There are two conditions for the cosmic microwave background radiation to be able to support life:
- It emits strongly enoughIt's partially composed of photons at wavelengths required by photosynthetic pigments.
- It's bright enough to transmit a useful amount of energy to any life out there.
If either of those conditions is not met, life cannot survive (asand as the first stars will notwon't be born for another 100 million years or so, there likely isn't another feasible energy source). It turns out that any photosynthetic life forms on your planemo will only have a few million years in which to evolve - not nearly enough. If they were magically transported there from the future, they might have a chance, but not if they formed in situon the planemo itself.
Let's first see what limits we can place on life based on the evolving temperature of the CMB. It makes sense to talk about the cosmic microwave background in terms of its redshift. For instance (which, as we lookin cosmology, tells us how far away something is, and therefore how far back in time we're looking). For instance, at some redshift $z$, the temperature of the CMB is
$$T(z)=T_0(1+z)$$
where $T_0\approx2.725\text{ K}$, its approximate temperature today (at $z=0$), and $z$ is the redshift. Similarly, the relationship between the age of the universe and redshift is approximately
$$t(z)=\frac{2}{3H_0\sqrt{\Omega_M}}\frac{1}{(1+z)^{3/2}}$$
where $H_0$ is the Hubble constant today and $\Omega_M$ is the dimensionless matter density of the universeuniverse; today, $\Omega_M\approx0.04$.
Now, theThe CMB was released during recombinationrecombination, wherewhen $T\approx3000\text{ K}$. This corresponds to a redshift of $z\sim1100$, and thus an agea time of about 375,000 years after the big bang. Now, 3,000 Kelvin isn't much - it's about half the surface temperature of the Sun. That means the radiation peaks at a wavelength of $\lambda\approx966\text{ nm}$, which is already outside the visible spectrum. However, the CMB will still consist partly of visible light until it reaches roughly the Draper point, which corresponds to an age of 2.7 million years. Given that photosynthetic pigments on Earth are largely sensitive to visible light - which will only compose a small part of the CMB, even after it's emitted - it seems that after a few million years, they'll be out of luck.
Since the cosmic microwave background is (approximately) a black body, we can analyze it in terms of Planck's law. Andersen et al. 2018 actually did this for the CMB at various times in the early universe, and determined what it would look like to a human observer. They found that it was actually too bright for human eyes up to 1.2 million years, and too dim after 5.3 million years - roughly twice as late as our simple calculation ofestimate using the Draper point.
##In the Dark Ages
As I said before, the first stars - massive, luminous objects called Population III stars - won't form until 100 million to 200 million years after the Big Bang. This means that there should be a very long period of time called the Dark Ages between when the CMB is formed and then becomes invisible to human eyes and when these stars form. It's called the Dark Ages because, well, it's dark. There's so significant source of visible light in the universe.
Life seems unlikely to evolve within 5 million years, as Serban Tanasa indicated, and I agree. If it did, though, an interesting question would be whether it could survive these hundreds of millions of years until the first stars are born, perhaps by going dormant in some way. (I don't really have an answer for this, to be honest; I doubt it but can't really prove it). One of the problems, of course, with thinking about the evolution of life under strange conditions is that we only have on planet of data - and the lifeforms on it are carbon-based. I'm not sure how well we can extrapolate life on Earth to the not-carbon-based life on this planemo.
##The impossible planemo
Something that I've ignored up until this point is that your planemo can't exist at all at this point in the universe. It would be composed of heavy elements and silicate compounds. Unfortunately, those compounds can only form after the first stars in the universe - Population III stars - are born and die (which. For the same reason, as I said beforeany life on this body could not be carbon-based, should happen about 100 or 200 million years after the Big Bang)as there wouldn't be any carbon to go around.