The not so distant star light problem

Can you explain specifically what is dubious about large distance measurements. It would be best if this could be expressed purely in terms of the techniques themselves and their possible weaknesses, not in terms of vague suspicions of the "scientific establishment".

Fair enough, this is reasonable.

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There is supposedly, what, one hundred billion stars in our galaxy alone? Hundreds of billions of galaxies all around ours?

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About four hundred billion stars in the Milky Way, and around 200 billion galaxies in the observable universe.If our current estimate of the number of extrasolar planets per star continue to hold for other galaxies, then that gives around 300 sectillion planets in the observable universe. Low enough to be statistically negligible when averaged over millions of stars. Space is extremely sparse. Think about it. Draw a sphere 10 light years in radius around the sun. The Sun (Sol), Alpha Centauri, Sirius, Bernard's star occupy around 0.0000016% of this volume. Hence the chance of objects being directly in line with each other is about one in fifty million. This error is swamped when one does a statistical analysis on millions of stars. Not really, black holes a relatively rare, there is only one per hundred thousand stars, so again that gives you a one in five hundred billion chance that a black hole will introduce an error into a distance measurement. In fact the chance is even less, since black holes aren't luminous, the stars light would have to align almost exactly with the black hole itself, which are usually only a few kilometers in diameter.Dark Matter is non-luminous and its only effect would be gravitational lensing, which distorts images, but wouldn't change calculations of their distance. Lensing doesn't magnify or shrink images. In either case the density of Dark Matter is too low to cause significant lensing.

Actually yes, but it doesn't really matter due to Liouville's theorem. Although the brightness of one star will increase, it's brightness viewed from another point will decrease in a way that compensates, so measuring from a few angles gives you the correct average.
Even more crucially, lensing travels as the object causing it moves, so it isn't too difficult to track that motion and "subtract off" the lensing contribution.Although I should say that my understanding of cosmological observations would not be the best. I know General Relativity and the observational techniques that are used, but I don't really have an understanding some of the deeper statistical principles and techniques applied in some of WMAP studies for example.