How long will last the CBMR? I am confused about the duration of the cosmic background microwave radiation.Will it eventually be replaced by same radiation from discrete directions?
 A: The current model of the expansion of the universe is dominated by $\Omega_{\Lambda}$. In the distance future
1-$\Omega_{\Lambda}$ << 1.
At this era the expansion of the universe will become exponential, and at some point the temperature of the CMB will become less than the temperature of Hawking radiation of black holes. When that happens, near a black hole the radiation will become mostly the Hawking radiation.
I will later calculate the time when this happens.
ADDED
Step 1. In the future, the scale factor (a) gets larger, and in the Friedmann equation with four $\Omega$s (with subscripts: R, M, k, and $\Lambda$) have the following approximate values.
$\Omega_{\Lambda}$ ~ 1, and the other 3 $\Omega$s << 1.
Assuming a=10 (z=-0.9) should be adequate for this purpose.  The corresponding time (since the big bang) is
(Eq 1) $t_1 = 51 Gyr$.
This is calculated using
http://www.astro.ucla.edu/~wright/CosmoCalc.html
with $H_0$=70 (km/s)Mpc (1/$H_0$~14 Gyr) and $\Omega_m$=0.315. Note: the corresponding value of $1/H(t_1)$ is
$$(Eq 2) \frac {1} {H(t_1)} = 0.8 Gyr.$$
So the Friedmann equation for $t > t_1$ becomes (approximately)
$$(Eq 3) H(t) = \frac {da/dt} {a} = \frac {1} {0.8 Gyr}$$
Step 2: (TO BE COMPLETED - NEEDS INPUT FROM @Janko Bradvica) Select a value for the mass of a black hole to use for the remainder of the discussion.
I want to develop the answer to the question considering only a single value for the mass of a black hole. Three are three plausible choices.  I would like Janko to choose one.
(a) The black hole at the center of the Milky Way. This is the simplest, but also the most likely to be unrealistic.
(b) A black hole with the total mass of the Milky Way (including dark matter). This is moderately realistic and only moderately complicated.
(c) A black hole with the total mass of the Local galaxy cluster (which contains the Milky Way including dark matter). This is the most realistic, but also the most complicated.
Step 3. The remainder of the work will involve solving (Eq 3) and calculating a value for $t_2$ (time when the black hole temperature is greater than the CMB temperature) and $a(t_2)$, and then corresponding values for the temperatures $T_{CMB}(t_2)$ and $T_{Hawking}(t_2)$.
