├── folps_logo.png
├── setup.py
├── README.md
├── notebooks
    ├── pkl_cb_m00_z05.dat
    └── pkl_cb_m04_z05.dat
└── FOLPSnu.py


/folps_logo.png:
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https://raw.githubusercontent.com/henoriega/FOLPS-nu/HEAD/folps_logo.png


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/setup.py:
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 1 | from setuptools import setup
 2 | 
 3 | 
 4 | package_basename = 'FOLPSnu'
 5 | 
 6 | 
 7 | setup(name=package_basename,
 8 |       version='1.0.0',
 9 |       author='Hernán E. Noriega & Alejandro Aviles',
10 |       author_email='',
11 |       description='Computation of the galaxy redshift space power spectrum for cosmologies containing massive neutrinos',
12 |       license='',
13 |       url='https://github.com/henoriega/FOLPS-nu',
14 |       install_requires=['numpy', 'scipy'],
15 |       py_modules=[package_basename])


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/README.md:
--------------------------------------------------------------------------------
 1 | <p align="center">
 2 |     <img src="https://github.com/henoriega/FOLPS-nu/blob/main/folps_logo.png" width="700" height="200">
 3 | </p>
 4 | 
 5 | # FOLPSν (aka Flops)
 6 | FOLPSν is a Python code that computes the galaxy redshift space power spectrum for cosmologies containing massive neutrinos. The code combines analytical modeling and numerical methods based on the FFTLog formalism. <!-- to speed up the calculations of loop integrals. -->
 7 | 
 8 | For version with JAX (x10 times faster!): [Folpsax](https://github.com/cosmodesi/folpsax)
 9 | 
10 | [![arXiv](https://img.shields.io/badge/arXiv-2208.02791-red)](https://arxiv.org/abs/2208.02791)
11 | 
12 | 
13 | ## Developers (code and (e)BOSS and DESI pipelines): 
14 | - [Hernán E. Noriega](mailto:henoriega@estudiantes.fisica.unam.mx)
15 | - [Alejandro Aviles](mailto:avilescervantes@gmail.com)
16 | 
17 | 
18 | *Special thanks to Arnaud de Mattia for helping with the [Jax](https://github.com/cosmodesi/folpsax) version of this code.* 
19 | 
20 | 
21 | 
22 | 
23 | 
24 | 
25 | 
26 | 
27 | 
28 | ## Run
29 | 
30 | **Dependences**
31 | 
32 | The code employs the standard libraries:
33 | - NumPy 
34 | - SciPy
35 | 
36 | We recommend to use NumPy versions ≥ 1.20.0. For older versions, one needs to rescale by a factor 1/N the [FFT computation](https://github.com/henoriega/FOLPS-nu/blob/main/FOLPSnu.py#L626). 
37 | 
38 | To run the code, first use git clone:
39 | 
40 | ```
41 | git clone https://github.com/henoriega/FOLPS-nu.git
42 | ```
43 | or install via pip by:
44 | 
45 | ```
46 | pip install git+https://github.com/henoriega/FOLPS-nu
47 | ```
48 | 
49 | Once everything is ready, please check the [Jupyter Notebook](https://github.com/henoriega/FOLPS-nu/blob/main/notebooks/Example.ipynb) which contains some helpful examples. 
50 | 
51 | 
52 | 
53 | Attribution
54 | -----------
55 | 
56 | Please cite <https://arxiv.org/abs/2208.02791> if you find this code useful in your research. 
57 | 
58 |     @article{Noriega:2022nhf,
59 |     author = "Noriega, Hern\'an E. and Aviles, Alejandro and Fromenteau, Sebastien and Vargas-Maga\~na, Mariana",
60 |     title = "{Fast computation of non-linear power spectrum in cosmologies with massive neutrinos}",
61 |     eprint = "2208.02791",
62 |     archivePrefix = "arXiv",
63 |     primaryClass = "astro-ph.CO",
64 |     month = "8",
65 |     year = "2022"
66 |     }
67 | 


--------------------------------------------------------------------------------
/notebooks/pkl_cb_m00_z05.dat:
--------------------------------------------------------------------------------
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775 | 52.3236	0.002937419419126098
776 | 53.3806	0.002798064109428232
777 | 54.45900000000002	0.002665314383320352
778 | 55.5591	0.002538872301833108
779 | 56.68150000000001	0.002418420585544997
780 | 57.826499999999996	0.0023036906846688574
781 | 58.9948	0.002194388175653033
782 | 60.1865	0.002090288909944992
783 | 61.402399999999986	0.00199111794211153
784 | 62.642699999999984	0.0018966640653819214
785 | 63.9082	0.0018066807889673088
786 | 65.1992	0.0017209707815457737
787 | 66.51640000000002	0.001639319678445138
788 | 67.86010000000002	0.001561548702323146
789 | 69.231	0.001487464342314995
790 | 70.62950000000001	0.001416899375391032
791 | 72.05630000000001	0.0013496797357287742
792 | 73.5119	0.0012856500675342624
793 | 74.99700000000001	0.0012246542134077439
794 | 76.51199999999999	0.001166556009963842
795 | 78.05769999999998	0.0011112107006731565
796 | 79.63449999999999	0.001058495238014725
797 | 81.24329999999999	0.0010082761171849597
798 | 82.88449999999999	0.0009604424434143896
799 | 84.55889999999998	0.0009148768387264602
800 | 86.267	0.000871476054132183
801 | 88.00979999999998	0.0008301296598680496
802 | 89.78780000000002	0.0007907450948592805
803 | 91.60159999999998	0.0007532314777255724
804 | 93.45199999999997	0.0007174982941452825
805 | 95.33989999999999	0.0006834581799868718
806 | 97.26590000000002	0.0006510336579769113
807 | 99.23080000000003	0.0006201475354447888
808 | 101.23499999999999	0.000590732283366089
809 | 103.28000000000004	0.0005627080321218294
810 | 105.36699999999999	0.0005360047178320165
811 | 107.49499999999999	0.0005105821427084335
812 | 109.66699999999997	0.0004863543313534391
813 | 111.88200000000002	0.00046328508354151037
814 | 114.14300000000001	0.00044129821711450884
815 | 116.44800000000004	0.00042036966080575763
816 | 118.801	0.0004004217396638659
817 | 121.20100000000005	0.0003814245868281072
818 | 123.64900000000006	0.0003633321726022111
819 | 126.14699999999995	0.00034609424534747896
820 | 128.69500000000002	0.00032967705179126383
821 | 131.29499999999996	0.00031403551498964094
822 | 133.94799999999995	0.00029913352587338573
823 | 136.65400000000002	0.00028494176260801334
824 | 139.41400000000004	0.00027142645010457754
825 | 142.23	0.0002585510615127064
826 | 145.10399999999993	0.0002462818099058453
827 | 148.03500000000005	0.00023459891842289693
828 | 151.02599999999998	0.00022346736752201768
829 | 154.07600000000005	0.00021286880932203835
830 | 157.18899999999996	0.00020276851361448474
831 | 160.36399999999998	0.00019315008401630656
832 | 163.604	0.00018398553356881848
833 | 166.90899999999996	0.0001752569976058494
834 | 170.281	0.00016694199598841887
835 | 173.72099999999998	0.00015902177064538957
836 | 177.23	0.00015147833157566382
837 | 180.81	0.0001442924913210324
838 | 184.46300000000008	0.0001374462932224046
839 | 188.18900000000005	0.00013092628289473472
840 | 191.99099999999993	0.00012471439605513747
841 | 195.86899999999991	0.00011879843351489864
842 | 199.826	0.00011316218063111494
843 | 203.86299999999997	0.00010779325661431751
844 | 207.9809999999999	0.00010267972779383332
845 | 212.1829999999999	0.00009780786079171187
846 | 216.46899999999997	0.0000931680877211022
847 | 220.842	0.0000887479996714091
848 | 225.3039999999999	0.00008453700906703081
849 | 229.8550000000001	0.00008052680629041941
850 | 234.49800000000002	0.0000767067787357857
851 | 239.23500000000004	0.00007306780752820594
852 | 244.06799999999998	0.00006960125451776559
853 | 248.99899999999994	0.00006629892934571237
854 | 254.0289999999999	0.00006315366376650028
855 | 259.161	0.0000601573919421834
856 | 264.3959999999999	0.00005730363321573624
857 | 269.7370000000001	0.00005458512674200887
858 | 275.186	0.000051995537790847424
859 | 280.7450000000001	0.0000495288349190438
860 | 286.41699999999986	0.0000471788725374954
861 | 292.2030000000001	0.00004494062916086433
862 | 298.105	0.00004280888051735315
863 | 304.12800000000004	0.00004077766772169871
864 | 310.27199999999993	0.000038843042776144945
865 | 316.5399999999999	0.000037000237151057464
866 | 322.9340000000001	0.00003524502402898305
867 | 329.45799999999986	0.00003357286565075054
868 | 336.11300000000006	0.000031980225735178734
869 | 342.90299999999985	0.000030463014891427294
870 | 349.83	0.000029017817757242544
871 | 356.8970000000001	0.000027641169604949627
872 | 364.10700000000014	0.000026329789879461478
873 | 371.4630000000001	0.000025080569443055325
874 | 378.96599999999995	0.00002389086543184673
875 | 386.6220000000002	0.000022757387872662292
876 | 394.43199999999996	0.000021677776359880107
877 | 402.40099999999995	0.000020649227873195728
878 | 410.53014375108813	0.000019669581185225265
879 | 418.8235092067095	0.000018736411180990628
880 | 427.28441390793176	0.00001784751290011416
881 | 435.916242415458	0.00001700078599059117
882 | 444.7224476635301	0.000016194229745929046
883 | 453.706552341184	0.000015425938377735777
884 | 462.87215030140806	0.000014694096512587795
885 | 472.2229079987683	0.00001399697490253666
886 | 481.7625659560772	0.00001333292633911008
887 | 491.49494026068896	0.000012700381761207518
888 | 501.42392409102644	0.000012097846547555367
889 | 511.5534892739427	0.000011523896985147164
890 | 521.8876878735451	0.000010977176905183927
891 | 


--------------------------------------------------------------------------------
/notebooks/pkl_cb_m04_z05.dat:
--------------------------------------------------------------------------------
  1 | #k, pkl
  2 | 0.00001	35.500613815779495
  3 | 0.000010104300000000008	35.857283700871854
  4 | 0.000010308399999999996	36.55484709506928
  5 | 0.000010516599999999992	37.265903371090744
  6 | 0.000010729100000000004	37.991114902130256
  7 | 0.000010945800000000003	38.730119591327956
  8 | 0.000011166999999999997	39.48391918877948
  9 | 0.000011392499999999993	40.25181067559011
 10 | 0.000011622699999999991	41.03513420334732
 11 | 0.000011857499999999998	41.83352665997935
 12 | 0.000012097000000000006	42.64730550107865
 13 | 0.000012341400000000009	43.47712685952146
 14 | 0.000012590699999999998	44.322966824001135
 15 | 0.000012845100000000007	45.18547958344109
 16 | 0.000013104499999999996	46.0643016201183
 17 | 0.000013369299999999998	46.96076278505737
 18 | 0.000013639300000000002	47.87416022482094
 19 | 0.000013914900000000006	48.805820906785854
 20 | 0.000014195999999999994	49.75537975316704
 21 | 0.000014482800000000001	50.72348527229951
 22 | 0.000014775299999999989	51.71010993925288
 23 | 0.000015073799999999991	52.7162373597987
 24 | 0.000015378300000000005	53.74183842944412
 25 | 0.00001568900000000001	54.78755726473339
 26 | 0.00001600590000000001	55.853363948901816
 27 | 0.0000163293	56.94023677010894
 28 | 0.0000166591	58.04780858103284
 29 | 0.0000169957	59.177390889425745
 30 | 0.00001733900000000001	60.3286155476131
 31 | 0.000017689299999999997	61.50245565177913
 32 | 0.000018046599999999987	62.69887754846904
 33 | 0.00001841119999999999	63.91885142016476
 34 | 0.000018783099999999986	65.16234218605605
 35 | 0.0000191626	66.43031715048072
 36 | 0.00001954970000000002	67.72273983916601
 37 | 0.000019944600000000006	69.04024120718199
 38 | 0.000020347500000000018	70.38345068048798
 39 | 0.00002075859999999999	71.75299616603454
 40 | 0.000021177900000000016	73.14883850801294
 41 | 0.00002160570000000001	74.57193659903656
 42 | 0.0000220422	76.02291457885578
 43 | 0.00002248750000000002	77.5020630608107
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 46 | 0.000023878	82.1140808167195
 47 | 0.000024360399999999995	83.71178540657228
 48 | 0.000024852500000000023	85.34042528663616
 49 | 0.000025354600000000016	87.00094652076612
 50 | 0.0000258668	88.69363209781798
 51 | 0.000026389300000000022	90.41909469463637
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 53 | 0.000027466299999999998	93.97178366276601
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 55 | 0.000028587199999999993	97.66388315335814
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 61 | 0.000032232	109.63387077142887
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 73 | 0.00004097490000000001	138.154701609648
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 99 | 0.00006892079999999996	228.00464439561298
100 | 0.00007031310000000001	232.4406870393466
101 | 0.00007173360000000007	236.963271020294
102 | 0.0000731827	241.57354010074135
103 | 0.0000746611	246.27358919569159
104 | 0.00007616929999999998	251.06487308632404
105 | 0.00007770800000000002	255.94947811377256
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126 | 0.00011826900000000003	383.5362477397427
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128 | 0.000123095	398.5690682965507
129 | 0.00012558200000000003	406.3119260353673
130 | 0.00012811899999999997	414.2179047886987
131 | 0.00013070699999999998	422.26142412586194
132 | 0.00013334700000000002	430.4519464645404
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135 | 0.00014159300000000005	456.03640197502995
136 | 0.00014445399999999994	464.882977859796
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141 | 0.00015964600000000008	511.7884791965993
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152 | 0.00019893100000000012	632.3099191370856
153 | 0.00020295000000000005	644.5828020943386
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719 | 16.734100000000005	0.04737623576114046
720 | 17.072099999999995	0.04512327870732562
721 | 17.417	0.042977021302573526
722 | 17.7689	0.040932668749013405
723 | 18.1278	0.03898613899818059
724 | 18.494	0.03713192654651671
725 | 18.8676	0.035365891042102617
726 | 19.2488	0.03368362479857545
727 | 19.6376	0.03208178470631041
728 | 20.0344	0.030555575476011995
729 | 20.439099999999996	0.029102380414510534
730 | 20.851999999999997	0.027718217144190278
731 | 21.273199999999996	0.026400011794099842
732 | 21.702999999999996	0.025144237199157957
733 | 22.1414	0.023948416606108254
734 | 22.5887	0.022809358299821107
735 | 23.045000000000005	0.021724560488599495
736 | 23.510499999999997	0.0206913923040186
737 | 23.985500000000002	0.01970716478117503
738 | 24.46999999999999	0.018769934878305923
739 | 24.964300000000005	0.017877254484414493
740 | 25.468700000000002	0.017026841773364366
741 | 25.983099999999997	0.016217173752587445
742 | 26.508	0.015445850566674118
743 | 27.043500000000005	0.01471121820541316
744 | 27.589900000000007	0.01401142465916073
745 | 28.147199999999998	0.013345080796347069
746 | 28.7158	0.012710380143706745
747 | 29.2959	0.012105852426357429
748 | 29.887700000000002	0.011530094808767448
749 | 30.49149999999999	0.010981680733917618
750 | 31.107500000000005	0.010459349745300845
751 | 31.73590000000001	0.009961896975931242
752 | 32.377	0.009488099248680291
753 | 33.0311	0.009036803098660619
754 | 33.6983	0.008607043871201005
755 | 34.379099999999994	0.008197650976026462
756 | 35.0736	0.007807760124049982
757 | 35.78210000000001	0.007436428049164695
758 | 36.50500000000001	0.007082714674826051
759 | 37.242399999999996	0.006745871838909854
760 | 37.994699999999995	0.006425046850955282
761 | 38.7623	0.006119440087193883
762 | 39.54529999999999	0.005828408150230982
763 | 40.3442	0.005551189332024189
764 | 41.15919999999999	0.0052871683406215545
765 | 41.99069999999999	0.005035693960423591
766 | 42.839000000000006	0.004796179948346011
767 | 43.70440000000001	0.004568067536485342
768 | 44.58720000000001	0.0043508239684043925
769 | 45.488	0.004143874557085745
770 | 46.406899999999986	0.003946789317977223
771 | 47.34439999999999	0.003759070346739555
772 | 48.300799999999995	0.0035802868165578853
773 | 49.27649999999999	0.0034100097307096467
774 | 50.272	0.003247816732182298
775 | 51.28759999999999	0.0030933399257349597
776 | 52.3236	0.0029462266287197856
777 | 53.3806	0.002806100408206846
778 | 54.45900000000002	0.0026726331074478937
779 | 55.5591	0.0025455235410492827
780 | 56.68150000000001	0.0024244511970704886
781 | 57.826499999999996	0.0023091446589319437
782 | 58.9948	0.002199306612155361
783 | 60.1865	0.0020947104639305048
784 | 61.402399999999986	0.001995078673499664
785 | 62.642699999999984	0.0019001978317379272
786 | 63.9082	0.0018098191708138713
787 | 65.1992	0.001723743386312115
788 | 66.51640000000002	0.0016417541447357334
789 | 67.86010000000002	0.0015636709240883888
790 | 69.231	0.0014892984792353049
791 | 70.62950000000001	0.001418468025400092
792 | 72.05630000000001	0.0013510039736392215
793 | 73.5119	0.001286749577048793
794 | 74.99700000000001	0.00122554734350136
795 | 76.51199999999999	0.0011672598990992206
796 | 78.05769999999998	0.001111741301578326
797 | 79.63449999999999	0.0010588674378377973
798 | 81.24329999999999	0.0010085037541906418
799 | 82.88449999999999	0.0009605384199436579
800 | 84.55889999999998	0.0009148531460511251
801 | 86.267	0.0008713438523044432
802 | 88.00979999999998	0.0008298993000732717
803 | 89.78780000000002	0.0007904262016325475
804 | 91.60159999999998	0.0007528329871692397
805 | 93.45199999999997	0.0007170284891131826
806 | 95.33989999999999	0.0006829247305983944
807 | 97.26590000000002	0.000650443671972658
808 | 99.23080000000003	0.0006195075878737057
809 | 101.23499999999999	0.0005900484599493373
810 | 103.28000000000004	0.000561985938316791
811 | 105.36699999999999	0.000535249522135629
812 | 107.49499999999999	0.0005097986352486256
813 | 109.66699999999997	0.0004855468941412991
814 | 111.88200000000002	0.0004624577661908387
815 | 114.14300000000001	0.0004404547212149148
816 | 116.44800000000004	0.0004195134001213883
817 | 118.801	0.00039955582338016424
818 | 121.20100000000005	0.0003805518667230162
819 | 123.64900000000006	0.00036245524961034354
820 | 126.14699999999995	0.000345215487080237
821 | 128.69500000000002	0.0003287986112546405
822 | 131.29499999999996	0.00031315934497124225
823 | 133.94799999999995	0.00029826139390102705
824 | 136.65400000000002	0.0002840752614931836
825 | 139.41400000000004	0.00027056701101593457
826 | 142.23	0.00025769996884104977
827 | 145.10399999999993	0.0002454402134520634
828 | 148.03500000000005	0.00023376783319227924
829 | 151.02599999999998	0.00022264769673646935
830 | 154.07600000000005	0.00021206133841991252
831 | 157.18899999999996	0.00020197393822797221
832 | 160.36399999999998	0.0001923689975074711
833 | 163.604	0.00018321845081173532
834 | 166.90899999999996	0.00017450434890566705
835 | 170.281	0.00016620414192826911
836 | 173.72099999999998	0.0001582990035462724
837 | 177.23	0.00015077088092199116
838 | 180.81	0.00014360053317853414
839 | 184.46300000000008	0.00013676995482090606
840 | 188.18900000000005	0.0001302656384586569
841 | 191.99099999999993	0.0001240694835591593
842 | 195.86899999999991	0.00011816924591927949
843 | 199.826	0.0001125486816527885
844 | 203.86299999999997	0.00010719537577113268
845 | 207.9809999999999	0.00010209736361467264
846 | 212.1829999999999	0.0000972408910641402
847 | 216.46899999999997	0.0000926163609656418
848 | 220.842	0.00008821134841278603
849 | 225.3039999999999	0.00008401524817138014
850 | 229.8550000000001	0.00008001972852247788
851 | 234.49800000000002	0.00007621416658454818
852 | 239.23500000000004	0.00007258943143965952
853 | 244.06799999999998	0.00006913687439956015
854 | 248.99899999999994	0.00006584829596133496
855 | 254.0289999999999	0.00006271651739901408
856 | 259.161	0.00005973346905797587
857 | 264.3959999999999	0.000056892662310011024
858 | 269.7370000000001	0.000054186834414912165
859 | 275.186	0.00005160964707328219
860 | 280.7450000000001	0.000049155066157130724
861 | 286.41699999999986	0.00004681694625379245
862 | 292.2030000000001	0.000044590262548717235
863 | 298.105	0.000042469790367634014
864 | 304.12800000000004	0.000040449576421490534
865 | 310.27199999999993	0.00003852566941379835
866 | 316.5399999999999	0.000036693303610917775
867 | 322.9340000000001	0.000034948253653153804
868 | 329.45799999999986	0.000033285986685589444
869 | 336.11300000000006	0.000031702966986741584
870 | 342.90299999999985	0.00003019511046107684
871 | 349.83	0.000028759004348928973
872 | 356.8970000000001	0.000027391188119091975
873 | 364.10700000000014	0.00002608838551593704
874 | 371.4630000000001	0.00002484749179434939
875 | 378.96599999999995	0.00002366586646680602
876 | 386.6220000000002	0.00002254022741624754
877 | 394.43199999999996	0.000021468216794269912
878 | 402.40099999999995	0.00002044703819273473
879 | 410.53014375108813	0.00001947453336921906
880 | 418.8235092067095	0.000018548282952960113
881 | 427.28441390793176	0.000017666086985521796
882 | 435.916242415458	0.00001682585014319165
883 | 444.7224476635301	0.000016025576760329746
884 | 453.706552341184	0.000015263366089418026
885 | 462.87215030140806	0.000014537407786551481
886 | 472.2229079987683	0.000013845977611649232
887 | 481.7625659560772	0.0000131874333331635
888 | 491.49494026068896	0.000012560210827614955
889 | 501.42392409102644	0.00001196282036455168
890 | 511.5534892739427	0.00001139384306829404
891 | 521.8876878735451	0.000010851927547921286
892 | 


--------------------------------------------------------------------------------
/FOLPSnu.py:
--------------------------------------------------------------------------------
   1 | #!/usr/bin/env python
   2 | # coding: utf-8
   3 | 
   4 | #===========================================================================================================================
   5 | #============= ============= ============= ============  FOLPSν  ============== ============== ============= =============== 
   6 | # This is a code for efficiently evaluating the redshift space power spectrum in the presence of massive neutrinos.
   7 | 
   8 | # We recommend to use NumPy versions ≥ 1.20.0. For older versions, one needs to rescale by a factor 1/N the FFT computation. (see: https://github.com/henoriega/FOLPS-nu).
   9 | #===========================================================================================================================
  10 | 
  11 | #standard libraries
  12 | import numpy as np
  13 | import scipy
  14 | from scipy import integrate
  15 | from scipy.interpolate import CubicSpline
  16 | from scipy import interpolate
  17 | from scipy.fft import dst, idst
  18 | from scipy.special import gamma
  19 | from scipy.special import spherical_jn
  20 | from scipy.special import eval_legendre
  21 | from scipy.integrate import quad
  22 | import sys
  23 | 
  24 | 
  25 | 
  26 | def interp(k, x, y):
  27 |     '''Cubic spline interpolation.
  28 |     
  29 |     Args:
  30 |         k: coordinates at which to evaluate the interpolated values.
  31 |         x: x-coordinates of the data points.
  32 |         y: y-coordinates of the data points.
  33 |     Returns:
  34 |         Cubic interpolation of ‘y’ evaluated at ‘k’.
  35 |     '''
  36 |     inter = CubicSpline(x, y)
  37 |     return inter(k)  
  38 |     
  39 | 
  40 | 
  41 | def Matrices(Nfftlog = None):    
  42 |     '''M matrices. They do not depend on the cosmology, so they are computed only one time.
  43 |     
  44 |     Args:
  45 |         if 'Nfftlog = None' (or not specified) the code use as default 'Nfftlog = 128'. 
  46 |         to use a different number of sample points, just specify it as 'Nfftlog =  number'.
  47 |         we recommend using the default mode, see Fig.~8 at arXiv:2208.02791. 
  48 |     Returns:
  49 |         All the M matrices.
  50 |     '''
  51 |     global M22matrices, M13vectors, bnu_b, N
  52 |     
  53 |     k_min = 10**(-7); k_max = 100.
  54 |     b_nu = -0.1;   #Not yet tested for other values
  55 |     
  56 |     if Nfftlog == None:
  57 |         N = 128
  58 |         
  59 |     else:
  60 |         N = Nfftlog
  61 |         
  62 |     
  63 |     #Eq.~ 4.19 at arXiv:2208.02791
  64 |     def Imatrix(nu1, nu2):
  65 |         return 1/(8 * np.pi**(3/2)) * ( gamma(3/2-nu1)*gamma(3/2-nu2)*gamma(nu1+nu2-3/2) )/( gamma(nu1)*gamma(nu2)*gamma(3-nu1-nu2) )
  66 | 
  67 |     
  68 |     #M22-type
  69 |     def M22(nu1, nu2):
  70 |         
  71 |         #Overdensity and velocity
  72 |         def M22_dd(nu1, nu2):
  73 |             return Imatrix(nu1,nu2)*(3/2-nu1-nu2)*(1/2-nu1-nu2)*( (nu1*nu2)*(98*(nu1+nu2)**2 - 14*(nu1+nu2) + 36) - 91*(nu1+nu2)**2+ 3*(nu1+nu2) + 58)/(196*nu1*(1+nu1)*(1/2-nu1)*nu2*(1+nu2)*(1/2-nu2))
  74 |         
  75 |         def M22_dt_fp(nu1, nu2):
  76 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2))*(-23-21*nu1+(-38+7*nu1*(-1+7*nu1))*nu2+7*(3+7*nu1)*nu2**2) )/(196*nu1*(1+nu1)*nu2*(1+nu2)*(-1+2*nu2))
  77 |         
  78 |         def M22_tt_fpfp(nu1, nu2):
  79 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-12*(1-2*nu2)**2 + 98*nu1**(3)*nu2 + 7*nu1**2*(1+2*nu2*(-8+7*nu2))- nu1*(53+2*nu2*(17+7*nu2))))/(98*nu1*(1+nu1)*nu2*(1+nu2)*(-1+2*nu2))
  80 |         
  81 |         def M22_tt_fkmpfp(nu1, nu2):
  82 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-37+7*nu1**(2)*(3+7*nu2) + nu2*(-10+21*nu2) + nu1*(-10+7*nu2*(-1+7*nu2))))/(98*nu1*(1+nu1)*nu2*(1+nu2))
  83 |         
  84 |         #A function
  85 |         def MtAfp_11(nu1, nu2):
  86 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2))*(-5+nu1*(-4+7*(nu1+nu2))))/(7*nu1*(1+nu1)*(-1+2*nu1)*nu2)
  87 |         
  88 |         def MtAfkmpfp_12(nu1, nu2):
  89 |             return -Imatrix(nu1,nu2)*(((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2))*(6+7*(nu1+nu2)))/(56*nu1*(1+nu1)*nu2*(1+nu2)))
  90 |         
  91 |         def MtAfkmpfp_22(nu1, nu2):
  92 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-18+3*nu1*(1+4*(10-9*nu1)*nu1)+75*nu2+8*nu1*(41+2*nu1*(-28+nu1*(-4+7*nu1)))*nu2+48*nu1*(-9+nu1*(-3+7*nu1))*nu2**2+4*(-39+4*nu1*(-19+35*nu1))*nu2**3+336*nu1*nu2**4) )/(56*nu1*(1+nu1)*(-1+2*nu1)*nu2*(1+nu2)*(-1+2*nu2))
  93 |         
  94 |         def MtAfpfp_22(nu1, nu2):
  95 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-5+3*nu2+nu1*(-4+7*(nu1+nu2))))/(7*nu1*(1+nu1)*nu2)
  96 |         
  97 |         def MtAfkmpfpfp_23(nu1, nu2):
  98 |             return -Imatrix(nu1,nu2)*(((-1+7*nu1)*(-3+2*(nu1+nu2))*(-1+2*(nu1+nu2)))/(28*nu1*(1+nu1)*nu2*(1+nu2)))
  99 |         
 100 |         def MtAfkmpfpfp_33(nu1, nu2):
 101 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2))*(-13*(1+nu1)+2*(-11+nu1*(-1+14*nu1))*nu2 + 4*(3+7*nu1)*nu2**2))/(28*nu1*(1+nu1)*nu2*(1+nu2)*(-1+2*nu2))
 102 |         
 103 |         #D function
 104 |         def MB1_11(nu1, nu2):
 105 |             return Imatrix(nu1,nu2)*(3-2*(nu1+nu2))/(4*nu1*nu2)
 106 |         
 107 |         def MC1_11(nu1, nu2):
 108 |             return Imatrix(nu1,nu2)*((-3+2*nu1)*(-3+2*(nu1+nu2)))/(4*nu2*(1+nu2)*(-1+2*nu2))
 109 |         
 110 |         def MB2_11(nu1, nu2):
 111 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2)))/(4*nu1*nu2)
 112 |         
 113 |         def MC2_11(nu1, nu2):
 114 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-1+2*(nu1+nu2)))/(4*nu2*(1+nu2))
 115 |         
 116 |         def MD2_21(nu1, nu2):
 117 |             return Imatrix(nu1,nu2)*((-1+2*nu1-4*nu2)*(-3+2*(nu1+nu2))*(-1+2*(nu1+nu2)))/(4*nu1*nu2*(-1+nu2+2*nu2**2))
 118 |         
 119 |         def MD3_21(nu1, nu2):
 120 |             return Imatrix(nu1,nu2)*((3-2*(nu1+nu2))*(1-4*(nu1+nu2)**2))/(4*nu1*nu2*(1+nu2))
 121 |         
 122 |         def MD2_22(nu1, nu2):
 123 |             return Imatrix(nu1,nu2)*(3*(3-2*(nu1+nu2))*(1-2*(nu1+nu2)))/(32*nu1*(1+nu1)*nu2*(1+nu2))
 124 |         
 125 |         def MD3_22(nu1, nu2):
 126 |             return Imatrix(nu1,nu2)*((3-2*(nu1+nu2))*(1-4*(nu1+nu2)**2)*(1+2*(nu1**2-4*nu1*nu2+nu2**2)))/(16*nu1*(1+nu1)*(-1+2*nu1)*nu2*(1+nu2)*(-1+2*nu2))
 127 |         
 128 |         def MD4_22(nu1, nu2):
 129 |             return Imatrix(nu1,nu2)*((9-4*(nu1+nu2)**2)*(1-4*(nu1+nu2)**2))/(32*nu1*(1+nu1)*nu2*(1+nu2))
 130 |     
 131 |         
 132 |         return (M22_dd(nu1, nu2), M22_dt_fp(nu1, nu2), M22_tt_fpfp(nu1, nu2), M22_tt_fkmpfp(nu1, nu2),
 133 |                 MtAfp_11(nu1, nu2), MtAfkmpfp_12(nu1, nu2), MtAfkmpfp_22(nu1, nu2), MtAfpfp_22(nu1, nu2), 
 134 |                 MtAfkmpfpfp_23(nu1, nu2), MtAfkmpfpfp_33(nu1, nu2), MB1_11(nu1, nu2), MC1_11(nu1, nu2), 
 135 |                 MB2_11(nu1, nu2), MC2_11(nu1, nu2), MD2_21(nu1, nu2), MD3_21(nu1, nu2), MD2_22(nu1, nu2), 
 136 |                 MD3_22(nu1, nu2), MD4_22(nu1, nu2))
 137 |     
 138 |     
 139 |     #M22-type Biasing
 140 |     def M22bias(nu1, nu2):
 141 |         
 142 |         def MPb1b2(nu1, nu2):
 143 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-4+7*(nu1+nu2)))/(28*nu1*nu2)
 144 |         
 145 |         def MPb1bs2(nu1, nu2):
 146 |             return Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(2+14*nu1**2 *(-1+2*nu2)-nu2*(3+14*nu2)+nu1*(-3+4*nu2*(-11+7*nu2))))/(168*nu1*(1+nu1)*nu2*(1+nu2))
 147 |         
 148 |         def MPb22(nu1, nu2):
 149 |             return 1/2 * Imatrix(nu1, nu2)
 150 | 
 151 |         def MPb2bs2(nu1, nu2):
 152 |             return Imatrix(nu1,nu2)*((-3+2*nu1)*(-3+2*nu2))/(12*nu1*nu2)
 153 | 
 154 |         def MPb2s2(nu1, nu2):
 155 |             return Imatrix(nu1,nu2)*((63-60*nu2+4*(3*(-5+nu1)*nu1+(17-4*nu1)*nu1*nu2+(3+2*(-2+nu1)*nu1)*nu2**2)))/(36*nu1*(1+nu1)*nu2*(1+nu2))
 156 | 
 157 |         def MPb2t(nu1, nu2):
 158 |             return Imatrix(nu1,nu2)*((-4+7*nu1)*(-3+2*(nu1+nu2)))/(14*nu1*nu2)
 159 | 
 160 |         def MPbs2t(nu1, nu2):
 161 |             return  Imatrix(nu1,nu2)*((-3+2*(nu1+nu2))*(-19-10*nu2+nu1*(39-30*nu2+14*nu1*(-1+2*nu2))))/(84*nu1*(1+nu1)*nu2*(1+nu2))
 162 | 
 163 |         
 164 |         return (MPb1b2(nu1, nu2), MPb1bs2(nu1, nu2), MPb22(nu1, nu2), MPb2bs2(nu1, nu2), 
 165 |                 MPb2s2(nu1, nu2), MPb2t(nu1, nu2), MPbs2t(nu1, nu2))
 166 |     
 167 |     
 168 |     #M13-type
 169 |     def M13(nu1):
 170 |         
 171 |         #Overdensity and velocity
 172 |         def M13_dd(nu1):
 173 |             return ((1+9*nu1)/4) * np.tan(nu1*np.pi)/( 28*np.pi*(nu1+1)*nu1*(nu1-1)*(nu1-2)*(nu1-3) )
 174 |         
 175 |         def M13_dt_fk(nu1):
 176 |             return ((-7+9*nu1)*np.tan(nu1*np.pi))/(112*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 177 |         
 178 |         def M13_tt_fk(nu1):
 179 |             return -(np.tan(nu1*np.pi)/(14*np.pi*(-3 + nu1)*(-2 + nu1)*(-1 + nu1)*nu1*(1 + nu1) ))
 180 |         
 181 |         # A function
 182 |         def Mafk_11(nu1):
 183 |             return ((15-7*nu1)*np.tan(nu1*np.pi))/(56*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1)
 184 |         
 185 |         def Mafp_11(nu1):
 186 |             return ((-6+7*nu1)*np.tan(nu1*np.pi))/(56*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1)
 187 |         
 188 |         def Mafkfp_12(nu1):
 189 |             return (3*(-13+7*nu1)*np.tan(nu1*np.pi))/(224*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 190 |         
 191 |         def Mafpfp_12(nu1):
 192 |             return (3*(1-7*nu1)*np.tan(nu1*np.pi))/(224*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 193 |         
 194 |         def Mafkfkfp_33(nu1):
 195 |             return ((21+(53-28*nu1)*nu1)*np.tan(nu1*np.pi))/(224*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 196 |         
 197 |         def Mafkfpfp_33(nu1):
 198 |             return ((-21+nu1*(-17+28*nu1))*np.tan(nu1*np.pi))/(224*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 199 | 
 200 |         
 201 |         return (M13_dd(nu1), M13_dt_fk(nu1), M13_tt_fk(nu1), Mafk_11(nu1),  Mafp_11(nu1), Mafkfp_12(nu1),
 202 |                 Mafpfp_12(nu1), Mafkfkfp_33(nu1), Mafkfpfp_33(nu1))
 203 | 
 204 |     
 205 |     #M13-type Biasing
 206 |     def M13bias(nu1):
 207 |         
 208 |         def Msigma23(nu1):
 209 |             return (45*np.tan(nu1*np.pi))/(128*np.pi*(-3+nu1)*(-2+nu1)*(-1+nu1)*nu1*(1+nu1))
 210 |         
 211 |         return (Msigma23(nu1))
 212 |     
 213 |     
 214 |     #Computation of M22-type matrices
 215 |     def M22type(k_min, k_max, N, b_nu, M22):
 216 |         
 217 |         #nuT = -etaT/2, etaT = bias_nu + i*eta_m        
 218 |         nuT = np.zeros(N+1, dtype = complex)
 219 |         
 220 |         for jj in range(N+1):
 221 |             nuT[jj] = -0.5 * (b_nu +  (2*np.pi*1j/np.log(k_max/k_min)) * (jj - N/2) *(N-1)/(N))
 222 |             
 223 |         #reduce time x10 compared to "for" iterations
 224 |         nuT_x, nuT_y = np.meshgrid(nuT, nuT) 
 225 |         M22matrix = M22(nuT_y, nuT_x)
 226 |         
 227 |         return np.array(M22matrix)
 228 |     
 229 |     
 230 |     #Computation of M13-type matrices
 231 |     def M13type(k_min, k_max, N, b_nu, M13):
 232 |            
 233 |         #nuT = -etaT/2, etaT = bias_nu + i*eta_m 
 234 |         nuT = np.zeros(N+1, dtype = complex)
 235 |         
 236 |         for ii in range(N+1):
 237 |             nuT[ii] = -0.5 * (b_nu +  (2*np.pi*1j/np.log(k_max/k_min)) * (ii - N/2) *(N-1)/(N))
 238 |         
 239 |         M13vector = M13(nuT)
 240 |             
 241 |         return np.array(M13vector)
 242 |     
 243 |     
 244 |     #FFTLog bias for the biasing spectra Pb1b2,...
 245 |     bnu_b = 15.1*b_nu
 246 |     
 247 | 
 248 |     M22T =  M22type(k_min, k_max, N, b_nu, M22)
 249 |     M22biasT = M22type(k_min, k_max, N, bnu_b, M22bias)
 250 |     M22matrices = np.concatenate((M22T, M22biasT))
 251 |     
 252 |     M13T = M13type(k_min, k_max, N, b_nu, M13)
 253 |     M13biasT = np.reshape(M13type(k_min, k_max, N, bnu_b, M13bias), (1, int(N+1)))
 254 |     M13vectors = np.concatenate((M13T, M13biasT))
 255 |     
 256 |     print('N = '+str(N)+' sampling points')
 257 |     print('M matrices have been computed')
 258 |   
 259 |     return (M22matrices, M13vectors)
 260 | 
 261 | 
 262 | 
 263 | 
 264 | def LinearRegression(inputxy): 
 265 |     '''Linear regression.
 266 |     
 267 |     Args:
 268 |         inputxy: data set with x- and y-coordinates.
 269 |     Returns:
 270 |         slope ‘m’ and the intercept ‘b’.
 271 |     '''
 272 |     xm = np.mean(inputxy[0])
 273 |     ym = np.mean(inputxy[1])
 274 |     Npts = len(inputxy[0])
 275 |     
 276 |     SS_xy = np.sum(inputxy[0]*inputxy[1]) - Npts*xm*ym
 277 |     SS_xx = np.sum(inputxy[0]**2) - Npts*xm**2
 278 |     m = SS_xy/SS_xx
 279 |     
 280 |     b = ym - m*xm
 281 |     return (m, b)
 282 | 
 283 | 
 284 | 
 285 | 
 286 | def Extrapolate(inputxy, outputx):
 287 |     '''Extrapolation.
 288 |     
 289 |     Args:
 290 |         inputxy: data set with x- and y-coordinates.
 291 |         outputx: x-coordinates of extrapolation.
 292 |     Returns:
 293 |         extrapolates the data set ‘inputxy’ to the range given by ‘outputx’.
 294 |     '''
 295 |     m, b = LinearRegression(inputxy)
 296 |     outxy = [(outputx[ii], m*outputx[ii]+b) for ii in range(len(outputx))]
 297 |     
 298 |     return np.array(np.transpose(outxy))
 299 | 
 300 | 
 301 | 
 302 | 
 303 | def ExtrapolateHighkLogLog(inputT, kcutmax, kmax):
 304 |     '''Extrapolation for high-k values.
 305 |     
 306 |     Args:
 307 |         inputT: k-coordinates and linear power spectrum.
 308 |         kcutmax: value of ‘k’ from which ‘inputT’ will be interpolated.
 309 |         kmax: value of ‘k’ up to which ‘inputT’ will be interpolated.
 310 |     Returns:
 311 |         extrapolation for high-k values (from ‘kcutmax’ to ‘kmax’) for a given linear power spectrum ‘ inputT’.
 312 |     '''
 313 |     cutrange = np.where(inputT[0]<= kcutmax)
 314 |     inputcutT = np.array([inputT[0][cutrange], inputT[1][cutrange]])
 315 |     listToExtT = inputcutT[0][-6:]
 316 |     tableToExtT = np.array([listToExtT, inputcutT[1][-6:]])
 317 |     delta = np.log10(listToExtT[2])-np.log10(listToExtT[1])
 318 |     lastk = np.log10(listToExtT[-1])
 319 |     
 320 |     logklist = [];
 321 |     while (lastk <= np.log10(kmax)):
 322 |         logklistT = lastk + delta;
 323 |         lastk = logklistT
 324 |         logklist.append(logklistT)
 325 |     logklist = np.array(logklist)
 326 |     
 327 |     sign = np.sign(tableToExtT[1][1])
 328 |     tableToExtT = np.log10(np.abs(tableToExtT))
 329 |     logextT = Extrapolate(tableToExtT, logklist)
 330 |     
 331 |     output = np.array([10**logextT[0], sign*10**logextT[1]])
 332 |     output = np.concatenate((inputcutT, output), axis=1)
 333 |         
 334 |     
 335 |     return output
 336 | 
 337 | 
 338 | 
 339 | 
 340 | def ExtrapolateLowkLogLog(inputT, kcutmin, kmin):
 341 |     '''Extrapolation for low-k values.
 342 |     
 343 |     Args:
 344 |         inputT: k-coordinates and linear power spectrum.
 345 |         kcutmin: value of ‘k’ from which ‘inputT’ will be interpolated.
 346 |         kmin: value of ‘k’ up to which ‘inputT’ will be interpolated.
 347 |     Returns:
 348 |         extrapolation for low-k values (from ‘kcutmin’ to ‘kmin’) for a given linear power spectrum ‘inputT’.
 349 |     '''
 350 |     cutrange = np.where(inputT[0] > kcutmin)
 351 |     inputcutT = np.array([inputT[0][cutrange], inputT[1][cutrange]])
 352 |     listToExtT = inputcutT[0][:5]
 353 |     tableToExtT = np.array([listToExtT, inputcutT[1][:5]])
 354 |     delta = np.log10(listToExtT[2])-np.log10(listToExtT[1])
 355 |     firstk = np.log10(listToExtT[0])
 356 |     
 357 |     logklist = [];
 358 |     while (firstk > np.log10(kmin)):
 359 |         logklistT = firstk - delta;
 360 |         firstk = logklistT
 361 |         logklist.append(logklistT)
 362 |     logklist = np.array(list(reversed(logklist)))
 363 |     
 364 |     sign = np.sign(tableToExtT[1][1])
 365 |     tableToExtT = np.log10(np.abs(tableToExtT))
 366 |     logextT = Extrapolate(tableToExtT, logklist)
 367 |     
 368 |     output = np.array([10**logextT[0], sign*10**logextT[1]])
 369 |     output = np.concatenate((output, inputcutT), axis=1)
 370 |         
 371 |     
 372 |     return output
 373 | 
 374 | 
 375 | 
 376 | 
 377 | def ExtrapolatekLogLog(inputT, kcutmin, kmin, kcutmax, kmax):
 378 |     '''Extrapolation at low-k and high-k.
 379 |     
 380 |     Args:
 381 |         inputT: k-coordinates and linear power spectrum.
 382 |         kcutmin, kcutmax: value of ‘k’ from which ‘inputT’ will be interpolated.
 383 |         kmin, kmax: value of ‘k’ up to which ‘inputT’ will be interpolated.
 384 |     Returns:
 385 |         combines extrapolation al low-k and high-k.
 386 |     '''
 387 |     output = ExtrapolateLowkLogLog(ExtrapolateHighkLogLog(inputT, kcutmax, kmax), kcutmin, kmin)
 388 |     
 389 |     return output
 390 | 
 391 | 
 392 | 
 393 | 
 394 | def Extrapolate_inputpkl(inputT):
 395 |     '''Extrapolation to the input linear power spectrum.
 396 |     
 397 |     Args:
 398 |         inputT: k-coordinates and linear power spectrum.
 399 |     Returns:
 400 |         extrapolates the input linear power spectrum ‘inputT’ to low-k or high-k if needed.
 401 |     '''
 402 |     kcutmin = min(inputT[0]); kmin = 10**(-5);
 403 |     kcutmax = max(inputT[0]); kmax = 200
 404 |     
 405 |     if ((kmin < kcutmin) or (kmax > kcutmax)):
 406 |         output = ExtrapolatekLogLog(inputT, kcutmin, kmin, kcutmax, kmax)
 407 |         
 408 |     else:
 409 |         output = inputT
 410 |         
 411 |     return output
 412 | 
 413 | 
 414 | 
 415 | 
 416 | def CosmoParam(h, ombh2, omch2, omnuh2):
 417 |     '''Gives some inputs for the function 'fOverf0EH'.
 418 |     
 419 |     Args:
 420 |         h = H0/100.
 421 |         ombh2: Omega_b h² (baryons)
 422 |         omch2: Omega_c h² (CDM)
 423 |         omnuh2: Omega_nu h² (massive neutrinos)
 424 |     Returns:
 425 |         h: H0/100.
 426 |         OmM0: Omega_b + Omega_c + Omega_nu (dimensionless matter density parameter)
 427 |         fnu: Omega_nu/OmM0
 428 |         Massnu: Total neutrino mass [eV]
 429 |     '''
 430 |                 
 431 |     Omb = ombh2/h**2;
 432 |     Omc = omch2/h**2;
 433 |     Omnu = omnuh2/h**2;
 434 |         
 435 |     OmM0 = Omb + Omc + Omnu; 
 436 |     fnu = Omnu/OmM0;
 437 |     Massnu = Omnu*93.14*h**2;
 438 |         
 439 |     return(h, OmM0, fnu, Massnu)
 440 | 
 441 | 
 442 | 
 443 | 
 444 | def fOverf0EH(zev, k, OmM0, h, fnu):
 445 |     '''Rutine to get f(k)/f0 and f0.
 446 |     f(k)/f0 is obtained following H&E (1998), arXiv:astro-ph/9710216
 447 |     f0 is obtained by solving directly the differential equation for the linear growth at large scales.
 448 |     
 449 |     Args:
 450 |         zev: redshift
 451 |         k: wave-number
 452 |         OmM0: Omega_b + Omega_c + Omega_nu (dimensionless matter density parameter)
 453 |         h = H0/100
 454 |         fnu: Omega_nu/OmM0
 455 |     Returns:
 456 |         f(k)/f0 (when 'EdSkernels = True' f(k)/f0 = 1)
 457 |         f0
 458 |     '''
 459 |     if (fnu > 0):
 460 |         eta = np.log(1/(1+zev))   #log of scale factor
 461 |         Neff = 3.046              #effective number of neutrinos
 462 |         omrv = 2.469*10**(-5)/(h**2 * (1 + 7/8*(4/11)**(4/3)*Neff)) #rad: including neutrinos
 463 |         aeq = omrv/OmM0           #matter-radiation equality
 464 |             
 465 |         pcb = 5/4 - np.sqrt(1 + 24*(1 - fnu))/4     #neutrino supression
 466 |         c = 0.7         
 467 |         Nnu = 3                                     #number of neutrinos
 468 |         theta272 = (1.00)**2                        # T_{CMB} = 2.7*(theta272)
 469 |         pf = (k * theta272)/(OmM0 * h**2)  
 470 |         DEdS = np.exp(eta)/aeq                      #growth function: EdS cosmology
 471 |             
 472 |         yFS = 17.2*fnu*(1 + 0.488*fnu**(-7/6))*(pf*Nnu/fnu)**2  #yFreeStreaming 
 473 |         rf = DEdS/(1 + yFS)
 474 |         fFit = 1 - pcb/(1 + (rf)**c)                #f(k)/f0
 475 |             
 476 |     else:
 477 |         fFit = np.full(len(k), 1.0)
 478 |             
 479 |         
 480 |     #Getting f0
 481 |     def OmM(eta):
 482 |         return 1/(1 + ((1-OmM0)/OmM0)* np.exp(3*eta) )
 483 |         
 484 |     def f1(eta):
 485 |         return 2 - 3/2 * OmM(eta)
 486 |         
 487 |     def f2(eta):
 488 |         return 3/2 * OmM(eta)
 489 |         
 490 |     etaini = -6;  #initial eta, early enough to evolve as EdS (D + \propto a)
 491 |     zfin = -0.99;
 492 |         
 493 |     def etaofz(z):
 494 |         return np.log(1/(1 + z))
 495 |         
 496 |     etafin = etaofz(zfin); 
 497 |         
 498 |     from scipy.integrate import odeint
 499 |         
 500 |     #Differential eqs.
 501 |     def Deqs(Df, eta):
 502 |         Df, Dprime = Df
 503 |         return [Dprime, f2(eta)*Df - f1(eta)*Dprime]
 504 |         
 505 |     #eta range and initial conditions
 506 |     eta = np.linspace(etaini, etafin, 1001)   
 507 |     Df0 = np.exp(etaini)
 508 |     Df_p0 = np.exp(etaini)
 509 |         
 510 |     #solution
 511 |     Dplus, Dplusp = odeint(Deqs, [Df0,Df_p0], eta).T
 512 |     
 513 |     Dplusp_ = interp(etaofz(zev), eta, Dplusp)
 514 |     Dplus_ = interp(etaofz(zev), eta, Dplus)
 515 |     f0 = Dplusp_/Dplus_ 
 516 |         
 517 |     return (k, fFit, f0)
 518 | 
 519 | 
 520 | 
 521 | 
 522 | def pknwJ(k, PSLk, h):
 523 |     '''Routine (based on J. Hamann et. al. 2010, arXiv:1003.3999) to get the non-wiggle piece of the linear power spectrum.    
 524 |     
 525 |     Args:
 526 |         k: wave-number.
 527 |         PSLk: linear power spectrum.
 528 |         h: H0/100.
 529 |     Returns:
 530 |         non-wiggle piece of the linear power spectrum.
 531 |     '''
 532 |     #ksmin(max): k-range and Nks: points
 533 |     ksmin = 7*10**(-5)/h; ksmax = 7/h; Nks = 2**16
 534 | 
 535 |     #sample ln(kP_L(k)) in Nks points, k range (equidistant)
 536 |     ksT = [ksmin + ii*(ksmax-ksmin)/(Nks-1) for ii in range(Nks)]
 537 |     PSL = interp(ksT, k, PSLk)
 538 |     logkpkT = np.log(ksT*PSL)
 539 |         
 540 |     #Discrete sine transf., check documentation
 541 |     FSTtype = 1; m = int(len(ksT)/2)
 542 |     FSTlogkpkT = dst(logkpkT, type = FSTtype, norm = "ortho")
 543 |     FSTlogkpkOddT = FSTlogkpkT[::2]
 544 |     FSTlogkpkEvenT = FSTlogkpkT[1::2]
 545 |         
 546 |     #cut range (remove the harmonics around BAO peak)
 547 |     mcutmin = 120; mcutmax = 240;
 548 |         
 549 |     #Even
 550 |     xEvenTcutmin = np.linspace(1, mcutmin-2, mcutmin-2)
 551 |     xEvenTcutmax = np.linspace(mcutmax+2, len(FSTlogkpkEvenT), len(FSTlogkpkEvenT)-mcutmax-1)
 552 |     EvenTcutmin = FSTlogkpkEvenT[0:mcutmin-2] 
 553 |     EvenTcutmax = FSTlogkpkEvenT[mcutmax+1:len(FSTlogkpkEvenT)]
 554 |     xEvenTcuttedT = np.concatenate((xEvenTcutmin, xEvenTcutmax))
 555 |     nFSTlogkpkEvenTcuttedT = np.concatenate((EvenTcutmin, EvenTcutmax))
 556 | 
 557 | 
 558 |     #Odd
 559 |     xOddTcutmin = np.linspace(1, mcutmin-1, mcutmin-1)
 560 |     xOddTcutmax = np.linspace(mcutmax+1, len(FSTlogkpkEvenT), len(FSTlogkpkEvenT)-mcutmax)
 561 |     OddTcutmin = FSTlogkpkOddT[0:mcutmin-1]
 562 |     OddTcutmax = FSTlogkpkOddT[mcutmax:len(FSTlogkpkEvenT)]
 563 |     xOddTcuttedT = np.concatenate((xOddTcutmin, xOddTcutmax))
 564 |     nFSTlogkpkOddTcuttedT = np.concatenate((OddTcutmin, OddTcutmax))
 565 | 
 566 |     #Interpolate the FST harmonics in the BAO range
 567 |     preT, = map(np.zeros,(len(FSTlogkpkT),))
 568 |     PreEvenT = interp(np.linspace(2, mcutmax, mcutmax-1), xEvenTcuttedT, nFSTlogkpkEvenTcuttedT)
 569 |     PreOddT = interp(np.linspace(0, mcutmax-2, mcutmax-1), xOddTcuttedT, nFSTlogkpkOddTcuttedT)
 570 |     for ii in range(m):
 571 |         if (mcutmin < ii+1 < mcutmax):
 572 |             preT[2*ii+1] = PreEvenT[ii]
 573 |             preT[2*ii] = PreOddT[ii]
 574 |         if (mcutmin >= ii+1 or mcutmax <= ii+1):
 575 |             preT[2*ii+1] = FSTlogkpkT[2*ii+1]
 576 |             preT[2*ii] = FSTlogkpkT[2*ii]
 577 |                 
 578 |         
 579 |     #Inverse Sine transf.
 580 |     FSTofFSTlogkpkNWT = idst(preT, type = FSTtype, norm = "ortho")
 581 |     PNWT = np.exp(FSTofFSTlogkpkNWT)/ksT
 582 | 
 583 |     PNWk = interp(k, ksT, PNWT)
 584 |     DeltaAppf = k*(PSL[7]-PNWT[7])/PNWT[7]/ksT[7]
 585 | 
 586 |     irange1 = np.where((k < 1e-3))
 587 |     PNWk1 = PSLk[irange1]/(DeltaAppf[irange1] + 1)
 588 | 
 589 |     irange2 = np.where((1e-3 <= k) & (k <= ksT[len(ksT)-1]))
 590 |     PNWk2 = PNWk[irange2]
 591 |         
 592 |     irange3 = np.where((k > ksT[len(ksT)-1]))
 593 |     PNWk3 = PSLk[irange3]
 594 |         
 595 |     PNWkTot = np.concatenate((PNWk1, PNWk2, PNWk3))
 596 |         
 597 |     return(k, PNWkTot)
 598 | 
 599 | 
 600 | 
 601 | 
 602 | def cmM(k_min, k_max, N, b_nu, inputpkT):
 603 |     '''coefficients c_m, see eq.~ 4.2 - 4.5 at arXiv:2208.02791
 604 |     
 605 |     Args:
 606 |         kmin, kmax: minimal and maximal range of the wave-number k.
 607 |         N: number of sampling points (we recommend using N=128).
 608 |         b_nu: FFTLog bias (use b_nu = -0.1. Not yet tested for other values).
 609 |         inputpkT: k-coordinates and linear power spectrum.
 610 |     Returns:
 611 |         coefficients c_m (cosmological dependent terms)
 612 |     '''
 613 |     #define de zero matrices
 614 |     M = int(N/2)
 615 |     kBins = np.zeros(N)
 616 |     c_m = np.zeros(N+1, dtype = complex)
 617 |         
 618 |     #"kBins" trought "delta" gives logspaced k's in [k_min,k_max] 
 619 |     for ii in range (N):
 620 |         delta = 1/(N-1) * np.log(k_max/k_min)
 621 |         kBins[ii] = k_min * np.exp((ii) * delta)
 622 |     f_kl = interp(kBins, inputpkT[0], inputpkT[1]) * (kBins/k_min)**(-b_nu)
 623 | 
 624 |     #F_m is the Discrete Fourier Transform (DFT) of f_kl
 625 |     #"forward" has the direct transforms scaled by 1/N (numpy version >= 1.20.0)
 626 |     F_m = np.fft.fft(f_kl, n = N, norm = "forward" )
 627 |         
 628 |     #etaT = bias_nu + i*eta_m
 629 |     #to get c_m: 1) reality condition, 2) W_m factor
 630 |     for ii in range(N+1):
 631 |         etaT = b_nu +  (2*np.pi*1j/np.log(k_max/k_min)) * (ii - N/2) *(N-1)/(N)
 632 |         if (ii - M < 0):
 633 |             c_m[ii] = k_min**(-(etaT))*np.conj(F_m[-ii+M])
 634 |         c_m[ii] = k_min**(-(etaT)) * F_m[ii-M]
 635 |     c_m[0] = c_m[0]/2
 636 |     c_m[int(N)] = c_m[int(N)]/2 
 637 |         
 638 |     return(c_m)
 639 | 
 640 | 
 641 | 
 642 | 
 643 | def NonLinear(inputpkl, CosmoParams, kminout=0.001, kmaxout=0.5, nk = 120, EdSkernels = False):
 644 |     '''1-loop corrections to the linear power spectrum.
 645 |     
 646 |     Args:
 647 |         If 'EdSkernels = True' (default: 'False', fk-kernels), EdS-kernels will be employed.        
 648 |         inputpkl: k-coordinates and linear power spectrum.
 649 |         CosmoParams: Set of cosmological parameters [z_pk, omega_b, omega_cdm, omega_ncdm, h] in that order.
 650 |                    z_pk: redshift.
 651 |                    omega_i = omega_i = Omega_i h², where i=baryons (b), CDM (cdm), massive neutrinos (ncdm).
 652 |                    h = H0/100. 
 653 |     Returns:
 654 |         list of 1-loop contributions for the wiggle and non-wiggle (also computed here) linear power spectra.
 655 |     '''
 656 |     global TableOut, TableOut_NW, f0, kTout, sigma2w, sigma2w_NW
 657 |     
 658 |     global z_pk, omega_b, omega_cdm, omega_ncdm, h
 659 | 
 660 |     #CosmoParams
 661 |     (z_pk, omega_b, omega_cdm, omega_ncdm, h) = CosmoParams
 662 |     
 663 |     k_min = 10**(-7); k_max = 100.
 664 |     b_nu = -0.1;   #Not yet tested for other values
 665 |     
 666 |           
 667 |     #Extrapolates the linear power spectrum if needed. 
 668 |     inputpkT = Extrapolate_inputpkl(inputpkl) 
 669 |     
 670 |     
 671 |     ################################ KTOUT ###########################################    
 672 |     
 673 |     #kminout = 0.001; kmaxout = 0.5;
 674 |     
 675 |     kTout = np.logspace(np.log10(kminout), np.log10(kmaxout), num=nk)
 676 | 
 677 |     
 678 |     ##################################################################################
 679 |     
 680 |     def P22type(kTout, inputpkT, inputpkTf, inputpkTff, M22matrices, k_min,
 681 |                 k_max, N, b_nu):
 682 |         
 683 |         (M22_dd, M22_dt_fp, M22_tt_fpfp, M22_tt_fkmpfp, MtAfp_11, MtAfkmpfp_12, 
 684 |          MtAfkmpfp_22, MtAfpfp_22, MtAfkmpfpfp_23, MtAfkmpfpfp_33, MB1_11, MC1_11, 
 685 |          MB2_11, MC2_11, MD2_21, MD3_21, MD2_22, MD3_22, MD4_22, MPb1b2, MPb1bs2, 
 686 |          MPb22, MPb2bs2, MPb2s2, MPb2t, MPbs2t) = M22matrices
 687 |         
 688 |         #matter coefficients 
 689 |         cmT = cmM(k_min, k_max, N, b_nu, inputpkT)
 690 |         cmTf = cmM(k_min, k_max, N, b_nu, inputpkTf)
 691 |         cmTff = cmM(k_min, k_max, N, b_nu, inputpkTff)
 692 |         
 693 |         #biased tracers coefficients
 694 |         bnu_b = 15.1*b_nu
 695 |         cmT_b = cmM(k_min, k_max, N, bnu_b, inputpkT)
 696 |         cmTf_b = cmM(k_min, k_max, N, bnu_b, inputpkTf)
 697 |         
 698 |         #creating the zeros of P22
 699 |         #Ploop
 700 |         P22dd, P22dt, P22tt = map(np.zeros,3*(len(kTout),))
 701 |         #Bias
 702 |         Pb1b2, Pb1bs2, Pb22, Pb2bs2, Pb2s2, Pb2t, Pbs2t = map(np.zeros,7*(len(kTout),))
 703 |         #A-TNS
 704 |         I1udd_1b, I2uud_1b, I3uuu_3b, I2uud_2b, I3uuu_2b = map(np.zeros,5*(len(kTout),))  
 705 |         #D-RSD
 706 |         I2uudd_1D, I2uudd_2D, I3uuud_2D, I3uuud_3D, I4uuuu_2D, I4uuuu_3D, I4uuuu_4D = map(np.zeros,7*(len(kTout),))
 707 |         
 708 |         #etaT = bias_nu + i*eta_m
 709 |         etamT = np.zeros(N+1, dtype = complex)
 710 |         etamT_b = np.zeros(N+1, dtype = complex)
 711 |         for jj in range(N+1):
 712 |             ietam = (2*np.pi*1j/np.log(k_max/k_min)) * (jj - N/2) *(N-1)/(N)
 713 |             etamT[jj] = b_nu + ietam 
 714 |             etamT_b[jj] = bnu_b + ietam
 715 |             
 716 |         for ii in range(len(kTout)):
 717 |             K = kTout[ii]
 718 |             precvec = K**(etamT) 
 719 |             vec = cmT * precvec
 720 |             vecf = cmTf * precvec
 721 |             vecff = cmTff * precvec
 722 |             
 723 |             precvec_b = K**(etamT_b)
 724 |             vec_b = cmT_b * precvec_b
 725 |             vecf_b = cmTf_b * precvec_b
 726 |             
 727 |             P22dd[ii] = (K**3 * vec @ M22_dd @ vec).real 
 728 |             P22dt[ii] = (2*K**3 * vecf @ M22_dt_fp @ vec).real 
 729 |             P22tt[ii] = K**3 *(vecff @ M22_tt_fpfp @ vec + vecf @ M22_tt_fkmpfp @ vecf).real
 730 |             
 731 |             Pb1b2[ii] = (K**3 * vec_b @ MPb1b2 @ vec_b).real 
 732 |             Pb1bs2[ii] = (K**3 * vec_b @ MPb1bs2 @ vec_b).real
 733 |             Pb22[ii] = (K**3 * vec_b @ MPb22 @ vec_b).real  
 734 |             Pb2bs2[ii] = (K**3 * vec_b @ MPb2bs2 @ vec_b).real
 735 |             Pb2s2[ii] = (K**3 * vec_b @ MPb2s2 @ vec_b).real
 736 |             Pb2t[ii] = (K**3 * vecf_b @ MPb2t @ vec_b).real
 737 |             Pbs2t[ii] = (K**3 * vecf_b @ MPbs2t @ vec_b).real
 738 |             
 739 |             I1udd_1b[ii] = (K**3 * vecf @ MtAfp_11 @ vec).real 
 740 |             I2uud_1b[ii] = (K**3 * vecf @ MtAfkmpfp_12 @ vecf).real
 741 |             I3uuu_3b[ii] = (K**3 * vecff @ MtAfkmpfpfp_33 @ vecf).real
 742 |             I2uud_2b[ii] = K**3 * (vecf @ MtAfkmpfp_22 @ vecf + vecff @ MtAfpfp_22 @ vec).real 
 743 |             I3uuu_2b[ii] = (K**3 * vecff @ MtAfkmpfpfp_23 @ vecf).real
 744 |             
 745 |             I2uudd_1D[ii] = K**3 * (vecf @ MB1_11 @ vecf + vec @ MC1_11 @ vecff).real
 746 |             I2uudd_2D[ii] = K**3 * (vecf @ MB2_11 @ vecf + vec @ MC2_11 @ vecff).real
 747 |             I3uuud_2D[ii] = (K**3 * vecf @ MD2_21 @ vecff).real 
 748 |             I3uuud_3D[ii] = (K**3 * vecf @ MD3_21 @ vecff).real 
 749 |             I4uuuu_2D[ii] = (K**3 * vecff @ MD2_22 @ vecff).real 
 750 |             I4uuuu_3D[ii] = (K**3 * vecff @ MD3_22 @ vecff).real 
 751 |             I4uuuu_4D[ii] = (K**3 * vecff @ MD4_22 @ vecff).real 
 752 |         
 753 |         return (P22dd, P22dt, P22tt, Pb1b2, Pb1bs2, Pb22, Pb2bs2, Pb2s2, 
 754 |             Pb2t, Pbs2t, I1udd_1b, I2uud_1b, I3uuu_3b, I2uud_2b, 
 755 |             I3uuu_2b, I2uudd_1D, I2uudd_2D, I3uuud_2D, I3uuud_3D,
 756 |             I4uuuu_2D, I4uuuu_3D, I4uuuu_4D) 
 757 |     
 758 |     def P13type(kTout, inputpkT, inputpkTf, inputpkTff, inputfkT, M13vectors, 
 759 |                 k_min, k_max, N, b_nu):
 760 |                 
 761 |         (M13_dd, M13_dt_fk, M13_tt_fk, Mafk_11, Mafp_11, Mafkfp_12, Mafpfp_12, 
 762 |          Mafkfkfp_33, Mafkfpfp_33, Msigma23) = M13vectors
 763 |         
 764 |         
 765 |         #condition for EdS-kernels or fk-kernels (default: fk-kernels)
 766 |         if EdSkernels == False:
 767 |             Fkoverf0 = interp(kTout, inputfkT[0], inputfkT[1])
 768 |         else:
 769 |             Fkoverf0 = np.full(len(kTout), 1.0)
 770 |             
 771 |         
 772 |         #matter coefficients 
 773 |         cmT = cmM(k_min, k_max, N, b_nu, inputpkT)
 774 |         cmTf = cmM(k_min, k_max, N, b_nu, inputpkTf)
 775 |         cmTff = cmM(k_min, k_max, N, b_nu, inputpkTff)
 776 |         
 777 |         #biased tracers coefficients
 778 |         bnu_b = 15.1*b_nu
 779 |         cmT_b = cmM(k_min, k_max, N, bnu_b, inputpkT)
 780 |         cmTf_b = cmM(k_min, k_max, N, bnu_b, inputpkTf)
 781 |         
 782 |         #creating the zeros of P13 
 783 |         #Ploop
 784 |         P13dd, P13dt, P13tt = map(np.zeros,3*(len(kTout),))
 785 |         #Bias
 786 |         sigma23 = np.zeros(len(kTout))
 787 |         #A-TNS
 788 |         I1udd_1a, I2uud_1a, I3uuu_3a = map(np.zeros,3*(len(kTout),))
 789 |         
 790 |         #etaT = bias_nu + i*eta_m
 791 |         etamT = np.zeros(N+1, dtype = complex)
 792 |         etamT_b = np.zeros(N+1, dtype = complex)
 793 |         for jj in range(N+1):
 794 |             ietam = (2*np.pi*1j/np.log(k_max/k_min)) * (jj - N/2) *(N-1)/(N)
 795 |             etamT[jj] = b_nu + ietam 
 796 |             etamT_b[jj] = bnu_b + ietam
 797 |         
 798 |         sigma2psi = 1/(6 * np.pi**2) * scipy.integrate.simps(inputpkT[1], inputpkT[0])
 799 |         sigma2v = 1/(6 * np.pi**2) * scipy.integrate.simps(inputpkTf[1], inputpkTf[0]) 
 800 |         sigma2w = 1/(6 * np.pi**2) * scipy.integrate.simps(inputpkTff[1], inputpkTff[0])
 801 |         
 802 |         for ii in range(len(kTout)):
 803 |             K = kTout[ii]
 804 |             precvec = K**(etamT) 
 805 |             vec = cmT * precvec
 806 |             vecf = cmTf * precvec 
 807 |             vecff = cmTff * precvec
 808 |             vecfM13dt_fk = vecf @ M13_dt_fk
 809 |             
 810 |             precvec_b = K**(etamT_b)
 811 |             vec_b = cmT_b * precvec_b
 812 |             vecf_b = cmTf_b * precvec_b
 813 |             
 814 |             P13dd[ii] = (K**3 * vec @ M13_dd).real - 61/105 * K**2 * sigma2psi
 815 |             P13dt[ii] = 0.5 *(K**3 * (Fkoverf0[ii] * vec @ M13_dt_fk + vecfM13dt_fk)).real - (23/21*sigma2psi * Fkoverf0[ii] + 2/21*sigma2v)* K**2  
 816 |             P13tt[ii] = (K**3 * Fkoverf0[ii] * (Fkoverf0[ii] * vec @ M13_tt_fk + vecfM13dt_fk ) ).real - (169/105*sigma2psi * Fkoverf0[ii] + 4/21 * sigma2v)* Fkoverf0[ii]* K**2 
 817 |             
 818 |             sigma23[ii] = (K**3 * vec_b @ Msigma23).real 
 819 |             
 820 |             I1udd_1a[ii] = K**3 * (Fkoverf0[ii] * vec @ Mafk_11 + vecf @ Mafp_11).real + (92/35*sigma2psi * Fkoverf0[ii] - 18/7*sigma2v)*K**2 
 821 |             I2uud_1a[ii] = K**3 * (Fkoverf0[ii] * vecf @ Mafkfp_12 + vecff @ Mafpfp_12).real - (38/35*Fkoverf0[ii] *sigma2v + 2/7*sigma2w)*K**2 
 822 |             I3uuu_3a[ii] = K**3 * Fkoverf0[ii] * (Fkoverf0[ii] * vecf @ Mafkfkfp_33 + vecff @ Mafkfpfp_33).real - (16/35*Fkoverf0[ii]*sigma2v + 6/7*sigma2w)*Fkoverf0[ii]*K**2 
 823 |         
 824 |         return (P13dd, P13dt, P13tt, sigma23, I1udd_1a, I2uud_1a, I3uuu_3a)
 825 |        
 826 |     
 827 |     #Evaluation: f(k)/f0 and linear power spectrums
 828 |     h, OmM0, fnu, Massnu = CosmoParam(h, omega_b, omega_cdm, omega_ncdm)    
 829 |     inputfkT = fOverf0EH(z_pk, inputpkT[0], OmM0, h, fnu)
 830 |     f0 = inputfkT[2]
 831 |     
 832 |     
 833 |     
 834 |     #condition for EdS-kernels or fk-kernels (default: fk-kernels)
 835 |     if EdSkernels == False:
 836 |         Fkoverf0 = interp(kTout, inputfkT[0], inputfkT[1])
 837 |     else:
 838 |         Fkoverf0 = np.full(len(kTout), 1.0)
 839 |         
 840 |         
 841 |     
 842 |     #Non-wiggle linear power spectrum
 843 |     inputpkT_NW = pknwJ(inputpkT[0], inputpkT[1], h)
 844 |     
 845 |     
 846 |     #condition for EdS-kernels or fk-kernels (default: fk-kernels)
 847 |     if EdSkernels == False:
 848 |         
 849 |         inputpkTf = (inputpkT[0], inputpkT[1]*inputfkT[1])
 850 |         inputpkTff = (inputpkT[0], inputpkT[1]*(inputfkT[1])**2)
 851 |         
 852 |         inputpkTf_NW = (inputpkT_NW[0], inputpkT_NW[1]*inputfkT[1])
 853 |         inputpkTff_NW = (inputpkT_NW[0], inputpkT_NW[1]*(inputfkT[1])**2)
 854 |     
 855 |     else:
 856 |         inputpkTf = (inputpkT[0], inputpkT[1])
 857 |         inputpkTff = (inputpkT[0], inputpkT[1])
 858 |         
 859 |         inputpkTf_NW = (inputpkT_NW[0], inputpkT_NW[1])
 860 |         inputpkTff_NW = (inputpkT_NW[0], inputpkT_NW[1])
 861 |           
 862 |         
 863 |         
 864 |     #P22type contributions 
 865 |     P22 = P22type(kTout, inputpkT, inputpkTf, inputpkTff, M22matrices, k_min, k_max, N, b_nu)
 866 |     P22_NW = P22type(kTout, inputpkT_NW, inputpkTf_NW, inputpkTff_NW, M22matrices, k_min, k_max, N, b_nu)
 867 |     
 868 |     #P13type contributions
 869 |     P13overpkl = P13type(kTout, inputpkT, inputpkTf, inputpkTff, inputfkT, M13vectors, k_min, k_max, N, b_nu)
 870 |     P13overpkl_NW = P13type(kTout, inputpkT_NW, inputpkTf_NW, inputpkTff_NW, inputfkT, M13vectors, k_min, k_max, N, b_nu)
 871 |     
 872 |     #Computations for Table
 873 |     pk_l = np.interp(kTout, inputpkT[0], inputpkT[1])
 874 |     pk_l_NW = np.interp(kTout,inputpkT_NW[0], inputpkT_NW[1])
 875 |         
 876 |     
 877 |     sigma2w = 1/(6 * np.pi**2) * scipy.integrate.simps(inputpkTff[1], inputpkTff[0])
 878 |     sigma2w_NW = 1/(6 * np.pi**2) * scipy.integrate.simps(inputpkTff_NW[1], inputpkTff_NW[0])
 879 | 
 880 |     Ploop_dd = P22[0] + P13overpkl[0]*pk_l
 881 |     Ploop_dt = P22[1] + P13overpkl[1]*pk_l
 882 |     Ploop_tt = P22[2] + P13overpkl[2]*pk_l
 883 |     
 884 |     Pb1b2 = P22[3];        
 885 |     Pb1bs2 = P22[4];          
 886 |     Pb22 = P22[5]-interp(10**(-10), kTout, P22[5]);    
 887 |     Pb2bs2 = P22[6]-interp(10**(-10), kTout, P22[6]);
 888 |     Pb2s2 = P22[7]-interp(10**(-10), kTout, P22[7]); 
 889 |     sigma23pkl = P13overpkl[3]*pk_l
 890 |     Pb2t = P22[8]; 
 891 |     Pbs2t = P22[9];
 892 |     
 893 |     I1udd_1 = P13overpkl[4]*pk_l + P22[10];
 894 |     I2uud_1 = P13overpkl[5]*pk_l + P22[11];
 895 |     I2uud_2 = (P13overpkl[6]*pk_l)/Fkoverf0 + Fkoverf0*P13overpkl[4]*pk_l + P22[13];
 896 |     I3uuu_2 = Fkoverf0*P13overpkl[5]*pk_l + P22[14];
 897 |     I3uuu_3 = P13overpkl[6]*pk_l + P22[12];
 898 |     
 899 |     I2uudd_1D = P22[15];   I2uudd_2D = P22[16];   I3uuud_2D = P22[17];
 900 |     I3uuud_3D = P22[18];   I4uuuu_2D = P22[19];   I4uuuu_3D = P22[20];
 901 |     I4uuuu_4D = P22[21];
 902 |     
 903 |     TableOut = (kTout, pk_l, Fkoverf0, Ploop_dd, Ploop_dt, Ploop_tt, 
 904 |                 Pb1b2, Pb1bs2, Pb22, Pb2bs2, Pb2s2, sigma23pkl, Pb2t, Pbs2t, 
 905 |                 I1udd_1, I2uud_1, I2uud_2, I3uuu_2, I3uuu_3, I2uudd_1D, 
 906 |                 I2uudd_2D, I3uuud_2D, I3uuud_3D, I4uuuu_2D, I4uuuu_3D, 
 907 |                 I4uuuu_4D, f0, sigma2w)
 908 |     
 909 |     
 910 |     ######################## Non- Wiggle ########################################
 911 |     
 912 |     Ploop_dd_NW = P22_NW[0] + P13overpkl_NW[0]*pk_l_NW;
 913 |     Ploop_dt_NW = P22_NW[1] + P13overpkl_NW[1]*pk_l_NW;
 914 |     Ploop_tt_NW = P22_NW[2] + P13overpkl_NW[2]*pk_l_NW;
 915 |     
 916 |     Pb1b2_NW = P22_NW[3];        
 917 |     Pb1bs2_NW = P22_NW[4];          
 918 |     Pb22_NW = P22_NW[5]-interp(10**(-10), kTout, P22_NW[5]);
 919 |     Pb2bs2_NW = P22_NW[6]-interp(10**(-10), kTout, P22_NW[6]); 
 920 |     Pb2s2_NW = P22_NW[7]-interp(10**(-10), kTout, P22_NW[7]);
 921 |     sigma23pkl_NW = P13overpkl_NW[3]*pk_l_NW;
 922 |     Pb2t_NW = P22_NW[8];        
 923 |     Pbs2t_NW = P22_NW[9];         
 924 |     
 925 |     I1udd_1_NW = P13overpkl_NW[4]*pk_l_NW + P22_NW[10];
 926 |     I2uud_1_NW = P13overpkl_NW[5]*pk_l_NW + P22_NW[11];
 927 |     I2uud_2_NW = (P13overpkl_NW[6]*pk_l_NW)/Fkoverf0 + Fkoverf0*P13overpkl_NW[4]*pk_l_NW + P22_NW[13];
 928 |     I3uuu_2_NW = Fkoverf0*P13overpkl_NW[5]*pk_l_NW + P22_NW[14];
 929 |     I3uuu_3_NW = P13overpkl_NW[6]*pk_l_NW + P22_NW[12];
 930 |     
 931 |     I2uudd_1D_NW = P22_NW[15];   I2uudd_2D_NW = P22_NW[16];   I3uuud_2D_NW = P22_NW[17];
 932 |     I3uuud_3D_NW = P22_NW[18];   I4uuuu_2D_NW = P22_NW[19];   I4uuuu_3D_NW = P22_NW[20];
 933 |     I4uuuu_4D_NW = P22_NW[21];
 934 |     
 935 |     
 936 |     TableOut_NW = (kTout, pk_l_NW, Fkoverf0, Ploop_dd_NW, Ploop_dt_NW, Ploop_tt_NW, 
 937 |                    Pb1b2_NW, Pb1bs2_NW, Pb22_NW, Pb2bs2_NW, Pb2s2_NW, sigma23pkl_NW, 
 938 |                    Pb2t_NW, Pbs2t_NW, I1udd_1_NW, I2uud_1_NW, I2uud_2_NW, I3uuu_2_NW, 
 939 |                    I3uuu_3_NW, I2uudd_1D_NW, I2uudd_2D_NW, I3uuud_2D_NW, I3uuud_3D_NW, 
 940 |                    I4uuuu_2D_NW, I4uuuu_3D_NW, I4uuuu_4D_NW, f0, sigma2w_NW)
 941 |     
 942 |     return (TableOut, TableOut_NW)
 943 | 
 944 | 
 945 | 
 946 | 
 947 | def TableOut_interp(k):
 948 |     '''Interpolation of non-linear terms given by the wiggle power spectra.
 949 |     
 950 |     Args:
 951 |         k: wave-number.
 952 |     Returns:
 953 |         Interpolates the non-linear terms given by the wiggle power spectra.
 954 |     '''
 955 |     nobjects = 25
 956 |     Tableout = np.zeros((nobjects + 1, len(k)))
 957 |     for ii in range(nobjects):
 958 |         Tableout[ii][:] = interp(k, kTout, TableOut[1+ii])
 959 |         Tableout[25][:] = sigma2w
 960 |     return Tableout
 961 | 
 962 | 
 963 | 
 964 | 
 965 | def TableOut_NW_interp(k):
 966 |     '''Interpolation of non-linear terms given by the non-wiggle power spectra.
 967 |     
 968 |     Args:
 969 |         k: wave-number.
 970 |     Returns:
 971 |         Interpolates the non-linear terms given by the non-wiggle power spectra.
 972 |     '''
 973 |     nobjects = 25
 974 |     Tableout_NW = np.zeros((nobjects + 1, len(k)))
 975 |     for ii in range(nobjects):
 976 |         Tableout_NW[ii][:] = interp(k, kTout, TableOut_NW[1+ii])
 977 |         Tableout_NW[25][:] = sigma2w_NW
 978 |     return Tableout_NW
 979 | 
 980 | 
 981 | 
 982 | 
 983 | def PEFTs(kev, mu, NuisanParams, Table):
 984 |     '''EFT galaxy power spectrum, Eq. ~ 3.40 at arXiv: 2208.02791.
 985 |     
 986 |     Args: 
 987 |         kev: evaluation points (wave-number coordinates).
 988 |         mu: cosine angle between the wave-vector ‘\vec{k}’ and the line-of-sight direction ‘\hat{n}’.
 989 |         NuisamParams: set of nuisance parameters [b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, ctilde, 
 990 |                                                   alphashot0, alphashot2, PshotP] in that order.
 991 |                     b1, b2, bs2, b3nl: biasing parameters.
 992 |                     alpha0, alpha2, alpha4: EFT parameters.
 993 |                     ctilde: parameter for NL0 ∝ Kaiser power spectrum.
 994 |                     alphashot0, alphashot2, PshotP: stochastic noise parameters.
 995 |        Table: List of non-linear terms given by the wiggle or non-wiggle power spectra.
 996 |     Returns:
 997 |        EFT galaxy power spectrum in redshift space.
 998 |     '''
 999 |     
1000 |     #NuisanParams
1001 |     (b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, 
1002 |                 ctilde, alphashot0, alphashot2, PshotP) = NuisanParams
1003 |     
1004 |     #Table
1005 |     (pkl, Fkoverf0, Ploop_dd, Ploop_dt, Ploop_tt, Pb1b2, Pb1bs2, Pb22, Pb2bs2, 
1006 |          Pb2s2, sigma23pkl, Pb2t, Pbs2t, I1udd_1, I2uud_1, I2uud_2, I3uuu_2, I3uuu_3, 
1007 |          I2uudd_1D, I2uudd_2D, I3uuud_2D, I3uuud_3D, I4uuuu_2D, I4uuuu_3D, I4uuuu_4D, sigma2w) = Table
1008 |     
1009 |     fk = Fkoverf0*f0
1010 |         
1011 |     #linear power spectrum
1012 |     Pdt_L = pkl*Fkoverf0; Ptt_L = pkl*Fkoverf0**2;
1013 |         
1014 |     #one-loop power spectrum 
1015 |     Pdd = pkl + Ploop_dd; Pdt = Pdt_L + Ploop_dt; Ptt = Ptt_L + Ploop_tt;
1016 |         
1017 |         
1018 |     #biasing
1019 |     def PddXloop(b1, b2, bs2, b3nl):
1020 |         return (b1**2 * Ploop_dd + 2*b1*b2*Pb1b2 + 2*b1*bs2*Pb1bs2 + b2**2 * Pb22
1021 |                    + 2*b2*bs2*Pb2bs2 + bs2**2 *Pb2s2 + 2*b1*b3nl*sigma23pkl)
1022 |         
1023 |     def PdtXloop(b1, b2, bs2, b3nl):
1024 |         return b1*Ploop_dt + b2*Pb2t + bs2*Pbs2t + b3nl*Fkoverf0*sigma23pkl
1025 |         
1026 |     def PttXloop(b1, b2, bs2, b3nl):
1027 |         return Ploop_tt
1028 |         
1029 |     #RSD functions       
1030 |     def Af(mu, f0):
1031 |         return (f0*mu**2 * I1udd_1 + f0**2 * (mu**2 * I2uud_1 + mu**4 * I2uud_2)
1032 |                     + f0**3 * (mu**4 * I3uuu_2 +  mu**6 * I3uuu_3)) 
1033 |         
1034 |     def Df(mu, f0):
1035 |         return (f0**2 * (mu**2 * I2uudd_1D + mu**4 * I2uudd_2D) 
1036 |                     + f0**3 * (mu**4 * I3uuud_2D + mu**6 * I3uuud_3D)
1037 |                     + f0**4 * (mu**4 * I4uuuu_2D + mu**6 * I4uuuu_3D + mu**8 * I4uuuu_4D))
1038 |         
1039 |         
1040 |     #Introducing bias in RSD functions, eq.~ A.32 & A.33 at arXiv: 2208.02791
1041 |     def ATNS(mu, b1):
1042 |         return b1**3 * Af(mu, f0/b1)
1043 |         
1044 |     def DRSD(mu, b1):
1045 |         return b1**4 * Df(mu, f0/b1)
1046 |         
1047 |     def GTNS(mu, b1):
1048 |         return -((kev*mu*f0)**2 *sigma2w*(b1**2 * pkl + 2*b1*f0*mu**2 * Pdt_L 
1049 |                                    + f0**2 * mu**4 * Ptt_L))
1050 |         
1051 |         
1052 |     #One-loop SPT power spectrum in redshift space
1053 |     def PloopSPTs(mu, b1, b2, bs2, b3nl):
1054 |         return (PddXloop(b1, b2, bs2, b3nl) + 2*f0*mu**2 * PdtXloop(b1, b2, bs2, b3nl)
1055 |                     + mu**4 * f0**2 * PttXloop(b1, b2, bs2, b3nl) + ATNS(mu, b1) + DRSD(mu, b1)
1056 |                     + GTNS(mu, b1))
1057 |         
1058 |         
1059 |     #Linear Kaiser power spectrum
1060 |     def PKaiserLs(mu, b1):
1061 |         return (b1 + mu**2 * fk)**2 * pkl
1062 |         
1063 |     def PctNLOs(mu, b1, ctilde):
1064 |         return ctilde*(mu*kev*f0)**4 * sigma2w**2 * PKaiserLs(mu, b1)
1065 |     
1066 |     # EFT counterterms
1067 |     def Pcts(mu, alpha0, alpha2, alpha4):
1068 |         return (alpha0 + alpha2 * mu**2 + alpha4 * mu**4)*kev**2 * pkl
1069 |     
1070 |     #Stochastics noise
1071 |     def Pshot(mu, alphashot0, alphashot2, PshotP):
1072 |         return PshotP*(alphashot0 + alphashot2 * (kev*mu)**2)
1073 |         
1074 |     return (PloopSPTs(mu, b1, b2, bs2, b3nl) + Pcts(mu, alpha0, alpha2, alpha4)
1075 |                 + PctNLOs(mu, b1, ctilde) + Pshot(mu, alphashot0, alphashot2, PshotP))
1076 | 
1077 | 
1078 | 
1079 | 
1080 | def Sigma2Total(kev, mu, Table_NW):
1081 |     '''Sigma² tot for IR-resummations, see eq.~ 3.59 at arXiv:2208.02791
1082 |     
1083 |     Args:
1084 |         kev: evaluation points (wave-number coordinates). 
1085 |         mu: cosine angle between the wave-vector ‘\vec{k}’ and the line-of-sight direction ‘\hat{n}’.
1086 |         Table_NW: List of non-linear terms given by the non-wiggle power spectra.
1087 |     Returns:
1088 |         Sigma² tot for IR-resummations.
1089 |     '''
1090 |     kT = kev; pkl_NW = Table_NW[0];
1091 |         
1092 |     kinit = 10**(-6);  kS = 0.4;                                  #integration limits
1093 |     pT = np.logspace(np.log10(kinit),np.log10(kS), num = 10**2)   #integration range
1094 |         
1095 |     PSL_NW = interp(pT, kT, pkl_NW)
1096 |     k_BAO = 1/104                                                 #BAO scale
1097 |         
1098 |     Sigma2 = 1/(6 * np.pi**2)*scipy.integrate.simps(PSL_NW*(1 - spherical_jn(0, pT/k_BAO) 
1099 |                                                 + 2*spherical_jn(2, pT/k_BAO)), pT)
1100 |         
1101 |     deltaSigma2 = 1/(2 * np.pi**2)*scipy.integrate.simps(PSL_NW*spherical_jn(2, pT/k_BAO), pT)
1102 |         
1103 |     def Sigma2T(mu):
1104 |         return (1 + f0*mu**2 *(2 + f0))*Sigma2 + (f0*mu)**2 * (mu**2 - 1)* deltaSigma2
1105 |         
1106 |     return Sigma2T(mu)
1107 | 
1108 | 
1109 | 
1110 | 
1111 | def k_AP(k_obs, mu_obs, qperp, qpar):
1112 |     '''True ‘k’ coordinates.
1113 |     
1114 |     Args: where ‘_obs’ denote quantities that are observed assuming the reference (fiducial) cosmology.
1115 |         k_obs: observed wave-number.
1116 |         mu_obs: observed cosine angle between the wave-vector ‘\vec{k}’ and the line-of-sight direction ‘\hat{n}’.
1117 |         qperp, qpar: AP parameters.
1118 |     Returns:
1119 |         True wave-number ‘k_AP’.
1120 |     '''
1121 |     F = qpar/qperp
1122 |     return (k_obs/qperp)*(1 + mu_obs**2 * (1./F**2 - 1))**(0.5)
1123 | 
1124 | 
1125 | 
1126 | 
1127 | def mu_AP(mu_obs, qperp, qpar):
1128 |     '''True ‘mu’ coordinates.
1129 |     
1130 |     Args: where ‘_obs’ denote quantities that are observed assuming the reference (fiducial) cosmology.
1131 |         mu_obs: observed cosine angle between the wave-vector ‘\vec{k}’ and the line-of-sight direction ‘\hat{n}’.
1132 |         qperp, qpar: AP parameters.
1133 |     Returns:
1134 |         True ‘mu_AP’.
1135 |     '''
1136 |     F = qpar/qperp
1137 |     return (mu_obs/F) * (1 + mu_obs**2 * (1/F**2 - 1))**(-0.5)
1138 | 
1139 | 
1140 | 
1141 | 
1142 | def Hubble(Om, z_ev):
1143 |     '''Hubble parameter.
1144 |     
1145 |     Args:
1146 |         Om: Omega_b + Omega_c + Omega_nu (dimensionless matter density parameter).
1147 |         z_ev: redshift of evaluation.
1148 |     Returns:
1149 |         Hubble parameter.
1150 |     '''
1151 |     return ((Om) * (1 + z_ev)**3. + (1 - Om))**0.5
1152 | 
1153 | 
1154 | 
1155 | 
1156 | def DA(Om, z_ev):
1157 |     '''Angular-diameter distance.
1158 |     
1159 |      Args:
1160 |         Om: Omega_b + Omega_c + Omega_nu (dimensionless matter density parameter).
1161 |         z_ev: redshift of evaluation.
1162 |     Returns:
1163 |         Angular diameter distance.
1164 |     '''
1165 |     r = quad(lambda x: 1. / Hubble(Om, x), 0, z_ev)[0]
1166 |     return r / (1 + z_ev)
1167 | 
1168 | 
1169 | 
1170 | 
1171 | def Table_interp(k, kev, Table):
1172 |     '''Cubic interpolator.
1173 |     
1174 |     Args:
1175 |         k: coordinates at which to evaluate the interpolated values.
1176 |         kev: x-coordinates of the data points.
1177 |         Table: list of 1-loop contributions for the wiggle and non-wiggle
1178 |     '''
1179 |     f = interpolate.interp1d(kev, Table, kind = 'cubic', fill_value = "extrapolate")
1180 |     Tableout = f(k) 
1181 | 
1182 |     return Tableout
1183 | 
1184 | 
1185 | 
1186 | 
1187 | def RSDmultipoles(kev, NuisanParams, Omfid = -1, AP = False):
1188 |     '''Redshift space power spectrum multipoles.
1189 |     
1190 |     Args:
1191 |         If 'AP=True' (default: 'False') the code perform the AP test.
1192 |         If 'AP=True'. Include the fiducial Omfid after ‘NuisanParams’.
1193 |         
1194 |         kev: wave-number coordinates of evaluation.
1195 |         NuisamParams: set of nuisance parameters [b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, ctilde, alphashot0, 
1196 |                                                   alphashot2, PshotP] in that order.
1197 |                    b1, b2, bs2, b3nl: biasing parameters.
1198 |                    alpha0, alpha2, alpha4: EFT parameters.
1199 |                    ctilde: parameter for NL0 ∝ Kaiser power spectrum.
1200 |                    alphashot0, alphashot2, PshotP: stochastic noise parameters.
1201 |     Returns:
1202 |        Redshift space power spectrum multipoles (monopole, quadrupole and hexadecapole) at 'kev'.
1203 |     '''
1204 |             
1205 |     #NuisanParams
1206 |     (b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, 
1207 |                 ctilde, alphashot0, alphashot2, PshotP) = NuisanParams
1208 |     
1209 |     if AP == True and Omfid == -1:
1210 |         sys.exit("Introduce the fiducial value of the dimensionless matter density parameter as ‘Omfid = value’.")
1211 |      
1212 |     if AP == True and Omfid > 0:
1213 |                 
1214 |         #Om computed for any cosmology
1215 |         OmM = CosmoParam(h, omega_b, omega_cdm, omega_ncdm)[1]
1216 |         
1217 |         #qperp, qpar: AP parameters.
1218 |         qperp = DA(OmM, z_pk)/DA(Omfid, z_pk) 
1219 |         qpar = Hubble(Omfid, z_pk)/Hubble(OmM, z_pk) 
1220 |         
1221 |         
1222 |     def PIRs(kev, mu, Table, Table_NW):
1223 |         
1224 |         if AP == True:
1225 |             
1226 |             k_true = k_AP(kev, mu, qperp, qpar)
1227 |             mu_true = mu_AP(mu, qperp, qpar)
1228 |             
1229 |             Table_true = Table_interp(k_true, kev, Table)
1230 |             Table_NW_true = Table_interp(k_true, kev, Table_NW)
1231 |             
1232 |             Sigma2T = Sigma2Total(k_true, mu_true, Table_NW_true)
1233 |             
1234 |             Fkoverf0 = Table_true[1]; fk = Fkoverf0*f0
1235 |             pkl = Table_true[0]; pkl_NW = Table_NW_true[0];
1236 |             
1237 |             
1238 |             return ((b1 + fk * mu_true**2)**2 * (pkl_NW + np.exp(-k_true**2 * Sigma2T)*(pkl-pkl_NW)*(1 + k_true**2 * Sigma2T) )
1239 |                 + np.exp(-k_true**2 * Sigma2T)*PEFTs(k_true, mu_true, NuisanParams, Table_true) 
1240 |                 + (1 - np.exp(-k_true**2 * Sigma2T))*PEFTs(k_true, mu_true, NuisanParams, Table_NW_true)) 
1241 |             
1242 |         else:
1243 |             
1244 |             k = kev; Fkoverf0 = Table[1]; fk = Fkoverf0*f0
1245 |             pkl = Table[0]; pkl_NW = Table_NW[0];
1246 |             Sigma2T = Sigma2Total(kev, mu, Table_NW)
1247 |             
1248 |             return ((b1 + fk * mu**2)**2 * (pkl_NW + np.exp(-k**2 * Sigma2T)*(pkl-pkl_NW)*(1 + k**2 * Sigma2T) )
1249 |                 + np.exp(-k**2 * Sigma2T)*PEFTs(k, mu, NuisanParams, Table) 
1250 |                 + (1 - np.exp(-k**2 * Sigma2T))*PEFTs(k, mu, NuisanParams, Table_NW))     
1251 |     
1252 |     
1253 |     if AP == True:
1254 |         
1255 |         Nx = 8                                         #Points
1256 |         xGL, wGL = scipy.special.roots_legendre(Nx)    #x=cosθ and weights
1257 |         
1258 |         def ModelPkl0(Table, Table_NW):
1259 |             monop = 0;
1260 |             for ii in range(Nx):
1261 |                 monop = monop + 0.5/(qperp**2 * qpar)*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)
1262 |             return monop
1263 |         
1264 |         def ModelPkl2(Table, Table_NW):    
1265 |             quadrup = 0;
1266 |             for ii in range(Nx):
1267 |                 quadrup = quadrup + 5/(2*qperp**2 * qpar)*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)*eval_legendre(2, xGL[ii])
1268 |             return quadrup
1269 |         
1270 |         def ModelPkl4(Table, Table_NW):
1271 |             hexadecap = 0;
1272 |             for ii in range(Nx):
1273 |                 hexadecap = hexadecap + 9/(2*qperp**2 * qpar)*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)*eval_legendre(4, xGL[ii])
1274 |             return hexadecap
1275 |     
1276 |     else:
1277 |         
1278 |         Nx = 8                                         #Points
1279 |         xGL, wGL = scipy.special.roots_legendre(Nx)    #x=cosθ and weights
1280 |         
1281 |         def ModelPkl0(Table, Table_NW):
1282 |             monop = 0;
1283 |             for ii in range(Nx):
1284 |                 monop = monop + 0.5*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)
1285 |             return monop
1286 |         
1287 |         def ModelPkl2(Table, Table_NW):    
1288 |             quadrup = 0;
1289 |             for ii in range(Nx):
1290 |                 quadrup = quadrup + 5/2*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)*eval_legendre(2, xGL[ii])
1291 |             return quadrup
1292 |         
1293 |         def ModelPkl4(Table, Table_NW):
1294 |             hexadecap = 0;
1295 |             for ii in range(Nx):
1296 |                 hexadecap = hexadecap + 9/2*wGL[ii]*PIRs(kev, xGL[ii], Table, Table_NW)*eval_legendre(4, xGL[ii])
1297 |             return hexadecap
1298 |         
1299 |     
1300 |     Pkl0 = ModelPkl0(TableOut_interp(kev), TableOut_NW_interp(kev));
1301 |     Pkl2 = ModelPkl2(TableOut_interp(kev), TableOut_NW_interp(kev));
1302 |     Pkl4 = ModelPkl4(TableOut_interp(kev), TableOut_NW_interp(kev));
1303 |     
1304 |     #print('Redshift space power spectrum multipoles have been computed')
1305 |     #print('')
1306 |     #print('All computations have been performed successfully ')
1307 |     
1308 |     return (kev, Pkl0, Pkl2, Pkl4)
1309 | 
1310 | 
1311 | 
1312 | 
1313 | def RSDmultipoles_marginalized_const(kev, NuisanParams, Omfid = -1, AP = False, Hexa = False):
1314 |     '''Redshift space power spectrum multipoles 'const': Pℓ,const 
1315 |       (α->0, marginalizing over the EFT and stochastic parameters).
1316 |     
1317 |     Args:
1318 |         If 'AP=True' (default: 'False') the code perform the AP test.
1319 |         If 'AP=True'. Include the fiducial Omfid after ‘NuisanParams’.
1320 |         If 'Hexa = True' (default: 'False') the code includes the hexadecapole.
1321 |         
1322 |         kev: wave-number coordinates of evaluation.
1323 |         NuisamParams: set of nuisance parameters [b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, ctilde, alphashot0, 
1324 |                                                   alphashot2, PshotP] in that order.
1325 |                    b1, b2, bs2, b3nl: biasing parameters.
1326 |                    alpha0, alpha2, alpha4: EFT parameters.
1327 |                    ctilde: parameter for NL0 ∝ Kaiser power spectrum.
1328 |                    alphashot0, alphashot2, PshotP: stochastic noise parameters.
1329 |     Returns:
1330 |        Redshift space power spectrum multipoles (monopole, quadrupole and hexadecapole) at 'kev'.
1331 |     '''
1332 |     
1333 |     #NuisanParams
1334 |     (b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, 
1335 |                 ctilde, alphashot0, alphashot2, PshotP) = NuisanParams
1336 |     
1337 |     if AP == True and Omfid == -1:
1338 |         sys.exit("Introduce the fiducial value of the dimensionless matter density parameter as ‘Omfid = value’.")
1339 |      
1340 |     if AP == True and Omfid > 0:
1341 |                 
1342 |         #Om computed for any cosmology
1343 |         OmM = CosmoParam(h, omega_b, omega_cdm, omega_ncdm)[1]
1344 |         
1345 |         #qperp, qpar: AP parameters.
1346 |         qperp = DA(OmM, z_pk)/DA(Omfid, z_pk) 
1347 |         qpar = Hubble(Omfid, z_pk)/Hubble(OmM, z_pk) 
1348 |         
1349 |         
1350 |     def PIRs_const(kev, mu, Table, Table_NW):
1351 |         
1352 |         #NuisanParams_const: α->0 (set to zero EFT and stochastic parameters)
1353 |         alpha0, alpha2, alpha4, alphashot0, alphashot2 = np.zeros(5)
1354 |         
1355 |         NuisanParams_const = (b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, 
1356 |                               ctilde, alphashot0, alphashot2, PshotP)
1357 |         
1358 |         
1359 |         if AP == True:
1360 |             
1361 |             k_true = k_AP(kev, mu, qperp, qpar)
1362 |             mu_true = mu_AP(mu, qperp, qpar)
1363 |             
1364 |             Table_true = Table_interp(k_true, kev, Table)
1365 |             Table_NW_true = Table_interp(k_true, kev, Table_NW)
1366 |             
1367 |             Sigma2T = Sigma2Total(k_true, mu_true, Table_NW_true)
1368 |             
1369 |             Fkoverf0 = Table_true[1]; fk = Fkoverf0*f0
1370 |             pkl = Table_true[0]; pkl_NW = Table_NW_true[0];
1371 |             
1372 |             
1373 |             return ((b1 + fk * mu_true**2)**2 * (pkl_NW + np.exp(-k_true**2 * Sigma2T)*(pkl-pkl_NW)*(1 + k_true**2 * Sigma2T) )
1374 |                 + np.exp(-k_true**2 * Sigma2T)*PEFTs(k_true, mu_true, NuisanParams_const, Table_true) 
1375 |                 + (1 - np.exp(-k_true**2 * Sigma2T))*PEFTs(k_true, mu_true, NuisanParams_const, Table_NW_true))
1376 |         
1377 |         
1378 |         else:
1379 |             
1380 |             k = kev; Fkoverf0 = Table[1]; fk = Fkoverf0*f0
1381 |             pkl = Table[0]; pkl_NW = Table_NW[0];
1382 |             Sigma2T = Sigma2Total(kev, mu, Table_NW)
1383 |             
1384 |             return ((b1 + fk * mu**2)**2 * (pkl_NW + np.exp(-k**2 * Sigma2T)*(pkl-pkl_NW)*(1 + k**2 * Sigma2T) )
1385 |                 + np.exp(-k**2 * Sigma2T)*PEFTs(k, mu, NuisanParams_const, Table) 
1386 |                 + (1 - np.exp(-k**2 * Sigma2T))*PEFTs(k, mu, NuisanParams_const, Table_NW))
1387 |         
1388 |         
1389 |     Nx = 8                                         #Points
1390 |     xGL, wGL = scipy.special.roots_legendre(Nx)    #x=cosθ and weights
1391 |     
1392 |     def ModelPkl0_const(Table, Table_NW):
1393 |         if AP == True:
1394 |             monop = 1/(qperp**2 * qpar) * sum(0.5*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW) for ii in range(Nx))
1395 |             return monop
1396 |         else:
1397 |             monop = sum(0.5*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW) for ii in range(Nx))
1398 |             return monop
1399 |         
1400 |         
1401 |     def ModelPkl2_const(Table, Table_NW):    
1402 |         if AP == True:
1403 |             quadrup = 1/(qperp**2 * qpar) * sum(5/2*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW)*eval_legendre(2, xGL[ii]) for ii in range(Nx))
1404 |             return quadrup
1405 |         else:
1406 |             quadrup = sum(5/2*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW)*eval_legendre(2, xGL[ii]) for ii in range(Nx))
1407 |             return quadrup
1408 |         
1409 |         
1410 |     def ModelPkl4_const(Table, Table_NW):
1411 |         if AP == True:
1412 |             hexadecap = 1/(qperp**2 * qpar) * sum(9/2*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW)*eval_legendre(4, xGL[ii]) for ii in range(Nx))
1413 |             return hexadecap
1414 |         else:
1415 |             hexadecap = sum(9/2*wGL[ii]*PIRs_const(kev, xGL[ii], Table, Table_NW)*eval_legendre(4, xGL[ii]) for ii in range(Nx))
1416 |             return hexadecap
1417 |         
1418 |         
1419 |     if Hexa == False:
1420 |         Pkl0_const = ModelPkl0_const(TableOut_interp(kev), TableOut_NW_interp(kev));
1421 |         Pkl2_const = ModelPkl2_const(TableOut_interp(kev), TableOut_NW_interp(kev));
1422 |         return (Pkl0_const, Pkl2_const)
1423 |     
1424 |     else:
1425 |         Pkl0_const = ModelPkl0_const(TableOut_interp(kev), TableOut_NW_interp(kev));
1426 |         Pkl2_const = ModelPkl2_const(TableOut_interp(kev), TableOut_NW_interp(kev));
1427 |         Pkl4_const = ModelPkl4_const(TableOut_interp(kev), TableOut_NW_interp(kev));
1428 |         return (Pkl0_const, Pkl2_const, Pkl4_const)        
1429 |     
1430 |     
1431 |     
1432 | 
1433 | def PEFTs_derivatives(k, mu, pkl, PshotP):
1434 |     '''Derivatives of PEFTs with respect to the EFT and stochastic parameters.
1435 |     
1436 |     Args:
1437 |         k: wave-number coordinates of evaluation.
1438 |         mu: cosine angle between the wave-vector ‘\vec{k}’ and the line-of-sight direction ‘\hat{n}’.
1439 |         pkl: linear power spectrum.
1440 |         PshotP: stochastic nuisance parameter.
1441 |     Returns:
1442 |         ∂P_EFTs/∂α_i with: α_i = {alpha0, alpha2, alpha4, alphashot0, alphashot2}
1443 |     '''
1444 |     
1445 |     k2 = k**2
1446 |     k2mu2 = k2 * mu**2
1447 |     k2mu4 = k2mu2 * mu**2
1448 | 
1449 |     PEFTs_alpha0 = k2 * pkl
1450 |     PEFTs_alpha2 = k2mu2 * pkl 
1451 |     PEFTs_alpha4 = k2mu4 * pkl 
1452 |     PEFTs_alphashot0 = PshotP
1453 |     PEFTs_alphashot2 = k2mu2 * PshotP
1454 |     
1455 |     return (PEFTs_alpha0, PEFTs_alpha2, PEFTs_alpha4, PEFTs_alphashot0, PEFTs_alphashot2)  
1456 | 
1457 | 
1458 | 
1459 | 
1460 | def RSDmultipoles_marginalized_derivatives(kev, NuisanParams, Omfid = -1, AP = False, Hexa = False):
1461 |     '''Redshift space power spectrum multipoles 'derivatives': Pℓ,i=∂Pℓ/∂α_i 
1462 |       (derivatives with respect to the EFT and stochastic parameters).
1463 |     
1464 |     Args:
1465 |         If 'AP=True' (default: 'False') the code perform the AP test.
1466 |         If 'AP=True'. Include the fiducial Omfid after ‘NuisanParams’.
1467 |         If 'Hexa = True' (default: 'False') the code includes the hexadecapole.
1468 |         
1469 |         kev: wave-number coordinates of evaluation.
1470 |         NuisamParams: set of nuisance parameters [b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, ctilde, alphashot0, 
1471 |                                                   alphashot2, PshotP] in that order.
1472 |                    b1, b2, bs2, b3nl: biasing parameters.
1473 |                    alpha0, alpha2, alpha4: EFT parameters.
1474 |                    ctilde: parameter for NL0 ∝ Kaiser power spectrum.
1475 |                    alphashot0, alphashot2, PshotP: stochastic noise parameters.
1476 |     Returns:
1477 |        Redshift space power spectrum multipoles (monopole, quadrupole and hexadecapole) at 'kev'.
1478 |     '''
1479 |             
1480 |     #NuisanParams
1481 |     (b1, b2, bs2, b3nl, alpha0, alpha2, alpha4, 
1482 |                 ctilde, alphashot0, alphashot2, PshotP) = NuisanParams
1483 |     
1484 |     if AP == True and Omfid == -1:
1485 |         sys.exit("Introduce the fiducial value of the dimensionless matter density parameter as ‘Omfid = value’.")
1486 |      
1487 |     if AP == True and Omfid > 0:
1488 |                 
1489 |         #Om computed for any cosmology
1490 |         OmM = CosmoParam(h, omega_b, omega_cdm, omega_ncdm)[1]
1491 |         
1492 |         #qperp, qpar: AP parameters.
1493 |         qperp = DA(OmM, z_pk)/DA(Omfid, z_pk) 
1494 |         qpar = Hubble(Omfid, z_pk)/Hubble(OmM, z_pk) 
1495 |         
1496 |     
1497 |     def PIRs_derivatives(kev, mu, Table, Table_NW):
1498 |         
1499 |         if AP == True:
1500 |             
1501 |             k_true = k_AP(kev, mu, qperp, qpar)
1502 |             mu_true = mu_AP(mu, qperp, qpar)
1503 |             
1504 |             Table_true = Table_interp(k_true, kev, Table)
1505 |             Table_NW_true = Table_interp(k_true, kev, Table_NW)
1506 |             
1507 |             Sigma2T = Sigma2Total(k_true, mu_true, Table_NW_true)
1508 |             
1509 |             Fkoverf0 = Table_true[1]; fk = Fkoverf0*f0
1510 |             pkl = Table_true[0]; pkl_NW = Table_NW_true[0];
1511 |             
1512 |             #computing PEFTs_derivatives: wiggle and non-wiggle terms
1513 |             PEFTs_alpha0, PEFTs_alpha2, PEFTs_alpha4, PEFTs_alphashot0, PEFTs_alphashot2 = PEFTs_derivatives(k_true, mu_true, pkl, PshotP)
1514 |             PEFTs_alpha0_NW, PEFTs_alpha2_NW, PEFTs_alpha4_NW, PEFTs_alphashot0_NW, PEFTs_alphashot2_NW = PEFTs_derivatives(k_true, mu_true, pkl_NW, PshotP)
1515 |             
1516 |             exp_term = np.exp(-k_true**2 * Sigma2T)
1517 |             exp_term_inv = 1 - exp_term
1518 |             
1519 |             #computing PIRs_derivatives for EFT and stochastic parameters
1520 |             PIRs_alpha0 = exp_term * PEFTs_alpha0 + exp_term_inv * PEFTs_alpha0_NW
1521 |             PIRs_alpha2 = exp_term * PEFTs_alpha2 + exp_term_inv * PEFTs_alpha2_NW
1522 |             PIRs_alpha4 = exp_term * PEFTs_alpha4 + exp_term_inv * PEFTs_alpha4_NW
1523 |             PIRs_alphashot0 = exp_term * PEFTs_alphashot0 + exp_term_inv * PEFTs_alphashot0_NW
1524 |             PIRs_alphashot2 = exp_term * PEFTs_alphashot2 + exp_term_inv * PEFTs_alphashot2_NW
1525 |             
1526 |             return (PIRs_alpha0, PIRs_alpha2, PIRs_alpha4, PIRs_alphashot0, PIRs_alphashot2)
1527 |         
1528 |         
1529 |         else:
1530 |             k = kev; Fkoverf0 = Table[1]; fk = Fkoverf0*f0
1531 |             pkl = Table[0]; pkl_NW = Table_NW[0];
1532 |             
1533 |             Sigma2T = Sigma2Total(kev, mu, Table_NW)
1534 |             
1535 |             #computing PEFTs_derivatives: wiggle and non-wiggle terms
1536 |             PEFTs_alpha0, PEFTs_alpha2, PEFTs_alpha4, PEFTs_alphashot0, PEFTs_alphashot2 = PEFTs_derivatives(k, mu, pkl, PshotP)
1537 |             PEFTs_alpha0_NW, PEFTs_alpha2_NW, PEFTs_alpha4_NW, PEFTs_alphashot0_NW, PEFTs_alphashot2_NW = PEFTs_derivatives(k, mu, pkl_NW, PshotP)
1538 |             
1539 |             exp_term = np.exp(-k**2 * Sigma2T)
1540 |             exp_term_inv = 1 - exp_term
1541 |             
1542 |             #computing PIRs_derivatives for EFT and stochastic parameters
1543 |             PIRs_alpha0 = exp_term * PEFTs_alpha0 + exp_term_inv * PEFTs_alpha0_NW
1544 |             PIRs_alpha2 = exp_term * PEFTs_alpha2 + exp_term_inv * PEFTs_alpha2_NW
1545 |             PIRs_alpha4 = exp_term * PEFTs_alpha4 + exp_term_inv * PEFTs_alpha4_NW
1546 |             PIRs_alphashot0 = exp_term * PEFTs_alphashot0 + exp_term_inv * PEFTs_alphashot0_NW
1547 |             PIRs_alphashot2 = exp_term * PEFTs_alphashot2 + exp_term_inv * PEFTs_alphashot2_NW
1548 |             
1549 |             return (PIRs_alpha0, PIRs_alpha2, PIRs_alpha4, PIRs_alphashot0, PIRs_alphashot2)
1550 |         
1551 |         
1552 |     Nx = 8    
1553 |     xGL, wGL = scipy.special.roots_legendre(Nx)    #x=cosθ and weights
1554 |     
1555 |     def ModelPkl0_derivatives(Table, Table_NW):
1556 |         if AP == True:
1557 |             monop = 1/(qperp**2 * qpar) * sum(0.5*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW)) for ii in range(Nx))
1558 |             return monop
1559 |         
1560 |         else:
1561 |             monop = sum(0.5*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW)) for ii in range(Nx))
1562 |             return monop
1563 |         
1564 |     
1565 |     def ModelPkl2_derivatives(Table, Table_NW):
1566 |         if AP == True:
1567 |             quadrup = 1/(qperp**2 * qpar) * sum(5/2*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW))*eval_legendre(2, xGL[ii]) for ii in range(Nx))
1568 |             return quadrup 
1569 |         
1570 |         else:
1571 |             quadrup = sum(5/2*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW))*eval_legendre(2, xGL[ii]) for ii in range(Nx))
1572 |             return quadrup
1573 |     
1574 |     
1575 |     def ModelPkl4_derivatives(Table, Table_NW):
1576 |         if AP == True:
1577 |             hexadecap = 1/(qperp**2 * qpar) * sum(9/2*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW))*eval_legendre(4, xGL[ii]) for ii in range(Nx))
1578 |             return hexadecap
1579 |         
1580 |         else:
1581 |             hexadecap = sum(9/2*wGL[ii]*np.array(PIRs_derivatives(kev, xGL[ii], Table, Table_NW))*eval_legendre(4, xGL[ii]) for ii in range(Nx))
1582 |             return hexadecap  
1583 |     
1584 |     if Hexa == False:   
1585 |         Pkl0_derivatives = ModelPkl0_derivatives(TableOut_interp(kev), TableOut_NW_interp(kev));
1586 |         Pkl2_derivatives = ModelPkl2_derivatives(TableOut_interp(kev), TableOut_NW_interp(kev));
1587 |         return (Pkl0_derivatives, Pkl2_derivatives)
1588 |     
1589 |     else:
1590 |         Pkl0_derivatives = ModelPkl0_derivatives(TableOut_interp(kev), TableOut_NW_interp(kev));
1591 |         Pkl2_derivatives = ModelPkl2_derivatives(TableOut_interp(kev), TableOut_NW_interp(kev));
1592 |         Pkl4_derivatives = ModelPkl4_derivatives(TableOut_interp(kev), TableOut_NW_interp(kev));
1593 |         return (Pkl0_derivatives, Pkl2_derivatives, Pkl4_derivatives)
1594 |     
1595 |     
1596 |     
1597 |     
1598 | #Marginalization matrices
1599 | 
1600 | def startProduct(A, B, invCov):
1601 |     '''Computes: A @ InvCov @ B^{T}, where 'T' means transpose.
1602 |     
1603 |     Args:
1604 |          A: first vector, array of the form 1 x n
1605 |          B: second vector, array of the form 1 x n
1606 |          invCov: inverse of covariance matrix, array of the form n x n
1607 |     
1608 |     Returns:
1609 |          The result of: A @ InvCov @ B^{T}
1610 |     '''
1611 |     
1612 |     return A @ invCov @ B.T 
1613 | 
1614 | 
1615 | 
1616 | 
1617 | def compute_L0(Pl_const, Pl_data, invCov):
1618 |     '''Computes the term L0 of the marginalized Likelihood.
1619 |     
1620 |     Args:
1621 |          Pl_const: model multipoles for the constant part (Pℓ,const = Pℓ(α->0)), array of the form 1 x n
1622 |          Pl_data: data multipoles, array of the form 1 x n 
1623 |          invCov: inverse of covariance matrix, array of the form n x n
1624 |          
1625 |     Return:
1626 |          Loglikelihood for the constant part of the model multipoles 
1627 |     '''
1628 |     
1629 |     D_const = Pl_const - Pl_data
1630 |     
1631 |     L0 = -0.5 * startProduct(D_const, D_const, invCov)             #eq. 2.4 notes on marginalization
1632 |     
1633 |     return L0
1634 | 
1635 | 
1636 | 
1637 | 
1638 | def compute_L1i(Pl_i, Pl_const, Pl_data, invCov):
1639 |     '''Computes the term L1i of the marginalized Likelihood.
1640 |     
1641 |     Args:
1642 |          Pl_i: array with the derivatives of the power spectrum multipoles with respect to 
1643 |                the EFT and stochastic parameters, i.e., Pℓ,i=∂Pℓ/∂α_i , i = 1,..., ndim
1644 |                array of the form ndim x n
1645 |          Pl_const: model multipoles for the constant part (Pℓ,const = Pℓ(α->0)), array of the form 1 x n
1646 |          Pl_data: data multipoles, array of the form 1 x n
1647 |          invCov: inverse of covariance matrix, array of the form n x n
1648 |     Return:
1649 |          array for L1i
1650 |     '''
1651 |     
1652 |     D_const = Pl_const - Pl_data  
1653 |     
1654 |     #ndim = len(Pl_i)
1655 |     
1656 |     #computing L1i
1657 |     #L1i = np.zeros(ndim)
1658 |     
1659 |     #for ii in range(ndim):
1660 |     #    term1 = startProduct(Pl_i[ii], D_const, invCov)
1661 |     #    term2 = startProduct(D_const, Pl_i[ii], invCov)
1662 |     #    L1i[ii] = -0.5 * (term1 + term2)
1663 |     
1664 |     L1i = - startProduct(Pl_i, D_const, invCov)
1665 |     
1666 |     return L1i
1667 | 
1668 | 
1669 | 
1670 | 
1671 | def compute_L2ij(Pl_i, invCov, sigma_prior = np.inf):
1672 |     '''Computes the term L2ij of the marginalized Likelihood.
1673 |     
1674 |     Args:
1675 |          Pl_i: array with the derivatives of the power spectrum multipoles with respect to 
1676 |                the EFT and stochastic parameters, i.e., Pℓ,i=∂Pℓ/∂α_i , i = 1,..., ndim
1677 |                array of the form ndim x n
1678 |          invCov: inverse of covariance matrix, array of the form n x n
1679 |     Return:
1680 |          array for L2ij
1681 |     '''
1682 |     
1683 |     #ndim = len(Pl_i)
1684 |     
1685 |     #Computing L2ij
1686 |     #L2ij = np.zeros((ndim, ndim))
1687 |     
1688 |     #for ii in range (ndim):
1689 |         #for jj in range (ndim):
1690 |             #L2ij[ii, jj] = startProduct(Pl_i[ii], Pl_i[jj], invCov)
1691 |     
1692 |     L2ij = startProduct(Pl_i, Pl_i, invCov)
1693 |             
1694 |     # Adding prior variances to L2ij
1695 |     if isinstance(sigma_prior, (int, float)):
1696 |         L2ij += 1 / (sigma_prior ** 2)
1697 |     else:
1698 |         L2ij += np.diag(1 / np.array(sigma_prior) ** 2)
1699 |             
1700 |     return L2ij 
1701 | 


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