How long does it take for a white dwarf to cool to a black dwarf?
I was reading on white dwarfs, and I came across this sentence—
Without energy sources, the white dwarf cools to a black dwarf in a few billion years.
However, when I looked into the Wikipedia page on White dwarf, it says
Because the length of time it takes for a white dwarf to reach this state is calculated to be longer than the current age of the universe (approximately 13.8 billion years), it is thought that no black dwarfs yet exist.
So which is true?
And what is the proper definition of a black dwarf?
 Introductory Astronomy and Astrophysics. Zellik, M. Gregory, S. 4th edition. Brooks/Cole. 1998
I think what you need is here on the Wikipedia. In section "Radiation and cooling," it says "The rate of cooling has been estimated ... After initially taking approximately 1.5 billion years to cool to a surface temperature of 7140 K, cooling approximately 500 more K ... takes around 0.3 billion years, but the next two steps of around 500 K ... take first 0.4 and then 1.1 billion years."
One takeaway is that the rate of cooling (giving a fixed change in temperature, i.e., every 500 K) is increasing non-linearly. This is because the cooling is governed by diffusion process. So, at low temperature, to cool down 500 K more would take very long time than what it did in the past.
As someone said in the comment, there is no precise definition of a black dwarf. So, I would not say who is right or wrong without understanding how they define the cutoff.
However, if you roughly define it to be at the level that its color temperature passes beyond the visible wavelength (i.e. >7000 A or < 4000 K), and if you follow the info mentioned above by extrapolating from about 5500 K and assuming the rate of changing 500 K is constant as what it did in the previous step (i.e., from 6000 to 5500K taking 1.1 billion years), approximately we get the upper limit for cooling from 5500 to 4000 K as 3 billion years. By adding about the previous 2 billion years from the initial temperature down to 5500 K, we have >5 billion years for a white dwarf from its initial state down to about 4000 K. Note that the 5 billion is a lower limit because we did not consider non-linearity.
(Note that you can also approximate the non-linearity effect by assuming an increment of 1 billion years in each step implying by the step 6000-5000 K. By doing this, the lower limit would be >7 billion years.)
Since the age of the universe is 13 billion years, whether you believe that a black dwarf exists or not depend on i) definition, ii) rate of cooling, and iii) variation (which means there might be a white dwarf that was born cool or living in the environment that supports better cooling than the typical population).
If it's cooling more slowly, that is a *decreasing* rate. The *time to cool a certain units of temperature* is what's increasing, not the rate.
The problem here is that this crude extrapolation of the cooling rate doesn't work if the heat capacity changes with temperature. Dense, high mass white dwarfs can enter the Debye regime where their heat capacity falls rapidly with temperature and they *can* radiate their remaining heat quickly. Plots shown in my answer.
Is Debye cooling applied to its core only, while the overall heat capacity and the surface temperature also involve the properties of the envelope which are still uncertain?