暴漲宇宙的初始條件及其在宇宙背景輻射上所形成之特徵

Initial Condition of the Inflationary Universe and Its Imprint on the Cosmic Microwave Background

林裕翔(Yu-Hsiang Lin)

指導教授 ： 陳丕燊

國立臺灣大學/理學院/物理學研究所/博士(2017年)

https://doi.org/10.6342/NTU201701823

摘要

從WMAP到Planck人造衛星的觀測結果中，我們持續看到宇宙微波背景輻射在長波長波段的頻譜振幅低於Lambda-CDM標準宇宙模型的預測。儘管統計上這項差異還不構成違反標準宇宙模型的決定性證據，我們仍然有強烈的動機去考慮暴漲前宇宙對頻譜抑制的效應。觀測顯示在暴漲使宇宙尺度膨脹最後的e的60次方倍左右的期間裡，暴漲場的能量尺度為10的16次方個十億電子伏特。如果在更高的能量尺度裡，物質處於不同的相，或者重力遭到修正，則暴漲前宇宙的幾何演化可能不同於暴漲時期的近似平時空內的de Sitter宇宙。在「恰好足夠」的暴漲宇宙中，我們今日所觀測到的大尺度微擾早期是在即將要進入暴漲時期的過渡期附近離開哈伯膨脹的視界地平線，因此這些大尺度微擾事實上攜帶著暴漲前宇宙的資訊，而我們也可能可以從大尺度的頻譜中一窺暴漲前宇宙的真相。
我們首先考慮一個簡單的理論模型。在這個模型中，暴漲前宇宙充滿了由疇壁或宇宙弦這兩種受挫拓撲缺陷所構成的網絡。在疇壁的情況下，暴漲前宇宙仍然是加速膨脹，但是膨脹的加速度比暴漲時期小。在宇宙弦的情況下，暴漲前宇宙則是等速膨脹。我們引入Chaplygin氣體作為現象學的模型，以連接暴漲前的受挫拓撲缺陷網絡時期和暴漲時期。利用近似的長波長微擾初始條件，我們發現這樣的暴漲前宇宙模型有助於解釋大尺度頻譜的抑制。之後透過對於初始真空態更完整和系統性的分析，我們又發現暴漲前的疇壁網絡其實擁有和這裡所使用的近似初始條件不同的真空態，因此並不能造成大尺度頻譜抑制。
我們接著進一步證明，暴漲結束時的大尺度頻譜，事實上是直接由暴漲場初始量子態在哈伯膨脹世界地平線外的頻譜來決定。透過研究在FLRW宇宙中一個由狀態方程式參數w所描述的純量場的曲率微擾頻譜，我們得出在宇宙暴漲結束時，曲率微擾的長波長頻譜直接反映出暴漲場初始量子態在波長大於哈伯半徑的波段的頻譜。如果宇宙起始於超暴漲(w < -1)或正壓力(w > 0)時期底下的絕熱真空態，暴漲結束時，長波長頻譜的振幅將會被抑制。然而在正壓力時期的條件下，並沒有合乎物理因果律的機制來形成這樣的絕熱真空態。另一方面，只要宇宙是起始於-1 < w < 0的絕熱真空態，即使在暴漲結束之前宇宙曾經經歷過一段正壓力的時期，暴漲結束時的長波長頻譜振幅也將會被增強，而非抑制。我們進一步考慮由一個雙場模型所產生的二階段宇宙暴漲，並計算出其頻譜，證明其結果與透過前述較為簡單的單場分析所獲得的結論一致。
我們在上面所發現的兩種能解釋頻譜抑制的可能性——暴漲前的超暴漲或正壓力時期——在作為宇宙的初始條件上都不是完全令人滿意。這個困難也許暗示了頻譜抑制並非由半古典物理所造成，而是一種量子重力的效應。我們考慮採用量子重力中的Wheeler-DeWitt方程的解：Hartle-Hawking的無邊界波函數，來作為宇宙的初始條件。在這樣的條件底下，我們發現大尺度頻譜的抑制可能來自於有質量的暴漲場，而暴漲場的初始真空態是在緊緻流形中的歐氏瞬子。我們計算微擾的初始能量頻譜，發現對於任何中等質量的暴漲場來說，長波長的頻譜都會受到壓抑。

關鍵字

宇宙微波背景輻射 ； 大尺度頻譜振幅壓抑 ； 拓撲缺陷 ； 疇壁 ； 宇宙弦 ； Chaplygin氣體 ； 宇宙學微擾理論 ； 絕熱真空態 ； 宇宙暴漲 ； 初始條件 ； 頻譜演化 ； 超暴漲 ； 雙場暴漲 ； Hartle-Hawking無邊界波函數 ； Wheeler-DeWitt方程式 ； 最小超空間模型

並列摘要

There is an apparent power deficit relative to the Lambda-CDM prediction of the cosmic microwave background spectrum at large scales, which, though not yet statistically significant, persists from WMAP to Planck data. It is well-motivated to consider such power suppression as the imprint of the preinflationary era. The observations show that about the last 60 e-folds of inflation operates at the energy scale of 10 to the 16th power GeV. If at the higher energy scales the matter content is in a different phase, or that the gravity is modified, the evolution of the geometry in the preinflationary universe may not be the de Sitter expansion in a spatially flat universe, as it is deep in the inflationary era. In the scenario of "just-enough" inflation, the perturbations at the largest scales we observe today exit the horizon around the transition time from preinflation to inflation era, hence carrying the characteristics of the preinflationary universe. The large-scale spectrum may therefore serve as the window to peek into the preinflationary universe.
We first present a simple toy model corresponding to a network of frustrated topological defects of domain walls or cosmic strings that exist previous to the standard slow-roll inflationary era of the universe. If such a network corresponds to a network of frustrated domain walls, it produces an earlier inflationary era that expands more slowly than the standard one does. On the other hand, if the network corresponds to a network of frustrated cosmic strings, the preinflationary universe would expand at a constant speed. Those features are phenomenologically modeled by a Chaplygin gas that can interpolate between a network of frustrated topological defects and a de Sitter-like or a power-law inflationary era. We show that these scenarios can alleviate the quadrupole anomaly of the cosmic microwave background spectrum, based on the approximate initial conditions for the long-wavelength perturbations. A more thorough and systematic analysis on the initial vacuum carried out later will show that the preinflationary domain wall dominated era has a different vacuum state from the approximate one and does not suppress the long-wavelength spectrum.
We then go further to show that the large-scale spectrum at the end of inflation reflects the super-horizon spectrum of the initial state of the inflaton field. By studying the curvature perturbations of a scalar field in the Friedmann-Lemaitre-Robertson-Walker universe parameterized by the equation of state parameter w, we find that the large-scale spectrum at the end of inflation reflects the superhorizon spectrum of the initial state. The large-scale spectrum is suppressed if the universe begins with the adiabatic vacuum in a superinflation (w < -1) or positive-pressure (w > 0) era. In the latter case, there is however no causal mechanism to establish the initial adiabatic vacuum. On the other hand, as long as the universe begins with the adiabatic vacuum in an era with -1 < w < 0, even if there exists an intermediate positive-pressure era, the large-scale spectrum would be enhanced rather than suppressed. We further calculate the spectrum of a two-stage inflation model with a two-field potential and show that the result agrees with that obtained from the ad hoc single-field analysis.
Neither of the two possibilities discovered earlier--the preinflationary superinflation and positive-pressure eras--that attempt to account for the power suppression is completely satisfactory as a realistic initial condition of the inflationary universe. This difficulty may be a hint that the origin of the power suppression does not lie in the semi-classical physics, but in the quantum theory of gravity. We consider the Hartle-Hawking no-boundary wave function, which is a solution to the Wheeler-DeWitt equation, as the initial condition of the universe. We find that the power suppression can be the consequence of a massive inflaton, whose initial vacuum is the Euclidean instanton in a compact manifold. We calculate the primordial power spectrum of the perturbations and show that, as long as the scalar field is moderately massive, the power spectrum is suppressed at the long-wavelength scales.

並列關鍵字

Cosmic microwave background ； large-scale power suppression ； topological defect ； domain wall ； cosmic string ； Chaplygin gas ； cosmological perturbation theory ； adiabatic vacuum ； inflation ； initial condition ； spectrum evolution ； superinflation ； two-field inflation ； Hartle-Hawking no-boundary wave function ； Wheeler-DeWitt equation ； minisuperspace model

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