Arguments for an infinite size of the universe or multiverse:
Theoretical considerations: Some cosmological models, such as eternal inflation or certain variations of string theory, propose the existence of a multiverse with an infinite number of universes. These models suggest that our universe is just one of many regions within a larger, infinite cosmic structure.
No evidence of boundaries: Observations of the universe, such as the cosmic microwave background radiation, have not revealed any signs of a boundary or edge to the universe. If the universe were finite, one might expect to see some observable effects near its boundaries, but no such evidence has been found.
Spatial flatness: Observations of the large-scale structure of the universe suggest that it is spatially flat. A flat universe can extend infinitely without curving back on itself, supporting the idea of an infinite size.
Arguments for a finite size of the universe or multiverse:
Cosmic horizon: The observable universe is limited by the distance light has been able to travel since the Big Bang, known as the cosmic horizon. Beyond this horizon, there could be regions of the universe that are causally disconnected from us. This suggests a finite extent to the part of the universe we can observe.
Cosmic topology: The overall shape of the universe could be finite but unbounded. Just as a 2D surface of a sphere is finite but has no edges, the 3D universe could have a similar topology. This would mean that if one were to travel far enough in one direction, they would eventually return to their starting point.
Quantum gravity effects: At extremely small scales, the effects of quantum gravity become important. Some theories, such as loop quantum gravity, suggest that these quantum effects could result in a discrete and finite structure for space itself, imposing a maximum size on the universe.
It's important to note that our current understanding of the universe is limited, and the debate between an infinite or finite size is ongoing. Further observations, experiments, and theoretical advancements are needed to provide more definitive answers in the future.