Anisotropic Bianchi type-V generalized ghost pilgrim dark energy cosmological model in self-creation gravity

1 Introduction

Recent astronomical observations (Mishra and Dua, 2021; Spergel et al., 2003; Tegmark et al., 2004) have confirmed that the cosmos is expanding and accelerating. These findings indicate that dark matter (DM) and dark energy (DE) are the two dark components that dominate the universe. Whereas DE, an exotic energy with tremendous negative pressure, is utilized to explain the universe’s cosmic acceleration, DM, a pressureless form of matter, is primarily used to explain galaxy curves and the universe’s structure building. Many definitions of dark energy have been proposed, and academics are actively investigating its nature. The most obvious candidate for DE is the cosmological constant, with the equation of state (EoS) parameter ω = p ρ = − 1, where ρ is DE’s energy density and p is the pressure. However, it suffers from cosmic coincidence and fine-tuning problems (Weinberg, 1989). To address the issue of DE, cosmologists have developed a variety of theories, including quintessence (Steinhardt et al., 1999; Caldwell, 2002), phantom (Caldwell, 2002; Noijiri et al., 2005), tachyon (Padmanabhan, 2002), dilation (Gasperini et al., 2001), and others.

A type of DE, known as Veneziano ghost DE, has been proposed to explain the current fast stage of the universe (Urban and Zhitnitsky, 2010; Ohta, 2011; Cai et al., 2011). When H is used in place of the Hubble parameter, the energy density of the vacuum ghost field is proportional to three Quantum Chromodynamics mass scale (QCDH) (Zhitnitsky, 2010; Holdom, 2011). Quantum Chromodynamics mass scale (QCDH) is referred to as the QCD (quantum chromodynamics) mass scale. This ghost dark energy (GDE) is of importance to researchers because its energy density DE relies linearly on the Hubble parameter H, as in DE = H, and is associated with the QCD mass scale. The strong interaction observed in nature is described by QCD. H + O H 2 is the general vacuum energy of the Veneziano ghost field in QCD (Zhitnitsky, 2012). In the history of the early universe, or the early DE, H 2 plays a significant role (Maggiore et al., 2011). Compared to the regular GDE, also known as generalized ghost dark energy (GGDE), incorporating the H² term can improve its agreement with observational evidence (Cai et al., 2012). The generalized model’s energy density can be calculated as ρ D E = τ H + η H 2, where η is a constant. Based on the idea that the type of DE’s strong repulsive force can stop the creation of black holes (BH), Wei (2012) presented a new dark energy model dubbed pilgrim DE (PDE). In the context of PDE, GGDE has been rewritten as ρ D E = τ H + η H 2 u, where u is a PDE parameter (Sharif and Jawad, 2014).

Scientists have established several models to better understand dark energy. These models are used to explain the expansion of the universe and its potential causes. VijayaSanthi et al. (2017) examined one of these models, known as generalized ghost pilgrim dark energy (GGPDE) in the Bianchi type I universe. Ramprasad et al. (2025) employed Bianchi type-VI₀ spacetime to investigate GGPDE in Saez–Ballester theory. In addition, Garg et al. (2021) used Saez–Ballester gravity theory to investigate cosmological models of GGPDE. Additionally, Prasanthi and Davuluri (2024) investigated Brans–Dicke theory’s GGPDE. Rao and DivyaPrasanthi (2016) have also investigated the Brans–Dicke theory of gravity’s Bianchi type-VI₀ generalized ghost pilgrim dark energy model.

Furthermore, the phantomic behavior of GGPDE models in self-creation theory (SCT) was investigated by Katore and Kapse (2022). In f(Q) gravity, generalized ghost dark energy was investigated by Sharif and Madiha (2024). A thermodynamical image of the interacting generalized ghost dark energy was presented by Fayaz (2016), who investigated the anisotropic universe. The Kantowski–Sachs modified holographic Ricci dark energy model in the Saez–Ballester theory of gravity was investigated by BhaskaraRao et al. (2018). The modified holographic Ricci dark energy model in self-creation cosmology was examined by BhaskaraRao et al. (2016). The Bianchi type with the bulk viscous string cosmological model in the self-creation theory of gravitation was investigated by Padmanabhan (2002) and BhaskaraRao et al. (2015).

In the context of a non-flat FRW universe, Jawad (2014) investigated an alternative model known as GGPDE. Garg et al. (2021) investigated GGPDE using Saez–Ballester gravity theory. Mishra and Dua (2021) estimated the FLRW universe in the Brans–Dicke gravity theory using a simple parameterization of the average scale factor a(t). The Bianchi-type model, which has an anisotropic background, provides one of the simplest explanations for the early phases of the universe’s evolution. Bianchi spacetimes are well suited for describing spatially homogeneous and anisotropic cosmologies because their field equations are relatively simple and straightforward to solve.

We examine the GGPDE model in Barber’s second self-creation theory of gravity within the anisotropic Bianchi type-V spacetime, motivated by the aforementioned factors and the recent work of other authors (Bali and Kumari, 2017; Luciano, 2023; Vinutha and Vasavi, 2022; Wei and Cai, 2009; Myung, 2007; BhaskaraRao et al., 2026; Amendola and Tsujikawa, 2010; Bamba et al., 2012; Srivani Madhu et al., 2026; Giarè et al., 2024; Mukhopadhyay et al., 2024). Physical considerations led to the selection of Bianchi type-V geometry, one of the most straightforward non-trivial anisotropic geometries that generalize the flat FRW universe by incorporating spatial anisotropy while maintaining spatial homogeneity. Observational anomalies in the CMB quadruple and large-scale structure have been suggested as potential indicators of a somewhat anisotropic early universe, although Bianchi cosmologies offer a well-studied and principled framework for evaluating dark energy models beyond the isotropic approximation. Because it admits negative spatial curvature and a broader class of exact solutions than the more straightforward type-I geometry, the Bianchi type-V geometry is highly appealing. Because Barber’s second SCT naturally incorporates a dynamically changing scalar field that modifies the effective gravitational coupling, it offers a self-consistent mechanism for time-varying dark energy without requiring a bare cosmological constant, making it preferable to general relativity. SCT is consistent with observations because it adheres to the weak equivalency principle and reduces to GR in the limit of a large coupling constant. The remainder of the paper is organized as follows: the metric and field equations are discussed in Section 2. Section 3 provides exact solutions to the field equations. Kinematical and physical factors are discussed in Sections 4, 5, 6 deal with the non-interacting and interacting models, respectively. Sections 7, 8 deal with stability analysis and energy conditions, respectively. Section 9 provides conclusion.

2 Metric and field equations

The spatially homogeneous and anisotropic Bianchi type-V spacetime is given aswhere K , L , M are the metric potentials, which are functions of cosmic time t.

ds 2 = ‐ dt 2 + k 2 dx 2 + e 2 x L 2 dy 2 + M 2 dz 2 , (1)

The gravitation field equations of Barber’s second SCT are given aswhere G j i is the Einstein tensor and T j i and T ¯ j i are the energy momentum tensors of dark matter and GGPDE, respectively.

G j i = R j i ‐ 1 2 δ j i R = ‐ 8 π ϕ T j i + T ¯ j i , (2)

The scalar field ϕ satisfies

ϕ ; k , k = 8 π 3 μ T , (3)

where μ is the coupling constant to be determined from experiments. The measurement of deflection of light restricts the value of the coupling constant to μ ≤ 0.1.

The energy momentum tensor of DM is given as

T j i = d i a g 0 , 0 , 0 , ρ m . (4)

The energy momentum tensor of GGPDE is given as

T ¯ j i = d i a g − p D E , − p D E , − p D E , ρ D E . (5)

The field Equation 2 for the metric given in Equation 1 using Equations 4, 5 is derived as where overhead dots represent differentiation with respect to t.

− 1 K 2 + L ¨ L + M ¨ M + L ˙ M ˙ L M = − 8 ϕ − 1 p D E , (6)

− 1 K 2 + K ¨ K + K ˙ M ˙ K M + M ¨ M = − 8 ϕ − 1 p D E , (7)

− 1 K 2 + K ¨ K + K ˙ L ˙ K L + L ¨ L = − 8 ϕ − 1 p D E , (8)

− 3 K 2 + K ˙ L ˙ K L + K ˙ M ˙ K M + L ˙ M ˙ L M = − 8 π ϕ − 1 ρ m + ρ D E , (9)

− 2 K ˙ K 3 + L ˙ L K 2 + M ˙ M K 2 = 0 , (10)

2 K ˙ K − L ˙ L − M ˙ M = 0 , (11)

The physical quantities of observational interests in cosmology are the volume V, expansion scalar θ, the mean anisotropy parameter A h, and shear scalar σ 2, which are defined as

V = a 3 t = K L M , (12)

θ = K ˙ K + L ˙ L + M ˙ M , (13)

σ 2 = 1 3 K ˙ K 2 + L ˙ L 2 + M ˙ M 2 − K ˙ L ˙ K L − K ˙ M ˙ K M − L ˙ M ˙ L M , (14)

H = θ 3 , (15)

A h = 1 3 H 1 − H H 1 2 + H 2 − H H 2 2 + H 2 − H H 2 2 . (16)

From Equation 11, we obtain

K 2 = L M . (17)

Using Equations 3, 4, 5, we obtain

ϕ ¨ + ϕ ˙ K ˙ K + L ˙ L + M ˙ M = 8 3 π μ ρ m − 3 p D E + ρ D E . (18)

The energy conservation equation is presented as

T j i ; i + T ¯ j i ; i = 0 . (19)

From Equation 19, we obtain

ρ ˙ m + ρ ˙ D E + 3 H ρ m + ρ D E + p D E = 0 . (20)

In this study, we have considered both interacting and non-interacting models. The continuity equations of DM and GGPDE through an interaction Q are where Q > 0 shows energy flows from GGPDE to DM, Q < 0 means energy flows from DM to GGPDE, and Q = 0 indicates a non-interaction model. Wei and Cai (2009) proposed the interaction term Q as

ρ ˙ m + 3 H ρ m = Q , (21)

ρ ˙ D E + 3 H ρ D E + p D E = − Q , (22)

Q = 3 b H ρ m , (23)

where b > 0 is a coupling constant.

The energy momentum tensor of GGPDE is given aswhere u is the PDE parameter.

ρ D E = τ H + η H 2 u , (24)

3 Solutions to the field equations

Equations 6–11 comprise a system of field equations K , L , M , ρ m , ρ D E , p D E, and ϕ. To solve these field equations, consider the following physical condition:

• Relationship between metric potentials proposed by Collins et al. (1980):
• The trace of the energy momentum tensor is zero, as proposed by Reddy et al. (2014):

From Equations 17, 26, we obtain

K 2 = M k + 1 . (27)

From Equations 7, 8, we obtain

K ˙ M ˙ K M + M ¨ M = K ˙ L ˙ K L + L ¨ L . (28)

From Equation 26, we obtain Equation 29 as

K ˙ M ˙ K M + M ¨ M + k M ˙ M 2 = 0 . (29)

From Equation 28, we obtain Equation 30 aswhere k 1 , k 2 denote the integration constants.

M = k 1 t + k 2 2 3 k + 3 , (30)

From Equation 26, we obtain

L = k 1 t + k 2 2 k 3 k + 3 . (31)

From Equation 28, we obtain

K = k 1 t + k 2 1 3 . (32)

The Bianchi type-V generalized ghost pilgrim dark energy cosmological model in self-creation theory of gravity is given as

d s 2 = − d t 2 + k 1 t + k 2 2 3 d x 2 + e 2 x k 1 t + k 2 4 k 3 k + 3 d y 2 + k 1 t + k 2 4 3 k + 3 d z 2 . (33)

From Equations 18, 27, we obtain

ϕ ¨ + ϕ ˙ K ˙ K + L ˙ L + M ˙ M = 0 . (34)

Using Equations 31, 32, 33, Equation 35 provides the scalar field as Equation 34

ϕ t = ln k 1 t + k 2 . (35)

4 Physical and kinematical parameters of the model

The parameters that play a vital role in describing the dynamics of the obtained model are as follows: the volume V(t) is a measure of the total spatial volume of the universe. Its steady increase with cosmic time confirms the continuous expansion without recollapse. The scalar expansion θ = 3H measures the rate at which all three spatial dimensions are expanding together, and a decreasing θ in time is indicative of the decrease in the rate of expansion as the universe transits from the matter to the dark energy domination. The mean anisotropy parameter Ah measures the deviation of the individual directional Hubble rates from the isotropic mean H: Ah = 0 corresponds to the isotropic FRW limit, whereas a nonzero value characterizes the anisotropic nature of the Bianchi type-V geometry. The shear scalar σ 2 measures the rate of shape distortion of the universe without the volume distortion; for physically viable models, σ²/H² → 0 at late times, consistent with the observed near-isotropy of the CMB. The Hubble parameter H describes the instantaneous expansion rate, and its evolution characterizes the transition from decelerated to accelerated cosmic expansion (Equations 12–16).

Volume of the universe V = k 1 t + k 2 , (36)

Scalar expansion θ = k 3 k 1 t + k 2 , (37)

Hubble parameter H = k 3 3 k 1 t + k 2 , (38)

Mean anisotropy parameter A h = 1 3 k 1 − k 3 k 1 2 + 2 k k 1 − k 3 k − k 3 2 k k 1 2 + 2 k 1 − k 3 k − k 3 2 k 1 2 , (39)

Average anisotropy parameter a t = k 1 t + k 2 1 3 , (40)

D eceleration parameter q = − a a ¨ a ˙ 2 = 2 , (41)

Jerk parameter j t = q + 2 q 2 − q ˙ H = 10 , (42)

S hear scalar σ 2 = 1 27 k 1 2 + 4 k 2 k 1 2 + 4 k 1 2 ‐ 6 k k 1 2 ‐ 2 k 1 2 k + 1 k + 1 2 k 1 t + k 2 2 . (43)

From Equations 24, 38, the energy density of GGPDE is given as

ρ D E = τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u . (44)

5 Non-interacting model

The energy conservation equation for DM is

ρ m ˙ + 3 H ρ m = 0 . (45)

From Equations 41, 38, we obtain

ρ m = k 1 t + k 2 − k 3 k 1 . (46)

The energy conservation of GGPDE is

ρ ˙ D E + 3 H ρ D E + p D E = 0 . (47)

From Equation 43, Equation 36, and ω D E = p D E ρ D E, the EoS parameter of GGPDE in the non-interacting case is obtained as Equation 47

ω D E = − 1 − 1 3 H ρ ˙ D E ρ D E = − 1 + u k 1 t + k 2 3 k 3 2 η k 3 2 k 1 + τ k 3 k 1 k 1 t + k 2 k 1 t + k 2 η k 3 2 + τ k 3 k 1 t + k 2 . (48)

The pressure of GGPDE obtained from Equation 28 is presented as Equations 44, 46

p D E = 1 3 ρ m + ρ D E = 1 3 k 1 t + k 2 − k 3 k 1 + τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u . (49)

The overall density parameter of the non-interacting model is

Ω = Ω m + Ω DE = 1 3 H 2 ρ m + ρ D E = 3 k 1 t + k 2 2 k 3 2 k 1 t + k 2 − k 3 k 1 + τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u . (50)

6 Interacting model

The energy conservation equation for DM is Equation 21

ρ ˙ m + 3 H ρ m = 3 b H ρ m . (51)

From Equations 38, 51, we obtain (Equation 20)

ρ m = k 1 t + k 2 k 3 b − 1 k 1 . (52)

The energy conservation equation for GGPDE is Equation 22

ρ ˙ D E + 3 H ρ D E + p D E = − 3 b H ρ m . (53)

Then, the EoS parameter of GGPDE in the interacting case is obtained as Equations 44, 52, 53

ω = − 1 − 1 3 H ρ ˙ D E ρ D E + 3 b H ρ m ρ D E ,

ω = − 1 − 2 η k 3 k 1 + τ k 1 k 1 t + k 2 η k 3 2 + τ k 3 k 1 t + k 2 + 3 b k 1 t + k 2 k 3 b − 1 k 1 τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u . (54)

The pressure of the GGPDE in the interacting case obtained from Equation 28 is given as

p D E = 1 3 ρ m + ρ D E = 1 3 k 1 t + k 2 k 3 b − 1 k 1 + τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u . (55)

The overall density parameter of the interacting model is

Ω = Ω m + Ω DE = 1 3 H 2 ρ m + ρ D E = 1 3 k 3 2 k 1 t + k 2 k 3 b − 1 k 1 + 2 + 3 τ + η k 3 3 k 1 t + k 2 u k 3 3 k 1 t + k 2 . (56)

7 Stability analysis

In this section, the stability of both interacting and non-interacting models is discussed. The squared adiabatic speed of sound v s 2 = ∂p_DE/∂ρ_DE is a standard diagnostic for classical stability in the context of linear homogeneous perturbations of the dark energy fluid. Physically, v s 2 > 0 means that small perturbations propagate as stable oscillatory modes rather than growing exponentially, whereas v s 2 < 0 signals a classical instability. The Hubble radius H⁻¹ is an important scale in the evolution of the dark energy perturbations, such that on sub-Hubble scales (k >> aH) the perturbations in the dark energy increase quasi-independently and the sign of v s 2 directly determines whether they are stable or not; on super-Hubble scales (k << aH), the perturbations in the dark energy are frozen by the causal horizon, and their growth is suppressed regardless of the sign of v s 2. Causality also requires v s 2 ≤ 1 (in units of c²).

To characterize the stability of models, the sign of v s 2 = p ˙ D E ρ ˙ D E is crucial. If v s 2 > 0, it shows a stable model, and if v s 2 < 0 it shows an unstable model (Myung, 2007). In addition, the causality condition must be satisfied, meaning that the speed of the sound is less than the speed of the light.

The square of the speed of sound v s 2 in the non-interacting model is given as Equation 49

v s 2 = p ˙ D E ρ ˙ D E = 1 3 − k 3 k 1 t + k 2 − k 3 k 1 − 1 + u τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u − 1 − τ k 3 k 1 3 k 1 t + k 2 2 − 2 η k 3 2 k 1 9 k 1 t + k 2 3 u τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u − 1 − τ k 3 k 1 3 k 1 t + k 2 2 − 2 η k 3 2 k 1 9 k 1 t + k 2 3 . (57)

The square of the speed of sound in the interacting model is given as Equation 50

v s 2 = p ˙ D E ρ ˙ D E = 1 3 k 3 b − 1 k 1 t + k 2 k 3 b − 1 k 1 − 1 + u τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u − 1 − τ k 3 k 1 3 k 1 t + k 2 2 − 2 η k 3 2 k 1 9 k 1 t + k 2 3 u τ k 3 3 k 1 t + k 2 + η k 3 3 k 1 t + k 2 2 u − 1 − τ k 3 k 1 3 k 1 t + k 2 2 − 2 η k 3 2 k 1 9 k 1 t + k 2 3 . (58)

8 Energy conditions

In this section, we discuss energy conditions of both interacting and non-interacting models. It is worth mentioning that in standard general relativity, accelerated expansion is only possible if the strong energy condition (SEC) is violated, i.e., ρ + 3p < 0. Our model, however, is formulated within Barber’s second self-creation theory, which is a scalar-tensor modification of gravity. In SCT, the scalar field adds extra terms to the effective energy momentum. The combined (matter + scalar field) contribution governs the expansion kinematics. Consequently, the matter fields evaluated in isolation can individually satisfy the SEC, whereas the contribution from the scalar field simultaneously drives accelerated expansion. Thus, the energy conditions used here are evaluated on the matter sector only and are a measure of the physical reasonableness of the matter content, not a direct constraint on the sign of acceleration.

• ρ D E ≥ 0 WEC,
• ρ D E + p D E ≥ 0 DEC, and
• ρ D E + 3 p D E ≥ 0 SEC.

9 Conclusion

The graphical analysis of the cosmological model reveals a physically consistent and mathematically well-behaved universe that is in strong agreement with modern cosmological observations. The universe undergoes a steady volumetric expansion (Figure 1), accompanied by a rapidly decaying Hubble parameter (Figure 2), Average anisotropy parameter (Figure 3) and scalar expansion (Figure 4) that together confirm the transition from an early hot, dense, rapidly expanding phase to a late-time dark-energy-dominated regime. Matter density (Figure 5) dilutes progressively with cosmic expansion, becoming subdominant at late times in accordance with the standard ΛCDM framework, while the 3D surface analysis (Figure 6) demonstrates attractor-like behaviour where the model’s sensitivity to initial parameters diminishes over time. The non-interacting dark energy component (Figure 7) resides firmly in the quintessence regime (Equation 48) (−1 ≤ ω_DE ≤ −0.3), whereas the introduction of a dark matter–dark energy interaction term (Figure 8) shifts the equation of state deep into the phantom regime (−1.65 ≤ ω_DE ≤ −1.35), opening the possibility of a phantom-driven Big Rip scenario in the far future (Equation 54). In the interacting model, the phantom behavior is driven by dark matter–dark energy coupling, and the approach to a Big Rip depends sensitively on the coupling constant b and the PDE parameter u (Equation 55). In both models, pressure and dark energy density (Figures 9, 10) decay rapidly to near-zero at late times, with the interacting model additionally exhibiting a rising overall (Equation 56) density (Figures 11, 12) driven by matter–dark energy transfer a self-sustaining mechanism that effectively replenishes the universe’s total energy budget. The stability analysis confirms that the non-interacting model (Figure 13) is classically stable and causally consistent throughout all epochs, whereas the interacting model (Figure 14), despite showing transient superluminal sound speed at early times, stabilises within physically acceptable bounds at late times. Most importantly, both models (Figures 15, 16) satisfy all three fundamental energy conditions—the Weak (WEC), Dominant (DEC), and Strong (SEC)—across the entire cosmic evolution, providing robust theoretical validation that the proposed cosmological framework is physically viable. We note that in the context of Barber’s second SCT, the simultaneous satisfaction of the SEC and accelerated expansion is not contradictory, as the scalar field contribution provides the additional impetus for acceleration beyond what the matter sector alone would yield. The present work has several important limitations that should be explicitly acknowledged: (i) the solutions are obtained under two auxiliary conditions—the Collins et al. metric relation and the vanishing trace of the energy-momentum tensor—which constrain the solution space; (ii) the parameter values used in the graphical analysis are physically motivated but not formally constrained by observational data; (iii) the growing anisotropy parameter at late times requires careful parameter tuning to remain within CMB-permitted bounds, and late-time isotropization remains an open question; and (iv) a formal statistical analysis involving MCMC fitting to Pantheon+, BAO, or Cosmic Chronometer datasets has not been performed, which is an important limitation for establishing the quantitative viability of the model. These are natural and important directions for future work. Specifically, observational constraint of the model parameters using Pantheon+ Type Ia Supernova data, Baryon Acoustic Oscillation measurements, and Cosmic Chronometer H(z) data remains a high-priority extension of the present work. Further extensions could include investigation of alternative interaction kernels beyond Q = 3bHρm, exploration of other Bianchi type geometries, and embedding of the GGPDE model in f(R) or f(Q) modified gravity frameworks.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Panayiotis Charalambos Stavrinos, National and Kapodistrian University of Athens, Greece

Reviewed by: Bikash Chandra Roy, University of North Bengal, India

Imran Hashim, University of Lahore, Pakistan