Hartle-Hawking wave function and large-scale power suppression of CMB
Dong-han Yeom







Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan


In this presentation, we first describe the Hartle-Hawking wave function in the Euclidean path integral approach. After we introduce perturbations to the background instanton solution, following the formalism developed by Halliwell-Hawking and Laflamme, one can obtain the scale-invariant power spectrum for small-scales. We further emphasize that the Hartle-Hawking wave function can explain the large-scale power suppression by choosing suitable potential parameters, where this will be a possible window to confirm or falsify models of quantum cosmology. Finally, we further comment on possible future applications, e.g., Euclidean wormholes, which can result in distinct signatures to the power spectrum.


Hartle-Hawking wave function and large-scale power suppression of CMB[Proceedings of the 13th International Conference on Gravitation, Astrophysics, and Cosmology & the 15th Italian-Korean Symposium on Relativistic Astrophysics: A Joint Meeting. Talk on July 7, 2017, Seoul, Republic of Korea]
    Dong-han Yeominnocent.yeom@gmail.com 
    December 30, 2023
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§ INTRODUCTION


In order to resolve and understand the initial singularity of our universe, we need to rely on quantum gravity. There are various approaches, but the most conservative approach is to quantize space and time by following the canonical approach. The corresponding master equation for the wave function of the universe is called by the Wheeler-DeWitt equation <cit.>.

The Hartle-Hawking wave function <cit.> is a proposal of the boundary condition of the Wheeler-DeWitt equation, where this can be presented by the Euclidean path integral

    Ψ[∂ g_μν,∂Φ] = ∫𝒟g_μν𝒟Φ e^-S_E [g_μν,Φ],

where S_E is the Euclidean action as a functional of the metric g_μν and the inflaton field Φ. This path integral is over all geometries and fields that have the boundary ∂ g_μν and ∂Φ. This path integral can be further approximated by using the steepest-descent approximation, i.e., this integral is presented by the sum over on-shell solutions:

    Ψ[∂ g_μν,∂Φ] ≃∑ e^-S_E^on-shell,

where these on-shell solutions are called by instantons. Since several years, dynamical and complexified instantons have been extensively investigated <cit.>.

The Hartle-Hawking wave function not only resolves the initial singularity problem but also gives implications for inflationary cosmology <cit.>. In this presentation, we first revisit the investigations of Halliwell and Hawking <cit.>. Then by using this, we can investigate physical implications of the Hartle-Hawking wave function.



§ HALLIWELL-HAWKING REVISITED


This section is a brief review of the historical work of Halliwell and Hawking <cit.>.

We consider the Einstein gravity with a scalar field:

    S_E = ∫√(g) d^4x [ R/16 π - 1/2∇_μΦ∇^μΦ - V(Φ)],

where R is the Ricci scalar and V(Φ) is the potential term of the inflaton field Φ. For simplicity, we consider the massive scalar field (with the rescaled scalar field ϕ = Φ√(4π/3))

    V(Φ) = 8πσ^2/3( 1 + m^2ϕ^2),

where σ is a constant and m is the mass parameter of the scalar field. It is useful to define the effective Hubble parameter H_0 = √(8π/3).

In order to consider perturbations to the homogeneous solution, we introduce the following metric and field ansatz (i, j = 1, 2, 3 and we assume the summation convention for the spherical coordinate indices n, l, m):

    ds^2   =   σ^2[ ( N^2 - N_iN^i) dτ^2 + 2 N_i dx^i dτ + a^2γ_ij dx^idx^j],
    
    N    =    N_0[ 1 + 1/√(6) g_nlmQ_nlm],
    
    N_i   =    a [ 1/√(6) k_nlm( P_i)_nlm + √(2) j_nlm( S_i)_nlm],
    γ_ij   =   [ 1 + √(6)/3 q_nlmQ_nlm] dΩ_3^2 + [ √(6) b_nlm(P_ij)_nlm. 
          . + √(2) c^o_nlm( S_ij^o)_nlm + √(2) c^e_nlm( S_ij^e)_nlm + 2 d^o_nlm( G_ij^o)_nlm + 2 d^e_nlm( G_ij^e)_nlm],
    Φ   =   √(3/4π)ϕ + 3π/2 f_nlmQ_nlm,

where N_0 is an arbitrary constant, a = H_0^-1sin H_0τ and ϕ = const. are the background solutions of the de Sitter space (or an approximate solution of the slowly rolling scalar field), g, k, j, q, b, c^o,e, d^o,e, f are time dependent mode functions and Q, P_i, S_i, P_ij, S_ij^o,e, G_ij^o,e are space dependent expansion basis. The detailed definitions for the expansion basis are in <cit.>.

In this work, we are focusing on the scalar field perturbations. In the slow-roll limit, the scalar field perturbation f_nlm is determined by the following equation <cit.>:

    f̈_nlm + 3 H_0ḟ_nlm + ( m^2 + n^2-1/a^2)f_nlm = 0.

From this, we can calculate the power spectrum P(n) as a function of n (after summing over l and m)

    P(n) = 3n^2/4π⟨ f_n^2⟩,

where we simply denote f_n since it only depends on n.

In order to calculate the expectation value of f_n^2, we need to present the wave function for each perturbation f_n. By plugging the on-shell condition of f_n, the wave function will look like

    Ψ[f_n] ∝exp[ -1/2a^3 f_nḟ_n].

There would be many ways to write ḟ_n as a function of f_n. However, if we give the Euclidean vacuum condition following Laflamme <cit.>, one can obtain the proper form:

    Ψ[f_n] ∝exp[ -ḟ̃̇_n/2 f̃_na^3 f_n^2],

where f̃_n and ḟ̃̇_n are f_n and ḟ_n which are evaluated at the horizon crossing time, respectively. By using this wave form, one can finally obtain a closed form of the power spectrum P(n):

    P(n) = 3n^2/4πf̃_n/2a^3ḟ̃̇_n.







§ SUPPRESSION OF CMB POWER SPECTRUM


By solving equation (<ref>), we can obtain f̃_nlm. There are two linearly independent solutions, but we need to impose the correct boundary condition at τ = 0. The physically viable solution can be presented by the following analytic form <cit.>:

    f̃_n = D (z-z^2)^(n-1)/2_2F_1(n-ν,n+ν+1;n+1;z),

where

    z    =   1-cos H_0τ/2,
    ν   =    -1/2±√(9/4 - m^2/H_0^2),
    
    D    =   2^n+1Γ (1 + n/2 + ν/2)/n! H_0Γ(1 - n/2 + ν/2).

The time parameter τ will be Wick-rotated to τ = π/2H_0 + it. The function f̃_n should be evaluated at the horizon crossing time, i.e., n = aH = ȧ = sinh H_0t which is the analog relation of k/aH ≃ 1 for the flat topology case.

The final result of the power spectrum P(n) is plotted in figure <ref> (more detailed discussions are in <cit.>). If the mass term is zero, then the large-scale power spectrum is enhanced (upper), which is consistent with quantum field theoretical expectations <cit.>. However, as one turns on the mass term, the large-scale power spectrum can be partly suppressed (lower). Of course, the detailed choices of parameters depend on the detailed inflation model, but it is enough to show that quantum cosmological calculations based on the Hartle-Hawking wave function can explain the large-scale power suppression. Note that for both cases, the power spectrum is scale-invariant for small-scales as we expected.



§ FUTURE PERSPECTIVES


Up to now, the observed CMB data has a tension with the theoretical expectations (based single field inflation and the ΛCDM model), especially for large length scales <cit.>. It is not clear whether this tension will be confirmed to be real or not for future experiments. However, if it is the case, then this tension with theoretical expectations is a good window to test quantum gravity.

In this paper, we show that the Hartle-Hawking wave function can easily explain this power suppression behavior. The Hartle-Hawking wave function is not the only way to explain the large-scale power suppression (e.g., <cit.>), but we can say that this approach is very well-established and conservative compared to the others.

We can extend this investigation to the other inflationary models <cit.> and also improve approximations for dynamics of the inflaton field <cit.>. Perhaps, we may study not only compact instantons but also non-compact instantons, e.g., Euclidean wormholes <cit.>. In the end, we hope that future experiments can confirm or falsify several models of quantum cosmology. Then, quantum gravity will be established on the experimental ground.





§ ACKNOWLEDGMENT


The author would like to thank Pisin Chen and Yu-Hsiang Lin for the stimulated collaborations of this project. The work is supported by Leung Center for Cosmology and Particle Astrophysics (LeCosPA) of National Taiwan University (103R4000).

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