Wednesday, October 24, 2012

CMB and SNe

are cosmic microwave background measurements enough to claim spatial flatness and supernova measurements sufficient for claiming accelerated expansion of a FLRW-metric?

bonus question: if a set contains $n$ elements, how many subsets can you form? can you prove this relation with combinatorial arguments and with induction?


  1. bonus question, take 1:
    there are ${n\choose i}$ possibilities for forming a set with $i$ elements from the initial set with $n$ elements, in particular there's ${n\choose 0}=1$ possibilities for forming the empty set and ${n\choose n}=1$ possibilities for choosing the entire set. summing over these gives
    \sum_i{n\choose i} = \sum_i{n\choose i} 1^i 1^{n-1} = (1+1)^n=2^n
    possibilities for forming sets with $i=0\ldots n$ elements.

  2. bonus question, take 2:
    let's show by induction that one can form $2^n$ subsets from a set with $n$ elements. clearly, if the initial set is empty, one can form $2^0=1$ subset, and if there's only one element to start with, one can form 2 sets: the empty set and a set containing that particular element.

    assume now that the $2^n$-relation is valid, and let's try out starting from a set with $n+1$ elements. without the additional element, you can form $2^n$ sets, with the additional element you can equally form $2^n$ sets, adding just that one element to all possibly subsets. this results in $2^n+2^n=(1+1)\times 2^n=2\times 2^n=2^{n+1}$ sets, which constitutes the proof by induction.

  3. CMB: Define "spatial flatness". With infinite accuracy and precision? Of course not. Flat enough to be interesting? Yes.

    SNIa: Yes, under reasonable assumptions.