HW #6. 1. Inflaton. (a) slow-roll regime. HW6.nb 1

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Marion Todd
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1 HW6.nb HW #6. Inflaton (a) slow-roll regime In the slow-roll regime, we neglect the kinetic energy as well as f ÿÿ term in the equation of motion. Then H = ÅÅÅ 8 p 3 G N ÅÅÅ m f, 3 H f ÿ + m f = 0. We use 8 p G N = M - Pl. Putting them together, m f 3 Å è!!! f ÿ + m f = 0, and hence d f = - "##### 3 m M Pl d t. The solution is simply fhtl = fh0l - "##### 3 m M Pl t. Using this solution, the kinetic energy is fÿ = 3 m M Pl, while the potential energy is m ÅÅÅ f. Therefore, if f p M Pl, we see that the kinetic energy is indeed smaller than the potential energy. At the same time, f ÿÿ = 0 for the above solution and indeed is negligible compared other non-vanishg terms. The scale factor can be obtained from the Friedmann equation H = ÅÅ Rÿ = ÅÅ R 3 M Pl m f, R ÿ R = Å è!!! m f, d log R = Å è!!! m f d t and hence log RHtL = Å RH0L è!!! mifh0l t - è!!! m M Pl t M 6 When f p M Pl, the first term in the parentheses dominates and the scale factor grows exponentially with time.

2 HW6.nb (b) oscillating regime In the oscillating regime f ` M Pl and m t p, we study the equation of motion f ÿÿ + 3 H f ÿ + m f = 0, H = ÅÅ 3 M i Pl jf ÿ + m f y z k { with the ansatz fhtl = fht L t ÅÅÅ Å cos mht - t t L. f ÿÿ = fht L t ÅÅÅ Å H-m cos mht - t t L + Å t L + Å L t The leading term in ê Hm tl is the first term in the parantheses which is canceled by m f term. To see if the next-leading term is also cancelled in the equation of motion, we work out f ÿ = fht L t ÅÅÅ Å H- t L - t cos mht - t LL, H = ÅÅ 3 M Pl fht L t ÅÅÅ Hm + Å t t L cos mht - t L + Å cos mht - t t LL. The equation of motion is then fht L t ÅÅÅ Å H Å t t L + Å L + t t 3 Å è!!! ÅÅÅ Å "################################ m ################################ ###################################### t + Å t L cos mht - t L + Å cos t mht - t L fht L t ÅÅÅ Å H- t L - t cos mht - t LL = 0. The next leading terms are fht L t ÅÅÅ Å H m t Å sin mht - t t t LL - 3 Å è!!! ÅÅÅ Å m fht t L t ÅÅÅ Å t L = 0. This is satisfied if - 3 Å è!!! t m fht L = 0. t = "##### M ÅÅ Pl 3 m fht L. With this choice, the equation of motion is satisfied both for the leading and the next-leading terms in the expansion in ê m t. The expansion rate to the leading order is H = ÅÅ 3 M Pl fht L t ÅÅÅ m = ÅÅ t 3 M Pl 4 M Pl ÅÅÅ Å m = 4 Å 3 m t 9 t This differential equation can be integrated easily and we find R = RHt L I Å t t M ê3. Therefore the energy density scales as r = 3 M Pl H = 3 M 4 Pl Å = 4 ÅÅÅ 9 t 3 t H RHt L ÅÅÅ R L3 Indeed, the energy density is that of matter-domnated universe. (c) numerical solution We solve the differential equation numerically. A word of caution is that this is actually a not very safe thing to do if the solution oscillates crazy like this one. The numerical errors may build up. Mathematica seems to handle it fairly well, though. I didn't specify the boundary conditions. Normally, we take the initial time derivative to vanish, but of course we don't really know what the right boundary conditions are.

3 HW6.nb 3 sol = NDSolveA9f ''@td + 3 H f'@td + m f@td ã 0 ê. 9H Ø $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ÅÅÅ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 MPl Hf'@tD + m f@td L = ê. 8MPl Ø, m Ø 0 - <, f@0d ã 00, f '@0D ã 0=, f, 8t, 0, 00000<E 88f Ø InterpolatingFunction@880., <<, <>D<< Plot@Evaluate@f@tD ê. sol@@ddd, 8t, 0, 0000<, PlotRange Ø 8-, 00<, PlotStyle Ø RGBColor@0,, 0DD The slow-roll solution does not have vanishing time derivative at the initial time. PlotAf 0 - $%%%%%% 3 m MPl t ê. 8f 0 Ø 00, m Ø 0 -, MPl Ø <, 8t, 0, 3000<, PlotRange Ø 8-, 00<, PlotStyle Ø RGBColor@, 0, 0DE

4 HW6.nb 4 Show@%, %%D The fact that they agree so well with each other is a demonstration that the inflation is quite insensitive to the initial values; the solution quickly approaches the slow-roll solution. Plot@Evaluate@f@tD ê. sol@@ddd, 8t, 0000, 00000<, PlotRange Ø 8-, <D Plot@Evaluate@f@tD ê. sol@@ddd, 8t, 0000, 00000<, PlotRange Ø 8-0., 0.<D Note that the time t in this solution is not the same as the time t in the analytic solution in the oscillating regime; they are offset by the contribution from the slow-roll regime. But the offset is quickly forgotten as time goes on m t p. By multiplying the amplitude by time, we can see that the oscillation is more or less ê t, consistent with the analytic approximate solution.

5 HW6.nb 5 multiplying the amplitude by time, we can see that the oscillation is more or less ê t, consistent with the analytic approximate solution. Plot@Evaluate@t f@td ê. sol@@ddd, 8t, 0000, 00000<D To compare the numerical and anlaytic solutions head-to-head, we need to do some more work. We fix the parameters by looking at one period in the numerical solution Plot@Evaluate@f@tD ê. sol@@ddd, 8t, 40000, 4000<D t 0 = t ê. FindRoot@f@tD ã 0 ê. sol@@dd, 8t, 40300<D@@DD EvaluateAf@tD ê. sol@@dd ê. 9t Ø t 0 + p Then the approximate analytic m = ê. 8m Ø 0- <E

6 HW6.nb 6 t PlotA t + t - t 0 - ÅÅÅ p ÅÅÅ m f CosAm Ht - t 0 L - p E ê. 9t Ø $%%%%%% 3 MPl Å m f = ê. 8m Ø 0 -, MPl Ø < ê. 8f -> `, t 0 -> `<, 8t, 40000, 4000<E Plot@Evaluate@t f@td ê. sol@@ddd, 8t, 40000, 50000<, PlotStyle Ø RGBColor@, 0, 0DD

7 HW6.nb 7 t PlotAt t + t - t 0 - ÅÅÅ p ÅÅÅ m f CosAm Ht - t 0 L - p E ê. 9t Ø $%%%%%% 3 8f -> `, t 0 -> `<, 8t, 40000, 50000<, PlotStyle Ø RGBColor@0,, 0DE MPl Å m f = ê. 8m Ø 0 -, MPl Ø < ê Show@%, %%D They agree very well with each other. Optional It is too cumbersome to write many equations in Mathematica. It will be provided as a separate PDF file.

HW5.nb 1. , we find d t = - ÅÅÅÅÅ

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