The term "Hubble horizon" refers to the boundary beyond which objects are receding from an observer at speeds exceeding the speed of light due to the expansion of the universe. It represents the distance at which the recession velocity of an object equals the speed of light.
In the context of the cosmic microwave background (CMB), the Hubble horizon is related to the observable universe. The CMB represents the radiation left over from the early universe and is observed in all directions. The CMB has a characteristic temperature, and its observation allows us to measure the size and expansion of the universe.
The Hubble horizon at near the speed of light (relative to the CMB) would depend on the specific values of the cosmological parameters, such as the Hubble constant and the density of matter and dark energy. However, it is important to note that the concept of "near c" or traveling at speeds close to the speed of light is already subject to significant relativistic effects.
When an object approaches the speed of light, length contraction and time dilation become significant according to special relativity. Length contraction means that objects in the direction of motion appear shortened, and time dilation means that time appears to pass more slowly for the moving object compared to a stationary observer.
In the context of the expanding universe, objects approaching the speed of light would experience these relativistic effects. As a result, their perception of the Hubble horizon and the expansion of the universe would be influenced by time dilation and length contraction. However, it's important to note that objects cannot reach or exceed the speed of light according to our current understanding of physics, so these effects are theoretical in nature.
In summary, the Hubble horizon and the effects of time dilation and length contraction near the speed of light are complex and depend on various cosmological parameters. It's not straightforward to determine a specific value for the Hubble horizon at such velocities, as the behavior of objects near the speed of light is governed by the principles of special relativity.