Preinflation from the multiverse: can it solve the quadrupole problem in the cosmic microwave background?
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Abstract
We analyze a quantized toy model of a universe undergoing eternal inflation using a quantumfieldtheoretical formulation of the Wheeler–DeWitt equation. This socalled third quantization method leads to the picture that the eternally inflating universe is converted to a multiverse in which subuniverses are created and exhibit a distinctive phase in their evolution before reaching an asymptotic de Sitter phase. From the perspective of one of these subuniverses, we can thus analyze the preinflationary phase that arises naturally. Assuming that our observable universe is represented by one of those subuniverses, we calculate how this preinflationary phase influences the power spectrum of the cosmic microwave background (CMB) anisotropies and analyze whether it can explain the observed discrepancy of the power spectrum on large scales, i.e. the quadrupole issue in the CMB. While the answer to this question is negative in the specific model analyzed here, we point out a possible resolution of this issue.
1 Introduction
In physics, we have been facing the situation that we are lacking a fully consistent quantum theory of gravity for more than 80 years. Nevertheless, there have been several proposals for a theory of quantum gravity and we need to decide which one of these is the most promising path to be investigated further. Hence, we need predictions that can be tested by experiment or observation. However, we have to deal with the crucial problem that effects from a theory of quantum gravity are expected to only become dominant at very large energies or very small scales that correspond to the Planck scale, i. e. an energy of \(10^{19}\) GeV corresponding to a length scale of \(10^{35}\) m. Because of this it is extremely difficult to find sizeable effects of quantum gravity. One thus has to look for scenarios in which high energies or large curvature are involved and in this context either black holes or the early universe come to mind.
Within the last mentioned scenario, for the very first instants of the evolution of our universe, we have the widely accepted theory of inflation that – apart from providing a solution to the flatness and horizon problem in our universe – also gives rise to the structure in the universe we observe today. More importantly, the features of inflation are encoded in the anisotropies of the cosmic microwave background (CMB) radiation that has been measured to a large accuracy by the satellites COBE, WMAP and most recently Planck [1]. Inflation is estimated to involve energies of the order of \(10^{5}\) times the Planck energy, far more than particle accelerators that operate at energies of up to \(10^{15}\) times the Planck energy.
The anisotropy spectrum of the cosmic microwave background can be described to a remarkable accuracy with a small set of parameters deduced from inflation. However, in recent data there is still a discrepancy at the largest scales that awaits further explanation: The power at these scales is smaller than expected [1]. Given that these largest scales exit the horizon at the earliest during inflation, they are also the ones to be influenced by the highest energies during inflation and thus are most likely to carry information about any quantumgravitational effect happening at or before the onset of inflation.
Furthermore, any theory of quantum gravity should lead to a new fundamental equation that recovers general relativity in a semiclassical limit. For the Wheeler–DeWitt equation it has been shown that a semiclassical approximation gives rise to both general relativity and quantum field theory in curved spacetime and in a subsequent step leads to a functional Schrödinger equation with quantumgravitational corrections [2, 3]. For an inflationary model these effects can be estimated to be of order \(10^{10}\) and concrete calculations have confirmed this estimate [4, 5, 6, 7, 8, 9, 10, 11]. The magnitude of these effects in thus larger than in other less energetic scenarios, however, due to the fact that these corrections have this magnitude only for largescale anisotropies – where also the inherent statistical uncertainty due to cosmic variance is most prominent – and drop off quickly for smaller scales, it does not seem realistic that such semiclassical effects can be measured – at least in the CMB power spectrum. Apart from that it has been discovered that several theories of quantum gravity like Loop Quantum Gravity give rise to a phase that precedes inflation [12, 13, 14, 15, 16]. Such a preinflationary phase could thus lead to effects that can be orders of magnitude larger than the effects arising from a semiclassical approximation and thus can overcome cosmic variance.
Another aspect when considering inflation is that most inflationary models lead to socalled eternal inflation, which means that the universe as a whole inflates forever while bubbles of spacetime regions form in which inflation eventually ends. These bubbles can be regarded as universes of their own causally disconnected from the others, such that one can speak of a certain kind of multiverse [17, 18] (see [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] for other models of the multiverse). It has been shown recently that describing the model of an eternally inflating universe using a quantumfieldtheoretical formulation of the Wheeler–DeWitt equation – also called third quantization – leads to subuniverses exhibiting a phase whose scale factor evolves like \(a^{6}\) before reaching an approximate de Sitter phase [32, 33, 34]. Hence, from the perspective of a single subuniverse the de Sitterlike inflationary phase is preceded by a preinflationary phase. Also modeling an interaction between universes in the third quantization picture leads to specific kinds of preinflationary phases [35].
The objective of this article is thus to analyze a specific model of an inflating universe in the third quantization picture with regard to the effect of the induced preinflationary phase on the CMB temperature anisotropies in the subuniverses and to see whether the observed suppression on large scales can be explained by this specific preinflationary phase. This is a natural scenario to look for a potential explanation for the low CMB quadrupole, as this preinflationary phase will affect the largest modes, i.e. those that have recently reentered the horizon and therefore give the largest contribution to the CMB quadrupole. However, before tackling these issues, let us first review what has been done before with regard to this.
There have been different approaches to explain the discrepancy between the theoretical prediction of inflation and the measured temperature anisotropies of the CMB at the largest scales. We list some of them in the following. Fast roll inflation prior to standard inflation [36, 37], bounces and cyclic universes [38, 39, 40], a radiation dominated era [41, 42] and a preinflationary matter era supported by primordial micro black holes remnants [43] have been considered in the context of the low quadrupole problem. More recently, slowroll inflation preceded by a topological defect phase [44] as well as compactification before inflation [45] were also suggested as potential ways to explain the low CMB quadrupole.
The article is structured as follows. In the next section, we will present our chosen universe model and explain how we apply the third quantization formalism to it. In Sect. 3, we will review the methods to calculate the power spectrum of scalar perturbations. In Sect. 4, we show our results. Then, in Sect. 5, we present our conclusions. Finally, in Appendix A, we present further analytical expressions for the cosmological background evolution of the considered model, while in Appendix B, we present an approximation for the evaluation of the power spectrum after horizon crossing.
2 Model
We reconsider a model similar to our previous work [34] that led to the appearance of an instanton with the crucial difference that we now consider a flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe with scale factor a instead of a closed one. We shall see that there will be no instanton in this model. As before, we introduce a minimally coupled scalar field \(\varphi \) with mass m, which follows the potential \(\mathcal {V}(\varphi ) = \frac{1}{2}\,m^2\varphi ^2\).
The interpretation here is that this WDW equation (2.7) describes the individual subuniverses, in which the scalar field takes a specific value leading to the inflationary scale \(H_{\mathrm {dS}}\) and the \(\varphi \)derivative term that is present in the parent WDW equation (2.5) leaves its trace as the term \(\hbar ^2 K^2/a^2\). Note that the factor \(\hbar ^2\) indicates that this term is clearly of quantum origin.
3 Scalar perturbations
4 Numerical results
4.1 Toy model
On the lefthand side panel of Fig. 2 we present the evolution of \(z''/z\) (red) and of the wavenumber of the Hubble horizon (blue) for the background model in Eq. (4.1). In the asymptotic regimes, when \(z''/z\) tracks the behavior of \((aH)^2\), we find that the mode functions evolve according to the solution (3.11), with \(\lambda =0\) during the initial “stiff” epoch and \(\lambda =(6\alpha )/(42\alpha )\) during the subsequent powerlaw inflation. In the transition between these two regimes the solution (3.11) is no longer valid and a numerical integration of the Mukhanov–Sasaki equation is required. We can, however, analyze the features of the potential during such a transition, where we find a reversal of sign accompanied by two “bumps”. Defining a characteristic wavenumber \(k_c\) through the maximum value of \(z''/z\) in this interval (cf. Fig. 2), we expect to find imprints of the transition in the modes with wavenumber \(k_1\lesssim k_c\). For wavenumbers \(k_2\gg k_c\), we expect the modes to have enough time to achieve the Bunch–Davies vacuum before exiting the Hubble horizon during inflation. As such, these modes should have no memory of the preinflationary evolution and are “blind” to the shape of the potential during the transition.
4.2 Numerical computations

For modes with \(k<10^2k_\mathrm {c}\), we set the initial conditions for the perturbations deep inside the kinetically dominated period, at \(N=N_{\mathrm {ini},1}\) (cf. the mode \(k_1\) in Fig. 1). The values of the integration variables are fixed using the solutions (3.11) for \(w=1\) and setting \(c_{1k}=0\) and \(c_{2k}=1\).

For modes with \(k>10^2k_\mathrm {c}\), we consider that the modes are not sensitive to the shape of \(z''/z\) during the transition and that they are in the ground state at the beginning of inflation (cf. the mode \(k_2\) in Fig. 1). We use the solutions (3.11) for \(w=1+\alpha /3\), \(c_{1k}=1\) and \(c_{2k}=0\), to specify the initial values of the integration variables some \( N_{\mathrm {ini},2}\) efolds before the moment of horizon crossing.
5 Conclusion
Despite the incredible fits that inflationary models give to the CMB temperature anisotropies, there are still some anomalies, like the lowquadrupole problem of the CMB, that might hint towards new preinflationary physics. In the present paper, we explore the possibility of solving the quadrupole problem in the CMB in the paradigm of the multiverse within the framework of the third quantization. For this goal, we use a toy model as the one presented in the second section and assume, as a first approach, power law inflation despite being aware of the shortcomings of this kind of inflation.
Given our simplified model, it turns out that while we can get a significant suppression of the power spectrum on the largest scales as shown in Fig. 4, it turns out that generically a new “bump” or extra peak appears on the power spectrum between the mode corresponding the pivot scale used in the data analysis of the Planck mission and the scale corresponding to the present Hubble horizon. Therefore, despite the possibility of obtaining a suppression on large scales, the existence of the extra peak with such a great amplitude rules out the present model unless the value of \(\tilde{K}\), defined just after Eq. (2.14), is set so low as to wash away any possible imprints in the visible range. Indeed, we point out that in order for the suppression on the low multipoles to be observed, the good fit to the observational points for \(\ell > 30\) has to be spoiled, as can be seen in Fig. 5. A possibility to overcome this problem is to consider an interacting multiverse, for example, as the one presented in [35], which could alleviate the CMB quadrupole problem along the lines of [41, 42, 43]. We will present those results in a forthcoming paper.
Footnotes
 1.
The normalization employed here is equivalent to \(\tilde{K}=2/(3\sqrt{3}) K/K_\mathrm {max}\) where the parameter \(K_\text {max}\) is defined in [34] (the same parameter is defined as \(k_m\) in [33]) as the maximum value of K for which a quantum tunnelling effect can occur in a universe with closed spatial geometry. In the present case, with a flat spatial section, \(K_\text {max}\) has no particular physical meaning aside from fixing the energy scale of preinflation.
 2.
The variable v is defined in the interval \(]1,2\sqrt{3}]\) where \(v(a=0)=2\sqrt{3}\) and \(v(a\rightarrow +\infty )=1\).
 3.
The variable \(\xi \) is defined in the interval \(]0,\arccos (\sqrt{3}2)]\) where \(\xi (a=0)=\arccos (\sqrt{3}2) \) and \(\xi (a\rightarrow +\infty )=0\).
 4.
For \(w=1/3\) the potential \(z''/z\) is constant, which leads to trivial oscillatory solutions for \(v_k\).
 5.
While strictly speaking the formula (3.18) is only valid for \(k\eta \ll 1\), i.e. after the mode exits the Hubble horizon, it can be shown that a good approximation to the power spectrum after inflation can be obtained by extending this solution to the moment of horizon crossing, when \(k\eta \approx 1\). For more details please see Appendix B.
 6.
While in the rest of the paper the wavenumber k is dimensionless, in Fig. 4 we follow the convention in the literature and display k in units of h Mpc\(^{1}\).
 7.
These normalized angular power spectra contain the scalar and tensorial sectors.
Notes
Acknowledgements
This article is based upon work from COST Action CA15117 “Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA)”, supported by COST (European Cooperation in Science and Technology). The research of M. B.L. is supported by the Basque Foundation of Science Ikerbasque. She and J. M. also would like to acknowledge the partial support from the Basque government Grant No. IT95616 (Spain) and the project FIS201785076P (MINECO/AEI/FEDER, UE). The research of M. K. was financed by the Polish National Science Center Grant DEC2012/06/A/ST2/00395 as well as by a Grant for the Short Term Scientific Mission (STSM) “Multiverse impact onto the cosmic microwave background and its relation to modified gravity” (COSTSTSMCA1511736137) awarded by the abovementioned COST Action. For their kind hospitality while part of this work was done, M. K. and J. M. would like to thank the Centro de Matemática e Aplicações of the Universidade da Beira Interior in Covilhã, Portugal and M. K. also thanks the Department of Theoretical Physics and History of Science of the University of the Basque Country (UPV/EHU). M. K. is also grateful to M. P. Dąbrowski for fruitful discussions. J. M. would like to thank UPV/EHU for a Ph.D. fellowship.
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