1. No category

Why We Need Dark Energy:
A Thermodynamical Perspective
Diego Pavón & Ninfa Radicella
Universidad Autónoma de Barcelona
Salamanca: April 19, 2011
Why we need dark energy
Abstract
Abstract
It is argued that dark energy (or something dynamically equivalent, as
modified gravity) is required if the Universe is to tend to a state of
thermodynamical equilibrium in the long run.
Why we need dark energy
Dark matter yielding to dark energy
Why we need dark energy
Introduction
The standard cold dark matter model was in good health until around
the last decade of the previous century when it became apparent that
the fractional density of matter falls well below the Einstein-de Sitter
value, Ωm = 1. The death blow came at the close of the century with
the discovery of the current cosmic acceleration, something the said
model cannot accommodate by any means. However, to account for
the acceleration in homogeneous and isotropic models one must
either introduce some exotic energy component with a huge negative
pressure (dubbed dark energy) or, more drastically, devise some
theory of gravity more general than Einstein relativity. Thus, both
solutions appear somewhat forced and not very aesthetical. Here we
argue that dark energy (or something equivalent) is demanded on
thermodynamic grounds. In other words, we provide what we believe
is a sound thermodynamic motivation for the existence of dark energy.
Why we need dark energy
The second law
Ordinary systems tend to thermodynamic equilibrium
S′ ≥ 0
S ′′ ≤ 0
and
The entropy is contributed by the Horizon entropy, SH , and the
Fluid entropy, Sf , within the horizon.
SH ∝ A
Hubble horizon, lH = H −1
A∝
1
3 1
=
H2
2G ρ
Why we need dark energy
The second law
Therefore,
A′ =
9 1+w
2G a ρ
phantom models violate the second law.
A′′ =
9
(1 + w) (2 + 3w)
2G a2 ρ
Accordingly, A′′ ≤ 0 for −1 ≤ w ≤ −2/3, and A′′ > 0 otherwise. Thus,
universes dominated by fluids at late times with constant equation of
state either of phantom type or larger than −2/3 do not tend to
thermodynamical equilibrium.
Why we need dark energy
The second law
Figure: Evolution of the Hubble horizon area with the scale factor. w = −5/6
(solid line), w = −1 (dashed), and w = −1.2 (dot-dashed).
Why we need dark energy
The second law
A
1.42
-
10
50
100
a
Figure: Evolution of the horizon area with the scale factor for a universe
dominated by CDM and cosmological constant, w = −1. From top to bottom
Ωk0 = −0.02, 0, and 0.009, respectively. In all three cases ΩΛ = 0.7.
Why we need dark energy
Evolution of A in terms of the deceleration parameter
Bearing in mind that
aH ′
q=− 1 +
H
⇒ A′ = 8π(1 + q)/H 2
q′
A = 2A (1 + q) (1 + 2q) +
a
′′
Therefore, for q > −1/2 ⇒ A′′ cannot evolve from positive to
negative values unless q ′ < 0. This excludes evolutions of q as
0.5
q
0
È
a0
-0.5
-1
a
Why we need dark energy
Incorporating the fluid entropy
What about the fluid entropy? Is the second law really fulfilled?
′
Sf′ + SH
≥ 0?
Sf′
3(w−1)/2
′ ∝a
SH
′
⇒ Sf′ + SH
≥ 0 for w < 1 when a → ∞
′′
Likewise, is Sf′′ + SH
≤ 0 in the long run?
Sf′′
3(w−1)/2
′′ ∝ a
SH
′′
⇒ Sf′′ + SH
≤0
for −1 < w < −2/3 when a → ∞
Why we need dark energy
w 6= constant
Barboza-Alcaniz model (PLB (2008))
w(z) = w0 + w1
z(1 + z)
1 + z2
(1 + z = 1/a)
The accessible range
w1 < 2/3
−1 < w0 < −2/3
∪
2/3 ≤ w1 < 1
−1 < w0 < −w1 ,
comes to be fully consistent with the observational constraints from
SNIa, BAO & CMB data
−1.35 ≤ w0 ≤ −0.86 ,
−0.33 ≤ w1 ≤ 0.91
Why we need dark energy
Chaplygin gas
p = −A/ρ
⇒ ρ=
p
a + (B/a6 )
The evolution of the Hubble’s horizon area, A ∝ 1/ρ, is akin to the
one depicted by the solid line in Fig.1.
Moreover,
′
SCh
→0 &
′
SH
′′
SCh
′′ → 0 when a → ∞
SH
Therefore
′
′
SCh
+ SH
>0 &
′′
′′
SCh
+ SH
<0
when
Why we need dark energy
a→∞
Modified Gravity Models: DGP
The Dvali-Gabdaze-Porrati model (PLB (2000)) presents
k
H + 2 =
a
2
SH
3π r˜2
= kB 2 A
ℓpl
s
r˜A
1+
rc
We found,
′
=
S ′′
=
S
1
1
ρ
+ 2 +
2
3MP l
4rc
2rc
r˜A = p
!2
1
H 2 + (k/a2 )
′
Sm
Sr′
1 + ′ + ′ → 0,
SA
SA
′′
Sm
Sr′′
′′
SA 1 + ′′ + ′′ → 0 .
SA
SA
′
SA
Why we need dark energy
!
Modified Gravity Models: DGP
Both relations, taken together, imply an upper bound on the current
number of dust particles
n0 <
1
kB
27kB Ωm0 rc H02
∼ 1038 cm−3
4 ℓ2P l c
Since it is fulfilled by a huge margin, the second law is satisfied and
S ′′ results negative in the long run.
Why we need dark energy
Modified Gravity Models: Cardassian
Spatially flat FRW, dust dominated, model with generalized
Friedmann equation (Freese et al., PLB (2002))
H2 =
8πG
ρ + Bρα
3
At the background level every cardassian model can be mapped onto
some dark energy dominated model with k = 0 satisfying
α=1 + w
and
B = (8πG/3) (ρx0 /ρα
m0 )
′′
As a consequence, SH
< 0 for sensible α values.
Also, it can be seen that
′
′
Sm
+ SH
> 0,
and
′′
′′
Sm
+ SH
<0
as
Why we need dark energy
a→∞
Modified Gravity Models: Torsion
Bengochea and Ferraro model (PRD (2009))
Z
√
1
d4 x −g (τ + f (τ )) + Imatter
I=
16πG
τ = −6 H 2
f (τ ) = −α (−τ )−n ,
0.0002
A
0.0001
0
1
2
3
4
a
Figure: Area of the apparent horizon vs. the scale factor for the best fit model
of Bengochea & Ferraro. It is seen that A′ > 0 and A′′ < 0 when a → ∞.
Why we need dark energy
Modified Gravity Models: Torsion
1HaHL3
0
1
2
3
4
a
Figure: Entropy of the non-relativistic matter vs. the scale factor. When
′
′′
a ≫ 1 ⇒ Sm
< 0 and Sm
> 0.
In this case,
′
′
|Sm
|/SH
→ ∞,
and
′′
′
Sm
/|SH
|→∞
when a ≫ 1.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
The Barboza-Alcaniz’ can for an substantial range of its
parameters.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
The Barboza-Alcaniz’ can for an substantial range of its
parameters.
Some modified gravity models, such as DGP & Cardassian
models, can. Bengochea & Ferraro’s model cannot.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
The Barboza-Alcaniz’ can for an substantial range of its
parameters.
Some modified gravity models, such as DGP & Cardassian
models, can. Bengochea & Ferraro’s model cannot.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
The Barboza-Alcaniz’ can for an substantial range of its
parameters.
Some modified gravity models, such as DGP & Cardassian
models, can. Bengochea & Ferraro’s model cannot.
Models in which the present accelerated stage of expansion is
only transitory appear to be excluded by the second law.
Why we need dark energy
Conclusions
Neither a radiation dominated nor a CDM dominated universe
can tend to thermodynamic equilibrium in the long run.
Dark energy dominated universes, with constant w in the range
−1 ≤ w < −2/3 can. The ΛCDM model also can.
Phantom models with w = constant, cannot.
Dark energy models with w 6= constant deserve a separate
analysis. In particular, the Chaplygin gas model can.
The Barboza-Alcaniz’ can for an substantial range of its
parameters.
Some modified gravity models, such as DGP & Cardassian
models, can. Bengochea & Ferraro’s model cannot.
Models in which the present accelerated stage of expansion is
only transitory appear to be excluded by the second law.
Finally, the existence of dark energy or -equivalently- some
modified gravity theory could have been expected on
thermodynamic grounds.
Why we need dark energy