arXiv:2408.13279v1 Announce Type: new

Abstract: The hypothesis of low entropy in the initial state of the universe usually explains the observed entropy increase is in only one time direction: the thermodynamic arrow of time. The Hamiltonian formalism is commonly used in the context of general relativity. The set of Lagrange multipliers are introduced in the formalism, and they are corresponding to the Hamiltonian constraints which are written in terms of “weak equality” – the equality is satisfied if the constraints hold. Follow the low-entropy hypothesis, we postulate a modeling mechanism – a weak equality (of modeling) that holds only on the subspace of the theory space of physical models defined by some modeling constraints. By applying the modeling mechanism, we obtain a specific model of modified gravity under specific modeling conditions. We derive a novel equation of modeling from the mechanism, that describes how different gravitational models emerge. The solution of the modeling equation naturally turns out to be the model of $R^2$-gravity (with additional terms) if ordinary matter is negligible. We also found that this mechanism leads to two models: large-field inflation and wave-like dark matter. Interestingly, the wave-like dark matter model is supported by the most recent observations of Einstein rings.

**Understanding the Low-Entropy Hypothesis in the Universe**

The hypothesis of low entropy in the initial state of the universe has been widely accepted as an explanation for the observed increase in entropy over time, also known as the thermodynamic arrow of time. To explore this concept further, the Hamiltonian formalism, commonly used in the context of general relativity, comes into play. One essential component of this formalism is the introduction of Lagrange multipliers, which correspond to the Hamiltonian constraints. These constraints are written in terms of “weak equality,” meaning that the equality is satisfied if the constraints hold.

Building upon the low-entropy hypothesis, we propose a modeling mechanism that operates through a weak equality in the subspace of the theory space of physical models defined by certain modeling constraints. By harnessing this modeling mechanism, we can derive a specific model of modified gravity under the specified modeling conditions. The outcome of our efforts is a novel equation of modeling that elucidates the emergence of different gravitational models.

**Roadmap for the Future:**

**1. Elucidating the Modeling Equation**

In order to fully understand the implications of the low-entropy hypothesis and the modeling mechanism, further exploration of the derived equation of modeling is essential. This equation holds the key to comprehending how various gravitational models arise, offering valuable insights into the fundamental workings of our universe.

**2. Exploring the $R^2$-Gravity Model**

In our investigation, we have identified that the solution of the modeling equation leads to the emergence of the $R^2$-gravity model, with additional terms, under the premise that ordinary matter can be disregarded. Further research is needed to delve into the intricacies of this particular model and examine its implications for our understanding of gravity and cosmology.

**3. Investigating Large-Field Inflation**

Additionally, our modeling mechanism has resulted in the identification of another intriguing model: large-field inflation. This phenomenon, which played a crucial role in the early universe, holds the potential to unveil crucial information about the history and evolution of our cosmos. Further investigations and observational data are required to corroborate and refine our understanding of this intriguing concept.

**4. Examining Wave-like Dark Matter**

We have also discovered that our modeling mechanism suggests the existence of wave-like dark matter, a novel concept that provides a potential explanation for the recent observations of Einstein rings. Further analysis and measurements are essential to validate this model and expand our understanding of the elusive nature of dark matter.

**Challenges and Opportunities:**

While the low-entropy hypothesis and the modeling mechanism offer exciting avenues for exploration, there are several challenges and opportunities that lie ahead. The following are key considerations:

**Theoretical Challenges:**Further theoretical investigations are required to validate and refine the modeling mechanism, ensuring its compatibility with existing frameworks and experimental data.**Experimental Verification:**Conducting experiments and observations to test the predictions and implications of the derived models will be crucial in determining their real-world validity.**Data Analysis:**Analyzing observational data, such as the observations of Einstein rings, will play a crucial role in verifying the wave-like dark matter model and strengthening our understanding of this peculiar form of matter.**Interdisciplinary Collaboration:**The study of the low-entropy hypothesis, modeling mechanisms, and derived models requires collaboration between physicists, cosmologists, and other related fields to foster a comprehensive understanding of the complex phenomena at play.

By addressing these challenges and harnessing the opportunities they present, we can continue to unravel the mysteries of the universe and propel our knowledge of fundamental physics and cosmology to new heights.