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Each day light has more time to reach our eyes from distant galaxies. In one day, light travels 2.59×10^10 km.

So our observable universe (assuming my simple math skills apply here and there isn't something about relativity to take into account), expands that much every day in all directions. So at what rate are new galaxies being encompassed by the expanding observable universe?

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You probably mean how many galaxies? Still, the observable universe horizon isn't limited by the age of the Universe, but its inflation. Number of observable galaxies is actually getting smaller, not larger. And the rate of change would not be linear with time either. –  TildalWave Jun 21 at 18:11
    
Edited. Someone has suggested closing this question, but it may be better to address the misunderstandings of the questioner in answers. –  Jeremy Jun 21 at 20:40

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What we see at the outer reaches of the observable universe is a surface of the cosmic microwave background (CMB), which is at the time when the universe, still quite homogeneous (perturbations were a few parts in $10^5$), cooled enough for the electrons to combine with the protons and make hydrogen atoms. Each day, we can see a light-day farther and so we slowly see a different (larger) surface of the CMB (but there were no galaxies at this epoch). Therefore, as we observe the CMB over many years, we will always see the same time epoch, but at different spherical shells. Will we see new galaxies? Yes. Galaxies are forming continuously in view at high redshifts (2-20?). In a few million years, the small perturbations that we see today in the CMB will develop into young galaxies within our view. One cannot provide numbers for how many we might be seeing because small galaxies merge into bigger ones and those merge again. At high redshifts there is a lot of merging going, so the rate is particularly ill defined there.

How is this changed by the acceleration from the cosmological constant (aka dark energy)? From the Friedmann equation one sees that the Hubble constant reaches a final constant value of $c\sqrt{\Lambda/3}$ or about 40% higher than its present value. Eventually, in billions of years, the CMB will shift from $3^oK$ to much lower and then the CMB will fade (redshift) away. After that, those new distant galaxies will also fade away. And then more and more of the universe will go dim.

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