Phenomenological analysis of quantum collapse as source of the seeds of cosmic structure

The standard inflationary version of the origin of the cosmic structure as the result of the quantum fluctuations during the early universe is less than fully satisfactory as has been argued in [A. Perez, H. Sahlmann, and D. Sudarsky, Classical Quantum Gravity 23, 2317 (2006).]. A proposal is made there of a way to address the shortcomings by invoking a process similar to the collapse of the quantum-mechanical wave function of the various modes of the inflaton field. This in turn was inspired by the ideas of R. Penrose about the role that quantum gravity might play in bringing about such a breakdown of the standard unitary evolution of quantum mechanics. In this paper we study in some detail the two schemes of collapse considered in the original work together with an alternative scheme, which can be considered as “more natural” than the former two. The new scheme assumes that the collapse follows the correlations indicated in the Wigner functional of the initial state. We end with considerations regarding the degree to which the various schemes can be expected to produce a spectrum that resembles the observed one.

Figure 1

Plots of the three collapse schemes. We could appreciate that $C_2$ (middle) and $C_{\mathrm{wigner}}$ have a similar behavior despite their dissimilar functional form.

Figure 2

Semilog plot of $|{\alpha }_{lm}{|}^2(C_1(k))$ for different values of $(A,B)$, representing how robust is the scheme of collapse when it departs from $z_k$ constant. The abscissa is $l$ until $l=2600$.

Figure 3

Semilog plot of $|{\alpha }_{lm}{|}^2(C_2(k))$ for different values of $(A,B)$, representing how robust the scheme of collapse is when it departs from $z_k$ constant. The abscissa is $l$ until $l=2600$.

Figure 4

Semilog plot of $|{\alpha }_{lm}{|}^2(C_{\mathrm{wigner}}(k))$ for different values of $(A,B)$, representing how robust the scheme of collapse is when it departs from $z_k$ constant. The abscissa is $l$ until $l=2600$.

Figure 5

Plot showing how the integral of $|{\alpha }_{lm}{|}^2(C_1)$ varies with respect to changes in $B$ (${10}^{-4}–10$), keeping $A$ fixed. Both axes $B$ and $l$ are in logscale. See the main text for a more extensive explanation.

Figure 6

Plot showing how the integral of $|{\alpha }_{lm}{|}^2(C_2)$ varies with respect to changes in $B$ (${10}^{-4}-10$), keeping $A$ fixed. Both axes $B$ and $l$ are in logscale. See the main text for a more extensive explanation.

Figure 7

Plot showing how the integral of $|{\alpha }_{lm}{|}^2(C_{\mathrm{wigner}})$ varies with respect to changes in $B$ (${10}^{-4}-10$), keeping $A$ fixed. Both axes $B$ and $l$ are in logscale. See the main text for a more extensive explanation.

Figure 8

Logarithmic plot in both axes of the collapse times $|{\eta }_k^c|$ (in seconds), for the three schemes, taking in account only the best values of $(A,B)$ in the range of ${10}^{-3}{\mathrm{Mpc}}^{-1}<k<1{\mathrm{Mpc}}^{-1}$. For these plots $h=0.7$.

Figure 9

Semilogarithmic plot of the number of e-foldings between $a_k^H$ and $a_k^c$ for the three schemes, taking in account only the best values of $(A,B)$ in the range of ${10}^{-3}{\mathrm{Mpc}}^{-1}<k<1{\mathrm{Mpc}}^{-1}$. For these plots $h=0.7$.