In this Letter we consider three future singularities in different
Friedmann-Lema\^{\i}tre-Robertson-Walker scenarios and show that the universe
departs more and more from thermodynamic equilibrium as the corresponding
singularity is approached. Though not proven in general, this feature may
characterize future singularities of homogeneous and isotropic cosmologies.Comment: Key words: mathematical cosmology, singularities, thermodynamics; 11
  pages, to be publised in Physics Letters 

Abramo

Barrow

Caldwell

Callen

Christiansen

del Campo

Diego Pavón

Ellis

Frampton

Hawking

Radicella

Steinhardt

Wald

Wang

Winfried Zimdahl

Physics Letters B

English

arXiv

Crossref

A thermodynamic characterization of future singularities?

AbstractIn this Letter we consider three future singularities in different Friedmann–Lemaître–Robertson–Walker scenarios and show that the universe departs more and more from thermodynamic equilibrium as the corresponding singularity is approached. Though not proven in general, this feature may characterize future singularities of homogeneous and isotropic cosmologies

Pavón, Diego

Zimdahl, Winfried

Elsevier - Publisher Connector 

A thermodynamic characterization of future singularities? 

Physics Letters B 708 (2012) 217–220Contents lists available at SciVerse ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A thermodynamic characterization of future singularities?
Diego Pavón a,∗, Winfried Zimdahl b
a Department de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
b Universidade Federal do Espírito Santo, Departamento de Física, Av. Fernando Ferrari, 514, Campus de Goiabeiras, CEP 2905-910, Vitória, Espírito Santo, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:
Received 4 January 2012
Received in revised form 27 January 2012
Accepted 30 January 2012
Available online 1 February 2012
Editor: T. Yanagida
Keywords:
Mathematical cosmology
Singularities
Thermodynamics
In this Letter we consider three future singularities in different Friedmann–Lemaître–Robertson–Walker
scenarios and show that the universe departs more and more from thermodynamic equilibrium as the
corresponding singularity is approached. Though not proven in general, this feature may characterize
future singularities of homogeneous and isotropic cosmologies.
© 2012 Elsevier B.V. Open access under CC BY license.1. Introduction
Cosmological singularities arise as displeasing features in math-
ematical models of the universe [1,2]; world lines terminate and/or
physical quantities diverge. It is usually argued that this results
because of the exceedingly simplicity of the models in question,
implying that when we reach a much better understanding of the
physical processes that take place under the most extreme condi-
tions we will be able to design realistic models free of singularities.
Perhaps the most worrying of all is the Big Bang singularity
which persists even if the standard cosmological model is cor-
rected with the addition of an era of inﬂationary expansion just
before the radiation dominated epoch. This explains the interest
raised by proposals of universes, such as bouncing [3] and emer-
gent [4], with no beginning at all.
Singularities are heralded, among other things, by the growing
without bound of key physical quantities, such as energy densi-
ties and pressures. Here we focus on future cosmic singularities in
Friedmann–Lemaître–Robertson–Walker (FLRW) universes and ask
ourselves if they may be characterized by some thermodynamic
distinctive feature. Our provisory answer is in the aﬃrmative. We
reach this tentative conclusion after considering three singularities,
linked to different cosmological models. As it turns out in the three
cases the total entropy of the systems does not tend to a maximum
as the singularity is approached. That is to say, the systems do not
move towards thermodynamical equilibrium but on the contrary:
* Corresponding author.
E-mail addresses: diego.pavon@uab.es (D. Pavón), winfried.zimdahl@pq.cnpq.br
(W. Zimdahl).0370-2693 © 2012 Elsevier B.V.
doi:10.1016/j.physletb.2012.01.074
Open access under CC BY license.the closer they get to the singularity, the further away from the
said equilibrium the systems depart. By “system” we mean the ap-
parent cosmic horizon plus the matter and ﬁelds enclosed by it.
At this point it seems suitable to recall that physical systems
tend spontaneously to some equilibrium state compatible with the
constraints imposed on them. This summarizes the empirical basis
of the second law of thermodynamics. Put brieﬂy, the latter estab-
lishes that isolated, macroscopic systems, evolve to the maximum
entropy state consistent with their constraints [5]. As a conse-
quence their entropy, S , cannot decrease, i.e., S ′  0, where the
prime means derivative with respect to the relevant, appropriate
variable. Further, S has to be a convex function of the said vari-
able, S ′′ < 0, at least at the last phase of the evolution.
The apparent horizon in FLRW spacetimes is deﬁned as the
marginally trapped surface with vanishing expansion of radius [6]
r˜ A = 1/
√
H2 + ka−2, (1)
where a and k are the scale factor of the metric and the spatial
curvature index, respectively, and it is widely known to have an
entropy which, leaving aside possible quantum corrections, is pro-
portional to its area
Sh ∝A= 4π r˜2A, (2)
which agrees nicely well with the holographic entropy derived
from considerations of the foamy structure of spacetime [7]. Be-
sides, this horizon appears to be the appropriate thermodynamic
boundary [8].
In its turn, the entropy of the ﬂuid enclosed by the horizon is
related to its energy density and pressure by Gibbs’ equation [5],
namely,
218 D. Pavón, W. Zimdahl / Physics Letters B 708 (2012) 217–220T dS f = d
(
ρ
4π
3
r˜3A
)
+ p d
(
4π
3
r˜3A
)
, (3)
where T stands for the ﬂuid’s temperature.
As said above, for the second law to be satisﬁed the inequality
dSh + dS f  0 must hold at all times, and d2(Sh + S f ) < 0 at least
at the last stage of evolution.
Application of this broad idea to cosmic scenarios leads to a
variety of interesting results. Among others, dark energy (or some
other agent of late acceleration) appears thermodynamically mo-
tivated: both in the case of Einstein gravity [9] and in modiﬁed
gravity [10]. In particular, ever expanding universes dominated
either by radiation or pressureless matter cannot approach ther-
modynamic equilibrium at late times. This is also true for those
phantom dominated universes whose equation of state parameter,
w = p/ρ , is a constant [9].
The target of this Letter is to study whether the universe gets
closer and closer to thermodynamic equilibrium as it approaches a
future singularity. We assume Einstein gravity and provide some
general relations in Section 2. Then we consider three speciﬁc
FLRW scenarios: the big crunch singularity (Section 3), a sudden
singularity (Section 4), and a “little rip” singularity (Section 5). Dis-
cussion and ﬁnal comments are presented in Section 6.
2. General relations
The ﬁeld equations for a spatially homogeneous and isotropic
universe are the Friedmann equation
3
(
H2 + k
a2
)
= 3
r˜2A
= 8πGρ (k = 0,±1), (4)
and
H˙ = −4πG(ρ + p) + k
a2
= −H2 − 1
2r˜2A
(
1+ 3 p
ρ
)
. (5)
Quite generally we ﬁnd
A′ = 12π r˜
2
A
a
(
1+ p
ρ
)
, (6)
where the prime denotes the derivative with respect to the scale
factor a, and
A′′ = 36π r˜
2
A
a2
(
1+ p
ρ
)[
2
3
(
1+ 3 p
ρ
)
− p
′
ρ ′
]
. (7)
Via the proportionality Sh ∝A (cf. (2)), the relations (6) and (7)
will allow us to obtain information about S ′h and S
′′
h , respectively,
for various choices of p/ρ and p′/ρ ′ .
To determine the derivatives of S f we must ﬁrst discern the
temperature evolution of the ﬂuid. From Euler’s relation nT s =
ρ + p, where n and s are the number density of particles in a
comoving volume and the entropy per particle, respectively, and
the conservation equations ρ ′ = −3(ρ + p)/a and n′ = −3n/a, we
ﬁnd
s′ = p′ − (ρ + p) T
′
T
. (8)
Taking into account the perfect ﬂuid condition s′ = 0, the temper-
ature behavior is governed by
T ′
T
= p
′
ρ + p . (9)
Straightforwardly one obtains the desired expressions,S ′f = 2π
r˜3A
a
ρ + p
T
(
1+ 3 p
ρ
)
(10)
and
S ′′f = 2π
r˜3A
a2
ρ + p
T
[
12
p
ρ
(
1+ 3
2
p
ρ
)
− 9 p
′
ρ ′
(
1+ p
ρ
)
+ 1
2
(
1+ 3 p
ρ
)2]
. (11)
Sections 3–5 apply the above set of formulas to three different
cosmological scenarios that harbor a future singularity.
3. Big crunch singularity
Here we focus on the radiation-dominated, spatially closed (k =
+1) FLRW universe and explore the entropy behavior of the appar-
ent horizon and radiation enclosed by it as the big crunch draws
close. Under such circumstances the temperature gets so high that
matter becomes extremely relativistic and behaves as radiation,
hence p = ρ/3 and ρa4 = const.
The scale factor and Hubble function are given by
a(t) = C
√
1−
(
1− ts − t
C
)2
and H ≡ a˙
a
= −
√
C2 − a2
a2
, (12)
where ts is the time at which a = 0 and C2 ≡ 8πG3 ρa4. The horizon
area is A = 4π
C2
a4. Either by direct calculation or as special cases
from (6) and (7) with p = ρ/3 it follows that
A′ = 16π
C2
a3 > 0 and A′′ = 48π
C2
a2 > 0, (13)
respectively. Thus, the graph of A increases and is concave for all
values of the scale factor.
Eq. (10) leads to S ′f ∝ a2 > 0 and Eq. (11) implies S ′′f ∝ a > 0.
Thus, the total entropy (that of the apparent horizon, Sh , plus
that of the radiation ﬂuid), fulﬁlls the generalized second law,
S ′h + S ′f > 0. However, because S ′′h + S ′′f > 0 the system does
not tend to thermodynamic equilibrium as the big crunch is ap-
proached.
4. Sudden singularity scenario
In this scenario the singularity occurs at ﬁnite time (say ts) and
is characterized by the divergence of the acceleration and pressure
while the energy density, scale factor and Hubble expansion rate
remain ﬁnite. As the singularity is approached, the dominant en-
ergy condition (|p| < ρ) is violated but all other energy conditions
are respected.
Let us consider the scale factor of the spatially ﬂat (k = 0) FLRW
metric as introduced in [11,12], namely:
a(t) =
(
t
ts
)α
(as − 1) + 1−
(
1− t
ts
)β
, (14)
where as = a(t = ts) > 1, and α and β are constant parameters
lying in the range (0,1] and (1,2), respectively. Obviously, this ex-
pression holds for 0< t < ts .
Since
a˙(t) = α
ts
(as − 1)
(
t
ts
)α−1
+ β
ts
(
1− t
ts
)β−1
> 0, (15)
the Hubble function H = a˙/a never becomes negative.
D. Pavón, W. Zimdahl / Physics Letters B 708 (2012) 217–220 219A subsequent derivation yields
a¨(t) = α
t2s
(α − 1)(as − 1)
(
t
ts
)α−2
− β
t2s
(β − 1)
(
1− t
ts
)β−2
.
(16)
In the limit t → ts the ﬁrst term, both in a(t) and a˙(t), dominates
over the second one while the latter dominates in the expres-
sion for the acceleration which becomes negative and diverges. As
a → as , H → Hs and ρ → ρs > 0 where as , Hs and ρs are all ﬁnite
but ps → ∞ via the ﬁeld equation 3a¨/a = −4πG(ρ + 3p).
In view of the above, in the said limit (t → ts) we can write
a˙(a) = α
ts
(as − 1)
(
a − 1
as − 1
)(α−1)/α
,
H(a) = α
ts
(as − 1)1/α (a − 1)
(α−1)/α
a
, (17)
where we have eliminated the cosmic time in favor of the scale
factor.
The area of the apparent horizon is given by
A= 4πH−2 	 4π
(
ts
α
)2
(as − 1)−2/α a
2
(a − 1)2(α−1)/α , (18)
where the second equality holds to leading order only. From (6)
one has
A′ > 0 (19)
always. This corresponds to H ′(a) < 0, which can also be checked
explicitly from the full expression of H(t). Since p′ > 0 (the pres-
sure diverges) and ρ ′ < 0, both terms in the square bracket on the
right-hand side of Eq. (7) are positive, consequently,
A′′ > 0 (20)
as well. Both A′ and A′′ diverge upon approaching the singularity.
Before concluding that in this sudden singularity scenario
the universe departs from thermodynamic equilibrium we must,
as before, examine the thermodynamic behavior of the gravity
source.
From (10) and (11) it follows immediately that with p > 0,
p′ > 0 and ρ ′ < 0 one has both S ′f > 0 and S
′′
f > 0. The inequality
S ′f > 0 together with A′ > 0 implies that the generalized second
law, S ′h + S ′f > 0, is fulﬁlled in this scenario. On the other hand,
the inequality S ′′h + S ′′f > 0 means that the total entropy is a con-
cave function of the scale factor and the universe does not tend to
thermodynamic equilibrium.
5. Little rip scenario
The expression “little rip” was coined as a contraposition to “big
rip” [13]. In this scenario the ratio p/ρ < −1 but it increases and
approaches −1 as time goes on. Although sooner or later all bound
structures rip apart, at variance with the usual big rip scenario
[13], neither the energy density nor the scale factor diverge at a
ﬁnite time. There is a future singularity but, because the expansion
rate approaches de Sitter, it is pushed to t → ∞ [14].
The equation of state of the gravity source is p = −ρ − Aρ1/2,
with A > 0. This alongside the conservation equation ρ ′ = −3(ρ +
p)/a produces
ρ = ρ0
[
3A
2
√
ρ
ln
(
a
a
)
+ 1
]2
(21)0 0(cf. Eq. (12) in [14]). Notice that ρ augments with expansion, a
typical feature of phantom dark energy, but logarithmically only
and diverges just when a → ∞. The zero subscript denotes some
reference time, we conveniently take it as the time at which the
dark energy overwhelms all other components (matter, radiation,
etc.) to the point that their dynamical inﬂuence can be safely ig-
nored.
Because the FLRW metric is spatially ﬂat, we have A =
4π/H2 ∝ 1/ρ . From (6) one veriﬁes A′ < 0 and (7) provides us
with A′′ > 0.
We next study the entropy evolution of the ﬂuid. The equation
of state p = −ρ − Aρ1/2 in (10) yields S ′f > 0. Since A′ and S ′f
have opposite signs, a closer look at the behavior of these quan-
tities as they approach the singularity is necessary. Explicitly, we
have
A′ = −12π A
aH2ρ1/2
(22)
and
S ′f = 2π
Aρ1/2
aH3T
(
2+ 3 A
ρ1/2
)
. (23)
To proceed, information about the dependence of the tempera-
ture on a is required. The general law (9) yields
T ′
T
= 3
a
(
1+ A
2ρ1/2
)
. (24)
This integrates to
T
T0
=
(
a
a0
)3[ 3A
2
√
ρ0
ln
(
a
a0
)
+ 1
]
=
(
a
a0
)3(
ρ
ρ0
)1/2
. (25)
Consequently,
S ′f = 2π A
a30ρ
1/2
0
T0
2+ 3A
ρ1/2
a4H3
. (26)
In approaching the singularity, ρ diverges and the term ∝ ρ−1/2
in the second numerator of (26) can be neglected. Since H ∝ ρ1/2,
one has S ′f ∝ 1/(a4ρ3/2) in the limit of large values of a. In the
same limit A′ in (22) behaves as A′ ∝ −1/(aρ3/2). Therefore, as
a → ∞ the ratio |A′/S ′f | diverges as a3 whence S ′h + S ′f < 0 in the
long run; i.e., the generalized second law is violated as the little
rip singularity gets closer and closer.
Despite we already know that in this scenario no thermody-
namic equilibrium is approached when nearing the singularity, we
shall determine whether the function Sh + S f is convex or con-
cave in that limit. From (7) we ﬁnd that A′′ ∝ 1/(ρ3/2a2) for
a → ∞ while (11) yields S ′′f ∝ −1/(ρ3/2a5) in the far-future limit.
The ratio |A′′/S ′′f | diverges with the same power, a3, as the corre-
sponding ratio of the ﬁrst derivatives, i.e., S ′′h + S ′′f > 0 as a → ∞.
6. Concluding remarks
Macroscopic systems moving by themselves away from thermo-
dynamic equilibrium is something far removed from daily experi-
ence. This is enshrined in the second law of thermodynamics that
introduces the entropy function and dictates its overall evolution.
In this Letter we have studied the behavior of three FLRW uni-
verses as they draw close to a future singularity. As long as the
entropy concept is related to the apparent cosmic horizon, none of
the three approaches equilibrium.
220 D. Pavón, W. Zimdahl / Physics Letters B 708 (2012) 217–220Against this some comments may be raised: (i) The proposed
universes look too academic; we should not wonder that unrealis-
tic systems do not comply with thermodynamics. (ii) The Universe
is a very particular and unique system; why should it obey the
thermodynamical laws? (iii) Everyone expects the breakdown of
physical laws at singularities. Then, why should the second law be
an exception?
The ﬁrst comment seems persuasive; however, take into ac-
count that not so academic models, as is the case of phantom
models with constant w , do not approach equilibrium as they
near the singularity [9]. Further, it is rather problematic to draw
a dividing line between “academic” and “non-academic” models
in cosmology. Regarding the second one, as far as we know, ev-
ery macroscopic portion of the Universe fulﬁlls the second law of
thermodynamics. Therefore, there is no compelling argument by
which realistic cosmological models should not fulﬁll it as well. As
for the third comment, one should not forget that we are dealing
not with the singularities themselves but with the behavior of the
models as they approach their respective singularities. During the
approach physical laws still hold.
From this we may conclude that the three models considered
here are unphysical, at least in the regime nearing the singularity.
The third one, the “little rip”, more particularly because it violates
the generalized second law and it should be ruled out.
Finally, the fact that these three models do not approach equi-
librium in the last stage of their evolution may be seen as a feature
characterizing future singularities. (This is shared by the phantom
model mentioned above.) Nevertheless, since – for the moment –
the existence of counter examples cannot be ruled out it would berather premature to assert the general validity of the said feature
in FLRW cosmologies.
Acknowledgements
DP thanks the “Departamento de Física” of the “Universidade
Federal do Espírito Santo” (Vitória), where part of this work was
done, for warm hospitality and ﬁnancial support. This research was
also supported by the CNPq and the Ministry of Science and Inno-
vation under Grant FIS2009-13370-C02-01, and the “Direcció de
Recerca de la Generalitat” under Grant 2009GR-00164.
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