If temperature is dependent on mass, how did the early universe have a temperature?

Question

I am near the end of the book The First Three Minutes by Steven Weinberg. I am reading it in order to get a better picture of the early universe in the Big Bang model. But one thing I am having trouble with and wanted to get some clarity. Namely:

Question

What is meant by "temperature" if the very early universe in the Big Bang model was mostly radiation, or pre-inflation, something more basic than radiation?

Explanation

Below I will try to explain why I am getting confused. Please let me know if it needs clarification (but that's also part of the question, I haven't been able to clearly understand temperature in this context yet).

Since the temperature equation is about the movement of massive particles:

$$T = {m\overline{v^2}\over 3 k_B}$$

How do you think of temperature when there is no mass, or when there is mostly radiation in the very early universe? Here is where I get confused.. Radiation is moving at the speed of light $c$, and since the speed of light is constant, and since radiation doesn't have mass, it seems like the universe would be "maximally" hot (particles moving at the speed of light). But intuitively that's not true because the temperature formula requires mass. But then if the early universe was mostly radiation, how does it have a temperature? If mass slowly increased to be the majority ~400,000 years after the Big Bang, it seems the temperature would increase rather than decrease. But then taking into account the expansion, it makes sense that even if more matter was forming, the temperature would decrease. But how could it decrease if there was less matter before?

To summarize, in trying to pinpoint temperature, I seem to be stuck in cyclic reasoning and am looking for a better way of understanding it.

I am working toward learning the purely mathematical form of these theories, but have a ways to go. For now, looking for a description to broadly make sense of "temperature" during the evolution of the few moments after the Big Bang.

Examples of how "temperature" is used

In the book above, Steven Weinberg talks a lot about the temperature and radiation of the early universe, and how the early universe was dominated by photons/radiation, and ~400,000 years later became dominated by mass-particles when the temperature cooled.

Some other quotes from the web include:

In the first moments after the big bang, the universe was extremely hot and dense. As the universe cooled, conditions became just right to give rise to the building blocks of matter – the quarks and electrons of which we are all made. (CERN)

...the temperature was so high that the four fundamental forces ... were one fundamental force. (Planck Epoch, Wikipedia)

...the temperature of the universe was still too high to allow quarks to bind together to form hadrons. (Quark Epoch, Wikipedia)

Answer 1

Really, your confusion is rooted in the fact that the equation you give relating temperature and particle mass assumes that the thing's whose temperature you're predicting is a gas of particular matter that obeys Newtonian mechanics.

Light, and really anything in the early universe, decidedly does not obey Newtonian mechanics. There are several ways one can arrive at a more fundamental definition of temperature, but at root, it amounts to "two systems are at the same temperature if they don't exchange energy when you put them into contact with each other. If they do exchange energy, the energy flow is from the hotter system to the colder system."

Mathematically, this can be made more precise if you introduce a definition of entropy and energy, but for the purposes of this question, it's good enough to say that you can define the temperature of radiation using this second definition combined with some knowledge of how radiation behaves (in particular, the knowledge that it's pressure is equal to its density).

Answer 2

The way to understand this is as follows.

Assume that the early-universe is radiation-dominated, and additionally assume that the early universe is of FLRW type with a single fluid that obeys a barotropic equation of state

$p = w \mu$ where:

• $w = 1/3$ for radiation
• $w=0$ for dust
  • $w=1$ for a stiff fluid, etc...

Now, the Einstein field equations imply the Friedmann equation:

$\frac{\dot{a}^2}{a^2} = \frac{\mu}{3} - \frac{k}{a^2}$.

Note that $k=1,0,-1$ depending on whether the type of FLRW model you are considering is spatially closed, flat, or open.

Now, one integrates the Friedmann equation above and uses the Stefann-Boltzmann law

$\mu \approx T^4$ (This is incidentally how you get around the temperature-mass dependency from standard old thermodynamics)

to obtain the temperature of the universe.

Just for your information, if you take the specific example of a radiation-dominated early universe, going through the above calculations, we get something like:

$T \approx t^{-1/2}$.

That is, as the universe ages, the temperature falls, which is consistent with observations.

UPDATED: An example of how to understand this is as follows:

I will consider for simplicity a spatially flat spatially homogeneous and isotropic universe, i.e., a $k=0$ FLRW model. Now, for such universes, Einstein's field equations give: The Raychaudhuri equation:

$\dot{\theta} + \frac{1}{3}\theta^2 + \frac{1}{2} \left(\mu + 3p\right) = 0$,

The local energy conservation equation:

$\dot{\mu} + \theta \left(\mu + p\right) = 0$,

and

The Friedmann equation: $\frac{1}{3} \theta^2 = \mu$.

Note that in these equations, $\theta$ is the expansion scalar and is typically denoted as $\frac{\dot{a}}{a}$, where $a(t)$ is the universe scale factor, while $\mu$ and $p$ are the matter energy density and pressure respectively. As mentioned above, we will assume that the matter in the universe obeys a barotropic equation of state, such that $p = w \mu$, where $-1 \leq w \leq 1$.

Now, substituting $p = w\mu$ into the above equations, we see that we have the Raychaudhuri equation:

$\dot{\theta} + \frac{1}{3}\theta^2 + \frac{1}{2} \left(\mu + 3w \mu\right) = 0$,

The energy conservation equation: $\dot{\mu} + \theta \left(\mu + w \mu\right) = 0$.

We can use the Friedmann equation to solve for $\theta$, and substitute this into the conservation equation above to obtain:

$\boxed{\dot{\mu} = -\sqrt{3} \left(1+w\right) \mu^{3/2}}$.

Solving this ODE with initial condition $\mu(0) = c$, we obtain the solution: $\boxed{\mu(t) = \frac{4 c}{3 c t^2 (w+1)^2-4 \sqrt{3} \sqrt{c} t (w+1)+4}}$.

Now, from the Stefann-Boltzmann law, we have the relationship that: $\mu \approx T^4 \Rightarrow T \approx \mu^{1/4}$, so that we may write

$\boxed{T \approx \sqrt{2} \left[{\frac{c}{3 c t^2 (w+1)^2-4 \sqrt{3} \sqrt{c} t (w+1)+4}}\right]^{1/4}}$.

I have plotted this function for some values of $w$ and over a range of initial conditions. In particular, for a radiation-dominated universe $w=1/3$:

For $w = 0$, which is pressureless dust:

Either way, you see that, as time goes on and as the universe expands, the temperature falls, but this is the mathematics in detail for one such example.

Note that observations actually show that we live in a universe that is close to $k=0$ FLRW, so this example is actually quite relevant!!

I hope this helps.