Extended General Relativity for a Curved Universe

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4

Mohammed. B. Al-Fadhli1*

1College of Science, University of Lincoln, Lincoln, LN6 7TS, UK.

Abstract: The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB). Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses the positive curvature of the early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with quantum mechanics and the necessity of dark matter and dark energy might indicate its incompleteness. By utilising the Einstein–Hilbert action, this study presents extended field equations considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which could remove singularities from the theory and facilitate its quantisation. The extended equations have been utilised to derive the evolution of the Universe with reference to the scale factor of the early Universe and its radius of curvature.

Keywords: Astrometry, General Relativity, Boundary Term.

1. INTRODUCTION

Considerable efforts have been devoted to Recent evidence by Planck Legacy recent release modifying gravity in form of Conformal Gravity, (PL18) indicated the presence of an enhanced lensing Loop Quantum Gravity, MOND, ADS-CFT, String amplitude in the cosmic microwave background theory, 퐹(푅) Gravity, etc. (Mannheim, 1997, 2001; (CMB), which is notably higher than that estimated Randall and Sundrum, 1999; Garriga and Tanaka, by the lambda cold dark matter model (ΛCDM). This 1999; Germani and Sopuerta, 2002; Nojiri and endorses the positive curvature of the early Universe Odintsov, 2006, 2010; Amendola et al., 2007; Appleby with a confidence level greater than 99% (Aghanim et and Battye, 2007; Bekenstein, 2007; Nelson, 2010). al., 2020; Di Valentino, Melchiorri and Silk, 2020). Motivation to modify General Relativity (GR) Besides, the gravitational lensing by substructures of aimed to elucidate possible existence or the form of several galaxy clusters is an order of magnitude more dark matter/energy, achieve a better description of than the ΛCDM estimation (Meneghetti et al., 2020; observation data, verify theoretical restrictions in the Umetsu, 2020). This evidence endorses a spatially strong curvature regime such as within back holes as curved Universe in spite of the spacetime flatness of well as to formulate Quantum Gravity (Appleby and the local/present Universe. Further, a closed Universe Battye, 2007; Nelson, 2010). To achieve an efficient can provide an agreement with the observed CMB action for quantum corrections, several theories have anisotropy (Efstathiou, 2003). been formulated on the modification of Lagrangian Accordingly, in this study, the background/pre- gravitational fields. Such modifications appear to be existing curvature has been incorporated into the inevitable, which included higher-order curvature Einstein–Hilbert action. However, to comply with the terms as well as non-minimally coupled scalar fields energy conservation law and the compatibility of the (Vilkovisky, 1992; Capozziello and Lambiase, 2000; action, a new modulus of deformation/curvature of Garattini, 2013). However, these modifications have spacetime continuum is utilised, which is formulated to be consistent with the energy conservation law. on the theory of elasticity (Landau, 1986). The paper is organised as follows. Section 2 discusses the origination of the spacetime continuum modulus of curvature and mathematical derivations of the extended field equations. Sections 3 discusses the derivations of referenced Friedmann equations

and boundary contributions. Section 4 summarises *E-mail: [email protected] the outcomes and conclusions. Section 5 suggests the future development of this work. 1

2. Extended Field Equations By extracting one boundary part from Eq. (5) as

휇휈 4 ∫ 푔 훿푅휇휈√−푔푑 푥 (6) To consider pre-existing/background curvatures signified by the scalar curvature ℛ, a new modulus The variation in the Ricci curvature tensor 훿푅푢푣 can of deformation/curvature of the spacetime continuum be written in terms of the covariant derivative of the 퐸 =(stress/strain) in (푁/푚2) is defined, which can be 퐷 difference between two Levi-Civita connections, the expressed using Einstein field equations as Palatini identity: 푇푣 − 푇훿푣/2 c4 휇 휇 휌 휌 퐸퐷 = 푣 = ℛ (1) 훿푅휇휈 = ∇휌(훿Γ 휇휈) − ∇휈(훿Γ 휇휌) (7) 푅휇/ℛ 8πG where the stress is signified by the stress-energy The Ricci curvature tensor variation with respect to 휇 푔휇휈 tensor 푇휈 of trace 푇, while the strain is signified by the inverse metric tensor can be obtained utilising 휇 the Ricci curvature tensor 푅휈 as the change in the the metric compatibility of the covariant derivative, 푣 휇휈 curvature divided by pre-existing curvature ℛ. 훿푢 is ∇휌푔 = 0 (S. M. Carroll, 2003) as

the Kronecker delta (Straumann, 2013). According to 휇휈 휇휈 휌 휇휈 휌 푔 훿푅휇휈 = ∇휌(푔 훿Γ ) − ∇휈(푔 훿Γ ) (8) the theory of elasticity, the modulus times the volume 휇휈 휇휌 equals internal energy of reversible systems (Landau, Thus, the boundary part as a total derivative for any 1986). Thus, 퐸퐷 could represent the internal energy tensor density can be transformed according to the density of the space, i.e., the vacuum energy density. Stokes' theorem with renaming dummy indices as 퐸퐷 is proportional to the fourth-order of the speed of 휎휈 휇 휎휇 휇 휇 light, resembling Quantum field theory (QFT) energy ∭ ∇휇(푔 훿Γ 휈휎 − 푔 훿Γ 휇휎)√−푔 푑푉 ≡ ∭ ∇휇퐴 √−푔 푑푉 푉 푉 cut-off predictions of vacuum energy density (Rugh (9) 휇 3 and Zinkernagel, 2000). According to conservation of = ∯ 퐴 . 푛̂푢√−푔 푑푆 = ∮ 퐾휖√|q| 푑 푥 energy law, the Einstein–Hilbert action is extended to 푆 휕푉 휇휈 퐸 푅 where 퐾 = 퐾푢푣푞 is the trace of extrinsic curvature as 4 퐷 4 푆 = ∫[푇 + ℒ푀 ] √−푔 푑 푥 = ∫ [ + ℒ푀 ] √−푔 푑 푥 (2) 2 ℛ the second fundamental form, 푞 is the determinant of the induced metric on the manifold boundary, and where 푇 is the kinetic term, ℒ푀 is the Lagrangian 휖 푛̂ matter density denoting the potential term, 푅 is the equals 1 when the normal 푢 on the boundary is a Ricci scalar curvature and 푔 is the determinant of the spacelike entity and equals -1 when the normal is a

metric 푔푢푣. According to the principle of least action, timelike entity (Dyer and Hinterbichler, 2009). Thus, Eq. (2) should hold for any variation in the metric as the action in Eq. (2) can be rewritten in parts as

−1 퐸 훿(푅ℛ −푔) 훿(ℒ −푔) 퐷 4 3 퐸퐷 √ 푀√ 휇휈 4 푆 + 푆 + 푆 = ∫ (푅 −푔 푑 푥 + 퐾휖√|푞| 푑 푥) 훿푆 = ∫ [ ( ) + ] 훿푔 푑 푥 (3) 푐 푏 푚 푐√ 2 훿푔휇휈 훿푔휇휈 2ℛ (10) 4 The differentiation yields + ∫ ℒ푀√−푔 푑 푥

훿푅 −푔 훿ℛ −푔푅 훿 −푔푅 퐸퐷 √ √ √ where 푆푐, 푆푏, and 푆푚 denote the contributions to the 훿푆 = ∫ [ ( 휇휈 − 2 휇휈 + 휇휈 ) 2 ℛ훿푔 ℛ 훿푔 ℛ훿푔 action from the curvature without the boundary, the (4) 훿 −푔ℒ 훿ℒ −푔 √ 푀 푀√ 휇휈 4 boundary alone and the matter fields respectively. + 휇휈 + 휇휈 ] 훿푔 푑 푥 훿푔 훿푔 To solve the parted action, firstly, the action By performing the differentiation of the determinant without boundary parts is reduced to the following 휇휈 according to the Jacobi's formula as 훿푔 = 푔푔 훿푔휇휈 , 훿푆푐 + 훿푆푚 = 휇휈 휇휈 훿√−푔 = −훿푔/2√−푔 while 훿푔 푔휇휈 = −훿푔휇휈푔 (S. M. 휇휈 휇휈 퐸퐷 푅휇휈훿푔 ℛ휇휈훿푔 푔휇휈 Carroll, 2003) in addition to the differentiation of the ( − 푅 − 푅) 2 ℛ훿푔휇휈 ℛ2훿푔휇휈 2ℛ 휇휈 (11) scalar curvature 푅 which equals 푅휇휈푔 as ∫ √−푔훿푔휇휈푑4푥

푔휇휈 훿ℒ푀 휇휈 푢푣 푅 훿푔 + 푔 훿푅 − ℒ푀 + 휇휈 휇휈 [ 2 훿푔휇휈 ] ℛ훿푔휇휈

퐸 휇휈 휇휈 퐷 ℛ휇휈훿푔 + 푔 훿ℛ휇휈 The stress energy-momentum tensor is proportional − 푅 2 ℛ2훿푔휇휈 to the Lagrangian term in the action by the definition

훿푆 = ∫ −푔훿푔휇휈푑4푥 (5) 푔휇휈 √ (Straumann, 2013) as − 푅 ( 2ℛ ) 2훿ℒ푀 푇푢푣: = 푔휇휈ℒ푀 − 푔 훿ℒ 훿푔휇휈 (12) − 휇휈 ℒ + 푀 2 푀 훿푔휇휈 [ ] Additionally, the energy-momentum tensor includes The lambda/pressure is not considered which might pressure terms according to the equation of state be incorporated implicitly into stress-energy tensors. (Vikman, 2005). 2

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By substituting Eqs. (1,12) in Eq. (11) and using the By combining action parts in Eqs. (13, 19) according principle of least action yield to Eq. (10), the new extended field equations are

ℛ휇휈 1 8휋퐺 1 푅 − ℛ 1 8휋퐺 푅 − 푅 − 푅푔 = 푇 (13) 푅 − 푅푔̂ + (퐾 − 퐾푞̂ ) = 푇 (20) 휇휈 ℛ 2 휇휈 푐4 휇휈 휇휈 2 휇휈 ℛ 휇휈 2 휇휈 푐4 휇휈

휇휈 here ℛ휇휈/ℛ = ℛ휇휈/ℛ휇휈푔̅ = 푔휇휈̅ corresponds to Weyl’s The boundary tensor/term is only significant at high- conformal transformation of the metric (Kozameh et energy limits such as with black holes (Dyer and al., 1985; Straub, 2006). Eq. (13) can be rewritten as Hinterbichler, 2009) and the early Universe, and can

1 8휋퐺 remove the singularities from the theory. 푅 − 푅푔̂ = 푇 (14) 휇휈 2 휇휈 푐4 휇휈 3. Universe Evolution Model where 푔̂휇휈 = 푔휇휈 + 2푔휇휈̅ is the conformal transformation of the metric tensor due to the fact that Einstein The Friedmann–Lemaîtree–Robertson–Walker spaces are a subclass of conformal spaces (Kozameh et (FLRW) metric model is the standard cosmological al., 1985). The conformal transformation of Einstein model, which assumes an isotropic and homogenous field equations could describe the tidal distortion of Universe (Ellis and van Elst, 1998; Lachì Eze-Rey and gravitational waves in absence of matter (Penrose, Luminet, 2003), where the isotropy and homogeneity 2005) and the galaxy rotation curve as they account of the early Universe based on the CMB are consistent for the Universe curvature evolution over its age or with this model. over the conformal time. Secondly, concerning the boundary action, the 3.1 Referenced FLRW Metric Model variation in the boundary part 퐾휖√|푞| 푑3푥 in respect 휇휈 to its inverse induced metric 푞 is According to the PL18 release which revealed 휇휈 휇휈 퐾휇휈훿푞 + 푞 훿퐾휇휈 푞휇휈 the positive curvature of a closed early Universe ∫ [( − 퐾 ) 휖] √|푞|훿푞휇휈푑3푥 (15) 훿푞휇휈 2 (Aghanim et al., 2020; Di Valentino, Melchiorri and Silk, 2020), the reference radius of curvature 푟 upon as 푞휇휈 = −푞 훿푞휇휈/훿푞 , Eq. (15) can be written as 푃 휇휈 휇휈 the emission of the CMB representing the early 퐾 훿푞휇휈 훿퐾 휇휈 퐾 휇휈 휇휈 3 Universe curvature radius and the corresponding ∫ [( 휇휈 − (푞휇휈 + 2푞휇휈 )) 휖] √|푞|훿푞 푑 푥 (16) 훿푞 2 훿푞휇휈퐾 early Universe scale factor 푎푃 at the reference time 푡푝 are incorporated to reference the FLRW metric model 훿퐾 /훿푞 퐾 where 휇휈 휇휈 could can resemble the conformal as shown in Figure 1. distortion of the boundary curvature as with Eq. (13) corresponding the Weyl’s conformal transformation, which can be expressed as a positive function Ω2 on the spacetime boundary manifold as follows 휇휈 퐾휇휈훿푞 퐾 ∫ [( − (푞 + 2푞 Ω2)) 휖] √|푞|훿푞휇휈푑3푥 푟 푎 훿푞휇휈 2 휇휈 휇휈 (17) 푝 푝 푟 a By substituting the variation in the boundary part in 휃 Eq. (17) into the full action in Eq. (5) according to Eq.

(10), the variation in boundary action is 휙

훿푆푏 = 퐸 퐷 2 휇휈 퐾휇휈훿푞 퐾 (18) − (푞 + 2푞̅ ) 휇휈 3 ∫ ℛ훿푞휇휈 2ℛ 휇휈 휇휈 √|푞|훿푞 푑 푥

휇휈 휖 −퐾휇휈훿푞 푅 퐾푅 2 휇휈 + 2 (푞휇휈 + 2푞̅휇휈) [( ℛ 훿푞 2ℛ ) ]

2 where 푞̅휇휈 is the conformal induced metric 푞휇휈Ω . By Figure 1. The hypersphere of a positively curved early chosen 휖 as a timelike entity, substituting Eq. (1) in Universe expansion upon the CMB emissions. 푟푝 is the Eq. (18) and using the principle of least action reference radius of the intrinsic curvature and 푎푝 is the reference scale factor of the early Universe at the 1 푅 1 푅 −퐾휇휈 + 퐾푞̂휇휈 + 퐾휇휈 − 퐾푞̂휇휈 (19) 2 ℛ 2 ℛ corresponding reference time 푡푝 . 푎/푎푃 denotes a new dimensionless scale factor of this referenced metric. 푟, 휙, 휃 where 푞̂휇휈 is the conformally transformed induced are the comoving coordinates. metric on the spacetime manifold boundary.

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The four-dimensional spacetime interval 푑푠 of the By integrating, the scale factor evolution is referenced metric is 푀 퐺 푐 ( ) 푝 2 2 푎 휂 /푎푝 = 2 (1 − 푐표푠 휂) (27) 푎 푑푟 푐 푟푝 푟푝 푑푠2 = 푐2푑푡2 − ( + 푟2푑휃2 푎 2 1 − (푟2/푟 2) 4 푝 푝 (21) 푀 = 휋휌 푟 3 where 푝 3 푝 푝 is the mass of the early Universe 2 2 2 + 푟 푠푖푛 휃 푑휙 ) plasma. The constant in Eq. (27) can be written in

terms of the modulus 퐸퐷 representing the vacuum where 푎/푎푝 is the new dimensionless scale factor and 푟, 휙, 휃 are the comoving coordinates. Accordingly, the energy density and the Universe energy density 퐸 metric components are using Eq. (1) as 퐸/6퐸퐷. Additionally, the evolution of the imaginary 푎2 푟2 푔 = 푐2, 푔 = − ( ) / (1 − ), time 휏(휂) can be obtained by integrating the spatial 푡푡 푟푟 푎 2 푟 2 푝 푝 퐻 (22) factor over the expansion speed 휂 while initiating 푎2 푎2 at the reference imaginary time 휏 and the 푔 = − ( ) 푟2, 푔 = − ( ) 푟2푠푖푛2휃. 푝 휃휃 푎 2 휙휙 푎 2 푝 푝 corresponding spatial factor 푎푝 . Thus, by rewriting Eq. (27) in terms of the Hubble parameter by its The Ricci curvature tensor 푅푢푣 is solved using definition at 휏 as 푑휏 = 푖 푑푎 as 푝 퐻푎 Christoffel symbols for 푔푢푣 in Eqs. (22) as well as the 푝 Ricci scalar curvature 푅 as follows 휏 휂 퐸 푐 ∫ 푑휏 = 푖 ∫ (1 − cos 휂) 푑휂 (28) 2 2 2 6퐻 퐸 푟 1 푎푎̈ 2푎̇ 2푐 푟 휏푃 0 휂 퐷 푝 푅푟푟 = 2 ( 2 + 2 + 2 ) / (1 − 2), 푐 푎푝 푎푝 푟푝 푟푝 By integration, the imaginary time evolution is 2 2 2 푟 푎푎̈ 2푎̇ 2푐 퐸 푐 푅 = ( + + ), 휃휃 2 2 2 2 휏(휂) = 푖 (휂 − sin 휂) + 휏푝 푐 푎푝 푎푝 푟푝 (29) (23) 6퐻휂퐸퐷 푟0 푟2푠푖푛2휃 푎푎̈ 2푎̇ 2 2푐2 푅 = ( + + ), where 휏푝 denotes the reference imaginary time. 휙휙 푐2 푎 2 푎 2 푟 2 푝 푝 푝 According to the energy conservation law and 2 2 2 3 3 푎̈ −6 푎̈ 푎̇ 푐 푎푝 because 휌푎 = 휌푝푎푝 =constant (Ryden, 2006) and by 푅푡푡 = −3 , 푅 = 2 ( + 2 + 2 2 ) . 푎 푐 푎 푎 푎 푟푝 substituting the spatial scale factor rate in Eq. (27) to 휌푎3 = The derivations are presented in Appendix A. No constant, the evolution of density is conformal transformation is applied for the metric 푐 −3 휌휂 = 퐷푝 (1 − cos 휂 ) (30) thus, its outcomes are comparable with literature. 푟푝 Besides Appendix B presents well-defined Christoffel where 퐷푃 is a constant. According to Eq. (25), the symbols of the conformally transformed metric 푔̂휇휈. Hubble parameter is dependent on the density; thus, by substituting Eq. (30) to Eq. (25) and initiating the 3.2 Referenced Friedmann Equations 푎̈ integration at 휏푝, thus, 퐻̇ = as 푎푝

By solving the field equations for a perfect fluid 퐻 휂 −3 −4휋퐺 퐷푝 푐 2 ∫ 퐻̇ = ∫ (1 − cos 휂) 푑휂 (31) given by 푇휇휈 = (휌 + 푃/푐 ) 푢휇푢휈 + 푃푔휇휈 (Straumann, 3 푟 퐻푝 0 푝 2013) and substituting Eqs. (22,23), Friedmann equations counting for the reference curvature radius By integration, the Hubble parameter evolution is

and the reference scale factor are 1 5 푐 2 3 푐 푐 퐻휂 = 퐻푎 ( cot 휂 + cot 휂 + cot 휂) + 퐻푝 (32) 2 2 2 5 2푟푝 3 2푟푝 2푟푝 푎̇ 8휋퐺 휌 푐 푎푝 퐻2 ≡ = − , (24) 푎2 3 푎2 푟 2 푝 where, 퐻푎 and 퐻푝 are the integration constants. By

푎̈ 4휋퐺 푃 combining Eqs. (27, 29-32) in complex plan results in 퐻̇ ≡ = − (휌 + 3 ). (25) 푎 3 푐2 the hyper-spherical spacetime wave function with where 퐻 , 푃 , and 휌 are the Hubble parameter, respect the reference curvature radius as

pressure, and density respectively. The equations do 퐸 푐 2 ⃗⃗⃗⃗ not show a conformal distortion. By utilising the 휓퐿(휂)/푟푝 = ∓ ((1 − cos 휂 ) ) 0<휂≤2휋 6퐸퐷 푟푝 (33) imaginary time 휏 = 푖푡 , the referenced Friedmann 푐 equations can be solved by rewriting Eq. (24) in terms 퐻휂푎푝(1−푐표푠 푟 휂 ) 1/2 푝 2 2 푖cot 푐 of the conformal time in its parametric form 푑휂 = 푐 푐 푐 (휂−푠푖푛 휂 ) + ((휂 − sin 휂) ) 푟푝 푎푝 2 2 푒 −푖 푑휏 휏 퐻휂 푎푝 푟푝 푎 at the reference imaginary time 푝 and the corresponding reference scale factor 푎푝 as where 퐸 /퐸퐷 is a dimensionless energy parameter as 휂 2휋 3 2 2 −1/2 8휋퐺 휌푝푎푝 푐 푎푝 2 the ratio of the Universe energy density to the ∫ 푑휂 = ∫ 푎푝 ( 푎 − 2 푎 ) 푑푎 (26) 0 0 3 푟푝 vacuum energy density. The positive and negative solutions of the wave function imply the evolution in 3 3 where 휌 = 휌푝푎푝 /푎 (Ryden, 2006). two directions. 4

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3.3 Early Universe Boundary Contribution 4. Conclusions

In this study, pre-existing universal curvatures For high energy limits, gravitational contributions of and boundary contributions were considered to early Universe plasma boundary can be obtained derive the extended field equations using the Einstein using the boundary term in the extended field –Hilbert action. The extended field equations consist equations in Eqs. (20); at the reference imaginary time of Einstein field equations corresponding to Weyl’s 휏푝, there is no conformal transformation. Therefore, conformal transformation of the metric in addition to the induced metric tensor on early Universe plasma the boundary term, which removes singularities from hypersphere 푞휇휈 is given in Eqs. (34), where 푅 is the the theory. extrinsic curvature radius. The extrinsic curvature The early Universe was modelled utilising the tensor is solved utilising the formula 퐾푢푣 = −푇⃗⃗⃗휇 . ∇휈푁⃗⃗⃗⃗휈 . referenced FLRW metric model. The extended field Due to the hypersphere similarity, the covariant equations were used to derive the evolution of the derivative reduces to the partial derivative as 퐾푢푣 = 휈 Universe in reference with the scale factor of the early −푇⃗⃗⃗휇 휕푁⃗⃗⃗ /휕⃗푇⃗⃗⃗ (Pavel Grinfeld, 2013) as Universe and its radius of curvature. The positive and c2 0 0 negative solutions of the wave function might imply

푎2 the evolution in two directions. The derived Hubble 0 푅 2 0 푞휇휈 = 2 , 푎푃 evolution shows acceleration/deceleration. This may 2 푎 2 2 resolve Hubble tension while the reference radius of 0 0 2 푅 푠푖푛 휃 (34) ( 푎푃 ) curvature might be the smallest possible radius of the 0 0 0 early Universe due to the boundary contribution.

푎2 0 − 푅 0 퐾휇휈 = 2 . 푎푃 5. Future Work 2 푎 2 0 0 − 2 푅 푠푖푛 휃 ( 푎푃 ) The conformal metric tensor will be to utilised

휇휈 investigate the conformal distortion of the Friedmann The trace of the extrinsic curvature is 퐾 = 퐾휇휈푞 = equations. −2/푅. The pre-existing curvature of early Universe

plasma boundary at the reference imaginary time 휏푝 Acknowledgement: I am grateful to the Preprints 2 is the Gaussian curvature ℛ푝 = 1/푟 푝 (Pavel Grinfeld, Editors Ms Mila Marinkovic and Ms Bojana Djokic for 2013). their rapid and excellent attention in the processing 푅 On the other hand, the Ricci scalar curvature of the submissions. at 휏 can be written as the difference between kinetic 푝 Conflicts of Interest: The author declares no conflict and potential energy densities whereby substituting of interest. Friedmann equations in Eqs. (24, 25) into the Ricci Funding: This research received no funding. scalar curvature 푅 in Eqs. (23) as

6퐺푝 4휋푃푝 4휋휌푝 Appendix A 푅 = ( − ) (35) 푝 푐2 푐2 3 The Ricci curvature tensor 푅푢푣 is solved using ℛ−푅 (퐾 − By solving the boundary term ℛ 휇휈 the Christoffel symbols of the second kind given by 1 8휋퐺 1 퐾푞 ) = 푝 푇 푇 = (휌 + 훤휌 = 푔휌휆(휕 푔 + 휕 푔 − 휕 푔 ) for the referenced 2 휇휈 푐4 휇휈 for a perfect fluid given by 휇휈 휇휈 2 휇 휆휈 휈 휆휇 휆 휇휈 푃 ) 푢 푢 + 푃푔 metric tensor 푔 in Eq. (22): 푐2 휇 휈 휇휈 (Straumann, 2013) whereas the normal 휇휈 on the boundary is chosen as a spacelike entity, and r 0 푎푎̇ 1 Γ = Γ 11 = 11 푟2 푟2 then substituting Eqs. (35-34) into the boundary term 푐2푎 2 (1 − ) 푟 2 (1 − ) 푝 푟 2 푝 푟 2 as follows 푝 푝

2 2 1 6퐺푝 4휋푃푝 4휋휌푝 0 푟 푎푎̇ 1 푟 2 − 2 ( 2 − ) 2 Γ 22 = 2 2 Γ 22 = −r (1 − 2) 푟 푝 푐 푐 3 c (36) 푐 푎푝 푟푝 2 ( 2) = 8휋퐺푝휌푝 1/푟 푝 푟 푝 푟2푎푎̇ 푠푖푛2휃 푟2 Γ0 = Γ1 = −r푠푖푛2휃 (1 − ) 푉 33 2 2 33 2 Multiplying by early Universe volume 푝 yields 푐 푎푝 푟푝

4퐺푝푃푝푉푝 푟 = 1 2 3 1 2 3 푎̇ 푝 4 (37) Γ = Γ = Γ = Γ = Γ = Γ = 푐 01 02 03 10 20 30 푎

The reference curvature radius 푟푝 > 0 because any 1 Γ2 = Γ2 = Γ3 = Γ3 = reduction in the volume causes an increase in the 12 21 13 31 푟

pressure. 2 3 3 Γ 33 = −푠푖푛 휃 푐표푠휃 Γ 23 = Γ 32 = 푐표푡 휃

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휆 The Ricci curvature tensor given by 푅휇휈 = 휕휆훤 휇휈 − The 휙 − 휙 component is 휕 훤휆 + 훤휌 훤휆 − 훤휌 훤휆 휈 휇휆 휇휈 휌휆 휇휆 휌휈 . The non-zero components 0 1 2 0 1 0 2 푅 = 푅33 = 휕0Γ 33 + 휕1Γ 33 + 휕2Γ 33 + Γ 33Γ 01 + Γ 33Γ 02 of the Ricci tensor are: 휙휙 1 1 1 2 3 0 + Γ 33Γ 11 + Γ 33Γ 12 − Γ 30Γ 33 The 푡 − 푡 component is 3 1 3 2 1 2 3 1 1 − Γ 31Γ 33 − Γ 32Γ 33 푅 푡푡 = 푅 00 = −휕0Γ 01 − 휕0Γ 02 − 휕0Γ 03 − Γ 01Γ 10 푟2푎푎̇ 푠푖푛2휃 푟2 − Γ2 Γ2 − Γ3 Γ3 2 02 20 03 30 푅휙휙 = 푅33 = 휕푡 2 2 − 휕푟 r푠푖푛 휃 (1 − 2) 푐 푎푝 푟푝 푎̇ 푎̇ 2 푎̈푎 − 푎̇ 2 푎̇ 2 푎̈ 푅 = −3휕 − 3 ( ) = −3 − 3 = −3 푟2푎푎̇ 푠푖푛2휃 푎̇ 푡푡 푡 푎 푎 푎2 푎2 푎 − 휕휃푠푖푛 휃 푐표푠휃 + 2 2 2 푐 푎푝 푎

The 푟 − 푟 component is 2 2 푟 r 0 2 3 0 2 0 3 − r푠푖푛 휃 (1 − 2) 2 푅푟푟 = 푅11 = 휕0Γ 11 − 휕1Γ 12 − 휕1Γ 13 + Γ 11Γ 02 + Γ 11Γ 03 푟푝 2 푟 푟푝 (1 − 2) 푟푝 − Γ1 Γ0 + Γ1 Γ2 + Γ1 Γ3 10 11 11 12 11 13 푟2 1 푎̇ 푟2푎푎̇ 푠푖푛2휃 2 푎푎̇ 1 푎푎̇ 푎̇ − r푠푖푛 휃 (1 − 2) − 2 2 푟푝 푟 푎 푐 푎푝 푅푟푟 = 휕푡 2 − 2휕푟 + 2 2 2 푟 푟 2 2 푟 푎 푐 푎푝 (1 − 2) 푐 푎푝 (1 − 2) 푟푝 푟푝 2 2 푟 1 r 1 1 + r푠푖푛 휃 (1 − 2) + 푠푖푛휃 푐표푠휃 co푡휃 푟푝 푟 + 2 2 − 2 2 2 푟 푟 푟 푟푝 (1 − 2) 푟푝 2 2 2 2 2 2 푟 푎푎̈푠푖푛 휃 푟 푎̇ 푠푖푛 휃 2 푟 2 2 푎푎̈ 푎̇ 푎̇ 푅휙휙 = 푅33 = 2 2 + 2 2 − 푠푖푛 휃 (1 + 3 2) 푐 푎푝 푐 푎푝 푟푝 푅푟푟 = 2 + 2 + 2 2 2 푟 2 2 푟 2 2 푟 푐 푎푝 (1 − 2) 푐 푎푝 (1 − 2) 푐 푎푝 (1 − 2) 푟푝 푟푝 푟푝 2 2 2 2 2 푟 푎̇ 푠푖푛 휃 2 + 푠푖푛 휃 − 푐표푠 휃 + 2 2 푐 푎푝 + 2 2 푟 푟푝 (1 − 2) 푟푝 2 2 푟 2 2 2 − 푠푖푛 휃 ( 2) + 푐표푠 휃 푎푎̈ 2푎̇ 2푐 푟푝 ( 2 + 2 + 2 ) 푎푝 푎푝 푟푝 푅 = 푟푟 푟2 푟2푎푎̈푠푖푛2휃 푟2푎̇ 2푠푖푛2휃 푟2 푐2 (1 − ) 2 푟 2 푅휙휙 = 푅33 = 2 2 + 2 2 2 + 2푠푖푛 휃 2 푝 푐 푎푝 푐 푎푝 푟푝

푟2푠푖푛2휃 푎푎̈ 2푎̇ 2 2푐2 푅휙휙 = 2 ( 2 + 2 + 2 ) The 휃 − 휃 component is 푐 푎푝 푎푝 푟푝

0 1 3 0 1 0 3 푅휃휃 = 푅22 = 휕0Γ 22 + 휕1Γ 22 − 휕2Γ 23 + Γ 22Γ 01 + Γ 22Γ 03 Thirdly, the inverse metric tensor 푔푢푣 is

1 1 1 3 2 0 + Γ 22Γ 11 + Γ 22Γ 13 − Γ 20Γ 22 1 2푎푝 2 0 0 (1 0 ) 0 푐 푎 − Γ2 Γ1 − Γ3 Γ3 2 21 22 23 32 푟 (1 − 2) 푟푝 2 2 2 푟 푎푎̇ 푟 푟 푎푎̇ 푎̇ 0 − 2 0 0 ( ) 푎 푅휃휃 = 휕푡 2 2 − 휕푟 r (1 − 2) − 휕휃 cot 휃 + 2 2 ( 2) 푐 푎푝 푟푝 푐 푎푝 푎 푎푝 푔푢푣 = −1 0 0 0 푟2 1 푎2 − r (1 − ) − cot2(휃) ( ) 푟2 2 푎 2 푟푝 푟 푝 −1 2 2 2 2 2 2 0 0 0 2 푟 푎푎̈ 푟 푎̇ 푟 푟 푎̇ 푎 2 2 푅 = + + 3 − 1 + csc2(휃) + ( 2) 푟 푠푖푛 휃 휃휃 2 2 2 2 ( 2 ) 2 2 ( 푎푝 ) 푐 푎푝 푐 푎푝 푟푝 푐 푎푝

푟2 2( ) − (1 − 2) − cot 휃 푢푣 푟푝 Finally, the Ricci scalar curvature 푅 = 푅푢푣푔 which equals the Ricci curvature tensor time the inverse 푟2푎푎̈ 푟2푎̇ 2 푟2 2( ) 2( ) metric tensor as follows 푅휃휃 = 2 2 + 2 2 2 + (2 2) − 1 + csc 휃 − cot 휃 푐 푎푝 푐 푎푝 푟푝 2 2 2 −6 푎̈ 푎̇ 푐 푎푝 2 2 2 푢푣 푟 푎푎̈ 2푎̇ 2푐 푅 = 푅푢푣푔 = 2 ( + 2 + 2 2 ) 푐 푎 푎 푎 푟푝 푅휃휃 = 2 ( 2 + 2 + 2 ) 푐 푎푝 푎푝 푟푝

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Appendix B

By utilising the conformal time and according to Eq. 0 푎̇ 푎푝 Γ 00 = 푎 (푎 − 2푎푝) (14), the conformally transformed metric 푔̂휇휈 is

푔̂ = (푔 + 2푔̅ ) = 0 1 r 푢푣 푢푣 푢푣 Γ 11 Γ = 11 푟2 푟 2 (1 − ) 2푎 푎푎̇ (푎 − 푎푝) 푝 푟 2 (1 − 푝) 푐2 0 0 0 = 푝 푟2 (푎 − 2푎 ) 푎 푐2푎 2 (1 − ) 푝 2 푝 푟 2 2푎 푎 푝 ( − 2) 푎푝 푎푝 0 0 0 푟2푎푎̇ (푎 − 푎 ) 푟2 2 0 푝 1 푟 Γ 22 = Γ 22 = −r (1 − ) (1 − 2) 푐2푎 2 (푎 − 2푎 ) 푟 2 푟푝 푝 푝 푝 2 2푎 푎 0 0 ( − ) 푟2 0 Γ0 2 2 33 푟 푎푝 푎푝 Γ1 = −r푠푖푛2휃 (1 − ) 33 푟 2 2푎 푎2 푟2푎푎̇ 푠푖푛2휃 (푎 − 푎 ) 푝 0 0 0 ( − ) 푟2푠푖푛2휃 푝 2 = 2 2 ( 푎푝 푎푝 ) 푐 푎푝 (푎 − 2푎푝)

1 2 3 1 2 3 푎̇ (푎 − 푎푝) Γ 01 = Γ 02 = Γ 03 = Γ 10 = Γ 20 = Γ 30 = Christoffel symbols of the second kind for the 푔̂휇휈 metric 푎 (푎 − 2푎푝) tensor are solves as given in the table 1 Γ2 = Γ2 = Γ3 = Γ3 = 12 21 13 31 푟

2 3 3 Γ 33 = −푠푖푛 휃 푐표푠휃 Γ 23 = Γ 32 = 푐표푡 휃

Data Availability: This study is purely based on mathematical derivations. All derivations are either presented in the main text or in appendices.

References

Aghanim, N. et al. (2020) ‘Planck 2018 results. V. CMB Noether symmetry approach’, General Relativity and power spectra and likelihoods’, Astronomy & Gravitation. Springer, 32(2), pp. 295–311. doi: Astrophysics. EDP Sciences, 89. doi: 10.1051/0004- 10.1023/A:1001935510837. 6361/201936386. Dyer, E. and Hinterbichler, K. (2009) ‘Boundary terms, Amendola, L. et al. (2007) ‘Conditions for the variational principles, and higher derivative modified cosmological viability of f(R) dark energy models’, gravity’, Physical Review D - Particles, Fields, Physical Review D - Particles, Fields, Gravitation and Gravitation and Cosmology, 79(2). doi: Cosmology. American Physical Society, 75(8), p. 083504. 10.1103/PhysRevD.79.024028. doi: 10.1103/PhysRevD.75.083504. Efstathiou, G. (2003) Is the low cosmic microwave Appleby, S. A. and Battye, R. A. (2007) ‘Do consistent background quadrupole a signature of spatial curvature?, $F(R)$ models mimic General Relativity plus Mon. Not. R. Astron. Soc. doi: 10.1046/j.1365- $\Lambda$?’, Physics Letters, Section B: Nuclear, 8711.2003.06940.x. Elementary Particle and High-Energy Physics, 654(1–2), Ellis, G. F. R. and van Elst, H. (1998) ‘Cosmological pp. 7–12. doi: 10.1016/j.physletb.2007.08.037. models (Carg\`{e}se lectures 1998)’. Available at: Bekenstein, J. D. (2007) ‘The modified Newtonian http://arxiv.org/abs/gr-qc/9812046 dynamics-MOND-and its implications for new physics’, Garattini, R. (2013) ‘Distorting general relativity: Contemporary Physics, 47(6), pp. 387–403. doi: Gravity’s rainbow and f(R) theories at work’, Journal of 10.1080/00107510701244055. Cosmology and Astroparticle Physics. IOP Publishing, Capozziello, S. and Lambiase, G. (2000) ‘Higher-order 2013(6), p. 017. doi: 10.1088/1475-7516/2013/06/017. corrections to the effective gravitational action from Garriga, J. and Tanaka, T. (1999) ‘Gravity in the Randall-

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 January 2021 doi:10.20944/preprints202010.0320.v4

Sundrum Brane World’, Physical Review Letters, 84(13). Physical Society, 83(23), pp. 4690–4693. doi: doi: 10.1103/PhysRevLett.84.2778. 10.1103/PhysRevLett.83.4690. Germani, C. and Sopuerta, C. F. (2002) ‘String Inspired Rugh, S. E. and Zinkernagel, H. (2000) ‘The Quantum Brane World Cosmology’, Physical Review Letters. Vacuum and the Cosmological Constant Problem’, Studies American Physical Society, 88(23), p. 4. doi: in History and Philosophy of Science Part B - Studies in 10.1103/PhysRevLett.88.231101. History and Philosophy of Modern Physics. Elsevier Ltd, Kozameh, C. et al. (1985) ‘Conformal Einstein spaces’, 33(4), pp. 663–705. Springer. Ryden, B. (2006) Introduction to Cosmology. San Lachì Eze-Rey, M. and Luminet, J.-P. (2003) ‘COSMIC Francisco, CA, USA: Addison Wesley, ISBN 0-8053- TOPOLOGY’, arXiv:gr-qc/9605010v2 9 Jan 2003. 8912-1. Landau, L. D. (1986) Theory of Elasticity, Theory of S. M. Carroll (2003) ‘Spacetime and Geometry: An Elasticity. Elsevier. doi: 10.1016/c2009-0-25521-8. Introduction to General Relativity’. doi: 10.1088/0264- Mannheim, P. D. (1997) ‘Are Galactic Rotation Curves 9381/22/20/B01. Really Flat?’, The Astrophysical Journal. American Straub, W. O. (2006) Simple Derivation of the Weyl Astronomical Society, 479(2), pp. 659–664. doi: Conformal Tensor, academia.edu. 10.1086/303933. Straumann, N. (2013) ‘General Relativity (Graduate Texts Mannheim, P. D. (2001) ‘Cosmic Acceleration as the in Physics)’, in Springer. Springer. Solution to the Cosmological Constant Problem’, The Umetsu, K. (2020) ‘Cluster–galaxy weak lensing’, Astrophysical Journal. American Astronomical Society, Astronomy and Astrophysics Review. Springer Science and 561(1), pp. 1–12. doi: 10.1086/323206. Business Media Deutschland GmbH, pp. 1–106. doi: Meneghetti, M. et al. (2020) ‘An excess of small-scale 10.1007/s00159-020-00129-w. gravitational lenses observed in galaxy clusters’, Science. Di Valentino, E., Melchiorri, A. and Silk, J. (2020) ‘Planck American Association for the Advancement of Science, evidence for a closed Universe and a possible crisis for 369(6509), pp. 1347–1351. doi: 10.1126/science.aax5164. cosmology’, Nature Astronomy. Nature Research, 4(2), pp. Nelson, W. (2010) ‘Restricting Fourth Order Gravity via 196–203. doi: 10.1038/s41550-019-0906-9. Cosmology’, Physical Review D - Particles, Fields, Vikman, A. (2005) ‘Can dark energy evolve to the Gravitation and Cosmology, 82(12). doi: phantom?’, Physical Review D - Particles, Fields, 10.1103/PhysRevD.82.124044. Gravitation and Cosmology. American Physical Society, Nojiri, S. and Odintsov, S. D. (2006) ‘Introduction to 71(2), p. 023515. doi: 10.1103/PhysRevD.71.023515. Modified Gravity and Gravitational Alternative for Dark Vilkovisky, G. A. (1992) ‘Effective action in quantum Energy’, International Journal of Geometric Methods in gravity’, Classical and Quantum Gravity. IOP Publishing, Modern Physics, 4(1), pp. 115–145. doi: 9(4), pp. 895–903. doi: 10.1088/0264-9381/9/4/008. 10.1142/S0219887807001928. Nojiri, S. and Odintsov, S. D. (2010) ‘Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models’, Physics Reports, 505(2–4), pp. 59– 144. doi: 10.1016/j.physrep.2011.04.001. Pavel Grinfeld (2013) Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. Penrose, R. (2005) The Road to Reality: A Complete Guide to the Laws of the Universe. Randall, L. and Sundrum, R. (1999) ‘An alternative to compactification’, Physical Review Letters. American