Cooling in the Universe

One of the questions in the cosmology courses is the cooling mechanism of cosmic fluid during it expansion according to classical concepts of the thermodynamics. In this short pedagogical paper, we quote the questions and give a natural approach dealing with this problem by measuring the dispersion velocity of the particles in the cosmic fluid by a comoving observer. We show that the thermal motion of the particles in the cosmic fluid deviates from the Hubble flow and follows the geodesics governed by the gravity of the homogeneous universe. The dynamics of the ”thermal peculiar velocity” of the particles leads in an expanding universe, momentum of the particles relax by the inverse of the expansion factor and the result is losing the energy of particles, hence cooling the universe. Cooling in the Universe 2 One of the main questions of students of the cosmology is the cooling mechanism of the universe during its expansion. According to the concepts from the classical thermodynamics the question are as: (i) Is our universe non-adiabatic (i.e. universe is connected to the cold thermal bath and losses its energy during the time) ? The answer to this question is trivial, there is nothing outside the universe, so thermodynamically our universe is adiabatic.(ii) So, if our universe expands adiabatically the particles of gas should loos their energy during the expansion of the universe by bouncing off from the physical boundary of the universe during the expansion which makes the cosmic fluid to cool down. In the standard cosmological text books [1, 2] and pedagogical articles [3, 4, 5, 6], the variation of temperature of the cosmic fluid during the expansion of the universe has been discussed in several ways. It has been shown that the momentum of particles in the cosmic fluid decrease with the inverse of scale factor of the expansion, however some of these approaches such as dealing the motion of particles with the special theory of relativity seems does not satisfy the students’s curiosity [1]. Here we use the natural general relativistic approach to revisit this problem. In the standard model of the cosmology the distance between any two points in an expanding homogeneous universe is given by r = ax where a is the expansion factor and x is defined as the comoving distance. As the universe expands, the scale factor increases and makes a larger distance between any two points in the universe. For an homogeneous universe the comoving distance does not change with time, so the velocity of expansion can be given by vh = xda/dt, so-called Hubble velocity. For the real universe, the density is not homogenous and due to the large structures and voids the density of matter deviates from the background density of universe. One of the consequences of this deviation is that there is no more a solid comoving coordinate system and it changes by time as the structures grow. So the velocity of cosmic fluid gets one more term of vpec = adx/dt, so-called peculiar velocity, due to the variation of comoving distance with time. We take a comoving observer freely follows the expansion of the universe. For this observer the global velocity of the cosmic fluid around him/here is zero. However due to none-zero temperature of the cosmic fluid, the observer will measure non-zero velocity due to the thermal motion of cosmic fluid, we call this velocity as thermal peculiar velocity. We should bear in mind that thermal peculiar velocity is different from that appear due to the perturbation in the gravitational potential of the large scale structures. The dynamics of the (thermal motion of) particles in the Friedman-Robertson-Walker (FRW) universe with the metric of ds = dt − adx is given by the geodesic equation as: ẍ + Γμνλẋ ν ẋ = 0, (1) where derivation is taken with respect to the comoving time of the particles, τ and Γμνλ is the Christoffel symbol in the (FRW) metric. Γi0 = 1/a× da/dt is the only non-zero spatial component of the Christoffel symbol in this metric and by substituting it in (1), Cooling in the Universe 3 the equation of motion of the particles in the comoving frame obtain as: ẍ + 2 a da dt × dt dτ ẋ = 0 (2) The solution of this equation implies: dx/dτ ∝ a. Using the chain rule to express this result in terms of differentiation with respect to t-component results: