exp}\left(-\frac{1}{2}\cBo^T \Sigma^{-1} \cBo\right)\; ,$$ where $d$ is the number of elements in $\cBo$, i.e. $d=24=8\times 3$; for 8 values of $r$ and 3 cartesian components. The $d\times d$ covariance matrix $\Sigma=\Sigma_{\rm o}+\Sigma_{\rm t}$, where $\Sigma_{\rm o} $ is the covariance of the errors on $\cBo$ and $\Sigma_{\rm t}$ describes the covariance of the underlying quantities $\cBt$. The dependence on the cosmological models comes through $\Sigma_{\rm t}$.

Consistency with $\Lambda$CDM7
------------------------------

We begin by assessing how well $\Lambda$CDM7 is consistent with the data. To do that we generated $10^7$ sets, ${\cal B}_{\rm rnd}$, each containing $d=18 $ numbers selected at random from a gaussian distribution given by (\[eq:pbo\]) computed with $\Sigma_{\rm t}$ for $\Lambda$CDM7. For each of those $10^7$ sets of ${\cal B}_{\rm rnd}$ we compute the corresponding $P({\cal B}_{\rm rnd})$ using (\[eq:pbo\]) and tabulate the negative of the log of the probability, $nlP_{\rm rnd}=-\ln P({\cal B}_{\rm rnd})$. We also compute $nlP_{\rm o}=-\ln P(\cBo)$ for the observed $\cBo$ also using $\Lambda$CDM7. We find that only 26% of the $10^7$ values of $nlP_{\rm rnd}$ exceed $nlP_{\rm o}$. Therefore, the $\Lambda$CDM7 cannot be rejected by the bulk flow results.

Independent constraints on $\sigma_8$ and $\gamma$
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The $\Lambda$CDM expected amplitude of the bulk 