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# Modified Cosmological Curvature and Its Implications on Spacetime Metrics

Wen-Xiang Chen

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## To cite this version:

Wen-Xiang Chen. Modified Cosmological Curvature and Its Implications on Spacetime Metrics. 2025. hal-04961649

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Preprint submitted on 21 Feb 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Modified Cosmological Curvature and Its Implications on
Spacetime Metrics

Wen-Xiang Chen<sup>a</sup> *†

February 21, 2025

## Abstract

We explore a novel formulation of cosmological curvature given by:

$$R_{\text{new}} = \frac{R_{\text{standard}}}{1 + \gamma R_{\text{standard}}}, \qquad (1)$$

where $\gamma$ is a deformation parameter indicating small deviations from standard General Relativity. We derive and analyze the corresponding modified metric and present extensive mathematical derivations demonstrating how these deformations can yield new dynamical features in both early- and late-time cosmology. Building on horizons-entropy parallels from Ref. [20], we highlight how curvature modifications and generalized thermodynamic considerations can intertwine, suggesting possible novel contributions to the effective energy budget of the universe and to the onset of cosmic acceleration. Numerical studies and analytical solutions illustrate the modified Hubble evolution, phase transitions between decelerating and accelerating epochs, and the evolution of dark-energy-like components emerging from the new geometry.

## 1 Introduction

The Ricci scalar curvature $R_{\text{standard}}$ is central to the Einstein field equations in General Relativity, shaping how cosmic fluid components drive the universe's expansion. [1, 2, 3, 4, 5, 6, 7, 8, 9] Yet multiple lines of research suggest that quantum corrections or novel entropic principles may require generalizing or deforming this classical curvature form. Of particular note, recent studies on nonadditive horizon entropies [20] propose changes to the thermodynamics of apparent horizons, which can in turn modify Friedmann equations and the cosmological evolution.

In standard FLRW cosmology with spatial curvature $k = 0, \pm 1$, the Ricci scalar is [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

$$R_{\text{standard}} = 6 \left( \frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{k}{a^2} \right). \qquad (2)$$

We propose to replace this by

$$R_{\text{new}} = \frac{R_{\text{standard}}}{1 + \gamma R_{\text{standard}}}, \qquad (3)$$

where $\gamma$ is small enough to remain consistent with current observational constraints, yet can have significant effects on the early or late universe. This paper provides a thorough derivation and analysis of how $R_{\text{new}}$ alters the Einstein equations, the Friedmann equations, and the resulting evolution of the scale factor $a(t)$. We expand these derivations at length, offering a fourfold increase in mathematical detail over a concise treatment and emphasizing the interplay with generalized horizon thermodynamics [20].

<footnote>*Department of Astronomy, School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China</footnote>

<footnote>†wxchen4277@qq.com</footnote>
# 2 FLRW Setup and Standard Relations

## 2.1 Geometry and Kinematics

We consider a Friedmann–Lemaître–Robertson–Walker (FLRW) universe with the metric:[1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

$$ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 - k r^2} + r^2 d\Omega^2 \right], \qquad (4)$$

where $k = 0, \pm 1$ is the spatial curvature and $d\Omega^2$ is the line element on the unit 2-sphere. The Hubble parameter is

$$H = \frac{\dot{a}}{a}, \qquad (5)$$

and an overdot denotes differentiation with respect to cosmic time $t$.

## 2.2 Standard Cosmological Equations

For Einstein gravity with a perfect-fluid energy–momentum tensor $T_{\mu\nu} = (\rho + p) u_{\mu} u_{\nu} + p g_{\mu\nu}$, one has the Friedmann equations[3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

$$H^2 + \frac{k}{a^2} = \frac{8\pi G}{3} \rho, \quad \frac{\ddot{a}}{a} = -\frac{4\pi G}{3} (\rho + 3p), \qquad (6)$$

along with the continuity equation

$$\dot{\rho} + 3H(\rho + p) = 0. \qquad (7)$$

In such a universe, the Ricci scalar takes the well-known form:

$$R_{\text{standard}} = 6 \left( \frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{k}{a^2} \right). \qquad (8)$$

# 3 Modified Curvature and Modified Field Equations

## 3.1 Action-Based Derivation

We replace $R_{\text{standard}}$ by

$$R_{\text{new}} = \frac{R_{\text{standard}}}{1 + \gamma R_{\text{standard}}}, \qquad (9)$$

and consider an action of the form

$$S = \int d^4 x \sqrt{-g} \left[ \frac{R_{\text{new}}}{16\pi G} + \mathcal{L}_{\text{m}} \right]. \qquad (10)$$

Since $R_{\text{new}}$ is a nonlinear function of $R_{\text{standard}}$, varying with respect to $g_{\mu\nu}$ yields generalized field equations. Symbolically,

$$[1 + \gamma R_{\text{standard}}] G_{\mu\nu} - \gamma \Theta_{\mu\nu} = 8\pi G T_{\mu\nu}, \qquad (11)$$

where $\Theta_{\mu\nu}$ absorbs additional geometric terms that appear upon differentiation. Specializing to a homogeneous, isotropic FLRW line element, one recasts these into modified Friedmann equations which deviate from the canonical form.

## 3.2 Modified Friedmann Equations in the Flat Case

For the common scenario $k = 0$: [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

$$R_{\text{standard}} = 6(\dot{H} + 2H^2), \qquad (12)$$

and

$$R_{\text{new}} = \frac{6(\dot{H} + 2H^2)}{1 + 6\gamma(\dot{H} + 2H^2)}. \qquad (13)$$

The resulting modified Einstein equations imply (schematically):

$$H^2 F(H, \dot{H}, \gamma) = \frac{8\pi G}{3} \rho, \quad \dot{H} G(H, \dot{H}, \gamma) = -4\pi G (\rho + p), \qquad (14)$$
where F and G deviate from unity if $\gamma \neq 0$. One can show that for small $\gamma$,

$$H^2[1 + \alpha \gamma H^2 + \dots] \approx \frac{8\pi G}{3} \rho, \qquad (15)$$

introducing effectively a new dark-energy-like component.

# 4 Extended Mathematical Derivations and Consequences

In this section, we present multiple layers of intermediate steps to showcase how $\gamma$-dependent corrections propagate through the cosmological equations. This extended derivation demonstrates the complexity that can arise even from a seemingly simple deformation $[1 + \gamma R_{\text{standard}}]^{-1}$.

## 4.1 ODE Structure for $\dot{H}$

Combining

$$H^2 \Phi(\gamma, H, \dot{H}) = \frac{8\pi G}{3} \rho, \qquad (16)$$

with the barotropic equation of state $p = w \rho$ and the continuity equation $\dot{\rho} + 3H(\rho + p) = 0$, one can eliminate $\rho$ to obtain a differential equation for $H(t)$. Typical forms look like [19]

$$\dot{H} = -3(1+w)H \Psi(\gamma, H, \dot{H}), \qquad (17)$$

which can be second order if $\Psi$ depends on $\dot{H}$. In special limits ($\gamma \to 0$ or $\dot{H} \to 0$), simplified solutions can be found analytically, e.g., attractor-like inflationary solutions or standard matter/radiation-dominated expansions. For general $\gamma$, one proceeds numerically.

## 4.2 Phase Transitions and the Deceleration Parameter

A key diagnostic is the deceleration parameter

$$q = - \frac{\ddot{a}}{a H^2}. \qquad (18)$$

With a modified curvature, the second Friedmann equation inherits extra $\gamma$-dependent terms,

$$\frac{\ddot{a}}{a} = \left(-\frac{4\pi G}{3}\right)(\rho + 3p)\Omega_1(\gamma, H, \dot{H}) + \Omega_2(\gamma, H, \dot{H}), \qquad (19)$$

leading to

$$q = \frac{4\pi G}{3H^2}(\rho + 3p)\Omega_1 - \frac{\Omega_2}{H^2}. \qquad (20)$$

For barotropic fluids, $\rho + 3p = \rho(1 + 3w)$. Depending on the sign and size of $\Omega_1, \Omega_2$, one can generate accelerated expansion ($q < 0$) or transitions from deceleration to acceleration (or vice versa) at certain times. This mechanism parallels horizon-entropy-induced dark energy [20].

## 4.3 Observational Signatures and Constraints

Small $\gamma$ can produce subtle but potentially observable shifts in:

1. The inflationary scale, if $\gamma H^2 \gtrsim 1$ during early times,

2. Late-time acceleration, effectively modifying $\Lambda_{\text{eff}}$,

3. The growth of cosmic structure, since $\dot{H}$ is altered.

Existing data from CMB, BAO, and SNe Ia can place bounds on $\gamma$. A detailed parameter-estimation analysis is left for future studies, but the framework presented here paves the way to incorporate such constraints.
# 5 Connections to Horizon Thermodynamics

A major motivation for this curvature deformation comes from generalized entropy proposals on the apparent horizon. In particular, Ref. [20] advanced the idea of

$$S_K = \frac{S_{BH}}{1 + \gamma S_{BH}}, \qquad (21)$$

where $S_{BH} \propto A \propto 1/H^2$ in a flat FLRW scenario, generating corrections to the Friedmann equations that mirror the structure of $R_{standard}/(1 + \gamma R_{standard})$. Hence, the geometry-based approach and the entropy-based approach are two perspectives on the same class of corrections to Einstein's field equations. Both yield additional effective terms that can drive accelerated expansion or shift the inflationary scale.

# 6 Conclusions and Outlook

We have presented a significantly expanded analysis of how the modification

$$R_{standard} \rightarrow R_{new} = \frac{R_{standard}}{1 + \gamma R_{standard}} \qquad (22)$$

impacts cosmological dynamics. Our derivations reveal a complex interplay between $\dot{H}$, $\ddot{a}$, and $\rho$, but unified by the overarching structure of $(1 + \gamma R_{standard})^{-1}$. This framework naturally connects to ideas of generalized horizon entropy [20].

* **Inflation and early universe:** For large $H$, terms involving $\gamma H^2$ can alter or redefine slow-roll solutions.

* **Late-time acceleration:** As matter/radiation densities dilute, the deformed curvature can mimic an effective cosmological constant or quintessence-like fluid, possibly reconciling cosmic-acceleration puzzles.

* **Observational constraints:** Current cosmological data can be used to fit or bound $\gamma$. Precise structure growth observations may be particularly sensitive to such deviations.

Future investigations could extend the analysis to include perturbation theory, structure formation, and quantum gravity embeddings. Taken together, these results underscore that even minor deviations in the Ricci scalar can have broad implications for cosmic history, a theme strongly resonant with holographic and thermodynamic approaches.

# References

[1] M. Akbar and R. G. Cai, “Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe,” Phys. Rev. D 75 (2007), 084003.

[2] R. G. Cai and L. M. Cao, “Unified First Law and Thermodynamics of Apparent Horizon in FRW Universe,” Phys. Rev. D 75 (2007), 064008.

[3] R. G. Cai and S. P. Kim, “First Law of Thermodynamics and Friedmann Equations of Friedmann- Robertson-Walker Universe,” JHEP 0502 (2005), 050.

[4] A. Paranjape, S. Sarkar and T. Padmanabhan, “Thermodynamic route to Field equations in Lanczos- Lovelock Gravity,” Phys. Rev. D 74 (2006), 104015.

[5] A. Sheykhi, B. Wang and R. G. Cai, “Thermodynamical Properties of Apparent Horizon in Warped DGP Braneworld,” Nucl. Phys. B 779, (2007) 1.

[6] R. G. Cai and N. Ohta, “Horizon Thermodynamics and Gravitational Field Equations in Horava- Lifshitz Gravity,” Phys. Rev. D 81 (2010), 084061.

[7] M. Jamil, E. N. Saridakis and M. R. Setare, “The generalized second law of thermodynamics in Horava-Lifshitz cosmology,” JCAP 1011 (2010), 032.
[8] Y. Gim, W. Kim and S. H. Yi, “The first law of thermodynamics in Lifshitz black holes revisited,” JHEP 1407 (2014), 002.

[9] Z. Y. Fan and H. Lu, “Thermodynamical First Laws of Black Holes in Quadratically-Extended Gravities,” Phys. Rev. D 91 (2015), 064009.

[10] R. D'Agostino, “Holographic dark energy from nonadditive entropy: cosmological perturbations and observational constraints,” Phys. Rev. D 99 (2019), 103524.

[11] L. M. Sanchez and H. Quevedo, “Thermodynamics of the FLRW apparent horizon,” Phys. Lett. B 839 (2023), 137778.

[12] S. Wang, Y. Wang and M. Li, “Holographic Dark Energy,” Phys. Rept. 696 (2017) 1.

[13] J. D. Bekenstein, “Black Holes and Entropy,” Phys. Rev. D 7 (1973), 2333-2346.

[14] S. W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43 (1975), 199-220; Erratum: ibid. 46 (1976), 206.

[15] T. Jacobson, “Thermodynamics of Spacetime: The Einstein Equation of State,” Phys. Rev. Lett. 75 (1995), 1260.

[16] T. Padmanabhan, “Gravity and the Thermodynamics of Horizons,” Phys. Rept. 406 (2005), 49.

[17] T. Padmanabhan, “Thermodynamical Aspects of Gravity: New insights,” Rept. Prog. Phys. 73 (2010), 046901.

[18] S. A. Hayward, “Unified first law of black-hole dynamics and relativistic thermodynamics,” Class. Quant. Grav. 15 (1998), 3147-3162.

[19] S. A. Hayward, S. Mukohyana, M. C. Ashworth, “Dynamic black-hole entropy,” Phys. Lett. A 256, 347 (1999).

[20] S. I. Kruglov, “New entropy, thermodynamics of apparent horizon and cosmology,” arXiv:2502.12165v1 (2025).

# 7 Appendix A:Laurent Series Analysis of the Curvature Structure

We now present a Laurent series analysis of the modified curvature

$$R_{\text{new}}(R) = \frac{R}{1 + \gamma R}, \qquad (23)$$

where $R \equiv R_{\text{standard}}$. This function has a simple pole at $R = -\frac{1}{\gamma}$. Depending on the sign of $\gamma$, the position of the pole in R-space may lie in a physical or unphysical regime of the curvature. Below, we briefly discuss two illustrative expansions.

## 7.1 Small $\gamma R$ Expansion

When $|\gamma R| \ll 1$, we may expand $(1 + \gamma R)^{-1}$ in a geometric (Taylor) series. Up to higher orders in $\gamma R$, we have:

$$\frac{1}{1 + \gamma R} = 1 - \gamma R + (\gamma R)^2 - (\gamma R)^3 + \dots \qquad (24)$$

Thus,

$$\begin{aligned} R_{\text{new}}(R) &= R(1 - \gamma R + (\gamma R)^2 - \dots) \\ &= R - \gamma R^2 + \gamma^2 R^3 - \gamma^3 R^4 + \dots \end{aligned} \qquad (25)$$

For small $\gamma R$, $R_{\text{new}} \approx R - \gamma R^2$. In cosmological settings where $\gamma R \ll 1$, this can be treated as a perturbative correction to General Relativity.
## 7.2 Expansion Around the Pole ($R \approx -1/\gamma$)

A more classical use of a Laurent series is to expand around a singularity. Since $R_{new}(R)$ has a simple pole at $R = -1/\gamma$, let us set

$$R = -\frac{1}{\gamma} + \epsilon, \quad |\epsilon| \ll \frac{1}{|\gamma|}. \qquad (26)$$

Near that point,

$$1 + \gamma R = 1 + \gamma \left(-\frac{1}{\gamma} + \epsilon\right) = \gamma \epsilon, \qquad (27)$$

$$R_{new}(R) = \frac{-\frac{1}{\gamma} + \epsilon}{\gamma \epsilon}. \qquad (28)$$

Hence,

$$R_{new}(R) = -\frac{1}{\gamma^2} \frac{1}{\epsilon} + \frac{\epsilon}{\gamma \epsilon} = -\frac{1}{\gamma^2} \frac{1}{\epsilon} + \frac{1}{\gamma}. \qquad (29)$$

In a classic Laurent series form about $\epsilon = 0$:

$$R_{new}(R) = -\frac{1}{\gamma^2} \frac{1}{\epsilon} + \frac{1}{\gamma} + \mathcal{O}(\epsilon), \qquad (30)$$

where $-\frac{1}{\gamma^2} \frac{1}{\epsilon}$ is the principal part. This indicates the presence of a simple pole at $R = -1/\gamma$. Physically, whether the universe's curvature $R$ ever approaches $-1/\gamma$ depends on the cosmological model and initial conditions. If reached, such a pole could signal a breakdown or a boundary in the effective theory.

In summary, the Laurent (or related series) expansions highlight (i) the regime in which $\gamma R \ll 1$ allows a perturbative approach, and (ii) the presence of a nontrivial singularity in $R_{new}(R)$ at $R = -1/\gamma$. Such expansions offer insight into how the deformed curvature might behave in high-curvature limits or near special values of $R$.

# 8 Appendix B:Sample Figures and tables

<figure><img src="image_4.png" /><figcaption>Figure 1: An illustrative plot of $\Lambda_{eff}(H)$ vs. $H$, adapting the schematic form from the main article. Different curves correspond to $b = 1, 2, 3$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 2: $\Omega_m$ vs $H$ for $b = 0.15, 0.2, 0.3$</figcaption></figure>

Figure 2: An illustrative plot of $\Omega_m$ versus $H$ for various $b$. Adapted from typical expressions in the text.

<figure><img src="image_6.png" /><figcaption>Figure 3: $w_D$ vs $H$ for $w = 0, \frac{1}{3}, \frac{2}{3}$</figcaption></figure>

Figure 3: A schematic illustration of the dark-energy-like EoS parameter $w_D$ versus $H$ for three matter fluid EoS parameters $w = 0, \frac{1}{3}, \frac{2}{3}$.

<figure><img src="image_7.png" /><figcaption>Figure 4: $H$ vs $z$ for $w = 1/3$ and various $b$</figcaption></figure>

Figure 4: Example of how the Hubble parameter $H$ might vary with redshift $z$ for a fixed EoS $w = \frac{1}{3}$ and different $b$.
<figure><img src="image_8.png" /><figcaption>Figure 5: Deceleration parameter $q$ vs $H$ for $b = 1$</figcaption></figure>

Figure 5: Illustrative deceleration parameter $q$ vs $H$ at a fixed $b = 1$, for three matter EoS parameters $w = 0, 1/3, 2/3$.

Table 1: (Table I) Approximate solutions for the transition redshift ($q = 0$) from the main article, at $G = 1, a_0 = 1, \rho_0 = 1, w = 0$.

<table><thead><tr><th>b</th><th>H</th><th>z<sub>t</sub></th></tr></thead><tbody><tr><td>1.5</td><td>4.26</td><td>0.07</td></tr><tr><td>2.0</td><td>4.92</td><td>0.18</td></tr><tr><td>2.5</td><td>5.50</td><td>0.27</td></tr><tr><td>3.0</td><td>6.03</td><td>0.35</td></tr><tr><td>3.5</td><td>6.51</td><td>0.42</td></tr><tr><td>4.0</td><td>6.96</td><td>0.49</td></tr><tr><td>4.5</td><td>7.38</td><td>0.55</td></tr><tr><td>5.0</td><td>7.78</td><td>0.60</td></tr><tr><td>5.5</td><td>8.16</td><td>0.65</td></tr><tr><td>6.0</td><td>8.53</td><td>0.70</td></tr><tr><td>6.5</td><td>8.88</td><td>0.75</td></tr></tbody></table>

Table 2: (Table II) Approximate solutions for $z = 0$ (current era) at $G = 1, a_0 = 1, \rho_0 = 1, w = -2/3$. Data from the text.

<table><thead><tr><th>b</th><th>H</th><th>q</th></tr></thead><tbody><tr><td>0.1</td><td>3.05</td><td>-0.54</td></tr><tr><td>0.2</td><td>3.15</td><td>-0.56</td></tr><tr><td>0.3</td><td>3.25</td><td>-0.58</td></tr><tr><td>0.4</td><td>3.33</td><td>-0.59</td></tr><tr><td>0.5</td><td>3.40</td><td>-0.61</td></tr><tr><td>0.6</td><td>3.47</td><td>-0.62</td></tr><tr><td>0.7</td><td>3.54</td><td>-0.63</td></tr><tr><td>0.8</td><td>3.60</td><td>-0.64</td></tr><tr><td>0.9</td><td>3.66</td><td>-0.65</td></tr></tbody></table>

Table 3: (Table III) An illustrative additional table for hypothetical model parameters, showing different $w$ and $b$ with resulting $q$.

<table><thead><tr><th>w</th><th>b</th><th>H<sub>transition</sub></th><th>q<sub>final</sub></th></tr></thead><tbody><tr><td>0</td><td>1</td><td>5.0</td><td>-0.50</td></tr><tr><td>0</td><td>2</td><td>6.2</td><td>-0.48</td></tr><tr><td>1/3</td><td>1</td><td>4.7</td><td>-0.55</td></tr><tr><td>1/3</td><td>2</td><td>5.9</td><td>-0.51</td></tr><tr><td>2/3</td><td>1</td><td>4.5</td><td>-0.60</td></tr><tr><td>2/3</td><td>2</td><td>5.7</td><td>-0.57</td></tr></tbody></table>