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The collapse of the wave function in the joint metric-matter quantization for inflation

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Abstract

It has been argued that the standard inflationary scenario suffers from a serious deficiency as a model for the origin of the seeds of cosmic structure: it can not truly account for the transition from an early homogeneous and isotropic stage to another one lacking such symmetries. The issue has often been thought as a standard instance of the “quantum measurement problem”, but as has been recently argued by some of us, that quagmire reaches a critical level in the cosmological context of interest here. This has lead to a proposal in which the standard paradigm is supplemented by a hypothesis concerning the self-induced dynamical collapse of the wave function, as representing the physical mechanism through which such change of symmetry is brought forth. This proposal was originally formulated within the context of semiclassical gravity. Here we investigate an alternative realization of such idea implemented directly within the standard analysis in terms of a quantum field jointly describing the inflaton and metric perturbations, the so called Mukhanov–Sasaki variable. We show that even though the prescription is quite different, the theoretical predictions include some deviations from the standard ones, which are indeed very similar to those found in the early studies. We briefly discuss the differences between the two prescriptions, at both, the conceptual and practical levels.

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Notes

  1. Strictly speaking that state is not the BD vacuum, simply because as the result of a slow-rolling inflaton field the space-time background cannot be exactly de Sitter. But here we will ignore that issue and refer to the state \(\vert 0\rangle \) as the BD vacuum, as it is often done. The relevant issue is that this state is as homogeneous and isotropic as the true BD vacuum.

  2. It is invariant under spatial translations \(\hat{T}(d_i)=\exp [i\hat{P}_i d_i]\) and rotations \(\hat{R}_x(\theta _i)=\exp [i\hat{L}(x)_i \theta _i]\), with \(\hat{P}_i\) and \(\hat{L}(x)_i\) the linear and the angular momentum operators, and \(d_i\) and \(\theta _i\) parameters labelling the transformations.

  3. It is straightforward to see that the evolution Hamiltonian commutes with the operators \(\hat{T}(d_i)\) and \(\hat{R}_x(\theta _i)\).

  4. In an abuse of notation we will be writing \(\fancyscript{H}=\prod _{\mathbf{{k}}}\fancyscript{H}_{\mathbf{{k}}}\), although technically the fact that we are working with an infinite number of degrees of freedom requires a construction known as the Fock space. The point is that, despite not being totally precise, this way of presenting things is more transparent.

  5. If we had relied on semiclassical gravity this issue did not arise at this point. There the Newtonian potential is a classical quantity and the classical to quantum connection occurs at the level of the Einstein equations, where one side is the classical Einstein tensor while the other side is the expectation value of the quantum energy-momentum operator, i.e. \(G_{\mu \nu }=8\pi G \langle \hat{T}_{\mu \nu } \rangle \).

  6. As such this quantity can not be measured, and is normally just estimated for the related quantity \(C_l \equiv \frac{1}{2l+1} \sum _{m } |\alpha _{lm} |^2\) to be given by the corresponding Gaussian value of \(C_l/\sqrt{l+ 1/2}\).

  7. In the standard approach the \(n\)-point correlation function for the field operator \(\hat{\psi }(x)\) is identified (without any apparent reason, see for instance the Refs. [1, 2]) with the average over an ensemble of classical anisotropic universes of the same correlation function, now for the Newtonian potential \(\psi (x)\). That is the reason for which they identify the expression (23) with

    $$\begin{aligned} \lim _{-k\eta _{R}\rightarrow 0}\langle 0\vert \hat{\psi }_{\mathbf{{k}}}(\eta _R)\hat{\psi }^{\dagger }_{\mathbf{{k}}^{\prime }} (\eta _R)\vert 0\rangle =\epsilon \frac{t_p^2H^2}{4k^3}\delta _{\mathbf{{k}}\mathbf{{k}}^{\prime }}. \end{aligned}$$
    (22)

    Note that in the corresponding expressions we have an ensemble average of the product of two 1-point functions in contrast with the 2-point function found in the usual approach.

  8. The standard amplitude for the power spectrum is usually presented as proportional to \(V/(\epsilon M_P^4) \propto H^2 t_p^2/\epsilon \), where \(V\) is the inflaton’s potential. The fact that \(\epsilon \) is in the denominator leads, in the standard picture, to a constraint scale for \(V\). However, in (24) the slow-roll parameter \(\epsilon \) is in the numerator. This is because we have not used (and in fact we will not) explicitly the transfer function \(T_{k} (\eta _R,\eta _D)\). In the standard literature it is common to find the power spectrum for the quantity \(\zeta (x)\), a field representing the curvature perturbation in the co-moving gauge. This quantity is constant for modes “outside the horizon” (irrespectively of the cosmological epoch), thus it avoids the use of the transfer function. The quantity \(\zeta \) can be defined in terms of the Newtonian potential as \(\zeta \equiv \psi + (2/3)(\mathcal{H }^{-1} \dot{\psi } + \psi )/(1+\omega )\), with \(\omega \equiv p/\rho \). For large-scale modes \(\zeta _k \simeq \psi _k [ (2/3) (1+\omega )^{-1} + 1]\), and during inflation \(1+\omega = (2/3)\epsilon \). For these modes \(\zeta _k \simeq \psi _k/\epsilon \) and the power spectrum is \(\mathcal{P }_{\zeta }(k) = \mathcal{P }_{\psi }(k) / \epsilon ^2 \propto H^2 t_p^2/\epsilon \propto V/(\epsilon M_P^4)\), which contains the usual amplitude. For a detailed discussion regarding the amplitude within the collapse framework see Ref. [26].

  9. We thank Prof. R. M. Wald for pointing this out.

  10. The selection of course refers to the fact that, according to the standard arguments, the resulting density matrix, after becoming essentially diagonal due to decoherence, represents an ensemble of universes, and our particular one corresponds to one of them. That one can be considered as selected by Nature to become realized. Alternatively, one might take the view that these other universes are also realized, and thus they also exits in realms completely inaccessible to us. In that case the selection corresponds to that universe in which we happen to exist.

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Acknowledgments

We are glad to acknowledge very useful discussions with Prof. R. M. Wald. The work of ADT is supported by a UNAM postdoctoral fellowship and the CONACYT grant No 101712. The work of GL and DS is supported in part by the CONACYT grant No 101712. GL acknowledges financial support by CONACYT postdoctoral grant. DS was supported also by CONACYT and DGAPA-UNAM sabbatical fellowships, UNAM-PAPIIT IN107412-3 grant, and gladly acknowledges the IAFE-UBA for the hospitality during a sabbatical stay.

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Author notes

  1. Alberto Diez-Tejedor

    Present address: División de Ciencias e Ingenierías, Departamento de Física, Campus León, Universidad de Guanajuato, 37150 , León, Mexico

  2. Gabriel León

    Present address: Department of Physics, University of Trieste, Strada Costiera 11, 34014 , Trieste, Italy

  3. Daniel Sudarsky

    Present address: Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 , Mexico, DF, Mexico

Authors and Affiliations

  1. Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, 04510 , Mexico, DF, Mexico

    Alberto Diez-Tejedor & Gabriel León

  2. Instituto de Astronomía y Física del Espacio, Universidad de Buenos Aires, Casilla de Correos 67, Sucursal 28, 1428 , Buenos Aires, Argentina

    Daniel Sudarsky

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  1. Alberto Diez-Tejedor
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Correspondence to Gabriel León.

Appendix: \(A_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}^c}), {B_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c), {C_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c)\) and  \({D_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c)\)  in  Eq. (16)

Appendix: \(A_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}^c}), {B_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c), {C_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c)\) and  \({D_{\mathbf{{k}}}}(\eta ,{\eta _{\mathbf{{k}}}}^c)\)  in  Eq. (16)

In a universe close to de Sitter the functions \(A_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}}^c), B_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}}^c), C_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}}^c)\) and \(D_{\mathbf{{k}}}(\eta ,\eta _{\mathbf{{k}}}^c)\) in Eq. (16) take the form

$$\begin{aligned} X_{\mathbf{{k}}} (\eta ,\eta _{\mathbf{{k}}}^c)&= X_{\mathbf{{k}}}^{(1)} (\eta ,\eta _{\mathbf{{k}}}^c)\cos \Delta _{\mathbf{{k}}}^c + X_{\mathbf{{k}}}^{(2)} (\eta ,\eta _{\mathbf{{k}}}^c)\sin \Delta _{\mathbf{{k}}}^c. \end{aligned}$$
(37)

Here \(X\) denotes \(A, B, C\) or \(D\), with the different \(X_{\mathbf{{k}}}^{(i)}(\eta ,\eta _{\mathbf{{k}}}^c)\) given by

$$\begin{aligned}&A_{\mathbf{{k}}}^{(1)} = 1-\frac{1}{s}\,C_{\mathbf{{k}}}^{(1)},\quad A_{\mathbf{{k}}}^{(2)} = \frac{1}{s_{\mathbf{{k}}}^c}-\frac{1}{s}\,C_{\mathbf{{k}}}^{(2)},\end{aligned}$$
(38a)
$$\begin{aligned}&B_{\mathbf{{k}}}^{(1)} =-C_{\mathbf{{k}}}^{(1)}C_{\mathbf{{k}}}^{(2)}, \quad B_{\mathbf{{k}}}^{(2)} =- 1+\frac{1}{s^2}+\frac{1}{s_{\mathbf{{k}}}^{c\,2}}-\frac{1}{ss_{\mathbf{{k}}}^c}C_{\mathbf{{k}}}^{(2)},\end{aligned}$$
(38b)
$$\begin{aligned}&C_{\mathbf{{k}}}^{(1)} = \frac{1}{s}-\frac{1}{s_{\mathbf{{k}}}^c},\quad \quad \; C_{\mathbf{{k}}}^{(2)} = 1+\frac{1}{ss_{\mathbf{{k}}}^c},\end{aligned}$$
(38c)
$$\begin{aligned}&D_{\mathbf{{k}}}^{(1)} = 1+\frac{1}{s_{\mathbf{{k}}}^c}\,C_k^{(1)}, \quad D_{\mathbf{{k}}}^{(2)} =-\frac{1}{s}\left(1-\frac{s}{s_{\mathbf{{k}}}^c}\,C_{\mathbf{{k}}}^{(2)}\right). \end{aligned}$$
(38d)

Remember that we are using \(s\equiv k\eta , s_{\mathbf{{k}}}^c\equiv k\eta _{\mathbf{{k}}}^c\) and \(\Delta _{\mathbf{{k}}}^c\equiv s-s_{\mathbf{{k}}}^c\).

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Diez-Tejedor, A., León, G. & Sudarsky, D. The collapse of the wave function in the joint metric-matter quantization for inflation. Gen Relativ Gravit 44, 2965–2988 (2012). https://doi.org/10.1007/s10714-012-1433-5

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