Modified holographic Ricci interacting dark energy models: dynamical system analysis and Bayesian comparison

Abstract

We perform a dynamical system analysis and a Bayesian model selection for a new set of interacting scenarios in the framework of modified holographic Ricci dark energy (MHR-IDE) models. The dynamical analysis shows a modified radiation epoch and a late-time attractor corresponding to an accelerated phase of holographic dark energy. In the Bayesian comparison, we use a combination of background data including type Ia supernovae, cosmic chronometers, the local value of the Hubble constant, baryon acoustic oscillations, and cosmic microwave background data. We find evidence against all the MHR-IDE scenarios studied in comparison with the $\varLambda$CDM when the full joint analysis is considered.

1 Introduction

Our universe is experiencing accelerated expansion [1]. This acceleration is attributed to dark energy, represented by a cosmological constant $\varLambda$ in the standard cosmological model. While the standard cosmological model, also known as Lambda cold dark matter ($\varLambda$CDM), is a suitable model for the current expansion and the phases observed in the universe’s evolution, it is plagued by several issues. These include the cosmological constant problem [2, 3], the coincidence problem [4,5,6], and discrepancies in the values of the Hubble parameter obtained from local measurements and those inferred from Planck’s data [7].

Over the past two decades, numerous models of dark energy have emerged to account for the current accelerated expansion of the universe (see for example Refs. [8,9,10,11] and references therein). Holographic dark energy (HDE) models, in particular, are built upon the holographic principle, as initially proposed by ’t Hooft [12]. In this framework, the authors of Ref. [13], inspired by the Bekenstein–Hawking entropy bound in black hole thermodynamics [14,15,16,17,18,19], postulate that the energy within a region of size L should not exceed the mass of a black hole of the same size, and thus $L^3\rho \le LM_P^{2}$. In a cosmological context, the largest allowable scale for L saturates this inequality. Since these seminal works, several dark energy models rooted in the holographic principle have been explored. For example, it has been demonstrated that selecting the scale L as the Hubble length [20] or the size of the particle horizon [21] does not lead to accelerated expansion. A successful model where the scale L is determined by the size of the future event horizon was proposed in Ref. [21]; however, this model faced criticism due to causality issues [22]. In Ref. [23], a generalized HDE model was proposed which also possesses a covariant description [24].

In Ref. [25], the holographic Ricci dark energy (HRDE) model was proposed, a model that circumvents the causality problems, where the dark energy density is proportional to the Ricci scalar. Subsequently, the authors of Ref. [26] indicated that the Jeans length of perturbations establishes a causal connection scale associated with the Ricci scalar, offering a physical motivation for the HRDE model. Numerous studies have delved into these models; see for example Refs. [27,28,29,30,31,32,33,34,35].

For example, the authors of Ref. [36] proposed a new scale (infrared cutoff) for the holographic dark energy model which includes a term proportional to $\dot{H}$,

$$\begin{aligned} \rho _x= 3(\alpha H^2+\beta \dot{H}), \end{aligned}$$

(1)

where $\alpha$ and $\beta$ are positive constants. This model, dubbed the modified holographic Ricci dark energy (MHRDE) model, has been widely studied (see for example Refs. [37,38,39,40]). In particular, the authors of Ref. [41] indicate that the scale L leading to (1) is a natural extension of the HDE model. Note that for $\alpha =2\beta$, the HRDE model is recovered from (1), for a flat scenario. In Ref. [23], it was demonstrated that any HDE, including HRDE, is a specific representative of the generalized HDE model; see for example Refs. [42, 43]. Inflationary holographic scenarios have also been explored in the context of the generalized HDE model [44,45,46].

Interacting holographic dark energy models have been studied extensively, originally with the aim of finding an accelerating scenario with $L=H^{-1}$ [6]. Over the years, many interacting holographic scenarios have been investigated in different contexts; see for example Refs. [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66].

On the other hand, Bayesian model selection is a powerful statistical tool for comparing the performance of cosmological models in light of the available data, and it has been widely used in cosmology [67,68,69,70,71,72]. For example, in Ref. [73], a Bayesian model selection of MHR-IDE models was performed with linear interaction. This work can be considered as an extension of Ref. [73], with five types of revised interactions.

This paper investigates the viability of five new types of interacting scenarios in modified holographic Ricci dark energy models. To assess the models’ viability, we characterize the asymptotic behavior of the MHR-IDE models through a dynamical system analysis and study their performance in fitting the known asymptotic behavior of our universe. We further conduct Bayesian model selection comparing the performance of MHR-IDE scenarios with the $\varLambda$CDM model, using background data including type Ia supernovae, cosmic chronometers, the local value of the Hubble constant, baryon acoustic oscillations, and the angular scale of the sound horizon at the last scattering.

The paper is structured as follows. In Sect. 2, we present the studied MHR-IDE scenarios. In Sect. 3, we develop the dynamical system analysis, including critical points, existence and stability conditions, and a description of the results for each scenario. In Sect. 4, we describe the data used and the methodology to assess the Bayesian comparison. In Sect. 5, we discuss the main results, and finally, in Sect. 6 we present final remarks on this work.

2 The modified holographic Ricci interacting dark energy models

We work in the framework of general relativity, considering a spatially flat Friedmann–Lemaître–Robertson–Walker metric. We assume a universe composed of baryons, radiation, dark matter, and dark energy, where each of these components is considered a barotropic perfect fluid. In this scenario, the Friedmann equations are written as

$$\begin{aligned}&3H^2 = \rho {,} \end{aligned}$$

(2)

$$\begin{aligned}&2\dot{H} + 3H^2 = -p{,} \end{aligned}$$

(3)

where $\rho = \rho _b + \rho _r + \rho _c + \rho _x$ and $p = p_b + p_r + p_c + p_x$ are the total energy density and pressure, respectively. We use the subscripts b, r, c, and x for baryons, radiation, cold dark matter, and dark energy, respectively. $H=\dot{a}/a$ is the Hubble expansion rate defined in terms of the scale factor a, where the dot represents a derivative with respect to the cosmic time, and we use units such as $8\pi G = 1$ and $c=1$. On the other hand, from the energy–momentum tensor conservation, we have

$$\begin{aligned} \dot{\rho } + 3H(\rho +p) = 0. \end{aligned}$$

(4)

Although the total energy density is conserved, this does not imply that each component is conserved separately, which allows us to assume that the dark components interact with each other through a phenomenological interaction term Q. Thus, considering a barotropic equation of state for all the components, $p_i = \omega _i\rho _i$, where $\omega _i$ is the state parameter, we separate Eq. (4) into the following equations:

$$\begin{aligned} \dot{\rho }_b + 3H\rho _b&= 0, \end{aligned}$$

(5)

$$\begin{aligned} \dot{\rho }_r + 4H\rho _r&= 0, \end{aligned}$$

(6)

$$\begin{aligned} \dot{\rho }_c + 3H\rho _c&= -Q, \end{aligned}$$

(7)

$$\begin{aligned} \dot{\rho }_x + 3H(1+\omega _x)\rho _x&= Q, \end{aligned}$$

(8)

where we assume that $\omega _b = \omega _c = 0, \omega _r = 1/3$, and $\omega _x$ is a variable dark energy state parameter. Using the change of variable $\eta = 3\ln a$ and defining $()' = d/d\eta$, Eqs. (5)–(8) are rewritten as

$$\begin{aligned} \rho '_b + \rho _b&= 0, \end{aligned}$$

(9)

$$\begin{aligned} \rho '_r + \frac{4}{3}\rho _r&= 0, \end{aligned}$$

(10)

$$\begin{aligned} \rho '_c + \rho _c&= -\varGamma , \end{aligned}$$

(11)

$$\begin{aligned} \rho '_x + (1+\omega _x)\rho _x&= \varGamma , \end{aligned}$$

(12)

where $\varGamma = Q/3H$. Note that for $\varGamma >0$, we have an energy transfer from cold dark matter to dark energy, and for $\varGamma <0$ we have the opposite energy transfer.

Taking into account the holographic motivation for the dark energy discussed in Sect. 1, we consider the ansatz (1) for the dark energy density written as

$$\begin{aligned} \rho _x = \alpha \rho + \frac{3\beta }{2}\rho ', \end{aligned}$$

(13)

where $\alpha$ and $\beta$ are considered as positive constants. For this scenario, the authors of Ref. [73] obtain a second-order differential equation by combining Eqs. (11)–(13),

$$\begin{aligned} \frac{3\beta }{2}\rho ''_d&+ \left( \alpha + \frac{3\beta }{2} - 1\right) \rho '_d + (\alpha -1)\rho _d\nonumber \\&+ \frac{1}{3}(2\beta - \alpha )\rho _{r0}e^{-4\eta /3} = \varGamma , \end{aligned}$$

(14)

where $\rho _d = \rho _c + \rho _x$ and $\rho _{r0}$ is the integration constant from Eq. (10).

In this work, we study five types of MHR-IDE models, defined as follows:

$$\begin{aligned} \varGamma _1= & {} \delta \rho '_c + \gamma \rho '_x,\quad \;\ \varGamma _2 = \delta \rho _d + \gamma \rho '_d,\quad \;\ \varGamma _3 = \gamma \rho ,\nonumber \\ \varGamma _4= & {} \gamma \rho ',\quad \;\, \varGamma _5 = \gamma q \rho , \end{aligned}$$

(15)

where $q = -(1+\dot{H}/H^2)$ is the deceleration parameter, $\rho$ is the total energy density, and $\delta$, $\gamma$ are constants associated with the interaction. In Ref. [73], an interaction proportional to the dark components was studied, $\varGamma =\delta \rho _c+\gamma \rho _x$, and no evidence supporting those MHR-IDE models was found when background data were used. In this work, we extend this analysis considering five new types of interactions, including an explicit change of sign in interaction $\varGamma _5$. This work aims to elucidate whether these new types of interactions modify the dynamics of the models, considering that previous results indicate a preference for $\varLambda$CDM when compared with previously studied MHR-IDE models.

Note that we can rewrite Eq. (14) as

$$\begin{aligned} \rho ''_d + b_1 \rho '_d + b_2 \rho _d + b_3 \varOmega _{b0}a^{-3} + b_4 \varOmega _{r0}a^{-4} = 0. \end{aligned}$$

(16)

In this form, this equation includes interactions $\varGamma _1-\varGamma _5$, so that the constants $b_1$, $b_2$, $b_3$, and $b_4$ result in different combinations of the holographic and interaction parameters (see Table 1).

Table 1 Definition of the constants $b_1$, $b_2$, $b_3$, and $b_4$ in terms of the parameters defined for each MHR-IDE scenario

The general solution of Eq. (16) takes the form

$$\begin{aligned} \rho _d(a) = 3H_0^2( \varOmega _{h1}a^{-4} +\varOmega _{h2}a^{-3} + \tilde{C}_1a^{3\lambda _1} + \tilde{C}_2a^{3\lambda _2}),\nonumber \\ \end{aligned}$$

(17)

where the coefficients and integration constants in (17) are given by

$$\begin{aligned} \varOmega _{h1}&= \frac{9b_4\varOmega _{r0}}{12b_1 - 9b_2 - 16},\quad \varOmega _{h2} = \frac{b_3\varOmega _{b0}}{b_1 - b_2 - 1},\nonumber \\ \lambda _{1,2}&= -\frac{1}{2}(b_1 \pm \sqrt{{b_1}^2 - 4b_2}), \end{aligned}$$

(18)

$$\begin{aligned} \tilde{C}_1&= \frac{\varOmega _{h2}(1+\lambda _2)}{\lambda _1 - \lambda _2} + \frac{\varOmega _{h1}(4+3\lambda _2)}{3(\lambda _1 - \lambda _2)} + \frac{2(\varOmega _{x0}-\alpha )}{3\beta (\lambda _1 - \lambda _2)} \nonumber \\&\quad + \frac{3\varOmega _{b0} + 4\varOmega _{r0} - 3\lambda _2(\varOmega _{c0} + \varOmega _{x0})}{3(\lambda _1 - \lambda _2)}, \end{aligned}$$

(19)

$$\begin{aligned} \tilde{C}_2&= -\varOmega _{h1}-\varOmega _{h2}+\varOmega _{c0} + \varOmega _{x0}-\tilde{C}_1, \end{aligned}$$

(20)

where $\varOmega _{c0}$ and $\varOmega _{x0}$ are the current values of the density parameters for cold dark matter and dark energy, respectively. Therefore, we can write the Hubble expansion rate associated to the total energy density as

$$\begin{aligned} H(a) = H_0\sqrt{\begin{array}{l}(\varOmega _{h1} + \varOmega _{r0})a^{-4} + {\varOmega _{b0}a^{-3} + \varOmega _{h2}a^{-3}}\\ \qquad +\tilde{C}_1a^{3\lambda _1} + \tilde{C}_2a^{3\lambda _2}\end{array}}. \end{aligned}$$

(21)

Equation (21) recasts the analytical solution for the expansion rate corresponding to the MHR-IDE models $\varGamma _1-\varGamma _5$ in a single expression. We note that there is a generic contribution to the radiation- and matter-dominated epochs arising from the holographic and interacting nature of the MHR-IDE models studied herein. These terms were also observed in Ref. [73], where a different interaction was analyzed within the context of MHR-IDE models. In particular, Table 1 shows that the contribution to radiation completely vanishes for $\alpha =2\beta$ in models $\varGamma _1$ and $\varGamma _2$ (this behavior is also noted in the model examined in Ref. [73]), whereas this contribution persists in models $\varGamma _3$, $\varGamma _4$, and $\varGamma _5$, even for $\alpha =2\beta$, where the contribution will depend on the interaction terms. On the other hand, the contribution to the matter epoch is absent in model $\varGamma _2$, which depends on the interaction contribution in models $\varGamma _3$, $\varGamma _4$, and $\varGamma _5$, and it vanishes for $2\alpha =3\beta$ in model $\varGamma _1$ (similar behavior to that observed in Ref. [73]).

In the following section, we present the dynamical system analysis of MHR-IDE models defined in Table 1, to investigate the global dynamics of these models and to compare this with the asymptotic behavior of the $\varLambda$CDM model.

3 Dynamical system analysis

We apply dynamical system methods [8, 11] to identify the relevant cosmological eras in the MHR-IDE models. By writing the system of Eqs. (9)–(12) in terms of the density parameters $\varOmega _i=\rho _i/3H^2$ and using the Friedmann Eq. (2) as a constraint among density parameters, i.e.,

$$\begin{aligned} \varOmega _r+\varOmega _b+\varOmega _c+\varOmega _x=1, \end{aligned}$$

(22)

we reduce the system of Eqs. (9)–(12) to three equations for $\varOmega _r'$, $\varOmega _c'$, and $\varOmega _x'$, where $\varOmega _r'$ has the same structure for all the MHR-IDE models and is given by

$$\begin{aligned} \varOmega _r'=\frac{2 \varOmega _r (\alpha -2 \beta -\varOmega _x)}{3 \beta }. \end{aligned}$$

(23)

The equations for $\varOmega _c'$ and $\varOmega _x'$ depend upon the specific MHR-IDE model, and their explicit forms are shown in Sects. 3.1–3.5.

The stability of the critical points representing the different cosmological epochs is found by calculating the eigenvalues of the linearized system at the critical points. For each MHR-IDE scenario, we consider the set of equations

$$\begin{aligned} \varOmega _i'=f_i(\varOmega _l), \end{aligned}$$

(24)

where $f_i$ is a function of the density parameters $\varOmega _r$, $\varOmega _c$, and $\varOmega _x$, and $i=r,c,x$. From Eq. (24) we find the critical points $\varOmega _l^*$ by calculating

$$\begin{aligned} f_i(\varOmega _l^*)=0. \end{aligned}$$

Then we linearize the set of Eq. (24) around the critical points

$$\begin{aligned} \delta \varOmega _i'=J_i^l(\varOmega _j^*)\varOmega _l, \end{aligned}$$

where $J_i^l=\frac{\partial f_i}{\partial \varOmega _l}$ is the Jacobian matrix, from which we can find different regions in the parameter space. In particular, we have unstable, saddle, or stable points when the real part of the eigenvalues of the Jacobian matrix are all positives, a mixture of positives and negatives, or all negatives, respectively.

In the following, we perform a dynamical system analysis of MHR-IDE models presented in Table 1. We study models $\varGamma _1$ and $\varGamma _2$ in four possible scenarios, models $\varGamma _{11}$, $\varGamma _{12}$, $\varGamma _{13}$, and $\varGamma _{14}$ representing particular cases of $\varGamma _1$ for $\delta =0$, $\gamma =0$, $\delta = \gamma$, and $\delta \ne \gamma$, respectively. Similarly, $\varGamma _{21}$, $\varGamma _{22}$, $\varGamma _{23}$, and $\varGamma _{24}$ are variants of $\varGamma _2$ for the same scenarios. In Tables 2, 3, and 4, we show the critical points and the corresponding stability conditions and classification, resulting from the dynamical system analysis for scenarios $\varGamma _1$, $\varGamma _2$, and $\varGamma _3-\varGamma _5$, respectively. The scenarios $\varGamma _{14}$ and $\varGamma _{24}$ are too intricate to yield significant results, so in the dynamical system analysis we limit ourselves to cases $1-3$ in models $\varGamma _1$ and $\varGamma _2$. We also observe that models $\varGamma _{13}$ and $\varGamma _{21}$ are the same model. To improve the visualization of the critical points, we have defined constants $C_1-C_7$ in Table 2, constants $D_1-D_7$ in Table 3, and constants $E_1-E_3$ in Table 4 as follows:

$$\begin{aligned} C_1&=\alpha -2\beta , \nonumber \\ C_{(2,3)}&=1+\alpha \gamma -C_4\pm \sqrt{(1+\alpha \gamma -C_4)^2-6\beta \gamma },\nonumber \\ C_4&=\frac{1}{2}(2\alpha -3\beta ), \nonumber \\ C_5&=1+2\beta +4\gamma , \nonumber \\ C_{(6,7)}&=-\gamma -C_4\pm \sqrt{(1+\gamma -C_4)^2-6\beta \gamma }, \end{aligned}$$

(25)

$$\begin{aligned} D_{(1,2)}&=C_4\mp \sqrt{(1-C_4)^2+6\beta \delta },\quad D_3=-\frac{(1-C_4)^2}{6\beta },\nonumber \\ D_4&=\frac{1}{2}(5-2\sqrt{6}),\nonumber \\ D_5&=\frac{1}{2}(2+2\gamma +3\beta ),\nonumber \\ D_{(6,7)}&=\gamma {+}C_4\pm \vert 1+\gamma -C_4\vert , \end{aligned}$$

(26)

$$\begin{aligned} E_1&=-\frac{1}{4}(1+2\beta ),\nonumber \\ E_{(2,3)}&=C_4\mp \sqrt{(C_4-1+\frac{3}{2}\gamma )^2+3\beta \gamma }. \end{aligned}$$

(27)

On the other hand, to analyze the type of the effective fluid for each critical point, in Tables 2, 3, and 4 we use the effective state parameter, defined as $\omega _{\text {eff}}=\frac{p}{\rho }$.

In the following, we describe the critical points and the existence conditions for each model. For the analysis of the stability conditions, we restrict ourselves to the parameter ranges

$$\begin{aligned} 0<\alpha<1,\quad 0<\beta<1,\quad -1<\gamma<1,\quad -1<\delta <1,\nonumber \\ \end{aligned}$$

(28)

which are consistent with the subsequent analysis in Sect. 5. Also, in order to analyze the weak energy condition at each critical point, we show the parameter conditions where the corresponding energy densities are defined as positive.

Table 2 Description of the critical points $\{\varOmega _{r}^{*},\varOmega _{c}^{*},\varOmega _{x}^{*}\}$ for scenario $\varGamma _{1}$. The intervals in (28) are considered as global constraints. To improve the visualization of the expressions we use the constants $C_1-C_7$ defined in (25)

3.1 Scenario $\varGamma _1$

For model $\varGamma _1$, in addition to Eq. (23), we have the following set of equations:

$$\begin{aligned} \varOmega _c'= & {} \frac{2 (\alpha -\varOmega _x) ((\delta -\gamma +1) \varOmega _c+\gamma )+\beta (\gamma (3 \varOmega _c-\varOmega _r+3 \varOmega _x-3)-3 \varOmega _c)}{3 \beta (\delta -\gamma +1)}, \end{aligned}$$

(29)

$$\begin{aligned} \varOmega _x'= & {} \frac{2 (\alpha -\varOmega _x) ((\delta -\gamma +1) (\varOmega _x-1)-\gamma )+\beta (-\delta (3 \varOmega _c-\varOmega _r +3 \varOmega _x-3) -3 \varOmega _x+\varOmega _r+3) }{3 \beta (\delta -\gamma +1)}.\nonumber \\ \end{aligned}$$

(30)

Note that Eqs. (29) and (30) are well defined for $\gamma \ne 1+\delta$. Inside the intervals (28), the constants $C_1-C_7$, defined in (25) and used in Table 2, are real numbers and we must have $\gamma \ne -\frac{1}{4}$ in order to the critical point $P_8$ exist. In the following, we describe the critical points $P_1-P_{11}$ in Table 2, corresponding to models $\varGamma _{11}$, $\varGamma _{12}$, and $\varGamma _{13}$.

For model $\varGamma _{11}$, we find three critical points, $P_1-P_3$ in Table 2. Point $P_1$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $P_1$ require $0<C_1<(1-4\gamma )^{-1}$ and $\gamma <0$ inside the intervals (28). In the limit $C_1\ll 1$ (or $\alpha -2\beta \ll 1$) we obtain the domination of the radiation term at $P_1$. Points $P_2$ and $P_3$ correspond to a combination of the dark components. At point $P_3$, the dark energy term dominates, with $\omega _{\text {eff}}<-1$ in the range (28). In the limit $\gamma \ll 1$ we have $\omega _{\text {eff}}\approx \frac{C_4 \gamma }{C_4-1}$ at $P_2$ and $\varOmega _x^*\approx 1$ at $P_3$. Positive energy densities at $P_1$, $P_2$, and $P_3$ require the additional condition $\alpha >\frac{3\beta }{2}$. Note that the condition $C_1>0$, together with the condition $0<\alpha <1$, implies $\beta <\frac{1}{2}$.

For model $\varGamma _{12}$, we find four critical points, $P_4-P_7$ in Table 2. Point $P_4$ is an effective fluid of radiation type, corresponding to a combination of the radiation component with the dark energy (DE) component. Positive energy densities at $P_4$ require $0<C_1<1$. $P_5$ is a saddle point corresponding to a combination of baryons and DE with $\omega _{\text {eff}}=0$, and for $C_4\approx 0$ this point corresponds to the domination of baryons. $P_6$ is a combination of the dark components, with $\omega _{\text {eff}}\approx 0$ in the limit $\delta \ll 1$; $P_7$ represents the domination of DE with $\omega _{\text {eff}}<-1$ in the range (28). Positive energy densities at $P_4$, $P_5$, $P_6$, and $P_7$ require the additional conditions $0<C_4<1$ for $\delta >0$ or $-\alpha \delta<C_4<1+(1-\alpha )\delta$ for $\delta <0$.

For model $\varGamma _{13}$ we find four critical points, $P_8-P_{11}$ in Table 2. Point $P_8$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $P_8$ require $0<C_1<1+4\gamma$ and $-\frac{1}{4}<\gamma <0$. $P_9$ represents a combination of the dark sector and baryons, and at this critical point we always have one negative energy density, and $\omega _{\text {eff}}=0$. For $C_4\approx 0$, this point corresponds to the domination of baryons. Points $P_{10}$ and $P_{11}$ are combinations of the dark sector. Point $P_{11}$ has $\omega _{\text {eff}}<-1$ in the range (28). In the limit $\gamma \ll 1$, we have $\omega _{\text {eff}}\approx \frac{\gamma }{C_4-1}$ at $P_{10}$ and $\varOmega _x^*\approx 1$ at $P_{11}$. Positive energy densities at $P_8$, $P_{10}$, and $P_{11}$ require the additional condition $C_4>-\alpha \gamma$.

In Fig. 1 we show the phase space diagrams corresponding to models $\varGamma _{11}$, $\varGamma _{12}$, and $\varGamma _{13}$, with fixed parameter values. We observe consistent behavior in the phase diagrams for the three models: a radiation-dominated source, a saddle point corresponding to a dark sector combination, and a final attractor corresponding to the domination of the holographic dark energy. A distinctive feature of the MHR-IDE models is the contribution of holographic dark energy during both the radiation epoch and the intermediate epoch dominated by dark matter, in contrast to the $\varLambda$CDM model.

Fig. 1

Phase space diagrams for MHR-IDE models: $\varGamma _{11}$ (left), $\varGamma _{12}$ (center), and $\varGamma _{13}$ (right). We have projected the diagrams in the plane $\varOmega _b=0$, and we use $\alpha =0.95$, $\beta =0.45$, and $\delta =\gamma =0.01$

Table 3 Description of the critical points $\{\varOmega _{r}^*,\varOmega _{c}^*,\varOmega _{x}^*\}$ for scenario $\varGamma _{2}$. The intervals in (28) are considered as global constraints. To improve the visualization of the expressions, we use the constants $D_1-D_7$ defined in (26)

3.2 Scenario $\varGamma _2$

For model $\varGamma _2$, in addition to Eq. (23), we have the following set of equations:

$$\begin{aligned}{} & {} \varOmega _c'=\frac{2 \alpha (\gamma +\varOmega _c)+\beta (\gamma (3 \varOmega _c-\varOmega _r+3 \varOmega _x-3) -3 (\delta \varOmega _c+\delta \varOmega _x+\varOmega _c))-2 \varOmega _x (\gamma +\varOmega _c)}{3 \beta }, \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \varOmega _x'=\frac{2 \alpha (\varOmega _x-1-\gamma )+\beta (3 (\delta -\gamma ) (\varOmega _c+\varOmega _x)+(\gamma +1) (\varOmega _r+3) -3 \varOmega _x)+2 \varOmega _x (\gamma -\varOmega _x+1)}{3 \beta }. \end{aligned}$$

(32)

In the range (28), the constants $D_1$ and $D_2$ at critical points $Q_3$ and $Q_4$ in Table 3 are real numbers under the conditions $\alpha +\sqrt{6\beta }\le 1+\frac{3\beta }{2}$ or $\alpha +\sqrt{6\beta }>1+\frac{3\beta }{2}$ and $\delta -D_3>0$. In addition, the existence condition for $Q_1$ is $\delta \ne \frac{1}{3}$. In the following, we describe critical points $Q_1-Q_{8}$ in Table 3, corresponding to models $\varGamma _{22}$ and $\varGamma _{23}$. Note that model $\varGamma _{21}$ is equivalent to model $\varGamma _{13}$.

For model $\varGamma _{22}$, we have four critical points, $Q_1-Q_4$ in Table 3. Point $Q_1$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $Q_1$ require $0<C_1<1-3\delta$ and $0<\delta <\frac{1}{3}$. Point $Q_2$ represents a combination of the dark sector and baryons, and at this critical point we always have one negative energy density, and $\omega _{\text {eff}}=0$. For $C_4\approx 0$, this point corresponds to the domination of baryons. Points $Q_3$ and $Q_4$ are combinations of the dark sector. Point $Q_4$ has $\omega _{\text {eff}}<-\frac{1}{3}$ in the range (28) for $\delta >D_3$. In the limit $\delta \ll 1$, we have $\omega _{\text {eff}}\approx \frac{\delta }{1-C_4}$ at $Q_3$ and $\varOmega _x^*\approx 1$ at $Q_4$. Positive energy densities at $Q_1$ and $Q_{3}$ are obtained with no additional conditions, but $Q_{4}$ has $\varOmega _c^*<0$ in the range where $Q_1$ and $Q_3$ have positive energy densities.

For model $\varGamma _{23}$, we have four critical points, $Q_5-Q_{8}$ in Table 3. Point $Q_5$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $Q_5$ require $0<C_1<1+\gamma$ and $-1<\gamma <0$. Point $Q_{6}$ represents a combination of the dark sector and baryons, and at this critical point we always have one negative energy density, and $\omega _{\text {eff}}=0$. For $C_4\approx 0$, this point corresponds to the domination of baryons. Points $Q_{7}$ and $Q_{8}$ are combinations of the dark sector. Point $Q_{8}$ has $\omega _{\text {eff}}<-1$ in the range (28) with $-1<\gamma <0$ and $\alpha <1+\gamma$. In the limit $\gamma \ll 1$, and for $C_4<1+\gamma$, we have $\omega _{\text {eff}}\approx 0$ at $Q_{7}$ and $\varOmega _x^*\approx 1$ at $Q_{8}$. Positive energy densities at $Q_5$, $Q_7$, and $Q_8$ require the additional condition $0<C_4<1$.

In Fig. 2 we show the phase space diagrams corresponding to models $\varGamma _{22}$ and $\varGamma _{23}$, with fixed parameter values. We observe consistent behavior in the phase diagrams for these models: a radiation-dominated source, a saddle point corresponding to a dark sector combination, and a final attractor corresponding to the domination of the holographic dark energy.

Fig. 2

Phase space diagrams for MHR-IDE models: $\varGamma _{22}$ (left) and $\varGamma _{23}$ (right). We have projected the diagrams in the plane $\varOmega _b=0$, and we use $\alpha =0.95$, $\beta =0.45$, $\delta =0.05$, and $\gamma =-0.05$

3.3 Scenario $\varGamma _3$

For model $\varGamma _3$ we have, in addition to Eq. (23), the autonomous equations

$$\begin{aligned} \varOmega _c'= & {} \frac{2 \alpha \varOmega _c-3 \beta (\gamma +\varOmega _c)-2 \varOmega _c \varOmega _x}{3 \beta }, \end{aligned}$$

(33)

$$\begin{aligned} \varOmega _x'= & {} \frac{2 \alpha (\varOmega _x-1)+\beta (3 \gamma +\varOmega _r-3 \varOmega _x+3)-2 \varOmega _x (\varOmega _x-1)}{3 \beta }. \nonumber \\ \end{aligned}$$

(34)

In the ranges (28), the constants $D_1$ and $D_2$ at critical points $R_2$ and $R_3$ in Table 4 are real numbers under the conditions $C_4\le 1-\sqrt{6\beta }$ or $C_4>1-\sqrt{6\beta }$ and $\gamma >D_3$. In the following, we describe the three critical points corresponding to model $\varGamma _3$, $R_1-R_3$ in Table 4.

Point $R_1$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $R_1$ require $0<C_1<1-3\gamma$ and $0<\gamma <\frac{1}{3}$. Points $R_2$ and $R_3$ are combinations of the dark sector. Point $R_3$ has $\omega _{\text {eff}}<-\frac{1}{3}$, considering the existence conditions for $D_1$ and $D_2$. In the limit $\gamma \ll 1$ we have $\omega _{\text {eff}}\approx \frac{\gamma }{1-C_4}$ at $R_2$ and $\varOmega _x^*\approx 1$ at $R_3$, for $C_4<1$. Positive energy densities at $R_1$ and $R_2$ are obtained with no additional conditions, but $R_3$ has $\varOmega _c^*<0$ in the range where $R_1$ and $R_2$ have positive energy densities.

3.4 Scenario $\varGamma _4$

For model $\varGamma _4$ we have, in addition to Eq. (23), the autonomous equations

$$\begin{aligned} \varOmega _c'= & {} \frac{2 \alpha (\gamma +\varOmega _c)-2 \varOmega _x (\gamma +\varOmega _c)-3 \beta \varOmega _c}{3 \beta }, \end{aligned}$$

(35)

$$\begin{aligned} \varOmega _x'= & {} \frac{-2 \alpha (\gamma -\varOmega _x+1)+2 \varOmega _x (\gamma -\varOmega _x+1)+\beta (\varOmega _r-3 \varOmega _x+3)}{3 \beta }. \nonumber \\ \end{aligned}$$

(36)

In the ranges (28) there are no additional existence conditions for critical points $R_4-R_6$. In the following, we describe critical points $R_4-R_6$ in Table 4.

Point $R_4$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $R_4$ require $0<C_1<1+4\gamma$ and $-\frac{1}{4}<\gamma <0$. Points $R_5$ and $R_6$ are combinations of the dark sector. Point $R_6$ has $\omega _{\text {eff}}<-1$ in the range (28). In the limit $\gamma \ll 1$ we have $\omega _{\text {eff}}\approx \frac{\gamma }{C_4-1}$ at $R_5$ and $\varOmega _x^*\approx 1$ at $R_6$, for $C_4<1$. Positive energy densities at $R_4$, $R_5$, and $R_6$ require the additional condition $C_4>-\alpha \gamma$.

3.5 Scenario $\varGamma _5$

For model $\varGamma _5$, we have, in addition to Eq. (23), the autonomous equations

$$\begin{aligned} \varOmega _c'= & {} \frac{\alpha (2 \varOmega _c-3 \gamma )+3 \gamma (\beta +\varOmega _x)-\varOmega _c (3 \beta +2 \varOmega _x)}{3 \beta }, \end{aligned}$$

(37)

$$\begin{aligned} \varOmega _x'= & {} \frac{\alpha (3 \gamma +2 \varOmega _x-2)+\beta (-3 \gamma +\varOmega _r-3 \varOmega _x+3)+\varOmega _x (-3 \gamma -2 \varOmega _x+2)}{3 \beta }. \end{aligned}$$

(38)

Note that the constants $E_2$ and $E_3$ are real numbers in the range (28). In the following, we describe the critical points $R_7-R_9$ in Table 4.

Point $R_7$ is a combination of radiation and the dark components, and it corresponds to an effective fluid of radiation type. Positive energy densities at $R_7$ require $0<C_1<1+3\gamma$ and $\gamma >0$. Points $R_8$ and $R_9$ are combinations of the dark sector. Point $R_9$ has $\omega _{\text {eff}}<-\frac{1}{3}$ in the range (28). In the limit $\gamma \ll 1$ we have $\omega _{\text {eff}}\approx \frac{\gamma }{2(1-C_4)}$ at $R_8$ and $\varOmega _x^*\approx 1$ at $R_9$, for $C_4<1$. Positive energy densities at $R_7$, $R_8$, and $R_9$ require the additional condition $C_4>\frac{3\gamma }{2}C_1$.

Table 4 Description of the critical points $\{\varOmega _{r}^*,\varOmega _{c}^*,\varOmega _{x}^*\}$ for scenarios $\varGamma _{3}-\varGamma _5$. The intervals in (28) are considered as global constraints. To improve the visualization of the expressions, we use the constants $E_1-E_3$ defined in (27)

Fig. 3

Phase space diagrams for MHR-IDE models: $\varGamma _{3}$ (left), $\varGamma _{4}$ (center), and $\varGamma _{5}$ (right). We have projected the diagrams in the plane $\varOmega _b=0$, and we use $\alpha =0.95$, $\beta =0.45$, and $\delta =\gamma =0.01$

In Fig. 3 we show the phase space diagrams corresponding to models $\varGamma _{3}$, $\varGamma _{4}$, and $\varGamma _{5}$, with fixed parameter values. We observe consistent behavior in the phase diagrams for the three models: a radiation-dominated source, a saddle point corresponding to a dark sector combination, and a final attractor corresponding to the domination of the holographic dark energy. We note the same distinctive features observed in the MHR-IDE models $\varGamma _1$ and $\varGamma _2$, namely a contribution of holographic dark energy during both the radiation epoch and the intermediate epoch dominated by dark matter, in contrast to the $\varLambda$CDM model.

Summarizing the dynamical system results for the MHR-IDE models $\varGamma _1-\varGamma _5$, we can state that the radiation epoch is always modified for these models, but by considering $\alpha -2\beta \ll 1$ and $\gamma \ll 1$ and/or $\delta \ll 1$, these modifications are ameliorated. A critical point corresponding to an attractor is present in each model, and it represents an accelerated phase. There is also a saddle point corresponding to a combination of the dark sector. Finally, the positiveness of the energy densities at the critical points is not guaranteed; it exists only in determined ranges of the parameters.

4 Observational analysis and model selection

To constrain the MHR-IDE models, we use several data sources, including type Ia supernovae from the Pantheon sample [74], cosmic chronometers [75], baryon acoustic oscillations [76,77,78], and the position of the angular scale of the sound horizon at last scattering [79]. We briefly present each dataset below.

4.1 Supernovae type Ia

We use the Pantheon sample, containing a set of 1048 spectroscopically confirmed SNe Ia in the redshift range $0.01<z<2.3$ [74]. This dataset contains measurements of the apparent magnitude $m_b$, related to the distance modulus by $\mu =m_b - M_b$, where the absolute magnitude $M_b$ is a nuisance parameter. In terms of the luminosity distance, $d_L$, the distance modulus is defined as

$$\begin{aligned} \mu = 5 \log {d_L} + 25, \end{aligned}$$

(39)

where

$$\begin{aligned} d_L = (1+z)\int _{0}^{z} \frac{\textrm{d}z'}{H(z')}. \end{aligned}$$

(40)

is measured in Mpc.

4.2 Cosmic chronometers

This dataset corresponds to 24 measurements of the expansion rate at different redshift values. These cosmic chronometers are obtained from the differential age method [80], from which we only consider data at redshift $z<1.2$ (see Refs. [75, 81]). Note that these data constitute the only method providing cosmology-independent, direct measurements of the expansion history of the universe [81].

4.3 Baryon acoustic oscillations (BAO)

We use three isotropic BAO measurements from the 6dF Galaxy Survey (6dFGS) [76], Main Galaxy Sample (MGS) [77], and Extended Baryon Oscillation Spectroscopic Survey (eBOSS) [78], and three data points from the BOSS DR12 [82], corresponding to BAO anisotropic measurements. All of these BAO measurements are given in terms of $D_V(z)/r_s$, $D_M(z)/r_s$, or $D_H(z)/r_s$, where $D_V$ is a combination of the line-of-sight and transverse distance scales defined in Ref. [83], $D_M(z)$ is the comoving angular diameter distance, which is related to the physical angular diameter distance by $D_M(z) = (1+z)D_A(z)$, and $D_H = c/H(z)$ is the Hubble distance. We define $D_V(z)$ and $D_A(z)$ as

$$\begin{aligned} D_V(z)= & {} \left( D_M^2(z)\frac{z}{H(z)} \right) ^{1/3}, \end{aligned}$$

(41)

$$\begin{aligned} D_A(z)= & {} \frac{1}{(1+z)}\int _{0}^z \frac{\textrm{d}z'}{H(z')}. \end{aligned}$$

(42)

The standard ruler length $r_s$ is the comoving size of the sound horizon at the drag epoch, defined as

$$\begin{aligned} r_s = \int _{z_d}^\infty \frac{c_s\, \textrm{d}z}{H(z)}, \end{aligned}$$

(43)

where $c_s = \frac{1}{\sqrt{3(1+\mathcal {R})}}$ is the sound speed in the photon-baryon fluid, $\displaystyle \mathcal {R} = \frac{3\varOmega _{b0}}{4\varOmega _{\gamma 0}(1+z)}$ [84], and $z_d$ the redshift at the drag epoch.

4.4 Cosmic microwave background

We consider one data point corresponding to the position of the angular scale of the sound horizon at last scattering, as background data coming from the early universe physics,

$$\begin{aligned} \textit{l}_a = \frac{\pi D_M(z_*)}{r_s(z_*)}, \end{aligned}$$

(44)

where $z_* = 1089.80$, in accord with Planck’s 2018 [79]. We compare our calculated value with the one reported by the Planck collaboration in 2015, $l_a = 301.63 \pm 0.15$ [85].

4.5 Bayesian model selection

The evaluation of a model’s performance in light of the data is based on the Bayesian evidence [86]. This is the normalization integral on the right-hand side of Bayes’ theorem, which is related to the posterior probability P for a set of parameters $\varTheta$, given the data $\mathcal {D}$, described by a model $\mathcal {M}$:

$$\begin{aligned} P(\varTheta \vert \mathcal {D},\mathcal {M}) = \frac{\mathcal {L}(\mathcal {D}\vert \varTheta ,\mathcal {M})\mathcal {P}(\varTheta \vert \mathcal {M})}{\mathcal {E}(\mathcal {D}\vert \mathcal {M})}, \end{aligned}$$

(45)

where $\mathcal {L},\mathcal {P}$, and $\mathcal {E}$ are the likelihood, prior distribution, and evidence, respectively. We can write the evidence for a continuous parameter space $\varOmega _{\mathcal {M}}$ as

$$\begin{aligned} \mathcal {E}(\mathcal {D} \vert \mathcal {M}) = \int _{\varOmega _{\mathcal {M}}} \mathcal {L}(\mathcal {D}\vert \varTheta ,\mathcal {M})\mathcal {P}(\varTheta \vert \mathcal {M}). \end{aligned}$$

(46)

In order to compare the performance of different models given a dataset, we use the Bayes’ factor defined as the ratio of the evidence of models $\mathcal {M}_0$ and $\mathcal {M}_1$ as

$$\begin{aligned} B_{01} = \frac{\mathcal {E}(\mathcal {D} \vert \mathcal {M}_0)}{\mathcal {E}(\mathcal {D} \vert \mathcal {M}_1)}, \end{aligned}$$

(47)

which we use to interpret the strength of the evidence according to Jeffreys’ scale given in Table 5 of Ref. [86]. This is an empirically calibrated scale, representing weak, moderate, or strong evidence. In our work, we consider the $\varLambda \text {CDM}$ model as reference model ($\mathcal {M}_{1}$); therefore, if $\ln B_{01}<0$, we will have evidence in favor of $\varLambda \text {CDM}$, whereas if $\ln B_{01}>0$, the evidence will favor the MHR-IDE model.

Table 5 The Jeffreys’ scale, empirical measure for interpreting the evidence in comparing two models $\mathcal {M}_0$ and $\mathcal {M}_1$ as presented in Ref. [86]. The probability column refers to the posterior probability of the favored model, assuming both models are equally likely and fill the entire model space

On the other hand, for a set of measurements contained in a vector $\mathcal {S}$, we have the $\chi ^2$ function defined as

$$\begin{aligned} \chi _{\mathcal {S}}^2 = [\mathcal {S}^{\text {o}bs}-\mathcal {S}^{\text {t}h}]^T \mathcal {C}^{-1}[\mathcal {S}^{\text {o}bs}-\mathcal {S}^{\text {t}h}] \end{aligned}$$

(48)

where $\mathcal {S}^{\text {obs}}$ represents the measured value, $\mathcal {S}^{\text {th}}$ is the theoretical value computed assuming a model with parameters $\varTheta$, T is the transposed vector, and $\mathcal {C}$ corresponds to the covariance matrix of the measurements contained in the vector $\mathcal {S}^{\text {obs}}$. In our case, the values in $\mathcal {S}^{\text {th} }$ represent the functions $\mu (z)$, H(z), and $l_a(z_*)$ for SNe-Ia, CC, and CMB data, respectively, and $D_V(z)/r_s$, $D_M(z)/r_s$, or $D_H(z)/r_s$ for BAO data.

The analyses for all samples were performed assuming a multivariate Gaussian likelihood of the form

$$\begin{aligned} \mathcal {L}(\mathcal {D}|\varTheta ) = \exp \left[ -\frac{\chi ^2(\mathcal {D}|\varTheta )}{2}\right] . \end{aligned}$$

(49)

To find the best-fit model parameters, we perform a joint analysis including all the data, and we use the overall $\chi ^2$ function defined as

$$\begin{aligned} \chi ^2 = \chi ^2_{\text {SNe-Ia}} + \chi ^2_{\text {CC}} + \chi ^2_{\text {BAO}} + \chi ^2_{\text {CMB}}. \end{aligned}$$

(50)

To calculate the evidence and estimate the cosmological parameters, we use the MULTINEST algorithm [87, 88], setting a tolerance of 0.01 as convergence criterion and working with a set of 1000 live points to improve the precision in the estimation of the evidence.

5 Analysis and results

We perform a Bayesian comparison of models $\varGamma _{11}$, $\varGamma _{12}$, $\varGamma _{13}$, and $\varGamma _{14}$ representing the variations of $\varGamma _1$ for the cases where $\delta =0$, $\gamma =0$, $\delta =\gamma$, and $\delta \ne \gamma$, respectively. Similarly, $\varGamma _{21}$, $\varGamma _{22}$, $\varGamma _{23}$, and $\varGamma _{24}$ are the variations of $\varGamma _2$ for the same cases. In this work, we used the priors shown in Table 6 and the combination of all the background data displayed in Sect. 4, where we have considered two different approaches in using the data with the purpose of clarifying whether there is a noticeable impact on our results when considering different priors for the Hubble parameter h. Considering a Gaussian prior is equivalent to including the local measure of $H_0$ [7] in the dataset. Note that for the parameters $\gamma$ and $\delta$, we have explored the analytical solutions of our models inside the parameter space (28), where we found an interval between $-$ 0.1 and 0.1 for these parameters, in which our functions are well defined.

We consider a joint analysis with the dataset Pantheon + CC + BAO + CMB, as described in Sect. 4, which is studied in two different scenarios distinguished by the assigned prior for the h parameter (see Tables 7 and 8). Our main results are summarized in Tables 7, 8, and 9.

Table 6 Priors on the free parameters of the MHR-IDE models. For the Gaussian prior we inform ($\mu$, $\sigma ^2$) and for the uniform prior, (a, b) represents $a< x < b$

In our analysis, we fix the following parameters, under the assumption that the variation in the radiation and baryonic components is not significant:

$$\begin{aligned} \varOmega _{r0}=\left( 1+\frac{7}{8}\left( \frac{4}{11}\right) ^{\frac{4}{3}}N_{\text {eff}}\right) \varOmega _{\gamma 0},\quad \varOmega _{b0}=0.02235,\nonumber \\ \end{aligned}$$

(51)

where $N_{\text {eff}} = 3.046$ [90], $\varOmega _{\gamma 0} = 2.469 \times 10^{-5}$, and $\varOmega _{b0}$ [91] correspond to the effective number of neutrinos, the photon density parameter, and the baryon density parameter, respectively.

Tables 7 and 8 present the best-fit parameters with their associated $1\sigma$ error for the MHR-IDE models studied in this work, for a Gaussian and a uniform prior for the h parameter, respectively. Table 9 shows the logarithm of the Bayesian evidence and the logarithm of the Bayes factor which is interpreted in terms of the Jeffreys scale (5). As a comparison, in Tables 7, 8, and 9, we also show the results for HRDE and MHRDE models.

We note that the strength of the Bayesian evidence for the Gaussian prior is weak/moderate for the MHR-IDE models in Table 1, whereas when considering the uniform prior, the evidence is strong for all models in Table 1, i.e., when we consider a uniform prior for the h parameter, the evidence better supports the $\varLambda$CDM model. However, independently of the prior used, the evidence favors $\varLambda$CDM for all the MHR-IDE studied.

Figures 5, 6, 7, 8, and 9 in the appendix show the contour plots for models $\varGamma _1-\varGamma _5$ with $1\sigma$ and $2\sigma$ confidence levels, where we have considered the joint analysis for each prior distribution of parameter h, the Gaussian prior in green and the uniform prior in blue. In general, in all the models but $\varGamma _{23}$, the parameters associated with interaction and holography coincide inside the $1\sigma$ region when the prior for h is changed. On the other hand, the parameters $\alpha$ and $\beta$ in general (all scenarios but $\varGamma _{23}$, $\varGamma _{24}$, and $\varGamma _5$) exhibit some degree of correlation approaching $\alpha \approx 2\beta$. Note that the analysis with a uniform prior on h selects as posterior for h a value compatible with early-universe measurements (and $\varLambda$CDM), while the Gaussian prior selects as h posterior a value more compatible with local measurements. On the other hand, we observe that the amount of dark matter decreases for all the MHR-IDE models when compared with $\varLambda$CDM. Furthermore, except for $\varGamma _{11}$, all the models in Tables 7 and 8 are compatible with null interaction inside the $1\sigma$ region.

In Fig. 4 we have used the best-fit parameters in Table 7 to show the evolution of the density parameters, $\omega _{\text {eff}}$ and $\varGamma /H^2$, for models $\varGamma _{14}$ (left) and $\varGamma _{24}$ (right). We observe that the dark energy contribution significantly affects the radiation- and matter-dominated epochs in both models. Furthermore, in the interaction in model $\varGamma _{14}$, a change of sign occurs around $z\approx 10^4$. Although models $\varGamma _{14}$ and $\varGamma _{24}$ were not studied with the dynamical system analysis in Sect. 3, in the evolution plots obtained from the analytical solutions in Sect. 2 and from the best-fit parameters obtained in Sect. 4, we observe the same behavior as in Sect. 3, namely a dark energy contribution to radiation- and matter-dominated epochs, which results in a significant alteration with respect to the $\varLambda$CDM model.

Fig. 4

Evolution of the density parameters $\varOmega _x$ (black), $\varOmega _c$ (blue), $\varOmega _b$ (brown), $\varOmega _r$ (orange), $\omega _{\text {eff}}$ (dot-dashed purple), and $\varGamma /H^2$ (dashed red), for models $\varGamma _{14}$ (left) and $\varGamma _{24}$ (right)

In Refs. [30, 34, 39, 48, 65, 66, 92, 93], the performance of holographic dark energy models in fitting cosmological data was assessed relative to the $\varLambda$CDM model, where several criteria were used, including $\chi ^2/dof$, Akaike information criterion (AIC) and Bayesian information criterion (BIC) [94], and Bayesian evidence [73]. The authors of Ref. [30] studied the HRDE model, finding evidence against its use compared with $\varLambda$CDM. In addition, in Ref. [92], the HRDE and MHRDE models were studied using expansion and growth data of structures, through the BIC. The authors found strong indications against the HRDE models when compared with $\varLambda$CDM. To our knowledge, no evidence in favor of any interacting holographic dark energy scenarios has been found to date.

Finally, we note that for a Gaussian prior on h, the MHRDE model presents inconclusive evidence. In this sense, the authors of Ref. [73] previously found weak evidence in favor of $\varLambda$CDM when they compared the MHRDE for the same dataset; however, they used different means for the Gaussian prior on h.

Table 7 Best-fit parameters for the joint analysis Pantheon + CC + BAO + CMB. These results consider the priors in Table 6 with the h prior Gaussian. The corresponding $\chi ^2_{\min }$ is shown for each model

Table 8 Best-fit parameters for the joint analysis Pantheon + CC + BAO + CMB. These results consider the priors in Table 6 with the h prior uniform. The corresponding $\chi ^2_{\min }$ is shown for each model

Table 9 Bayes’ evidence (ln $\mathcal {E}$) and interpretation for the joint analysis Pantheon + CC + BAO + CMB in the MHR-IDE models presented in Table (1). Note that ln$B_i=$ln$\mathcal {E}_i-$ln$\mathcal {E}_{\varLambda \text {CDM}}<1$ favors the $\varLambda$CDM model

6 Final remarks

Given that no previous MHR-IDE models have been successful in fitting the data better than $\varLambda$CDM, we were motivated to conduct a more exhaustive study to assess a wider range of interactions in the holographic context. In this work, we studied five new types of modified holographic Ricci models including new interactions in the dark sector (MHR-IDE) not previously considered.

In Sect. 2 we presented the MHR-IDE models and found analytical solutions. In Sect. 3 we performed the dynamical system analysis for models $\varGamma _{11}$, $\varGamma _{12}$, $\varGamma _{13}$, $\varGamma _{22}$, $\varGamma _{23}$, $\varGamma _{3}$, $\varGamma _{4}$, and $\varGamma _{5}$. This analysis indicated that the radiation epoch is always modified for these models, but by considering $\alpha -2\beta \ll 1$ and $\gamma \ll 1$ and/or $\delta \ll 1$ for interacting parameters, the modifications were ameliorated. A critical point corresponding to an attractor is present in each model, and it represents an accelerated phase. The evolution of density parameters and effective state parameter for models $\varGamma _{14}$ and $\varGamma _{24}$ in Fig. 4 showed the same behavior for these models as that observed in the dynamical system analysis in Sect. 3.

The studied models were fitted using Bayesian inference techniques for a joint analysis of the dataset Pantheon + CC + BAO + CMB. We investigated whether these new MHR-IDE models were competitive against the $\varLambda$CDM model, in the framework of Bayesian comparison.

As an overall result, we found that the Bayesian comparison favored the $\varLambda$CDM model, irrespective of the interaction considered and independently of the prior assigned to the parameter h (Gaussian or uniform).

Note that in our analysis, the MHRDE model presents inconclusive evidence in Table 9 when a Gaussian prior in h is used, in contrast to the strong evidence against this model when a uniform prior is considered. This change in the evidence may be artificial and due to the tension between the CMB data used and the local measurement of $H_0$ we are using. Certainly, further analysis is needed to thoroughly assess this observation.

Finally, we conclude that the new MHR-IDE models studied do not contribute to improving the Bayesian evidence or parameter estimation with respect to the existing holographic dark energy models [73]. Most of the interacting scenarios revised in this work are compatible with a null interaction in light of the Bayesian inference techniques employed and present certain correlations in the holographic parameters ($\alpha \approx 2\beta$), accounting for the fact that the Bayesian analysis seems to indicate that MHR-IDE scenarios are compatible with the MHRDE model.

Appendix A: Contour plots

See Figs. 5, 6, 7, 8 and 9.

Fig. 5

Contour plots with $1\sigma$ and $2\sigma$ regions are shown for $\varGamma _{11}$ in the left panel and $\varGamma _{12}$ in the right panel. We considered the full joint analysis with Pantheon + CC + BAO + CMB. We show the contours for a uniform prior (UP) and a Gaussian prior (GP)

Fig. 6

Contour plots with $1\sigma$ and $2\sigma$ regions are shown for $\varGamma _{13}$ in the left panel and $\varGamma _{14}$ in the right panel. We considered the full joint analysis with Pantheon + CC + BAO + CMB. We show the contours for a uniform prior (UP) and a Gaussian prior (GP)

Fig. 7

Contour plots with $1\sigma$ and $2\sigma$ regions are shown for $\varGamma _{22}$ in the left panel and $\varGamma _{23}$ in the right panel. We considered the full joint analysis with Pantheon + CC + BAO + CMB. We show the contours for a uniform prior (UP) and a Gaussian prior (GP)

Fig. 8

Contour plots with $1\sigma$ and $2\sigma$ regions are shown for $\varGamma _{24}$ in the left panel and $\varGamma _{3}$ in the right panel. We considered the full joint analysis with Pantheon + CC + BAO + CMB. We show the contours for a uniform prior (UP) and a Gaussian prior (GP)

Fig. 9

Contour plots with $1\sigma$ and $2\sigma$ regions are shown for $\varGamma _{4}$ in the left panel and $\varGamma _{5}$ in the right panel. We considered the full joint analysis with Pantheon + CC + BAO + CMB. We show the contours for a uniform prior (UP) and a Gaussian prior (GP)