## Emails and cosmology

Let us say that on average I receive *N* emails per day, where *N* is a relatively large number, say 80 or so. On average I manage to read only a part of that quantity of emails, say *kN*, where *k* is usually much less than 1. The rest remains unread in my inbox. Since Murphy's law says that those unread messages are the most urgent and the senders of those emails will get upset because I didn't read them, they will re-send them the next day.

So after the first day I have *x _{0} = N - kN = N(1-k)* emails in my inbox. After the second day I'll have

*x*and after the third day I'll have

_{1}= 2 x_{0}+ N(1-k) = 2 N(1-k) + N(1-k) = 3 N(1-k)*x*.

_{2}= 2 x_{1}+ N(1-k) = 7 N(1-k)
If we substitute *W=N(1-k)* and write a recursive formula for the number of emails in my inbox on the *i*th day we get the following result: *x _{i}=2 x_{i-1} + W*. It can then be proved that the non-recursive formula for

*x*is

_{i}*x*.

_{i}=W (2^{(i+1)}-1)
The highest estimate for the number of atoms in the universe today (including possible dark matter) is somewhere around 10^{81}. A simple calculation shows that for a very optimistic value of *k=0.6*, the number of stored emails on my inbox will rise above the number of atoms in the universe in *n* days, where *n* is roughly equal to *(log _{2} (10^{81} / N(1-k) + 1) - 1 = 266.4*.

If we apply the anthropic principle to this result, we arrive to the conclusion that the universe must be rapidly expanding. In the other case all existing atoms in the cosmos would be used for storing emails and none would be left to form carbon-based human beings which could then argue about cosmology and full inboxes. However if we disregard the anthropic principle we can arrive at a different conclusion with much more serious consequences. If the flow of emails will not stop until August 2006 the universe as we know it will cease to exist when it will no longer be capable of archiving arriving emails.

**Acknowledgements**

I would like to thank Tina for a very interesting and deeply philosophical discussion we had a few hours ago that led me to the conclusions written above. I would also like to thank all CS students who need help writing various GA projects and several high-volume mailing list for providing crucial statistical data for this research. And no, the number of unread emails that have accumulated recently in my inbox had *absolutely* nothing to do with this line of thoughts.