#### Singularities in particle-like description of FRW cosmology

Eur. Phys. J. C
Singularities in particle-like description of FRW cosmology
Marek Szydłowski 0 1
Aleksander Stachowski 1
0 Mark Kac Complex Systems Research Centre, Jagiellonian University , ul. Łojasiewicza 11, 30-348 Kraków , Poland
1 Astronomical Observatory, Jagiellonian University , Orla 171, 30-244 Kraków , Poland
In this paper, we apply the method of reducing the dynamics of FRW cosmological models with a barotropic form of the equation of state to the dynamical system of the Newtonian type to detect the finite scale factor singularities and the finite-time singularities. In this approach all information concerning the dynamics of the system is contained in a diagram of the potential function V (a) of the scale factor. Singularities of the finite scale factor make themselves manifest by poles of the potential function. In our approach the different types of singularities are represented by critical exponents in the power-law approximation of the potential. The classification can be given in terms of these exponents. We have found that the pole singularity can mimic an inflation epoch. We demonstrate that the cosmological singularities can be investigated in terms of the critical exponents of the potential function of the cosmological dynamical systems. We assume that the general form of the model contains matter and some kind of dark energy which is parameterised by the potential. We distinguish singularities (by an ansatz involving the Lagrangian) of the pole type with the inflation and demonstrate that such a singularity can appear in the past.
1 Introduction
The future singularity seems to be of fundamental
importance in the context of the observation acceleration phase of
the expansion of the current universe. While the
astronomical observations support the standard cosmological model,
CDM, we are still looking for the nature of dark energy and
dark matter. In the context of an explanation of the
conundrum of acceleration it appears that different theoretical ideas
might be relevant as regards the substantial form of dark
energy and a modification of the model of gravity [
1
]. For
cosmological models with a different form of dark energy it
is possible to define some form of effective equation of state
peff = peff(ρeff), where ρeff is the effective energy density.
For such a model we have the coefficient of the equation of
peff , which is very close to the value of −1,
corstate weff = ρeff
responding to the cosmological constant. In consequence in
the future, in the evolution of the universe can appear some
new types of singularities. It was discovered by Nojiri et al.
[
2
] that phantom/quintessence models of dark energy, for
which weff −1, may lead to one of four different
finitetime future singularities. Our understanding of the finite scale
factor singularity is the following. The singularities at which
aa assumes a finite value, we call a finite scale factor. The
appearance of future singularities is a consequence of the
violation of the energy condition and may arise in
cosmologies with phantom scalar fields, models with interaction of
dark matter with dark energy, and modified gravity theories
[
3,4
].
All types of finite late-time singularities can be classified
into five categories, following the divergences of the
cosmological characteristics [
2,4
]:
– Type I (big-rip singularity): As t → ts (finite), the scale
factor diverges, a → ∞, and the energy density as well
as the pressure also diverges, ρ → ∞, | p| → ∞. They
are classified as strong [
5,6
].
– Type II (typical sudden singularity): As t → ts (finite),
a → as (finite), ρ → ρs, | p| → ∞. Geodesics are not
incomplete in this case [
7–9
].
– Type III (big freeze): As t → ts, a → as, and ρ diverges,
ρ → ∞, as well as | p| → ∞. In this case there is no
geodesic incompleteness and these models can be
classified as weak or strong [
10
].
– Type IV (generalised sudden singularity). As t → ts,
a → as (finite value), ρ → ρs, | p| → ps. Higher
derivatives of the Hubble function diverge. These singularities
are weak [
11
].
– Type V (w singularities): As t → ts, a → ∞, ρ → ∞,
| p| → 0 and w = ρp diverges. These singularities are
weak [
12–14
].
It is interesting that singularities of type III appear in
vector-tensor theories of gravity [
15,16
], while singularities
of type II can appear in the context of a novel class of vector
field theories basing on generalised Weyl geometries [17].
The problem of obtaining constraints on
cosmological future singularities from astronomical observations was
investigated for all five types of singularities: type I in [
18
],
type II in [
19
], type III in [
20
], type IV in [
21
], type V in [
16
].
In this paper, we propose complementary studies of future
singularities in the framework of cosmological dynamical
systems of the Newtonian type. For the FRW cosmological
models with the fluid, which are described by the effective
equation of state peff = peff(ρeff) and ρeff = ρeff(a), the
dynamics of the model, without loss of generality, can be
reduced to the motion of a particle in the potential V =
V (a) [
22
]. In this approach, a fictitious particle mimics the
evolution of a universe and the potential function is a single
function of the scale factor, which reconstructs its global
dynamics.
Our methodology of searching for singularities of the
finite scale factor is similar to the method of detection of
singularities by Odintsov et al. [
2,12,20,23–25
], by
postulating the non-analytical part in a contribution to the Hubble
function. In our approach we assume that singularities are
related with the lack of analyticity in the potential itself or its
derivatives. Additionally we postulate that in the
neighbourhood of the singularity, the potential as a function of the scale
factor mimics the behaviour of the poles of the function. The
advantage of our method is connected strictly with the
additive non-analytical contribution to the potential with energy
density of fluids, which is caused by lack of analyticity of the
scale factor or its time derivatives. This contribution arises
from dark energy or dark matter.
In the paper, we also search pole types singularities in
FRW cosmology models in the pole inflation model. These
types of singularities become manifest by the pole in the
kinetic part of the Lagrangian. In this approach in searching
for singularities, we take an ansatz on the Lagrangian.
The aim of the paper is twofold. Firstly (Sect. 2) we
consider future singularities in the framework of the
potential function. Secondly (Sect. 3) we consider singularities
in the pole inflation approach. In Sect. 4 we summarise our
results.
2 Future singularities in the framework of potential of
dynamical systems of Newtonian type
2.1 FRW models as dynamical system of Newtonian type
We consider a homogeneous and isotropic universe with a
spatially flat space-time metric of the form
where˙ ≡ ddt , H ≡ aa˙ is the Hubble function.
We assume that ρ(t ) = ρ(a(t )) and p(t ) = p(a(t ))
depend on the cosmic time through the scale factor a(t ).
From Eqs. (2) and (3) we get the conservation equation in
the form
Equation (2) can be rewritten in the equivalent form
ds2 = dt 2 − a2(t ) dr 2 + r 2(dθ 2 + sin2 θ dφ2) ,
where a(t ) is the scale factor and t is the cosmological time.
For the perfect fluid, from the Einstein equations, we have
the following formulae for ρ(t ) and p(t ):
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
3a2
ρ = 3H 2 = a˙2 ,
2a¨ a˙ 2 ,
p = − a − a2
ρ˙ = −3H (ρ + p).
a˙ 2 = −2V (a),
where
where ρeff = ρm + and ρm = ρm,0a−3. From Eqs. (2) and
(3), we can obtain the acceleration equation in the form
a¨ 1
a = − 6 (ρ + 3 p).
∂ V
a¨ = − ∂a .
An equivalent form of the above equation is
Due to Eq. (9), we can interpret the evolution of a universe,
in dual picture, as the motion of a fictitious particle of unit
mass in the potential V (a). The scale factor a(t ) plays the
role of a positional variable. The equation of motion (9) has
a form analogous to the Newtonian equation of motion.
From the form of effective energy density, we can find
the form of V (a). The potential V (a) determines the whole
dynamics in the phase space (a, a˙ ). In this case, the
Friedmann equation (5) is the first integral and determines the
phase space curves representing the evolutionary paths of
The lines x22 + V (a) = − k2 represent possible evolutions
of the universe for different initial conditions.
We can identify any cosmological model by the form of
the potential V (a). For the dynamical system (11)–(12) all
critical points correspond to vanishing of its right-hand sides
x0 = 0, ∂V∂a(a) a=a0 = 0 .
From the potential function V (a), we can obtain
cosmological functions, such as
t =
a
da
√−2V (a)
,
the Hubble function
1 d ln(−V )
h(t ) ≡ −(q(t ) + 1) = − 2 d ln a
(note that if V (a) = − 6a2 , h(t ) = 0), the effective matter
density,
the cosmological models. The diagram of potential V (a) has
all the information which is needed to construct a phase space
portrait. Here, the phase space is two-dimensional,
(a, a˙ ) : a˙22 + V (a) = − k2 ,
and the dynamical system can be written in the following
form:
(20)
(21)
(22)
(23)
(24)
(25)
the first derivative of an effective pressure with respect of
time
p˙eff =
2√−2V (a)
a
d2V (a)
da2
−
2V (a)
a2
and the Ricci scalar curvature (1)
R =
The finite-time singularities can be detected using Osgood’s
criterion [
26
]. We can simply translate this criterion into the
language of cosmological dynamical systems of the
Newtonian type. Goriely and Hyde formulated necessary and
sufficient conditions for the existence of the finite-time
singularities in dynamical systems [
27
].
As an illustration of these methods used commonly in the
context of integrability, a one-degree freedom Hamiltonian
system is considered with a polynomial potential. Such a
system can be simply reduced to the form of the
dynamical system of the Newtonian type [
28
]. It is interesting that
the analysis of the singularities of this system is
straightforward when one considers the graph of the potential functions.
These systems can possess a blow-up of the finite-time
singularities.
Following Osgood’s criterion, a solution a(t ) of the
equation
a˙ =
−2V (a(t )),
with an initial problem
a(t0) = a0,
∞
a0
da
√−2V (a)
< ∞.
blows up in the finite time if and only if
Let us assume that solutions become, in a finite time ts,
for which a = φ (t ) at t = ts diverges, φ (ts) = ∞, where
ts < ∞. Then we have
ts =
∞
a0
da
√−2V (a) + t0 < ∞.
For example, let
(26)
t n
a(t ) = A + B 1 − t
s
t n
⇒ B 1 − t
s
= (a − as). (30)
Moreover, a solution a = φ (t ) is unique if
φ(t)
da
√−2V (a) = ∞.
If the potential assumes a power law, V = V0(as − a)α,
the integral (25) does not diverge if only 1 − α2 is positive.
In the opposite case, as a → as, this integral diverges, which
is an indicator of a singularity of a finite time t → ts and
a → ∞.
In our further analysis we will postulate the form of the
additive potential function V (a) with respect to the effective
energy density ρeff (the interaction between the fluids is not
considered)
ρma2
6
,
where ρm = ρm,0a−3(1+w) and pm = wρm, w = const
and the choice of f (a) is related with the assumed form of
the dark energy: ρde = − 6 fa(2a) . Numerically one can simply
detect these types of singularities. An analytical result can
be obtained only for special choices of the function f (a). In
this context Chebyshev’s theorem is especially useful [
29
].
Following Chebyshev’s theorem [
30,31
] for rational
numbers p, q, r (r = 0) and non-zero real numbers α, β, the
integral
I =
x p α + β xr q dx
is elementary if and only if at least one of the quantities
p+r1 , q, p+r1 + q is an integer.
It is a consequence that the integral (28) may be rewritten
as
1
= p + 1
× 2 F1
I = r1 α p+r1 +q β− p+r1 By
1 + p
r
, q − 1
α p+r1 +q β− p+r1 y 1+r p
1 + p
r
, 2 − q,
1 + p + r
r
; y ,
β xr and By
where y = α
1+r p , q − 1 is an incomplete
beta function and 2 F1
geometric function.
For the second distinguished singularity of a finite scale
factor, Chebyshev’s theorem can also be very useful. In the
detection of these types of singularities, a popular
methodology is to start from some ansatz on the function a(t ), which
is near the singularity.
1+r p , 2 − q, 1+ p+r ; y is a
hyperr
.
(31)
(32)
(33)
(34)
(35)
(36)
(37)
Because a(t = ts) = as, A = as, the basic dynamical
equation a˙ 2 = −2V (a) reduces to
d n B
dt (a − as) = − ts
t
1 − ts
n−1
n B
= − ts
a − as
B
Therefore
n
−2V (a) = − ts
B n (a − as)1− n1 ,
1
where n < 0 i.e. V (a) ∝ (as − a)α.
On the other hand if we postulate the above form of the
potential one can integrate the equation of motion,
a˙ 2 = −2V (a) = V0(as − a)α,
i.e.
as − a =
Therefore
V0 (ts − t ) 2−2α .
a(t ) = as + C (ts − t )n,
where n = 2−2α .
This approach was considered in [
21,32
]. We propose a
similar approach, but we consider additionally the baryonic
matter and the ansatz is defined by the potential V (a).
In our approach to the detection of future singularities it is
more convenient ansatz for the form of the potential function
rather than for directly for a(t ) function. We propose two
ansatzes.
The first ansatz has the following form:
a˙ 2 = 31 ρm,0a−1−3w + V0(|as − a|)α.
The form of the potential is assumed as a sum of the potential
for the barotropic matter satisfying the equation of state p =
wρ and the potential of dark energy. We assume that the
energy density of the matter behaves regularly, but the energy
density of dark energy has a pole for some finite value a = as.
The second ansatz is given by
x˙2 = B x m + C x n ,
where x = |as − a|. This ansatz describes the cosmic
evolution with the potential near the singularity a = as (we assume
m or n is negative) when effects of matter are negligible. For
as = 0, the two ansatzes are the same.
We may postulate a form of the potential, which is an
additive function with respect different components of the fluid.
We distinguish a part which arises from the barotropic
matter and an additional part which gives the behaviour of the
potential in the neighbourhood of poles (or its Padé
approximants). Our approach to the singularity investigation has its
origin in Nojiri and Odintsov’s paper [
23
].
Let us integrate Eq. (37) with the help of Chebyshev’s
theorem,
t (x ) =
1
= √B
x
dx
√B x m + C x n
x
m
x − 2
Let us introduce the new variable u,
m
x − 2 dx = du
⇒
x =
C x n−m
1 + B
For finding Chebyshev’s first integral I (Eq. (29)) we
check whether p+r1 = 2(2n−−mm) , q = − 21 , p+r1 + q = 2(2n−−nm) .
For example for the general case if m = −1 − 3w then
n = 3(1+w)−22kk(1+3w) , where k ∈ Z. When the above
conditions are obeyed then the solution of Eq. (40) has the
following form:
1
t (a) − ts = √
1
m |as − a|1− m2
B 1 − 2
×2 F1 1n −− mm2 , 3/2, 1 −n 3−2mm+ n ; CB |as − a|n−m .
For the special case of (36) for as ≈ 0, the solution (41)
gives the following expression:
t (a) − ts =
1
1
3 ρm,0
1
3+3w
2
a 3+23w
×2 F1
3+3w
2
α + 1 + 3w
, 3/2,
− 1+23w + α 3V0 aα+1+3w .
× α + 1 + 3w ; ρm,0
V0 (as − a)α−2 x (t )
2.3 Singularities for the potential
V = − 61 ρm,0a−1−3w − V20 (|as − a|)α
The potential V (a) for ansatz (36) is given by the following
formula:
1 V0
V = − 6 ρm,0a−1−3w − 2 (|as − a|)α.
Dabrowski et al. [
33
] assumed that singularities can appear
in the future history of the universe. The singularities can
appear also as the higher than second derivatives of the scale
factor blow up (Fig. 1). Such a singularity may not be visible
on the phase space (a, a˙ ). The potential V (a) for the best
fit value (see Sect. 3) is presented in Fig. 2. In this case,
the generalised sudden singularity appears. We also show
the diagram of the potential when the big freeze singularity
appears (see Fig. 3). In this case, dynamical system (11)–(12)
has the form
(1 + 3w)
3
ρm,0a−2−3w + α2 V0 (as − a)α−1 ,
for as < a. The phase portrait for the above dynamical system
for the best fit value (see Sect. 3) is presented in Fig. 4.
Because the dynamical systems (44)–(45) and (46)–(47)
are insufficient for introducing the generalised sudden
singularity, the above dynamical system can be replaced by a
three-dimensional dynamical system of the following form:
1.5
1.0
...
a
If we investigate the dynamics in terms of the geometry
of the potential function then a natural interpretation can be
given. It means a lack of analyticity of the potential itself
(in consequence da/dt blows up) or its derivatives (higher
order derivatives of the scale factor blow up). The
singularities are hidden beyond the phase plane (a˙ , a) and we are
looking for it in the enlarged phase space. In our approach
to the detection of different types of finite scale factor
singularities we explore information contained in the geometry of
the potential function, which determines all characteristics
of the singularities. This function plays an analogous role to
the function h(t ) in the standard approach.
k
1
k
1
x
k
1
k
1
1
4
a
8
6
2
0
2
From the potential (43) we can obtain a formula for the
Hubble parameter,
H˙ = ±∞.
The second derivative of the Hubble function is
for as < a. Note that in the singularity a = as if α < 1 then
H
¨ = H
3(1+w)2
2
ρm,0a−3−3w +
αV0
+ 2a2
(as − a)α−1 + a
αV0
2V0
+ a2 (as − a)α ,
α(α −1)V0
2
(as − a)α−1
(as − a)α−2
(54)
(55)
(56)
(57)
for as < a. Note that in the singularity a = as if α < 2 then
H¨ = ±∞.
The effective matter density is given by
and the effective pressure has the following form:
for as < a. The first derivative of the effective pressure has
the following form:
1 V 3/2(as − a)−2+ 32α (2as2 − 4aas + a2(2 + α − α2))
p˙eff = 3√2a3 0
(58)
(59)
(60)
(61)
(62)
Big
feeeze
for as > a and
1 V 3/2(a − as)−2+ 32α (2as2 − 4aas + a2(2 + α − α2))
p˙eff = 3√2a3 0
(63)
for as < a. Note that for the singularity a = as if α < 1
then peff = ± ∞ and if α > 1 then p = 0.
The type of singularity with respect to the parameter α is
presented in Fig. 5.
For classification purposes we take into account
singularities located at a constant, non-zero value of the scale factor
(we do not consider a singularity at a = 0). This
classification covers the last five cases from Dabrowski’s paper
[
33
]. Due to such a representation of singularities in terms
of a critical exponent of the pole one can distinguish generic
(typical) cases from non-generic ones. The classification of
the finite scale factor singularities for the scale factor a > 0
and the potential V = − 16 ρm,0a−1−3w − V20 (|as − a|)α [
33
]
is presented in Table 1. Singularities are called generic if the
corresponding value of the parameter α for such
singularities is of a non-zero measure. In the opposite case, as the
parameter α assumes a discrete value, such singularities are
fine-tuned.
It is interesting that in the case without matter, a
wsingularity appears for a special choice of the parameter α
(α = 4/3). Let us note that all singularities without the
wsingularity are generic.
2.4 Padé approximant for the potential V (a)
The standard methodology of searching for singularities
based on the Puiseux series [
34
]. We proposed, instead of
the application of this series, use of the Padé approximant
for parametrisation of the potential which has poles at the
singularity point.
The second derivative of the non-analytical part of the
potential V (a), which we denote as V¨˜ (a), in the
neighbourhood of a singularity, can be approximated by a Padé
approximant. The Padé approximant of order (k, l), where k > 0
and l > 0, is defined by the following formula:
Pkl (x ) =
c0 + c1x + · · · + cl xl
1 + b1x + · · · bk x k .
The coefficients of the Padé approximant can be found by
solving the following system of equations:
Pkl (x0) = f (x0),
Pkl (x0) = f (x0),
.
.
.
Pk(lk+l)(x0) = f (k+l)(x0),
(64)
(65)
(66)
(67)
2V˜ (3)(a0)2 − V¨˜ (a0)V˜ (4)(a0) a
+ (2V˜ (3)(a0) + V˜ (4)(a0)a0)
× 1 −
V˜ (4)(a0)
(2V˜ (3)(a0) + V˜ (4)(a0)a0)
a
−1
,
(68)
where derivation is with respect to time, a0 is the value of
a for which the coefficients of the Padé approximant are
calculated. For a > as
V¨˜ (a) = −
V¨˜ (a) = −
V¨˜ (a) = −
V0α(α − 1) (a − as)α−2,
2
V0α(α − 1)(α − 2)
2
V0α(α − 1)(α − 2)(α − 3)
2
(a − as)α−3,
and for a < as
V¨˜ (a) = −
V¨˜ (a) =
V¨˜ (a) = −
V0α(α − 1) (as − a)α−2,
2
V0α(α − 1)(α − 2)
2
V0α(α − 1)(α − 2)(α − 3)
2
(as − a)α−3,
(a − as)α−4,
(as − a)α−4.
In this case, for the Padé approximant, a singularity appears
when a = as.
V a
0.5
In Fig. 6 it is shown how the Padé approximant can
approximate the potential V¨˜ (a) in the neighbourhood of a
singularity.
The Padé approximant is not only used for a better
approximation of the behaviour of the potential or time derivatives
near the singularity. It can be used directly in the basic
formula da/dt = √−2V (a) for defining non-regular parts of
the potential. Therefore in our approach we can apply just
this ansatz instead of the ansatz for a(t ) like in the standard
approach. Against the background of Padé exponents, we can
make the following assumption:
a˙ 2 = −2V = −2Vm + Pkl
= −
ρm,0a2
6
c0 + c1a + · · · + cl al .
+ 1 + b1a + · · · + bk ak
(69)
3 Singularities in the pole inflation
Let us concentrate on pole types singularities in the FRW
cosmology models. These types of singularities are
manifest by the pole in the kinetic part of the Lagrangian. We
also distinguish pole inflation singularities in the following
for a(t ) > as and
a¨ (t ) =
[
35–37
]. In this approach in searching for singularities, we
take an ansatz on the Lagrangian rather than the scale factor
postulated in the standard approach.
We consider dynamics of cosmological model reduced to
2
the dynamical system of the Newtonian type, i.e., ddt2a =
− ddVa , where a(t ) is the scale factor, t is the cosmological
time. Then the evolution of the universe is mimicked by a
motion of a particle of a unit mass in a potential which is a
function of the scale factor only.
By pole singularities we understand such a value of the
scale factor a = as for which the potential itself jumps
to infinity or its kth-order derivatives (k = 2, 3, . . .) with
respect to the scale factor (in consequence we obtain jump
discontinuities in the behaviour of the time derivatives of the
scale factor).
In the pole inflation approach, beyond the appearance of
inflation, the kinetic part of the Lagrangian has a pole or the
derivatives have a pole. We use the pole inflation approach
in the form which was defined in [
35–37
].
The Lagrangian has the following form [
36
]:
1
L = −3aa˙ 2 − 2 apρ− pρ˙2a3 − V0(1 − cρ)a3,
(70)
d .
where ap, p, V0 and c are model parameters and ˙ ≡ dt
The form of the Lagrangian is inspired by the corresponding
Lagrangian in Galante et al.’s paper [
36
]. The Lagrangian
(70) describes the evolution FRW cosmologies with the
scalar field in the Einstein frame. In the original Lagrangian,
the quantity ρ plays the role of the scalar field. In the above
Lagrangian, we can distinguish the kinetic part ( 21 KE(ρ)ρ˙2)
and the potential of the scalar field (V0(1 − cρ)). The
original function KE(ρ) is given by a Laurent series. But for
our consideration we cut off the Laurent series at the first
term (KE(ρ) = ρapp ). Here, the pole in the kinetic part of the
Lagrangian can appear when ρ = 0.
Let ρ = |a −as|n. Then the Lagrangian (70) can be
rewritten as
n2
L = − 3aa˙ 2 − 2 ap|a − as|3n− p−2a˙ 2a3
− V0(1 − c|a − as|n)a3.
(71)
After variation with respect to the scale factor a we get
the acceleration equation, which can be rewritten as
1
a¨ (t ) = −3a˙ (t )2 + 2 a(t )2(6V0 − 2cV0(a(t ) − as)n−1
×((3 + n)a(t ) − 3as) − apn2(a(t ) − as)3n− p−3
×((1 + 3n − p)a(t ) − 3as)a˙ (t )2) / [6a(t )
+apn2a(t )3(a(t ) − as)3n− p−2
(72)
c = afi−nn
(80)
H (t ) =
H (t ) =
√2V0√1 − c|a(t ) − as|n
6 + apn2a(t )2|a(t ) − as|3n− p−2
.
V0
3
√1 − c|a(t ) − as|n
1 − ca(t )2|a(t ) − as|n−2
,
Let ap = − n6c2 and p = 2n. Then the first integral (74)
has the form
which guarantees the inflation behaviour when as/a(t ) 1.
The slow roll parameters can be used to find the value of
the model parameters. These parameters are defined as
H
˙
= − H
and
η = 2 − 2H˙ .
The following relation exists between the scalar spectral
index and the tensor-to-scalar ratio and the slow roll
parameters:
ns − 1 = −6 + 2η
and r = 16 .
Let as a(t ). If we use Eqs. (76) and (77), then we get
the equations for the parameters c and ap,
c = afi−nnr (16(3n − p)(1 + n − ns − p)
+4(−1 − 5n + ns + 2 p)r + r 2) / 768n3
−16n2(40 p + r ) + r (4 p + r )(4(−1 + ns + p) + r )
+4n(32 p2 − 8 pr + (4 − 4ns − 3r )r ) ,
(78)
ap =
6afi−n3n+ pr (4(−1 + n + ns) + r ) / n2(32(6n2
−5np + p2) + 4(−1 − 5n + ns + 2 p)r + r 2) , (79)
where afin is the value of the scale factor in the end of the
inflation epoch. Because we assume ap = − n6c2 and p = 2n,
we get
(81)
and the tensor-to-scalar ratio r is given by
r = 4(1 − ns).
The best fit of the scalar spectral index ns is equal to 0.9667
[
38
]. In consequence, r = 0.1332.
Because in this model the singularity is in the beginning
of the inflation and we also assume that the number of e-folds
is equal to 50, the value of as is e−50afin ≈ 1.93 × 10−22afin.
Up to now, the inflation has the methodological status of a
very interesting hypothesis added to the standard
cosmological model. Note that in the context of pole singularities, the
following question is open: Can pole singularities be treated
as an alternative for inflation?
The type of singularities in the model is dependent on the
value of the parameter n. If n > 2 then the singularity in
the model represents the generalised sudden singularity. The
typical sudden singularity appears when n < 2 (see Fig. 7).
Figure 8 presents the evolution of H / V0 in the pole
inflation model by way of an example value of n = −1. Note
that, in this case, a typical sudden singularity appears. In this
singularity, the value of the Hubble function is equal to zero.
Figure 9 presents the evolution of the scale factor in the pole
inflation model for the example of our value n = −1.
We can use this approach for a description of inflation in
the past. This model also can be used for a description of the
behaviour of dark energy in the future. But it is possible only
in the case when the generalised sudden singularity appears
(n > 2). In the case when the typical sudden singularity
appears, the bounce appears in the singularity. As a result,
this case (n < 2) cannot be considered as a model of the
behaviour of dark energy in the future.
Our result is in agreement with the general statement that
physically reasonable cosmological models with the eternal
inflation possess an initial singularity in the past [
39, 40
]. In
typical
sudden
singularity
generalized
sudden
singularity
1
0
1
2
3
4
5
Fig. 7 The diagram presents the generic types of singularity in the pole
inflation model with respect to the value of the parameter n
0.6
0.5
20
40
60
80
100
Fig. 8 The diagram presents the evolution of H/ V0 in the pole inflation
model for an example value of n = −1. The dimensionless time τ is
equal to V0t . Here, τ = 100 corresponds to the end of the inflation
period (t = 10−32 s)
20
40
60
80
100
Fig. 9 The diagram presents the evolution of the scale factor in the
pole inflation model for an example value n = −1. The dimensionless
time τ is equal to V0t . Here, τ = 100 corresponds to the end of the
inflation period (t = 10−32s)
the standard approach to the classification of singularities in
the future, Borde and Vilenkin elided the fact that inflation
in the past history takes place. Of course, if we assumed that
the inflation epoch was in the past, corresponding results
obtained without this ansatz should be corrected.
4 Conclusions
In this paper, we study the finite scale factors using the
method of reducing dynamics of FRW cosmological
models to the particle moving in the potential as a function of
the scale factor. In the model we assume that the universe is
filled by matter and dark energy in a general form, which is
characterised by the potential function. The singularities in
the model appear due to a non-analytical contribution in the
potential function. Near the singularity point the behaviour
of the potential is approximated by poles.
Using the potential method we detected the scale factor
singularities. In the detection of singularities of the finite
scale factor we used a methodology similar to the detection
of singularities of the finite time [
2
]. An advantage of this
method is that the additional contribution to the potential is
additive and is strictly related with the form of the energy
density of dark energy.
Using the method of the potential function gives us a
geometrical framework of the investigation of singularities.
The dynamics is reduced to the planar dynamical system
in the phase space (a, da/dt ). The system possesses a first
integral energy like for a particle moving in the potential
21 ddat 2 + V (a) = E = const . The form of the potential
uniquely characterises the model under consideration.
We demonstrated that finite scale factor singularities can
be investigated in terms of the critical exponent α of the
approximation of the potential near the singularity point
a = as : V = V0(a − as)α . The classification of singularities
can be given according to the value of the parameter α. For
the class of singularities under consideration the effects of
visible matter near singularities are negligible in comparison
with the effects of dark energy modelled by the non-regular
potential.
For a better approximation of the behaviour of the
potential near the singularities we apply the method of Padé
approximants.
In the general the behaviour of the system is approximated
by the behaviour of the potential function near the poles. The
singularities appear as a consequence of the lack of
analyticity of the potential or its derivatives with respect to the scale
factor. In consequence, the time derivatives of the scale factor
with respect the cosmological time blows up to infinity.
For the generalised sudden singularity under
consideration da/dt , d2a/dt 2, and the Hubble parameter are regular
and the third derivative with respect to time blows up. Of
course, this type of singularity cannot be visualised in the
phase space (a, da/dt ), because higher dimensional
derivatives are non-regular. Therefore we construct a higher
dimensional dynamical system in which the non-regular behaviour
of d3a/dt 3 can be presented. Finally, the dynamical system
in which one can see this type of singularity has dimension
three.
Our general conclusion is that the framework of the
particle like reducing cosmological dynamical systems can be
useful in the context of singularities in FRW cosmology
with a barotropic form of the equation of state. Different
types of singularities have different and universal values of
exponents in a potential approximation near the
singularities. We believe that this simple approach reveals a more
fundamental connection of the singularity problem with an
important area in physics—critical exponents in complex
systems.
It is interesting that the generalised sudden singularity is
a generic feature property of modified gravity cosmology [
8
]
as well as brane cosmological models [
41
].
Our method has heuristic power, which helps us to
generalise some types of singularities. The advantage of the
proposed method of singularities detection seems to be its
simplicity. Our ansatz involves rather the potential of
cosmological system than the scale factor, in order for the potential
function to be an additive function of the matter contribution
ρeff in opposition to the scale factor.
Let us consider a w-singularities case discovered by
Dabrowski and Denkiewicz [
12
]. After simple calculations
one can check that the potential of the form V = (a − as )4/3
admits generalised w-singularities when both ρeff and H are
zero, d p/dt goes to zero, and w diverges. Let us note that
in the case of a non-zero cosmological constant this type of
singularity disappears automatically.
It was proposed to constrain the position of singularities
based directly on the ansatz of an approximation for the scale
factor near the singularity [
12, 20, 23–25
]. It is a model
independent approach as it is based only on the mathematics of
singularity analysis. Then this scale factor approximation is
used in cosmological models to determine a type of
singularities and estimate model parameters. An alternative approach
which we believe is methodologically proper is to consider a
cosmological model and prove the existence of singularities
in it. Of course, such singularities are model dependent. Then
we estimate the redshift corresponding to the singularity and
determine the type of singularity. This approach has been
recently applied by Alam et al. [
42
]. In their paper the
position of possible future singularities is taken directly from the
brane model, and after constraining the model parameters one
can calculate the numerical value of singularity redshift. Note
that, in the brane model, the generalised sudden singularity
can appear in the future history of the universe [
41, 43, 44
].
For these singularities the potential function jumps
discontinuously following the corresponding pole singularity.
In the standard approach of probing of singularities, the
ansatz for prescribing of the asymptotic form of the scale
factor a(t ) is considered. In our investigation, we search some
special types of pole singularities, called pole inflation
singularities. In the study of the appearance of these types of
singularities, we make the ansatz by the Lagrangian of the
model. This Lagrangian contains a regular part as well as
jump discontinuities. The jump discontinuities can appear in
the kinetic part of the Lagrangian. Our estimation of the slow
roll parameters shows the existence of pole inflation in the
past history of the universe.
In our paper, we demonstrated that inclusion of the
hypothesis of the inflation in the past evolution of the universe can
modify our conclusions about their appearance and position
during cosmic evolution. In standard practice, the
information as regards the inflation in the past is not included in the
postulate of a prescribed asymptotic form of the scale
factor a(t ). The situation can be analogical to the situation in
Vilenkin [
39, 40
] when the eternal inflation determines the
singularity of the big bang in the past.
Acknowledgements We thank Dr. Adam Krawiec and Dr. Orest
Hrycyna for insightful discussions.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
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to the original author(s) and the source, provide a link to the Creative
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