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Register a new userVector and tensor contributions to the curvature perturbation at second order
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We derive the evolution equation for the second order curvature perturbation using standard techniques of cosmological perturbation theory. We do this for different definitions of the gauge invariant curvature perturbation, arising from different splits of the spatial metric, and compare the expressions. The results are valid at all scales and include all contributions from scalar, vector and tensor perturbations, as well as anisotropic stress, with all our results written purely in terms of gauge invariant quantities. Taking the large-scale approximation, we find that a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is also negligible. We also find that the version of the gauge invariant curvature perturbation which is exactly conserved is the one defined with the determinant of the spatial part of the inverse metric.
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We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the `separate universe approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions.
Working with perturbations about an FLRW spacetime, we compute the gauge-invariant curvature perturbation to second order solely in terms of scalar field fluctuations. Using the curvature perturbation on uniform density hypersurfaces as our starting point, we give our results in terms of field fluctuations in the flat gauge, incorporating both large and small scale behaviour. For ease of future numerical implementation we give our result in terms of the scalar field fluctuations and their time derivatives.
The galaxy number density is a key quantity to compare theoretical predictions to the observational data from current and future Large Scale Structure surveys. The precision demanded by these Stage IV surveys requires the use of second order cosmological perturbation theory. Based on the independent calculation published previously, we present the result of the comparison with the results of three other groups at leading order. Overall we find that the differences between the different approaches lie mostly on the definition of certain quantities, where the ambiguity of signs results in the addition of extra terms at second order in perturbation theory.
The anisotrpy of the redshift space bispectrum $B^s(mathbf{k_1},mathbf{k_2},mathbf{k_3})$, which contains a wealth of cosmological information, is completely quantified using multipole moments $bar{B}^m_{ell}(k_1,mu,t)$ where $k_1$, the length of the largest side, and $(mu,t)$ respectively quantify the size and shape of the triangle $(mathbf{k_1},mathbf{k_2},mathbf{k_3})$. We present analytical expressions for all the multipoles which are predicted to be non-zero ($ell le 8, m le 6$ ) at second order perturbation theory. The multipoles also depend on $beta_1,b_1$ and $gamma_2$, which quantify the linear redshift distortion parameter, linear bias and quadratic bias respectively. Considering triangles of all possible shapes, we analyse the shape dependence of all of the multipoles holding $k_1=0.2 , {rm Mpc}^{-1}, beta_1=1, b_1=1$ and $gamma_2=0$ fixed. The monopole $bar{B}^0_0$, which is positive everywhere, is minimum for equilateral triangles. $bar{B}_0^0$ increases towards linear triangles, and is maximum for linear triangles close to the squeezed limit. Both $bar{B}^0_{2}$ and $bar{B}^0_4$ are similar to $bar{B}^0_0$, however the quadrupole $bar{B}^0_2$ exceeds $bar{B}^0_0$ over a significant range of shapes. The other multipoles, many of which become negative, have magnitudes smaller than $bar{B}^0_0$. In most cases the maxima or minima, or both, occur very close to the squeezed limit. $mid bar{B}^m_{ell} mid $ is found to decrease rapidly if $ell$ or $m$ are increased. The shape dependence shown here is characteristic of non-linear gravitational clustering. Non-linear bias, if present, will lead to a different shape dependence.
The perturbation method is an approximation scheme with a solvable leading-order. The standard way is to choose a non-interacting sector for the leading order. The adaptive perturbation method improves the solvable part in bosonic systems by using all diagonal elements of the Fock state. We consider the harmonic oscillator with the interacting term, $lambda_1x^4/6+lambda_2x^6/120$, where $lambda_1$ and $lambda_2$ are coupling constants, and $x$ is the position operator. The spectrum shows a quantitative result, less than 1 percent error, compared to a numerical solution when we use the adaptive perturbation method up to the second-order and turn off the $lambda_2$. When we turn on the $lambda_2$, the deviation becomes larger, but the error is still less than 2 percent error. Our qualitative results are demonstrated in different values of coupling constants, not only focused on a weakly coupled region.
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