scholarly journals Asymptotic Solutions of a Generalized Starobinski Model: Kinetic Dominance, Slow Roll and Separatrices

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 500
Author(s):  
Elena Medina ◽  
Luis Martínez Alonso
Keyword(s):  
Asymptotic Expansions ◽  
Hubble Parameter ◽  
Initial Conditions ◽  
Asymptotic Solutions ◽  
Inflationary Model ◽  
Series Solutions ◽  
Fixed Amount ◽  
Inflaton Field ◽  
Slow Roll ◽  
Jacobi Formalism

We consider a generalized Starobinski inflationary model. We present a method for computing solutions as generalized asymptotic expansions, both in the kinetic dominance stage (psi series solutions) and in the slow roll stage (asymptotic expansions of the separatrix solutions). These asymptotic expansions are derived in the framework of the Hamilton-Jacobi formalism where the Hubble parameter is written as a function of the inflaton field. They are applied to determine the values of the inflaton field when the inflation period starts and ends as well as to estimate the corresponding amount of inflation. As a consequence, they can be used to select the appropriate initial conditions for determining a solution with a previously fixed amount of inflation.

scholarly journals Harmonic hybrid inflation

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Federico Carta ◽  
Nicole Righi ◽  
Yvette Welling ◽  
Alexander Westphal
Keyword(s):  
Domain Wall ◽  
Symmetry Breaking ◽  
Initial Conditions ◽  
Scalar Potential ◽  
Field Range ◽  
Slow Roll ◽  
Minimal Form ◽  
Wide Range ◽  
Hybrid Inflation ◽  
Type Iib String Theory

Abstract We present a mechanism for realizing hybrid inflation using two axion fields with a purely non-perturbatively generated scalar potential. The structure of the scalar potential is highly constrained by the discrete shift symmetries of the axions. We show that harmonic hybrid inflation generates observationally viable slow-roll inflation for a wide range of initial conditions. This is possible while accommodating certain UV arguments favoring constraints f ≲ MP and ∆ϕ60 ≲ MP on the axion periodicity and slow-roll field range, respectively. We discuss controlled ℤ2-symmetry breaking of the adjacent axion vacua as a means of avoiding cosmological domain wall problems. Including a minimal form of ℤ2-symmetry breaking into the minimally tuned setup leads to a prediction of primordial tensor modes with the tensor-to-scalar ratio in the range 10−4 ≲ r ≲ 0.01, directly accessible to upcoming CMB observations. Finally, we outline several avenues towards realizing harmonic hybrid inflation in type IIB string theory.

scholarly journals Reliable Iterative Method for solving Volterra - Fredholm Integro Differential Equations

2021 ◽  
Vol 26 (2) ◽  
Author(s):  
Samaher Marez
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Iterative Method ◽  
Iterative Process ◽  
Initial Conditions ◽  
High Accuracy ◽  
Series Solutions ◽  
Math Program ◽  
The Right

  The aim of this paper, a reliable iterative method is presented for resolving many types of Volterra - Fredholm Integro - Differential Equations of the second kind with initial conditions. The series solutions of the problems under consideration are obtained by means of the iterative method.  Four various problems are resolved with high accuracy to make evident the enforcement of the iterative method on such type of integro differential equations. Results were compared with the exact solution which exhibit that this technique has compatible with the right solutions, simple, effective and easy for solving such problems. To evaluate the results in an iterative process the MATLAB is used as a math program for the calculations.

scholarly journals Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints

Universe ◽  
2020 ◽  
Vol 6 (11) ◽  
pp. 199
Author(s):  
Igor Fomin ◽  
Sergey Chervon
Keyword(s):  
Scalar Field ◽  
Power Law ◽  
Hubble Parameter ◽  
Gravity Models ◽  
Observational Constraints ◽  
Slow Roll ◽  
Correct Model ◽  
Scalar Spectral Index ◽  
Wide Range ◽  
Exponential Power

We investigate the ability of the exponential power-law inflation to be a phenomenologically correct model of the early universe. We study General Relativity (GR) scalar cosmology equations in Ivanov–Salopek–Bond (or Hamilton–Jacobi like) representation where the Hubble parameter H is the function of a scalar field ϕ. Such approach admits calculation of the potential for given H(ϕ) and consequently reconstruction of f(R) gravity in parametric form. By this manner the Starobinsky potential and non-minimal Higgs potential (and consequently the corresponding f(R) gravity) were reconstructed using constraints on the model’s parameters. We also consider methods for generalising the obtained solutions to the case of chiral cosmological models and scalar-tensor gravity. Models based on the quadratic relationship between the Hubble parameter and the function of the non-minimal interaction of the scalar field and curvature are also considered. Comparison to observation (PLANCK 2018) data shows that all models under consideration give correct values for the scalar spectral index and tensor-to-scalar ratio under a wide range of exponential-power-law model’s parameters.

Investigating inflation driven by DBI-essence scalar field

2020 ◽  
Vol 29 (12) ◽  
pp. 2050087
Author(s):  
Gargee Chakraborty ◽  
Surajit Chattopadhyay
Keyword(s):  
Scalar Field ◽  
Early Universe ◽  
Inflationary Model ◽  
Type Iii ◽  
Ultraviolet Cutoff ◽  
Big Rip ◽  
Slow Roll ◽  
Limiting Case

Motivated by the work of Nojiri et al., Phys. Lett. B 797, 134829 (2019), the present study demonstrates inflation driven by holographic DBI-essence scalar field. Considering a simple correction due to the Ultraviolet cutoff, we have studied the slow-roll parameters. It has been observed that the role of the UV-cutoff is not negligible and in the limiting case of [Formula: see text] the inflationary model is characterized by Type-III singularity but can avoid Big-Rip singularity. Finally, it has been observed that the trajectories in [Formula: see text] are compatible with the observational bound found by Planck. It has been concluded that the tensor to scalar ratio for this model can explain the primordial fluctuation in the early universe as well. However, under the purview of [Formula: see text] inflation, although the DBI-essence scalar field can explain primordial fluctuation, the holographic DBI-essence scalar field does not lead to [Formula: see text] trajectory satisfying the Planck’s observational bound.

scholarly journals Light of Planck-2015 on Noncanonical Inflation

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kh. Saaidi ◽  
A. Mohammadi ◽  
T. Golanbari
Keyword(s):  
Scalar Field ◽  
Observational Data ◽  
Power Law ◽  
Hubble Parameter ◽  
Field Model ◽  
Kinetic Term ◽  
Inflationary Scenario ◽  
Slow Roll ◽  
Power Law Function ◽  
Free Parameters

Slow-roll inflationary scenario is considered in noncanonical scalar field model supposing a power-law function for kinetic term and using two formalisms. In the first approach, the potential is picked out as a power-law function, that is, the most common approach in studying inflation. Hamilton-Jacobi approach is selected as the second formalism, so that the Hubble parameter is introduced as a function of scalar field instead of the potential. Employing the last observational data, the free parameters of the model are constrained, and the predicted form of the potential and attractor behavior of the model are studied in detail.

scholarly journals Chameleon field dynamics during inflation

2018 ◽  
Vol 27 (04) ◽  
pp. 1850041 ◽  
Author(s):  
Nasim Saba ◽  
Mehrdad Farhoudi
Keyword(s):  
Scalar Field ◽  
Equation Of State ◽  
State Parameter ◽  
Big Bang ◽  
Late Time ◽  
Inflationary Model ◽  
Slow Roll ◽  
Equation Of State Parameter ◽  
The Common ◽  
The Universe

By studying the chameleon model during inflation, we investigate whether it can be a successful inflationary model, wherein we employ the common typical potential usually used in the literature. Thus, in the context of the slow-roll approximations, we obtain the e-folding number for the model to verify the ability of resolving the problems of standard big bang cosmology. Meanwhile, we apply the constraints on the form of the chosen potential and also on the equation of state parameter coupled to the scalar field. However, the results of the present analysis show that there is not much chance of having the chameleonic inflation. Hence, we suggest that if through some mechanism the chameleon model can be reduced to the standard inflationary model, then it may cover the whole era of the universe from the inflation up to the late time.

Asymptotic solutions of the Orr—Sommerfeld equation

1975 ◽  
Vol 280 (1295) ◽  
pp. 271-316
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Properties ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Asymptotic Approximations ◽  
General Kind ◽  
Work Done ◽  
Sommerfeld Equation ◽  
Special Case

A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

Transience to instability in a liquid sheet

2010 ◽  
Vol 666 ◽  
pp. 358-390 ◽  
Author(s):  
N. S. BARLOW ◽  
B. T. HELENBROOK ◽  
S. P. LIN
Keyword(s):  
Asymptotic Stability ◽  
Weber Number ◽  
Convective Instability ◽  
Initial Conditions ◽  
Fourier Integral ◽  
Liquid Sheet ◽  
Series Solutions ◽  
Integral Response ◽  
Gaussian Perturbation ◽  
Ambient Gas

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow liket1/3. If only the position is perturbed, we find that the sheet is stable and decays liket−2/3at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in theabsenceof ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow liket1/2with a convective instability.

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