Dark Energy predictions from GREA: Background and linear perturbation theory

Juan Garcı́a-Bellido1, ∗
1
Instituto de Fı́sica Teórica UAM-CSIC, Universidad Autonóma de Madrid, Cantoblanco 28049 Madrid, Spain
(Dated: May 28, 2024)
General Relativistic Entropic Acceleration (GREA) theory provides a covariant formalism for
out-of-equilibrium phenomena in GR, extending the Einstein equations with an entropic force that
behaves like bulk viscosity with a negative effective pressure. In particular, the growth of entropy
associated with the homogeneous causal horizon can explain the present acceleration of the Uni-
verse, without introducing a cosmological constant. The dynamics of the accelerated universe is
characterized by a single parameter α, the ratio of the causal horizon to the curvature scale, which
provides a unique history of the Universe distinguishable from that of ΛCDM. In particular, we
arXiv:2405.02895v2 [astro-ph.CO] 25 May 2024

explain the coincidence problem and the Hubble tension by shifting the coasting point to higher
redshifts. All background observables are correlated among themselves due to their common depen-
dence on α. This scenario gives a specific evolution for the effective equation of state parameter,
w(a). Furthermore, we study the linear growth of matter perturbations in the context of a homo-
geneous expanding background driven by the entropy of the causal horizon. We find that the rate
of growth of matter fluctuations in GREA slows down due to the accelerated expansion and allevi-
ates the σ8 tension of ΛCDM. We compute the growth function of matter fluctuations, the redshift
space distortions in the galaxy correlation function, as well as the redshift evolution of the baryon
acoustic oscillation scale, and find that the integrated Sachs-Wolfe effect is significantly larger than
in ΛCDM. It is interesting to note that many of the tensions and anomalies of the standard model
of cosmology are alleviated by the inclusion of this transient period of acceleration of the Universe
based on known fundamental physics. In the near future we will be able to constrain this theory
with present data from deep galaxy surveys.

I. INTRODUCTION In this paper, we perform a detailed calculation of the

background and linear matter perturbations with the as-
General Relativistic Entropic Acceleration (GREA) sumption that the entropy growth responsible for the ac-
theory [1] gives a covariant formalism for out of equilib- celerated expansion comes from the homogeneous cosmo-
rium dynamics in the context of general relativity. A con- logical horizon. We show that, in this case, the dynamics
sequence of GREA is the explicit breaking of the time re- can be described in terms of just one parameter, α, the
versal invariance when there is entropy production. This ratio of the spatial curvature to the horizon distance,
drives an entropic force that behaves effectively like bulk which gives the size of the cosmological horizon and the
viscosity with a negative effective pressure. The natural rate of growth of the horizon entropy. A given value of α
consequence of this formalism is that the source for space- defines a unique history of the universe, with a dynam-
time curvature is the Helmholtz free energy, F = U −T S, ics that differs from ΛCDM and gives an explicit predic-
and not just matter and radiation [2, 3]. tion for all background observables, like the values of the
When the entropy production is associated with the matter and dark energy content of the universe, the rate
growth of horizons one can describe their thermodynam- of expansion today, the coasting point, the value of the
ical effects as a GHY boundary term [4, 5]. In the cos- equation of state of dark energy and its derivative, the
mological context, this theory predicts that the present angular diameter and luminosity distances, etc.
acceleration of the universe could arise from the growth of In order to compare with the plethora of cosmologi-
entropy associated with cosmic [6] and black hole hori- cal data, we need not just the background evolution but
zons [2], without the need to introduce a cosmological also the linear growth of matter fluctuations. We de-
constant, providing an alternative dynamics to that of rive a second order differential equation in GREA for
the Standard Cosmological Model (ΛCDM). the amplitude of matter perturbations in a homogeneous
Already before the recent evidence of significant devi- background, and compare with the corresponding one in
ations from the standard model of cosmology through ΛCDM. The solutions match those during the matter
state-of-the-art observations by DES-Y5 SNe [7] and era, but start deviating around redshift one, when the
DESI-Y1 BAO [8], GREA predictions gave a better de- entropic forces begin to take over and start accelerating
scription of cosmological observations than ΛCDM [9]. In the universe.
particular, GREA explains the coincidence problem and In Section II we describe the extended Einstein field
can alleviate the Hubble tension by shifting the coasting equations in the presence of out-of-equilibrium phenom-
point (when the universe transitioned from matter domi- ena associated with the growth of horizons. We recall
nation to acceleration) to higher redshifts and extending from Ref. [1] that the first law of thermodynamics can
the period of acceleration. be incorporated into Einstein’s equations as an effective
 2

bulk viscosity, which drives an effective negative pressure fluid [1, 10], with Θ = Dλ uλ the trace of the congruence
as a consequence of the second law of thermodynamics. of geodesics,
In Section III we explore the background phenomenol-
ogy of GREA in the context of entropic forces arising fµν = ζ Θ (gµν + uµ uν ) = ζ Θ hµν , (2)
from the cosmological horizon. In Section IV we derive such that the covariantly-conserved energy-momentum
the linear perturbation theory of GREA and compare tensor has the form of a perfect fluid tensor,
cosmological observables with the corresponding ones in
ΛCDM. Finally, we conclude in Section V. T µν = p g µν + (ρ + p)uµ uν − ζ Θ hµν
= p̃ g µν + (ρ + p̃)uµ uν , (3)
II. THE EINSTEIN FIELD EQUATIONS IN GREA with p̃ = p + pS , and the bulk viscosity coefficient ζ can
be written as [1]
We give here a short summary of the covariant formal-
ism developed in Ref. [1] that describes the extended Ein- T dS
ζ= . (4)
stein field equations in the presence of out-of-equilibrium Θ dV
phenomenon associated with the growth of horizons. The In the case of an expanding universe, Θ = dtd
ln V = 3H
gravitational action of the Standard Model of Cosmology 2 3
and the coefficient becomes ζ = T Ṡ/(9H a ), see [6],
(ΛCDM) is given by
with S the entropy per comoving volume of the Universe.
1
Z
√ h i 1Z √ Entropy production therefore implies ζ > 0.
S= d4 x −g R + 2κLm − d4 x −g Λ , Note that the matter contribution to the Hamiltonian
2κ M κ M
constraint arises from variations of the matter action
where Lm is the matter Lagrangian, κ = 8πG and Λ is with respect to the lapse function (and afterwards set-
the cosmological constant. In the context of GREA we ting N (t) = c),
substitute the bulk term of the cosmological constant for
√
Z Z
the GHY boundary term of the horizon [6], Sm = d4 x −g Lm = − dt N (t) Vc a3 ρ , (5)
1
Z
4 √
h i 1Z √
S= d x −g R + 2κLm + d3 x h K , where Vc is the comoving volume,
2κ M κ H
4π r2 dr
Z
2π
where K is the extrinsic curvature and h the 3D metric Vc = √ = × (6)
1−kr 2 (−k)3/2
of the cosmological horizon, H = ∂M. h p p  p i
The main difference of GREA with respect to ΛCDM ln 1 − k r2 − −k r2 + −k r2 (1 − k r2 ) ,
is the breaking of time-reversal invariance at the level of
the action by the entropic force term that arises due to for a given radial coordinate r. The conservation of the
the growth of the cosmic horizon. This introduces a cos- total energy momentum tensor is derived from the first
mological arrow of time. Most of the evolution of the uni- law of thermodynamics,
verse has been quasi-adiabatic since these entropic forces
acted only during inflation, reheating after inflation and dU + pdV − T dS = 0 ⇒ (7)
at the present time, when the universe is old enough and
Vc d(ρ a3 ) + p d(a3 ) − T dSc = 0
 
⇒ (8)
the cosmic horizon associated with radiation and mat-
ter has grown sufficiently large for its entropy growth to T Ṡc
ρ̇ + 3H(ρ + p) = , (9)
dominate over the attraction of matter and induce the a3
present acceleration. Eventually, even this period of ac-
where Sc is the entropy per comoving volume. For adi-
celeration will end as the entropic term gets diluted and
abatic expansion, or constant Sc , we recover the usual
we will end in an empty Minkowsky space-time [6].
conservation equation [1].
Therefore, the Einstein field equations, extended to
out-of-equilibrium phenomena, can be written as [1],
The causal cosmological horizon
1
Gµν = Rµν − R gµν = κ (Tµν − fµν ) ≡ κ Tµν , (1)
2
Let us describe here the entropic forces induced by
where fµν arises from the first law of thermodynamics, the causal cosmological horizon of a FLRW universe [6].
and introduces an effective negative pressure associated We start by considering an arbitrary comoving 2-sphere
with the growth of entropy, pS = −T dS/dV < 0, accord- around the origin of coordinates. Then the trace of its
ing to the second law of thermodynamics. This extra extrinsic curvature is given by (we set c = 1 everywhere)
component to the Einstein equations can be interpreted √ p
as an effective bulk viscosity term of a real (non-ideal) hK = −2N (t) r a 1 − kr2 sin θ (10)
 3

90

5 GREA
80

a H(a) [km/s/Mpc]
ΛCDM
2
dH (a)

70
1 ΛCDM

0.5
60 GREA

0.2
50
0.2 0.5 1 2 0 1 2 3 4

a z

FIG. 1: The horizon distance as a function of the scale factor

in GREA (color lines), compared with ΛCDM (black dot- 0.4
dashed line) and matter dominated (dotted blue line). It is
clear that the ratio of the curvature scale to the horizon dis-
tance is very approximately one, for all realizations of GREA.
0.2
GREA
√
√ 1 + ω (a)
where r = sinh η −k / −k along the lightcone, and η
is the conformal time. The boundary term for the causal
0.0
cosmological horizon, dH = a η, can be written as [6]
ΛCDM
1
Z
a  √ 
SGHY = − dt N (t) √ sinh 2η −k (11)
2G −k -0.2
Z Z
= − dt N (t) TH SH ≡ − dt N (t) Vc a3 ρH ,
0.0 0.5 1.0 1.5 2.0

where TH is the temperature and SH the entropy associ- a

ated with the causal cosmological horizon [6]
FIG. 2: The upper panel shows the evolution of the inverse
√
comoving horizon with the corresponding coasting points
ℏ a sinh 2η −k kB π d2H
kB TH = 2
√ , SH = . (12) (dashed lines) for each value of α (in color). The lower panel
2π dH −k ℏ G shows the evolution of the effective DE equation of state w(a).

The fact that we can naturally assign a temperature and

an entropy to a hypersurface is a signal of the existence of
an underlying quantum description of gravity and ther-
modynamics [11]. This is made explicit by the appear-
ance of ℏ in both quantities. Their product, however,
does not depend on ℏ and leads to a classical emergent
phenomenon, the acceleration of the universe. Note also
that variation w.r.t. the lapse function will give an ex-
tra contribution to the Hamiltonian constraint coming
from the boundary term [4], whose origin is related to force arising from the causal horizon. We will assume
the quantum degrees of freedom of the horizon, a phe- here a concrete scenario of open inflation [6] in which the
nomenon which has connections with the Holographic causal horizon coincides with the boundary of a bubble
principle [12, 13]. wall separating our flat FLRW universe from an open
empty space, therefore matching conditions require that
both the scale factor and its derivative should be con-
III. GREA PHENOMENOLOGY 2
tinuous at that hypersurface, i.e. 3H√ in = κ ρ(ain ) and
2 2
Hout = −k/ain , which determines −k = a0 H(a0 ) to-
We can then solve the dynamical Friedmann equations day. We write the Hamiltonian constraint in confor-
for the accelerated universe in the context of an entropic mal time (where primes denote derivatives w.r.t. τ =
 4
√ 1
0.80

−k η = a0 H0 η) as 0.80

★ ★

2 2
a′
   
a 4π a sinh(2τ ) 0.75 ★
0.75
GREA
★
= ΩM + , (13)
a0 a0 3 a0 (−k)3/2 Vc ★ ★

H(0)
H(0)
0.70
0.70 GREA
where the comoving volume (6) is now ΛCDM ★ ★
ΛCDM
h  √  √ i 0.65 ★ 0.65 ★
(−k)3/2 Vc = π sinh 2 −k η0 − 2 −k η0 . (14)
★ ★

0.60 0.60
We can then fix η0 and determine the ratio between the 0.5 0.6 0.7

zc
0.8 0.9 1.0 -1.10 -1.05 -1.00

ω0
-0.95 -0.90

causal horizon today and the curvature scale (c = 1),

0.80

a0 η0 √ ★
0.0
ΛCDM
α H(a0 ) dH (a0 ) ≡ √ = −k η0 . (15)
a0 / −k 0.75 ★ -0.1

GREA
This ratio can be used to parametrize the different GREA ★ -0.2

H(0)

ωa
0.70
scenarios and compare with ΛCDM, see Fig. 1, where
★ ΛCDM
-0.3 GREA
★
★
1 1 7 ΩM − 1 3
 
2 3/2 0.65 ★ ★ ★ ★ ★

H0 dH (a) = √ a · 2 F1 , , , a , -0.4

ΩM 2 6 6 ΩM ★
(16) 0.60
0.20 0.25 0.30 0.35 0.40
-0.5
-1.10 -1.05 -1.00 -0.95 -0.90

is the expression for dH (a) in ΛCDM. ΩM ω0

Solving the Hamiltonian constraint equation (13) with

initial conditions set by the Cosmic Microwave Back- FIG. 3: Different GREA scenarios in the plane of ΩM , the
ground, deep in the matter era, where ai (τ ) = τ 2 ΩM /4, coasting point zc , the present rate of expansion H(0) (in units
of 100 km/s/Mpc) and the EOS parameters (w0 , wa ). We also
with ΩM = 0.31, ΩK = 0 and H0 = 67.8 km/s/Mpc [14], plot the ΛCDM values (black dashed lines).
we find an effective dark energy contribution ΩΛ ≃ 0.70
today, for a narrow range of values of α ≃ 1, see Table I.
precisely today?, in the case of GREA the growth of the
α ΩM ΩΛ h0 w0 wa zc cosmic horizon during the matter era is responsible for
1.139 0.368 0.632 0.786 -0.908 -0.315 0.931 the transition to a dominance of the entropic repulsive
1.119 0.336 0.664 0.749 -0.937 -0.331 0.840 forces associated with the acceleration of the universe.
1.099 0.307 0.693 0.714 -0.966 -0.345 0.757 The moment at which this happens depends on the ac-
1.081 0.279 0.721 0.682 -0.996 -0.354 0.678 tual value of the spatial boundary today in units of the
1.062 0.254 0.746 0.651 -1.028 -0.365 0.603 causal horizon, as can be seen in Fig. 2a, where we show
1.043 0.231 0.769 0.622 -1.060 -0.367 0.533 the coasting points for the different realizations of GREA.
0.310 0.690 0.678 -1.000 -0.000 0.645
In this scenario, the value of α depends on how far is our
worldline from the bubble wall, as pictured in Fig. 1 of
TABLE I: The GREA values of the DM and DE content, the Ref.[6]. If the bubble wall would have been further away,
rate of expansion today, H(0) = 100 h0 km/s/Mpc, the ef- the onset of acceleration would have happened later for
fective EOS parameters (w0 , wa ) and the redshift zc of the the same value of ΩM inside our universe. However, in an
corresponding coasting point, for different values of the pa- open universe the largest volume is near the bubble wall,
rameter α. From top to bottom, the values of α correspond so it is expected that most observers will lie far from the
to the (red, orange, yellow, green, blue, purple) colors in the
rest of the figures. We compare those values with the corre-
center of the inflated bubble and near its edges. There-
sponding ones for ΛCDM (bottom row). fore the finetuning associated with the coincidence prob-
lem is strongly alleviated. Note also that different GREA
scenarios populate the parameter space of the cosmologi-
It is important to emphasize that GREA has ex-
cal background in a way that can be easily distinguished
actly the same number of free parameters as ΛCDM, i.e.
from ΛCDM, see Fig. 3, with multiple panels, showing
(α, ΩM , H0 ) versus (ΩΛ , ΩM , H0 ), and nevertheless has
that of every 2D parameter space there is always a value
very different dynamics, both at the present times and
of α that can accommodate a universe like that of ΛCDM,
in the far future. Moreover, while in ΛCDM there is a
except for the value of wa , which is always large and neg-
coincidence (“why now”) problem, i.e. why is ΩΛ ≃ ΩM
ative, in agreement with recent observations.
Furthermore, in GREA the value of the rate of expan-
sion today is not fixed at the CMB, like in the case of
1 Note that there is a difference w.r.t. Ref. [6], and we can now ΛCDM. Here, depending on α, the present rate of expan-
set ΩK = 0 inside the causal patch. sion, H(0), could be larger or smaller than the ΛCDM
 5

value derived from the CMB, see Fig. 4a and Table I. In

fact, for certain values of the spatial curvature one can 1.15

H(z) ratio (GREA / ΛCDM)

resolve in a natural way the so-called Hubble tension [15].
Finally, while in ΛCDM, or its variant w0 wa ΛCDM, 1.10
the value of the equation of state of Dark Energy is given
by w(a) = w0 + wa (1 − a), as an expansion around the
1.05
present value, in GREA we have the whole function w(a)
to compare with observations, as can be seen in Fig. 2b. ΛCDM
The evolution of the scale factor in GREA allows one to 1.00

compute the effective equation of state of dark energy,

−d ln H 2 (a)
  a 3 
0
0.95 GREA
1 + w(a) = − ΩM
d ln a3 H02 a
0.90
0 1 2 3 4
−d ln sinh(2τ )
 
= . (17) z
d ln a3 a2 (τ )
√
In the matter dominated era, τ ∝ a, and therefore 5
1 + w(a) → 1/2 as a → ai , see Fig. 2b. In the far
future, a combination of curvature and decaying entropy a- 1
production gives a dependence 1+w(a) → 2/3 as a → ∞. 2
The rate of expansion therefore will decay as curvature
in the far future, diluting the universe, and ending in
flat Minkowski space-time, see Fig.4b. Note that this is H(a) 1 a - 3 2
very different from the standard ΛCDM scenario with the ΛCDM
asymptote to empty de Sitter. 0.5

In fact, in GREA theory there is no freedom to

chose (ΩM , H(0), w0 , wa ) independently, as it happens GREA
in w0 wa ΛCDM. The actual values of (w0 , wa ) are pre- 0.2

cisely given by the theory, just like the present rate of

0.5 1 5 10
expansion, H(0) = 100 h0 km/s/Mpc, once you fix the
ratio α, see Fig. 3 for some examples. As a consequence, a
if the cosmological data favours values that are far away
from ΛCDM (w0 = −1, wa = 0), then we can test the FIG. 4: The upper panel shows the ratio of the evolution of
whole theory by comparing those values with the corre- the rate of expansion in GREA (color lines) versus ΛCDM
sponding predictions for both H(0) and (ΩM , ΩΛ ), as can (black dot-dashed line). The lower panel shows the actual
evolution in GREA and ΛCDM, with the past (matter) and
be seen in Table I. future (curvature) asymptotics (dotted blue lines).
We note in Fig. 5 that the GREA scenario is not in ten-
sion with the recent DESI-BAO [8] + DES-SNIa [7] data,
although a proper analysis, including the background and matter perturbation we need to develop a linear pertur-
linear perturbations of GREA, has to be performed with bation theory in the accelerating background of GREA,
the full data set (CMB, SNIa, LSS, BAO, RSD, ISW, H0 , which will drive a rate of growth that may be distin-
etc.). In fact, the amount of information that we can ex- guishable in principle from that of ΛCDM. Furthermore,
tract from present observations is larger than what can be in order to compare with observations, this linear growth
inferred from their projection on to the plane (w0 , wa ), has to be properly characterized as a function of the free
and future data will be even more constraining at differ- parameters of the theory.
ent redshifts. The equations driving the evolution of matter fluctu-
ations can be derived from the covariant conservation of
the total energy-momentum tensor (3). The background
IV. LINEAR PERTURBATION THEORY IN
metric in conformal time is that of a curved FLRW with
GREA
a Newtonian potential Φ and curvature fluctuation Ψ,

ds2 = a2 (η) −(1 + 2Φ) dη 2 + (1 − 2Ψ)γij dxi dxj .

 
So far the description of GREA has been done at the
level of the background evolution, while we have ignored (18)
the small matter fluctuations that give rise to galaxies In the absence of shear and vorticity (i.e. for a spatially-
and the large scale structure (LSS) of the universe. In symmetric matter energy-momentum tensor), the New-
order to understand the effect of GREA on the growth of tonian potential and curvature fluctuation satisfy Φ = Ψ.
 6

1.0

ΛCDM
0.8 ΛCDM
0
0.6
GREA

δ(a)
°1 0.4
wa

GREA 0.2

°2
0.0
DESI BAO + CMB + PantheonPlus 0.0 0.2 0.4 0.6 0.8 1.0

DESI BAO + CMB + Union3 a

DESI BAO + CMB + DESY5
°3 1.1
°1.0 °0.8 °0.6 °0.4
w0
1.0

FIG. 5: The plot shows the parameter space (w0 , wa ). Cos-

mological data seemed to prefer GREA from ΛCDM at the Φ(a) / Φ0 0.9 ΛCDM
2σ level, already in 2021, see Ref. [9]. Nowadays, the SN-Ia
from DES Y5, and the BAO from DESI Y1, give compelling
evidence that ΛCDM is excluded at close to 4σ, while GREA 0.8
is within the 2σ contours. Figure adapted from Ref. [8]. GREA

0.7
Then the Bardeen equation [16] for the gauge invariant
curvature fluctuation in flat space (K = 0) with a cos-
mological constant and entropic pressure canP
be written,
P
0.6
0.0 0.2 0.4 0.6 0.8 1.0
in terms of a general equation of state w = p i / ρi ,
as a

Φ′′ (η) + 3HΦ′ (η) − 3wH2 Φ = c2s ∇2 Φ , (19) FIG. 6: We show the damped evolution of the density contrast
(upper panel) and the Newtonian potential (lower panel), as a
function of the scale factor for ΛCDM (black dot-dashed line)
where the Poisson equation
and GREA (colors correspond to those of Fig. 2). The gray
line corresponds to the growth in an open matter dominated
∇2 Φ = 4πGρ a2 δ (20) cosmology. It is clear that both ΛCDM and GREA perturba-
tions deviate significantly from a matter-only scenario after
relates the curvature fluctuation Φ to the gauge-invariant a ∼ 0.2 or z ∼ 4.
density contrast, δ. Expanding Bardeen’s equation (19)
for a matter fluid with negligible speed of sound, c2s ≃ 0,
we find

Φ′′ (η) + 3HΦ′ (η) + (Λa2 − 8πG a2 pS )Φ(η) = 0 , (21)

with pS = −ρH is the entropic pressure (13). We can

then rewrite the Bardeen equation, using the Poisson where τ = a0 H0 η, and the time evolution is determined
equation, Φ ∝ δ/a, and the identity by eq.(13). We can now integrate with initial condition
deep in the matter era δi (τ ) ∝ ai (τ ) ∝ τ 2 . Once the
3 2 ΩM solution is found for the density contrast, we can derive
H′ + 2H2 = H + Λa2 − 8πG a2 pS ,
2 0 a the time evolution of the Newtonian potential Φ via the
Poisson equation for matter fluctuations (20).
to finally write

a′ ′ 3 ΩM
δ ′′ (τ ) + δ (τ ) − δ(τ ) = 0 , (22)
a 2 a(τ ) For flat ΛCDM, the solution to eq. (22) can be written
 7

0.5 0.7

ΛCDM 0.6
0.4

GREA
0.5
0.3
H0 dA (z)

f(z) σ8 (z)
GREA
0.4

0.2 ΛCDM
0.3

0.1
0.2

0.0 0.1
0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

z z
0.570
FIG. 7: We show the angular diameter distance for ΛCDM
(black dot-dashed line) and GREA (colors correspond to those 0.565
of Fig. 2).
0.560

explicitly in terms of Hypergeometric functions,

0.555 GREA
γ(z)
0.550
1 11 ΩM − 1 3
 
δ(a) = δ0 a · 2 F1 1, , , a , (23) 0.545
3 6 ΩM
ΛCDM
√ 0.540
1 1 7 ΩM − 1 3
 
2 a
τ (a) = √ · 2 F1 , , , a , (24)
ΩM 2 6 6 ΩM 0.535

1 11 ΩM − 1 3
 
0.530
Φ(a) = Φ0 · 2 F1 1, , , a . (25) 0 1 2 3 4
3 6 ΩM
z
In the case of GREA, we must solve first the background
evolution (13), and then the linear perturbation eq. (22). FIG. 8: We show the growth function f (z) σ8 (z) and growth
index γ(z) for ΛCDM (black dot-dashed line) and GREA (col-
We compare in Fig. 6 the time evolution of both the ors correspond to those of Fig. 2).
density contrast δ(a) and the Newtonian potential Φ(a)
for ΛCDM and GREA for the values of α in Table I.
fined as the power of ΩM (a) in the growth function,
 −5/6 h −1/2 i 
The growth function and index 1 1 P−1/6 ΩM (a)
γ(a) = + ln  h i  . (27)
2 ln ΩM (a) P
−5/6
Ω
−1/2
(a)
1/6 M
In Large Scale Structure, the redshift-space distortions
in the two-point correlation function ξ(r) depends on a We have shown in Fig. 8 the growth index for ΛCDM
particular combination of the density contrast that ap- and GREA for different values of ΩK as a function of
pears in the velocity perturbation known as the growth redshift. Future surveys should be able to distinguish
function [17], f (a) = d ln δ(a)/d ln a, where δ(a) is the between these two scenarios.
linear matter perturbation. In the case of ΛCDM the
growth function has a compact expression [18],
h i The BAO scale and the S8 tension
−5/6 −1/2
P−1/6 ΩM (a)
1/2
f (a) = ΩM (a) h i ≡ ΩγM (a) , (26) One of the most important physical quantities that
−5/6 −1/2
P1/6 ΩM (a) one can measure with deep galaxy surveys is the BAO
scale, corresponding to the size of the sonic horizon as a
where ΩM (a) = ΩM /a3 H 2 (a) and Pnm (z) are the associ- function of redshift. The angular BAO scale depends
ated Legendre polynomials. The growth index γ is de- on both the sonic horizon at the baryon drag epoch,
 8

χs (zdrag ) = 110 Mpc/h, and the angular diameter dis- 1.3

tance, while the radial BAO scale depends also on the
rate of expansion,

ISW ratio (GREA / ΛCDM)

1.2

σ8 ratio (GREA / ΛCDM)

χs (zdrag )
θBAO (z) = , (28)
dA (z) 1.1

H(z)
rBAO (z) = χs (zdrag ) . (29)
c 1.0

The angular diameter distance for flat ΛCDM is given

0.9
explicitly in terms of τ (a) of eq. (24)

τ (0) − τ (z) 0.8

H0 dA (z) = . (30)
1+z
We show in Fig. 7 the angular diameter distance as a 0.7
1.06 1.08 1.10 1.12
function of redshift.
α
The luminosity distance needed to make connection
with the SN-Ia data is simply related to the angular di-
FIG. 9: We show the ratio of the amplitude of fluctuations
ameter distance via the Etherington relation, (red) at 8 Mpc/h and the ISW effect (blue) between GREA
and ΛCDM, as a function of α. There is a value for which
dL (z) = (1 + z)2 dA (z) . both ratios are one, for α = 1.092.

Another cosmological observable which is derived from

weak lensing is the magnitude of the fluctuations to- along the line of sight from the last scattering surface,
day on a scale of 8 Mpc/h. For ΛCDM and GREA we see eq. (31), and compare with the predicted value in
have different predictions and one can write the ratio ΛCDM. We show the ratio in Fig. 9. While the effect of
as a function of α, as shown in Fig. 9. For those val- GREA on σ8 can be at most a few percent, in the case
ues of α that resolve the H0 tension, predicting today of the ISW effect, the correction can reach 20 or 30%.
H(0) = 73 ± 1 km/s/Mpc [15], the σ8 value predicted by Note that we have studied here only the homogenous
GREA is around 4% lower than that of ΛCDM, resolving ISW effect, due to the global expansion of the universe.
also the S8 tension [15]. There is a much more pronounced ISW effect towards
We should be cautious here, since in order to claim deep voids in the cosmic web, where photons travers-
a resolution of the so-called H0 and S8 tensions within ing large voids get an extra decrement in the ISW tem-
GREA requires a full CMB, weak lensing and Cepheid- perature contrast due to the growth of the Newtonian
TRGB analysis. In fact, recent claims of resolution of potential as the void becomes emptier. Such an effect
each of these tensions have been discussed recently and is known to be significantly larger and cannot be ac-
it could be that they are associated with unknown sys- counted for by the evolution of the LSS around large
tematics yet to be understood. voids in ΛCDM [19]. However, GREA can account for
this extra acceleration due to the production of entropy
The ISW effect
associated with the formation of the cosmic web and the
growth of voids as a consequence of gravitational collapse
of large structures like filaments, sheets and superclus-
An important observable that can be obtained from the ters [2]. This effect is not computed here and may be
CMB maps is the Integrated Sachs-Wolfe (ISW) effect, relevant for the comparison with observations in the fu-
which arises because of the decrease in the Newtonian ture.
potential Φ due to the presence of an accelerating ex-
pansion, as photons travel since decoupling and feel the
redshift induced by the expansion of the Universe. This V. CONCLUSIONS
corresponds to a temperature decrement given by
Z η0
δT General Relativistic Entropic Acceleration Theory is a
=2 dη Φ′ (η0 − η) . (31) covariant framework that can explain the present accel-
T 0
eration of the universe from the entropic force associated
The equation for Φ(η) can be derived in flat ΛCDM as with the growth of the cosmological horizon, without the
Φ′′ (τ ) + 3(a′ /a) Φ′ (τ ) + Λ a2 Φ(τ ) = 0, see eq. (21). need to introduce a fundamental cosmological constant,
For GREA the Newtonian potential evolves according i.e. Λ = 0 always. The cosmic entropic acceleration
to the density contrast, see Fig. 6b. One can integrate associated with the homogeneous expansion of the cos-
 9

mological causal horizon can be described in terms of just subject of a future publication.
one parameter α, the ratio between the horizon distance In summary, it is encouraging that most of the ten-
and the spatial curvature. sions and anomalies of the standard model of cosmol-
The entropic acceleration due to the causal horizon ogy are resolved, or at least alleviated, by the inclusion
starts to be important only recently, thanks to the of this transient period of acceleration of the universe
sinh(2τ ) dependence in Eq. (13), alleviating the coinci- based on known fundamental physics. We leave for a
dence problem. It is the fact that the universe is old and future publication the detailed comparison of the predic-
big that GREA started to dominate the expansion of the tions of GREA theory with the present cosmological data
universe only recently. In the past, such a term was ex- from deep and wide galaxy surveys like DES, DESI and
ponentially negligible, for ratios α ≃ 1. One only has Euclid.
to live near the edge of the bubble, where most of the
volume is. For larger horizon distances the acceleration
would have started later for the same amount of matter. ACKNOWLEDGEMENTS
The recent evidence for deviations from the ΛCDM
paradigm seems to agree remarkably well with the pre-
The author thanks Julien Lesgourgues for constructive
dictions of GREA at the homogeneous background level.
criticisms on a preliminary draft that has helped improve
In particular, it resolves the Hubble tension by shift-
the paper. He also acknowledges support from the Span-
ing the coasting point zc to higher redshifts and extend-
ish Research Project PID2021-123012NB-C43 [MICINN-
ing the period of accelerated expansion with respect to
FEDER], and the Centro de Excelencia Severo Ochoa
ΛCDM. Nevertheless, this entropic acceleration doesn’t
Program CEX2020-001007-S at IFT.
last for ever and eventually the GREA entropic term is
diluted just like matter, so that the universe ends in a
flat Minkowski space-time, instead of de Sitter.
We find that some background observables in GREA,
like ΩM and the effective equation of state parameter to- ∗
juan.garciabellido@uam.es
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