Abstract
The cosmic microwave background (CMB) radiation, the relic afterglow of the Big Bang, has become one of the most useful and precise tools in modern precision cosmology. In this article, we employ Tsallis non-extensive statistical framework to calculate the cosmic microwave background (CMB) temperature and its probability distribution by utilising a recently proposed blackbody radiation inversion (BRI) technique and the cosmic background explorer/ far infrared absolute spectrophotometer (COBE/FIRAS) dataset. Here, we have used the best-fit values of q = 0.99888 ± 0.00016 and q = 1.00012 ± 0.00001, obtained by fitting COBE/FIRAS data with two different versions of non-extensive models. We compare the results with the more conventional extensive statistical analysis i.e. for q = 1.
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Acknowledgements
The authors are grateful to the Department of Physics, Birla Institute of Technology, Mesra, Ranchi, for allotting an excellent research environment during the research work. The authors thank Koustav Konar and Soumen Karmakar for their help and support. The authors would also like to thank Balendu Pathak for encouraging this research. S. Dhal thanks UGC (Savitribai Jyotirao Phule Single Girl Child with grant number - UGCES-22-GE-ORI-F-SJSGC-3962) for the fellowship to carry out research.
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S. D. has performed the analysis, figure, and computational work and wrote the manuscript with text and figure. R. K. P. conceived the idea, wrote the manuscript, analysed and supervised the overall work for the final manuscript.
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Appendices
Appendix 1
All the original intensities and reconstructed intensities along with their uncertainties relevant to Fig. 6 are listed below.
Frequency (GHz) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−22 W m−2 Hz−1) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−20 W m−2 Hz−1) |
|---|---|---|---|---|
Original | Reconstructed | |||
68.1 | 2.007 | 1.4 | 2.008 | 3.0 |
81.6 | 2.495 | 1.9 | 2.497 | 4.1 |
95.4 | 2.930 | 2.5 | 2.932 | 5.3 |
108.9 | 3.277 | 2.3 | 3.280 | 6.5 |
122.4 | 3.540 | 2.2 | 3.543 | 7.6 |
136.2 | 3.720 | 2.1 | 3.724 | 8.7 |
149.7 | 3.814 | 1.8 | 3.818 | 9.6 |
163.5 | 3.834 | 1.8 | 3.838 | 10.3 |
177.0 | 3.789 | 1.6 | 3.792 | 10.9 |
190.5 | 3.688 | 1.4 | 3.687 | 11.3 |
204.3 | 3.540 | 1.3 | 3.544 | 11.5 |
217.8 | 3.362 | 1.2 | 3.366 | 11.7 |
231.3 | 3.160 | 1.1 | 3.164 | 11.6 |
245.1 | 2.932 | 1.0 | 2.942 | 11.4 |
258.6 | 2.714 | 1.1 | 2.717 | 11.0 |
272.4 | 2.482 | 1.2 | 2.486 | 10.5 |
285.9 | 2.259 | 1.4 | 2.262 | 10.1 |
299.4 | 2.043 | 1.6 | 2.046 | 9.6 |
313.2 | 1.832 | 1.8 | 1.835 | 8.9 |
326.7 | 1.638 | 2.2 | 1.641 | 8.3 |
340.2 | 1.457 | 2.2 | 1.460 | 7.7 |
354.0 | 1.288 | 2.3 | 1.290 | 7.1 |
367.5 | 1.135 | 2.3 | 1.138 | 6.4 |
381.3 | 0.994 | 2.3 | 0.996 | 5.9 |
394.8 | 0.870 | 2.2 | 0.872 | 5.3 |
408.3 | 0.758 | 2.1 | 0.760 | 4.8 |
422.1 | 0.657 | 2.0 | 0.659 | 4.3 |
435.6 | 0.570 | 1.9 | 0.571 | 3.9 |
449.1 | 0.492 | 1.9 | 0.493 | 4.0 |
462.9 | 0.422 | 1.9 | 0.424 | 3.1 |
476.4 | 0.363 | 2.1 | 0.364 | 2.7 |
490.2 | 0.310 | 2.3 | 0.311 | 2.4 |
503.7 | 0.265 | 2.6 | 0.266 | 2.1 |
517.2 | 0.226 | 2.8 | 0.227 | 1.8 |
531.0 | 0.192 | 3.0 | 0.193 | 1.6 |
544.5 | 0.163 | 3.2 | 0.164 | 1.4 |
558.3 | 0.138 | 3.3 | 0.139 | 1.2 |
571.8 | 0.117 | 3.5 | 0.118 | 1.1 |
585.3 | 0.099 | 4.1 | 0.099 | 0.9 |
599.1 | 0.083 | 5.5 | 0.084 | 0.7 |
612.6 | 0.070 | 8.8 | 0.071 | 0.7 |
626.1 | 0.058 | 15.5 | 0.059 | 0.6 |
639.9 | 0.045 | 28.2 | 0.050 | 0.5 |
Appendix 2
All the original intensities and reconstructed intensities along with their uncertainties relevant to Fig. 7 are listed below.
Frequency (GHz) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−22 W m−2 Hz−1) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−20 W m−2 Hz−1) |
|---|---|---|---|---|
Original | Reconstructed | |||
68.1 | 2.007 | 1.4 | 2.006 | 4.8 |
81.6 | 2.495 | 1.9 | 2.494 | 6.6 |
95.4 | 2.930 | 2.5 | 2.928 | 8.5 |
108.9 | 3.277 | 2.3 | 3.276 | 10.3 |
122.4 | 3.540 | 2.2 | 3.538 | 12.0 |
136.2 | 3.720 | 2.1 | 3.719 | 13.7 |
149.7 | 3.814 | 1.8 | 3.812 | 15.1 |
163.5 | 3.834 | 1.8 | 3.832 | 16.3 |
177.0 | 3.789 | 1.6 | 3.786 | 17.2 |
190.5 | 3.688 | 1.4 | 3.685 | 17.9 |
204.3 | 3.540 | 1.3 | 3.538 | 18.2 |
217.8 | 3.362 | 1.2 | 3.36 | 18.4 |
231.3 | 3.160 | 1.1 | 3.158 | 18.1 |
245.1 | 2.932 | 1.0 | 2.936 | 17.9 |
258.6 | 2.714 | 1.1 | 2.712 | 17.4 |
272.4 | 2.482 | 1.2 | 2.480 | 16.8 |
285.9 | 2.259 | 1.4 | 2.257 | 16.0 |
299.4 | 2.043 | 1.6 | 2.041 | 15.1 |
313.2 | 1.832 | 1.8 | 1.831 | 14.1 |
326.7 | 1.638 | 2.2 | 1.637 | 13.1 |
340.2 | 1.457 | 2.2 | 1.457 | 12.2 |
354.0 | 1.288 | 2.3 | 1.287 | 11.5 |
367.5 | 1.135 | 2.3 | 1.135 | 10.3 |
381.3 | 0.994 | 2.3 | 0.994 | 9.5 |
394.8 | 0.870 | 2.2 | 0.870 | 8.4 |
408.3 | 0.758 | 2.1 | 0.759 | 7.6 |
422.1 | 0.657 | 2.0 | 0.657 | 6.9 |
435.6 | 0.570 | 1.9 | 0.570 | 6.1 |
449.1 | 0.492 | 1.9 | 0.492 | 5.4 |
462.9 | 0.422 | 1.9 | 0.423 | 4.8 |
476.4 | 0.363 | 2.1 | 0.363 | 4.2 |
490.2 | 0.310 | 2.3 | 0.311 | 3.7 |
503.7 | 0.265 | 2.6 | 0.266 | 3.3 |
517.2 | 0.226 | 2.8 | 0.227 | 2.9 |
531.0 | 0.192 | 3.0 | 0.193 | 2.6 |
544.5 | 0.163 | 3.2 | 0.164 | 2.2 |
558.3 | 0.138 | 3.3 | 0.138 | 1.9 |
571.8 | 0.117 | 3.5 | 0.117 | 1.7 |
585.3 | 0.099 | 4.1 | 0.099 | 1.4 |
599.1 | 0.083 | 5.5 | 0.083 | 1.2 |
612.6 | 0.070 | 8.8 | 0.070 | 1.1 |
626.1 | 0.058 | 15.5 | 0.059 | 0.9 |
639.9 | 0.045 | 28.2 | 0.050 | 0.8 |
Appendix 3
All the original intensities and reconstructed intensities along with their uncertainties relevant to Fig. 8 are listed below.
Frequency (GHz) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−22 W m−2 Hz−1) | Intensity (× 10−18 W m−2 Hz−1) | Uncertainties (× 10−20 W m−2 Hz−1) |
|---|---|---|---|---|
Original | Reconstructed | |||
68.1 | 2.007 | 1.4 | 2.020 | 3.6 |
81.6 | 2.495 | 1.9 | 2.513 | 4.8 |
95.4 | 2.930 | 2.5 | 2.953 | 6.3 |
108.9 | 3.277 | 2.3 | 3.305 | 7.6 |
122.4 | 3.540 | 2.2 | 3.573 | 9.0 |
136.2 | 3.720 | 2.1 | 3.758 | 10.2 |
149.7 | 3.814 | 1.8 | 3.855 | 11.2 |
163.5 | 3.834 | 1.8 | 3.878 | 12.1 |
177.0 | 3.789 | 1.6 | 3.834 | 12.9 |
190.5 | 3.688 | 1.4 | 3.735 | 13.3 |
204.3 | 3.540 | 1.3 | 3.589 | 13.6 |
217.8 | 3.362 | 1.2 | 3.411 | 13.7 |
231.3 | 3.160 | 1.1 | 3.209 | 13.6 |
245.1 | 2.932 | 1.0 | 2.986 | 13.4 |
258.6 | 2.714 | 1.1 | 2.760 | 12.8 |
272.4 | 2.482 | 1.2 | 2.526 | 12.3 |
285.9 | 2.259 | 1.4 | 2.301 | 11.8 |
299.4 | 2.043 | 1.6 | 2.083 | 11.2 |
313.2 | 1.832 | 1.8 | 1.870 | 10.4 |
326.7 | 1.638 | 2.2 | 1.673 | 9.8 |
340.2 | 1.457 | 2.2 | 1.490 | 9.0 |
354.0 | 1.288 | 2.3 | 1.317 | 8.3 |
367.5 | 1.135 | 2.3 | 1.163 | 7.6 |
381.3 | 0.994 | 2.3 | 1.019 | 6.9 |
394.8 | 0.870 | 2.2 | 0.892 | 6.2 |
408.3 | 0.758 | 2.1 | 0.779 | 5.6 |
422.1 | 0.657 | 2.0 | 0.675 | 5.1 |
435.6 | 0.570 | 1.9 | 0.585 | 4.6 |
449.1 | 0.492 | 1.9 | 0.506 | 4.1 |
462.9 | 0.422 | 1.9 | 0.435 | 3.5 |
476.4 | 0.363 | 2.1 | 0.374 | 3.1 |
490.2 | 0.310 | 2.3 | 0.320 | 2.8 |
503.7 | 0.265 | 2.6 | 0.274 | 2.5 |
517.2 | 0.226 | 2.8 | 0.234 | 2.2 |
531.0 | 0.192 | 3.0 | 0.199 | 1.9 |
544.5 | 0.163 | 3.2 | 0.169 | 1.6 |
558.3 | 0.138 | 3.3 | 0.143 | 1.4 |
571.8 | 0.117 | 3.5 | 0.121 | 1.2 |
585.3 | 0.099 | 4.1 | 0.102 | 1.1 |
599.1 | 0.083 | 5.5 | 0.086 | 0.9 |
612.6 | 0.070 | 8.8 | 0.073 | 0.7 |
626.1 | 0.058 | 15.5 | 0.061 | 0.7 |
639.9 | 0.045 | 28.2 | 0.051 | 0.5 |
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Dhal, S., Paul, R.K. A study of cosmic microwave background using non-extensive statistics. Exp Astron 57, 25 (2024). https://doi.org/10.1007/s10686-024-09943-x
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DOI: https://doi.org/10.1007/s10686-024-09943-x