ΛCDM+S - Thermodynamic Cosmology: Simulating The Universe's Expansion Without Dark Energy

One of the biggest scientific discoveries of the 20th century was that the universe's expansion is accelerating. It has now been 27 years since that groundbreaking discovery, and the physical nature of dark energy, the main component theorized to be responsible for the universe’s accelerated expansion, remains unknown. This lack of understanding represents a great challenge in fundamental physics, leading to some of the biggest unsolved mysteries of modern physics.

The current standard model used to describe the universe is the ΛCDM (Lambda-Cold Dark Matter) model. In this model, Λ is responsible for the accelerated expansion of the universe as a cosmological constant, while “CDM” refers to cold dark matter. ΛCDM has been very successful at explaining a wide range of cosmological phenomena. However, it fails to reconcile the precise current expansion rate of the universe across different datasets. Furthermore, 1701 Type Ia Supernovae show systematic luminosity deviations from the predictions made by the ΛCDM Model.

Fifty years ago, physicists discovered that black holes have entropy (the thermodynamic measure for the disorder of a system), and it is proportional to the area of its event horizon. This led me to a broader question: if the observable universe has a horizon, could similar thermodynamic principles apply to it? I hypothesize that the accelerated expansion is explained by a combination of gravity and entropy. In my project, I developed an extended ΛCDM model accounting for entropy growth. I then compared this model with the ΛCDM model and astrophysical measurements.

My research began from a literature review of a variety of books and textbooks published about relevant topics by authors such as Steven Weinberg, Adam Reiss, and Stephen Hawking, alongside roughly 75 peer-reviewed papers and datasets from journals such as Astronomy and Astrophysics and The Astrophysics Journal.

Mathematical Basis

Based on this literature review, I defined an initial mathematical framework by combining the solutions of two Friedmann Equations (Friedmann, A. 1922):

A matter dominated universe (gravitationally-dominated).

A derived entropy-driven universe model (thermodynamically dominated, dark energy-like).

The two solutions were numerically evaluated in Python, gathering a total of 972,000 data points. An analysis of these evaluations showed that the gravity-dominated solution closely matches the early universe's expansion history, while the thermodynamically dominated solution aligns with the late universe's expansion history. To account for this transition, I added a Sigmoid function to the mathematical framework, producing a model consistent with the early and late universe dynamics.

Bayesian Inference via Monte Carlo Method

The Sigmoid function model introduced 4 new parameters (priors). Since the exact values of these parameters are unknown, I modelled them through Gaussian distributions to capture a range of possible values and model uncertainty. Then, I ran 500 Monte Carlo simulations (3.5M data points) over the transitioning model to explore the prior predictive distribution for the evolution of the universe. Based on these simulations, I obtained a likelihood function (how well my simulations predict observed datasets) by comparing the information from the simulations to 48 million samples from observed cosmic datasets. Lastly, I refined my priors using a Metropolis-Hastings Markov Chain Monte Carlo (MCMC) with 64,000 samples. The MCMC produced posterior distributions with reduced uncertainty, completing Bayes’ Theorem.

Main Results

The transition model's best run achieved an agreement of 93.4% relative to ΛCDM's prediction for the area evolution of the universe, including a 92% agreement in the early universe and 96.7% in the late universe. The model also revealed a Chi-squared difference of 0.19 relative to ΛCDM, showing that even over 3.5 million data points, the model's curves remain consistent with ΛCDM.

Observational Data Tests

To evaluate the validity of the ΛCDM+S Model, I compared its predictions against 48 million samples of observational data.

The median Root Mean Square Error (RMSE) corresponds to a deviation in the cosmic luminosity distance of 8.6% during redshifts 0.1 to 0.5. This implies that Type Ia Supernovae should appear roughly 0.15 mags brighter than predicted by ΛCDM, which aligns with the unexplained 1701 SNe Ia residuals mapped in the Pantheon+ Dataset.

The residuals relative to ΛCDM aligns with the observed Hubble tension, a disagreement in the value of the current expansion rate of the universe. Specifically, the difference between the early universe's prediction (Planck Dataset, = 67 km/s/mpc) and the late universe's prediction (SH0ES Pantheon+ Dataset, = 73 km/s/mpc) is precisely the residual between ΛCDM+S and ΛCDM. The ΛCDM+S model has a preference towards higher Hubble Constant values due to the systematic underprediction of the universe's area in the recent past, thus aligning with the late universe's prediction.

Because the model aligns with late-universe observations, it was tested with 48 million samples from the DESI and SDSS datasets to ensure consistency with the early-universe observables. Overall, the model showed a good agreement:

98% agreement to the Acoustic Scale of the Cosmic Microwave Background Radiation (CMB)

99% agreement to the Distance Ruler of the Baryon Acoustic Oscillations (BAO)

However, there are some minor tensions, particularly with the CMB due to the lack of a full Boltzmann Code in ΛCDM+S. These values were derived using chi-squared statistics and Bayesian analysis.

Bayesian Analysis Remarks

ΛCDM+S deals with 5 parameters, 3 more than traditional ΛCDM. Therefore, it is systematically statistically disfavoured relative to ΛCDM in tests such as the BIC and AIC due to its higher complexity.

It was also noticed that the median of the Posterior distributions remained quite similar (deviations of <5%) and that uncertainty decreased by 35-50%, showing that the datasets used were informative and relevant, and that the mathematical framework used to derive the Sigmoid function is self-consistent.

Limitations

The Chi-Squared test used to compare ΛCDM+S to ΛCDM is a Monte Carlo ensemble propagation of parameter uncertainty, which slightly differs from traditional Chi-Squared tests.

This model does not statistically outperform ΛCDM in Bayesian statistics, but rather provides a potential extension to explain the accelerated expansion of the universe, the SNe Ia Residuals and Hubble Tension.

ΛCDM+S treats Pantheon+ residual as purely due to cosmology. However, there could be unknown systematic errors such as dust, Malmquist bias, calibration drifts, that mimic the same signal.

These results suggest that incorporating a transition from a gravitationally dominated universe to a thermodynamically dominated universe could provide an extension to current cosmology. Indeed, my model produces the predictions of ΛCDM to a significant degree of accuracy, while also better modelling certain astrophysical observations. More specifically, the results of this model indicate that ΛCDM could be overestimating the universe's size in the recent past by roughly 8.6%, which would be consistent with the SNe Ia Residuals and the Hubble Tension without requiring the addition of unknown parameters such as fluid dark energy. Thus, although ΛCDM+S is statistically disfavoured relative to ΛCDM in Bayesian analysis due to the three extra parameters, it is capable of fitting and providing a potential physical mechanism for observations not fully explained by ΛCDM. This better agreement with observed data may indicate that thermodynamics plays a role in fundamental physics, and suggests that the universe's behaviour can be modelled using principles of black hole thermodynamics, possibly hinting at a deep relationship between black holes and the universe as a whole. Overall, the results of this project support the idea that the accelerated cosmic expansion may not arise from a constant energy density, but instead from the universe’s tendency to evolve toward higher entropy states.

From this, I learned that different physical interpretations can produce similar predictions, meaning that agreement with observational data may not uniquely determine the underlying physics.

Following CWSF, I plan to continue developing this thermodynamic cosmology framework (see proposed 1-year research plan in next figure). My immediate priority will be to review my work and rigorously justify certain assumptions, such as the Gaussian priors. Then, I will focus on enhancing my simulations’ framework, by implementing a full Boltzmann Code, allowing me to capture measurable deviations from the CMB. I will also focus on drafting a research paper for peer review and publication, as well as publish my code on a GitHub repository and my data on a Streamlit app for reproducibility and public viewing.

I would like to thank several people who have supported me throughout my journey to CWSF the past year.

I would like to thank Samuel Prince and Kamelia Maguemoun, and Dr. Alex Wright for their extensive feedback regarding my project. I would also like to thank the FLASF and CWSF organisations as a whole, for providing me this wonderful opportunity.

Lastly, I would like to thank my family for encouraging me to continue to reveal the mysteries of the universe.

Observational Data Sources

[1] Planck Collaboration. (2020). Planck 2018 Results. VI. Cosmological Parameters. Astronomy & Astrophysics, 641, A6. https://doi.org/10.1051/0004-6361/201833910

[2] Scolnic, D., et al. (2022). The Pantheon+ Analysis: The Full Data Set and Light-Curve Release. The Astrophysical Journal, 938(2), 113. https://doi.org/10.3847/1538-4357/ac8b7a

[3] Brout, D., et al. (2022). The Pantheon+ Analysis: Cosmological Constraints. The Astrophysical Journal, 938(2), 110. https://doi.org/10.3847/1538-4357/ac8e04

[4] Riess, A. G., et al. (2022). A Comprehensive Measurement of the Local Value of the Hubble Constant. The Astrophysical Journal Letters, 934(1), L7. https://doi.org/10.3847/2041-8213/ac5c5b

[5] Suyu, S. H., et al. (2017). H0LiCOW – I: Program Overview. Monthly Notices of the Royal Astronomical Society, 468(3), 2590–2604. https://doi.org/10.1093/mnras/stx483

[6] Dey, A., et al. (2019). Overview of the Dark Energy Spectroscopic Instrument. The Astronomical Journal, 157(5), 168. https://doi.org/10.3847/1538-3881/ab089d

[7] Sloan Digital Sky Survey (SDSS) Collaboration. (2000). The Sloan Digital Sky Survey: Technical Summary. The Astronomical Journal, 120, 1579–1587. https://doi.org/10.1086/301513

[8] Eisenstein, D. J., et al. (2005). Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies. The Astrophysical Journal, 633(2), 560–574. https://doi.org/10.1086/466512

Mathematical and Theoretical Framework

[9] Bekenstein, J. D. (1973). Black Holes and Entropy. Physical Review D, 7(8), 2333–2346. https://doi.org/10.1103/PhysRevD.7.2333

[10] Bousso, R. (2002). The Holographic Principle. Reviews of Modern Physics, 74(3), 825–874. https://doi.org/10.1103/RevModPhys.74.825

[11] Carroll, S. M. (2004). Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press.

[12] Friedmann, A. (1922). On the Curvature of Space. Zeitschrift für Physik, 10, 377–386.

[13] Hawking, S. W. (1975). Particle Creation by Black Holes. Communications in Mathematical Physics, 43, 199–220. https://doi.org/10.1007/BF02345020

[14] Padmanabhan, T. (2010). Thermodynamical Aspects of Gravity: New Insights. Reports on Progress in Physics, 73(4), 046901. https://doi.org/10.1088/0034-4885/73/4/046901

[15] Weinberg, S. (2008). Cosmology. Oxford University Press.