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2.5 Post-Newtonian corrections to the inspiral chirp

0 A quick start: Most GRASP users probably generate chirps by calling make_filters() (Sec. [*]). This is all they will need to know:
  1. If you are using the routine make_filters() (Sec. [*]) to generate templates and you wish to include the 2.5 post-Newtonian corrections in your chirp calculations, simply set ${\tt order}=5$ when you call make_filters(). The chirps returned will be 2.5 post-Newtonian chirps.
  2. If you do not want the 2.5 post-Newtonian corrections - they will slow down your chirp calculations - set ${\tt order} \le 4$ when you call make_filters(). This is probably what you have been doing, so you won't need to change anything.
  3. The behavior of the post-Newtonian series does not get better as you go to higher order: if anything, it gets worse. Therefore, if you use 2.5 post-Newtonian order templates in your search, the admonition in Sec. [*] about checking the ``corners ''of the filter-bank space hold in spades at higher order

Now, for a more thorough explanation: The 2.5 post-Newtonian corrections to the orbital frequency and phase have been calculated by Blanchet [9]. These include corrections of O[$(v/c)^5$] beyond the quadrupole approximation in the phase and frequency evolution. The expressions are

$\displaystyle f(t)$ $\textstyle =$ $\displaystyle {M_\odot \over 16 \pi T_\odot m_{\rm tot}}
\biggl\{ \Theta^{-3/8}...
...7729 \over 21504 } + {3 \over 256 } \eta \right) \pi \Theta^{-1}
\;\biggr\} \;,$ (6.9.53)

$\displaystyle \phi (t)$ $\textstyle =$ $\displaystyle \phi_c -{1\over\eta} \biggl\{ \Theta^{5/8} +
{\rm ...}
-\left( {3...
...eta \right)
\pi \log \left( { \Theta \over \Theta_o } \right) \; \; \biggr\}\;.$ (6.9.54)

Here $\Theta$ is the dimensionless time variable given by Eq. ([*]). The ellipses represent the second post-Newtonian terms already given in Eqs. ([*]) and ([*]), as well as the spin correction given in Eqs. ([*]) and ([*]). The constant $\Theta_o$ is arbitrary; changing its value shifts the phase by a constant. In the code, it is set to the value of the time parameter $\Theta$ at the beginning of the chirp; this insures that the argument of logarithm is close to unity throughout the chirp. The value of $\phi_c$ is then chosen so the phase is zero at the start time, i.e. when the orbital frequency is equal to Initial_Freq.

Computing the logarithm is slow, therefore the code is designed to logically step over the 2.5 post-Newtonian corrections unless they are explicitly called for. Perhaps, in the future, we will write some optimized code to speed up the log calculation.

How to (not) include the 2.5-post-Newtonian corrections to the waveform in your chirp calculations: As we stated above, simply changing the value of the parameter ${\tt order}$ is all that is needed in make_filters(). However, if you are making direct calls to the underlying routines phase_frequency() or chirp_filters() (as opposed to having make_filters() do it for you) you need to set n_phaseterms=6, and phaseterms[5] =1.0. This will turn on the 2.5 post-Newtonian corrections. To illustrate this, here is how the code block in the examples phase_evoltn() and filters() has to be modified to include the 2.5 post-Newtonian corrections.

   /* post-Newtonian [O(1/c^n)] terms you wish to include (or suppress)
      in the phase and frequency evolution: */
   phaseterms[0] =1.;       /* The Newtonian piece           */
   phaseterms[1] =0.;       /* There is no O(1/c) correction */
   phaseterms[2] =1.;       /* The post-Newtonian correction */
   phaseterms[3] =1.;       /* The 3/2 PN correction         */
   phaseterms[4] =1.;       /* The 2 PN correction           */
   phaseterms[5] =1.;       /* The 5/2 PN correction         */

Notice that n_phaseterms=6 and phaseterms[5] =1.0. Nothing else needs to be changed in the examples.

next up previous contents
Next: Function: make_filters() Up: Additional contributions to the Previous: Spin Effects   Contents
Bruce Allen 2000-11-19