Cosmological dynamics with modified induced gravity on the normal DGP branch
ABSTRACT In this paper we investigate cosmological dynamics on the normal branch of a
DGP-inspired scenario within a phase space approach where induced gravity is
modified in the spirit of $f(R)$-theories. We apply the dynamical system
analysis to achieve the stable solutions of the scenario in the normal DGP
branch. Firstly, we consider a general form of the modified induced gravity and
we show that there is a standard de Sitter point in phase space of the model.
Then we prove that this point is stable attractor only for those $f(R)$
functions that account for late-time cosmic speed-up.
- Dynamical systems and differential equations: proceedings of the Third International Conference" Dynamical Systems and Differential Equations": Kennesaw State University, Georgia, May 2000. 01/2001;
- Duke Mathematical Journal 01/1957; · 1.70 Impact Factor
arXiv:1008.4240v5 [gr-qc] 8 Jan 2012
Cosmological dynamics with modified induced gravity
on the normal DGP branch
Department of Physics, Faculty of Basic Sciences,
University of Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN
In this paper we investigate cosmological dynamics on the normal branch of a DGP-
inspired scenario within a phase space approach where induced gravity is modified in the
spirit of f(R)-theories. We apply the dynamical system analysis to achieve the stable
solutions of the scenario in the normal DGP branch. Firstly, we consider a general form
of the modified induced gravity and we show that there is a standard de Sitter point in
phase space of the model. Then we prove that this point is stable attractor only for those
f(R) functions that account for late-time cosmic speed-up.
PACS: 04.50.-h, 95.36.+x, 98.80.-k
Key Words: Dark Energy, Braneworld Cosmology, Curvature Effects, Dynamical Sys-
tem, Induced Gravity
There are many astronomical evidences supporting the idea that our universe is currently
undergoing a speed-up expansion . Several approaches are proposed in order to explain the
origin of this novel phenomenon. These approaches can be classified in two main categories:
models based on the notion of dark energy which modify the matter sector of the gravitational
field equations and those models that modify the geometric part of the field equations generally
dubbed as dark geometry in literature [2,3]. From a relatively different viewpoint (but in the
spirit of dark geometry proposal), the braneworld model proposed by Dvali, Gabadadze and
Porrati (DGP)  explains the late-time cosmic speed-up phase in its self-accelerating branch
without recourse to dark energy . However, existence of ghost instabilities in this branch of
the solutions makes its unfavorable in some senses . Fortunately, it has been revealed recently
that the normal, ghost-free DGP branch has the potential to explain late-time cosmic speed-up
if we incorporate possible modification of the induced gravity in the spirit of f(R)-theories .
This extension can be considered as a manifestation of the scalar-tensor gravity on the brane.
Some features of this extension are studied recently [8,9].
Within this streamline, in this paper we study the phase space of the normal DGP cosmology
where induced gravity is modified in the spirit of f(R)-theories. We apply the dynamical system
analysis to achieve the stable solutions of the model. To achieve this goal, we firstly consider
a general form of the modified induced gravity. We obtain fixed points via an autonomous
dynamical system where the stability of these points depends explicitly on the form of the
f(R) function. There are also de Sitter phases, one of which is a stable phase explaining the
late-time cosmic speed-up. Secondly, in order to determine the stability of critical points and
for the sake of clarification, we specify the form of f(R) by adopting some cosmologically viable
models. The phase spaces of these models are analyzed fully and the stability of critical points
are studied with details.
2 DGP-inspired f(R) gravity
2.1The basic equations
Modified gravity in the form of f(R)-theories are derived by generalization of the Einstein-
Hilbert action so that R (the Ricci scalar) is replaced by a generic function f(R) in the action
where Lmis the matter Lagrangian and κ2= 8πG. Varying this action with respect to the
µν + T(f)
= diag(ρ,−p,−p,−p) is the stress-energy tensor for standard matter, which is
assumed to be a perfect fluid and by definition f′≡
µνis the stress-energy tensor
of the curvature fluid that is defined as follows
f(R) − Rf′?
+ f′ ;αβ(gαµgβν− gαβgµν).
By substituting a flat FRW metric into the field equations, one achieves the analogue of the
Friedmann equations as follows 
f(R) − Rf′?
−2f′˙H = κ2ρm+˙R2f′′′+ (¨R − H˙R)f′′,
where a dot marks the differentiation with respect to the cosmic time. In the next step, following
 we suppose that the induced gravity on the DGP brane is modified in the spirit of f(R)
gravity. The action of this DGP-inspired f(R) gravity is given by
where gABis the five dimensional bulk metric with Ricci scalar R, while qabis induced metric
on the brane with induced Ricci scalar R. The Friedmann equation in the normal branch of
this scenario is written as 
(infra-red) behavior of the DGP model. The Raychaudhuri’s equation is written as follows
2κ2 is the DGP crossover scale with dimension of [length] and marks the IR
˙R2f′′′+ (¨R − H˙R)f′′
To achieve this equation we have used the continuity equation for ρ(f)as
˙ ρ(f)+ 3H
where the energy density and pressure of the curvature fluid are defined as follows
f(R) − Rf′?
f(R) − Rf′??
After presentation of the required field equations, we analyze the phase space of the model
fully to explore cosmological dynamics of this setup.
2.2 A dynamical system viewpoint
The dynamical system approach is a convenient tool to describe dynamics of cosmological
models in phase space. In this way, we rewrite equation (7) in a dimensionless form as
In the present study, we firstly consider a generic form of the f(R) function, so that one can
define the dynamical variables independent on the specific form of the f(R) function as follows
(see for instance Ref. )
Also we define the following quantities
r ≡ −dlnf
We note that a constant value of m leads to the models with f(R) = ξ1+ ξ2R1+mwhere the
parameter m shows the deviation of the background dynamics from the standard model and ξ1
and ξ2are constants. However, in general the parameter m depends on R and R itself can be
expressed in terms of the ratio r =x4
Based on the new variables, the Friedmann equation becomes a constraint equation so that we
can express one of these variables in terms of the others. Introducing a new time variable
τ = lna = N and eliminating x1(by using the Friedmann constraint equation) we obtain the
following autonomous system
dN= x2(x5+ x4+ 2),
dN= (x2+ x5)(x5+ x4) + 1 − 3x3− 5x4− 2x2,
x1≡ Ωm= 1 − x2− x3− x4− x5.
The deceleration parameter which is defined as q = −1 −
q = 1 + x4,
x3. This means that m is a function of r,that is, m = m(r).
+ x3(2x4+ x5+ 4),
+ x4(2x4+ 4),
H2, now can be expressed as
and the effective equation of state parameter of the system is defined by
ωeff= −1 −2˙H
Figure 1: The effective equation of state parameter (left panel) and the deceleration parameter (right
panel) of the critical point E versus m. The shaded region denotes a non-phantom acceleration era
which occurs in the parametric space with 0.45 < m < 0.67. A slightly phantom-like behavior exists
for m < 0.45 and when m → 0, it reaches to a strong phantom-like phase. For negative values of m
the EoS parameter is stiff-like.
2.3Critical points and their stability
The critical points of the scenario and some of their properties are listed in table 1. In this
table, Γ is defined as
4m2− 9m + 2 ±√−160m4+ 272m3− 111m2+ 4m + 4
2m2− 3m + 1
We consider only the plus sign of this equation in our forthcoming arguments. The minus sign
does not create suitable cosmological behavior since it leads to weff< −10 or weff> 0.7 for
In table 1, the critical points A, B and C are independent of the form of f(R). Nevertheless,
the stability of these points depends on the form of f(R) explicitly. The critical curve D exists
just for f(R) models with m(r = −1
that m is defined as m(r) = −n(1+r)
of the effective equation of state parameter corresponding to this critical curve depends on the
evolution (for example, a formal de Sitter-type era occurs for curve D at x∗
phase space behavior of the point E (with m(r) = 1 + r) depends on the form of f(R) via m,
we treat it separately in section 3. In which follows, we classify the important subclasses in
order to see their dynamical behaviors.
, the critical curve D exists just for n =
2(for instance, in models of the form f(R) = R+γR−n
2). The value
2. Then, different intervals on this curve describe different era of the universe
4). Since the
a) The radiation dominated era
The points A and B demonstrate effectively the radiation dominated epoch of the universe
stability of these points is investigated in which follows.
3with a(t) ∝ t
2). This phase can be realized also by the curve D for x∗
b) The matter dominated era
The matter era (ωeff = 0) could be existed for models with m(r = −1
ized point x∗
expansion with scale factor a(t) = a0(t − t0)
E with m(r = −0.13) = 0.87.
2by the local-
8on the curve D. Note that this matter era is properly described by a cosmic
3. This era also can be realized in localized point
c) The de Sitter era
The de Sitter phase (ωeff = −1) in the normal branch of this DGP-inspired f(R) model is
realized by the curve C of critical points. It is important to point out that the mentioned de
Sitter solution is the standard de Sitter phase just for x∗
matter density parameter vanishes (Ωm= 0). Also this phase can be realized from the curve
of the non-localized points D with x∗
case of the curve C in the localized point x∗
standard de Sitter era since its Ωmis non-vanishing (in this case Ωm= 0 occurs at x∗
2= 2, since in this localized point the
4(this de Sitter point can be regarded as the special
4). But one can see that this point gives no
d) Transition from q > 0 to q < 0
The critical point E depends explicitly on the form of f(R) via m and this is the case also
for stability of this point. The point E describes a phase transition of the universe from decel-
eration to the acceleration era at ωeff= −1
Also the mentioned feature for f(R) models with m(r = −1
curve D) occurs at the fixed point x∗
is that it clearly realizes the late-time acceleration of the universe in its normal branch for
0 < m ≤ 0.67. In figure 1 the localized point E represents the deceleration phase for m > 0.67
and a non-phantom accelerating phase for 0.45 ≤ m ≤ 0.67. For m < 0.45, since q < −1, the
model realizes an effective phantom phase with possibility of future big rip singularity which is
characteristics of a non-canonical (phantom) field dominated universe. Similarly, the effective
phantom behavior for the curve D occurs at x∗
3and q = 0 in this model for m(r = −0.33) = 0.67.
2. So, one of the important feature of this model
2(corresponding to the
In the next step we determine the stability of the critical points under small perturbations.
The stability of these points is determined by the eigenvalues of the Jacobian matrix. For a
general f(R) term on the brane, stability of the critical points depends on the form of f(R) (or
equivalently on the parameter m). It is obvious that in general m = m(r) is not a constant;
it is a function of other variables so that one can expand this function of curvature about any
of the fixed points. The results of our investigation for stability of critical points mentioned in
Table 1: Location, effective EoS and deceleration parameter of the critical points for the normal
branch of a general DGP-inspired f(R) scenario. The fixed point D exists only for those f(R)
models that m(r = −1
m − 1
3(11 − 4x∗
3(1 + 4x∗
table 1, are summarized as follows:
• Point A
As has been mentioned, this point is a radiation dominated era. In this case the eigenvalues
4(m − 1)
, −4 ± i.
Note that around this point, m ≡ mA = m(r = 0). Hence this point is a spiral attractor
if 0 < m(r = 0) < 1, otherwise it is a saddle point. The corresponding 2D phase space for
arbitrary m = m(r) is shown in figure 2 (left panel). Note that the Jacobian matrix in this
point has no dependence on the m′(r) since this point lies around r = 0. Here a prime denotes
derivative with the respect to r.
• Point B
This point is also a radiation dominated era in which the eigenvalues are as follows
λ3,4=3m2− m + m′r(1 + r)
(m2− m)2+ 2m′rm2(1 − r) − 2mm′r(1 + r) + m′2(r + r2)2
where the parameter m should be expanded around the point B (that is, m = mB). Here
the stability issue depends also on m and m′, so that this point can be either a saddle or a
spiral attractor. The corresponding 2D phase space for arbitrary m = m(r) is shown in figure
2 (right panel). An important point should be emphasized here: by setting x4= 0 in plotting
figures 2, their dependence on m(r) is wasted. However, the situation is different if one plots
K 0.5 0.0 0.51.0
Figure 2: The phase subspace x3- x5of our setup at x2= x4= 0 (left panel) and x2=5
(right panel). The critical point shown in the left panel is point A and in the right panel it is point
B. Point A in the mentioned subspace is a spiral attractor and point B is a saddle point. Note that in
the 4D phase space these are either spiral attractor or saddle point depending on the form of f(R).
4and x4= 0
the 3D subspace x3- x4- x5for these points. In this case, one should determine the form of the
function m(r =x4
which indicates that the point B is a saddle point for constant values of m.
x3). For a constant m, the eigenvalues reduce to
• Curve C
Generally, if a nonlinear system has a critical curve, the Jacobian matrix of the linearized
system at a critical point on the line has a zero eigenvalue with an associated eigenvector tangent
to the critical curve at the chosen point. When dynamical variables are not independent, some
eigenvalues of the Jacobian matrix are zero. In this case, the phase space of the nonlinear
system reduces to a lower dimensional phase space. The stability of an specific critical point on
the curve can be determined by the nonzero eigenvalues, because near this critical point there
is essentially no dynamics along the critical curve (i.e., along the direction of the eigenvector
associated with the zero eigenvalue). So, the dynamics near this critical point may be viewed
in a reduced phase space obtained by suppressing the zero eigenvalue direction. On the other
hand, such curves are actually normally hyperbolic [11,12]. We consider a point on the curve
C with coordinates (2, 1, −2, 0). This point is a standard de Sitter phase. The eigenvalues
corresponding to this point are as follows
where χ is defined as
−460m + 54 + 3√3√8019m2− 1840m + 108
K 202468 10
x3- x5for x4= −2 (right panel). The critical point which is shown in the left panel is a center and is
related to the standard de Sitter phase. The critical points shown in the right panel which lie on the
2D phase space x3- x5for x2= 2 and x4= −2 (left panel) and the 3D phase space x2-
3(11 − 4x2) are spiral attractors. These figures are plotted for m(r = −2) =1
The parameter m should be expanded around the standard point of C as defined previously
with mC= m(r = −2). Here the stability issue depends only on m. It is a stable spiral in the
subspace of the last two eigenvalues when 0 < m. On the other hand, the second eigenvalue is
negative for m < 0 and 0.09 ≤ m. So the standard de Sitter point in the 4D phase space is a
spiral attractor if
m ≥ 0.09.
This point for other values of m is a saddle point. Figure 3 shows the 2D and 3D phase spaces
of the critical curve C. We note that in the 2D subspace, the de Sitter curve is reduced to a de
Sitter point (as figure 3 shows, this point is a center). Therefore, the center manifold theory
is required to investigate its stability [13,14]. In 3D subspace (right panel), the non-localized
points in the x2- x3plane lie on the line x3=1
satisfies (27), therefore this curve is a spiral de Sitter attractor.
3(11−4x2). Figure 3 is plotted for m =1
3Analytical results for some specific models
As we have mentioned previously, m is a function of r, that is, m = m(r). Since the ΛCDM
model defined with f(R) = R − 2Λ, corresponds to m = 0, we can say that the quantity
m characterizes the deviation of the background dynamics from the standard ΛCDM model.
Now we consider two specific model of f(R) in order to obtain more obvious results. We also
focus on the cosmological viability of these models. A cosmologically viable scenario contains
an early time radiation dominated era followed by a matter dominated era that reaches to a
standard de Sitter phase which is a stable attractor. We focus here only on the last two stages:
matter domination and then a stable de Sitter attractor. At the first stage, the existence of
a matter dominated era (weff = 0) constrains our DGP-inspired f(R) model only to those
f(R) functions that m(r = −1
there exist two de Sitter phases: the localized point x∗
de Sitter point, and also the de Sitter curve C which realizes a standard de Sitter point at the
localized point x∗
connection between the unstable matter dominated era and the stable, standard de Sitter era is
necessary condition for cosmological viability of a scenario. This connection can be investigated
in the m − r plane.
2and m(r = −0.13) = 0.87 (see table 1). In the first case,
4on the curve D which is a formal
2= 2. The second case is associated just to the de Sitter curve C. A correct
A) f(R) = R + γR−n
For this model, the parameter m takes the following form
m(r) =−n(1 + r)
which is independent on γ . On the other hand, the point E of table 1 is characterized by the
m(r) = r + 1.
Equations (28) and (29) give two solutions m1= 0 and m2= 1−n for m. So, for the mentioned
f(R) function, point E is characterized by the following relation
2n(1 − n)
E(1−n):0,Γ(1−n)+ 4(1 − n)
, −Γ(1−n)+ 4(1 − n)
2(1 − n)
,ωeff= −1 −
3(1 − n),
where Γ(1−n)≡ Γ|m=1−n.
Note that the critical point E for m = 0 (that is, E(0)) is indefinite and therefore we exclude
it from our considerations. Now we investigate the stability of the critical point E(1−n). Since
the EoS parameter corresponding to this point varies with m, in order to determine the stability
of this point, one has to fix the value of m.
In figure 4 (left panel), we have shown the behavior of the parameter m as a function of r for a
special f(R) model given as f(R) = R +γRnwith n = 0.13. As this figure shows, this model
contains a connection between the matter era and the standard de Sitter era which is located
at m = −0.065. However, this model is not cosmologically viable since it reaches an unstable
standard de Sitter era. In figure 4 (right panel), we plotted the curve m(r) for f(R) = R+γRn
with n =
which indicates that the standard de Sitter era is unstable (see Eq. (27)). So, this model is
not cosmologically viable too. Note that in the right panel of figure 4, the point A lies also on
the dashed line which is corresponding to the critical point E. This feature indicates that this
model contains the critical point E with m(r = −1
phase (compare this case with figure 1 that m =1
2. The matter dominated era evolves to the standard de Sitter era at m = −0.25
2lies in the shaded region).
2which is a non-phantom acceleration
Figure 4: The solid curves illustrate the diagram m − r of f(R) = R + γR−nwith n = 0.13 (left
panel) and with n =1
2(right panel). The dashed-line is corresponding to the point E of table 1. The
intersection point A shows a matter domination phase. The standard de Sitter phase is corresponding
to the line r = −2 and dotted (blue) line indicates the line of stability.
B) f(R) = R
In this model the parameter m is defined as
m(r) = −n + r(2 + r)
which is independent on η. As has been pointed out previously, the matter dominated era can
be achieved from critical point E in which the relative parameter is given by equation (29).
Equating (29) and (31), we obtain two solutions for m as m =
point E gives a matter dominated phase (with m(r = −0.13) = 0.87) if we set n = 0.35,
that is to say, if f(R) = R
m(r = −1
E plays the role of a non-phantom acceleration era. In figure 5 (left panel), the curve m(r) is
plotted for f(R) = R
Therefore, this point is stable since it belongs to the region defined by relation (27). Finally,
we plot the m(r) curve for f(R) = Rexp(η
there is an acceptable connection between the matter dominated era and the standard de Sitter
phase since the standard de Sitter point at m =1
this DGP-inspired f(R) model there is a non-phantom acceleration phase which emerges from
the point E with m(r = −1
. Now, the critical
R). In the model with f(R) = Rexp(η
R) (models with
2), the matter dominated era is realized by the curve D and the critical point
R). This model reaches the standard de Sitter phase at m = 0.2.
R) as shown in the right panel of figure 5. In this case
2is a stable attractor. We note also that in
2. This is corresponding to the point A of figure 5 (right panel).
4Summary and Conclusion
In this paper we investigated cosmological dynamics of the normal DGP setup in a phase space
approach where the induced gravity is modified in the spirit of f(R)-theories. The motivation
Figure 5: The diagram m − r for f(R) = Rnexp(η
shows the m(r) function of the critical point E. The intersection point A shows a matter domination
phase. This model reaches to the standard de Sitter phase at B with m = 0.2 which is a stable point.
In this model the matter dominated era can be realized only by E.
R) with n = 0.35 (left panel). The dashed-line
for this study within a dynamical system approach lies in the fact that recently it has been
revealed that the normal, ghost-free DGP branch has the potential to explain late-time speed-
up if we incorporate possible modification of the induced gravity in the spirit of f(R)-theories.
In this respect, a phase space analysis of the scenario would be interesting to reveal some
aspects of this late-time behavior. Especially the stability of this late-time de Sitter phase is
important to have a cosmologically viable solution. We applied the dynamical system analysis
to achieve the stable solutions of the scenario in the normal DGP branch. We have shown
that generally there are some fixed points that one of those is the standard de Sitter phase.
Therefore, the normal branch of this DGP-inspired braneworld scenario realizes the late-time
acceleration phase of the universe expansion. However, the stability of this point depends on the
form of f(R) via the parameter m ≡
these setups. A cosmologically viable scenario contains an early time radiation dominated era
followed by a matter dominated era that reaches a standard de Sitter phase which is a stable
attractor. Here we focused only on the last two stages: matter domination era followed by a
stable de Sitter attractor. To be more specific, we considered two models with f(R) = R+γRn
and f(R) = Rnexp(η
domination era restricted us to consider two cases n = 0.13 and n =
and two cases n = 0.35 and n = 1 for the second one. On the other hand, it is shown that
the standard de Sitter phase is stable just for m ≥ 0.09. So, since the first model reaches the
standard de Sitter phase (which is determined by the line r = −2) at m < 0.09, it is not a
cosmologically viable model. However, since the second model reaches this phase at m > 0.09,
it is a cosmologically viable model for n = 0.35 and n = 1.
dlnR. Then, we investigated the cosmological viability of
R) in our DGP-inspired setup. The condition for existence of the matter
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