ball1.gif MAX 4

Devlin et al. (MAX 4 $\gamma$ Ursae Minoris) and Clapp et al. (MAX 4 $\sigma$ Herculis and $\iota$ Draconis) use 3.5, 6, and 9 cm$^{-1}$ data from the balloon-borne MAX (Millimeter-wave Anisotropy eXperiment) 4 experiment to constrain CMBR anisotropy. Ganga et al. (1998) summarize the experiment.

The FWHM of the beams, assumed to be gaussian, are $\sigma_{\rm FWHM} =
0.55^\circ \pm 0.05^\circ$ at 3.5 cm$^{-1}$ and $\sigma_{\rm FWHM} =
0.75^\circ \pm 0.05^\circ$ at 6 or 9 cm$^{-1}$ (both one standard deviation uncertainty). The zero-lag window function of the smooth-scan, sinusoidally-chopped experiment is

$\displaystyle W_\ell = N^2  e^{-\sigma_{\rm G}{}^2 (\ell + 0.5)^2}$ $\textstyle {}$ $\displaystyle \sum_{k=0}^\ell
{\Gamma (k + 0.5) \Gamma (\ell - k + 0.5) \pi \ov...
...\left((\ell - 2k) \Delta\phi/2\right) \over
(\ell - 2k) \Delta\phi/2} \right]^2$  
  $\textstyle {}$ $\displaystyle      \times
\left[ J_1 \left( (\ell - 2k) \Phi_0 \right) \right]^2 ,$ (3)

where $\sigma_{\rm G} = \sigma_{\rm FWHM}/\sqrt{8 {\rm ln} 2}$, $J_1$ is a Bessel function of the first kind, $\Phi_0 = 0.7^\circ$ is half of the peak-to-peak chop angle, and the bin size $\Delta\phi = (6/21)^\circ$. The normalization $N^2 = 1.138$ for the 3.5 cm$^{-1}$ $W_\ell$ and $N^2 = 1.313$ for the 6 or 9 cm$^{-1}$ $W_\ell$'s. It is fixed by the membrane (step-function sky) calibration technique used.

The first column in the window function file is $\ell$, which runs from 2 to 1000. The second column is the MAX 4 3.5 cm$^{-1}$ zero-lag $W_\ell$ and the third column is the MAX 4 6 or 9 cm$^{-1}$ zero-lag $W_\ell$.


Table: MAX 4 zero-lag Window Function Parameters
  $\ell_{e^{-0.5}}$ $\ell_{\rm e}$ $\ell_{\rm m}$ $\ell_{e^{-0.5}}$ $\sqrt{I(W_\ell)}$
3.5 cm$^{-1}$ 80 132.7 145 224 1.51
6 or 9 cm$^{-1}$ 70 113.7 127 196 1.41

The quoted $\gamma$ Ursae Minoris bandtemperature values are from a gaussian autocorrelation analysis by S. Tanaka (private communication 1995), with 10% added in quadrature to the statistical 1 $\sigma$ error bars to account for calibration uncertainty. The quoted $\sigma$ Herculis and $\iota$ Draconis bandtemperature values are from Ganga et al. (1998). They were computed assuming a flat bandpower spectrum and, following Ganga et al. (1997), account for the MAX 4 absolute calibration uncertainty of 10% as well as the beamwidth uncertainty.

Ganga et al. (1998) and Ratra et al. use the $\sigma$ Herculis and $\iota$ Draconis data to constrain cosmological parameters.

Fig.: MAX 4 zero-lag window functions. (Postscript version here.) win_MAX-4.gif

REFERENCES

ball23.gifLink to the experiment webpage.

A.C. Clapp, et al., ``Measurements of Anisotropy in the Cosmic Microwave Background Radiation at Degree Angular Scales Near the Stars Sigma Herculis and Iota Draconis", Astrophys. J. Lett. 433, L57 (1994).

M.J. Devlin, et al., ``Measurements of Anisotropy in the Cosmic Microwave Background Radiation at $0.{\!\!}^\circ{5}$ Angular Scales Near the Star Gamma Ursae Minoris", Astrophys. J. Lett. 430, L1 (1994).

K. Ganga, B. Ratra, J.O. Gundersen, and N. Sugiyama, ``UCSB South Pole 1994 Cosmic Microwave Background Anisotropy Measurement Constraints on Open and Flat-$\Lambda$ Cold Dark Matter Cosmogonies", Astrophys. J. 484, 7 (1997).

K. Ganga, B. Ratra, M.A. Lim, N. Sugiyama, and S.T. Tanaka ``MAX 4 and MAX 5 Cosmic Microwave Background Anisotropy Measurement Constraints on Open and Flat-$\Lambda$ Cold Dark Matter Cosmogonies", Astrophys. J. Supp. 114, 165 (1998).

B. Ratra, R. Stompor, K. Ganga, G. Rocha, N. Sugiyama, and K.M. Górski, ``Cosmic Microwave Background Anisotropy Constraints on Open and Flat-$\Lambda$ Cold Dark Matter Cosmogonies from UCSB South Pole, ARGO, MAX, White Dish, and SuZIE Data", Astrophys. J. 517, 549 (1999).


Bharat Ratra and Tarun Souradeep
Department of Physics, Kansas State University
Last updated: 2000-08-31