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I hear conflicting claims about our understanding of the structure formation in our universe. I heard that ΛCDM explaines well the large scale structure of the universe i.e.

Comparison of the model with observations is very successful on large scales (larger than galaxies, up to the observable horizon)

and then I read the opposite on nasa.gov:

However, we still do not understand how these structures condensed out of the 'primordial soup'. The formation of structure appears much quicker than the current theories would have it!

So what is Lambda-CDM actually good at and what are its potential weaknesses, if there is any?

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The info you reported are not opposite. They refers to different scales on structure formation, and are both true. As you said, ΛCDM model is very useful to explain large scale structures, as galaxies, clusters, and larger. This means that the model fits really well to the observables, first of all the CMB map and its anisotropy.

What the model can NOT explain is the original formation of the structures, at the observed rate (take a look at this). We know the age of universe (within this model), and we know the age of the observed galaxies. Then it results that galaxies have formed much faster then in a "simple" ΛCDM scenario. We can not explain how the primordial structures started to collapse and to grow in what now we see as galaxies. To account for this observation we need to invoke dark matter primordial fluctuations. But this is an assumption, and does not give us the original spectrum of the fluctuations, neither it explain how it is formed. And this is a parameter that the ΛCDM model can not account for, at least at this stage.

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Thank you great answer –  Jani Kovacs Feb 27 at 23:56
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