε ψ ψ
g′(ψ)+g(ψ)csc(ψ)= sin cos13 , (34)
is higher for denser media since the correction increases 6k2 2 2
with ρ0 as ∆t/tGR √ρ0. This behavior is significantly (cid:18) (cid:19) (cid:18) (cid:19)
∝
differentwithrespecttoothermodelsaswewillseeinthe whose full solution is
7
ψ 1 ε 33ψ 165sin(ψ) 11sin(2ψ) 121sin(3ψ) 25sin(4ψ)
g(ψ) = c cot + +
1 2 6k2 2048 8192 − 8192 − 24576 − 8192
(cid:18) (cid:19) (cid:18)
43sin(5ψ) 5sin(6ψ) sin(7ψ) ψ
cot .
− 40960 − 24576 − 57344 2
(cid:19) (cid:18) (cid:19)
(35)
1.0
`
g@ΨD
0.8 AG
`
g@ΨD+AG
0.6
0.4
Ε=-I1´10-42M4GeV4
k= 8.38´10-76GeV2
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Ψ
Figure 3: Analogous representation to the one shown in Fig. 1, Figure 4: Analogous to Figure 2 for model 2. The limit of the
whichincludestheGRsolution,themodificationgivenby(35)and region depicted inyellow shows the proposed value ε=−µ4, and
thesumofthetwo. Inthisfiguregˆ(ψ)=1019g(ψ)inordertomake µ=10−42 GeV accordingto[27].
themodificationobservable.
(ε < µ4 and negative). The situation changes for
In Figure 3, it is possible to see the behavior of the ε| <| µ4 (ε > µ4 and positive). This behavior can
modification for ε = µ4, and µ = 10−42 GeV as it was − | |
be observed in Figure 4, where the validity of the linear
−
the value originally proposed in reference [27].
regime is shown to decrease for higher values of ε and
The series expansion of (35) around ψ = π reads, in | |
lower densities. As we will see in the following example,
this case:
thisisageneralpropertyoff(R)modelsthatprovideac-
celerated cosmologies, at least, for densities higher than
11πε(ψ π)
g(ψ) − . (36) the vacuum energy. Results in Figure 4 can be under-
≃− 8192k2
stoodby estimating the correctionofthe collapsingtime
Once again the intersection between the particular so- inthe linearregime as itis determined by equation(30).
lution of (34) and the GR solution AG can be deter- This f(R) model provides a relative difference for the
mined in the ψ π limit, with help of equation (31). collapsing time in GR value given by
→
g(ψ) = A implies
G
| | | | ∆t 113ε3π2
. (38)
11π ε t ≃ − 3k6234
ψ π | |. (37) GR
≃ − 2048k2
We observe that the correction is more negligible for
As it can be seen in Figure 3, the difference between denser objects. This unexpected fact can be understood
the modified ψ and ψ is not distinguishable for since GR modification to the scale factor is proportional
C C,GR
ε = µ4 if density is higher than the standard dark to g ε/k2 whereas k ρ . According to this depen-
0
− ∝ ∝
energy density. The same result is found for ε > µ4 dence, a stellar object with a higher density will suffer
−
8
a less important modification and vice versa. The rela- to be ofthe orderof the presently observedeffective cos-
tivetimemodificationislowerfordensermediasincethe mological constant3. With such parameter choice, this
correction decreases with ρ as ∆t/t ρ−6. model is a viable dark energy candidate. The relation
0 GR ∝ 0
between λ and R in vacuum is given by H2 = λR /6
0 0 0
accordingto[28],whereH isthepresentHubbleparam-
0
Model 3: f(R)=λR 1+ R2 −n−1 eter (see [29] for recent WMAP data) and the proposed
0(cid:20)(cid:16) R20(cid:17) (cid:21) value for λ=0.69.
Thelastf(R)modeltobeanalyzedinthepresentwork
is the well-known Starobinsky model proposed in Refer- For the sake of simplicity, let us choose n=1. In this
ence [28]. For this model, n,λ> 0 and R is considered case, the equation (24) may be rewritten as follows:
0
3 ItisimportanttoremarkthatR0 inthiscontextisaparameter
ofthemodelandnottheinitialscalarcurvature.
9kλR sin3(ψ)csc4 ψ 72kλR sin4 ψ csc(ψ)
g′(ψ) g(ψ)csc(ψ) 0 2 R2+3k2 + 0 2 R2+3k2sec12 ψ
− − − 32(9k2+R02)2(cid:16) (cid:17) 0 R2+9k2se(cid:16)c12(cid:17) ψ 2 (cid:18) 0 (cid:18)2(cid:19)(cid:19)
(cid:0) (cid:1) 0 2
(cid:16) (cid:16) (cid:17)(cid:17)
9kλR3tan ψ sec4 ψ R2 27k2sec12 ψ
0 2 2 0− 2
=0. (39)
− (cid:16) (cid:17) (cid:16) (cid:17)(cid:16) 3 (cid:16) (cid:17)(cid:17)
2 9k2sec12 ψ +R2
2 0
(cid:16) (cid:16) (cid:17) (cid:17)
Unlike the other two cases, we are not able to find an- Thisasymptoticlimitofthelinearcorrectiondependson
alytical solution for equation (39). Thus, specific values a higher powerof (ψ π) than the GR solutiongivenby
−
fork andλparametersandR arerequiredto finda nu- eq. (31). This fact implies that we cannot estimate the
0
merical solution. This solution is plotted in Figure 5. In validityofthelinearregimebyusingtheψ π asinthe
→
anycase,equation(39)canbe studiedin the asymptotic previous cases. The modification is more important at
limits ψ 0 and ψ π. Thus, the correspondingseries intermediates values of ψ, as it can be observed in Fig-
→ →
expansion of (39) in the ψ 0 becomes ure 5. The numerical results are showed in Figure 6. In
→
asimilarwaytothesecondmodel,forλ<λ themodifi-
g(ψ) 9kλR 3k2 R2 ψ 0
g′(ψ) 0 − 0 =0, cationofthecollapseisalwayslinearandnotimportant.
− − ψ − 4(9(cid:0)k2+R02)2(cid:1) Thesituationisdifferentforλ>λ0,wherethecollapseis
(40) severely modified at densities closer to the vacuum one.
We have checked numerically that denser environments
whose analytical solution is
are less affected by this gravitational model.
g(ψ) = c1 + 3kλR0 R02−3k2 ψ2 . (41)
ψ 4(9(cid:0)k2+R02)2(cid:1)
V. CONCLUSIONS
Since the homogeneousequationdoes notdepend onthe
f(R) model, the condition c = 0 is also necessary in
1
order to have a finite solution. When the considered Inthisworkwehavestudiedthe gravitationalcollapse
asymptotic limit is ψ π, equation (39) approximately in f(R) gravity theories. These theories provide correc-
→
becomes tions to the field equations that modify the evolution of
g(ψ) 9kλR (ψ π)3 3k2+R2 gravitational collapse with respect to the usual General
−g′(ψ)+ ψ−π + 032(−9k2+(cid:0)R02)2 0(cid:1) =0, mReulsattipvriotyvidreesusilmtsi.larInretshuilstscfoonrtethxte,cvoilalabpleseft(iRm)esmtoodtehles
(42)
values obtained in General Relativity. In addition, col-
whose analytical solution is lapse times must be much shorter than the age of the
universe and long enough to allow matter cluster.
3kλR (ψ π)4 3k2+2R2
g(ψ)=c (ψ π)+ 0 − 0 . Theanalyzedf(R)modelspresentbothimportantdif-
1 − 32(9k2+(cid:0)R02)2 (cid:1) ferent quantitative and qualitative behaviors when com-
(43) pared with General Relativity collapses. In fact, all of
9
1.0
`
g@ΨD
0.8
AG
`
g@ΨD+AG
0.6
0.4 Λ=0.69
k= 8.38´10-76GeV2
R0= 2.01´10-83 GeV2
0.2
0.0
-0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Ψ
Figure 5: The plotted lines are analogous to the ones in Fig. 1 Figure 6: Analogous to Figure 2 for studied model 3. The limit
and Fig. 3 but with the solution given by (39). In this figure oftheregiondepictedinyellowshowstheproposedvalueλ=0.69
gˆ(ψ)=109g(ψ)inordertomakethemodificationobservable. [28].
we may getan idea of the importance of the corrections.
themshowacollapsinginitialepochwithhighercontrac-
Inside these objects, pressure is the same order of mag-
tion than in General Relativity. This result is expected
nitude as density, and it is expected to introduce an im-
since f(R) theories modify the gravitational interaction
portant modification into star evolution and dynamics.
bytheadditionofanewscalarmediator. Itiswell-known
Therefore, it is enough to take into account the typi-
that a scalar force is always attractive and can only re-
cal value of the density of a neutron star, approximately
duce the time of gravitational collapse. This result is
10−3GeV4 to estimate if correction will be important.
interesting since observations of structures at high red-
Althoughthisvalueis35ordersofmagnitudelargerthan
shift introduce some tension with the standard ΛCDM
the dust density used above, the results do not change
model [30], and the tendency of f(R) models to increase
dramatically. A direct extrapolation suggests that we
the gravitational attraction at early times can alleviate
may expect even more negligible modifications for f(R)
this problem.
modelsthatpresentdarkenergyscenarios(models2and
Although this general behavior is shared by the three
3). In models with higher powers of the scalar curva-
models analyzed throughout this investigation, they
ture, the correction to General Relativity will be more
present significant differences when the modifications to
important but still negligible (as for model 1).
the General Relativity collapse leave the linear regime.
On the one hand, the R2 model has a modification
that increases with the density of the collapse object:
Acknowledgments. This work has been supported
(∆tc/tGR) √ρ. The opposite behavioris foundfor the
R−1 model∝, where this modification decreases with den- by MICINN (Spain) project number FPA 2008-00592
sity as (∆t /t ) ρ−6. Finally, a similar situation is and Consolider-Ingenio MULTIDARK CSD2009-00064.
c GR
∝ AdlCD also acknowledges financial support from NRF
reproduced numerically in the Starobinsky model. The
departurefromthelinearcollapseisabletoexcludeinter- and URC research fellowships (South Africa) and kind
hospitality of UCM, Madrid while elaborating part of
esting parameters regions of these models that support
the manuscript.
late-time accelerationas seen in Figures 2, 4 and 6.
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