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Phys. Scr. 99 (2024)065050 https://doi.org/10.1088/1402-4896/ad49e4
PAPER
Cosmologies in f(R,
m
) theory with non-minimal coupling between
geometry and matter
Sergio Bravo Medina
1
, Marek Nowakowski
2,∗
, Ronaldo V Lobato
3
and Davide Batic
4
1
Departamento de Física, Pontificia Universidad Javeriana, Cra.7 No.40-62, Bogotá, Colombia
2
ICTP-South American Institute for Fundamental Research, Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 São Paulo, SP Brazil
3
ICRANet, Piazza della Repubblica 10, Pescara, 65122, Italy
4
Department of Mathematics, Khalifa University of Science and Technology, Sas Al Nakhl Campus, PO Box 2533 Abu Dhabi, United Arab
Emirates
∗
Author to whom any correspondence should be addressed.
E-mail: sergiobravom@javeriana.edu.co,marek.nowakowski@ictp-saifr.org,contact@rvlobato.com and davide.batic@ku.ac.ae
Keywords: cosmology in modified gravity, direct matter-geometry coupling, (
)
f
R,mtheory
Abstract
Among the recent extensions to standard General Relativity,
(
)
f
R,
mgravity has risen an interest
given the possibility of coupling between geometry and matter. We examine the simplest model with
non-minimal coupling in the context of cosmology. We pay special attention to the question of how
far this model could reproduce the observational fact of our universe.
1. Introduction
Since the discovery of the accelerated expansion of the Universe, cosmologists have developed the ΛCDM
model, a standard cosmological model rooted in Einstein’s general relativity [1]. This model incorporates a
positive cosmological constant Λ[2], the Friedmann-Robertson-Walker metric [3], and the concept of Dark
Matter (DM)[4]. The otherwise robust model is not completely without problems. To start with, despite
numerous efforts, a candidate for DM has not yet been found. The second problem is coined as Hubble tension,
careful phrasing of the fact that different measurements of the Hubble constant yield different results [5]. In the
future, as we seek to explain these issues, we may require a new theoretical model. This could involve modifying
Einstein’s gravity [6], replacing the cosmological constant with different models of Dark Energy (DE)[7], and
exploring alternative candidates for DM [8]. The number of extensions of general relativity as well as the number
of DE models is, of course, quite large [9–30].
Despite the remarkable success of the ΛCold Dark Matter (ΛCDM)model in explaining a wide array of
cosmological observations, from the cosmic microwave background (CMB)anisotropies to the large-scale
structure of the Universe, several significant challenges remain unresolved. The nature of dark energy,
epitomized by the cosmological constant Λ, poses a profound theoretical conundrum, known as the
‘cosmological constant problem’[2]. The astonishingly small value of Λ, as required to explain the observed
acceleration of the Universe, stands in stark contrast to theoretical predictions from quantum field theory,
typically larger by many orders of magnitude [2].
Moreover, the ΛCDM model does not escape from the Hubble tension, a growing discrepancy between the
values of the Hubble constant, H
0
, measured directly from local astronomical observations and those inferred
from the CMB under the ΛCDM framework [31]. These inconsistencies point to potential physics beyond the
standard model, either in the form of new particles or fields, or through modifications to the theory of general
relativity itself.
In response to these challenges, a plethora of modified gravity theories have been proposed as alternatives to
general relativity, aiming to provide a more comprehensive theoretical framework that can naturally incorporate
the phenomena attributed to dark energy and dark matter. Theories such as f(R)gravity, where the Ricci scalar R
in the Einstein-Hilbert action is replaced by a function f(R), offer promising ways to explain cosmic acceleration
without the need for a cosmological constant [9,10].
RECEIVED
14 March 2024
REVISED
29 April 2024
ACCEPTED FOR PUBLICATION
10 May 2024
PUBLISHED
28 May 2024
© 2024 IOP Publishing Ltd
Expanding further, f(R,T)theories introduce a coupling between matter, represented by the trace of the
energy-momentum tensor T, and geometry, providing a framework to explore the effects of such couplings on
the dynamics of the Universe [11]. The
()
f
R,
mmodels we focus on in this paper extend this idea by
incorporating the matter Lagrangian
m
directly into the gravitational action, potentially offering new insights
into the interaction between dark matter and dark energy, and their impact on the evolution of cosmic
structures [32].
Our motivation for selecting
()
f
R,
mgravity stems from its ability to unite the geometrical and material
sectors of the Universe in a single, coherent theoretical framework. This approach not only allows for the
exploration of the cosmic acceleration and dark matter problems from a new angle but also provides a platform
for testing the limits of Einstein’s general relativity on cosmological scales. By investigating the cosmological
implications of
()
f
R,
mgravity, we aim to contribute to the ongoing discussion on viable alternatives to
ΛCDM, exploring whether these theories can offer a more satisfactory explanation of observational phenomena
without some of the fine-tuning issues that plague the standard model.
It is therefore a priori not clear which class of models is theoretically preferred over the others. We think that
simplicity of the modification or, in other words, Occam’s razor, could serve us here as guiding principle. Within
a range of models, this possibility seems quite likely. Among the wide class of models based on a Lagrangian
function of the type f(R), where Ris the Ricci scalar [33], one could choose to focus on R
2
gravity [34]and the
resulting cosmological models [35]. Recently, another class has been examined, based on
()
f
R,
mwhere
m
[32,36,37]represents the matter Lagrangian, often taken to be the energy density ρ, pressure p, or the trace of the
energy-momentum tensor T-with all these quantities being diffeomorphically invariant. In this class, an
additional term in the Einstein-Hilbert Lagrangian of the form
s
Rmwould be considered a mild extension of
the previous model. Moreover, it provides a straightforward coupling of the geometry encoded in Rwith matter
represented astrophysically by
m
. The astrophysical and other implications of such a theory have been
examined in [38–41]with consequences for Neutron star and White Dwarf physics. Some cosmological aspects
with specific choices of
m
have been considered in [42–47]. Except for [43], the choice of the function f(R,L
m
)
is different from ours. The
()
f
R,
mtheories have their own theoretical peculiarities that require careful
examination. One notable aspect is that choosing r=
pT,,
mdoes not lead, through standard metric
variation to the energy-momentum tensor of a perfect fluid T
μν
[48–51]. On the other hand, a constrained
variation based on the Lagrangian with constraints can result in the prefect fluid energy-momentum tensor.
However, this comes at the cost of T
μν
no longer being conserved. We show that explicitly using the new
Friedmann equation in
()
f
R,
m.
In solving the new Friedmann equations, we try to make the model resemble our present universe. While it
seems possible to achieve this by choosing adequate initial values, the evolution into the past and future can
bring some surprises, as will be shown below. One of the simplest and model independent feature of our
universe is the lower limit on its age as given for example by the redshift of the oldest galaxies [52]. A second
constraint independent of the model comes from uranium decays [53]. If a cosmological model, as appealing as
it may be, cannot reproduce these facts, it is then certainly not a good candidate to describe our universe. We will
pay a special attention to these questions while examining the details of the model under discussion.
The paper is organized as follows. In section 2, we outline the basics of
()
f
R,
mtheory with minimal
coupling, denoted as
s
Rm. In this section, we also present the new Friedmann equations with a cosmological
constant in a dimensionless form suitable for numerical integration. In section 3, we present the generalization
of one of the Friedmann equations. In its standard form, it reads Ω
Λ,0
+Ω
m,0
=1. However, in the context of
()
f
R,
mtheory, it generalizes to Ω
Λ,0
+Ω
m,0
+Ω
σ,0
=1. In the same section, we discuss the possible initial
values, a necessary undertaking, since the new Friedmann equation contains the second derivative of the density.
We believe it is necessary to provide a brief overview of the standard cosmological model in its analytical form
(see section 4). This is important because in section 5, we will present the numerical results and compare the new
model with the standard one. In section 6, we will draw our conclusions.
2. The (
)
f
R,mtheory
As a generalization of the standard Einstein-Hilbert Lagrangian in General Relativity, i.e.
()
k
=+R
1
2,1
GR m
where κ=8πG
N
, a new Lagrangian has been proposed by making use of the invariants Rand
m
. It makes use of
a general function,
()
f
R,
m, and reads simply with an action [32,36,37]
2
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
() ()
ò
=-SdxgfR,. 2
m
4
From the formal variation of the action (δ
g
S=0)with respect to the metric, the following field equations are
obtained [44,54]
()() ()+--- =
mn mn mn mn mn
fR g f f f g f T
1
2
1
2,3
RRL
mL
mm
where =¶
¶
f
R
f
R,
=
¶
¶
f
L
f
L
mm
and the energy-momentum tensor T
μν
is given by
() ()
d
d
=-
-
-
mn mn
Tg
g
g
2.4
m
An alternative form of these field equations may be obtained by taking the trace of (3)and solving for ,f
R
,
namely
() ()=+--ffTffL fR
1
6
2
3
1
3.5
RL L
mR
mm
When replacing it back into (3), we obtain the equation
⎛
⎝⎞
⎠
() ()++- - - -=
mn mn mn mn mn
fG f fR f L g f T Tg f
1
6
1
2
1
30. 6
RRL
mLR
mm
Here,
=-
mn mn mn
G
RgR
1
2represents the Einstein tensor. As mentioned in the Introduction, the particular
form of
()
f
R,
mwe wish to work with is the same as the one presented in [39]
() ()
ks=++fR RR,2,7
mmm
where σis the parameter which determines the coupling between matter and geometry.
With respect to the choice of the matter Lagrangian
m
that reproduces the perfect fluid Energy-
Momentum tensor there is a current debate [49,50]. While some authors make the choice
r
=
m, others
suggest =
P
m
[48]with particular values for the polytropic index. With the choice in [39], i.e. =-
P
m
,
the field equations read
⎛
⎝⎞
⎠[] () ()
kss ks-- -- =+
mn mn mn mn mn
PR g P Rg RT
1
2
1
4
1
21, 8
or in the alternative form
⎛
⎝⎞
⎠
() () ()ks ks ks ks-+--+-+=
mn mn mn mn mn mn
PG Rg PRg R T Tg P12 1
33 11
320,9
which is consistent with the equations displayed above. The conservation of the energy-momentum tensor is
given in general by [54]
[( )] ()= ¶
¶
mmn m
mn
TfR
g
2ln , . 10
Lmm
m
This suggests that choosing the matter Lagrangian to be proportional to ρor Pleads to the conservation of the
energy-momentum tensor. However, this is only the case when relying directly on equation (4). Unfortunately,
given the choices of the matter Lagrangian as mentioned above, it does not reproduce the perfect fluid energy-
momentum tensor. To illustrate this point, if we start with
r
=
m, the energy-momentum tensor defined in
(4)will not contain the pressure term. On the other hand, by using a constrained variation in the metric
[48,50,51]or alternatively, a constrained Lagrangian formalism, we can explicitly show that, for example,
choosing the energy density as the matter Lagrangian yields a perfect fluid energy-momentum tensor. However,
its conservation no longer holds, as we introduce the metric through the constraints.
There is a certain confusion on this issue in the literature [43]. This is a point of divergence between our
approach and others. Therefore, we explicitly demonstrate the non-conservation of T
μν
. The equations of
motion, as presented in (3),(8), and (9), remain valid when using the constrained formalism. In this formalism,
T
μν
is the energy-momentum tensor of perfect fluid, which, in general, is not conserved and is defined as
() ()r=+ -
mn m n mn
TPuuPg.11
As mentioned earlier, the model under discussion displays a non-conservation of the energy-momentum tensor,
specifically ∇
μ
T
μν
≠0(at least in general). This non-conservation is also a characteristic of other gravity models
such as Rastall Gravity [55,56], and has been advocated in modular gravity models [57,58]. From the field
equations (3)we can obtain the modified Friedmann equations in the flat (k=0)Friedmann–Lemaitre–
Robertson–Walker metric
3
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
()( ) ( )qqf==-++
mn mn
ds g dx dx dt a t dr r d r dsin . 12
222222222
By taking the 0-0 components of the new field equations, one obtains [44]
() ()
r+-- + =Hf f f R f Hf f31
231
2,13
RRL
mRL
2
mm
where His the Hubble parameter given by
=Haa
. The space components (i.e. μ=i,ν=j)give
() ()
+--+ -=Hf H f f Hf f f f P33
1
2
1
2,14
RRRRL
mL
2
mm
where the
f
R
term comes from the double derivatives in the field equations (,and ∇
μ
∇
ν
). For the model we
have chosen, we have
()
ks s sr sr=+ =+ = =ffRff
1
2,1, , . 15
RmLRR
m
Together with the choice
r
=
m, the new Friedmann equations read
[] ()
krsk r r r=+ + -HHHH
346 2, 16
22
()[() ]()
krskr rr=- + - + + + +
a
aPHP HP H
63362. 17
2
In general relativity, the two Friedmann equations lead to the conservation law
()
r
r++=HP30
.Itis
known that the same conservation law follows from ∇
μ
T
μν
=0[1]. It is straightforward to derive from (16)and
(17)the following relation
() [( )( ) ()]()
rr s rr rr++=- ++ ++ ++HP HHP HHHP H36311362,18
23
which clearly gives back the conservation law (and therefore, the conservation of the energy-momentum tensor)
in the case σ=0. The nature of the non-conservation of the energy-momentum tensor is explicitly proved here.
It should have been addressed also in [43,45,46]where the authors consider cosmological models of f(R,L
m
)
theories.
Commonly, equations (16)and (17)are known as (modified)Friedmann equations. In some papers [59]
equation (17)is also called Raychaudhuri equation. Both equations are related by (18).
In rewriting the Friedmann equations in a dimensionless form, specific choices were made to align the
theoretical framework with observational benchmarks and simplify the numerical analysis. By introducing
c=r
rcrit
, we scale the density to the critical density, which is pivotal in determining the curvature of the
Universe. Similarly, scaling the Hubble parameter by its current value H
0
through =h
H
H
2
2
0
2allows us to examine
the evolution of the expansion rate relative to its present value. The introduction of ˜
s
s=H0
2helps in assessing
the impact of the coupling constant σwithin the typical energy scales of the current universe. Lastly, by using
η=H
0
t, time derivatives are normalized to the current expansion rate, making them dimensionless and easier to
handle numerically. These transformations are not only mathematically convenient but also provide a direct
link between theoretical predictions and observable quantities. To this end, we introduce
˜()cr
r
kr ssh r kº= º º = =
HhH
HHHt H
3,,,,3,19
crit
crit 0
2
2
2
0
20
200
2
where ρ
0
is a reference value of ρat the present epoch. Similar comment applies to the Hubble parameter H
0
.In
addition, we write the time derivatives in terms of derivatives with respect to the dimensionless parameter ηby
means of
()
◦
hh
º= =AdA
d
dA
dt
dt
dAH
1.20
0
In the context of our model, the barotropic index γcharacterizes the equation of state (EOS)of the
cosmological fluid, namely P=(γ−1)ρ. This parameter is crucial for defining the thermodynamic properties of
the Universe’s contents, where γ=1 corresponds to a universe dominated by non-relativistic matter (the choice
we make here for the calculations),γ=4/3 to one dominated by radiation, and γ=0 to the cosmological
constant scenario with vacuum energy. Each value of γdistinctly affects the evolution dynamics of the Universe,
as reflected in the modified Friedmann equations. By means of the aforementioned (EOS), equations (16),(17)
and (18)take the form
˜[] ()
◦◦
cs c c c=+ + -hhhh18 12 6 , 21
22
()
˜[( ) ( ) ] ( )
◦◦
◦◦ ◦
gcsg c g cc c+=- - - - + - + +hh h h h
1
232 332 18 1 3 6, 22
2 2
4
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
˜[( ) ( ) ] ( )
◦◦◦◦◦◦
cgc sg cc gc c+=- + + ++ +hhhhhhh3 9 6 9 24 3 3 18 6 . 23
23
Since we want to compare the modified cosmological model with standard cosmology we should also include
the cosmological constant Λ. This can be done without much effort by considering the action
⎛
⎝⎞
⎠
() ()
()
ò
k
=- -
L
+L L
SdxgfR,24
fR m,4
m
with
() ()
ks=++
L
fR RR,2.25
mmm
This gives the new Einstein equations
()() ()
k
+--- +
L=
mn mn mn mn mn mn
fR g f f f L g g f T
1
22
1
226
RRL
mL
mm
as well as the generalized Friedmann equations
() ()
kr+- - + -
L=Hf f f f R Hf f31
232
1
2,27
RL
mRR L
2
mm
() ()
k
+--+ -+
L=Hf H f f Hf f f f P33
1
22
1
2.28
RRRRL
mL
2
mm
With the explicit choice of f
Λ
the modified Friedmann equations simplify to
[] ()
krsk r r r=+ + - +
L
HHHH
346 2 3,29
22
()[() ]()
krskr rr=- + - + + + + + La
aPHP HP H
63362
3.30
2
Their dimensionless form can be written by using the dimensionless parameter Ω
Λ
=ρ
vac
/ρ
crit
as follows
˜[] ()
◦◦
cs c c c=+ + - +W
L
hhhh18 12 6 , 31
22
()
˜[( ) ( ) ] ( )
◦◦
◦◦ ◦
gcsg c g cc c+=- - - - + - + + +W
L
hh h h h
1
232 332 18 1 3 6 . 32
2 2
These equations are now suitable for numerical integration.
Normally, the inclusion of Λin a modified gravity theory might not be well justified as the modification itself
can account for Dark Energy. But it appears that this is a too global statement and the role of Λcan vary from
model to model. In the section on numerical results we will demonstrate that the cosmological constant does not
affect the characteristic feature of the model, i.e., its shortcoming to produce a reasonable lifetime of the
Universe.
3. The initial values and the ∑
i
Ω
i
=1 relation
The Friedman equations are now first order in hand second order in χ. Therefore, we need three initial values,
which we provide at the time t
0
corresponding to the present epoch. We have
() () () () ( )
◦◦ ◦◦
cc kr cc kr
º== º= º= º =hht H
HtHhht H
HtH
1, 3,,
3.33
00
0
0
000
0
200
0
0
2000
0
3
We look for model independent measurements for the density of the Universe (ρ
0
), the Hubble parameter (H
0
)
and its first and second derivatives (
H
0
,
H0). For
H
we use the deceleration parameter q, given by [60]
() ()
º- = - +qaa
a
H
Hq,1 34
22
and thus,
() ()
=- +HH q1, 35
00
2
0
where q
0
=q(t=t
0
)has been obtained in a model independent way [61]. The jerk parameter will be useful for
H0appearing in the second modified Friedmann equation. This is given by [60,62]
()
==++ja
aH jH
H
H
H
31 36
332
5
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
and therefore,
() ( ) ()
=- - =++Hj HHHjq H13 2. 37
000
300 00 0
3
To ensure that our exploration of the modified
()
f
R,
mgravity model remains grounded in observational
reality, we adopt standard cosmological parameters that are widely accepted within the community. This choice
allows us to rigorously test whether the modifications introduced by the theoretical framework can provide a
viable alternative to the ΛCDM model, particularly in light of recent tensions and discrepancies such as those in
measurements of the Hubble constant. By using these parameters, the model’s predictions can be directly
compared with those derived from both local and cosmological scales, providing a comprehensive evaluation of
its empirical adequacy. For the numerical values, it is important to note that due to recent discrepancies in
measurements of the Hubble constant, known as the Hubble Tension [5], there are several values for H
0
. For
example we have
=
+
H073,52 ±1,62 km/s/Mpc and
=
-
H
067,4 ±0,5 km/s/Mpc, respectively. As for the
current energy density of the Universe and the deceleration parameter, we use values found in [42,45,61]:
c==
r
r0.285 0.012
0
0
crit
,q
0
=−0.545 ±0.107, j
0
=1.30 ±0.37 with the value for the Hubble parameter
taken as H
0
=71.34 ±1.74 km s
−1
Mpc
−1
. It is worth noting that these are specific parameter choices, but other
sets of parameters are available in [61]. Possible values for these parameters are displayed in table 1. This
particular choice of parameters that as of today we are as close as possible to the standard cosmology. For a
standard reference value of ◦
c
0
, we consider the standard cosmological model and use the continuity
equation (23)with
˜
s
=0
, even in the presence of Λ≠0. This leads to
()
◦
cgc+=h3 9 0. 38
For the initial value we then obtain
()
◦
cgcgc=- =-h33. 39
0000
Since h
0
=1, in the case of dust γ=1, we have
()
◦
cc=-3. 40
00
In mathematics, when solving differential equations, one typically assumes that all parameters and functions
entering the equation, as well as the initial values are known and given. However, in physics, the situation can
become more complex because these parameters and initial values may not be precisely known. When
considering a differential equation at the point where the initial values are applied, it leads to a relationship
between these initial values and the parameters of the equation. The Friedmann equations provide a notable
example of this phenomenon.
The first modified Friedmann equation can be expressed in terms of the density parameters Ω
i
. To this
purpose, let us start by defining the Ω
m
parameter as
()
r
r
kr
W= = H3.41
m
crit
2
This, in turn, implies that χ=Ω
m
and
r
r=W = W
k
m
H
crit
3
2. Thus, our first Friedmann equation can be
reformulates as follows
Table 1. Specific choices of the parameters according to [5,42,45,61].
Density parameters
Data SNIa Hubble
Ω
m,0
0.285 ±0.012 0.239 ±0.015
Ω
Λ,0
0.715 0.761
Δ0.012 0.015
Hubble data
Model H
exp
GP GA ΛCDM
H
0
73.88 ±1.34 73.44 ±1.40 71.34 ±1.74 72.08 ±1.06
q
0
−1.070 ±0.093 −0.856 ±0.111 −0.545 ±0.107 −0.645 ±0.023
j
0
3.00 ±0.62 1.30 ±0.37 0.52 ±0.24 1.00
Pantheon data
Model H
exp
GP GA ΛCDM
H
0
71.13 ±0.46 71.92 ±0.38 71.81 ±1.14 71.84 ±0.22
q
0
−0.616 ±0.105 −0.558 ±0.040 −0.466 ±0.244 −0.572 ±0.018
j
0
1.56 ±0.74 0.85 ±0.12 0.55 ±1.65 1.00
6
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
() () ()
ssskr=W +W +W - + L
HHHH HHH18 12 3.42
mm m
2222 2
Recognizing this equation as
⎡
⎣⎤
⎦
˜()
skr
W+W+ W + W - =
L
HH
H
18 12 1, 43
mmm
2
we can express it in a more compact form. Finally, defining the term proportional to σas Ω
σ
, we arrive at
()W+W+W=
sL
1, 44
m,0 ,0 ,0
where we have taken the values at t=t
0
. Making use of (35)and (40)the above equation reduces to
˜() ()sW+W+W - =
Lq15 12 1. 45
mm,0 ,0 ,0 0
Equation (45)establishes a relationship between initial values and parameters within the modified Friedmann
equation. Moreover, it introduces the deceleration parameter q
0
, which is a measured quantity, though not
strictly necessary for solving the first order modified Friedmann equation in H. Consequently, there are generally
two approaches to solve the new Friedmann equations. The first one involves specifying h
0
=1, χ
0
,◦
c
0
, and Ω
Λ
while varying ˜
s
. Equation (45)can then be used to determine q
0
. The second approach relies on the measured
value of q
0
and fixes the other parameters as previously described. In this case, equation (45)provides the
parameter ˜
s
. In particular, we note that in the case without coupling between Rand
m
, the relative densities
today read
()W+W=
L
1. 46
m,0 ,0
Numerically, however, there is an uncertainty in the obtained values, reflected in the relation
Ω
m,0
+Ω
Λ,0
=1±Δ, where Δrepresents this uncertainty. We can exploit it in our model by setting, as shown
in equation (44),
˜() ()s= D
W-q15 12 .47
m,0 0
4. The standard cosmology
Before embarking on a discussion of the numerical solutions it is convenient to briefly outline the analytical
picture of the standard cosmology. This makes sense since we will be comparing the two scenarios.
The dimensionless solution for h
Λ+
for the standard cosmological model can be obtained by taking
˜
s
=0
in
equations (21)and (22)with γ=1, namely
()c=+W
L
h,48
2
()
◦
c+=- +W
L
hh 1
2.49
2
Combining the above equations, we arrive at a Riccati equation of the form
()
◦
=- + W
L
hh
3
2
3
2.50
2
We can solve the differential equation above by setting
() ()
òò
h
¢
W- ¢ =¢
hh
hh
¢=
¢=
L¢=
¢=
dh
hd
3
2,51
h
hh
12
0
where η
0
=H
0
t
0
is the time at which the Hubble parameter is H
0
, which, in turn, is related to the present values
of the parameters used. To solve the integral above, we redefine
=
¢
WL
x
hand write
() ()
òhh
W-=-
L=W
=W
L
Ldx
x
1
1
3
2.52
x
xh
120
One possible solution for the integral on the left-hand side is
⎛
⎝⎞
⎠()
ò
-=+
-
dx
x
x
x1
1
2ln 1
1.53
2
However, such a solution is valid for −1<x<1. This would imply that <W
L
hand thus, χ=h
2
−Ω
Λ
<0,
which would suggest an unphysical negative density. Another solution is
7
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
⎛
⎝⎞
⎠()
ò
-=+
-
dx
x
x
x1
1
2ln 1
1.54
2
It is valid whenever |x|>1. Moreover, it implies h
2
>Ω
Λ
and, consequently χ=h
2
−Ω
Λ
>0, ensuring a
positive and physically meaningful density. If we also use the fact that
⎛
⎝⎞
⎠()=+
-
-
xx
x
coth 1
2ln 1
155
1
and solve for the integral (52), we end up with the result
⎜⎟ ⎜⎟
⎡
⎣
⎢⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎤
⎦
⎥() ()hh
WW
-W=-
L
-
L
-
L
h1coth coth 13
2.56
11
0
Solving for hin the γ=1 case leads to
⎜⎟
⎡
⎣
⎢⎛
⎝
⎞
⎠
⎤
⎦
⎥
() ( ) ( )hhh=W W - + W
L+ L L -
L
hcoth 3
2coth 1.57
0
1
For the general γcase, the equation above reads
⎜⎟
⎡
⎣
⎢⎛
⎝
⎞
⎠
⎤
⎦
⎥
() ( ) ( )hghh=W W - + W
L+ L L -
L
hcoth 3
2coth 1.58
0
1
In the standard notation of the scale factor a
Λ
(t)and the Hubble parameter H
Λ
(t), we get
⎜⎟
⎛
⎝⎞
⎠
() ( ) ( )mn=-W
W+
L+ L
L
g
g
at t
1sinh , 59
1
32
3
() ( ) ( )mn=L+
L+
Ht t
3coth , 60
where
⎜⎟
⎛
⎝
⎞
⎠()mgnm=L=+W
-W -
L
L
t
3
2,1
2ln 1
1
61
0
for a positive cosmological constant Λ. For completeness, we also give the solution for Λ<0 which reads
⎡
⎣
⎢⎤
⎦
⎥
() ()
() ()
ab
ab
=-
-
L-
g
at a t
t
cos
cos ,62
0
0
2
3
() ( ) ( )ab=- -
L- L
Ht ttan 63
3
with
⎜⎟
⎛
⎝⎞
⎠()ab b g
=+ -
L=-L
-
tHtan 3,3
2.64
010
Without loss of generality we can set t
0
=0. Then from (64)we can infer that 0 <α<π/2 and hence
()a>cos 0. To ensure that the scale factor is positive it suffices to choose
[]Î
ap
b
ap
b
-+
t
,
22
corresponding
to Big Bang and collapse.
We will explore later how this simple solutions evolves when we introduce a non-zero σ.
Two important model-independent restrictions on a realistic cosmological model relate to the Universe
lifetime. In the standard cosmological model, we can calculate the lifetime (neglecting the short radiation
period)by setting the argument of the Hubble function to zero, which results in a singularity. Choosing η
0
=0,
we obtain
⎜⎟
⎛
⎝
⎞
⎠
∣∣ ()h=W
+W
-W
L
L
L
1
3
ln 1
1
,65
univ
we get |η
univ
|;0.96 for Ω;0.7. This lifetime is not in conflict with model-independent estimates, such as those
from the oldest galaxies [52](which indicate the existence of galaxies some 400 million years after the Big Bang)
and [53], which points to a lower limit of 12.5 Gy.
8
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
5. Numerical results
Assuming σ≠0, we have converted the system under discussion into a first order system which we solved by
standard numerical methods. As an independent check, we used MAPLE18 routines and found that the results
coincide.
When solving the modified Friedmann equations numerically, the question arises: What values should we
take for ˜
s
? The standard cosmology has two constants, the Newtonian one G
N
and the cosmological constant.
Interestingly, the dimension of σis the same as that of the Newtonian constant. If we take σas nG
N
with nsome
number of order 1, ˜
s
would turn out extremely small. Therefore, it makes sense to initially try more moderate
values for ˜
s
. In the subsequent figures, the blue curve represents the standard cosmological model prediction.
In figure 1, we have chosen the initial values as discussed in section III, with Ω
Λ
=0.7. The chosen values of ˜
s
are relatively small to illustrate that the dimensionless Hubble function closely follows the one of the standard
cosmological model around ηclose to zero. Several notable features can be observed in the plot, which are likely
to persist for other values of ˜
s
. First of all, the Hubble function reaches the singularity much earlier than in the
standard model. This poses a challenge if we aim to adapt the model under discussion as a viable cosmological
model of our universe. As mentioned earlier, one of the constraints on any cosmological model is its ability to be
roughly compatible with a lower limit on the lifetime of the Universe. This constraint arises from evidence such
as the existence of galaxies just 400 million years after the Big Bang [52]. Therefore, any cosmological model
must strive to be in agreement with the estimated lifetime of the Universe as predicted by the standard
cosmological model. Secondly, we notice another interesting feature. The Hubble function becomes negative
which implies a contraction of the Universe. Therefore, the dimensionless density χstarts increasing again. This
is known as the Big Crunch scenario which happens also in f(R,L
m
)models with the simple choice L
m
=ρ. The
Big Crunch here is a result of the model and cannot be avoided.
In figure 2, we have plotted the same functions for two values of ˜
s
appearing also in figure 1, but over a larger
scale of time. One can appreciate the steep rise of hat a relatively modest negative values of η.
In figure 3, we have increased the values of ˜
s
but there is no visible effect on the time of the Big-Bang.
However, for larger values of the parameter, the density becomes unphysically negative.
This is confirmed in figure 4which shows the same functions, but over a larger timescale. The differences
between the models become then visible at large η.
Increasing the values of ˜
s
does not bring any new insight, but for the first time there appears a cusp for
˜
s
=5
. MAPLE18 interpretes this as singularities. This is demonstrated in figure 5.
In figure 6, we return to relative small values of ˜
s
to show that the cusp behavior is quite common in the
model. This is interesting from a purely theoretical point of view, but leaves doubts about the viability of the
model over large time scales.
Figure 1. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor a choice of the
parameter ˜
s
.
9
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
Finally, we turn our attention the the behaviour of hand χfor negative values of ˜
s
which is depicted in
figure 7,8, and 9.
In figure 7, we encounter a universe whose contraction slows down to turn into a expanding universe at η
around zero. Again many cusps accompany this behavior which could well mean that universe undergoes a
singularity there.
For small and very small values of negative ˜
s
we have again negative densities.
We have also explored a second approach for selecting the parameter ˜
s
. More precisely, we utilized (44)and
considered the error bars associated with measured values in standard cosmology. Figure 10 provides an
example where we calculated ˜
s
with high precision, staying within the range of uncertainty. This allowed us to
Figure 2. The same plot as in 1, but over a large timescale.
Figure 3. The same plot as in 1, but for large values of ˜
s
.
10
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
examine how sensitive the model is to the choice of parameters. The lifetime of the Universe comes out
approximately |η
univ
|;0.4, which is too small to account for the existence of the first galaxies in our universe.
The Big Crunch appears to be unavoidable again.
In the third approach, we used equation (45)while keeping the standard value for Ω
m,0
, but varying Ω
Λ
and
q
0
. We then solved the equation to find ˜
s
. An example is illustrated in figure 11. This variation led to a slight
increase in the Universe lifetime. Motivated by this observation, we explored scenarios with a negative
cosmological constant. Two such examples are noteworthy. Setting q
0
=−0.5 and Ω
Λ
=−50 resulted in
˜
s
10 and a lifetime of |η
univ
|;0.51. Alternatively, when choosing a deceleration parameter of q
0
=−1, we
obtained
˜
s
7.8
and a lifetime of 0.56. However, it appears challenging to significantly extend the model
lifetime beyond these values.
Figure 4. The same plot as in 2but form larger values of ˜
s
.
Figure 5. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor large values the
parameter ˜
s
.
11
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
In summary, this model represents a mild extension of Einstein gravity, particularly when viewed from the
perspective of its Lagrangian. Surprisingly, the cosmology given by this model for a spatially flat, homogenous
universe is quite different from the standard case based on Einstein equations.
6. Conclusions
Proposing a new gravity theory which goes beyond Einstein leads inevitably to the examination of the new model
in a cosmological context. In this context, theoretical advancements revealing new phenomena are equally
important as empirical efforts to evaluate the model potential to supersede the standard cosmological
Figure 6. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor small values of
the parameter ˜
s
.
Figure 7. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor large negative
values of the parameter ˜
s
.
12
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
framework. We have emphasized both aspects studying the extension
()
f
R,
mtheory with an explicit coupling
between geometry and matter in the form σRρwith ρbeing the energy density. This appears as a natural choice
used also in astrophysical context. We explored the model over a wide range of the coupling constant σwith
initial data mimicking our present universe. Given that the model predicts a universe lifespan that is drastically
shorter than the observed age of our universe (approximately 13.8 billion years), it fails to meet a fundamental
criterion for a viable cosmological model. This discrepancy leads us to conclude that the model, in its current
form, cannot adequately describe the observed universe. As such, the potential occurrence of a Big Crunch,
while theoretically interesting, becomes a secondary concern because the model is already ruled out based on its
inability to account for the observed age of the Universe.
Figure 8. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor small negative
values of the parameter ˜
s
.
Figure 9. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor very small values
of the parameter ˜
s
.
13
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
All in all, the findings highlight challenges in adapting the model to realistically represent the Universe,
particularly regarding the Universe lifetime and the issue of negative densities in some scenarios.
Before comparing our work with existent literature, let us point out that once we derive the field equations
from the Lagrangian principle, the Lagrangian should contain a matter part L
m
from which the perfect energy-
momentum tensor can be derived. The latter seems only possible via a constrained variation with respect to
the metric. The conservation of the energy-momentum tensor in theories of the type f(R,L
m
)is not guaranteed
and requires special attention. In [43]the choice of the Lagrangian is f(R,L
m
)=R/2+L
m
+σRL
m
with
L
m
=−p. It is then difficult to study the dust scenario with p=0 as it implies L
m
=0. Furthermore, the
energy-momentum tensor is not conserved in this choice in contrast to the authors’claim. In [46]the choice is
Figure 10. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηfor values of ˜
s
resulting from equation (44).
Figure 11. The dimensionless Hubble function hand the dimensionless density χversus the dimensionless time ηusing equation (45)
with values of the parameters as indicated in the figure.
14
Phys. Scr. 99 (2024)065050 S Bravo Medina et al
()=+
a
f
RL R L,2
mmwith αbeing some number. It is not clear how we can derive from that a
hydrodynamical energy-momentum tensor and, as explained above, whether we obtain a conservation law. The
choice of the Lagrangian in [46]is also quite different from ours as it lacks the direct coupling between geometry
(represented by R)and matter (represented by L
m
). This shows that such coupling has special consequences. In
[45]the Lagrangian reads () ab=+-
f
RL R L,2
mm
nwith αand βconstants. Remarks similar to the case of
[46]would apply also here.
Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
ORCID iDs
Sergio Bravo Medina https://orcid.org/0000-0003-4399-1281
Marek Nowakowski https://orcid.org/0000-0001-8716-4670
Davide Batic https://orcid.org/0000-0002-7619-4900
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