Effects of Anisotropic Stress in Interacting Dark Matter  Dark Energy Scenarios
Abstract
We study a novel interacting dark energy dark matter scenario where the anisotropic stress of the large scale inhomogeneities is considered. The dark energy has a constant barotropic state parameter and the interaction model produces stable perturbations in the large scale of the universe. The resulting picture has been constrained using different astronomical data in a spatially flat FriedmannLemaîtreRobertsonWalker (FLRW) universe. We perform different combined analyses of the astronomical data to measure the effects of the anisotropic stress on the strength of the interaction and on other cosmological parameters as well. The analyses from several combined data show that a nonzero interaction in the dark sector is favored while a noninteraction scenario is still allowed within 68% confidencelevel (CL). The anisotropic stress measured from the observational data is also found to be small, and its zero value is permitted within the 68% CL. The constraints on the dark energy equation of state, , also point toward its ‘’ value and hence the resulting picture looks like a noninteracting CDM as well as CDM cosmology. However, from the ratio of the CMB TT spectra, we see that the model has a deviation from the standard CDM cosmology which is very hard to detect from the CMB TT spectra only. Although the deviation is not much significant, but from the present astronomical data, we cannot exclude such deviation. Overall, we find that the model is very close to the CDM cosmology. Perhaps, a more accurate conclusion can be made with the next generation of surveys that are not so far. We also argue that the current tension on might be released for some combinations of the observational data. In fact, the allowance of in the phantom region is found to be more effective to release the tension on .
pacs:
98.80.k, 95.36.+x, 95.35.+d, 98.80.EsI Introduction
A remarkable revolution in the dynamical history of the universe has been witnessed in the last several years. Around 20 years back, the distant Supernovae of Type Ia (SNIa) first indicated an accelerating expansion of the universe and thereafter a lot of distinct astronomical observations have strengthened such observational prediction. To interpret this acceleration a hypothetical fluid with negative pressure became necessary and subsequently, cosmological constant was revived into the picture. The cosmological constant, , has a negative equation of state, and together with colddarkmatter, the joint scenario CDM has been found to be the best cosmological model, at least according to a series of astronomical measurements. In CDM scenario, both the cosmological constant and the colddarkmatter remain conserved separately, as if they are two disjoint sectors. Such model of the universe is widely referred to as the noninteracting cosmological scenario. Unfortunately, the cosmological constant suffers from the finetuning problem (also known as the cosmological constant problem), where being timeindependent, it reports an unimaginable difference (of the order of ) in its value determined in the Planck and low energy scales. The problem associated with the cosmological constant is not a new detection, it is persisting since long back ago Weinberg1989 , even before the lateaccelerating phase. Thus, attempts have been made aiming to provide with a reasonable justification on the finetuning problem Wetterich:1994bg . While on the other hand, people have tried to bypass this problem through the introduction of dark energy models Copeland:2006wr ; AT ; Bamba:2012cp (we also refer to some specific scalar field dark energy models that may also account for the early scenarios of the universe deHaro:2016hpl ; deHaro:2016cdm ). However, although the introduction of dark energy models relieves the cosmological constant problem but they raised another serious issue which is widely known as the cosmic coincidence problem Zlatev:1998tr . Such coincidence problem led to another class of cosmological theories which is the theory of nongravitational interaction between dark matter and dark energy.
The nongravitational interaction in the dark sector, precisely between dark matter and dark energy is a phenomenological concept that was originally thought to explain the different values of the timeindependent cosmological constant Wetterich:1994bg , but later on, such concept was found to be very useful to explain the cosmic coincidence problem Amendola:1999er ; Chimento:2003iea ; Cai:2004dk ; Hu:2006ar ; delCampo:2008sr ; delCampo:2008jx . Certainly, this led to a large amount of investigations towards this direction where the dark sectors have direct interaction Billyard:2000bh ; Barrow:2006hia ; Chimento:2009hj ; Quartin:2008px ; Valiviita:2009nu ; Thorsrud:2012mu ; dfm1 ; dfm2 ; dfm3 ; dfm4 ; dfm5 ; Yang:2014hea ; Yang:2014gza ; yang:2014vza ; Faraoni:2014vra ; Duniya:2015nva ; Valiviita:2015dfa ; Pan:2012ki ; Pan:2016ngu ; Mukherjee:2016shl ; Sharov:2017iue (also see Pan:2013rha ; Chen:2011cy ; Pan:2014afa ; Cai:2017yww ; Kumar:2017bpv ; Odintsov:2017icc ). Such interacting scenarios have good motivation if the particle physics theory is considered, because from the particle physics view, mutual interaction between any two fields, is a natural phenomenon, irrespective of the nature of the fields. Although the interacting dynamics is complicated and a generalized cosmic scenario, but it recovers the noninteraction cosmology as a special case. Thus, the theory of nongravitational interaction between dark matter and dark energy is a generalized version of the noninteracting dark matter and dark energy cosmologies. Interestingly enough, the observational data at recent time found that the direct interaction between dark matter and dark energy cannot be excluded Salvatelli:2014zta ; Nunes:2016dlj ; Yang:2016evp ; Kumar:2016zpg ; vandeBruck:2016hpz ; Yang:2017yme ; Kumar:2017dnp ; Yang:2017zjs ; Yang:2017ccc . Moreover, very recently, it has been reported that the current tension on the local Hubble constant can be alleviated with the introduction of dark matter and dark energy interactions DiValentino:2017iww ; Kumar:2016zpg . Additionally, the crossing of phantom barrier has also been found to be an easy consequence of the nongravitational interaction. Thus, the theory of interacting dark energy might be considered to be an appealing field of research and indeed a hot topic for the next generation of the astronomical surveys.
The current work presents a general interacting scenario where aside from the nongravitational interaction between dark matter and dark energy, we include the possibility of an anisotropic stress. The anisotropic stress appears when the first order perturbation is considered, and in most of the cases, it is generally neglected. From both theoretical and the observational grounds, the possibility of an anisotropic stress cannot be excluded at all. Although the dimension of the resulting parameters space is increased, but, due to advancements of the astronomical data, the measurement of the anisotropic stress becomes important. The most important thing is to measure the effect of this quantity on the large scale structure evolution of the concerned cosmological scenario. This is the primary motivation of this work. In particular, looking at the perturbation equations (see section II) one can realize that a nonzero value of the anisotropic stress, can affect the temperature anisotropy in the cosmic microwave background spectra and also on the matter power spectra as well. Thus, for a detailed understanding of the interacting scenario in its large scale structure, the anisotropic stress plays a significant role.
Thus, following the above motivation, in a spatially flat FriedmannLemaîtreRobertsonWalker universe, we study an interacting dark matterdark energy scenario in presence of an anisotropic stress and then constrain this model using a series of latest astronomical data from cosmic microwave background radiation, Joint Light Curve analysis (JLA) sample of Supernovae Type Ia, baryon acoustic oscillations (BAO) distance measurements, Hubble parameter measurements from cosmic chronometers (CC), weak gravitational lensing and the local Hubble constant value from the Hubble Space Telescope (HST). The analyses are based on the use of publicly available markov chain monte carlo package cosmomc where the convergence of the cosmological parameters follows the well known GelmanRubin statistics.
The work has been organized in the following way. In section II we describe the background and the perturbation equations for the coupled dark energy in presence of the mattersourced anisotropic stress. Section III describes the observational data that we use to analyze the present models. In section IV we discuss the constraints on the current model. Finally, section V closes the work with the main findings of this investigation.
Ii The interacting universe
In this section we shall describe an interacting cosmological scenario both at background and perturbative levels. To do this, we assume the most general metric for the underlying geometry of the universe which is characterized by the FriedmannLemaîtreRobertsonWalker (FLRW) line element. We also assume that the metric is spatially flat. The evolution equations for a pressureless dark matter and a dark energy fluid in this universe obey the following conservation equations
(1)  
(2) 
where the prime denotes the differentiation with respect to the conformal time (i.e. “”); , is the conformal Hubble parameter; , are respectively the energy densities of cold dark matter and dark energy; is the barotropic equation of state of the dark energy, that means , here is the pressure of the dark energy fluid and as usual zero pressure is attributed to colddarkmatter sector. The quantity in the right hand sides of (1) and (2) is the energy transfer rate between the dark sectors and depending on its sign the direction of energy flow is determined. To be precise, a positive interaction rate () assigns the energy flow from dark matter to dark energy while the negative interaction rate reverses its direction of energy flow. In addition, we consider the presence of nonrelativistic baryons () and relativistic radiation () which follow the standard evolution equations, that means they do not take part in the interaction. The dynamics of the spatially flat universe is thus constrained by the Hubble equation
(3) 
where , is the total energy density of the universe. Hence, if the energy transfer rate, , is specified, then the evolution equations for and can fully be determined using the conservation equations (1) and (2) together with the Hubble constraint (3). However, the presence of interaction in the dark sector may significantly affect the large scale structure of the universe and hence it becomes necessary to consider the evolution equations for the interacting model at the perturbative levels. Thus, in order to do so we consider the perturbed FLRW metric Mukhanov ; Ma:1995ey ; Malik:2008im
(4) 
Here, the quantities appearing in the above metric (4), namely, , , and represent the gaugedependent scalar perturbations. For this metric one can find the perturbation equations and following Valiviita:2008iv ; Majerotto:2009np , the perturbed energy and momentum balance equations for the interacting dark matter and dark energy scenario can be found to be
(5) 
(6) 
where is the characterization symbol for any fluid (dark matter or dark energy) and prime is the representation of the differentiation with respect to the conformal time, mentioned earlier. The relation between the peculiar velocity potential and the local volume expansion rate is, in the Fourier space with mode Mukhanov ; Ma:1995ey ; Malik:2008im ; Valiviita:2008iv ; Kodama:1985bj . Here by , we mean the density perturbation for the fluid , and momentum transfer potential has been assumed to be the simplest physical choice, that gives its value to zero in the rest frame of dark matter Valiviita:2008iv ; Koyama:2009gd ; Clemson:2011an . Hence, the momentum transfer potential takes the expression Yang:2014gza .
Now, the pressure perturbation , for any fluid , is related as Valiviita:2008iv which for the dark energy fluid turns out to be .
The evolution of fluid perturbations could be described by the adiabatic speed of sound . In this adiabatic case, the relation between the perturbations of and is related by . However, for an entropic fluid, the pressure might not be a unique function of the energy density . Therefore, there would be another degree of freedom to describe the microproperties of a general fluid. That is the physical speed of sound in the rest frame (‘’ represents the restframe) which is defined in the comoving frame of the fluid. Now, we note that when the entropic perturbation vanishes, the physical sound speed and the adiabatic sound speed vanishes, that means, . Hence, in the case of entropic fluid such as scalar fields, one needs both its equation of state and its sound speed, to have a complete description of dark energy and its perturbations. However, in order to fully describe a dark energy fluid and its perturbations, one should also consider the possibility of an anisotropic stress, even in an isotropic and homogenous FLRW universe, where the anisotropic stress , can be taken as a spatial perturbation.
In the synchronous gauge (, , and ), the evolution equations for density perturbations and velocity perturbations equations for dark energy and dark matter respectively read
(7) 
(8) 
(9)  
(10) 
where the factor includes the perturbation term of Hubble expansion rate Gavela:2010tm . In the next section, the anisotropic stress of dark energy would be assumed as an appropriate expression.
Following Hu Hu:1998kj , the evolution of the anisotropic stress, , is considered to be
(11) 
where is the viscous speed of sound which controls the correspondence between the velocity (or metric) shear and the anisotropic stress. In particular, for a relativistic fluid, , while for a general dark energy fluid, is a free parameter and it can be constrained through the observational data Huey:2004jz ; Koivisto:2005mm ; Mota:2007sz ; Koivisto:2008ig ; Cardona:2014iba ; Amendola:2013qna ; Saltas:2010tt ; Kunz:2006ca ; Song:2010rm .
The anisotropic stress can be linked to some either the overdensity of matter (dark matter () baryons ()) or dark energy as shown in Cardona:2014iba . The anisotropic stress that is linked to the overdensity of dark matter is also known as the mattersourced anisotropic stress model having the form
(12) 
while when the anisotropic stress is linked to the overdensity of dark energy (similarly, it might be dubbed as the darkenergysourced anisotropic stress model)
(13) 
where , is the gauge invariant density perturbations for matter () and dark energy (), respectively. The above relations are established on the fact that the anisotropic stress and the overdensity of dark matter (or, dark energy) may modify the gravitational slip in an effective way Song:2010rm . The latest analysis on the observational constraints of dark energy with anisotropic stress can be found in Chang:2014mta ; Chang:2014bea .
Let us come to the interaction model that we wish to study in this work. Before taking any typical interaction model we recall that the interaction function directly enters into the pressure perturbation for dark energy as Valiviita:2008iv
(14) 
According to the qualitative analysis on the largescale instability in the dark sector perturbations during the early radiation era Valiviita:2008iv , in the pressure perturbation of dark energy (14), the coupling term in the pressure perturbation can lead to a driving term which includes the factor , and it becomes very large if is close to ‘’. This causes rapid growth of . Qualitatively, this is the source of the instability: in the presence of energymomentum transfer in the perturbed dark fluids, momentum balance requires a runaway growth of the dark energy velocity. In order to avoid the perturbation instability, and based on the phenomenological consideration, we assume the constant equation of state in the interacting dark energy with the energy transfer rate . The presence of the factor in the interaction function does not bother with the dark energy equation of state, and hence the stability of the interaction model in the large scale structure of the universe rests on the coupling parameter of the interaction. The perturbation equations (7) to (10) for the specific interaction model turn out to be
where the mattersourced anisotropic stress is, . In this work, we consider the mattersourced model with , that means, , as the simplest case in such complicated interacting dynamics. Although there is no such strict restriction to exclude the possibility of anisotropic stress sourced by dark energy, but, however, since the cluster effects of dark energy is smaller in compared to the dark matter, the effects of anisotropic stress sourced by dark energy must be weaker in respect to the anisotropic stress sourced by dark matter. As a result, the anisotropic stress sourced by dark matter might be more relevant in this context.
Iii Observational data sets and the statistical technique
In this section we describe the main observational data that we have used to constrain the cosmological scenarios and also we outline the statistical methodology. We use various astronomical data ranging from low redshifts to high redshifts, for our analysis. Below we summarize the data sets with their corresponding references.

Cosmic Microwave Background Radiation: The full Planck 2015 low temperaturepluspolarization and the high likelihood (“Planck TT, TE, EE + lowTEB”) Adam:2015rua ; Aghanim:2015xee have been used. For the interacting dark energy with mattersourced anisotropic stress, the amplitude of CMB at low multipole () is very sensitive to the values due to the fact that the anisotropic stress of dark energy is proportional to the velocity of dark energy directly. The summation of the Newtonian potentials becomes
(15) where is the gauge invariant density contrast and is related to the anisotropic stress via . Thus, an extra contribution to the integrated SachsWolfe (ISW) effect due to the existence of anisotropic stress of dark energy is

Joint LightCurve Analaysis: The Joint Lightcurve Analysis (JLA) sample Betoule:2014frx containing 740 Supernovae Type Ia in the lowredshift range have been considered.

Baryon Acoustic Oscillations Distance Measurements: For baryon acoustic oscillations (BAO) data, we mainly use four different data points. In particular, we use the CMASS and LOWZ samples from the latest Data Release 12 (DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS) respectively at the effective redshifts and GilMarin:2015nqa . In addition, we include the 6dF Galaxy Survey (6dFGS) measurement at Beutler:2011hx , and the Main Galaxy Sample of Data Release 7 of Sloan Digital Sky Survey (SDSSMGS) at Ross:2014qpa .

Redshift Space Distortion Data: We employ the redshift space distortion (RSD) measurements from two disctinct galaxy samples, the one which includes the CMASS sample with an effective redshift of GilMarin:2016wya while the other includes the LOWZ sample with an effective redshift of GilMarin:2016wya .

Hubble Parameter Measurements: We also employ the recently released cosmic chronometers (CC) data with measurements of the Hubble parameter values in the redshift interval Moresco:2016mzx . The cosmic chronometers are basically some galaxies which evolve passively and are the most massive. An accurate measurement of the differential age evolutions of such galaxies together with the spectroscopic estimation of with high accuracy yield the Hubble parameter value through . For more on the CC, we refer the readers to Moresco:2016mzx .

from the Hubble Space Telescope: The present Hubble constant value yielding Riess:2016jrr from the Hubble Space Telescope (HST) has been used. We label this value as HST.

Weak Gravitational Lensing Data: Finally, we also use the weak gravitational lensing data (WL) along with the previous data sets. The sample is taken from the CanadaFranceHawaii Telescope Lensing Survey (CFHTLenS) which spans 154 square degrees in five optical bands. In this survey, 21 sets of cosmic shear correlation functions linked to six redshift bins have been presented, see Refs. Heymans:2013fya ; Asgari:2016xuw for details. The tomographic correlation functions measured measured from the blue galaxy sample and consistent with zero intrinsic alignment nuisance parameter has been named as and we have used this for the present work. From the likelihood analysis of the CFHTLenS data one can extract the information of our universe. Here, the true inverse covariance matrix takes the form in which , and is the measured covariance matrix. The inclusion of the anisotropic stress of dark energy certainly modifies the summation of potentials given in Eq. (15). Moreover, for the presence of anisotropic stress, the lensing potential gains an extra contribution leading to the convergence power spectrum at angular wave number as
(17) where is the comoving distance; is the angular diameter distance out to and it depends on the curvature scalar . We note that in the present work we have assumed . The quantity is the horizon distance, and represents lensing efficiency function for the redshift bin , see Heymans:2013fya ; Asgari:2016xuw for more discussions.
Now, for the interacting dark energy with mattersourced anisotropic stress, the amplitude of CMB at low multipole () is very sensitive to the values of due to the fact that the anisotropic stress of dark energy is proportional directly to the velocity of dark energy fluid. The summation of the Newtonian potentials becomes
(18) 
where is related to the anisotropic stress through the relation . For the influence of WL, the convergence power spectrum will also be modified by the anisotropic stress in the same way, but in the spatial part of the Newtonian potentials.
The likelihood for our analysis is, , where is, . We modify the code CAMB Lewis:2002ah which is freely available and here we implement a numerical algorithm. This numerical algorithm is called to solve the background equations and after that corresponding to each data set we calculate the values. Finally, we call another code known as cosmomc, a markov chain monte carlo package together with a convergence diagnostic GelmanRubin that is used to extract the cosmological parameters. The parameters space for our present model is, (ninedimensional space). Here, is the cold dark matter density, is the baryon density, is the ratio of sound horizon to the angular diameter distance, is the optical depth, is the scalar spectra index, is the amplitude of the initial power spectrum and the remaining , , are the model parameters described earlier. Certainly, the inclusion of both interaction (in terms of the strength ) and the anisotropic stress () extends the parameters space in compared to the minimum number of parameters space in CDM, see Barrow:2014opa for a detailed discussion.
The priors of specific model parameters are displayed in Table 1.
Parameter  Prior 

Parameters  CMB  CMB+BAO+RSD  CMB+BAO+RSD+HST  CMB+BAO+WL+HST  


























Iv Results and analysis
The interacting scenario in presence of the mattersourced anisotropic stress is the main focus of the work. However, we have also constrained the interacting scenario where no matter sourced anisotropic stress is present. The motivation of the second analysis is to see how the presence of matter sourced anisotropic stress affects the cosmological dynamics. In order to constrain both the interacting scenarios we have used the following observational data:

CMB (Planck TTTEEE+lowTEB),

CMB+BAO+RSD,

CMB+BAO+RSD+HST,

CMB+BAO+WL+HST,

CMB+BAO+RSD+WL+HST+JLA+CC.
For the interacting scenario with matter sourced anisotropic stress, we have presented the observational summary in Table 2 where the constraints on the model parameters are shown at 68% and 95% confidence levels. In Fig. 1 we display the onedimensional posterior distributions for some selected model parameters for the above observational data. Let us now analyze the observational constraints on the model parameters.
From Table 2 we see that the observational data favor a nonzero interaction between dark matter and dark energy. However, observing the errorbars of the coupling parameter, , one can readily conclude that is allowed by almost all observational data. That means within confidencelevel, a noninteracting CDM model is still allowed. From the the dark energy equation of state we find that only CMB and the combined analysis CMB+BAO+RSD hint for its quintessence nature. One can see that the CMB data alone constrain the dark energy equation of state, at 68% CL ( at 95% CL) while from the combination CMB+BAO+RSD, we find at 68% CL ( at 95% CL). One may notice that the addition of BAO and RSD to CMB decreases the error bars in the dark energy equation of state, that means the parameter space for gets reduced. Interestingly enough, when the prior from the HST is included to the other data sets, the dark energy equation of state moves toward the cosmological constant boundary. The combined analysis CMB+BAO+RSD+HST shows that at 68% CL ( at 95% CL). The last two analyses, namely, CMB+BAO+WL+HST and CMB+BAO+RSD+WL+HST+CC+JLA infer the same about the dark energy equation of state, see the last two columns of the Table 2. Thus, from the results, one can identify that the dark energy resembles with the cosmological constant. Hence, one may conclude that, although a nonzero deviation from the CDM cosmology is favored by the observational data but effectively, such deviation is very minimal and hence the model is close to the CDM model. In Fig. 2 we display the 68% and 95% confidencelevel contour plots for the combinations (, ) and (, ) and (, ) using different combined analyses performed in this work. From the left panel of Fig. 2, we find that for lower values of , the dark energy equation of state, has a shifting nature toward the quintessence regime while from the middle panel of this figure we observe that, as increases, that means when it shifts towards the quintessence regime, the density parameter for cold dark matter increases. From the right panel of Fig. 2, we show the dependence of coupling parameter with the dark energy equation of state, , from which making any decisive conclusion between the dependence of with looks very hard, in fact, the parameters and look uncorrelated.
The inclusion of prior from HST also affects other cosmological parameters. For instance, from the constraints on the anisotropic stress displayed in Table 2 one can find the considerable changes in its constraints. The magnitude of the anisotropic significantly changes. The only CMB data constrains at 68% CL ( at 95% CL) while from CMB+BAO+WL+HST, at 68% CL ( at 95% CL) and from CMB+BAO+RSD+WL+HST+JLA+CC, it is at 68% CL ( at 95% CL). One may observe that there is no such significant changes in the error bars in the anisotropic stress. In Fig. 3, we present the 68% and 95% confidencelevel contour plots where we show the effects of the anisotropic stress on some selected cosmological parameters.
We now focus on the dynamics of the univese on the large scales for the current cosmological scenario. In Fig. 5, we have plotted the CMB TT power spectra (see the left panel of Fig. 5) and the ratio of the CMB TT power spectra (see the right panel of Fig. 5) for different values of the anisotropic stess and compared the analyses with the base CDM model. It is quite clear from this figure that at low angular scales, for large anisotropic stress, the model deviates vastly from the CDM model while as the angular scale increases, the deviation reduces from the CDM and at high angular scales, the anisotropic stress does not produce any effective changes in the power spectra. However, the right panel of Fig. 5 says something more which is not visible from the left panel of Fig. 5. From the ratio of CMB TT spectra displayed in the right panel of Fig. 5, one can see that the model still shows a slight deviation from CDM in the large angular scale and even if a nonzero value of the anisotropic stress is allowed. Thus, the model has a slight difference from CDM and such a difference is very small. However, we have a very interesting observation from Fig. 6 displaying the CMB TT spectra and the ratio of the CMB TT spectra for different strengths of the coupling parameters. From the left panel of Fig. 6 one can see that the a slight deviation of the stressed interacting scenario from the CDM model is observed for a large value of the coupling parameter (, which is a very big value in compared to the observational estimation summarized in Table 2) while from the right panel of Fig. 6, it is quite clear that the model definitely has a deviation from the base CDM for any . However, the deviation is not much significant.
Parameters  CMB  CMB+BAO+RSD  CMB+BAO+RSD+HST  CMB+BAO+WL+HST  
























iv.1 Comparison with noanisotropic stress
In the previous section IV we have studied the effects of the anisotropic stress on the cosmological parameters when the dark fluids are interacting with each other. A question that immediately appears in this context is, how the contribution of anisotropic stress affects the large scale structure of the universe and also in the estimation of the cosmological parameters? To answer these questions, we perform similar analyses making in the evolution equations with the same priors presented in Table 1. It is quite evident that the analysis with noanisotropic stress will effectively present a quantitative and qualitative differences on the cosmological parameters. The observational constraints for this particular scenario have been shown in Table 3. The 68% and 95% confidence level contour plots for some selected parameters have been presented in Fig. 7 where we also show their onedimensional posterior distributions. From both the analyses, one can clearly notice that except for CMB data only, the exclusion of the anisotropic stress lowers the strength of the coupling parameter. This is an interesting observation in this work which clearly demonstrates that the addition of is the measure of increment in the coupling strength, . However, from the analyses presented in Table 2 and Table 3, a clear conclusion that one might draw is, in both the scenarios (with and without anisotropic stress), the coupling strength recovers its zero value within the 68% CL, that means the model CDM may recover the noninteracting CDM cosmology within this 68% CL and this model has a close resemble with that of the CDM cosmology. This might be considered to be a common behavior of the models. Furthermore, it should be mentioned that only the combination CMB+BAO+RSD with , does not recover the limit in anyway and thus, this particular combination always indicates for a nonzero interaction in the dark sector.
Probably the most interesting observation comes from the relative deviation of the CMB TT and matter power spectra shown in Fig. 8. A quick look says that both the scenarios are close to CDM but there is something more that we would like to describe here. Let us focus on the left panel of Fig. 8 where the relative deviation of the CMB TT spectra has been shown. One can notice that for , the scenario CDM significantly differs from CDM, and with the increase of , the difference between the scenarios decreases. We find a particular where the difference between the models becomes zero, but after that, for some certain , the difference again increases where the model CDM stays far from CDM in compared to CDM. Further, we again notice that, the model CDM approaches toward CDM and become closer in compared to the CDM. However, for large , both the models become indistinguishable from one another. We note that, for all , the quantity that reports the difference of the model from the CDM is very small.
Now we concentrate on the relative deviation in the matter power spectra, . We remark that for all , the qunatity is very small informing the closeness toward the CDM model, but however, we notice some additional features. We find that for very small , almost for , the quantity for the model CDM is slightly greater in compared to the model CDM, however, for , the reverse scenario is observed, and finally, both the models look indistinguishable.
Thus, overall, one may coclude that indeed the scenarios CDM and CDM maintain differences amongst each other but for large (for CMB TT spectra) and large (matter power spectra), both the scenarios effectively approach toward the cosmology with almost no difference between them.
Parameters  CMB  CMB+BAO+RSD  CMB+BAO+RSD+HST  CMB+BAO+WL+HST  
() 

()  

Parameters  








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