Final answer:
The temperature of the universe is inversely proportional to the cosmic scale factor for dark matter because as the universe expands, the wavelength of photons gets stretched, lowering the energy and the temperature. Dark matter's temperature, which was once in equilibrium with radiation, behaves similarly and decreases with expansion.
Step-by-step explanation:
The relationship between the temperature of the universe and the cosmic scale factor for dark matter is rooted in the laws of thermodynamics and cosmology. In an expanding universe, the cosmic scale factor a(t) describes how distances between non-gravitating objects increase with time.
According to the standard cosmological model, as the universe expands, the wavelength of photons gets stretched with the scale factor, leading to a decrease in the energy of the photons, which in turn implies a lowering of temperature.
The temperature of the cosmic microwave background radiation (CMB) can be used as a proxy for the universe's temperature. This temperature, T, is inversely proportional to the scale factor: T ∑ 1/a. Since dark matter does not interact with light (or any electromagnetic radiation), we assume that its temperature behaves similarly to that of the CMB. Thus, as the universe expands (the scale factor increases), the temperature decreases, confirming the inverse relationship.
To demonstrate this, we can use the fact that the radiation density, which is proportional to T^4, scales like 1/a^4. Assuming dark matter was in thermal equilibrium with the radiation at some point in the early universe, its temperature, T_dm, would also scale as 1/a. Hence, this is a conceptual demonstration as the detailed derivations would require more extensive calculations based on the cosmological models.