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The cosmological constant problem and topological gravity motivated by renormalization group

Mori, Taisaku 森, 大作 名古屋大学

2020.04.02

概要

一般相対性理論と場の量子論は素粒子のように非常に小さなスケールの現象から宇宙のような広大なスケールでのダイナミクスに渡る様々な実験及び観測により、高い精度で検証され、成功を収めてきた。一方で宇宙論等には未解決の問題も依然として存在し、それを解決するための様々な試みも行われている。この問題の 1 つが階層性問題の類である宇宙項問題と呼ばれる問題である。すなわち、量子論における繰り込みの手法を宇宙項に用いた時、観測された値と合わせようとすると、理論のパラメータに不自然な微調整が必要となってしまう。この宇宙項問題に対するアプローチの 1つとしてユニモジュラー重力と呼ばれる理論が知られている。

ユニモジュラー重力はもともとアインシュタインが提案した理論で、一般相対性理論に対して新たな拘束条件を課したものである。一般相対性理論は一般の座標変換の下での共変性を持つが、この拘束条件は体積要素の大きさを一定に保つものを除いてこの共変性を破る。それにも拘らず、この理論で得られる方程式は一般相対性理論のものと同等になる。ただし、この理論では宇宙項に含まれる宇宙定数が方程式を導出する際に積分定数として現れるため、宇宙定数は理論を特徴づけるパラメータではなく、初期条件等によって決まる。この性質によりこの理論では宇宙項問題が解決される可能性がある。

申請者は上記のユニモジュラー重力の拡張として提案された模型の一つで、位相的場の理論とみなせるものについて研究を行った。この模型では宇宙定数及び重力結合定数は定数ではなく、動的なスカラー場となる。この理論には、このスカラー場が低エネルギーで定数となる安定な解が存在し、これにより現在の宇宙の加速膨張解を再現することが出来るが、この定数は初期条件等により決まる。このようにこの理論では量子論としての宇宙項の問題が古典論としての初期条件の問題に置き換わるため、微調整の問題が緩和されることが期待される。

申請者は、この模型では宇宙定数を含む重力理論に現れる結合定数に対応するスカラー場が宇宙のエネルギースケールに依存していることに着目した。このような結合定数のスケール依存性は、スケール変換への応答を見るくりこみ群方程式と類似性がある。そこで申請者はこの模型を変形し、模型に現れる方程式を繰り込み群方程式に見立て、低エネルギーと高エネルギーの両方で固定点を持つような模型を構築した。このように変形した模型では、高エネルギーでの固定点が、宇宙初期のインフレーションを実現する一方で、低エネルギーの固定点が現在の宇宙の加速膨張を再現する。更に申請者は上記の変形した模型で、適切なポテンシャル項を仮定し、インフレーションと現在の宇宙の加速膨張を再現する固定点が現れる条件を求め、なおかつ、2 つの固定点を繋ぐような解が存在するパラメータ領域を特定した。

以上のことは、位相的場の理論を使い、宇宙初期のインフレーションと現在の宇宙の加速膨張を再現するとともに宇宙項問題を解く模型を構築するための足掛かりとなる可能性がある。

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