Filtering out the cosmological constant in the Palatini formalism of modified gravity

0.62 MB·English

by  Florian Bauer

Filtering out the cosmological constant in the Palatini
formalism of modified gravity
Florian Bauer
High Energy Physics Group, Dept. ECM, and Institut de Ciències del Cosmos (ICC)
Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Catalonia, Spain
Email: [email protected]
1 Abstract
1
0 According to theoretical physics the cosmological constant (CC) is expected to be
2 muchlargerinmagnitudethanotherenergydensitiesintheuniverse,whichisinstark
n contrast to the observed Big Bang evolution. We address this old CC problem not
a by introducing an extremely fine-tuned counterterm, but in the context of modified
J gravity in the Palatini formalism. In our model the large CC term is filtered out, and

0 it does not prevent a standard cosmological evolution. We discuss the filter effect in
2
the epochs of radiation and matter domination as well as in the asymptotic de Sitter

future. The final expansion rate can be much lower than inferred from the large CC
]
c withoutusingafine-tunedcounterterm. Finally,weshowthattheCCfilterworksalso
q
in the Kottler (Schwarzschild-de Sitter) metric describing a black hole environment
-
r with a CC compatible to the future de Sitter cosmos.
g
[

1 Introduction
2
v
6
The starting point of this work is a CC or equivalently a vacuum energy density Λ of
4
5 enormous magnitude. This expectation is suggested by contributions to the CC coming
2 from phase transitions in the early universe, zero-point energy in quantum field theory
.
7 or even from quantum gravity. In general, all these parts are of different magnitude and
0
probably unrelated to each other. Hence, the sum Λ of all terms is dominated by the
0
largest contribution. Since other energy sources dilute with the expansion of the universe,
1
: the CC will eventually take control over the cosmos. Depending on its sign the CC would
v
i induce in the very early universe either a Big Crunch or an eternal de Sitter phase with
X
a very high Hubble rate H ∝ Λ. Obviously, the standard Big Bang evolution does not
r
a happen in this case.
The simplest way to avoid this problem is the introduction of a CC counterterm Λ ,
ct
which makes the sum |Λ+Λ | smaller than the currently observed critical energy dens-
ct
ity ρ ∼ 10−47GeV4. For concreteness let us assume that Λ ∼ −M4 were related to the
c0 ew
electroweak phase transition at the energy scale M ∼ 102GeV. Then the counterterm
ew
must be extraordinarily close to (−Λ) requiring an enormous amount of fine-tuning,
(cid:12) (cid:12)
(cid:12)(cid:12)1+ Λct(cid:12)(cid:12) < (cid:12)(cid:12)ρc0(cid:12)(cid:12) ∼ 10−55. (1)
(cid:12) Λ (cid:12) (cid:12) Λ (cid:12)
Apartfromthefine-tuningoftheclassicalcounterterm, thesituationisevenmoreinvolved
when quantum corrections are included, cf. Ref. [1] for an elaborated discussion in the
1
contextoftheelectroweaksectorofthestandardmodelofparticles. Moreover,theproblem
worsens when Λ is dominated by higher energy scales, possibly originating from grand
unified theories where Λ ∼ (1016GeV)4 or quantum gravity with Λ ∼ (1019GeV)4 for
instance. Summingup, thefine-tuningoftheCCisconsideredtobeoneofthemostsevere
problems in theoretical physics [2, 3]. In addition, the current accelerated expansion of the
universe [4, 5, 6] can be explained very well by a tiny CC of the same magnitude as the
energy density of matter, giving rise to the so-called coincidence problem. For the latter
problem many explanations have been proposed [7, 8, 9, 10, 11], which induce late-time
accelerated expansion. However, most of these models tacitly assume that the large Λ has
been fine-tuned away and thus they do not address the big CC problem.
Without fine-tuning we have to accept the existence of the presumed huge CC, and
we have to find a way to neutralise its effects in order to obtain a reasonable cosmological
evolution. Along this line, several proposals have been made, e.g. relaxation models for a
large CC in the context of matter with an inhomogeneous equation of state (EOS) [12],
or in the LXCDM framework [13] with a variable cosmological term [14, 15], see also [16,
17, 18, 19, 20, 21]. Removing or filtering out vacuum energy has been investigated e.g. in
Refs.[22,23,24,25], anditisafeatureinunimodulargravity[26,27,28,29,30]. Recently,
aCCrelaxationmodelhasbeendiscussedinthecontextofmodifiedgravitywithanaction
functional f(R,G) involving the Ricci scalar R and the Gauß-Bonnet invariant G in the
metric formalism [31, 1], where the action is varied with respect to the metric g only.
ab
In this work, we also consider a modified gravity model with an action functional f
in terms of the Ricci scalar R and the squared Ricci tensor Q = R Rab. However, here
ab
we apply the Palatini formalism, where the metric g and the connection Γa are treated
ab bc
independently by the variation principle. In contrast, the metric formalism requires from
the beginning that
1
Γa [g] = gad(g +g −g ) (2)
bc 2 dc,b bd,c bc,d
is the Levi-Civita connection of g , whereas the Palatini connection depends also on the
ab
matter sector. As we will show in this paper, this allows the construction of a filter for
a large CC, and the results will be similar in effect to unimodular gravity. However, in
our setup the CC is not eliminated completely, but it appears in suppressed corrections.
Furthermore, we investigate the filter effect from the early universe till the asymptotic
future in addition to black hole environments. It turns out that finite vacuum energy
shifts originating e.g. from phase transitions can be neutralised, too.
There is an interesting conceptual difference to the CC relaxation models in the metric
formalism, wherethe large vacuum energyis not filteredout from the total energycontent,
but gravity is modified such that Λ does not induce large curvatures. This happens on the
level of the Einstein equations, i.e. by solving differential equations to obtain concrete low-
curvaturesolutions. InthePalatiniframeworkofthispaper,wewillshowthatthelargeCC
canberemovedalreadyinanalgebraicwaybeforesolvingdifferentialequations. Moreover,
thelatterareonlyofsecondorder, whereasthemetricversionofmodifiedgravitygenerally
involvesahigherdifferentialorder,signallingtheexistenceofnewdegreesoffreedom,which
can be the source of new instabilities and other problems. More differences between both
formalisms will become visible in the forthcoming discussions in this paper.
Some work on the Palatini formalism can be found e.g. in Refs. [32, 33, 34, 35, 36, 37],
and comparisons with the metric and other formalisms were made in Refs. [38, 39, 40, 41,
42, 43, 44]. Non-trivial properties in Palatini models have been investigated in Refs. [45,
46, 47, 48], and finally the resolution of the Big Bang singularity has been proposed in this
2
context [49, 50], possibly in connection with loop quantum gravity [51].
The paper is organised as follows, in Sec. 2 we briefly reproduce how to solve f(R,Q)
modified gravity models. In Sec. 3 we present our model which filters out the large CC.
The results will be applied to cosmology in Sec. 4 discussing the radiation, matter and
late-time de Sitter era. Finally, in Sec. 5 we show that the CC filter works also for the
Kottler (Schwarzschild-de Sitter) solution describing a black hole in the presence of a CC.
We conclude in Sec. 6 and give an outlook to future developments.
InthisworkthespeedoflightcandthePlanckconstant(cid:126)aresettounity,thesignature
of the metric is (−1,+1,+1,+1).
2 f(R,Q) modified gravity in the Palatini formalism
The action of our setup is given by
ˆ
(cid:20) (cid:21)
(cid:112) 1
S = d4x |g| f(R,Q) +S [g φ], (3)
mat ab,
2
whereg isthe“physical” metric,onwhichthematterfieldsφinS propagate. TheRicci
ab mat
scalar R and the squared Ricci tensor Q depend on both the metric and the connection Γa
bc
while the Ricci tensor R is defined only in terms of the latter:
ab
R [Γ] = Γe −Γe +Γe Γf −Γe Γf (4)
ab ab,e eb,a ab fe af eb
R[g,Γ] = Ra = gabR (5)
a ab
Q[g,Γ] = RabR = gacgbdR R . (6)
ab ab cd
Note that in general these quantities are different from their metric versions, and the
absence of torsion implies that the connection is symmetric. Moreover, we restrict our
discussion to the case of a symmetric Ricci tensor R in the action (3). This property is
ab
not automatic even for symmetric connections, which was shown recently in Ref. [52].
With these preliminaries the variation 2δS/δgab = 0 of the action functional S with
respect to g yields the modified Einstein equations
ab
1
f R n+2f R aRn− δ nf = T n, (7)
R m Q m a 2 m m
where the energy-momentum tensor T n emerges from the term S , and f and f are
m mat R Q
partial derivatives of f with respect to the scalars R and Q. Moreover, we obtain from
δS/δΓa = 0 the equation of motion (EOM) for the Palatini connection,
bc
(cid:104)(cid:112) (cid:105)
∇ |g|(f gmn+2f Rmn) = 0, (8)
a R Q
where ∇ denotes the covariant derivative in terms of the yet unknown connection Γa .
a bc
At this point we should remark that in the metric formalism the term Q yields EOMs
with higher-order derivatives and problematic extra degrees of freedom. Generally, it is
difficult to avoid instabilities, e.g. of the Ostrogradski-type1 [53]. In contrast, the Palatini
1In the metric approach this type of instability can be avoided in gravity actions depending only on
the Ricci scalar R and the Gauß-Bonnet term G. It would be interesting to compare both approaches in
the context of f(R,G) models. However, G contains the squared Riemann tensor, and to our knowledge
no method to solve the corresponding Palatini EOMs has been found yet.
3
formalism provides second-order EOMs for our scenario just as standard general relativity,
and problems from extra degrees of freedom do not occur.
In the following, we work along the lines of Ref. [49], where a procedure for solving the
Palatini EOMs is discussed. The strategy is as follows: the formal solutions to both EOMs
in (7) and (8) relate in an algebraic way the geometrical scalars R and Q with the physical
metric g and the energy-momentum tensor T n, respectively. Thus, one can express R
ab m
and Q in terms of the matter content alone by solving this set of equations. Subsequently,
the results are plugged back into the formal solutions to obtain the explicit form of the
Palatini connection Γ and all the quantities derived from it. Since the results will involve
themetricg anditsderivatives, itispossibletorelatethecosmicexpansionratewiththe
ab
matter energy density, for instance. In the rest of this section, we explain the procedure
for general f(R,Q) models.
First, let us write Eq. (7) in matrix form by introducing the 4×4-matrices Pˆ = Rn =
m
R gan and Tˆ = T n, whose entries are just the components of the corresponding tensors
ma m
components. With the identity matrix Iˆwe obtain
1
f Pˆ +2f (Pˆ)2− fIˆ= Tˆ, (9)
R Q
2
and the trace of this equation reads
f R+2Qf −2f = T, (10)
R Q
where R = tr(Pˆ), Q = tr(Pˆ2) and T = T m.
m
For determining the Palatini connection Γ we introduce the auxiliary metric h and
mn
consider the equation
(cid:104)(cid:112) (cid:105)
∇ [Γ] |h|hmn = 0, (11)
a
where ∇ [Γ] is the covariant derivative in terms of Γ. Consequently, in order to solve this
a
equation the connection Γ must be compatible with h , i.e. it has to be the Levi-Civita
mn
connection of h ,
mn
1
Γa [h] = had(h +h −h ), (12)
bc 2 dc,b bd,c bc,d
just as in general relativity. Hence, we convert Eq. (8) into (11) by defining hmn in the
following way,
(cid:112)|h|hˆ−1 = (cid:112)|g|gˆ−1Σˆ with Σˆ = (cid:16)f Iˆ+2f Pˆ(cid:17), (13)
R Q
where the metric h (and analogously g ) has been written in matrix notation as hˆ =
mn mn
h with its inverse hˆ−1 = hmn. Calculating the determinant of both sides, we find
mn
h2h−1 = h = gdet Σˆ, which allows to eliminate (cid:112)|h| and finally yields
gˆ−1Σˆ (cid:113)
hmn = hˆ−1 = , h = hˆ = |detΣˆ|Σˆ−1gˆ. (14)
(cid:113) mn
|detΣˆ|
Since the connection in (12) solves Eq. (11), the formal solution of Eq. (8) is also given
by Γa [h] in (12) if h is defined as in (14).
bc mn
TheremainingstepistofindPˆ whichrequiresanexplicitformfortheenergy-momentum
tensor T n. Here, we assume that the matter sector can be described by a perfect fluid,
m
T n = (ρ+p)u un+(p−Λ)δ n, (15)
m m m
4
where ρ and p are the energy density and pressure of (ordinary) matter, including e.g.
dust and incoherent radiation. And u denotes the corresponding 4-velocity vector of the
m
matter field. Λ represents the energy density corresponding to the cosmological constant,
and it contains all vacuum energy contributions. Thus we require p (cid:54)= −ρ without loss of
generality. Next, let us write the matrix expression (9) in the following way
(2f )2Mˆ2 = x2Iˆ+µu(cid:92)un (16)
Q m
1f
Mˆ := Pˆ + R Iˆ (17)
4f
Q
1
x2 := 2f (p−Λ)+f f + f2 (18)
Q Q 4 R
µ := 2f (ρ+p). (19)
Q
By explicit calculation one can check that
c ·2f Mˆ = xIˆ+yu(cid:92)un (20)
a Q m
with
(cid:112)
−x+c · x2+µ(u um)
b m
y := (21)
(u um)
m
is a solution to Eq. (16), which yields Pˆ. The constants c = ±1 and the sign convention2
√ a,b
for x = x2 will be fixed later by consistency considerations.
Now, we have to determine the scalars R and Q, which follow from the trace equa-
tion (10) and the trace of (20),
c (2f R+2f ) = 4x−y, (22)
a Q R
where u um = −1 will be used from here on. In the last equation we eliminate all roots
m
by squaring twice, which results to
(cid:16) (cid:17)2
(2f R+2f )2+8x2+µ = 36x2(2f R+2f )2, (23)
Q R Q R
where c = ±1 and odd powers of x have dropped out. Solving this algebraic equation
a,b
together with (10) is the tough part of the Palatini formalism. Once this task is achieved,
R,QandPˆ = Rn in(20)aregivenasfunctionsofρ,p,Λonly,andcanbeusedtocalculate
m
the connection Γ. For this purpose, let us write the matrix Σˆ in (13) as
(cid:18) (cid:19)
1 L
Σˆ = 2f Mˆ + f Iˆ= L Iˆ+L u(cid:92)un = L Iˆ+ 2 u(cid:92)un , (24)
Q R 1 2 m 1 m
2 L
1
1
with L := c x+ f , L := c y. (25)
1 a R 2 a
2
Using det(Iˆ+a(cid:91)bn) = 1+bma we find det(Σˆ) = L3(L −L ), and the inverse matrix
m m 1 1 2
reads
1 L
Σˆ−1 = Iˆ− 2 u(cid:92)un.
m
L L (L −L )
1 1 1 2
2Changing the sign of x in Eqs. (20) and (21) by x → −x corresponds to c → −c , and therefore it
a a
(cid:112) (cid:112)
does not yield a new solution. Here, we use the convention x2+µ(u um)=x 1+µ(u um)/x2, and
√ m m
the second possibility −x 1+··· would just mean c →−c .
b b
5
Finally, we have all ingredients to write down the explicit form of the auxiliary metric3
from Eq. (14)
(cid:18) (cid:19)
L
2
h = Ω g − u u (26)
mn mn m n
L −L
1 2
(cid:18) (cid:19)
L
hmn = Ω−1 gmn+ 2umun , (27)
L
1
where
(cid:113)
|detΣˆ| (cid:112)|L3(L −L )|
Ω := = 1 1 2 . (28)
L L
1 1
Once h is known, the connection Γa follows directly from Eq. (12), and subsequently
mn bc
the Ricci tensor (4) and the scalars in Eqs. (5) and (6) can be calculated.
3 Relaxing the CC with a filter
In this section we study a modified gravity model to relax the CC in the Palatini f(R,Q)
framework. Motivated by earlier work in the metric formalism [31, 1] we consider the
following ansatz for the gravity action
Rn
f(R,Q) = κR+z with z := β , B = R2−Q, (29)
Bm
where κ and β are non-zero constant parameters and n and m positive numbers. In the
metric formalism model [31] the structure of the function B was enforcing the universe
to expand like a matter dominated cosmos even when the matter energy density ρ was
m
much smaller in magnitude than the vacuum energy density Λ. However, in the Palatini
framework the function B is not known in the beginning, and it is necessary to investigate
under which circumstances a relaxed cosmological expansion behaviour can be obtained.
Moreover, we will see in the following that z is not a correction to the Einstein-Hilbert
term but a crucial part of the action functional f. Therefore, one should refrain from
considering the limit z → 0.
From Eq. (29) we find
κR+nz z
f = −2f R, f = m , (30)
R Q Q
R B
and the trace of the stress tensor (15) reads
T = −4Λ+3p−ρ. (31)
As a result, Eq. (10) provides the first equation for finding R and B (or Q),
γz = κR−4Λ+3p−ρ, γ := (n−2−2m), (32)
while the second one is given in Eq. (23), explicitly
0 = f3R2S +f2 S +f R−2S (33)
Q 3 Q 2 Q 1
3The relation in Eq. (26) is called a disformal transformation [54], which is used e.g. in MOND theor-
ies [55] and scalar field models for dark energy [56].
6
(cid:20) (cid:21)
1
S := 72 κR+ρ+p+ z(2+2n+γ) (34)
3
3
S := 4(cid:2)z2(cid:0)−17n2+4n(2+γ)+4(2+γ)2(cid:1)
2
+ 12z(ρ+p)(2−n+γ)+6zκR(4−5n+2γ)
+ 9(ρ+p)2−9(κR)2(cid:3) (35)
S := 24(cid:2)z(κR+nz)2(n−2−γ)(cid:3). (36)
1
Apparently, this set of equations is quite complicated, thus we will solve it approximately.
First, we consider in this work only the epochs when the large CC Λ dominates over all
other energy sources. Then Eq. (32) implies the relation z = βRn/Bm = O(Λ) suggesting
thatB/R2 andf−1 = B/(mz)arerelativelysmallquantities,whichmaybeusedasexpan-
Q
sion parameters in Eq. (33). At this point we cannot prove this suggestion quantitatively
because R and B are not known yet. However, it will be confirmed later by Eqs. (43)
and (45). As a result of assuming that f−1 is sufficiently small, the term S proportional
Q 3
to f3 in (33) will be the most important one, and a good zero-order solution can be found
Q
by neglecting the other terms f2 S +f R−2S and solving only S = 0.
Q 2 Q 1 3
Our goal is a relaxed universe, i.e. one which is not dominated by the large CC term,
and it can be realised by requiring that the Ricci scalar R following from S = 0 is
3
free from large O(z) contributions. This happens when the parameter n is restricted by
2+2n+γ = 3n−2m = 0, which eliminates the O(z) term in Eq. (34). We will apply this
condition from now on, hence Eq. (33) can be written as
0 = κR+r (37)
(cid:18) (cid:19)(cid:20) (cid:21)
− 2mz B 1+ 3 (3κR+2r)+ 9 (cid:0)(κR)2−r2(cid:1)
9 R2 mz 4(mz)2
8 (cid:18) B (cid:19)2(cid:18) 3κR (cid:19)2
+ mz 1+ with r := ρ+p.
27 R2 2(mz)
FromthefirstlineinthelastequationoneclearlyobservesthatCCtermswithEOSp = −ρ
do not contribute to the Ricci scalar at leading order. In other words, the CC is filtered
out from r = ρ+p, which describes the matter sector. Note that z = O(Λ) still appears
in suppressed correction terms.
Inthefollowing,wewillsolveEq.(37)insituationsrelevantforcosmology. Afirstorder
correction to S = 0 can be found by keeping only the term z(B/R2) from the second line
3
in (37), which leads to
(cid:18) (cid:19) (cid:18) (cid:19)1
κR+r = 2mz B = 2mz β m R−43, (38)
9 R2 9 z
where we used z = βRn/Bm from Eq. (29) with 3n = 2m in the last step. Thus, we obtain
a 7th-order polynomial equation in R,
(cid:32) (cid:18) (cid:19)1 (cid:33)3
(κR+r)3(κR)4 = L := κ4 2mz β m , (39)
3
9 z
which clearly shows that R is a function of ρ and p only. Finally, we find the approximate
solution
(cid:18)D2(cid:19) (cid:18)L3(cid:19)13
κR = −r+D+O with D := , (40)
r r4
7
when the matter-related quantity r lies in the range |D| (cid:28) r (cid:28) |Λ|, which we refer to as
the early-time limit or the limit of large energy density in r. Moreover, κR < 0 because
r > 0 for ordinary matter, and κR will remain negative for decreasing ρ because κR → 0
is not a solution of Eq. (39).
In the opposite limit, r → 0, which is called the late-time limit in the following, we
obtain from Eq. (39)
3 (cid:18)r2(cid:19) 1
κR = ρ − r+O with ρ := L7, (41)
e 7 ρ e 3
e
indicating that κR approaches the negative constant ρ for vanishing matter. For a given
e
value of Λ or z the parameter β must be chosen adequately for L = ρ7 < 0. In Sec. 4.2 we
3 e
will show that (−ρ ) is close to the critical energy density in the asymptotic future, which
e
corresponds to the tiny observed value of the effective CC in Eq. (83). The parameter
dependence of the solution set of the full equation (37) might further constrain β, however
thishastobedeterminednumerically. InFig.1weshowanexampleform = 3andρ < 0,
e
whichnicelydemonstratesthevalidityofourapproximations. Notethatnotallvaluesofm
mightallowphysicallyreasonablesolutions. Wewilldiscusssomeexamplesforβ attheend
of Sec. 4.2. However, in this work we concentrate on analytical results, and the complete
parameter dependence as well as the case L > 0 will be investigated elsewhere.
3
Viaz = β(R2/3/B)m fromEq.(29)itisstraightforwardtoobtainB andB/R2 fromthe
approximate solution of R. In the following we denote subdominant corrections by ε (cid:28) 1.
Accordingly, at early times Eq. (40) yields
(cid:114)L (cid:18)2 (cid:19)−1 (cid:18) 2D (cid:19)
B = 3 mz (−κ)−2 1− +O(ε2) , (42)
D 9 3 r
(cid:18) (cid:19)
B D D
= 1− +O(ε2) , (43)
R2 (cid:0)2mz(cid:1) r
9
whereas from Eq. (41) we obtain the corresponding late-time results
(cid:18)2 (cid:19)−1 (cid:18) 2 r (cid:19)
B = ρ3 mz (−κ)−2 1− +O(ε2) , (44)
e 9 7ρ
e
B (cid:18)2 (cid:19)−1(cid:18) 4 r (cid:19)
= ρ mz 1− +O(ε2) . (45)
R2 e 9 7ρ
e
These results confirm that B/R2 is sufficiently small to justify that we neglected some
terms in Eq. (37), hence, we have found consistent solutions for R and B. More support
forthevalidityoftheapproximationsisprovidedbythenumericalsolutionofthecomplete
Eq. (37) in Fig. 1.
With R and B as functions of only the matter-related term r, it is possible to calcu-
late L in (25). First, we have to define the square root of x2 in (20). Since we work
1,2
in the limit |R2/B| (cid:29) 1, which implies |f R| = |mzR/B| (cid:29) z/R, we identify f R to be
Q Q
the dominant term in Eqs. (30) and (18). Thus f ≈ −2f R and x2 ≈ 1f2 ≈ (f R)2,
√ R Q 4 R Q
and we choose the convention x = f R 1+···, where ··· denotes the remaining terms in
Q
Eq. (18) divided by (f R)2. Next, we plug x ≈ −1f ≈ f R into the trace equation (22)
Q 2 R Q
(cid:113)
c (2f R+2f ) = 3x+c ·x 1−(2f r)x−2, (46)
a Q R b Q
8
8
7
exact result
6 early time
late time
5
κR/ρ 4
e
3
2
1
0
0 1 2 3 4 5 6 7 8
-r/ρ
e
Figure 1: Numerical solutions for κR < 0 as a function of r = ρ with p = 0. The plot
showstheexactresultinEq.(37)(blacksolidcurve)andtheapproximationsatearlytimes
(red dashed) from Eq. (40) as well as at late times (blue dashed-dotted) from Eq. (41),
respectively. In this example with m = 3 the parameter β < 0 has been chosen such that
−|Λ| ≈ 358ρ only for numerical reasons. A more realistic ratio of |Λ/ρ | would be much
e e
larger, however the results would not differ qualitatively, which is true also for other values
of the EOS p/ρ as long as r (cid:28) |Λ|. In any case, κR approaches −r (dotted diagonal line)
in the region −ρ (cid:28) r (cid:28) |Λ|.
e
9
which gives in leading order c (−2)f R = f R(3+c ). Obviously, c = c = −1 is the
a Q Q b a b
only solution4 in the large |f R| limit, and we find
Q
1
L = −x+ f
1 R
2
(cid:18) 3 1 (cid:19) (mz)2
= −2f R+R−1 mz+ κR− r + (1+ε), (47)
Q 4 4 6f R3
Q
(cid:113)
L = −x−x 1−(2f r)x−2
2 Q
(cid:18)4 1 (cid:19) (mz)2
= −2f R+R−1 mz+ (κR+r) + (1+ε), (48)
Q 3 2 3f R3
Q
where we used a series expansion of the root in
(cid:18) (cid:19)
2
f = −2f R+R−1 κR+ mz , (49)
R Q
3
(cid:115)
8mz+3κR−3r (2mz+3κR)2
x = f R 1− + . (50)
Q 6f R2 36f2R4
Q Q
Above and from here on ε denotes small terms of the order κR/z, r/z or B/R2. Moreover,
Λ has been eliminated in favour of
3
z = − (κR−4Λ+3p−ρ). (51)
6+4m
Finally, we write down the series expansion of
(cid:18) (cid:19)
1mz 1 B
L −L = − 1+ (1+ε) , (52)
1 2 3 R 2R2
which appears in the auxiliary metric h . Now, we have all ingredients available for
ab
discussing solutions to the Palatini field equations in the next sections.
4 Cosmology
For investigating the cosmological evolution the physical metric is of the spatially flat
Friedmann-Lemaître-Robertson-Walker (FLRW) type with g = −1 and g = a2(t) in
00 ii
Cartesian coordinates, where a(t) is the scale factor as a function of cosmological time t.
The matter component is at rest (um = −δm) in these coordinates. Accordingly, the
0
auxiliary metric h in Eq. (26) is completely defined by the diagonal elements h = Ωg
ab ii ii
and h = Ω(g −L /(L −L )), which follow explicitly from Eqs. (47), (48) and (52),
00 00 2 1 2
(cid:112) (cid:114)
Ω = |L1(L1−L2)| = sgn(L ) |2(mz)2B−1|(cid:0)1+ε2(cid:1), (53)
1
L 3
1
−L (cid:18) R2(cid:19)(cid:18) B (cid:19)
1
h = Ω = Ω −6 1− (1+ε) . (54)
00 L −L B R2
1 2
4Notethatourscenariowithc =−1reliesfromtheverybeginningontheexistenceofmatter,because
b
otherwise the vector u in Eq. (16) were absent and c = +1 would be required in Eq. (21). However,
m b
matter is a fact of reality, which singles out c =−1 in our model. Consequently, Eq. (41) is the correct
b
solution in the r→0 limit.
10