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Hi all. Say we have a background inflaton field ##\varphi## and that we've integrated the background equation for ##\varphi##, ##H(\eta)##, and ##a(\eta)## up to the number of e-folds of inflation corresponding to ##\epsilon = 1## in the slow-roll parameter. We then wish to solve for the ##k## modes of the tensor perturbation in the transverse-traceless gauge using ##\ddot{h}+ 2\frac{\dot{a}}{a}\dot{h} + k^2 h = 0## by plugging in the ##a(\eta)## solved from the background (the tensor modes don't couple to scalar perturbations ##\delta \varphi##) where a dot means derivative with respect to conformal time. To do this we would need to know the initial values of ##h## and ##\dot{h}## at conformal time ##\eta(N = 0)## where ##N## is e-folds of inflation. In Dodelson, it is noted that at early enough times all ##k## modes are subhorizon which reduces the equation ##\ddot{v} + (k^2 - \frac{\ddot{a}}{a})v = 0## for the coefficient ##v## of the creation operator ##\hat{a}_k## associated with a given mode to the equation for a simple harmonic oscillator. This gives us boundary conditions on ##h## and ##\dot{h}## at early times, including the initial time of inflation.
But is there a more rigorous, possibly model-dependent, but experimentally based method of obtaining the initial conditions for the ##k## modes of the tensor perturbations? For example when solving for ##\varphi## from the Klein-Gordon equation, one can use the approximately constant energy scale of inflation to set the initial value for ##\varphi## and use the slow-roll condition to set the initial value for the background inflaton ##\dot{\varphi}## and tune these initial values based on agreement with experiment. However for ##h## the choices of initial ##h## and ##\dot{h}## are based solely on the harmonic oscillator boundary conditions so I was wondering if, in practice, one chooses these initial conditions in a way more akin to the way one would choose the initial conditions for ##\varphi##.
I would also appreciate any references on the general framework of choosing initial conditions for background fields and perturbations in inflation.
As an aside, any references discussing the range of physical modes ##\frac{\Delta k}{a(\eta)}## at the present time that one would be interested in for tensor perturbations would be quite helpful as well. Thanks in advance.
But is there a more rigorous, possibly model-dependent, but experimentally based method of obtaining the initial conditions for the ##k## modes of the tensor perturbations? For example when solving for ##\varphi## from the Klein-Gordon equation, one can use the approximately constant energy scale of inflation to set the initial value for ##\varphi## and use the slow-roll condition to set the initial value for the background inflaton ##\dot{\varphi}## and tune these initial values based on agreement with experiment. However for ##h## the choices of initial ##h## and ##\dot{h}## are based solely on the harmonic oscillator boundary conditions so I was wondering if, in practice, one chooses these initial conditions in a way more akin to the way one would choose the initial conditions for ##\varphi##.
I would also appreciate any references on the general framework of choosing initial conditions for background fields and perturbations in inflation.
As an aside, any references discussing the range of physical modes ##\frac{\Delta k}{a(\eta)}## at the present time that one would be interested in for tensor perturbations would be quite helpful as well. Thanks in advance.