Table Of ContentEddington-inspired Born-Infeld gravity: astrophysical and cosmological constraints
P.P. Avelino∗
Centro de Astrof´ısica da Universidade do Porto,
Rua das Estrelas, 4150-762 Porto, Portugal and
Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias,
Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
(Dated: January 13, 2012)
In thisletter we computestringent astrophysical and cosmological constraints on a recently pro-
posed Eddington-inspiredBorn-Infeld theoryof gravity. Wefind,usingageneralized version of the
Zel’dovich approximation, that in this theory a pressureless cold dark matter fluid has a non-zero
2 effective sound speed. We compute the corresponding effective Jeans length and show that it is
1 approximately equal to the fundamental length of the theory R∗ =κ1/2G−1/2, where κ is the only
0 additionalparameteroftheorywithrespecttogeneralrelativityandGisthegravitationalconstant.
2 Thisscaledeterminestheminimumsizeofcompact objects whichareheldtogetherbygravity. We
n alsoestimatethecriticalmassabovewhichpressurelesscompactobjectsareunstabletocolapseinto
a a black hole, showing that it is approximately equal to the fundamental mass M∗ = κ1/2c2G−3/2,
J and we show that the maximum density attainable inside stable compact stars is roughly equal to
2 the fundamental density ρ∗ = κ−1c2, where c is the speed of light in vacuum. We find that the
1 mereexistenceofastrophysical objects ofsize Rwhichareheld togetherbytheirowngravity leads
to the constraint κ< GR2. In the case of neutron stars this implies that κ< 10−2m5kg−1s−2, a
] limit which is stronger by about 10 orders of magnitude than big bang nucleosynthesis constraints
O
and by more than 7 orders of magnitudethan solar constraints.
C
.
h I. INTRODUCTION [17]theEiBItheoryhasalsobeenshowntosupportcom-
p pact stars made of pressureless dust. Astrophysical con-
-
straints on the single extra parameter of the theory κ
o Recently, a new Eddington-inspired theory of gravity
r with a Born-Infeld like structure [1] has been proposed have been determined considering the physics of astro-
t
s by Banados and Ferreira [2] (see also [3–15] for other nomical objects such as neutron stars [17] and the Sun
a [18].
relevantstudies ofBorn-Infeldtype gravitationalmodels
[
and [16] for a recent review on alternative theories of In this letter we compute both astrophysical and cos-
1 gravity). A key feature of the Eddington-inspired Born- mological constraints on the EiBI theory of gravity. In
v Sec. IIbasicresultsfortheevolutionoftheUniversedur-
Infeld(EiBI)theoryofgravityintroducedin[2]isthatit
4 ingtheradiationeraarereviewedandaconstraintonthe
is equivalent to Einstein’s general relativity in vacuum.
4
value of κ is derivedfrom primordialnucleosynthesis. In
In [2] it was shown that, in the non-relativistic limit,
5
Sec. III the Zel’dovich approximation is generalized to
2 the EiBI theory of gravity leads to a modified Poisson
account for the modification to the Poisson equation in
. equation given by
1 Eq. (1) which it is shown to lead to an effective funda-
0
2 κ 2 mental effective length, in the case of pressureless cold
2 φ=4πGρ + ρ , (1)
1 ∇ m 4∇ m dark matter. We use this result to compute the min-
: imum size (allowed by the theory) of compact objects
v whereφisthegravitationalpotentialandρm isthe mat- which are held together by gravity and to determine the
i
X ter density. The gravitational constant G, κ and the maximumdensityattainableinsidestablecompactstars.
speedoflightinvacuumcaretheonlyparametersofthe Wealsoobtainthecriticalmassabovewhichpressureless
r
a theory. With these parameters it is possible to define a compactstarscannot exist. We then use these results to
fundamental length, time, mass and density respectively derive stringentastrophysicalconstraintson the value of
by κ. We conclude in Sec. IV.
κ κ κc4 c2
R = , t = , M = , ρ = , (2)
∗ G ∗ Gc2 ∗ G3 ∗ κ II. BACKGROUND EVOLUTION OF THE
r r r
UNIVERSE
whosephysicalinterpretationwillbe revealedinthis let-
ter. In [2] it was shown that the EiBI theory of grav- The EiBI theory of gravity leads to modifications to
ity significantly changes the universe dynamics at early the dynamicsofthe Universeatearlytimes. In[2]it has
times, leading to a non-singular cosmological model. In been shown that, in the radiation era, the Friedmann
equation is given by
2
8πGc
2
∗Electronicaddress: [email protected] H = f(ρ˜), (3)
3κ ×
2
where we shall show that the constraint in Eq. (7) might be
(1+ρ˜)1/2(3 ρ˜)3/2 improved by several orders of magnitude by taking into
f(ρ˜) = ρ˜ 1+ − account the mere existence of compact round astronom-
− 33/2 ×
(cid:18) (cid:19) ical objects.
(1+ρ˜)(3 ρ˜)2
− , (4)
× (3+ρ˜2)2
III. LINEAR EVOLUTION OF DENSITY
ρ˜ = κρ/c2, H = a˙/a is the Hubble parameter, a is the
PERTURBATIONS
cosmological scale factor and a dot represents a deriva-
tive with respect to the physical time t. Two stationary
In this section we shall consider the non-relativistic
points with H = 0 at ρ˜= 3 and ρ˜= 1 were identified,
− regimeand generalizethe Zel’dovichapproximation[20],
for positive andnegativeκ, respectively. For small ρ˜one
for the evolution of cold dark matter density perturba-
recovers the usual dynamics with
tions in an expanding background, to account for the
H2 =8πGρ/3, (5) modification to the Poisson equation in Eq. (1). Here
we shall consider times much later than t and conse-
nuc
butforvaluesofρ˜oforderunitythedynamicsisseverely quently the backgroundevolutionof the Universe is that
modified. In this letter we shall assume that κ > 0, so of the standard cosmologicalmodel.
that ρ˜>0,in orderto avoidinstabilities associatedwith Thetrajectoryofthecolddarkmatterparticlesinaho-
an imaginary effective sound speed (see Sec. III). mogeneous and isotropic Friedmann-Robertson-Walker
backgroundcan be written as
A. Primordial nucleosynthesis constraint
~r =a(t) ~q+ψ~(~q,t) . (8)
h i
Less than one second after the big bang protons and
neutrons were in thermal equilibrium. However, about where~qistheunperturbedcomovingpositionandψ~(~q,t)
one second after the big bang the temperature T drops is the comoving displacement. Calculating the first
below 1MeV and the neutron-proton inter-conversion derivative of Eq. (8) with respect to the physical time
rate( G2T5,whereG istheFermicouplingconstant) one obtains
∼ F F
becomes smaller than expansion rate (H). This leads to
~v =H~r+~v , (9)
theapproximatefreezeoutoftheratiobetweenthenum- pec
berofneutronsandthenumberofprotonst 1safter
thebigbang,exceptforfreeneutrondecay.nAuclt∼houghthe where~v =~r˙ and~v =aψ~˙ is the peculiar velocity.
pec
formationof 4He is delayeduntil the temperature of the The gravitational acceleration felt by the cold dark
Universe is low enough for deuterium to form (at about matter particles is equal to
T =0.1MeV),itiscrucialthatthedynamicsofthe Uni-
verseatthestartoftheprimordialnucleosynthesisepoch ~a r¨= φ. (10)
≡ −∇
(t 1s) is very close to the standard one in order
nuc
∼
that the good agreement between primordial nucleosyn- where φ is given by a generalized Poisson equation
thesispredictionsandobservedlightelementabundances
κ
is preserved (see [19] for a recent review). This implies 2φ=4πG(ρ¯+3p¯+δρ )+ 2ρ , (11)
m m
that the following very conservative constraint must be ∇ 4∇
satisfied
where δρ ρ ρ¯ , ρ is the matter density, ρ¯ is
m m m m m
κρ ≡ −
nuc the average matter density, ρ¯is the average density and
ρ˜ = <3, (6)
nuc c2 p¯ is the average pressure. Eq. (11) generalizes Eq. (1)
to account for a homogeneous cosmological background
with ρ = ρ(t ). Hence, taking into account that
nuc nuc
ρ 3H2 /(8πG), Eq. (6) implies that with an arbitrary equation of state. However, the two
nuc ∼ nuc equations are equivalent if p¯=0.
κ<8πGR2 6 108m5kg−1s−2, (7) Mass conservation requires that
Hnuc ∼ ×
where Hnuc = H(tnuc), the Hubble radius is defined by ρ¯ma3d3q =ρmd3r, (12)
R =cH−1,anditsvalueatthenucleosynthesisepochis
H
RHnuc R⊙ 2light seconds (R⊙ is the solar radius). where the infinitesimal volume elements d3r and d3q are
∼ ∼
The constraint in Eq. (7) may be improved by almost related by
an order of magnitude by requiring that the variation
of the Hubble parameter at tnuc ∼ 1s, with respect to d3r = ∂~r =a3 1+ ψ +...
tnhueclsetoasnydnathrdesmisopdreolvvidaeluset,hiessltersosntgheasnt c1o0s%m.olPogriicmaolrcdoina-l d3q (cid:12)(cid:12)∂~q(cid:12)(cid:12) Xi i,i !∼
straintonthe EiBItheory ofgravity. Inthe nextsection (cid:12)a3(1(cid:12)+a ψ~), (13)
(cid:12) (cid:12)
∼ ∇·
3
a commadenotes a partialderivativewith respectto the A. Astrophysical constraints
comoving coordinates and the approximation is valid up
to first order in the comoving displacement. Hence An effective Jeans length can be defined for the cold
dark matter as
δ a ψ~, (14)
∼− ∇· 2πa π πκ
λ = =c = R , (22)
Jeff seff ∗
whereδ ≡δρm/ρ¯m. IntegratingEq. (11)oneobtainsthe kJeff rGρ¯m r4G ∼
first order solution
where k is the value of k for which the term within
Jeff
4πG(ρ¯+3p¯) κρ¯ bracketsin Eq. (21) is equalto zero. The effective Jeans
φ= ~r 4πGρ¯ aψ~ + m δ, (15)
∇ 3 − m k 4 ∇ lengthforthecolddarkmatterintheEiBItheoryofgrav-
ity defines the critical scale bellow which the collapse of
with ψ~ =ψ~ +ψ~ where ψ~ and ψ~ are the irrotational pressureless dust is no longer possible (for wavelengths
k ⊥ k ⊥
λ<λ R matter fields oscillate coherently). Con-
and divergence parts of the comoving displacement, re- Jeff ∗
∼
spectively ( ψ~ =~0 and ψ~ =0). A more general sequently, λJeff determines the minimum scale of com-
∇× k ∇· ⊥ pact objects which are held together by gravity (note
solution may be obtained by adding, to the right hand
that including pressure increases the Jeans scale). The
side of Eq. (11), the term ϕ where ϕ is an arbitrary
scalarfieldsatisfyingtheLap∇laceequation 2ϕ=0. For demonstration that λJeff is independent of the matter
∇ density being approximately equal to the fundamental
simplicity,weshallignorethatextratermsinceitwillnot
scale of the theory R is one of the key results of this
haveanyimpactonourconclusions. UsingtheRachaud- ∗
letter.
hury equation
By requiring that λ is equal to the schwarzchild
Jeff
radius r ,
a¨ 4πG(ρ¯+3p¯) s
= , (16)
a − 3 πκ 2GM
λ = = =r , (23)
Jeff 4G c2 s
togetherwithEqs. (8), (10)and(15)onemayshowthat r
one obtains the critical mass above which pressureless
ψ~¨+2Hψ~˙ 4πGρ¯ ψ~ = κρ¯m δ. (17) compact stars are unstable to collapse into a black hole
− m k − 4a ∇
√π
DecomposingEq. (17)intoitsparallelandperpendicular M = κ1/2c2G−3/2 M∗, (24)
4 ∼
components one finds
whichisessentiallyequaltothe fundamentalmassofthe
ψ~¨ + 2Hψ~˙ 4πGρ¯ ψ~ = 1κ ρ , (18) theory M∗. The fundamental density, given by
k k− m k −a4∇ m
ψ~¨ + 2Hψ~˙ =0. (19) ρ = M∗ = c2 c2λ−2 , (25)
⊥ ⊥ ∗ R3 κ ∼ G eff
∗
Integrating Eq. (19) one finds that ~v = aψ~˙ provides an estimate of the maximum density attainable
a−1, which simply expresses the conserpveac⊥tion of apnecg⊥ula∝r inside stable compact stars (note that adding pressure
momentum. Ontheotherhand,usingEqs. (14)and(17) increases the Jeans scale, leading to a decrease of the
one finally obtains an equation for the evolution of the maximum density with respect to the pressureless case).
cold dark matter density perturbations ByrequiringthatλJeff issmallerthanthesolarradius
R one obtains the following conservative constraint
⊙
δ¨+2Hδ˙ 4πGρ¯ δ =c2 2δ, (20)
− m seff∇ κ< 4GR2 3 107m5kg−1s−2, (26)
wherec2 =κρ¯ /4istheeffectivesoundspeedsquared π ⊙ ∼ ×
seff m
of the cold dark matter in the EiBI theory of gravity. whichisabouttwoordersofmagnitudeweakerthanthat
In Fourier space one obtains derived in [18], where a detailed model for the structure
of the sun has been considered. However,much stronger
k2c2 constraintson κ may be obtainedby consideringsmaller
δ¨~k+2Hδ˙~k− 4πGρ¯m− as2eff!δ~k =0. (21) astTrohperheysaircealseovbejreacltns.aturalsatellitesintheSolarSystem
which are massive enough to relax to a rounded shape
where~kisthecomovingwavenumberandk = ~k . Inthis throughtheirinternalgravity. Someofthemhaveradiiof
letter we assume that c2 > 0, or equivalen|tly| κ > 0, theorderof100km[21]. SubstitutingR byR=100km
seff ⊙
in order to avoid unwanted instabilities associated with inEq. (26)improvestheκconstraintbymorethan7or-
an imaginary effective sound speed. ders of magnitude. On the other hand, neutron stars
4
[22] with a typical radius of about R 12km (nearly sureless cold dark matter fluid has a non-zero effective
NS
∼
5ordersofmagnitude smallerthanR )arealsoheldto- Jeans length. We used this result to provide a physical
⊙
gether by gravity. They have core densities which are interpretation of the fundamental length R , mass M
∗ ∗
predicted to be larger than ρ 1017kgm3. By re- anddensityρ ofthe theoryandto obtainstringentlim-
c ∗
quiring that ρ > 1017kgm3 one∼obtains the constraint its on κ, the only additional parameter of theory with
∗
κ<1m5kg−1s−2 whichissimilartotheoneobtainedin respect to general relativity.
[17]consideringarelativisticmodelforinternalstructure The cosmologicaland astrophysicalconstraintscanbe
of the neutron star. An even tighter constraint may be summarized as κ < GR2, where R is either the Hub-
obtainedbyrequiringthattheminimumscaleofcompact ble radius at the∼onset of primordial nucleosynthesis
objects which are held together by gravity λJeff ∼ R∗ (tnuc ∼1s)orthescaleofcompactobjectswhichareheld
is smaller than 12km. Substituting R⊙ by R=12kmin together by gravity. The strongest astrophysical limit
Eq. (26) one obtains the constraint (κ < 10−2m5kg−1s−2) is about 10 orders of magnitude
strongerthanbigbangnucleosynthesisconstraints,yield-
κ<10−2m5kg−1s−2, (27) ing a constraint on the fundamental mass (M < 5M )
∗ ⊙
and density (ρ >9 1018kgm−3) of the theory. These
∗
which is more than 9 orders of magnitude stronger than ×
limits imply that large changes with respect to the dy-
the limit given in Eq. (26) and more than 7 orders of
namics of the standard cosmological model in the early
magnitude tighter than the solar constraints obtained in
Universe are expected in the context of the EiBI theory
[18]. Eq. (27) also strengthens the constraint given in of gravity but only for cosmic times t<10−5s.
[17] by two orders of magnitude.
IV. CONCLUSIONS Acknowledgments
In this letter we determined astrophysical and cosmo- We thank Margarida Cunha for useful com-
logicalconstraints ona recently proposedEiBI theory of ments on the manuscript. This work is partially
gravity. We have shown, using a generalized version of supported by FCT-Portugal through the project
the Zel’dovichapproximation,that in this theorya pres- CERN/FP/116358/2010.
[1] M. Born and L. Infeld, Proc.Roy.Soc.Lond. A144, 425 0910.4693.
(1934). [13] I.Gullu,T.C.Sisman,andB.Tekin,Class.Quant.Grav.
[2] M. Banados and P. G. Ferreira, Phys.Rev.Lett. 105, 27, 162001 (2010), 1003.3935.
011101 (2010), 1006.1769. [14] M. Alishahiha, A. Naseh, and H. Soltanpanahi,
[3] S. Deser and G. Gibbons, Class.Quant.Grav. 15, L35 Phys.Rev. D82, 024042 (2010), 1006.1757.
(1998), hep-th/9803049. [15] I. Gullu, T. C. Sisman, and B. Tekin, Phys.Rev. D82,
[4] D. N. Vollick, Phys.Rev. D69, 064030 (2004), gr- 124023 (2010), 1010.2411.
qc/0309101. [16] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis
[5] M. N. Wohlfarth, Class.Quant.Grav. 21, 1927 (2004), (2011), 1106.2476.
hep-th/0310067. [17] P.Pani,V.Cardoso,andT.Delsate,Phys.Rev.Lett.107,
[6] J. Nieto, Phys.Rev. D70, 044042 (2004), hep- 031101 (2011), 1106.3569.
th/0402071. [18] J. Casanellas, P. Pani, I. Lopes, and V. Cardoso, Astro-
[7] D. Comelli, Phys.Rev. D72, 064018 (2005), gr- phys.J. 745, 15 (2012), 1109.0249.
qc/0505088. [19] K. Nakamura and P. D. Group, Journal of Physics G:
[8] D. N. Vollick, Phys.Rev. D72, 084026 (2005), gr- Nuclear and Particle Physics 37, 075021 (2010).
qc/0506091. [20] Y. B. Zel’Dovich, Astron. & Astrophys.5, 84 (1970).
[9] Y. M. Zinoviev, Journal of High Energy Physics 2006, [21] B. J. and Buratti, in Encyclopedia of Physical Science
009 (2006). and Technology (Third Edition), edited by E. in Chief:
[10] R.FerraroandF.Fiorini,Phys.Rev.D78,124019(2008), Robert A. Meyers (Academic Press, New York, 2003),
0812.1981. pp. 329 – 355, third edition ed.
[11] F. Fiorini and R. Ferraro, Int.J.Mod.Phys. A24, 1686 [22] J. Lattimer and M. Prakash, Science 304, 536 (2004),
(2009), 0904.1767. astro-ph/0405262.
[12] R. Ferraro and F. Fiorini, Phys.Lett. B692, 206 (2010),
