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Weekly Report

Weekly Report. Chien -I Chiang 江建誼. The Issue I’m Tackling and what I have done. The Issue: Trans- Planckian Problem Can there be signals other than power spectrum to discuss this problems?

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Weekly Report

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  1. Weekly Report Chien-I Chiang 江建誼

  2. The Issue I’m Tackling and what I have done • The Issue: Trans-Planckian Problem • Can there be signals other than power spectrum to discuss this problems? • What I have done this week:1. Review the classical cosmological perturbation.2.Finally understand what is the “initial condition” of the vacuum state.(May be clearer on this point in the following slides)

  3. The Issue I’m Tackling and what I have done this week • 3. Relation between power spectrum and transfer function. • Most of my study is based on TASI lectures on Inflation. The videos and lecture note can be found in this website:http://physicslearning2.colorado.edu/tasi/tasi_2009/tasi_2009.htm

  4. Non-uniqueness of vacuum state • Consider a SHO with mass . In the Heisenberg picture we can expand the position operator as where is complex mode function satisfies the classical EOM . • In order to satisfy the commutator , , . Let , we then have .

  5. Non-uniqueness of vacuum state • Hence we have: • And the vacuum state is defined as • So far the solution is arbitrary, except the normalization condition. Different solution Different annihilation and creation operator Different vacuum state

  6. Non-uniqueness of vacuum state • So we need one more boundary condition. • The requirement that be the eigenstate of Hamiltonian provides the B.C we need.

  7. Non-uniqueness of vacuum state

  8. Zero-Point Fluctuation in the Ground State

  9. Scalar Perturbations • Consider single-field slow-roll models of inflation defined by the action • Choose the gauge: • Expand the action to second order

  10. Scalar Perturbations • Defining Mukhanov variableand transfer to conformal time • Variation to obtain EOM and Fourier transform

  11. Scalar Perturbations • Solution in the de Sitter Limit:General solution:Note that in the far past, i.e Same as SHO

  12. Power Spectrum and Observation • After we calculated the power spectrum of we need the transfer function to relate the primordial spectra to the observational quantities, such as temperature fluctuation. Observational quantity Transfer function The time of horizon exit.

  13. My Question:The primordial perturbation is frozen when the scale is super-horizon. To relate this spectra to the observable quantities, we need to calculate the transfer function. Different observable has different transfer function, but the same primordial power spectra. So it seems that though we have many kinds of experiment, the only probable evidence for Trans-Planckian physics is the power spectrum?

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