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There were extensive searches in the early and mid 1990's for astronomical micro-lensing events, and these concluded that either the Milky Way is anomalous or there just aren't enough MACHOs to be a big part of the missing mass.

MACHO? That's MAssive Compact Halo Object. Anything small, heavy, cool and gravitationally bound to a galaxy. Cool neutron stars would count.

Micro-lensing? If a MACHO passes very precisely between a telescope and a distant star there is a detectable change in the intensity of the light from the star due to gravitational lensing, and that change follows a predictable curve depending on the intervening mass and the impact parameter.

So, with a sensitive (you need to be able to image down to 20th magnitude or better to get much data), computerized telescope you watch a nearby galaxy (say the Magellanic clouds) every night and count. With enough data you can make a pretty good estimate of the total mass of these things in the galaxy.

Aside: When I was an undergrad I worked on a small, cooled CCD telescope for a physics prof. And we'd try to image micro-lensing events reported by IAU telegrams. We were struggling with the tracking software, so it was a struggle to get imaging below 17th magnitude, but we did have a good run where we got a light curve that matched the big boys pretty well.

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The inner temperature of a newly formed neutron star is around $10^{11}-10^{12}$ Kelvin degrees. It emits lots of neutrinos and cools to $10^{6}$ Kelvin within years. That's a high enough temperature behind the fact that neutron stars emit thermal X-rays. The temperature may decrease a little bit but getting to $2.7$ Kelvin is, well, unrealistic.

But more generally, it's a good idea that the dark matter is pretending to be cosmic microwave radiation. You could replace neutron stars by small black holes that emit at the CMB temperature, too. The temperature would have to be fine-tuned to the right temperature (otherwise we would see it as extra anisotropy in WMAP pictures) - and unlikely to keep it as the Universe expands and the CMB temperature cools down. ;-) If this were true, the Universe would have to fine-tuned to confuse observers who happen to live 13,730,002,011 years after the Big Bang. :-)

Oh, no, I am actually wrong. It's not excessively fine-tuned because objects near the CMB temperature don't cool down anymore. But it's still difficult for matter in any reasonable state to cool down that much.

By the way, even if you don't consider neutron stars but more general large objects as an explanation of dark matter - they're referred to MACHO, RAMBO etc. - they seem to be disfavored as a solution to the dark matter. It seems that most of the dark matter has to be cold dark matter, and WIMP (Weakly Interacting Massive Particle) arguably represents a majority of the mass of the dark matter.

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@Luboš, what is the source of neutrinos emitted by neutron star? –  voix Feb 6 '11 at 11:47
    
@voix, when the star is becoming a neutron star, the atoms - including protons and electrons - have to become neutrons. That goes via $p+e^- \to n +\nu$ where $\nu$ is neutrino. For every electron in the original star, you emit a neutrino. The neutrinos' energies may be substantial. –  Luboš Motl Feb 6 '11 at 11:51
    
@Luboš, but does newly formed neutron star emit neutrinos? –  voix Feb 6 '11 at 12:05
    
@Luboš, the question of dark matter nature is still open. MACHO Project excluded only low mass MACHOs (brown dwarfs and Jupiter-sized objects). –  voix Feb 6 '11 at 12:21
    
@Lubos, I think neutrino emissions is related to temperature density. At high emough temperatures, there should be considerable churning between the different isotopes, and some of these reactions emit neutrinos. –  Omega Centauri Feb 6 '11 at 16:12
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