[Pr. snoerkel (OP)]

During the Big Bang, as I understand it, particles spontaneously formed when the Universe had expanded and cooled sufficiently to allow formation of matter.

On the surface this seems likely, but on second thought this scenario seems to contradict basic thermodynamics:
Before the condensation, the Universe had a high, but uniform temperature, and it would have been impossible to construct a working Carnot machine. After, energy imbalances came into existing, and the construction of a Carnot machine became possible. Thus, entropy must have decreased.

We already know that matter can convert into energy. Now my question is: Do we have any proof that matter can form from energy? If that is possible. it should be possible to construct a perpetuum mobile by compressing photons

[flr]

The entropy increased during Big Bang because the volume increased and therefore the number of possible arrangements increased.

Nuclear explosions, nuclear fusion in stars are proof that matter can be converted in energy.

It is not possible to build a perpetuum mobile from photons only. Let's say one somehow have lots of photons utilized as fuel, and the engine can extract this energy of photons for propulsion of the ship. What happens to photons as you travel is the following: The photons loose energy as this energy is used propel the ship, and their wavelength became larger and larger to the point they are radio-waves and then later (as you travel more) their wavelength became so large (like CMB) that you won't be able to extract useful work from them (i.e. or , if you wish, the extracted work will be bellow the threshold of thermal fluctuations in the ship (sqrt(N_molecules_in_ship)).

[Pr. snoerkel (OP)]

A Carnot machine works by taking in energy from a reservoir at higher temperature and deposing that energy in a reservoir with a lower temperature. In the process part of the  heat energy is converted into mechanical energy. In the process the entropy increases. If all temperatures are uniform, there will be no colder reservoir in which to deposit the energy and no mechanical energy can be created. This the entropy cannot increase and is thus at its maximum.

That matter can be converted to energy is an everyday experience - what is really challinging is the opposite process.

Since there is no maximum wavelength of a photon (at least what we know of) in principle you could use any wavelength for propulsion (ignoring the practical problems, of course)

[evan_au]

The extremely high and fairly uniform temperature of the Big Bang fell to a fairly low and fairly uniform temperature as the Universe expanded. As you say, these small temperature variations (as seen in the mottled CMB temperature map) would not support a very powerful Carnot engine (heat engine).

However, when the temperature fell enough to allow gravitational collapse of hydrogen/deuterium clouds, a new source of large temperature differences became possible - hydrogen fusion.

With this, you can create a pretty powerful Carnot engine... but entropy will increase through use of this engine, so no perpetual motion machine here, I'm afraid!

[Pr. snoerkel (OP)]

 Yes, with the formation of matter, the construction of a Catnot engine became possible. But since it was not possible before,  as far as I can see, this means that the entropy decreased during the formation of the hydrogen/deuterium. So it seems that there is a contradiction between the thermodynamic laws and the Big Bang theory here?