Z. Haba
We add a thermal noise to Starobinsky equation of slow roll inflation. We calculate the number of e-folds of the stochastic system. The power spectrum and the spectral index are evaluated from the fluctuations of e-folds using an expansion in the quantum and thermal noise terms.
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Certain models of cosmic inflation produce strong cosmological perturbations at short length scales, which may later collapse into primordial black holes. To find the statistics of these strong perturbations and the ensuing black holes, it is necessary to go beyond linear perturbation theory. Stochastic inflation provides a way to take the leading non-linear effects into account. In this contribution, I discuss recent progress in numerical computations of stochastic inflation. A numerical approach can include more of the non-linearities than an analytical calculation, and can be applied to sin…
We present a complete numerical treatment of inflationary dynamics under the influence of stochastic corrections from sub-Hubble modes. We discuss how to exactly model the stochastic noise terms arising from the sub-Hubble quantum modes that give rise to the coarse-grained inflaton dynamics in the form of stochastic differential equations. The stochastic differential equations are solved event-by-event on a discrete time grid. We then compute the power spectrum of curvature perturbations that can be compared with the power spectrum computed in the traditional fashion using the Mukhanov-Sasaki …
Numerical stochastic inflation constrained by frozen noise
针对单场原初黑洞模型中的强暴胀扰动，本文利用模式冻结将随机方程降为一维，提出 constrained stochastic inflation，并结合半解析与重要性采样实现尾部计算 $10^9$ 倍加速。
The mechanism of thermal inflation, a relatively short period of accelerated expansion after primordial inflation, is a desirable ingredient for a certain class of particle physics models if they are not to be in contention with the cosmology of the early Universe. Though thermal inflation is most simply described in terms of a thermal effective potential, a thermal environment also gives rise to thermal fluctuations that must be taken into account. We numerically study the effects of these thermal fluctuations using lattice simulations. We conclude that though they do not ruin the thermal inf…
Curvaton distribution from stochastic inflation
本文基于随机暴胀形式体系，将检验场识别为曲率子，解析给出曲率扰动分布，并通过研究衰变率与质量参数空间，讨论了曲率子对曲率扰动产生不可忽略贡献的概率。
We demonstrate that there is no controlled description of stochastic eternal inflation consistent with the refined swampland de Sitter conjecture.
Stochastic constant-roll inflation and primordial black holes
本文求解了随机常滚暴胀系统，发现大第二慢滚参数下的随机效应可将形成小行星质量原初黑洞所需的曲率功率谱高度从$10^{-2}$降至$10^{-3}$。
Small noise expansion of stochastic inflation
本文引入小噪声展开技术，将完全非线性随机暴胀系统重新表述为无限维纳过程，并指出Starobinsky测试场近似仅适用于线性区域，标量场非高斯性仅出现在慢滚参数的领先阶。
We extend the formalism of stochastic inflation to the setup of non-attractor inflation with a sound speed $c_s$. We obtain the Langevin equations for the superhorizon perturbations and calculate the stochastic corrections to curvature perturbation power spectrum. It is shown that the fractional stochastic corrections in mean number of e-folds and power spectrum are at the order of power spectrum. We also calculate the boundary crossing and the first hitting probabilities in a hypothetical dS space with two boundaries in field space. Furthermore, the stochastic corrections in power spectrum in…
Stochastic constant-roll inflation beyond the hilltop with the spectral method
利用谱方法求解二次山顶势（对应常数滚动暴胀）的福克-普朗克方程，发现穿越山顶并在反射边界附近缓慢隧穿的罕见轨迹主导平均首通时间，提出用中位数代替均值描述暴胀背景，并计算粗粒化ΔN分布。
By making a suitable generalization of the Starobinsky stochastic inflation, we propose a classical phase space formulation of stochastic inflation which may be used for a quantitative study of decoherence of cosmological perturbations during inflation. The precise knowledge of how much cosmological perturbations have decohered is essential to the understanding of acoustic oscillations of cosmological microwave background (CMB) photons. In order to show how the method works, we provide the relevant equations for a self-interacting inflaton field. For pedagogical reasons and to provide a link t…
Primordial black hole compaction function from stochastic fluctuations in ultra-slow-roll inflation
本研究针对超慢滚暴胀中随机效应显著的原初黑洞形成问题，利用ΔN形式与常滚近似数值求解致密函数，发现其呈尖峰形态，会大幅提升PBH丰度并拓宽质量分布，需重新开展相关数值模拟。
We show how importance sampling can be used to reconstruct the statistics of rare cosmological fluctuations in stochastic inflation. We have developed a publicly available package, PyFPT, that solves the first-passage time problem of generic one-dimensional Langevin processes. In the stochastic-$δN$ formalism, these are related to the curvature perturbation at the end of inflation. We apply this method to quadratic inflation, where the existence of semi-analytical results allows us to benchmark our approach. We find excellent agreement within the estimated statistical error, both in the drift-…
We consider primordial fluctuations in thermal inflation scenario. Since the thermal inflation drives about 10 $e$-folds after the standard inflation, the time of horizon-exit during inflation corresponding to the present observational scale shifts toward the end of inflation. It generally makes the primordial power spectrum more deviated from a scale-invariant one and hence renders some models inconsistent with observations. We present a mechanism of generating the primordial curvature perturbation at the end of thermal inflation utilizing a fluctuating coupling of a flaton field with the fie…
Stochastic inflation is an effective theory describing the super-Hubble, coarse-grained, scalar fields driving inflation, by a set of Langevin equations. We previously highlighted the difficulty of deriving a theory of stochastic inflation that is invariant under field redefinitions, and the link with the ambiguity of discretisation schemes defining stochastic differential equations. In this paper, we solve the issue of these "inflationary stochastic anomalies" by using the Stratonovich discretisation satisfying general covariance, and identifying that the quantum nature of the fluctuating fie…
We apply Starobinsky's formalism of stochastic inflation to the case of a minimally coupled scalar field with linear self-interaction potential. We solve the corresponding Fokker-Planck equation exactly, and obtain analytical expressions for the stochastic expectation values.
Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production
本文首次利用ΔN形式对超慢滚暴涨中的曲率扰动进行随机计算，揭示其高度非高斯尾部，并发现随机效应使原初黑洞丰度比高斯近似增加约10^5倍。
We compute the power spectrum of curvature perturbations in stochastic inflation. This combines the distribution of first crossing times through the end-of-inflation surface, which has been previously studied, with the distribution of the fields value at the time when a given scale crosses out the Hubble radius during inflation, which we show how to compute. This allows the stochastic-$δN$ formalism to make concrete contact with observations. As an application, we study how quantum diffusion at small scales (arising e.g. in models leading to primordial black holes) affects the large-scale pert…
The relative probability to decay towards different vacua during inflation is studied. The calculation is performed in single-field slow-roll potentials using the stochastic inflation formalism. Various situations are investigated, including falling from a local maximum of the potential and escaping from a local minimum. In the latter case, our result is consistent with that of Hawking and Moss, but is applicable to any potential. The decay rates are also computed, and the case of a generic potential with multiple minima and maxima is discussed.
More fields are different: Stochastic view of multi-field inflationary scenario
该研究从随机视角探讨多场暴胀，发现小噪声区域下场的数量决定暴胀终止与否，极多场的暴胀或无法形成可观测宇宙，揭示场数量对暴胀结局的关键影响。
Thermal curvature perturbations in thermal inflation
该研究计算热膨胀中曲率扰动功率谱，发现flaton场与辐射浴的热平衡增强场方差，抑制曲率扰动及原初黑洞形成，并缩短快速滚动膨胀阶段。
Stochastic inflation as a superfluid
本文提出用超流体方程描述暴胀超视界涨落，通过函数性薛定谔方法和粗粒化建立框架，揭示波函数相位信息并兼容Starobinsky随机暴胀方法。
A safe way to reheat the universe, in models of natural and quadratic inflation, is through shift symmetric couplings between the inflaton $ϕ$ and the Standard Model (SM), since they do not generate loop corrections to the potential $V(ϕ)$. We consider such a coupling to SM gauge fields, of the form $ϕF\tilde{F}/f$, with sub-Planckian $f$. In this case gauge fields can be exponentially produced already {\it during inflation} and thermalize via interactions with charged particles, as pointed out in previous work. This can lead to a plasma of temperature $T$ during inflation and the thermal mass…
The formalism of stochastic inflation is a powerful tool for analyzing the backreaction of cosmological perturbations, and making precise predictions for inflationary observables. We demonstrate this with the simple $m^2ϕ^2$ model of inflation, wherein we obtain an effective field theory for IR modes of the inflaton, which remains coupled to UV modes through a classical noise. We compute slow-roll corrections to the evolution of UV modes (i.e. quantum fluctuations), and track this effect from the UV theory to the IR theory, where it manifests as a correction to the classical noise. We compute …
We present a review on the state-of-the-art of the mathematical framework known as stochastic inflation, paying special attention to its derivation and giving references for the readers interested on results coming from the application of the stochastic framework to different inflationary scenarios, especially to those of interest for primordial black hole formation. During the derivation of the stochastic formalism, we will emphasise two aspects in particular: the difference between the separate universe approach and the true long wavelength limit of scalar inhomogeneities and the generically…
Stochastic Inflation in Numerical Relativity
本文基于随机膨胀理论，采用数值相对论方法重新推导了3+1方程，并在慢-roll和超慢-roll场景中验证了其有效性，展示了理论和数值的鲁棒性。
We generalize the stochastic approach to quasi-power-law inflationary Universes,obtain the corresponding Langevin and Fokker-Planck equations for the scalar field driving inflation and find stationary solutions to the above FP equation.
We study the Brownian motion of a field where there are boundaries in the inflationary field space. Both the field and the boundary undergo Brownian motions with the amplitudes of the noises determined by the Hubble expansion rate of the corresponding dS spacetime. This setup mimics models of inflation in which curvature perturbation is induced from inhomogeneities generated at the surface of end of inflation. The cases of the drift dominated regime as well as the diffusion dominated regime are studied in details. We calculate the first hitting probabilities as well as the mean number of e-fol…
We investigate non-thermal gravitino production after tribrid inflation in supergravity, which is a variant of supersymmetric hybrid inflation where three fields are involved in the inflationary model and where the inflaton field resides in the matter sector of the theory. In contrast to conventional supersymmetric hybrid inflation, where non-thermal gravitino production imposes severe constraints on the inflationary model, we find that the "non-thermal gravitino problem" is generically absent in models of tribrid inflation, mainly due to two effects: (i) With the inflaton in tribrid inflation…
Itô, Stratonovich, and zoom-in schemes in stochastic inflation
本文在随机暴胀中教学式地比较Itô与Stratonovich积分方法，提出交替缩放方案，证明其在马尔可夫极限下等价于Itô方法，并阐明与线性扰动理论和ΔN形式论的关系。
I derive the stochastic equation for the perturbations of the metric for a gauge - invariant energy - momemtum - tensor (EMT) in stochastic inflation. A quantization for the field that describes the gauge - invariant perturbations for the metric is developed. In a power - law expansion for the universe the amplitude for these perturbations on a background metric could be very important in the infrared sector.
Hybrid inflation is a two field model where inflation ends due to an instability. In the neighborhood of the instability point, the potential is very flat and the quantum fluctuations dominate over the classical motion of the inflaton and waterfall fields. In this article, we study this regime in the framework of stochastic inflation. We numerically solve the two coupled Langevin equations controlling the evolution of the fields and compute the probability distributions of the total number of e-folds and of the inflation exit point. Then, we discuss the physical consequences of our results, in…
We examine the effect of the thermal vacuum on the power spectrum of inflation by using the thermal field dynamics. We find that the thermal effect influences the CMB anisotropy at large length scale. After removing the divergence by using the holographic cutoff, we observe that the thermal vacuum explains well the observational CMB result at low multipoles. This shows that the temperature dependent factor should be considered in the study of power spectrum in inflation, especially at large length scale.
The phenomenon of stochastic eternal inflation is studied for a chaotic inflation potential in a Bianchi Type I spacetime background. After deriving the appropriate stochastic Klein-Gordon equation, we give details on the conditions for eternal inflation. It is shown that for eternal inflation to occur, the amount of anisotropy must be small. In fact, it is shown that eternal inflation will only take place if the shear anisotropy variables take on values within a small region of the interior of the Kasner circle. We then calculate the probability of eternal inflation occurring based on techniq…
After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter $ε_1$ if and only if $ε_2^2\ll 6/ε_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $ΔN$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.
We consider thermal, trapped and chromo-natural inflation in light of the swampland criteria and the Trans-Planckian Censorship Conjecture (TCC). Since thermal inflation occurs at energies low compared to those of Grand Unification, it is consistent with the TCC, and it is also consistent with the refined swampland conditions. Trapped and chromo-natural inflation are candidates for primordial (high energy scale) inflation. Since in both of these scenarios there are effective damping terms in the scalar field equation of motion, the models can easily be consistent with the swampland criteria. T…
Non-Gaussian tails without stochastic inflation
本文证明超慢滚暴胀中曲率扰动的非高斯尾部可由δN形式论产生，无需随机暴胀，且考虑动量扰动后两种方法高度一致，尾部形状依赖暴胀相空间。
I consider a new semiclassical expansion for the inflaton field in the framework of warm inflation scenario. The fluctuations of the matter field are considered as thermally coupled with the particles of the thermal bath. This coupling parameter depends on the temperature of the bath. The power spectrum remains invariant under this new semiclassical expansion for the inflaton. However, I find that the thermal component of the amplitude for the primordial field fluctuations should be very small at the end of inflation.
We revisit the stochastic effects in the model of anisotropic inflation containing a $U(1)$ gauge field. We obtain the Langevin equations for the inflaton and gauge fields perturbations and solve them analytically. We show that if the initial value of the electric field is larger than its classical attractor value, then the random stochastic forces associated with the gauge field is balanced by the frictional damping (classical) force and the electric field falls into an equilibrium (stationary) regime. As a result, the classical attractor value of the electric field is replaced by its station…
We revisit the model of axion inflation in the context of stochastic inflation and investigate the effects of the stochastic noises associated to the electromagnetic fields. Because of the parity violating interaction, one polarization of the gauge field is amplified inducing large curvature perturbation power spectrum. Taking into account the stochastic kicks arising from the short modes at the time of horizon crossing we obtain the corresponding Langevin equations for the long modes of the electromagnetic and axion fields. It is shown that a mean-reverting process governs the dynamics of the…
It is known that the Higgs potential in the Standard Model can drive a successful inflation as long as the Higgs field couples non-minimally to gravity. It is then inevitable to take into account the loop corrections of the Standard Model particles to the Higgs potential in the Higgs inflation. In this paper, we discuss the one-loop corrections at finite temperature to the curvature perturbation generated during the Higgs inflation. We find that the thermal loop effects can suppress the power of the curvature perturbation at large scales, thus resulting in a low quadrupole of the temperature a…
We present a novel mechanism for the production of primordial black holes (PBHs). The mechanism is based on a period of thermal inflation followed by fast-roll inflation due to tachyonic mass of order the Hubble scale. Large perturbations are generated at the end of the thermal inflation as the thermal inflaton potential turns from convex to concave. These perturbations can lead to copious production of PBHs when the relevant scales re-enter horizon. We show that such PBHs can naturally account for the observed dark matter in the Universe when the mass of the thermal inflaton is about $10^6\,$…
Cosmic Strings from Thermal Inflation
本文研究热暴胀末期对称性破缺相变中形成的宇宙弦，发现其对称性破缺尺度上界低于阿贝尔希格斯弦，且弦环分布下截断由尖点湮灭而非引力辐射决定。
STOLAS: STOchastic LAttice Simulation of cosmic inflation
本文开发了宇宙暴胀的STOLAS C++程序包，基于stochastic-δN形式主义直接计算曲率扰动三维图，验证了与标准微扰理论的一致性，并利用重要性采样成功实现了原初黑洞丰度的非微观数值采样。
Stochastic scalar-tensor inflation and beyond
提出一种规范无关的粗粒化方法，将随机暴胀形式扩展到广义标量-张量理论，通过有效场论系数映射得到各类引力理论的随机运动方程，并展示了多场暴胀等扩展应用。
I study a stochastic approach for warm inflation considering back - reaction of the metric with the fluctuations of matter field. This formalism takes into account the local inhomogeneities fo the spacetime in a globally flat Friedmann - Robertson - Walker metric. The stochastic equations for the fluctuations of the matter field and the metric are obtained. Finally, the dynamics for the amplitude of these fluctuations in a power - law expansion for the universe are examined.
Hybrid Inflation is a two-field model where inflation ends by a tachyonic instability, the duration of which is determined by stochastic effects and has important observational implications. Making use of the recursive approach to the stochastic formalism presented in Ref. [1], these effects are consistently computed. Through an analysis of back-reaction, this method is shown to converge in the valley but points toward an (expected) instability in the waterfall. It is further shown that quasi-stationarity of the auxiliary field distribution breaks down in the case of a short-lived waterfall. I…
We derive the stochastic description of a massless, interacting scalar field in de Sitter space directly from the quantum theory. This is done by showing that the density matrix for the effective theory of the long wavelength fluctuations of the field obeys a quantum version of the Fokker-Planck equation. This equation has a simple connection with the standard Fokker-Planck equation of the classical stochastic theory, which can be generalised to any order in perturbation theory. We illustrate this formalism in detail for the theory of a massless scalar field with a quartic interaction.
I consider the COBE data coarse-grained field that characterize the now observable universe for a model of warm inflation which takes into account the thermal coupled fluctuations of the scalar field with the thermal bath. The power spectrum for both, matter and metric fluctuations are analyzed. I find that the amplitude for the fluctuations of the metric when the horizon entry, should be very small for the expected values of temperature.
We propose functional approach to the stochastic inflationary universe dynamics. It is based on path integral representation of the solution to the differential equation for the scalar field probability distribution. In the saddle-point approximation scalar field probability distributions of various type are derived and the statistics of the inflationary-history-dependent functionals is developed.
Nonperturbative stochastic inflation in perturbative dynamical background
本文研究了包含瞬时超慢-roll阶段的膨胀模型，通过量子场论在弯曲时空中的框架，推导出准de Sitter时空中的随机方程，并验证了随机描述与分析结果的一致性。
Recent Developments in Stochastic Inflation
本文回顾随机暴胀形式论从标量势模型到费米子、光子及导数相互作用理论的推广，并讨论其无法捕捉的长期环路修正。
The standard field-theoretical approach to the slow-roll inflation is introduced. We then show as, in order to calculate the mean square of the canonical gauge invariant quantum fluctuations associated to a generic field, the logarithm of the scale factor has to be used as the time variable in the Fokker-Planck equation in the stochastic approach. Then we compute the growth of different test fields with a small effective mass during slow-roll inflationary models, comparing the results with the one for the gauge invariant canonical fluctuation associated to the inflaton, the Mukhanov variable. …
Primordial thermal gravitational waves from warm inflation
本文首次研究暖暴胀产生热引力波的可能性，发现热压缩真空态下的引力子携带热特征并降低CMB的BB模式振幅，结合Planck和BICEP2/Keck数据为热引力子的存在提供了证据。
We study the quantum backreaction from inflationary fluctuations of a very light, non-minimally coupled spectator scalar and show that it is a viable candiate for dark energy. The problem is solved by suitably adapting the formalism of stochastic inflation. This allows us to self-consistently account for the backreaction on the background expansion rate of the Universe where its effects are large. This framework is equivalent to that of semiclassical gravity in which matter vacuum fluctuations are included at the one loop level, but purely quantum gravitational fluctuations are neglected. Our …
We propose a new approach for calculating the curvature perturbations produced during inflation in the stochastic formalism. In our formalism, the fluctuations of the e-foldings are directly calculated without perturbatively expanding the inflaton field and they are connected to the curvature perturbations by the $δN$ formalism. The result automatically includes the contributions of the higher order perturbations because we solve the equation of motion non-perturbatively. In this paper, we analytically prove that our result (the power spectrum and the nonlinearity parameter) is consistent with…
Towards Stochastic Inflation in Higher-Curvature Gravity
本文研究了高曲率项非最小耦合至inflaton时的随机膨胀，聚焦二次曲率不变量，突出Gauss-Bonnet项对背景和标量模式演化的影响，并讨论了其在慢-roll和超慢-roll极限下的随机演化。
We prove that the stochastic and standard field-theoretical approaches produce exactly the same results for the amount of light massive scalar field fluctuations generated during inflation in the leading order of the slow-roll approximation. This is true both in the case for which this field is a test one and inflation is driven by another field, and the case for which the field plays the role of inflaton itself. In the latter case, in order to calculate the average of the mean square of the gauge-invariant inflaton fluctuation, the logarithm of the scale factor $a$ has to be used as the time …
Chaotic inflation on the brane is considered in the context of stochastic inflation. It is found that there is a regime in which eternal inflation on the brane takes place. The corresponding probability distributions are found in certain cases. The stationary probability distribution over a comoving volume and the creation probability of a de Sitter braneworld yield the same exponential behaviour. Finally, nonperturbative effects are briefly discussed.
The perturbative approach to stochastic inflation is used to determine the spectrum of density fluctuations and gravitational waves due to the coarse grained field. The amplitude of the curvature fluctuation spectrum, the spectral index and the running of the spectral index are in general found to be smaller than in the standard approach to inflation. Furthermore, the amount of non-gaussianity due to the second order perturbation is estimated.
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