How far is it to a sudden future singularity of pressure?
Mariusz P. Da̧browski∗
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
Tomasz Denkiewicz†
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and
Fachbereich Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany
Martin A. Hendry‡
Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
(Dated: November 12, 2018)
We discuss the constraints coming from current observations of type Ia supernovae on cosmo-
logical models which allow sudden future singularities of pressure (with the scale factor and the
energy density regular). We show that such a sudden singularity may happen in the very near
future (e.g. within ten million years) and its prediction at the present moment of cosmic evolution
cannot be distinguished, with current observational data, from the prediction given by the standard
quintessence scenario of future evolution. Fortunately, sudden future singularities are characterized
by a momentary peak of infinite tidal forces only; there is no geodesic incompletness which means
that the evolution of the universe may eventually be continued throughout until another “more
serious” singularity such as Big-Crunch or Big-Rip.
PACS numbers: 98.80.Cq; 98.80.Hw
Over the past decade observations of high-redshift
Type Ia supernovae (SNIa) have provided strong evi-
dence that the expansion of the universe is accelerating,
driven in the standard paradigm by some form of dark
energy[1, 2]. Current data[2] continue to leave open the
possibility that dark energy exists in the form of phan-
tom energy, which may violate all energy conditions [3]:
the null (̺c2 + p ≥ 0), weak (̺c2 ≥ 0 and ̺c2 + p ≥ 0),
strong (̺c2 + p ≥ 0 and ̺c2 + 3p ≥ 0), and dominant
energy (̺c2 ≥ 0, −̺c2 ≤ p ≤ ̺c2) conditions (where
c is the speed of light, ̺ is the mass density in kgm−3
and p is the pressure). Phantom matter may dominate
the universe in the future and drive it towards a Big-Rip
(BR) singularity in which all matter will be dissociated
by gravity [4]. This is dramatically different from the
standard picture of future cosmic evolution which sug-
gests an asymptotically empty de-Sitter state driven by
the cosmological constant or quintessence [5] and leading
to the violation of the strong energy condition only.
Phantom-driven scenarios have encouraged the study
of other exotic possibilities for the future evolution of
the universe. One of these possibilities appears in those
models which do not assume any explicit form for the
equation of state p = p(̺), leaving the evolution of the
energy density and pressure unconstrained. This free-
dom may result in a so-called sudden future singularity
(SFS) of pressure [6] which violates only the dominant
energy condition. The nature of a sudden future singu-
larity is different from that of a standard Big-Bang (BB)
singularity, and also from a Big-Rip singularity, in that it
does not exhibit geodesic incompletness and the cosmic
evolution may eventually be extended beyond it [7, 8].
The only physical characteristic of these singularities is
a momentarily infinite peak of the tidal forces in the uni-
verse. In more general models this peak may also appear
in the derivatives of the tidal forces. It is interesting to
note that these types of singularity are in a way simi-
lar to yet another type, which were termed finite density
singularities [9]. However, the crucial difference is that
finite density singularities occur as singularities in space
rather than in time, which means that even at the present
moment of cosmic evolution they could exist somewhere
in the Universe [10]. We will not discuss in detail finite
density singularities in this paper since they basically ap-
pear in cosmological models without homogeneity. On
the other hand, it is worth mentioning that the sudden
future singularities are quite generic since they may arise
in both homogeneous [11] and inhomogeneous [12] mod-
els of the universe.
In order to obtain a sudden future singularity consider
the simple framework of an Einstein-Friedmann cosmol-
ogy governed by the standard field equations
, (1)
p = − c
, (2)
where the energy-momentum conservation law
˙̺ = −3 ȧ
, (3)
is trivially fulfilled due to the Bianchi identity. Here a(t)
is the scale factor, G is the gravitational constant, and
the curvature index k = 0,±1. What is crucial to ob-
tain a sudden future singularity is that no link between
http://arxiv.org/abs/0704.1383v2
the energy density and pressure (the equation of state) is
specified. This allows us to integrate (3) only by quadra-
tures as
̺a3 = exp
3p(t′)
c2̺(t′)
ln a(t′)
p(t′)
̺(t′)
ln a(t′)dt′
Of course (4) reduces to the standard expression for
energy conservation, ̺a3(w+1) = const., provided a
barotropic equation of state, p = w̺c2 for constant w,
is assumed. (The condition for phantom models, for ex-
ample, is w < −1).
From equations (1)-(2) one can easily see that a pres-
sure singularity p → ∓∞ occurs when the acceleration
ä → ±∞, no matter that the value of the energy density
̺ and the scale factor a(t) are regular. Since in that case
| p |> ̺, it is clear that the dominant energy condition
is violated. This condition can be achieved if the scale
factor takes the form [6]
a(t) = as [1 + (1− δ) ym − δ (1− y)n] , y ≡
with the appropriate choice of the constants δ, ts, as,m, n.
Moreover, we can see that the r-th derivative of the scale
factor (5) is given by
a(r) = as
m(m− 1)...(m− r + 1)
(1− δ) ym−r
+ (−1)r−1δ
n(n− 1)...(n− r + 1)
(1− y)n−r
and is related to the appropriate pressure derivative
p(r−2). Thus, in general, it is possible that one has a
pressure derivative p(r−2) singularity which accompanies
the blow-up of the r-th derivative of the scale factor a(r).
Observationally this could be manifested in, for example,
the blow-up of the characteristics known as statefinders ,
such as jerk, snap etc. [14]. The pressure derivative sin-
gularity p(r−2) appears when
r − 1 < n < r r = integer , (7)
and for any r ≥ 3 it fulfills all energy conditions. These
singularities are called generalized sudden future singu-
larities (GSFS) and are possible, for example, in theories
with higher-order curvature quantum corrections [13].
Let us now return to the case of r = 2, for which 1 <
n < 2 and we obtain sudden future singularity models
of pressure (and obviously all of its higher derivatives)
which lead to violation of the dominant energy condition.
In such models, expressed in terms of the scale factor (5),
the evolution begins with the standard BB singularity at
t = 0 for a = 0, and finishes at SFS for t = ts where a =
as ≡ a(ts) is a constant. (Note that we have changed the
original parametrization of Ref. [6] for the scale factor
(5) using A = δas).
The standard Friedmann limit (i.e. models without an
SFS) of (5) is achieved when δ → 0; hence δ becomes
the “non-standardicity” parameter of SFS models. Ad-
ditionally, notwithstanding Ref. [6] and in agreement
with the field equations (1)-(2), we assume that δ can be
both positive and negative leading to a deceleration or
an acceleration (cf. (6)) of the universe, respectively.
It is important to our discussion that the asymptotic
behaviour of the scale factor (5) close to the BB singu-
larity at t = 0 is given by a simple power-law aBB = y
simulating the behaviour of flat k = 0 barotropic fluid
models with m = 2/[3(w+1)] . This allows us to preserve
all the standard observed characteristics of early universe
cosmology – such as the cosmic microwave background,
density perturbations, nucleosynthesis etc. – provided we
choose an appropriate value of m. On the other hand,
close to an SFS the asymptotic behaviour of the scale
factor is non-standard, aSFS = as [1− δ (1− y)n ], show-
ing that aSFS = as for t = ts (i.e. y = 1) at the SFS.
Notice that one does not violate the energy conditions if
the parameter m lies in the range
0 < m ≤ 1 (w ≥ −1/3), (8)
This range of values is, in fact, equivalent to a standard
(neither quintessence-like nor phantom-like) evolution of
the universe. However, with no adverse impact on the
field equations (1)-(2), one could also extend the val-
ues of m to lie in the complementary ranges [7] m > 1
(i.e. −1 < w < −1/3) for quintessence, and m < 0 (i.e.
w < −1) for phantom, although these ranges may lead
to violation of the strong and weak energy conditions
respectively.
We will next calculate the luminosity distance as a
function of redshift, and hence the redshift-magnitude
relation, for SFS models. This will allow us to estab-
lish whether these models are a realistic possibility for
the future evolution of the universe, and more specifi-
cally whether current cosmological observations of high
redshift supernovae are consistent with values of the con-
stant n in the range 1 < n < 2, as required in order that
the scale factor will display an SFS (or, more generally,
a GSFS for r − 1 < n < r). We will then explore the
range of values for the other SFS model parameters which
are consistent with current observational constraints on
standard cosmology, and thus determine limits on how
far into the future an SFS might occur. In fact, as we
will see below, we need to consider only two further pa-
rameters: δ and y0 = t0/tS , where t0 is the current age
of the Universe in the SFS model. Notice that, in view of
(8), it is reasonable to take m = 2/3 as for the standard
dust-dominated evolution. This implies that, at early
times, our SFS model reduces to the Einstein-de-Sitter
universe.
We proceed within the framework of Friedmann cos-
mology, and consider an observer located at r = 0 at
coordinate time t = t0. The observer receives a light ray
emitted at r = r1 at coordinate time t = t1. We then
have a standard null geodesic equation
1− kr2
, (9)
with the scale factor a(t) given by (5). Using (5) again,
the redshift is given by
1 + z =
a(t0)
a(t1)
δ + (1− δ) ym0 − δ (1− y0)
δ + (1− δ) ym1 − δ (1− y1)
n , (10)
where y0 = y(t0) and y1 = y(t1). The luminosity distance
is defined as
DL = r1a(t0) (1 + z) . (11)
Neglecting extinction and k−corrections, the observed
and absolute magnitudes of a source at redshift z and
luminosity distance DL are related by
m(z) = M − 5 log10 H0 + 25 + 5 log10 DL(z), (12)
which, with the help of the equation (9), (10) and (11),
allows a redshift-magnitude relation for SFS cosmologi-
cal models to be constructed. It is obvious that equation
(9) has to be integrated numerically in order to estab-
lish the relation between t0 and t1, which can then be
inserted into (10) and (11) to constrain the SFS model
parameters. As a first step we determine the dependence
on the SFS model parameters of the Hubble law, which
replaces equation (12) for z ≈ 0, i.e. cz ≈ H0DL, where
H0(kms
Mpc−1) =
3.09× 1019
t0(sec)y0
m (1− δ) ym−10 + nδ (1− y0)
δ + (1− δ) ym0 − δ (1− y0)
is the present value of the Hubble parameter, which we
can take as 72kms−1Mpc−1 [1].
Similarly we could derive an expression, in terms of
the SFS model parameters, for the deceleration param-
eter q0 = −(äa/ȧ2)0. However, in order to search the
parameter space for models which are admissible by cur-
rent observations, we write the product of H0 and q0 as
q0H0 = −
= (14)
m(m− 1)(1− δ)ym−20 − δn(n− 1) (1− y0)
m(1− δ)ym−10 + nδ (1− y0)
In order to obtain an accelerated universe at the present
moment of the evolution, this product should be nega-
tive. Fig. 1 shows an example plot of the product H0q0
as a function of δ and y0, with the other parameters fixed
atm = 2/3, n = 1.9993, t0 = 13.2457 Gyr. From the plot
we see that there are large regions of the parameter space
which admit cosmic acceleration. We have explored the
parameter space further with various configurations of
m,n, δ, y0, t0, q0, and H0, and obtained the general con-
clusion that there is a large class of SFS models which
are compatible with current acceleration.
H0*q0
FIG. 1: Parameter space (H0q0, δ, y0) for fixed values of
m = 2/3, n = 1.9993, t0 = 13.3547 Gyr of the sudden future
singularity models. There are large regions of the parameter
space which admit cosmic acceleration.
Out of these admissible models we then searched for
those which are compatible with the redshift-magnitude
relation (12) observed for recent SNIa data [2], and hence
with the derived parameters of the standard ‘Concor-
dance cosmology’ (CC). We were able to identify SFS
models that are in remarkably tight agreement with cur-
rent SNIa data. As an illustrative example Fig. 2 shows
luminosity distance as a function of redshift for the CC
model with H0 = 72kms
−1Mpc−1, Ωm0 = 0.26 and
ΩΛ0 = 0.74, and an SFS model with parametersm = 2/3,
y0 = 0.99936, δ = −0.471, n = 1.9999. We see that the
SFS model mimics the CC model very closely over a wide
range of redshifts. In particular, it is clear that recent
SNIa data from the Tonry at al. ‘Gold’ sample [1] and
SNLS sample [2] cannot yet discriminate between the CC
and SFS models.
Taking the current age of the universe in the SFS model
to be equal to the age of the CC model, i.e. t0 = 13.6Gyr,
we find that the time to the sudden singularity is ts−t0 ≈
8.7Myr, which is amazingly close to the present epoch.
In that context there is no wonder that these singularities
are called “sudden”. We have also checked that the larger
the value of r in (7) the later in future a GSFS appears.
It means that the strongest of these singularities which
violates the dominant energy condition (i.e. an SFS) is
more likely to become reality.
Our remark about the effect of the sudden pressure
singularity seems in agreement with the result of Ref.
FIG. 2: The distance modulus µL = m − M for the con-
cordance cosmology (CC) model with H0 = 72kms
−1Mpc−1,
Ωm0 = 0.26, ΩΛ0 = 0.74 (dashed curve) and sudden fu-
ture singularity (SFS) model for m = 2/3, n = 1.9999, δ =
−0.471, y0 = 0.99936 (solid curve). Also shown are the ‘Gold’
(open circles) and SNLS (filled circles) SNIa data. Taking the
age of the SFS model to be equal to that of the CC model,
i.e. t0 = 13.6 Gyr, one finds that an SFS is possible in only
8.7 million years.
[16] which showed that the dominant energy condition is
now violated and that it became violated quite recently
(at redshift z ∼ 0.2). Of course this violation may also
be due to phantom energy [3].
In conclusion, we have shown that a sudden future
singularity may happen in the comparatively near fu-
ture (e.g. within ten million years) and its prediction
at the present moment of cosmic evolution cannot be
distinguished, with current observational data, from the
prediction given by the standard quintessence scenario
of future evolution in the Concordance Model. Fortu-
nately, sudden future singularities are characterized by a
momentary peak of infinite tidal forces only; there is no
geodesic incompletness which means that the evolution
of the universe may eventually be continued beyond the
SFS until another “more serious” singularity such as a
Big-Crunch or a Big-Rip. One could then consider, more
generally, a scale factor of the form [7, 15]
a(t) = A+ [(as −A)−D(tr − ts)p − Etos] ym (15)
− (A+Dtpr) (1− y)n +D(tr − tsy)p + Etosyo ,
where the constants m, o, p, A,D,E are chosen so that
the universe begins with a Big-Bang at t = 0 where a = 0,
next faces a sudden future singularity at t = ts where
a(ts) = as, and then eventually continues to a Big-Rip
at t = tr where a(tr) → ∞. All of the matter sources
may be involved since the constants in (15) can be taken
as: 0 < m ≤ 1 (quintessence), p < 0 (phantom), and
o > 1 (standard positive matter pressure).
Whether the universe will end in a Big-Rip or a Big-
Crunch is an open question. Moreover, unlike a sudden
future singularity, both a Big-Rip and Big-Crunch sin-
gularity would represent the real end of the universe.
Fortunately, as was shown in Refs. [4, 17], a Big-Rip
singularity is not possible in the very near future: in or-
der to reach it one must wait about the same time as
the current age of the universe. Apart from that, it is
still possible to avoid it due to a negative tension brane
contribution in a turnaround cyclic cosmology [18].
ACKNOWLEDGEMENTS
M.P.D. and T.D. acknowledge the support of the Polish
Ministry of Education and Science grant No 1 P03B 043
29 (years 2005-2007).
∗ Electronic address: mpdabfz@sus.univ.szczecin.pl
† Electronic address: atomek@sus.univ.szczecin.pl
‡ Electronic address: martin@astro.gla.ac.uk
[1] J.L. Tonry et al., Astroph. J. 594, 1 (2003); M. Tegmark
et al., Phys. Rev. D 69, 103501 (2004).
[2] P. Astier et al., Astron. Astrophys. 447, 31 (2006);
A.G. Riess et al., astro-ph/0611572; J.D. Neill et al.
astro-ph/0701161; J. Guy et al. astro-ph/0701828; M.W.
Wood-Vasey et al. astro-ph/0701041.
[3] R.R. Caldwell, Phys. Lett. B 545, 23 (2002); M.P.
Da̧browski, T. Stachowiak and M. Szyd lowski, Phys.
Rev. D 68, 103519 (2003); P.H. Frampton, Phys. Lett.
B 562 (2003), 139; H. Štefančić, Phys. Lett. B586, 5
(2004); S. Nojiri and S.D. Odintsov, Phys. Lett. B595, 1
(2004).
[4] R.R. Caldwell, M. Kamionkowski, and N.N. Weinberg,
Phys. Rev. Lett. 91, 071301 (2003).
[5] R.R. Caldwell, R. Dave, and P.J. Steinhardt, Phys. Rev.
Lett. 80, 1582 (1998).
[6] J.D. Barrow, Class. Quantum Grav. 21, L79 (2004).
[7] L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev.
D70, 121503(R) (2004); L. Fernandez-Jambrina and R.
Lazkoz, Phys. Rev. D74, 064030 (2006).
[8] M.P. Da̧browski and A. Balcerzak, Phys. Rev. D73,
101301(R) (2006).
[9] M.P. Da̧browski, Journ. Math. Phys. 34, 1447 (1993).
[10] M.P. Da̧browski and M.A. Hendry, Astroph. Journ. 498,
67 (1998); R.K. Barrett, C.A. Clarkson, Class. Quantum
Grav. 17, 5047 (2000).
[11] J.D. Barrow and Ch. Tsagas, Class. Quantum Grav. 22,
1563 (2005).
[12] M.P. Da̧browski, Phys. Rev. D71, 103505 (2005).
[13] S. Nojiri, S.D. Odintsov and S. Tsujikawa, Phys. Rev. D
71,063004 (2005).
[14] U. Alam et al., Mon. Not. R. Astron. Soc. 344, 1057
(2003), M.P. Da̧browski, Phys. Lett. B 625, 184 (2005).
[15] C. Cattoen and M. Visser, Class. Quantum Grav. 22,
4913 (2005).
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[17] A.A. Starobinsky, Grav. Cosmol. 6, 157 (2000).
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(2007).
mailto:mpdabfz@sus.univ.szczecin.pl
mailto:atomek@sus.univ.szczecin.pl
mailto:martin@astro.gla.ac.uk
http://arxiv.org/abs/astro-ph/0611572
http://arxiv.org/abs/astro-ph/0701161
http://arxiv.org/abs/astro-ph/0701828
http://arxiv.org/abs/astro-ph/0701041
http://arxiv.org/abs/astro-ph/0702728

  We discuss the constraints coming from current observations of type Ia
supernovae on cosmological models which allow sudden future singularities of
pressure (with the scale factor and the energy density regular). We show that
such a sudden singularity may happen in the very near future (e.g. within ten
million years) and its prediction at the present moment of cosmic evolution
cannot be distinguished, with current observational data, from the prediction
given by the standard quintessence scenario of future evolution. Fortunately,
sudden future singularities are characterized by a momentary peak of infinite
tidal forces only; there is no geodesic incompletness which means that the
evolution of the universe may eventually be continued throughout until another
``more serious'' singularity such as Big-Crunch or Big-Rip.

How far is it to a sudden future singularity of pressure?
Mariusz P. Da̧browski∗
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
Tomasz Denkiewicz†
Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland and
Fachbereich Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany
Martin A. Hendry‡
Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK
(Dated: November 12, 2018)
We discuss the constraints coming from current observations of type Ia supernovae on cosmo-
logical models which allow sudden future singularities of pressure (with the scale factor and the
energy density regular). We show that such a sudden singularity may happen in the very near
future (e.g. within ten million years) and its prediction at the present moment of cosmic evolution
cannot be distinguished, with current observational data, from the prediction given by the standard
quintessence scenario of future evolution. Fortunately, sudden future singularities are characterized
by a momentary peak of infinite tidal forces only; there is no geodesic incompletness which means
that the evolution of the universe may eventually be continued throughout until another “more
serious” singularity such as Big-Crunch or Big-Rip.
PACS numbers: 98.80.Cq; 98.80.Hw
Over the past decade observations of high-redshift
Type Ia supernovae (SNIa) have provided strong evi-
dence that the expansion of the universe is accelerating,
driven in the standard paradigm by some form of dark
energy[1, 2]. Current data[2] continue to leave open the
possibility that dark energy exists in the form of phan-
tom energy, which may violate all energy conditions [3]:
the null (̺c2 + p ≥ 0), weak (̺c2 ≥ 0 and ̺c2 + p ≥ 0),
strong (̺c2 + p ≥ 0 and ̺c2 + 3p ≥ 0), and dominant
energy (̺c2 ≥ 0, −̺c2 ≤ p ≤ ̺c2) conditions (where
c is the speed of light, ̺ is the mass density in kgm−3
and p is the pressure). Phantom matter may dominate
the universe in the future and drive it towards a Big-Rip
(BR) singularity in which all matter will be dissociated
by gravity [4]. This is dramatically different from the
standard picture of future cosmic evolution which sug-
gests an asymptotically empty de-Sitter state driven by
the cosmological constant or quintessence [5] and leading
to the violation of the strong energy condition only.
Phantom-driven scenarios have encouraged the study
of other exotic possibilities for the future evolution of
the universe. One of these possibilities appears in those
models which do not assume any explicit form for the
equation of state p = p(̺), leaving the evolution of the
energy density and pressure unconstrained. This free-
dom may result in a so-called sudden future singularity
(SFS) of pressure [6] which violates only the dominant
energy condition. The nature of a sudden future singu-
larity is different from that of a standard Big-Bang (BB)
singularity, and also from a Big-Rip singularity, in that it
does not exhibit geodesic incompletness and the cosmic
evolution may eventually be extended beyond it [7, 8].
The only physical characteristic of these singularities is
a momentarily infinite peak of the tidal forces in the uni-
verse. In more general models this peak may also appear
in the derivatives of the tidal forces. It is interesting to
note that these types of singularity are in a way simi-
lar to yet another type, which were termed finite density
singularities [9]. However, the crucial difference is that
finite density singularities occur as singularities in space
rather than in time, which means that even at the present
moment of cosmic evolution they could exist somewhere
in the Universe [10]. We will not discuss in detail finite
density singularities in this paper since they basically ap-
pear in cosmological models without homogeneity. On
the other hand, it is worth mentioning that the sudden
future singularities are quite generic since they may arise
in both homogeneous [11] and inhomogeneous [12] mod-
els of the universe.
In order to obtain a sudden future singularity consider
the simple framework of an Einstein-Friedmann cosmol-
ogy governed by the standard field equations
, (1)
p = − c
, (2)
where the energy-momentum conservation law
˙̺ = −3 ȧ
, (3)
is trivially fulfilled due to the Bianchi identity. Here a(t)
is the scale factor, G is the gravitational constant, and
the curvature index k = 0,±1. What is crucial to ob-
tain a sudden future singularity is that no link between
http://arxiv.org/abs/0704.1383v2
the energy density and pressure (the equation of state) is
specified. This allows us to integrate (3) only by quadra-
tures as
̺a3 = exp
3p(t′)
c2̺(t′)
ln a(t′)
p(t′)
̺(t′)
ln a(t′)dt′
Of course (4) reduces to the standard expression for
energy conservation, ̺a3(w+1) = const., provided a
barotropic equation of state, p = w̺c2 for constant w,
is assumed. (The condition for phantom models, for ex-
ample, is w < −1).
From equations (1)-(2) one can easily see that a pres-
sure singularity p → ∓∞ occurs when the acceleration
ä → ±∞, no matter that the value of the energy density
̺ and the scale factor a(t) are regular. Since in that case
| p |> ̺, it is clear that the dominant energy condition
is violated. This condition can be achieved if the scale
factor takes the form [6]
a(t) = as [1 + (1− δ) ym − δ (1− y)n] , y ≡
with the appropriate choice of the constants δ, ts, as,m, n.
Moreover, we can see that the r-th derivative of the scale
factor (5) is given by
a(r) = as
m(m− 1)...(m− r + 1)
(1− δ) ym−r
+ (−1)r−1δ
n(n− 1)...(n− r + 1)
(1− y)n−r
and is related to the appropriate pressure derivative
p(r−2). Thus, in general, it is possible that one has a
pressure derivative p(r−2) singularity which accompanies
the blow-up of the r-th derivative of the scale factor a(r).
Observationally this could be manifested in, for example,
the blow-up of the characteristics known as statefinders ,
such as jerk, snap etc. [14]. The pressure derivative sin-
gularity p(r−2) appears when
r − 1 < n < r r = integer , (7)
and for any r ≥ 3 it fulfills all energy conditions. These
singularities are called generalized sudden future singu-
larities (GSFS) and are possible, for example, in theories
with higher-order curvature quantum corrections [13].
Let us now return to the case of r = 2, for which 1 <
n < 2 and we obtain sudden future singularity models
of pressure (and obviously all of its higher derivatives)
which lead to violation of the dominant energy condition.
In such models, expressed in terms of the scale factor (5),
the evolution begins with the standard BB singularity at
t = 0 for a = 0, and finishes at SFS for t = ts where a =
as ≡ a(ts) is a constant. (Note that we have changed the
original parametrization of Ref. [6] for the scale factor
(5) using A = δas).
The standard Friedmann limit (i.e. models without an
SFS) of (5) is achieved when δ → 0; hence δ becomes
the “non-standardicity” parameter of SFS models. Ad-
ditionally, notwithstanding Ref. [6] and in agreement
with the field equations (1)-(2), we assume that δ can be
both positive and negative leading to a deceleration or
an acceleration (cf. (6)) of the universe, respectively.
It is important to our discussion that the asymptotic
behaviour of the scale factor (5) close to the BB singu-
larity at t = 0 is given by a simple power-law aBB = y
simulating the behaviour of flat k = 0 barotropic fluid
models with m = 2/[3(w+1)] . This allows us to preserve
all the standard observed characteristics of early universe
cosmology – such as the cosmic microwave background,
density perturbations, nucleosynthesis etc. – provided we
choose an appropriate value of m. On the other hand,
close to an SFS the asymptotic behaviour of the scale
factor is non-standard, aSFS = as [1− δ (1− y)n ], show-
ing that aSFS = as for t = ts (i.e. y = 1) at the SFS.
Notice that one does not violate the energy conditions if
the parameter m lies in the range
0 < m ≤ 1 (w ≥ −1/3), (8)
This range of values is, in fact, equivalent to a standard
(neither quintessence-like nor phantom-like) evolution of
the universe. However, with no adverse impact on the
field equations (1)-(2), one could also extend the val-
ues of m to lie in the complementary ranges [7] m > 1
(i.e. −1 < w < −1/3) for quintessence, and m < 0 (i.e.
w < −1) for phantom, although these ranges may lead
to violation of the strong and weak energy conditions
respectively.
We will next calculate the luminosity distance as a
function of redshift, and hence the redshift-magnitude
relation, for SFS models. This will allow us to estab-
lish whether these models are a realistic possibility for
the future evolution of the universe, and more specifi-
cally whether current cosmological observations of high
redshift supernovae are consistent with values of the con-
stant n in the range 1 < n < 2, as required in order that
the scale factor will display an SFS (or, more generally,
a GSFS for r − 1 < n < r). We will then explore the
range of values for the other SFS model parameters which
are consistent with current observational constraints on
standard cosmology, and thus determine limits on how
far into the future an SFS might occur. In fact, as we
will see below, we need to consider only two further pa-
rameters: δ and y0 = t0/tS , where t0 is the current age
of the Universe in the SFS model. Notice that, in view of
(8), it is reasonable to take m = 2/3 as for the standard
dust-dominated evolution. This implies that, at early
times, our SFS model reduces to the Einstein-de-Sitter
universe.
We proceed within the framework of Friedmann cos-
mology, and consider an observer located at r = 0 at
coordinate time t = t0. The observer receives a light ray
emitted at r = r1 at coordinate time t = t1. We then
have a standard null geodesic equation
1− kr2
, (9)
with the scale factor a(t) given by (5). Using (5) again,
the redshift is given by
1 + z =
a(t0)
a(t1)
δ + (1− δ) ym0 − δ (1− y0)
δ + (1− δ) ym1 − δ (1− y1)
n , (10)
where y0 = y(t0) and y1 = y(t1). The luminosity distance
is defined as
DL = r1a(t0) (1 + z) . (11)
Neglecting extinction and k−corrections, the observed
and absolute magnitudes of a source at redshift z and
luminosity distance DL are related by
m(z) = M − 5 log10 H0 + 25 + 5 log10 DL(z), (12)
which, with the help of the equation (9), (10) and (11),
allows a redshift-magnitude relation for SFS cosmologi-
cal models to be constructed. It is obvious that equation
(9) has to be integrated numerically in order to estab-
lish the relation between t0 and t1, which can then be
inserted into (10) and (11) to constrain the SFS model
parameters. As a first step we determine the dependence
on the SFS model parameters of the Hubble law, which
replaces equation (12) for z ≈ 0, i.e. cz ≈ H0DL, where
H0(kms
Mpc−1) =
3.09× 1019
t0(sec)y0
m (1− δ) ym−10 + nδ (1− y0)
δ + (1− δ) ym0 − δ (1− y0)
is the present value of the Hubble parameter, which we
can take as 72kms−1Mpc−1 [1].
Similarly we could derive an expression, in terms of
the SFS model parameters, for the deceleration param-
eter q0 = −(äa/ȧ2)0. However, in order to search the
parameter space for models which are admissible by cur-
rent observations, we write the product of H0 and q0 as
q0H0 = −
= (14)
m(m− 1)(1− δ)ym−20 − δn(n− 1) (1− y0)
m(1− δ)ym−10 + nδ (1− y0)
In order to obtain an accelerated universe at the present
moment of the evolution, this product should be nega-
tive. Fig. 1 shows an example plot of the product H0q0
as a function of δ and y0, with the other parameters fixed
atm = 2/3, n = 1.9993, t0 = 13.2457 Gyr. From the plot
we see that there are large regions of the parameter space
which admit cosmic acceleration. We have explored the
parameter space further with various configurations of
m,n, δ, y0, t0, q0, and H0, and obtained the general con-
clusion that there is a large class of SFS models which
are compatible with current acceleration.
H0*q0
FIG. 1: Parameter space (H0q0, δ, y0) for fixed values of
m = 2/3, n = 1.9993, t0 = 13.3547 Gyr of the sudden future
singularity models. There are large regions of the parameter
space which admit cosmic acceleration.
Out of these admissible models we then searched for
those which are compatible with the redshift-magnitude
relation (12) observed for recent SNIa data [2], and hence
with the derived parameters of the standard ‘Concor-
dance cosmology’ (CC). We were able to identify SFS
models that are in remarkably tight agreement with cur-
rent SNIa data. As an illustrative example Fig. 2 shows
luminosity distance as a function of redshift for the CC
model with H0 = 72kms
−1Mpc−1, Ωm0 = 0.26 and
ΩΛ0 = 0.74, and an SFS model with parametersm = 2/3,
y0 = 0.99936, δ = −0.471, n = 1.9999. We see that the
SFS model mimics the CC model very closely over a wide
range of redshifts. In particular, it is clear that recent
SNIa data from the Tonry at al. ‘Gold’ sample [1] and
SNLS sample [2] cannot yet discriminate between the CC
and SFS models.
Taking the current age of the universe in the SFS model
to be equal to the age of the CC model, i.e. t0 = 13.6Gyr,
we find that the time to the sudden singularity is ts−t0 ≈
8.7Myr, which is amazingly close to the present epoch.
In that context there is no wonder that these singularities
are called “sudden”. We have also checked that the larger
the value of r in (7) the later in future a GSFS appears.
It means that the strongest of these singularities which
violates the dominant energy condition (i.e. an SFS) is
more likely to become reality.
Our remark about the effect of the sudden pressure
singularity seems in agreement with the result of Ref.
FIG. 2: The distance modulus µL = m − M for the con-
cordance cosmology (CC) model with H0 = 72kms
−1Mpc−1,
Ωm0 = 0.26, ΩΛ0 = 0.74 (dashed curve) and sudden fu-
ture singularity (SFS) model for m = 2/3, n = 1.9999, δ =
−0.471, y0 = 0.99936 (solid curve). Also shown are the ‘Gold’
(open circles) and SNLS (filled circles) SNIa data. Taking the
age of the SFS model to be equal to that of the CC model,
i.e. t0 = 13.6 Gyr, one finds that an SFS is possible in only
8.7 million years.
[16] which showed that the dominant energy condition is
now violated and that it became violated quite recently
(at redshift z ∼ 0.2). Of course this violation may also
be due to phantom energy [3].
In conclusion, we have shown that a sudden future
singularity may happen in the comparatively near fu-
ture (e.g. within ten million years) and its prediction
at the present moment of cosmic evolution cannot be
distinguished, with current observational data, from the
prediction given by the standard quintessence scenario
of future evolution in the Concordance Model. Fortu-
nately, sudden future singularities are characterized by a
momentary peak of infinite tidal forces only; there is no
geodesic incompletness which means that the evolution
of the universe may eventually be continued beyond the
SFS until another “more serious” singularity such as a
Big-Crunch or a Big-Rip. One could then consider, more
generally, a scale factor of the form [7, 15]
a(t) = A+ [(as −A)−D(tr − ts)p − Etos] ym (15)
− (A+Dtpr) (1− y)n +D(tr − tsy)p + Etosyo ,
where the constants m, o, p, A,D,E are chosen so that
the universe begins with a Big-Bang at t = 0 where a = 0,
next faces a sudden future singularity at t = ts where
a(ts) = as, and then eventually continues to a Big-Rip
at t = tr where a(tr) → ∞. All of the matter sources
may be involved since the constants in (15) can be taken
as: 0 < m ≤ 1 (quintessence), p < 0 (phantom), and
o > 1 (standard positive matter pressure).
Whether the universe will end in a Big-Rip or a Big-
Crunch is an open question. Moreover, unlike a sudden
future singularity, both a Big-Rip and Big-Crunch sin-
gularity would represent the real end of the universe.
Fortunately, as was shown in Refs. [4, 17], a Big-Rip
singularity is not possible in the very near future: in or-
der to reach it one must wait about the same time as
the current age of the universe. Apart from that, it is
still possible to avoid it due to a negative tension brane
contribution in a turnaround cyclic cosmology [18].
ACKNOWLEDGEMENTS
M.P.D. and T.D. acknowledge the support of the Polish
Ministry of Education and Science grant No 1 P03B 043
29 (years 2005-2007).
∗ Electronic address: mpdabfz@sus.univ.szczecin.pl
† Electronic address: atomek@sus.univ.szczecin.pl
‡ Electronic address: martin@astro.gla.ac.uk
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mailto:mpdabfz@sus.univ.szczecin.pl
mailto:atomek@sus.univ.szczecin.pl
mailto:martin@astro.gla.ac.uk
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