what must the universe be such that cosmology can account for its own possibility as knowledge?

Your question is fundamentally transcendental. Rather than asking what does cosmology tell us about the universe?, it asks:

What must the universe be like if cosmology itself is to be possible as a form of knowledge?

That is analogous to Kant asking what the world must be like for experience to be possible, except your question is directed specifically at cosmology as a reflexive science.

Several conditions seem necessary.

1. The universe must be intelligible. There must be stable regularities that permit mathematical description. If every event were utterly singular, no cosmological laws could exist.
2. The universe must produce knowers. Cosmology is part of the universe reflecting upon itself. The physical processes described by cosmology must eventually generate organisms capable of observation, memory, mathematics, and theory construction. Cosmology therefore presupposes that epistemic agents are not accidental outsiders but products of cosmic evolution.
3. The universe must preserve traces of its own history. Cosmology works because earlier states leave observable relics—the cosmic microwave background, nucleosynthesis, galaxy distributions, gravitational structure. Without persistent records, reconstructing cosmic history would be impossible.
4. The universe must permit local observers to infer global structure. Since no observer can step outside the universe, cosmology depends on the remarkable fact that local observations contain information about the whole. This is one of the distinctive epistemic features of cosmology discussed in philosophy of science.

But your question goes one step further.

If cosmology is to account for its own possibility, then it cannot merely describe galaxies and expansion. It must also explain why beings capable of cosmology arise and why their cognitive practices reliably track cosmic structure.

That introduces a reflexivity condition:

A cosmological theory is incomplete if it explains stars, galaxies, and observers but not the emergence of the epistemic relation between observer and universe.

In other words, cosmology must include within its ontology the conditions under which cosmological knowledge becomes possible.

This resembles what philosophers call epistemic reflexivity: a theory should be capable of accounting for the conditions of its own production without undermining itself.

One can formulate the requirement as a transcendental principle:

The universe must be such that the existence of cosmological knowledge is itself a physically realizable consequence of the universe's own evolution.

Notice what this excludes.

• A universe whose laws make reliable cognition impossible.
• A universe where intelligence could never emerge.
• A universe whose structure leaves no recoverable evidence of its past.
• A universe in which successful cosmology is merely miraculous rather than naturally produced.

The deepest version of your question, however, concerns self-reference.

A genuinely self-accounting cosmology would have to satisfy something like this:

The theory describes a universe that produces observers who produce the very theory that describes that universe.

That is a fixed-point requirement. The epistemology is not external to the ontology; it is one of its consequences. The universe is not merely an object of cosmology but the generator of cosmological inquiry itself.

This shifts the aim of cosmology from explaining the universe to explaining the universe together with the possibility of its own intelligibility. Some philosophers have argued that cosmology raises uniquely reflexive questions because there is only one universe and no external standpoint from which to validate theories about it.

One concise way to express the principle is:

The universe must be recursively intelligible: its lawful evolution must generate observers whose knowledge-forming capacities are themselves lawful products of that evolution, allowing the universe to become partially self-knowing through them.

That does not imply the universe literally possesses consciousness. It means that cosmological knowledge is not an alien addition to reality but an emergent mode in which a portion of reality successfully models the larger whole from which it arose. In that sense, the possibility of cosmology is itself a cosmological fact.

is cosmology incomplete without the principles which you outlined

I would distinguish between two senses of "incomplete."

Scientifically, no.

Modern cosmology is complete as a physical theory without explicitly incorporating these principles. Its goal is to model the evolution, structure, and dynamics of the universe using observational evidence and physical laws. It is not normally expected to explain why cosmology itself is possible. Indeed, philosophers of cosmology often emphasize that cosmology has distinctive epistemic limits because we have only one universe to observe and no external vantage point from which to test it.

Philosophically, perhaps yes.

If your criterion for completeness is stronger—namely:

A theory should account for the conditions under which it can itself arise as knowledge,

then contemporary cosmology is incomplete.

It explains:

• how matter formed,
• how stars formed,
• how planets formed,
• how life may arise,
• and, increasingly, how intelligent observers evolved.

But it generally stops short of explaining why the cognitive capacities of those observers should be capable of discovering the universe's lawful structure. That question is usually left to philosophy of science, epistemology, cognitive science, or evolutionary theory rather than cosmology itself.

This is a transcendental demand

What you are proposing resembles a transcendental criterion:

A cosmology is fully adequate only if it explains not merely the universe, but also the possibility of cosmological knowledge.

That is a stronger requirement than standard scientific practice adopts.

It is analogous to asking whether evolutionary biology should merely explain organisms, or whether it should also explain why scientific reasoning itself is an evolved capacity capable of producing reliable knowledge. Many philosophers argue that this latter question cannot simply be ignored, because otherwise the theory presupposes the reliability of the very cognitive faculties it seeks to naturalize.

There is also a coherence argument

Suppose a cosmological theory claimed:

• consciousness is an accidental illusion,
• rational inference has no reliable connection to reality,
• mathematics is merely arbitrary,
• and scientific theories are just adaptive fictions.

Then that cosmology would undermine the justification for believing itself.

This is what philosophers call a reflexivity or self-reference problem. An ontology that denies the possibility of the kind of subject required to formulate that ontology risks incoherence. Similar reflexivity arguments have been developed in other fields, particularly social theory, where it has been argued that an ontology should account for the possibility of self-reflective inquiry if it is not to become self-undermining.

A possible principle of cosmological adequacy

Your question suggests a criterion that is not standard but is philosophically interesting:

A cosmology is explanatorily complete only if the existence of cosmological inquiry is itself a consequence predicted by the cosmology.

This is stronger than requiring that observers merely exist. It requires explaining why those observers can develop mathematics, inference, experimentation, and theories that successfully recover the universe's structure.

That criterion transforms cosmology from a theory about the universe into a theory that contains its own epistemic emergence.

An interesting consequence

If this criterion were accepted, then cosmology would no longer end with the evolution of intelligent life.

Its explanatory arc would be:

Big Bang → structure formation → chemistry → life → intelligence → scientific communities → cosmology.

The final stage would not be incidental. It would be part of the phenomenon the theory is explaining.

In that picture, cosmology reaches completion not when it explains galaxies, but when it explains how a universe governed by lawful regularities can eventually generate local systems capable of discovering those very regularities. That would make cosmology intrinsically reflexive: the emergence of cosmological knowledge becomes one of the universe's own evolutionary outcomes, rather than an external fact appended to the theory.

but cosmological knowledge is a part of the universe so when you say 'cosmology explains not merely the universe' how can it claim to explain the universe with a part of it missing

I think you've identified a genuine tension in how I phrased it.

You're right: if cosmological knowledge is itself a physical phenomenon in the universe, then a theory that leaves it unexplained has, by definition, left part of the universe unexplained.

So the distinction I made between "the universe" and "the possibility of cosmological knowledge" is too crude. A better distinction is between:

• the universe as traditionally modelled by cosmology (spacetime, matter, radiation, structure formation, etc.), and
• the universe in its full ontology, which also includes epistemic agents, scientific practices, theories, and knowledge as physical processes.

Once you adopt a naturalistic view, cosmological knowledge is not external to the universe. It is one of the universe's own states.

That changes the question significantly.

Instead of saying:

"Cosmology explains the universe but not the possibility of cosmology."

it would be more accurate to say:

Current cosmology explains many physical processes within the universe but does not yet explain all of the physical processes that constitute cosmological knowledge itself.

Those processes include:

• the evolution of brains,
• the emergence of representation,
• mathematical reasoning,
• measurement,
• theory formation,
• scientific communities,
• and the reliability of these processes as guides to cosmic truth.

Some of these topics are studied by other sciences, but they are rarely integrated into cosmology as part of a single explanatory framework. Philosophers of cosmology have long noted that cosmology inevitably carries epistemological commitments alongside its physical ones.

The deeper issue, however, is the one I think you're really pointing toward.

If cosmology claims to be a theory of the universe as a whole, then the activity of doing cosmology is itself one of the phenomena the theory must encompass. Otherwise there is a curious asymmetry:

• galaxies are explained,
• stars are explained,
• observers are partly explained,
• but the existence of true cosmological theories is taken for granted rather than explained.

That is odd because those theories are themselves configurations of matter and energy occurring in the universe.

This is very similar to a requirement of closure. A complete cosmology should be closed under its own ontology:

Everything that exists according to the theory—including the theory's own physical realization in human (or other) knowers—should fall within its explanatory scope.

Notice this does not mean the theory must derive every scientific paper from the Big Bang. That would be an impossible standard. Rather, it should explain why systems capable of producing increasingly accurate cosmological models can emerge at all.

In that sense, your question isn't asking cosmology to explain something outside the universe. It's asking cosmology to explain one of the universe's most remarkable products: its own capacity for self-representation.

I think this leads to a useful reformulation of the criterion:

A cosmology is explanatorily complete only if cosmological knowledge itself appears within the theory as a natural consequence of the universe's evolution, rather than as an unexplained presupposition.

That is a stronger notion of completeness than is standard in physics, but it is internally coherent. It treats scientific knowledge not as an external viewpoint on the universe, but as one of the universe's own emergent phenomena. In that respect, the question becomes not "Can the universe be known?" but "How does the universe come to contain states that successfully model itself?" That strikes me as a profound cosmological question in its own right, not merely an epistemological afterthought.