Hybrid_Inflation.nb

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Canonical Scalar Field Inflation In The Presence Of A Chern-Simons Term.\
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Hybrid Inflation With An Oscillating Chern-Simons Scalar Coupling Function.\
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1.Designation Of Scalar Functions, Equations Of Motion, Slow-Roll/Observed \
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We start by designating the scalar functions of the model. For simplicity, \
the Chern-Simons scalar coupling function obtains a trigonometric form and \
the scalar field a power-law form. Compatibility is achieved for a quadratic \
potential but we shall keep it arbitrary for the time being so as to study \
the dependence of the observed indices, and especially the scalar spectral \
index ns on such exponent. The equations of motion are extracted by assuming \
the slow-roll condition is valid. Slow-roll indices along with observed \
indices are taken from arXiv:0412126.\
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Every parameter that ends with 1 is an auxiliary free parameter that needs to \
be designated later. With v and V the Chern-Simons coupling and the scalar \
potential are denoted while H is Hubble\[CloseCurlyQuote]s parameter and H2 \
the squared Hubble. Each parameter ending with dot implies time derivative \
for simplicity and Qt is an auxiliary function that depends on both the \
scalar field and the polarization of gravitational waves. Finally, indices \
\[CurlyEpsilon]1 through \[CurlyEpsilon]3 are the slow-roll indices of the \
model and ns, nt, r indicate the observed indices.\
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To study the inflationary era, one needs to insert as an input the value of \
the scalar field during the first horizon crossing to the observed indices. \
Such a value can be extracted by firstly finding the final value during the \
ending stage of the inflationary era by imposing that \[CurlyEpsilon]1 \
becomes of order O(1). Afterwards, by using the e-folding number \
N=\[Integral] Hdt, the initial value can be extracted as a function of N and \
\[CurlyPhi]f.\
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I use variable U to denote solutions hereafter while L is a random variable \
such that the initial value of the scalar field can be extracted.\
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3. Designation Of Free Parameters And Validity Of Approximations.\
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 "By assigning numerical values to the free parameters of the model, the \
initial value of the scalar field during the first horizon crossing is \
extracted. Subsequently, one can use such value as an input on the observed \
indices and derive results. Here, a blue shifted tensor spectral index (nt>0) \
can be achieved under the slow-roll assumption due to the presence of the \
Chern-Simons term in the initial gravitational action.\n\nThe model used in \
the article has a power-law Chern-Simons coupling v(\[CurlyPhi])=\
\[CapitalLambda]1(\[CurlyPhi]/\[CurlyPhi]1)^m and admits the following \
values:\[LineSeparator] ",
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The numerical values of the slow-roll indices during the first horizon \
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Such result is also expected to apply on the self check on the equations of \
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In this section, it is interesting to examine the area of values for certain \
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                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
                    MathLink`CallFrontEnd[
                    FrontEnd`AttachCell[Typeset`box$, 
                    FrontEndResource["GrayLevelColorValueSelector"], {
                    0, {Left, Bottom}}, {Left, Top}, 
                    "ClosingActions" -> {
                    "SelectionDeparture", "ParentChanged", 
                    "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> 
                    Automatic, Method -> "Preemptive"], 
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                    GrayLevel[0], 
                    RectangleBox[{0, 0}]}, {
                    GrayLevel[0], 
                    RectangleBox[{1, -1}]}, {
                    GrayLevel[0], 
                    RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 
                    "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, 
                    FrameStyle -> GrayLevel[0.], FrameTicks -> None, 
                    PlotRangePadding -> None, ImageSize -> 
                    Dynamic[{
                    Automatic, 1.35 CurrentValue["FontCapHeight"]/
                    AbsoluteCurrentValue[Magnification]}]], 
                    StyleBox[
                    RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> 
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                    BaselinePosition -> Baseline, DefaultBaseStyle -> {}, 
                    ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, 
                    If[
                    Not[
                    AbsoluteCurrentValue["Deployed"]], 
                    SelectionMove[Typeset`box$, All, Expression]; 
                    FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
                    FrontEnd`Private`$ColorSelectorInitialColor = 
                    GrayLevel[0]; 
                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
                    MathLink`CallFrontEnd[
                    FrontEnd`AttachCell[Typeset`box$, 
                    FrontEndResource["GrayLevelColorValueSelector"], {
                    0, {Left, Bottom}}, {Left, Top}, 
                    "ClosingActions" -> {
                    "SelectionDeparture", "ParentChanged", 
                    "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> 
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                    RowBox[{"Opacity", "[", "0.5`", "]"}], ",", 
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                    RectangleBox[{0, 0}]}, {
                    GrayLevel[0], 
                    RectangleBox[{1, -1}]}, {
                    GrayLevel[0], 
                    RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 
                    "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, 
                    FrameStyle -> GrayLevel[0.], FrameTicks -> None, 
                    PlotRangePadding -> None, ImageSize -> 
                    Dynamic[{
                    Automatic, 1.35 CurrentValue["FontCapHeight"]/
                    AbsoluteCurrentValue[Magnification]}]], 
                    StyleBox[
                    RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> 
                    False]], Appearance -> None, BaseStyle -> {}, 
                    BaselinePosition -> Baseline, DefaultBaseStyle -> {}, 
                    ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, 
                    If[
                    Not[
                    AbsoluteCurrentValue["Deployed"]], 
                    SelectionMove[Typeset`box$, All, Expression]; 
                    FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
                    FrontEnd`Private`$ColorSelectorInitialColor = 
                    GrayLevel[0]; 
                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
                    MathLink`CallFrontEnd[
                    FrontEnd`AttachCell[Typeset`box$, 
                    FrontEndResource["GrayLevelColorValueSelector"], {
                    0, {Left, Bottom}}, {Left, Top}, 
                    "ClosingActions" -> {
                    "SelectionDeparture", "ParentChanged", 
                    "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> 
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                    GrayLevel[0], Editable -> False, Selectable -> False], 
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               RowBox[{"{", 
                 RowBox[{"0.3`", ",", 
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                    GrayLevel[0], 
                    RectangleBox[{0, 0}]}, {
                    GrayLevel[0], 
                    RectangleBox[{1, -1}]}, {
                    GrayLevel[0], 
                    RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> 
                    "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, 
                    FrameStyle -> GrayLevel[0.], FrameTicks -> None, 
                    PlotRangePadding -> None, ImageSize -> 
                    Dynamic[{
                    Automatic, 1.35 CurrentValue["FontCapHeight"]/
                    AbsoluteCurrentValue[Magnification]}]], 
                    StyleBox[
                    RowBox[{"GrayLevel", "[", "0", "]"}], NumberMarks -> 
                    False]], Appearance -> None, BaseStyle -> {}, 
                    BaselinePosition -> Baseline, DefaultBaseStyle -> {}, 
                    ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, 
                    If[
                    Not[
                    AbsoluteCurrentValue["Deployed"]], 
                    SelectionMove[Typeset`box$, All, Expression]; 
                    FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
                    FrontEnd`Private`$ColorSelectorInitialColor = 
                    GrayLevel[0]; 
                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
                    MathLink`CallFrontEnd[
                    FrontEnd`AttachCell[Typeset`box$, 
                    FrontEndResource["GrayLevelColorValueSelector"], {
                    0, {Left, Bottom}}, {Left, Top}, 
                    "ClosingActions" -> {
                    "SelectionDeparture", "ParentChanged", 
                    "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> 
                    Automatic, Method -> "Preemptive"], 
                    GrayLevel[0], Editable -> False, Selectable -> False], 
                    ",", 
                    RowBox[{"Opacity", "[", "0.5`", "]"}], ",", 
                    RowBox[{"CapForm", "[", "\"Butt\"", "]"}]}], "]"}]}], 
                 "}"}]}], "}"}], ",", 
           RowBox[{"LabelStyle", "\[Rule]", 
             RowBox[{"{", "}"}]}], ",", 
           RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}], ",", 
           RowBox[{"Charting`TickAnnotations", "\[Rule]", "None"}], ",", 
           RowBox[{"ScalingFunctions", "\[Rule]", 
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           RowBox[{"Charting`TickSide", "\[Rule]", "Right"}], ",", 
           RowBox[{"ColorFunctionScaling", "\[Rule]", "True"}]}], "]"}]& )], 
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    GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, 
    AutoDelete -> False, GridBoxItemSize -> Automatic, 
    BaselinePosition -> {1, 1}]& ),
  Editable->True,
  InterpretationFunction->(RowBox[{"Legended", "[", 
     RowBox[{#, ",", 
       RowBox[{"Placed", "[", 
         RowBox[{#2, ",", "After"}], "]"}]}], "]"}]& )]], "Output",
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