Hawking radiation and cosmic microwave background radiation

[Orion1]

In order for the black hole to evaporate it must have a temperature greater
than that of the present-day black-body radiation of the Universe.

Cosmic microwave background radiation temperature:
$$T_u = 2.725 \; \text{K}$$

Hawking radiation temperature:
$$T_H = \frac{\hbar c^3}{8 \pi G M k_B}$$

Hawking radiation temperature is greater than or equal to cosmic microwave background radiation temperature:
$$T_H \geq T_u$$

$$\frac{\hbar c^3}{8 \pi G M k_B} \geq T_u$$

Hawking total black hole mass:
$$M_H \leq \frac{\hbar c^3}{8 \pi G k_B T_u} \leq 1.226 \cdot 10^{23} \; \text{kg}$$
$$\boxed{M_H \leq 1.226 \cdot 10^{23} \; \text{kg}}$$

Earth total mass:
$$M_{\oplus} = 5.9722 \cdot 10^{24} \; \text{kg}$$

$$\frac{M_H}{M_{\oplus}} = 0.007539 = 0.754 \; \text{%}$$

According to reference 4 - p. 2, eq. 1:
$$\frac{M_H}{M_{\oplus}} = 0.8 \; \text{%}$$

Are these equations correct?

Reference:
Cosmic microwave background radiation - temperature - Wikipedia
Hawking radiation - black hole evaporation - Wikipedia
Earth mass - Wikipedia
The Last Eight Minutes Of A Primordial Black Hole - Joseph Kapusta

 

[phinds]

Your premise is incorrect. Hawking radiation will occur regardless of the temperature of the universe. This has been discussed on this forum several times.

The total EFFECT of the Hawking radiation will be affected by the temperature of the universe, since the incident radiation from the CMB may more than offset the loss to the BH due to Hawking radiation, but that does NOT mean that Hawking radiation does not occur.

 

[Chronos]

A black hole always radiates, but, the present background thermal energy is far greater than the temperature of any black hole a solar mass [or larger], so, black holes are actually absorbing more energy than they radiate at this time in cosmic history.

 

[Orion1]

The premise is 'In order for the black hole to evaporate', not 'in order for a black hole to radiate', perhaps you are confusing evaporation with radiation...

It does seem improbable that small black holes under that mass threshold exist in the present epoch of the Universe.

If quantum or micro black holes ever existed, they would have evaporated already a long time ago, in the very earliest epochs of the Universe.

 

[phinds]

The premise is 'In order for the black hole to evaporate', not 'in order for a black hole to radiate', perhaps you are confusing evaporation with radiation...

OOPS. You're right, I did. Thanks for the correction.

 

[mfb]

It does seem improbable that small black holes under that mass threshold exist in the present epoch of the Universe.

There are theories that small black holes (something like a big mountain) formed shortly after the Big Bang. They could evaporate today.