11institutetext: Université Paris-Saclay, UVSQ, CNRS, CEA, Maison de la
Simulation, 91191, Gif-sur-Yvette, France 22institutetext: Université Paris-
Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette France
22email<EMAIL_ADDRESS>33institutetext: Ecole Normale Supérieure de
Lyon, CRAL, UMR CNRS 5574, 69364 Lyon Cedex 07, France 44institutetext:
Astrophysics Group, University of Exeter, EX4 4QL Exeter, UK
44email<EMAIL_ADDRESS>
# Reevaluating the cosmological redshift: insights into inhomogeneities and
irreversible processes

P. Tremblin 1122 G. Chabrier 3344

###### Abstract

Aims.  Understanding the expansion of the Universe remains a profound
challenge in fundamental physics. The complexity of solving General Relativity
equations in the presence of intricate, inhomogeneous flows has compelled
cosmological models to rely on perturbation theory in a homogeneous FLRW
background. This approach accounts for a redshift of light encompassing
contributions from both the cosmological background expansion along the
photon’s trajectory and Doppler effects at emission due to peculiar motions.
However, this computation of the redshift is not covariant, as it hinges on
specific coordinate choices that may distort physical interpretations of the
relativity of motion.

Methods. In this study, we show that peculiar motions, when tracing the
dynamics along time-like geodesics, must contribute to the redshift of light
through a local volume expansion factor, in addition to the background
expansion. By employing a covariant approach to redshift calculation, we
address the central question of whether the cosmological principle alone
guarantees that the averaged local volume expansion factor matches the
background expansion.

Results.  We establish that this holds true only in scenarii characterized by
a reversible evolution of the Universe, where inhomogeneous expansion and
compression modes mutually compensate. In the presence of irreversible
processes, such as the dissipation of large-scale compression modes through
matter virialization and associated entropy production, the averaged expansion
factor becomes dominated by expansion in voids that cannot be compensated
anymore by compression in virialized structures. Furthermore, a Universe where
a substantial portion of its mass has undergone virialization, adhering to the
background evolution on average leads to significant violations of the second
law of thermodynamics. Our approach shows that entropy production due to
irreversible processes during the formation of structures plays the same role
as an effective, time-dependent cosmological constant, i.e. dynamical dark
energy, without the need to invoke new unknown physics. Our findings
underscore the imperative need to reevaluate the influence of inhomogeneities
and irreversible processes on cosmological models, shedding new light on the
intricate dynamics of our Universe.

###### Key Words.:

##  1 Introduction

The comprehension of the recessional velocity of extragalactic objects hinges
upon the redshift-distance relation derived from homogeneous solutions of
Einstein’s field equations within the framework of the Friedmann-Lemaître-
Robertson-Walker (FLRW) metric. Within this coordinate system, the spatial
coordinates are comoving with the matter that fills the Universe, attributing
the cosmological redshift to the expansion of “space” along the path of
photons. However, observations in the nearby Universe have revealed a notable
scatter around the linear redshift-distance relation, primarily driven by the
presence of peculiar velocities of sources, inducing a Doppler shift upon
photon emission (Peebles, 1993; Peacock, 1998). The debate in the literature
regarding whether cosmological and Doppler shifts are equivalent phenomena is
ongoing (Bunn & Hogg (2009), see also Peacock (1998); Whiting (2004);
Chodorowski (2007); Peacock (2008)). As noted in Bunn & Hogg (2009), the
interpretation of space expansion is not entirely satisfactory from a
relativistic standpoint, as it relies on a specific choice of coordinates,
lacking covariance. Conversely, Räsänen (2009) introduced a covariant approach
to redshift calculation for an inhomogeneous dust universe. While this
approach represents a significant advancement in understanding the nature of
redshift in a manner consistent with the covariant framework of General
Relativity, it has been relatively overlooked. This is largely because, on
average, the results do not seem to deviate from the background FLRW expansion
in a statistically homogeneous and isotropic universe (Räsänen, 2009; Rasanen,
2010).

This paper aims to reevaluate the cosmological redshift by addressing the
influence of inhomogeneities and irreversible processes. In Sect. 2, we begin
by providing a detailed analysis on the standard interpretation of the
redshift within the FLRW metric and discuss the limitations of this approach
by using non-comoving coordinates. In the subsequent section (Sect. 3),
building on Räsänen (2009), we introduce a covariant approach to the redshift
calculation, emphasizing the role of peculiar motions and local volume
expansion factors. We propose a new approach to the redshift calculation by
introducing a tracer of time-like geodesics, the cosmological redshift is
associated to the local expansion rate along these time-like geodesics while
the Doppler shifts are associated to the potential non-geodesic motions of the
source and observer. Following this, we present in Sect. 4 our findings on the
impact of irreversible processes on the cosmological redshift. We argue that
in scenarios characterized by irreversible evolution, such as the dissipation
of large-scale compression modes through matter virialization and associated
entropy production, the averaged expansion factor is dominated by expansion in
voids. This leads to significant violations of the second law of
thermodynamics if the Universe’s evolution adheres to the background evolution
on average. Lastly, we discuss in Sect. 5 the broader implications of our
findings for cosmological models and observations. We conclude in Sect. 6 on
the necessity of reevaluating the influence of inhomogeneities and
irreversible processes to better understand the complex dynamics of the
Universe.

##  2 Redshift in a homogeneous expanding universe in non-comoving coordinates

We begin with a standard flat FLRW metric with a scale factor, $a(t)$, such
that

$ds^{2}=-c^{2}dt^{2}+a^{2}(dr^{2}+r^{2}d\Omega^{2}).$ (1)

The proper distance $R$ is

$R=a(t)r,$ (2)

and the total velocity is then

$\frac{dR}{dt}=H(t)R+a(t)\frac{dr}{dt},$ (3)

with $H\equiv\dot{a}/a$. Within the total velocity, one can discern a
contribution originating from the recession velocity due to the expansion of
space, and another contribution arising from peculiar velocities
$v_{p}=a(t)dr/dt$. In the late evolution of the Universe, peculiar motions are
non-relativistic in the weak field limit. Perturbation theory then leads to
the following formulation of the redshift with a cosmological contribution
from space expansion and a contribution from the Doppler shift associated to
peculiar motions

$1+z\approx\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}Hdt\right)\left(1+\frac{v_{p}}{c}\right),$
(4)

assuming for simplicity that the observer at time $t_{\mathrm{dst}}$ has no
peculiar motion, and the source at time $t_{\mathrm{src}}$ a peculiar motion
$v_{p}$ at emission. With this approach, the inhomogeneities have no
significant impact on the redshift, provided that the peculiar velocities
remain small $v_{p}\ll c$.

We now assume a perfectly homogeneous FLRW Universe with no peculiar motion
$v_{p}=0$. We wish to apply a local coordinate transformation in a non-
comoving frame, characterized by an expansion factor $a_{s}(t)$ different from
$a(t)$ such that

$\displaystyle r_{d}$ $\displaystyle=$ $\displaystyle ra_{d}(t),$ (5)
$\displaystyle t_{d}$ $\displaystyle=$ $\displaystyle t+\alpha(r,t),$ (6)

with $a_{s}=a/a_{d}$ and $\alpha(r,t)$ a function that will be specified
later, such that $t_{d}\approx t$ at leading order in $O(\dot{a}_{d}r/c)$. We
use such a coordinate transformation, locally, such that $\dot{a}_{d}r/c$ is
indeed much smaller than unity. With this transformation, the proper distance
$R$ is

$R=a(t)r=a_{s}(t)r_{d},$ (7)

with $a_{s}=a/a_{d}$. The total velocity is then

$\frac{dR}{dt}=H_{s}(t)R+a_{s}(t)\frac{dr_{d}}{dt_{d}},$ (8)

with $H_{s}\equiv\dot{a_{s}}/a_{s}$. Within the total velocity, one can
discern now a contribution originating from the recession velocity due to the
expansion of “space”, characterized by the scale factor $a_{s}$, and another
contribution arising from non-comoving velocities $v_{p,r}=a_{s}H_{d}r_{d}$.

We choose the function $\alpha(r,t)$ such that the metric in $(r_{d},t_{d})$
coordinates has only perturbative deviations from a homogeneous metric with a
scale factor $a_{s}$. The coordinate transformation verifies

$\displaystyle dr_{d}$ $\displaystyle=$ $\displaystyle
a_{d}dr+r\dot{a}_{d}dt,$ (9) $\displaystyle dt_{d}$ $\displaystyle=$
$\displaystyle(1+\dot{\alpha})dt+\alpha^{\prime}dr,$ (10)

with $\dot{\alpha}$ the derivative with respect to $t$ and $\alpha^{\prime}$
the derivative with respect to $r$. This system can be inverted to give

$\displaystyle a_{d}dr$ $\displaystyle=$
$\displaystyle((1+\dot{\alpha})dr_{d}-r\dot{a}_{d}dt_{d})/(1+\dot{\alpha}-rH_{d}\alpha^{\prime}),$
(11) $\displaystyle dt$ $\displaystyle=$
$\displaystyle(dt_{d}-\alpha^{\prime}/a_{d}dr_{d})/(1+\dot{\alpha}-rH_{d}\alpha^{\prime}),$
(12)

with $H_{d}=\dot{a_{d}}/a_{d}$. By injecting these equations into $ds^{2}$,
the non-diagonal metric coefficient is

$g_{r_{d}t_{d}}=2(c^{2}\alpha^{\prime}/a_{d}-a_{s}^{2}(1+\dot{\alpha})r\dot{a}_{d})/(1+\dot{\alpha}-rH_{d}\alpha^{\prime})^{2}$
(13)

We impose

$\alpha(r,t)=\frac{1}{2}\frac{a^{2}H_{d}r^{2}}{c^{2}},$ (14)

such that the metric in $(r_{d},t_{d})$ coordinates is given by

$ds^{2}=-c^{2}Adt_{d}^{2}+a_{s}^{2}(Bdr_{d}^{2}+r_{d}^{2}d\Omega^{2})-2cdt_{d}a_{s}dr_{d}C,$
(15)

with,

$\displaystyle A$ $\displaystyle=$
$\displaystyle(1-a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})/(1+\dot{\alpha}-a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})^{2},$
(16) $\displaystyle B$ $\displaystyle=$
$\displaystyle((1+\dot{\alpha})^{2}-a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})/(1+\dot{\alpha}-a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})^{2},$
(17) $\displaystyle C$ $\displaystyle=$
$\displaystyle\dot{\alpha}a_{s}H_{d}r_{d}/c/(1+\dot{\alpha}-a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})^{2}$
(18)

At leading order in $O(a_{s}H_{d}r_{d}/c)$, we get

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
O(a_{s}^{2}H_{d}r_{d}^{2}/c^{2})$ (19) $\displaystyle\dot{\alpha}$
$\displaystyle=$ $\displaystyle O(a_{s}^{2}H_{d}^{2}r_{d}^{2}/c^{2})$ (20)
$\displaystyle\alpha^{\prime}$ $\displaystyle=$ $\displaystyle
O(a_{s}^{2}H_{d}r_{d}/c^{2})$ (21) $\displaystyle A$ $\displaystyle=$
$\displaystyle 1+O(a_{s}^{2}H_{d}^{2}r^{2}/c^{2})$ (22) $\displaystyle B$
$\displaystyle=$ $\displaystyle 1+O(a_{s}^{2}H_{d}^{2}r^{2}/c^{2})$ (23)
$\displaystyle C$ $\displaystyle=$ $\displaystyle
O(a_{s}^{3}H_{d}^{3}r_{d}^{3}/c^{3})$ (24)

The metric in $(r_{d},t_{d})$ coordinates has therefore only perturbative
deviations from a homogeneous metric with a scale factor $a_{s}$. In this
coordinate system and using the same computation of the redshift as usually
done for weak perturbations around a homogeneous metric, at the leading order
in $O(a_{s}H_{d}r_{d}/c)$, the redshift of light emitted by a source at
$r_{d}=a_{d}r_{\mathrm{src}},t_{d}=t_{\mathrm{src}}$ toward an observer at
$r_{d}=0,t_{d}=t_{\mathrm{dst}}$ can be expressed as follows:

$1+z\approx\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}H_{s}dt\right)\left(1+\frac{H_{d}a_{s}a_{d}r_{\mathrm{src}}}{c}\right),$
(25)

with two contributions: one stemming from the “space” expansion along the
photon path of the homogeneous metric with scale factor $a_{s}(t)$ and another
originating from a Doppler term from non-comoving velocities only at emission.
At leading order in $O(H_{s}r/c)$, this is equivalent to
$z\approx(H_{s}+H_{d})a_{s}a_{d}r_{\mathrm{src}}/c=Har_{\mathrm{src}}/c$ only
when $H_{S}$ is independent of time. This method of calculating redshift is
not covariant, as it essentially converts part of the cosmological redshift
along the photon’s path into a Doppler shift that occurs solely at the point
of emission.

##  3 Covariant computation of the redshift in an inhomogeneous universe

Following Räsänen (2009), let us begin by defining a set of observers who are
tracing time-like geodesics, characterized by their four-velocity denoted as
$u^{\alpha}$. Consequently, these observers satisfy the conditions
$u^{\beta}\nabla_{\beta}u^{\alpha}=0$ and $u^{\alpha}u_{\alpha}=-1$. It is
important to note that these observers may not necessarily move in tandem with
the matter filling the Universe, as this matter can experience acceleration
due to non-gravitational forces and may not strictly adhere to time-like
geodesics. However, especially on large scales and within the context of dark
matter, we can consider the dark matter fluid as a useful tracer of these
time-like geodesics.

To further elucidate, we introduce the projection tensor, denoted as
$h_{\alpha\beta}$, which operates within the tangent space orthogonal to
$u^{\alpha}$. It is defined as $h_{\alpha\beta}\equiv
g_{\alpha\beta}+u_{\alpha}u_{\beta}$. With this tensor, we can proceed to
decompose the covariant derivative of $u^{\alpha}$

$\nabla_{\beta}u_{\alpha}=\frac{1}{3}h_{\alpha\beta}\theta+\sigma_{\alpha\beta}+\omega_{\alpha\beta}$
(26)

with $\theta=\nabla_{\alpha}u^{\alpha}$, the volume expansion rate,
$\sigma_{\alpha\beta}=\nabla_{(\alpha}u_{\beta)}-h_{\alpha\beta}\theta/3$ the
shear tensor, and $\omega_{\alpha\beta}=\nabla_{[\beta}u_{\alpha]}$, the
vorticity tensor. For simplicity, we will assume in the rest of this study
that the shear and vorticity tensors can be neglected.

We now define $k^{\alpha}$, the tangent vector of the null geodesics that
satisfies $k^{\beta}\nabla_{\beta}k^{\alpha}=0$ and $k^{\alpha}k_{\alpha}=0$.
The tangent vectors $u^{\alpha}$ and $k^{\alpha}$ are parallel propagated with
respect to the time-like and null geodesics, respectively, but not with
respect to each other. Consequently, the photon momentum changes along the
time-like geodesics and the redshift is defined by

$1+z=\frac{E_{\mathrm{src}}}{E_{\mathrm{dst}}}$ (27)

with $E_{\mathrm{src}}$ the photon energy at emission by the source and
$E_{\mathrm{dst}}$ the photon energy at the location of the observer. The
energy can be computed from $E=-u_{\alpha}k^{\alpha}$ and following Räsänen
(2009) we decompose $k^{\alpha}$ into a component parallel and a component
orthogonal to the time-like geodesics $k^{\alpha}=E(u^{\alpha}+e^{\alpha})$
with $u_{\alpha}e^{\alpha}=0$ and $e_{\alpha}e^{\alpha}=1$ (and consequently
$h_{\alpha\beta}e^{\alpha}e^{\beta}=1$). The evolution of the energy along the
null geodesic can then be followed with the affine parameter $\lambda$,

$\displaystyle\partial_{\lambda}E$ $\displaystyle\equiv$ $\displaystyle
k^{\alpha}\nabla_{\alpha}E$ (28) $\displaystyle=$
$\displaystyle-k^{\alpha}k^{\beta}\nabla_{\alpha}u_{\beta}$ (29)
$\displaystyle=$
$\displaystyle-E^{2}e^{\alpha}e^{\beta}\nabla_{\alpha}u_{\beta}$ (30)
$\displaystyle=$ $\displaystyle-\frac{E^{2}}{3}\theta$ (31)

Without vorticity, the hypersurfaces of constant proper time are orthogonal to
$u^{\alpha}$, and $t(\lambda)$ is monotonic. We can then invert the relation
between $\lambda$ and $t$ to obtain $d\lambda=dt/E$, hence

$1+z=\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}\frac{\theta(t,\@vec{x}(t))}{3}dt\right)$
(32)

A more detailed demonstration including vorticity and shear can be found in
Räsänen (2009). The main difference here is that we take a set of observers
following time-like geodesics that are not necessarily co-moving with matter
filling the universe. For a homogeneous metric with scale factor
$\tilde{a}(t)$, $\tilde{H}(t)=\dot{\tilde{a}}/\tilde{a}$, and for small three-
velocity $\@vec{\tilde{u}_{g}}$, the volume expansion factor is given by
$\theta\approx 3\tilde{H}+\@vec{\nabla}\cdot\@vec{\tilde{u}_{g}}$. Applying
eq. 32 to the non-comoving coordinates defined above, we get

$\displaystyle 1+z$ $\displaystyle\approx$
$\displaystyle\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}\frac{3H_{s}+\@vec{\nabla}\cdot(dr_{d}/dt_{d}\@vec{e}_{r_{d}})}{3}dt\right)$
(33) $\displaystyle\approx$
$\displaystyle\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}(H_{s}+H_{d})dt\right).$
(34)

With this covariant approach, “space-time” geometry characterized by the
expansion rate $H_{s}$ and non-comoving velocities of observers following
time-like geodesics characterized by $H_{d}r_{d}$ contribute both as an
expansion factor along the trajectory of the photons. For a homogeneous FLRW
solution with a Hubble expansion rate $H$, the comoving coordinate system
would lead to $\tilde{H}=H$ and $\@vec{\tilde{u_{g}}}=0$ and the non-comoving
coordinate system defined in Sect. 2 to $\tilde{H}=H_{s}$ and
$\@vec{\tilde{u_{g}}}=dr_{d}/dt_{d}\@vec{e}_{r_{d}}$. In both coordinate
systems, a consistent calculation of the cosmological redshift is obtained
with $\theta=3H$.

As a simple example, let us consider a universe with $G=0$ such that $H=0$ and
$a(t)=1$. In such a universe the cosmological redshift should be exactly zero,
since there is no expansion along the photon path and there should be only a
Doppler shift from the velocity difference between the source and the
observer. Let us assume for simplicity that they are both at rest such that
the total redhift is zero. Let us also consider an arbitrary coordinate system
with a scale factor $a_{s}(t)$ and an expansion rate $H_{s}(t)$, we have thus
$a_{d}(t)=1/a_{s}(t)$ and $H_{d}(t)=-H_{s}(t)$. In such a coordinate system,
the perturbative redshift calculation gives

$\displaystyle 1+z$
$\displaystyle\approx\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}H_{s}(t)dt\right)\left(1-\frac{H_{s}(t_{\mathrm{src}})r_{\mathrm{src}}}{c}\right),$
(35) $\displaystyle 1+z$
$\displaystyle\approx\left(1+\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}(H_{s}(t)-H_{s}(t_{\mathrm{src}}))dt\right)$
(36)

which demonstrates that the Doppler shift at emission cannot compensate the
expansion of the coordinate system along the photon path as soon as this
expansion is time dependent. However, with the covariant calculation, the
redshift is exactly zero since
$\theta=3H_{s}(t)-\@vec{\nabla}(H_{s}(t)\@vec{r}_{d})=0$. This further
demonstrates that the perturbative formulation of the redshift with the Hubble
expansion rate $H$ instead of the local expansion rate $\theta$ along time-
like geodesics is not covariant.

Lastly, it is imperative to highlight the precise physical entity that should
be traced by the local volume expansion rate and $\@vec{\tilde{u_{g}}}$. It is
crucial to note that the local volume expansion rate should not mirror the
fluid velocity but rather correspond to the velocity field of a hypothetical
set of free-falling observers following time-like geodesics (Räsänen, 2009).
The matter filling the universe can only serve as a tracer of these time-like
geodesics if the acceleration due to non-gravitational forces can be deemed
negligible. Hence, we advocate for the use of expressions that avoid the term
”expansion of space”, as it lacks covariance, and instead, we will refer to
the expansion along time-like geodesics when discussing cosmological redshift.
Then, the Doppler shifts can be defined as originating from the potential non-
geodesic motion of the source and observer at emission and reception.

In a inhomogeneous universe with an arbitrary expanding coordinate system
defined by $\tilde{a}(t),\tilde{H}(t)$, the general expression of the redshift
can therefore be defined as follows

$\displaystyle 1+z=$
$\displaystyle\exp\left(\int_{t_{\mathrm{src}}}^{t_{\mathrm{dst}}}\frac{\theta(t,\@vec{x}(t))}{3}dt\right)\times$
(37)
$\displaystyle\left(1+\frac{v_{\mathrm{Doppler}}(t_{\mathrm{src}},\@vec{x}(t_{\mathrm{src}}))}{c}-\frac{v_{\mathrm{Doppler}}(t_{\mathrm{dst}},\@vec{x}(t_{\mathrm{dst}}))}{c}\right)$
(38)

with

$\displaystyle\theta(t,\@vec{x})=3\tilde{H}(t)+\@vec{\nabla}\cdot\@vec{\tilde{u}}_{\mathrm{g}},$
(39) $\displaystyle
v_{\mathrm{Doppler}}(t,\@vec{x})=\tilde{a}(t)(\@vec{\tilde{u}}_{\mathrm{fluid}}-\<EMAIL_ADDRESS>(40)

Here, we define the cosmological redshift as the integral along the line of
sight of the local expansion rate along time-like geodesics, and the Doppler
shifts as the Doppler terms from non-geodesic motions of the source and the
observer in the non-relativistic limit. These definitions are now fully
covariant as $\theta$ and the velocity difference in the Doppler shift are
invariant under a change of the coordinate system. Fundamentally, one can see
that the cosmological contribution should not transform into a Doppler one or
vice-versa under a change of the coordinate system. We consider only the
expansion rate in Eq. 37 since vorticity does not contribute to the redshift
and shear is negligible on large scale if structures have no preferred
orientation (Räsänen, 2009).

At large scale and especially with dark matter, the matter within the Universe
is free-falling and can be taken as a tracer of the dynamics along time-like
geodesics. However, in the cosmological standard model, the coordinate system
is comoving with the FLRW background and is not comoving with these time-like
geodesics. Consequently, the conventional redshift calculation overlooks the
inhomogeneous dynamics along these geodesics, attributing a Doppler shift
solely to peculiar motions at the point of emission, mirroring our example
using non-comoving coordinates within a homogeneous FLRW universe. In
contrast, the covariant calculation of the redshift highlights that peculiar
motions responsible for the formation of cosmic structures do indeed
contribute to the redshift along the photon path through the local expansion
rate.

This insight leads us to reconsider the influence of inhomogeneities on the
cosmological redshift. Back-reaction is not a question of small peculiar
velocities relative to the speed of light, but requires a deeper inquiry:
whether, in a statistically homogeneous and isotropic universe, the averaged
local volume expansion rate along time-like geodesics provides a close
approximation to the background expansion $3H$, which might not be the case
even if the Doppler shifts of the source and observer relative to time-like
geodesics are negligible.

##  4 Impact of irreversible processes on the cosmological redshift

The condition $\langle\theta\rangle\approx 3H$ can be expressed equivalently
as $\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle\approx 0$. When we start from
a homogeneous universe where $\@vec{u_{g}}=0$ and consider a reversible
evolution, one might naturally expect e.g. based on hydrodynamics equations,
that the condition $\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle\approx 0$
would hold . Essentially this implies that inhomogeneous expansion and
compression modes evolve smoothly and mutually compensate on average,
resulting in $\langle\theta\rangle\approx 3H$ (see top panel of Fig. 1). It is
a common belief that the cosmological principle i.e. homogeneity and isotropy
of the Universe at large scales, is sufficient to ensure such a condition. In
previous investigations addressing the back-reaction problem (Buchert &
Ehlers, 1997; Buchert, 2008), which examines the impact of inhomogeneities on
the cosmological redshift, it has been generally assumed that this condition
holds (Buchert & Ehlers, 1997; Räsänen, 2009). Consequently, it would appear
that back-reaction is negligible in the non-relativistic limit (Ishibashi &
Wald, 2006) (but see also Buchert et al. (2015)). However, we aim to
reevaluate this conclusion since structure formation does not have
inhomogeneous expansion and compression modes that mutually compensate on
average: compression modes are dissipated by virialization and the associated
entropy production and cannot compensate the expansion of the voids. Hence in
the presence of irreversible processes, $\langle\theta\rangle\neq 3H$. The
link between expansion and entropy production can be made explicit by using
the second principle of thermodynamics assuming that a quasi-equilibrium
defined by a temperature $T$ and a stress tensor with pressure $\sigma$ has
been reached by virialization,

$\displaystyle TdS=dE+\sigma dV,$ (41) $\displaystyle Tds=de+Zk_{b}Td(\log
V),$ (42)

for a closed system of N particles (dN=0) with an entropy per particle
$s=S/N$, an internal energy per particle $e=E/N$, and a compressibility factor
$Z=\sigma V/Nk_{b}T$. Assuming the evolution of an irreversible system from an
initial state and the evolution of a reversible system that reaches the same
internal energy $e$, temperature $T$, compressibility factor $Z$, and has the
same entropy as the initial state,

$\displaystyle T\Delta s$ $\displaystyle=\Delta
e+Zk_{b}T\log(V_{\mathrm{irrev}}/V_{\mathrm{ini}}),$ (43) $\displaystyle 0$
$\displaystyle=\Delta e+Zk_{b}T\log(V_{\mathrm{rev}}/V_{\mathrm{ini}}),$ (44)

implies that

$\displaystyle\frac{V_{\mathrm{irrev}}}{V_{\mathrm{ini}}}=\frac{V_{\mathrm{rev}}}{V_{\mathrm{ini}}}\exp\left(\frac{\Delta
s}{k_{b}Z}\right).$ (45)

We recall that the expansion rate can be directly linked to the variation of
volume by using the equation of mass conservation with a Lagrangian time
derivative

$\displaystyle\frac{d\log V}{dt}=\theta,$ (46)

which implies the following expression of the redshift for motions that follow
time-like geodesics in an irreversible system (assuming spatial ergodicity
along the photon path),

$\displaystyle 1+z$
$\displaystyle=\left(\frac{V_{\mathrm{irrev}}(t_{\mathrm{dst}})}{V_{\mathrm{ini}}(t_{\mathrm{src}})}\right)^{1/3},$
(49)
$\displaystyle=\left(\frac{V_{\mathrm{rev}}}{V_{\mathrm{ini}}}\right)^{1/3}\exp\left(\frac{\Delta
s}{3k_{b}Z}\right),$
$\displaystyle=\frac{a(t_{\mathrm{dst}})}{a(t_{\mathrm{src}})}\exp\left(\frac{\Delta
s}{3k_{b}Z}\right)$

where $a(t)$ is the scale factor of the reversible background evolution. Eq.
49 demonstrates the direct link between entropy production and the
cosmological redshift in an irreversible system. The second principle of
thermodynamics states that the entropy production is always positive with
$\Delta s>0$, the cosmological principle is therefore not sufficient to ensure
that the expansion of an irreversible system follows the background expansion
without violation of the second principle.

Figure 1: Top: reversible evolution with expansion and compression modes.
Down: irreversible evolution when compression modes are dissipated by
virialization through accretion shocks.

In an adiabatic system, the occurrence of an irreversible process and the
subsequent creation of entropy is possible through the dissipation of large
scale motions into microscopic motions. This process, driven by virialization,
is referred to as “violent relaxation” within the context of structure
formation in the Universe, in the presence of non-collisional dark matter
(White, 1996). When a significant portion of the large-scale compression modes
dissipates due to virialization, they can no longer compensate the expansion
occurring in cosmic voids, inevitably leading to the conclusion that
$\langle\theta\rangle\neq 3H$ (see bottom panel of Fig. 1). From numerical
simulations (Haider et al., 2016), we can estimate that 50 % of the dark
matter mass is completely virialized in haloes and 45 % virialized in two
directions in filaments (Eisenstein et al., 1997). A realistic universe has
therefore virialized most of its mass and almost completely the initial
compression modes that appeared during the formation phase of the large scale
structures. Furthermore, it is crucial to emphasize that an inhomogeneous
system undergoing irreversible processes cannot, on average, precisely mimic
the behavior of the FLRW background. The background evolves akin to a
reversible system (constant entropy), while an inhomogeneous system subject to
entropy generation would essentially need to violate the second law of
thermodynamics with processes generating negative local entropy to align its
averaged entropy with the background entropy. Additionally, the process of
dark matter virialization entails the acquisition of microscopic stress
through microscopic virialized motions. In this context, the dark matter fluid
ceases to be truly pressureless after virialization, resulting in a non-zero
stress tensor within the Jeans equations. This can be seen as a phase
transition to a bound state similar to the formation of a liquid but in the
context of statistical mechanics for non-ideal self-gravitating fluids
(Tremblin et al., 2022). A central point to note is that the equilibrium
observed in large-scale structures is not a balance between gravitational
attraction and stresses induced by non-gravitational interactions. Dark
matter, by definition, is a tracer for dynamics along time-like geodesics.
Therefore, its virialization marks the dissipation of large-scale dynamics
along these geodesics into microscopic motions that continue to follow
geodesic trajectories in free-fall. This fundamental aspect underscores why
the virialization of large-scale structures inevitably influences the
cosmological redshift.

The presence of entropy creation, characterized by
$\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle\neq 0$, necessarily implies the
existence of discontinuities in the three-velocity field. Remarkably, dark
matter virialization occurs subsequent to shell crossing, resulting in the
formation of caustics, which are essentially discontinuities in the three-
velocity field. In the context of entropy creation through violent relaxation,
these discontinuities can be likened to non-collisional accretion shocks (see
Parks et al. (2012, 2017) for non-collisional shocks in plasmas) and are
associated to a microscopic stress tensor that is also discontinuous across
the shocks/caustics. The nature of such discontinuities within the framework
of general relativity may initially appear unclear. We propose to characterize
these discontinuities by utilizing exact solutions of Einstein’s field
equations in spherical symmetry. These solutions are locally comoving with the
time-like geodesics and are known as generalized Lemaître-Tolman-Bondi
solutions, when accounting for a microscopic stress tensor and virialization
(Lasky & Lun, 2006).

The metric for such solutions with a fluid characterized by an energy density
$\rho$ and pressure $\sigma$ is given by

$ds^{2}=-N(r,t)^{2}dt^{2}+\frac{1}{1+2E(r,t)}R^{\prime
2}dr^{2}+R(r,t)^{2}d\Omega^{2},$ (50)

with $R^{\prime}$ the derivative with respect to $r$ and with a unit choice
such that $c=1$. $N(r,t)$ is the lapse function and $E(r,t)$ the local
curvature. The following Hamiltonian constraint equation can be derived from
the ADM formalism (Lasky & Lun, 2006)

$\frac{1}{2}u^{2}=\left(\frac{GM}{R}+E\right)$ (51)

with $u\equiv\dot{R}/N$, $\dot{R}$ the derivative with respect to $t$ and
$M(r,t)=4\pi\int_{0}^{R(r,t)}\rho\tilde{r}^{2}d\tilde{r}$, with the following
evolution equations

$\displaystyle\frac{\dot{E}}{N}$ $\displaystyle=$
$\displaystyle-\frac{1+2E}{\rho+\sigma}\frac{1}{R^{\prime}}\frac{\partial\sigma}{\partial
r}u,$ (52) $\displaystyle\frac{\dot{\rho}}{N}$ $\displaystyle=$
$\displaystyle-N(\rho+\sigma)\frac{1}{R^{2}R^{\prime}}\frac{\partial(R^{2}u)}{\partial
r},$ (53) $\displaystyle\frac{\dot{u}}{N}$ $\displaystyle=$
$\displaystyle-\frac{GM}{R^{2}}-4\pi G\sigma
R-\frac{1}{R^{\prime}}\frac{1+2E}{\rho+\sigma}\frac{\partial\sigma}{\partial
r}.$ (54)

The Euler equation provides the following relation between the lapse $N$ and
the pressure $N^{\prime}/N=-\sigma^{\prime}/(\rho+\sigma)$. It is evident at
this stage that the local curvature $E(r,t)$ undergoes an evolution
reminiscent of the behavior of kinetic energy in classical Newtonian
hydrodynamics. This similarity suggests that the curvature can be linked to
microscopic dissipation through the presence of discontinuities. To establish
this connection, we proceed under the assumption that the fluid can be
characterized by an internal microscopic energy, denoted as $e$, such that the
total energy density $\rho$ can be expressed as $\rho=\rho_{m}+\rho_{m}e$,
with $\rho_{m}$ representing the mass density. It is important to emphasize
that our analysis holds validity for both collisional and non-collisional
systems. The microscopic internal energy can be defined for a non-colliosional
system as the microscopic stress tensor in the context of the Jeans equations.
In the non-collisional limit, we do not have simple closure relations that
establish a connection between the internal energy and the stress tensor with
other variables. However, it is worth noting that such a relation is not a
requisite element within our analysis. In the non-relativistic limit $E,e\ll
1$, $\sigma\ll\rho$. In this limit, and defining $\tau=1/\rho_{m}$,

$\displaystyle\dot{E}$ $\displaystyle=$
$\displaystyle-\frac{\tau}{R^{\prime}}\frac{\partial\sigma}{\partial r}u,$
(55) $\displaystyle\dot{\tau}$ $\displaystyle=$
$\displaystyle\frac{\tau}{R^{\prime}}\frac{1}{R^{2}}\frac{\partial(R^{2}u)}{\partial
r},$ (56) $\displaystyle\dot{e}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho_{m}R^{\prime}}\frac{1}{R^{2}}\frac{\partial(R^{2}u)}{\partial
r}\sigma,$ (57) $\displaystyle\dot{u}$ $\displaystyle=$
$\displaystyle-\frac{GM}{R^{2}}-\frac{\tau}{R^{\prime}}\frac{\partial\sigma}{\partial
r}.$ (58)

By employing the transformation $dm=\rho_{m}R^{2}dr/R^{\prime}$, where $m$
represents the so-called ”mass” variable, one can readily identify the
hydrodynamics equations in Lagrangian coordinates (see page 8 of Godlewski &
Raviart (1996) or page 16 of Després (2017)) with an additional equation
governing the evolution of curvature. From this system, we can derive the
following conservative equation that establishes a coupling between curvature
and internal energy:

$\dot{E}+\dot{e}=-\frac{\partial}{\partial m}\left(\sigma uR^{2}\right).$ (59)

We assume now that a discontinuity is located at $r_{i}$ with
$R_{i}(t)=R(r_{i},t)$, with the possibility for $u$ and $E$ to jump. By using
standard Rankine-Hugoniot relations,

$\dot{m}_{i}\left(E_{l}+e_{l}-E_{r}-e_{r}\right)=\left(\sigma_{l}u_{l}-\sigma_{r}u_{r}\right)R^{2}_{i},$
(60)

with $\dot{m}_{i}(t)$, the mass flow through the discontinuity. It is apparent
from this relation that the behavior of curvature bears a resemblance to that
of kinetic energy. The second law of thermodynamics, encompassing the
dissipation of large-scale motions and the concomitant generation of entropy,
thus dictates that the evolution of curvature follows the condition $\Delta
E<0$. By applying the Hamiltonian constraint and using the continuity of
$GM/R$ across the discontinuity, we can also derive the following
relationship:

$\frac{1}{2}\left(\dot{R}_{l}^{2}-\dot{R}_{r}^{2}\right)=E_{l}-E_{r}.$ (61)

Under the assumption that the solution tends to become homogeneous on each
side of the interface, we arrive at the following jump condition concerning
the local volume expansion rate:

$\frac{1}{18}\left(\theta_{l}^{2}-\theta_{r}^{2}\right)=\frac{1}{R_{i}^{2}}\left(E_{l}-E_{r}\right).$
(62)

It is worth highlighting that these co-moving solutions can find utility in a
broader context not necessarily tied to gravitational dynamics. While we have
primarily considered particles following time-like geodesics, one can also
investigate co-moving solutions with non-geodesic motions in the limit as $G$
approaches zero. In this scenario, the solution becomes a spherical shock
discontinuity akin to the Sedov solution in classical hydrodynamics. Within
this context, $\theta_{l}=\@vec{\nabla}\cdot\@vec{u_{g}}$ remains
approximately constant within the volume undergoing expansion, while
$\theta_{r}=0$ outside of it. These shocks dissipate large-scale motions into
microscopic motions, accompanied by entropy creation and a distinctive jump in
the local curvature, the evolution of which conforms to $\Delta E<0$.
Commencing with a homogeneous system characterized by zero curvature and
transitioning to a locally comoving space-time with time-like geodesics, we
find that virialization leads to the generation of local negative curvature.
Although the connection between virialization and negative curvature has been
discussed in existing literature (Roukema et al., 2013; Roukema, 2018), our
contribution lies in linking the curvature production by virialization to the
second law of thermodynamics. This implies that the averaged space-time
curvature of a co-moving solution that traces the motion of time-like
geodesics, cannot be zero in the presence of irreversible processes.

Figure 2: Top: Illustration of a discontinuous velocity profile that impacts
the redshift-distance relation. Bottom: equivalent apparent velocity profile
by integration of the gradients along the line-of-sight.

##  5 Link with observations and quantitative estimates

DESI Collaboration et al. (2023, 2024a) present the release of the first
science data from the DESI (Dark Energy Spectroscopic Instrument) project,
which includes data from commissioning and Survey Validation phases conducted
between December 2020 and June 2021\. The authors highlight that the DESI
collaboration has successfully validated its survey design and observing
strategy, ensuring the data’s quality and completeness and that future
releases will facilitate detailed studies of the universe’s large-scale
structure, providing critical insights into dark energy and the fundamental
physics governing the universe’s evolution. Following the initial data
release, the DESI collaboration has unveiled its first-year analysis results
of baryon acoustic oscillations (BAO) based on extensive observations of
galaxies, quasars, and the Lyman-$\alpha$ forest (DESI Collaboration et al.,
2024b). While the DESI data alone align with the $\Lambda$CDM model, the
results deviate when generalized to the $\omega_{0}\omega_{a}$CDM model. The
DESI data combined with cosmic microwave background (CMB) and supernova data,
suggests a rejection of the $\Lambda$CDM model in favor of the
$\omega_{0}\omega_{a}$CDM model with significant confidence levels between 2
and 4$\sigma$ (DESI Collaboration et al., 2024b). The combined data indicate a
preference for $\omega_{0}>-1$ and $\omega_{a}<0$, where
$\omega(a)=\omega_{0}+\omega_{a}(1-a)$ represents the equation-of-state
parameter of dark energy with $a(t)$ the scale factor of the FLRW cosmology.
If confirmed, these findings could have profound implications for our
understanding of the Universe’s dynamics. In that context, Tada & Terada
(2024) suggest that the nature of dark energy might be explained by a
quintessential scalar field, although this field could potentially become a
phantom in the past. They conclude that the DESI data challenges the
$\Lambda$CDM model and supports a dynamic dark energy scenario, thereby
contributing to our understanding of cosmic acceleration and the potential
need for new physics beyond the standard cosmological model.

However, due to the covariance of General Relativity, the current approach in
the standard cosmological model can be reformulated as follows: the choice of
an expanding background coordinate system should be arbitrary and introducing
a cosmological constant to drive the dynamics of the coordinate system has no
physical meaning other than selecting an expanding coordinate system with
$a_{\mathrm{eff}}(t)$, $H_{\mathrm{eff}}(t)$, and three velocities
$\@vec{u_{\mathrm{eff,g}}}$ for time-like geodesics such that
$\langle\theta\rangle\approx 3H_{\mathrm{eff}}$ and
$\langle\@vec{\nabla}\cdot\@vec{u_{\mathrm{eff,g}}}\rangle\approx 0$. This
choice compensates in the coordinate system the averaged expansion rate along
time-like geodesics present in the three-velocity field. If one prefers to use
the FLRW solution of a fictive homogeneous universe without a cosmological
constant, where the Hubble expansion rate is $H(t)$ (with $H^{2}=8\pi
G\bar{\rho}/3$), and the three velocities for time-like geodesics are
$\@vec{u_{g}}$, then
$\langle\theta\rangle=3H+\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle$ and
$\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle\neq 0$ as soon as virialization
and entropy production occur. We highlight that virialization during structure
formation is likely to replace dark energy because it precisely mimics the
counter-intuitive anti-gravity effect needed for dark energy. The dissipation
of large-scale compression modes by virialization stabilizes large-scale
gravitational collapse (anti-gravity) while remaining a purely gravitational
effect, as virialized dark matter continues to be in free-fall.

In this effective coordinate system, we can use the second principle of
thermodynamics to infer an effective evolution equation that takes into
account entropy production,

$\displaystyle\frac{d(\log V_{\mathrm{eff}})}{dt}=\frac{d(\log
V_{\mathrm{rev}})}{dt}+\frac{1}{Zk_{b}}\frac{ds}{dt},$ (63) $\displaystyle
H_{\mathrm{eff}}(t)=H(t)+\frac{1}{3Zk_{b}}\frac{ds}{dt},$ (64)

which can be rewritten as,

$\displaystyle H_{\mathrm{eff}}(t)^{2}=\frac{8\pi
G\bar{\rho}}{3}+\frac{\Lambda_{\mathrm{eff}}(t)c^{2}}{3},$ (65)
$\displaystyle\Lambda_{\mathrm{eff}}(t)=\frac{2H}{c^{2}Zk_{b}}\frac{ds}{dt}+\frac{1}{3c^{2}Z^{2}k_{b}^{2}}\left(\frac{ds}{dt}\right)^{2}.$
(66)

In this context, the effective cosmological constant appears to vary with
time, as suggested by the early DESI data release. It is important to note,
however, that this does not imply the existence of new physics; rather, the
effective cosmological constant serves as a proxy to account for the impact of
entropy production on the averaged expansion rate along time-like geodesics.
Entropy production is expected to begin during structure formation at the
first shell crossing, peak during the virialization of structures, and
gradually decline to the present day. This behavior could simulate the need
for $\omega(a)<-1$ in the past and $\omega(a)>-1$ in the present. As noted in
Tada & Terada (2024), The increase of $\omega(a)$ is correlated with the
decrease of the Hubble constant, which is the opposite direction to solve the
Hubble tension. The Hubble Tension refers to the significant discrepancy
between two different methods of measuring the Hubble constant. The first
method involves local measurements, such as observing the distances and
redshifts of nearby galaxies. The second method involves the Cosmic Microwave
Background (CMB) and large-scale structure data, which rely on $\Lambda$CDM
model and measurements from the early Universe (Kamionkowski & Riess, 2023).
The local measurements consistently yield a higher value for the Hubble
constant (around 73 km/s/Mpc) compared to the CMB-based measurements (around
67 km/s/Mpc). Although the decline phase of entropy production after the
virialization of structures cannot explain the Hubble tension, the initial
increase phase after the first shell crossing could play an important role in
resolving the current tension. However, precise estimates of the evolution of
$\Lambda_{\mathrm{eff}}(t)$ require dedicated numerical simulations to
accurately constrain the evolution of entropy production during structure
formation and subsequent evolution.

We can, however, use existing numerical simulations to provide a more
quantitative order-of-magnitude estimate. Let us consider the illustrative
example depicted in the top panel of Fig. 2. In this scenario, we assume that
within the bound and virialized large-scale structures with a characteristic
size of $d_{\mathrm{lss}}$, the local volume expansion rate, denoted as
$\theta$, is approximately negligible ($\theta\approx 0$). Conversely, in the
expanding cosmic voids with a typical size of $d_{\mathrm{void}}$, $\theta$ is
substantially higher, with $\theta\approx 6u_{\mathrm{pre-
shock}}/d_{\mathrm{void}}\gg 3H$. The bottom panel of Fig. 2 shows an
equivalent apparent velocity profile by integrating the gradients along the
line-of-sight. Both profiles in Fig. 2 provide the same integrated redshift
using eq. 32 and illustrate how peculiar velocities resulting from structure
formation can mimic a Hubble flow. Without discontinuities in Fig. 2, the
velocity field divergence in the structures and in the voids would exactly
compensates such that $\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle=0$.
However, this is not the case with discontinuities: in the extreme case for
which $\@vec{\nabla}\cdot\@vec{u_{g}}$ is strictly positive everywhere (i.e.
with expanding motions in the lss regions in Fig. 2),
$\langle\@vec{\nabla}\cdot\@vec{u_{g}}\rangle$ is clearly strictly positive on
average. In a realistic configuration for the large scale structures with
$\theta\approx 0$, the dominant contributor to the redshift arises from the
expansion occurring within the voids, accounting for a fraction of
$d_{\mathrm{void}}/(d_{\mathrm{lss}}+d_{\mathrm{void}})$. This expansion in
the voids cannot be counterbalanced by compression within the structures, as
it has dissipated due to violent relaxation and virialization processes.
Assuming typical structure sizes of approximately $d_{\mathrm{lss}}\approx$ 3
Mpc and void sizes around $d_{\mathrm{void}}\approx$ 30 Mpc, the average
expansion rate, represented by $\langle\theta\rangle/3$, attains values on the
order of 70 km/s/Mpc with pre-shock velocities of roughly $u_{\mathrm{pre-
shock}}\approx 1200$ km/s. This estimation falls within the same order of
magnitude as the infall velocities observed in N-body numerical simulations
(Zu & Weinberg, 2013).

The forthcoming generation of high-resolution cosmological simulations on
exascale supercomputers will be essential in advancing our understanding of
entropy production through caustics (i.e., velocity discontinuities or non-
collisional shocks). These simulations will enable more precise estimates of
the averaged local expansion rate along time-like geodesics. Such estimates
are crucial for interpreting dynamic dark energy scenarios, where dark energy
could be replaced by entropy production during structure formation. These
insights will be essential for analyzing future data releases from the DESI
collaboration and for interpreting observations from the Euclid satellite
(Amendola et al., 2018). By combining Euclid’s weak lensing data, redshift
surveys (including baryon acoustic oscillations, redshift distortions, and the
full power spectrum shape), and CMB data from Planck, we can constrain dynamic
dark energy scenarios more effectively.

##  6 Conclusions

In conclusion, this paper highlights the intricate relationship between the
cosmological redshift and entropy production due to structure formation. We
emphasize the need of the covariant calculation of the cosmological redshift,
which accounts for peculiar motions with a local expansion rate along time-
like geodesics.

Our analysis suggests that virialization and the subsequent entropy production
during structure formation play a crucial role in this dynamic. The time-
varying nature of the effective cosmological constant, as indicated by recent
DESI data, may be a manifestation of these processes rather than new physics.
This perspective necessitates a reassessment of dark energy models, proposing
that what is currently attributed to dark energy might instead be the result
of entropy production during structure formation.

Future high-resolution cosmological simulations on exascale supercomputers
will be vital for refining our understanding of these phenomena. These
simulations will help produce more accurate estimates of the averaged local
expansion rate along time-like geodesics, providing essential insights for
interpreting dynamic dark energy scenarios and forthcoming observational data
from DESI and the Euclid satellite. This research opens new avenues for
understanding the fundamental forces shaping our universe, emphasizing the
need for continuous advancements in computational and observational cosmology.

###### Acknowledgements.

We thank Quentin Vigneron, Thomas Guillet, and Pascal Wang for all the
discussions that lead to this paper. We also thank David Elbaz and Matthias
González for helpful comments on this manuscript.

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# Nonexistence of spectral gaps in Hölder spaces for continuous time dynamical
systems

Ian Melbourne Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
UK Nicolò Paviato Mathematics Institute, University of Warwick, Coventry, CV4
7AL, UK Dalia Terhesiu Mathematisch Instituut, University of Leiden, Niels
Bohrweg 1, 2333 CA Leiden, Netherlands

(27 February 2021)

###### Abstract

We show that there is a natural restriction on the smoothness of spaces where
the transfer operator for a continuous dynamical system has a spectral gap.
Such a space cannot be embedded in a Hölder space with Hölder exponent greater
than $\frac{1}{2}$ unless it consists entirely of coboundaries.

##  1 Introduction

Decay of correlations (rates of mixing) and strong statistical properties are
well-understood for Axiom A diffeomorphisms since the work of [2, 9, 10].
Mixing rates are computed with respect to any equilibrium measure with Hölder
potential. Up to a finite cycle, such diffeomorphisms have exponential decay
of correlations for Hölder observables. In the one-sided (uniformly expanding)
setting, this is typically proved by establishing quasicompactness and a
spectral gap for the associated transfer operator $L$. Such a spectral gap
yields a decay rate $\|L^{n}v-\int v\|\leq C_{v}e^{-an}$ for $v$ Hölder, where
$\|\;\|$ is a suitable Hölder norm and $a$, $C_{v}$ are positive constants.
Decay of correlations for Hölder observables is an immediate consequence of
the decay for $L^{n}$. This philosophy has been extended to large classes of
nonuniformly expanding dynamical systems with exponential [13] and
subexponential decay of correlations [14].

For continuous time dynamical systems, the usual techniques [5, 7, 8] bypass
spectral gaps; the only exceptions that we know of being Tsujii [11, 12].
However, the result in [11] is for suspension semiflows over the doubling map
with a $C^{3}$ roof function, where the smoothness of the roof function is
crucial and very restrictive. A spectral gap for contact Anosov flows is
obtained in [12]; unfortunately it seems nontrivial to extend this to
nonuniformly hyperbolic contact flows (or uniformly hyperbolic contact flows
with unbounded distortion), see [1] which proves exponential decay of
correlations for billiard flows with a contact structure but does not
establish a spectral gap. Indeed, apart from [11, 12], there are no results on
spectral gaps of transfer operators for semiflows and flows.

The results of Tsujii [11, 12] provide a spectral gap in an anisotropic Banach
space. In this paper we obtain a restriction on the Banach spaces that can
yield a spectral gap. We work in the following general setting:

Let $(\Lambda,d)$ be a bounded metric space with Borel probability measure
$\mu$, and let $F_{t}:\Lambda\to\Lambda$ be a measure-preserving semiflow. We
suppose that $t\to F_{t}$ is Lipschitz a.e. on $\Lambda$. Let
$L_{t}:L^{1}(\Lambda)\to L^{1}(\Lambda)$ denote the transfer operator
corresponding to $F_{t}$ (so
$\int_{\Lambda}L_{t}v\,w\,d\mu=\int_{\Lambda}v\,w\circ F_{t}\,d\mu$ for all
$v\in L^{1}(\Lambda)$, $w\in L^{\infty}(\Lambda)$, $t>0$). Let $v\in
L^{\infty}(\Lambda)$ and define $v_{t}=\int_{0}^{t}v\circ F_{r}\,dr$ for
$t\geq 0$.

###### Theorem 1.1

Let $\eta\in(\frac{1}{2},1)$. Suppose that $L_{t}v\in C^{\eta}(\Lambda)$ for
all $t>0$ and that $\int_{0}^{\infty}\|L_{t}v\|_{\eta}\,dt<\infty$. Then
$v_{t}$ is a coboundary:

$v_{t}=\chi\circ F_{t}-\chi\quad\text{for all $t\geq 0$, a.e.\ on $\Lambda$}$

where $\chi=\int_{0}^{\infty}L_{t}v\,dt\in C^{\eta}(\Lambda)$. In particular,
$\sup_{t\geq 0}|v_{t}|_{\infty}<\infty$.

Here, $|g|_{\infty}=\operatorname{ess\,sup}_{\Lambda}|g|$ and
$\|g\|_{\eta}=|g|_{\infty}+\sup_{x\neq y}|g(x)-g(y)|/d(x,y)^{\eta}$.

Theorem 1.1 implies that any Banach space admitting a spectral gap and
embedded in $C^{\eta}(\Lambda)$ for some $\eta>\frac{1}{2}$ is cohomologically
trivial. However, for (non)uniformly expanding semiflows and (non)uniformly
hyperbolic flows of the type in the aforementioned references, coboundaries
are known to be exceedingly rare, see for example [3, Section 2.3.3]. Hence,
Theorem 1.1 can be viewed as an “anti-spectral gap” result for such continuous
time dynamical systems.

##  2 Proof of Theorem 1.1

Let $v\in L^{\infty}(\Lambda)$, with $L_{t}v\in C^{\eta}(\Lambda)$ for all
$t>0$ and $\int_{0}^{\infty}\|L_{t}v\|_{\eta}\,dt<\infty$ where
$\eta\in(\frac{1}{2},1)$. Following Gordin [6] we consider a martingale-
coboundary decomposition. Define $\chi=\int_{0}^{\infty}L_{t}v\,dt\in
C^{\eta}(\Lambda)$, and

$v_{t}=\int_{0}^{t}v\circ F_{r}\,dr,\qquad m_{t}=v_{t}-\chi\circ F_{t}+\chi,$

for $t\geq 0$. Let ${\mathcal{B}}$ denote the Borel $\sigma$-algebra on
$\Lambda$.

###### Proposition 2.1

(i) $t\to m_{t}$ is $C^{\eta}$ a.e. on $\Lambda$.

(ii) ${\mathbb{E}}(m_{t}|F_{t}^{-1}{\mathcal{B}})=0$ for all $t\geq 0$.

###### Proof.

(i) For $0\leq s\leq t\leq 1$ and $x\in\Lambda$,

$\displaystyle|m_{s}(x)-m_{t}(x)|$
$\displaystyle\leq|v_{s}(x)-v_{t}(x)|+|\chi(F_{s}x)-\chi(F_{t}x)|$
$\displaystyle\leq|s-t||v|_{\infty}+|\chi|_{\eta}\,d(F_{s}x,F_{t}x)^{\eta}.$

Since $t\mapsto F_{t}$ is a.e. Lipschitz, it follows that $t\mapsto m_{t}$ is
a.e. $C^{\eta}$.

(ii) Let $U_{t}v=v\circ F_{t}$, and recall that $L_{t}U_{t}=I$ and
${\mathbb{E}}(\cdot|F_{t}^{-1}{\mathcal{B}})=U_{t}L_{t}$. Then

$\displaystyle L_{t}m_{t}$ $\displaystyle=\textstyle
L_{t}(v_{t}-U_{t}\chi+\chi)=\int_{0}^{t}L_{t}U_{r}v\,dr-\chi+\int_{0}^{\infty}L_{t}L_{r}v\,dr$
$\displaystyle\textstyle=\int_{0}^{t}L_{t-r}v\,dr-\chi+\int_{0}^{\infty}L_{t+r}v\,dr=\int_{0}^{t}L_{r}v\,dr-\chi+\int_{t}^{\infty}L_{r}v\,dr=0.$

Hence ${\mathbb{E}}(m_{t}|F_{t}^{-1}{\mathcal{B}})=U_{t}L_{t}m_{t}=0$. ∎

Proof of Theorem 1.1 Fix $T>0$, and define

$M_{T}(t)=m_{T}-m_{T-t}=m_{t}\circ F_{T-t},\quad t\in[0,T].$

Also, define the filtration ${\mathcal{G}}_{T,t}=F_{T-t}^{-1}{\mathcal{B}}$.
It is immediate that $M_{T}(t)=m_{t}\circ F_{T-t}$ is
${\mathcal{G}}_{T,t}$-measurable. Also, for $s<t$ we have
$M_{T}(t)-M_{T}(s)=m_{T-s}-m_{T-t}=m_{t-s}\circ F_{T-t}$, so

${\mathbb{E}}(M_{T}(t)-M_{T}(s)|{\mathcal{G}}_{T,s})={\mathbb{E}}(m_{t-s}\circ
F_{T-t}|F_{T-s}^{-1}{\mathcal{B}})={\mathbb{E}}(m_{t-s}|F_{t-s}^{-1}{\mathcal{B}})\circ
F_{T-t}=0$

by Proposition 2.1(ii). Hence $M_{T}$ is a martingale for each $T>0$. Next,

$|M_{T}(t)|_{\infty}=|m_{t}\circ
F_{T-t}|_{\infty}\leq|m_{t}|_{\infty}\leq|v_{t}|_{\infty}+2|\chi|_{\infty}\leq
T|v|_{\infty}+2|\chi|_{\infty}.$

Hence $M_{T}(t),\,t\in[0,T]$, is a bounded martingale.

By Proposition 2.1(i), $M_{T}$ has $C^{\eta}$ sample paths. Since
$\eta>\frac{1}{2}$, it follows from general martingale theory that
$M_{T}\equiv 0$ a.e. Taking $t=T$, we obtain $m_{T}=0$ a.e. Hence
$v_{T}=\chi\circ F_{T}-\chi$ a.e. for all $T>0$ as required.

For completeness, we include the argument that $M_{T}\equiv 0$ a.e. We require
two standard properties of the quadratic variation process
$t\mapsto[M_{T}](t)$; a reference for these is [4, Theorem 4.1]. First,
$[M_{T}](t)$ is the limit in probability as $n\to\infty$ of

$S_{n}(t)=\sum_{j=1}^{n}(M_{T}(jt/n)-M_{T}((j-1)t/n))^{2}.$

Second (noting that $M_{T}(0)=0$),

$[M_{T}](t)=M_{T}(t)^{2}-2\int_{0}^{t}M_{T}\,dM_{T},$

where the stochastic integral has expectation zero. In particular,
${\mathbb{E}}([M_{T}])\equiv{\mathbb{E}}(M_{T}^{2})$.

Since $M_{T}$ has Hölder sample paths with exponent $\eta>\frac{1}{2}$, we
have a.e. that

$|S_{n}(t)|=O(t^{\eta}n^{-(2\eta-1)})\to 0\quad\text{as $n\to\infty$.}$

Hence $[M_{T}]\equiv 0$ a.e. It follows that ${\mathbb{E}}(M_{T}^{2})\equiv 0$
and so $M_{T}\equiv 0$ a.e. ∎

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