Observational constraints on sign-changeable interaction models and alleviation of the $H_0$ tension
Supriya Pan Weiqiang Yang Chiranjeeb Singha Emmanuel N. Saridakis










supriya.maths@presiuniv.ac.in
Department of Mathematics, Presidency University, 86/1 College Street, 
Kolkata 700073, 
India


d11102004@163.com
Department of Physics, Liaoning Normal University, Dalian, 116029, P. R. 
China


cs12ip026@iiserkol.ac.in
Department of Physical Sciences, 
Indian Institute of Science Education and Research Kolkata,
Mohanpur 741246, West Bengal, India

 
msaridak@phys.uoa.gr
Department of Physics, National Technical University of Athens, Zografou
Campus GR 157 73, Athens, Greece
Department of Astronomy, School of Physical Sciences, University of Science 
and Technology of China, Hefei 230026, P.R. China








We investigate various scenarios which include interaction forms between dark matter and 
dark energy that exhibit sign reverse, namely where the 
transfer of energy between the dark fluids changes sign during   evolution. We study 
the large-scale inhomogeneities in such interacting scenarios and we confront them with  
the latest astronomical data. 
Our analysis shows that the sign-changeable interaction models are able to produce stable 
perturbations. Additionally, the  data seem to slightly favor a non-zero 
interaction, however, within 1σ confidence level (CL) 
the scenarios cannot be distinguished from non-interacting cosmologies. We find 
that the best-fit value of the dark-energy equation-of-state parameter lies in the 
phantom 
regime, while the quintessence region  is also allowed nevertheless at more than  
2σ CL. Examining the effect of the interaction on the CMB TT and matter 
power spectra we show that while from the simple spectra  it is hard to 
distinguish the interacting case from ΛCDM  scenario, in the residual graphs  
 the interaction is indeed traceable. Moreover, we find that sign-changeable interaction 
models can reconcile the H_0 tension, however the σ_8 tension is still 
persisting. Finally,  we examine the validity of the laws of thermodynamics and we 
show that  the generalized second law is 
always satisfied, while the second derivative of the total entropy becomes 
negative at late times which implies that the universe tends towards thermodynamic equilibrium.



98.80.-k, 95.36.+x, 95.35.+d, 98.80.Es




Observational constraints on sign-changeable interaction models and alleviation of the H_0 tension
    Emmanuel N. Saridakis
    
==================================================================================================




§ INTRODUCTION


Recent observations of various origin suggest that around 69% of the universe consists 
of the dark energy sector, while around  26% constitutes of the cold dark matter one  
<cit.>. One usually assumes that these two sectors do not mutually
interact, and the resulting scenario is capable of describing very efficiently the 
various 
sets of independent observations. Nevertheless, since the underlying microscopic theory 
of both these sectors is unknown, there is not any field-theoretical argument against the 
consideration of a possible mutual interaction. Furthermore, such an interaction could 
alleviate the known coincidence problem <cit.>, namely why are the energy 
densities of  dark  matter and dark energy  currently of the same order although they 
follow completely different scaling laws during evolution. Hence, in the literature one
can find many such interacting scenarios (see <cit.> for review and 
references therein), independently of the specific dark-energy nature, namely whether
it arises from fields   <cit.> or
through a gravitational modification 
<cit.>.
The interacting scenarios can be very efficient in describing late-time universe, and 
moreover they seem to be slightly favored comparing to non-interacting ones
<cit.>. 
  Additionally, they seem to be efficient in addressing the H_0 tension 
<cit.> as well 
as the 
σ_8 tension 
<cit.>. 
  
  
In the above scenarios the interaction term, that determines  the interaction rate 
and thus the flow of energy between the dark matter and dark energy sectors, is 
introduced phenomenologically. Although there is a large variety in such choices, 
they are usually assumed to have the same sign, namely the energy flow is from one 
component to the other during the whole universe evolution. However, an interesting 
question arises, namely what would happen if  we allow for a sign change of the 
interaction term during the cosmological evolution. 
 Such a consideration might be further useful to investigate the 
dark sectors' physics.
Hence, in the present work we desire to investigate this possibility and in particular to 
confront the obtained scenarios with different observational data coming from probes like 
the cosmic microwave background radiation (CMB), supernovae type Ia (SNIa), baryon 
acoustic 
oscillations (BAO), and Hubble parameter measurements. 


We organize the present work in the following way. In Section <ref> we provide the 
cosmological equations at background and perturbative levels, in presence
of arbitrary coupling in the dark sector. In Section <ref> 
we present the models that we wish to study in this work. In Section <ref> 
we describe the observational datasets and the fitting methodology. In Section 
<ref> we 
provide the constraints on the models, and we perform a Bayesian analysis   
 in comparison with ΛCDM cosmology.  Finally, we close the present work in 
Section <ref> with a brief summary of the results. 




§ INTERACTING COSMOLOGY



In this Section we briefly review interacting cosmology. We consider a homogeneous 
and isotropic flat Friedmann-Robertson-Walker (FRW)
line element of the form

    ds^2 = -dt^2+a^2(t) δ_ij dx^i dx^j,

where a (t) is the expansion scale factor of the universe.
Furthermore, we consider   the universe to be filled 
with baryons, cold dark matter, radiation, and the dark energy sector (which may be of 
effective origin or not), all of which considered as barotropic perfect fluids.
 Thus, the Friedmann equations that determine the 
 universe evolution    are written as

    H^2   = 8π G/3ρ_t,
           2Ḣ + 3 H^2  = - 8 π G  p_t,

 with  G the Newton's constant and H=ȧ/a the Hubble 
function (dots denote derivatives with respect to t). In the 
above equations we have 
introduced the total energy density and pressure respectively  as
ρ_t = ρ_r +ρ_b +ρ_c 
+ρ_x and p_t = p_r + 
p_b + p_c + p_x, with the subscripts r,  b,  c,  x denoting  
radiation, baryon, cold dark matter and   dark energy.
 
Although the Bianchi identities lead to the conservations of the total energy momentum 
tensor, they do not imply anything for the separate sectors, and thus one can assume that 
some of them mutually interact <cit.>. In this work we allow the dark 
matter and dark energy sectors to interact, while radiations and baryonic matter are 
considered to be conserved. In particular, we assume that

    ρ̇_b + 3 H ρ_b =0
          ρ̇_r + 4 H ρ_r =0
          ρ̇_c + 3 H  ρ_c = -Q,
          ρ̇_x + 3 H (1+w_x)ρ_x = Q,

where w_x is the dark-energy equation-of-state parameter (for baryonic and dark matter 
we consider the dust case w_b=w_c= 0, while for radiation as usual w_r=1/3).
The introduced quantity Q is a phenomenological descriptor of the 
interaction, and its form is considered arbitrarily. If Q>0 then the energy transfer is 
from cold dark matter (pressureless dark matter) to dark energy, while if Q<0 then it 
is from dark energy to dark matter. Moreover, as usual the conservation equations for 
baryonic matter and radiation give ρ
_b=ρ _b0a^-3 and   ρ _r=ρ 
_r0a^-
4  respectively, with ρ _i0 (i=r,b)   the value of ρ
_i at present time. 
 In summary,  if the interaction function is given  then the Friedmann equation 
(<ref>) 
alongside the conservation  equations 
(<ref>) and (<ref>), can determine the evolution of the
universe.   

 

One can see that the conservation equations (<ref>) and (<ref>) can be 
written in 
an alternative way as 

    ρ̇ _c+3H( 1+w_c^𝚎𝚏𝚏) ρ
    _c =0, 
          ρ̇ _x+3 H( 1+w_x^𝚎𝚏𝚏) ρ
    _x =0,

where   w_c^𝚎𝚏𝚏, and 
w_x^𝚎𝚏𝚏 
are the effective equation-of-state parameters for cold dark matter and dark energy, 
given as

    w_c^𝚎𝚏𝚏 =Q /3 Hρ _c, 
          w_x^𝚎𝚏𝚏 =w_x-Q /3 Hρ _x.

Hence, as we observe, the interaction  affects the  equation of state 
of these components. In particular, dark matter may depart from dust while dark 
energy may be quintessence or phantom like even if the initial w_x is fixed to one 
regime.

We proceed to the investigation of the above scenarios at the level of perturbations.
We consider    scalar perturbations around an FRW metric given by 
<cit.>

    ds^2 =a^2(τ ){-(1+2ϕ )dτ ^2+2∂ _iBdτ 
    dx^i.
                  .
    +[(1-2ψ )δ _ij+2∂ _i∂ _jE]
    dx^idx^j},

where τ represents the conformal time and the quantities ϕ, B, 
ψ, E, denote the gauge-dependent scalar perturbations. Thus, in the case of 
interacting cosmology with w_x ≠ -1 the perturbation equations in 
the 
synchronous gauge (ϕ
=B=0, ψ =η, and k^2E=-h/2-3η), with k the Fourier mode and h, η, 
being the 
metric scalar perturbations <cit.>,   
are written as  
<cit.>:
 
    δ _x^' =-(1+w_x)( θ _x+h^'/2) -3ℋw_x^'
    θ _x/k^2
           -3ℋ(c_sx^2-w_x)[ δ _x+3ℋ
    (1+w_x)θ _x/k^2] 
    
          +aQ /ρ _x[ -δ _x+δ Q  /Q  +3ℋ
    (c_sx^2-w_x)θ _x/k^2] ,
    
          θ _x^' =-ℋ(1-3c_sx^2)θ 
    _x+
    c_sx^2/(1+w_x)k^2δ _x
          
    +aQ /ρ _x[ 
    θ _c-(1+c_sx^2)θ _x/1+w_x] , 
          δ _c^' =-( θ _c+h^'/2) +
    aQ  /ρ _c( δ _c-δ Q  /Q  ) , 
    
          θ _c^' =-ℋθ _c.

In the above expressions we have introduced the overdensities  δ _i=δρ 
_i/ρ _i,  as well as the velocity perturbations θ_i, with 
    primes denoting   derivatives with respect to the conformal time τ, with 
ℋ=a'/a   the conformal Hubble function. Moreover, c_sx^2 is the 
adiabatic sound speed, which in the following will be set to 1
<cit.> (the adiabatic sound speed for 
the dark matter in the dust case is c_sc^2 =0).  

 Now, for  the case where w_x=-1, namely in the case of interacting 
vacuum 
scenario, 
the perturbation equations are slightly different. In the synchronous gauge the momentum 
conservation equation for dark matter reduces to θ̇_c =0 <cit.>, 
while the density perturbations can be recast into <cit.> 

    δ̇_c = -ḣ/2 + Q/ρ_cδ_c .

However, in this gauge  the vacuum energy is spatially 
homogeneous, i.e. δρ_x = 0.  

 Finally, it is important to remark that although in the present work, as 
well as 
in many works in the literature, the spatial curvature of the universe is not 
considered,  a series of works   shows that the observational data 
do not exclude the non-zero 
curvature  
<cit.>, and hence the general universe 
picture should include the spatial curvature as a free parameter. We will address such a 
complete investigation in a separate work.    

  



§ SIGN-CHANGEABLE INTERACTION MODELS



Since the appearance of interacting cosmology,  a variety of phenomenological coupling 
functions Q have been introduced and  investigated  in the literature. In general, most 
of the coupling functions indicate a particular direction of energy transfer, namely
either the transfer of energy from dark matter  to dark energy or vice versa. This 
includes   functions of the form Q ∝ H ρ_c, Q
∝ H ρ_
x, Q  ∝ H (ρ_c + ρ_x), Q  ∝ H (ρ_c ρ_x)/(ρ_c + 
ρ_x),  Q  ∝ Hρ_x^2/(ρ_c + ρ_x), nonlinear forms, etc
<cit.>.

However, in principle one could also have the case in which the interaction 
function, i.e. Q, changes sign during the evolution, namely the energy transfer between 
the interacting sectors changes direction <cit.>. 
In the following subsections we examine two of such models separately.



 §.§ Interaction function  Q = 3 H ξ (ρ_c - ρ_x)


 
The first interacting model in which the interacting function changes sign during 
evolution is 

    Q (t) = 3 H ξ (ρ_c - ρ_x),

where ξ is the coupling parameter. Note the interesting feature that  when ρ_c = 
ρ_x then the interaction becomes zero even if ξ≠ 0. This kind of feature was 
observed 
in a different interaction model 
<cit.>. 


Here, we consider two choices for 
the dark energy equation-of-state parameter w_x, namely the cosmological constant 
case w_x= -1 (which corresponds to the interacting vacuum scenario, from now on 
model IVS1), and the case w_
x ≠ -1 (from now on model IDE1).  The latter case is interesting since it allows 
to extract constraints on the dark energy equation-of-state parameter and examine whether 
it lies in the quintessence    or in the 
phantom regime. 


We proceed by numerically elaborating 
the 
background cosmological equations, using as independent variable the redshift, z = 
a_0/a-1, 
setting the present scale factor a_0 to 1. Moreover, we introduce the density 
parameters of the various components through Ω_i=8π G ρ_i/(3H^2) and we 
set their current values as Ω_x0≈0.69, Ω_c0≈0.25, 
Ω_b0≈0.05 and Ω_r0≈10^-4 in agreement with observations 
<cit.>. 

In order to obtain a picture for the above  interaction form,
in Fig. <ref> we first depict the evolution of Q normalized by Q_0 = H_0 
ρ_t0 (with H_0 and ρ_t0 the present values of the Hubble parameter and the 
total energy density 
ρ_t respectively), for various values of ξ. As we can see, the sign-change 
direction and its exact moment depends on ξ, while  
the qualitative features of the interaction function do not change for  w_x> -1 
or 
w_x < -1.



 


In  Fig. <ref> we depict the evolution of the deceleration
parameter for model IDE1, for w_x > -1 and w_x < -1, and choosing various values 
of the coupling parameter ξ. As we observe, the value of ξ may make faster or 
delay the transition from deceleration to acceleration, which is a significant advantage 
in 
fitting observations precisely. Additionally, note that  the 
qualitative features  do not change for either w_x> -1 or 
w_x < -1. A similar graph can be obtained in the case of model IVS1, namely when w_x= 
-1.

We mention that in this scenario one can extract the solution for the ratio 
ρ_x/ρ_c   analytically. In particular, inserting (<ref>) into  
(<ref>) and (<ref>) gives 

    ρ_x/ρ_c=-w_x/2ξ-√(1+w_x^2/4 
    ξ^2)(1-ba^3√(4 ξ^2+  w_x^2)/1+ba^3√(4 ξ^2+  w_x^2)),

where 

    b≡-2ξΩ_x0/Ω_c0+w_x+√(4 ξ^2+  
    w_x^2)/2ξΩ_x0/Ω_c0+w_x-√(4 ξ^2+  w_x^2).


We proceed by investigating the coincidence parameter r  = 
ρ_c/ρ_x for 
this interaction model. As it is known, alleviation of the coincidence problem requires a 
non-zero value for r at late times. The evolution of the coincidence parameter for the model IDE1 is 
presented in Fig. <ref>, for different dark energy equation of 
states and setting  a typical 
value of ξ = 0.01 (for different values of ξ the qualitative nature of 
the curves remains unaffected). Moreover, for comparison we also depict the 
results for some well known interaction scenarios of the literature, namely Q = 3H ξρ_x, Q = 3H ξρ_c and Q = 3 H ξ 
(ρ_c+ρ_x).
As one can see,   as z → 0 all curves 
tend to  a non-zero value, which implies that the coincidence problem is alleviated. 
Additionally, although at low redshifts all models present the same behavior,   
at high redshifts the sign-changeable interaction models considered in this work are 
distinguishable from the known interacting models of the literature.

 Finally, concerning the perturbations in this scenario, they are determined by  
(<ref>)-(<ref>), with 

    δ 
    Q/Q=δ_cρ_c-δ_xρ_x/ρ_c-ρ_x+2θ+h'/6 
    ℋ,

 with θ≡θ_μ^μ the volume expansion of the total fluid.
 
  
 
 

 
 

 §.§ Interaction function  Q = 3 H (αρ_c - βρ_x)


 We proceed with the investigation of the interacting model

    Q  = 3 H (αρ_c - βρ_x),

where α, β are the coupling parameters considered to have the same sign. 
In the case where α = β = ξ the above model coincides with the one of the 
previous subsection, namely (<ref>). We consider two choices for 
 w_x, namely  w_x= -1 (from now on model IVS2), and the case w_
x ≠ -1 (from now on model IDE2), which will be further  divided 
into w_x > -1 and w_x < -1.  
 
In Fig. <ref> we   depict the evolution of Q normalized by Q_0 = H_0 
ρ_t0, for various values of   the coupling 
parameters α and β.  Additionally, in Fig. <ref> we present the 
evolution of the deceleration
parameter for   model IDE2, for w_x > -1 and w_x < -1 and choosing various values 
of   α and β. As we observe, the values of the 
coupling parameters significantly determine the exact transition redshift from 
deceleration to 
acceleration, while the exact value of  w_x does not have a significant effect.
 Similar graphs can be obtained in the case of  IVS2 model, namely when w_x= 
-1. 


Inserting (<ref>) into  
(<ref>) and (<ref>) provides as the analytical solution for the ratio 
ρ_x/ρ_c, namely

    ρ_x/ρ_c=α-β-w_x/2 β-√(
    (α-β-w_x)^2+4 
    αβ)/2 β
                          ·(1-ba^3√((α-β-w_x)^2+4 αβ)/1+ba^3√((α-
    β-w_x)^2+4 αβ)),

with

    b≡ - 2βΩ_x0/Ω_c0-(α-β-w_x)+√(
    (α-β-w_x)^2+4 αβ)/2βΩ_x0/Ω_c0-(α-β-w_x)-√(
    (α-β-w_x)^2+4 
    αβ).

 
We now investigate the evolution of the coincidence parameter r  = ρ_c/ρ_x 
for this 
interaction model. In Fig. <ref>, we depict the evolution of r
for 
various values of the coupling parameters  α and β and  
for different dark energy equation of 
states. Furthermore, for comparison
we also depict the 
results for some well known interaction scenarios, namely Q = 3H ξρ_x, Q = 3H 
ξρ_c and Q = 3 H ξ 
(ρ_c+ρ_x).
Similarly to model IDE1 above, for the present model IDE2 we can see that for z 
→ 0 all curves 
tend to  a non-zero value, which implies that the coincidence problem is alleviated. 
Additionally, although at low redshifts all models present similar behavior,   
at high redshifts the sign-changeable interaction models considered here are 
distinguishable from the known interacting models of the literature.    
 
 




 Lastly, concerning the perturbations, in this scenario they are determined by  
(<ref>)-(<ref>), with 

    δ Q/Q= 
    αδ_cρ_c-βδ_xρ_x/αρ_c-βρ_x+
    2θ +h^'/6ℋ.

 
 
 
 




§ THE DATA AND METHODOLOGY



In this section we briefly  describe the observational datasets and the methodology we 
follow in order to  constrain the aforementioned interaction models.  





  * Cosmic microwave background (CMB) radiation: We consider the CMB  data from Planck 
2015 measurements <cit.>, and in particular  we use 
the high-ℓ  and   low-ℓ  temperature and polarization 
   data from  <cit.>.  



  * Baryon acoustic oscillation (BAO): We use data from  BAO distance  
measurements from the following sources. Data from 6dF
Galaxy Survey (6dFGS) (redshift measurement at z_eff=0.106
) <cit.>, data from Main Galaxy Sample of Data Release 7 of Sloan
Digital Sky Survey (SDSS-MGS) (z_eff=0.15) <cit.>, and   data from CMASS and LOWZ samples of the latest Data Release 12
(DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS) (z_eff
=0.57) <cit.> and (z_eff=0.32) <cit.>.


  * Supernovae Type Ia (SNIa): We use the most latest compilation of SNIa,
consisting of 1048 data points spanned over the redshift interval z ∈ [0.01, 2.3] 
known as the Pantheon sample <cit.>.     



  * Cosmic Chronometers (CC): We use the Hubble parameter measurements from the  cosmic 
chronometers. The total number of data is 30 and the measurements are spanned over the 
redshift 
interval 0< 
z < 2 <cit.>. For technical details we further refer to 
<cit.>. 
 




In order to perform the analysis and extract the observational constraints, we   
use the Markov chain Monte Carlo package  <cit.> where a convergence diagnostic by Gelman-Rubin is included 
<cit.>, which 
in addition supports the
Planck 2015 likelihood code <cit.>[See the publicly available 
code  at <http://cosmologist.info/cosmomc/>. Note that although 
Planck 
2018 cosmological parameters <cit.> are now available, the Planck 2018 
likelihoods  are still not available publicly, and thus we use the   Planck 2015 
ones. ].
In the case of IDE1 model  we have the eight-dimensional parameter space: 
𝒫_1 ≡{Ω_bh^2, Ω_ch^2, 100 
θ_MC, τ, w_x, n_s, log[10^10A_S], ξ},   while  
for IVS1 we have one parameter less since w_x  =-1.  
For IDE2  we have the 
nine-dimensional parameter space 𝒫_2 ≡{Ω_bh^2, 
Ω_ch^2, 100 θ_MC,τ, w_x, n_s, log[10^10A_S],α, β}, 
and 
similarly for IVS2 we have one parameter less. In the above expressions  
 Ω_bh^2 is the physical baryons density, Ω_c
h^2 is the cold dark matter density, 100 θ_MC is the ratio of sound horizon to 
the angular diameter distance, τ is the
optical depth, w_x is the equation-of-state parameter for dark energy, n_s is the 
scalar spectral index, and A_S is the amplitude of the initial power spectrum.   
The 
remaining parameters in 𝒫_1 and 𝒫_2, namely ξ, α and
β are 
the coupling parameters for the interaction models. 
Finally, in Table <ref> we present the flat priors imposed on the 
free parameters of the prescribed interacting scenarios. 






                            

                       
                

               
 
                            





                            

                 
                     

                                                                              
 
     
 

 
  
 
 
 




 


 




§ OBSERVATIONAL CONSTRAINTS



In this section we provide the observational constraints on the sign-changeable scenarios 
and we 
discuss their consequences.  In order to acquire a complete picture we consider three 
different 
combinations of the observational
datasets described above, namely CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+CC. 

  


 §.§ Interaction function  Q = 3 H ξ (ρ_c - ρ_x)



The observational summary for IDE1 model is presented in Table 
<ref>. Additionally, in   Fig. <ref> we provide the 
corresponding 2D contour plots of various parameters at  1σ and 2σ  
confidence level (CL), alongside the 1D marginalized posterior distribution.
From Table <ref>   we can notice  that 
the 
addition of Pantheon or Pantheon+CC to the observational combination CMB+BAO, slightly 
improves the constraints by reducing their error bars, nevertheless the improvement is 
not   significant.  Additionally, from 
 Fig. <ref> we can see that the parameters (H_0, w_x) and (H_0, 
Ω_m0) are negatively correlated to each other. 

Concerning the dark energy equation-of-state parameter w_x, from 
Fig. <ref> we infer that a phantom nature of w_x is clearly preferred at 
more than 2σ CL.  Additionally, concerning the coupling  ξ and the dark 
energy equation-of-state parameter w_x, from Table  <ref>,  
it is clearly observed that the evidence for an interaction is  small (ξ =  
-0.00014_- 
   0.00004^+    0.00014 at 1σ for CMB+BAO, ξ =  
-0.00013_-    0.00003^
+  0.00013 at 1σ for  CMB+BAO+Pantheon, ξ = -0.00013_-    0.00003^+    
0.00013  at 1σ for CMB+BAO+Pantheon+CC). Hence, within 1σ the 
non-interacting cosmology is   allowed.
In summary, our observational confrontation shows that this interaction 
model at the background level essentially mimicks a w_xCDM-type cosmology with w_x < 
-1 at more than 2σ CL.

 
 We now proceed by examining the tensions on the two main parameters, namely H_0 and 
σ_8.  As we observe from both Table  <ref> and  Fig. 
<ref> (specifically from the posterior distribution of H_0 which is the 
extreme right plot of the bottom 
panel), H_0 acquires slightly higher values compared to  Planck <cit.>, 
with 
slightly higher error bars. Although the local estimation of the Hubble constant 
obtained by Riess et al. <cit.>, i.e. H_0 = 73.24± 1.74, is certainly 
greater than the 
estimated mean values of H_0  for this interaction model, due to the increased error 
bars  the tension is reduced  to the level of 2σ CL.  Thus,   the interaction 
between the dark components  provides a way   to reduce the tension from 
> 3 σ CL. Nevertheless, concerning  σ_8 tension
we deduce that the  present interaction model  is not able to alleviate it.


Let us now examine the effect  of the interaction on the large scale observables, and 
mainly on the CMB TT and matter power spectra. In the 
upper graph of Fig. <ref> we show how the interaction affects the CMB TT 
spectra, considering the constraints on the parameters extracted from  all   
observational datasets, namely CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+CC, in 
which for completeness  we add the non-interacting case of
ΛCDM cosmology. From this graph it is hard to distinguish the interacting case 
from ΛCDM  scenario. However, in the lower graph of Fig. 
<ref> we depict the corresponding residual plot (with reference to 
ΛCDM model), and one can indeed trace a distinction between   interacting and 
non-interacting cosmologies, mainly  in the lower multipoles. 
Similarly, we investigate the effects of the interaction through the matter power spectra 
presented in Fig. <ref>. Although in the upper graph the distinction 
between   interacting and non-interacting cosmologies cannot be observed, in the lower 
graph the deviation from the non-interacting ΛCDM cosmology is clear.  This is 
one 
of the main results of the
present work. 
 

We close the analysis of this model by focusing on the case where w_x = -1, thus w_x 
is not a free parameter and is fixed to the cosmological constant value,  namely we 
examine the interacting vacuum 
scenario. In this case we summarize the results in 
Table <ref>, and in Fig. <ref>  we present the corresponding 
contour plots. In this model the case of no-interaction seems to be favored, while 
H_0 acquires smaller values, and therefore the H_0 tension is not alleviated. 
Concerning σ_8 tension we deduce that it  cannot be released either. 
Finally, at large scales this specific interacting scenario 
does not return different results in the CMB TT and matter power spectra comparing to  
ΛCDM cosmology, and therefore we do not explicitly present the corresponding 
plots.  
  
 
                    



                            

             
                 

                                              


                             



 

 
  
          
          
          
          




 §.§ Interaction function  Q = 3 H (αρ_c - βρ_x)



The observational summary  of   model IDE2 is shown in Table <ref>,
while the corresponding 2D contour plots are presented in Fig. <ref>. 
Similarly to model IDE1,  one can notice that the addition of Pantheon or Pantheon+CC to 
the combined analysis CMB+BAO, improves the parameters space only slightly. Moreover, 
from 
 Fig. <ref> we can see that the parameters (H_0, w_x) and (H_0, 
Ω_m0) are negatively correlated to each other. 
 

          
 Concerning the coupling parameters    α and β, our analysis shows 
that  the 
zero values are allowed within 1σ. Additionally, the dark-energy 
equation-of-state parameter prefers the phantom regime 
for all   datasets, namely we see that w_x < -1 at more 
than 2σ. Furthermore, similarly to IDE1 model, in the present IDE2 scenario  we 
also find that the estimations of H_0 are slightly higher compared to the 
ΛCDM-based 
Planck estimation <cit.>, and due to the higher error bars on H_0 the 
relevant tension can be slightly reconciled due to the interaction.
However, concerning σ_8 we see that the tension is not released. 

          

We proceed by  investigating the effect of the interaction  on the CMB TT and matter 
power 
spectra. In the upper graph of Fig. <ref> we depict the CMB TT spectra 
considering the constraints on the parameters extracted from  all observational 
datasets, namely CMB+BAO, CMB+BAO+Pantheon, and CMB+BAO+Pantheon+CC, in which for 
completeness  we add the non-interacting case of ΛCDM cosmology. Moreover, in the 
lower graph of Fig. <ref> we present the corresponding residual plot (with 
reference to ΛCDM model).
As we observe,  this interaction model is distinguished from the non-interacting 
ΛCDM cosmology at both lower and higher multipoles. 
We further investigate the effects of the interaction on the matter 
power spectra, depicted in Fig. <ref>. Although in the upper graph the 
distinction between the interacting and non-interacting cosmologies cannot be observed, 
in 
the lower graph the deviation from the non-interacting ΛCDM cosmology is clear
even for the small values of the coupling parameters α, β that were obtained 
from the  three different observational datasets. This is one of the main results of the 
present work. 
               
                            

                    
                       

                
 





 
We proceed by analyzing the case w_x = -1, namely the interacting vacuum scenario. The 
results of the analyses are presented in Table <ref>, while in Fig. 
<ref> we depict the corresponding contour plots. From these we deduce that 
the 
coupling parameter α acquires the zero value 
within 1σ CL irrespectively of 
the datasets, while β has a tendency towards non-zero values nevertheless the 
value zero is allowed within 1σ except for the final combination 
CMB+BAO+Pantheon+CC. Additionally, concerning  H_0, for 
CMB+BAO+Pantheon and CMB+BAO+Pantheon+CC datasets we find that its estimations are 
relatively high compared to  ΛCDM-based Planck estimation <cit.>, 
while the error bars on H_0 are also increased. Hence, this enables H_0 to acquire  
values close to its local estimation <cit.>, and thus the tension on H_0 
is weakly resolved. However, the σ_8 tension  cannot be alleviated. 

                     



 §.§ Bayesian analysis

 

We close the observational confrontation by   presenting   observational viabilities 
of the models using the Bayesian evidence. The Bayesian analysis is an important part of 
the cosmological model selection that quantifies the fitting results  
compared to a reference scenario. The 
computation of Bayesian evidence is  performed with the code  
<cit.>[See the freely available code in
https://github.com/yabebalFantay
e/MCEvidencegithub.com/yabebalFantaye/MCEvidence.], which directly computes the 
evidences of the model   with respect to the reference  
ΛCDM scenario.  

In Bayesian analysis one needs to calculate the posterior probability of the model 
parameters θ, subject to a particular observational dataset x and any prior 
information for 
the underlying model M. Recalling the Bayes theorem one can write that  

    p(θ|x, M) = p(x|θ, M) π(θ|M)/p(x|M),

in which  p(x|θ, M) is the likelihood function (depending on the 
model parameters θ with the given data set), and where π(θ|M) refers to 
the prior information. The quantity p(x|M) is  the 
Bayesian evidence. Given two models, namely M_i and M_j, where 
M_i is the model under investigation and M_j is the reference model (here the 
ΛCDM scenario), the posterior probability is given by 

    p(M_i|x)/p(M_j|x) = π(M_i)/π(M_j) p(x| M_i)/p(x|M_j) = 
    π(M_i)/π(M_j)  B_ij.

The quantity B_ij = p(x| M_i)/p(x|M_j) is the Bayes factor of the considered 
model M_i relative to the reference model M_j, and quantifies 
how the observational data support the model M_i over M_j.  In 
Table <ref> we show the corresponding classification following  
<cit.>.  
 
                                                                               
         
            


                 
   


  

In Table <ref> we present the computed values of ln B_ij for all   
sign-changeable interacting scenarios considering all   observational datasets. As we 
can see, the values of |ln B_ij| for IDE2 are greater than the values of   |ln 
B_ij| for IDE1 and this is true for all   datasets. This was expected since
IDE2 scenario has one extra free parameter compared to IDE1. Similarly,  the values of 
|ln B_ij| for model IVS2 are greater than those IVS1. Nevertheless, overall,  
ΛCDM cosmology   is still favored over the present interacting models.  



                                              


§ LAWS OF THERMODYNAMICS IN SIGN-CHANGEABLE INTERACTION MODELS




In this section we shall investigate the thermodynamical laws in a universe governed 
by  sign-changeable interacting dark energy.
In order to investigate the thermodynamical properties of a specific 
cosmological model one assumes that the universe is a thermodynamical system bounded by 
a cosmological horizon, and then he applies arguments from  black hole 
thermodynamics <cit.>. In particular, 
one considers that the universe is bounded 
by the apparent horizon with radius r_h= (H^2+ k/a^2  )^-1/2 
<cit.> which, for a spatially flat universe  becomes the Hubble horizon r_h = 
1/H. Hence, one can show that the 
first law of thermodynamics  can lead to the first Friedmann equation  
<cit.>. 

We proceed by examining the validity of the generalized second law of thermodynamics, 
which states that the total entropy of the universe, namely the entropy of the various 
cosmological fluids plus the entropy of the horizon, should be a non-decreasing function 
of time <cit.>. 
Concerning the entropy of the various fluids that constitute the universe one has
S_r+ S_b 
+S_c+ S_x,
 where S_i (i = r, b, c, x) denotes the entropy of the i-th fluid. Thus, the
first law of thermodynamics for each individual fluid becomes
<cit.>.

    T dS_r    = dE_r + p_r dV ,
    
    T d S_b    = dE_b + p_b dV = dE_b,
    
    T dS_c    = dE_c + p_c dV =  dE_c,
    
    T d S_x    = dE_x + p_x dV,

where V= 4π r_h^3/ 3 is the volume of the universe, E_i 
stands for the internal energy of the i-th fluid given by E_i = 4/3 π  
r_h^3 ρ_i, and p_i is the corresponding pressure. Note that the various fluids 
are considered to have the same temperature. However, we mention that  
over the entire cosmic evolution the temperatures of different cosmic fluids 
are not the same <cit.>,  as they evolve differently. In 
particular, the temperatures of radiation and dark energy remain different from 
the horizon temperature for a long period of time, while for the 
non-relativistic matter its temperature becomes and remains equal with the 
horizon one. Nevertheless, if at some epoch in the universe evolution the 
horizon temperature comes close or become equal to  
the temperature of the dark energy sector, then they will remain the same for 
most of the expansion of the universe. Therefore, the assumption, that the 
temperatures of different cosmic fluids are the same, is not unjustified 
<cit.>. 




The entropy of the horizon is  
taken to be that of a black hole, namely
S_h= k_B 𝒜/(4  l_pl^2),
with k_B   the Boltzmann's constant, ħ  the Planck's constant, and l_pl = 
(√(ħ G/c^3))  the Planck's length, however taking the area to be that 
corresponding to the horizon, i.e.  𝒜= 4 π r_h^2 <cit.>. Therefore, in units where ħ= k_B= c= 8 π G = 1, the horizon entropy 
(<ref>) reduces to

    S_h = 8 π^2 r_h^2.

Moreover, concerning the temperature of the apparent horizon one can use the black hole result but 
with the apparent horizon instead of the black-hole one, namely T_h = 
1/(2 π r_h)  <cit.>.


Using all the above relations together with the assumption that the temperature of 
the fluids  T should be equal to T_h  one can 
find that 

    Ṡ= Ṡ_r+ Ṡ_b+ Ṡ_c + Ṡ_x + Ṡ_h 
          = 4 π^2 H r_h^6 [ρ_r (1+w_r) +ρ_b + ρ_c + (1+ w_x)ρ_x  ]^2,

where  we have replaced the involved ṙ_h=-Ḣ/H^2 using the Friedmann 
equations (<ref>),(<ref>). As we can see, the total entropy is always a
non-decreasing function, and thus the generalized second law of thermodynamics is 
satisfied. Hence, although the sign change of the interaction between dark matter and dark 
energy could lead to a local entropy decrease at the microscopic level, the total entropy 
of the universe plus the one of the horizon is always non-decreasing.
The above relation for the first law of thermodynamics in interacting cosmology  has been 
established earlier in  <cit.>. 


Now, let us examine whether the universe with sign-changeable interaction in the dark 
sectors will result to thermodynamic equilibrium. As it is discussed extensively in the literature 
(see for instance <cit.>), in order for this to be 
achieved one needs to have a total entropy whose first derivative is positive, while at 
late times its second derivative should be negative <cit.>, and thus the entropy asymptotically 
tends towards a constant value. Using x= ln a as the independent variable the above 
require that
S^'≥ 
0 and S^'' < 0, where S=S_r+ S_b 
+S_c+ S_x+S_h, with
primes denoting 
differentiation with respect to x.

Equation 
(<ref>) can   be rewritten as 

    S^' = 16 π ^2/H^4 ( H^')^2,

and thus it leads to 

    S^'' = 2 S^' (  H^''/H^'- 
    2 H^'/H) = 2 S^' (  h^''/h^'- 2 h^'/h)= 2 S^'Δ,

where h= H/H_0 and Δ= (  h^''/h^'- 
2 h^'/h). 
In 
Fig. <ref> we depict the evolution of S^''  for both  IDE1 
  and IDE2 models. As we observe, in both models at late times, namely at  present epoch 
(ln a = 0) and in the future,  the total entropy 
is convex, i.e.  S^'' < 0, and hence the universe tends towards 
thermodynamic equilibrium. 
This can also be seen by the fact that asymptotically S^'' 
goes to zero. These features hold also for the two models IVS1 and IVS2.  Moreover, note 
that the individual curves  are practically 
indistinguishable.

 


Hence, in summary, we conclude that for cosmological scenarios with sign 
changeable interaction functions the thermodynamical laws remain 
valid. 






 










§ CONCLUDING REMARKS



Interacting cosmology has attracted the interest of the literature, since on   one 
hand it cannot be excluded from the field theoretical point of view, and on the other 
hand it may offer a solution to the coincidence problem. However, almost all 
phenomenologically introduced interacting functions have constant sign, namely the energy 
flow maintains its direction throughout the whole universe evolution.

In the present work we investigated sign-changeable interacting scenarios, in 
which the 
interaction function, and thus the energy flow, changes sign during the evolution of the 
universe, since there is not any theoretical reason of not considering such forms. 
We  considered various   models and we extracted the involved 
equations at both the background and perturbation levels. Then we used various data 
combinations  from cosmic microwave background (CMB), 
baryon acoustic oscillations (BAO),  Supernovae Type Ia (SNIa) and cosmic chronometers 
(CC) in order to constrain the model parameters. Finally, we performed a Bayesian 
analysis in order to compare the fitting efficiency of the examined models with the 
reference  ΛCDM paradigm.


For both examined sign-changed interacting  models, namely Q = 3 H ξ (ρ_c - 
ρ_x) and  Q = 3 H (αρ_c - βρ_x), we found that the dark-energy 
equation-of-state parameter w_x  prefers the phantom regime for all   datasets, at 
more than 2σ. Concerning the coupling parameters we saw that although the best-fit 
 values might be non-zero, the zero value, namely no interaction, is included within 
1σ. Moreover, we showed that this results is maintained if we impose w_x to take 
 the cosmological constant value -1, namely considering the interacting vacuum model. 

 We proceeded by examining the effect of the interaction on the CMB TT and matter 
power spectra. As we showed, while from the simple CMB TT spectra  it is hard to 
distinguish the interacting case from ΛCDM  scenario, from the residual plot 
(with reference to ΛCDM model) one can indeed trace a distinction between the 
interacting and non-interacting cosmologies, mainly  in the lower multipoles. Similarly, 
the simple matter power spectra cannot be used to examine the interaction, however 
using the corresponding residual graphs we showed that the deviation from the 
non-interacting ΛCDM cosmology is clear.  The fact that the residual spectra 
plots can be used to distinguish the models from  ΛCDM paradigm, even if at the 
background level the latter is allowed withing 1σ CL, is one of the main results 
of the present work.
 
 
Concerning  H_0, we saw that in all cases its obtained values are slightly 
higher compared to the ΛCDM-based Planck estimation, and thus the H_0 
tension seems to be alleviated  as a result of the interaction, although only 
partially. Hence, we deduce that 
although the interaction may be small, it is adequate to alleviate the H_0 tension. 
Nevertheless, concerning   σ_8 tension   we found that the present sign 
changeable interaction models are not able to 
release it. These features show that sign-changeable interacting scenarios might be 
worthy for  further investigations.                  
                    

Finally, we examined the validity of the laws of thermodynamics in a universe with 
sign-changeable interactions in the dark sector. As we saw, the generalized second law is 
always satisfied, namely the total entropy, constituting from all fluids as well as from 
the horizon entropy, is always increasing. Additionally, its second derivative becomes 
negative at late times, which implies that the universe tends towards thermodynamic equilibrium.

 
Last but not least, it is interesting to study the sign changeable 
models in extended parameter spaces, allowing the spatial  curvature to be non-zero. The 
inclusion of massive neutrinos could also be appealing. These projects are left for 
future investigation.                   
                  


 The authors would like to thank an anonymous referee for important comments 
that 
helped   to improve the manuscript 
substantially. SP acknowledges the research funding through the Faculty Research 
and 
Professional 
Development Fund 
(FRPDF) Scheme of Presidency University, Kolkata, India. 
WY acknowledges the support from the National Natural Science Foundation of China under 
Grants No. 
11705079 and No. 11647153.
This article is based upon work from CANTATA COST (European Cooperation in Science and 
Technology) 
action CA15117, 
EU Framework Programme Horizon 2020.



 


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