diff --git "a/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt" "b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/39E2T4oBgHgl3EQfOAbJ/content/tmp_files/load_file.txt" @@ -0,0 +1,926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf,len=925 +page_content='Inflation in Weyl Scaling Invariant Gravity with R3 Extensions Qing-Yang Wanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Yong Tanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' and Yue-Liang Wua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='e aUniversity of Chinese Academy of Sciences (UCAS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China bSchool of Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' UCAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hangzhou 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China cInternational Center for Theoretical Physics Asia-Pacific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing/Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China dNational Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing 100101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China eInstitute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' China (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2023) Abstract The cosmological observations of cosmic microwave background and large-scale structure indicate that our universe has a nearly scaling invariant power spectrum of the primordial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, the exact origin for this primordial spectrum is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here, we propose the Weyl scaling invariant R2 + R3 gravity that gives rise to inflation that is responsible for the primordial perturbation in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We develop both analytic and numerical treatments on inflationary observables, and find this model gives a distinctive scalar potential that can support two different patterns of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first one is similar to that occurs in the pure R2 model, but with a wide range of tensor-to-scalar ratio r from O(10−4) to O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The other one is a new situation with not only slow-roll inflation but also a short stage of oscillation-induced accelerating expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Both patterns of inflation have viable parameter spaces that can be probed by future experiments on cosmic microwave background and primordial gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='03744v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='CO] 10 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INTRODUCTION Inflation is a hypothetical epoch of exponential expansion introduced in the very early universe to solve the cosmological horizon and flatness problems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is also a reasonable scheme to explain the origin of primordial density perturbations, which plays the role of the seeds that formed the structure of current universe [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In recent years, the precise measurement of cosmic microwave background (CMB) presents us with an almost scale invariant spectrum of primordial perturbations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This result is usually explained by an approximate de Sitter spacetime of the very early universe [5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, it is theoretically explored that there is a more profound and basic principle behind the phenomenon, namely, local Weyl scaling invariance of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This symmetry is first proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Weyl in the attempt of understanding gravity and electromagnetism in a unified framework [10, 11], and after a century of development, it has been applied extensively to particle physics, cosmology [12–30] and gauge theory of gravity [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Lately, inflation in the Weyl scaling invariant theory of gravity, especially induced by a quadratic curvature term R2, has been of many concern [35–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Comparing with the conventional R2 model, which is also called Starobinsky model [46–49], the scaling invariant version not only allows a viable inflation scenario with good observational agreement, but also provides a framework to comprehend another fundamental puzzles, such as hierarchy problem [37, 40, 50] and dark matter candidates [41, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, inflation with only quadratic scalar curvature might be just a simplistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' From the viewpoint of effective field theory, any higher-order curvature effects may exist and play a role in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence it is reasonable to evaluate their impacts on inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, the extensions with high-order tensors, like RµνRµν or RµνρσRµνρσ, can result in unacceptable ghost degrees of freedom [51], while the terms of arbitrary functions of the Ricci scalar are known to be safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, in this paper, we consider a minimal extension of Ricci scalar beyond the R2 model with Weyl scaling invariance, namely a cubic term coupled with an extra scalar field as denominator R3/ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We will show that even if this term is extremely small, it will have an essential impact on inflation, which even open up a completely different inflationary scenario from Weyl R2 and conventional R2 + R3 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' II, we develop the analytic formalism of Weyl R2 + R3 model and derive the effective scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show that in some cases, the 2 potential has two different kinds of global minima, leading to two distinctive inflationary pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' III, we investigate the inflation in the pattern of evolving to the side minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We calculate the spectral index ns and tensor-to-scalar ratio r of the inflationary perturba- tions, and give the preferred parameter space allowed by the latest observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical treatments are developed for more transparent, physical understanding of the asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' IV, we investigate the pattern of evolving to the center minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A special process called “oscillating inflation” is considered in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Finally, conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We adopt the following conventions: metric ηµν = (−1, +1, +1, +1), natural unit ℏ = c = 1 and MP ≡ 1/ √ 8πG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='435 × 1018 GeV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' WEYL SCALING INVARIANT R2 + R3 MODEL We start with the following Lagrangian for metric field gµν, scalar field ϕ, and Weyl gauge field Wµ ≡ gWwµ with local scaling symmetry L √−g = 1 2 � ϕ2 ˆR + α ˆR2 + β ϕ2 ˆR3 � − ζ 2DµϕDµϕ − 1 4g2 W FµνF µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) Here g is the determinant of metric, α, β and ζ are constant parameters, Dµ = ∂µ − Wµ is the covariant derivative associated with scaling symmetry, gW is the coupling constant, Fµν ≡ ∂µWν − ∂νWµ defines the invariant field strength of Wµ, and ˆR is the Ricci scalar defined by the local scaling invariant connection ˆΓρ µν = 1 2gρσ [(∂µ + 2Wµ)gσν + (∂ν + 2Wν)gµσ − (∂σ + 2Wσ)gµν] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (2) Explicit calculation shows the relation between ˆR and usual R defined by metric field gµν, ˆR = R − 6WµW µ − 6 √−g∂µ(√−gW µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (3) It is straightforward to verify the invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) under the following Weyl scaling transformation metric : gµν → g′ µν = f 2(x)gµν, scalar : φ → φ′ = f −1(x)φ, Ricci scalar : ˆR → ˆR′ = f −2(x) ˆR, Weyl vector : Wµ → W ′ µ = Wµ − ∂µ ln f(x), (4) where f(x) is an arbitrary positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3 The purpose to explore the Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Theoretically, such a ˆR3 term constitutes as a simple extension of the ˆR2 theory, motivated from perspective of effective field theories and also quantum loop corrections in more fundamental theories [31– 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phenomenologically, it is worthwhile to explore how such a term would modify the cosmological observations related to inflation, and evaluate the likelihood and robustness of the predictions in the lowest-order theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Formalism in Einstein frame General f(R) gravity is equivalent to the Einstein gravity with a scalar field [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [41], we have extended the proof in general scaling invariant F( ˆR, ϕ) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can explicitly show that by introducing an auxiliary scalar field χ and rewrite the high-order curvature terms as F( ˆR, ϕ) ≡ ϕ2 ˆR + α ˆR2 + β ϕ2 ˆR3 = F ˆR( ˆR → χ2, ϕ)( ˆR − χ2) + F( ˆR → χ2, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (5) Here F ˆR denotes the derivative over ˆR, F ˆR = ∂F( ˆR, ϕ)/∂ ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can verify that the equiv- alence relation χ2 = ˆR can be obtained from the Euler-Lagrange equation, δL δχ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substi- tuting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1), we find L √−g = 1 2 � ϕ2 + 2αχ2 + 3β ϕ2 χ4 � ˆR − 1 2 � αχ4 + 2β ϕ2 χ6 � − ζ 2DµϕDµϕ − 1 4g2 W FµνF µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (6) Now we have demonstrated that linearization of ˆR has led to the non-minimal coupling of the scalar field, χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can transform the above Lagrangian into the Einstein frame by making a Weyl or conformal transformation of the metric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, we note that scaling invariance is still preserved in our model with χ → χ′ = f −1(x)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, we can directly normalize the coefficient before the Ricci scalar as ϕ2 + 2αχ2 + 3βχ4/ϕ2 = 1, (7) due to the scaling invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is equivalent to making a Weyl transformation with f(x) = � ϕ2 + 2αχ2 + 3βχ4/ϕ2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Further dropping the total derivative term 4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (3) due to its null surface integral, we can write the Lagrangian as L √−g =1 2R − ζ 2DµϕDµϕ − V (ϕ) − 1 4g2 W FµνF µν − 3W µWµ =R 2 − ∂µϕ∂µϕ 2/ζ + ϕ2/3 − V (ϕ) − 1 4g2 W FµνF µν − 6 + ζϕ2 2 � Wµ − ∂µ ln |6 + ζϕ2| 2 �2 , (8) with the scalar potential V (ϕ) = α 2 χ4 + β ϕ2χ6 = α 6β � ϕ4 − ϕ2� + α3ϕ4 27β2 �� 1 − 3β α2 � 1 − ϕ−2��3/2 − 1 � , (9) where we have solved χ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (7) χ2 = αϕ2 3β �� 1 − 3β α2 (1 − ϕ−2) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (10) It is now clear that we have a minimally-coupled scalar ϕ with a non-canonical kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To further simplifying the theoretical formalism, we introduce the following redefini- tions for the scalar and the Weyl gauge field ϕ2 ≡ � � � � � 6 |ζ| sinh2 � ±Φ √ 6 � for ζ > 0, 6 |ζ| cosh2 � ±Φ √ 6 � for ζ < 0, (11) ˜Wµ ≡ Wµ − 1 2∂µ ln |6 + ζϕ2| ≡ gW ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (12) Then the final Lagrangian turns into a more compact form L √−g = 1 2R − 1 2∂µΦ∂µΦ − V (Φ) − 1 4g2 W ˜Fµν ˜F µν − 1 2m2(Φ) ˜W µ ˜Wµ, (13) with the mass term of Weyl gauge field m2(Φ) = � � � � � +6 cosh2 � Φ √ 6 � for ζ > 0, −6 sinh2 � Φ √ 6 � for ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (14) We should note that m2(Φ) is negative when ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, to avoid Weyl gauge boson becoming tachyonic in this case, it requires some other mechanisms to obtain a real mass, for example, introducing other scalar field, which we do not explore in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For viable inflation, both positive and negative are possible, as we shall show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In the above discussion, we have demonstrated that Weyl scaling invariant ˆR2+ ˆR3 model can be written equivalently as the Einstein gravity coupled with a self-interacting scalar Φ 5 and a massive vector ˜Wµ with a field-dependent mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This conclusion is also true for any Weyl scaling invariant model of gravity with high-order curvature ˆRn as the above formalism applies straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is also worth to point out that Weyl vector boson can serve as a dark matter candidate [27, 28, 41], with details of the relic abundance being discussed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In this paper, we shall concentrate on the scalar potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) and discuss the viable inflation scenarios with the presence of ˆR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective scalar potentials There are two necessary requirements for the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first one is ϕ2 > 0 since ϕ is a real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The other is 1 − 3β α2 � 1 − 1 ϕ2 � ≥ 0, otherwise an imaginary potential will emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Consequently, there are some constraints on the parameters and the viable value of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can rewrite the second requirement as sinh �±Φ √ 6 � ≥ or ≤ � |ζ| 6 − 2α2/β , for ζ > 0, cosh �±Φ √ 6 � ≥ or ≤ � |ζ| 6 − 2α2/β , for ζ < 0, (15) where “ ≥ ” for β < α2 3 and “ ≤ ” for β ≥ α2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For convenience, we define λ ≡ � |ζ| 6−2α2/β and γ ≡ 3β α2, then discuss the possible ranges of the potential corresponding to different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results are listed in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To ensure the theoretical stability, we require that Φ can only evolve within these ranges where the potential is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 shows some instances of the scalar potential for several values of ζ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first discuss the case of positive ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ = 0, it is a hill-top-like potential with two minima at Φ = ± √ 6 sinh−1 � ζ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, as long as there is a tiny cubic curvature, whether positive or negative, the shape of potential will be affected significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ > 0, the potential turns to decrease near Φ = 0, and a third vacuum can form there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This behavior is transparent, because when ζ > 0, Φ = 0 corresponds to ϕ2 = 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11), then substituting it in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) will obtain V |Φ=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When γ < 0, the potential turns to rise near Φ = 0 and become imaginary and unphysical in − √ 6 sinh−1 λ < Φ < √ 6 sinh−1 λ, which has been listed in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we switch to the case of negative ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1 that when ζ < 0 and |ζ| or |γ| is relatively small, the modification of ˆR3 term on the Weyl R2 potential is moderate, 6 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective potential range of the Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ζ γ or β real V (ϕ) ζ > 0 γ ≥ 1 |Φ| ≤ √ 6 sinh−1 λ 0 ≤ γ < 1 fully real γ < 0 |Φ| ≥ √ 6 sinh−1 λ −6 < ζ < 0 γ > 1 1+ζ/6 fully imaginary 1 < γ ≤ 1 1+ζ/6 |Φ| ≤ √ 6| cosh−1 λ| γ ≤ 1 fully real ζ ≤ −6 γ ≥ 1 |Φ| ≤ √ 6| cosh−1 λ| 1 1+ζ/6 < γ < 1 fully real γ ≤ 1 1+ζ/6 |Φ| ≥ √ 6| cosh−1 λ| unlike the dramatic change near Φ = 0 in the case of positive ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is because the mapping of Φ ⇒ ϕ2 does not cover the interval of ϕ2 < 1 for ζ < 0 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words, for negative ζ with modest |γ|, Φ → 0 does not lead to ϕ2 → 0, which brings the violent behavior of the potential around here in the case of ζ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, when ζ is excessively negative or |γ| is large enough, the violent variation will reappear to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For γ > 0, the potential will return to a downward trend near Φ = 0, albeit there is no true vacuum formed (but a false vacuum is formed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' And for excessively negative γ, the imaginary potential will reappear in the range of − √ 6| cosh−1 λ| < Φ < √ 6| cosh−1 λ|, which we have listed this situation in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' I (see ζ ≤ −6 with γ ≤ 1 1+ζ/6 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, inflation takes place when the potential is flat and Φ evolves to the vacuum (Φ|V =0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The cosmological observations would restrict the potential and the initial value Φi when inflation starts, here the Φi is defined as the value when the comoving horizon of the inflationary universe shrinks to the same size as today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For ζ > 0 and γ > 0, the scalar potential contains three separate vacua, one lying at the center and the other two at both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore, there are two different viable inflationary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' One pattern refers to the evolution into the central minimum, and the other into the side minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can calculate the value of Φ which corresponds to the hill-top of the 7 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 10 5 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 2 10-10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Effective potentials of Weyl R2 + R3 model with α = 109 and various γ and ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here we only depict the real ranges of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' potential in this case Φh = ± √ 6 sinh−1 � ζ 12 √3γ − 2γ 3 − 4γ , (16) which is the critical point of two inflationary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Neglecting the velocity, if the initial value of inflation field satisfies |Φi| > |Φh|, it will evolve towards the side vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' If |Φi| < |Φh| at the beginning, the inflation field will evolve towards the central vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For other cases of ζ and γ, there are only the global side minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence the only feasible inflationary pattern is that Φ evolves to either one of the side minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The initial value Φi has to correspond to a real potential, and when there is a false vacuum in ζ < 0 case, it requires a large enough |Φi| outside two local maxima of the potential to ensure the gradient of V (Φi) towards the true vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we are going to discuss the inflation in these two patterns respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 8 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INFLATION TO THE SIDE In this inflation pattern, ϕ2 (defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11)) is usually not very close to 0, and as we shall show later, observations generally would require an extremely small cubic curva- ture, namely |γ| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore in many cases, |γ(1 − ϕ−2)| ≪ 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Under this condition, we are able to have analytical treatment and expand the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (9) as V (ϕ) =ϕ4 − ϕ2 2αγ + ϕ4 3αγ2 � −3γ 2 � 1 − 1 ϕ2 � + 3γ2 8 � 1 − 1 ϕ2 �2 + γ3 16 � 1 − 1 ϕ2 �3 + O �γ4 ϕ8 �� = 1 8α � 1 − ϕ2�2 � 1 + γ 6 � 1 − 1 ϕ2 � + O �γ2 ϕ4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (17) Then with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11), we derive V (Φ) = � � � � � 1 8α � 1 − 6 |ζ| sinh2 � Φ √ 6 ��2 � 1 + γ 6 � 1 − |ζ| 6 csch2 � Φ √ 6 �� + O(γ2) � for ζ > 0, 1 8α � 1 − 6 |ζ| cosh2 � Φ √ 6 ��2 � 1 + γ 6 � 1 − |ζ| 6 sech2 � Φ √ 6 �� + O(γ2) � for ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (18) The first term is exactly the effective potential of Weyl ˆR2, which has been shown in [41, 45], and the rest originates from the cubic curvature term ˆR3, to the leading order of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next we shall calculate the inflationary physical quantities, the spectral index ns and tensor-to- scalar ratio r, and contrast them with the latest observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first give an analytical calculation for two limiting cases, then show the full numerical results for general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical approach of γ → 0 case We first discuss the γ → 0 case and show how ζ affects ns and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The slow-roll parameters in this case can be derived as ϵ ≡ 1 2 �V ′(Φ) V �2 = 12 sinh2 � 2Φ √ 6 � � |ζ + 3| − 3 − 6 sinh2 � Φ √ 6 ��2, (19) η ≡ V ′′(Φ) V = 12 cosh � 4Φ √ 6 � − 4|ζ + 3| cosh � 2Φ √ 6 � � |ζ + 3| − 3 − 6 sinh2 � Φ √ 6 ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (20) Generally, the slow-roll inflation occurs when ϵ and |η| is small enough, and it will end when any of them evolves to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the situation we are concerned with, ϵ breaks the slow-roll 9 limit before the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus we derive the value of Φ when inflation ends according to ϵ = 1 Φe = � 3 2 ln � 2 � |ζ + 3|2 + 3 √ 3 − |ζ + 3| + � 7 3|ζ + 3|2 − 4|ζ + 3| √ 3 � |ζ + 3|2 + 3 + 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (21) When |ζ| > O(102), which is a preferred range by the observational constraints as we will show shortly, the above equation can be approximated as Φe ≃ � 3 2 ln � 1 √ 3 � 2 + � 7 − 4 √ 3 − √ 3 � |ζ + 3| � ≃ � 3 2 ln (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (22) It is now clear that when |ζ| is large enough, Φe will be almost independent of the sign of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we calculate initial value Φi, which is defined when the size of comoving horizon during inflation shrinks to the present size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We first focus on the e-folding number of the slow-roll inflation N ≡ ln ae ai ≃ � Φe Φi dΦ √ 2ϵ, (23) where ai/e ≡ a(Φi/e) is the cosmic scale factor when inflation starts/ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (19) into it, we find N = (|ζ + 3| − 3) ln � tanh � Φ √ 6 �� − 6 ln � cosh � Φ √ 6 �� 4 ����� Φe Φi = |ζ + 3| − 3 4 ln � �tanh � 1 2 ln(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) � tanh � Φi √ 6 � � � − 3 2 ln � �cosh � 1 2 ln(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3|) � cosh � Φi √ 6 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (24) For the circumstances we are concerned with, namely N ∼ (50, 60) and |ζ| > O(102), the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (24) is much smaller than the first term, and it can be estimated as ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus we derive Φi ≃ √ 6 tanh−1 �� 1 − 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3094|ζ + 3| + 1 � e −4(N+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3) |ζ+3|−3 � ≡ √ 6 tanh−1 Ω(ζ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (25) Here we have defined Ω(ζ, N) for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When |ζ| ≫ 4N, it can be further approximated as Φi ≃ � 3 2 ln |ζ| 2N+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (25) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (19) and (20), we find ϵi = 48Ω2 [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, (26) ηi =4 [(Ω4 − 1)|ζ + 3| + 3(Ω4 + 6Ω2 + 1)] [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (27) 10 As a result, the tensor-to-scalar ratio r and spectral index ns of inflationary perturbations in the γ → 0 limit are finally calculated as r = 16ϵi = 768Ω2 [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2, (28) ns = 1 − 6ϵi + 2ηi = 1 + 8(Ω4 − 1)|ζ + 3| + 24(Ω4 − 6Ω2 + 1) [(Ω2 − 1)|ζ + 3| + 3(Ω2 + 1)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (29) For N ∼ (50, 60) and |ζ| > O(102), We can approximate the expressions as r ≃ r∗ − 54 ζ2 , (30) ns ≃ n∗ s − 11N ζ2 , (31) where r∗ ≃ 12 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2, n∗ s ≃ 1 − 2 N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55 − 3 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2 (32) are the predictions of Starobinsky model (see Appendix A for an analytical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Thus it is evident that the predictions of inflationary perturbations in our model will converge to that of Starobinsky model when γ → 0 and ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' As |ζ| decreases, the value of r and ns will also decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this trend as the pink area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' According to the latest observation [54], the lower limit of ns has been constrained to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959, hence it requires |ζ| > 270 in this γ → 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Analytical approach of ζ → ∞ case Now we discuss the ζ → ∞ case and show how γ affects r and ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' When ζ is large enough, the potential is greatly widened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The side vacua are far away from 0 and so are Φi and Φe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=', Φi ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4MP, Φe ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='8MP for ζ = 104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (11) can be approximated as ϕ2 = 6 |ζ| � eΦ/ √ 6 ± e−Φ/ √ 6 2 �2 ≃ e √ 2/3 � Φ−√ 3/2 ln(2|ζ|/3) � ≡ e √ 2/3(Φ−Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (33) Here and after, without losing generality, we may choose to evolve in the positive Φ region, and denote Φ0 as the minimum in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (17), we have the scalar potential for Φ ≫ 0 V (Φ) = 1 8α � 1 − e √ 2/3(Φ−Φ0)�2 � 1 + γ 6 � 1 − e−√ 2/3(Φ−Φ0)� + O(γ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (34) 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The predictions of spectral index ns combined with tensor-to-scalar ratio r in the Weyl R2 + R3 model with e-folding number N ∼ (50, 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The pink area shows the results in the γ → 0 case with various ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The yellow and green areas respectively show the ζ → ∞ and ζ = −650 cases with various γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The red line is the result with both γ → 0 and ζ → ∞, which is equivalent to the Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The blue area is the latest observation constraint given by the BICEP/Keck collaboration [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ignoring the O(γ2) terms, we give an approximate expression for the slow-roll parameters ϵ ≡ 1 2 �V ′(Φ) V �2 ≃ � γe √ 2/3(Φ−Φ0) − 2(γ + 6)e √ 8/3(Φ−Φ0) + γ �2 3 � e √ 2/3(Φ−Φ0) − 1 �2 � γ − (γ + 6)e √ 2/3(Φ−Φ0)�2, (35) η ≡ V ′′(Φ) V ≃ 6(γ + 4)e √ 8/3(Φ−Φ0) − 8(γ + 6)e √ 6(Φ−Φ0) + 2γ 3 � e √ 2/3(Φ−Φ0) − 1 �2 � γ − (γ + 6)e √ 2/3(Φ−Φ0)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (36) In this case, the slow-roll inflation also ends at ϵ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' To find the expression of Φe, we further approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (35) as ϵ ≃ e−√ 8/3(Φ−Φ0) � γ − 12e √ 8/3(Φ−Φ0)�2 108 � e √ 2/3(Φ−Φ0) − 1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (37) 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' → 60= -650 95% CL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='03 68% CL 5×1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01 500 X = 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='003 3x 10 3 × 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='975 ns0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0← N = 50 8个VThen Φe can be derived as Φe = Φ0 − � 3 2 ln �√ 3 γ �� 2(2 + √ 3)γ + 9 − 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (38) If γ is extremely small, we will find Φe ≃ Φ0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='94MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Next, we derive the analytic formula for Φi in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The e-folding number of the slow-roll inflation can be calculated with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (37) as N = − �27 4γ tanh−1 �� γ 12e−√ 2 3 (Φ−Φ0) � − 3 8 ln � 12 − γe−√ 8 3 (Φ−Φ0)� − √ 6 4 (Φ − Φ0) ���� Φe Φi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (39) Considering N ∼ (50, 60) and γ < O(10−3), the first term of the integral is dominant, while the rest are the marginal terms which can be approximately treated as a constant, ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence we have N ≃ �27 4γ � tanh−1 �� γ 12e−√ 2/3(Φi−Φ0) � − tanh−1 �� γ 12e−√ 2/3(Φe−Φ0) �� − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7, (40) and derive Φi = Φ0 − � 3 2 ln ����� �12 γ tanh � tanh−1 �� γ 12e−√ 2/3(Φe−Φ0) � + � 4γ 27(N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7) ������ ≃ Φ0 − � 3 2 ln ����� �12 γ tanh � tanh−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='622√γ) + � 4γ 27(N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='7) ������ ≡ Φ0 − � 3 2 ln Θ(γ, N), (41) where we have defined Θ(γ, N) for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Then substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (35) and (36), we find ϵi = [γΘ(1 + Θ) − 2(γ + 6)]2 3 [1 − Θ]2 [γΘ − (γ + 6)]2, (42) ηi = 2γΘ3 + 6(γ + 4)Θ − 8(γ + 6) 3 [1 − Θ]2 [γΘ − (γ + 6)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (43) Finally, we derive r and ns of the inflationary perturbations in the ζ → ∞ limit r = 16ϵi = 16 [γΘ(1 + Θ) − 2(γ + 6)]2 3 [1 − Θ]2 [γΘ − (γ + 6)]2 , (44) ns = 1 − 6ϵi + 2ηi = 1 − 2 [γΘ(1 + Θ) − 2(γ + 6)]2 [1 − Θ]2 [γΘ − (γ + 6)]2 + 4γΘ3 + 3(γ + 4)Θ − 4(γ + 6) 3 [1 − Θ]2 [γΘ − (γ + 6)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (45) 13 If γ is extremely small, smaller than O(10−4), the above expressions can be linearly approx- imated as r ≃ r∗ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4γ, (46) ns ≃ n∗ s − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='42γN, (47) where r∗ and n∗ s have been defined in the last paragraph of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can see that compared with the predictions of Starobinsky model, a positive γ will reduce both r and ns, while a negative γ will increase them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this trend as the yellow area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is manifest that the observations have constrained |γ| ≲ 5 × 10−4 in this ζ → ∞ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, this result agrees with other numerical investigations of the R3-extended Starobin- sky model [55–61], since the potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (34) is the same as the R3-extended Starobinsky model with a vacuum shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (30) and (31), we note that the predictions of r and ns in the γ → 0 case is similar to that of the ζ → ∞ and γ > 0 case with a simple replacement of γ ↔ 24 ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This can be seen more clearly from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2, where the pink area overlaps with the yellow area with γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' General cases Now we discuss the general cases with various ζ and γ by numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here the parameter ranges satisfying observational constraints (see blue area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2) are marked with colored areas, where the color gradient from blue to red corresponds to ascending value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The gray areas represent that the potential defined by these parameters cannot support an adequate inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words, their maximal e-folding number is unable to reach N = 50 or 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The white areas are the parameter ranges that can give rise to ample inflation, but their prediction of ns or r has been excluded by the observation constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here we mark two dotted lines to distinguish the boundaries of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Beyond the pink one indicates a large ns that exceeds the observational upper limit, while beyond the green one signifies a too small prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Let us focus on the colored parameter ranges that are allowed by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In the |ζ| ≫ 1000 case, the result is roughly equivalent to the analytical calculation shown in the last subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The prediction of r is limited to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='002 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, distinctive situations appear when |ζ| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' First, when −1000 < ζ < −200, the restrictions on γ is relaxed, which can stand |γ| ∼ 6 × 10−3 at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Besides, the upper limit of r is 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Possible parameter space for Weyl R2 + R3 model when Φ evolves to the side vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The colored areas are the parameter ranges allowed by the latest observations of BICEP/Keck collaboration [54], where the color gradient from blue to red corresponds to r increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 to the observational upper limit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The dotted lines are the boundaries that ns exceeds the observational upper (pink line) or lower (green line) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The gray areas represent the parameter ranges with inadequate inflation, namely, the maximal e-folding number of inflation cannot reach N = 50 or 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' greatly expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' There is even a small parameter range that gives r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show an example as the green area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It clearly shows a distinguishable feature from the Weyl R2 model and the R3-extended Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' If the next generation experiment of CMB B-mode polarization detects the primordial gravitational waves with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='01, it may support Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Another notable feature emerges at 0 < ζ < 200, where the 15 r0 2 ns > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='974 4 inadequate e-folds 6 3000 2000 1000 0 1000 2000 × 10-3 4 N = 60 2 ns < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959 0 2 ns > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='974 4 inadequate e-folds 9- 3000 2000 1000 0 1000 20000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='015 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='001 3000×10-3 4 N = 50 2 ns < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='959negative γ, even if very small, can greatly affect the predictions of primordial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, there are some cases with small positive ζ and small negative γ can give proper r and ns that match the observation constraints, and generally, r is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For instance, when ζ = 80, γ = −4 × 10−8, and N = 60, we have ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='963 and r = 3 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' INFLATION TO THE CENTER As we mentioned earlier, the third vacuum appears at Φ = 0 in the case of ζ > 0 and γ > 0, and if the initial value satisfies |Φi| < |Φh| (Φh is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (16)), inflation can happen in the evolution of Φ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Actually, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A process called “oscillating inflation” [62–74] will continue immediately after the end of slow-roll inflation because the scalar potential in this case is a non-convex function in the region close to the vacuum, which means there is d2V dΦ2 < 0 when Φ nears 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' for such a non-convex potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' despite the slow-roll conditions (ϵ ≪ 1 and |η| ≪ 1) has been violated during the bottom oscillation of the inflaton potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' the universe can keep accelerating expansion until the average amplitude of the inflaton’s oscillation becomes lower than the borderline of d2V dΦ2 from negative to positive (if there is a rounded transition in a small enough ∆Φ at the bottom to connect the left and right sides of the potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' see [62]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' or until the contribution of the radiation produced in reheating process becomes non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is helpful to understand the behavior of oscillating inflation from the perspective of the effective equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For an oscillating scalar field Φ, its effective equation of state in one oscillating period is defined as ⟨w⟩ ≡ ⟨p⟩ ⟨ρ⟩ = ⟨ ˙Φ2 − ρ⟩ ⟨ρ⟩ = ⟨ ˙Φ2⟩ Vm − 1 = ⟨Φ dV dΦ⟩ Vm − 1 = 1 − 2⟨V ⟩ Vm , (48) where ⟨⟩ means the average value in one oscillation period, and Vm represents the maximal potential of this oscillation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The accelerating expansion of the universe requires ⟨w⟩ < − 1 3, which is equivalent to the following relation U ≡ ⟨V − ΦdV dΦ⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (49) In fact, U amounts to the intercept of the tangent to the potential at a certain Φ, shown as the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' As long as the intercept is positive and the contribution of radiation is insignificant, the accelerating expansion will proceed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is the reason why a non-convex potential can bring about oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 16 For the process with oscillating inflation, the definition of e-folding number should be replaced to ˜N ≡ ln afHf aiHi ≡ ln aeHe aiHi + ln aoHo aiHi ≃ N + No, (50) where the subscripts i and e have been defined in the last section, af and Hf represent the cosmic scale and Hubble parameter when the full inflationary period ends, ao and Ho represent their multiple of increase or decrease during the oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It indicates that the new definition is equivalent to adding a correction No based on the e-folding number of slow-rolling period if we take He ≈ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Generally, No is related to the shape of potential near its vacuum, reheating efficiency, and the scale of the aforementioned rounded bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Given that our model does not possess an explicit rounded bottom, No depends only on the first two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the shape of potential, actually, our model has the following approximate form near the center vacuum V (Φ) ≃ ξ(Φ4 − Φ2) 2α + ξ2Φ4 3α �� 1 + 1 ξΦ2 �3/2 − 1 � , (51) where ξ ≡ α2 3βζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Since α determines the height of the potential, which has been fixed for each set of ζ and β according to the observation result of ∆2 s ∼ V 24π2ϵ ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 × 10−9 [75], the shape of the potential is essentially determined by ξ in the oscillatory region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For reheating efficiency, we consider a constant transfer rate Γ and the transferred energy all turns to radiation ρr ¨Φ + (3H + Γ) ˙Φ + dV dΦ = 0, (52) ˙ρr + 4Hρr − Γ ˙Φ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (53) Then No is substantially related to the parameters ξ and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We numerically solve the above equations, and visualize in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is transparent that if ξ ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1, oscillating inflation will bring appreciable correction to the e-folding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Because an inefficient reheating process will postpone the end of the oscillating inflation, we can see a smaller Γ corresponds to a larger No for a certain ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' However, No will tend to a fixed value as Γ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This property can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We can prove that the potential has a quasi-linear form when Φ → 0 V |Φ→0 ≃ √ξ 3α |Φ|, (54) 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4 10-2 100 102 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Oscillating inflation in the center-evolving pattern of Weyl R2 + R3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The upper part is a diagram for visualizing the condition of oscillating inflation, where the effective equation of state ⟨w⟩ < − 1 3 is equated with that the intercept U of the tangent to a certain point on the potential corresponding to the average amplitude is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The lower part shows the increased e-folding number during the oscillating inflation for various ξ and reheating efficiency Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' which implies that U|Φ→0 → 0 according to its definition as the intercept of the tangent to the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence ⟨w⟩ will quickly converge to − 1 3 as the oscillation proceeds, and No will soon grow to a nearly constant maximum if Γ is too small to make the universe promptly produce enough radiation to stop the oscillating inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' This is the reason why No has an extreme for each ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Now we consider the reheating is inefficient, that is to adopt No with Γ → 0, to derive the slow-roll e-folding number N corresponding to ˜N ∼ (50, 60), and then to calculate ns 18 103 104 105 106 107 108 10-7 10-5 10-3 103 104 105 106 107 108 10-7 10-5 10-3 <10-4 10-3 10-2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Possible parameter space for Weyl R2 + R3 model when Φ evolves to the center vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Here the total e-folding number ˜N ≡ N + No is considered with Γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The meaning of markers is the same as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3, except for the color correspondence of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' and r for various parameters ζ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The viable parameter space is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 5, where the meaning of markers is the same as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 3, except for the scale of color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' It is evident that the observation constraint on ns limits the parameters to ζ > 103 and γ < 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' r has an upper limit ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006, but no lower limit in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' CONCLUSIONS Cosmological observations have suggested that our universe has a nearly scaling invariant power spectrum of the primordial density perturbation, which motivates the scaling sym- 19 metry as the possible feature of the underlying fundamental theories that lead to inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We present the theoretical formalism of the Weyl scaling invariant gravity, ˆR2 + ˆR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We show this model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (1) can be rewritten equivalently to the Einstein gravity coupled with a massive gauge boson, and a scalar field as the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We further discuss the viable ranges of the scalar potential according to the requirement for reality and demonstrate how the R3 term would affect the shape of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Compared with the Weyl R2 inflationary potential [41, 45] with two side minima, the R3 extension brings an additional minimum at center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Hence, there are two viable scenarios for the inflation in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The first is to roll towards the side minima, while the other is a new situation of rolling towards the center minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Both scenarios allows viable parameter spaces that be probed by future experiments on cosmic microwave background and primordial gravitational wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the first scenario, we calculate the spectral index ns and tensor-to-scalar ratio r of primordial perturbations both analytically and numerically, and contrast the parameter spaces with the latest observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results manifest that the level of cubic curvature is limited to |γ| < 6×10−3, and the prediction of r in this pattern has a wide range from O(10−4) to the upper limit of the observations, O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' These results are significantly different from the R3-extended Starobinsky model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' For the second scenario, a special process called oscillating inflation emerges after the familiar slow-roll inflation because the potential near the center minimum is a non-convex function that can lead to a sufficiently negative value of average equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' We calculate the correction of e-folding number in the oscillating inflation stage, and then derive the viable parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The results indicate that the parameters are limited to γ < 5 × 10−4 and ζ > 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Moreover, r has an upper limit ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='006, but no lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' ACKNOWLEDGMENTS QYW and YT thank Shi Pi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' YT is supported by National Key Re- search and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2021YFC2201901), and Natural Sci- ence Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11851302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' YLW is supported by the Na- tional Key Research and Development Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='2020YFC2201501, and NSFC under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11690022, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 11747601, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 12147103, and the Strategic Prior- ity Research Program of the Chinese Academy of Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' XDB23030100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 20 Appendix A: Analytical treatment of Starobinsky inflation We give an analytical calculation of the tensor-to-scalar ratio r and spectral index ns in the Starobinsky inflationary model, namely, the Einstein gravity modified by a R2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The effective scalar potential can be written as V (φ) = 1 8α � 1 − e−√ 2/3φ�2 , (A1) where α is the coefficient of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' The relevant two slow-roll parameters are computed as ϵ = 4 3 1 � e √ 2/3φ − 1 �2, η = −4 3 e √ 2/3φ − 2 � e √ 2/3φ − 1 �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A2) Since inflation ends when ϵ ∼ 1 is reached first (η ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='15), we have φe = � 3 2 ln � 1 + 2 √ 3 � ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='94MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A3) Then according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (23), the e-folding number is N = � 3 4 � e √ 2/3φ − � 2 3φ ��φe φi = 3 4 � e √ 2/3φi − e √ 2/3φe − � 2 3(φi − φe) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A4) For N ∼ (50, 60), we find that approximately φi ≃ � 3 2 ln �4 3(N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A5) Substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A2), we finally derive r = 16ϵ = 12 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2, (A6) ns = 1 − 6ϵ + 2η = 1 − 2 N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55 − 3 (N + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='55)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' (A7) These results are shown as the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Guth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 23, 347-356 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Feldman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Brandenberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 215, 203-333 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 21 [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Akrami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [Planck], Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 641, A10 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 59, 101-133 (1919).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Smolin, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' B 160, 253-268 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} 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Steinhardt and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Turok, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 89, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='4, 043515 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Quiros, [arXiv:1401.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Noller and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Ross, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' D 100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content='12, 123516 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E2T4oBgHgl3EQfOAbJ/content/2301.03744v1.pdf'} +page_content=' 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