Possible combinations of early and late time cosmologies through BAO scales

The theoretical scenarios of early universe and late time accelerated expansion constitute an open problem in cosmology. Relative Baryon Acoustic Oscillation (BAO) scales measured by 6dFGS, SDSS+2dFGRS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm SDSS+2dFGRS$$\end{document}, BOSS DR11 Lyα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and BOSS DR11 QSO Lyα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} are used to investigate the parameter spaces of combinations of pre-recombination physics and late time cosmology. For pre-recombination physics we fix Neff=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm N_{eff}=3$$\end{document} and 4 and incorporate an Early Dark Energy (EDE) component. For late time cosmologies we consider curved Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM model, Dynamical Dark Energy (DDE) model with CPL parameterisation (ω0,ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0,\omega _1$$\end{document}) of equation of state, Coupled Dark Energy (CDE) model with constant coupling and curved DGP brane-world scenario. Combination of a curved Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}CDM (ΩkΛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\textrm{k}}\Lambda $$\end{document}CDM) model with Neff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm N_{eff}$$\end{document}-EDE cosmology is found to be possible. For DDE, the relative BAO scales are achievable through ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document} preferring the region ω1<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1< 0$$\end{document}. For Neff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm N_{eff}$$\end{document}-EDE-CDE, the BAO scales are generated for both dark energy-dark matter and dark matter-dark energy conversion, however with exceptionally large value of dark energy density parameter and constant equation of state with ω0>-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0 > -1$$\end{document}. Curved DGP cosmology (Ωk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\textrm{k}$$\end{document}DGP) is found to join with Neff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\textrm{eff}}$$\end{document}-EDE cosmology through curvature parameter Ωk(0)∈[-0.002,-0.1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\mathrm{k(0)}}\in [-0.002, -0.1]$$\end{document} and cross-over parameter Ωrc∈[0.08,0.45]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\mathrm{r_c}} \in [0.08, 0.45]$$\end{document}. Cosmic ages are calculated for the derived parameters of ΩkΛ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\textrm{k}}\Lambda $$\end{document}CDM, flat DDE and Ωk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\textrm{k}}$$\end{document}DGP models and compared with the ages of few globular clusters available in literature. Curved cosmologies are found to be promising.


Introduction
The goal of cosmology has been to sketch the evolution of the universe since its birth to present epoch with little understanding of its composition and primordial state. Whereas the initial state of the universe is a regime of quantum cosmology with major modification of general relativity, the late time cosmic history is governed by a quintessential component known as dark energy whose nature or physics is still not well understood. It is a primary component for current cosmological models. Observations of the type Ia supernovae [1][2][3], Cosmic Microwave Background (CMB [4], large scale clustering of galaxies [5] and Baryon Acoustic Oscillation (BAO) scales [6][7][8] have shown us that the cosmological model with a cosmological constant ( ) as a candidate of dark energy and Cold Dark Matter (CDM) as the source of large scale clustering of matter is a good approximation to explain the observable universe. The CDM model, being the standard or concordance model of modern cosmology is inclusive of several cosmological phenomena such as the initial singularity, cosmic inflation leading to primordial matter perturbation, big bang nucleosynthesis, formation of the first atoms and flat spatial geometry of the universe [9]. Despite its success in explaining the universe, continuously growing precision in measurements of expansion rate, CMB and BAO has revealed that cosmic expansion history may greatly differ from what is expected from the base model. The current expansion rate measured from the Cosmic Microwave Background (CMB) is H 0 = 67.4 ± 0.5 km/s/Mpc [10] whereas the distance ladder technique used by SH0ES (Supernova H0 for the Equation of State) gives H 0 = 73.5±1.4 km/sec/Mpc [11]. It is in 4.2σ tension with the early universe. The Hubble parameter measured by the time delay in strong lensing technique by the H0LiCOW collaboration [12] and STRIDES (Strong-Lensing Insights into Dark Energy Survey) [13] team is H 0 = 73.3 ± 1.8 km/sec/Mpc and H 0 = 74.2 ± 3.0 km/sec/Mpc respectively and are again in tension with the CMB measurement. However, the Carnegie-Chicago Hubble Program measured H 0 = 69.8 ± 0.8 km/sec/Mpc by using the tip of the red giant branch (TRGB) technique which is 1.2σ in agreement with the CMB measured value [14]. Apart from the H 0 tension, Battye and Moss [15] reported the discrepancy in the number of cluster counts measured from the CMB and lensing data. The lensing data point towards a lesser clumpines of matter than preferred by the CMB fitted to the CDM model. The discrepancy between early universe and late universe (low redshift) measurements has tempted us to think beyond the concordance model, perhaps to include "new physics". Generally suggested remedy such as increased local expansion due to our being situated inside an underdense region of the cosmic web has been ruled out [16,17]. Although a phantom dark energy with equation of state ω = p/ρc 2 < −1 is also eligible for explaining the enhanced local expansion rate, it has been disfavoured by intermediate redshift measurements [9]. A promising approach is to alter the composition and physics of the pre-recombination era [18]. These are inclusion of additional relativistic particle species (say, effective number of neutrino species) in the early universe [19] and the Early Dark Energy (EDE) scenarios [20] which can elevate cosmic expansion rate to the level of the demands of late time measurements. The EDE scenario invokes a scalar field constructed in such a way that its energy density increases the CMB inferred value of the expansion rate. This component operates only over a narrow epoch in the expansion historybetween matter -radiation equality and recombination. Irrespective of the H 0 tension, presence of exotic dark energy in the late universe (as a result of its time evolution, generally encapsulated by the redshifted equation of state, ω(z) or its equivalent scalar fields such as quintessence [21][22][23], phantom [24], quintom [25][26][27]), alteration of theory of gravity on large scale producing the accelerated expansion and hence mimicking the dark energy phenomenon and departure from spatial flatness (see its indication in [28,29]) cannot be ruled out.
These are yet open problems in theoretical cosmology and constitute interesting late time effects.
There is yet another astrophysical tension with the CDM model. It is the problem of cosmic age which depends on H 0 and cosmological model parameters (density parameters, i (contributed by matter, dark energy and curvature) and equation of state of dark energy ω) as t 0 = H −1 0 f( i , ω). Due to large value of H 0 implied by local universe measurements the CDM based age of the universe becomes smaller than the expected value of 13.8 Gyrs derived from relatively small value of H 0 implied by early universe measurements. Another manifestation of the age tension is the observational report of few old globular clusters in M31 galaxy with ages > 15 Gyrs [30,31]. Clearly the standard CDM cosmology with simple set of parameters and a derived age of about 13.8 Gyrs cannot accommodate these stars. Attempts have been made to accommodate these old clusters with interacting dark energy models [32]. But the case was not strong in elevating the age. Holographic and ghost dark energy models were also introduced to address the cosmic age problem by considering some other cases of old cosmic objects [33]. Therefore, one wonders whether a cosmic combination of early and late time cosmologies beyond the standard paradigm can address the cosmic age problem.
In this work we try to investigate combination of prerecombination (early universe) new physics and late time cosmologies to see if any one combination (pre-recombination physics+late time cosmology) is eligible to constitute a good approximation. The relative BAO scale r BAO (z) which is defined as the ratio of sound horizon at drag epoch (r S (d)) and the volume average angular diameter distance (D V (z)) provides us with the tool for the above study. Quite often, the scale is also expressed by the ratio of angular diameter distance d A (z) to the sound horizon at the drag epoch (we express it as s BAO (z)). We select the BAO scale as it provides one with low redshift observations for understanding the cosmological parameters associated with expansion history of the universe [34]. Whereas pre-recombination physics fixes the sound horizon, the late time effects are encoded in the angular diameter distance D V (z). Allowed ranges of parameters of the late time cosmological models are extracted from available relative BAO scales (r BAO and s BAO ) to examine possible combination of pre-recombination physics and late time cosmology. Our considerations of pre-recombination physics encompass one EDE component coming from scalar field and effective number of relativistic species N eff = 3, 4. Whereas 3 degrees of freedom for neutrinos is standard physics of the early universe, existence of the fourth flavour (particularly the sterile neutrinos) has been indicated (see [15] for an interesting requirement). For late time cosmologies we consider (i) curved CDM model, (ii) Dynamical Dark Energy model with CPL parameterisation of equation of state, (iii) Coupled Dark Energy (CDE) with constant coupling in the dark sector and (iv) curved DGP extradimensional cosmology. We extend the cosmology beyond the concordance CDM with inclusion of spatial curvature ( k ), dynamics of dark energy with equation of state parameters ω 0 -ω 1 , dark energy-dark matter coupling (δ) and crossover scale parameter ( r c ) of extra-dimensional cosmology. These are the models which are often invoked for resolving the existing H 0 tension [35]. That extended parameter space gives idea of a better cosmology can be found in the constraints of dark energy obtained through a 12 parameter extension of the concordance cosmology [36]. Focus of our analysis is to derive parameter space or preferred region of the dark universe cosmology in conjunction with early universe physics.
The EDE density parameter defines the sound horizon at drag epoch through expansion rate at early time (ET), H ET (z) = H 0 E ET (z). The volume average angular diameter distance is governed by the late time (LT) expansion history H LT (z) = H 0 E LT (z). Here E(z) is the dimensionless hubble function parametrized by the cosmological parameters. These parameters encapsulate cosmological densities, dark energy physics and modified gravity effects. The relative BAO scale becomes independent of the Hubble parameter H 0 and is governed by physics of dark energy, cosmological parameters and the background theory of gravity.
The paper is organised as follows. Section 2 introduces the BAO scale and its role in contemporary cosmology. The BAO data used in this work are also highlighted. Section 3 describes the formalism of EDE used in the work as new pre-recombination physics and also highlights the late time expansion histories in base CDM model, curved lambda CDM model, dynamical dark energy models, coupled dark matter-dark energy scenario and the DGP brane cosmology. In Sect. 4 we address the cosmic age problem with k CDM and k DGP models. Results and discussions are presented in Sect. 5. Section 6 concludes.

The BAO scale and cosmology
The Baryon Acoustic Oscillation (BAO) which refers to primordial acoustic oscillations in the baryon-photon fluid, has been detected in the clustering of galaxies [37,38]. It is a great triumph for cosmology as it is robust probe for determining cosmic expansion history H(z) and the cosmological parameters related to it. Discovery of the scale of the comoving sound horizon has been an opportunity to constrain dark energy and spatial curvature with galaxy surveys within z ∼ 0.5 carried out through the Sloan Digital Sky Survey (SDSS) and 2dF redshift surveys [37][38][39][40]. Joint fits of BAO scale with CMB have been eligible to constrain dark energy scenarios [41]. Extracting BAO scale from large scale clustering of galaxies is still a thriving area of research in contemporary cosmology.
Since the first detection of the BAO scale, deep surveys such as Baryon Oscillation Spectroscopic Survey (BOSS) and 6dF Galaxy Survey (6dFGS) have measured the BAO scale over a wide range of redshifts [41]. BAO has been measured for BOSS quasars at redshift z = 2.3-2.4 [42][43][44]. Comparison of the high redshift measurements with available low redshift measurements gives us clue about true expansion history of the universe. The extended BOSS (eBOSS) mission reported measurement of BAO scale using clustering of QSOs at 0.8 < z < 2.2 [45]. The relative BAO scale data used in this work are shown in Table 1.
Addison et al. [41] reported usefulness of the BAO scale in testing the H 0 disagreement realised in the most precise CMB fits (Planck Collaboration [4]) and local distance ladder estimations [49]. Addition of the BAO scale measurements to WMAP, Atacama Cosmology Telescope or South Pole Telescope for CMB has shown that the value of H 0 predicted from early universe physics of the base CDM model is at least 2.4-3.1σ lower than the local universe value and is not owed to any systematic effect associated with CMB measurements. It has established that we are at cross roads in understanding the cosmic expansion. Possibilities of new physics or cosmology of the early and late universe are open to us. The relative BAO scale, r s (d)/D V (z eff ) and d A (z eff )/r s (d) measured respectively at low and high redshifts provide one with the tools to test what combination of early universe physics and late universe expansion dynamics are preferred. Given a pre-recombination physics we investigate the allowed ranges of parameters of the late time cosmologies which are capable of reproducing the measured relative BAO scales at z = 0.10-2.36 (Table 1).

Pre-recombination physics and late time expansion dynamics
In the standard Hot Big Bang cosmology, baryons and photons were tightly coupled forming a single fluid prior to the recombination epoch. Primordial perturbations caused a sound wave travelling across the fluid. The sound horizon at which baryons were released from the Compton drag of the photons defines the drag epoch (z d ). The sound horizon at the drag epoch is given by: Here, c s = c/ √ 3 + R is the sound speed which is governed by the ratio of baryon to photon density The early time expansion rate H ET (z) is defined for z d < z < ∞, where the drag epoch is given by z ≈ 1020 [50]. H ET (z) is assumed to be determined by matter, relativistic particles and the EDE component. We ignore the curvature density term (1/a 2 ) in relation to matter density (1/a 3 ), radiation density (1/a 4 ) and EDE density (1/a 4.5 ). Therefore, the Friedmann equation for early time expansion rate is expressed as, where, m(0) , rel(0) are present density parameters of matter and relativistic species and EDE is the density parameter of the EDE component. We adopt the following param-  [ 48] N eff = 3, 4 and γ (0) ≈ 10 −5 , the present density parameter for photons. Sterile neutrino as fourth degree of freedom can affect the expansion rate of the universe, in particular, it can contribute to the total radiation energy density at early times and affect the measurement of the Hubble constant. So, this possibility of a fourth degree of freedom from sterile neutrinos can have implications for our understanding of the universe, and its expansion rate. Battye and Moss [15] have suggested the mass of the sterile neutrino to be m eff v,sterile = 0.450 ± 0.124 eV that resolves the discrepancy of galaxy count through Sunyaev Zeldovich method and that preferred by the CMB data.
EDE refers to a component resulting from new physics of the early universe whose density decreases steeply compared to radiation (ρ EDE ∝ a −m , with m > 4) and hence becomes dominant only in deep radiation era and becomes subdominant by the epoch of big bang nucleosynthesis and subsequent evolutionary phases. In the very early universe (a → 0) EDE behaves as a cosmological constant. Originally proposed by Poulin et al. [20] the EDE model resolves the H 0 tension by increasing the expansion rate of the early universe. Commonly adopted candidate is some scalar fields [50]. [20] used presence of scalar field (φ) with potential of the type V(φ) ∝ (1 − cos(φ/f)) n in the early universe as a possible resolution to the Hubble tension. Here f is an unknown energy scale and n > 0. In the model used by Poulin et al. [20] the EDE component is a scalar field which behaves as a cosmological constant in very early time (a → 0) and then starts oscillating above a critical scale factor a c thus becoming a dynamical dark energy fluid. The energy density parameter is given by critical scale factor a c [20] where 1 + z = 1/a. Here, with "n" being the exponent appearing in the potential V(φ). This refers to equation of state of EDE once the scalar field behaves as a dynamical fluid. The critical scale factor above which the scalar field becomes dynamical is chosen as a c = 10 −5 − 10 −3 (see [20] for a reasonable range). In this work, a c = 10 −4 has been considered. φ (a c ) is related to Total (a c ) = 1. For n > 2 which refers to EDE component diluting much faster than radiation (see Eq. (5)), f EDE ≈ 0.05 (5%) has been shown to alleviate the existing tension between early and late time measurements of the Hubble parameter [20]. Thus φ (a c ) has been set as 0.05 and n is set as n = 3 so that ρ φ ∼ a −4.5 , a component diluting faster than radiation. This value of n is preferred as it gives H 0 = 70.6 ± 1.3 km/s/Mpc solving the H 0 tension within the 2σ confidence level. [20].
In terms of cosmological redshift, Eq. (4) is used in Eq. (3) to compute the early time expansion rate that finally enters into the sound horizon (Eq. (1)).
The relative BAO scales are expressed as and The volume average angular diameter distance is given by [50] Here is the comoving angular diameter distance with d A satisfying the distance duality relation with luminosity distance d L (z) as: The luminosity distance is expressed in terms of the late time expansion history H ET (z) as where sin n = x, sin(x) and sinh(x) respectively for flat, closed and open universes. The late time expansion rate for several cosmological models are discussed below. Apart from curved CDM and DGP model the late time expansion is assumed to be that of flat spatial section. In flat cosmology, the late time expansion rate with dark energy equation of state ω(z) is expressed as In presence of spatial curvature, the term k(0) (1 + z) 2 is added under the square root of equation (11). Here dark energy models are encapsulated in the equation of state parameter, ω(z). Two distinct scenarios occur for constant and redshift (time) dependent equation of state of dark energy. The late time cosmological models whose parameter spaces are investigated are highlighted below.
(i) CDM universe: This is the concordance model of cosmology which is based on the assumption of correctness of general relativity on cosmological scales and validity of the maximally symmetric Friedmann-Lemaitre-Robertson-Walker metric. For the CDM model, ω(z) = −1 (constant) so that equation (11) becomes In presence of curvature, the CDM model is extended as This is k CDM model. Measurements of the CMB power spectrum indicate possibility of a closed universe at more than 3σ confidence [51]. This creates trouble for the standard CDM cosmology based on inflationary flatness condition. Simple extension of this model through spatial curvature has been analysed in several studies of combination of datasets such as Planck, CMB lensing, Type Ia supernova luminosity distance, weak lensing, joint light curve analysis and local measurements of the Hubble constant, H 0 [36,51]. A dark energy component with constant equation of state (ω = −1) has been ruled out at more than 95% confidence level (C.L.) by Planck+local H 0 data. This is found to be valid when CMB lensing, Type Ia supernovae and weak lensing data are included. But the tension remains when BAO data is included. It still allows the cosmological constant between 68% and 95% C.L. under the combination Planck + local H 0 + BAO [36]. Surprisingly, inclusion of spatial curvature stabilises this result. We re-examine these inferences through the available BAO scales used as probes against N eff -EDEk CDM model. (ii) Dynamical dark energy (DDE): Dynamical dark energy scenarios are expressed by parameterized equation of state. The most widely studied and observationally constrained parameterization is the CPL parameterization [52,53] which is expressed as Here ω 0 , ω 1 are parameters constrained by observations. With (12.1) and (10) the late time expansion history in dynamical dark energy model becomes The CPL parameterisation has been extensively used in phenomenological studies employed to understand the evolution of dark energy. Kumar et al.2014 [54] reported constraints on (ω 0 , ω 1 ) through SNLS3, BAO and Planck + WMAP9 + WiggleZ measurements of matter power spectrum. Menci et al.2020 [55] reported constraints on the 'look back time derivative' of equation of state, ω 1 through abundance of massive galaxies at high z. Du et al.2019 [56] reported 68% and 95% confidence level (C.L.) bounds on these two parameters through the combined datasets -CMB + BAO, CMB + BAO + JLA (Joint Light Curve Analysis) and CMB + BAO + JLA + CC (Cosmic Chronometers). Zhao et al.2020 [34] forecasted allowed ranges of these parameters through addition of the Fast Radio Burst (FRB) data to the CMB datasets. The derived ranges of the equation of state parameters are discussed in Sect. 4. (iii) Coupled dark energy model (CDE): In addition to the CDM and DDE scenarios where dark energy and dark matter retain their identities, there are interesting models where dark energy couples to the dark matter component. This is motivated by the fact that dark energy and dark matter are of the same order in the present epoch. This cosmic coincidence (as it is known) suggests that there must be hidden coupling in the dark sectors [50]. In recent years interacting dark energy model where energy transfer occurs between dark matter and dark energy has been found to be appealing for resolving the Hubble tension [51]. Interaction between quintessence dark energy (some canonical scalar fields) and dark matter was discussed in [57,58]. This is intrinsic to scalar-tensor theories of gravity [28,[59][60][61]. But it is not yet certain if dark energy is really some form of scalar fields. One approach to treat interaction between dark energy and dark matter is to parameterize the coupling without assuming scalar fields. The approach is as follows. In flat FLRW cosmology, the dark energy-dark matter interaction is parameterised as [62], Here T μν m and T μν de denote stress tensors for dark matter and dark energy and Q ν is a 4-vector governing energy-momentum transfer between dark matter and dark energy. Usually Q ∝ Hρ, H being the Hubble parameter and ρ being the energy density of the dark sectors. One may have three choices [63]: ρ = ρ m , ρ = ρ de , ρ = ρ m + ρ de . In absence of details of dark energy-dark matter interaction the model is parameterised as Q = δ Hρ m = ρ m , where is the rate of interaction and δ is coupling strength [64][65][66]. The evolution equations for densities of dark matter and dark energy are expressed as, δ > 0 signifies transfer of energy from from dark energy to dark matter and δ < 0 signifies transfer of energy from dark matter to dark energy. Solution of (17) is then given by For constant δ models, Eq. (19) becomes For constant ω X , solution of (18) is With (20) and (21), the late time expansion history is The late time expansion rate for coupled dark energy depends on the three parameters: δ, ω X , X(0) . We estimate these parameters to obtain the r BAO and s BAO scales for low redshifts and high redshift respectively. (iv) DGP Extra-dimensional cosmology: The models discussed above are based on general relativity, where Einstein field equations for expansion are assumed to be valid with exotic matter sources such as field (dynamical equation of state) or negative pressure fluid. The possibility that late time cosmic acceleration is not caused by a stress-energy tensor (dark energy field or fluid) rather by modification in the gravity sector was proposed in [67,68]. This is the DGP braneworld model which is a 5 dimensional low energy modification to general relativity. The idea is based on leakage of gravity off a 3 D brane at a cross-over scale r c . It has been realised that alternative gravitational theories are eligible to alleviate the Hubble tension through weakening of gravity at intermediate scales [69]. Many of them are based on geometrical alternative to general relativity. Here we consider the braneworld scenario. Some astrophysically appealing features of DGP model have been realised. The Hubble parameter in the braneworld is indistinguishable from that in the CDM model. This model also explains older age of the universe and observed quasar abundance for redshift greater or similar to 4 [70]. The Friedmann equation in DGP model is modified as For a flat universe, k = 0 the late time (ρ → 0) self acceleration with a(t) ≈ exp(t/r c ) is evident from equation (23). This self accelerating model, however, has been ruled out by observations and by presence of ghost (see [50,71] for review). An alternative approach is the normal-DGP model (NDGP) which is non-self accelerating and ghost free [72,73]. It is obtained by the replacement, r c → −r c . Late time acceleration is achieved by adding a cosmological constant. The modified Friedmann equation in NDGP (henceforth called as k DGP) model is expressed as Defining a dimensionless density parameter of the cross-over scale as r c = c 2 /4H 2 0 r 2 c , the density parameters are related as Here, (0) = c 2 /3H 2 0 , k(0) = −kc 2 /a 2 0 H 2 0 . In terms of redshift and the cosmological density parameters, Eq. (21) is written as Defining H 2 (z) − H 2 0 k(0) (1 + z) 2 = p, the solution of the quadratic equation (25) is obtained as p 2 > 0 is demanded as it is the case with standard GR with p > 0. Therefore we choose the negative root in (26) so that It gives the late time expansion rate Closed DGP model was found to be satisfactory with background expansion rate data [74] and it allows (0) > 1. Evolution of late time expansion in closed curvature models with novel dark energy equation of state has been recently discussed by [75]. Consistent transition redshift and interesting future behaviour of expansion were obtained. Spatial curvature with k(0) = −0.01, −0.1 were found to be interesting in the context of ever accelerating universe. The parameter space in curved DGP model required for generating the relative BAO scales is constructed by r c and k(0) . For the analysis, the Hubble function E(z) = H LT /H 0 required for estimating angular diameter distances and hence the relative BAO scales is calculated in low redshift (z eff < 1) regime through a Taylor expansion.

Prospects for curved cosmologies k CDM and k DGP
We emphasize on further implications in cosmology of the two curved cosmologies. After highlighting few theoretical aspects of curved cosmologies we present one astrophysical prospect of k CDM and k DGP models. Inflationary expansion with k < 0 has been studied earlier [76][77][78]. Closed models with inflation gives a large scale cut off density fluctuation around the curvature scale, R c = cH −1 0 | k(0) | −1/2 [51]. Closed models can also severely challenge the eternal inflation [79,80]. Creation of open universe k > 0 from de-sitter space was discussed by [81]. Evolution of density perturbations with k > 0 inflation was studied by Lyth and Stewart [82]. Bucher, Goldhaberard and Turok [83] presented a natural emergence of open universe from inflation with any value of total between 0 and 1 and hence positive values of k . Their calculation showed that generation of k > 0 ( < 1) universe from inflation does not call for much fine tuning.
In addition to the theoretical scenarios, cosmology beyond CDM has become inevitable for addressing certain problems emerging from recent cosmological observation. These are H 0 tension [11][12][13], σ 8 tension [84][85][86] and the cosmic age problem [87][88][89]. Here we address the cosmic age problem realised through discrepancy between local universe values and CMB derived values of the Hubble parameter (H 0 ). For the standard set of cosmological parameters ( (m(0)) ≈ 0.3, (de(0)) ≈ 0.7, k(0) = 0) whereas local universe values of the parameter (larger) produce a young universe, the CMB derived values (as small as 67 km/s/Mpc) Fig. 2 Derived ages of the universe in k DGP model Fig. 3 Derived ages of the universe in DDE model produce a relatively old universe. Another manifestation of the cosmic age problem is the set of 9 globular clusters (see Table 2 with references) whose ages are more than the cosmic age derived by the Planck measurements of the CMB on the basis of standard CDM cosmology. With the allowed ranges of the parameters of k CDM and k DGP models derived from BAO scale we calculate the cosmic ages (t 0 ) for the three values of the present Hubble parameter, H 0 = 73.04 km/s/Mpc, 67.4 km/s/Mpc, and 69.84 km/s/Mpc measured respectively by using local universe probes (Cepheids and Type Ia supernovae) [90], CMB anisotropy [10] and the Tip of the Red Giant Branch (TRGB) stars [14].
The present cosmic age is given by, where E(z) = H(z)/H 0 and it has been substituted in equation (30) for the two models. The derived ages (in Gyr) for the parameters of the two models are shown in Figs. 1 and 2.
The ages are compared with ages of few old globular clusters available in literature (see Table 2). We find that both k CDM and k DGP models are eligible to accommodate the old globular clusters. We also calculate the cosmic age in flat DDE model with the derived ranges of the equation of state parameters ( ω 0 and ω 1 ). It is found that the ages are between (9-11) Gyrs, well below the CDM model (see Fig. 3) for DDE. These two curved models, therefore, can serve as serious alternatives to CDM model. The vertical coloured bars on the right hand side of Figs. 1, 2 and 3 represents cosmic ages in years. At the top of the Figs. 1, 2  and 3, the values of the Hubble parameter derived by different groups are highlighted.

Results and discussion
Combinations of pre-recombination physics and late time cosmologies that yield observed relative BAO scales at low and high effective redshift have been reported. Fixing the prerecombination expansion dynamics with N eff = 3, 4 and an Early Dark Energy component of [20] type, we investigate the allowed ranges of cosmological parameters appearing in the late time expansion rate. The matter density and CMB photon density are pivoted at m(0) = 0.3, γ = 10 −5 . CDM with spatial curvature (N eff -EDEk CDM), Dynamical   (ii) N eff -EDE-DDE: The DDE cosmology joins with N eff -EDE scenario for wide ranges of the equation of state parameters ω 0 , ω 1 with ω 0 lying mostly in the phantom regime ω 0 < −1. This is at par with the inference that a phantom like equation of state at present epoch is eligible for resolving the H 0 tension. However, quintessence-like regimes ω 0 > −1 are also not disfavoured. We discard a major portion of the parameter space for z eff = 0.106 and 2 derived for N eff = 3 (see top two panels of Fig. 5). The derived ranges of these two parameters are compared with 95% C.L. bounds arising from CMB + BAO, CMB + BAO + JLA and CMB + JLA + BAO + CC and reported in [56] (see Table 3). It is found that all the ranges of the look back time derivative (ω 1 ) of equation of state obtained for N eff = 3, 4 are within the 95% C.L. intervals derived from above mentioned joint probes. Moreover, the preference of ω 1 for occupying the regions of ω 1 < 0 is consistent with earlier report that dynamical dark energy with "downward-going" model, ω 1 > 0 (ω becoming more negative with decrease in cosmological redshift) is not favoured by a 12 parameter extension of existing cosmology [36]. As ω 1 represents time evolution of dark energy we consider this consistency of the extracted range of it as a marker for a possible match between DDE and N eff -EDE. The Lambda lines ω 0 = −1, ω 1 = 0 are displayed in Fig. 5 [92]. These data sets give posterior peak of δ ∼ 10 −3 and ω < −1 (see Table 6 and Figure 5 of [92] by noting that the parameter 1 is same as δ). Therefore, we rule out the parameter space of this cosmology as a possible universe. (iv) N eff -EDEk DGP: In the curved DGP model the parameter space that is used to generate relative BAO scales are the cross-over scale density parameter r c (and the curvature density parameter k with the pivoting m(0) = 0. Given the present measurements of the Hubble parameter (local universe, [11][12][13][14] Table 3 Measurements of relative BAO scales at low and high effective redshifts, z eff which are used in this work for examining possible combination of pre-recombination physics and late time cosmology Effective no. of Derived ranges of 95% C.L. on (ω 0 , ω 1 ) degrees of freedom ω 0 , ω 1 (From [56])  realised to be soft enough for understanding the integrated cosmic expansion history. We find that k CDM and k DGP models are eligible to address the cosmic age problem by elevat-ing the present age beyond the one predicted by the standard paradigm. It is seen that k CDM model produces age within (13.5-15.5) Gyrs. k DGP model further elevates the age as (16.6-17.0) Gyrs. The flat DDE model is unable to generate cosmic ages compatible with the old stars. Thus these two curved cosmologies accommodate the globular clusters having age (t GC = 13.99 − 16.00) Gyrs. This can be treated as an astrophysical test of the two curved cosmologies. In addition, the large scale cut off of curvature scale for the derived range k ≈ (−0.002) − (+0.1) (for the two curved models) is found to be in the range (12.99-99.53) Gpc (see Table 4) which is consistent with the lower bound of low quadrupole anisotropy of the CMB [77,93].

Conclusions
In this work, we have searched for possible combination of early universe physics and late universe cosmology by reproducing the observed relative BAO scale at low and high redshift. In the early universe physics, we considered standard and enhanced degrees of freedom of relativistic species (N eff = 3, 4) along with an early dark energy (EDE) component. Parameter spaces of N eff -EDEk CDM, N eff -EDE-DDE, N eff -EDE-CDE and N eff -EDEk DGP cosmologies were extracted. Whereas both closed and open spatial curvature in non-standard CDM cosmology joins with the early universe physics, the curvature parameter for closed model, k ∼ −0.06 is at par with Planck data. Preference of closed DGP brane cosmology is evident through reproduction of the BAO scales with relatively soft curvature, k ∼ −0.002 − (−0.01). Both k CDM and k DGP models are found to elevate the cosmic age and accommodate the old globular clusters reported in literature. The flat DDE model is inconsistent with the old clusters. If cosmic age problem requires departure from the standard paradigm the curved cosmologies within and outside general relativity with mild spatial curvature may provide with a possible expansion history. We wish to infer that the BAO scale alone is capable of probing non-standard cosmologies that connect early and late universe evolution. We close with the perspective remark that there is enough opportunity for models beyond general relativity to address cosmological conundrums including expansion history and the age problem.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The authors have mentioned the observational data used in the paper as tables in the main manuscript. The algorithm and theoretical data that has been generated in developing this article will be shared on request to the corresponding author.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 . SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development.