Wednesday, February 13, 2013

scale-invariance of perturbations

can you find an intuitive argument why the variance of density fluctuations seeded by inflation is proportional to the wave number? what would be the corresponding relation if we lived in a universe with 4 spatial dimensions instead of 3?

bonus question: can you show that if a square real-valued matrix $A$ is skew-symmetric, $A^t=-A$, then the matrix exponential $\exp(A)$ of that matrix is orthogonal?

1 comment:

  1. bonus question: for showing the orthogonality of $U=\exp(A)$, $U^{-1}=U^t$, it is enough to show that $U^tU=\mathrm{id}$, which can be done with the Cauchy-product,
    \begin{equation}
    U^tU
    = \sum_i\frac{(A^i)^t}{i!}\times\sum_j\frac{A^j}{j!}
    = \sum_i\sum_j\frac{(A^j)^t}{j!}\frac{A^{i-j}}{(i-j)!}
    \end{equation}
    by substituting the binomial formula,
    \begin{equation}
    \ldots = \sum_i\frac{1}{i!}\sum_j {i\choose j}(-A)^j A^{i-j}
    \end{equation}
    one arrives at the exponential series,
    \begin{equation}
    \ldots = \sum_i\frac{1}{i!}(A-A)^i
    = \exp(0)
    = \mathrm{id}.
    \end{equation}

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