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In a situation where you have people traveling between parallel Earths, where they dropped in with little ability to tell which Earth they are on, could checking cosmic background radiation be a sensible way to ascertain if this is a universe they had visited before?

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No. Interesting parallel Earths branched from ours within the last million years or maybe billion years if you want to look at totally alien life. The background radiation is coming to us from 14 billion years, with just a little modification from very large things in the way. If the timeline diverges a million years ago, the background radiation will still be from the time before that so be exactly the same.

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This is a very clever idea. Kudos to you for coming up with it.

The answer depends on a few things. First, the temperature of the cosmic microwave background is related to the redshift by the formula $$T(z)=T_0(1+z)$$ where $T_0$ is the temperature we measure today, at redshift $0$. Noterdaeme et al. (2010), incidentally, remark that $T_0=2.725\pm 0.002\text{ K}$.

Cosmology and Particle Astrophysics, by Bergstrom and Goobar, mention that there is a redshift-time relation: $$t(z)~\frac{2}{3H_0\sqrt{\Omega_M}}\frac{1}{(1+z)^{3/2}}$$ where $H_0$ is the current Hubble constant and $\Omega_M$ is the density of matter in the universe (part of the density parameter). Inverting the relation, we find that $$z(t)=\left(\frac{2}{3H_0\sqrt{\Omega_M}}\frac{1}{t}\right)^{2/3}-1$$ You can then find the temperature as a function of time, matter density, and the Hubble constant: $$T(t)=T_0\left(\frac{2}{3H_0\sqrt{\Omega_M}}\frac{1}{t}\right)^{2/3}$$ Let's say that your intrepid travelers always visit each universe at around the same point in time - that is, they aren't bounding around from the Big Bang to the end of the universe and back. Then there's one basically one factor to measure, assuming all universes have the same age: $$T\propto T_0H_0^{-2/3}\Omega_M^{-1/3}$$ If you can measure this factor, you can probably differentiate one universe from another.

Also, don't limit yourself to the cosmic microwave background! There's a gravitational wave background and a cosmic neutrino background, too. We can use information from the cosmic neutrino background (C$\nu$B), too; the mean velocity of a neutrino in the background is given by (see Safdi et al. (2014)) $$\langle v\rangle=160 \left(\frac{m_{\nu} c^2}{{\rm eV}}\right)^{-1} \ (1+z)\ \ \ {\rm km/s}=160 \left(\frac{m_{\nu} c^2}{{\rm eV}}\right)^{-1}\left(\frac{2}{3H_0\sqrt{\Omega_M}}\frac{1}{t}\right)^{2/3}\ \ \ {\rm km/s}$$ In other words, $$\langle v\rangle\propto v_0 m_\nu^{-1}H_0^{-2/3}\Omega_M^{-1/3}$$ which is a similar relation.

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  • $\begingroup$ Thus you proved the CMB can be used to differentiate between universes that differ significantly in age, expansion rate or mean density. In other words, between universes so different that there would hardly be any trace of Earth in them :-) $\endgroup$ – Radovan Garabík Oct 17 '15 at 10:40
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    $\begingroup$ @RadovanGarabík Well, yeah. :-) I do think that precision experiments could differentiate between universes where some things were changed by only a tiny bit. The trouble is, changing things by a tiny bit can cause big problems (think of fine-tuning). $\endgroup$ – HDE 226868 Oct 18 '15 at 15:19
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After reading your title and question, I remembered an article I read about something along these lines. Basically, there is a theory that the cosmic background not only provides hints that there was a universe before the big bang that it also shows "bruising" where it has collided with other universes.

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I am afraid that it would not be very practicall unless the situation were very specific.

Note: I use terminology that is original according to Wikipedia, because I find it more logical: the Universe is everything and multiverses are big parts of it (in this case looking like the classical image of the Universe in modern cosmology).

Mean background radiation temperature depends on the age of the multiverse and expansion rate and for this temperature to change significantly, the whole multiverse would have to be so much different that the existence of Earth-like life could be hard to justify.

More precise measurements of the mean temperature and deviations from it require advenced satelites collecting data for a longer time unless you assume a new, very advenced technology. Besides, if the multiverses we visit were globally similar, even such measurements would not show differences - such differences would mean that totally different galaxies exist now far away. If you assume that only the Earth is similar and for example jump technology allows to jump to almost Earth planets that can appear in very different multiverses (it could justify even very different multiverses from the previous paragraph), we could distinguisch them by looking at other planets in the solar system, stars (very easy at night if already the visible constelations are different and the traveller knows them) or other galaxies (with a telescope) unless you declare that all near space is similar enough that it is the background radiation that is easiest to distinguish with available technology.

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