I Cosmological Scalar Field Density Dilution

AI Thread Summary
A homogeneous free scalar field with mass m and a quadratic potential exhibits oscillations at frequency m when the mass is much greater than the Hubble parameter (m ≫ H). The discussion emphasizes the importance of neglecting Hubble friction, allowing the scalar field to behave as an underdamped oscillator. The energy density of the scalar field is derived from the Klein-Gordon equation, leading to the conclusion that it dilutes as a^{-3} after inflation ends. This relationship is confirmed by averaging the energy density over time and applying the appropriate equations of motion. The analysis highlights the dynamics of scalar fields in cosmological contexts, particularly during inflationary phases.
Samama Fahim
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Consider a homogeneous free scalar field ##\phi## of mass m which has a
potential

$$V(\phi) = \frac{1}{2}m^2\phi^2$$

Show that, for ##m ≫ H##, the scalar field undergoes oscillations with
frequency given by $m$ and that its energy density dilutes as
##a^{−3}##.

This is from Modern Cosmology, Scott Dodelson, Chapter 6.

For the part "Show that its energy density dilutes as ##a^{−3}##", following is my attempt:

In the equation ##\frac{\partial \rho}{\partial t} = -3H(P+\rho)##, put ##P = \frac{1}{2} \dot{\phi}^2-V(\phi)## and ##\rho=\frac{1}{2} \dot{\phi}^2+V(\phi)## to get

$$\frac{\partial \rho}{\partial t} = -3\frac{\dot{a}}{a}\dot{\phi}^2,$$

where ##H = \dot{a}/a##. I am not sure how to proceed or proceed or whether this is the correct approach. Should I use Friedmann's equation instead? But it involves densities of other species as well, and there is no assumption here whether one species dominates. Or Should I convert ##d\rho/dt## to ##d \rho/ da##?

Kindly provide a hint as to how to proceed.
 
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Sorry, going through my replies and must have missed this at the time...
The paper goes into some detail but you can proceed in a more simple way.

Recall from the Klein Gordon equation for the inflation,$$\ddot{\phi} + 3H\dot{\phi} + V' = 0$$The key assumption is to neglect the Hubble friction, ##3H\dot{\phi} \ll 1##, i.e. you have an underdamped oscillator when ##m## is large. We are given that the potential is quadratic (this would also apply for small oscillations around any quadratic minimum), so you have ##V' = m^2\phi## and the equation of motion ##\ddot{\phi} + m^2 \phi = 0## is just SHM with frequency ##m##. So write, for example,$$\phi = \phi_0 \cos{m t}$$For the next part, you have the right idea,$$\dot{\rho} = -3H(\rho + P) = -3H \dot{\phi}^2 $$Finally you just look at the time average. Note ##\langle \dot{\phi}^2 \rangle = \frac{1}{2} m^2 \phi_0^2 = \langle \rho \rangle##, so you just get$$d \ (\log \langle \rho \rangle) = -3 \ d (\log a)$$which then spits out the ##\rho \sim a^{-3}## dependence after inflation ends.
 
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